Solar Magnetism [1st ed. 2023] 9819917581, 9789819917587

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Hongqi Zhang

Solar Magnetism

Solar Magnetism

Hongqi Zhang

Solar Magnetism

Hongqi Zhang Chinese Academy of Sciences National Astronomical Observatories Beijing, China

ISBN 978-981-99-1758-7 ISBN 978-981-99-1759-4 (eBook) https://doi.org/10.1007/978-981-99-1759-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

To gratefully thank the contribution of my colleagues and students in the cooperative study of solar physics

Introduction

Grau, teurer Freund, ist alle Theorie, Und grün des Lebens goldner Baum. Johann Wolfgang von Goethe Tao gives birth to one, two, three and everything. Lao Zi

Solar physics is a fundamental branch of astrophysics, which has a long history. People have noticed the activity of the sun for a long time. For example, as early as 3000 years ago, there were records about sunspots in oracle bone inscriptions of the Shang Dynasty in China. One of the earliest records about sunspots recognized by the world is in the first year of Heping (28BC) of emperor Cheng of the Western Han Dynasty in China, “日出黄, 有黑气, 大如钱, 居日中央 (Translation: The sun is sick, it carries black gas, the size is an order of the coin in the center of the solar disk).” The so-called “black gas” here is the sunspot. In ancient China, sunspots were observed and recorded more than 1000 years earlier than other countries in the world. The modern scientific observation of sunspots can be traced back to Galileo Galileo (Galileo, 1564–1642). In 1908, American astronomer G. E. Hale (1868– 1938) first used the Zeeman effect of the solar spectrum to find that the magnetic field of sunspots in active regions can be as strong as 3–4 kilogauss (Hale, 1908). In 1952, the American solar physicist H. D. Babcock (1882–1968) and his son successfully discovered the existence of the full solar disk magnetic field with the photoelectric magnetograph (Babcock & Babcock, 1955). The discovery of solar magnetic field and the observation of solar spectrum from X-ray, ultraviolet, and visible light to infrared make a qualitative leap in the research of solar physics. However, the research of solar magnetic fields has been playing a leading role in the field of solar physics. In 1956, the Japanese astrophysicist W. Unno first discussed the radiative transfer of the polarized spectrum in the solar atmospheric magnetic field, and then the former

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Soviet Union solar physicist D. N. Rachkovsky (1962a,b) analyzed the influence of the magneto-optical effect on the measurement for the solar magnetic field, which laid a basic theory for the further analysis of measurement results of the solar magnetic fields. In 1942, H. O. G. Alfvén (1908–1995) put forward the concept of magnetic freezing in the process of using the law of electromagnetic induction to study solar plasma. More importantly, he found a new wave pattern—Alfvén wave (Alfvén, 1942). This discovery indicates that people have an essential understanding of the interaction between conductive fluid and magnetic fields. A lot of research work has led to the birth of magnetohydrodynamics as a new discipline and provided an important means for the study of the physical processes related to the magnetic field in the solar atmosphere. The following is a brief introduction to the outline of this book and the main contents we hope to explore: This book starts with the problem of solar magnetic field measurement. It involves the basic mechanism of the formation of magnetic-sensitive spectral lines in the solar atmosphere. Taking two spectral lines of photospheric FeIλ5324.19Å and chromospheric Hβλ4861.34Å as examples, we discuss the basic mechanism, possible problems, and uncertainties of spectral line formation in the lower solar atmospheric magnetic field. The basic principle of solar magnetic field measurement using magnetographs is discussed. As examples, we especially introduce the method for the measurements of the local and full disk solar magnetic field and the relevant problems. Although it is only a few examples, its basic principle should be universal. Then we try to discuss when the magnetic field in the solar atmosphere is observed by polarized spectrum analysis, the basic question is what the observed basic structure of the sun is, how to judge the extension of the magnetic field from the solar photosphere to the higher atmosphere through the observation, and what important information of the observed chromospheric magnetic field data provide to us. Based on the basic principles of plasma physics, how to explore the basic relationship between the observation data of solar magnetic fields. The observation and study of the magnetic field in the solar active region is an attractive topic. The usual solar eruptions are closely related to it. From the photospheric vector magnetograms in the solar active regions in this book, we can see that the magnetic fields in the solar eruptive regions are usually strongly distorted. It is characterized by severe magnetic shear, locally strong current, and magnetic helicity density. How the non-potential magnetic field is formed and its internal relationship with solar flare-coronal mass ejections are often discussed and debated by many solar physicists. Although this book introduces the magnetic field measurement results of the solar photosphere and chromosphere, the more precise measurement of the solar upper atmospheric magnetic field still needs to be further studied. It relates to the extended form of non-potential magnetic energy in the solar upper atmosphere. It is naturally associated with solar eruptions. We also introduce the basic forms of the extrapolation of the solar photospheric magnetic field by means of the force-free field and provide some discussed results.

Introduction

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The long-term continuous observation of the solar magnetic field is of great significance to understand the periodicity of solar activity. In this book, we focus on the variation of magnetic helicity with the solar cycles from the perspective of different kinds of the solar magnetic field observed data. Observations show that although the distribution of magnetic helicity in the solar surface displays the statistical characteristics of negative (positive) signs in the Northern (Southern) Hemisphere, it also does the complexity with the temporal and spatial distribution. The systematic study of magnetic helicity in the solar atmosphere actually reveals the generated process of the magnetic field inside of the sun. This process can be very complex. Theoretical research shows that magnetic helicity is an important parameter in the theory of solar dynamo. It provides a window to explore the formation of the sun’s internal magnetic field. In this book, based on the turbulent dynamo theory, we explore the possible mechanism of the formation of the solar magnetic field and its internal relationship with magnetic helicity. In the last part of the book, we present some questions in the study of the solar magnetic field. Of course, different people may have different views and understandings. These are very natural because many problems and contexts are related to the solar magnetic field, which do not clear still. We know that many aspects involved in the study of the solar magnetic field need to be in-depth. This book does not cover all aspects of the issues. I only hope that readers can enter a more advanced level after reading this book. For the basic content and knowledge of solar physics, please refer to a series of books in English, such as the books by Zirin (1988); Stix (2002), and the books about the solar and cosmic magnetohydrodynamics in Moffatt (1978); Parker (1979a); Krause & Rädler (1980); Zeldovich et al. (1983); Rüdiger (1989); Priest (2014). In addition, there are more Chinese books, such as solar physics monographs such as Hu et al. (1983); Zhang (1992); Lin (2000); Fang, Ding & Chen (2008); Yang & Jing (2015), and books related to the solar and cosmic magnetohydrodynamics such as Hu (1987); Mao (2013). In this book, we do not concern with the theory of plasma dynamics and waves in the solar magnetic field. Readers who are interested in these can read the relevant parts of the above books, such as Wu & Chen (2021).

Contents

1 Measurements of Solar Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 General Radiative Transfer of Light . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 The Equation of Radiative Transfer . . . . . . . . . . . . . . . . . . . 1.2 Radiation and Polarization of Spectral Lines in Solar Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Representations of Polarized Light . . . . . . . . . . . . . . . . . . . 1.2.2 Spherical Vectors and Decoupling of Component Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Evaluation of the Spherical Vector Components . . . . . . . . 1.3 Spectral Line Broadening and Thermodynamic Equilibrium Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Spectral Line Broadening . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Thermodynamic Equilibrium and Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Ionization Equilibrium Equations . . . . . . . . . . . . . . . . . . . . 1.4 Introduction to Quantum Field Theory of Polarized Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Zeeman Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Stark Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Electric Dipole Approximation . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Beyond the Electric Dipole Approximation . . . . . . . . . . . . 1.4.5 Complex Refractive Index and Broadening . . . . . . . . . . . . 1.5 Polarized Radiative Transfer of Spectral Lines . . . . . . . . . . . . . . . . . 1.5.1 General Description of Radiative Transfer . . . . . . . . . . . . . 1.5.2 Formal Solutions of Polarized Radiative Transfer Without Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Magneto-Optical Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4 Numerical Calculation of Radiative Transfer of Stokes Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 2 3 5 7 8 8 11 14 15 15 18 20 21 22 26 26 28 34 36

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1.6

1.7

1.8

1.9

1.10

1.11

1.12

1.13

1.5.5 Formation Layers of Spectral Lines . . . . . . . . . . . . . . . . . . . 1.5.6 Nonlinear Least-Squares Fitting . . . . . . . . . . . . . . . . . . . . . . Solar Magnetic Field Telescope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Lyot Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Solar Magnetic Field Telescope . . . . . . . . . . . . . . . . . . . . . . Observations of Vector Magnetic Field and Comparing with Different Vector Magnetographs . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Vector Magnetograms at Huairou, Mitaka and Mees Observatories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 Vector Magnetograms at Huairou and Yunnan Observatories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diagnosis of Faraday Rotation with Video Vector Magnetograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 Stokes Parameters at Different Wavelengths of Spectral Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2 Nonlinear Least-Squares Fitting . . . . . . . . . . . . . . . . . . . . . . 1.8.3 Qualitative Estimations of Magneto-Optical Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Instruments for Measurement of Full Solar Disk Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.1 Magnetic Field Measurements . . . . . . . . . . . . . . . . . . . . . . . 1.9.2 Hα Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.3 Derivation of Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . Effects of Polarization Crosstalk and Solar Rotation on Measuring Full-Disk Solar Photospheric Vector Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.1 Correction for Polarization Crosstalk: Method One . . . . . 1.10.2 Correction for Polarization Crosstalk: Method Two . . . . . Stokes Profile Analysis on Measuring Full-Disk Solar Photospheric Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.11.1 Spatial Integration Scanning Spectra of Filter Magnetograph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solar Model Atmospheres with Non-local Thermodynamic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12.1 General form of Statistical Equilibrium . . . . . . . . . . . . . . . 1.12.2 Model Chromosphere with Non-local Thermodynamic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . Formation of Hβ Line in Solar Chromospheric Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13.1 Radiative Transfer of Hβ Line . . . . . . . . . . . . . . . . . . . . . . . 1.13.2 Broadening of Hβ Line in Magnetic Field Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13.3 Numerical Calculation of Hβ Line in Atmospheric Model of Quiet Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13.4 Formation Layers of Hβ Line . . . . . . . . . . . . . . . . . . . . . . . .

39 41 47 47 50 53 53 56 58 58 60 64 65 66 68 69

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1.14 Formation of Hβ Line in Solar Umbral Magnetic Atmosphere . . . 1.15 Coronal Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15.1 Resonance Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15.2 Explicit Formulae for the Stokes Parameters of the Scattered Radiation in Magnetic-Dipole Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15.3 Coronal Magnetic Field Measurements . . . . . . . . . . . . . . . . 1.16 Brief Overview of the History of Solar Optical Instruments by Ai (1993a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.17 Discussions for Some Challenges on Measurements of Solar Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic Structures of Solar Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Description of Astrophysical Plasma . . . . . . . . . . . . . . . . . . . . 2.1.1 Microscopic Description of Plasma . . . . . . . . . . . . . . . . . . . 2.1.2 Equation for Zeroth Moment . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Momentum Conservation Law . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Energy Conservation Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.5 Derivation of Basic Equations of Ohm’s Law . . . . . . . . . . 2.2 Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Magnetic Fields in Quiet Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Photospheric Magnetic Features in Quiet Sun . . . . . . . . . . 2.3.2 Extending Magnetic Field from Lower Layer of Quiet Sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Basic Configuration of Sunspots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Magnetic Field of Sunspots . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Models of Sunspot Magnetic Fields . . . . . . . . . . . . . . . . . . 2.4.3 Penumbral Fine Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Decay of Sunspots and Magnetic Fields . . . . . . . . . . . . . . . 2.4.5 Moving Magnetic Features . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6 Moving Magnetic Features from High-Resolution Magnetograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Chromospheric Magnetic Fields in Active Regions . . . . . . . . . . . . . 2.5.1 Measurements of Chromospheric Magnetic Field . . . . . . . 2.5.2 Possible Extension of Hβ Chromospheric Magnetic Field from Photosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Reversal Features in Hβ Chromospheric Magnetograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Magnetic Field of Dark Filaments in Quiet Sun . . . . . . . . 3 Solar Magnetic Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Magnetic Energy, Shear, Gradient, and Electric Current in Solar Active Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Magnetic Energy, Shear, and Gradient . . . . . . . . . . . . . . . . 3.1.2 Relationship of Magnetic Shear and Gradient with Electric Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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107 110 112 114 117 117 117 118 119 120 121 123 123 123 126 131 131 133 136 137 138 140 143 143 145 146 148 151 151 154 167

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3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.1.3 Ratio Between Different Components of Current . . . . . . . 3.1.4 Evolution of Current with Flares in Active Regions . . . . . Current Helicity in Solar Active Regions . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Magnetic Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Observational Evidence of Magnetic Chirality . . . . . . . . . 3.2.3 Fine Features of Magnetic Field, Electric Current and Helicity in Solar Active Regions . . . . . . . . . . . . . . . . . . 3.2.4 Questions from Vector Magnetic Fields to Electric Current and Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlations Between Subsurface Kinetic Helicity and Photospheric Current Helicity in Active Regions . . . . . . . . . . . 3.3.1 Subsurface Kinetic Helicity . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Correlations Between Kinetic and Current Helicity . . . . . Magnetic Helicity and Tilt Angle Evolution in Active Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Twisted Magnetic Field and Helicity . . . . . . . . . . . . . . . . . . 3.4.2 Helicity Injection and Tilt Angle in Newly Emerging Flux Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Two Typical Solar Active Regions . . . . . . . . . . . . . . . . . . . . Magnetic Field, Horizontal Motion and Helicity . . . . . . . . . . . . . . . 3.5.1 Evolution of Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Transport of Magnetic Helicity . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Evolution of Current Helicity Density . . . . . . . . . . . . . . . . . Magnetic Helicity and Energy Spectra of Solar Active Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Implication of Magnetic Spectrum of Active Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Comparisons Among Magnetic Helicity, Energy and Velocity Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.3 Comparison of Magnetic Spectrum of Active Regions from Huairou and HMI Vector Magnetograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of Magnetic Helicity and Energy Spectra . . . . . . . . . . . . 3.7.1 Magnetic Helicity and Energy Spectra of Individual Active Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Field in Flare-CME Active Regions . . . . . . . . . . . . . . . . . 3.8.1 Change of Vector Magnetic Fields Associated with Flares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2 A Survey of Flares and Current Helicity in Active Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.3 Injective Magnetic Helicity and Flare-CMEs . . . . . . . . . . . 3.8.4 Powerful Flares and Dynamic Evolution of Magnetic Field at Solar Surface . . . . . . . . . . . . . . . . . . . .

171 172 174 174 177 179 182 183 183 184 187 187 190 191 191 191 193 194 195 196 201

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4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Force-Free Field Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Methods for Extrapolation of Solar Magnetic Fields . . . . 4.2 Linear Force Free Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Chiu–Hilton Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Alissandrakis Fourier Transform . . . . . . . . . . . . . . . . . . . . . 4.2.3 Fast Fourier Analysis Method of Linear Force-Free Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Study on Two Methods for Nonlinear Force-Free Extrapolation Based on Semi-analytical Field . . . . . . . . . . . . . . . . . 4.3.1 Theories and Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Calculations for Different Methods . . . . . . . . . . . . . . . . . . . 4.3.3 Comparisons of Different Methods . . . . . . . . . . . . . . . . . . . 4.4 Extrapolated Magnetic Fields with Observations . . . . . . . . . . . . . . . 4.5 Eruptions of Flare-Coronal Mass Ejections . . . . . . . . . . . . . . . . . . . . 4.5.1 Flares with Filament Eruption . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Ribbon Separation of Eruptive Flares with Possible Electric Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Magnetic Properties of Flare-CME Productive Active Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Magnetic Helicity Budget of Solar Active Regions and Coronal Mass Ejections . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.5 Statistical Analysis Between Magnetic Field and Coronal Mass Ejections . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Formation of Magnetic Reconnection . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Magnetic Reconnection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Helical Magnetic Field and Solar Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Distribution of Magnetic Helicity with Solar Active Cycles . . . . . . 5.1.1 Statistical Analysis of Current Helicity of Active Regions from Observed Vector Magnetograms . . . . . . . . . 5.1.2 Hemispheric Distribution of Current Helicity of Active Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Butterfly Diagram of Current Helicity of Solar Active Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Helicity with Solar Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Relationship Between Twist and Tilt of Magnetic Fields in Active Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Evolved Helicity Comparison Among Magnetographs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4 Accuracy of Measured Vector Magnetic Fields . . . . . . . . . 5.2.5 Comparison Between Current and Subsurface Kinetic Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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219 219 220 223 224 225 226 228 229 236 240 242 246 246 248 251 252 255 259 259 265 265 266 269 270 271 272 275 276 277

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Contents

5.2.6

5.3

5.4 5.5

5.6

5.7

5.8

Magnetic Helicity and Energy Spectra of Active Regions with Solar Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.7 Resolution Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.8 Summary for Evolution of Magnetic Spectrum . . . . . . . . . Statistical Study of Transequatorial Loops . . . . . . . . . . . . . . . . . . . . 5.3.1 Distribution of Tansequatorial Loops . . . . . . . . . . . . . . . . . 5.3.2 Helicity Patterns of Active Regions Connected by Transequatorial Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . Long Term Injection of Magnetic Chirality of Active Regions from Solar Subsurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Large-Scale Soft X-ray Loops and Magnetic Chirality in both Hemispheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Magnetic Chirality of Soft X-Ray Loops Related to Solar Active Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Hemispheric Distribution of Helical Soft X-Ray Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Handedness of Large-Scale Soft X-Ray Loops and Magnetic (Current) Helicity . . . . . . . . . . . . . . . . . . . . . Observational Cross-Helicity on Solar Surface . . . . . . . . . . . . . . . . . 5.6.1 The Cross-Helicity Conservation Law . . . . . . . . . . . . . . . . 5.6.2 Correlation of Data from SOHO/MDI and SDO/ HMI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Distribution of Cross-Helicity with Latitude . . . . . . . . . . . Transfer of Magnetic Helicity Flux with Solar Cycles . . . . . . . . . . . 5.7.1 Observational Data Analysis of Magnetic Helicity . . . . . . 5.7.2 Long Term Transfer of Magnetic Helicity . . . . . . . . . . . . . Statistical Studies on Photospheric Magnetic Nonpotentiality of Active Regions and Associated Flares with Solar Cycles . . . . . . 5.8.1 Magnetic Nonpotentiality and Complexity Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.2 Statistical Analysis and Results . . . . . . . . . . . . . . . . . . . . . . 5.8.3 Summary for Different Nonpotential Magnetic Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Magnetic Helicity with Solar Dynamo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Solar Dynamo and Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Mean-Field Solar Dynamo . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Equation for Magnetic Helicity . . . . . . . . . . . . . . . . . . . . . . 6.2 Radial Distribution of Magnetic Helicity in Solar Convective Zone: Observations and Dynamo Theory . . . . . . . . . . . . . . . . . . . . . 6.2.1 Velocity Structure of Solar Convection Zone . . . . . . . . . . . 6.2.2 Current Helicity Data Obtained at Huairou Solar Observing Station . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Dynamo Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 The Nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

279 282 284 285 285 288 292 295 296 297 300 301 301 303 305 306 307 307 312 313 316 323 325 326 326 328 329 329 331 333 335

Contents

6.3

6.4

6.5

6.6

xvii

6.2.5 Numerical Implementation and Nonlinear Solution . . . . . 6.2.6 Helicity Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.7 Appendix: Quenching Functions . . . . . . . . . . . . . . . . . . . . . Current Helicity of Active Regions as a Tracer of Large-Scale Solar Magnetic Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 The Role of Helicities in Magnetic Field Evolution . . . . . 6.3.2 An Estimate for Current Helicity in Active Regions . . . . . 6.3.3 Dynamo Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Simulated Butterfly Diagrams for Current Helicity . . . . . . 6.3.5 Appendix: Current Helicity Versus Magnetic Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reversal Helicity and Solar Dynamo Possibility . . . . . . . . . . . . . . . 6.4.1 Possibilities on Reversal Helicity in Subatmosphere . . . . . 6.4.2 Appendix for Reversal Helicity and Solar Dynamo Possibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulent Cross-Helicity in Mean-Field Solar Dynamo Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Transformation Symmetry Properties and Cross-Helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Mean-Field Theory of Cross-Helicity . . . . . . . . . . . . . . . . . 6.5.3 The u r br Patterns by Dynamo Models . . . . . . . . . . . . . . . . 6.5.4 Discussion for Cross-Helicity . . . . . . . . . . . . . . . . . . . . . . . . Estimates of Current Helicity and Tilt of Solar Active Regions and Joy’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 Estimate for Tilt of Sunspots . . . . . . . . . . . . . . . . . . . . . . . . 6.6.3 Estimates for Current Helicity and Twist In Dynamo Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 More Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Measurements of Solar Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . 7.2 Basic Configuration of Solar Magnetic Fields . . . . . . . . . . . . . . . . . . 7.3 Solar Magnetic Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Space Weather . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

339 342 343 344 345 346 347 349 351 353 353 356 360 361 362 366 367 368 370 373 374 377 378 378 379 380

Postscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

Chapter 1

Measurements of Solar Magnetic Field

1.1 General Radiative Transfer of Light Radiative transfer is the physical phenomenon of energy transfer in the form of electromagnetic radiation. The propagation of radiation through a medium is affected by absorption, emission, and scattering processes. The equation of radiative transfer describes these interactions mathematically. In astrophysical research, radiation transfer process is an important means to explore the universe.

1.1.1 Definitions In terms of the spectral radiance, Iν , the energy flowing across an area element of area da , located at r in time dt in the solid angle d about the direction n in the frequency interval ν to ν + dν is d E ν = Iν (r, n, t) cos θ dν da d dt

(1.1)

where θ is the angle that the unit direction vector nˆ makes with a normal to the area element. The units of the spectral radiance are seen to be energy/time/area/solid angle/frequency. In MKS units this would be W· m−2 · sr−1 · Hz−1 (watts per squaremetre-steradian-hertz).

1.1.2 The Equation of Radiative Transfer The equation of radiative transfer is used to describe that as a beam of radiation travels, it loses energy to absorption, gains energy by emission process, and redistributes energy by scattering. The differential equation of radiative transfer is © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Zhang, Solar Magnetism, https://doi.org/10.1007/978-981-99-1759-4_1

1

2

1 Measurements of Solar Magnetic Field

1 ∂ 1 Iν +  · ∇ Iν + (kν,s + kν,a )ρIν = jν ρ + kν,s ρ c ∂t 4π

 

Iν d

(1.2)

where c is the speed of light, jν is the emission coefficient, kν,s and kν,a are the 1 kν,s Iν d scattering and the absorption opacity, ρ is the mass density, and the 4π  relates to the radiation scattered from other directions onto a surface, respectively. As scattering has been ignored, a generally steady-state solution in terms of the emission and absorption coefficients may be written:  s  Iν (s) = Iν (s0 )e−τν (s0 ,s) + jν (s  )e−τν (s ,s) ds  (1.3) s0

where τν (s1 , s2 ) is the optical depth of the medium between positions s1 and s2 :  τν (s1 , s2 ) =

s2

ρ kν,a (s) ds.

(1.4)

s1

For a medium in the conditions of local thermodynamic equilibrium (LTE), the emission coefficient and absorption coefficient are functions of temperature and density only and are related by: jν = Bν (T ) kν,a

(1.5)

where Bν (T ) is the black body spectral radiance at temperature T. The solution to the equation of radiative transfer is then:  s  −τν (s0 ,s) Iν (s) = Iν (s0 )e + Bν (T (s  ))kν,a (s  )e−τν (s ,s) ds  (1.6) s0

Knowing the temperature profile and the density profile of the medium is sufficient to calculate a solution to the equation of radiative transfer.

1.2 Radiation and Polarization of Spectral Lines in Solar Atmosphere The magnetic field of the Sun, as a star, can be obtained through the diagnosis of solar polarized light. Normally the atoms in a solar atmosphere will absorb certain frequencies of energy in the electromagnetic spectrum, producing characteristic dark absorption lines in the spectrum. When the atoms are within a magnetic field, however, these lines become split into multiple, closely spaced lines. The energy also becomes polarized with an orientation that depends on the orientation of the magnetic field. Thus, the strength and direction of the solar magnetic field can be determined

1.2 Radiation and Polarization of Spectral Lines in Solar Atmosphere

F0

F1

F2

unpolarized

0

45

3

F3

right−handed circular polarization

Fig. 1.1 Polarized states of the beam

by detecting the state of the spectrum under the Zeeman effect. Moreover, the Hanle effect of spectral lines also can be used to diagnose the magnetic field, especially in the higher solar atmosphere (cf. Zirin, 1988; Stix, 2002).

1.2.1 Representations of Polarized Light A complete description of polarized light needs four parameters. Four typical light beams are shown in Fig. 1.1. F0 is an unpolarized light beam. F1 and F2 are linear polarization with the electric vector at position angles 0◦ and 45◦ , respectively, and F3 transmits right-handed circular polarization (Stenflo, 1994). At any fixed point in space the electric vector E can be decomposed as E = Re(E 1 e1 + E 2 e2 ), where

E k = E 0k e−iωt ,

(1.7)

k = 1, 2

(1.8)

and E 0k are complex amplitudes. Since both amplitudes and phases are involved, the two complex numbers represent four parameters characterizing the light. The oscillating phase factor e−iωt can be ignored for the polarized problems, since it is the same for both components for the electric vector, and it disappears when forming observable quantities (Stenflo, 1994). The Jones vector J is simply defined as  J=

 E1 . E2

(1.9)

The interaction with a medium can be described by a matrix w operating on J: J = wJ.

(1.10)

4

1 Measurements of Solar Magnetic Field

The Jones matrices for the four filters of Fig. 1.1 are 

   10 10 , w1 = , w0 = 01 00     1 11 1 1 i , w3 = . w2 = 2 11 2 −i 1

(1.11)

To derive w1 and w2 for the linearly polarizing filters, one needs to use a linear polarization basis e1 and e2 , while for the derivation of w3 a circular polarization basis is the appropriate one. (Note that filter F3 does not represent a λ/4 plate + linear polarizer, as is normally used in practical instruments to detect circular polarization since the light that emerges would then be linearly instead of circularly polarized. Another λ/4 plate has to be added to make the transmitted light have the same right-handed circular polarization as the incident beam, to represent the function of F3 .) The 2 × 2 coherency matrix D of the radiative field is directly obtained from Jones vector by   E 1 E 1∗ E 1 E 2∗ D = JJ† = , (1.12) E 1∗ E 2 E 2 E 2∗ where J† denotes the adjoint of J (transposition and complex conjugation of J). The intensity I of a light beam is proportional to |E0 |2 , the square of the amplitude of the electric vector. As the constant of proportionality is unimportant for the description of the polarization state, it is convenient in the context of polarization theory to choose it to be unity. Thus, we define I = |E01 |2 + |E02 |2 ,

(1.13)

I = TrD,

(1.14)

which implies

where Tr means the trace (sum over the diagonal matrix elements). The 2 × 2 coherency matrix can also be represented in the form of a fourdimensional vector Dv , defined as ⎞ ⎛ D11 ⎜ D12 ⎟ ⎟ (1.15) Dv = ⎜ ⎝ D21 ⎠ D22 where Di j are the components of D. The Pauli spin matrix σk is defined as  σ0 =

 10 , 01

 σ1 =

 1 0 , 0 −1

 σ2 =

 01 , 10

 σ3 =

 0 i . −i 0

(1.16)

1.2 Radiation and Polarization of Spectral Lines in Solar Atmosphere

5

The Jones matrices of the four filters can now be expressed in the conveniently compact form 1 (1.17) wk = (σ0 + σk ), k = 0, 1, 2, 3. 2 The coherency matrix and Stokes formalisms will allow us to obtain a deeper physical understanding of these various 2 × 2 matrices. Inserting the formula (1.17) for wk , we get an expression that subsequently can be reduced to the simple form Ik =

1 [I + T r (σk D)], 2

k = 0, 1, 2, 3,

(1.18)

by noting that σk† = σk , D † = D (as a consequence of the definition (1.12)), σ0 D = D, σk2 = σ0 ,and T r (σ D) = T r (Dσ). The Stokes parameters Sk can be defined operationally in terms of the intensity measurements Ik with the four filters of Fig. 1.1 as Sk = 2Ik − I0 .

(1.19)

S0 thus represents the ordinary intensity, S1 and S2 the amount of linear polarization along with position angles 0 and 45◦ , and S3 the amount of right-handed circular polarization. Using expression (1.18) in Eq. (1.19), we obtain the relation between the Stokes parameters and the coherency matrix: Sk = Tr(σk D).

(1.20)

The Stokes parameters, which are often denoted I , Q, U and V instead of Sk , form a 4-vector be obtained as ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ I S0 E 1 E 1∗ + E 2 E 2∗ ⎜ S1 ⎟ ⎜ Q ⎟ ⎜ E 1 E ∗ − E 2 E ∗ ⎟ 1 2 ⎟ ⎟ ⎜ ⎟ ⎜ (1.21) S=⎜ ⎝ S2 ⎠ = ⎝U ⎠ = ⎝ E 1∗ E 2 + E 1 E 2∗ ⎠ . S3 i E 1∗ E 2 − i E 1 E 2∗ V

1.2.2 Spherical Vectors and Decoupling of Component Equations The complete equation for the classical oscillator is (Stenflo, 1994) x¨ +

e 2 (˙x × B) + γ x˙ + ω02 x = − E. m m

(1.22)

6

1 Measurements of Solar Magnetic Field

The complex spherical unit vectors eq , q = 0, ±1, can be defined in terms of the Cartesian linear unit vectors ex , e y , and ez as e0 = ez

√ e± = ∓(ex ± ie y )/ 2,

(1.23)

where we have used the more compact notation e± instead of e±1 (cf. Shore & Menzel, 1968). Now let the Cartesian components of the vector E be E x , E y , and E z , while the corresponding spherical vector components are denoted E q , q = 0, ±1. We define these spherical vector components such that their relations to the Cartesian components are formally the same as the relations between the corresponding unit vectors: E0 = Ez

√ E± = ∓(Ex ± iE y )/ 2.

(1.24)

Then, with these definitions, the spherical vector decomposition of the real, linear vector E can be done in the following forms: E=



E q∗ eq =

q



E q eq∗ =

q



(−1)q E q e−q .

(1.25)

q

A scalar product becomes a·b=

 q

aq bq∗ =

 (−1)q aq b−q .

(1.26)

q

The fundamental role of the spherical vectors is seen if we choose a coordinate system such that the z-axis is along the direction of the magnetic field B, and express the momentum equation (1.22) in terms of the spherical vector components. The troublesome v × B term then becomes  vq eq∗ (1.27) v ×B = iB q

where B = |B|. Equation (1.27) shows that the different components of v and B are no longer coupled to each other (different q values do not mix in the products), in contrast to the Cartesian case. This means that the momentum vector equation (1.22) can be expressed as three independent, scalar equations,

  d2 eB e d 2 − qi −γ + ω0 xq = − E q , q = 0, ±1, dt 2 m dt m

(1.28)

1.2 Radiation and Polarization of Spectral Lines in Solar Atmosphere

7

These three equations describe three independent, damped harmonic oscillators, which have different oscillation frequencies because of the q-dependent term in Eq. (1.28). In the limit of vanishing magnetic field B, the three frequencies coincide.

1.2.3 Evaluation of the Spherical Vector Components If we decompose the linear polarization unit vector eα in its Cartesian components, and then convert it to the spherical vector components, we easily find (Stenflo, 1994) 1 εα± = ∓ √ (cos γ cos α ± i sin α)e±iφ 2 α ε0 = − sin γ cos α.

(1.29)

For the absorption-dispersion problems that we are considering here, the azimuth phase factor e±iφ is of no consequence. Therefore, we are free to choose any value of φ, e.g., zero. Let us now choose a linear polarization basis e1 and e2 , such that the projection B⊥ of the magnetic-field vector makes an angle χ (in the counter-clockwise direction) with e1 , χ − π/2 with e2 . Thus, α = π − χ for e1 , π − (χ − π/2) for e2 . If we further make use of our freedom to choose φ = 0, we obtain from Eq. (1.29) ε10 = sin γ cos χ, ε20 = sin γ sin χ,

√ ε1± = ±(cos γ cos χ ∓ i sin χ)/ 2 √ ε2± = ±(cos γ sin χ ± i cos χ)/ 2.

(1.30)

For the special case that χ = 0 this system reduces to ε10 = sin γ, ε20 = 0,

√ ε1± = ± cos γ/ 2 √ ε2± = i/ 2.

(1.31)

8

1 Measurements of Solar Magnetic Field

1.3 Spectral Line Broadening and Thermodynamic Equilibrium Relations 1.3.1 Spectral Line Broadening 1.3.1.1

Doppler and Pressure Broadening

In many cases, and assuming the radiator (or absorber) velocities to be non-relativistic and their distribution to be Maxwellian, i.e., the relevant one-dimensional velocity distribution to be Gaussian, the corresponding normalized line shape function is also Gaussian, namely (Grim, 1997, p. 54) √ L D (ω) = exp[−(ω/ω D )2 ]/ πω D

(1.32)

with the Doppler broadening parameter given by  ωD =

2kT Mc2

1/2 ω0

(1.33)

in means of the radiator’s kinetic temperature T , radiator mass M, and the frequency detuning ω from the rest frame transition frequency ω0 . Pressure broadening is less conducive to any general statement, except for classification according to either the underlying physical mechanism or the basic approximation used in the line profile calculations. The corresponding line shape functions have normally no simple analytic form, except for the impact approximation for isolated lines, i.e., lines which are not overlapping the transitions in the same spectrum. These line shapes are Lorentz profile, normally L(ω) =

w/π w 2 + (ω − d)2

(1.34)

in terms of half-width (HWHM) w and the shift d. For theoretical purposes, it is more convenient to use w, rather than the FWHM width 2w = γ. A schematic representation of Doppler and Lorentzian profile equal FWHM width is shown in Fig. 1.2, together with a combined profile obtained by the convolution  L c (ω) =

+∞ −∞

L D (ω  )L(ω − ω  )dω 

which is, in this case, a (Voigt, 1912) profile. See Sect. 1.4.5 for details.

(1.35)

1.3 Spectral Line Broadening and Thermodynamic Equilibrium Relations

9

Fig. 1.2 Normalized Gauss (Doppler) and Lorentz (impact) profile of equal half widths (FWHM width = 2 in x-units). Also shown is the Voigt profile resulting from the convolution of these two profiles. From Griem (1997)

1.3.1.2

Ion Microfields

Equally important as cross sections and related quantities for line broadening calculations are the ion-microfield (F) distributions. The archetype of such distributions was derived by Holtsmark (1919) in the ideal gas limit for perturbing ions. Because of isotropy, we can write (Griem, 1997) W (F) = 4π F 2 P(F)d F,

(1.36)

if P(F) describes the probability of finding the field vector in d Fx d Fy d Fz and W (F) is the distribution of field strength magnitudes. The general expression for P(F) is  P(F) =

 ···

δ(F −

n 

F j ) p(r1 , r2 · · · rn )dr1 dr2 · · · drn

(1.37)

j=1

in terms of the fields produced by n ions j which are at positions r j . Holtsmark assumed unshielded Coulomb fields and uniform distributions p(r1 , ...) of the positions, i.e., p = V −n for a normalization volume V . In this limit, it is easy to calculate the Fourier transform A(k) = A(k) of P(F), which is obtained by multiplication with exp(ik · F)dF and integration over the field. Actually, A(k) is only a function of k, and the 3-n dimensional integral in position space becomes the n-th power of the integral for, say, ion 1. Except for the final inverse transform, all calculations can be done analytically, leading to

10

1 Measurements of Solar Magnetic Field

W H (F) = H0 (β)/F0

(1.38)

β = F/F0

(1.39)

with the reduced field strength

and Holtsmark’s normal field strength  F0 = 2π

4 15

2/3

z p e 2/3 z p e 2/3 N p ≈ 2.603 N 4π0 4π0 p

(1.40)

in terms of ion charge z p and density N p . Figure 1.3 contains a plot of the Holtsmark function H0 (β), together with distribution functions more appropriate for dense plasmas to be discussed now. Please refer to the discussion on the broadening of hydrogen lines in Sect. 1.13.1. The other field strength distributions shown in Fig. 1.3 include two improvements of the physical model. The configuration space distribution functions can be corrected for correlations between ions by introducing (Mozer & Baranger, 1960) two particle correlation functions g(|ri − rk |) = g(rk ) into p(r1 , r2 , · · · , rn ) = V −n [1 +



g(r jk ) + · · · ]

j 1000G, the LS method has been used.

54

1 Measurements of Solar Magnetic Field

May 5 06:52:52 UT

May 5 06:51:18 UT

Fig. 1.23 The white light and 171 Å images observed by the TRACE satellite on 5 May 1999 in active region NOAA 8525. The size of the images is 1. 85 × 1. 85. The north is the top and the east is at the right. From Zhang et al. (2003b)

The Imaging Vector Magnetograph (IVM) at Mees Solar Observatory is yet another Stokes profile analyzing magnetograph. It has been in operation since 1992. The magnetograph includes a dedicated 28-cm aperture telescope, a polarization modulator, a tunable Febry–Pérot filter, CCD cameras, and control electronics. It takes images of areas on the Sun and records the polarization and wavelength in sequences (Mickey et al., 1996). The data reduction was described by LaBonte et al. (1999). The typical spectral line used by IVM is FeIλ 6302.5 Å. Figure 1.23 shows the white light and 171 Å images in the active region NOAA 8525 observed by the TRACE satellite on 5 May 1999. It is found that the active region consisted of a main sunspot and some small pores located northwest of the main spot. The active region is located at N22, E6 near the center of the solar disk. It is clearly seen that fibrils extended out from the center of the main sunspot in 171 Å. These provide some basic morphological information on the magnetic field in the active region atmosphere. Figure 1.24 shows three sets of the photospheric vector magnetograms overlaid by the white light and 171 Å images. We found a basic consistency of the vector magnetic field among the magnetograms of Huairou (HRM), Mitaka (MTK), and HSP/Mees (SPM). This active region was an αp region. The direction of transverse magnetic fields is roughly parallel to 171 Å fibril features and is also consistent with that of penumbral features in the white light image in Figs. 1.23 and 1.24. For the stronger transverse magnetic field (larger than 200 G) the mean error angle of the transverse magnetic field is −3.◦ 6. A similar case is found in the comparison of vector magnetograms between SPM and HRM. The mean error angle of the transverse field (larger than 200 G) between S and HRM is −12.◦ 8, while that between SPM and MTK the mean error angle is −17.◦ 5. It means that the transverse field observed by the SPM tends to be rotated counterclockwise, as one compares with that of MTK and HRM. The relative intensity of white light images is about 50 in the umbra and 250 in the quiet Sun region near the sunspot in Fig. 1.24. It is found that the scatter distribution of stronger magnetic field normally occurs where the relative intensity of white light is about 200; this is just

1.7 Observations of Vector Magnetic Field and Comparing …

55

May 5 (HR) 06:08--06:35 UT

May 5 (MTK) 01:28 UT

May 4 (SPM) 20:19--22:24 UT

May 5 06:52:52 UT

May 5 06:51:18 UT

Fig. 1.24 The vector magnetograms observed at Huairou (HR), Mitaka (MTK) and with HSP/Mees (SPM) in active region NOAA 8525. The arrows mark the directions of transverse magnetic field. The solid (dashed) contours correspond to positive (negative) fields of ±50, 200, 500, 1000, 1800, 3000 G. From Zhang et al. (2003b)

the penumbral region of the sunspot in active region NOAA 8525. As one notices that the sunspot relates to the strong longitudinal magnetic field, this actually reflects the correlation between the transverse and longitudinal field in the active region. The relationship between the transverse and longitudinal magnetic field, and also the transverse field observed at different wavelengths from the FeIλ5324.19 Å line center, were analyzed by Zhang (2000) from this active region. The observational result by HRM is consistent with the interpretation that the magneto-optical effect causes the counter-clockwise rotation of the linear polarized light of the spectral lines

56

1 Measurements of Solar Magnetic Field

Table 1.5 Active region 8525 Dif. Mag. ϕ MTK–HR HR–SPM MTK–SPM

σϕ

σT

Tnum

Snum

−3.◦ 6

20.◦ 5

−12.◦ 8 −17.◦ 5

14.◦ 5 10.◦ 2

213.6G 199.9G 177.2G

3600 255 255

1100 29 30

in the positive polarity regions. It is noticed that one can not exclude the influence of Faraday effects on the measurement of MTK vector magnetograms, which are observed at −0.08 Å of FeIλ6302.5 Å line, even if it is weaker than that obtained at the line center. The statistical results on the comparison of the transverse magnetic field in active region 8525 observed by different vector magnetographs can be found in Table 1.5. As one removes the difference on the calibration of the transverse magnetic field at different observatories and compares the correlation of the transverse components of vector magnetograms obtained by different vector magnetographs, the good statistical correlations can be found between the vector magnetograms obtained by different magnetographs, such as SPM and IVM, SPM and MTK, HRM and MTK.

1.7.2 Vector Magnetograms at Huairou and Yunnan Observatories By fitting the 2D scanned data of the solar active region by Solar Stokes Spectral Telescope (S 3 T ) at Yunnan Astronomical Observatory, distributions of the vector magnetic field has been studied by Liang et al. (2006) in Fig. 1.25, which profiles of Stokes I , Q, U , V have been shown in Figs. 1.8 and 1.9. The maximum magnetic field strength is about 2100G and locates at the south part of the umbra, somewhat away from its center. A limb-darkening function (Pierce & Slaughter, 1977) indicated that the continuum intensity dropped off quickly beyond the solar limb. For the limb-darkening, the northeast part of the AR is darker than the southwest one and the observed umbra deviates from the real one toward the solar limb. Correspondingly, the maximum value of the magnetic strength deviates from the observed umbra toward the solar disk center. From the maximum, the strength decreases monotonously. The strength is about 1400 G at the interface of the umbra and penumbra, and about 400 G at the edge of the sunspot. Using magnetic field intensity B, inclination γ and azimuth χ, Liang et al. (2006) calculate the longitudinal and transverse magnetic fields of the AR, shown in left of Fig. 1.25. Before calculating transverse magnetic fields, the 180◦ ambiguity of azimuth χ has been emendated by the methods described by Canfield et al. (1993). The contours denote the longitudinal magnetic field, while the white arrows present the transverse field. It is shown that the transverse magnetic field is radial distribution, and its maximum of about 1600 G locates at the center of the umbra. Figure 1.25

1.7 Observations of Vector Magnetic Field and Comparing …

57

Fig. 1.25 Left: White light image of the sunspot overlaid with the contours of the longitudinal magnetic field and the transverse magnetic field (denoted by white arrow). Right: FeI 5324 Å image overlaid by the transverse magnetic field denoted by black arrows. The field-of-view is 60" × 60". The north is up, and the west on the right side. After Liang et al. (2006)

Fig. 1.26 The relationship between azimuthal angles of the transverse magnetic field observed by S 3 T and HR (left) and the corresponding intensity of transverse magnetic field (right). After Liang et al. (2006)

also shows FeI 5324 Å image overlaying the transverse magnetic field obtained by Huairou Solar Observing Station of Huairou/Beijing at 02:34UT on November 17, 2002 (Ai & Hu, 1986; Zhang et al., 2003b), which is proximate to the time (02:50UT) of S 3 T observation. Every pixel stands 0.35" × 0.35" in the Huairou magnetogram (HRM) and 4.5" × 2.5" in S 3 T magnetogram respectively. To facilitate the contrast, the HRM magnetogram has been resized to 4.5" × 2.5" per pixel by linear interpolation. It is found that two magnetograms give a basic consistency of the transverse magnetic fields. The transverse magnetic field of HRM is almost radial distribution, too. The relationships between HRM and S 3 T magnetograms in the AR10197 are shown in Fig. 1.26. In the left figure, the X-axis and Y-axis

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1 Measurements of Solar Magnetic Field

denote the azimuthal angles of the transverse magnetic field observed by HR and S 3 T respectively. The linear fit of the relationship between these azimuthal angles is represented by the solid line Y = 12.6 + 0.88X with a correlation coefficient ρ Azimu = 0.86. In the right figure, the intensity of both transverse magnetic fields is denoted by X-axis and Y-axis respectively. The linear fit of the relationship between these intensities is represented by the solid line Y = 39.5 + 0.965X with a correlation coefficient ρ Bt = 0.883. The two correlation coefficients, ρ Azimu = 0.86 and ρ Bt = 0.883 indicate that the transverse magnetic fields obtained by S 3 T and HR have a strong correlation. Therefore, the features of S 3 T and HRM magnetograms are similar.

1.8 Diagnosis of Faraday Rotation with Video Vector Magnetograph 1.8.1 Stokes Parameters at Different Wavelengths of Spectral Line Figure 1.27 shows Stokes parameters Q and U observed at the different wavelengths −0.0, −0.075, −0.12, −0.15 Å from the center of FeIλ5324.19 Å line in the active region on 1999 May 5. A corresponding vector magnetogram has been shown in Fig. 1.28, where the transverse components of the field are inferred by Stokes parameters Q and U observed at −0.0 and −0.15 Å from the center of FeIλ5324.19 Å line, and Stokes parameter V at −0.075 Å.

0.0 Stokes Q

-0.075

0.0 Stokes U

-0.075

-0.12

-0.15

-0.12

-0.15

Fig. 1.27 Stokes images Q (left) and U (right) in the blue wing (0.0, −0.075, −0.12, −0.15 Å) of the FeIλ5324.19 Å line in the active region on 1999 May 5 observed at Huairou Solar Observing Station. The white (black) contours mark the positive (negative) signals. The size of maps is 0. 9 × 0. 9

1.8 Diagnosis of Faraday Rotation with Video Vector Magnetograph Fig. 1.28 A vector magnetogram inferred by the Stokes parameters Q, U and V , where Q and U are observed at the line center (blue arrows) and at −0.15 Å (red arrows) from the FeIλ5324.19 Å center, and Stokes parameter V at −0.075 Å in an active region on 1999 May 5. The solid (dashed) contours correspond to positive (negative) field. The size of maps is 0. 9 × 0. 9

59

0.00 -0.15

Figure 1.29 shows the scatter plot on the azimuthal angle correlation of “transverse components” of the magnetic field between that obtained at the line center and at the different wavelengths −0.04, −0.075, −0.12, −0.15 Å from the center of FeIλ5324.19 Å line in the active region on 1999 May 5. It is found that the difference of the azimuthal angles of the “transverse components” of the magnetic field increases with the increase of the interval of the observational wavelengths, which is the evidence of the magneto-optical effect, by comparing with the result of radiative transfer discussed above. Some of the data scatter probably also come from the following possibilities: (1) The errors of the position adjustment of the magnetograms and (2) the observational noise in the magnetograms due to the lower signals of the Stokes Q and U of the line. For analyzing these problems, we pay attention to the distribution of the transverse field in the active region. The main results are following: (1) The magneto-optical effect is a notable problem for the measurement of the transverse field with FeIλ5324.19 Å line. Although the magneto-optical effect can be neglected in the general study of the transverse magnetograms obtained by the Huairou Magnetograph (Wang et al., 1992), while its influence is obvious near the line center than that in the far wing. (2) The difference between “azimuthal angles” of the transverse field obtained at different wavelengths in the wing of FeIλ5324.19 Å line can be found in some areas of the active region. It provides good evidence of the magneto-optical effect for the measurement of the transverse field. West & Hagyard (1983) demonstrated that, for the Video Vector Magnetograph at the Marshall Space Flight Center (MSFC) system, Faraday rotation can be neglected for field strength less than 1800 G and field inclination greater than 45◦ . Hagyard & Pevtsov (1999) have shown that azimuth rotation can have a pronounced effect on the

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1 Measurements of Solar Magnetic Field

1.5

0.5

0.5

0.0 -200

2.0

-100

0 degree

100

2.0

-0.12

0 degree

100

2.0

-0.15

Bt (1000G) -100

0 degree

100

200

0.0 -200

-100

0 degree

100

0.0 -200

200

2.0

-0.12

1.0

0 degree

100

200

0.0 -200

0 degree

100

200

-100

0 degree

100

200

-0.15

1.0

0.5

0.5

-100

-100

1.5

1.5

1.0

1.0

0.5

0.0 -200

200

0.5

0.5

0.0 -200

-100

1.5

1.0

1.0

0.5

0.0 -200

200

1.5 Bt (1000G)

1.0

-0.075

1.5 Bt (1000G)

1.0

2.0

-0.04

1.5 Bt (1000G)

Bt (1000G)

Bt (1000G)

1.5

2.0

-0.075

Bt (1000G)

2.0

-0.04

Bt (1000G)

2.0

-100

0 degree

100

200

0.0 -200

Fig. 1.29 Left: The relationship between the transverse magnetic field and the “errors of azimuthal angles” of the transverse field in the active region on 5 May 1999. Right: The data is selected for that the longitudinal field is larger than 200 G and the transverse field is larger than 500 G. The ordinate marks intensity of the transverse field (obtained at the FeIλ5324.19 Å line center and the unit is 1000 G) and abscissa marks angle differences of the “transverse field” inferred by the Stokes parameters Q and U obtained at the line center and −0.04, −0.075, −0.12, −0.15 Å from the line center. From Zhang (2000)

calculation of vertical electric currents and the force-free parameter α. By studying the FeI λ5250.22 Å line, Hagyard et al. (2000) found that the Faraday rotation of the azimuth will be a significant problem for observations taken near the center of a spectral line for fields as low as 1200 G and inclinations of the field in the range 20◦ –80◦ . They also found that it is difficult to separate Faraday rotation effect from the π − σ rotation effect. To study the influence of Faraday rotation more accurately, we have attempted to recover the vector magnetic field parameters and other physical parameters such as Doppler width, opacity ratio, damping constant, and the line-center wavelength, using the inversion technique from the Stokes profiles obtained by scanning each wavelength of the FeI λ5324.19 Å line with the vector magnetograph system (HSOS). Except for the vector magnetic field parameters, others resulting from the unconstrained nonlinear least-squares inversion fitting could, in some instances, be physically unrealistic. This results from treating these parameters as independent (Balasubramaniam & West, 1991).

1.8.2 Nonlinear Least-Squares Fitting Ignoring magneto-optic effects, Auer (1977a, b) suggested that the application of nonlinear least-squares fit of the observed Stokes profiles to the analytical solutions to the radiative transfer equations for polarized radiation given by Unno (1956). The

1.8 Diagnosis of Faraday Rotation with Video Vector Magnetograph

61

analytical solutions of Stokes absorption spectral line profiles under the influence of Zeeman broadening, under the restrictive assumptions of a plane-parallel Milne– Eddington model atmosphere, have been derived by Landolfi & Landi Degl’Innocenti (1982). Thus, the fitting procedure has been enhanced to include magneto-optic effect and the line of sight velocities (Skumanich et al., 1985; Skumanich & Lites, 1987). The inversion procedure of Lites & Skumanich (1990) is robust enough to permit people to analyze Stokes profile data in an automated way. They can also correct the combined effects of stray or scattered light and the filling factor for the magnetic elements.

1.8.2.1

Linear Least-Squares Fitting

A linear fit of Stokes parameters is necessary and sufficient to estimate the crosstalk of different polarimeters. The fit between Stokes Q and U can be written in the form Q b − Q r = a1 + a2 (Vb − Vr ), Ub − Ur = a3 + a4 (Vb − Vr ).

(1.179)

A scatter plot of the standard deviation in the polarization signals as a function of their respective polarization intensity in an active region is shown in Fig. 1.30 for the sunspot penumbra. The spectral scan data were corrected as instrumental polarization

Fig. 1.30 Polarization crosstalk introduced by the polarimeter. Sample scatter plots derived from one set of ±60 mÅ data. The scatter plot of Stokes (Qb -Qr )/I versus (Vb -Vr )/I is shown on the top plot and the scatter plot for (Ub -Ur )/I versus (Vb -Vr )/I is shown on the second. The solid lines in two plots represent the linear fits to the data. From Su & Zhang (2004b)

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1 Measurements of Solar Magnetic Field

crosstalk (as determined by using one of the standard deviation data and assuming it to be independent of wavelength). The theoretical analysis shows that it is insignificant the influence of magnetooptical effects in the far wing of the line. So Faraday rotation should not be the main reason that leads to more serious differences in the azimuthal angles of the transverse components of magnetograms (Su & Zhang, 2004a, b, 2005). We explain the problem from the following two aspects: (1) one can’t expect a perfect result when applying the linear relations to correct the azimuths and magnitudes of every pixel of vector magnetograms. The correction method is suitable for most pixels, but invalid for some cases. (2) for the low signal-to-noise ratio and the predominant vertical field, the observed azimuth in the umbra is scattering and less significant for our data analysis (Hagyard et al., 2000). So we omit the data of the inner umbra in the following data reduction. As for the second problem, the umbra spectral line FeI λ5324.19 Å is asymmetric in the two wings obtained from Kitt Peak observational results (Wallace et al., 2000). The spectral line asymmetry is caused by velocity gradients in the spot (Skumanich et al., 1985). Moreover, there exist the disturbances of other lines in the wings of FeI λ5324.19 Å.

1.8.2.2

Nonlinear Least-Squares Fitting

Balasubramaniam & West (1991) applied the method to the low-resolution Stokes Q, U, and V profiles observed with the MSFC instrument and unemployed the Stokes I profile. We use the fitting method of Balasubramaniam & West (1991) to study the Faraday rotation. The 8 parameters used to optimally fit the analytical profiles to the observed profiles are line center (λ0 ), Doppler width (λ D ), damping constant (a), the slope of the source function (uB1 ), opacity ratio (η0 ), total magnetic field strength (H), the inclination of the magnetic field vector (ψ), and the azimuth of the magnetic field vector (φ). All the above parameters are treated as independent parameters. We also assume that the spectral lines are symmetric. The summation of all the wavelength points is taken care of by the i index. The a j refers to all the parameters that enter the Stokes Q, U, and V profiles, see Eq. (1.168). The weighting functions σ refer to the standard deviation (see Fig. 1.31). Detailed presentation of fitting method refers to the paper of Balasubramaniam & West (1991). Observed sample Stokes Q/I, U/I, and V/I profiles and the fitting ones from Q/I and U/I, representing a spatial point in the umbra, are depicted in Fig. 1.15. Dotted lines are the profiles fitted using the eight parameters. In our analysis, the errors in the magnetic field parameters are about 1.6%, 1.7%, and 1.0%, for the magnetic field strength, the inclination, and the azimuth, respectively. The errors of the other parameters derived from the fits shown in Fig. 1.15 are as follows: Damping constant ≈1.2%, line center wavelength ≈0.71%, opacity ratio ≈4.5%, Doppler width ≈2%, and slope of source function ≈5.8%. A detailed discussion of the accuracy of the fitting process for all the parameters is presented in William et al. (1992, Chap. 15). The Stokes profiles represented in Fig. 1.15 are convolved with the filter transmission function. There are two reasons why there are some fluctuations of the images at

1.8 Diagnosis of Faraday Rotation with Video Vector Magnetograph

63

Fig. 1.31 Scatter plots of the standard deviation in the linear and circular polarization as a function of their respective polarization intensity, shown here for the penumbra. The solid lines in the figure represent the linear fits. The standard deviation is symmetric about zero for the negative polarization. From Su & Zhang (2004b)

some wavelength locations. One is the bad seeing conditions, another is the different phases of the 5 min oscillations since the spectrum was scanned for about two hours (Balasubramaniam & West, 1991).

1.8.2.3

Theoretical Analysis of Faraday Rotation and π − σ Rotation

It is known that measurements of the orientation of linearly polarized light will yield the azimuth of the transverse magnetic field (within the constraints of the 180◦ ambiguity). However, since the linearly polarized light at the line center is perpendicular to the direction of the linearly polarized light in the wings, one must be careful in deriving the azimuth φ from the equation φ = 21 tan−1 (U/Q), where U and Q are the Stokes intensities for linearly polarized light. Hagyard et al. (2000) called this effect as the π − σ rotation.Thus, depending on the strength of the magnetic field, the magnetic sensitive line used and the spectral resolution of the measuring instrument, the azimuth of the transverse field may rotate abruptly by 90◦ from its orientation measured near the central wavelength when measured in the wings of line. Hagyard et al. (2000) argued that they could not separate the magneto-optical effect and π − σ rotation effect and designated their net effect as Zeeman–Faraday (Z-F) rotation. If S(λ) stands for the line profile of any Stokes parameter, and T(λ, λ ) is the transmission profile of the filter, where λ is the bandpass location relative to the line center, then the intensity of the transmitted Stokes parameter becomes

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1 Measurements of Solar Magnetic Field



S(λ ) =



2λ −2λ



S(λ)T (λ, λ )dλ



2λ −2λ

T (λ, λ )dλ,

(1.180)

where λ is large enough so that the integration covers the whole Stokes line profile sufficiently. We can see that the observed Stokes parameter value, S(λ ) with filter system, is a kind of averaged effect in some wavelength range, (from −2λ to 2λ), using the Stokes profile S(λ) as weight at each wavelength (λ ). Thus, we can conclude confidently that, in the profile of azimuth versus wavelength offset from a spectral line center, the ‘split’ made by π − σ rotation effect can be either diminished even eliminated. Moreover, we find that when the ‘split’ is enlarged, the azimuth is suffered from Z-F rotation more seriously than when the ’split’ is diminished or eliminated completely. For the FeIλ5324.19 Å, its Lande´ factor g = 1.5. In the solar magnetic atmosphere, its Zeeman split is less than that of line FeI 5250.22 Å, and the latter has a larger Lande´ factor g = 3. So we can deduce that in observations taken near the line center of FeI 5250.22 Å, the affection of Z-F rotation to the azimuth is more significant than that to the azimuth obtained with the FeI λ5324.19 Å line center.

1.8.3 Qualitative Estimations of Magneto-Optical Rotation Using the nonlinear least-squares inversion technique, we can obtain the best fit between observations and theory to recover the vector magnetic field parameters, etc. Thus, the observed and calculated azimuths of magnetic field versus λ derived from Stokes profiles, Q/I and U/I could also be plotted. Regarding the recovered parameter, φ as the ‘true’ azimuth, we define the observed azimuth error due to Faraday rotation φ as the difference between the ‘true’ azimuth and the model azimuth value in the line center. More quantitatively we see the following results: (1) The largest azimuth rotation is occurring in the umbra region. (2) Since these pixels span the penumbral region or lie near the umbra region, it appears that large rotations can occur there. (3) For most of the pixels, the Faraday rotation is less than 30◦ , the correlation plots show that the maximum rotation varies with H and ψ smoothly and linearly except for those pixels depicted in discussions (1) and (2). A comparison of maximum Faraday rotation as a function of field strength between HSOS and MSFC is given in Table 1.6, where the data of MSFC is acquired from the paper of Hagyard et al. (2000), and the data of HSOS by Su & Zhang (2004b). Maybe this comparison is not precise. It is only a quantitative estimation for different active regions for both magnetic sensitive lines used at MSFC and HSOS. Tables 1.7 and 1.8 show the mean value (φ) of maximum Faraday rotation as a function of field strength and inclination respectively.

1.9 Instruments for Measurement of Full Solar Disk Magnetic Fields

65

Table 1.6 Comparisons of pixels’ number of maximum Faraday rotation as a function of field strength obtained by two video vector magnetographs at HSOS and MSFC Range HSOS MSFC φ > 60◦ 50◦ < φ < 60◦ 30◦ < φ < 50◦ φ < 30◦

3/79 0 5/79 71/79 ≈ 90%

18/80 5/80 12/80 45/80 ≈ 56%

Table 1.7 The mean value (φ) of maximum Faraday rotation as a function of field strength in a sample of active regions H (G) 1000 1500 2000 2500 φ (Degree) 7.4 12.7 18.1 23.5

Table 1.8 The mean value (φ) of maximum Faraday rotation as a function of field inclination in a sample of active regions ψ (Degree) 80.0 60.0 40.0 20.0 φ (Degree) 5.9 10.8 15.7 20.5

1.9 Instruments for Measurement of Full Solar Disk Magnetic Fields Solar Magnetism and Activity Telescope (SMAT) is a new project at Huairou Solar Observing Station, National Astronomical Observatories of China started in 2003 (Zhang et al., 2007). The major directions of scientific research for SMAT are mainly included: (a) The diagnostic of Stokes parameters in the solar magnetic atmosphere with the measurements of full disk magnetic fields by the wide field of view optic system in the video vector magnetograph; (b) The evolution of magnetic field in the solar surface, especially the development of non-potential magnetic field in active regions and the interactions of the magnetic field between different active regions, especially the non-potentiality of global solar magnetic field and the triggers of flare-CMEs; (c) To understanding the large-scale vector magnetic field in the solar surface and the relationship with the generations of magnetism inside of the Sun, which concerns the formation of the large-scale magnetic field by the emergence of magnetic flux from the subatmosphere and its disappearance in the solar atmosphere, and also the formation of magnetic helicity from the solar subatmosphere; (d) The forecast the solar activities and space-weather from the observational large-scale solar vector magnetic field. The Solar Magnetism and Activity Telescope (SMAT) comprises two telescopes. One is for the measurements of full-disk video vector magnetic field and another is for full disk Hα observations. This telescope works gradually after the first light

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1 Measurements of Solar Magnetic Field

Fig. 1.32 The Solar Magnetism and Activity Telescope (SMAT) at Huairou Solar Observing Station of the National Astronomical Observatories of China. From Zhang et al. (2007) 20cm

1

filter 6562.81 2

3

filter 5324.19 10cm

1

2

3

Fig. 1.33 Optical scheme of full disk vector magnetograph (bottom) and full disk Hα telescope (top). 1—collimator lens, 2—focus lens, 3—CCD camera. From Zhang et al. (2007)

at the end of 2005. To perform the measurements of global magnetic configuration and the relationship with solar activities together as described above, we put two telescopes on a single mounting in Figs. 1.32 and 1.33.

1.9.1 Magnetic Field Measurements Full-disk video vector magnetograph in Fig. 1.33 is performed by a telescope with a telecentric optical system of 10 cm aperture and 77.086 cm effective focal length. It is noticed that in the telecentric optics, all of the points in the field of view are treated equally, even if the e-rays and o-rays in the birefringent filter give rise to

1.9 Instruments for Measurement of Full Solar Disk Magnetic Fields

67

slightly different focus position, while in the collimated optic system the wide field of view causes the variation of the bandpass of the filter in different positions of the image plane. Because the wide field of view is a basic problem in the design phase of a full disk vector magnetograph, the telecentric optics has been decided for the measurements of solar vector magnetic field with the narrow bandpass of the birefringent filter. A birefringent filter for the measurement of vector magnetic field is centered at 5324.19 Å and its bandpass is 0.1 Å. Its internal configuration is shown in Fig. 1.34 and the parameters of the optical elements are summarized in Table 1.9. Seven calcite elements are wide-fielded by inserting half-wave plates at the middle and a quartz element has been mounted in the front of calcite elements (Evens, 1949). The

Q C4a C4b HIG KD*P

IF HIG

PG1

G

G Q1 C2 PG C7a

HQ p

QH p

C7b

H p

C8a

Q H p

C8b

H

C6a

QH p

C6b

H

C5a C5b

QH

H p

HIG

C3a C3b

QH H QH p p

H QH p

Q

Fig. 1.34 Design of birefringent filter at 5325.19 Å for the measurements of the magnetic field. IF denotes interference filter, c1–c7 denote calcite, Q1 denotes quartz, G denotes glass, p denote polarizers, G denote protecting glass, HIG denotes quarter-wave plate, H and Q denote half and quarter-wave plates respectively. From Zhang et al. (2007) Table 1.9 Optical parameters of birefringent filter for the measurement of magnetic field (center wavelength = 5324.19 Å, passband = 0.1 Å, clear aperture = 37 mm) Half width w (Å) Thickness d (mm) Retardation order Material Construction n 0.1 0.2 0.4 0.8 1.6 3.2 6.4 12.8

35.196 × 2 17.598 × 2 8.799 × 2 4.399 × 2 2.200 × 2 1.100 × 2 1.100 × 1 10.440 × 1

11520 × 2 5760 × 2 2880 × 2 1440 × 2 720 × 2 360 × 2 360 × 1 180 × 1

Calcite Calcite Calcite Calcite Calcite Calcite Calcite Quartz

Wide field Wide field Wide field Wide field Wide field Wide field No wide field No wide field

68

1 Measurements of Solar Magnetic Field C4a C4b Q1a Q1b C3a C3b HIG

IF

HIG

G

G PG1 PG

HQ H HQH HQ H HQ p p p p p

C6a

C6b

C5a C5b

H

HQ p

H

Q2a

HQ p

HIG

HIG

Q2b PG2

H

p

Fig. 1.35 Design of birefringent filter at 6562.81 Å for Hα observations. The characters are similar to Fig. 1.33. From Zhang et al. (2007)

temperature of the filter is set at 42◦ and is controlled with an accuracy of 0.01◦ . The center wavelength of the filter can be tuned within ±0.5 Å. The magnetic analyzer comprises KD*P crystals sandwiched between transparent electrodes. If the voltage is so chosen so that the KD*P modulator gives a quarterwave retardation, the Stokes V (the longitudinal component of magnetic field) can be detected, while as a quarter-wave plate located in the front of the KD*P modulator and it is parallel (or at 45◦ with respect) to the axis of the entrance polaroid of the birefringent filter, Stokes U (or Q) can be used to diagnose the transverse components of the magnetic field. The retardation of the KD*P is a function of applied voltage. After the tests, the voltage of KD*P modulator has been chosen at 1050V for the measurements of magnetic field in this system. The estimation and analysis of the crosstalk are important in the diagnosing of solar vector magnetic fields. The effect of crosstalk on Stokes Q and U from Stokes V, due to the much weaker of the signals of Q and U relative to V, is a basic problem for the measurements of the transverse components of magnetic field in the regions far from the center of the solar disk in the vector magnetograms, due to a wide field of view in the optical system, relative to the normal vector magnetograph for the measurement of local solar vector magnetic field in the small field of view. The effect of crosstalk is caused by the errors of the telecentric optical system and the magnetic analyzer in the wide field of view of magnetograph mainly. The frame rate of the CCD camera is 30 frames/s and its maximum transmission rate is 60 Mbyte/s. According to the design, the spatial resolution of the full disk vector magnetograms is less than 5 and the temporal resolution for observing a full disk vector magnetogram is an order of (or less) than 10 min.

1.9.2 Hα Observations Full solar disk Hα (6562.81 Å) telescope is performed by a collimated optics of 20 cm aperture and 180.0 cm effective focal length in Fig. 1.33. A birefringent filter for the measurement of the full solar disk Hα is centered at 6562.81 Å and its bandpass is 1/4 Å. Its internal configuration is shown in Fig. 1.35

1.9 Instruments for Measurement of Full Solar Disk Magnetic Fields

69

Table 1.10 Optical parameters of birefringent filter for Hα observation (center wavelength = 6302.81 Å, passband = 0.25 Å, clear aperture = 37 mm) Half width w (Å) Thickness d (mm) Retardation order Material Construction n 0.25 0.5 1.0 2.0 4.0 8.0 16.0

23.0 × 2 11.5 × 2 5.75 × 2 2.875 × 2 1.4375 × 2 13.536 × 2 13.536 × 1

5952 × 2 5952 2976 1488 744 372 186

Calcite Calcite Calcite Calcite Calcite Calcite Quartz

Wide field Wide field Wide field Wide field Wide field Wide field No wide field

and the parameters of the optical elements are summarized in Table 1.10. Six calcite elements are wide-fielded by inserting half-wave plates at the middle and a quartz is mounted in the front of calcite elements. The temperature of the filter is set at 42◦ and is controlled with an accuracy of 0.01◦ . The center wavelength of the filter can be tuned within ±2. Å from the Hα line center. The image size of the telescope is 9 mm × 9 mm, and the size of CCD is 2029 × 2044 pixels. The frame rate is 2.1 frame/s and the maximum transmission rate is 20 Mbyte/s. According to the design, the spatial resolution of full disk Hα filtergrams is less than 2 and series of images can be observed continuously.

1.9.3 Derivation of Magnetic Field As the birefringent filter has been used for the measurements of the magnetic field in the video vector magnetograph, the measured signals are integral effects relative to the transmitted rate of the filter at different wavelengths. It can be written in the form  (1.181) Ia,b (λo ) = i a,b (λo )T (λo − λ)dλ, where i a (λ) and i b (λ) relate to the right and left circularly polarized components of light relative to the measurements of the longitudinal magnetic field, and opposite linear polarized components of Q or U light to the measurements of transverse one respectively. T (λo − λ) is the transmitted profile at the filter centered at wavelength λo . Ia (λ) and Ib (λ) are integral components of polarized lights. The normalized Stokes s(q, u, and v) can be performed by the following formulae: s(λ) =

 Ia (λ) −  Ib (λ) S(λ) = . I (λ)  Ia (λ) +  Ib (λ)

(1.182)

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Fig. 1.36 The comparison between the full disk longitudinal magnetograms observed form MDI of SOHO satellite (left) and vector magnetograph at Huairou (right) on 2006 May 22. From Zhang et al. (2007)

Figure 1.36 shows the comparison between both longitudinal magnetograms obtained by MDI of SOHO satellite and SMAT on 2006 May 22. It can be found the basic correlation of both magnetograms. This means that the observational longitudinal magnetograms of SMAT are successful. Some slight differences of both magnetograms mainly come from the noise, seeing condition, and also the observing and data reducing methods probably. Figure 1.37 shows a photospheric vector magnetogram obtained by Solar Magnetism and Activity Telescope (SMAT) on 2006 April 28. For displaying the full disk vector magnetogram well, the local area of the vector magnetogram has been shown in Fig. 1.38. The transverse components extend from the centers of active regions. It is found that the sensitivity of the longitudinal component of magnetic fields is about or lesser than 5 G and the transverse one is about 100 G in Fig. 1.39. Figure 1.40 shows a Hα filtergram obtained by SMAT at Huairou Solar Observing Station.

1.10 Effects of Polarization Crosstalk and Solar Rotation on Measuring Full-Disk Solar Photospheric Vector Magnetic Fields Observations of full-disk solar photospheric vector magnetic fields are very important for the studies of the global properties and evolution of solar magnetism. Such observations can allow us to investigate the magnetic connectivities between active regions and the non-potentiality of magnetic fields with a much wider field of view (FOV) than do local-region observations.

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Fig. 1.37 Full disk vector magnetogram observed from vector magnetograph on 2006 April 28. From Zhang et al. (2007)

1.10.1 Correction for Polarization Crosstalk: Method One For convenience, we define Q0 and U0 to be the real, linear polarization signals, Q  and U  the contaminated, linear polarization signals, and Cq and Cu the fractions of circular polarization V crosstalk in the linear polarization signals Q0 and U0 , respectively. When there is a fraction Cq of V crosstalk in the Q0 signal, we have the Q  expressions at the red and blue wings of FeI 5324.19 Å line Q λ0 +δλ1 = Q 0λ0 +δλ1 + Cq Vλ0 +δλ1 , Q −λ0 +δλ2 = Q 0−λ0 +δλ2 + Cq V−λ0 +δλ2 ,

(1.183)

where the subscript λ0 is the actual filter position. δλ1 and δλ2 refer to the wavelength shifts relative to λ0 and −λ0 due to Doppler velocity, respectively, and generally δλ1 = δλ2 for the full-disk observations. Equation (1.183) yield:

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Fig. 1.38 Local area of full disk vector magnetogram on 2006 April 28. The bars mark the transverse magnetic field. From Zhang et al. (2007)

Fig. 1.39 The intensity distribution of longitudinal and transverse magnetic field along the dashed beeline in the local area of full disk magnetogram on 2006 April 28 (bottom left), which vector magnetogram is shown in Fig. 1.38. From Zhang et al. (2007)

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Fig. 1.40 Full disk Hα filtergram on 2006 April 22. From Zhang et al. (2007)

Q λ0 +δλ1 − Q −λ0 +δλ2 = (Q 0λ0 +δλ1 − Q 0−λ0 +δλ2 ) + Cq (Vλ0 +δλ1 − V−λ0 +δλ2 ). (1.184) When the second term is far greater than the first term on the right-hand side of Eq. (1.184), then Q 0λ0 +δλ1 − Q 0−λ0 +δλ2 could be omitted and we get a equation on the fraction Cq of V crosstalk in Q signal Q λ0 +δλ1 − Q −λ0 +δλ2 = Cq (x, y)(Vλ0 +δλ1 − V−λ0 +δλ2 ).

(1.185)

A similar equation on the fraction Cu of V crosstalk in U signal is  = Cu (x, y)(Vλ0 +δλ1 − V−λ0 +δλ2 ). Uλ 0 +δλ1 − U−λ 0 +δλ2

(1.186)

where Cq (x, y) and Cu (x, y) are explicitly written as the function of spatial position (x, y) and thus they may be called the correcting distributions for polarization crosstalk. For local-region observations, because the wavelength shifts caused by solar rotation are small and uniform, the correcting distributions are relatively uniform and largely free from the spatial position function (West & Balasubramaniam, 1992). At each spatial position (xi , yi ), the above distributions Cq (xi , yi ) and Cu (xi , yi ) are estimated by making the difference in polarization measurements between the red (R) and blue (B) wings (Q B − Q R )/I versus (VB − VR )/I , and (U B − U R )/I versus (VB − VR )/I at the filter positions ±0.06 Å, respectively, over a range of 5 × 5 pixels. Here, the script B refers to the wavelength −λ0 + δλ1 and the script R the wavelength λ0 + δλ2 of the line. A linear fit through the data is used to estimate

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Fig. 1.41 The V, Q, and U images observed at the filter position +0.06 Å with the full-disk vector magnetograph of SMAT. A comparison between the polarized V and U signals in two squares indicates that the circular polarization has been intermixed into the linear polarization signal. Left is east and right to the west. From Su & Zhang (2007)

the crosstalk. The top row of Fig. 1.42 shows the full-disk correcting distributions Cq (x, y) and Cu (x, y). Their values are in the range of −0.5 to 0.5, and the values near the limb are greater than those near the disk center. The mean values of |Cq (xi , yi )| and |Cu (xi , yi )| are 14% and 23%, respectively. Additionally, there is a regular shape in the Cq and Cu distributions. This is a result of the 5 × 5 binning. To eliminate any possible correcting error introduced by it, we further smooth the maps over 80 × 80 pixels, which is equal to 2. 6 × 2. 6. Using the above Cq and Cu maps, we can get the approximately ’real’ linear polarization images through the expressions Q 0 = Q  − Cq ∗ V and U 0 = U  − Cu ∗ V . The bottom row of Fig. 1.42 shows the corrected Q  and U  images. Comparing them

Fig. 1.42 Top row: the full-disk correcting distributions Cq and Cu for V-Q(U) polarization crosstalk. A smoothing over 80 × 80 pixels will be applied before they are used to correct crosstalk. Bottom row: the corrected Stokes Q and U images with the smoothed Cq and Cu distributions. From Su & Zhang (2007)

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with those in Fig. 1.41, we can see that the contaminated U signals located in the square seem to disappear. It is found that the mean, absolute values of Q and U signals in Fig. 1.41 are 0.00145 and 0.00161, respectively, while those in Fig. 1.42 are 0.00146 and 0.00169. There are nearly no changes for the magnitudes of the corrected Q and U signals. The correlations of the Q and U signals before corrections in the square with the corresponding V signals are 17% and 56%, respectively. However, after crosstalk corrections, they change into 3% and −36%. This indicates that the correction for U signals introduces negative V-signal crosstalk in them while the correction for Q signals is relatively better. The next subsection will continue to discuss this problem.

1.10.2 Correction for Polarization Crosstalk: Method Two Equation (1.183) may be added together to give Q λ0 +δλ1 + Q −λ0 +δλ2 = Q 0λ0 +δλ1 + Q 0−λ0 +δλ2 + Cq (Vλ0 +δλ1 + V−λ0 +δλ2 ). (1.187) In the same way, a similar equation can be obtained for U:  0 = Uλ00 +δλ1 + U−λ + Cu (Vλ0 +δλ1 + V−λ0 +δλ2 ). Uλ 0 +δλ1 + U−λ 0 +δλ2 0 +δλ2

(1.188)

Can we find a pair of wavelengths in the line, where Vλ0 +δλ1 ≈ −V−λ0 +δλ2 ? Once δV = Vλ0 +δλ1 + V−λ0 +δλ2 → 0, we no longer need to estimate the correcting distributions Cq and Cu . It is well known that the net circular polarization, Stokes V, shows an antisymmetric profile shape, while its two sides have approximately symmetric shapes relative to wavelength offsets ±λ0 . Therefore, Vλ0 +δλ1 ≈ Vλ0 −δλ1 and V−λ0 −δλ2 ≈ V−λ0 +δλ2 . For Vλ0 +δλ1 = −V−λ0 −δλ2 , then Vλ0 +δλ1 ≈ V−λ0 +δλ2 . In other words, two particular offsets happen to approximately balance the two V signals at wavelength offsets: λ0 + δλ1 and −λ0 − δλ2 . After a series of comparisons, at last, we determine that two such offsets are around ±0.12 Å from the line center. Figure 1.43 shows the simulations of the ratios of the second to the first term on the right-hand side of Eqs. (1.187) and (1.188) versus the proportions of V crosstalk in the linearly polarized Q and U signals. The simulations are carried out with the following four pairs of wavelengths: (−0.16 Å, +0.08 Å), (−0.14 Å, +0.10 Å), (−0.10 Å, +0.02 Å), and (−0.08 Å, +0.04 Å). The first and the last two pairs are used to simulate the filter positions ±0.12 Å and ±0.06 Å, respectively. In addition, the first and the third pairs correspond to Doppler velocity ξ of 2.25 km/s and the others to 1.12 km/s. We can see that δV at the filter positions ±0.12 Å is much closer to zero than at ±0.06 Å. At ±0.12 Å, when ξ < 1.12 km/s, the above two ratios are close to zero even when Cq and Cu are up to 25%. For example, when ξ = 1.12 km/s and Cq = Cu = 10%, the ratio of Cq (VB + VR )/(Q 0B + Q 0R ) is 0.046 and Cu (VB + VR )/(U B0 + U R0 ) is 0.006. But when ξ = 2.25 km/s and Cq = Cu = 10%, these two ratios are 0.81 and

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Fig. 1.43 Simulation of Cq (VB + V R )/(Q 0B + Q 0R ) versus Cq and of Cu (VB + V R )/(U B0 + U R0 ) versus Cu . The solid and short-dashed lines are for the filter positions ±0.06 Å while the long-dashed and long-dashed-dot lines for the filter positions ±0.12 Å. From Su & Zhang (2007) Table 1.11 The simulated error of transverse field caused by the residual δV = Vλ0 +δλ1 + V−λ0 +δλ2 B (G)

BT (G)

ξ = 1.12 km/s δV (pos. = 30 degr )

δ BT (G)

ξ = 2.25 km/s δV (pos. = 75 degr )

δ BT (G)

500 1000 1500 2000 2500 3000

250 500 750 1000 1250 1500

−0.0004 −0.0007 −0.0009 −0.0007 −0.0004 0.0003

30 27 21 15 6.0 4.0

−0.0011 −0.0019 −0.0019 −0.0009 0.0012 0.0045

75 60 39 15 18 54

Note In this simulation, the vector magnetic field inclination γ = 30◦ , transverse field azimuth χ = 30◦ , and Cq = Cu = 10%. The filter positions are at ±0.12 Å

0.05, respectively. In Table 1.11, we give the theoretical errors of transverse field, Bt , at the offsets ±0.12 Å, due to the conversion of a fraction of δV into linear polarized signals. They greatly depend on the Doppler velocity and weakly depend on the field strength. This method is not advantageous for correcting weak transverse fields. For example, when Bt = 250 G, the error ratio δ Bt /Bt is 12% if ξ = 1.12 km/s, but this ratio increases to 30% if ξ = 2.25 km/s. In Fig. 1.44, the left and the middle column show the Q and U images at −0.12 Å and +0.12 Å filter positions, respectively. The last column shows the half-summation of the Q and U images at the two filter positions. A qualitative comparison among the corrected Stokes U signals in the square indicates that this correcting method for crosstalk seems to work as well as the correcting method one mentioned above. A quantitative analysis shows that before corrections, the correlations of the Q and U signals at filter −0.12 Å in the square with the corresponding V signals are 12% and 37%; while after corrections, they decrease to 9% and 17%, respectively. It indicates that the second method does not introduce negative V signals as the first one does.

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Fig. 1.44 Left and middle columns: the linear polarization images obtained at the filter positions −0.12 Å and +0.12 Å, respectively. Right column: the half-summation of linear polarization images obtained at the two filter positions. The square marks the region where the linearly polarized signal was contaminated by the circularly polarized signal. From Su & Zhang (2007)

In summary, method two is more effective for correcting the linear polarization signals obtained at the filter positions ±0.12 Å. For us, it is much more convenient than the first method because the circular polarization signals are not employed in the process of data reductions. The linear polarization signals can also avoid being contaminated again by the circular polarization signal. However, the drawback is that there will be a large correcting error in the weak transverse field region (Bt < 250 G), where the Doppler velocity is greater than 1.12 km/s. Although the signal-to-noise (s/n) ratio of SMFT is relatively better than that of SMAT, we find a basic consistency of the polarization signals taken with the two magnetographs. To make the comparisons more quantitative in the regions with strong field strengths, in Fig. 1.46 we plot the slice signals of the polarized images of Fig. 1.45. The signals are selected on the white lines L1 and L2 which run across the positive and negative magnetic fluxes, respectively, in the V image shown at the bottom-left corner of the first panel of Fig. 1.46. The signals on L1 are in the top row and those on L2 in the bottom row. The long-dashed lines are for the SMFT signals and the solid and short-dashed lines for the SMAT signals. Although the polarized signals of both instruments are obtained with the same integration time (summation of 256 frames), the SMFT magnetograph exhibits much higher sensitivity in measurements of vector magnetic fields than that of SMAT. This may be due to the following reason: the diameter of SMFT is 35 cm and that of SMAT 10 cm. The former thus can collect about 12 times as much light. Therefore, the detection sensitivity of SMFT is 3.5 times that of SMAT. Table 1.12 seems to indicate this point too. It shows the ratio of the summation of the absolute polarized signals of SMFT (f) on the line L1 (L2) to that of SMAT (a). t|V | is the summation of all absolute, circularly polarized signals and t|Q| (t|U | ) for that of all absolute, linearly polarized signals on L1 or L2, a1 the magnetic signal is

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Fig. 1.45 Left column: the polarization images taken by the SMAT at the filter position +0.06 Å. Middle column: the polarization images taken by the SMFT at the filter position −0.075 Å for Stokes V image and the line center for Stokes Q and U images. Right column: the half-summation of polarization images taken by the SMAT at the filter positions ±0.12 Å. The size of the images is 3 .75 × 2 .81. From Su & Zhang (2007)

Fig. 1.46 Comparisons between the SMFT and the SMAT polarized signals which are located on the lines L1 and L2 marked in the longitudinal magnetogram at the bottom-left corner of the first panel. The signals on L1 are in the bottom row and those on L2 in the top row. Long-dashed lines are for the SMFT polarized signals while solid and short-dashed lines are for the SMAT polarized signals. From Su & Zhang (2007)

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Table 1.12 The ratio of SMFT (f) magnetic signal in Fig. 1.46 to that of SMAT (a) Ratio t|V f | /t|Va1 | t|V f | /t|Va2 | t|Q f | /t|Q a1 | t|Q f | /t|Q a2 | t|U f | /t|Ua1 | t|U f | /t|Ua2 | L1 L2 Mean

2.35 1.60 1.98

4.09 2.48 3.28

1.56 2.57 2.06

1.87 3.08 2.47

3.32 3.11 3.20

3.81 3.18 3.49

Note t|V | is the summation of all absolute, circularly polarized signals on L1 or L2 and t|Q| (t|U | ) is the summation of all absolute, linearly polarized signals; a1 is the signal taken at the filter position +0.06 Å and a2 at ±0.12 Å Table 1.13 The calibration coefficients C L and C T for vector magnetic fields Offset(Å) −0.20 −0.18 −0.16 −0.14 −0.12 −0.10 −0.08 −0.06 −0.04 −0.02 0.00 CL CT

50836 33433 22753 16389 12652 10633 9952 21117 15922 11859 9876 8559 7469 6837

10626 13626 24812 2.7e07 6492 6336 6291 6289

taken at the filter position +0.06 Å and a2 at the filter positions ±0.12 Å. The average values of the ratios are in the range of 2 to 3.5. It should be noted that the SMFT magnetograph is tuned at −0.075 Å to measure circularly polarized signals and to line center to measure linearly polarized signals. The above comparisons should have been carried out at the same filter position. Table 1.13 lists the mean values of C L and C T for the longitudinal and transverse components of vector magnetic field by SMAT over 14 pixels at some wavelengths in the line. The coefficient C T is relatively small and changes smoothly near the line center in the table. This information indicates that the line center is an optimum place, with both good linearity and sensitivity suitable for the measurement of the transverse magnetic field. Figure 1.47 shows the good correlation between the vector magnetograms observed by SMFT and SMAT in the region marked with the white box on the polarized V image at the bottom-right corner of the first panel. The relationships of the longitudinal components, the transverse components, and the azimuthal angles of both vector magnetic fields are shown in the first two rows; the vector magnetograms in the last row. The filter wavelength is marked in each panel. In the scatter correlation plots, a 2.5 factor has been multiplied to the circular polarization and a 3.0 factor to the linear polarization signals of the SMAT. To remove the effect of the 180◦ -ambiguity on the azimuthal angle, about 17% (50) of total pixels are eliminated from the plots of azimuthal angle correlation. Table 1.14 lists the comparisons more quantitatively for the region shown in the white box of Fig. 1.47. We give the definitions of longitudinal and transverse magnetic field errors between two sm f t instruments as |B L ,T |=|B L ,T − B Lsmat ,T |. A similar definition for azimuthal angle sm f t smat error is χ=χ −χ . The mean magnitude differences of the vector magnetic fields are not so large: several tens Gauss for B L and about 100 G for BT . The mean error angle of the transverse fields of SMFT (center) and SMAT (+0.06 Å) is −1.1◦ , while that of SMFT (center) and SMAT (±0.12 Å) is −4.0◦ in the table. This means that the transverse field observed by the SMFT tends to be rotated clockwise. Generally speaking, although the SMFT is 2–3 times more sensitive than the SMAT, we found a high consistency of the vector magnetic fields taken by two magnetographs.

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Fig. 1.47 The relationship between the vector magnetograms observed by the SMFT and the SMAT in the region marked with the white box in the V image at the bottom-right corner of the first panel. The relationships of the longitudinal components, the transverse components and the azimuthal angles of the vector magnetic fields of both instruments are in the first two rows, the vector magnetograms in the last row. The wavelengths of both instruments are marked in each panel. In the scatter correlation plots, a 2.5 factor has been applied to the circularly polarized signal and a 3.0 factor to the linearly polarized signals of the SMAT. From Su & Zhang (2007)

Therefore, we can confirm the polarization Q, U, and V signals measured by the SMAT. In Fig. 1.48, we present the simulated calibration curves (solid and diamond lines) for vector magnetic fields, and the long- and short-dashed lines are the linear fits to them. The simulation is carried out at −0.12 Å off the line for the longitudinal field and at both −0.12 Å off the line and the line center for the transverse field. The theoretical calibration curve for the transverse field at the line center behaves with better linearity than at −0.12 Å off the line. The large differences between the simulation and the linear fit occur at ∼2000 G for B L calibration and ∼1000 G for BT calibration. Table 1.15 lists the linear calibration errors at different wavelengths in the line for longitudinal field B L = 2165 G and transverse field BT = 1250 G. We can see that the weak-field approximations are adopted in the inversion of vector field measurement of B L at −0.12 Å off the line introduces the least error, while the measurement of BT at the line center can introduce the least error. It thus indicates that we should carry out the measurements of vector magnetic fields at these two wavelengths to minimize the calibration errors. However, we should note that to

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Table 1.14 The statistical results comparing vector magnetograms observed by two vector magnetographs Dif. mag. |B L | σ BL |BT | σ BT χ σχ Tnum SMFT32.0 G SMAT(a1) SMFT48.1 G SMAT(a2)

23.1 G

81.2 G

79.0 G

−1.1◦

21.2◦

300

36.1 G

98.0 G

86.8 G

−4.0◦

26.7◦

300

Note a1 and a2 have the same meanings as in Table 1.12. |B L |, |BT |, and χ are the mean errors of longitudinal field, transverse field, and azimuthal angle, respectively. σ BL , σ BT , and σχ are their corresponding root mean square errors. For convenience, the vector magnetograms are compressed to reduce the data pixels. Tnum is the total point numbers of |B L | and |BT |. Please notice that to remove the 180◦ -ambiguity effect on the azimuthal angle errors, 50 pixels are eliminated from the statistical number of χ

Fig. 1.48 Simulation of the calibration curves for the longitudinal (solid line) and the transverse (solid and diamond line) components of vector magnetic fields, where the Stokes V signal is obtained at the wavelength offset −0.12 Å off the line and Stokes Q and U signals at both the wavelength offset −0.12 Å off the line and the line center. The dashed lines are the linear fits to the calibration curves. From Su & Zhang (2007)

avoid Faraday rotation, the measurement of the transverse magnetic field must be taken at wavelengths far from the line center (West & Hagyard, 1983; Hagyard et al., 2000; Su & Zhang, 2004b; Su et al., 2006). Therefore, in daily observations, we obtain the azimuth of the transverse field at the far line wing to largely avoid the Faraday rotation and obtain the magnitude of the transverse field at the line center to largely avoid the calibration error. At about 05:20 UT on 2006 September 16, we obtained two sets of Stokes Q, U, and V images at filter positions ±0.06 Å, respectively, each of which is the sum of 1024 frames. First, we remove the circular polarization crosstalk in the linear polarization signals using correction method one introduced in Sect. 1.10.1. Then we selected two MDI Dopplergrams obtained at 00:30 UT and 17:41 UT, respectively. After being smoothed over 2 × 2 pixels, an average of two MDI images is used to calculate the wavelength shifts in the polarization images. Finally, a vector magnetogram is built

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Table 1.15 Linear calibration errors when B L = 2165 G and BT = 1250 G Wavelength (Å) −0.12 −0.10 −0.08 −0.06 −0.04 −0.02 δ B L /B L (%) δ BT /BT (%)

5.20 13.2

7.6 14.1

10.0 14.5

12.4 13.2

14.3 9.5

15.7 5.3

0.00 3.6

Fig. 1.49 The longitudinal magnetograms of 2006 September 16 before (left) and after (right) calibration. The image is the sum of 1024 frames. From Su & Zhang (2007)

up from one set of images. Figure 1.49 shows the longitudinal magnetograms before (left) and after (right) calibrations. Toward the east limb, the signals were weakened in comparison with the west limb due to the wavelength shift towards the line center by the solar rotation. After calibration, we can see clearly that the weakened signals are recovered. As to the transverse magnetogram, there are no obvious differences between the images before and after calibration, so we don’t show them.

1.11 Stokes Profile Analysis on Measuring Full-Disk Solar Photospheric Magnetic Fields Based on the spectral profile analysis of filter magnetograph and the application of astronomical data analysis technique, the full-disc distribution of Stoke parameter crosstalk in the Solar Magnetism and Activity Telescope (SMAT) has been acquired quantitatively by Wang et al. (2008, 2010), who produce a correction template to remove the full-disc Stokes parameter crosstalk. This template is important to the accuracy of full-disc longitudinal magnetogram. Then an observational calibration method under the weak-field approximation has been proposed on the basis of the scanning spectrum of filter magnetograph. The following main results have been

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obtained by Wang et al. (2010): For the original Stokes V images, there is about −1.67 × 10−3 of Stokes I crosstalking into V at all wavelength positions and this crosstalk presents itself as a horse-saddle surface distribution on full-disc. After being calibrated into magnetic flux density, the Stokes I-to-V crosstalk has a amplitude about 50 G, but introduces negative and positive spurious magnetic field into the blue wing and red wing magnetograms respectively.

1.11.1 Spatial Integration Scanning Spectra of Filter Magnetograph Traditionally, the crosstalk among four polarization parameters, the Stokes I, Q, U, and V, should be express as Iobs = M · I the Mueller Matrix multiplying with the Stokes vector (Stenflo, 1994; Keller et al., 2003). When we only consider the crosstalk in the Stokes V measurement, a general expression has been written from the fourth line of Iobs = M · I , Vobs = V + M41 ·I + M42 ·Q + M43 ·U,

(1.189)

where the Stokes Iobs ≈ 0.5I and M44 ≈ 1 are assumed for the intensity. As the usual case for the measurement of filter magnetograph, the observed Stoke V is normalized by the intensity, and then spatially averaged within the unresolved pixel or “manmade-defined” area. In the following integration, S0 expresses the integral area and also the normalizing denominator.  S0

Vobs d S · = I S0

 

V Q U + M41 + M42 · + M43 · I I I

 ·

dS . S0

(1.190)

S0

When the formula (1.190) is applied to the wavelength scanning data set, the spatial coordinate variables (x, y) are disappear after integration. Hence the right part of formula (1.190) are three types of spectra (circular polarization spectrum from the Sun, crosstalk spectrum from intensity, and crosstalk spectrum from linear polarization). But they are not the same magnitude in many cases. Since the intrinsic degree of polarization (Q 2 + U 2 + V 2 )1/2 /I is usually  1 on the Sun except in sunspots and spatially resolved magnetic flux tubes, it is the crosstalk from the Stokes I into the other Stokes parameters that is the dominating effect (Stenflo, 1994, p. 326). For the first spatial integration type, we choose a 22 × 22 solar center region in which there is a plage region and no sunspots. As the transverse field is weak for this region, the last two crosstalk terms are neglected but the Stokes I crosstalk term reserves itself in formula (1.190). Then the spatial averaged spectrum of this region is decomposed into two parts as follows:

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Fig. 1.50 Two types of spatial integration scanning spectra from 01:09 to 01:58 UT on 2008 December 31. The left curve is spatially sampled of local region 22” × 22” near the center of solar disk. The right one is the full-disc spatial average of scanning spectrum. The horizontal axis in two pictures begin from −0.6 Å, but the scanning points begin from −0.4 Å. This scanning range is the maximum range for present SMAT. The dashed give the level of polarization √ √ lines in two pictures zero-point. The error bar definitions are σ/ n in (a) and 10œ/ n in (b), where σ is the standard deviation and n is CCD pixels used during the spatial average. From Wang et al. (2010)

 Sc

Vobs d S · = I Sc

 

 V dS V + M41 · =  Sc + M41  Sc , I Sc I

(1.191)

Sc

where the symbol  Sc means average inside area Sc . The first part V /I  Sc is a spectrum antisymmetry to the spectral line center (Fig. 1.50a). The second part (the

M41  Sc ≈ −1.67 × 10−3 is a constant along the dimension of wavelength √ dashed line in Fig. 1.50a). The standard deviation of mean value (σmean = œ/ n) is adopted as the error bar definition of the scatter points V /I  Sc in Fig. 1.50a, where σ is the standard deviation. The second term M41  Sc just makes the spectrum

V /I  Sc shift upward or downward in Fig. 1.50a and it does not change the shape of the Stokes V. Because this circular polarization signal has such a baseline (dashed line in Fig. 1.50a), we call M41  Sc a name “polarization zero-point” of solar center region. The polarization zero-point problem has been mentioned by Ulrich et al. (2002), Demidov (1996), and (Stenflo, 1994, p. 290). However, both the meaning and the reason of “zero-point” are not be completely the same. For the second spatial integration type, the integration area is enlarged to the full-disc (S f means full-disc area). In this case, the opposite polarities in circular and linear polarization from the Sun compensate each other, and the three terms containing V, Q and U should be zero in Eq. (1.190) after the full-disc average. The only surplus term is the Stokes I crosstalk if M41 remains the plus or minus sign in FOV. Of course, we must point out that the Sun has a nonzero mean field if the integration area is a projected disc having the normal line along the line of sight rather than a close surface (Scherrer et al., 1977). However, it is assumed here that SMAT could not measure the solar mean field of the full-disc Stokes V, as its magnetic

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Fig. 1.51 The top panel are the magnetograms before template correction and the lower panel are their corresponding magnetograms after the template correction. The greyscale in the whole picture is limited to 50 G to strengthen the display contrast. Please note their crosstalk correction template are the same. Due to different calibration signs, the blue wing magnetogram appears dark and the red one appears bright before correction. The solar originated Doppler field affects the magnetic sensitivity and the calibration coefficients’ distribution on the full-disc magnetogram of a single wavelength position. From Wang et al. (2010)

sensitivity does not reach 10−4 Ic . Here, we use the convention as Lin et al. (2000) to express the polarization sensitivity. In our data reduction, we find the obvious integration average term  Sf

Vobs d S · = I Sf

 M41 ·

dS = M41  S f , Sf

(1.192)

Sf

where M41  S f is also −1.67 × 10−3 . The antisymmetric Stokes V profile disappears in Fig. 1.50b as a result of the mixture of opposite polarities . That is to say, the polarization zero-point of solar center in FOV is very close to the polarization zeropoint of full-disc average in FOV. Therefore, the full-disc spatial average of Vobs /I almost becomes a√horizontal straight line in Fig. 1.50b. The error bar display in Fig. 1.50b is 10œ/ n, which is 10 times of the error bar display in Fig. 1.50a. It is because averaging the data set on full-disc has less statistical error than averaging the data set in local region. Figure 1.51 shows the magnetograms from different wavelength positions: −12 m Å, +12 m Å, −7 mÅ, and +7 m Å. The observing times are also different: December of 2008 and June of 2009. The original magnetograms are shown in the upper panel of Fig. 1.51 and their corresponding corrected magnetograms are shown in the lower panel. It is just used one correction template for all of them. The correction effect in Fig. 1.51 is strengthened as one limited the grayscale of all the magnetograms to ±50 G. Table 1.16 lists some calibration coefficients at ten wavelength positions,

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1 Measurements of Solar Magnetic Field

Table 1.16 SMAT calibration coefficients of longitudinal magnetic field (Unit: Gauss), From Wang et al. (2010) Blue wing −0.15 Å −0.12 Å −0.10 Å −0.05 Å −0.02Å Cali. coeff. 24806 19728 16402 14316 30568 M41 −15∼−67 −12∼−53 −10∼−44 −9∼−39 −19∼−83 correction Red wing +0.02Å +0.05Å +0.10 Å +0.12Å +0.15 Å Cali. coeff. −26764 −12386 −12249 −16393 −21197 M41 16∼73 8∼34 7∼33 10∼45 13∼58 correction

and also the minimum and maximum I-to-V crosstalk correction corresponding to these observing wavelength positions by Wang et al. (2010).

1.12 Solar Model Atmospheres with Non-local Thermodynamic Equilibrium It should be said that the solar model atmosphere of the nonlocal thermal dynamic equilibrium (NLTE or Non-LTE) is an important method or a better approximation to study the solar upper atmosphere. Here we mainly introduce the basic concept of nonlocal thermal dynamic equilibrium.

1.12.1 General form of Statistical Equilibrium We let Rabs denote that absorption rate from the lower levels μ and μ , while R abs is the absorption rate to the upper levels r and r  in the atomic statistical equilibrium equation. Similar Rstim and R stim represent the transition rates of stimulated emission to lower and from upper levels, respectively, and correspondingly for the spontaneous transition rates Rspon and R spon . For completeness, we also introduce the collisional excitation and deexcitation rates Rcoll.exc , R coll.exc , Rcoll.deexc and R coll.deexc with the relationship (Stenflo, 1994) iωm,m  ρm,m  = Rabs + R stim + R spon + Rcoll.exc + R coll.deexc − R abs − Rstim − Rspon − R coll.exc − Rcoll.deexc .

(1.193)

For the diagonal terms of the density matrix with m  = m, the left-hand side of Eq. (1.193) vanishes, since ωmm  = 0. Contributions from the small off-diagonal terms

1.12 Solar Model Atmospheres with Non-local Thermodynamic Equilibrium

87

to the diagonal ρmm may usually be neglected so that we only have to deal with a system of equations relating the diagonal elements to each other. For the special case of a nearly isotropic, unpolarized radiation field and complete redistribution among the substates of the levels different from the Jm level, we get the unpolarized statistical equilibrium equations ⎡



⎣

Jμ

+





 ϕν Jν dν + C Jm Jr

B Jm Jr

+

(2Jm + 1)ρmm 

 A Jr Jm + B Jr Jm

Jr



ϕν Jν dν + C Jm Jμ



Jr

=



 A Jm Jμ + B Jm Jμ

ϕν Jν dν + C Jr Jm ρrr Jr 

 B Jμ Jm

(1.194)

ϕν Jν dν + C Jμ Jm ρμμ Jμ .

Jμ

The left-hand side contains the loss terms, the right-hand side the gain terms concerning the sublevel population ρmm . Equation (1.194) follows that the assumption of complete redistribution among the sublevels of the Jμ and Jr levels leads to complete redistribution for the Jm level as well. In Eq. (1.194) we have retrieved the standard form of the statistical equilibrium equations in the theory of unpolarized non-LTE radiative transfer. We can introduce the source function in the form Sν =

N j A ji , Ni Bi j − N j B ji

(1.195)

and the relationship Ci j gj = e Ei j /kT and g j B ji = gi Bi j , C ji gi (1.196) where Bi j , B ji and A ji are Einstein coefficients, Ci j and C ji are collisional coefficients, E i j is the transition energy. NLTE population departure coefficients bi are defined as: (1.197) bl = Nl /NlL T E , bu = Nu /NuL T E A ji =

2hν 3 B ji , Ni Ci j = N j C ji , c2

with N the actual population and N L T E the Saha–Boltzmann values for the lower and upper level, respectively. Expressed in departure coefficients the general source function (1.132) of the spectral lines becomes

88

1 Measurements of Solar Magnetic Field

Sν =

and

1 2hν 3 , c2 bi −hν/κT e −1 bj

bi g j −hν/κT Ni = e . Nj b j gi

(1.198)

(1.199)

l may now be derived directly, for the transition as a The line source function Sν0 whole, from the condition of statistical equilibrium for the two-level atoms (Rutten, 2003): d N2 = N1 R12 − N2 R21 = 0, (1.200) dt

where Ri j = Ai j + Bi j Jν0 + Ci j . Now transform C12 into C21 , divide by A21 and use (1.196) for B21 /A21 : 2hν03 /c2 [(B21 /A21 )Jν0 + (C21 /A21 ) exp(−hν0 /kT ) 1 + C21 /A21 − (C21 /A21 ) exp(−hν0 /kT ) = (1 − ν0 )Jν0 + ν0 Bν0 ,

l Sν0 =

(1.201)

where Bν0 =

1 2hν 3 −hν/κT 2 c e −1

and

ν0 =

C21 . C21 + A21 + B21 Bν0

(1.202)

1.12.2 Model Chromosphere with Non-local Thermodynamic Equilibrium To study the formation of the hydrogen Hβ line in the solar magnetic atmosphere, we select different solar atmospheric models. One is the atmospheric model of the quiet Sun and another is that of sunspot umbrae in active regions to compare the difference between them. Figure 1.52 shows the distributions of the temperature T, hydrogen density Nh , and electronic density Ne with height in the atmospheric model of the quiet Sun provided by Vernazza et al. (1981), and that of the sunspot umbra by Ding & Fang (1991). The temperature in the umbral model is lower than that of the quiet Sun at the low height and increases faster with the increment of the height at the high solar atmosphere. Similar sunspot umbral atmospheric models have also been proposed by Stellmacher & Wiehr (1970, 1975), and Allen (1973). These umbral models normally show a similar tendency on the distribution of the temperature, hydrogen density, and electronic density with height.

1.12 Solar Model Atmospheres with Non-local Thermodynamic Equilibrium

89

Fig. 1.52 The distribution of temperature T, hydrogen density Nh and electronic density Ne with height in the atmospheric model of the quiet Sun by Vernazza et al. (1981) marked with VAL and the sunspot umbra by Ding & Fang (1991) marked with DF. From Zhang (2020)

The non-local thermodynamic equilibrium (Non-LTE) population departure coefficients bi are defined as: bl = Nl /NlL T E , bu = Nu /NuL T E ,

(1.203)

LT E the Saha–Boltzmann values for with N(l,u) being the actual population and N(l,u) the lower and upper levels, respectively. The treatment of Non-LTE is suitable for the analysis of the formation of the spectral lines in the upper solar atmosphere due to the weakening contribution of collisions, while the approximation of local thermodynamic equilibrium (LTE) is normally acceptable in the lower solar atmosphere due to the high denseness of the populations, where the departure coefficients bi ≈ 1 in Eq. (1.203). Figure 1.53a shows the distributions of the departure coefficients b2 and b4 of the hydrogen atomic levels n = 2 andn = 4 with the continuum optical depth in the model of the quiet Sun, which is provided by Vernazza et al. (1981). It is found that the departure coefficients bi in the lower solar atmosphere are of the order of unity, but they increase drastically with the decrement of the populations of hydrogen density Nh and electronic density Ne , and the increment of temperature T at low optical depths, and then change rapidly at very thin depths. Assuming the same thermodynamic mechanism in the quiet Sun and sunspot umbrae, the departure coefficients bi of hydrogen atomic levels in the sunspot atmosphere can be phenomenologically provided by comparing with the tendency of the departure coefficients bi in the quiet Sun. Figure 1.53b shows that the population departure coefficients bi ≈ 1 in the deep umbral model atmosphere, and they increase

90

1 Measurements of Solar Magnetic Field

Fig. 1.53 The distribution of temperature T, hydrogen density Nh , electronic density Ne , and the departure coefficients (b2 and b4 ) of Hydrogen levels with the continuum optical depth marked in lg τc scale in the atmospheric model of the quiet Sun (top) by Vernazza et al. (1981) (VAL, 1981) and the sunspot umbra (bottom) by Ding & Fang (1991). From Zhang (2020)

in the interval of lg τ = [−6, −7.5] and then decrease in lg τ = [−7.5, −8.5] gradually. In the thermodynamic equilibrium (see, Eq. (1.194)), atoms are distributed over their bound levels according to the Boltzmann excitation equation (Mihalas, 1978). The continuum normally forms at the lower solar atmosphere, thus the Planck function B(T, ν) can be used as its source function

1.13 Formation of Hβ Line in Solar Chromospheric Magnetic Field

B C (T, ν) =

1 2hν 3 , c2 exp(hν/kT ) − 1

91

(1.204)

and the source function of the hydrogen Hβ line is S L (Hβ ) =

1 2hν 3 , c2 b2 exp(hν/kT ) − 1 b4

(1.205)

where the values of b2 and b4 are shown in Fig. 1.53. The self-consistency of these departure coefficients of the Hβ line with the observations can be presented in the following sections.

1.13 Formation of Hβ Line in Solar Chromospheric Magnetic Field The study of the solar chromospheric magnetic field is important for understanding solar intrinsical properties. It is believed that solar active phenomena often associate with the complex configurations of chromospheric magnetic fields. The observations of chromospheric magnetograms were made at the Crimean, Kitt Peak, and Huairou Observatories (Severny & Bumba, 1958; Tsap, 1971; Garicia et al., 1980; Zhang et al., 1991). The Hβ line is a working line of the Huairou Magnetograph of National Astronomical Observatories of China (Zhang & Ai, 1986). The formation of the Stokes parameters of chromospheric lines, such as the Hβ line, is a notable problem for the measurement of the chromospheric magnetic field. The solar Hβ line profile is shown in Fig. 1.54. Some photospheric lines overlap in the wing. The wavelength of the Hβ line is 4861.34 Å and its equivalent width is 4.2 Å. The core of the line is formed at a height of about 1900 km (Allen, 1973). It should be pointed out that different formation heights of the Hβ line were obtained by various authors. Its oscillator strength is 0.1193 and the residual intensity at the core is 0.128 (Grossmann-Doerth & Uexkull, 1975). It is mainly Doppler broadening in the core of the Hβ line, and resonance damping and Stark broadening in the wing. The Hβ line is composed of 7 lines and all located within a width of nearly 0.1 Å. We were not able to find published values for all the parameters, so we used the formulae method given by Bethe & Salpeter (1957) and calculated, for each component line, the wavelength shift, the normalized oscillator strength, and damping constant, shown in Table 1.17. In comparison with the results given by Allen (1973) and Garicia & Mark (1965), these available parameters showed a slight difference in the mean wavelengths. When the magnetic field alone is present, the Hβ line shows the anomalous Zeeman effect. In the solar atmosphere, both magnetic field and interatomic microscopic electric field are present, the wave function of the hydrogen atom energy levels become degenerate, and different wave functions correspond to complicated energy shifts. Here we omit the transitions of the electric-quadrupole and the magnetic dipoles.

92

1 Measurements of Solar Magnetic Field

Fig. 1.54 Top: Hβ lines in the quiet Sun (solid line) and sunspot umbra (dotted line) observed by the National Solar Observatory of USA (L. Wallace, K. Hinkle, and W. C. Livingston, http://diglib. nso.edu/ftp.html). Bottom: d I /dλ inferred by the corresponding Hβ lines above Table 1.17 Parameters of Hβ line Transition

Wavelength (Å)

γ R × 108

gf

g fw

2 p1/2 − 4d3/2 2s1/2 − 4 p3/2 2 p1/2 − 4s1/2 2s1/2 − 4 p1/2 2 p3/2 − 4d5/2 2 p3/2 − 4d3/2 2 p3/2 − 4s1/2

4861.279 4861.287 4861.289 4861.298 4861.362 4861.365 4861.375

6.5452 0.8328 6.3122 0.8328 6.5452 6.5452 6.3122

0.2436 0.13702 0.00609 0.0685 0.4385 0.0487 0.0122

0.2552 0.1435 0.0068 0.0718 0.45935 0.0510 0.01278

Note γ R is the damping parameter, g f is the oscillator strength and g f w is the normalized oscillator strength

1.13.1 Radiative Transfer of Hβ Line When we study the formation of polarized light in the magnetic field with Stokes parameters, Unno–Rachkovsky equations of polarized radiative transfer of spectral lines in the solar atmospheric magnetic field take the form of Eq. (1.130). As a

1.13 Formation of Hβ Line in Solar Chromospheric Magnetic Field

93

magnetic field is present, the broadening of the hydrogen line, such as Hβ, should be the joint effect of the magnetic field and the microscopic electric field inclined at various angles and distributed according to the Holtsmark statistics. When we calculated the physical parameters of the absorption coefficient of the polarized light, the effect of broadening by the electric field, the Doppler broadening, the radiative damping, and the resonance damping should be integrated over whole directions and probabilities. It is assumed that the microscopic electric field is isotropic in the solar atmosphere, and their contribution to the polarization of the emergent light of the Hβ line is negligible, thus the angular dependence and polarization properties of the π and σ components of the transitions of the Hβ line are same as those obtained with the classical theory. It is consistent with the analysis of weak field approximation for the Hβ line by Stenflo et al. (1984). In comparison with formulae of the non-polarized hydrogen line proposed by Zelenka (1975), one can introduce the absorption coefficient of Hβ line in the magnetic field atmosphere takes the form: η p,l,r = ηa





W (β, r0 /D)dβ

0

i

+

βl

f wi H (ai , vi (m))

  2π  π  ∞  λ − λ j (θ, χ, β, m) 1  ηa W (β, r0 /D)dβ sin θdθdχ, f wj H aj, 4π λ D 0 0 βl j

(1.206) and ρ p,l,r = 2ηa



 f wi F(ai , vi (m))

W (β, r0 /D)dβ

0

i

+

βl

  2π  π  ∞  λ − λ j (θ, χ, β, m) 1  W (β, r0 /D)dβ sin θdθdχ, ηa f wj F aj, 2π λ D 0 0 βl j

(1.207) √

where ηa =

   Nl bu πe2 λ20 hc 1 − , b exp − l mc2 λ D K c bl kλT ρ

(1.208)

and K c is the continuum absorption coefficient at 5000 Å, Nl is the particle number on the lower level of the hydrogen line under LTE. f wi and f wj are normalized oscillator strengths of polarized subcomponents of the Hβ line. W (β, r0 /D) is Holtsmark distribution function of the microscopic electric field. βl measures the relative size of the shift caused by the electric field, relative to the fine structure of the hydrogen line. If βl is very small, then the effect of the electric field can be negligible. For the state of the principal quantum number n, the shift caused by the fine structure is on the order of

94

1 Measurements of Solar Magnetic Field

T1 =

8π 4 me8 . c3 h 5 n 3

(1.209)

This is in units of cm −1 . The shift caused by the electric field F is on the order of T2 =

3h Fn . 8π 2 mec

(1.210) 2/3

For Hβ line, we chose βl = 0.1189/F0 = 9.512 × 1012 Ne . Then, for the photosphere the second terms on the right of formulae (1.206) and (1.207) are the main terms and the effect of the interatomic electric field is all-important. As we go up in the chromosphere, the density of charged ions falls and the first term becomes more and more important. H(a,v) and F(a,v) in formulae (1.206) and (1.207) are the Voigt and Faraday–Voigt functions, suffixes p, l, r in η and F refer to the π component (m = 0) and both σ components (m = ±1), respectively. The θ and χ are the inclination and azimuth angles of the electric field, which will be discussed again. The first terms in the formulae (1.206) and (1.207) reflect the contribution under the weak microscopic electric field, thus the approximation of Zeeman splitting is correct. There vi (m = 1), vi (m = 0) and vi (m = −1) connected with Zeeman splitting are following, for k = 1, 2, 3, vi (m = 2 − k) = v − vm(k) ,

(1.211)

and vm(k) =

eλ2 H [(G − G  )M − G  (2 − k)]. 4πm 2e λ D

(1.212)

Here, M is the magnetic quantum number of the lower energy level, G and G  are the lande´ factors for the lower and upper levels. f wi is the normalized oscillator strengths of the Zeeman sub-components of the Hβ line and it can be obtained by eq. (6.33) of Stenflo (1994). While the second terms reflect the common contribution of the magnetic field and strong microscopic electric field to the line broadening in the formulae (1.206) and (1.207). λ j is the shift of the polarized j-component of the spectral line caused by the magnetic and strong electric field. f wj is the normalized oscillator strength of the polarized sub-components of the Hβ line in the magnetic and microscopic electric field. The formulae of the broadening of the different polarized components of the spectral line will be discussed in the next subsection. The damping parameter of i component of the line is ai =

γi λ20 , 4πcλ D

(1.213)

1.13 Formation of Hβ Line in Solar Chromospheric Magnetic Field

95

where γi = (γradiation + γr esonance + γelectr on )i .

(1.214)

Here, we followed Zelenka (1975) and represented the electric broadening by an additional damping factor, this is only an approximation.

1.13.2 Broadening of Hβ Line in Magnetic Field Atmosphere The Stark broadening of hydrogen lines in the stellar atmosphere has been studied for a long time (Griem, 1964; Vidal et al., 1970, 1973). The line broadening under the joint action of a strong magnetic field and the microscopic electric field has been done (Nguyen-Hoe et al., 1967; Mathys, 1983). The solar magnetic field is not very strong, being on the order of 10–1000 G. In general, the effect of the electric spin and coupling between the spin and the orbital angular momentum can not be negligible (Casini & Landi Degl’Innocent, 1993). Now, a hydrogen atom and a perturbing, charged particle are considered in the extra magnetic and electric field. The Hamiltonian of the hydrogen atom can be written as Hˆ = Hˆ 0 + Vˆ H + Vˆ F ,

(1.215)

where the first term on the right is the unperturbed Hamiltonian, the second and third terms represent the perturbation of the magnetic field and electric field, respectively. The perturbation equation is ( Hˆ 0 + Vˆ H + Vˆ F ) | ψ = (E 0 + E  ) | ψ,

(1.216)

where E 0 and E  are energy eigen value and perturbation, respectively. For the simplification, we suppose the magnetic field in the z-direction, the electric field at an angle θ to z-axis, and its projection in the x-y plan at an angle χ to x-axis. The electric field perturbing term is Vˆ F = F · r =F cos θz + F sin θ(cos χx + sin χy) =F0 [cos θz + sin θ(cos χx + sin χy)]β,

(1.217)

2

where β = F/F0 and F0 = 1.25 × 10−9 Ne3 esu, Ne being the electric density. We have

n, l, j, m j | Vˆ F | n, l  , j  , m j  = F0 β[cos θ n, l, j, m j | z | n, l  , j  , m j + sinθ n, l, j, m j | (cos χx + sin χy) | n, l  , j  , m j ],

(1.218)

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1 Measurements of Solar Magnetic Field

where all the symbols have their usual meanings and their specific form can be found from Slater (1960). For the magnetic perturbing term, we have (Ter Haar, 1960), if j = j’, eH m j g j δll  δm j m j , (1.219)

n, l, j, m j | Vˆ H | n, l  , j  , m j  = 2m e gj =

2l + 2 , 2l + 1

for

j = l + 0.5,

gj =

2l , 2l + 1

for

j = l − 0.5,

and, if j=j’, eH 

n, l, j, m j | Vˆ H | n, l  , j  , m j  = − 2m e

1

[(l + 1/2)2 − m 2j ] 2 2l + 1

δll  δm j m j .

(1.220)

Agai, all the symbols have their usual meanings. Left-multiply Eq. (1.220) by ψ |, and use the identity

n, l, j, m j | Vˆ H + Vˆ F | n, l  , j  , m j  ≡ K q,q  ,

(1.221)

the suffixes q and q  in K refer to various possible wave function states. Assume the perturbed wave function to be a linear combination of unperturbed wave functions: 2n  2

| ψ >=

cq  | ψq  ,

(1.222)

q  =1

the Eq. (1.219) then becomes 2n  2

(K q,q  − E  δqq  )cq  = 0,

(1.223)

q  =1

where q = 1, 2, . . . , 2n 2 . Normally, the shifts of the energy levels depend on the distribution of the magnetic and electric field nearby the hydrogen atom. The condition of non-zero solution in Eq. (1.223) is Det | K q,q  − E  δqq  |= 0.

(1.224)

Through Eq. (1.224), we can find the 2n 2 possible splits of the upper and lower energy levels and the corresponding 2n 2 probabilities of wave functions at various directions and intensities of the electric field for each sub-levels of the hydrogen line. One can calculate the shifts λ j of the sub-components of the spectral line. Normally,

1.13 Formation of Hβ Line in Solar Chromospheric Magnetic Field

97

the general solution of the eigenvalue problem for hydrogen lines is regardless of the direction of the magnetic field and microscopic electric field. The choice of the coordinate system for the calculation of the Hermitian matrix of the perturbing Hamiltonian was made by Casini & Landi Degl’Innocent (1993). As the microscopic electric field in the solar atmosphere is isotropic, we assume that the polarized states of the sub-components of the Hβ line still rely on the magnetic field. Because the perturbed wave function is the linear combination of unperturbed wave functions and the polarized states of the sub-components of the line also relate to the magnetic quantum number m, e.g., m = 0 for the π component and m = ±1 for both σ components. The normalized oscillator strengths relate to the distribution possibilities of the wave functions, f wj (n, l, j, m −→ n  , l  , j  , m  ) ∼ c2 (n, l, j, m)c2 (n  , l  , j  , m  ) and

m = m − m  = 0, ±1.

(1.225)

(1.226)

The coefficients c(n, l, j, m) and c(n  , l  , j  , m  ) are shown in the formula (1.225) and connect with the perturbed wave function of the low and high levels of the Hβ line. This is only a simplification of the calculation of the radiative transfer of the Hβ line. Thus, the analysis of the radiative transfer of the Stokes parameters of the Hβ line in the solar atmosphere becomes possible. Once the spectral shifts, normalized strengths, and polarized states of the various transitions of the Hβ line have been determined, one can obtain the absorption coefficient for the different polarization components. In the upper layer of the solar atmosphere, the effect of the electric field distribution function can be omitted. The above formulae (1.206) and (1.207) simplify into the usual expressions for the absorption coefficient in the magnetic field. We used the anomalous Zeeman splitting formula only for 7 component lines of Hβ and considered only the Doppler broadening, the radiation damping, and resonance damping. It is a good expression for the Hβ line.

1.13.3 Numerical Calculation of Hβ Line in Atmospheric Model of Quiet Sun Under the conditions of the solar magnetic field atmosphere, we numerically solved the quantum mechanical perturbation matrix and found the shifts in the energy levels and the corresponding absorption coefficient for the Hβ line. Because the expression for the absorption coefficient is more complicated, to calculate its value at a given wavelength at a point in the atmosphere, we picked out 20 points and evaluated the absorption between the initial optical depth and ln τc = −14.0, where τc is the

98 1.0

12

a

10

b

0.8 8 rq (10- 4)

0.6 ri

Fig. 1.55 Profiles of Stokes parameters I /Ic , Q/Ic , U/Ic and V /Ic of the Hβ line calculated for the VAL model atmosphere, a homogeneous magnetic field intensity 1000–3000 G of inclination ψ = 30◦ , azimuth ϕ = 22.5◦ and μ = 1. Ic is the continuum. From Zhang (2019)

1 Measurements of Solar Magnetic Field

0.4

3000

6

2000

4 2

1000 & 3000

0.2

1000

0 0.0 0.0

0.2

0.4 0.6 0.8 Wavelength (0.1nm)

-2 0.0

1.0

10

c

0.4 0.6 0.8 Wavelength (0.1nm)

1.0

d

3000

3000 3

6

rv (10- 2)

8 ru (10- 4)

0.2

4

12

2000

4 2

1000

2

2000

1

1000

0 -2 0.0

500 0.2

0.4 0.6 0.8 Wavelength (0.1nm)

1.0

0 0.0

0.2

0.4 0.6 0.8 Wavelength (0.1nm)

1.0

continuum optical depth at 5000 Å. Then, in the course of integrating with Unno– Rachkovsky equations, the calculated values were interpolated and the calculation was continued till ln τc = −20.0. To find the magnetic-optical effect in the line formation, we also solved radiative transfer equations with ρ Q = ρU = ρV = 0. For analyzing the formation of the Hβ line, we used the VAL mean quiet chromospheric model C (Vernazza et al., 1981) and relevant non-LTE departure coefficients. We also assumed that, when the degeneracy of energy levels disappears under the action of the magnetic field and microscopic electric field, each magneton energy level keeps its original departure coefficient. It is only an approximation. The emergent Stokes profiles of the Hβ line are shown in Fig. 1.55. The calculation results show that the influence of the magneto-optical effect for Stokes parameter V is insignificant, but that for Stokes parameters Q and U in the Hβ line center is significant, for example, the error angle of the transverse field is about 7◦ . The variation of normalized calibration values of Stokes V ( in the λ = 0.3 Å from the line center) and Q (in the line center) with the magnetic field strength is shown in Fig. 1.56. We can see that the calibration values for the longitudinal and transverse components of the magnetic field of the Hβ line are basically linear. This means that the weak field approximation (Eq. 1.143) is correct for the Hβ line, 1 ∂2 I λ2H sin2 ψ sin 2ϕ, 4 ∂λ2 1 ∂2 I λ2H sin2 ψ cos 2ϕ, U ≈− 4 ∂λ2 ∂I V ≈ − λ H cos ψ, ∂λ Q≈−

(1.227)

1.13 Formation of Hβ Line in Solar Chromospheric Magnetic Field

99

2 (right diagram) Fig. 1.56 V (B)/B (left diagram) at λ = 0.3 Å from the line center and Q(B)/B⊥ ◦ at the Hβ line center calculated for the VAL model atmosphere. ψ = 30 , ϕ = 22.5◦ and μ = 1. The computational values are marked by circles and normalized to unity for B = 300 G

which was analyzed by Stenflo (1994) also. Figure 1.56 shows Stokes V of the Hβ line deduced by Stokes I through the weak field approximation in the quiet region. The amplitude of Stokes V of the Hβ line in Fig. 1.56 is almost as same as the numerical one calculated with Unno–Rachkovsky equations. However, obvious discrepancies of Stokes Q between the numerical result with Unno–Rachkovsky equations and that of weak field approximation can be found in Fig. 1.57, especially near the Hβ line center. Under the weak field approximation, the extreme value of the Stokes Q does not occur at the center of the Hβ line, but it does at the λ = 0.15 Å from the line center. It probably reflects that the real chromosphere is more complex than our estimation with the VAL model atmosphere.

1.13.4 Formation Layers of Hβ Line The contribution functions of Stokes parameters of the Hβ line calculated with the VAL model atmosphere (Vernazza et al., 1981), a homogeneous magnetic field intensity 1000 G of inclination ψ = 30◦ , azimuth ϕ = 22.5◦ and μ = 1, are shown in Fig. 1.58. We can see that peaks of the contribution functions of Stokes parameters at −0.45 Å from the Hβ line center occur in the photosphere. As the wavelength shifts toward the Hβ line center, these peaks decrease gradually and other peaks, formed in the chromosphere, increase gradually. This means that the emergent Stokes parameters of the Hβ line mainly contribute from in both layers besides the temperature minimum zone. The emergent Stokes parameters near the Hβ line center mainly form in the chromosphere and that in the wing mainly form in the photosphere, even the Hβ line form in a relative wide layer in the solar atmosphere. The average formation depth of Stokes parameters of the Hβ line is shown in Fig. 1.58. It only gives us a rough estimation of the formation layers of the observed magnetic field

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1 Measurements of Solar Magnetic Field

Fig. 1.57 Stokes profiles Q of the Hβ line. The numerical solution (left) for the VAL model atmosphere, where Q = U for no magneto-optical effect (triangle) with B = 3000 G, μ = 1, ψ = 30◦ , azimuth ϕ = 22.5◦ . As the magneto-optical effect is considered, Q (circle) is not equal to U (square). Stokes profiles Q (right) deduced by the observed I through the weak field approximation for no magneto-optical effect, where triangles mark Stokes Q deduced by the average Hβ line profile I at the center of supergranule near the center of the disk (Grossmann-Doerth & Uexkull, 1975), and stars (squares) mark that in the blue (red) wing of Hβ line (Kurucz et al., 1984). Ic is the continuum

with the Hβ line in the solar atmosphere. After comparing the relationship between the atmospheric optical depth and height in VAL model atmosphere, we see that in our calculation the emergent Stokes parameters of the Hβ line center almost form in the higher atmosphere (1500–1600 km), but that in the wing form at the lower atmosphere, for example, the Stokes parameters at −0.45 Å away from the Hβ line center reflects the information of the photospheric field (300 km). The formation height of the Balmer lines in the solar atmosphere excluding the magnetic field was analyzed also by several authors (cf. Gibson, 1973). From the calculated results, we can see that the formation heights of Stokes parameters Q, U , and V are the almost same as I . As compared with the model atmosphere of the Sun (Vernazza et al., 1976) in Fig. 1.14, it is consistent that the formation depth of Stokes parameters in the far wing of the Hβλ4861.34 Å line is mainly below the solar temperature minimum region (about τ500nm = 10−4 ∼ 10−3 ) and is located at the photosphere, while the line center forms in the chromosphere dominantly. We need to point out that real formation layers of Stokes parameters of the Hβ line are more complex than theoretical cases. Numerical results of the equations of radiative transfer depend on the selection of atmospheric models and parameters of spectral lines. For example, the formation height near the line center depends on the selection of the value of absorption coefficient of the line and in the wing of the line that also depends on the amplitude of the line broadening. The formation layers of different kind structures observed in the Hβ images in the solar atmosphere may be different, such as sunspot umbrae, penumbrae, and dark filaments, even though, these features are observed at the same wavelength in the wing of the chromospheric line.

1.14 Formation of Hβ Line in Solar Umbral Magnetic Atmosphere 1.0

3

a

b

0.00 0.06 0.15 0.25 0.45

0.8

2

0.6 Cq

Ci (104)

101

1

0.4 0 0.2

0.0 -7

-6

-5 -4 -3 -2 lgτc (Optical depth)

-1

-1 -7

0

-6

-5 -4 -3 -2 lgτc (Optical depth)

-1

0

-6

-5 -4 -3 -2 lgτc (Optical depth)

-1

0

4

3

c

d 3

Cv (102)

Cu

2

1

2

1

0 0

-1 -7

-6

-5 -4 -3 -2 lgτc (Optical depth)

-1

0

-1 -7

Fig. 1.58 Contribution function of Stokes parameters I , Q, U and V of the numerical solution of Hβλ4861.34 Å line calculated for the VAL model atmosphere at the wavelengths λ = 0.45 (cross), 0.25 (star), 0.15 (diamond), 0.06 (delta) and 0.0 (block) Å from the Hβ line center. B = 1000 G, ψ = 30◦ , azimuth ϕ = 22.5◦ and μ = 1. τc is continuum optical depth at 5000 Å. The horizontal coordinate is at the logarithm. From Zhang (2019)

However, the computation of the formation layers of the line enables us to estimate the fundamental information of the possible spatial distribution of the magnetic field.

1.14 Formation of Hβ Line in Solar Umbral Magnetic Atmosphere In Fig. 1.59, we can see that the residual intensity is significantly weakened at the Hβ line center in the observed sunspot umbra. In Fig. 1.60, it is noticed that the numerically calculated residual intensity at the core of the Stokes I profile is of the order 0.5 in the umbral model atmosphere, and the profile of Stokes I is roughly consistent with the observed umbral one in Fig. 1.59 if the blended lines are ignored. The calculated result is also marked by plus symbols in Fig. 1.59. There are no obvious differences in the Stokes I values between the umbral magnetic field strengths of 1000 and 3000 G. The amplitude of the Stokes V is of the order 10−2 , and those

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Fig. 1.59 Hβ lines in the quiet Sun (solid line) and sunspot umbra (red dotted line) observed by the National Solar Observatory of USA (L. Wallace, K. Hinkle, and W. C. Livingston, http://diglib.nso. edu/ftp.html). The blue plus symbols mark the results of the numerical calculation for the umbra. From Zhang (2020) 1.0

8

a

6

b

3000

0.8 4

1000 & 3000

2000 2 rq (10- 4)

0.6 ri

Fig. 1.60 Profiles of Stokes parameters ri , rq , ru , and rv (i.e., I /Ic , Q/Ic , U/Ic , and V /Ic ) of the Hβ line calculated for the umbral atmospheric model by Ding & Fang (1991), a homogeneous magnetic field intensity 1000–3000 G of inclination ψ = 30◦ , azimuth ϕ = 22.5◦ and μ = 1. Ic is the continuum. From Zhang (2020)

0.4

1000 0 -2 -4

0.2 -6 0.0 0.0

0.2

0.4 0.6 0.8 Wavelength (0.1nm)

-8 0.0

1.0

8 6

0.2

0.4 0.6 0.8 Wavelength (0.1nm)

1.0

4

c

d

3000

3000 4

3

2000 1000

rv (10- 2)

ru (10- 4)

2 0

2

2000

1

1000

-2 -4 -6 -8 0.0

500 0.2

0.4 0.6 0.8 Wavelength (0.1nm)

1.0

0 0.0

0.2

0.4 0.6 0.8 Wavelength (0.1nm)

1.0

of Stokes Q and U are 10−4 . These values are similar to those of the quiet Sun (see Fig. 1.55 and 1.57). This means measuring the transverse components of the magnetic fields of sunspots is still a technological challenge due to their very weak signals (Zhang, 1995d). Moreover, the peak values of Stokes V are located at 0.2 Å from the line center as calculated using the umbral model atmosphere in Fig. 1.60, and at 0.3 Å for the quiet Sun (see Fig. 1.55). Similar tendencies for Stokes Q and U can be found as

1.14 Formation of Hβ Line in Solar Umbral Magnetic Atmosphere 2.0

3

a

2

Cq

Ci (104)

b

0.00 0.06 0.15 0.25 0.45

1.5

1.0

0.5

1

0

0.0 -8

-7

-6

-5 -4 -3 -2 lgτc (Optical depth)

-1

-1 -8

0

-7

-6

-5 -4 -3 -2 lgτc (Optical depth)

-1

0

-7

-6

-5 -4 -3 -2 lgτc (Optical depth)

-1

0

4

3

c

d 3

Cv (102)

2

Cu

Fig. 1.61 Contribution functions Ci , Cq , Cu , and Cv of Stokes parameters I , Q, U , and V of the numerical solution of Hβλ4861.34 Å line calculated for the umbral atmospheric model by Ding & Fang (1991) at the wavelengths λ = 0.45 (cross), 0.25 (star), 0.15 (diamond), 0.06 (delta) and 0.0 (black) Å from the Hβ line center. B = 1000 G, ψ = 30◦ , azimuth ϕ = 22.5◦ and μ = 1. τc is continuum optical depth at 5000 Å. The horizontal coordinate is in the logarithmic scale. From Zhang (2020)

103

1

2

1

0 0

-1 -8

-7

-6

-5 -4 -3 -2 lgτc (Optical depth)

-1

0

-1 -8

well. This reflects that the contribution of the Holtsmark broadening in the umbral atmosphere, which depends on the density of the charged particles and is relatively weaker than that in the quiet Sun. This result is consistent with the abundance of the electrons Ne in both models. Figure 1.61 shows the contributions of the Stokes parameter I , Q, U , and V in the umbra model at 0.0, 0.06, 0.15, 0.25, and 0.45 Å from the center of the Hβ line. It is noticed that the contributions of the emergent Stokes parameters in the higher solar atmosphere (the continuum optical depth lg τc ≈ −7) are much weaker than that in the lower atmosphere (lg τc ≈ −1), even though there are some differences in the Stokes parameters at different wavelengths from the center of the Hβ line. This result implies that the upper umbral atmosphere is transparent to the Hβ line. The signals of the emergent Stokes parameters of the Hβ line in the umbrae formed mainly in the lower solar atmosphere, while those near the line center in the quiet Sun form in the chromosphere (see Fig. 1.58). This result also can be found by comparing the residual intensities at the center of the Hβ line for the quiet Sun and the umbra in Fig. 1.54 due to the different absorptions of the line. It means that the formation of the Hβ line in the umbrae shows an obvious contrast with that in the quiet Sun. A similar observed result on the formation depth of the sunspots in the white light has been named the Wilson effect. Bray & Loughhead (1965) contended that the true explanation of the Wilson effect lies in the higher transparency of the spot material compared to the photosphere. The morphological evidence can be found in Fig. 1.62. Similar patterns of a sunspot umbra occur both in the photospheric and the Hβ filtergrams, even if the bright ribbons of the Hβ flare are probably attached to the umbrae in the Hβ filtergram. The blended lines from the deep layer cause the reversed signals in the umbral areas of the Hβ magnetograms as indicated by

104

1 Measurements of Solar Magnetic Field 04:09 UT, May 10, 1991

01:54 UT

04:05 UT

04:46 UT

Fig. 1.62 Active region NOAA 6619 on May 10, 1991. Top left: Photospheric FeIλ 5324.19 Å filtergram. Bottom left: Photospheric vector magnetogram. Top right: Hβ filtergram. Bottom right: Hβ longitudinal magnetogram. White (black) is positive (negative) polarity in the magnetograms. The size of each figure is 2. 8 × 2. 8. From Zhang (2019)

Zhang (1993, 2019). The blended lines formed in the deep solar atmosphere cause the reversed signals in the umbral areas of the Hβ magnetogram in Fig. 1.62. It will be discussed in the Sect. 2.5.3 again. A simple schematic sketch of the formation heights of the Stokes parameters observed by the Hβ line is proposed in Fig. 1.63. The red dashed lines show the rough detectable heights in the Hβ magnetograms. In the umbrae, the observed Stokes parameters mainly form in deep layers in Fig. 1.61, although weak contributions also arise from the chromosphere in Fig. 1.63. This means that the Hβ line in the sunspot umbrae forms at the continuum optical depth lg τc ≈ −1 for our calculations, i.e., about 300 km in the photospheric layer. The sunspot umbral atmosphere is more transparent for the Hβ line. The departure coefficients of the hydrogen atom in the model umbral atmosphere under the NLTE hypothesis were presented and compared with their distributions for the quiet Sun. The locations of the peak values of the Stokes V of the Hβ line were close to the line center in the model umbral atmosphere as compared to those of the quiet Sun. This result is consistent with the weaker contribution of the micro-electric field broadening the Hβ line in the solar umbral atmosphere than in the quiet Sun. It is worth noting that, because the sunspot umbra exhibits lower temperature characteristics compared to the quiet Sun, the umbral atmosphere is generally transparent to the spectral lines that require higher temperature excitation, such as the Hβ line. Moreover, the ratio of the emergent Stokes parameters of spectral lines from different depths of the solar

1.15 Coronal Magnetic Fields

105

Chromosphere

Chromosphere

Umbra Fig. 1.63 A schematic for the estimation of the formation height of Stokes parameters in the measurements of magnetic fields with Hβ line in the sunspot regions. The red thick dashed lines mark the major contribution heights of Stokes parameters of the Hβ line, while the yellow thin dashed line does the weak contribution of the line in the chromosphere. From Zhang (2020)

atmosphere depends on the determination of the parameters used in the radiative transfer equations in the magnetic fields and the solar atmospheric models (Bai et al., 2013). In comparison with these, accurate observations are still necessary and important.

1.15 Coronal Magnetic Fields The temperature of the corona is on the order of one million degrees, and the particle number density is about the order of 1015 /m 3 . The scattering of light plays a major role. It is the optically thin outer atmosphere of the sun. Understanding the static and dynamic properties of the solar corona is one of the great challenges of modern solar physics. Magnetic fields are believed to play a dominant role in shaping the solar corona. Current theories also attribute reorganization of the coronal magnetic field and the release of magnetic energy in the process as the primary mechanism that drives energetic solar events. Next, we will briefly discuss the measurement of the coronal magnetic field from the perspective of light scattering polarization. For a detailed introduction to the polarization and scattering of light in the corona, please refer to Stenflo (1994), Landi Degl’Innocenti & Landolfi (2004).

1.15.1 Resonance Scattering To determine the radiation emitted by the classical model atom we have just to recall some important results from classical electrodynamics. It is well-known (see e.g. Jackson, 1962) that an oscillating, monochromatic dipole p(t) produces in the radiation zone an electromagnetic wave whose frequency is the same as that of the dipole,

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1 Measurements of Solar Magnetic Field

and whose electric field is described by the equation (see Landi Degl’Innocenti & Landolfi, 2004) E(r, , t) = k 2

eikr eikr ( × p(t)) ×  = k 2 p⊥ (t), r r

(1.228)

where k is the wavenumber, r is the distance from the dipole,  is a unit vector in the direction of propagation, and p⊥ = p − ( · p) is the component of the dipole in the plane perpendicular to . From this equation, it is possible to determine the polarization of the electromagnetic wave emitted by a classical dipole in any direction. If we now introduce two mutually orthogonal unit vectors perpendicular to the direction  e∗ i · e j = δi j , e∗ i ·  j = 0 (i, j = 1, 2), (1.229) we can write

Eγ =



∗  Cγi Ei ,

(1.230)

i  are given by where the direction cosines Cγi  Cγi = u · ei∗ .

(1.231)

In this case, the incident radiation (isotropic and unpolarized) at the resonant frequency ν0 is characterized by the polarization tensor Iij =

ν02 k B T δi j , c2

(1.232)

where we have used the classical expression for the Planck function (The Rayleigh– Jeans law), where T is the absolute temperature, and k B is the Boltzmann constant. We can write for the frequency-integrated radiation emitted per unit time in the solid angle d by an atomic oscillator embedded in a magnetic field  ∞  3 πe02  d  ∗  γ d Cαi Cβ∗ j Cαk Cβl Ikl ( ) Fα (ν)∗ Fβ (ν)dν, d I˜i j () = 2 mc 4π ∞ kl αβ

(1.233) where we have extracted the factor ν04 from the integral since the Fourier transforms Fα (ν) are substantially non-zero only for ν  ν0 . The frequency-integrated Stokes parameters di S˜i () scattered in the solid angle d by an atomic oscillator are given by d S˜i () =

 (σi )nm d I˜mn (), nm

with σi the Pauli spin matrices defined in Eqs. (1.16) and (1.20).

(1.234)

1.15 Coronal Magnetic Fields

107

It is similar to Eq. (1.104) and ν = ω/2π, we can write the dispersion relation  Fα (ν) =

∞

γ

e−2πi(ν0 −ανL −ν) e− 2 t dt = −

0

1 i , 2π (ν0 − αν L − ν) − i

(1.235)

where, = γ/4π. The frequency integral can be easily evaluated with the help of the residue theorem  ∞ 1 1 , (1.236) Fα (ν)∗ Fβ (ν)dν = γ 1 + i(α − β)H ∞ where H=

2πν L e0 B = and α, β = 0, ±1. γ 2mcγ

(1.237)

1.15.2 Explicit Formulae for the Stokes Parameters of the Scattered Radiation in Magnetic-Dipole Transitions Most commonly, the forbidden lines that are observed in the solar corona (e.g., the green line of Fe XIV at 5304 Å, the red line of Fe X at 6374 Å, and the infrared lines of Fe XIII at 10747 and 10798Å) are optically thin emission lines that originate in the M1 transitions |J | = 1 within the ground term, i.e., transitions of the form (α0 J ) → (α0 J0 ), where α0 specifies the atomic configuration of the ground term (the particular kind of coupling is not of concern here). These lines are excited in the corona both by isotropic collisions with charged particles (through both direct excitation and cascade processes) and by anisotropic radiation from the underlying photosphere, which is reemitted in all directions in the process known as “resonance scattering”. The observed lines then carry a polarization signature that is typical of this scattering process. If a magnetic field is present, each level J is split into a series of 2J + 1 Zeeman substates, distinguished by the magnetic quantum number, M. In that case, the polarization signature is modified with respect to the field-free case, and in principle, it carries a complete information on the local vector magnetic field (apart from the well-known 180◦ ambiguity in the determination of the magnetic-field orientation on the plane of the sky (POS)). Because only the transition (α0 J → α0 J0 ) contributes significantly to the scattered radiation from the atom within the lines spectral range, we can safely adopt the expression for the emission coefficients valid in the approximation of the twolevel atom. We can define that N is the number density of the radiating ion; A(α0 J → α0 J0 ) M1 is the Einstein coefficient for the spontaneous magnetic dipole (M1) transition (α0 J → α0 J0 ); ρ KQ (α0 J ) are the irreducible spherical components ˆ M1 of the density matrix (or statistical tensor) of the excited level (α0 J ); T QK (i, k) are the irreducible spherical components of suitable geometric tensors, specifying ˆ and the orientation of the reference the geometry of the observer (i.e., the LOS, k,

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1 Measurements of Solar Magnetic Field

direction for linear-polarization measurement) with respect to the frame of reference adopted; and (ω0 − ω) = φ(ω0 − ω) + iψ(ω0 − ω) is the complex line profile centered on ω0 (in general, φ is the Voigt profile and ψ is the associated Faraday–Voigt dispersion function), see Eq. (1.116). The multipole moments of the density matrix are defined by the expression 

 J J K = (−1) 2K + 1 ρ(αJ M, α J  M  ). M −M  −Q  MM (1.238) Once the frequency and the Einstein coefficient for the spontaneous transition (α0 J ) → (α0 J0 ) are given and the values of ρ KQ (α0 J ) (for K = 0, 2) are derived from the solution of the statistical-equilibrium equation (see Landi Degl’Innocenti & Landolfi, 2004), the four Stokes parameters of the scattered radiation can be calculated directly from equations by Casini & Judge (1999). In this section, the explicit formulae for the four Stokes parameters of the scattered radiation will be derived in order to clarify their dependence on the different diagnostic quantities. In order to simplify the notation, we introduce the population density of the excited level (see Landi Degl’Innocenti (1984), Eq. [43]), ρ KQ (αJ, α J  )



J −M

√

√ Nα0 J = N 2J + 1ρ00 (α0 J ),

(1.239)

the “reduced” statistical tensor for that level (see Landi Degl’Innocenti (1984), Eq. [39]), ρ KQ (α0 J ) , (1.240) σ QK (α0 J ) = 0 ρ0 (α0 J ) and the effective Landé factor of the transition (α0 J ) → (α0 J0 ) (e.g., Landolfi & Landi Degl’Innocenti (1982)), g¯α0 ,J α0 J0 =

1 1 (gα0 J + gα0 J0 ) + (gα0 − gα0 J0 )[J (J + 1) − J0 (J0 + 1)]. (1.241) 2 4

Since the orientation of the atomic system is negligible, we find, after some Racah’s algebra, the explicit formulae for the four Stokes parameters of the scattered radiation ˆ ˆ M1 ] ¯ − ω)[1 + D J J0 σ02 (α0 J )T02 (0, k) ε(0) 0 (ω, k) = C J J0 φ(ω ˆ = C J J0 φ(ω¯ − ω)D J J0 σ02 (α0 J )T02 (i, k) ˆ M1 , (i = 1, 2) εi(0) (ω, k)  2 ˆ ˆ M1 , ω L C J J0 φ (ω¯ − ω)[g¯α0 J,α0 J0 + E J J0 σ02 (α0 J )]T01 (3, k) ε(1) 3 (ω, k) = − 3 (1.242) where ω Nα J A(α0 J → α0 J0 ) M1 , (1.243) C J J0 = 4π 0

1.15 Coronal Magnetic Fields

   1 1 2 , D j j0 = (−1)1+J +J0 3(2J + 1) J J J0

109

(1.244)

⎡     √ 1 2 1 1 1 1 E J J0 = 3 2J + 1 ⎣(−1) J −J0 gα0 J J (J + 1)(2J + 1) J J J J J J0 ⎧ ⎫⎤ ⎨ 1 2 1⎬  gα0 J0 J0 (J0 + 1)(2J0 + 1) J0 J 1 ⎦ . ⎩ ⎭ J0 J 1 (1.245) The coefficients D J J and E J J0 of Eqs. (1.244) and (1.245) can be given as functions of J or J0 only, if we explicitly take into account that |J | = 1 for all lines of interest. We then find, for J = J0 − 1 and J = 0  1 =√ 10 (J  1 (2J =√ 5

J (2J − 1) , + 1)(2J + 3)

− 1)(2J + 3) 6J (2J + 3) − 12 g¯α0 J,α0 J +1 − gα0 J +1 , E J,J +1 J (J + 1) (2J + 1)(2J + 3) (1.246) whereas, for J = J0 + 1, D J,J +1

D J0 +1,J0 E J0 +1,J0

 (J0 + 2)(2J0 + 5) 1 , =√ 10 (J0 + 1)(2J0 + 1) 

6J0 1 (2J0 + 1)(2J0 + 5) g¯α0 J0 +1,α0 J0 − gα0 J0 , =√ 2J0 + 2 5 ((J0 + 1)(2J0 + 1)

(1.247)

the geometric tensors in Eq. (1.242) are given by 1 ˆ M1 = √ T02 (0, k) (3 cos2  B − 1), 2 2 3 ˆ M1 = √ T02 (1, k) cos 2γ B sin2  B , 2 2 3 ˆ M1 = − √ T02 (2, k) sin 2γ B sin2  B , 2 2  3 1 ˆ T0 (3, k) M1 = cos  B , 2

(1.248)

where we indicated with  B the inclination angle of the magnetic field on the line-of-sight (LOS) and with γ B the position angle of the reference direction for linear-polarization measurements in the B frame (see Fig. 1.64). Therefore, if the reference direction for linear-polarization measurements is set parallel to the

110

1 Measurements of Solar Magnetic Field

Fig. 1.64 Geometry of the magnetic field, B, and the propagation vector of the scattered radiation, k, relative to the local vertical of the Sun, x3 . Relative geometry of the magnetic field, B, the propagation vector of the scattered radiation, k, and the local vertical of the Sun at the observed point (l.v.s.). The LOS, coincident with k, is contained in the x1 x2 plane of the S frame [which has x3 ≡ (x3 )s ]. The angles ϑ B and ϕ B represent, respectively, the polar and azimuthal angles of the magnetic field in the S frame, whereas  B and  B are the corresponding angles in the frame of reference of the LOS [which has x3 ≡ (x3 )k ]. position angles γ and γ B determine the reference direction for linear-polarization measurements (r.d.l. p.), respectively, in the S frame and in the B frame [which x3 ≡ (x3 ) B ]. After Casini & Judge (1999)

projection of the magnetic field onto the plane of the sky (POS) (γ B = 0, π), from Eq. (1.248), and recalling Eq. (1.242), we see that the U-polarization vanishes, whereas the Q-polarization has the same sign of the “alignment factor,” σ02 (α0 J ) = ρ20 (α0 J )/ρ00 (α0 J ), (For E1 transitions, the Q-polarization would show the opposite sign in that case, see Sahal-Brèchot (1974), Eq. [27].) This result was originally derived by Charvin (1965). If the sign of the alignment factor cannot be assessed a priori, the conditions Q > 0 and U = 0 determine the direction of B in the POS with an ambiguity of 90◦ . Landi Degl’Innocenti & Landolfi (2004) believe that the interpretations of polarized observations in the coronal forbidden line still involve some basic problems and difficulties, such as the approximation of atomic models, magnetic dipole transitions, and low values of Einstein coefficients in weak magnetic fields. The corona, as observed in the forbidden lines mentioned above, does not present, in general, remarkable condensations. Therefore, the diagnostic content of observations in these lines is non-local, and one must ultimately rely either on coronal models of the density and magnetic field or on tomographic techniques based on the hypothesis of a rigid or quasi-rigid rotation.

1.15.3 Coronal Magnetic Field Measurements The Solar Observatory for Limb Active Regions and Coronae (SOLARC) is a 0.46 m off-axis (unobscured) reflecting coronagraph at the summit of Haleakala on Maui. It combines a large aperture (by present solar telescope standards) with a fully achro-

1.15 Coronal Magnetic Fields

111

matic, low scattered light optical design and was built to demonstrate and explore coronal Zeeman magnetometry (Kuhn et al., 2003). A long-standing solar problem has been to measure the coronal magnetic field. Lin et al. (2000) believe it determines the coronal structure and dynamics from the upper chromosphere out into the heliospheric environment. It is only recently that Zeeman splitting observations of infrared coronal emission lines have been successfully used to deduce the coronal magnetic flux density. Here they extend this technique and report first results from a novel coronal magnetometer that uses an off-axis reflecting coronagraph and optical fiber-bundle imaging spectropolarimeter. Lin et al. (2000) determine the line-of-sight magnetic flux density and transverse field orientation in a two-dimensional map with a sensitivity of about 1 G with 20 spatial resolution after 70 min of integration. These full-Stokes spectropolarimetric measurements of the forbidden Fe xiii 1075 nm coronal emission line reveal the line-of-sight coronal magnetic field 100 above an active region to have a flux density of about 4 G. The corona was observed on 2004 April 6 near the west limb passage of NOAA Active Region 10581. Figure 1.65 shows the direction and linear polarization magnitude obtained from each of the pointings near NOAA AR 10581. These results are qualitatively similar to comparable resolution Fe xiii linear polarimetry obtained during eclipse conditions. The data are superposed over a EUV Imaging Telescope (EIT) 28.4 nm Fe xv image. We might expect the polarization to trace the local coro-

Fig. 1.65 Left: Measured linear polarization of Fe xiii for all of the pointings near NOAA AR 581, superposed on an EIT 28.4 nm Fe xv image. The plotted lines show the magnitude and direction of the linear polarization. The overlapping telescope pointings and fiber positions cause the nonuniform spatial sampling. Right: Fitted Stokes V profiles and data corrected for Q/U crosstalk for the mean of the central 4 pixels in each of the 10 fiber bundle columns closest to the west solar limb. Observed and fitted Stokes I profiles are overplotted and correspond to the scale on the right. After Lin et al. (2000)

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1 Measurements of Solar Magnetic Field

nal magnetic field since it is this magnetized plasma that scatters photospheric light. This expectation is weakened by the Van Vleck effect (see Landi Degl’Innocenti & Landolfi 2004, p. 189), which rotates the polarization perpendicular to the field when the angle between the local B field at the scattering center and the local solar radial direction are larger than the Van Vleck angle (about 55◦ ). When this angle is close to the Van Vleck angle, the polarization amplitude approaches zero. Certainly, the tops of loops should exhibit polarization perpendicular to the projected field direction while other parts of a loop should scatter polarized light with a varying but generally nonradial polarization that follows the loop. We see from Fig. 1.65 that the observed polarization does tend to “respond” to the underlying loop structure, although it is also clear that this tendency is more complicated than the simple correspondence implied if the visible loops are a fair tracer of the dominant local magnetic field direction. Note that the degree of linear polarization generally increases away from the limb as the incident scattering radiation becomes more anisotropic and as collisional excitation becomes less important with declining coronal electron density. The mean least-squares fits to crosstalk corrected apparent V profiles are illustrated for each height above the limb in Fig. 1.65. Liu & Lin (2008) constructed a theoretical coronal magnetic field model of active region AR 10582 observed by the SOLARC coronagraph in 2004 by using a global potential field extrapolation of the synoptic map of Carrington Rotation 2014. Synthesized linear and circular polarization maps from thin layers of the coronal magnetic field model above the active region along the line of sight are compared with the observed maps. Liu & Lin (2008) found that the observed linear and circular polarization signals are consistent with the synthesized ones from layers located just above the sunspot of AR 10582 near the plane of the sky containing the Sun center.

1.16 Brief Overview of the History of Solar Optical Instruments by Ai (1993a) Ai (1993a) pointed out: “in 1611 the Galilean telescope opened the door to modern astronomy. The development of optical instruments for solar physics during the following about 400 years is shown in Fig. 1.66. The vertical axis characterizes the complexity of the radiation parameters analysed, from white light, spectrum or monochromatic light, polarized monochromatic light, to Stokes profiles. The horizontal indicates the spatial dimension and simultaneous field of view, from point, line, plane, to cube. From Fig. 1.66 it is clear that there is a logical relation between the advance of the complexity of the radiation parameter analyzed and the improvements of the simultaneous field of view. The pioneering measurements of solar magnetic fields were done by Hale (1908). He invented the first generation of instruments to measure the solar magnetic field by the Zeeman effect in 1908 and observed the strong magnetic fields of sunspots. The second generation was also invented at Mount Wilson Observatory in 1952 by Babcock (1953), who developed a photoelectric device

1.16 Brief Overview of the History of Solar Optical Instruments …

113

Fig. 1.66 Development of solar optical instruments From Ai (1993a)

to measure the weaker fields. Later a system to measure the vector magnetic field was developed at the Crimean Astrophysical Observatory (Severny, 1962), but all these systems could only record one spatial point at a time. In 1971 Livingston & Harvey (1971) developed the third array 512 × 1 pixels. This powerful instrument is used to record daily full-disk magnetograms. The recording efficiency was increased 512 times as compared with the Babcock system, and it allowed higher spatial and temporal resolutions. The fourth-generation instrument is the video magnetograph, in which a narrow-band birefringent filter replaces the traditional grating spectrograph, and a CCD camera at video rates is used. This system has been implemented in the 1980s (Hagyard et al., 1982; Zirin,1985; Ai & Hu, 1986). It has a very high recording efficiency with 512 × 512 or 1024 × 1024 pixels. In China, the video magnetograph is called the Solar Magnetic Field Telescope, which was propped in 1966 based on the considerations of Fig. 1.66, and which was completed in1987 after 20 years of research and construction.” “In the past decade, the video magnetograph has become a major tool for investigations of solar magnetic fields. Its great power combining about 0.5” angular resolution with about one-minute temporal resolution and high sensitivity has allowed many new results to be obtained, e.g., the fine feature of the magnetic field in active and quiet regions. Round the clock observations (at Big Bear and Huairou) to explore the long-time evolution of the magnetic field, magnetic shear, etc., because of its high quality and relatively low cost, the video system has become popular worldwide.”

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“With the beginning of the 1990s, two developments have taken place. The first concerns the multi-channel video magnetograph. The multi-channel birefringent filter (Ai & Hu, 1987a, b, c) allows several spectral lines from different solar layers to be observed simultaneously to study the 3D structure of the vector magnetic field and electric field of the solar atmosphere. It may be called the fifth-generation magnetograph due to its cubic field of view. The second concerns the investigation of stokes profiles of magnetically sensitive lines. The main aim is to accurately determine the vector magnetic field.”

1.17 Discussions for Some Challenges on Measurements of Solar Magnetic Fields Starting from the basic background of the transfer process of the magneto-sensitive spectral lines in the solar magnetic atmosphere, we have introduced the basic methods and problems in the measurements of the solar magnetic fields with the observations of the magnetic fields at the Huairou Solar Observing Station of the National Astronomical Observatories of the Chinese Academy of Sciences. After analysis, it is found that there are still some inherent frontier topics in the quantitative analysis of the solar magnetic fields that need to be further explored. Some of them are as follows: • Radiative transfer of magnetic sensitive lines: We have presented the radiative transfer equations of magnetically sensitive lines under the different approximations, whether ignoring the scattering or the assumption of thermodynamic equilibrium, such as the numerical and analytical forms (in Sect. 1.5.2), which bring some ambiguity for the accurate results. The analytical solution has normally been used to inverse the photospheric vector magnetic fields. This means that the varieties of solar atmosphere models, such as the sunspot, facula, and quiet Sun, have been ignored. From the calculated results, we can find the different magnetic sensitivities of polarized spectral lines for the different kinds of solar atmospheres, such as the quiet Sun and sunspots in Fig. 1.12. It is also noticed that the different sensitivity for the measurements of the longitudinal and transverse components of the magnetic field can be easily found from Eqs. (1.143), i.e., the longitudinal and transverse fields relate to the first- and second-order derivative of the profiles of spectral lines, respectively. In the condition Eqs. (1.145) and (1.146) of the approximation of the weak fields, the influence of magneto-optical effects has been neglected for the transverse components of the magnetic fields. We can find that these provide the constraint condition on the observations of solar magnetic fields by different approximate methods or situations.

1.17 Discussions for Some Challenges on Measurements of Solar Magnetic Fields

115

• Solar model atmosphere: We have presented the observations of photospheric vector magnetic fields with the FeIλ5324.19 Å line and chromospheric magnetic fields with the Hβλ4861.34 Å line based on the analysis of the radiative transfer equations for polarized radiation by Unno (1956) and Rachkovsky (1962a, b) under the assumption of local thermodynamic equilibrium or their extension to non-local thermodynamic equilibrium at Huairou Solar Observing Station. Even if the numerical calculation can be used in the analysis of the radiative transfer Eq. (1.130) of Stokes parameters and provides some important information on the formation of polarized spectral lines in the different atmosphere models, it still rarely has been used in the real inversion of the magnetic field, due to some difficulty and arbitrariness in the accurate selections of inputted atomic and solar parameters. It is noticed that the different formation heights of magnetic sensitive lines occur in the quiet Sun and sunspots, such as the Wilson effect of sunspots, due to the transparency of sunspots, for the measurements of the photospheric magnetic field, and the different heights for different chromospheric features, such as the prominence, fibrils, and plages, etc. This means that the observed magnetograms probably do not always provide the information of magnetic fields at the same height in the solar atmosphere, even if at the similar optical depths of the working spectral lines. A notable question is the detection of the configuration of magnetic fields in the solar eruptive process. Some studies with Stokes parameters of spectral lines can be found by Chen et al. (1989); Hong et al. (2018). • Measurements of magnetic fields in the corona: The Hanle effect is a reduction in the polarization of light when the atoms emitting the light are subject to a magnetic field in a particular direction, and when they have themselves been excited by polarized light. The diagnostic of Hanle effects with polarized light in the corona is a notable question for the measurements of the coronal magnetic fields (cf. Lin et al., 2000; Liu & Lin, 2008; Qu et al., 2009; Raouafi et al., 2016; Li et al., 2017). The analysis of the polarized lights due to the Hanle effect for the diagnostic of the coronal magnetic field is still a challenging topic (Landi Degl’Innocenti & Landolfi, 2004). • Diagnostics of electric fields in the solar atmosphere: The Stark effect is notable in the broadened wings of hydrogen and helium lines. It has been used in the analysis of the radiative transfer of the Hβ line for the measurements of magnetic fields at Huairou Solar Observing Station, see Eqs. (1.206) and (1.207).

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The Stark effect is the shifting and splitting of spectral lines of atoms and molecules due to the presence of an external electric field. It is the electric-field analog of the Zeeman effect, where a spectral line is split into several components due to the presence of the magnetic field. Normally, the isotropic plasma is statistically electrically neutral, due to the Debye shielding effect. The electric fields in the solar flares were presented by Zhang & Smartt (1986) based on the spectral analysis of the linear and quadratic components of the Stark broadening. Some presentations on the electric field inferred from the photospheric magnetic can be found in Sect. 3.8.4. The measurements of the polarized light due to the Stark effect (or electric fields) in the anisotropic plasma is also a challenging topic. • The limited spatial resolution for the measurements of magnetic fields: It is normally believed that the mean free path of photons emerging from the solar photosphere and chromosphere is about 100 km (Stenflo, 1973; Zuccarello, 2012; Judge et al., 2015). This relates to the detection of the magnetic flux tubes in the solar atmosphere and their possible distribution (Stenflo, 1973, 1994; Wang et al., 1985). • Influence of the Doppler motion for the measurements of solar magnetic fields: The influence for the measurements of magnetic fields from the Doppler velocity field due to the solar rotation was discussed (such as Wang et al. (1996)), and some for Doppler measurements of the velocity field in the solar photosphere and implications for helioseismology can be found in the paper by Rajaguru et al. (2006). Based on the above discussions, we can find that the diagnostic of solar magnetic fields in the solar atmosphere still is a fundamental topic. Although we have made some achievements in the analysis of solar vector magnetic fields based on the theory of the radiative transfer of spectral lines from observations, the quantitative study and the corresponding assessment on its accuracy remain some to be made.

Chapter 2

Basic Structures of Solar Magnetic Fields

Observations of the solar magnetic field enable humans to successfully extend the knowledge of electromagnetic physical processes beyond Earth. The discussion of the basic configuration and evolutionary form of the solar magnetic field has extremely important scientific significance in understanding the cosmic magnetic fluid framework.

2.1 Basic Description of Astrophysical Plasma Plasma is the main state of cosmic matter. Plasma astrophysics mainly studies the role of electromagnetic processes in the dynamics of cosmic matter in the gaseous state and high conductivity. It is the lack of understanding of cosmic phenomena and the characteristics of cosmic plasma that can explain the slow development of plasma astrophysics and the challenge of this research. From the pioneering work of Hannes Alfvén (1942), it became an independent branch of physics. Therefore, he received the Nobel Prize in physics in 1970.

2.1.1 Microscopic Description of Plasma The averaged Liouville equation or kinetic equation gives us a microscopic (though averaged in a statistical sense) description of the plasma state’s evolution (see, Somov, 2006; Boyd & Sanderson, 2003). Let us consider the way of transition to a less comprehensive macroscopic description of a plasma. We start from the kinetic equation for particles of kind k in the form:

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Zhang, Solar Magnetism, https://doi.org/10.1007/978-981-99-1759-4_2

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∂ f k (X, t) ∂ f k (X, t) Fk,α (X, t) ∂ f k (X, t) + vα + = ∂t ∂rα mk ∂vα



∂ fˆk ∂t

 .

(2.1)

c

Here, m k is the mass of a particle of kind k and the statistically averaged force is Fk,α (X, t) =



Fkl,α (X, X 1 ) fl (X 1 , t)d X 1

(2.2)

X1

l

and the collisional integral 

∂ fˆk ∂t

 =− c

∂ Jk,α (X, t), ∂vα

(2.3)

where the flux of particles of kind k Jk,α (X, t) =

 l

X1

1 Fkl,α (X, X 1 ) f kl (X, X 1 , t)d X 1 mk

(2.4)

in the six-dimensional phase space X = {r, v}. The second moment of the distribution function is defined to be (k) 

 (r, t) = m k

αβ

Here, we have introduced

v

(k) vα vβ f k (r, v, t)d 3 v = m k n k u k,α u k,β + pαβ .

vα = vα − u k,α

(2.5)

(2.6)

2.1.2 Equation for Zeroth Moment Let us calculate the zeroth moment of the kinetic equation  v

∂ fk 3 d v+ ∂t



∂ fk 3 vα d v+ ∂rα v

 v

Fk,α ∂ f k 3 d v= m k ∂vα

  v

∂ fˆk ∂t

 d 3 v.

(2.7)

c

We interchange the order of integration over velocities and the differentiation to time t in the first term and coordinates rα in the second one. Under the second integral vα

∂ fk ∂vα ∂ ∂ = (vα f k ) − f k = (vα f k ) − 0, ∂rα ∂rα ∂rα ∂rα

(2.8)

2.1 Basic Description of Astrophysical Plasma

119

since r and v are independent variables in phase space X. Taking into account that the distribution function quickly approaches zero as v → ∞, the integral of the third term is taken by parts and is equal to zero. Thus, by integration of Eq. (2.7), the following equation is found to the result: ∂ ∂n k + n k u k,α = 0. ∂t ∂rα

(2.9)

This is the usual continuity equation expressing the conservation of particles of kind k or (that is the same, of course) conservation of their mass: ∂ρk ∂ + ρk u k,α = 0. ∂t ∂rα

(2.10)

ρk (r, t) = m k n k (r, t).

(2.11)

Here,

is the mass density of particles of kind k.

2.1.3 Momentum Conservation Law Now let us calculate the first moment of the kinetic equation (2.1) multiplied by the mass m k :    ∂ fk ∂ fk 3 ∂ fk 3 3 vα d v + m k vα vβ d v + m k vα Fk,α d v mk ∂t ∂r ∂v α α v v v    (2.12) ∂ fˆk = m k vα d 3 v. ∂t v c

With allowance made for the definitions, we obtain the momentum conservation law ∂ ∂ (k) c (m k n k u k,α ) + (m k n k u k,α u k,β + pαβ ) − Fk,α (r, t)v = Fk,α (r, t)v . ∂t ∂rβ (2.13) (k) is the pressure tensor. Here, pαβ The mean force acting on the particles of kind k in a unit volume (the mean force per unit volume) is:  Fk,α (r, t)v =

v

Fk,α (r, v, t) f k (r, v, t)d 3 v.

(2.14)

This should not be confused with the statistical mean force acting on a single particle. The statistically averaged force is under the integral in the formula.

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In the particular case of the Lorentz force, we rewrite the mean force per unit volume as follows:   1 1 q q (2.15) Fk,α (r, t)v = n k ek E α + (uk × B)α = ρk E α + (jk × B)α . c c q

q

Here, ρk and jk are the mean densities of electric charge and current, produced by the particles of kind k. However, note that the mean electromagnetic force couples all the charged components of cosmic plasma together because the electric and magnetic fields, E and B, act on all charged components and, at the same time, all charged components contribute to the electric and magnetic fields according to Maxwell equations. If there are several kinds of particles, and if each of them is in the state of thermodynamic equilibrium, then the mean collisional force can conventionally be expressed in terms of the mean momentum loss during the collisions of a particle of kind k with the particles of other kinds: Fk,α (r, t)v = −

 m k n k (u k,α − u l,α ) l=k

τkl

.

(2.16)

Here, τkl−1 = νkl is the mean frequency of collisions between the particles of kinds k and l. This force is zero, once the particles of all kinds have identical velocities. The mean collisional force, as well as the mean electromagnetic force, tends to make astrophysical plasma a single hydrodynamic medium.

2.1.4 Energy Conservation Law The second moment Eq. (2.5) of a distribution function f k is the tensor of momentum flux density (k) αβ . In general, in order to find an equation for this tensor, we should multiply the kinetic Equation by the factor m k vα vβ and integrate over velocity space v. To derive the energy conservation law, we multiply Eq. (2.1) by the particle’s kinetic energy m k vα2 /2 and integrate over velocities, taking into account that vα = u k,α + vα , and

vα v = 0,

(2.17)

vα2 = u 2k,α + (vα )2 + 2u k,α vα .

(2.18)

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121

A straightforward integration yields ∂ ∂t



 2

 uk ρk u 2k ∂ k ρk u k,α + ρk εk + + εk + pk,β + qk.α 2 ∂rα 2  q (rad) = ρk (E · uk ) + Fk(c) · uk + Q(c) (r, t). k (r, t) + Lk

(2.19)

Here, m k εk (r, t) =

1 nk

 v

m k (vα )2 mk f k (r, v, t)d 3 v = 2 2n k

 v

(vα )2 f k (r, v, t)d 3 v

(2.20)

is the mean kinetic energy of chaotic (non-directed) motion per single particle of kind k. Thus the first term on the left-hand side of Eq. (2.19) represents the time derivative of the energy of the particles of kind k in a unit volume, which is the sum of the kinetic energy of a regular motion with the mean velocity u k and the so-called internal energy.

2.1.5 Derivation of Basic Equations of Ohm’s Law Let us write the momentum-transfer Eq. (2.13) for the electrons and ions, taking proper account of the Lorentz force (2.15) and the friction force (2.16). We have two following equations:

  ∂ (e) 1 ∂ αβ − en e E + (ue × B) + m e n e νei (u i,α − u e,α ), m e (n e u e,α ) = ∂t ∂rβ c α (2.21)

(i)   ∂ αβ 1 ∂ m i (n i u i,α ) = − Z i en i E + (ui × B) + m e n e νei (u e,α − u i,α ). ∂t ∂rβ c α (2.22) The last term in (2.21) represents the mean momentum transferred, from ions to electrons. It is equal, with the opposite sign, to the last term in (2.22). It is assumed that there are just two kinds of particles, their total momentum remaining constant under the action of elastic collisions. Suppose that the ions are protons (Z i = 1) and electrical neutrality is observed: n i = n e = n.

(2.23)

Let us multiply Eq. (2.21) by −e/m e and add it to Eq. (2.22) multiplied by e/m i . The result is

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2 Basic Structures of Solar Magnetic Fields



 ∂ e e 1 1 Eα [n e (u i,α − u e,α )] = Fi,α − Fe,α + e2 n + ∂t mi me me mi 



 ue ui + e2 n ×B + ×B me mi α α   me − νei en (u i,α − u e,α ) + (u i,α − u e,α ) . mi Here, Fe,α = −

∂

(e)

αβ

∂rβ

and Fi,α = −

∂

(2.24)

(i)

αβ

∂rβ

.

(2.25)

Let us introduce the velocity of the center-of-mass system (m i  m e ) u = ui +

me ue ≈ ui mi

(2.26)

On treating Eq. (2.24), we neglect the small terms of the order of the ratio m e /m i . On rearrangement, we obtain the equation for the current j = en(ui − ue )

(2.27)

in the system of coordinates (2.26). This equation is   1 e  ∂j e2 n E+ u×B − = (j × B) ∂t me c mec e e Fi − Fe . − νei j + mi me

(2.28)

The prime designates the electric current in the system of moving plasma, i.e., in the rest frame of the plasma. Let Eu denote the electric field in this frame of reference, i.e., 1 (2.29) Eu = E + u × B. c Now, we divide Eq. (2.28) by νei and represent it in the form j =

1 e2 n ω (e) 1 ∂j + Eu − B j × n − m e νei νei νei ∂t νei



e e Fi − Fe mi me

(2.30)

where n = B/B and ω (e) B = eB/mc is the electron gyro-frequency. Thus, we have derived a differential equation for the current j . As the electric current j has to be written in the form of general Ohm’s law

2.3 Magnetic Fields in Quiet Sun

and then

123

1 j=σ E+ u×B , c

(2.31)

 c ∂B = ∇× u × B − j . ∂t σ

(2.32)

2.2 Magnetohydrodynamics Magnetohydrodynamics (MHD) (magneto-fluid dynamics or hydromagnetics) is the study of the dynamics of electrically conducting fluids. The fundamental concept behind MHD is that magnetic fields can induce currents in a moving conductive fluid, which in turn creates forces on the fluid and also changes the magnetic field itself. The set of equations that describe MHD is a combination of the Navier–Stokes equations of fluid dynamics and Maxwell’s equations of electromagnetism. The complete set of the MHD equations for the ideal medium has the form Somov (2006): ∇p 1 ∂v + (v · ∇)v = − − B × (∇ × B), ∂t ρ 4πρ ∂B (2.33) = ∇ × (v × B), ∇ · B = 0, ∂t ∂ρ ∂s + ∇ · (ρv) = 0, + (v · ∇)s = 0, p = p(ρ, s), ∂t ∂t where s is the entropy per unit mass and the thermodynamic identities dε = T ds +

p 1 dρ and dw = T ds + dp, 2 ρ ρ

(2.34)

where ε is the kinetic energy per unit mass and w is the specific enthalpy. From MHD Eq. (2.33), it is convenient to introduce the Alfvén speed in the form: B0 uA = √ . 4πρ0

(2.35)

2.3 Magnetic Fields in Quiet Sun 2.3.1 Photospheric Magnetic Features in Quiet Sun The distribution of the magnetic field in the solar atmosphere is a very interesting topic, which concerns the possible spatial configuration of the magnetic field and its

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Fig. 2.1 Continuum intensity for the central 40 in the quiet Sun shown as a grayscale, with the L . Provided by B. Lites (Lites et al., 2008) corresponding longitudinal magnetic field of Bapp

evolution (Zhang et al., 1998). Figure 2.1 shows the observations of very quiet Sun using the Solar Optical Telescope/Spectro-Polarimeter (SOT/SP) aboard the Hinode spacecraft. It allows us to see the details of the complex distribution of the magnetic field on the solar surface.

2.3.1.1

Relationship Between Magnetic Features and Photospheric Filigrees

Time series of filtergrams and corresponding magnetograms of a quiet region near the center of the solar disk was obtained with the Swedish Vacuum Solar Telescope during the period of 15:52–17:38 UT on 17 September 1995. In the Swedish magnetograph at the La Palma, a birefringent filter is mounted in front of the CCD receiving system. The bandpass of the filter is 0.15 Å and the observing wavelength of the filter is at −0.06 Å from FeIλ5250.2 Å line center for the measurement of the magnetic field. The spatial non-uniformity of the intensity, caused by the optical system, is removed by flat fielding. The spatial resolution of magnetograms is the same as photospheric filtergrams. The magnetic features in the magnetograms are located near the emission features in Fig. 2.2, which shows a photospheric filtergram and corresponding longitudinal magnetogram. We can find that the strong magnetic field consists of small-scale magnetic features with single polarity. Some small-scale magnetic features are about 0.3 –0.4 . By comparing the magnetic field with bright structures, we can find that the magnetic field almost coincides with the distribution of bright structures in the photospheric filtergram. Since the exposure of a frame is

2.3 Magnetic Fields in Quiet Sun

125

Fig. 2.2 A series of local longitudinal magnetograms in the quiet Sun (top) and corresponding photospheric filtergrams (bottom) at 15:59:53, 16:00:33, 16:02:08, and 16:02:41 UT on 17 September 1995 (from the left to right). The size of these local magnetograms and filtergrams is 9 × 9 , respectively. The white (black) areas in the magnetograms correspond to positive (negative) fields. From Zhang et al. (1998)

Fig. 2.3 The relationship between the intensity of photospheric features and the signals of longitudinal magnetic field in the quiet Sun at 16:02:08 UT on 17 September 1995. From Zhang et al. (1998)

about 0.2 s, the signal of the weaker magnetic fields is insignificant and almost the same as the noise level. Figure 2.3 shows the statistical distribution of the photospheric pixel intensity with the circularly polarized values in the observational region. We can see that strong photospheric magnetic features tend to occur in the photospheric bright features in

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the quiet region. But we also find that the relationship is not simple, for example, some bright pixels do not correspond to the strong magnetic field. We see that the magnetic field shows network-like features and most photospheric bright features are located near the position of magnetic features. Anyway, some photospheric bright features do not have corresponding magnetic features. size of some magnetic features is in order of 1.0 × 103 km does not have a corresponding bright feature, which supports Keller’s result (1992) that the relative larger magnetic features are darker.

2.3.1.2

Lifetime of Magnetic Network Elements in Quiet Sun

Using the 155 h coordinated magnetograph data of Huairou and Big Bear Solar Observatories, Liu et al. (1994) have studied the evolution and lifetime of magnetic network elements in an enhanced network region. Both statistical and counting methods give a mean lifetime of network elements of 50 h in Fig. 2.4. The network elements are divided into two categories according to their evolution: “breakup” and “merging”. They have similar average lifetimes, which is also consistent with the result by Wang et al. (1989). It is also found that the number of elements that disappear by merging is about twice that by the breakup. This may indicate that the creation and disappearance of magnetic network elements are balanced.

2.3.2 Extending Magnetic Field from Lower Layer of Quiet Sun A possible model of the magnetic field in the quiet Sun was proposed by Gabriel (1974, 1976) in Fig. 2.5, who suggested that the structure of the quiet solar atmosphere is dominated by effects due to the magnetic field distribution produced by field concentrations at the borders of supergranulation cells. It is normally believed that the

Fig. 2.4 Time histogram to show the distribution of network lifetime. From Liu et al. (1994)

2.3 Magnetic Fields in Quiet Sun

127

Fig. 2.5 A preliminary model for the supergranule structures showing the primary (x’) and secondary (dashed) transition regions. From Gabriel (1974)

magnetic field shows a significant horizontal component in the chromosphere. Similar models in the active regions were proposed by Giovanelli (1980) and Giovanelli and Jones (1982) using the photospheric and chromospheric magnetograms. Figure 2.6 shows a high-resolution image of a local region in the quiet chromospheric networks on June 3, 2022, observed by the Daniel K. Inouye Solar Telescope in the National Solar Observatory of Unite States. This image shows a 36300 km (50”) wide area with a resolution of 18 km. The image was taken at 4861.3 Å using the Hβ line of the Hydrogen Balmer series. We can see The chromophore fibers extend upward from the bright filigrees, and the solar granulations are faintly visible. This is consistent with our previous discussion on the formation of the Hβ line. Through “deep integration” observations of a video magnetograph at Huairou Solar Observing Station of National Astronomical Observatories of China, the photospheric vector magnetic field was systematically measured near the solar south polar region on April 12, 1997, when the Sun was in the minimal phase between the 22nd and 23rd solar cycle. Deng et al. (1999) found that the polar magnetic field deviated from the normal of the solar surface by about 42.2◦ ± 3.2◦ , a stronger magnetic element may have a smaller inclination, and that within the polar cap above heliolatitude of 50◦ , the unsigned and net flux densities were 7.8 G and −3.4 G, respectively, and consequently, the unsigned fluxes were about 5.5 ×1022 Mx. The high-resolution photospheric (FeIλ5324.19 Å line) and chromospheric (Hβ line) magnetograms in the quiet regions were observed simultaneously at Huairou using the deep integration technique (Zhang & Zhang, 1998, 1999a). In the preliminary results, it is found that the magnetic features keep almost the same sizes in both layers near the center of the solar disk. To determine the configuration of the magnetic field in the quiet regions, the observational results in different (center to limb) positions in the solar disk were also analyzed, because the limb observation provides information on the horizontal components of magnetic flux tubes in the chromosphere and photosphere. Thus, how to explain this observational phenomenon becomes an interesting problem, which also concerns the formation layers of these working lines used at the Huairou Magnetograph of National Astronomical Observatories of China, especially for the chromospheric magnetograms.

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Fig. 2.6 High-resolution monochromatic image (50” × 50”) of hydrogen Hβ in the local quiet region of the sun. Provided by the National Solar Observatory of Unite States

There, we pay attention to the possible configuration of the magnetic field in the quiet regions under the estimation of the formation ranges of the spectral lines used in the magnetograph. Figure 2.7 shows two magnetograms in the quiet region near the center of the solar disk. One is observed at the −0.075 Å from the center of the FeIλ5324.19 Å line and another is made at the −0.24 Å from the Hβ line center. The observation and data reduction of the photospheric and chromospheric magnetograms were presented by Zhang & Zhang (1998). In both magnetograms, the network and some internetwork magnetic fields can be found well. We can find that the distribution of the magnetic features in the chromosphere is the same as the photospheric ones. In Fig. 2.8, we show the relationship between the chromospheric and photospheric magnetic signals point by point in both magnetograms (Zhang & Zhang, 2000a). The average signal in the photospheric magnetogram is about 5.4 times the chromospheric one, and both signals show a roughly linear relationship. If we consider the transmitted profile of the filter at the Huairou magnetograph (Ai et al., 1982), it is found that the ratio of the Stokes V /I between the FeIλ5324.19 Å and Hβ lines at the working wavelengths of the Huairou magnetograph is about 5.2– 5.9, which is inferred by the numerical calculation with different observational line

2.3 Magnetic Fields in Quiet Sun

129

Fig. 2.7 The photospheric (a) and chromospheric (b) magnetograms in the quiet region near the center of solar disk. The white (black) areas are positive (negative) polarity. The bright (white) contours correspond to positive (negative) field of 10, 20, 40, and 80 G. The size of the magnetograms is 4 .62 × 3 .40. From Zhang & Zhang (2000a) Fig. 2.8 The ratio of magnetic signals of Stokes parameter V /I between Hβ line (ordinate) and FeIλ5324.19 Å line (abscissa) at observing wavelengths of the Huairou magnetograph. The Hβ magnetic signal is timed by 5.4. From Zhang & Zhang (2000a)

profiles in the model atmosphere of the quiet Sun (Ai et al., 1982; Zhang & Ai, 1987) and the observational calibration (Wang et al., 1996). Due to the deep integration of the Stokes parameter V signal in the measurement of the magnetic field, the noise level is about 3 G in the photospheric magnetogram and 5 G in the chromospheric one, and the real spatial resolution of the magnetic features in both magnetograms is about 3 –4 . By comparing with the analysis of the line formation in the solar atmosphere, it can be obtained that the magnetic strengths at the formation layers of FeIλ5324.19 Å and Hβ lines do not change obviously. This means that the magnetic field extends up from the photosphere to the average formation layer of the Hβ magnetograms, which is at 1000–1500 km high above the photosphere, i.e., the magnetic features in the chromosphere also keep the similar configuration to that in the photosphere. These results not only include the network magnetic features but also the internetwork ones.

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Moreover, the sub-arcsecond resolution photospheric magnetograms in the quiet Sun were observed by Keller (1992) and Zhang et al. (1998), while the observational evidence of the spatial configurations of the magnetic field with the corresponding sub-arcsecond resolution magnetograms observed by the chromospheric lines has not been got yet, due to the low magnetic sensitivity of these chromospheric lines. Our observation represents that the magnetic flux tubes in the quiet sun probably do not extend significantly with the strong horizontal component and keep almost the same vertical flux in the low chromosphere if we consider the limitation of the angular resolution in the magnetograms. The observed results taken by Trace satellite show that the structures in the soft X-ray images consist of thin loop structures. This probably provides information on the magnetic flux tubes in the high solar atmosphere and it is also consistent with our observed magnetograms in the photosphere and chromosphere (Zhang and Zhang, 2000a; Zhang & Zhang, 1998, 1999a, b, 2000b). After analysis of the formation of the chromospheric Hβ line and photospheric FeIλ 5324.19 Å line in the magnetic field and also the corresponding magnetograms in the quiet region, the possible configuration of the magnetic flux tubes can be estimated. The magnetic field probably extends to the chromosphere keeping almost the same vertical magnetic flux from the photosphere. Neither the magnetic flux tubes merge in the quiet corona nor not, we can obtain the result that the magnetic field is highly concentrated in the chromosphere in the limits of the spatial resolution of Huairou chromospheric magnetograms. This is consistent with the observational results of Hα and CaII (H and K) filtergrams, which outline the magnetic features and their extension along the fibrils in the quiet chromosphere. The twisted magnetic flux tubes with the evolution of the footpoints proposed by Parker (1972, 1983) can be found in Fig. 2.9. It provides a fundamental idea on

Fig. 2.9 Topology of magnetic fluxtubes that are twisted by random walk footpoint motion (left; Parker, 1972), leading to a state where fluxtubes are wound among its neighbors (right; Parker, 1983)

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the evolution of the magnetic flux tubes with the reconnection of magnetic fields to cause some of the topological dissipation. Parker (1983) estimated the build-up of the magnetic stress energy B0 Bt /4π of a field line with longitudinal field B0 and transverse component Bt = B0 vt/l, B0 Bt B 2 v2 t dW − v= 0 , dt 4π 4πl

(2.36)

and estimated an energy build-up rate of dW/dt = 107 (erg cm− 2s−1 ), based on B0 = 100G, v = 0.4kms −1 , l = 1010 cm, and assuming that dissipation is sufficiently slow that magnetic reconnection does not begin to destroy Bt until it has accumulated random motion stress for 1 day (see also, Aschwanden, 2006).

2.4 Basic Configuration of Sunspots 2.4.1 Magnetic Field of Sunspots The nature of sunspots is a strong magnetic field, where the magnetic pressure is much high than the fluid pressure. It inhibits the upward transport of energy by convection and results that the brightness and temperature in sunspots are lower than the whole photosphere. In Hα monochromatic images, the bridge structures connect the positive and negative polarities of sunspots. Sunspot groups generally are consisted of magnetic pairs of positive and negative polarities. It is because the magnetic field is source-free, that the total magnetic flux through the solar surface is always zero. Hale (1908) firstly discovered the sunspot magnetic field. Hale & Nicholson (1938) subsequently found that all of the sunspot magnetic fields have a typical bipolar sunspot magnetic polarity, which sequence changes with the solar cycle. In a particular solar cycle, the leading sunspots of the bipolar sunspot groups are normally positive in the Northern Hemisphere, while negative ones in the Southern Hemisphere. Then, in the next (former) solar cycle, the leading negative in the Northern Hemisphere, while the positive in the Southern Hemisphere. This rule is known as the Hale polarity law. The maxima of the sunspot magnetic field in the photosphere is normally located in the center of the sunspot umbra with the magnetic induction intensity of about 1800– 3700 G (Livingston, 2002). This peak value shows a substantially linear relationship with the radius of sunspots. For the large sunspots, the field strength in the center of sunspots is about twice of the small one, and the magnetic flux is ∼30 times, but on the whole, the average field strength in terms of sunspots (1200–1700 G), that of large sunspots are only 1.5 times of small sunspots. The strength of the magnetic field becomes weak with the radius of sunspots. In the outer boundary of the penumbra, the field is about 700–1000 G. Although, in the umbra and penumbra, the light intensity difference is large, with a clear boundary, but the magnetic field from the umbra center to the outer boundary of penumbra

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changes continuously, not as obvious as the light intensity jump. For the isolated circular sunspots (Zurich H type), its internal field distribution shows substantially (Lin, 2000):

r −1 B(r ) = 1+ r ≤ r0 , B0 r0 where B is the field, r is the polar radius (the distance to the geometric center of sunspots), r0 is the radius of the spot penumbra, and B0 is the field strength at the center of umbra. This ratio relates to the type and radius of sunspots (Skumanich et al., 1994). The schematic diagram of a sunspot is shown in Fig. 2.10. In the umbra center the magnetic field is basically perpendicular to the photosphere, while, from the umbra center outward, the magnetic field inclines gradually (see Fig. 2.11). As the magnetic field is perpendicular to the photospheric surface, the inclination is 0; as the magnetic field is in the horizontal direction along the photospheric surface, the angle is 90◦ . In the sunspot outer boundary, the angle between the magnetic field and the radial direction is about 10–30◦ (Lites & Skumanich,

Fig. 2.10 Sketch shows the interlocking-comb structure of the magnetic field in the filamentary penumbra of a sunspot (after Thomas et al., 2002). The bright radial filaments, where the magnetic field is inclined (at approximately 40◦ to the horizontal in the outer penumbra), alternate with dark filaments in which the field is nearly horizontal. Within the dark filaments, some magnetic flux tubes (i.e., bundles of magnetic field lines) extend radially outward beyond the penumbra along an elevated magnetic canopy, while other, “returning” flux tubes dive back below the surface. The sunspot is surrounded by a layer of small-scale granular convection (squiggly arrows) embedded in the radial outflow associated with a long-lived annular supergranule (the moat cell) (large curved arrow). The submerged parts of the returning flux tubes are held down by turbulent pumping (vertical arrows) by the granular convection in the moat

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Fig. 2.11 Left: the angle ζ between the magnetic field of NOAA 10930 and the vertical direction of the solar surface. When the magnetic field is in the horizontal direction of the photosphere, the inclination is 90◦ . The polarity of the main sunspot in the figure is negative, and the inclination is mostly between 90◦ and 180◦ , which is represented by black. Right: the relationship between the magnetic field inclination of the sunspot and the radial distance. The curve is the inclination of the magnetic field on the line crossing the sunspot in the east–west direction in the left figure. It can be seen that the magnetic field is perpendicular to the photosphere at the center of the umbra (180◦ ), and close to the horizontal 90◦ , at the outer boundary of the penumbra. The inclination changes gradually from the center of the umbra to the outer boundary of the penumbra

1990; Skumanich et al., 1994). The angle relates to the scale and decaying stage of sunspots. Some observers believe that the field in the sunspot boundary is horizontal. In the sunspot regions, the transverse field is basically along the radius direction, however, there may be 10◦ rotation angles (Lites & Skumanich, 1990). The observations suggest that the maximum field strength at the center of regular sunspots decreases with height, but still has values of 1000–1800 G in the upper chromospheric to lower coronal layers.

2.4.2 Models of Sunspot Magnetic Fields The models of sunspot magnetic fields in magnetohydrostatic theory have been proposed by some authors (Schluter & Temesvary, 1958; Deinzer, 1965; Yun, 1970, 1971; Low, 1980; Osherovich, 1979, 1980, 1982). There we introduce the model of sunspot magnetic field by Osherovich (1982). The plasma equilibrium equation in a uniform gravity is 1 (∇ × B) × B = ∇ P − ρg, (2.37) 4π For axisymmetric configurations, it may be reduced to the scalar equation (Low, 1975)

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∂2ψ ∂ +r ∂z 2 ∂r



1 ∂ψ r ∂r

+ IA

d IA ∂ = −4πr 2 P(ψ, z), dψ ∂ψ

(2.38)

where P(ψ, z) is gas pressure and the magnetic field components are Br = −

∂ψ /r, ∂z

Bz =

∂ψ /r, ∂r

Bφ = I A /r.

(2.39)

The function ψ related to magnetic flux φ through a horizontal surface by 

R

φ = 2π

Bz r dr = 2π[ψ(R, z) − ψ(0, z)].

(2.40)

0

The “Similarity” assumption writes in the terms of only one parameter α: ψ(r, z) = ψ(α),

(2.41)

α = ξ(z)r.

(2.42)

where

Most authors have used

ψ(α) = ψ0 e−α , 2

(2.43)

where ψ0 = const. (Schluter & Temesvary, 1958; Deinzer, 1965; Yun, 1970, 1971; Osherovich, 1979; Low, 1980). Equation (2.38) may there be reduced to the one-dimensional differential equation (cf. Fang et al., 2008; Mao, 2013) f yy  − y 4 + y 2 K 2 f = −8πP.

(2.44)

y 2 = B(z)

(2.45)

where

is the magnetic field strength in the sunspot center, P = P(∞, z) − P(0, z),

(2.46)

and f and K are constants. Equation (2.44) was obtained from (2.38) in a general case, taking into account the azimuthal field. With Bφ = 0 we have the original Schluter–Temesvary equations f yy  − y 4 = −8πP. For the case P = P0 −

B 2 ψ , 2

(2.47)

(2.48)

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I A = 0, where P0 = const., Eq. (2.38) is linear. It has an exact analytic solution

r 2 (r 2 /8λ2 ) −(z/λ) (r 2 /8λ2 ) , + X 2e e ψ = ψ0 e 8λ

(2.49)

1 2 where ψ0 , λ, X are constants, and B = 64 λ . The Eq. (2.49) is a superposition of two solutions. the first describes the magnetic lines which go out of the plane z = 0 and do not return to it. The second describes the lines that return in their original plane. In other words, the contribution of the second part to the magnetic flux through the plane z = 0 is equal to zero according to the formula (2.40). Equation (2.39) may be used to obtained Bz , Br , and tan ϕ = Br /Bz . In the plane z = 0, we can find 2 (2.50) Bz = 2ψ0 ξ 2 (−1 + X (1 − α2 ))e−α ,



 √ 2X ξ tan ϕ = α − 2 + , ξ −1 + X (1 − α2 )

(2.51)

where primes denote derivations to the depth (z), see Fig. 2.12. As a particularly simple case (Osherovich & Flaa, 1983), we have Yun’s sunspot model (Yun, 1971) with Bφ ∼ Br , which corresponds to I A (α) = I A (0)α2 e−α . 2

(2.52)

It is called the Schluter–Temesvary–Yun model (ST-Y). Another model, in which twisting increases linearly with distance from the center of the sunspot, is the Schluter–Temesvary-Linear model (ST-L): I A (α) = I A (0)α3 e−α . 2

(2.53)

Fig. 2.12 The picture of magnetic field lines: a In Schluter–Temesvary theory; b in the theory of Osherovich (1982). After Osherovich (1982)

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Fig. 2.13 The horizontal distribution for the components of the magnetic field at z = 0 for return flux model with twist field. The dash–dot line represents the vertical distribution on the sunspot axis Bz (0, z). The dashed line represents Bz (0, z) for the return flux model without twisted field. The curves are obtained for S = −0.8. After Osherovich & Flaa (1983)

It is noticed that in the ST-L model the pressure and temperature profiles are, in principle, different from those in the ST-Y model. For the return flux model (Osherovich & Flaa, 1983) a simple choice of I A (ψ), I A (ψ) = (ψ − ψ0 )ξ(0)S.

(2.54)

Figure 2.13 shows twisted magnetic lines in two planes, z = 0 and z = 500 km for the return flux model. This figure indicates a decrease in twisting with height.

2.4.3 Penumbral Fine Features The most obvious characteristic in the sunspot penumbra is covering by the arranged radially alternating light and dark penumbral fibers in Fig. 2.11 (left). Generally, bright filaments are the location on the outside and relatively vertical flux tubes, while the dark filaments are more nearly horizontal flux tubes (Solanki & Montavon, 1993; Solanki, 2003). They constitute a “not carded” (uncombed) or “interlock” (interlocking) the interleaved structure. Angle difference between the two 30–40◦ , bright filaments can enter the corona and across a great distance in the solar surface, and dark ones at low altitudes. High spatial resolution movies of sunspots taken at the Swedish Solar Observatory on La Palma reveal that the Evershed effect is time-dependent (Shine et al., 1994). Outward proper motions are visible in both the continuum and Dopplergrams. These can be tracked over most of the width of the penumbra and overlap regions that show inward moving penumbral grains. The radial spacing between the moving structures is about 2000 km, and they exhibit irregular repetitive behavior with a typical interval of 10 min. These are probably the cause of 10 min oscillations sometimes seen in penumbral power spectra. Higher velocities are spatially correlated with the rela-

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tively darker regions between bright filaments. Regions with a strong variation in the Doppler signal show peak-to-peak modulations of 1 km/s on an average velocity of about 3–4 km/s.

2.4.4 Decay of Sunspots and Magnetic Fields 2.4.4.1

Decay of Sunspots

Sunspots emerging and aging are two very asymmetric processes. The new emerging sunspots usually require only a few days to reach their maximum state, while their aging process occupies most of their life. Small-scale, irregular shape, and no moat sunspots decay fast, mainly in the form of brokenness, the brokenness is a harbinger that the bright and light bridge appear in the penumbra (Zwaan, 1987). In the aging process of sunspots, the area and the magnetic flux of sunspots gradually reduce (van Driel-Gesztelyi, 1998). Petrovay & van Driel-Gesztelyi (1997) studied over four hundred sunspots aging process and found that the relationship between the rate of aging of sunspots D and radius r , the maximum ratio of the radius r0 (r/r0 ) of sunspots (once in their lifetime appears), while the life of sunspots T is proportional its maximum area of A0 : T ∼ A0 . D ∼ r/r0 , The magnetic fields of the following sunspots in the bipolar sunspot groups are relatively weak and decay rapidly.

2.4.4.2

Lifetime of Sunspot Magnetic Fields from Coordinated Videomagnetograph Observations

Wang et al. (1989) show the evolution of an active region, which was observed jointly with the videomagnetographs of the Huairou and Big Bear Solar Observatories from 1987 September 24–27, with only four interruptions of 6–8 h each. The data show the longer term evolution of the magnetic fields in Fig. 2.14. It is found that the dominant sunspot mainly ejects the magnetic field of opposite sign, the surrounding plage fields steadily contract and retreat inward toward the umbra. The overall result is shrinking and weakening of spot and plage. The extent of the most was reduced by 50% in 75 h. The principal loss of flux appears to be due to “cancellation” at the main neural line (e.g., Martin et al., 1985). Some flux disappears through fragmentation, which makes the elements fall below our threshold, while only a tiny loss due to diffusion can be detected.

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Fig. 2.14 Left: The large-scale velocity pattern of the decaying active region. To avoid long arrows, the outflow in the moat, which is in the order of 0.5 kms−1 , is not scaled properly. Right: The size of the moat as the function of time. From Wang et al. (1989)

2.4.5

Moving Magnetic Features

2.4.5.1

Discovery of Moving Magnetic Features

Sheeley (1969), Vrabec (1971), and Harvey & Harvey (1973) firstly discovered the moving magnetic features (MMFs). In recent decades, detailed observations have been carried out by continuous high temporal and spatial resolution observations of magnetic fields based on ground-based and space-based magnetographs. The observed MMFs move with a range of speeds around 1 km/s. The mean lifetimes of MMFs range from 1 to 8 h. Their paths in the moat are almost radially outward from the sunspot along with the continuation of dark filaments. Various observations (Harvey & Harvey, 1973; Vrabec, 1974; Brickhouse & Labonte, 1988; Lee, 1992; Zhang et al., 1992; Yurchyshyn et al., 2001; Sainz Dalda & Martínez Pillet, 2005) show that MMFs exhibit a range of lifetimes (18 h), sizes, and net flux (1–25 ×1018 M x), and bipole separations. MMFs show a range of sizes 1–2 and carry a flux of the order of 1019 Mx. Most of these “steady flow of bright points” (Sheeley, 1969) move at about 1 km s−1 along paths that are almost radially outward from the penumbrae and form a magnetic flux outflow (Vrabec, 1974). MMFs tend to originate from certain azimuths around parent sunspots and follow particular paths. MMFs may first appear at or 0–8 Mm beyond the outer edge of the parent sunspot (Harvey & Harvey, 1973; Li et al., 2009), many of them have precursors inside the penumbra and originate at 1–8” inside the penumbra fibrils and move to cross the penumbral boundary into the moat region (Sainz Dalda & Martínez Pillet, 2005; Ravindra, 2006).

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They may also appear as bright features in G-band and Ca II images (Shine and Title 2001). In multi-wavelength observations MMFs are almost invisible in H-alpha images, indicating that they are low-lying loops. 2.4.5.2

Physical Models of Moving Magnetic Features

MMFs are considered to be the intersections of the photosphere and the tiny flux tubes detached from sunspots (Harvey & Harvey, 1973). To describe MMFs’ physical nature, theoretical and schematic interpretations (e.g., “-loop”, “U-loop”, and “Oloop” models) have been proposed and discussed, see Figs. 2.15 and 2.16. The first theoretical model for MMFs proposed by Harvey & Harvey (1973) suggested that flux tubes be twisted and separated from the main body of sunspots and swept to the network by the interaction of the magnetic field with the supergranulation or granulation. These loose, twisted field lines are moved away from the sunspots by the supergranulation velocity fields. Wilson (1986) explained the outward moving of MMFs with an oscillatory velocity field that generates the new flux loops near the sunspots limb. Spruit et al. (1987) suggested that MMFs are small pieces of large U-loops that rise to the surface from the convection zone and disintegrate. Thomas et al. (2002) explained MMFs as the submergence of penumbral magnetic flux by convective pumping.

Fig. 2.15 Top: Harvey & Harvey (1973) MMF model was first proposed. Twisted magnetic field flux tubes are separated from the main body of the sunspot and swept to the network by the supergranulation velocity field. Twists in the tubes erupt through the visible layer to produce a large number of opposite-polarity pairs of MMF. Bottom: Bernasconi et al. (2002) found the structure and behavior of peculiar moving dipolar features (MDFs) in the emerging flux. In longitudinal magnetograms, the MDFs appeared to be small dipoles flowing into sunspots

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Fig. 2.16 Top: Wilson (1986) velocity field with concussion In the sunspot boundary generating new “O”-type magnetic flux tube ring to explain the observed phenomena bipolar MMF. Bottom left: If O-ring tightly around its core, relatively independent of the surrounding magnetic field, then it is in the ball of light formed on both positive and negative magnetic elements equal brightness, Dipole magnetic flux outflow does not cause the outflow. Bottom right: If O-ring is wrapped around the magnetic field, drag and squeeze each other so that the magnetic flux dipole poles are not equal, it will take away the outflow flux

2.4.6 Moving Magnetic Features from High-Resolution Magnetograms High-resolution observations of small-scale magnetic features moving around sunspots in the photospheric and chromospheric layers contribute to a better knowledge of sunspot structure, and to the understanding of the transport of magnetic energy and mass throughout the chromosphere. In many well-developed sunspots, a mean outflow is observed in both the penumbra and the surrounding moat region. In the penumbra, the magnetic field is strong and inclined, while in the moat it is weak. On magnetogram time series, Moving Magnetic Features (MMFs) are observed to originate from the outer penumbra, become isolated magnetic elements, move radially outward into the moat regions, and eventually disappear among the network fields. A series of achievements on MMFs (Harvey & Harvey, 1973; Vrabec, 1974; Brickhouse & Labonte, 1988; Lee, 1992; Zhang et al., 1992; Shine & Title, 2001; Yurchyshyn et al., 2001; Bernasconi et al., 2002; Zhang et al., 2003; Hagenaar & Shine, 2005; Sainz Dalda & Martínez Pillet, 2005; Li et al., 2006b; Ravindra, 2006; Ryutova & Hagenaar, 2007; Li et al., 2009, 2010; Lim et al., 2012; Criscuoli et al., 2012; Sainz Dalda et al., 2012) are summarized as the following:

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1. Most of these “steady flow of bright points” (Sheeley, 1969) move at about 0.3– 1.0 km s−1 along paths that are almost radially outward from the penumbra and form magnetic flux outflow. 2. MMFs tend to originate from certain azimuths around parent sunspots and follow particular paths. MMFs may first appear at 0–10 beyond the outer edge of the parent sunspot. Many of them have precursors that originate inside the penumbra. 3. MMFs exhibit a range of lifetimes (0.2–8 h), radii (0.5–2 ), net fluxes (1–25×1018 Mx), and bipole separations. They may also appear as bright features in G − band and CaiiH images. 4. Previous researches have categorized MMF according to their parity (unipolar or bipolar), polarity (same or opposite to the parent sunspot), the direction of motion (outward or inward), and dipole orientation (whether the inner/outer footpoint has the same polarity as the sunspot). The complex interaction of turbulent convection, radiation, and magnetic field makes sunspots hard to model. The monolithic model (Cowling, 1957) regards the sunspot as a magnetostatic and tight bundle of magnetic field lines. In the cluster model (Parker, 1979b, c), the bundles of magnetic flux tubes are loosely confined by subsurface converging downflows in the surrounding plasma. Multi-wavelengths observations (Zhang et al., 1992; Penn & Kuhn, 1995; Zuccarello et al., 2009) have been performed in Hα and CaiiH filtergrams. MMFs are almost invisible in Hα images, indicating that they are low-lying features. Zhang et al. (1998) analyzed the relationship between the fine structures of granulation and magnetic fields using a time series of Fei λ5250.2 Å photospheric filtergrams and corresponding magnetograms in a quiet region. They found that, although most bright filigree features in photospheric filtergrams are related to corresponding magnetic features, they are generally not co-spatial. It is also found that some bright features and their corresponding photospheric magnetic fields show fast changes within several minutes. Modern high-resolution observational efforts and techniques, especially space-based observatories, provide more details of MMFs than ever before. Kubo et al. (2007) studied the isolated and non-isolated MMFs using coordinated Dunn Solar Telescope (DST) and Solar and Heliospheric Observatory (SOHO ) observations. The magnetograms obtained by the Hinode observatory provide an opportunity to study the MMFs’ motion in and out of sunspot penumbral fibrils (Ravindra, 2006) and small-scale reconnection events (Guglielmino et al., 2010; Murray et al., 2012; Su et al., 2012). Li & Zang (2013) categorize MMFs into two categories and four types according to the location of their first appearances, and the source of their initial magnetic flux. Penumbral MMFs: They originate inside or immediately outside the outer penumbral boundary, separate from penumbra fibrils, and move outward. They are differentiated into two types by their polarities. α-MMF: Having the polarity opposite to the parent sunspot. β-MMF: Sharing the parent sunspot’s polarity. Moat MMFs: They originate outside the penumbra outer boundary, and first appear among moat granulations, with no obvious connection with penumbra fibrils. They

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are also differentiated into two types, according to whether they are attached to other MMFs when they first appear. γ-MMF: Emerge in the moat, with no obvious affiliation to other magnetic objects. θ-MMF: A fragment separated from a magnetic object in the moat. To demonstrate the dynamic nature of the moat, as examples, we examined the contacts made by a particular MMF with neighboring magnetic elements. Figure 2.17a shows the evolution of a big, negative polarity β-MMF. When it left the penumbra, its initial flux was only 3.2 × 1018 Mx. During its 14-h long lifetime, it gained flux

(a)

a MMF in the Moat 5 4 3 2 1 0

Flux Gain Flux Loss

30 25

Magnetic Flux Flux Density Ca II Light Flux

20

400 300

15

200

10 5 0 0

500

100 0 200

400 600 Time (minute)

Center Flux Density (Gs)

Magnetic Flux (10^18 Mx)

(c)

800

Fig. 2.17 a The evolution of a negative polarity β-MMF. Its lifetime is nearly 14 h. After leaving the penumbra, this MMF contacted with more than one hundred smaller magnetic elements of both polarities. b The black (white) arrows indicate the contact events during which flux was gained (lost). The lengths of the arrows are proportional to the square root of the amount of flux transferred. The azimuth locations of the arrows indicate the approximate contact points. The background is a time-averaged, MMF-centered magnetogram. c lower panel: The evolution of this MMF’s flux (solid), flux density (dotted), and CaiiH light flux (dashed, in arbitrary units). upper panel: The plus and minus symbols indicate the flux gain and loss during the MMF’s contacts with neighboring magnetic elements. From Li & Zang (2013)

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143

mostly by merging magnetic elements of the same polarity. It lost flux partly by fragmentation and partly by cancellation with the opposite-polarity magnetic elements. We count 59 contact events during which the MMFs gained flux. The total flux gained (56.9 ×1018 Mx) more than doubles the MMF’s maximum flux (25.1 ×1018 Mx). These events are indicated by the black, inward-pointing arrows in Fig. 2.17b. There were also 53 contact events in which the MMF lost 49.8 ×1018 Mx of flux in total. In Fig. 2.17b they are indicated by the white arrows that point outward. The lower panel of Fig. 2.17c compares the MMF’s evolution of flux (solid curve) and flux density (dotted curve) with the CaiiH light flux (dashed curve), where the light flux is the two-dimensional integration of image intensity in arbitrary units. The upper panel plots the amounts of flux gained (plus sign) and lost (minus sign) in the contact events. Some of the flux cancellation events appear as bright points on CaiiH images. During the MMF’s rise phase, the conglomeration process was the dominant mechanism, while flux cancellation was dominant in the decline phase. Li & Zang (2013) present the following major results regarding MMFs’ origins, motion, evolution patterns, and their impacts on lower chromospheric layers: (1) Fifty percent of MMFs originate from or among the penumbra fibrils. (2) The evolution of MMFs that are fragments from a parent magnetic area is usually a decay and fragmentation process. The decay process starts once they become isolated features. (3) Most of the CaiiH bright points are caused by cancellations between opposite polarity magnetic areas. Bright points may also be triggered by MMFs’ severance from the penumbra, fragmentation, and merging. Fast-moving moat MMFs appear as bright dots in the low chromosphere, while big magnetic elements appear as dark patches in filtergrams. Li et al. (2015) find that the magnetic elements move along the connection lines between the pores and sunspots’ penumbrae, converge and cause the pores’ formation and destruction. Pores also receive flows produced by small-scale bipolar emergence and merge dissociated elements in the moat. The MMF inflows that are diminishing to the pore’s magnetic flux often trigger chromospheric bright points. During their decay, the pores produce the outflow of magnetic elements, in directions other than the inflow from sunspots. An example of the evolution of the magnetic features nearby the sunspot can be found in Fig. 2.18. Chen et al. (2015) also demonstrated two kinds of cancellations between opposite polarity magnetic fluxes inducing the generation of recurrent jets nearby the major spot.

2.5 Chromospheric Magnetic Fields in Active Regions 2.5.1 Measurements of Chromospheric Magnetic Field Metcalf et al. (1995) took the observations of the NaIλ5896 Å line, made with the Stokes polarimeter at Mees Solar Observatory, to measure the chromospheric vector magnetic field in active regions and calculated the height dependence of the net

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2 Basic Structures of Solar Magnetic Fields (a)

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Fig. 2.18 a Serial maps of longitudinal magnetogram showing the appearance, outward motion, and vanishing of pore (marked with a cross). The flux density saturates at ±50 G. The solid ellipse accentuates magnetic elements produced by sunspots and moves toward pore, while the elements circled by the dashed ellipse move away from the pore. b Temporal evolution of pore (marked with a box). From Li et al. (2015)

Lorentz force in the photosphere and low chromosphere. It is found the magnetic field is not force-free in the photosphere but becomes force-free roughly 400 km above the photosphere. Choudhury et al. (2001) analyzed the three-dimensional magnetic field structure of solar active regions by comparing the observed and computed chromospheric magnetograms. The best correlation between the observed and the model chromospheric magnetograms was found at the height of 800 km, which also corresponds to the height of the line formation for CaIIλ8542 Å and converges to a potential field configuration later. Deng et al. (2010) offered explanations or speculations to the observed discrepancies between the photospheric and chromospheric lines in terms of the three-dimensional structure of the magnetic and velocity fields and emphasized the importance of spectropolarimetry using chromospheric lines. At Huairou Solar Observing Station of National Astronomical Observatories in the Chinese Academy of Sciences, a great of observational data of chromospheric magnetic field with the high spatial resolution has been obtained, and a series of works in the areas has been made. Figure 2.19 shows the photospheric and chromospheric filtergrams and corresponding longitudinal magnetograms in a solar active region.

2.5 Chromospheric Magnetic Fields in Active Regions

145

01:53UT, 1991 May 3

02:02UT

Fig. 2.19 Photospheric and Hβ chromospheric images (left) and corresponding longitudinal magnetograms (right) in a solar active region (Zhang, 2006a)

2.5.2 Possible Extension of Hβ Chromospheric Magnetic Field from Photosphere Figure 2.20 shows a set of Hβ filtergrams at different wavelengths with an interval of 0.04 Å, between −0.40 Å and −0.12 Å from the line center (Zhang, 1996a). In the filter gram at −0.40Å, the chromospheric features are not distinct. It is similar to the photospheric image. We notice that as the observed wavelength in the set of filter grams approaches the Hβ line center, the chromospheric features become significant. The dark filaments extend and their lengths increase from outside the sunspot penumbra in the west of the sunspot. These filaments constitute the super penumbral features. On the east side of the sunspot, there is no evidence of filamentlike features. The possibility is that the spatial resolution of these images is lower, so chromospheric fine features cannot be found. It is found that the magnetic field of active regions extends up non-uniformly from the photosphere into the chromosphere and consists of the fibril-like features in the chromosphere (Zhang et al., 1991). Figure 2.20 also shows a set of chromospheric longitudinal magnetograms. Filament-like magnetic features in the magnetograms show a good correlation with features in the chromospheric filtergrams. These chromospheric features thus reflect information on the magnetic field in the corresponding

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2 Basic Structures of Solar Magnetic Fields

-0.40

-0.36

-0.40

-0.36

-0.32

-0.28

-0.32

-0.28

-0.24

-0.20

-0.24

-0.20

-0.16

-0.12

-0.16

-0.12

Fig. 2.20 Hβ filtergrams (left) and corresponding Hβ longitudinal magnetograms (right) at different wavelength from −0.4 to −0.12 Å in a solar active region (Zhang, 1996a)

atmospheric layers. A set of chromospheric magnetograms obtained at various wavelengths from the wing of the Hβ line to the center can be used to infer the variation of line-of-sight components of the magnetic field with height in the chromosphere.

2.5.3 Reversal Features in Hβ Chromospheric Magnetograms By analyzing Hβ chromospheric magnetograms, we can find that in some areas the difference between the chromospheric and photospheric magnetograms can be detected. It probably results from the extension of the magnetic field from the photosphere (Zhang et al., 1991). Chen et al. (1989) pointed out that some reversal structures, relative to the photospheric one, probably exist nearby the regions of the strong magnetic field in the chromospheric magnetograms obtained at the wavelength of Hβ-0.24 Å (such as, at −0.32 Å, −0.28 Å, and −0.24 Å in Fig. 2.20). It is normally called CAZJ reversion of the chromospheric magnetic field (Almeida, 1997). A similar case has been presented by Wang & Shi (1992). The observational results have been systematically analyzed, such as some data analysis has been made by Li et al. (1994). By analyzing the formation of the Hβ line, several possibilities for the reversal structures in the chromospheric magnetograms can be inferred: (a) the real reversal magnetic structures in the solar atmosphere; (b) the reversal sign of the Stokes

2.5 Chromospheric Magnetic Fields in Active Regions

147

parameter V caused by the reversion of the Hα line profile in the local areas of the chromosphere (Almeida, 1997); (c) the disturbance of the blended lines in the wing of the Hβ (Zhang, 1996a). The blended lines in the wing of the Hβ line probably cause the reversed sign in the magnetograms obtained at the single wavelength of the Hβ line. It was presented by a series of Hβ chromospheric magnetograms obtained at various wavelengths near the line center with the vector video magnetograph at Huairou Solar Observing Station as a diagnostic of chromospheric magnetic structures (Zhang, 1993). The 2D distribution of the circular polarization light of the Hβ line with its blended lines at the various wavelength in active regions can be obtained (cf. Fig. 2.20), which consists of the analyses of Stokes’ profile V of this line in Fig. 2.21. Due to the disturbance of the photospheric blended line FeIλ4860.98 Å for the measurement of the chromospheric magnetic field, the reversal in the chromospheric magnetograms relative to the photospheric ones occurs in the sunspot umbrae significantly. It is the same with the reversal of Stokes V nearby the line center in Fig. 1.54, i.e., in the weak approximation Eqs. (1.139) and (1.145), the strength of the longitudinal magnetic field is proportional to the variation of the line profile (Stenflo et al., 1984). V ∼ d I /dλ.

(2.55)

It is noticed that in the quiet plage regions, even penumbrae, the influence of the photospheric blended FeIλ4860.98 Å line is not significant relatively. As regards the observation of the Hβ chromospheric magnetograms, we can select the working wavelength between −0.20 and −0.24 Å from the line core of Hβ to avoid the wavelengths of the photospheric blended lines in the wing of Hβ. After the spectral analysis of chromospheric magnetograms, we conclude that the distribution of the chromospheric magnetic field is similar to the photospheric field, especially in

Fig. 2.21 Statistical distribution of Stokes parameter V in the blue wing of Hβ line in umbrae of a solar active region (Zhang, 1993)

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2 Basic Structures of Solar Magnetic Fields

the umbrae of the sunspots. Similar evidence on the reversal features in Hα magnetograms in active regions has been diagnosed in detail by (Balasubramaniam & West, 1991; Hanaoka, 2005).

2.5.4 Magnetic Field of Dark Filaments in Quiet Sun Solar quiescent dark filament (prominence) is a remarkable subject. Here, we only introduce the results of the measurements of the dark filaments at Huairou Solar Observing Station. Bao & Zhang (2003) observed the line-of-sight magnetic field in the chromosphere and photosphere of a large quiescent filament on the solar disk on September 6, 2001, using the Solar Magnetic Field Telescope in Huairou Solar Observing Station. The chromospheric and photospheric magnetograms together with Hβ filtergrams of the filament were examined. The filament was located on the neutral line of the large-scale longitudinal magnetic field in the photosphere and the chromosphere in Fig. 2.22. The lateral feet of the filament were found to be related to magnetic structures with opposite polarities. Two small lateral feet are linked to

Fig. 2.22 Hβ filament and the corresponding magnetograms and Dopplergram on Sep. 6, 2001. Top left: Hβ filtergram. Top right: photospheric longitudinal magnetogram, the red (blue) contours correspond to positive (negative) fields of 100 and 200 G. Bottom left: Hβ Dopplergram, purple (green) is down (up) word. Bottom right: Hβ longitudinal magnetogram, the purple (blue) contours correspond to positive (negative) fields of 40 and 80 G. Provided by Xinming Bao and Bao & Zhang (2003)

2.5 Chromospheric Magnetic Fields in Active Regions

149

weak parasitic polarity. There is a negative magnetic structure in the photosphere under a break of the filament. At the location corresponding to the filament in the chromospheric magnetograms, the magnetic strength is found to be about 40–70 G (measuring error of about 39 G). The magnetic signal indicates the amplitude and orientation of the internal magnetic field in the filament. Bao & Zhang (2003) discussed several possible causes which may produce such a measured signal. A twisted magnetic configuration inside the filament is suggested.

Chapter 3

Solar Magnetic Activities

The effects of geomagnetic storms on power-grids are much more dramatic. On March 13, 1989, 7 days after a strong X-ray flare from our Sun, ground currents from geomagnetic storms brought down Canada’s entire Hydro Quebec power system. The origin of the flare was an impressively large sunspot known as AR 5395, it was 54 times larger than the Earth but yet still covered less than one half of a percent of the solar disc. Transformers which are used to manipulate the current and voltage flows through power-grids are designed to operate with alternating current (AC). Ground-induced currents are direct currents (DC) and they caused the transformers in the Hydro Quebec power system to catch on fire and breakdown. While such transformers alone cost millions of dollars, the total estimated cost of this blackout was placed at hundreds of millions of dollars (From: http://astronomy.swin.edu.au/ cosmos/g/geomagnetic+storms). This means solar magnetic activities are notable topics in the study of astrophysics and solar-terrestrial physics.

3.1 Magnetic Energy, Shear, Gradient, and Electric Current in Solar Active Regions The active region (NOAA 5395) of 1989 March was a prolific flare-producing region. Some powerful flares occurred in this highly sheared active region, which was observed at the Huairou Solar Observing Station of the National Astronomical Observatories of China (Zhang, 1995c). Figure 3.1 shows the vector magnetogram and Dopplergram of active region NOAA5395 on March 11, 1989. The 180◦ -ambiguity of the transverse magnetic field is resolved. The removal of the 180◦ ambiguity of the observed transverse photospheric magnetic field is important to analyze the vector magnetic field and relevant electric current inferred from it. The procedures were presented by some authors (Wu & Ai, © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Zhang, Solar Magnetism, https://doi.org/10.1007/978-981-99-1759-4_3

151

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3 Solar Magnetic Activities

March 11, 1989 01:52

01:57 March 11, 1989 01:53

01:57 Fig. 3.1 Top: The transverse component of a photospheric vector magnetogram resolved the 180◦ ambiguity in the active region (NOAA 5395) at 01:57 UT on March 11, 1989, which is overlaid by a Hβ filtergram at 01:52 UT. The arrows mark the directions of the transverse magnetic field. Bottom: A corresponding Hβ Dopplergram of the active region at 01:53 UT with longitudinal components of magnetic field (the same as in the top). The white (black) indicates up-(down-)ward follow. The red (blue) contours correspond to positive (negative) fields of ±50, 200, 500, 1000, 1800, and 3000 Gauss. The size is 5. 23 × 3. 63. The north is top and the east is at left

1990; Cuperman et al., 1990; Metcalf, 1994; Wang et al., 1994a; Moon et al., 2003; Georgoulis et al., 2004; Li et al., 2007, 2009). It is noticed that Ai et al. (1991) found that the flares occurred on the redshift side of the inversion lines of the Hβ Doppler velocity fields observed half to two hours before in super active regions. A similar case can be confirmed in Fig. 3.1, while in

3.1 Magnetic Energy, Shear, Gradient, and Electric Current in Solar Active Regions

a, 02:26 UT, Feb 10 1989

b, 01:57 UT, Mar 11

c, 02:02 UT, Apr 11

d, 00:40 UT, May 05

153

Fig. 3.2 A series of observed photospheric vector magnetograms corresponding to the super active region NOAA 5395. Grey structures mark the positions of sunspots. Arrows mark the directions of the transverse magnetic fields. The red (blue) contours correspond to positive (negative) fields of ±20, 100, 200, 400, 1000, and 2000 Gauss. Figure b shows the distribution of the vector magnetic field in the superactive region (NOAA 5395) at 01:57 UT on March 11, 1989 (to Fig. 3.1). The size is 5. 23 × 3. 63. North is up, east is left

this Dopplergram the distortion near the flare sites can not be neglected, due to the almost same period between this Dopplergram and the flare. In Fig. 3.2, we show a series of vector magnetograms associated with the active region of solar NOAA 5395 (Schmieder et al., 1993; Zhang et al., 1994). It shows the developments and changes of the active region at the solar north latitude 30◦ in the months before and after the series of solar rotations. This indicates that a large amount of magnetic field energy emerges from the interior of the Sun and is released into the solar-terrestrial space. The horizontal scale of the above magnetograms is about 15 times the diameter of the Earth. It is entirely conceivable that such a rapidly changing magnetic energy would be released to produce the event that would destroy the entire Quebec power system in Canada. Studies have shown that large flares are usually accompanied by coronal mass ejections, which can instantaneously release up to 1032 ergs of energy. Particles, including electrons, protons, and heavy nuclei, are heated and accelerated in the solar atmosphere as magnetic energy is released. This energy is 10 million times that released by a volcanic eruption. On the other hand, it is less than one-tenth of the total energy released by the Sun every second. The evolution and related analysis of non-potential magnetic fields in the solar active regions will be gradually expanded below.

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3 Solar Magnetic Activities

3.1.1 Magnetic Energy, Shear, and Gradient Some pieces of evidence in the powerful flare-producing regions can be found, such as in Figs. 3.1 and 3.2: (1) The twisted transverse magnetic field occurred in baylike magnetic main poles of opposite polarity in the active region. (2) The newly sheared vector magnetic structures formed near the magnetic neutral line between the collided magnetic main poles of opposite polarity. With the emergence of new magnetic flux, the change in sheared angles of the horizontal magnetic field between the magnetic main poles of opposite polarity change in comparison with daily vector magnetograms, and the distribution of the intensity of the vertical current inferred from the horizontal magnetic field evolved only gradually. (3) The flare sites occurred near the magnetic islands and bays of opposite polarity and were associated with the change in the vector magnetic field. Although some of the flare sites are located near the peak areas of the vertical electrical current density, their corresponding relationship is insignificant. We will present these evidences in the following.

3.1.1.1

Magnetic Energy Density with Shear

The energy released by solar flares and other explosive events relies on the accumulation of the free magnetic energy (non-potential magnetic energy) that is defined as the difference between the total magnetic energy (E o ) and potential magnetic energy (E p ): E = E o − E p .

(3.1)

This means that the free energy is defined concerning the energy of the potential field that matches the distribution of the observed field’s normal component. This potential field has the lowest possible magnetic energy consistent with the boundary condition (for example see Priest 2014 and also in Sect. 4.1.1.1). The total magnetic energy of a field B is given in the form  E=

B2 d V, 8π

(3.2)

which means that the magnetic energy is a three-dimensional integral quantity, such as in the solar atmosphere. Hagyard et al. (1981) defined the source field to describe the non-potentiality of magnetic field on the photosphere: Bn = Bo − B p ,

(3.3)

where Bo is the observed vector magnetic field, B p denotes the potential field extrapolated from the vertical component of Bo , and Bn is the so-called source field that

3.1 Magnetic Energy, Shear, Gradient, and Electric Current in Solar Active Regions

155

Bn Bo

Bp Bph

γ ψ o

ϕ

θ

Bnh

Boh

Fig. 3.3 A schematic of the relationships among the observed magnetic field Bo , the potential field B p and the non-potential field Bn

is the non-potential component of the magnetic field. Using Eq. (3.3), it is found  Bn = Bo2 + B 2p − 2Bo B p cos ψ, where ψ is the inclined angle between Bo and B p in Fig. 3.3. 

Now, we can introduce the magnetic energy density ρ, as in E =

(3.4) 

B2 dV = 8π

ρd V , where the field’s energy density ρ ≡ B 2 /8π. Analogously, for E o and E p , the observed and potential-field energy densities are ρo ≡ Bo2 /8π and ρ p ≡ B 2p /8π, respectively. We also notice that the free energy density integrated over a volume V yields energy, and the free energy density integrated over the photosphere yields a quantity with dimensions of energy per unit length. This quantity is not, strictly speaking, free energy—but it is plausibly related to the free energy present in the volume above the photosphere. The observation of photospheric vector magnetograms provides a chance to analyze the distribution of magnetic energy density in the low solar atmosphere. We deal exclusively with 2D arrays of magnetic energy from photospheric magnetic fields in the following. An energy density parameter of the non-potential magnetic field defined by Lü et al. (1993) is proportional to B2n : ρn =

(Bo − B p )2 B2 = n. 8π 8π

(3.5)

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3 Solar Magnetic Activities

It follows that the definition of the real free energy density (Eq. (3.1)) is ρ f r ee = ρo − ρ p =

B2o − B2p 8π

=

(Bn + B p )2 − B2p 8π

,

(3.6)

and it can be written 1 2 B + 8π n 1 2 B + = 8π n

ρ f r ee =

1 1 2 1 Bn · B p = Bn + Bn B p cos γ 4π 8π 4π    Bo − B p cos ψ 1 −1 , Bn B p cos ψ + cos 4π Bn

where γ and ψ are spatial angles, and cos−1 can find the relationship



Bo −B p cos ψ Bn



(3.7)

= γ − ψ, in Fig. 3.3. We

   Bo − B p cos ψ cos ψ + cos−1 Bn    Boh − B ph cos θ , (3.8) = sin ϑ p sin ϑn + cos ϑ p cos ϑn cos θ + cos−1 Bnh where ϑn is the inclination angle between the vector of the non-potential field Bn and its horizontal component Bnh , and ϑ p is that between the vector of the potential field the angle B p and its horizontal component B ph , while  θ is defined as the horizontal B −B

cos θ

= ϕ − θ, in Fig. 3.3. angle between Boh and B ph , and cos−1 oh Bnhph Because we cannot directly measure the vertical component of the non-potential magnetic field in the photospheric layer from the observational vector magnetograms, we normally require a model field to match the observed vertical field, whether in the analysis of non-potential fields or potential fields. For potential models, this choice corresponds to the Neumann boundary condition (see, for instance, Sakurai, 1982), although observations of changes in vertical magnetic fields associated with flares demonstrate that the observed vertical field is not generally consistent with the field’s lowest-energy state. This means that the vertical component of the magnetic field does not really contribute to the magnetic free energy density from the photospheric vector magnetograms. For observations near the disk center, one can approximate the line-of-sight field as the vertical component of the observed field (Sakurai, 1982). We choose to neglect the vertical component of the non-potential magnetic field in the following analysis. As the three-dimensional shear of the magnetic field has been ignored (Leka & Barnes, 2003), the magnetic free energy density contributed by the horizontal components of these magnetic fields is ρfh =

   1 2 − B 2 ) = 1 B 2 + 1 B B cos θ + cos−1 Boh − B ph cos θ (Boh , nh ph ph 8π 8π nh 4π Bnh

(3.9)

3.1 Magnetic Energy, Shear, Gradient, and Electric Current in Solar Active Regions

157

where Boh , Bnh , and B ph are the horizontal components of observational, nonpotential, and potential magnetic field respectively, and horizontal angles θ and ϕ are defined in Fig. 3.3. Equation (3.9) implies that ρ f h is not necessarily positive, because the second term in Eq. (3.9) can be negative and its absolute value may be larger than the first one in some cases. From Eqs. (3.7) and (3.9), we also find the inclined angles between the observed vector magnetic field and the potential one do not simply reflect the status of free energy density in the low solar atmosphere, regardless of whether we consider the contribution of the vertical component of the vector magnetic field. The difference between the free energy density ρ f h and non-potential parameter ρnh contributed by horizontal components of magnetic fields is ρeh = ρ f h − ρnh =

1 Bnh · B ph . 4π

(3.10)

This means that the contribution of the parallel component of the non-potential field relative to the potential one is non-negligible, and the difference ρeh between two differently defined magnetic energies will only vanish where the potential components of the magnetic field are perpendicular to the non-potential ones. It is noticed that Boh · B ph relates to the normally defined magnetic shear and also the potential field, where Boh · B ph = Bnh · B ph + B 2ph . This means that Boh · B ph contains the contribution of magnetic potential energy (B 2ph ) of solar active regions. One can find the shear angle θ

θ = cos

−1

Bnh B ph cosϕ + B 2ph Boh B ph

 = cos −1



Bnh cosϕ + B ph Boh

 .

(3.11)

From the above discussion, one can find that the normally defined shear angle θ cannot be used to determine information about free magnetic energy in the lower solar atmosphere completely because Eq. (3.11) partly relates to the term B 2ph also. This is consistent with the idea that the magnetic shear contains information about the energy density contributed from the horizontal component of the potential field. The magnetic shear is an important parameter to measure the non-potentiality of the magnetic field in solar active regions (cf. Severny, 1958; Hagyard et al., 1984; Lü et al. 1993; Schmieder et al., 1993; Wang et al., 1994a; Zhang et al., 1994; Chen et al., 1994; Li et al., 2000), while the non-potential field can also be measured from the strong magnetic gradient of active regions, which is strongly correlated with active region flare-CME productivity (cf. Severny, 1958; Falconer, 2001). It is found that the vertical component of the non-potential field relative to the potential field with the shear angle θ can be written in a series expansion as the angle is small

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3 Solar Magnetic Activities

Bnh⊥

  θ3 θ5 + − ······ = Bonh sin θ = Bonh θ − 3! 5!   θ2 θ4 = Bonh θ 1 − + − ······ . 3! 5!

(3.12)

This means the shear angle provides some message of the non-potential field, and the parallel component of the non-potential field is Bnh = B ph − B0 cos θ.

(3.13)

This also means that the magnetic shear angle θ cannot reflect the non-potentiality of the magnetic field simply. As one knows that the transverse components of the potential magnetic field are not an observed quantity, but have to be inferred by the extrapolation of longitudinal components of the observational magnetic field (cf. Hagyard & Teuber, 1978; Sakurai, 1982). This is based on the assumption of the extension of magnetic lines of force from photospheric magnetic charges. This means that one cannot accurately estimate the contribution of the longitudinal component of non-potential magnetic fields of solar active regions from observations. Figure 3.4 shows the relationship among the observational highly twisted vector magnetic fields, the calculated potential, non-potential magnetic fields, and corresponding magnetic free energy densities in the superactive region NOAA 65806619-6659 in 1991. We can find in a row (a) of Fig. 3.4 that highly twisted transverse magnetic fields form in the active region. Row (b) of Fig. 3.4 shows the calculated transverse components of potential magnetic fields inferred by the observational longitudinal magnetic fields of the active region. Row (c) of Fig. 3.4 shows the nonpotential components of magnetic fields inferred by Eq. (3.3). In Fig. 3.4 row (d), the magnetic free energy densities in the active region are inferred by ρ f h in Eq. (3.9) contributed by the transverse components of fields only. We find that the free energy density is negative in some areas (highly sheared magnetic neutral lines) of 1 1 2 Bnh + 4π Bnh · B ph ) < 0 in Eq. (3.7), where B ph is inferred the active region, i.e., ( 8π by the observed longitudinal magnetic field. These areas of negative sign, labeled A and B, can be defined as “negative energy regions” in a row (d) of Fig. 3.4. Figure 3.5 shows a schematic of the process of development of the observed highly twisted vector magnetic field in the super active region NOAA 6580-6619-6659 in 1991. With the emergence of the new magnetic flux, the highly sheared magnetic field formed near the magnetic neutral line of the active region, and the transverse component of the magnetic fields gradually became parallel to the magnetic neutral line. This relates to the storage of free magnetic energy and the formation of “negative energy regions” in the active region. Figure 3.5 also shows the relationship among the observational magnetic field Bo , the potential field B p , and the non-potential field Bn near the magnetic neutral line in the active region as viewed from above. This provides a possibility for the 1 1 Bn2 < − 4π Bn · B p . occurrence of “negative energy regions” in the active regions as 8π

3.1 Magnetic Energy, Shear, Gradient, and Electric Current in Solar Active Regions 04:02 UT, Apr 14

01:14 UT, May 11 A

A B

159

05:29 UT, June 9 A

B

B

a

A

A B

A

B

B

b

A

A B

A

B

B

c A

A B

B

A

B

d Fig. 3.4 The vector magnetic fields in active regions NOAA 6580 (N28, W12) on 1991 April 14 (left), 6619 (N29, E09) on 1991 May 11 (middle) and 6659 (N32, E05) on 1991 June 9 (right). Top row (a): The arrows mark the observed transverse component of field Bo . Second row (b): The arrows show the transverse components of potential field B p inferred from the longitudinal components of observational magnetic fields. Third row (c): The arrows show the transverse components of the non-potential field Bn inferred from Eq. (3.3). The contours indicate the longitudinal magnetic field distribution of ±50, 200, 500, 1000, 1800, and 3000 gauss. Bottom row (d): The contours show the distribution of free magnetic energy densities ρ f h of ±1, 5, 10, 20, 50, 100, 200, and 400 (×103 Mx2 /cm4 ) for the quantity shown in grayscale, where solid (dashed) lines correspond to positive (negative) sign

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3 Solar Magnetic Activities

Bn ϕ EFR

Bo

θ

Bp

Fig. 3.5 A schematic on the development of active region NOAA 6580-6619-6659 in 1991 April– June, and also the relationship among the observational magnetic field Bo , the potential field B p and the non-potential field Bn as viewed from above

The relevant results can be found in Fig. 3.4d row. We can find that in these “negative energy regions” the inclined angles ϕ between the potential and non-potential com1 Bn · B p is ponents of transverse fields in the active region are larger than 90◦ , i.e., 4π negative in Figs. 3.4 and 3.5. For comparison, Fig. 3.6 shows the distribution of different kinds of magnetic energy density parameters in active region NOAA 6580-6619-6659 in 1991. In the calculation of photospheric energy density, only transverse components of the magnetic fields have been used, because the longitudinal components do not change in this analysis. Row (a) of Fig. 3.6 shows magnetic energy densities inferred from the observational transverse magnetic fields. Row (b) of Fig. 3.6 shows magnetic energy density inferred from the potential transverse magnetic fields, which is calculated using only the longitudinal components of magnetic fields. Row (c) of Fig. 3.6 shows the energy density parameters ρn of non-potential magnetic fields. There are obvious differences among the magnetic energy density parameters inferred from the observational transverse components of field Boh , from the potential transverse components of field B ph , and also from the non-potential transverse com1 Bnh · B ph ponents of field Bnh . Notice that row (d) of Fig. 3.6 shows the difference 4π between magnetic energy density ρ f h (Eq. (3.9)) and energy density parameter ρn of non-potential transverse magnetic fields in Eq. (3.5). The large-scale negative sign areas A and B reflect where components of the potential field show large inclined angles to the non-potential one in the active region NOAA 6580-6619-6659. Figure 3.7 shows the evolution of vector magnetic field, 171 Å images, magnetic 1 Bnh · B ph in the middle of free energy density ρ f h and the difference quantity 4π active region NOAA 11158 to analyze the evolution of these quantities before X2.2 flare on February 15. It is found that the large-scale negative energy density parameter 1 B · B ph tends to extend along the direction of EUV 171 Å loops and the highly 4π nh sheared major magnetic neutral line between large-scale opposite polarities in the active region. As compared with the evolution of magnetic energy density, it is found that the maximum value of free magnetic energy density C weakens and the channel

3.1 Magnetic Energy, Shear, Gradient, and Electric Current in Solar Active Regions 04:02 UT, Apr 14

01:14 UT, May 11 A

A B

161

05:29 UT, June 9 A

B

B

a

A

A B

A

B

B

b

A

A B

A

B

B

c A

A B

B

A

B

d Fig. 3.6 The magnetic energy density parameters in active region NOAA 6580 on 1991 April 14 (left), 6619 on 1991 May 11 (middle) and 6659 on 1991 June 9 (right). Top row (a): Inferred from the observed transverse component of field Bo . Second row (b): Inferred from the transverse components of potential field B p calculated by the longitudinal components of observational magnetic fields. Third row (c): Inferred from the transverse components of non-potential field Bn . Bottom row 1 (d): The difference Bnh · B ph between magnetic free energy density ρ f h and energy density 4π of non-potential field ρnh in (Eq. 3.10). The contours indicate ±1, 5, 10, 20, 50,100, 200, 400 (×103 Mx2 /cm4 ) for the quantity shown in grayscale, where solid (dashed) lines correspond to positive (negative) sign

162

3 Solar Magnetic Activities 13:00 UT, 2011 Feb 14

D

C

C

D

17:00 UT, 2011 Feb 14

D

C

C

D

20:48 UT, 2011 Feb 14

D

C

C

D

01:00 UT, 2011 Feb 15

D

C

C

D

Fig. 3.7 A local area of active region NOAA 11158 in 2011 February 14–15. From left to right: the longitudinal magnetic fields, 171 Å images, the magnetic free energy density ρ f h , and quan1 tity Bnh · B ph . The contours of magnetic energy density indicate ±1, 5, 10, 20, 50, 100, 200, 4π 400×103 (Mx2 /cm4 ). The red (green) ones relate to the positive (negative) sign

1 where 4π Bnh · B ph is negative (marked by D) tends to increase near the magnetic neutral line in the active region NOAA 11158 before the X2.2 flare.

3.1.1.2

Free Energy Contributed from Non-potential Magnetic Fields

Now we analyze the free magnetic energy contributed by different components of magnetic fields (p117 Priest 2014) Wo =

1 8π

 Bo2 d V =

1 8π



1 (Bn2 + 2Bn · B p + B 2p )d V = 8π

 (Bn2 + 2Bn · B p )d V + W p ,

(3.14)

3.1 Magnetic Energy, Shear, Gradient, and Electric Current in Solar Active Regions

163

where Wo is the observed magnetic energy, W p is the potential magnetic energy, Bo is the observed magnetic field, Bn and B p are the non-potential and potential components of the magnetic field, respectively. As we set B p = ∇ψ, it is funding the free energy W f = Wo − W p  1 1 2 = ψBn · ds. Bn d V + 8π 4π S

(3.15)

If the vertical component Bn⊥ = 0 in the surface S of the close volume, it is found 1 Wf = 8π

 Bn2 d V.

(3.16)

This means that the free magnetic energy relates to non-potential field only in the

1 special condition of Bn⊥ = 0 in the surface S. The integral 4π [∇ · (ψBn )d V is an observable quantity in the solar photosphere by magnetographs. We also notice   d ∂ B2 2 d V + B 2 u · dS B dV = dt ∂t  ∂ 2 (B p + Bn2 + 2B p · Bn )d V + (B 2p + Bn2 + 2B p · Bn )u · dS, = ∂t (3.17) dW f d = (Wo − W p ) dt dt   ∂ψBn ∂ Bn2 dV + 2 · dS + [Bn2 + 2∇ · (ψBn )]u · dS (3.18) = ∂t ∂t    ∂ψBn ∂ Bn2 2 dV + + [Bn2 + 2∇ · (ψBn )]u · dS, = ∂t ∂t where  2



∂ [∇ · (ψBn ) − ψ∇ · Bn ]d V ∂t  ∂ ∂ ∇ · (ψBn )d V = 2 (ψBn ) · dS. =2 ∂t ∂t

∂ (B p · Bn )d V = 2 ∂t

(3.19)

As Bn ⊥dS, one can find dW f = dt



∂ Bn2 dV + ∂t

[Bn2 + 2∇ · (ψBn )]u · dS.

(3.20)

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3 Solar Magnetic Activities

It is similar to Low (1982), from Eq. (3.14), one can find the force-free fields Wo =

1 8π



Bo2 d V 1 =− (Bo · r)Bo · dS 4π 1 =− [(Bn + B p ) · r](Bn + B p ) · dS, 4π  1 1 Wp = B 2p d V = − (B p · r)B p · dS, 8π 4π  1 W = (Bo2 − B 2p )d V 8π 1 =− {[(Bn + B p ) · r](Bn + B p ) − (B p · r)B p } · dS 4π 1 =− {[(Bn + B p ) · r]Bn + (Bn · r)B p } · dS. 4π As Bn ⊥dS,

1 W = − [(Bn · r)B p ] · dS 4π  [(Bnx X + Bny Y )B pz ]d xd y. =

(3.21)

(3.22)

(3.23)

(3.24)

z=0

As F(r, t) is a vector field at the spatial position x at time t, d dt



 (t)

F(x, t) · ds =



(t)

−

∂(t)

 ∂F(x, t) + [∇ · F(x, t)v] · ds ∂t

(3.25)

[v × F(x, t)] · dl,

where  is a moving surface in three-space bounded by the closed curve ∂, ds is a vector element of the surface , and dl is a vector element of the curve ∂. Let F = (Bn · r)B p , then

and

∂B p ∂F(x, t) ∂(Bn · r) = B p + (Bn · r) ∂t ∂t ∂t    ∂B p ∂r ∂Bn · r + Bn · B p + (Bn · r) = ∂t ∂t ∂t

(3.26)

∇ · F(x, t) = ∇ · [(Bn · r)B p ] = [−(r · ∇)Bn + r × (∇ × Bn ) + (Bn · ∇)r] · B p .

(3.27)

3.1 Magnetic Energy, Shear, Gradient, and Electric Current in Solar Active Regions

165

If the contribution of the second term in Eq. (3.17) has been ignored, from the induction equation ∂B = ∇ × [v × B − η(∇ × B)], (3.28) ∂t the variation of the free energy can be written dW f 1 = dt 4π

∂ D(t)

[(vn × Bn ) × (Bn + B p ) + (v p × B p ) × Bn

− η(∇ × Bn ) × (Bn + B p )] · dS (3.29)  1 + [(∇ × Bn ) · (vn × Bn + v p × B p ) − η(∇ × Bn )2 ]d V, 4π D(t) where η is the magnetic diffusivity, vn and v p are the corresponding velocity fields relative to Bn and B p , respectively. As the potential field is fixed (v p = 0) and the magnetic diffusivity η = 0, then Eq. (3.29) becomes simple.

3.1.1.3

Conclusions and Discussions for Magnetic Energy Density

We have presented the magnetic energy density parameters inferred from photospheric vector magnetograms of the recurrent active region NOAA 6580-6619-6659 and 11158. This provides a chance to analyze the storage and evolution of magnetic energy in the active regions. After this analysis, the main results are as follows. (1) Observations of photospheric vector magnetic fields provide important information on the distribution of free magnetic energy density in the lower solar atmo1 2 B and sphere. The free magnetic energy density is comprised of two terms 8π nh 1 Bnh · B ph . The first term relates to non-potential magnetic fields and the second 4π one to the relationship between the non-potential and potential fields. This means that the change in photospheric free magnetic energy density not only depends on the non-potential component of the magnetic field but also on the relationship with the potential field. 1 Bnh · B ph is an important quantity for understanding the degree of (2) The term 4π 1 Bnh · B ph tends to occur magnetic shear in the active regions. The negative sign of 4π in the areas of highly sheared magnetic fields in the active regions. It is also noticed 1 1 2 + that in the strongly sheared areas ( Bnh Bnh · B ph ) < 0 relative to the highly 8π 4π sheared magnetic fields defined as “negative energy region” can form in delta active regions. If the inclination angles between potential and non-potential magnetic lines of force decrease with height at the solar atmosphere of the active regions, the term 1 Bnh · B ph will change the sign from negative to positive with the increase of the 4π

166 Fig. 3.8 A simplified 1 schematic of Bnh · B ph 4π with height above the photospheric magnetic neutral line in the highly sheared active region as viewed from the side. The red (blue) shade shows the positive (negative) sign area. The red arrow indicates the photospheric magnetic neutral line in the active region

3 Solar Magnetic Activities

_

+ BL=0

height near the magnetic neutral lines in the active regions. This is consistent with the idea that some free magnetic energy is stored in the high solar atmosphere of active regions where powerful flares might be triggered. A very simplified picture of 1 Bnh · B ph with height in active regions is shown in Fig. 3.8. the relationship 4π (3) It is found that the mean photospheric magnetic energy in the active regions changes obviously before some powerful solar flares. This might reflect the release of stored free magnetic energy powering flares, even if the free energy density can be negative in the photosphere in the local areas of some active regions. From the analysis of magnetic energy density of active regions in the lower solar atmosphere, we still would like to discuss the following questions. The analysis of magnetic shear in active regions is normally based on the inclination angle between observed and potential transverse magnetic fields. It is found that the magnetic shear provides some information about the triggering of solar flares and CMEs, although it cannot provide all information about free magnetic energy in flare/CME-producing regions even in the lower solar atmosphere. This is because the real configuration of magnetic fields of active regions is more complex. The transverse and longitudinal components of magnetic fields are inferred from the Stokes parameters Q, U, and V, respectively, with different calibration coefficients. The accuracy of measurements of different components of the vector magnetic fields is still a basic problem for the determination of photospheric free magnetic energy density in solar active regions. It should be noted that the transverse potential field is a fictitious quantity inferred by the observational longitudinal field, so its strength is a reference value in the calculation of the magnetic energy density of active regions in the photosphere. The transverse components of potential magnetic fields can be calculated by the extrapolation of the longitudinal components of the observed magnetic fields. The estimation of non-potential components of the fields also obviously depends on the

3.1 Magnetic Energy, Shear, Gradient, and Electric Current in Solar Active Regions

167

measured results of longitudinal fields, and the choice of extrapolation methods for the horizontal components of potential fields.

3.1.2 Relationship of Magnetic Shear and Gradient with Electric Current In contact with Fig. 3.4, the evolution of the photospheric vector magnetic field in active region NOAA6659 in the heliographic image is shown in Fig. 3.9. The transverse field twisted and turned parallel to the magnetic neutral lines gradually with the development of the active region. The area of the active region extended. These reflect the intensive emergence of the magnetic flux of opposite polarity. The new magnetic flux emerged and pushed the old magnetic flux away. This means that the possible growth and evolution of the photospheric vertical current accompanied the emergence and evolution of magnetic flux. Figure 3.10 shows the distribution of magnetic shear in the active region NOAA 6659. The shear angle can be weighted by the transverse magnetic field (cf. Hagyard et al., 1984) Boh · B ph , (3.30) θT = Boh · cos−1 Boh B ph where Boh and B ph are the observed transverse field and that calculated from the magnetic charges in the approximation of potential field. The amplitude of the shear angle in Eq. (3.30) reflects the non-potentiality of the active region. The gradient of the photospheric longitudinal magnetic field in active regions can be inferred from (cf. Leka & Barnes, 2003)

Fig. 3.9 Vector magnetograms with 180c ir c ambiguity-resolved transverse components at 00:50 UT on 1991 June 6, at 03:53 UT on 1991 June 9, and at 043:8 UT on 1991 June 12 (from left to right) transformed from the image plane to heliographic coordinates. The contours indicate the longitudinal magnetic field distribution, ±20, 160, 640, 1280, 1920, 2240, 2560, and 2880 G. Zhang (1996b)

168

3 Solar Magnetic Activities June 9, 1991 05:29 UT

June 9, 1991 05:29 UT

Fig. 3.10 The distribution of magnetic shear (left) and gradient (right) in active region NOAA 6659 on 1991 June 9. The red arrows mark the observed transverse field and the blue arrows show the transverse components inferred from the calculation of magnetic charges. From Zhang (2006a)

 |∇(Bz )| =

∂ Bz ∂x

2

 +

∂ Bz ∂y

2 .

(3.31)

The distribution of the corresponding magnetic shear and gradient in active region NOAA 6659 inferred from the photospheric vector magnetogram is shown in Fig. 3.10. The main contribution of magnetic shear in the active region comes from the deviation of the transverse field from the potential field inferred by magnetic charges in the photosphere, while the magnetic gradient comes from the non-uniformity of the longitudinal field. As one knows that the non-potential magnetic field in the solar active regions implies the electric current in the atmosphere of active regions also. It is also an important index for analyzing solar flare activities (such as Zhang & Wang, 1994). Thus, the analysis of the current in the photosphere of active regions is also important, but the question is what is the relationship between the magnetic shear and electric current in solar active regions. The relationship between the magnetic field and current is 1 ∇ × B, (3.32) J= μ0 where J is in units of Am−2 and μ0 = 4π × 10−3 GmA−1 is the permeability in free space. The vertical component of the current in the cylindrical coordinates is (in Fig. 3.3)   ∂ 1 1 ∂ (r Bnϕ ) − (Bnr ) [∇ × (B p + Bn )]z = Jz = μ0 μ0 r ∂r ∂ϕ   (3.33) ∂ Bnh cosϕ ∂ Bnh sinϕ 2Bnh + − , = μ0 r ∂r μ0 r ∂ϕ

3.1 Magnetic Energy, Shear, Gradient, and Electric Current in Solar Active Regions

169

or in the integral form 1  (∇ × B)z · dSz μ0 S 1 1 = (Br eˆ r + Bϕ eˆ ϕ ) · (dr eˆ r + r dϕˆeϕ ) = Br dr + Bϕr dϕ. μ0 C μ0 C

Iz =

(3.34)

It can be found that the current implicates the change in magnetic shear and gradient in the non-potential magnetic field. The relationship between the non-potential field and magnetic shear has been provided in Eqs. (3.12) and (3.13). As letting B = Bb and b is the unit vector along the direction of the magnetic field, the current may be written in the form J=

1 B ∇ × b + (∇ B) × b. μ0 μ0

(3.35)

It is found that the electric current in solar active regions relates to the properties of chirality and gradient of the magnetic field. The first term in Eq. (3.35) connects with the twist of unit magnetic lines of force and intensity of the field. The second term in Eq. (3.35) connects with the heterogeneity and orientation of the magnetic field. According to the formula of current helicity density (Zhang & Bao, 1999), one can obtain (3.36) h c = B · ∇ × B = B 2 b · ∇ × b. As assuming that the equilibrium is force-free, we can write B∇ × b + (∇ B) × b = αBb

(3.37)

and dotting this with b gives α = b · ∇ × b.

(3.38)

By comparing Eqs. (3.36–3.38) with Eq. (3.35), it is found that the basic information on the magnetic shear and gradient are contained in the shear term of the current. The definition of the angle of observed magnetic shear in Eq. (3.35) relative to the direction of transverse field inferred from the potential field is slightly different from the orientation of transverse field relative to ∇ B, but it reflects the difference of the reference frames in analyzing the non-potentiality of the magnetic field only. It is found that the dominant contribution of the shear component of current occurs near the highly sheared magnetic neutral line in the active region. It is consistent with the contribution of magnetic shear and gradient in Fig. 3.10. This means that the shear component of the current actually provides both magnetic shear and gradient of the field in active regions. The directions of magnetic shear and gradient are lost in Eqs. (3.30–3.31). Moreover, it is also noticed that the contribution of the twist component of the current is not contained in the analysis of the magnetic shear of the transverse

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3 Solar Magnetic Activities

Fig. 3.11 The photospheric vertical electric current Jz in the active region (NOAA 6659) at 05:29 UT on June 9, 2000, inferred by the vector magnetogram in Fig. 3.5. The arrows mark the transverse field. The solid (dashed) contours correspond to the upward (downward) flows of vertical current of ±0.002, 0.008, 0.02, 0.04, 0.075, 0.12 Am−2 . The size of view is 2. 72 × 2. 72. From Zhang (2001a)

June 9, 1991 05:29 UT

A

B

field and the gradient of the longitudinal field in the active regions. This means that the electric current is relatively a more complete quantity than the magnetic shear and gradient in the study of the non-potential field. The magnetic shear or gradient are not enough to describe the magnetic non-potentiality simply. It is normally believed that the high gradient of longitudinal magnetic field and the shear of the transverse magnetic field near the magnetic neutral line in solar active regions are two important indexes for analyzing solar flare activities in the solar atmosphere. The solar flares normally connect with the non-potential magnetic field. The normal morphological description of photospheric magnetic shear in solar active regions can be found in Fig. 3.5. The shear term in Eq. (3.35) actually provides a quantitative description of the relationship between the shear and gradient of the magnetic field, and this term can be written in the corresponding form μ10 cosθ(∇ B)⊥ , where θ is the shear angle of the photospheric transverse magnetic field and (∇ B)⊥ is the horizontal gradient of the magnetic field. The distribution of the electric current in the active region NOAA 6659 is shown in Fig. 3.11. It is found that the electric current flows outward from the center of the active region and downward at the surrounding areas. By analyzing the distribution of the electric current, one also can write the vertical current in the form, which connects with Eq. (3.35), B Jz = μ0



∂b y ∂bx − ∂x ∂y

where bx =

Bx B



1 + μ0

and



∂B ∂B by − bx ∂x ∂y

by =

By . B

 ,

(3.39)

(3.40)

Figure 3.12 shows the correlative relationship between both terms in the right side of Eq. (3.39) in the active region NOAA 6659.

3.1 Magnetic Energy, Shear, Gradient, and Electric Current in Solar Active Regions Fig. 3.12 The scatter of the vertical current density in the active region NOAA 6659 on June 9, 1991. The ordinate marks the vertical current contributed by μB0 (∇ × b)z and abscissa marks that by 1 μ0 ((∇ B) × b)z . The unit is Am−2 . From Zhang (2001a)

171

Current of chirality

0.05

0.00

-0.05 -0.05

0.00 Current of hetergeneity

0.05

3.1.3 Ratio Between Different Components of Current We now define a deviation rate R from the chiralical magnetic field, which can be written in the form R=

shear magnitude | (∇ B) × b | = . helicit y magnitude | B∇ × b |

(3.41)

The mean deviation rate Rz for the vertical current in active region NOAA 6659 can be estimated by standard errors of the current Rz =

3.34 ≈ 0.57. 5.81

(3.42)

This means that the helicity term in Eq. (3.35) contributes to the electric current of active region NOAA 6659 dominantly. As two samples, we analyze both kinds of electric current at two special places in Fig. 3.11. A in Fig. 3.11 is located near and B is away from the magnetic neutral line in the active region. Figure 3.13 shows the scatter of both kinds of vertical current of Eq. (3.35) near A and B in active region NOAA 6659 on June 9, 1991. It is found that standard errors near A are 9.59 for the helicity term and 5.52 for the shear term (the unit is 10−3 Gm−2 ). Rz A ≈ 0.58. While near B the standard errors are 4.44 and 1.07 for both terms. Rz B ≈ 0.24. These quantitatively provide that the vector magnetic field in the active regions is roughly consistent with the approximation of force-free equilibrium in the photosphere, while the influence of the non-force free field probably is not negligible in the vicinity of the magnetic neutral line. In the theoretical analysis of the magnetic field, the above results mean that the magnetic field solar active regions are mainly force-free and it is not excessively twisted (Bellan, 2001). After the analysis, the main results are the following: (1) The electric current in the solar active regions can be separated into two parts. One relates to the distribution of the chirality and intensity of the magnetic field.

172

3 Solar Magnetic Activities 0.05

a Current of chirality

Current of chirality

0.05

0.00

-0.05 -0.05

0.00 Current of hetergeneity

0.05

b

0.00

-0.05 -0.05

0.00 Current of hetergeneity

0.05

Fig. 3.13 The scatter of the vertical current density in the active region NOAA 6659 on June 9, 1991 near A (a) and B (b) of Fig. 3.11. The ordinate marks the vertical current current contributed by μB0 (∇ × b)z and abscissa marks that by μ10 ((∇ B) × b)z . The unit is Am−2 . From Zhang (2001a)

The other connects with the heterogeneity of the magnetic field intensity and the orientation of the field. (2) It is found that in principle the magnetic shear does not immediately relate to the force-free magnetic field. As an example, in the powerful flare producing region NOAA 6659, it is found that the electric current mainly is contributed by that of magnetic chirality.

3.1.4 Evolution of Current with Flares in Active Regions 3.1.4.1

Evolution of Current

As one sets c is a constant vector and with the formula ∇ · (a × c) = (∇ × a) · c − a · (∇ × c), it is found   d V [c · (∇ × a)] = d V [∇ · (a × c)] = dS · (a × c) = c · (dS × a), V

V

S

(3.43)





then

S

d V (∇ × a) = V

dS × a.

(3.44)

S

It is easy to find that the change in the total electric current I can be written dI = dt

 dV V

1 ∂J = ∂t μ0

 dV V

1 ∂ (∇ × B) = ∂t μ0

dS × S

∂B . ∂t

(3.45)

3.1 Magnetic Energy, Shear, Gradient, and Electric Current in Solar Active Regions

173

This means that the evolution of the magnetic field in the solar surface actually reflects the change in the electric current in the solar atmosphere.

3.1.4.2

Evolution of Current with Flares in Active Region NOAA 5747

NOAA 5747 was a flare-productive active region during its transit across the solar disk in October 1989. After the resolution of the 180◦ -ambiguity of the transverse field synthetically, and transformation of vector magnetograms from the image plane to the heliographic frame, Wang et al. (1994b) have determined the distribution of the photospheric vertical electric current density in the active region. By analyzing the evolution of vector magnetograms and vertical current over a 6-day period (October 17–22) in the active region, Wang et al. (1994b) found the following results: (1) Two magnetic fluxes of opposite polarities emerged synchronously with their separating motion, one of which converged with an old magnetic structure and caused a number of flares. (2) There appeared a new current system, with the emergence of the fluxes. (3) The initial Hβ bright kernels occurred in the vicinity of the neutral line of vertical current (Jz = 0) with a steep gradient, but not just on the sites of vertical current peaks. (4) The flares were probably triggered by the interaction between the newly emerging electric current system and the old current system. Figure 3.14 shows the evolution of vertical electric current density and the relationship with flares in active region NAOO 5747.

Fig. 3.14 A series of maps of vertical electric current density in active region NAOO 5747. Contour levels are ±0.2, 0.4, 0.8, 1.2, 1.4, 1.6, 2.0 ×10−2 . Bold contours indicate the patterns of flares at the corresponding times. From Wang et al. (1994b)

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3 Solar Magnetic Activities

3.2 Current Helicity in Solar Active Regions Magnetic (current) helicity is an important physical quantity in magnetohydrodynamics. It involves the basic problem of the topological connectivity of magnetic field force lines. It quantifies the twisting and knotting characteristics of the magnetic fields. In recent years, it has been widely valued in the research of solar physics and astrophysics. We discuss the subject of magnetic helicity with the observation of the solar magnetic field.

3.2.1

Magnetic Helicity

The linking number L X Y of the two curves can be found from the Gauss linkage formula (Berger & Field, 1984): L XY = −

1 4π



ds

ds 

  r dX(s) dY(s  ) · 3× ds r ds 

(3.46)

where r = X(s) − Y(s  ). When integrated over all space, the helicity integral can be expressed in a similar form (Moffatt, 1969; 1981; Arnold, 1974). In Coulomb gauge the vector potential A is given by 1 c

A=



d3x

J(x  ) , r

(3.47)

where r = x − x . Using J = (c/4π) ×B and integrating by parts, we find that 1 A(x) = − 4π which gives 1 Hm = − 4π



 3

d x



d3x

r × B(x  ), r3

d 3 x  B(x) ·

r   × B(x ) . r3

(3.48)

(3.49)

The twist number is given by 1 Tw = − 2π

ds

ˆ d U(s) ˆ · [U(s) × X(s)], ds

(3.50)

ˆ = U/|U| (in general, (3.50) will not work for a field line Y that travels with U ˆ perpendicular or backwards with respect to X, because then U(s) would not be single-valued. However, Tw can still be defined for ill-behaved Y-curves via (3.52) below.) Finally, the writhing number is the Gauss linkage integral applied to the axis:

3.2 Current Helicity in Solar Active Regions

Wr = −

1 4π



ds

ds 

175

  r dX(s) dX(s  ) . · 3× ds r ds 

(3.51)

where r = X (s) − X(s  ). Wr can be shown to equal the signed sum of crossovers exhibited by the axis curve, averaged over all projection angles (Fuller, 1978). White (1969) has proved that (3.52) L X Y = Tw + Wr (an earlier, more restricted version of this theorem was found by Calugareanu (1959)). Of these quantities, only Tw can be defined for a subsection of a ribbon, as L X Y and Wr are given by double integrals. On the other hand, only L X Y is topologically invariant, i.e., unchanged by deformations of X and Y that do not let the two curves cross each other. Furthermore, the writhing number has the unique property of depending only upon the geometry of the axis curve X. ∂B = As we study the variation of the magnetic helicity by using Faraday’s law ∂t −c∇ × E, the evolution of the magnetic helicity density h m can be derived (Berger & Field, 1984), ∂h m = −2cE · B − c∇ · (E × A + φB), (3.53) ∂t where φ is the scalar potential. Integrating this over the volume V , the magnetic helicity satisfies the evolution equation  d Hm = −2 E · Bd V + (φB + A × E) · nd S = −2ηC, (3.54) dt V ∂V

where C = V J · Bd V is the current helicity. The vector potential A of the magnetic field is involved in the general definition of the formula of the magnetic helicity (3.48). According to the requirement of gauge invariance, when the potential is gauge transformed, all physical quantities and physical laws remain unchanged. The formula (3.49) should also satisfy the gauge invariance (A + ∇φ → Hm = 0), but it is only a closed magnetic field at the integral boundary, where the gauge invariance is satisfied only when the surface (closed field) or the whole space at infinity (convergent field) is satisfied. Wang & Zhang (2005) pointed out that when studying the magnetic activity of the solar atmosphere, two boundaries are usually defined: the photosphere surface and a certain level of the corona above the photosphere (or infinity, depending on the actual problem). Therefore, the general definition of magnetic helicity in the actual solar atmosphere does not satisfy the requirement of gauge invariance. From the “practical” point of view for the observables, Berger & Field (1984) partially solves the problem of boundary conditions (or avoids closed boundary conditions) by introducing a reference field, thus improving the above problems to a certain extent. The difference between the introduced imaginary reference field and the real field is that the region above the photosphere is replaced by a potential

176

3 Solar Magnetic Activities

field (no body current distribution, but surface current distribution must exist on both sides of the photosphere boundary surface if the tangential component of the magnetic field jumps). The difference between the magnetic helicity of the real field and the magnetic helicity of the reference field is gauge-invariant, and the only boundary condition is the normal continuity of the magnetic field on both sides of the photosphere. (3.55) H R (Va ) ≡ H R (Ba , Bb ) − H R (Pa , Bb ), ˆ S = Bb · n| ˆ S. Ba · n|

(3.56)

In the formulas (3.55) and (3.56), the subscript a represents the space above the photosphere, b represents the space inside the Sun below the photosphere, the spherical boundary surface R represents the reference value, and P represents the reference field of a potential field state above the photosphere. Berger & Field (1984) proves that when the real field and the reference field are consistent with the magnetic field configuration of the b space below the photosphere, the specific configuration of the b space magnetic field does not affect the value of the relative magnetic helicity. Of course, from the geometrical point of view of magnetic helicity, it reflects the geometric topological properties of the magnetic field. Therefore, the magnetic helicity difference of any two fields should be determined by their magnetic field difference, that is, determined by the field configuration in the spatial region where the magnetic fields are different. The evolution of the current helicity density can be derived, ∂h c = −2c∇ × E · ∇ × B − c∇ · [(∇ × E) × B] , ∂t

(3.57)

or in the form ∂h c = 2(u × B) · [∇ × (∇ × B)] + ∇ · [2(u × B) × (∇ × B) − B × (∇ × (u × B))]. ∂t

(3.58)

As the volume V is fixed and one can integrates Eq. (3.57) over V and ∂V is the surface of V , it is found that   d Hc = −2c (∇ × E) · (∇ × B)d V − c (3.59) [(∇ × E) × B] · nd S. dt V ∂V

It is found that the current helicity is not conserved. If the ideal Ohm’s law applies, 1 then E = − V × B. We can find that the time variation of the current helicity c depends on the twisted motion and variation of the magnetic field.

3.2 Current Helicity in Solar Active Regions

177

3.2.2 Observational Evidence of Magnetic Chirality  Although the determination of the real total current helicity Hc =

 hcd V =

B·

(∇ × B)d V is complex in the solar atmosphere, while the marks of the current helicity in the photosphere can be detected by the photospheric vector magnetograms. We can analyze the distribution of the current helicity density and its evolution in the photosphere. We notice that the current helicity density can be written into two parts, h c = B · (∇ × B) + B⊥ · (∇ × B)⊥ .

(3.60)

The first term on the right side of Eq. (3.60) is observable, and can be inferred by photospheric vector magnetograms (Abramenko et al., 1997). While the second term is difficult to obtain as one renounces more assumptions, due to no observational data of the vector magnetic field in the other layers of the solar atmosphere. If one neglects the second term (the combined one between the transverse magnetic field and transverse current in the current helicity) in the right of Eq. (3.60), one can analyze the chirality of the magnetic field near the areas where the longitudinal component of the magnetic field and electric current are more dominant than the transverse ones. It is important to provide the basic property of the twisted magnetic field near strong magnetic poles in the active regions because one normally believes that the magnetic poles vertically extend up from the deep atmosphere (Lites & Skumanich, 1990). The observational part of the current helicity density is  h c (obs) = B · (∇ × B) = Bz

∂ By ∂ Bx − ∂x ∂y

 .

(3.61)

The derivatives are approximated by a four-point differencing scheme; the current helicity is computed at each intersection of four magnetogram pixels and the calculated helicity smoothed to eliminate the small-scale fluctuation of the observing data (Wang et al., 1994b). In the approximation of the force-free field, the α factor can be obtained by the formula (Pevtsov et al., 1994) α=

(∇ × B) · B = B2



∂ By ∂ Bx − ∂x ∂y

 /Bz .

(3.62)

As we compare the distribution of the α factor with B · (∇ × B) in the photosphere, we find that both parameters show also the same sign distribution, while the α factor brings more information on the highly sheared magnetic field (such as near the magnetic neutral line) and also no any real information on B⊥ · (∇ × B)⊥ . The resolution of the 180◦ ambiguity of the highly sheared transverse magnetic field

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Fig. 3.15 The vector magnetograms (left) and corresponding current helicity parameter B · ∇ × B (right) in the active region on 5 May 1999. The transverse components of the field are observed at 0.0 and −0.15 Å from the 5324.19 Å line center, respectively. In the left, the solid (dashed) contours correspond to the positive (negative) longitudinal field, and the arrows indicate the transverse field. In the right, the solid (dotted) contours/white (black) intensity correspond to positive (negative) helicity at −0.15 and 0.0 Å, and the contours correspond to ± 0.0025, 0.01, 0.025, 0.05, 0.09, 0.15 (G 2 m −1 ). The size of the maps is 1. 84 × 1. 84. From Zhang (2000)

normally is difficult in areas near the magnetic neutral lines in the flare-producing active regions. The magneto-optical effect is another notable problem for the measurement of the transverse magnetic field near center of the sunspots where the field is strong and with the smaller inclination of the magnetic field to the line of sight (Landolfi & Landi Degl’Innocenti, 1982). The observational error (about 10◦ for Huairou Mgnetograph) of the azimuthal angles of transverse field, due to the magneto-optical effect, probably causes a change in the observed mean current helicity value to influence the basic information of the current helicity significantly for some active regions (Bao et al., 2000a; Zhang, 2000). When one calculates the current helicity density by the observational photospheric vector magnetograms obtained at different wavelengths from the FeIλ5324.19 Å line center to the wing, the magneto-optical effect probably also influences the calculated distribution of the photospheric current helicity density in the active region. For analyzing the influence of the magneto-optical effect on the calculation of the current helicity in the active regions, in Fig. 3.15 we show the distribution of the transverse magnetograms, which magnetic field is obtained at 0.0 and −0.15 Å from the FeIλ5324.19 Å line center and the corresponding current helicity density parameter B · (∇ × B) in the active region on 1999 May 5 discussed above. The slightly different distribution of the current helicity parameter B · (∇ × B) in the active region can be found due to the difference in the amplitudes of Stokes Q and U observed at different wavelengths in the wing of the line.

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This is an example of the influence of the magneto-optical effect for determining the current helicity density in active regions, even if the magneto-optical effect also exists in the far wing of the line. It is consistent with the comparison of datasets (a series of magnetograms of active region NOAA 5747) obtained at Huairou and Mees Solar Observatories (Bao et al., 2000a). The transverse magnetic field in the active region (NOAA 5747) obtained at both Observatories show the same twist tendency, while the mean values of current helicity density parameter B · (∇ × B) are slightly different. The vector magnetograms observed at Mees Solar Observatory are reversed by a routine procedure that the Faraday rotation has been considered (Ronan et al., 1992). This means that the influence of the magneto-optic effect probably is a notable problem for the analysis of basic properties (such as the mean current helicity) of active regions using the Huairou vector magnetograms, even if in some times it has been neglected (relative to the other problems, such as that on the resolution of the 180◦ ambiguity of the transverse magnetic field, etc.). Moreover, the diagnosis of the magnetic field, including the magneto-optical effect, in solar active regions has also be discussed in more detail in Chap. 1.

3.2.3 Fine Features of Magnetic Field, Electric Current and Helicity in Solar Active Regions It is noticed, Parker (1984, 2002) pointed out that the remarkable fibril structure of the magnetic fields at the surface of the sun (with fibrils compressed to 1,000–2,000 gauss) lies outside existing statistical theories of magnetohydrodynamic turbulence. The nature of the elemental structures of the solar magnetic field is one of the most important mysteries in solar physics (Deng et al. 2009). The basic configuration of magnetic flux (the magnetic fluxes, intrinsic field strengths, area factors are discussed and the thermodynamic properties of fluxtubes) in the solar atmosphere was inferred from Stokes profiles by Stenflo et al. (1984). Lites et al. (2007, 2008) analyze Fe I 630 nm observations of the quiet Sun at disk center taken with the spectropolarimeter of the Solar Optical Telescope aboard the Hinode satellite. It is found that network areas exhibit a predominance of kG field concentrations. A similar result was presented by Jin et al. (2009). Su et al. (2009) presented Hinode vector magnetograms and G-band data to study the distributions of local twist α and current helicity in the active region NOAA 10930. It is found that the patches of positive and negative helicities were intermixed showing a mesh pattern in the umbra and a thread pattern in the penumbra. For its main stable sunspot, there was a positive-helicity patch accounting for 43% of the umbra area surrounding the inner umbra, which had a predominantly negative helicity. The fine distributions of α and helicity on a penumbral filament indicated that it may be possible for the two opposite helicities to coexist in a filament and their magnitudes were nearly equivalent.

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20:53:15-21:05:50 UT, Dec 12

Fig. 3.16 The image (left) and vector magnetogram (right) in active region NOAA 10930 in 2006 October 12. Arrows show the transverse component and the white (black) areas indicate positive (negative) polarity of the field. North is at the top, west is at the right. The size of images is 32 × 32 . From Zhang (2010)

NOAA 10930 was a fast-developing active region. The magnetic shear formed near the magnetic neutral line of the active region with some powerful flares (Magara et al., 2008; Inoue et al., 2008; Tan et al., 2009). We take on the analysis of the fine features of vector magnetic field in the active region NOAA 10930 on 2006 December 12, which vector magnetograms were observed by SP of Hinode in Fig. 3.16. It is found in Fig. 3.16 that the magnetic field in active region NOAA 10930 extends in the form of fibril features from the main magnetic poles of active regions almost along the direction of the transverse component of the field. The width of the typical fibril magnetic features is an order of 0.5 and the length is about 5 . The relationship between the magnetic fibril field and electric current is notable for analyzing the property of the active regions. It provides an important chance to analyze the possible configuration of the fine features of the electric current system. The vertical component of the electric current can be inferred from the observational magnetic 1 ∇ × B, where μ0 = 4π × 10−3 GmA−1 . field according to the Ampere role J = μ0 Figure 3.17b shows the vertical component of the electric current inferred from a part of vector magnetograms in Fig. 3.16. It is found the electric current in the active region shows the fibril features also. The size of fine features of electric current is similar to the magnetic ones. However, the correlation between the current density and Jsh⊥ and Jt⊥ are different from the large scale when the same analyses are applied to fibrils. Figure 3.18 shows the amplitudes of the magnetic field, current, both components of current and current helicity density along the vertical solid line in the magnetogram in Fig. 3.17. It provides the morphological configuration of electric current, its components, and helicity density across the magnetic fibrils. It is found there is no significant relationship in the intensity between the (longitudinal and transverse components of)

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a

b

c

d

Fig. 3.17 a The local vector magnetic field in active region NOAA 10930 is marked by the box in Fig. 3.16. The white (black) areas indicate the intensity distribution of the transverse field. The red and green contours show the longitudinal components of the magnetic field of ±200, 500, 1000, 2000, 4000 Gauss. b–d The corresponding vertical current density J⊥ and its twist Jt⊥ and shear Jsh⊥ components of current density. The white (black) indicates the up (down) flow of the current. From Zhang (2010)

magnetic field and other vertical current parameters, while the peaks of current and current helicity density tend to occur near the high variation of the magnetic field. One of the high amplitude of magnetic field is marked by the horizontal dash lines in Fig. 3.18. The position marked by the dash lines in Fig. 3.18 corresponds to the cross points of straight lines in Fig. 3.17. The maximum of the vertical current (about 0.4 Am−2 in Fig. 3.18b) relates to the high gradient of the magnetic field in Fig. 3.18a. It is found that the high morphological correlation between the vertical current density in Fig. 3.18b and its shear component in Fig. 3.18d. There is no significant relationship between the high gradient of the magnetic field and the amplitude of the twisted component of current density in Fig. 3.18a, c. For the current inferred from the vector magnetogram of Fig. 3.17, the ratio between the shear and twist components on the vertical current of Eq. (3.41) is R total = 0.629, while for the current along the vertical line in Fig. 3.17 is R line = 1.265. The difference between Rtotal and Rline means that the twist component is dominant in the contribution of fine features of the electric current to the total current in the active region, while the shear component is dominant for the highly sheared magnetic features in the penumbra of sunspots.

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Fig. 3.18 a The solid line indicates the intensity distribution of the longitudinal component of the magnetic field along the vertical line in Fig. 3.17a. The dotted line indicates the relative fluctuation of the transverse magnetic field and the zero value is set at the intensity value of its large-scale smooth transverse field. b–e The corresponding intensity of the vertical component of the current J⊥ , chirality component Jt⊥ , heterogeneity component Jsh⊥ of current, and the longitudinal component h z of current helicity density. The horizontal dash lines indecate the cross points between the vertical and horizontal lines. From Zhang (2010)

3.2.4 Questions from Vector Magnetic Fields to Electric Current and Helicity Based above presentation on the observations of the vector magnetic field of the Sun and also the electric current and helicity inferred from the magnetograms, one can find some fundamental questions: • The resolution of the 180◦ ambiguity of transverse fields is a difficult question due to the property of polarized light with Zeeman effects. Several basic assumptions and approaches have been used to resolve the 180◦ ambiguity of transverse components of magnetic fields, such as comparing the observed field to a reference field or direction, minimizing the vertical gradient of the magnetic pressure, minimizing the vertical current density, minimizing some approximation to the total current density, and minimizing some approximation to the field’s divergence (Metcalf et al., 2006; Georgoulis, 2012). Which of these treatments is prioritized is still questionable. • The projection of vector magnetic fields from the solar disk to the heliospheric coordinates relates to the transform of the different components of the magnetic field, such as in Fig. 3.9, while the different Stokes parameters relate to different

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sensitivities and noise levels (cf. Eq. (1.143)). This leads to the inconsistency in the transformation of different components of magnetic fields. It also causes degradation in spatial resolution of the vector magnetic fields after the inevitable smoothing fields from the image plane to heliographic coordinates in Fig. 3.9. • The observations of the chromospheric magnetic field are important for diagnosing the spatial configuration of magnetic lines of force as comparing with the photospheric one, while the disturbance of photospheric blended lines in the wing of the Hβ line also bothers the detection of the magnetic field in the high solar atmosphere in Fig. 2.21. The lower sensitivity of Stokes Q and U of the Hβ line cripples its importance on the observations of the transverse components of chromospheric magnetic fields. • The electric current helicity density h c = εi jk bi ∂bk /∂x j contains six terms, where bi is components of the magnetic field. Due to the observational limitations, only four of the above six terms can be inferred from solar photospheric vector magnetograms. By comparing the results for simulation we distinguished the statistical difference of the above six terms for isotropic and anisotropic cases (Xu et al., 2015). This means that the electric current and magnetic (current) helicity do not contain completeness in the theoretical sense.

3.3 Correlations Between Subsurface Kinetic Helicity and Photospheric Current Helicity in Active Regions Over a long time, the kinetic helicity kept unmeasurable since the observation is limited above the photosphere. The tool of helioseismology has recently made the direct investigation of kinetic helicity in the upper convection zone come true (Zhao, 2004; Kosovichev, 2006). The kinetic helicity in the interior of the solar convection zone, if detectable, is certainly a more direct parameter for tracking the α−effect than the photospheric current helicity. However, at present, the information of kinetic helicity is also limited at the shallow layer of solar subsurface, particularly 0–12 Mm in Zhao (2004).

3.3.1 Subsurface Kinetic Helicity Following Zhao (2004), the kinetic helicity is defined as: αv = v · (∇ × v)/|v|2 , where v is three-dimensional subsurface velocity derived from time–distance helioseismology inversions, as described by Zhao & Kosovichev (2003). In particular, he used a component of αv corresponding to the vertical component of the velocity and vorticity.

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αvz = vz (∂v y /∂x − ∂vx /∂ y)/(vx2 + v 2y + vz2 )

(3.63)

Here we adopt the original denotations of “αv1 ” and “αv2 ” to stand for the subsurface kinetic helicity at the depth of 0–3 and 9–12 Mm, respectively. Local helioseismology provides a unique tool to determine the sub-photospheric flows of active regions. A statistical study by Zhao (2004) showed that the subsurface kinetic helicity inside active regions observed by the Solar Heliospheric Observatory/Michelson Doppler Imager (SOHO/MDI) observations seemed to have a hemispheric preponderance, like what magnetic (or current) helicity observations had shown (Pevtsov et al., 1995; Bao & Zhang, 1998). Gao et al. (2009) analyzed the connection between the photospheric current helicity, calculated from vector magnetograms observed by Huairou Solar Observing Station, and the subsurface kinetic helicity measured from MDI observations in 38 solar active regions. Although there was an opposite hemispheric trend between the sign of current helicity and that of subsurface kinetic helicity near the solar surface, the result did not support that the subsurface kinetic helicity had a cause and effect relation with the photospheric current helicity at the depth of 0–12 Mm. A similar result was reported by Maurya et al. (2011) as well. Now, Solar Dynamics Observatory/Helioseismic and Magnetic Imager (SDO/HMI; Scherrer et al., 2012; Schou et al., 2012) observations provide an unprecedented opportunity to investigate the connection between subsurface kinetic helicity and current helicity, as both subsurface flow velocity and photospheric vector magnetic field are available at the same time (Fig. 3.19). Subsurface kinetic helicity can be computed from subsurface flow velocities, which are routinely processed through the HMI time–distance analysis pipeline (Zhao et al., 2012). Photospheric current helicity can be computed from the photospheric vector magnetograms (Hoeksema et al., 2014).

3.3.2 Correlations Between Kinetic and Current Helicity The main purpose of this work is to study whether there is a correlation between the photospheric current helicity and subsurface kinetic helicity inside active regions. Current helicity is defined as Hc = B · (∇ × B). Similar to what was employed by Bao & Zhang (1998), the averaged value of the vertical component density of the current helicity, denoted as Hcz , was used in this study. Gao et al. (2012) compute the weighted Hcz , defined as Hcz divided by the total magnetic field strength, i.e., Hcz /|B2 |. Similarly, kinetic helicity is defined as Hk = v · (∇ × v), and take the averaged value of its vertical component, Hkz , and the value weighted by the speed, Hkz /|v2 |, for further studies. To compute both values of Hcz and Hkz , we only use the areas where |Bz | > 50 Gs. The top panel of Fig. 3.20 shows evolutions of the weighted current-helicity density and weighted kinetic-helicity density obtained at a depth of 0–1 Mm for AR 11158. The time series from 2011 February 13 to 17 are analyzed. The weighted

3.3 Correlations Between Subsurface Kinetic … 140

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Fig. 3.19 a Example of the subsurface velocity field of NOAA AR 11158 at a depth of 0–1 Mm. The background shows the vertical component of the velocity and arrows show the horizontal components. The maximum vertical velocity is 285 m s−1 and the longest arrow represents a horizontal speed of 441.8 m s−1 . The spatial sampling of the vertical component is 2.016 ×2.016 , and the field of view is 3.02 ×2.02 . b Example of vector magnetogram of NOAA AR 11158, with white showing positive polarity and black showing negative. The longest arrow represents a transverse field of 2356 G, and the transverse field lower than 300 G is not shown. From Gao et al. (2012)

current-helicity density with a 12-min cadence is averaged over a 4-hr period to compare with the weighted kinetic-helicity density, which is computed from the subsurface velocity obtained from an 8-hr data sequence with a 4-hr time step. Error bars are plotted for the current helicity with corresponding standard deviations. The two helicity curves show very similar varying tendencies and the correlation coefficient attains 0.67. Furthermore, we separate the data series into two sections. For the decreasing phase before 06:00UT of 2011 February 14, the correlation coefficient between the kinetic-helicity density and the current-helicity density is as high as 0.84. But for the rising phase after 06:00UT of 2011 February 14, the correlation coefficient drops to 0.52. The decrease of the correlation may be related to the X2.2 flare event that occurred at 01:44 UT of 2011 February 15. The bottom panel of Fig. 3.20 shows the evolutions of the unweighted currenthelicity density and unweighted subsurface kinetic-helicity density. Different from the similar evolutionary tendency of the weighted parameters, the two unweighted parameters seem to evolve out of phase. As shown in this panel, the unweighted kinetic helicity seems to have a decreasing phase about 8 hr earlier than that of the

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Fig. 3.20 Top: Evolution of the weighted current helicity (marked with star, corresponding to the left vertical axis) and the weighted subsurface kinetic helicity (marked with cross, corresponding to the right vertical axis) for AR 11158. Bottom: Same as the top panel but for the unweighted current helicity and unweighted subsurface kinetic helicity. Dashed lines, marked as “A” and “B”, correspond to 10:00 UT and 14:00 UT of 2011 February 14, respectively. From Gao et al. (2012)

3 Solar Magnetic Activities

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current helicity before 10:00 UT of February 14, and have an increasing phase about 4 hr behind that of the current helicity after 18:00 UT of February 14. The snapshots in Fig. 3.21 show maps of the weighted and unweighted Hkz as well as the weighted and unweighted Hcz for some selected periods. It can be found that both of the kinetic-helicity parameters are more fragmented than the current-helicity parameters, but there is no clear correspondence between the sign distributions of the kinetic helicity and the current helicity. The subsurface kinetic helicity used in our analysis is presumably corresponding to the twisting inside active regions in the upper convection zone. The in-phase evolution of the subsurface kinetic helicity and the current helicity probably indicate that the twisting beneath the active region surface indeed plays an important role to shape the current helicity distribution observed in the photosphere. On the other hand, perhaps it is not surprising that these two helicities evolve in phase, because beneath the photosphere magnetic field lines are frozen with plasma and the evolution of the photospheric magnetic field somehow reflects the subsurface motions. Another fact we cannot ignore is that despite the high positive correlation in the evolutionary curves of the two helicities, the signs of the two helicities do not often stay the same. In particular, for AR 11158, the signs of the two helicities are more often opposite than same. Therefore, although the evolution of the two helicities is in positive correlation, it is quite likely that we get a negative or no correlation if we pick just one random snapshot. This may help explain why the study by Gao et al. (2009) (see Sect. 5.2.5), which statistically analyzed the correlation of the two helicities using snapshots from many different active regions, did not find a correlation.

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Fig. 3.21 Snapshots showing weighted Hkz (upper left) and unweighted Hkz (upper right) at 18:00 UT of 2011.02.15, and weighted Hcz (bottom left) and unweighted Hcz (bottom right) at 19:00 UT of 2011.02.15. The parameters are displayed relative to the corresponding standard deviations, and color bars at the left lower corner of each panel show this scale. From Gao et al. (2012)

3.4 Magnetic Helicity and Tilt Angle Evolution in Active Regions The magnetic helicity of a solar flux tube can be decomposed into the twist around the flux tube axis and the writhe of the helical flux tube axis. The twist and writhe are both geometric properties of flux tubes and they can interconvert casting doubt on magnetic helicity conservation. If a twisted flux tube without initial writhe emerges from the convective zone and its writhe is caused by kink instability, the magnetic helicity and writhe should have the same sign (Longcope et al., 1998). The accumulated relative helicity can be considered as the total magnetic helicity including twist and writhe. So what is the relation between the accumulated helicity and the writhe of active regions? We will introduce it in some of the following sections. Figure 3.22 shows a possible twisted magnetic flux loop in the solar subsurface.

3.4.1 Twisted Magnetic Field and Helicity The change in magnetic helicity in the solar atmosphere relates to the motion of footpoints of the magnetic field in the solar surface (Berger & Field, 1984) d Hm = −2 dt

∂V

[(Vt · A p )Bn − (A p · Bt )Vn ]ds,

(3.64)

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Fig. 3.22 Horizontal flux tube whose axis is deformed into a left-handed -loop. The field lines twist about the axis in the right-handed sense The tube was initially straight and untwisted. From Longcope et al. (1998)

where the magnetic field B and velocity field V are observable in the solar atmosphere. The first term in Eq. (3.64) is the contribution from the twisting motion of footpoints of the magnetic field in the solar surface, while the second term does that from the emergence of twisted magnetic flux from the subatmosphere. The emergence of highly twisted magnetic flux tubes from the subatmosphere has a similar contribution to the helicity injection from the shear motion of magnetic flux tubes in the photosphere. The contribution of the second term in the right side of Eq. (3.64), such as the emerging motion of magnetic flux, to the magnetic helicity in the solar atmosphere was discussed by Kusano et al. (2003). While, an important improvement in this project was demonstrated by Demoulin & Berger (2003) with a relatively simple form, who presented that the horizontal motions, deduced by tracking the photospheric cut of magnetic flux tubes, include the effect of both the emergence and the shearing motions whatever the magnetic configuration complexity is. According to the analysis of Demoulin & Berger (2003), one can obtain that d Hm = −2 dt

∂V

(U · A p )Bn ds,

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Vn Bt . Bn

(3.65)

(3.66)

This implies that one can not exclude the contribution of emerging flux in the horizontal motion of magnetic footpoints in the solar surface. Similar to Eq. (3.64), the local change rate of magnetic helicity density can be written in the form (Berger & Field, 1984) dh m = −2∇ · [(Vt · A p )B − (A p · Bt )V]. dt

(3.67)

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It means that the change in magnetic helicity density can be detected in principle using the local change in the moving magnetic field. Because it is difficult for the quantitative measurement of moving magnetic fields in a close surface in the solar atmosphere, the local injected rate of magnetic helicity density can not be obtained completely. On the other hand, as the highly sheared or twisted magnetic flux emerges from the solar subatmosphere, one probably can approximately get the basic observational message on the local injection of magnetic helicity density and find its relevant relationship with the observed current helicity density in the solar surface (Zhang 2001a). This means that the local change rate of magnetic helicity density can be written in the form, dh m = f (G), dt

where

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(3.68)

even if the locally injected helicity does not remain in the lower atmosphere completely, because a part of magnetic helicity density probably transfers along the magnetic lines of force into the corona and interplanetary space (Demoulin & Berger, 2003; Longcope & Welsch, 2000). For a bundle of flux tubes with the uniformly distributed magnetic helicity density, the change rate of magnetic helicity density from the local motion of footpoints of the magnetic field in the solar surface can be estimated approximately G dh m = , (3.69) dt l where l is the equivalent length of flux tubes. This means morphologically that the rotation of footpoints of magnetic flux (or the emergence of twisted magnetic flux) causes the twist of magnetic field above the photosphere, and a part of the twisted field remains in the lower layer of the solar atmosphere probably as the helicity has not ejected from the lower atmosphere completely. According to Eq. (3.65), the relationship between the mean magnetic helicity density h m density and the mean current helicity density h cz can be obtained 2 hm = − Sc L c

 Tc

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(3.70)

where Tc is the typical relaxation time, and Sc and L c are the typical horizontal and vertical spatial scale of emerging magnetic flux in the solar atmosphere before the transport of helicity into the interplanetary space, and k is a correlative parameter.

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Fig. 3.23 Integrated helicity flux as a function of total AR magnetic flux (H0 = 1041 M X 2 , 0 = 1021 M X ). From Yang et al. (2009)

3.4.2 Helicity Injection and Tilt Angle in Newly Emerging Flux Regions Figure 3.23 depicts integrated magnetic helicity fluxes (Hmax − Hmin ) versus the total magnetic flux of the ARs by Yang et al. (2009). The magnetic flux m is onehalf of the maximum sum of the unsigned positive and negative magnetic fluxes. The best linear fit line (solid line in Fig. 3.23) shows the relation between integrated magnetic helicity fluxes and magnetic fluxes log

m Hmax − Hmin = a log +b H0 0

(3.71)

where a = 1.85, b = −0.41, H0 = 1041 M X 2 and 0 = 1021 M X . This indicates that the accumulated magnetic helicity is proportional to the exponent of magnetic flux (|H | ∝ 1.85 ). Jeong & Chae (2007) found an exponent of 1.3 for the coefficient “a” from four active regions. Labonte et al. (2007) found a similar value of 1.8 for “a”. If we suppose that the coronal part of each AR is represented by a single semicircular loop, the average twist Tw in these loops would be 10b H0 /20 = 0.039 turns. Nindos et al. (2003) found values between 0.01 and 0.17. Labonte et al. (2007) found a value of 0.022. Tian & Alexander (2008) found a similar value, 0.03. For a typical coronal loop with diameter d = 100 Mm at the photosphere foot-points, the twist rate q (radians per unit length) would be T w/(πd/2) = 2.48 × 10−12 cm−1 . This result is similar to that in the survey of Labonte et al. (2007), who obtained 1.4 × 10−12 cm−1 for 48 X-flaring regions and 345 non-X-flaring regions. However, these results are all one order smaller than the average twist rates of 10−11 cm−1 in AR loops by using vector magnetograms to find αbest (Pevtsov et al., 1995). Note that the αbest is related to the twist rate q by αbest = 2q for the thin flux rope model (Longcope et al., 1998) and a debate that whether such an assumption is valid for one active region still exists (e.g., Leka et al., 2005).

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3.4.3 Two Typical Solar Active Regions To better understand the accumulation of magnetic helicity and tilt angle evolution in newly emerging ARs, Yang et al. (2009) have divided our samples into two groups A and B, according to the resulting relation between accumulated magnetic helicity and the rotation of the tilt angles. Yang et al. (2009) neglect the evolution detail of magnetic helicity and tilt angle, and focus on the final accumulated helicity H (t) and the change in tilt angle T a = T a(t) − T a(0). Group A: H · T a > 0. The accumulated helicity of ARs is negative (positive) when the tilt angle rotates clockwise (counter-clockwise). Group B: H · T a < 0. The accumulated helicity of ARs is positive (negative) when the tilt angle rotates clockwise (counter-clockwise). There are 43 ARs (74%) of the samples in Group A. 19 ARs are in the Northern Hemisphere and 24 ones are in the Southern Hemisphere. 15 ARs (26%) of the samples are in group B. 6 ARs are in the Northern Hemisphere and 9 are in the Southern Hemisphere. All active regions in our samples obey the Hale–Nicholson law (Hale & Nicholson, 1925): the leading polarities are always positive (negative) in the Northern (Southern) Hemisphere in the 23rd solar cycle. Thus one active region will satisfy Joy’s law when the leading polarities are closer to the equator than the trailing polarities (Hale et al., 1919) if its final tilt angle when we stop to follow the AR is 90◦ < T a(t) < 180◦ (180◦ < T a(t) < 270◦ ) in the Northern (Southern) Hemisphere according to the definition of tilt angle in our study. In the Southern Hemisphere, among the 22 ARs which follow the Joy’s law, 17 (77%) belong to group A, while among the 11 ARs which do not follow Joy’s law, 7 (63%) belong to group A. In the Northern Hemisphere, among the 14 ARs which follow Joy’s law, 11 (79%) belong to group A, while among the 11 ARs which do not follow Joy’s law, 8 (73%) belong to group A. The probability is 78% for an AR to be in group A if it follows Joy’s law, which is higher than the probability of 68% if it does not follow Joy’s law. Moreover, the theoretical estimation on the tilt, current helicity, and twist of magnetic fields in solar active regions was proposed by Kleeorin et al. (2020), Kuzanyan et al. (2020).

3.5 Magnetic Field, Horizontal Motion and Helicity 3.5.1 Evolution of Magnetic Field The evolution of the magnetic field of active region NOAA 10488 is presented in Fig. 3.24. In the first column of Fig. 3.24, Liu & Zhang (2006) shows the vector magnetograms after projection-effect correction. As the AR was not far from the central meridian between October 27−30, the correction of the projection effect of the vector magnetic field is quite small. The negative spot marked “c” always had a

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Fig. 3.24 Vector magnetograms of HSOS (left) and computed horizontal velocity vectors being superposed on MDI longitudinal magnetograms (right). The maximum arrow length measures the transverse magnetic field of 1200 G and velocity of 0.8 km s−1 , respectively. The field of view is 225 × 168 . From Liu & Zhang (2006)

left-handed twist. The leading positive spot had right-handed twist on October 27 and 28. The upper part of the leading positive spot where marked “a” held a right-handed twist on October 29 and 30, while the lower part marked “b” had a weak left-handed twist at that time. Further discussion about the twist of the AR will be presented later. The computed horizontal velocity vectors are shown in the second column of Fig. 3.24 The spiral directions of the velocity vectors in most areas of the AR are opposite to those of the transverse magnetic fields. For example, on October 27, the velocity vectors direct to clockwise, opposite to the rotation direction of the transverse field. On October 28, the velocity vectors both in the lower part of the leading positive spot marked “B” and in most areas of the negative following spot marked “C” direct oppositely to the transverse fields. The conditions on October 29 and 30 are similar to that of 28. On October 29 and 30, the velocity vectors in the upper part of the leading positive spot marked “A” have opposite rotation to that of the transverse field, too. However, they have a spiral pattern similar to that of the transverse field on October 28. The transverse magnetic fields can be regarded as the horizontal projections of the spiral magnetic fluxes. The transverse velocities computed by the LCT method represent the horizontal motions of the footpoints of the magnetic field lines. So when a magnetic flux tube emerges from the sub-photosphere, the footpoints of the magnetic field should rotate in opposite direction than the spirals of the transverse

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Fig. 3.25 Left: Gray-scale maps of G ≡ −2(u · A p )Bz of one-hour averages. The white and black colors indicate the positive and negative signs of G, respectively. The white rectangles mark the areas of maxima of G which correlate with strong shear motion (SSM). The dashed and full contours represent longitudinal magnetic field strengths of ±200 G, the field of view is 400 × 300 . Right top: a Time profile of the AR’s longitudinal magnetic field flux derived from full-disk MDI images. b Time profile of the rate of helicity injected by horizontal motions. c Time profile of the accumulated change in helicity H(t) calculated from the measured dH/dt (thick line) and the estimated H(t) if a spline interpolation is used for the determination of the missing dH/dt values (thin line). Right bottom: Ratio of the coronal helicity accumulation to the square of the magnetic flux. The dashed line indicates the time separating the rotation and shear phases. From Liu & Zhang (2006)

magnetic field lines. This is confirmed by the observational result shown in Fig. 3.24. As for the contrary case of “A” area on October 28, it is probably due to the fast expansion of the magnetic flux during its fast emergence.

3.5.2 Transport of Magnetic Helicity Figure 3.25b, c display the temporal variation of the rate of helicity changes, dH/dt, deduced from horizontal motions, and of the accumulated change in helicity, H (t),

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calculated from the measured dH/dt, separately. There are time intervals when no dH/dt measurements are available owing to the lack of high-cadence MDI data. Following Nindos & Zhang (2002), we estimate these missing dH/dt values by spline interpolation. The resulting H curve is presented in Fig. 3.25c with a thin line. The temporal variations of dH/dt and H show that during the AR’s development the rate of helicity change is negative, and the absolute value of the accumulated change in magnetic helicity is increasing. October 29 is a critical time for the magnetic helicity change. Before about 8:00 UT on October 29, that is, in the rotation phase, the dH/dt is small and the helicity change is rather slow. In the shear phase, the dH/dt is rather large and the helicity change becomes fast and significant. In approximately equal time duration, the accumulated change in the magnetic helicity in the shear phase is about 3 times larger than that of the rotation phase. That the strong shear motion (SSM) brings more magnetic helicity into the corona than the twist one implies that the interaction of two different flux systems brings more helicity into the upper atmosphere than the twist of a single flux system in this AR. An easy way to evaluate the physical significance of the observed flow on the transport of magnetic helicity is to examine the distribution of G ≡ −2(u · A p )Bz . It is a measure of the local contribution of the foot-point motion to the rate of the transport of magnetic helicity (Chae, 2001). Figure 3.25 shows the gray-scale map of G at specific times. Since we are interested in the large-scale trends of the helicity variability, the maps of G are one hour averages around the times indicated in each map. In the rotation phase, the maxima of G are located mainly in the rotating or twisting area of the main positive spot. In the shear phase, the maxima of G are located mainly near the magnetic neutral line between the new emerging positive spot and the old negative spot where intense shearing motions took place (marked by the white rectangle). This further confirms our above conclusion.

3.5.3 Evolution of Current Helicity Density The vertical component of current helicity density provides some information on the local twisting of the magnetic field in the photosphere. The gray-scale maps of the current helicity density, hc, for the areas of B > 200G are shown in the first column of Fig. 3.26. The maps of hc are one or few hours averages around the times indicated in each map. This can be confirmed by the helicity evolution of AR 10488, as is shown in Fig. 3.26. From October 28, the intensity of the local current helicity density increases on the following day in most areas where the G and hc have the same sign, and decrease in most areas where the G and hc have opposite signs. For example, on October 28, a part of the preceding component of the main sunspots has positive G and positive hc (marked “A” and “a” in the maps of the two parameters separately in Fig. 3.26), and the current helicity density in this area increases in absolute value on October 29. For the area marked “B” and “b” separately which has opposite sign of G and hc, the current helicity density decreases from being positive on October 28

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Fig. 3.26 Gray-scale maps of the time averages of current helicity density hc (left) and the time averages of G ≡ −2(u · A p )Bz (right). The white and black contours represent longitudinal magnetic field strengths of 200 and −200 G, respectively. The field of view is 225 × 168 . From Liu & Zhang (2006)

to a weak negative value on October 29. Most parts of both preceding and following components of the main sunspots have the opposite signs of G and hc on October 29, therefore the intensity of current helicity density on October 29 decreases obviously to the weaker distribution on October 30. Most parts of the main spots remain to have opposite signs of G and hc on October 30, so we can predict that the intensity of the current helicity density there will probably decrease on the next day.

3.6 Magnetic Helicity and Energy Spectra of Solar Active Regions We have applied a novel technique to estimate the magnetic helicity spectrum using vector magnetogram data at the solar surface. We have made use of the assumption that the spectral two-point correlation tensor of the magnetic field can be approximated by its isotropic representation.

196 Fig. 3.27 Photospheric vector magnetograms (left) and plots of Jz Bz (right) for the active region NOAA 11158 between 11–15 February 2011. The arrows show the transverse component of the magnetic field. Light (dark) shades indicate positive (negative) values of Bz on the left and Jz Bz on the right

3 Solar Magnetic Activities 2011.02.11_23:59:53_TAI

2011.02.12_23:59:54_TAI

2011.02.13_23:59:54_TAI

2011.02.14_23:59:54_TAI

2011.02.15_23:59:54_TAI

3.6.1 Implication of Magnetic Spectrum of Active Regions We have analyzed data from the solar active region NOAA 11158 during 11–15 February 2011, taken by the Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory (SDO). The pixel resolution of the magnetogram is about 0.5 , and the field of view is 250 × 150 . Figure 3.27 shows photospheric vector magnetograms (left) and the corresponding distribution of h C(z) = Jz Bz (right) from the vector magnetograms of that active region on different days. It turns out that the mean value of the current helicity density, HC(z) = h C(z) , is positive and ≈ 2.7 G2 km−1 . Furthermore, as a proxy of the force-free α parameter, we determine α = Jz /Bz , which is on the average α ≈ 2.8 × 10−5 km−1 . For future reference, let us estimate the current helicity normalized to its theoretical maximum value, henceforth referred to as relative helicity. This is not to be confused with the gauge-invariant magnetic helicity relative to that of an associated potential field (Berger, 1984). Thus, we consider the ratio  1/2 rC = Jz Bz / Jz2 Bz2

(3.72)

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as an estimate for the relative current helicity. For the active region NOAA 11158 we find rC = +0.034. This value is based on one snapshot, but similar values have been found at other times. Let us now turn to the two-point correlation tensor, Bi (x, t)B j (x + ¸, t) , where x is the position vector on the two-dimensional surface and angle brackets denote ensemble averaging or, in the present case, averaging over annuli of constant radii, i.e., |ξ| = const. Its Fourier transform with respect to ¸ can be written as 

 Bˆ i (k, t) Bˆ ∗j(k , t) = i j (k, t)δ2 (k − k ),

(3.73)

where Bˆ i (k, t) = Bi (x, t) eik·x d 2 x is the two-dimensional Fourier transform, the subscript i refers to one of the three magnetic field components, the asterisk denotes complex conjugation, and ensemble averaging will be replaced by averaging over concentric annuli in wavevector space. Following Matthaeus et al. (1982), it is possible to determine the magnetic helicity spectrum from the spectral correlation tensor i j (k, t) by making the assumption of local statistical isotropy. At the end of this presentation, we consider the applicability of this assumption in more detail. Considering that k defines the only preferred direction in i j , and that ki Bˆ i = 0, the only possible structure of i j (k, t) takes the form (cf. Moffatt, 1978) i j (k, t) =

2E M (k, t) i HM (k, t) (δi j − kˆi kˆ j ) + εi jk kk , 4πk 4πk

(3.74)

where kˆi = ki /k is a component of the unit vector of k, k = |k| is its modulus with k 2 = k x2 + k 2y , and E M (k, t) and HM (k, t) are the magnetic energy and. magnetic helicity spectra, normalized such that E M (t) ≡

1 2 B = 2



∞

E M (k, t) dk,  0∞ H M (t) ≡ A · B = HM (k, t) dk.

(3.75)

0

Note that the mean energy density in erg/cm3 is E M /4π. We emphasize that the expression for i j (k, t) differs from that of Moffatt (1978) by a factor 2k, because we are here in two dimensions, so the differential for the integration over shells in wavenumber space changes from 4πk 2 dk to 2πk dk. Note that the magnetic vector potential is not an observable quantity, so the magnetic helicity might not be gauge-invariant. However, if the spatial average is over all space, or if the magnetic field falls off sufficiently rapidly toward the boundaries, both H M (t) and HM (k, t) are gauge-invariant. Indeed, with the present analysis, HM (k, t) is manifestly gauge-invariant, because it has been computed directly from the magnetic field as obtained through the photospheric vector magnetogram. The components of the correlation tensor of the turbulent magnetic field can be referenced in Eq. (3.74).

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In the following, we present shell-integrated spectra. However, because we consider here two-dimensional spectra, they correspond to the power in annuli of radius k and are obtained as   2E M (k) = 2πk Re x x +  yy + zz φk ,   (3.76) k HM (k) = 4πk Im cos φk  yz − sin φk x z φk , where the angle brackets with subscript φk denote averaging over annuli in wavenumber space. The realizability condition (Moffatt, 1969) implies that k|HM (k, t)| ≤ 2E M (k, t).

(3.77)

It is therefore convenient to plot k|HM (k, t)| and 2E M (k, t) on the same graph, which allows one to judge how helical the magnetic field is at each wavenumber. Furthermore, to assess the degree of isotropy, we also consider magnetic energy (h) (v) (k) and E M (k) based respectively on the horizontal and vertical magnetic spectra E M field components, defined via   (h) (k) = 4πk Re x x +  yy φk , 2E M (v) (k) = 4πk Re zz φk . 2E M

(3.78)

(h) (v) Under isotropic conditions, we expect E M (k) ≈ E M (k) ≈ E M (k). We now consider magnetic energy and helicity spectra for the active region NOAA 11158. The calculated region of the field of view is 256 × 256 (i.e. 512 × 512 pixels) or L 2 = (186 Mm)2 . We present first the results for NOAA 11158 at 23:59:54UT on 13 February 2011; see Fig. 3.28a. It turns out that the magnetic energy spectrum has a clear k −5/3 range for wavenumbers in the interval 0.5 Mm−1 < k < 5 Mm−1 . The magnetic helicity spectrum is predominantly positive at intermediate wavenumbers, but we also see that toward high wavenumbers the magnetic helicity is fluctuating strongly around small values. To determine the sign of magnetic helicity at these smaller scales, we average the spectrum overbroad, logarithmically spaced wavenumber bins; see the lower panel of Fig. 3.28. This shows that even at smaller length scales the magnetic helicity is still positive, again consistent with the fact that this active region is at southern latitudes. To calculate the relative magnetic helicity, we define the integral scale of the magnetic field in the usual way as

 lM =

k

−1

 E M (k) dk

E M (k) dk.

(3.79)

The realizability condition of Eq. (3.77) can be rewritten in the integrated form (e.g., Kahniashvili et al., 2013) as

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Fig. 3.28 a 2E M (k) (solid line) and k|HM (k)| (dotted line) for NOAA 11158 at 23:59:54UT on 13 February 2011. Positive (negative) values of HM (k) are indicated by open (closed) symbols, respectively. (v) 2E M (k) (red, dotted) and (h) 2E M (k) (blue, dash-dotted) are shown for comparison. b Same as the upper panel, but the magnetic helicity is averaged over broad logarithmically spaced wavenumber bins. After Zhang et al. (2014)

 HM =

 HM dk ≤ 2

k −1 E M (k) dk ≡ 2l M E M .

(3.80)

In particular, we have |H M (t)| ≤ 2l M E M (t). This allows us then to define a normalized relative magnetic helicity, r M = H M /2l M E M ,

(3.81)

which obeys |r M | ≤ 1. Again, this quantity is not to be confused with the gaugeinvariant helicity of Berger (1984). For the active region NOAA 11158 at 23:59:54 UT on 13 February 2011 we have l M ≈ 5.8 Mm, H M ≈ 3.3 × 104 G2 Mm, and E M ≈ 6.7 × 104 G2 , so r M ≈ 0.042. The relative magnetic helicity has thus the same sign as the relative current helicity. The corresponding magnetic column energy in the twodimensional domain of size L 2 is L 2 E M /4π ≈ 1.8 × 1024 erg cm−1 , which is about three times larger than the values given by Song et al. (2013). The magnetic column helicity is L 2 H M ≈ 1.1 × 1033 Mx2 cm−1 . Several estimates of the gauge-invariant magnetic helicity of NOAA 11158 using time integration of photospheric magnetic helicity injection (Vemareddy et al., 2012; Liu & Schuck, 2012) and nonlinear forcefree coronal field extrapolation (Jing et al., 2012; Tziotziou et al., 2013) suggest magnetic helicities of the order of 1043 Mx2 . This value would be comparable to ours

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Fig. 3.29 Similar to Fig. 3.28, showing E M (k, t) (upper panel) and k|HM (k, t)| (lower panel) for the other days. After Zhang et al. (2014)

if the effective vertical extent were ≈100 Mm. We should remember, however, that there is no basis for such a vertical extrapolation of our two-dimensional data. (h) (v) (k) and E M (k) based respectively Interestingly, the magnetic energy spectra E M on the horizontal and vertical magnetic field components agree remarkably well at wavenumbers below k = 3 Mm−1 , corresponding to length scales larger than 2 Mm. This suggests that our assumption of isotropy might be a reasonable one. The mutual (h) (v) (k) and E M (k) at larger wavenumbers could in principle be departure between E M a physical effect, although there is no good reason why the magnetic field should be mostly vertical only at small scales. If it is indeed a physical effect, it should then (h) (v) (k) and E M (k) in the future be possible to verify that this wavenumber, where E M depart from each other, is independent of the instrument. Alternatively, this departure might be connected with different accuracies of horizontal and vertical magnetic field measurements (Zhang et al., 2012). If that is the case, one should expect that with future measurements at the better resolution the two spectra depart from each other at larger wavenumbers. In that case, our spectral analysis could be used to isolate potential artifacts in the determination of horizontal and vertical magnetic fields. In Fig. 3.29 we show 2E M (k) and k|HM (k)| for different days. It turns out that on small scales the spectra are rather similar in time, and that there are differences in

3.6 Magnetic Helicity and Energy Spectra of Solar Active Regions

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Fig. 3.30 Unsigned current helicity spectrum, |HC (k)|. After Zhang et al. (2014)

the amplitude mainly on large scales. Also, the sign of HM (k) remains positive for the different days. We find that the mean spectral values of magnetic energy of the active region at the solar surface is consistent with a k −5/3 power law, which is expected based on the theory of Goldreich & Sridhar (1995) and consistent with spectra from earlier work on solar magnetic fields (Abramenko, 2005; Stenflo, 2012), ruling out the k−3/2 spectrum suggested by Iroshnikov (1963) and Kraichnan (1965). Under isotropic conditions, the current helicity spectrum, HC (k, t), is related to the magnetic helicity spectrum via HC (k, t) ∼ k 2 HM (k, t).

(3.82)

 It is normalized such that

HC (k) dk = J · B . In Fig. 3.30 we show |HC (k)|

obtained in this way. For k  1 Mm−1 the current helicity spectrum shows a k −5/3 spectrum, which is consistent with numerical simulations of helically forced hydromagnetic turbulence (Brandenburg & Subramanian, 2005b; Brandenburg, 2009), and indicative of a forward cascade of current helicity. Similar spectra have also been obtained for the kinetic helicity (André & Lesieur, 1977; Borue & Orszag, 1997), which implies that the relative helicity decreases toward smaller scales; see the corresponding discussion on p. 286 of Moffatt (1978).

3.6.2 Comparisons Among Magnetic Helicity, Energy and Velocity Spectra Figure 3.31 shows the Doppler velocity field and the corresponding longitudinal magnetogram in active region NOAA 11158. The Dopplergram in Fig. 3.31 is an average of 29 Dopplergrams observed in 00:00-00:20UT, 2011 February 14 by the

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Fig. 3.31 Doppler velocity field (left) and longitudinal magnetic field (right) of active region NOAA11158. In the Dopplergram, the blue shows moving up and yellow does moving down. In the magnetogram, yellow shows positive polarity and blue does negative one. After Zhang & Brandenburg (2018)

Solar Dynamics Observatory/Helioseismic and Magnetic Imager (SDO/HMI). The contribution of the oscillation component of the velocity has been removed basically. It is found that the Evershed flow occurs in the strong magnetic structures in the active region and the small-scale velocity field forms nearby the active region. This means the velocity field nearby the active region contributes from the active region and the quiet Sun. Figure 3.32 shows the spectrum of the velocity field of the active region inferred by the Dopplergram in Fig. 3.31. The red dashed line shows the spectrum of the velocity field in the active region, which includes that of the quiet sun also. A similar result had been shown by Zhao & Chou (2013) with the continuous high spatial resolution Doppler observation of the Sun by SDO/HMI. The red dotted line shows the spectrum of the active region relative to the magnetic structures only. We can find that the bulge near the 2–5 Mm−1 has been removed and the slope of the spectrum of velocity energy is consistent with that of magnetic energy (−5/3). The spectrum of velocity field in 2–5 Mm−1 reflects just the typical scale of the that in the quiet Sun, which includes the contribution of the granulations etc. This probably can provide some messages: It is found the similar scale distributions between the magnetic field and velocity patterns in active regions, for example, the mass flow along with the magnetic fields. This reflects the energy exchanges between the magnetic fields and velocity fields, and also helicity probably. The contribution of the velocity fields in the quiet Sun to active regions can be neglected in the analysis of the interaction between the velocity and magnetic field.

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Fig. 3.32 The spectrum of magnetic energy E M (k) (black solid line), magnetic helicity k HM (k) (black dotted line) and velocity energy E V aq (k) (red dashed line, the velocity in the quiet Sun has been included also) and E V a (k) (red dotted line, the velocity related with the magnetic features only) in active region NOAA 11158. After Zhang & Brandenburg (2018)

3.6.3 Comparison of Magnetic Spectrum of Active Regions from Huairou and HMI Vector Magnetograms Figure 3.33 shows the distribution of vector magnetograms of active region NAOO11890 observed by Huairou Observing Station and HMI, and the corresponding spectrums of magnetic energy E M (k), and helicities Hc (k) and HM (k). It is found the different spatial resolution between Huairou and HMI vector magnetograms. The slopes of magnetic fields of both magnetograms are different in the high wavenumber, which reflect the difference on observational small-scale magnetic fields. Figure 3.34 shows that the values of the Huairou magnetic fields are weaker than HMI one statistically. This means that the calibrated strength of the fields of the Huairou magnetogram shows non-linearity relative to HMI one. It is the reason on the difference between Huairou and HMI spectrums of magnetic fields due to the different spatial resolution and calibrations of vector magnetograms.

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04:50:39 UT, 2013 Nov 08

04:48:00 UT, 2013 Nov 08

Fig. 3.33 Left: Huairou (top) and HMI (bottom) vector magnetograms of active region NOAA11890. Right: The corresponding spectrum of magnetic energy E M (k) (solid lines), magh (k) (red netic helicity HM (k) (dotted lines), and current helicity HC (k) (blue dashed lines). E M dotted-dashed lines) and E vM (k) (yellow dotted-dashed lines) show the horizontal and vertical parts of magnetic energy, which relate to the transverse and longitudinal components of fields

Fig. 3.34 The statistical correlation of the longitudinal and transverse components of fields between Huairou and HMI vector magnetograms of active region NOAA11890 in Fig. 3.33

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3.7 Evolution of Magnetic Helicity and Energy Spectra 3.7.1 Magnetic Helicity and Energy Spectra of Individual Active Regions 3.7.1.1

Active Region NOAA11158

We have analyzed data from the solar active region NOAA 11158 during 12–16 February 2011, taken by the Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO). The pixel resolution of magnetograms is about 0.5 , and the field of view is 250 × 150 . In our study, 600 vector magnetograms in the active region have been used. We now consider magnetic energy and helicity spectra for the active region NOAA 11158. The calculated region of the field of view is 256 × 256 (i.e. 512 × 512 pixels). We present the energy and helicity in active region NOAA 11158 on 12–16 February 2011 by means of the method of Zhang et al. (2014). For analyzing the basic properties of the spectrums of magnetic energy and corresponding helicity in the active region, Fig. 3.35 shows average values of spectrums inferred from 600 vector magnetograms of active region NOAA11158 on 11–15 February 2011. These are comparable with that of Zhang et al. (2014), and the fluctuations in the calculation of individual samples have been removed. This provides a basic estimation of the spectral distribution of magnetic energy and helicity in the active region.

Fig. 3.35 Mean spectrums of 2E M (k) (solid line), k|HM (k)| (dotted line) and |HC (k)| (deash line) averaging over 600 vector magnetograms of active region NOAA11158 on 11–15 February 2011. After Zhang et al. (2016)

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Fig. 3.36 Evolution of photospheric magnetic helicity HM (t) (solid line) and magnetic energy E M (t) (dotted line) of active region NOAA 11158. After Zhang et al. (2016)

Figure 3.36 shows the evolution of mean helicity and energy inferred by Eq. (3.75) in the photosphere of active region NOAA 11158. It is found that the magnetic helicity and energy change with the development of the active region. A decrease of magnetic helicity of the active region occurred on 2011 Feb. 14, while the magnetic energy did not. This means the helicity in the active region does not have a simple changing relationship with the magnetic energy monotonously. It is also consistent with the evolution tendency of the current helicity and kinetic helicity in the active region NOAA 11158 carried out by analyzing vector magnetograms and subsurface velocities by Gao et al. (2012). In the estimation in the hydrodynamic turbulence by Kolmogorov (1941a), Obukhov (1941) and also Batchelor (1953) the scale exponent α value of k −α power law is about 5/3 (about 1.67). Similar results of 3/2 have been proposed by Iroshnikov (1963) and Kraichnan (1965) in the turbulence with magnetic fields. It was calculated by Abramenko et al. (1997) and Stenflo (2012) based on the observational solar magnetic fields. It turns out the mean scale exponent α values of the magnetic energy and magnetic helicity spectrum for wavenumbers in the interval 1 Mm−1 < k < 6 Mm−1 . Figure 3.37 shows the evolution of mean scale exponent α values of the magnetic energy spectrum for wavenumbers in the active region NOAA 11158. It is found that the minimum α(|E M |) value of |E M | is 1.1, the maximum value is 2.0, and the mean value is about 1.67 as the active region developed. Under isotropic conditions, the current helicity spectrum HC (k, t) is related to the magnetic helicity spectrum via HC (k, t) ∼ k 2 HM (k, t). Figure 3.37 shows the evolution of mean scale exponent α values of the current helicity spectrum for wavenum-

3.7 Evolution of Magnetic Helicity and Energy Spectra

207

Fig. 3.37 Evolution of the scale exponent α of photospheric current helicity spectrum (solid line) and magnetic energy spectrum (dotted line) of active region NOAA 11158. After Zhang et al. (2016)

bers in the active region NOAA 11158. It is found that the minimum α(|HC |) value of |HC | is 0.9, the maximum value is 1.7, and the mean value is about 1.6 as the scale of active region developed. It implies that the value of α(|HC |) of this active region is an order of 5/3 and consistent with our previous estimation (Zhang et al. 2014). This means that the mean spectral values of magnetic energy and current helicity of this active region at the solar surface is roughly consistent with a k −α power law by Kolmogorov (1941a). The spectrum of magnetic helicity has been calculated also. The minimum α(k|HM |) value of k|HM | is 1.9, the maximum value is 2.8, and the mean value is 2.65 as the active region developed. The scaling exponent α values of magnetic helicity and energy in the active region actually reflect the characteristic distribution of different scale elements of helicity and energy in the solar surface, while do not reflect the complexity of the distribution of the magnetic field with different polarities completely. The evolution of r M and l M in active region NOAA 11158 is shown in Fig. 3.38. The average value of r M is about 0.05, while that of l M is about 6 Mm in the developed stage of the active region. It is found that the relative magnetic helicity r M shows a relatively complex relationship with the development of the active region and it shows a similar tendency with HM in Fig. 3.36.

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Fig. 3.38 Evolution of the photospheric relative magnetic helicity (dotted line r M ) and integrated scale of magnetic field (solid line l M ) for active region NOAA 11158. After Zhang et al. (2016)

3.7.1.2

Active Region NOAA11515

For presenting the evolution of the spectrum of magnetic field and the corresponding helicity in solar active regions, we have also analyzed data from the solar active region NOAA 11515 during 30 June–6 July 2012, taken by the Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory (SDO). The pixel resolution of magnetograms is about 0.5 , and the field of view is 250 × 150 . In our study, about 840 vector magnetograms have been used. Figure 3.39 shows photospheric vector magnetograms and the corresponding distribution of h C(z) = Jz Bz from the vector magnetograms of that active region on different days. It provides the spatial distribution of magnetic field and current helicity density of this active region in the solar surface. For analyzing the basic properties of the spectrums of magnetic energy and corresponding helicity in the active region, Fig. 3.40 shows average values of spectrums inferred from about 840 vector magnetograms of active region NOAA11515 on 30 June–6 July 2012. These are comparable with the result of Zhang et al. (2014) and the average values of NAOO 11158 in Fig. 3.35. As comparing active regions NOAA 11158 and 11515, it is found that the mean spectral configuration of magnetic energy and helicity is slightly different for different active regions. It turns out the mean scale exponent α values of the magnetic energy and magnetic helicity spectrum for wavenumbers in the interval 1 < k < 6 (Mm−1 ). Figure 3.41 shows the evolution of mean scale exponent α values of the magnetic energy spectrum for wavenumbers in the active region NOAA 11515. It is found that the minimum α(|Hc |) value of |Hc | is 1.2, the maximum value is 2.7, and the mean value is about 2.0. It is found that the minimum α(|E M |) value of |E M | is 2.0, the maximum value is 2.6, and the mean value is about 2.4. As one comparing with active region NOAA11158, it is found such values is larger than that of NOAA11158. This means the values of

3.7 Evolution of Magnetic Helicity and Energy Spectra

209

2012.07.01_00:00:08_TAI

2012.07.02_00:00:08_TAI

2012.07.03_00:00:08_TAI

2012.07.04_00:00:08_TAI

2012.07.05_00:00:08_TAI

Fig. 3.39 Photospheric vector magnetograms (left) and plots of Jz Bz (right) for the active region NOAA 11515 on 30 June–4 July 2012. The arrows show the transverse component of the magnetic field. Light (dark) shades indicate positive (negative) values of Bz on the left and Jz Bz on the right. After Zhang et al. (2016)

the magnetic energy spectrum of this active region are higher than the estimation by the theory of Kolmogorov (1941a) in our analyzed spectral range. The evolution of r M and l M in active region NOAA 11515 is shown in Fig. 3.42. The average value of r M is about 0.22, while that of l M is about 8 Mm with the

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3 Solar Magnetic Activities

Fig. 3.40 Mean spectrums of 2E M (k) (solid line), k|HM (k)| (dotted line) and |HC (k)| (dash line) averaging over 840 vector magnetograms of active region NOAA11515 on 30 June–6 July 2012. After Zhang et al. (2016)

Fig. 3.41 Evolution of the scale exponent α of photospheric current helicity spectrum (solid line) and magnetic energy spectrum (dotted line) of active region NOAA 11515. After Zhang et al. (2016)

evolution of the active region. We can find that the values of the relative magnetic helicity and scale of magnetic energy decrease with the evolution of the active region, even the area of the active region increases.

3.8 Magnetic Field in Flare-CME Active Regions

211

Fig. 3.42 Evolution of the photospheric relative magnetic helicity (dotted line r M ) and integrated scale of magnetic field (solid line l M ) of active region NOAA 11515. After Zhang et al. (2016)

3.8 Magnetic Field in Flare-CME Active Regions It is generally believed that the magnetic field is generated near the bottom of the convective zone and emerges to the solar surface where it forms solar active regions. Both magnetic energy and helicity are brought into the corona with the magnetic flux as the field emerges. The magnetic energy released in solar active phenomena, such as flares and CMEs, is provided by the non-potential components of the magnetic field in ARs. There are some possibilities for the accumulation of non-potential magnetic energy in the solar atmosphere. One is that the twisted magnetic flux emerges to form the observed twisted magnetic ropes or magnetic knots in delta ARs (Tananka, 1991; Wang et al., 1994b; Leka et al., 1996; Liu & Zhang, 2001; Ruan et al., 2014). Another possibility is magnetic shear or squeeze between different magnetic structures of ARs in the photosphere which reflect the interaction of different magnetic flux systems with free magnetic energy accumulating in the solar corona (Hagyard et al., 1984; Chen et al., 1994; Zhang, 2001a, 2001b; Deng et al., 2001; Dun et al., 2007). The interaction of different magnetic flux systems actually arises from the emergence of new magnetic flux.

3.8.1 Change of Vector Magnetic Fields Associated with Flares The detection of the change in vector magnetic field in the period of powerful flare in the active regions is a notable project and was presented by Chen et al. (1989) with the video vector magnetograph firstly. They indicated that Initial bright points of the flare are related to the high shear region of the magnetic field and the emergence of

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new flux. The gradient of the longitudinal magnetic field decreased after the flares near the neutral line, and the shear angle of the transverse field changes relative to the neutral line.

3.8.1.1

Change of Vector Magnetic Fields Associated with Flares, Case 1

The active region of 1991 June, NOAA 6659, was a powerful flare-producing region (Zhang, 1993). About five white-light flares were observed in the active region. Zhang et al. (1994) presented the observational results on photospheric vector magnetic and velocity fields of this region obtained at the Huairou Solar Observing Station of National Astronomical Observatories of China and discussed the evolution of the magnetic fields and its relation to the powerful (white-light) flares. Figure 3.43 shows the change in transverse components of vector magnetic fields in the active region and the relationship with sites of white-light flare. After analysis of a series of monochromatic images of Stokes parameters Q, U, and V with high spatial resolution, the following newly observed phenomena were found by Zhang et al. (1994): (1) In this δ-region, some of the magnetic flux showed inverted polarity, and the longitudinal components of the emerging flux were aligned in fibril-like features whose small angles were relative to the horizontal components of the field on the solar surface, where the lines of force were highly sheared. (2) Before and after powerful (white-light) flares, the obvious change in vector magnetic fields occurred at the magnetic neutral line near the sites of these flares. The whitelight flares probably are caused by the violent motion of the photospheric lines of force. Lin et al. (1996) presented a solar flare in the FeIλ5324 Å line on 24 June 1993, and this flare occurred as an indicator of the excitation in the low photosphere. Song et al. (2020) discussed the possible white-light flare powered by magnetic reconnection in the lower solar atmosphere.

3.8.1.2

Change of Vector Magnetic Fields Associated with Flares, Case 2

The Bastille Day flare on 2000 July 14 was well observed by several space- and ground-based observatories and studied extensively by many researchers. It is discovered by Wang et al. (2005) that a large fraction of X-class flares is associated with a very interesting evolutionary pattern in sunspots: part of the outer spot structure decays rapidly after major flares; in the meantime, central umbral and/or penumbral structure becomes darker. These changes take place in about 1 hour and are permanent. It is found that the active region NOAA AR 9077 has sunspot structure change similar to that associated with the 2000 July 14 X5.7 flare in Fig. 3.44. Wang et al. (2005) found that the new evidence includes the following: (1) the Evershed velocity of decayed penumbral segments was weakened significantly

3.8 Magnetic Field in Flare-CME Active Regions

213

Fig. 3.43 The change in transverse components of vector magnetic fields in active region NOAA 6659 on 1989 June 6 and the relationship with sites of white-light flare. Bottom right: The shaded areas show the reversed polarity region of the corresponding longitudinal component of the magnetic field from 00:50 to 01:27 UT. The solid (dashed) lines mark the regions of positive (negative) polarity. From Zhang et al. (1994)

following the flare, indicating an actual weakening of penumbral structure; (2) based on available vector magnetograms before and after the flare, the transverse field strength decreased at the areas of penumbral decay and increased significantly near the flaring neutral line; (3) a new electric current system is found near the flare neutral line after the flare; and (4) the center-of-mass positions of opposite magnetic polarities converged toward magnetic neutral line immediately following the onset of the flare, and magnetic flux of the active region decreased steadily following the flare. It is proposed that a simple quadrupolar magnetic reconnection model may explain most of the observations: two magnetic dipoles join at the configuration before the flare; magnetic reconnection creates two new sets of loops: a compact flare loop and a large-scale expanding loop that might be the source of the CME. The outer penumbral fields become more vertical due to this reconnection, corresponding to the penumbral decay. Following initiation of magnetic reconnection associated with the flare, reconnected fields near the magnetic neutral line are first enhanced, then gradually weakened as it submerges. However, this model is questionable from one aspect of the observations: it is failed to identify two far-end footpoints of this quadrupolar magnetic reconnection.

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3 Solar Magnetic Activities

Fig. 3.44 Left: Two MDI WL images at 10:02 and 11:13 UT that were taken before and after the flare, respectively, and the difference image between the post and preflare images. Middle: MDI magnetogram, TRACE WL image during the flare, and intensity of difference image overplotted on the TRACE WL image. The contour levels are 1, 2, 3, and 4% above (or below) the preflare photosphere intensity of each area. The field of view is 19 × 13 . Right: Two averaged MDI Dopplergrams, before and after the flare, respectively, to demonstrate the evolution of Evershed flows. From Wang et al. (2005)

3.8.2 A Survey of Flares and Current Helicity in Active Regions The spatial and temporal relationship between chromospheric Hβ flares and photospheric current helicities in active regions have been examined by Bao et al. (1999). All of the data were obtained by the vector magnetograph system at Huairou Solar Observing Station of Beijing Astronomical Observatory. It is focused the analysis on NOAA Active Region 6233, which was observed on 30 August 1990. The result shows that rapid and substantial changes in the distribution of current helicity in an area or in its vicinity are most likely to trigger flares, but no compelling correlation between peaks of current helicity and flare sites. It is found that the time variations of current helicity in the active regions with highly productive flares are more significant than those of the poorly flare-productive active regions, and that the magnitude of current helicity does not always decrease after flares. Therefore it concludes that

3.8 Magnetic Field in Flare-CME Active Regions

215

Fig. 3.45 Variations of the average current helicity h c of the photospheric magnetic fields as a function of time in four flare-productive active regions (NOAA 6233, NOAA 6891, NOAA 7321 and NOAA 7773). Arrows indicate start times of flares. From Bao et al. (1999)

the rate of variation of current helicity may be considered as an indicator of flare activity. Note that the average current helicity h c for a whole active region changes as obviously as the current helicity imbalance ρh =

h c (i, j) 100% |h c (i, j)|

(3.83)

in flare-productive active regions, as shown in Fig. 3.45. From this figure, it can be found that the magnitude of current helicity does not always come down after a flare. Flaring activity seems to be globally associated with the rate of variations in h c . However, there is not a one-to-one relation between flare activity and variation of h c . This result does not agree with that of Pevtsov et al. (1995). Bao et al. (1999) argue that the rate of variation of current helicity in active regions is more closely related to solar flares, and it may better characterize the non-potentiality of active regions rather than the values of current helicity.

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3 Solar Magnetic Activities

3.8.3 Injective Magnetic Helicity and Flare-CMEs After the maturity of the main bipolar spots of AR 10488 in Figs. 3.24, 3.25 and 3.26, it is found that the average of dH/dt is about −5 × 1041 Mx2 hr−1 . Till 22:00 UT on October 31, the accumulated change in helicity has reached −4 × 1043 Mx2 , and the magnetic flux has reached about ±3 × 1022 Mx. If the accumulated change in helicity keeps increasing in absolute value at the speed of that after the maturity of the main bipolar spots, it will amount to −6 × 1043 Mx2 at 00:00 UT on November 3. This represents the high level of helicity in an AR. Chae et al. (2004) produced lower helicity of 8 × 1042 Mx2 for another AR. The main reason for this difference seems to be the difference in the magnetic fluxes of the studied ARs. Nindos & Andrews (2004) indicated that the amount of the stored pre-flare coronal helicity was small for flares without CMEs than for flares with CMEs, and that the maximal absolute coronal helicity for flares with CMEs is about 7 × 1043 Mx2 and the average is about 2.68 ± 1.81 × 1043 Mx2 . So the coronal helicity inferred from horizontal motion in AR 10488 is sufficient for one or two flares with CMEs before 00:00 UT on November 3. There were 17 C-class, 6 M-class, and 2 X-class flares that occurred in this AR during its disk passage. Identifying the time of flares and CMEs and scanning the movies of them obtained by Large Angle and Spectrometric Coronagraph Experiment, we have found that there were 2 CMEs (occurred at 01:59 and 10:06 UT on November 3 separately) associated with the flares (X2.7 and X3.9 flares occurred at 01:09 and 09:43 UT on November 3 separately). It should be pointed out that the helicity of a single CME or MC depends on the length of the MC flux tube adopted (using the MC helicity computation as a proxy to the CME helicity). If a shorter length of 0.5 AU is adopted, the mean helicity of a single CME or MC is typically 2 × 1042 Mx2 (DeVore, 2000).

3.8.4 Powerful Flares and Dynamic Evolution of Magnetic Field at Solar Surface The strength of the magnetic field and the speed of its evolution are two vital parameters in the study of the change in magnetic field in the solar atmosphere. Liu et al. (2008) propose a dynamic and quantitative depiction of the changes in the complexity of the active region (AR): E = u × B, where u is the velocity of the foot-point motion of the magnetic field lines and B is the magnetic field. E represents the dynamic evolution of velocity field and magnetic field shows the sweeping motions of magnetic foot points, exhibits the buildup process of current, and relates to the changes in non-potentiality of the AR in the photosphere. It is actually the induced electric field in the photosphere. It can be deduced observationally from velocities computed by the local correlation tracking (LCT) technique and vector magnetic fields derived from the vector magnetograms.

3.8 Magnetic Field in Flare-CME Active Regions

217

Fig. 3.46 Gray scale maps of E z . The white and black contours represent longitudinal magnetic field strengths of 200 and −200 G, respectively. The field of view is 225 × 168 . From Liu & Zhang (2006)

It is found that: (1) The initial brightenings of flare kernels are roughly located near the inversion lines where the intensities of E are very high. (2) The daily averages of the mean densities of E and its normal component (Ez ) decrease after flares for most cases we studied, while those of the tangential component of E (Et ) show no obvious regularities before and after flares. (3) The daily averages of the mean densities of Et are always higher than those of Ez , which can not be naturally deduced by the daily averages of the mean densities of Bz and Bt . Figure 3.46 shows gray-scale maps of E z in AR NOAA 10488. In the rotation phase, the maxima of E z are located near the rotating or twisting areas of the main positive spot. In the shear phase, the maxima of E z shift to the “strong shear area” which is between the new positive and the old negative spots showing SSM. The maxima of E z amount to 0.1 ∼ 0.2 V cm−1 , which are comparable with the electric field in the slow magnetic reconnection stage of the evolution of a two-ribbon flare (0.1 V cm−1 ) reported by Wang et al. (2003). The positions of the maxima of E z imply that the parameter E z possibly relates to magnetic non-potentiality in the solar atmosphere in this AR. In order to see more clearly the relationship between E z and the non-potentiality, we have investigated a flare in this AR. On October 30, an M1.6 flare was observed by the HSOS Hα telescope. It began at 01:56 UT, reached its maximum at 02:07 UT, and ended at 02:29 UT. Figure 3.47 shows the Hα filtergram of this flare at 02:03 UT, superposed with the contours of E z = ±0.12 V cm−1 (the thick lines). From the figure, we find that there is no obvious correlation between the maxima of E z and the bright kernels. However, in the “strong shear area”, E z is intense, and the Hα filtergram presents a less bright strip (see the mark “f” in Fig. 3.47). This phenomenon remains for a few hours around the flare time and deserves further investigation. Recalling the X-class

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3 Solar Magnetic Activities

Fig. 3.47 An M1.6 Hα flare observed at HSOS on October 30, the thick white and black contours represent the E z = ±0.12 V cm−1 , and the thin ones longitudinal magnetic field strengths of ±200G, the field of view is 225 × 168 . From Liu & Zhang (2006)

flare kernels being located right between the new positive and the old negative spots at 01:25 UT on November 3, we suggest that the magnetic free energy is accumulated persistently by the SSM in this “strong shear area”. We need full and accurate data of more flares, especially the powerful flares, to investigate the relationship between E z and flare.

Chapter 4

Spatial Magnetic Configurations of Solar Active Regions and Eruptions

The study on the spatial structure and properties of the solar magnetic field is an important aspect of solar physics, and it is also a necessary condition to fully understand the solar magnetic active phenomenon. However, due to the observed limitations, it is currently unable to get information about the spatial configuration of the magnetic field above the photosphere in detail. It is particularly difficult to measure the coronal magnetic field because the coronal spectral lines are too wide and too dim, and the magnetic fields are too weak as measuring them by means of the Zeeman splitting spectral method, even if the Hanle effect has been proposed to measure the coronal fields. So far the research of coronal magnetic fields is still mainly based on the observations of coronal morphological configurations and compared with the extrapolated magnetic field lines from the photospheric fields. Of course, there are still problems in this conceptual basis itself, mainly because the coronal morphological structure is not completely equal to the configuration of the magnetic field.

4.1 Force-Free Field Approximation Because in the corona and chromosphere, the macroscopic distribution of plasma is essentially controlled by the magnetic field, the observed structures in the corona and chromosphere (even the photosphere) have been used as inspection standards of the theoretical extrapolated magnetic fields. If the theoretical extrapolated magnetic field lines are consistent with the observed coronal and chromospheric structure, we normally believe that the extrapolated magnetic field reflects basically the real one in the solar atmosphere. The equilibrium between the magnetic field and plasma structure can be described by the following magnetic hydrostatic equations © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Zhang, Solar Magnetism, https://doi.org/10.1007/978-981-99-1759-4_4

219

220

4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

−∇p +

1 (∇ × B) × B + ρg = 0 , μ0

(4.1)

where p is the fluid pressure, B is the magnetic field, ρ is the density, g is the gravitational acceleration, μ0 = 4π × 10−7 H m −1 is the vacuum permeability. In the chromosphere and the low corona, the magnetic pressure is much larger than the gas pressure and gravity (i.e., β is relatively small, β=2μ0 p/B 2 ), then Eq. (4.1) can be approximated in the form (∇ × B) × B = 0,

(4.2)

i.e., the current is parallel to the magnetic field, the Lorentz force is zero. Formula (4.2) can also be written ∇ × B = α(r, t)B , (4.3) where α(r, t) is a function of the spatial location r and time t, called the force-free factor. The force-free magnetic field should satisfy the condition of divergence free, namely ∇ · B = 0. (4.4) Equations (4.3) and (4.4) are the force-free field equations.

4.1.1 Methods for Extrapolation of Solar Magnetic Fields 4.1.1.1

Approximation of Potential Field

The observational solar magnetic field can be inferred as the current system formed in the subphotosphere. Above the photosphere, there may also exist the current especially in active regions. Therefore, the photospheric magnetic field should be the superposition of two current systems. If we ignore the current above the solar atmosphere with the remained current system below the photosphere only, then the magnetic field above the photosphere performs a potential field. For the potential field approximation ∇ × B = 0, one can introduce a scalar potential φ. The magnetic field is expressed as B = −∇φ,

(4.5)

Taking the divergence on the above equation φ = 0,

(4.6)

In the potential field approximation, scalar potential φ satisfies the Laplace Eq. (4.6). If given the appropriate boundary conditions, Eq. (4.6) has a unique solution. In the

4.1 Force-Free Field Approximation

221

Dirichlet problem, the boundary condition is φ in the boundary, while in the Neumann problem, the boundary condition is the directional derivative of the boundary potential function φ. The latter is just applicable to the solar magnetic field, when is the longithe study area is located near the center of the solar surface, Bn = − ∂φ ∂n tudinal component Bz of the solar magnetic field. In a potential field approximation, B = −∇φ can be obtained by solving the following boundary value problem: −n · ∇φ = Bn (z = 0) ,

φ = 0 (z > 0),

(4.7)

where n is the vertical unit vector in the photospheric surface (z = 0), and as requiring z → ∞, B → 0. 4.1.1.2

Magnetic Charge Method

The magnetic charge simulation method was proposed by Schmidt in 1964. It replaces the spheric approximation by the plane, which only can be used to calculate the magnetic field distribution in a small area of the solar corona (e.g., active regions). The calculation is convenient and without the limit of spatial resolution. This method is similar to the calculation of the electrostatic potential, the vertical component of the magnetic field in the solar photosphere has to be analogized as the magnetic charge. If the studied region is located near the center of the solar surface, the longitudinal component Bz of the magnetic field in the photosphere (and as parallel to the line of sight) can be used as the vertical component of the field. The photospheric (z = 0) magnetic charge distribution actually reflects that of the magnetic field (z > 0). Of course, the imaginary magnetic field generates in the subphotosphere (z < 0) without the physical meaning. Lets the photospheric surface density of magnetic charge σ , the magnetic field strength is B, the unit normal vector is n in the photosphere, then the potential function φ(r ) at any point P above the photosphere can be expressed as  φ(r ) =

σ ds = ρ



σ ds , |r − r  |

(4.8)

where ρ is the distance P from the area element ds in the photosphere, r  is the coordinate of area element. The relationships between σ and B is determined by Ostrogradski-Gaussian Theorem: 2π σ = Bn = Bz ,

(4.9)

where Bz is the photospheric longitudinal field, so Eq. (4.8) can be written φ(r ) =

1 2π



Bz ds , |r − r  |

then the magnetic field is deterred by Eq. (4.5).

(4.10)

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4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

Potential field approximation now becomes almost a routine extrapolation method used for the observational study, which is used to establish the initial conditions in the MHD simulation. In addition, it can be used to solve the photospheric transverse magnetic field 180◦ uncertainty; used to calculate the magnetic shear angle to check the non-potential extent of magnetic field.

4.1.1.3

Spheric-Harmonic Method

Newkirk et al. (1968) proposed using the spherical harmonics method to solve the Laplace equation, and developed by Alissandrakis (1981). It can be applied to the large area, where the spherical curvature must be considered, even if the global magnetic field above the photosphere, but the effects of the solar wind must be considered. This method is theoretically more complete. To achieve 2 × 2 spatial resolution, the polynomial must be calculated to as many as N = 90. For the Laplace equation (4.6), when r > R time (R is the solar radius), the solution is φ(r, θ, φ) = R

N  l 

fl (r )Plm (θ )[glm cos(mϕ) + h lm sin(mϕ)] ,

(4.11)

l=1 m=0

where f (r ) =

(rω /r )l+1 − (r/rω )l . (rω /r )l+1 − (R/rω )l

(4.12)

The boundary conditions are: when r = rω and f (r ) = 0, the magnetic field is a radial one. The physical meaning of this boundary condition is that when r = rω , the energy of solar wind is dominant, it will force the frozen magnetic field lines along the direction of flow freely. From the photos of the total solar eclipse, you can see this case occurred roughly rω = 2.6R. Therefore, r = rω is also known as the source of the spherical surface (source surface), because it is equivalent with r  rω spatial current. The contribution of this current to the spatial potential field at R < rω must of course considered. The normalization condition of function fl (r ) is fl (r ) = 1. The coefficients glm and h lm in Eq. (4.11) can be fitted with the observed photospheric magnetic field (r = R) with the least-squares method. The high accuracy of the calculated magnetic field relates to the large Polynomial number N . The third solution seeking potential field method is the use of the general forcefree field described solution in the following, and then set α = 0, it is the potential field solution. It is generally used to calculate the structure of the magnetic field in solar activity regions, but also commonly used in the study of the large-scale structure of the coronal magnetic field. Potential field calculations need to use the vertical (Sun’s radial) component of the photospheric magnetic field, i.e., the line of sight component (called the longitudinal field) measured by magnetographs. So only as the observed area is located near the

4.2 Linear Force Free Field

223

center of the solar surface, the longitudinal magnetic field can be approximately as the vertical component of the field, so when a study area is not located near the center of the solar surface to calculate the magnetic field, you can only use the magnetic field as boundary conditions, which region is measured just a few days before or a few days later in the day near the center plane. The problem is that the magnetic field changes in this period, namely, the simultaneous problems between the observed magnetic field and the theoretically calculated coronal structures. And when calculating the full surface magnetic field, due to the need for the whole photospheric magnetic field as boundary conditions, it takes through a solar rotation cycle. In such a long period, the magnetic field itself changes obviously. So to get the whole photospheric field on the surface in a solar rotation cycle as a boundary to calculate magnetic the field configurations, it only is regarded as the average results of this period. It means that the potential field approximation is only applicable to extrapolate the magnetic field with its slow evolution. It mainly has been used to understand the large-scale structure of the coronal magnetic field.

4.2 Linear Force Free Field The current in the solar atmosphere is not necessarily zero, especially in the above active regions there exist generally current, so the potential use of the non-forcefree field is more reasonable. The simplest linear force-free field has gained more in-depth research. Linear force-free field of the classical approach is to solve a scalar into Helmholtz equation, whose solution is mainly divided into two categories: the Fourier series method (Nakagawa & Raadu, 1972; Seehafer, 1978; Alissandrakis, 1981; Gary, 1989; Aly, 1992; Démoulin et al., 1997; Song & Zhang, 2004) and Green’s function method (Chiu & Hilton, 1977; Semel, 1988; Yan, 1995). In a Cartesian coordinate system, it can be proven that the constant α linear forcefree field B (∇ × B = αB) can use a scalar function P to express as B = ∇ × ∇ × (P eˆ z ) + α∇ × (P eˆ z ) ,

(4.13)

or  B=

∂P ∂2 P +α ∂ x∂z ∂y



 eˆ x +

∂2 P ∂P −α ∂ y∂z ∂x



 eˆ y −

∂2 P ∂P + 2 2 ∂x ∂y

 eˆ z ,

(4.14)

where P satisfies the Helmholtz scalar equation (∇ 2 + α 2 )P = 0.

(4.15)

Some commonly used methods, such as Nakagawa–Raadu, Chiu–Hilton, etc., the resulting different expressions of the magnetic field are commonly derived from the (4.14) and (4.15). Here is some commonly used constant α linear force-free field

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4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

formula in the following. These formulas are generally applied to the calculation of the magnetic field in the specific solar active regions.

4.2.1 Chiu–Hilton Method Chiu and Hilton (1977) presented a method for solving the constant-α linear forcefree field. It takes a Cartesian coordinate system, at z = 0 plane rectangular area (0 ≤ x ≤ L x , 0 ≤ y ≤ L y ) in, Bz (x, y, 0) from the observed values. Outside the region Bz (x, y, 0) = 0, it is the boundary value problem without horizontal borders. In cylindrical coordinates, the Helmholtz equation (4.15) in the problem of the semiinfinite space the general solution is 

∞ 

P(ρ, φ, z) =

m=−∞



α

+ 0

∞

eimφ α

√

[Am (k)e−

k 2 −α 2 z

+ Bm (k)e

√ k 2 −α 2 z

]Jm (kρ)dk

   2 2 2 2 [Am (k) cos( α − k z) + Cm (k) sin( α − k z)]Jm (kρ)dk . 

(4.16)

Considering the solution of semi-infinite space, by decaying conditions: lim P(ρ, φ, z) = 0,

(4.17)

z→∞

one can introduced formula (4.16) with z in the coefficients of the exponential growth Bm (k) = 0.

(4.18)

And Am (k), A m (k), Cm (k) is undetermined coefficients, k is a horizontal wave number, Jm (kρ) for the Bessel function. Because of the uncertain boundary, the arbitrary finite integrable function Cm (k) makes the formula (4.16) satisfy the boundary value Bz (x, y, 0), and the solution is not unique. So by means of the existing observational data limitations (only z = 0 a level of longitudinal magnetic field observations more reliable), one can only assume Cm (k) is zero. Bessel function orthonormality and addition formulas and identities 

∞

√

0

kdk k 2 − α2

{exp[−(k 2 − α 2 )1/2 z]}J0 (k R) =

exp[∓iα(R 2 + z 2 )1/2 ] , (4.19) (R 2 + z 2 )1/2

Green’s function can be obtained using the expression to represent the magnetic field: Bi =

1 2π

 0

Lx



Ly 0

d x  dy  G i (x, y, z; x  , y  )Bz (x  , y  , 0),

i = x, y, z,

(4.20)

4.2 Linear Force Free Field

225

where

x − x  ∂ y − y + α , R ∂z R y − y  ∂ x − x Gy = − α , R ∂z R

∂ − , Gz = − ∂R R

Gx =

the assisted functions

z 1 cos(αr ) − cos(αz), Rr R  R = (x − x  )2 + (y − y  )2 .

=

(4.21)

(4.22) (4.23)

√ r = R 2 + z 2 is the distance from the seek point (x, y, z) to a point in the photosphere (x  , y  , 0). Considering when the radius r → ∞, the flux in the hemispherical surface should be zero, it is required that in the study region the net flux is zero.

4.2.2 Alissandrakis Fourier Transform Alissandrakis (1981) used the Fourier analysis method for solving linear force-free field Eqs. (4.3). The vertical component of the magnetic field is expressed as Ny Nx  

Bz =

Bn x ,n y exp(−lz)

(4.24)

n x =1 n y =1

k x2 + k 2y − α 2 , k x = 2π n x /L, k y = 2π n y /L, L is the length of the horizontal coordinates of the calculated volume, Bn x ,n y is the Fourier amplitudes of (n x , n n y ) order calculated by observing the photospheric longitudinal magnetic field. The same method can be used to obtain two horizontal components Bx and B y of the magnetic field Ny Nx   −i(n x l − n y α) Bn x ,n y Bx = exp(−lz) (4.25) 2π(n 2x + n 2y ) n =1 n =1 where, l =

x

By =

y

Ny Nx   n x =1 n y

where i is the imaginary unit.

−i(n x l + n y α) exp(−lz), Bn x ,n y 2π(n 2x + n 2y ) =1

(4.26)

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4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

4.2.3 Fast Fourier Analysis Method of Linear Force-Free Field Song and Zhang (2004) proposed the fast Fourier analysis and calculation method for the linear force-free field. They believe that Alissandrakis’s Fourier analysis using the integral in infinite two-dimensional space is too abstract and does not easy to apply, and in his works the force-free condition (4.3) has been used only, while the conditions of the divergence free have not completely. Fast Fourier analysis of the linear force-free field calculation crystallizes Alissandrakis’s Fourier analysis, and improve to satisfy Eq. (4.4). Fast Fourier analysis and calculation method firstly set linear force-free field observation field Bz at x direction N1 sampling points in the y direction N2 sampling points, here N1 = 2m 2 , N2 = 2m 2 , m 1 , m 2 are positive integers easy made FFT calculation procedures, if the observed sampling points is not the form of the exponential of 2, then add some points in this point set Bz = 0 is greater then the calculated field observation domain, so that its benefits are that the calculation on the border gradually reduced to zero. The observation field of two-dimensional finite Fourier expansion is: Bz = f (x, y) =

N 1 −1 N 2 −1   i=0

Ck1 ,k2 e I (i x+ j y) ,

(4.27)

j=0

√ where I = −1 is the imaginary unit, f is a complex function, the real part is the observed value of Bz , the imaginary part is zero. Ck1 ,k2 can be obtained by the following equation: Ck1 ,k2 =

1 N1 N2

N1 −1,N 2 −1

 f

λ,m=0

λ m , N1 N2

 e

−2π I (k1 Nλ +k2 Nm ) 1

2

.

(4.28)

Formula (4.28) can be calculated using the fast Fourier rapid Ck1 ,k2 . Similar to Eq. (4.27), to write the magnetic field in a three-dimensional space below: N1 −1,N 2 −1 Bx (u, v, z)e I (ux+vy)2π , Bx (x, y, z) = u,v=0

B y (x, y, z) =

N1 −1,N 2 −1

B y (u, v, z)e I (ux+vy)2π ,

(4.29)

u,v=0

Bz (x, y, z) =

N1 −1,N 2 −1

Bz (u, v, z)e I (ux+vy)2π ,

u,v=0

assuming complex Fourier coefficients have the following simple dependencies with z:

4.2 Linear Force Free Field

227

Bx (u, v, z) = Bx (u, v, 0)e−kz , B y (u, v, z) = B y (u, v, 0)e−kz , Bz (u, v, z) = Bz (u, v, 0)e

−kz

(4.30)

,

complex k is not a constant, and it depends on the wave number u, v and α. It can be proved that formulas (4.29) and formulas (4.30) are self-consistent, just so that the magnetic field can satisfy the force-free and divergence-free conditions. Tanking the partial derivative of Eq. (4.29), and into force-free Eq. (4.3), the following linear algebraic equations can be obtained: α Bx (u, v, 0) − k B y (u, v, 0) − (I 2π v)Bz (u, v, 0) = 0,

(4.31)

k Bx (u, v, 0) + α B y (u, v, 0) + (I 2π u)Bz (u, v, 0) = 0,

(4.32)

(I 2π v)Bx (u, v, 0) − (I 2π u)B y (u, v, o) + α Bz (u, v, 0) = 0.

(4.33)

The solution of linear equations with nonzero Bx , B y , Bz requires that the coefficient determinant is zero:    α −k −2π v I    k α −2π u I  = −α[4π 2 (u 2 + v 2 ) − α 2 − k 2 ] = 0.   2π v I −2π u I α  This determinant determines the complex wave number k spreading up with the magnetic field configuration. Taking the determinant of the results into Eqs. (4.31) and (4.32) obtained Bx , B y , Bz substituting (4.29) and (4.30) where they will have a linear force-free field expressions (4.34): Bx (x, y, z) =

N1 −1,N 2 −1 u,v=0

B y (x, y, z) =

N1 −1,N 2 −1 u,v=0

−I (ku − vα) −kz e Bz (u, v, 0)e I (ux+vy)2π , 2π(u 2 + v 2 ) −I (kv + uα) −kz e Bz (u, v, 0)e I (ux+vy)2π , 2π(u 2 + v 2 )

Bz (x, y, z) =

N1 −1,N 2 −1

(4.34)

e−kz Bz (u, v, 0)e I (ux+vy)2π ,

u,v=0

 N1 − 1 N2 − 1 −α 2 + 4π 2 (u 2 + v 2 ), 0 ≤ u ≤ ,0≤v≤ . Bz (u, v, 0) N1 N2 can be obtained by (4.28). Since formula (4.34) is the solution of Eqs. (4.31–4.33) to get, so this inability to meet the conditions. By the formula (4.34) can be obtained where k =

228

4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

∂ By ∂ Bz ∂ Bx + + = ∂x ∂y ∂z

N1 −1,N 2 −1

ku 2 − vαu + kv 2 + αuv −kz (ux+vy)2π e e = 0. u 2 + v2 u,v=0 (4.35) So formula (4.34) satisfies the conditions with the divergence-free. Formula (4.34) and (4.35) at u = 0, v = 0 are not hold. For this situation, one can find a particular solution of (4.31), (4.33), and (4.35), then finally got a rigorous self-consistent solution of the linear force-free field. The method starts from Alissandrakis’s Fourier analysis and tries to make up for the deficiencies of Alissandrakis’s Fourier analysis. Fourier analysis of the specific Alissandrakis whether there is still a theoretical flaw in the debate. Such as citation (Li et al., 2004) in the commentary: “satisfy the linear force-free field (∇ × B = αB) in the case, no loose condition (∇ · B = 0) naturally satisfied, so that the fast Fourier analysis and calculation method for the linear force-free field correction on the divergence-free is entirely superfluous, so Song & Zhang’s Fourier analysis on Alissandrakis Law review is not appropriate.” Meanwhile in citation (Jing et al., 2008), it is pointed that: the coefficients Bz (u, u, 0) in Eq. (4.34) must be as u = 0 , v = 0 is satisfied Bz (0, 0, 0) = 0, otherwise the condition of divergence-free is not satisfied, so in theory in (4.34), only prescriptive u = 0 v = 0 is zero for the corresponding terms, no need to look particular solution.

4.3 Study on Two Methods for Nonlinear Force-Free Extrapolation Based on Semi-analytical Field Based on the force-free assumption, the magnetic field in the solar chromosphere and corona can be obtained from the photospheric magnetic field extrapolation. For the potential field (α = 0), the precise analytical solutions can be obtained from the Eqs. (4.3) and (4.4), and several methods are proposed to solve Eqs. (4.3) and (4.4) for a potential field (e.g., Schmidt, 1964; Newkirk et al., 1968). While for the linear force-free (α = constant) extrapolation method, the analytical solutions can also be obtained from the Eqs. (4.3) and (4.4), but the uncertain constant α must be given for the calculations. The value of α can be decided by the comparisons between the distribution of extrapolated magnetic field lines and the observed data (e.g., Wiegelmann et al., 2005). Several linear force-free extrapolation methods have been used to extrapolate the magnetic field (e.g., Nakagawa & Raadu, 1972; Chiu & Hilton, 1977; Seehafer, 1978; Alissandrakis, 1981; Gary, 1989; Aly, 1992; Yan, 1995; Song & Zhang, 2005, 2006). So far the potential field and linear force-free field extrapolations have been developed well, but they describe the magnetic field above the photosphere roughly. While using the nonlinear force-free model is more reasonable.

4.3 Study on Two Methods for Nonlinear Force-Free Extrapolation …

229

4.3.1 Theories and Algorithms Recently several models for nonlinear force-free extrapolation have been proposed (e.g., Sakurai, 1981; Chodura & Schlueter, 1981; Wu et al., 1990; Roumeliotis, 1996; Amari et al., 1997, 1999; Yan & Sakurai, 2000; Wheatland et al., 2000; Valori et al., 2005; Wiegelmann, 2004; Song et al., 2006). Although some differences exist among these methods (Schrijver et al., 2006; DeRosa et al., 2009), some extrapolated fields can give the results that are consistent with the observations (e.g. Régnier & Amari, 2004; Wiegelmann et al., 2006 and Régnier & Prist, 2007), so the magnetic field extrapolation provides a promising tool for us to understand the magnetic field in the chromosphere and corona approximatively, then to predict solar activity. Since all the above methods can be used to extrapolate the magnetic field in the chromosphere and corona from the photospheric magnetic field, and these methods have been used to describe the magnetic field lines and topology in active regions (e.g., Song et al., 2006, 2007; He et al., 2008; Wang et al., 2008; Guo et al., 2009), testing the validity of these models become an imperative subject. Generally speaking, the extrapolated results of force-free models are either compared to the analytical field or compared to some observed data such as X-ray, EUV, etc. The semi-analytical solutions of force-free and divergence-free equations given by Low and Lou (1990) can provide 3D magnetic field easily, which creates axial symmetry numerical solutions satisfying the force-free and divergence-free equations in the spherical coordinates. Thus, a part of the analytical field can be used for testing the validity of the nonlinear force-free magnetic field extrapolation method (Liu et al., 2012).

4.3.1.1

Boundary Integral Equation (BIE) Method

The boundary integral equation (BIE) technique was proposed by Yan and Sakurai (1997, 2000), subsequently developed by Yan and Li (2006), He and Wang (2008). The magnetic field can be obtained by directing the integration of the magnetic field on the bottom boundary surface. There is a force-free field in the half-space  above the photospheric surface . The boundary condition is B = B0

on

B ∈ ,

(4.36)

where B0 is the photosphere vector magnetic field. At infinity, an asymptotic constrain condition should be included to ensure a finite energy content in the half-space  above when r −→ ∞, (4.37) B = O(r −2 ) where r is the radial distance. The method uses the two constraint conditions ((4.3) and (4.4) and two boundary conditions (4.36) and (4.37) to calculate the magnetic field above the photosphere.

230

4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

The reference function Y is introduced in this method (Yan & Li, 2006), Y =

cos(λρ) cos(λρ  ) , − 4πρ 4πρ 

(4.38)

where ρ = [(x − xi )2 + (y − yi )2 + (z − z i )2 ]1/2 is the distance between a variable point (x, y, z) and the given field point (xi , yi , z i ), ρ  = [(x − xi )2 + (y − yi )2 + (z + z i )2 ]1/2 , and λ is a factor dependent on the location of point i. After a series of derivations the magnetic field B can be obtained from the following formula:  B(xi , yi , z i ) =

z i [λr sin(λr ) + cos(λr )]B0 (x, y, 0) d xd y, 2π [(x − xi )2 + (y − yi )2 + z i2 ]3/2

(4.39)

where r = [(x − xi )2 + (y − yi )2 + z i2 ]1/2 and B0 is the magnetic field of photospheric surface and λ = λ(xi , yi , z i ) can be calculated from equation 

 

(Y ∇ 2 B − B∇ 2 Y )d =



Y (λ2 B − α 2 B − α × B)d = 0,

(4.40)

where  is half-space above the photospheric surface , and  is the same as  but cut a small neighborhood where the point calculated is included (c. f. Yan & Li, 2006). Unlike α, λ is not constant along the magnetic field line, it is a function of position. The theories of BIE are described in detail in the paper of Li et al. (2004) and Yan and Li (2006). BIE method is to find the best λ through iteration and to obtain the magnetic field at the same time. When the magnetic field satisfies the following conditions (4.43) and (4.44), the corresponding λ is a good value: |J × B| , |J| |B|

(4.41)

|∇ · B| Vi |δBi | = , |Bi | |B| σi

(4.42)

f i (λx , λ y , λz ) = gi (λx , λ y , λz ) = and

f i (λ∗x , λ∗y , λ∗z ) = min( f i (λx , λ y , λz )), gi (λ∗x , λ∗y , λ∗z ) = min(gi (λx , λ y , λz )).

(4.43)

We set the following constraints: f i (λ∗x , λ∗y , λ∗z ) ≤  f ,

gi (λ∗x , λ∗y , λ∗z ) ≤ g ,

(4.44)

where  f and g are sufficiently small thresholds. In fact, f i (λx , λ y , λz ) indicate the angles between B and J, if f i (λx , λ y , λz ) = 0 there is no Lorentz force and Eq. (4.3) is satisfied. Where gi (λx , λ y , λz ) stand for the divergence of B, Eq. (4.4) can be satisfied

4.3 Study on Two Methods for Nonlinear Force-Free Extrapolation …

231

when gi (λx , λ y , λz ) = 0. BIE method uses a simple downhill technique to find λ, then calculate B that satisfies Eqs. (4.44).

4.3.1.2

Approximate Vertical Integration (AVI) Method

In the vertical integration method (Wu et al., 1990; 1990), a finite difference scheme has been used to solve the height-dependent mixed elliptic-hyperbolic partial differential equations (4.45)–(4.48), which can be deduced from Eqs. (4.3) and (4.4): ∂ Bz ∂ Bx = + α By , ∂z ∂x

(4.45)

∂ By ∂ Bz = − α Bx , ∂z ∂y

(4.46)

∂ By ∂ Bz ∂ Bx =− − , ∂z ∂x ∂y

(4.47)

α Bz =

∂ By ∂ Bx − . ∂x ∂y

(4.48)

But they are ill-posed problems and they have the problems of divergence when the height increases (Démoulin et al., 1992; Cuperman et al., 1990). Song et al. (2006) proposed the approximate vertical integration (AVI) method and tried to avoid those ill-problems that the vertical integration method contains. In this method, at first, they constructed the magnetic field by the following formulas, supposing the solutions with second-order continuous partial derivatives in a certain height range, 0< z 0) the force-free equations are fulfilled when L is equal to zero. This method involves minimizing L by optimizing the solution function B(x, t) through states that are increasingly force- and divergence-free, where t is an artificial time-like parameter introduced.

4.3.1.4

Nonlinear Force-Free Magnetic Field Solutions

Low and Lou (1990) describe a special class of nonlinear force-free fields, which satisfy Eqs. (4.3) and (4.4), written as a second-order partial differential equation (4.53) in the spherical coordinate system 1 B= r sin θ



 1 ∂A A ˆ ˆ rˆ − θ + Q φ , r ∂θ r

(4.53)

where A and Q are two scalar functions. The force-free condition requires Q to be a strict function of A with dQ (4.54) α= dA and

∂ 2 A 1 − μ2 ∂ 2 A dQ = 0, + +Q ∂r 2 r 2 ∂μ2 dA

(4.55)

where μ = cosθ . Mathematically the equation is a variable separable differential equation and the two separable solutions are A=

P(μ) , rn

Q(A) = a A1+1/n ,

(4.56) (4.57)

where a and n are constants and the Legendre polynomial function P satisfies the nonlinear differential equation

234

4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

(1 − μ2 )

1 + n 1+2/n d2 P P + n(n + 1)P + a 2 = 0. dμ2 n

(4.58)

For the magnetic field vanishing at infinity, it implies that P=0

at

μ = −1, 1.

(4.59)

It can be seen that Eq. (4.53) describes an axial symmetry magnetic field. Low and Lou pointed out that an arbitrary position of a plane determined by two parameters l (the distance between the plane surface boundary and the point source) and φ (the angle between the normal direction of surface and the z-axis associated with the spherical coordinate system) may represent an active region on the solar photosphere. So it can be taken as the boundary conditions for the magnetic field extrapolation. When n >0, the boundary equation (4.59) creates a discrete infinite set of eigenvalues, 2 2 , m = 0, 1, 2, 3,..., and αn,0 = 0, and the corresponding which can be denoted by αn,m eigenfunctions is Pn,m (μ). So for different n and m, it can give different distributions of the magnetic field in the spherical coordinate system, which meet the requirements of divergence-free and force-free equations. Then we choose specified l and φ for coordinate transforming since the magnetic field in Cartesian coordinate is needed in our extrapolation. We choose two of these solutions as test fields: π SAF1: the semi-analytical field with n = 1, m = 1, l = 0.3, and φ = , set x ∈ 4 [−0.5, 0.5], y ∈ [−0.5, 0.5] and z ∈ [0, 1] in Cartesian coordinate system. 4π , set SAF2: the semi-analytical field with n = 3, m = 1, l = 0.3, and φ = 5 x ∈ [−0.5, 0.5], y ∈ [−0.5, 0.5] and z ∈ [0, 1] in Cartesian coordinate system. The mesh is 64 pixel × 64 pixel for those two solutions. Note again: In the spherical polars (r, θ, φ), the axisymmetric field can be written in the form B= We can write

1 r sin θ



1 ∂A ∂A , − , Q(A) . r ∂θ ∂r

∂ 2 A 1 − μ2 ∂ 2 A + + α Q = 0, ∂r 2 r 2 ∂μ2

(4.60)

(4.61)

for the force-free magnetic field with ∇ × B = αB, and 1 ∂Q ∂2 A 1 1 1 ∂2 A ∂Q − + + 2 2 r ∂μ r 1 − μ ∂φ∂r r ∂φ∂μ ∂r α= . 1 ∂A ∂A + r ∂μ ∂r As

∂A = 0, then ∂φ

(4.62)

4.3 Study on Two Methods for Nonlinear Force-Free Extrapolation …

∂Q 1 ∂Q + dQ r ∂μ ∂r . α= = 1 ∂A ∂A dA + r ∂μ ∂r

235

(4.63)

 Proof: We can use the vector formula in spherical coordinates     1 ∂ 1 1 ∂ Ar ∂ ∂ Aθ ∇ ×A= (Aφ sin θ ) − rˆ + − (r Aφ ) θˆ r sin θ ∂θ ∂φ r sin θ ∂φ ∂r   ∂ Ar 1 ∂ ˆ (r Aθ ) − + φ. r ∂r ∂θ (4.64) We can obtain     ∂ Q sin θ ∂ 1 ∂A 1 − − rˆ ∇ ×B= r sin θ ∂θ r sin θ ∂φ r sin θ ∂r     1 ∂ 1 1 ∂A ∂ rQ 1 − θˆ + r sin θ ∂φ r sin θ r ∂θ ∂r r sin θ     r ∂A ∂ 1 1 ∂A 1 ∂ − − φˆ + r ∂r r sin θ ∂r ∂θ r sin θ r ∂θ



1 ∂Q 1 ∂2 A 1 1 1 ∂Q ∂2 A 1 + rˆ + − = θˆ r sin θ r ∂θ r sin θ ∂φ∂r r r 2 sin2 θ ∂φ∂θ sin θ ∂r   1 ∂2 A 1 ∂A 1 ∂ 1 ˆ + φ. − r sin θ ∂r 2 r 2 ∂θ sin θ ∂θ (4.65) From the relationship of force-free field, α ∇ × B = αB = r sin θ then



1 ∂A ∂A ˆ ˆ rˆ − θ + Q(A)φ , r ∂θ ∂r



1 ∂Q 1 ∂2 A 1 1 ∂A 1 + =α r sin θ r ∂θ r sin θ ∂φ∂r r sin θ r ∂θ

2 ∂ A 1 1 ∂Q 1 ∂A 1 − = −α r r 2 sin2 θ ∂φ∂θ sin θ ∂r r sin θ ∂r   1 ∂2 A 1 ∂A 1 1 1 ∂ =α Q. − + r sin θ ∂r 2 r 2 ∂θ sin θ ∂θ r sin θ

(4.66)

(4.67)

236

4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

We can find

∂2 A 1 ∂Q 1 1 ∂A + =α r ∂θ r sin θ ∂φ∂r r ∂θ 2 ∂ A ∂Q ∂A 1 − = −α r 2 sin θ ∂φ∂θ ∂r ∂r   2 2 1 ∂A ∂ A sin θ ∂ = −α Q. + 2 ∂r 2 r sin θ ∂θ sin θ ∂θ

(4.68)

1 1 1 ∂A 1 ∂Q ∂2 A − =α 2 r ∂μ r 1 − μ ∂φ∂r r ∂μ 1 ∂2 A ∂Q ∂A + =α 2 r ∂φ∂μ ∂r ∂r 2 2 2 ∂ A 1−μ ∂ A + = −α Q. ∂r 2 r 2 ∂μ2

(4.69)

∂ 2 A 1 − μ2 ∂ 2 A + + α Q = 0, ∂r 2 r 2 ∂μ2

(4.70)

1 1 1 ∂2 A ∂Q ∂2 A 1 ∂Q − + + r ∂μ r 1 − μ2 ∂φ∂r r 2 ∂φ∂μ ∂r α= . 1 ∂A ∂A + r ∂μ ∂r

(4.71)

set μ = cos θ then

We can obtain

where

 End proof.

4.3.2 Calculations for Different Methods 4.3.2.1

Metrics Comparisons

Like Schrijver et al. (2006), Amari et al. (2006), and Valori et al. (2007), Liu et al. (2011, 2012) also calculated some metrics that have been employed for checking the performance of the extrapolation field. In the following, we will introduce them one by one. Cvec is used to quantify the vector correlation, which is defined as Cvec =

 i

Bi · bi /(

 i

|Bi |2

 i

|bi |2 )1/2 ,

(4.72)

4.3 Study on Two Methods for Nonlinear Force-Free Extrapolation …

237

where Bi and bi are the field vector of the semi-analytical field and the extrapolated field at the grid point i, respectively. If the vector fields are identical, then Cvec ≡ 1; if Bi ⊥ bi , then Cvec ≡ 0. Ccs is based on the Cauchy–Schwarz inequality and mostly used to measure the differences of the vector fields: Ccs =

1  1  Bi · bi = cos θi , M i |Bi ||bi | M i

(4.73)

where M is the total number of vectors in the volume to be calculated, and θi is the angle between Bi and bi . Ccs = 1 indicates Bi and bi are parallel; and contrarily, Ccs = −1 indicates Bi and bi are anti-parallel; Ccs = 0 indicates Bi and bi are mutually perpendicular. E n is a normalized vector error: En =



|bi − Bi |/



|Bi |.

(4.74)

1  |bi − Bi |/|Bi |, M i

(4.75)

i

i

A mean of the above-normalized vector error: Em =

when E m = E n = 0, the agreement is perfect, which is different from the first two metrics. However, in this work, we will use E  m(n) = 1 − E m(n) instead of E m(n) for the comparisons. The last one measures the degree of the energy of extrapolated field normalized to that of the semi-analytical field:  |bi |2 .  = i 2 i |Bi |

(4.76)

For these metrics, if Bi and bi are identical, Cvec , Ccs , , E  n and E  m should be equal to unity. E  n and E  m are based on the differences between the semi-analytical field vectors and the extrapolated field vectors. They thus include the information on the agreement of two vectors both in direction and magnitude. While Cvec and Ccs are relatively more influenced by the directional differences between the semianalytical field vectors and the extrapolated field vectors (Schrijver et al., 2006). Moreover, Cvec and Ccs are less sensitive to the errors of the extrapolated field than    the normalized vector error E n , and the mean vector error E m , especially E n is a sensitive and reliable indicator of extrapolation accuracy (Valori et al., 2007). Results of these metrics for two semi-analytical fields (SAF1 and SAF2) are shown in Table 4.1. It also shows the results of the extrapolated fields of BIE, original AVI, and improved AVI methods. We find the consistencies between the semi-analytical fields and the corresponding extrapolated fields of the improved AVI method are

238

4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions 



Table 4.1 The metrics of Cvec , Ccs , E n , E m and  for the extrapolated fields of BIE, original AVI and improved AVI methods in the calculated box (64 × 64 × 64)   Cvec Ccs En Em  Low & Lou (SAF1) BIE AVI (Improved) AVI (Original) Low & Lou (SAF2) BIE AVI (Improved) AVI (Original)

1.00 0.978 0.983 0.956 1.00 0.959 0.958 0.939

1.00 0.956 0.979 0.969 1.00 0.873 0.864 0.858

1.00 0.770 0.803 0.661 1.00 0.658 0.651 0.402

1.00 0.721 0.722 0.189 1.00 0.567 0.403 0.214



1.00 0.990 0.943 0.825 1.00 0.978 0.728 0.678



better than those of the original AVI method. Especially for E n and E m , they display that the singular point globally affects the accuracy of extrapolation due to only the  finite points used in the improved AVI method. The evident improvements on E n  and E m also indicate that they are sensitive to the errors of extrapolated fields. For each method, the consistencies between SAF1 and the corresponding extrapolated fields are better than those between SAF2 and the corresponding extrapolated fields. These results are consistent with those obtained by other authors (Schrijver et al., 2006; Amari et al., 2006; Valori et al., 2007). Just as Amari et al. (2006) pointed out that SAF2 may be considered as a theoretical challenge to push the methods to their limits. In addition, we can find the order of amplitude of these metrics are comparable to those of other methods (e.g., Schrijver et al., 2006; Amari et al., 2006; Valori et al., 2007). Note that for our extrapolations, only the bottom boundary of the semianalytical fields are used, and these metrics only show their overall performance of the extrapolation methods.

4.3.2.2

SAF1

The correlations of Bxs, Bys, Bzs, and azimuths (φ) between SAF1 and the extrapolated fields of the BIE method, the original AVI and improved AVI methods are shown in Fig. 4.1, where the solid, dotted, and dashed lines indicate the BIE method, the original AVI, and improved AVI methods, respectively. x- and y-axis present the extrapolated height and the correlation coefficient, respectively. For the BIE method, the correlation coefficients of Bx, By and φ are greater than 95%, and those of Bz are greater than 98% below z = 10. This suggests the results of the BIE extrapolated

0.95

Correlation coefficent

1.00

Bx

0.90 0.85 0

2

4

Z

6

8

10

1.00 0.95

Correlation coefficent

Correlation coefficent

Correlation coefficent

4.3 Study on Two Methods for Nonlinear Force-Free Extrapolation …

Bz

0.90 0.85 0

2

4

Z

6

8

10

239

1.00 0.95

By

0.90 0.85 0

2

4

2

4

Z

6

8

10

6

8

10

1.00 0.95 0.90 0.85 0

Z

Fig. 4.1 Correlations of Bxs, Bys, Bzs, and azimuths (φ) between SAF1 and the extrapolated fields (of the BIE, original, and improved AVI methods) at different heights. From Liu et al. (2011)

field are reliable for z < 10. Relative to the original AVI method, it can be seen that there are evident increases in the correlation coefficients between SAF1 and the extrapolated field of the improved AVI method. For example, there is an increase of 4% for the correlation coefficient of Bx at z = 10. Furthermore, it can be found that for the improved AVI method, the correlation coefficients of By and Bz between SAF1 and the extrapolated field are greater than 95% below z = 10; while the correlation coefficients of Bx and φ are less than 95% below z = 10. It may be reliable for the extrapolated field of the improved AVI when the height is below z < 10 for the fact that all its correlation coefficients are greater than 90% within this height range. The distributions of the selected magnetic field lines for SAF1 and the extrapolated fields of each method are shown in Fig. 4.2, where the red and blue lines are for the closed and open magnetic field lines, respectively. It can be seen that the extrapolated field line distributions basically coincide with those of SAF1 at the lower height, but the differences become evident as the height increases, especially for the open magnetic field lines of the AVI method.

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4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

Fig. 4.2 Distributions of the selected magnetic field lines for SAF1 and the corresponding extrapolated fields of the BIE, original, and improved AVI methods. From Liu et al. (2011)

4.3.2.3

SAF2

In Fig. 4.3, the distributions of the selected magnetic field lines for SAF2 and the extrapolated field of each method are all drawn and their disagreements are evident. For example, the open field lines of three extrapolated fields all can reach higher relative to those of SAF2. The differences of the closed field lines between SAF2 and the extrapolated fields, either the BIE method or the AVI method, become more evident as the height rises. Note that we cannot see more improvements in the field lines of the improved AVI method.

4.3.3 Comparisons of Different Methods In the AVI method, as the differential technique is used to extrapolate the magnetic field, the singular points cannot be removed completely. In the study of Liu et al. (2012), some small-scale smoothing has been used to solve the problem and get better results than those obtained by Song et al. (2006). It is found that the reliable results are only limited at the lower height and new improvements are still needed for this method. On the other hand, the benefit of this method is time-saving in carrying out the extrapolation.

4.3 Study on Two Methods for Nonlinear Force-Free Extrapolation …

241

Fig. 4.3 Distributions of the selected magnetic field lines of SAF2 and the corresponding extrapolated fields of different methods. From Liu et al. (2011)

In the BIE method, it uses the Green function to construct Helmholtz equations, then the integral is applied to solve the Helmholtz equations and extrapolate the magnetic field. So the problem of singular points in differential equations can be avoided. While the key problem for the BIE method is whether a reasonable λ can be found or not, which may strongly affect the extrapolated results. For example, if we can improve our computer power, better results could be obtained by modifying the iteration and computational accuracy. From the metrics listed in Table 4.1 and the correlation analyses to each magnetic field component, it can be found that the improved AVI method is better than the original AVI method evidently. It can also be found that poor results are obtained for the Bx extrapolation in comparison with SAF1 and for the By extrapolation in comparison with SAF2. It may be that the strong current density near or in the weak field regions has a strong influence on the solutions of nonlinear force-free extrapolation. Through comparisons, it is found that there are evident differences between the semi-analytical field and the extrapolated fields. But for the lower height, the two extrapolation methods can give reliable results. Last, it should be noted that using the semi-analytical field of Low and Lou (1990) to test the validity of the extrapolation method may have theoretical disadvantages as only the finite bottom boundary data are used in the extrapolations, but the semi-analytical field presents the global magnetic configurations.

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4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

4.4 Extrapolated Magnetic Fields with Observations Figure 4.4 shows an example of flare-producing active region NOAA 6619 on 1991 May 10. A series of flares occurred in this region. It is found that the extrapolated magnetic lines of force connect the opposite polarities under the approximation of the nonlinear force-free field, which is based on the optimization method by Wheatland et al. (2000) and Wiegelmann (2004). In the observational photospheric vector magnetogram, the highly sheared transverse component is formed along the magnetic neutral line between the opposite magnetic polarities marked by N and S in Fig. 4.4. This provides some basic information on the spatial configuration of the magnetic field in the active region, even if some ambiguity on the topology of extrapolated magnetic lines of force with observations in detail can be found. As comparing the relationship between the Hβ magnetogram and photospheric transverse field, it is found a tendency on the basic consistency between the distribution of chromospheric magnetic fibrils and direction of observational photospheric transverse field resolved 180◦ -ambiguity near the center of the solar disk. The reversal structure in the umbra N in the chromospheric magnetograms probably is caused by the disturbance of photospheric blended lines in the wing of the Hβ line (Zhang, 1996a). We need to point out that the general formation height of the Hβ the line is less than 2000 km from the photosphere, while its formation height in the umbrae is much lower than that in the quit Sun as indicated in Sect. Hbumb. As the size of our analyzed reversal magnetic configuration is larger than the height difference between the photosphere and chromosphere, the conservation of the magnetic flux in both layers should be considered (Zhang, 1995b):  B · ds = 0,

(4.77)

S

where B is the magnetic field strength and S is the area passed by the magnetic flux in the photosphere and chromosphere. If this deformed magnetic configuration is relatively stable, it should also be consistent with the static equilibrium condition of the magnetohydrodynamics. On the other hand, it is difficult to guess that the chromospheric reversal magnetic structures form above the sunspot umbrae because the umbral magnetic field normally is vertical to the solar surface. Thus, if the chromospheric reversal magnetic structures exist really, the size of these kinds of twisted magnetic lines of force would be limited and can be reflected in the photospheric vector magnetograms. As pointed by Alfvén and Falthammer (1963), the condition of the energy decrement for the twisted magnetic rope is 

a 0

 Bφ2 r dr > 2

a 0

Bz2 r dr,

(4.78)

4.4 Extrapolated Magnetic Fields with Observations

243

04:05UT May 10 1991 04:46 UT

N S

Fig. 4.4 Active region NOAA 6619 on 1991 May 10 overlaid by the extrapolated magnetic lines of force (top). The light (heavy) contours mark the positive (negative) polarities of the photospheric magnetic field. The red (blue) lines mark the close (open) magnetic lines of force. The Hβ chromospheric longitudinal magnetic field and photospheric transverse magnetic field (bottom). The white (black) marks the positive (negative) polarity

where Bz is the axial field, Bφ is toroidal one and a is the radius of the magnetic rope. This means that as the density of magnetic energy of the toroidal field is larger than twice of the axial one, the magnetic rope becomes unstable. Song et al. (2007) present a spherical nonlinear force-free field (NLFFF) reconstructing method based on the photospheric vector magnetograms. The importance of this method is its ability to reveal the NLFFF configurations, which is necessary for understanding the physical mechanisms of the initiation of the large-scale solar eruptions, such as coronal mass ejections and sympathetic flares. Using smooth continuous functions, the basic NLFFF-governing partial differential equations in

244

4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

Fig. 4.5 Left: Height variations of the NFFF configurations computed by use of the vector magnetograms of the active regions NOAA 10486 and NOAA 10488 on 2003 October 29. Br is shown by contours, while Bθ /B1 and Bφ /B1 by vectors. a r = R0 + 0.10, b r = R0 + 4.0, and c r = R0 + 8.0, where z = r − R0 . Right: Configurations of the NFFF lines computed by use of the vector magnetograms of NOAA 10486 and 10488. Top: θ1 = 30◦ , φ1 = 225◦ , and l1 = 60, 000. Bottom: θ1 = 0◦ , φ1 = 270◦ , and l1 = 60, 000. The units for θ and φ are degrees. After Song et al. (2007)

spherical coordinates reduce to a set of tractable ordinary differential equations. The numerical scheme used there is similar to the recent local nonlinear force-free one developed by Song and coworkers (Song & Zhang, 2004, 2005, 2006). To illustrate this method, they give some test examples. One is to compute a well-known NLFFF analytical solution given by Low and Lou (1990). The other is for two active regions NOAA 10486 and NOAA 10488 observed on 2003 October 29 in Fig. 4.5, where θ is the solar latitude and φ is the usual central meridian longitude. The results show that the transequatorial magnetic loops are revealed and coincided with some EUV Imaging Telescope loops. Figure 4.6 shows the spatial configuration of the magnetic field above the photosphere inferred from the photospheric vector magnetogram at 02:50UT on 2000 July 13 by the method of upward integration of the nonlinear force-free field (Song et al., 2006). It is found that the magnetic lines of force twisted above the photosphere and the direction of lines of force in the lower layer of the atmosphere are almost consistent with the tendency of the photospheric transverse components of the field in Fig. 4.6. The helical configuration of the magnetic field shows left-handedness. It is

4.4 Extrapolated Magnetic Fields with Observations

245

Fig. 4.6 The extrapolation of magnetic lines of force in active region NOAA 9077 inferred from the photospheric vector magnetogram at 02:50UT on 2000 July 13 in the top (left) and side (right) of view. From Zhang (2008)

consistent with the above-analyzed results of the magnetic helicity from photospheric vector magnetograms and the developing process of helical configuration of soft-Xray features above the photosphere in the active region 9077. It can be inferred that the helical spatial magnetic configuration formed above the photosphere was contributed by the development of magnetic shear near the photospheric magnetic neutral line between N 1 and S1 before the powerful “Bastille day” flare-CME on 2000 July 14 in Figs. 4.6 and also 4.7, even if the magnetic shear of the photospheric magnetic field and the photospheric mean current helicity density decayed before the flare-CME in the active region (Deng et al., 2001; Liu & Zhang, 2001, 2002; Zhang, 2002). It means that the magnetic field did not connect the poles N 1 and S1 of opposite polarities immediately, as they closed together due to the shearing motion between them with the emergence of large-scale magnetic flux in the development of the powerful flare-producing region. It provides the transfer form of the non-potential magnetic field from the subatmosphere into the corona. The consistency between the potentiallike spatial configuration of the magnetic field and the 171Å flare post loops after the trigger of “Bastille day” flare-CME observed by TRACE has been demonstrated by Zhang (2002). It can be inferred that the amount of magnetic helicity transported into the interplanetary space with the flare-CME. As pointed by Solanki et al. (2003) in the study of the spatial configuration of solar active regions, the current sheets located in the corona or upper chromosphere have long been thought to act as an important source of coronal heating, requiring their location in the corona or upper chromosphere. The observation of chromospheric magnetic fields just provides information on the distribution of complex magnetic fields above the photosphere. Zhang et al. (2006), Zhang and Flyer (2008) pointed out that the loss of magnetic field is the cause of coronal mass ejection (CME). The magnetic energy storage mechanism is crucial for understanding how the magnetic field possesses enough free energy to overcome the Aly limit and turn it on (Aly, 1984). The accumulation of magnetic helicity in the corona plays an important role in storing magnetic energy. They put forward a hypothesis that there is an upper bound on the total

246

4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

magnetic helicity that the forceless field can contain. This is directly related to the hydromagnetic property that the forceless field in unbounded space must be selfconfining. They provide mathematical evidence and numerical values that suggest that an upper bound on magnetic field helicity may exist. When the magnetic helicity accumulates beyond this upper limit, it will induce a non-equilibrium state, leading to coronal mass ejections, the natural product of coronal evolution. Monotonic accumulation of magnetic helicity can lead to the formation of magnetic flux ropes suitable for kink instability. This means that coronal mass ejections initiated by exceeding the helicity limit and kink instability may both be the result of the accumulation of coronal helicity. This provides insights into the observed associations of CMEs with the magnetic features at their solar surface origins.

4.5 Eruptions of Flare-Coronal Mass Ejections 4.5.1 Flares with Filament Eruption Many researchers studied the electric field in the reconnecting current sheet observationally (Qiu et al., 2002; Wang et al., 2003; Jing et al., 2005. It is generally believed that the flare brightenings in the chromosphere are due to the progressive magnetic reconnection in the corona. Wang et al. (2003) pointed out that the sweeping motion of the magnetic footpoint physically corresponds to the rate of the magnetic flux convected into the diffusion region at the reconnection point in the corona, where the reconnecting current sheet is generated. So there is quite possibly some relationship between E in the photosphere and electric field in the reconnecting current sheet. Figure 4.7 shows the active region NOAA 9077 produced one of the largest solar flares (3B/X5.7) seen On July 14, 2000, associated with a violent halo coronal mass ejection (CME). It had a βγ δ magnetic classification and the morphology of the sunspot group and magnetic field changed every day. We present here the relationship between the large-scale motions of the spots and the major flare on July 14, based on precise measurements of the proper motions. It is found that: (1) The special magnetic morphology and quick, successive fragmentation caused the active region to be always in a high shear configuration; (2) There is a good spatial correspondence between the direction of the movement of one spot group and the place where the lament was cut off and activated; (3) The motion characteristics of the rapidly emerging flux system showed a good correlation between spot motions and the largest flares, suggesting that the initiation of the two-ribbon flare on July 14 was promoted by the successive emergence of the flux systems. The intensive major flare was always connected strongly with a newly emerging magnetic flux system. This confirms that δ-configurations and dynamical processes are important in large flares. Figure 4.8 shows a typical coronal mass ejection (CME) observed on 2003 February 18, by various space and ground instruments, in white light, Hα, EUV, and X-ray (Bao et al., 2006). The Hα and EUV images indicate that the CME started with the

4.5 Eruptions of Flare-Coronal Mass Ejections

247

Fig. 4.7 Top left: A time sequence of white-light observations of AR 9077 from TRACE. Bottom left: Proper motions of the spots. Right: A series of Hβ (left column) and TRACE images (right column, with 195Å at 10:12:33, others are in 171Å), obtained almost at the same time, which show the quick changes near the point “D” before and during the major flare took place on July 14. North is up and east is to the left. From Liu and Zhang (2001)

eruption of a long filament located near the solar northwest limb. The white light coronal images show that the CME initiated with the rarefaction of a region above the solar limb and followed by the formation of a bright arcade at the boundary of the rarefying region at height 0.46R above the solar surface. The rarefying process synchronized with the slow rising phase of the eruptive filament, and the CME leading edge was observed to form as the latter started to accelerate. The lower part of the filament brightened in Hα as the filament rose to a certain height and parts of the filament were visible in the GOES X-ray images during the rise. These brightenings imply that the filament may be heated by the magnetic reconnection below the filament in the early stage of the eruption. It is estimated by Bao et al. (2006) that a possible mechanism that leads to the formation of the CME leading edge and cavity is the magnetic reconnection which takes place below the filament after the filament has reached a certain height.

248

4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

Fig. 4.8 Evolution of the 2003 February 18 event observed in various wavelengths. Top row panels: Composite of the Hα images of the limb and the disk. The white arrows indicate the emission features appearing in the lower part of the filament. Middle row panels: EIT 195Å images, the white arrows indicate the flare ribbons. Bottom row panels: GOES Soft X-ray Imager (SXI) images, the white arrows indicate the brightenings in the top of the eruptive filament at 02:01 UT and flare ribbons at 02:09 UT and 02:31 UT, respectively. From Bao et al. (2006)

The mechanism of magnetic reconnection is an important topic in the study of solar eruptions. We will give a brief introduction in Sect. 4.6.

4.5.2 Ribbon Separation of Eruptive Flares with Possible Electric Fields Wang et al. (2003) presented a detailed study of a two-ribbon flare in the plage region observed by Kanzelhohe Solar Observatory (KSO), which is one of the stations in our global Hα network. They select this event due to its very clear filament eruption, two-ribbon separation, and association with a fast coronal mass ejection (CME). Figure 4.9 is the time sequence of KSO Hα images showing the evolution of the event. It is a classical two-ribbon flare accompanied by the filament eruption. It is found that the separation motion consisted of a fast stage of rapid motion at a speed of about 15 km s−1 in the first 20 min and a slow stage with a separation speed of about 1 km s−1 lasting for 2 h. Wang et al. (2003) then estimate the rate of the magnetic reconnection in the corona, as represented by the electric fields E c in the reconnecting current sheet, by measuring the ribbon motion speed and the magnetic fields obtained from MDI. As seen in Fig. 4.10a, at this time, the displacement between the filament and CME is about 1.4-2 solar radii, taking into account the projection effects. This gives roughly

4.5 Eruptions of Flare-Coronal Mass Ejections

249

Fig. 4.9 Sequence of Hα images showing the evolution of the flare and the disappearance of filament. The field of view is 512 × 512 . From Wang et al. (2003)

Fig. 4.10 Left: Time profiles of the flare ribbon separation, filament heights measured from KSO and EIT images, and CME heights measured by S. Yashiro. Lines: Least-squares fits hyperbolic functions. Right: Velocity profiles of the ribbon separation, filament, and CME derived from the fits of the height profiles. From Wang et al. (2003)

the scale of the expanding system at that time. As estimated from the velocity profiles in Fig. 4.10b, the average acceleration rate was 260 m s−2 . It is found that there were two stages as well in evolution of the electric fields: E c = 1 V cm−1 averaged over 20 min in the early stage, followed by E c = 0.1 V cm−1 in the subsequent 2 h. The two stages of the ribbon motion and electric fields coincide with the impulsive and decaying phases of the flare, respectively, yielding clear evidence that the impulsive flare energy release is governed by the fast magnetic reconnection in the corona.

250

4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

Wang et al. (2003) also measured the projected heights of the erupting filament from KSO Hα and SOHO/EIT images. The filament started to rise 20 min before the flare. After the flare onset, it was accelerated quickly at a rate of 300 m s−1 , and in 20 min, reached a speed of at least 540 km s−1 , when it disappeared beyond the limb in the EIT observations. The acceleration rate of the CME is estimated to be 58 m s−2 during the decay phase of the flare. The comparison of the height and velocity profiles between the filament and CME suggests that fast acceleration of mass ejections occurred during the impulsive phase of the flare, when the magnetic reconnection rate was also large, with E c = 1 V cm−1 . As the rate of magnetic reconnection can be estimated by the speed of the ribbon separation and longitudinal magnetic field, Xie et al. (2009) found similar time profiles between the speeds of the ribbons and the rates of the magnetic reconnection, and confirm the results of Wang et al. (2003), providing evidence that the rate of magnetic reconnection in the corona determines the speed of flare ribbon separation. They also find that the separation speed is not as high as expected in the strong magnetic field. This demonstrates that the expansion of the flare ribbons is restrained by the strong magnetic field. The temporal evolution of Er ec has multiple peaks, likely indicating multiple bursts of magnetic reconnection during the flare. Along with the flare ribbons, Er ec fluctuates in a small range except near the HXR source. The localized enhancement of the reconnection rate corresponds to the position of the HXR source during the flare peak, but this is not the case before and after the peak times. The spatial distributions of Er ec are inhomogeneous along with the ribbon, and strong Er ec is found where the HXR emission is enhanced. This is consistent with the results of Temmer et al. (2007) and also Lin et al. (2005). These variations in Er ec are probably due to the three-dimensional structure of energy release. Jing et al. (2007) demonstrate the physics of nonuniformity at different stages of the flare. They concluded that it is due to the process of sigmoid-to-arcade transformation in the framework of the tether-cutting model (Moore et al., 2001). Fig. 4.11 displays the rapid development of the eruption during a period of 2 min by Liu et al. (2003). Hot plasma emissions were seen rising up over the magnetic neutral line at a high speed. The white arrows in the difference images (left and middle panels) show the sites of the frontier of the erupting plasma. The area around p was the initial site for the eruption. The right panels give velocity measurements from MDI. It should be noted that at the initiation of the eruption, there was no obvious brightening enhancement at p (some places nearby even got dimmer); therefore, there is no direct evidence of driving the eruption from below. The white circles and ellipses outline two new flare ribbons that followed the plasma eruption at 20:04:29 UT. In the hump of the curvy erupting frontier, several strands of flaring plasma were moving upward in an upper right direction. An obvious twisted fine structure in the plasma is exhibited by the black arrows in this figure. It is interesting that the velocity map at 20:04 UT clearly illustrates the first flare with its ribbons (marked 1 and 2), while the one at 20:05 UT reflects an instance of the second flare with its compact ribbons (marked 3 and 4).

4.5 Eruptions of Flare-Coronal Mass Ejections TRACE/1600

251 MDI/velocity

erupting plasma

p

p

1 20:03:48-20:03:10

p

2 20:04:03-20:03:25

p

3

p

20:04:00

p

4

20:04:29-20:03:51

20:05:17-20:04:40

20:05:00

Fig. 4.11 Development of the magnetic flux rope in 2 min. Left and middle panels: Running difference images. Right panels: MDI velocity maps. The flare ribbons are marked with numbers 1–4. The regions in circles and ellipses demonstrate the evolution of the driven flare. The white arrows point to the expanding frontier of the plasma, and the black arrows to the twisted structures

4.5.3 Magnetic Properties of Flare-CME Productive Active Regions Zhang et al. (2000b) examined the observational soft X-ray flares and the relationship with photospheric vector magnetograms in the active region (NOAA 7070). They analyzed the soft X-ray flare on Feb. 24–25, 1992, especially the pre-flare and the relationship with the highly sheared photospheric vector magnetic field near the photospheric magnetic neutral line. They found that the initial reconnection of the magnetic field in the flare on Feb. 24–25, 1992 probably occurs near the magnetic neutral line in the lower atmosphere of the active region, where the highly sheared magnetic flux erupts up and triggers the reconnection of the large-scale magnetic field. The possible process of the magnetic reconnection of the limb flare on Feb. 20–21, 1992 in this active region is proposed also based on the analogy with the flare on Feb. 24–25 near the center of the solar disk in Figs. 4.12 and 4.13.

252

4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

Fig. 4.12 A time sequence of the soft X-ray flare on 20-21 Feb. 1992 (left). The reconnection process of the magnetic lines of force in the period of the soft X-ray flare is inferred by homologous flares in the active region in February 1992 (right). The arrow lines show the magnetic lines of force above the photosphere in the active region. The short arrows indicate the moving direction of magnetic lines of force. a and b mark the topological correspondence of magnetic lines of force. From Zhang et al. (2000b)

Fig. 4.13 The longitudinal magnetogram on 21 Feb. 1992 (left). The photospheric longitudinal magnetogram of 25 Feb. 1992 is rotated to the east limb of the sun and the magnetic field lines are extrapolated under the approximation of the potential magnetic field (right). The north is top and the east is at left. From Zhang et al. (2000b)

4.5.4 Magnetic Helicity Budget of Solar Active Regions and Coronal Mass Ejections Nindos et al. (2003) computed the magnetic helicity injected by transient photospheric horizontal flows in six solar active regions associated with halo coronal mass ejections (CMEs) that produced major geomagnetic storms and magnetic clouds

4.5 Eruptions of Flare-Coronal Mass Ejections

253

(MCs) at 1 AU. The velocities are computed using the local correlation tracking (LCT) method. Their computations cover time intervals of 110–150 h, and in four active regions, the accumulated helicities due to transient flows are factors of 8–12 larger than the accumulated helicities due to differential rotation. As was first pointed out by Demoulin and Berger (2003), we suggest that the helicity computed with the LCT method yields not only the helicity injected from shearing motions but also the helicity coming from flux emergence. They compare the computed helicities injected into the corona with the helicities carried away by the CMEs using the MC helicity computations as proxies to the CME helicities. If we assume that the length of the MC flux tubes is l = 2 AU, then the total helicities injected into the corona are a factor of 2.9–4 lower than the total CME helicities. If we use the values of l determined by the condition for the initiation of the kink instability in the coronal flux rope or l = 0.5 AU then the total CME helicities and the total helicities injected into the corona are broadly consistent. Their study, at least partially, clears up some of the discrepancies in the helicity budget of active regions because the discrepancies appearing are much smaller than the ones reported in previous studies. However, they point out the uncertainties in the MC/CME helicity calculations and also the limitations of the LCT method, which underestimates the computed helicities. These results are for initial configurations that are cylindrically symmetric, and therefore the application of these results to active region fields is far from being straightforward. Here we adopt a critical value of about nk ≈ 2 and the relation Hk ≈ 2F 2

(4.79)

have been used. In Table 4.2 we also give the accumulated helicity HLC T computed with the LCT method (other than differential rotation), the number of CMEs associated with the active region, and the total helicity carried away by the CMEs, by using the three different values of l. The CME helicities derived with l = 2 AU are a factor of 2.9–10.1 higher than the total helicity accumulated by both the computed horizontal motions and differential rotation. The larger helicity deficit is associated with AR 9201; we remind the reader, however, that in the MC associated with AR 9201 the best fit B0 and R have been judged to be poor. For the remaining active regions, the CME helicities are a factor of 2.9–4 higher than the total helicity accumulated by both the computed transient motions and differential rotation. The CME helicities determined with l derived from equation (4.79) and l = 0.5 AU are broadly consistent with the sums of HLC T + Hr ot (again except AR 9201). This picture shows better in columns (9)–(11) in Table 4.2, in which we give the helicity deficit between the CMEs and all computed motions.

1178

−1697

276

133

+

+

−

+

−

−

(1)

8210..........................

8375..........................

9114,9115,9122d ........

9182..........................

9201..........................

9212,9113,9118d ........

−477

(×1040 Mx2 )

−666

−428

34

−257

334

324

(4)

Hrot

(×1040 Mx2 )

3

1

4

3

9

9

(5)

CMEs

−3582

−2980

1176

−5712

4428

16596

(6)

l = 2AU

tot b (×1040 Mx2 ) HCME

l = lk

−1245

−1922

404

−2052

1296

9270

(7)

l = 0.5AU

−8959

−745

294

−1428

1107

4149

(8)

68

90

74

66

65

75

(9)

l = 2AU

8

84

23

5

−17

56

(10)

l = lk

−28

60

−5

−37

−36

1

(11)

l = 0.5AU

tot − H tot c (HCME LC T − Hr ot )/HCME (%)

b

Active region’s chirality Cols. (6)–(8) refer to the total helicity ejected byx the CMEs derived using l = 2 AU, Eq. (4.79), and l = 0.5 AU, respectively, for the MC helicity computation c In cols. (9)–(11) the total helicity ejected by the CMEs has been derived using l = 2 AU, Eq. (4.79), and l = 0.5 AU, respectively, for the MC helicity computation d The values of H LCT and Hrot refer to the whole complex of active regions

a

3784

(2)

NOAA active region

(3)

αARa

HLCT

Table 4.2 From Nindos et al. (2003)

Helicity budgets

254 4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

4.5 Eruptions of Flare-Coronal Mass Ejections

255

4.5.5 Statistical Analysis Between Magnetic Field and Coronal Mass Ejections The properties of the magnetic field of flare-coronal-mass-ejection (flare-CME) productive active regions and their statistic correlations with CME speed have been presented by Guo et al. (2007), used a sample of 86 flare-CMEs in 55 solar active regions. Four measures, including the tilt angle (T ilt), total flux (Ft), length of the strong-field and strong-gradient main neutral line (Lsg), and effective distance (d E ), are used to quantify the properties of the magnetic field of flare-CME productive active regions.

4.5.5.1

Definition of Nonpotential Magnetic Parameters

(1) Ft, the total flux of an active region, is a quantitative measure of an active region’s size, which is well correlated with its overall productivity for energetic events (Giovanelii, 1939; McIntosh, 1990; Canfield et al., 1999; Tian et al., 2002). (2) T ilt is defined as the angle between the direction of the polarity axis of an active region and the local latitude, which could be computed with the equation tan(T ilt)=δy/δx, where δx and δy are Cartesian coordinate differences between the leading and following polarities of an active region in the heliographic plane. This definition of tilt is similar to the one used by Tian et al. (2002). The T ilt of solar active regions relates to the writhe (spatial deformation or turning of the axis) of the flux tube, which is one of the measurable parameters of solar active regions that give us information about subsurface physical processes associated with the creation and subsequent evolution of magnetic flux tubes inside the Sun (Holder et al., 2004). (3) d E , effective distance, is a structural parameter proposed by Chumak and Chumak (1987). This parameter is defined as (d E = (Rn + Rs)/Rsn), where Rs = (N s/π )−1/2 , Rn = (N n/π )−1/2 , N s(N n) is total area of the negative (positive) polarity, Rs(Rn) is the equivalent radius of negative (positive) polarity and Rsn is the distance between the flux-weighted centers of opposite polarities. (4) Lsg, the length of the portion of the main neutral line on which the potential transverse field is strong (>150 G) and the horizontal gradient of the longitudinal magnetic field is sufficiently steep (>50 G/Mm) (Falconer et al., 2003). The neutral line separates opposite polarities of the longitudinal magnetic field. High gradient and strong shear usually appear in the vicinities of neutral lines, where flares frequently occur (Zirin & Liggett, 1987; Zirin, 1988; Hagyard & Rabin, 1986; Hagyard, 1988; Zhang et al., 1994; Zhang, 2001b). As an example, Fig. 4.14 shows a typical data set that have been compiled for a CME-productive active region NOAA 10720. According to the SEC Solar Event Weekly Report, this region produced an X3.8 flare on Jan. 17, 2005. The flare started at 06:59, reached a maximum at 09:52, and ended at 10:07. Associated with this flare, a complex NW-directed full-halo CME at a speed of 2547 Km/s appeared at C2 at about 09:54 (the top left panel in Fig. 4.14), so one chooses longitudinal magnetograms of NOAA 10720 at 06:24, about 35 min before the flare onset, to measure the 4 parameters. T ilt and d E of this magnetogram are sketched in the top

256

4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

Fig. 4.14 T op le f t: A complex NW-directed full-halo CME initiating in NOAA 10720 and obtained by LASCO/C2 at 09:54 on Jan 17, 2005. T op right: The longitudinal field with contour levels of ±100, 500, 1000 Gauss. The angle b-o-c is the tilt angle for this magnetogram, about 58.06◦ . Bottom le f t: A sketch of d E for this magnetogram (d E = 3.55). The circle with a center at “N” (“S”) represents the positive (negative) polarity. The line S-N is the distance between the fluxweighted centers of opposite polarities. The line with an arrow S-E (N-F) is the equivalent radius of negative (positive) polarity. Bottom right: The strong-field and strong-gradient main neutral line. Note the last three panels are for NOAA 10720 at 06:24 UT with a field of view of 338 × 242 . The spot marked with “S” (“N”) indicates the flux-weighted center of negative (positive) polarity in the T op right and Bottom le f t panels. From Guo et al. (2007)

right panel and bottom left panel in Fig. 4.14, respectively. Calculating both of them relates to flux-weighted centers of two polarities. “S” indicates the flux-weighted center of negative polarity and “N” indicates the flux-weighted center of positive polarity. For T ilt, b-o joins the flux-weighted centers “S” and “N”, c-o is a parallel line of the local latitude, and the angle b-o-c is the tilt angle for this magnetogram. It is about 58.06◦ . For d E , the cycle with a center at “S” and with the same area as the negative polarity represents the photosphere cross section of the flux rope of negative polarity, and the cycle with a center at “N” and with the same area as the positive polarity represents the photosphere cross section of the flux rope of positive polarity. The two simplified polarities are located so close to each other that they largely overlap each other. This geometric representation could be regarded as a sketch map of the δ configuration, which is defined as the umbra of opposite polarity lying in a common penumbra. By calculating the ratio of the sum of radii of two opposite polarities to the distance between the flux-weighted centers of the two opposite polarities, d E could roughly quantify the magnetic configuration of this active region. The bottom right panel of Fig. 4.14 shows the gradient on the main neutral line on which the potential transverse field is strong (> 150 G) and the gradient of the line-of-sight field is sufficiently steep (> 50 G/Mm). The parameter

4.5 Eruptions of Flare-Coronal Mass Ejections

257

Lsg is the length of the main neutral line. For this image, Lsg is 523 arcsec, which is much longer than the threshold of 50 arcsec.

4.5.5.2

Four Parameters Versus Speeds of CMEs (VC M E )

To present some correlations among Ft, d E , and Lsg for each relationship, Guo et al. (2007) estimate Pearson’s linear correlation coefficient (CC Pl ) and Spearman’s rank correlation coefficient (CC Sr ) with its two-sided significance (SSr ) for its deviation from zero. It is shown the speed of 86 CMEs versus the parameters of magnetograms of CMEassociated active regions in Fig. 4.15. The estimated correlation coefficients corresponding to the relationships are shown in Tables 4.3 and 4.4. It is found that, although the estimated Pearson’s linear correlation coefficients between d E and VC M E , Ft, and VC M E decrease a bit compared to the results of Guo et al. (2006), the correlation is still there: d E correlate better with VC M E than T ilt and Ft in Table 4.3. Furthermore, it should be noticed that the correlation coefficient of Ft vs. CME speed is affected less by the number of the sample. For the 86 events, It is examined the correlations between VC M E and two combined parameters, Ft×d E and Ft×Lsg (Fig. 4.16). From Table 4.4, it is found that any

Fig. 4.15 T op: VC M E versus T ilt for the 86 CMEs. The solid line denotes a least-square quadratic polynomial fit to the data points. T he second panel: Same as the top panel, but for VC M E versus Ft. T he thir d panel: Same as the top panel, but for VC M E versus Lsg. Bottom: Same as the top panel, but for VC M E versus d E . From Guo et al. (2007)

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4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

Table 4.3 VC M E and the parameters CC Pl CC Sr VC M E VC M E VC M E VC M E

versus T ilt versus Ft versus Lsg versus d E

−0.0406 0.3337 0.289 0.3888

−0.1256 0.3386 0.2370 0.3499

Table 4.4 VC M E and combined parameters CC Pl VC M E versus Ft×d E 0.3665 VC M E versus Ft×Lsg 0.2495

SSr

CC P

0.2491 0.0014 0.0280 9.6e-4

0.1290 (d E ) 0.1752 (Lsg) −0.0239 (Ft) 0.0325 (d E ) 0.2463 (Ft) 0.2730 (Lsg)

CC Sr

SSr

0.3815 0.2842

2.9e-4 0.0080

Fig. 4.16 T op: VC M E versus Ft×d E for the 86 CMEs. The solid line denotes a least-square quadratic polynomial fit to the data points. Bottom: Same as the top panel, but for VC M E versus Ft×Lsg. From Guo et al. (2007)

kind of correlation coefficient between CME speed and Ft×d E is better than it is between VC M E and Ft×Lsg. However, the correlation between VC M E and either of the two combined parameter is not as good as expected. After detailed analysis it is found it is due to the appearance of six events: five slow CMEs from NOAA 9393, a huge active region with a very strong magnetic field (large Ft), and a very complex magnetic configuration (large Lsg and large d E ), and one slow CME from NOAA 8674, a region with large Ft and large Lsg. Without the six events, the correlation quantities between VC M E and two combined parameters will be CC Pl = 0.6557, CC Sr = 0.5613 with SSr = 6.11e-8 for Ft×d E and CC Pl = 0.5520, CC Sr = 0.4565 with SSr = 2.08e-5 for Ft×Lsg. It should be noticed that Ft×d E correlates better with VC M E than Ft×Lsg either with or without the six special events. These indicate that active regions with large Ft and large d E tend to produce faster CMEs than do active regions with large Ft and large Lsg.

4.6 Formation of Magnetic Reconnection

259

4.6 Formation of Magnetic Reconnection Magnetic reconnection is the breaking and rejoining of magnetic field lines in a highly conducting plasma. It is also called magnetic field annihilation, and it is a very important process of rapid energy release in astrophysics. Reconnection converts magnetic energy into kinetic energy, thermal energy, and particle acceleration energy. Here we briefly introduce the basic theoretical framework of magnetic reconnection.

4.6.1 Magnetic Reconnection According to simple resistive magnetohydrodynamics (MHD) theory, reconnection happens because the plasma’s electrical resistivity near the boundary layer opposes the currents necessary to sustain the change in the magnetic field. The resistivity of the current layer allows magnetic flux from either side to diffuse through the current layer, canceling out flux from the other side of the boundary. In the ideal MHD, either these forces are opposite by a plasm pressure gradient maintaining equilibrium or plasma and field lines will move together until these forces are in balance. However, with the introduction of finite resistivity, no matter how small, the field is no longer frozen into the plasm and slippage of field lines of opposite polarity (Boyd & Sanderson, 2003). The Sweet–Parker model consists of a simple diffusion region of length 2L and with 2l, say, lying between opposite directed magnetic fields (Fig. 4.17), for which an order-of-magnitude analysis may be conducted as follows. For a steady state, the

Fig. 4.17 Sweet–Parker reconnection. The diffusion region is shaded. The plasma velocity is indicated by solid-headed arrows and the magnetic field lines by light-headed arrows. From Priest and Forbes (2000)

260

4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

plasma mast carries the field lines in at the same speed as they are trying to diffuse outward, as a magnetic field Bi enters the diffusion layer at a speed vi , so that vi =

η . l

(4.80)

This expression comes directly from Ohm’s Law (Ohm) as follows. Conservation of mass implies that the rate (4ρ Lvi ) at which mass is entering the sheet from both sides must equal the rate (4ρ Lvo ) (Priest & Forbes, 2000, P. 120). ηvo In dimensionless variables vi2 = may be rewritten L √ vo /v Ai Mi = √ , (4.81) Rmi where Mi =

vi v Ai

(4.82)

is the inflow Alfvén number (or dimensionless reconnection rate) and Rmi =

Lv Ai η

(4.83)

is the magnetic Reynolds number based on the inflow Alfvén speed. The outflow magnetic field strength (Bo ) is determined from flux conservation vi Bi = vo Bo .

(4.84)

Petschek (1964) realized that a slow-mode shock provides another way (as well as a diffusion region) of converting magnetic energy into heat and kinetic energy. He suggested that four such shocks would stand in the flow when a steady state is reached. Petschek suggested that the mechanism chokes itself off when Bi becomes too small, and so, by putting Bi = 21 Be , he estimated a maximum reconnection rate (Me∗ ) of π . (4.85) Me∗ ≈ 8 log Rme This lies in practice between 0.1 and 0.01, and so is much faster than Sweet–Parker. The inflow region in Petschek’s mechanism is a diffuse fast-mode expansion, in which the pressure and field strength decrease and the flow converge as the magnetic field is carried in. A fast-mode disturbance has plasma and magnetic pressure increasing or decreasing together, while a slow-mode disturbance has them changing in the opposite sense. An expansion makes the pressure decrease while a compression makes it increase, even in the incompressible limit.

4.6 Formation of Magnetic Reconnection

261

Fig. 4.18 a The different regimes of fast reconnection for different values of b. b Reconnection rate (Me = ve /v Ae ) as a function of Mi = VI /V AI for different b. From Priest and Forbes (2000)

Priest and Forbes (1986) wanted to explore different types of inflow and were puzzled at many strange features of numerical reconnection experimented. By evaluating the magnetic field at the entrance to the diffusion region in terms of Be , they then showed that 

Me Mi

2

 −1   4Me (1 − b) 4Rme m e 1/2 m i 1/2 ≈ . 0.834 − loge tan π π

(4.86)

Priest and Forbes (2000) pointed that the introduction of the new parameter b has a remarkable effect. It produces a whole range of different regimes: when b = 0, Petschek’s regime (a weak fast-mode expansion) is recovered; but when b = 1, the inflow field on the y-axis (the vertical direction in Fig. 4.17) is uniform, so we have a Sonnerup-like solution with a weak slow-mode expansion across the whole inflow region. In general, there is a continuum of solutions for other values of the parameter b, which can be determined by the nature of the flow on the inflow boundary. Figure 4.18a shows the relationship between the fast reconnection and the values of b, and the positions of Petschek and Sonnerup-like solution. The way that the reconnection rate (Me ) varies with Mi and b for a given Rme is shown in Fig. 4.18b. When b > 0 the reconnection rate is faster than the Petschek rate for the same Mi (although the analysis is only accurate for M  l). For b = 1, Me increases linearly with Mi , while for b = 0 the Petschek maximum can be seen. All regimes with b < 1 also possess a maximum reconnection rate, although when b < 0 it is slower than Petschek’s. When b > 1, there is no maximum rate, within the limitations of the theory.

262

4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

Fig. 4.19 Magnetic field lines (solid) and streamlines (dashed) in the upper half-plane (y > 0) for several different regimes of almost-uniform reconnection: a slow compression (b < 0), b Petschek (b = 0), c hybrid expansion (0 < b < 2/π ), (d) hybrid expansion (2/π < b < 1), e Sonnerup (b = 1), (f) flux pile-up (b > 1). From Priest and Forbes (2000)

The field lines and streamlines for these cases are shown in Fig. 4.19. When b < 0 the streamlines near the y-axis are converging and thus tend to compress the plasma, thereby producing a slow-mode compression. When b > 1 the streamlines diverge and so tend to expand the plasma, producing a slow-mode expansion. We refer to this type of reconnection as the flux pile-up regime, since the magnetic field lines come closer together and the field strength increases as the field lines approach the diffusion region. The intermediate range 0 < b < 1 gives a hybrid family of slowand fast-mode expansions, with the fast-mode regions tending to occur at the sides. Another feature is that the central diffusion regions are much larger for the flux pileup regime than the Petschek regime. The slow-mode compression regime also has central diffusion regions that are much larger than in Petschek’s regime, but these solutions are rather slow—slower even than the Sweet–Parker regime. The main results from the above analysis are that the type of reconnection regime and the rate of reconnection depend sensitively on parameter b which characterizes the inflow boundary conditions. Petschek (b = 0) and Sonnerup-like (b = 1) solutions are just particular members of a much wider class.

4.6.1.1

The Resistive Tearing Mode

The resistive instabilities were first derived in the seminal paper of Furth et al. (1963). The term “tearing instability” or “tearing mode” (implying the unstable mode) refers to a spontaneous reconnection process, which may occur in any sheared magnetic configuration (Biskamp, 2000). The double tearing mode is characterized by the presence of two (or more) resonant surfaces, the coupling of which may give rise to faster dynamics than in the standard tearing mode. It is useful at this point to introduce the linear displacement vector

4.6 Formation of Magnetic Reconnection

263

ξ , ∂t ξ = γ ξ = v1 , which is generally used in ideal MHD stability theory, and more specifically the component ξx in the direction of the equilibrium gradient, ξx = −ik y φ1 /γ . In the vicinity of x = xs one has B0 (x)  (x − xs )B0 and in terms of a power series expansion in the parameter k 2 xs2 , ξ = ξ (0) + ξ (1) + ... and lowest order, dξ (0) /d x = 0 except at the resonant surfaces x = ±xs , one can be written 1 dξ (1) λH 1 =− ξ0 d x π (x − xs )2 where λ H = −π

k2 B02



xs 0

B02 d x

(4.87)

(4.88)

is a measure of the free magnetic energy of the mode. We can define  λH = λ H (k B0 /η)1/3 . (a) For strong ideal instability  λ H  0 use of the asymptotic properties of the γ = λ H , i.e.,

-function, (z + a)/ (z + b)  z a−b , gives the ideal growth rate  γ = λ H k B0 ;

(4.89)

γ 3/2 − 1)/4], (b) Marginal instability  λ H = 0 corresponds to the first pole of [( which is  γ = 1 or (4.90) γ = (ηk 2 B02 )1/3 ; λ H |  1 corresponding (c) For deeply ideally stable M H D conditions  λ H < 0, | to small growth rate  γ  1, it gives  γ =

 1  4/5

2

43 

η3/5 (k B0 )2/5 |λ H |−4/5 .

(4.91)

4

Hence in the M H D-stable regime the mode reduces, as expected, to the standard tearing mode.

4.6.1.2

Coalescence Instability

Island coalescence in a resistive plasma has been studied in a corrugated neutral-sheet equilibrium, illustrated in Fig. 4.20 (Biskamp 2000): ψ(x, y) =

B∞ ln[cosh(kx) +  cos(ky)]. k

(4.92)

264

4 Spatial Magnetic Configurations of Solar Active Regions and Eruptions

Fig. 4.20 Islated sheet pinch solution (4.93) with  = 0.71. From Biskamp (2000)

B∞ is the field intensity for |x| → ∞ and √  is a measure of the island width w I given by cosh(w I k/2) = 1 + 2, w I /c  4  for w I k  1. This configuration belongs to a general class of solutions of the equilibrium equation ∇ 2 ψ = k 2 e−ψ , which can be written in the form ψ = − ln{8| f  |2 /(1 + | f |2 )2 }, where f (z) is an arbitrary complex function √ of z = x + i y (Fadeev et al., 1965). Solution (4.92) results from the choice f = 2k/B∞ ( + ekz ) where the corresponding current density is j = ∇2ψ =

1 − 2 B∞ . k [cosh(kx) +  cos(kh)]2

(4.93)

The current maxima are located at the minima of ψ, i.e., at the O-points of the islands along the y-axis, y = (2n + 1)π . Hence the configuration corresponds to a chain of current density blobs, which if perturbed from their equilibrium positions, attract each other. The equilibrium is ideally unstable with respect to pairwise attraction for any island width (Pritchett & Wu, 1979).

Chapter 5

Helical Magnetic Field and Solar Cycles

In recent years, a large number of observational data based on the solar vector magnetic field have been obtained. They have been used to investigate changes in the non-potential magnetic field and the helicity with the solar cycles. From the point of view that the vector magnetic field in the solar surface with different characteristic scales changes with the sun, the magnetic helicity in the solar surface atmosphere is not only a tangled component in the properties of magnetic turbulence, but also its statistical distribution characteristics carry its formation inside the sun. The main information is often interesting.

5.1 Distribution of Magnetic Helicity with Solar Active Cycles “Whirling storms in the earth’s atmosphere, whether cyclones or tornadoes follow a well-known law which is said to have no exceptions: the direction of whirl in the Northern hemisphere is left-handed or anti-clockwise, while in the Southern hemisphere it is right-handed or clockwise” (Hale et al., 1919). He recognized a similarity with the above-cited polarity rule now known as Hale’s polarity rule for sunspots: the sign of magnetic field is anti-symmetric over the solar equator and changes with every 11-year cycle. The helical properties of sunspots with solar cycles were analyzed statistically by Ding et al. (1987), and it has been confirmed in recent about thirty years in the study of magnetic helicity of active regions (Seehafer, 1990; Pevtsov et al., 1995; Abramenko et al., 1997; Bao & Zhang, 1998; Hagino & Sakurai, 2005; Xu et al., 2007). The consistency of the magnetic helicity was also analyzed by some authors (cf. Georgoulis & LaBonte, 2007; Georgoulis et al., 2009; Zhang et al., 2010a).

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Zhang, Solar Magnetism, https://doi.org/10.1007/978-981-99-1759-4_5

265

266

5 Helical Magnetic Field and Solar Cycles

5395

6659 6619

6891

7440

6615 7321

Fig. 5.1 Butterfly diagram of the electric current helicity. The mean density of the current helicity of active regions is marked by size of the circles for grades: 0, 1, 3, 5, 7 (10−3 G 2 m −1 ). The solid and hollow circles mark the negative and positive sign of helicity. The dashed-dotted line marks the average value of current helicity and the dashed line marks the average value of the imbalance of current helicity after the data smooth. From Zhang and Bao (1998)

Figures 5.1 and 5.2 show the distribution of mean current helicity density of active regions in 1988–1997 inferred from vector magnetograms observed at Huairou Solar Observing Station. Moreover, the reversal sign of helicity in these statistical works has also attracted much attention (Bao et al., 2000b; Zhang et al., 2002; Kuzanyan et al., 2003).

5.1.1 Statistical Analysis of Current Helicity of Active Regions from Observed Vector Magnetograms In Fig. 5.3, 984 active regions (6205 magnetograms in all) have been selected from 1988 to 2005, in which 431 active regions are in the 22nd solar cycle and 553 active regions are in the 23rd solar cycle (Gao et al., 2008). We limited the latitudes of active regions under the 40◦ and most of them are under 35◦ (only several are between 35◦ and 40◦ ). So the projection effects of active regions are neglectable in our work. Furthermore, we applied the linearly interpolated result of the average matrix to correct the azimuth rotation in our statistical work. Net electric current passing through the solar surface should be zero, so statistically, there must be a plus or minus electric current with similar magnitudes in active regions. However, positive spurious

5.1 Distribution of Magnetic Helicity with Solar Active Cycles

267

A

B

C

E

F

G H

Fig. 5.2 Distribution of the mean density of the current helicity parameter of active regions with solar rotation cycles, which is marked by size of the circles for grades : (0, 1, 3, 5, 7)×10−3 G 2 m −1 . Signs of current helicities are the same as Fig. 5.1. (Top) Mean density of current helicity in the northern hemisphere. (Bottom) Mean density of current helicity in the southern hemisphere. From Zhang and Bao (1999)

currents will appear at the local maxima of magnetic fields for either positive or negative polarity active regions if there’s Faraday rotation existing in the magnetograms (Hagino & Sakurai, 2004). In our result The average values of net electric current distribute on both sides of zero electric currents uniformly, there’s no spurious net current, which is evidence for the correction of Faraday rotation (Fig. 5.3). By the way, we have an upper limit of Bz to 900 G in our statistical work because we correct the azimuth errors caused by Faraday rotation at present corresponding to the area where the longitudinal magnetic field is less than 900 G (see Table 5.1). We compare the distribution of active regions according to Hc and their errors caused by Faraday rotation, see Fig. 5.4. From the figure we know the maximal

268

5 Helical Magnetic Field and Solar Cycles

a.

b.

Fig. 5.3 The statistical distribution of electric current in solar active regions in the 22nd and 23rd solar cycle(Bz > 500G) after the correction of Faraday rotation. Figure a corresponds to 22nd solar cycle, Figure b corresponds to 23rd solar cycle. From Gao et al. (2008) Table 5.1 Average distribution of azimuth rotation with longitudinal magnetic field and inclination Bz (G) Inclination(◦ )

85

75

250

0.00

0.17

65 0.13

55

45

5.45 10.74

35 0.00

25

15 0.00

5 0.00

0.00

350

0.00

1.38

0.67

6.68 15.24 14.17

0.00

0.00

0.00

450

0.00

0.00

3.08

5.33

0.00

0.00

0.00

7.03 13.86

550

0.00

0.00

18.89 17.95 15.81 10.36

4.66

−19.15

0.00

650

0.00

0.00

0.00 28.88 11.81 13.46

18.96

−26.80

0.00

750

0.00

0.00

0.00

2.51

4.98

8.10

14.67

−6.23

0.00

850

0.00

0.00

0.00

0.00

6.05

3.21

−0.62

29.91

0.00

950

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

1050

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

Fig. 5.4 Distribution of active regions according to the values of Hc0 and Hc1 − Hc0 , solid line corresponds to the former, dash line corresponds to the latter. Hc0 is the value of current helicity density before the correction of Faraday rotation, and Hc1 is that after the correction

5.1 Distribution of Magnetic Helicity with Solar Active Cycles

269

fluctuation of current helicity density comes from Faraday rotation is about 0.1 G 2 m −1 , so for those active regions with larger absolute values of current helicity density than 0.1 G 2 m −1 there is little probability to change signs because of the Faraday rotation. Although there should be variation with similar magnitudes for values of all active regions, the changed part will be comparable to the absolute value of real helicity for those less than 0.1 G 2 m −1 and far less than the value of real helicity for those greater than 0.1 G 2 m −1 systematically. So the distribution of the helicity for those active regions greater than 0.1 G 2 m −1 with the latitude should reflect the basic trend of helicity even if there’s no correction of Faraday rotation.

5.1.2 Hemispheric Distribution of Current Helicity of Active Regions Gao et al. (2008) presented the latitudinal distribution of Hc and αav in two solar cycles (see Fig. 5.5). Both parameters show the HSR holds in two solar cycles well. Table 5.2 shows the rate of change of current helicity with solar latitude. In Table 5.2, values in most years that are negative show a consistency with the HSR, and the emphasized parts are results from relatively larger data samples.

a.

b.

c.

d.

Fig. 5.5 Latitudinal distribution of Hc and αav . In which Figures a and b are for Hc corresponding to 22nd and 23rd solar cycles respectively. Figures c and d are for αav corresponding to 22nd and 23rd solar cycles, respectively

270

5 Helical Magnetic Field and Solar Cycles

Table 5.2 List of gradient of helicity to latitude Year d Hc /dθ dαav /dθ (10−5 G 2 m −1 deg −1 ) (10−10 m −1 deg −1 ) 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

−1.8 ± 0.8 0.2 ± 0.8 −1.6 ± 0.5 −4.7 ± 1.2 0.1 ± 1.1 −0.9 ± 1.5 −2.3 ± 1.0 −0.3 ± 1.1 −2.3 ± 1.5 −0.5 ± 1.0 −1.1 ± 0.5 −0.5 ± 0.3 −1.3 ± 0.4 −0.8 ± 0.3 0.0 ± 0.3 −0.1 ± 1.1 0.7 ± 1.0 −1.7 ± 2.9

−2.9 ± 2.0 0.7 ± 1.3 −3.0 ± 1.1 −8.3 ± 1.9 −1.3 ± 1.4 0.2 ± 2.2 −6.3 ± 3.0 −2.0 ± 3.4 −3.4 ± 4.2 −0.7 ± 1.8 −2.4 ± 1.0 −1.2 ± 0.6 −2.5 ± 0.7 −2.0 ± 1.1 −1.9 ± 1.1 −1.4 ± 3.3 0.7 ± 4.0 −9.8 ± 5.5

Active region numbers 25 56 69 67 78 65 39 24 8 20 55 102 142 117 74 18 17 8

5.2 Butterfly Diagram of Current Helicity of Solar Active Regions It is noticed that the twist or current helicity distribution is also affected by the interchange of twist and writhe due to helicity conservation (cf. Zeldovich et al., 1983; Brandenburg & Subramanian, 2005a) and the evidence in solar active regions (cf. Holder et al., 2004). For the emergence of helical magnetic flux, Longcope et al. (1998) pointed out the emerging flux tube twist resulting from helical turbulence (Sigma-Effect). The magnetic helicity was proposed relative to the solar dynamo process (e.g. Frisch et al., 1975; Brandenburg & Subramanian, 2005a). It is generated, according to the mean-field solar dynamo model, due to the effects of solar differential rotation and the action of Coriolis force on turbulent motions of plasma in the solar convection zone (e.g. Berger & Ruzmaikin, 2000; Kuzanyan et al., 2000; Kleeorin et al., 2003; Choudhuri et al., 2004; Nandy, 2006; Zhang et al., 2006).

5.2 Butterfly Diagram of Current Helicity of Solar Active Regions

271

5.2.1 Helicity with Solar Cycles Figure 5.6 shows the distribution of the average helical characteristics of the magnetic field in solar active regions in the form of butterfly diagrams (latitude-time) for 1988–2005 (which covers the most of 22nd and 23rd solar cycles). These results are inferred from photospheric vector magnetograms recorded at Huairou Solar Observing Station after statistical reduction of the influence of magneto-optical (or Faraday) effects in the measurements of magnetic field (Su & Zhang, 2004b; Gao et al., 2008). This longest available systematic dataset covering the period of two solar cycles comprises 6205 vector magnetograms of 984 solar active regions (most of the large solar active regions of both solar cycles). Of these, 431 active regions belong to the 22nd solar cycle and 553 to the 23rd one. We have limited the latitudes of active regions to ±40◦ and most of them are below 35◦ . The helicity values of the active regions have been averaged over latitude by intervals of 7◦ of solar latitude, and over overlapping 2-year periods (i.e., 1988–1989, 1989–1990, ..., 2004–2005). By this way of averaging, we were able to group at least 30 data points to make error bars (computed as 95% confidence intervals) reasonably small. In this sampling, we find that 66% (63%) of active regions have negative (positive) mean current helicity in the northern (southern) hemisphere over the 22nd solar cycle and 58% (57%) in the 23rd solar cycle. The message which we infer from this butterfly diagram is as follows: 1. The helicity and twist patterns are in general anti-symmetric to the solar equator. This result confirms the hemispheric rule obtained in studies of 11-year observational data sets (Kleeorin et al., 2003; Pevtsov et al., 1994, 1995; Bao & Zhang, 1998). 2. The helicity pattern is more complicated than Hale’s polarity law for sunspots. Our results revealed specific latitudes and times on the butterfly diagram where the hemispheric helicity law is inverted. So, we found areas of the “wrong” sign at the end of the butterfly wings. This is a challenge for dynamo theory. We can interpret this phenomenon as penetration of the activity wave from one hemisphere into the other “wrong” hemisphere. An analogous pattern can be recognized in sunspot data at the end of the Maunder minimum (Sokoloff, 2004; Sokoloff et al., 2006). The other domain of the “wrong” helicity sign located just at the beginning of the wing has been predicted (Chumak et al., 2004) as a result of additional twisting of magnetic tubes arising to form a sunspot group. 3. There is an approximately 2-year time lag between the sunspots and current helicity and twist patterns: the helicity and twist patterns come after the sunspot pattern. Moreover, the maximum value of helicity, at the surface at least, seems to occur near the edges of the butterfly diagram of sunspots. This is an unexpected result that poses another challenge for dynamo theory. The theory predicts (Parker, 1955a) a lag of the opposite sign (helicity and twist pattern should appear some 2.7 years before the sunspot pattern). Conventional dynamo models, however, ignore the fact that helicity needs time to follow the magnetic field.

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A

αav (10-9m-1)

40°

Heliographic latitude

30° 20° 10° 0° -10° -20° -30° -40° 1988

15

-1

Hc (10 G m )

40°

Heliographic latitude

30° 20° 10° 0° -10° -20° -30° -40° 1988

20

leading sunspot is negative

Fig. 5.6 Top: the distribution of the averaged twist αff ; and bottom: electric current helicity HCz of solar active regions in the 22nd and 23rd solar cycles. Superimposed, the underlying colored “butterfly diagram” shows how sunspot density varies with latitude over the solar cycle. The vertical axis gives the latitude and the horizontal gives the time in years. The circle sizes give the magnitude of the displayed quantity. The bars to the right of the circles show the level of error bars computed as 95% confidence intervals, scaled to the same units as the circles. 72 out of 88 groups for current helicity (82%) as well as 67 out of 88 groups for the twist (74%) have the error bars lower than the signal level. From Zhang et al. (2010b)

5.2.2 Relationship Between Twist and Tilt of Magnetic Fields in Active Regions Using vector magnetic field data from the HMI/SDO (Helioseismic and Magnetic Imager instrument onboard the Solar Dynamics Observatory), Liu et al. (2022) study

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the signs of helicity (magnetic twist αav , z-component of current helicity Hc ) and tilt angle of 85 sample active regions (ARs) that appeared on the central solar disk (within ±45◦ from disk center) between December 2018 and November 2020. This time range spans the exchange period of Solar Cycle 24 and 25. Figure 5.7a shows the distribution of αav as a function of latitude and time for the 85 ARs during the time period from December 2018 to November 2020. It can be seen that the change of latitude of ARs with time displays a trend that largely follows the butterfly diagram within 24 months of investigation. Liu et al. (2022) separate Solar Cycle 24 and 25 with two inclined lines whose slopes are similar

Fig. 5.7 a: Distribution of the observational twist αav in the 85 sample ARs as a function of latitude and time for the exchange period between solar cycles 24 and 25 from 2018 December to 2020 November. The data between/beyond the two solid inclined lines belong to Solar Cycle 24/25. The vertical dashed line marks the minimum of the 13-month smoothed sunspot number. The sizes of the circles indicate the corresponding typical values of observational twist αav (blue for negative and red for positive). The values of αav corresponding to certain circle sizes are shown in the middle of the figure. b: Average twist αav as a function of the latitude of the 85 sample ARs (red for Solar Cycle 24 and blue for Solar Cycle 25). The red/blue solid line is the least-squares fit to the observational data of the 38/47 sample ARs which belong to Solar Cycle 24/25, and the black line is that of the whole sample ARs. c: Similar to panel (a), but the sizes of the circles indicate the corresponding typical values of observational tilt angle. d: Similar to panel (b), but for average tilt angle as a function of the latitude. From Liu et al. (2022)

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Table 5.3 Percentages of ARs following Joy’s law/Hale’s law/Hemispherical rule Class of ARs Sample size Joy’s lawa Hale’s lawb Hemisph rulec All Northern Southern Cycle 24 Cycle 25

85 45 40 38 47

57(67%) 28(62%) 29(73%) 20(53%) 37(79%)

80(94%) 43(96%) 37(93%) 33(87%) 47(100%)

53(62%) 30(67%) 23(58%) 22(58%) 31(66%)

a,b,c The numbers and percentages of ARs obeying Joy’s law, Hale’s law, and hemispherical rule, respectively

to the ones presented by Usoskin and Mursula (2003) and McClintock and Norton (2014). As shown in Fig. 5.7a, the sample ARs show a tendency of following the hemispherical rule, i.e., with negative/positive twist signs in the northern/southern hemisphere. Figure 5.7b shows the average twist αav as a function of the latitude of the ARs (red for Solar Cycle 24 and blue for Solar Cycle 25). The slopes of the solid lines should be negative for “dominant” twists which are negative/positive in the northern/southern hemisphere. From Fig. 5.7b we can see that all slopes are negative. These results imply that most of the sample ARs obey the hemispherical rule, no matter throughout the whole studied exchange period or during the end/beginning of the solar Cycle 24/25. Figure 5.7c is similar to Fig. 5.7a, but the blue/red circles represent ARs with negative/positive tilt angle. The size of the circle is proportional to the value of the tilt angle as shown in the middle of the figure. It shows that ARs have negative tilt angles in the northern hemisphere and positive ones in the southern. Figure 5.7d is similar to Fig. 5.7b, but for average tilt angle as a function of the latitude. It is found that all the slopes of the least-squares that fit the tilt and latitude are negative. These results imply that most of the ARs obey Joy’s law. Table 5.3 shows the proportions of ARs obeying Joy’s law, Hale’s law, and hemispherical rule, respectively. From the last column, we can see that 62% of ARs follow the hemispherical rule. Among them, 67% of ARs in the northern hemisphere have negative twists, and 58% in the southern hemisphere have positive twists. And the hemispherical preference is more obvious for the ARs at the beginning of Solar Cycle 25 (66%) than for those at the end of Solar Cycle 24 (58%). Considering most of the ARs of Solar Cycle 24/25 are located in low/high latitudes, this result means that the hemispherical tendency of twist increases with latitude. Figure 5.8a shows the scatter plot of αav (X-axis) and the average tilt angle (Y-axis) of the sample of 85 ARs. The red solid line is the least-squares fit of the 38 ARs of Solar Cycle 24, and the blue one is that of the 47 ARs of Cycle 25. Figure 5.8b shows the scatter plot of αav (X-axis) and Writhe (Y-axis). We can see that the data points are very scattered and the correlation coefficients are small which implies a very poor correlation between Twist and Tilt/Writhe. When a flux tube rises through the convection zone, it can be deformed by Coriolis force or convective turbulence. And this deformation produces magnetic writhe and an equivalent twist of the opposite

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Fig. 5.8 a: Scatter plot of αav (X-axis) and average tilt angle (Y-axis) of the sample of 85 ARs. The red/blue solid line is the least-squares fit to the observational data of the 38/47 sample ARs belonging to Solar Cycle 24/25. The notes “CC25” and “CC24” mark the corresponding correlation coefficients. b: Similar to Panel (a), but for scatter plot of αav and writhe. From Liu et al. (2022)

sign in the tube to conserve helicity (Berger, 1984; Longcope et al., 1998; Berger & Ruzmaikin, 2000; Liu et al., 2014. In this scenario, one can be expected that the sign of the acquired writhe of an AR is opposite to that of the twist. Interestingly, the result by Liu et al. (2022) is different from this point of view. This may be due to the fact that the twist could be present from the flux tube formation deep down. The dynamo process is one possibility to generate a twist not (or weakly) linked to the tilt/writhe. Compared with the statistical correlation results between magnetic helicity and tilt angle on emerging active regions in Sect. 3.4.3 (Yang et al., 2009), the discussion of the problem may still need to be further explored.

5.2.3 Evolved Helicity Comparison Among Magnetographs Xu et al. (2007) have compared vector magnetograms of 228 active regions observed by Solar Magnetic Field Telescope (SMFT) at Huairou (HR) Solar Observing Station and the Solar Flare Telescope (SFT) at Mitaka (MTK) of the National Astronomical Observatory of Japan from 1992 to 2005 and 55 active regions observed by SFT and Haleakala Stokes Polarimeter (HSP) at Mees Solar Observatory, the University of Hawaii from 1997 to 2000 in Fig. 5.9. Two helicity parameters, current helicity density h c and α ff coefficient of linear force-free field are calculated. From this comparison, it is concluded: (1) the mean azimuthal angle differences of transverse fields between HR and MTK data are systematically smaller than that between MTK and Mees data; (2) there are 83.8% of h c and 78.1% of α ff for 228 active regions observed at HR and MTK agree in sign, and the Pearson linear correlation coefficient between these two data sets is 0.72 for h c and 0.56 for α ff . There are 69.1% of h c and 65.5% of α ff for 55 active regions observed at MTK and Mees agree in sign, and the Pearson linear correlation coefficient between these two data sets is 0.63 for

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Fig. 5.9 Left: Time variation of the slope of linear fit calculated from the latitudinal profile of helicity for each year over 1992–2005. The solid and dashed lines represent HR data and MTK data, respectively. Right: Time variation of the slope of linear fit calculated from the latitudinal profile of helicity for each year over 1997–2000. The solid and dashed lines represent Mees data and MTK data, respectively. The upper panel is dh c /dθ and the lower is dα ff /dθ. Error bars represent one sigma uncertainty of slope of the linear fit. From Xu et al. (2007)

h c and 0.62 for α ff ; (3) there is a basic agreement on the time variation of helicity parameters in active regions observed at HR, Mees, and MTK.

5.2.4 Accuracy of Measured Vector Magnetic Fields In addition to some fundamental achievements in discovering some important properties of solar magnetic activities, it is also needed to notice that the magnetic fields and relevant parameters, such as helicity, etc., observed with different instruments not only have shown the basic same tendency, but also with some differences (Wang et al., 1992; Bao et al., 2000a; Pevtsov et al., 2006; Xu et al., 2016). As a sample, the discrepancies on the statistical distribution of the current helicity parameters h c and α of solar active regions obtained by different solar vector magnetographs at Huairou in China and Mitaka in Japan can be found by Xu et al. (2016) in Fig. 5.10. This relates to a basic question on the accuracy for the measurements of the solar magnetic fields through solar vector magnetograph and the corresponding observing theories of solar magnetic fields. Some restrictions on the calibration of the magnetic field from Stokes parameters have been presented due to the nonlinearity and simplicity in the above calculation of the radiative transfer equations of the spectral lines in the solar magnetic atmosphere. It means that there are still some questions on the theories of the solar observations relative to the diagnostic with the radiative transfer

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Fig. 5.10 Correlations in a h c and b αav obtained from SMFT (Huairou) and SFT (Mitaka) data. The values of correlation coefficients (CC) are shown on the panels. From Xu et al. (2016)

of spectral lines, even if some problems of solar instrumental techniques have been ignored.

5.2.5 Comparison Between Current and Subsurface Kinetic Helicity Gao et al. (2009) made a comparison between photospheric current helicity and subsurface kinetic helicity in solar active regions. Four parameters are employed: average current helicity Bz · (∇ × B)z  (Hc ), average force-free field factor  (∇ × B)z · sign[Bz ]/ |Bz | (αave ) and mean subsurface kinetic helicity v · (∇ × v)/|v|2  (αv ), which is denoted as two different parameters αv1  and αv2  according to different depths beneath the solar surface. A total of 38 active regions are investigated. The results show that the signs of Hc  and αave have typical hemispheric distribution feature. In contrast, the sign of αv1  presents the opposite feature to the above two parameters. For αv2 , there is not the obvious preponderance of the sign in each hemisphere as the other three parameters. Although there is opposite hemispheric preponderance between the sign of current helicity and that of kinetic helicity at 0–3 Mm beneath the solar surface, the weak correlations between Hc  and αv1 , αave and αv1 , Hc  and αv2 , αave and αv2  (The correlation coefficients are −0.095, 0.118, −0.102, −0.179 respectively) do not support that the photospheric current helicity has a cause and effect related to the kinetic helicity at 0–12 Mm beneath the solar surface. Furthermore, the hemispheric and latitudinal distributions of four parameters are shown in Fig. 5.11. The figure clearly shows the negative slopes of Hc  and αave and the positive slope of αv1 , numerically, they are −3.8 × 10−6 G 2 m −1 deg −1 , -2 × 10−12 m −1 deg −1 and 2.7 × 10−6 ms −2 deg −1 . The trend for the αv2  is not clear. The value of the slope is -5.6 × 10−6 ms −2 deg −1 , which presents an opposite trend with the original trend of 88 data sets. Nevertheless, the trend is very weak.

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Fig. 5.11 Latitudinal distributions of photospheric current helicity and subsurface kinetic helicity. Left Top: current helicity Hc ; Right Top: subsurface kinetic helicity at the depth of 0–3 Mm; Left bottom: average force-free field factor αave ; Right bottom: subsurface kinetic helicity at the depth of 9–12 Mm. From Gao et al. (2009)

Fig. 5.12 Correlation between photospheric current helicity and subsurface kinetic helicity. Left Top: Hc  and αv1 ; Right Top: Hc  and αv2 ; Left bottom: αv1  and αave ; Right bottom: αv2  and αave . From Gao et al. (2009)

From the above, we can obtain the opposite (but uncertain) preponderance between signs of current helicity and those of subsurface kinetic helicity at 0–3 Mm (9–12 Mm) beneath the solar surface. There are 24 (63.2%) active regions that have opposite signs between Hc  and αv1 . For αave and αv1 , there are 20 (52.6%) active regions that have opposite signs. Some numerical simulations of α−effect due to magnetic buoyancy seem to be supported (e.g., Brandenburg and Schmitt 1998). To be clearer, we show the correlations among these parameters in Fig. 5.12. The weak correlations among these parameters do not support that the photospheric current helicity has a cause and effect related to the kinetic helicity at 0–12 Mm beneath the solar surface, i.e., there may be not any correlation between the current helicity on the solar surface and the kinetic helicity at such shallow depth of the solar convection zone.

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However, it is worthwhile pointing out that the subsurface kinetic helicity and the photospheric current helicity are computed using observations taken from different instruments at different times, although quite similar.

5.2.6 Magnetic Helicity and Energy Spectra of Active Regions with Solar Cycles The redistribution of magnetic helicity contained within different scales was argued to be the interchange of twist and writhe due to magnetic helicity conservation (cf. Zeldovich et al., 1983; Kerr & Brandenburg, 1999). Furthermore, the spectral magnetic helicity distribution is important for understanding the operation of the solar dynamo (Brandenburg & Subramanian, 2005a). The statistical analyses of vector magnetograms at Huairou Solar Observing Station have been achieved in recent years (such as Bao & Zhang, 1998; Gao et al., 2008; Zhang et al., 2010b). It also provides a chance to analyze the evolution of the global distribution of the spectrum of magnetic fields of active regions and the relationship with solar active cycles, because the collectivity effect of active regions is also important in the analysis of solar cycles (Zhang et al., 2016). Figure 5.13 shows the average spectrums of magnetic helicity k HM (dotted line), current helicity Hc (solid line), and magnetic energy E M (dashed line) inferred by over 6629 Huairou vector magnetograms of solar active regions observed in 1988– 2005. The calculated method is similar to the individual active regions in above. For the consistency in the calculation of the long-term evolution of the magnetic field of solar active regions by means of a series of Huairou vector magnetograms, the spatial resolution of Huairou vector magnetograms has been compressed to 2 × 2 , therefore, the influence of the different seeing condition in the observations has been removed mostly. Due to the relatively low spatial resolution, Fig. 5.13 does not show more messages at high wave numbers. It is found that the shallow slope of the spectrum of magnetic energy at the high wave number is mainly from the observational errors of transverse components of the magnetic field, after the data analysis. For analyzing the evolution of statistical helicity and energy spectrum of solar active regions, Fig. 5.14 shows the distribution of the average scale exponent α of the spectrum of magnetic helicity, current helicity, and magnetic energy with solar cycles in the spectral within 0.2 < k < 0.6 (Mm −1 ). One can find that the slopes of the spectrums do not change obviously with the latitudes, as one averages the spectrums of active regions in 1988–2005. The mean scale exponent α value of k −α power law of k HM is about 2.2, that of HC is about 1.1 and that of E M is about 1.4. Figure 5.14 also shows the temporal variation of the slopes of the spectrums of magnetic energy and helicities of active regions. These slopes show high correlation with sunspot numbers significantly. The high values occur in 1990–1992 and 2000– 2003, and low values in 1995. These are consistent with the periods of solar maximum

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Fig. 5.13 The average spectrums of magnetic helicity (doted line), current helicity (dashed line), and magnetic energy (solid line) inferred by 6629 Huairou vector magnetograms of solar active regions in 1988–2005. After Zhang et al. (2016)

and minimum. The statistical correlation coefficient between sunspot numbers and slopes of the magnetic energy is 0.827, and that between sunspot numbers and current helicity is 0.730. This means the statistical variation of the spectrum of active regions shows the same periodicity as solar cycles. It is also relevant that the magnetic strength of sunspots in the solar maximum is high. We also notice that the correlation coefficient between sunspot numbers and slopes of current helicity changes to 0.831, as one takes the sunspot numbers to one year delay. It is constant with the observational result by Zhang et al. (2010b), in which the maximum of mean current helicity of active regions tends to delay than that of sunspot numbers. Similar evidence is that the complex magnetic configuration of active regions tends to occur in the decaying phase of the solar cycle 23 (after 2002) (Guo et al., 2010). Figure 5.15 shows the statistical distribution of the integral scale l M of magnetic energy of solar active regions inferred by 6029 Huairou vector magnetograms in 1988-2005. The correlative coefficient between the integral scale of magnetic energy (inferred by Eq. 3.79) and sunspot numbers is 0.802. The average value of the integral scale of magnetic energy is about 8 Mm in the solar maximum and 6 Mm in the Minimum for our calculated active regions. This is consistent with that the largescale magnetic patterns of active regions tend to occur near the maximum of solar cycles. Figure 5.15 presents also that the average photospheric relative magnetic helicity r M of active regions by Eq. (3.81) shows a tendency with the change of solar cycles (marked by sunspot numbers), except after 2003. The high sway of the mean relative magnetic helicity in 2003–2005 is roughly consistent with the high complexity of magnetic fields of active regions obtained by Guo et al. (2010) based on the analysis of MDI longitudinal magnetograms, even if the different data sets are used. It is

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Fig. 5.14 Top: The distribution of average scale exponent α of magnetic helicity (dotted line), current helicity (solid line), and magnetic energy (dashed line) with the latitude inferred by 6629 Huairou vector magnetograms of solar active regions in 1988-2005. Bottom: The distribution of average scale exponent α of current helicity (solid line) and magnetic energy (dashed line) with the time. The dotted line at the bottom shows the sunspot numbers. The error bars are 0.3σ . After Zhang et al. (2016)

noticed that a series of Huairou vector magnetograms of super active regions (such as NOAA 10484, 10486, and 10488) near the end of 2003 have been used in the study. These active regions show high nonpotentiality (cf. Liu & Zhang, 2006; Zhou et al., 2007; Zhang et al., 2008a). It is the reason for the high normalized magnetic helicity |r M | that occurred statistically in this period.

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Fig. 5.15 The solid line shows the statistical distribution of integral scale l M of magnetic energy and the dashed line does that of the photospheric relative magnetic helicity r M inferred by 6029 Huairou vector magnetograms of solar active regions in 1988–2005. The error bars are 0.3σ . The dotted line shows the sunspot numbers. After Zhang et al. (2016)

5.2.7 Resolution Dependence In the study, the HMI and Huairou vector magnetograms have been used to estimate the spectra of magnetic energy and helicity of solar active regions. In addition, temporal changes of the magnetic energy spectra of active regions and the evolution with the solar cycle have been found. Since we use vector magnetograms of different spatial resolutions to analyze the evolution of the spectral distributions of magnetic energy at different times, we now address the possible uncertainty regarding the relationship between the observational resolution of the magnetic field and the spectral shape at large wavenumbers. The lower spatial resolution of vector magnetograms of ground-based observations implies a source of error in the spectrum of the magnetic field at high wavenumbers. To estimate the possible errors in the calculation of the magnetic spectrum due to the low spatial resolution of the observational magnetic fields by the Huairou vector magnetograms, Fig. 5.16 shows the mean spectra of magnetic energy as well as magnetic and current helicity. Figure 5.17 shows the evolution of the spectral indices αC and α E for wavenumbers in the active region NOAA 11158, whose pixel size of the analyzed region of the HMI vector magnetograms have been downsampled from 512 × 512 to 128 × 128. The pixel resolution is 2 × 2 , which is almost the same as that of the Huairou vector magnetograms. The same tendency is found for the magnetic energy spectra as in Fig. 3.35. The high noise in the time series of αC and α E in Fig. 3.37 is now reduced. From 2011 February 12 to 16, the mean value of α E is about 1.82 and that of αC is about 1.34 for 0.4 Mm−1 < k < 2.0 Mm−1 , while the

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Fig. 5.16 Same as Fig. 3.35, but with pixels compressed from 512 × 512 to 128 × 128. After Zhang et al. (2016)

Fig. 5.17 Same as Fig. 3.37, but with pixels compressed from 512 × 512 to 128 × 128. After Zhang et al. (2016)

values obtained for the original resolution in Fig. 3.37 are 1.52 and 1.62, respectively, (the lower value of α E is due to including the rapid growth during the emerging stage of the active region on 2011 February 12). For the detailed analysis, we also have reversed the HMI vector magnetograms to Stokes parameters (Q, U , and V ) in the approximation of the weak field with Gaussian smoothing for reducing to the forms of the lower spatial resolution, compressing them to the lower pixel resolution of the Stokes parameters, and then reverting to vector magnetograms again. We found almost the same tendency for the spectrum of magnetic fields such as shown

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in Fig. 5.16, although the amplitudes of the slopes of the spectra of the magnetic fields changed slightly. We notice that we still cannot imitate the real case of the lower observational spatial resolution completely, such as the shallow slope of the spectra of magnetic energy at high wavenumbers in Fig. 5.13. The difference with the degraded data implies that the resolution of the observational vector magnetograms might still be problematic in the diagnostics of the spectra of the magnetic field in the detailed study. This may affect some of the analyses regarding the changes of the spectral slopes with the solar cycle when using the Huairou vector magnetograms, although one should keep in mind that our conclusions from the temporal variations are compatible with those found for individual active regions.

5.2.8 Summary for Evolution of Magnetic Spectrum We have applied the technique of Zhang et al. (2014) to estimate magnetic energy and helicity spectra using vector magnetogram data at the solar surface. We have made use of the assumption that the spectral two-point correlation tensor of the magnetic field can be approximated by its isotropic representation. We have analyzed the evolution of magnetic energy and helicity spectra in active regions and have also analyzed the changes with the solar cycle. Our major results are the following. 1. The values of α E and αC of solar active regions are of the order of 5/3, although αC is slightly lower than α E , i.e., the current helicity spectrum is slightly shallower than the magnetic energy spectrum. We have also found a systematic change of α E and αC with the development of active regions, which reflects their structural changes. 2. There is no obvious relationship between the change of the photospheric normalized magnetic helicity r M and the integral scale of the magnetic field l M of individual active regions. This means that the increase of the mean scale of magnetic structures does not imply that the magnetic helicity in the active regions increases. 3. We have found that there is a correlation between the variation of the spectra of magnetic energy and helicity of solar active regions with solar cycles. This reflects that the characteristic scales and the intensity of the magnetic fields of active regions change with the solar cycle. 4. Interestingly, even though the mean α E and αC of active regions vary with the cycle and increase with increasing mean magnetic energy density, they do not change with latitude even though the mean magnetic energy density does change with latitude. This suggests that the underlying magnetic field represents a part that is independent of the global cyclic magnetic field and possibly a signature of what is often referred to as local small-scale dynamo. Values of α E and αC of the order of 5/3 are roughly compatible with a Kolmogorovlike forward cascade (Kolmogorov, 1941a; Obukhov, 1941), which is expected from

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the theory of nonhelical hydromagnetic turbulence when the magnetic field is moderately strong (Goldreich & Sridhar, 1995). However, for decaying turbulence, Lee et al. (2010) found that the scaling depends on the field strength and takes on a shallower Iroshnikov–Kraichnan k −3/2 spectrum (Iroshnikov, 1963; Kraichnan, 1965) for weaker fields and a steeper k −2 weak-turbulence spectrum for stronger fields; see Brandenburg and Nordlund (2011) for the respective phenomenologies in the three cases. The steeper k −2 spectrum has recently also been found in decaying turbulence simulations where the flow is driven entirely by the magnetic field (Brandenburg et al., 2015). It is thus tempting to associate the changes in the values of α E and αC with corresponding changes between different scaling laws.

5.3 Statistical Study of Transequatorial Loops Transequatorial loops (TLs) are one type of coronal structures which connect different regions in the opposite hemispheres (Chen et al., 2006). The first direct evidence of TLs was observed by Skylab (Chase et al., 1976; Švestka et al., 1977). A TL is not an occasional phenomenon; it was found by Pevtsov (2000) that as many as one-third of all active regions exhibit TLs in soft X-ray images. The regions connected by TLs tend to have the same chirality. Canfield et al. (1996) studied the chirality of active regions on the opposite sides of the equator and found that the regions with the same chirality form transequatorial loops, but those with the opposite chirality do not. This was further confirmed by Fárník et al. (1999) and Pevtsov (2000).

5.3.1 Distribution of Tansequatorial Loops 5.3.1.1

Relation with Solar Cycle

(Chen et. al., 2020) in Fig. 5.18a shows the number of TLs in different years. In the declining phase of the solar cycle, the number of TLs decreases; in the rising phase, it increases. The number of TLs is large in high solar activity (1992, 2000), and is small in low activity (1996, 1997). The relation between the numbers of TLs and active regions in each year is shown in Fig. 5.18b. The dotted line shows that the average value of the ratio is only about 10%, which is different from Pevtsov’s (2000) result (about 30%). We obtain the number of active regions in each year by calculating the NOAA region number of the last day of the year minus the region number of the first day of the year. The SXT data are incomplete in 1991 and 2001; in these two years, the active region numbers are calculated only during the period in which SXT had observations.

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Fig. 5.18 a The number of TLs in different years. b The ratio in the number of TLs to active regions in each year. The dotted line shows the average value. Years 1991–1996 belong to solar cycle 22 and years 1996–2001 belong to solar cycle 23. From Chen et al. (2006)

5.3.1.2

Three Parameters of TLs

Using the footpoint positions, three parameters of TLs are computed; separation of footpoints, tilt angle, and asymmetry in latitude. Figures 5.19a, 5.20a, and 5.21a show the distribution of TLs in terms of these three parameters. The curves show the polynomial fit, and the dotted lines indicate the average values of these parameters. Figures 5.19b, 5.20b, and 5.21b show the variations of these parameters in different years. They also show an error at 1σ level.

5.3.1.3

Separation of Footpoints

The separation of footpoints means the distance between the two footpoints of TLs. From Fig. 5.19, we can see that the average value of separation is about 27◦ , which is close to Pevtsov (2000)’s result and different from Chase et al. (1976) value of about 20◦ . The period of 1991–1995 is the declining phase of solar cycle 22 and the period of 1997-2001 is the increasing branch of solar cycle 23. From Fig. 5.19b we can see that the mean value of separation decreases gradually year by year from 1991 to 1995 and it also decreases from 1997 to 2001. The mean value of separation in solar cycles 22 and 23 is also calculated. In solar cycle 22, the mean value is about 22◦ ; in solar cycle 23, the mean value is about 31◦ . In 1996, a change of magnetic polarity took place happened, but the mean value of separation for this year is near to the average value of the whole TLs. Spörer’s law shows the solar latitude at which new sunspots appear gradually decreases, from high latitudes at the beginning of a solar cycle to low latitudes at the end of the cycle. The footpoint separation of TLs is consistent with Spörer’s law from Fig. 5.19b. It decreases following the solar cycle and the mean value is lower in the declining phase (1991–1995) than in the ascending part (1997–2001).

5.3 Statistical Study of Transequatorial Loops

287

Fig. 5.19 Separation of TLs. a The number distribution of footpoint separation. The curve shows a polynomial fit. The dotted line shows the mean value. b The separation in different years. The triangle shows the average value in each year. The error bar is for 1σ interval. The dotted line shows the overall average value. From Chen et al. (2006)

Fig. 5.20 Tilt angle of TLs. a The number distribution of tilt angles. The curve shows a polynomial fit. The dotted line shows the mean value. b The tilt angle in different years. The triangle shows the average value in each year. The error bar is for 1σ interval. The dotted line shows the overall average value. From Chen et al. (2006)

5.3.1.4

Tilt Angle

A tilt angle is defined as the angle between a transequatorial loop and the equator. We can obtain the tangent value of the tilt angle using the difference in footpoint latitudes divided by the difference in footpoint longitudes. When the northern footpoint is to the west of the southern footpoint, the tilt angle is defined to be less than 90◦ (acute angle). On the contrary, if the northern footpoint is to the east of the southern footpoint, the tilt angle is defined to be more than 90◦ (obtuse angle). Figure 5.20a shows TLs numbers distribution in different tilt angles. The average value is near 90◦ . Figure 5.20b shows the tilt angles in different years. The error bar is for 1σ interval. In Fig. 5.20b, In Fig. 5.20b, it can be seen that the angles between the transequatorial loops and the equator do not vary significantly with the solar cycles. It is similar to Pevtsov (2004)’s result.

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Fig. 5.21 Asymmetry in the latitude of TLs. a The number distribution in asymmetry. The curve shows a polynomial fit. The dotted line shows the mean value. b Asymmetry in different years. The triangle shows the average value in each year. The error bar is for 1σ interval. The dotted line shows the overall average value. From Chen et al. (2006)

5.3.1.5

Asymmetry in Latitude

The asymmetry here is defined as the latitude of the northern footpoint subtracted by the absolute value of the latitude of the southern footpoint. If the latitude value of the northern footpoint is larger (smaller) than that of the southern footpoint, the result is positive (negative), respectively. We found that the mean value of asymmetry in latitude is near 0◦ , namely, the transequatorial loops are almost symmetry in latitude. Figure 5.21b presents the latitude asymmetry in different years. The average value in each year is also ∼ 0◦ . We see no obvious difference in the TLs asymmetrical in latitude in different years and nor in the decreasing phase of solar cycle 22 and the ascending phase of solar cycle 23.

5.3.2 Helicity Patterns of Active Regions Connected by Transequatorial Loops

5.3.2.1

Parameters and Calculating Methods of Current Helicity

For the selected active regions at Huairou, the vector magnetic field data having favorable seeing conditions and within ±35◦ from the disk center in longitudinal degree are included. There are 43 pairs of active regions having the vector magnetograms which satisfy the above conditions. The number of TLs in an active region may be multiple (Chen et al., 2006), so there are 81 active regions for the selected 43 pairs of active regions including more than 800 vector magnetograms. Figure 5.22a is a Yohkoh soft X-ray image on January 19, 1999, in this figure a transequatorial loop is viewed. Figure 5.22b is the corresponding Kitt Peak full

5.3 Statistical Study of Transequatorial Loops

289

Fig. 5.22 a A Yohkoh soft X-ray image presents a transequatorial loop on Jan 19, 1999. The two circles exhibit the regions that the transequatorial loop connected. b Corresponding Kitt Peak full disk longitudinal magnetogram. “P” represents the positive magnetic polarity and “N” shows the negative magnetic polarity. From Chen et al. (2007) Table 5.4 The helicity values of AR 8440 and AR 8439 AR number αbest ρh (%) 8440 8439

1.16 0.04

31 21

αbest (MP)

ρh (MP) (%)

1.17 (P) 0.80 (N)

30 (P) 11 (N)

Note the units of αbest is 10−8 m −1 . MP represents the magnetic polarity regions; “P” shows the positive magnetic polarity region, “N” shows the negative magnetic polarity region

disk longitudinal magnetogram. From the figures, we can see that the transequatorial loop connected the positive magnetic polarity of AR 8440 in the northern hemisphere and the footpoint existed in the negative polarity region of AR 8439 in the southern hemisphere. After that, the corresponding vector magnetograms of the two active regions at Huairou Solar Observing Station are found. For the two active regions, we select the magnetograms with good seeing conditions and near the central meridian. The vector magnetograms are handled according to the processes described. The mean values of αbest and ρh of them are calculated, the signs are all positive, the values of them are exhibited in Table 5.4. The results show that the pair of active regions (AR 8440, AR 8439) have the same chirality and the pair of magnetic polarity regions connected by this TL also have the same handedness. Figure 5.23 shows the distribution of magnetic field lines of a transequatorial loop in the corona. They are calculated by the approximation of the constant-α force-free magnetic field. We obtain that the α value is 1.7 × 10−3 Mm −1 . The TLs twist sign is the same as the region pairs which are connected by TLs.

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Fig. 5.23 Force-free field extrapolation of the longitudinal magnetic field. Yohkoh Soft X-ray image, overlaid by the contour of Kitt Peak longitudinal magnetogram. α = 1.7 × 10−3 Mm −1 . From Chen et al. (2007)

5.3.2.2

19-JAN-99 14:40:55

Statistical Distribution of Helical Transequatorial Loops

Figure 5.24 exhibits the helicity correlation of the 43 pairs of active regions which are connected by transequatorial loops. From the figure, we can’t see the obvious relationship between them. For the parameter αbest , 22 pairs (51%) of active regions have the same helicity patterns and 21 pairs (49%) of active regions own the opposite chirality. Applying the parameter ρh , we obtain 26 pairs (60%) of active regions showing the same signs and 17 pairs (40%) showing the opposite signs. The results of both proxies exhibit that some active region pairs connected by transequatorial loops haven’t the same chirality. Figure 5.25 presents the helicity patterns relation of 52 pairs of the magnetic polarity regions connected by 43 TLs. The results are almost similar to the active region pairs, the two parameters show that the current helicity of the pairs of the magnetic polarity regions has no obvious correlation. Among these pairs, 28 pairs (54%) show the same helicity signs and 24 pairs (46%) show the opposite chirality using αbest , 32 pairs (62%) show the same handedness through the parameter ρh . The force-free field extrapolation is made for these transequatorial loops using the observed longitudinal magnetic field. Some TLs have no corresponding magnetograms and some TLs haven’t definite signs through fitting the soft X-ray image. There are only 30 TLs that are extrapolated. There are 15 TLs that the twist signs are positive, 11 TLs that the twist signs are negative and 4 TLs of the extrapolation value are zero. TLs number distribution in different twist values is presented in Fig. 5.26. From the figure, we can obtain that the average value is near to 0. The TLs are extrapolated using the force-free field model, and the parameter αbest of loops are also calculated, these two quantities are comparable, so the correlation

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291

Fig. 5.24 Left: Correlation of αbest of the active region pairs connected by transequatorial loops. Right: Relationship of ρh of the active region pairs connected by TLs. X-axis presents the helicity values of active regions in the southern hemisphere, Y-axis shows the values in the northern hemisphere. Error bars (when present) correspond to 1σ of the mean helicity values from multiple magnetograms of the same active region. Points without error bars correspond to active regions represented by a single magnetogram. From Chen et al. (2007)

Fig. 5.25 Left: Correlation of αbest of the magnetic polarity pairs connected by transequatorial loops. Right: Relationship of ρh of the magnetic polarity pairs connected by TLs. X-axis presents the helicity values of magnetic polarity regions in the southern hemisphere, Y-axis shows the values in the northern hemisphere. Error bars (when present) correspond to 1σ of the mean helicity values from multiple magnetograms of the same magnetic polarity region. Points without error bars correspond to magnetic polarity regions represented by a single magnetogram. From Chen et al. (2007)

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Fig. 5.26 Distribution of the twist values of transequatorial loops. The dotted line shows the average value. From Chen et al. (2007)

can be analyzed. For the ρh , although it can represent the current helicity at a certain degree, it has a different physical meaning as the extrapolation value. Active region pair maybe connect one TL or host two TLs. If the active region pairs are only connected by one TL, we consider the helicity relationship between the TLs and the magnetic polarity region pairs; if the active region pairs are connected by two TLs, we look for the chirality correlation between the TLs and the active region pairs. These TLs are divided into two sections according to whether the two regions connected by the TLs having the same chirality or not.

5.4 Long Term Injection of Magnetic Chirality of Active Regions from Solar Subsurface The transfer of magnetic chirality in solar active regions is a relatively long-term process, which relates to the emerging magnetic flux ropes generated in the subatmosphere. This analysis has been presented based on the calculation on the injection of magnetic helicity in the solar surface. For analyzing the possible reversal magnetic helicity in both hemispheres, Fig. 5.27 shows the 195Å solar active region NOAA10484, 10486, and 10488, and the corresponding magnetogram observed by SOHO satellite. These super active regions occurred on the solar disk from 18 October to 4 November 2003 and produced an amount of unexpected eruptive events. It is found that the inverse sigmoid configuration in the difference of 195Å images in AR 10486 at 10:14–10:36UT on 2003 October 28. It is an index of magnetic helicity in the active region. This is consistent with the analysis by Zhang (2008). Active region NOAA 10484, 10486, and 10488 were new developing active regions in the solar rotation cycle in the chart of 2003 November 4, and AR 10484 and 10486 decayed significantly in the subsequent solar rotation cycles in the chart of 2003 December 1 in Fig. 5.28. The emerging sequence of these active regions

5.4 Long Term Injection of Magnetic Chirality of Active Regions …

10:14:11UT, Oct 28 AR10488

10:24:34-10:36:10UT

AR10484

AR10488

AR10486

293

09:36:03UT

AR10484

AR10486

AR10488

AR10484

AR10486

Fig. 5.27 The EIT 195Å image (left), the EIT 195Å image difference (middle), and corresponding MDI magnetogram (right, grayscale) on 2003 October 28. The top is north and right is at west. From Zhang (2012) Fig. 5.28 The local synoptic charts of MDI magnetogram in 2003 September, October, and December. The sine of latitude is ±0.5 and the range of longitude is 200◦ for the interesting region. From Zhang (2012)

synoptic 03.09.10

220

260

300

340

20

60

synoptic 03.11.04 AR10488 AR10484 AR10486 220

260

300

340

20

60

synoptic 03.12.01 AR10507 AR10501 AR10508 220

260

300

340

20

60

synoptic 03.12.28 AR10525 AR10520 AR10523+10524 220

260

300 340 Longitude (degree)

20

60

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5 Helical Magnetic Field and Solar Cycles

in late October was presented by Zhou et al. (2007). Active region 10484 (L355, N4) and 10488 (L288, N8) formed in the northern hemisphere and the transverse component of the magnetic field in these active regions rotated counterclockwise as their passage in the solar surface. Liu and Zhang (2006) found the negative magnetic helicity transported from the subatmosphere into the corona in AR 10488. AR 10486 (L293, S15) formed in the southern hemisphere. The transfer of magnetic helicity from AR 10486 was analyzed by Zhang et al. (2008a), who detected the negative magnetic helicity from the active region with the strong anti-clockwise rotation of sunspots in 2003 Oct. 25–30. It is found that the magnetic flux of AR 10486 was about 20% relative to the total flux of the Sun inferred from the MDI synoptic chart on 2003 November 4. It is comparable with the total flux of AR 10484 and 10488 in the northern hemisphere. The relationship between the accumulation of magnetic helicity and the sign of current helicity in AR10486 and 10488 has been analyzed by Liu and Zhang (2006) and Zhang et al. (2008a), who found the mean current helicity of these active regions shows the negative sign also. Active region NOAA 10484, 10486, and 10488 show the negative sign of magnetic helicity, and they are located in both solar hemispheres. The injection of magnetic helicity in these active regions is calculated by the local correlative tracker (LCT) method (Chae, 2001) with MDI 96min magnetograms. Our results for active region NOAA 10488 are slightly different from that of Liu and Zhang (2006) because they did the helicity accumulation with the high time cadence data sequence. Even though these, the same information on the evolution of injective magnetic helicity in the active regions can be found basically. Figure 5.29 shows the transfer of magnetic helicity in active region NOAA 10484, 10501, and 10520 in the southern hemisphere. These active regions occurred at almost the same location in the solar surface at continuous solar rotation cycles, which can be found in Fig. 5.28. It is found the amount of negative magnetic helicity injected from the solar subatmosphere, −5.5 × 1043 M x 2 in 20-27 Oct. 2003, −2.2 × 1043 M x 2 in 16–23 Nov. 2003, and 0.16 × 1043 M x 2 in 12–20 Dec. 2003. Figure 5.30 shows the transfer of magnetic helicity in active regions NOAA 10486, 10508, and 10523-10524 in the southern hemisphere. These active regions occurred at almost the same location in the solar surface at continuous solar rotation cycles, which can be found in Fig. 5.28. It is found the amount of negative magnetic helicity injected from the solar subatmosphere, −8.7 × 1043 M x 2 in 20–27 Oct. 2003, −4.3 × 1043 M x 2 in 16–23 Nov. 2003 and 1.2 × 1043 M x 2 in 12–20 Dec. 2003. Figure 5.31 shows the transfer of magnetic helicity in active regions NOAA 10488, 10507, and 10525 in the northern hemisphere. These active regions occurred at almost the same location in the solar surface at continuous solar rotation cycles, which can be found in Fig. 5.28. It is found the amount of magnetic helicity injected from the solar subatmosphere, −1.8 × 1043 M x 2 on 26 Oct.–1 Nov. 2003, 5.5 × 1043 M x 2 in 21–28 Nov. 2003, and 0.4 × 1043 M x 2 in 19–25 Dec. 2003. Figures 5.29–5.31 show the long-term injection of magnetic helicity from active regions. The magnetic helicity injects with the same sign (negative) in active region NOAA 10484-10501-10520 and 10486-10508-10523-10524 basically in Figs. 5.29 and 5.30, while NOAA 10488-10507-10525 does the opposite signs of helicity in

5.5 Large-Scale Soft X-ray Loops and Magnetic Chirality in both Hemispheres

295

Fig. 5.29 The injection rate (left) and accumulation (right) of magnetic helicity in the regions of Fig. 5.28 in the northern hemisphere. From Zhang (2012)

the different periods obviously in Fig. 5.31. It can be found injection processes of magnetic helicity from the subatmosphere show two tendencies: the monotonic and mixed one at the same regions in the solar surface.

5.5 Large-Scale Soft X-ray Loops and Magnetic Chirality in both Hemispheres The magnetic chirality in the solar atmosphere has been studied based on soft X-ray and magnetic field observations. It is found that some large-scale twisted soft X-ray loop systems occur for several months in the solar atmosphere, before the disappearance of the corresponding background large-scale magnetic field. It provides the observational evidence of the helicity of the large-scale magnetic field in the solar atmosphere and the reverse one relative to the helicity rule in both hemispheres with solar cycles. Using photospheric vector magnetograms from the Haleakala Stokes Polarimeter and coronal X-ray images from the Yohkoh Soft X-Ray Telescope (SXT), Pevtsov

296

5 Helical Magnetic Field and Solar Cycles

Fig. 5.30 The injection rate (left) and accumulation (right) of magnetic helicity in the regions of Fig. 5.28 in the southern hemisphere. From Zhang (2012)

et al. (1997) inferred values of the force-free field parameter α at both photospheric and coronal levels within 140 active regions. They found that both values are well correlated.

5.5.1 Magnetic Chirality of Soft X-Ray Loops Related to Solar Active Regions Figure 5.32 shows a sample of the twisted large-scale soft X-ray configuration in the solar northern hemisphere from the Yohkoh Soft X-Ray Telescope (SXT). It is found that the twisted large-scale soft X-ray configuration remained in the solar atmosphere for several months in the period of 2000 June–September, even if the topology of the soft X-ray configuration changed gradually. It is normally believed that the soft Xray configuration in the solar atmosphere provides the basic information of magnetic field, as one believes that the field is bound up in the ionized plasma.

5.5 Large-Scale Soft X-ray Loops and Magnetic Chirality in both Hemispheres

297

Fig. 5.31 The injection rate (left) and accumulation (right) of magnetic helicity in the regions of Fig. 5.28 in the northern hemisphere. From Zhang (2012)

5.5.2 Hemispheric Distribution of Helical Soft X-Ray Loops For analyzing the distribution of magnetic chirality in the solar atmosphere, in Table 5.5 we presented statistics on 753 large-scale soft X-ray loop systems from 1991 to 2001 observed by Yohkoh satellite. The handedness of soft X-ray loops can be inferred by their twist or sigmoid configuration. It is found that the handedness of soft X-ray loops statistically obeys the hemispheric sign rule. Most of them possess left (right) handedness in the northern (southern) hemisphere. It is found that the handedness for about 31% soft X-ray loops cannot be identified because their configurations are not too far from the approximation of the potential field or cannot be identified as sigmoid or twist configurations. This lack of identification does significantly not influence the trend in the ratio of the handednesses of soft X-ray loops between the northern and southern hemispheres. As these unidentified soft X-ray loop systems are ignored, one can find that the portion of the systems which are in accord with the hemispheric rule is 77.3% in the northern hemisphere and 81.5% in the southern hemisphere. It is roughly consistent with results calculated from the

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Fig. 5.32 Soft X-ray images of a part of the Sun (left), where the large-scale soft X-ray loops twist clockwise, and the corresponding photospheric magnetograms (right) in the period of 2000 June–September. The white (black) indicates the positive (negative) polarity in the magnetograms. The top is north and right is at west. From Zhang et al. (2010a)

vector magnetograms (Seehafer, 1990; Pevtsov et al., 1995; Abramenko et al., 1996; Bao & Zhang, 1998; Hagino & Sakurai, 2005; Xu et al., 2007). Figure 5.33 shows the proportion of soft X-ray loops following the hemispheric handedness rule for helicity in the northern and southern hemispheres. It is found the change of the proportion of soft X-ray loops following the hemispheric handedness rule of helicity and also their imbalance of chirality in both hemispheres. The relatively high tendency of the reverse magnetic helicity has occurred in 1991, 1992,

5.5 Large-Scale Soft X-ray Loops and Magnetic Chirality in both Hemispheres

299

Table 5.5 The statistics of handedness of soft X-ray loops in the northern and southern hemispheres Year 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 Total Nn 4 Pn 6 Qn 7 Ps 13 Ns 7 Qs 3 T otal 40

38 24 27 23 8 12 132

25 5 22 24 10 4 90

31 4 14 13 4 5 71

5 3 5 7 2 7 29

5 1 5 6 1 4 22

16 2 11 19 2 5 55

23 3 15 26 2 22 91

32 9 18 16 2 14 91

24 6 13 11 3 6 63

22 5 7 27 1 7 69

225 68 144 185 42 89 753

N is the number of soft X-ray loops with left-handedness P is the number of soft X-ray loops with right-handedness Q is the number of unidentified soft X-ray loops The subscript n and s indicate the northern and southern hemispheres, respectively 100 90 Rule trend of soft X-loops (%)

Fig. 5.33 The proportion of soft X-ray loops following the hemispheric handed rule of helicity in the northern and southern hemispheres. From Zhang et al. (2010a)

80 70 60 50 Northern hemisphere Southern hemisphere

40 30 1989

1991

1993

1995 1997 Year

1999

2001

2003

and 1995 in the northern hemisphere, while it has not been significant in the southern hemisphere. Figure 5.34 shows the statistical latitudinal distribution of soft X-ray loops in Fig. 5.33. It is found the trends of the mean latitude of soft X-ray loops to migrate toward the equator with the phase of the solar cycles following sunspots in the butterfly diagram. Because there were very few soft X-ray loops in 1991, 1995, and 1996 which are included in our statistics, the deviation from the butterfly diagram in these years can be noted. Most of the large-scale soft X-ray loops show the left (right) handedness in the northern (southern) hemisphere, which follows the handedness rule for current helicity of solar active regions, while the statistical distribution of the reverse soft X-ray loops shows left (right) handedness in the southern (northern) hemisphere) as one can see in Fig. 5.34.

300 40

Left handedness

20 Latitude (degree)

Fig. 5.34 The mean latitudinal distribution of soft X-ray loops with the left and right handedness. σ -error bars are shown by vertical lines. From Zhang et al. (2010a)

5 Helical Magnetic Field and Solar Cycles

0

-20

-40 1989

1991

1993

40

1995 1997 Year

1999

2001

2003

Right handedness

Latitude (degree)

20

0

-20

-40 1989

1991

1993

1995 1997 Year

1999

2001

2003

5.5.3 Handedness of Large-Scale Soft X-Ray Loops and Magnetic (Current) Helicity It is noticed that the synthetical analysis on the accumulation of magnetic helicity and the relationship with mean current helicity density (and also mean force-free α) is an important way to understand the basic information on dynamics of magnetic helicity in solar active regions (Zhang, 2006b). It can be used to analyze the relationship between the handedness of large-scale soft X-ray loops and the corresponding magnetic helicity in the solar atmosphere, while the helicity of large-scale soft X-ray loops is probably contributed from the nearby solar active regions and also enhanced networks. Table 5.6 shows the mean current helicity density and force-free α parameter of solar active regions calculated by vector magnetograms observed at Huairou Solar Observing Station, National Astronomical Observatories of China. These active regions show the negative sign of current helicity and they are the same sign with accumulated magnetic helicity. Active region NOAA 9033 was a fast-developing active region on 2000 June 13 in the right of Fig. 5.32, it was NOAA 9070 in the next solar rotation and it became the large-scale enhanced magnetic networks in the magnetogram of August 6. Active region NOAA 9114 located in the left of magnetogram on August 6 and active region NOAA 9149 near the right bottom on September 3

5.6 Observational Cross-Helicity on Solar Surface

301

Table 5.6 The mean current helicity density and α parameter inferred from vector magnetograms of active regions observed at Huairou relative to Fig. 5.32 Date AR Lat (deg) h c (10−2 G 2 /m) α(10−7 /m) 2000 Jun 12–15 2000 Jul 06 2000 Aug 09–11 2000 Aug 31–Sep 06

9033 9070 9114 9149

15.8 10.1 10.3 6.7

−0.329 −0.184 −0.085 −0.059

−0.209 −0.413 −0.125 −0.113

in Fig. 5.32. It can be estimated that the large-scale soft X-ray loops are mainly contributed from active region NOAA 9033 and its following rotated regions in the solar disk, while the contribution from other active regions, such as NOAA9114 and 9149, also cannot be neglected probably. This shows the consistency between the accumulation of magnetic helicity and remained handedness of the magnetic field in the solar atmosphere.

5.6 Observational Cross-Helicity on Solar Surface The cross-helicity is a quantity to measure the correlation between the velocity and magnetic fields. Woltjer (1958a) and Moffatt (1969) obtained the conservation of cross-helicity, if the fluid is inviscid, barotropic, electrically perfectly conducting, and the external body forces are conservative. The cross-helicity is normally defined  as Hχ = V B · UdV , where B and U are magnetic and velocity fields respectively. In recent years, the interest in cross-helicity has been increasing. The studies of Yoshizawa (1990) indicate that the saturation level of the induced mean magnetic field is determined through the alignment with the mean velocity under the crosshelicity effect and the evolution equation for the cross-helicity is useful for exploring mean-field dynamo properties (Yoshizawa et al., 1999, 2000). Kuzanyan et al. (2007), Pipin et al. (2011d) and Rüdiger et al. (2012) investigated the h χ =u · b, where u and b are small-scale fluctuations in velocity and magnetic fields, respectively. ... is the ensemble average of the quality under mean-field theory. They applied the mean-field method to the cross-helicity and acquired the butterfly diagram of the turbulent crosshelicity under different assumptions about the mechanisms governing the generation of the large-scale poloidal field of the Sun.

5.6.1 The Cross-Helicity Conservation Law According to Moffatt (1969), the cross-helicity conservation law can be derived briefly. For a continuous inviscid fluid, the external body forces are conservative and

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the barotropic condition p = f (ρ) is taken into account, equation of motion can be written in the form DU 1 = −∇(h + ) + (j × B) , (5.1) Dt ρ  where h = d p/ρ and  is the potential of the conservative body forces and other parameters that have general meanings. In the condition of electrically perfectly conducting, the induction equation can be written in the form B D B = · ∇U . Dt ρ ρ

(5.2)

Calculating the scalar products of Eq. (5.1) with B/ρ, and Eq. (5.2) with U, and summing them up, one can obtain a differential form of the cross-helicity conservation law     B 1 2 D U·B = · q −h− , (5.3) Dt ρ ρ 2 where q2 = U · U. Let S be any surface enclosing a volume V and moving with the fluid, and let  I =

V

B · UdV .

(5.4)

According to the conservation of mass, it can be found  B·U ρdV ρ V    1 2 q − h −  dV = (B · ) 2 V    1 2 q − h −  dS . = (n · B) 2 S

dI = dt



D Dt



(5.5)

Therefore, the cross-helicity conservation law can be obtained as the fluid is inviscid, barotropic, electrically perfectly conducting and the external body forces are conservative:  B · U = const , (5.6) V

provided that B · n = 0 on the surface of volume V, where B is the induction vector and U is the velocity field.

5.6 Observational Cross-Helicity on Solar Surface

303

5.6.2 Correlation of Data from SOHO/MDI and SDO/HMI We analyzed the correlation of the magnetograms and Dopplergrams for SOHO/MDI and SDO/HMI, respectively, and found the full disk correlation coefficient of the Dopplergrams is more than 0.85 while that of the magnetograms is about 0.47. The result is shown in Fig. 5.35. Then we calculate the correlation in different solar domains and find that near the active region the correlation is strong and in the quiet region, the correlation is weak. Therefore, the weak correlations between the full-disk magnetograms from SOHO/MDI and SDO/HMI subject to the noise in the quiet region. The noise level of line-of-sight magnetic flux is 20 G for SOHO/MDI and 10 G for SDO/HMI. As a result, the full disk correlation gotten from a single magnetogram is influenced by the noise significantly. Figure 5.36 shows the correlations of the active regions on the solar disk on September 1st, 2010. Due to the result from Korzennik et al. (2004) the position angle given by SOHO/MDI has an error of 0.25◦ . Zhao et al. (2014) do not know whether there is a similar effect for SDO/HMI, so we change the position angle to find where the correlation reaches its maximum value. The results are shown in Fig. 5.37. The results indicate that 0.4◦ is a suitable modification in the correlation analysis between the magnetograms from SOHO/MDI and SDO/HMI. Zhao et al. (2014) analyzed the correlation of the magnetograms in different smoothing scales. After reducing the resolution of the data from SDO/HMI by smoothing over 4×4 pixels, we further smoothed the magnetograms from

Fig. 5.35 The correlation of magnetograms and Dopplergrams from SOHO/MDI and SDO/HMI at 23 : 02U T on September 1st in 2010. The first column shows the magnetogram and Dopplergram from SOHO/MDI. The second column shows the magnetogram and Dopplergram from SDO/HMI. The third column shows the correlation. From Zhao et al. (2014)

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Fig. 5.36 The correlation of magnetograms and Dopplergrams from SOHO/MDI and SDO/HMI at 23 : 02U T on September 1st in 2010. The first row shows the correlation between SOHO/MDI and SDO/HMI for NOAA 11102 and the second row shows that for NOAA 11101. From Zhao et al. (2014)

Fig. 5.37 The variance of the correlation coefficients with the position angle for NOAA 11102 (a) and NOAA 11101 (b). From Zhao et al. (2014)

SOHO/MDI and SDO/HMI at a scale of 19.8 × 19.8 . Then we shrink the size of the magnetogram of SDO/HMI to 1024 × 1024 and calculated the correlation of the smoothed magnetograms, which makes a description of the large-scale correlation on the solar surface. The results are shown in Fig. 5.38. The correlation coefficient of the magnetograms is 0.91, which increases significantly compared to that in Fig. 5.36. We analyzed the variation of the correlation coefficients with the smoothing scales, which is shown in Fig. 5.39. The results show a convex monotone increasing curve as expected. When the scale is over 19.8 × 19.8 , the magnetograms are strongly correlated between SOHO/MDI and SDO/HMI.

5.6 Observational Cross-Helicity on Solar Surface

305

Fig. 5.38 The same as Fig. 5.35, but the magnetograms and Dopplergrams are smoothed with a scale of 19.8 × 19.8 . From Zhao et al. (2014)

Fig. 5.39 The variation of the correlation coefficients with the smoothing scales. The left is for magnetograms between the data from SOHO/MDI and SDO/HMI, which are observed at 23 : 02U T on September 1st in 2010. The right is for Dopplergrams. From Zhao et al. (2014)

5.6.3 Distribution of Cross-Helicity with Latitude Following the former study of the cross-helicity (Zhao et al., 2011) based on full disk Dopplergrams and magnetograms (Scherrer et al., 1995), we calculate the crosshelicity using SDO/HMI to compare with the results from SOHO/MDI. The latitude distributions of u · b gotten from the full disk data of SOHO/MDI and SDO/HMI show different tendencies. Because the full disk correlation of the magnetograms is not very high, this kind of difference is acceptable. The distributions of b and u have the same tendencies between SOHO/MDI and SDO/HMI, while it is different for the distribution of u · b obvious. Beware that the latitude distribution of u · b

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Fig. 5.40 The latitude distributions of b, u, and u · b from left to right in the period from 23 : 00 : 00 UT to 24 : 00 : 00 UT on September 1st in 2010. The top is the results √ from SOHO/MDI and the bottom is the results from SDO/HMI. The error bar is defined as 3σ/ N , where σ is the standard deviation and N is the number of the averaged pixels. From Zhao et al. (2014)

cannot be estimated by the distributions of b and u intuitively because of (u · b) = u · b. Then we recalculated the distribution of u · b with latitude using the smoothed magnetograms and Dopplergrams (19.8 × 19.8 arcsec2 ). And the difference between the distributions gotten from SOHO/MDI and SDO/HMI decreases. The correlation coefficient of the curves of the latitude distribution is 0.628, and the maps before averaged spatially have a correlation coefficient of 0.780. The results are shown in Fig. 5.40. The patterns indicate that u and b tend to have clear tendencies in different intervals.

5.7 Transfer of Magnetic Helicity Flux with Solar Cycles By comparison with the statistical analysis of the mean current helicity density of solar active regions, the study on the long-term injection of magnetic helicity in the solar surface (including active regions and magnetic networks) probably is also an important way to diagnose the source of magnetic helicity from the subatmosphere. We present the transfer of magnetic helicity from the subatmosphere based on the evolution of magnetic fields with solar cycles.

5.7 Transfer of Magnetic Helicity Flux with Solar Cycles

307

5.7.1 Observational Data Analysis of Magnetic Helicity Michelson Doppler Imager (MDI) boarded on Solar and Heliospheric Observatory (SOHO) spacecraft measures the line-of-sight velocity, line and continuum intensity, and magnetic field of the Sun (Scherrer et al., 1995). The standard observables are computed from sets of five 1024×1024 filtergrams equally spaced by 75 mÅ near a Ni I spectral line at 6768Å and the line-of-sight magnetograms are obtained from Doppler shifts measured separately in right and left circularly polarized light (Liu et al., 2004). To analyze the contribution of magnetic helicity with solar active cycles, a series of 96min MDI full disk magnetograms in 1996–2009 has been used, which includes about all of the relevant data in these years. This work is taken on the calculation of the injective magnetic helicity in a window ±475 in N-S direction and ±460 in W-E direction from the center of the solar disk, which is about ±30◦ latitude and also longitude in the solar surface in Fig. 5.41. We can find that the most of solar active regions in the solar surface are included and the projective effect of magnetic fields is negligible.

5.7.2 Long Term Transfer of Magnetic Helicity The distribution of magnetic field in the relatively quiet Sun on 2003 October 13 and that of super active regions NOAA 10484, 10486, and 10488, on 2003 October 29 have been shown in Fig. 5.41. It is noticed that the maximum amplitude of magnetic helicity flux in the relatively quiet regions of the Sun is an order of 1037 M x 2 s −1 in Fig. 5.42, which includes the contribution from enhanced networks and some active

11:11:03UT, 2003 Oct. 13

14:27:03UT, 2003 Oct. 29

AR10487 AR10488

AR10484

AR10486

Fig. 5.41 MDI full disk magnetograms on 2003 October 13 and 29. The white (black) shows the positive (negative) polarity of the magnetic field. From Zhang and Yang (2013)

308 3 2 dH/dt (1038 Mx2 s-1)

Fig. 5.42 Injective magnetic helicity flux from the northern and southern hemisphere on 2003 October–November. From Zhang and Yang (2013)

5 Helical Magnetic Field and Solar Cycles

1 0 -1 -2 -3 10-01

10-11

10-21

10-31

11-10

11-20

11-30

10-11

10-21

10-31 Date (MM-DD)

11-10

11-20

11-30

3

dH/dt (1038 Mx2 s-1)

2 1 0 -1 -2 -3 10-01

regions. This value is larger than the estimation by Welsch and Longcope (2003) that the injective helicity flux is an order of 2.9 × 1034 M x 2 s −1 in the typical quiet region of the Sun. From their result, the hemispheric mutual-helicity flux from the quiet Sun is an order of 1043 M x 2 for a whole solar cycle. The contribution of magnetic helicity in the northern hemisphere in Fig. 5.42a mainly comes from active regions NOAA 10482, 10484, 10487, and 10488 on October 18–November 3, and NOAA 10501 and 10507 on November 18-30. While the contribution of magnetic helicity in the southern hemisphere in Fig. 5.42b mainly comes from active region NOAA 10471, 10473, 10476 on October 2–8, NOAA 10483, 10485 and 10486 on October 18–November 3, and following active regions on November 20–30. It is found that the extreme value of magnetic helicity flux is −2.6 × 1038 M x 2 s −1 . As samples, the contribution of the injective helicity from active region NOAA 10486, 10487, and 10486 at the end of 2003 October has been marked in Fig. 5.42a. The helicity contribution with the morphological evolution of magnetic fields in fast-developing super active region NOAA 10486 and 10488 has been analyzed by Liu and Zhang (2006) and Zhang et al. (2008a). It is found that the strong emergence, shear, and twist of the magnetic field in these active regions cause the helicity injection significantly. Moreover, because a window of ±30◦ latitude and also longitude in the center of the solar surface has been used to calculate the injective magnetic helicity, it means that the contribution of injective helicity in a fixed location of solar surface for about 5 days only can be counted, and it cannot be tracked near the east to west limb in the solar surface for studying the long-term evolution. Figure 5.43a shows the injection of net magnetic helicity flux from both (northern and southern) hemispheres in October–November. The mean value of net injective magnetic helicity flux is −0.18 × 1038 M x 2 s −1 . It is consistent with the analysis of the injection of magnetic helicity by Liu and Zhang (2006) and Zhang et al. (2008a)

5.7 Transfer of Magnetic Helicity Flux with Solar Cycles Fig. 5.43 Net (top) and normal (bottom) injective magnetic helicity flux from both hemispheres on 2003 October–November. From Zhang and Yang (2013)

309

3

dH/dt (1038 Mx2 s-1)

2 1 0 -1 -2 -3 10-01

10-11

10-21

10-31

11-10

11-20

11-30

10-11

10-21

10-31 Date (MM-DD)

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11-20

11-30

Norm. dH/dt (10-4 s-1)

2

1

0

-1

-2 10-01

for some of the super active regions in October, and Zhang et al. (2012) for the comparison of these active regions. For analyzing the helical properties of the magnetic field, the normal injective magnetic helicity flux is defined as FmN =

d Hm 4 , (|φ+ | + |φ− |)2 dt

(5.7)

where φ+ and φ− are positive and negative magnetic flux. Figure 5.43b shows the injection of normal net magnetic helicity flux from both (northern and southern) hemispheres in 2003 October–November. It is found that the mean value of normal injective helicity flux normalized by the magnetic flux is −0.36 × 10−5 s −1 . Figure 5.44 shows the injection of magnetic helicity from the subatmosphere in the northern and southern hemispheres in 2003. The mean value of injective magnetic helicity flux in the northern hemisphere is −0.15 × 1037 M x 2 s −1 and that in the southern one is 0.23 × 1037 M x 2 s −1 . It is consistent with the hemispheric rule of magnetic helicity (Seehafer, 1990). It is found that the mean value of net injective magnetic helicity flux in 2003 is 0.79 × 1036 M x 2 s −1 , and the mean value of normal injective helicity flux is 0.22 × 10−6 s −1 . This means that the net injective helicity in 2003 shows the opposite sign relative to that in 2003 October–November. Figure 5.45 shows the injection of magnetic helicity flux in 1996–2009 after the average in each solar rotation cycle for providing its long-term evolution in the solar surface. It is found the extreme values (about −2.0 × 1037 M x 2 s −1 ) of injective negative helicity flux from the solar surface occurred in 2001–2002 and (about 1.6 × 1037 M x 2 s −1 ) of positive one in 2002–2003 in the northern hemisphere, while that (about 2.0 × 1037 M x 2 s −1 ) in 2000–2001 and (about −1.7 × 1037 M x 2 s −1 ) in

310 3 2 dH/dt (1038 Mx2 s-1)

Fig. 5.44 Injective magnetic helicity flux from the northern and southern hemispheres in 2003. From Zhang and Yang (2013)

5 Helical Magnetic Field and Solar Cycles

1 0 -1 -2 -3 01-01 02-01 03-01 04-01 05-01 06-01 07-01 08-01 09-01 10-01 11-01 12-01 01-01 3

dH/dt (1038 Mx2 s-1)

2 1 0 -1 -2 -3 01-01 02-01 03-01 04-01 05-01 06-01 07-01 08-01 09-01 10-01 11-01 12-01 01-01 Date (MM-DD)

2003–2004 in the southern hemisphere. The net injective magnetic helicity from both hemispheres did not show a significant tendency, while the extreme values (about −2.0 × 1037 M x 2 s −1 ) and (about 2.2 × 1037 M x 2 s −1 ) of injective helicity flux from the solar surface occurred in 2000–2001. It is found that the major contribution of negative helicity flux tends to occur in 1997–2002, after the long-term smooth of net magnetic helicity flux in Fig. 5.45. This is roughly consistent with the result on the injection of large-scale magnetic helicity inferred from the MDI synoptic charts of the magnetic field (Yang & Zhang, 2012). The mean injective flux is −1.76 × 1036 M x 2 s −1 in the northern hemisphere and 1.41 × 1036 M x 2 s −1 in the southern hemisphere in 1996–2009. The contribution of negative helicity flux is 73% relative to the total helicity flux in the northern hemisphere and positive one is 62% in the southern hemisphere. It is consistent with the hemispheric sign rule of magnetic (current) helicity statistically (Seehafer, 1990; Pevtsov et al., 1995; Abramenko et al., 1996; Bao & Zhang, 1998; Hagino & Sakurai 2005; Xu et al., 2007). Berger and Ruzmaikin (2000) analyzed the helicity injection with solar cycles and found that throughout the 22 year cycle studied(1976–1998) the helicity production in the interior by differential rotation had the correct sign compared to observations of coronal structures - negative in the north and positive in the south. The net helicity flow into each hemisphere over this cycle was approximately 4 × 1046 M x 2 . Georgoulis et al. (2009) that accounting for various minor underestimation factors, they estimated a maximum helicity injection of 6.6 × 1045 M x 2 for solar cycle 23. Figure 5.46 shows the injective rate of the total absolute value of magnetic helicity flux in both hemispheres and the relationship with sunspot numbers in 1996–2009. The injective rate of the total absolute value of helicity flux is the sum of the absolute value of positive and negative ones. The mean injective rate of total helicity is

5.7 Transfer of Magnetic Helicity Flux with Solar Cycles

311

Fig. 5.45 Injective magnetic helicity flux from the northern (top left) and southern (top right) hemisphere, and net injective helicity from both hemispheres (bottom) in 1996–2009. The dotted line marks the net injective helicity after the smooth of 48 solar rotation cycles. The shadow areas mark the period without the relevant helicity calculation from the magnetograms. From Zhang and Yang (2013)

Fig. 5.46 Total magnetic helicity flux (thin solid line) and sunspot numbers (thin dotted line) in 1996–2009, while the black (red) thick dot-dashed line shows the total helicity flux (sunspot numbers) after smoothed significantly

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5 Helical Magnetic Field and Solar Cycles

2.40 × 1037 M x 2 s −1 in the calculated region in the solar disk. This provides a fundamental estimation of total injective magnetic helicity flux in about 5.0 × 1046 M x 2 in the 23rd solar cycle, which is a similar order with the estimation by Berger and Ruzmaikin (2000) and the calculation by means of a series of MDI magnetic synoptic charts by Yang and Zhang (2012). As the projective effects of the magnetic field in the solar surface and the contribution of helicity flux from the high latitudes have been estimated, the injective total magnetic helicity flux is an order of or larger than 5.0 × 1046 M x 2 in the solar cycle 23. As comparing the evolution of mean current helicity density of active regions with solar cycle (Yang et al., 2012), the relationship between the injective magnetic helicity flux in the solar surface and sunspot numbers with the solar cycle is shown in Fig. 5.46. It is found that the injection of total magnetic helicity flux is roughly consistent with the variation of sunspot numbers in the solar atmosphere. This also means that the magnetic fields of solar active regions bring the magnetic helicity from the subatmosphere dominantly. The correlative coefficient between both is 0.899 in our calculation. For quantitatively analyzing the correlation between magnetic helicity flux and sunspot numbers in detail, two years’ annual correlative coefficients between them in 2000–2001, 2002–2003, 2004–2005, and 2006–2007 are calculated. These are 0.53, 0.83, 0.76, 0.70, and 0.67, respectively. 2000–2001 is the solar maximum and 2006–2007 is the solar minimum in the 23rd solar cycle. This means that the low correlation probably tends to occur near the solar maximum of solar cycle 23. It is also noticed that the total magnetic helicity flux tends to delay than the total sunspot numbers, as comparing both smoothed ones in Fig. 5.46. The maxim of sunspot numbers occurs in 2001, while that of current helicity in 2002 after the smooth. This is consistent with the that the maxim of butterfly diagram of calculated current helicity of solar active regions delays than that of sunspot numbers in Fig. 5.6. It also means that the sunspot numbers do not reflect the relevant contribution of magnetic helicity from the subatmosphere completely. Corresponding evidence is that the magnetic helicity flux is not linearly proportional to the magnetic flux in solar active regions as one normalized the injective magnetic helicity by the magnetic flux in Fig. 5.43.

5.8 Statistical Studies on Photospheric Magnetic Nonpotentiality of Active Regions and Associated Flares with Solar Cycles A statistical study is carried out on the photospheric magnetic nonpotentiality in solar active regions and its relationship with associated flares. Yang et al. (2012) select 2173 photospheric vector magnetograms from 1106 active regions observed by the Solar Magnetic Field Telescope (SMFT) at Huairou Solar Observing Station (HSOS) from 1988 to 2008, which covers most of the 22nd and 23rd solar cycles. Of all the vector magnetograms, only one magnetogram is picked up for an AR each day. Most of the selected samples are observed during 1:00 to 7:00 UT when

5.8 Statistical Studies on Photospheric Magnetic Nonpotentiality …

313

Table 5.7 Number distributions of selected magnetograms (MGs) and active regions (ARs) from 1988 to 2008 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 MGs ARs MGs ARs

10 8 1999 166 88

37 21 2000 361 166

47 30 2001 285 136

108 58 2002 233 123

138 65 2003 144 77

97 52 2004 67 34

74 39 2005 101 50

56 25 2006 46 22

11 6 2007 20 11

54 32 2008 4 2

114 62 Total 2173 1106

HSOS is in relatively better seeing and atmospheric conditions. 2173 observed vector magnetograms containing 1106 ARs from June 1988 to March 2008 are chosen as samples. Table 5.7 lists the selected magnetograms and ARs each year in the two cycles. Various types of AR are included in these samples, from simple unipolar regions to complex δ ARs. The one-to-one match between Huairou and NOAA AR numbers is manually done for analyzing the relationship between the magnetic characteristics of an AR and associated flares. Thus the flare records taken by Geostationary Operational Environmental Satellite (GOES) can be used in the study.

5.8.1 Magnetic Nonpotentiality and Complexity Parameters 5.8.1.1

Magnetic Shear Angle

The magnetic shear is one of the typical parameters to describe the nonpotentiality of magnetic fields in the solar photosphere. Hagyard et al. (1984) and Lü et al. (1993) introduced the planar and three-dimensional magnetic shear angles, respectively. The three-dimensional magnetic shear angle ψ is the angle between the directions of the observed vector magnetic field Bo and its corresponding potential field Bp . The planar magnetic shear φ, which can be considered as the projection of ψ onto the plane of the sky, is defined as the azimuthal difference between the observed field and the potential field. ψ is a more direct measure of deviation from a potential field than φ. Here φ and ψ are angles between the directions of two vectors: φ = (B to , Bt p ) = arccos

ψ = (B o , B p ) = arccos





 Bto · Bt p , |Bto ||Bt p |

(5.8)

 Bo · B p , |Bo ||B p |

(5.9)

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5 Helical Magnetic Field and Solar Cycles

where the subscripts o and p indicate the observed and potential magnetic fields, respectively, and t refers to the transverse component of the magnetic field.

5.8.1.2

Vertical Current Density, Current Helicity Density, and Twist Parameter α

According to Ampére’s law and ignoring the effect of the electric displacement current, the electric current density is described by J = 1/μ0 (∇ × B), and the vertical component of electric current density is obtained by Jz =

1 1 (∇ × B)z = μ0 μ0



∂ By ∂ Bx − ∂x ∂y

 zˆ,

(5.10)

where Bx and B y are the two perpendicular components of a horizontal magnetic field, and μ0 = 4π × 10−6 G km A−1 is the  permeability in free space. The current helicity is defined as Hc = V B · (∇ × B) dV , and the vertical current helicity density can then be calculated by  (h c )z = Bz (∇ × B)z = Bz

∂ By ∂ Bx − ∂x ∂y

 .

(5.11)

So far there is no observational method to obtain the transverse component of current helicity density (h c )t , and the symbol h c is usually used to represent its vertical component. Based on the force-free field assumption, the Lorentz force is equal to zero, i.e. the electric current is parallel to the local magnetic field: (∇ × B) × B = 0,

or

∇ × B = αB.

(5.12)

If the magnetic field is approximated to be a linear force-free field, the local twist α is expressed as α=

(∇ × B)z . Bz

We use αav (Hagino & Sakurai, 2004) to characterize the dominated twist status of an AR:  (∇ × B)z · sign[Bz ]  . (5.13) αav = |Bz |

5.8 Statistical Studies on Photospheric Magnetic Nonpotentiality …

5.8.1.3

315

Free Magnetic Energy Density

The energy released through solar flares and other explosive events relies on the accumulation of the free magnetic energy (non-potential magnetic energy), which is defined as the difference between the total magnetic energy (E) and potential magnetic energy (E p ): E = E − E p . Hagyard et al. (1981) introduced the source field to describe the nonpotentiality of a magnetic field on the photosphere: Bs = Bo − Bp , where Bo is the observed vector magnetic field, Bp denotes the potential field extrapolated from the longitudinal component of Bo , and Bs is the so-called source field which is the non-potential component of the magnetic field. The analogous magnetic energy density defined by Lü et al. (1993) is proportional to B2s : ∗ ρenergy =

(Bo − B p )2 B2s = . 8π 8π

(5.14)

As one replaces the free energy density by Eq. (5.14) as the same as Lü et al. (1993), it is deduced that   Bo B p (Bo − B p )2 ∗ 2 ψ + sin , ρ f r ee = 8π 2π 2 in which Bo = |Bo |, Bp = |Bp |, and ψ is the shear angle between Bo and Bp . This parameter has been used in the following analysis of solar magnetic activities with solar cycles (Yang et al., 2012). It should be noticed that the definition of the real free energy density (Eq. 3.1) is ρfree = E o − E p =

Bo2 − Bp2 8π

,

which is different from that of Eq. (5.14).

5.8.1.4

Effective Distance

Effective distance (dE ), a structural parameter of an AR, proposed by (Chumak and Chumak 1987), presented a distinction between flare-quiet and flare-imminent ARs (Chumak et al. 2004). As a quantified magnetic complexity, dE depicts the degree of

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5 Helical Magnetic Field and Solar Cycles

the isolation or mutual penetration of the two polarities of an AR in a geometrical sense (Guo et al. 2006 and see Sect. 4.5.5). dE is calculated from dE =

Rp + Rn , Rpn

(5.15)

where  Rp =

Ap , Rn = π



An . π

Ap (An ) is the total area of positive (negative) polarity, Rp (Rn ) is the equivalent radius of positive (negative) polarity, respectively, and Rpn is the distance between the flux-weighted centers of the two opposite polarities (Guo et al. 2006, 2007; Guo and Zhang 2007; Guo et al. 2010). Considering the limitation of the parameter dE to uniquely characterize the complexity of an AR, an additional factor |Bz | is multiplied to dE . The modified effective distance dEm is defined as dEm = dE |Bz | =

Rp + Rn |Bz |. Rpn

(5.16)

dE is dimensionless, but |Bz | provides dEm some practical physical meaning to some extent. Here, the dimension of dEm is the same as that of the magnetic field, in the units of Gauss. dEm also reflects the complexity of an AR and could be considered as the weighted complexity of the entire AR in terms of the averaged strength of the sunspot magnetic field.

5.8.2 Statistical Analysis and Results 5.8.2.1

Strength Distribution of Nonpotentiality During Solar Cycles 22–23

According to the magnetic nonpotentiality and complexity measures described, the mean values of the planar magnetic shear angle (φ), shear angle of the vector magnetic field (ψ), unsigned vertical current density (|Jz |), unsigned current helicity density (|h c |), and free magnetic energy density (ρ f r ee ), and also the unsigned twist parameter (|αav |), effective distance of the longitudinal magnetic field (d E ), and the modified effective distance (d Em ) are calculated for each photospheric vector magnetogram. In each of the magnetograms, only the areas where the strength of longitudinal magnetic field is greater than 20 G are used in the calculations for the parameters |Jz |, |h c |, |αav |, ρfree , and dEm . The two mean shear angles φ and ψ are obtained on the pixels with both the transverse magnetic field greater than 200

5.8 Statistical Studies on Photospheric Magnetic Nonpotentiality …

317

G and the strength of longitudinal field greater than 20 G, which are distributed at the sunspot penumbra and along the polarity inversion lines. The longitudinal field strength of 80 G is chosen as the lower threshold for obtaining dE . The active samples (i.e., flare-productive ARs) are defined as the ARs with the equivalent flare strength (i.e. flare index FI defined in Sect. 5.8.2.2) greater than a typical value (such as M1.0 here) within the same subsequent time window. The rest of them belong to the quiet samples (i.e., flare-quiet ARs). Figure 5.47 shows the yearly mean values of the active and the quiet samples for each parameter. The error

60

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Fig. 5.47 Yearly mean values of φ, ψ, |Jz |, |h c |, |αav |, ρfree , dE , and dEm of AR samples during 1988–2008. Dots represent the yearly mean values of the samples that did not produce flares with FI ≥ 10.0 in the following 24 h (flare-quiet samples). Diamonds denote the yearly mean values of the samples that produced flares with FI ≥ 10.0 in the following 24 h (flare-productive samples). The monthly mean sunspot numbers during the same period are overlapped (dashed line) for reference. From Yang et al. (2012)

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bars indicate the standard deviations of the yearly mean values. The plot of monthly mean sunspot numbers (dashed line) is overlaid in each panel for reference. φ and ψ have similar distributions. The linear correlation coefficient between them is 0.875. Neither of the two shear angles varies with the solar cycle. Most of their values are distributed in the range of [10 ◦ , 40 ◦ ]. The points of ψ are more concentrated in this range. The mean value of ψ is 22.0 ◦ with the standard deviation of 6.6 ◦ during 1988–2008, while φ is 23.4 ◦ ± 7.6 ◦ . Combining with Fig. 5.47, there is no obvious fluctuation of the parameter |Jz | with solar cycles, in which most of AR samples have |Jz | ranging in 1.0–3.0 × 103 A km−2 . Some active samples at solar maximum have larger |Jz |. The samples with |Jz | larger than 2.5 × 103 A km−2 are likely connected with eruptive active regions. During the period from 2001 to 2003, which is the beginning of the declining phase of solar cycle 23, the values of |Jz | are mostly lower than the general level of 2.0 × 103 A km−2 (91.7%). After the year 2003, the values of |Jz | rise to the average level again. The distributions of |Jz | are very similar in form to the two shear angles, φ and ψ. In the panel of |αav |, the values mostly concentrate in the range of 0–1.5 × 10−5 km−1 . There are no obvious changes in the parameter |αav | with the solar cycle. The difference of |αav | between flare-productive and flare-quiet samples is insignificant except in the years 2004 and 2005. The parameters |h c |, ρfree , and dEm follow nicely the variation of solar activity cycle. There are two bell-shaped parts in the corresponding panels during these two solar cycles. The overall level of these parameters rises gradually toward the solar maximum; while near the solar minimum they all fall down. Comparing with the monthly mean sunspot numbers, the troughs of 1989–1990 may be caused by the lack of the available samples or by some unknown uncertainties or evolution processes, but this result is consistent with Bao and Zhang (1998) who have surveyed the evolution of the average current helicity in solar cycle 22. Table 5.8 lists the linear correlation coefficients between these eight parameters and the mean sunspot numbers in the yearly and monthly bases. These coefficients indicate that there is a much closer relationship between the mean sunspot number and |h c |, ρfree , and dEm . Comparing the two parameters dE and dEm representing complexity in magnetic configuration, dE of some ARs in the activity maximum periods is higher than that in

Table 5.8 Linear correlation coefficients between eight parameters and the mean sunspot number (SSN), in yearly and monthly bases φ ψ |Jz | |h c | |αav | ρfree dE dEm Mean SSN (yearly) Mean SSN (monthly)

0.379

0.010

0.093

0.594

−0.176

0.666

0.799

0.814

0.242

0.085

0.054

0.448

−0.067

0.474

0.367

0.530

5.8 Statistical Studies on Photospheric Magnetic Nonpotentiality …

319

the minimum periods, but the number of those ARs with higher dE are not many in the samples, and most of the samples are distributed in a narrow band of ± 0.93 around dE = 1.04. Combined with the average strength of magnetic fields, dEm reflects the magnetic complexity of an AR more accurately, while dE is only related to the morphology but limited physics. Both dE and dEm change with the solar cycle, suggesting that there are more complex ARs in the activity maximum periods and less complex ARs in the activity minimum periods. In Fig. 5.47, there can be seen a small peak in 2005 in each of the subplots. Checking the monthly mean sunspot number and the monthly 10.7cm solar radio flux, there is no abnormal variation around 2005. There are only a few more flares in 2005 and at the end of 2004. Table 5.9 shows a comparison between the declining phases of solar cycles 22 and 23. We choose two years with roughly the same mean number of sunspots to compare the corresponding parameters, which means the solar activity level is similar in the two years of each pair. The exception is the mean sunspot number of 2002 and 2003 to compare with the number of 1992. The quantities in Table 5.9 represent the relative differences between the values of cycle 23 and cycle 22. The total activity level in the declining phase of cycle 23 is generally lower than that in cycle 22, and coincidentally the parameters in Table 5.9 follow the same tendency, that is, the parameter values of cycle 23 are lower than of cycle 22. However, all the parameters of the active samples have larger positive differences between 2005 and 1994. This shows that the nonpotentiality in flare-productive ARs is strong in 2005. A non-monotonic decline of cycle 23 might be regarded as one of the precursors to the long and deep minimum between solar cycles 23 and 24.

5.8.2.2

Nonpotentiality Associated with Flares

One of the generally accepted flare classifications is the SXR classification. From 1975 to the present time, the GOES satellite has been recording the whole-Sun X-ray fluxes at 0.5–4 Å (hard channel) and 1–8 Å (soft channel) wavelength bands. The SXR flare classification (B, C, M, and X classes) utilizes the flux in the 1–8 Å range based on the order of magnitude of the peak burst intensity. For analyzing the relationship between the nonpotentiality and solar flares, a time window τ has been defined as the period forward in time (toward later times) from the observed vector magnetogram. Within a fixed time window τ , the SXR flare index summed by weighting the flares of different classes is used to measure the flaring capability of an AR: FI = 100

 τ

IX + 10

 τ

IM +



IC ,

(5.17)

τ

where IX , IM , and IC represent the indexes of X-, M-, and C-class SXR flare events, respectively, (Antalova, 1996; Abramenko, 2005). Because the X-ray background in the solar maximum is too high to detect B-class flares (Feldman et al., 1997; Joshi et al. 2010, cf.), the records of B-class SXR flares are not included in the computation.

|αav |

|Jz |

φ

02 & 03 versus 92 04 versus 93 05 versus 94 06 versus 95 07 versus 96 02 & 03 versus 92 04 versus 93 05 versus 94 06 versus 95 07 versus 96 02 & 03 versus 92 04 versus 93 05 versus 94 06 versus 95 07 versus 96

Year

−0.057 0.082 0.407 0.165 0.295 −0.358 0.092 0.591 0.212 −0.032 0.891 1.576 6.398 1.980 0.192

AS −0.398 −0.335 −0.183 −0.170 0.031 −0.560 −0.272 0.136 0.086 −0.244 −0.135 −0.236 1.325 0.280 −0.042

QS −0.329 −0.298 −0.101 −0.126 0.070 −0.509 −0.240 0.199 0.102 −0.212 0.032 −0.074 2.028 0.502 −0.008

TS

ρfree

|h c |

ψ 02 & 03 versus 92 04 versus 93 05 versus 94 06 versus 95 07 versus 96 02 & 03 versus 92 04 versus 93 05 versus 94 06 versus 95 07 versus 96 02 & 03 versus 92 04 versus 93 05 versus 94 06 versus 95 07 versus 96

Year 0.009 0.244 0.747 0.372 0.450 −0.603 −0.505 0.074 −0.208 −0.493 −0.609 −0.415 0.687 0.027 −0.565

AS −0.269 −0.259 0.218 0.068 0.129 −0.906 −0.609 −0.442 −0.336 −0.538 −1.001 −0.696 −0.407 −0.467 −0.502

QS

(continued)

−0.215 −0.214 0.291 0.108 0.177 −0.753 −0.600 −0.370 −0.320 −0.531 −0.768 −0.671 −0.256 −0.402 −0.511

TS

Table 5.9 Comparison of yearly averaged parameters between the declining phases of solar cycles 22 (1992–1996) and 23 (2002–2007). The tabulated values are relative differences between the values of cycle 23 and cycle 22. AS stands for active samples, QS for quite samples, and TS for total samples. Each of the paired years has comparable yearly mean sunspot numbers

320 5 Helical Magnetic Field and Solar Cycles

dE

02 & 03 versus 92 04 versus 93 05 versus 94 06 versus 95 07 versus 96

Year

Table 5.9 (continued)

0.165 −0.099 0.454 −0.350 −0.267

AS

TS −0.212 −0.189 −0.320 −0.288 −0.193

QS −0.282 −0.197 −0.444 −0.278 −0.180 dEm

02 & 03 versus 92 04 versus 93 05 versus 94 06 versus 95 07 versus 96

Year −0.105 −0.425 0.319 −0.432 −0.441

AS −0.661 −0.491 −0.637 −0.455 −0.384

QS

−0.521 −0.485 −0.505 −0.452 −0.393

TS

5.8 Statistical Studies on Photospheric Magnetic Nonpotentiality … 321

322

5 Helical Magnetic Field and Solar Cycles

In Fig. 5.47, the yearly mean values of each parameter for flare-productive ARs are higher than those for flare-quiet ones in the solar maximum periods, especially in the panels of |h c |, ρfree , and dEm . However, the twist factor |αav | shows an insignificant difference between the active samples and quiet samples, but shows more apparent difference with much higher uncertainty in 2005. For the same τ , the yearly mean values of each parameter for the flare-productive ARs increase as the threshold of FI is increased, and the differences between flare-productive ARs and flare-quiet ARs are gradually enlarged accordingly. In reverse, for the same threshold of FI, the yearly mean values of each parameter for the flare-productive ARs decrease as the time window τ is increased, and the differences between flare-productive and flarequiet ARs are correspondingly a little reduced. Setting a series of τ and different thresholds of FI, the yearly mean values will show basically similar distributions and evolution trends for each parameter. It suggests that the flare-productive ARs are more likely to have relatively strong nonpotentiality and great complexity, and those parameters characterizing nonpotentiality may be applied as indicators in the flare prediction (Yang et al., 2013). Furthermore, to study the flaring probability of an AR with certain properties of nonpotentiality and complexity, the solar flare productivity is adopted (Cui et al., 2006, 2007; Cui & Wang, 2008; Park et al., 2010), which is defined as P(X ) =

NA (≥ X ) , NT (≥ X )

(5.18)

where NT (≥ X ) is the total number of samples with the values of each parameter greater than its threshold X , and NA (≥ X ) is the number of active samples with the values exceeding the same X . The Boltzmann sigmoid function is used for fitting the data of flare productivity with the eight gradually varied parameters. φ and ψ have nearly the same tendency. |Jz | is also similar to the above two but rises faster at the value around 2.5 × 103 A km−2 . |h c | and |αav | have a similar increasing tendency in this plot. The shape of the fitting curve of ρfree bears a resemblance to that of dEm . When the value of dE is greater than 1.0, there are almost no distinct differences in the flaring capability. These plots reveal different flare productivity levels and their respective variation tendencies, as the result of the distributions and variations of the eight parameters in Figs. 5.47. The linear correlation coefficients between nonpotentiality parameters and complexity parameters d E and d Em are listed in Table 5.10. dEm has a relatively close relation with |h c | and ρfree . dEm has relatively higher positive correlation with nonpotentiality than dE , while dE have almost no relation with non-potential parameters. Nonpotentiality and complexity depict ARs from different aspects, thus it is normal that there is a weak correlation between them. But both of them do have positive correlations with the eruption probability of ARs.

5.8 Statistical Studies on Photospheric Magnetic Nonpotentiality …

323

Table 5.10 Linear correlation coefficients between nonpotentiality and complexity parameters φ ψ |Jz | |h c | |αav | ρfree dE d Em

0.259 0.330

0.186 0.241

0.162 0.357

0.220 0.566

0.076 0.069

0.242 0.618

5.8.3 Summary for Different Nonpotential Magnetic Parameters By calculating eight parameters (φ, ψ, |Jz |, |h c |, |αav |, ρfree , dE , and dEm ) of nonpotentiality and complexity for 2173 photospheric vector magnetograms in 1106 ARs associated with flares, we found the main results as follows: (1) Two mean magnetic shear angles φ and ψ, mean absolute vertical current density |Jz |, absolute twist factor |αav |, and effective distance dE in ARs do not change significantly with the global solar activity level. However, it is more likely that these parameters show higher values in the solar maximum than in the solar minimum. (2) The mean unsigned current helicity density |h c |, mean free magnetic energy density ρfree , and modified effective distance dEm show high positive correlation with the mean sunspot number, and these parameters also have relatively close relationship with each other. The Pearson linear correlation coefficients of the above three with the yearly mean sunspot numbers are larger than 0.59. They can be used to characterize the solar activity level as well as the traditional sunspot number. (3) The nonpotentiality and complexity parameters between flare-productive ARs and flare-quiet ones are useful to understand the evolution of flare-productive ARs and the relationship with the magnetic activity levels of the cycles. These nonpotentiality and complexity parameters may be synthetically applied as indicators to predict solar flares with some weight. (4) Due to the loss of the information of magnetic field strength in the parameter of effective distance dE , the modified effective distance dEm (including the strength of the magnetic field) turns out to be much better in indicating the magnetic activities of ARs.

Chapter 6

Magnetic Helicity with Solar Dynamo

The research on solar dynamo can be said to begin with the discovery of the 11year cycle from the statistical observation of sunspot activity and to explore the reasons for its formation. The observation results of the full solar disk magnetic field have enabled to discover the overall migration law of the large-scale longitudinal magnetic field on the surface of the sun for 22 years (see Fig. 6.1). The 11-year sunspot cycle is half of Babcock–Leighton’s recommended 22-year solar dynamo cycle, which corresponds to an oscillatory exchange of energy between toroidal and poloidal solar magnetic fields. At this point in the dynamo cycle, it is believed that within the convection zone forces emergence of the toroidal magnetic field through the photosphere, giving rise to pairs of sunspots, roughly aligned east-west with opposite magnetic polarities. The magnetic polarity of sunspot pairs changes every solar cycle, and this phenomenon is called the Hale cycle. In recent years, a large number of observations of the vector magnetic field in solar active regions have allowed people to re-discuss the theoretical basis and observational perspectives of solar generators from the perspectives of the non-potential, current, and magnetic helicity of the solar magnetic field. Discuss the shortcomings or deficiencies of solar generator theory with longitudinal field observation data. In 1955, Parker suggested a dynamo mechanism in the solar interior based on the combined action of differential rotation and cyclonic convective vortices as a viable way to generate magnetic fields capable of driving the activity cycle. We can quantify the differential rotation from the motion of large-scale magnetic fields at the solar surface and helioseismology in the solar convection zone. However, because of the opacity of the solar atmosphere, knowledge of the action of convective vortices can only be obtained from the magnetic helicity inferred from available observations of magnetic fields. This means that the magnetic fields in the solar surface observed using vector magnetographs just provide an important chance as the discovery of the solar dynamo process.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Zhang, Solar Magnetism, https://doi.org/10.1007/978-981-99-1759-4_6

325

326

6 Magnetic Helicity with Solar Dynamo

Fig. 6.1 The butterfly diagram for the magnetic fields (from Hathaway, 2010)

6.1 Solar Dynamo and Helicity 6.1.1 Mean-Field Solar Dynamo In mean-field theory one solves the Reynolds averaged equations, using either ensemble averages, toroidal averages, or in cases in cartesian geometry with periodic boundary conditions, two-dimensional (e.g., horizontal) averages (Krause and Rädler, 1980; Brandenburg and Subramanian, 2005a). We thus consider the decomposition U0 = U + u,

H = B + b,

A0 = A + a,

(6.1)

here U, B, and A are the mean velocity, magnetic fields, and magnetic potentials, while u, b, and a are their fluctuating parts. These averages satisfy the Reynolds rules, U1 + U2 = U1 + U2 , U = U, Uu = 0, U1 U2 = U1 U2 , ∂U/∂t = ∂U/∂t,

∂U/∂x = ∂U/∂x.

Some of these properties are not shared by several other averages; for Gaussian filtering U = U, and for spectral filtering U U = UU, for example. Note that U = U implies that u = 0. In the remainder, we assume that the Reynolds rules do apply. Averaging ∂H = ∇ × (U0 × H − η∇ × H), ∂t

(6.2)

yields then the mean-field induction equation,

and

∂B = ∇ × (U × B + E − η∇ × B), ∂t

(6.3)

∂b = ∇ × (U × b + u × B + G − η∇ × b), ∂t

(6.4)

6.1 Solar Dynamo and Helicity

327

where E = u × b and G = u × b − u × b.

(6.5)

We also can obtain the equation of magnetic potential as H = ∇ × A0 ,‘

then

and

∂A0 = U0 × H − η∇ × H + ∇φ, ∂t

(6.6)

∂A = U × (∇ × A) + E + η∇ 2 A, ∂t

(6.7)

∂a = U × (∇ × a) + u × A + G − η∇ × a, ∂t

(6.8)

where η is the magnetic diffusivity coefficient and φ is the scalar potential. If ∇(U · A) − A × (∇ × U) − (A · ∇)U = 0, we can obtain ∂A + (U · ∇)A = E + η∇ 2 A. ∂t

(6.9)

We obtain the mean-field dynamo equations and ignore the overline on the mean values of physical quantities for the simplification in the following sometimes ∂B + (U · ∇)B =(B · )U + ∇ × E + η∇ 2 B, ∂t ∂A + (U · ∇)A =E + η∇ 2 A. ∂t

(6.10)

According to mean-field dynamo theory, the electromotive force E averaged over convective eddies has a component parallel to the magnetic field, E = αB + · · · , where the pseudo-scalar α is related to kinetic and electric current helicities 1 α = − 13 τ0 (u · (∇ × u) − μρ b · (∇ × b)) + · · · . The quantity b · (∇ × b) can be statistically detected from vector magnetograms in the solar surface. Here · · ·  means averaging, and τ0 is an order of the correlation time. The fundamental point is that the α-effect includes two contributions (Pouquet et al., 1976), a hydrodynamical contribution as discussed above (αv ) associated with helicity of convective vortices, and also a contribution from the helicity of the magnetic field itself (αm ). The hydrodynamic helicity is determined by a correlation between the convective velocity u and its curl, i.e., u · (∇ × u), and so its observational determination requires knowledge of all three components of velocity while the Doppler effect gives a line-of-sight velocity component only. The magnetic part of the α-effect, αm , can be related to what has become known as the current helicity, proportional to b · (∇ × b), where b is the small-scale magnetic field.

328

6 Magnetic Helicity with Solar Dynamo

For a detailed introduction to the theory of mean-field dynamos, please refer to the monographs (such as, Krause and Rädler, 1980; Zeldovich et al., 1983; Rüdiger and Hollerbach, 2004; Brandenburg and Subramanian, 2005a). It’s not too much to expand here.

6.1.2 Equation for Magnetic Helicity Now ones multiply Eq. (6.4) by a and Eq. (6.8) by b, add them and average over the ensemble of turbulent fields. This yields an equation for the magnetic helicity χm = a p (x)b p (x) Kleeorin and Rogachevskii (1999) ∂χm ¯ = −2u × b · B − 2ηb · (∇ × b) − ∇ · F, ∂t

(6.11)

where F¯ p = V p χm − χmpn Vn + a × u × B − ηa × (∇ × b) + a × (u × b) − bφ. Electromotive force for anisotropic turbulence is given by ˆ − η∇ ˆ × B, u × b = V D M × B + αB

(6.12)

where ηˆ ≡ ηˆmn = (η pp δmn − ηmn )/2, and ηmn = ηδmn + η¯mn , and η¯mn = τ u m u n , and (VD M ) = −∇m η¯mn /2 is the velocity caused by the turbulent diamagnetism, and (v) (B) (v) (B) + αmn , then tensors αmn and αmn are given by αˆ = αmn = αmn (v) = −[ m ji τ u i (x)∇n u j (x) + n ji τ u i (x)∇m u j (x)]/2, αmn (B) = [ m ji τ bi (x)∇n b j (x) + n ji τ bi (x)∇m b j (x)]/(2μ0 ρ). αmn

(6.13) (6.14)

Substituting Eq. (6.12) into Eq. (6.11) ones obtain after simple manipulations an equation for the magnetic helicity:  2 m  ∂ χ ∂χm = −2η + 2ηmn Bm (∇ × B)n − 2αˆ mn Bm Bn − ∇ · F, ∂t ∂x p ∂ y p (r →0) (6.15)   where ones used an identity b · (∇ × b) = ∂ 2 χm /∂x p ∂ y p (r →0) and r = x − y. The second and third terms in Eq. (6.15) describe the sources of magnetic helicity. Therefore, the mean magnetic field B, the mean electric current ∝ ∇ × B and the hydrodynamic helicity are the sources of the magnetic helicity. The first term in Eq. (6.15) determines the relaxation of the magnetic helicity with the characteristic time T which depends on the molecular magnetic diffusion. This time is given by T

−1

2η =− m χ



∂ 2 χm ∂x p ∂ y p

 . (r →0)

(6.16)

6.2 Radial Distribution of Magnetic Helicity in …

329

The characteristic relaxation time T of the magnetic helicity is T ∼ τ0 Rm, i.e., it is much longer than the correlation time τ0 = l0 /u 0 of the turbulent velocity field, where u 0 is the characteristic turbulent velocity in the maximum scale of turbulent motions l0 . Magnetic Reynolds Number: Rm = l0 u 0 /η. The last term in Eq. (6.15) describes the turbulent flux F of the magnetic helicity. Equation (6.15) in the case of isotropic turbulence coincides with that derived in (Kleeorin and Ruzmaikin, 1982) (see also Gruzinov and Diamond, 1995; Kleeorin et al., 1995).

6.2 Radial Distribution of Magnetic Helicity in Solar Convective Zone: Observations and Dynamo Theory 6.2.1 Velocity Structure of Solar Convection Zone The convection zone is the outer-most layer of the solar interior. It is generally believed that the thickness of the solar convection zone is about 200,000 kilometers (about 0.3 solar radius). At the base of the convection zone the temperature is about 2,000,000◦ C. The research results show that the hydrogen in the convection zone is continuously ionized, increasing the specific heat of the gas, destroying the hydrostatic balance, and causing the hot gas to rise. The fluid element is almost in adiabatic state; once the cold fluid element drops, the temperature is lower than the surroundings and the density is higher, and it continues to fall. This forms convection. The fluid expands and cools as it rises. At the visible surface the temperature has dropped to 5,700 K and the density is only 0.0000002 gm/cm. Through observational studies of helioseismology, there are large-scale macroscopic flows within the solar convection zone, including the differential rotation of the convective zone and the meridian circulation. Since the internal magnetic field of the sun is strongly restricted by the flow field, this large-scale flow field plays an extremely important role in the study of the formation and evolution of the internal magnetic field of the sun. The Sun has an equatorial rotation speed of 2 km/s; its differential rotation implies that the angular velocity decreases with increased latitude. The poles make one rotation every 34.3 days and the equator every 25.05 days. Figure 6.2 shows the radial distribution of the angular velocity in the convective zone inside the sun predicted by helioseismological observations Gilman and Howe (2003). By means of helioseismological observations, Zhao et al. (2013) carry out two sets of inversions with and without the mass-conservation constraint. For the results inverted from only the helioseismic measurements without applying the massconservation constraint, a two-dimensional cross-sectional view of the meridionalflow velocity and flow profiles at some selected depths and latitudes are displayed in Fig. 6.3. These helioseismological observations provide important constraints on the formation of the magnetic field inside the Sun.

330

6 Magnetic Helicity with Solar Dynamo

Fig. 6.2 Isolines of the angular velocity in the outer part of the Sun after Gilman and Howe (2003). Note that no informations exist for the polar regions. The isolines are radial only in the higher midlatitudes. From Moffatt (1978)

Fig. 6.3 Meridional flow profile, obtained by inverting the measured acoustic travel times. Panel a shows a cross-sectional view of the meridional-flow profile, with the positive velocity directing northward. Panels b and c show the inverted velocity as functions of latitude averaged over several depth intervals. Panels d and e show the velocity as functions of depth averaged over different latitudinal bands. Horizontal error bars represent the widths of averaging kernels in the latitudinal direction (panels b and c) and the radial direction (panels d and e). From (Zhao et al., 2013)

6.2 Radial Distribution of Magnetic Helicity in …

331

Table 6.1 Current helicity Hc for active regions binned by depth, threshold σ = 0.5 Here and below Hc is measured in units of 10−3 G2 m−1 Depth N N∗ Hc N ∗ /N North s m d d+m South s m d d+m

47 5 34 39

1 1 8 9

−0.6 ± 0.2 −0.2 ± 0.7 −1.0 ± 0.7 −0.9 ± 0.6

0.02 ± 0.04 0.20 ± 0.35 0.24 ± 0.14 0.23 ± 0.13

41 6 38 44

5 2 11 13

0.5 ± 0.6 0.3 ± 1.5 0.6 ± 0.4 0.6 ± 0.4

0.12 ± 0.10 0.33 ± 0.38 0.29 ± 0.14 0.3 ± 0.13

6.2.2 Current Helicity Data Obtained at Huairou Solar Observing Station Our research is based on the current helicity data accumulated during 10 consequent years (1988–97) of observations at the Huairou Solar Observatory Station of the National Astronomical Observatories of China (Bao and Zhang, 1998) which were processed further by Zhang et al. (2002). Based on the distribution of the solar internal angular rotation rate with the depth of the convection zone in Fig. 6.2 and following Kuzanyan et al. (2003), we divide the active region into 4 groups, i.e., shallow, middle and deep active regions, as well as a group for which the depth cannot be estimated satisfactorily (Table 6.1). The separation of the active regions into three groups is based on the result of helioseismology (Schou et al., 1998) that the angular rotation rate growth monotonically with radius at least for the domain between fractional radii 0.65 and 0.95 and latitudes below 30–35◦ (for details see, Kuzanyan et al., 2003). The Solar Geophysical Data records, which can be obtained from the NOAA (USAF-MWL) database, provide us with several tens of longitudinal locations (in terms of the Carrington coordinate system) for each active region under investigation, for several consequent days. Therefore, we attempt to calculate partial, or “individual”, angular rotation rates to the Carrington rotation. For some active regions we can find a certain trend in the evolution of their Carrington coordinates with time. From the total sampling of data, which contain 410 active regions, we select subsamples for which this trend in Carrington longitude versus time has significant correlation. We determined the subsamples for which the correlation coefficient σ is greater than 0.5 and 0.6 respectively. These samples contain 178 and 134 active regions (or 43% and 33% of the available data), respectively. Because the current helicity is expected to be of opposite sign in northern and southern hemispheres, (Zhang et al., 2006) subdivide these groups between the two hemispheres and average the data in each group over all latitudes as well as cycle

332

6 Magnetic Helicity with Solar Dynamo

Table 6.2 Current helicity for active regions binned by depth, threshold σ = 0.6 North Depth s m d d+m South s m d d+m

N 33 2 28 30

N∗ 1 1 7 8

Hc −0.6 ± 0.3 −0.3 ± 9.4 −1.0 ± 0.8 −1.0 ± 0.8

N ∗ /N 0.03 ± 0.06 0.5 ± 0.69 0.25 ± 0.16 0.27 ± 0.16

33 3 29 32

4 2 9 11

0.6 ± 0.7 −0.2 ± 4.8 0.4 ± 0.4 0.4 ± 0.4

0.12 ± 0.11 0.7 ± 0.53 0.31 ± 0.17 0.34 ± 0.16

Hc

N ∗ /N

−0.8 ± 0.2 0.6 ± 0.2

0.16 ± 0.05 0.22 ± 0.05

Table 6.3 Current helicity for all 410 active regions Hemisphere N N∗ North South

193 217

30 47

phases. The result of averaging Hc is given in Table 6.1 for active regions with identified depth. Here d is a depth identifier, with “s” meaning shallow, “m” middle and “d” deep active regions. Because the number of active areas of middle depth appears to be quite low, and insufficient to estimate the sign of helicity, we combine quite arbitrarily the data for the middle and deep active regions into a single group, i.e., “d+m”. N is the number of active regions included in each group. For Table 6.1, we use the threshold σ = 0.5. To demonstrate the stability of the selection procedure to the threshold value, we give in Table 6.2 similar results for the threshold value σ = 0.6. In agreement with theoretical expectations, the data for Hc are remarkably antisymmetric in respect to the solar equator. Note that the same kind of antisymmetry was recognized in the averaging over latitude or time undertaken in Kleeorin et al. (2003). We note however that there are a significant number of active regions that violate this polarity law. The number of such active regions is given in Tables 6.1 and 6.2 as N ∗ . We present in Table 6.3 averaged values of the helicities for all 410 active regions for which the observations of helicity are available. These active regions follow the same polarity rule as the active regions with known depth, and again some active regions violate this rule. Their number is given as N ∗ . The number of active regions with current helicity that violate the polarity rule can be calculated for both hemispheres (Table 6.4). Note that it is not appropriate to average the current helicity over both hemispheres because the data in the northern and southern hemispheres cancel. We conclude from Table 6.4 that the deep (and

6.2 Radial Distribution of Magnetic Helicity in …

333

Table 6.4 Number of active regions with current helicity violating the polarity rule, binned by depth, threshold σ = 0.5 Depth N N∗ N ∗ /N s d+m

88 83

6 22

0.07 ± 0.05 0.27 ± 0.09

Table 6.5 Number of active regions with current helicity violating the polarity rule ordered by date, threshold σ = 0.5 Years N N∗ N ∗ /N 1988–89 1990–91 1992–93 1994–96

87 126 121 69

23 20 18 13

0.26 ± 0.09 0.16 ± 0.06 0.15 ± 0.06 0.18 ± 0.09

Table 6.6 Number of active regions with current helicity violating the polarity rule, ordered by latitude , threshold σ = 0.5 Latitude (deg) N N∗ N ∗ /N 24 ≤  ≤ 32 16 ≤  ≤ 24 12 ≤  ≤ 16 8 ≤  ≤ 12 −8 ≤  ≤ 8 −12 ≤  ≤ −8 −16 ≤  ≤ −12 −24 ≤  ≤ −16 −32 ≤  ≤ −24

18 53 36 48 65 58 46 67 12

4 10 5 8 6 12 8 19 3

0.22 ± 0.19 0.19 ± 0.11 0.14 ± 0.11 0.17 ± 0.11 0.08 ± 0.06 0.21 ± 0.10 0.17 ± 0.11 0.28 ± 0.11 0.25 ± 0.25

middle) active regions contain several times more cases of parity rule violations than the shallow active regions, and even slightly more than the active regions without definite estimation of depth. We were unable to recognize any clear trend in the number of active regions violating the polarity rule selected according to latitude or the cycle phase. However, we present the relevant data below (Tables 6.5 and 6.6).

6.2.3 Dynamo Model We use here a dynamo model which is an extension of the simplified model of Kleeorin et al. (2003). In particular, the present model includes an explicit radial

334

6 Magnetic Helicity with Solar Dynamo

coordinate and takes into account the curvature of the convective shell, and also quenching of turbulent magnetic diffusivity. We start from the general mean-field dynamo equations (see, e.g., Moffatt, 1978; Krause and Rädler, 1980). Using spherical coordinates r, θ, φ we describe an axisymmetric magnetic field by the azimuthal component of magnetic field B, and the magnetic potential of the poloidal field, A. Following Parker (1955a) we consider dynamo action in a convective shell. However, we retain a radial dependence of A and B in the dynamo equations and we do not neglect the curvature of the convective shell. The equations for A˜ = r sin θ A and B˜ = r sin θ B read    2˜ V A ∂ A˜ 1 ∂ A˜ ∂ A ∂ A˜ ∂ A˜ sin θ ∂ + θ + VrA = Cα α B˜ + η A + (6.17) ∂t r ∂θ ∂r ∂r 2 r 2 ∂θ sin θ ∂θ     ˜ sin θ ∂ VθB B˜ ∂(VrB B) ∂ B˜ ∂ ∂ ˜ + + = sin θ G r − Gθ A ∂t r ∂θ sin θ ∂r ∂θ ∂r     η B ∂ B˜ ∂ ∂ B˜ sin θ ∂ + ηB (6.18) + 2 ∂r ∂r r ∂θ sin θ ∂θ

where

Gr =

∂ ∂ , Gθ = . ∂r ∂θ

Here we measure lengths in units of the solar radius R and time in units of a diffusion time based on the solar radius and the turbulent magnetic diffusivity η. When estimating this timescale we use the “basic” (assumed uniform) value of the turbulent magnetic diffusivity, unmodified by the magnetic field. We consider the fractional radial range 0.64 < r < 1, where r = 0.64 corresponds to the bottom of the convective zone and r = 1 corresponds to the solar surface. The “convection zone” proper can be thought of as occupying 0.7 ≤ r ≤ 1.0, with 0.64 ≤ r ≤ 0.7 being a tachocline/overshoot region. The rotation law includes radial shear (proportional to G r ) and a latitudinal dependence (proportional to G θ ). At the surface r = 1 we use vacuum boundary conditions on the field, i.e., B = 0 and the poloidal field fits smoothly onto a potential external field. At the lower boundary, r = r0 = 0.64, B = 0 = Br = 0. At both r = r0 and r = 1, ∂χc /∂r = 0, where χc is the current helicity (see. Eq. (6.19)). Of course, these equations, although more elaborate than those often used to study the solar cycle, are still oversimplified. However, they appear adequate to reproduce the basic qualitative features of solar (and stellar) activity. Taking into account the exploratory nature of the approach, we use the simplest profiles of dynamo generators compatible with symmetry requirements and with producing a magnetic butterfly diagram that is concentrated toward low latitudes (see also Rüdiger et al., 1995; Moss and Brooke, 2000). Thus α(B = 0) = sin2 θ cos θ and Cα < 0 (this determines the sign value of the hydrodynamic α effect, see below in Sect. 6.2.4). The points θ = 0

6.2 Radial Distribution of Magnetic Helicity in …

335

and θ = 180◦ correspond to the North and South poles, respectively. See (Kleeorin et al., 2003) for further discussion of this approach. As a new feature of Eqs. (6.17) and (6.18), compared with the dynamo model exploited by Kleeorin et al. (2003), we retain here the possibility of including a contribution from the dynamo generated magnetic field in the turbulent diffusion coefficients (η A and η B ), and the meridional circulation (VθA , VrA , VθB , and VrB ). However, we do not consider fully here the role of meridional circulation. √ The magnetic field is measured in units of the equipartition field Beq = u 4πρ, and the vector potential of the poloidal field A is measured in units of R Beq . The density ρ is normalized to its value at the bottom of the convective zone, and the basic scales of the turbulent motions l and turbulent velocity u at the scale l are measured in units of their maximum values through the convective zone. Because turbulent diffusivity and α-effect depend on the magnetic field, we use their initial values in the limit of the very small mean magnetic field to obtain the dimensionless form of the equations. To emphasize this, we do not introduce the dynamo number in an explicit form here however use it below when convenient.

6.2.4 The Nonlinearities We present below a model for the nonlinear dynamo saturation. The model is based as far as possible on first principles and is similar to that used in the derivation of the equations of mean-field electrodynamics by Krause and Rädler (1980). As an important technical point, we use here a quasi-Lagrangian approach in the framework of Wiener path integrals to derive the dynamical equation for the evolution of the magnetic helicity including magnetic helicity flux (see Kleeorin and Rogachevskii, 1999). We also used the τ -approximation (Orszag’s third-order closure procedure) to determine the nonlinear mean electromotive force (see Rogachevskii and Kleeorin, 2000; Rogachevskii and Kleeorin, 2004). Here we note some important features of the model only. Note that a rigorous investigation of the turbulent diffusion of current helicity looks possible in principle. It would require at least the application of Orszag’s fourth-order closure procedure to derive the turbulent diffusion helicity flux. We stress again that the model analyzed is derived from the first principles as far as possible. The scope of the model is however obviously limited and does not include all possible physical mechanisms which could in principle contribute to dynamo saturation. In particular, we do not include the buoyancy of the magnetic field. Some other limitations are mentioned below. Bearing in mind t the natural limitations of the model, we introduce several numerical coefficients C1 , C2 , C3 , which we consider to be free parameters of order unity (see also Kleeorin et al., 2003 and Eq. (6.24) below).

336

6.2.4.1

6 Magnetic Helicity with Solar Dynamo

The α-effect

The key idea of the dynamo saturation scenario exploited below (as well as by Kleeorin et al. (2003)) is the splitting of the total α-effect into its hydrodynamic and magnetic parts. The calculation of the magnetic part of the α-effect is based on the idea of magnetic helicity conservation and the link between current and magnetic helicities and gives (see Kleeorin et al., 2000; Kleeorin et al., 2003—Appendix 6.2.7) α = χv φv +

φm c χ . ρ(z)

(6.19)

Here χv and χc are the hydrodynamic and current helicities, respectively, φv and φm are quenching functions which describe a link between the helicities and the corresponding contribution to the α-effect, e.g., for a kinematic dynamo φv is determined by the turnover time τ of the vortices of the largest scale. The analytical form of the quenching functions φv (B) and φm (B) is given in Appendix 6.2.7. In contrast to Kleeorin et al. (2003), we consider here the radial helicity profiles in an explicit form and so we keep in Eq. (6.44) the radial profile of density ρ(z) normalized by the density at the bottom of the convective zone. Based on Baker and Temesvary (1966) and Spruit (1974), we choose for ρ(z) the analytical approximation ρ(z) = exp[−a tan(0.45π z)]

(6.20)

where z = 1 − μ(1 − r ) and μ = (1 − R0 /R )−1 . Here a ≈ 0.3 corresponds to a tenfold change of the density in the solar convective zone, a ≈ 1 by a factor of about 103 , etc. However in the majority of our investigations, we took ρ = const., but we did also consider cases with a = 0.3. The conservation of magnetic helicity density can be written in the form χc 1 ∂χc + = (E · B +  · F) + κ∇ 2 χc , ∂t T 9πηρ

(6.21)

Ei = αi j B j + (V× B)i − ηi j (∇ × B) j ,

(6.22)

∂ Bl . ∂xk

(6.23)

where

or Ei = αi j B j + εi jk V j Bk − ηi j ε jkl

6.2 Radial Distribution of Magnetic Helicity in …

337

The equation for χ˜ c = r 2 sin2 θχc is     χ˜ c 2R 2 1 η B ∂ A˜ ∂ B˜ ∂ χ˜ c + = ∂t T l Cα r 2 ∂θ ∂θ   ˜ ˜ 1 ∂ A˜ ∂A ∂B sin θ ∂ ∂ 2 A˜ ˜ − ηA B 2 − η A B˜ 2 +η B ∂r ∂r r ∂θ sin θ ∂θ ∂r  ˜ ˜ ˜ ∂A B ∂A + (VθA − VθB ) − α B˜ 2 +(VrA − VrB ) B˜ ∂r r ∂θ

 sin θ ∂ F˜ θ ∂ F˜r − − , ∂r r ∂θ sin θ

(6.24)

where F˜ = r 2 sin2 θF, and the flux of the magnetic helicity is chosen in the form F = C1 η A (B) B 2 ∇[χv φv (B)] + C2 χv η A (B) φv (B) B 2 ρ + C3 κ∇χc ,

(6.25)

with ρ = −∇ρ/ρ. Eq. (6.24) is a generalization of Eq. (A.3) of Kleeorin et al. (2003) to the case considered here. R /l is the ratio of the solar radius to the basic scale of solar convection (we take (2R /l)2 = 300), T = (1/3)Rm(l/R )2 is the dimensionless relaxation time of the magnetic helicity, Rm = l u/η0 is the magnetic Reynolds number and η0 is the “basic” magnetic diffusion due to the electrical conductivity of the fluid. Note that T = 5 and Rm = 106 at the depth h ∗ = 108 cm (measured from the top of the convective zone), T = 150 and Rm = 3 × 107 at the depth h ∗ = 109 cm, and T = 107 and Rm = 2 × 109 at the depth h ∗ = 2 × 1010 cm. We appreciate that various estimates for the magnetic Reynolds number for the solar convective zone have been suggested and so we investigate below the robustness of our results with respect to T . Note also that if we average the parameter T over the depth of the convective zone, we obtain T ∼ 5 (see Kleeorin et al., 2003). 6.2.4.2

Turbulent Diffusivity

The simplest order-of-magnitude estimates for magnetic field turbulent diffusion suggest that it affects all magnetic field components similarly. Of course, this does not preclude that a more detailed parameterization of the turbulent transport coefficients could result in different estimates for the turbulent diffusion η B of toroidal and η A of poloidal magnetic field components, and Rogachevskii and Kleeorin (2004) provide the following estimates for the coefficients η B and η A for the cases of the weak and strong magnetic fields (remember that we measure magnetic field strength in units of equipartition value and that for the Parker migratory dynamo the toroidal magnetic field is much stronger than the poloidal). For the case of the weak magnetic field, the diffusion coefficients are ηA = 1 −

96 2 B , η B = 1 − 32B 2 , 5

(6.26)

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6 Magnetic Helicity with Solar Dynamo

while for strong magnetic fields the scaling is ηA =

1 , 8B 2

1 ηB = √ 3 2B

(6.27)

where the magnetic field is measured in units of the equipartition field Beq . Unsurprisingly, the coefficient of turbulent diffusion of magnetic helicity κ also has a dependence on B, namely, κ(B) = 1 − 24B 2 /5 for weak magnetic field and   3π 1 1+ (6.28) κ(B) = 2 40B in the strong-field limit. The theory gives more general formulae for these asymptotical expressions (see Rogachevskii and Kleeorin, 2004 and Appendix 6.2.7) We note that the turbulent diffusion estimates depend on the details of magnetic field evolution during which the magnetic helicity accumulated. In particular, the initial ratio between magnetic and kinetic energy appears in the complete equations of Rogachevskii and Kleeorin (2004). We appreciate the importance of this factor which is almost unaddressed in existing papers in dynamo theory. We accept (rather arbitrarily) that dynamo action starts in an (almost) non-magnetized medium. Also, we neglect the effects of possible inhomogeneities in the background turbulence. 6.2.4.3

Nonlinear Advection

Our model contains an inhomogeneous nonlinear suppression of turbulent magnetic diffusion, which causes turbulent diamagnetic (or paramagnetic) effects, i.e., a nonlinear advection of magnetic field which is not the same for the toroidal and poloidal parts of the magnetic field. The corresponding velocities were calculated by Rogachevskii and Kleeorin (2004) yielding   32 2 er + cot θ eθ (B) + 3ρ − B 5 r   32 2 er + cot θ eθ B 3ρ − VB = 5 r VA =

for a weak magnetic field, and   5 1 er + cot θ eθ (B)  +2 + VA = − √ ρ r 16B 2 3 8B 5 4 er + cot θ eθ + VB = √ ρ r 16B 2 3 8B for strong fields. Here (B) = (B 2 )/B 2 , er and eθ are unit vectors in the r and θ directions of spherical polar coordinates, [ρ ]r = −d ln ρ/dr, and [ B ]r = d ln B 2 /dr.

6.2 Radial Distribution of Magnetic Helicity in …

6.2.4.4

339

Rotation Law

In the region 0.7 ≤ r ≤ 1, we used an interpolation on the rotation law derived from helioseismic inversions. This was extended to include a tachocline region by interpolating between the helioseismic form at r = 0.7 and solid body rotation at r = r0 (see also Moss and Brooke, 2000). Our choice r0 = 0.64 gives a rather broad tachocline, but simplifies the numerics.

6.2.5 Numerical Implementation and Nonlinear Solution We simulate the model described in the meridional cross section of the spherical shell with 0 ≤ θ ≤ 180◦ and 0.64 ≤ r ≤ 1. We divide the region (rather arbitrarily) into 3 domains, namely, 0.64 ≤ r < 0.7, 0.7 ≤ r ≤ 0.8 and 0.8, ≤ r ≤ 1, and identify them with the domains of the deep, middle, and shallow active regions of Sect. 6.2.3. We attempt to identify the relative volume occupied by the current helicity of the “improper” sign with N ∗ /N . Our simulations show that the dynamo model leads to a steadily oscillating magnetic configuration for a quite substantial domain in the parameter space. These parameters seem acceptable when compared with current ideas in solar physics. We present here as a typical model with steady oscillations the case Cα = −5, Cω = 6 · 104 , i.e., D = −3 × 105 , C1 = C2 = 1, C3 = 0.5, T = 5 and (2R/l)2 = 300. Of course, the level of our knowledge concerning helicity transport inside the Sun is very far from being adequate to determine the numerical value of these parameters. The parameter set chosen gives a realistic time scale for the cycle period (about 10 years), but with a rather small nominal value of the diffusivity coefficient η0 , i.e., this is how we choose to resolve the well-known problem with the length of the solar cycle in the context of mean-field dynamo models. The value |Cα | (and |D|) chosen is perhaps larger than expected because we use the profile χv = sin2 θ cos θ, which significantly reduces the mean value of χv over the domain compared to that with the “standard” χv = cos θ. We note that the results are quite robust to the introduction of a radial dependence of χv and a substantial increase in T . We demonstrated that our results are robust at least in respect to the parameter T which is associated with magnetic Reynolds number. For T = 0.5 (i.e., smaller by a factor of 10 than in the basic run described above) we still obtain regular oscillations and the magnetic energy increases by a factor of 2 or 3 only. For still smaller values of T the solution becomes irregular. For this solution, the magnetic energy E m measured in the units of its equipartition value oscillates near the level E m ≈ 0.12, and the amplitude of the oscillations is about 0.035. This means that the averaged magnetic field strength is about 40% of the equipartition value. The magnetic configuration can be described as a system of activity waves which can be presented in the corresponding butterfly diagrams. In Fig. 6.4 we show the near-surface butterfly diagram (at r = 0.94). Here, a pair of activity waves migrate from the middle latitudes toward the solar equator, while

340

6 Magnetic Helicity with Solar Dynamo 90

latitude

latitude

90

0

−90

8.01

8.03

8.05

8.07

0

−90

8.09

8.01

8.03

8.05

time

8.07

8.09

time

Fig. 6.4 The near-surface (r = 0.94, left) and above the interface (r = 0.70, right) butterfly diagram of the mean magnetic field. Contours are equally spaced: solid represent positive values, broken negative, and the zero contour is shown as dotted. From Zhang et al. (2006)

another pair migrates from the middle latitudes toward the poles. We present in Fig. 6.4 butterfly diagrams for the region just above the interface (at r = 0.70). Here, both pairs of activity waves are much less pronounced in comparison to the structure shown in Fig. 6.4. The equatorward branch is however more pronounced compared to the polar in Fig. 6.4. From these synthetic plots, it seems plausible that the observed butterfly diagram can be mimicked adequately. The magnetic field structure found in the simulations is also quite consistent with expectations. As a typical example, we give in Fig. 6.5 the toroidal magnetic field distribution for an instant soon after the minimum of magnetic energy. The current helicity distribution at the same time is given in Fig. 6.6. Here the dotted line indicates the zero contour of current helicity. The helicity distribution is anti-symmetric with respect to the solar equator, but changes sign inside each hemisphere. If the helicity is basically positive in a given hemisphere, a region of negative helicity can be isolated

1

radius

Fig. 6.5 The toroidal magnetic field distribution at an instant just after the minimum of magnetic activity. Contours are equally spaced: solid represent positive values, broken negative, and the zero contour is shown as dotted. From Zhang et al. (2006)

0.9 0.8 0.7 0.64

0

100

theta

180

6.2 Radial Distribution of Magnetic Helicity in …

341

Fig. 6.6 The current helicity distribution. The dotted line here indicates the zero level of current helicity. Contours are equally spaced: solid represent positive values, broken negative, and the zero contour is shown as dotted. From Zhang et al. (2006)

90

latitude

Fig. 6.7 The butterfly diagram for the current helicity for the region just above the interface (r = 0.70). Contours are equally spaced: solid represent positive values, broken negative. From Zhang et al. (2006)

0

−90

8.01

8.03

8.05

8.07

8.09

time

near the equator at the base of the convective zone. The other region of opposite polarity in the helicity distribution is located near the poles. Near the bottom of the convection zone, the helicity pattern migrates in a similar way to the toroidal field, and the corresponding butterfly diagram is given in Fig. 6.7. Quite unexpectedly, the helicity pattern near the surface does not demonstrate any pronounced migration. The normalized local nonlinear dynamo number D N = α(B)/[η A (B)η B (B)] at every point of the computational grid, as a function of the mean magnetic field, is shown in Fig. 6.8. Here α(B) is normalized by the local value of α(B = 0). The nonlinear dynamo number decreases with the increase of the mean magnetic field. The latter dependence implies the saturation of the growth of the mean magnetic field in the nonlinear mean-field dynamo. Note that the dynamo number which is based on the hydrodynamic part αv of the α-effect increases with the mean magnetic field.

342

6 Magnetic Helicity with Solar Dynamo

Fig. 6.8 The normalized local nonlinear dynamo number at all grid points, as a function of the mean magnetic field. From Zhang et al. (2006)

This shows the very important role of the magnetic part αm of the α-effect, which causes the saturation of the growth of the mean magnetic field.

6.2.6 Helicity Distribution We need to reduce the numerical data from our modeling to a form comparable with the observations available. The important point is that the resolution of the mean helicity from the observations is very substantially lower than that of the sunspot data, not to mention that of the dynamo simulations. We apply the following procedure to reduce the resolution of the numerical data, and so allow us to make a meaningful comparison with the observations. We isolate a region 60◦ < θ < 120◦ , i.e., a ±30◦ –belt centered on the equator, because helicity data are available for this equatorial domain only. We separate it into a deep region, 0.64 < r < 0.8, and a shallow region with r > 0.8, and consider one hemisphere only, say the Northern (the simulated data are strictly anti-symmetric with respect to the solar equator). Let D+ and D− be the volumes inside each region a positive and negative sign, respectively. We calculate the values I+ = χ has where c c χ d V dt and I = χ d V dt, where Tc is the half length of the activity − Tc D+ Tc D− cycle (note that I− is negative). From our basic run, we obtain the following values of helicity integrals. For the “deep” region (0.64 ≤ r ≤ 0.8), we obtain I− = −5.4 × 10−5 and I+ = 2.1 × 10−5 ,

6.2 Radial Distribution of Magnetic Helicity in …

343

while for the “shallow” region 0.8 ≤ r ≤ 1.0 we obtained I− = −2.2 × 10−4 and I+ = 0. The clear difference in helicity distribution between deep and shallow regions remains robust when a is reduced to 0.5. For the deep region (0.64 ≤ r ≤ 0.8), we then obtain I+ = −I− = 4.4 × 10−5 (of course, the equality is a pure coincidence) while for 0.8 ≤ r ≤ 1 we obtained I− = −2.3 × 10−4 and I+ = 0. Note that the choice of the latitudinal and radial belts in which the helicity integrals are calculated significantly affects the numbers above. For the basic run, calculating the helicity integrals for the whole northern hemisphere we obtain I− = −2.7 × 10−3 and I+ = 7.2 × 10−4 for 0.7 ≤ r ≤ 0.8, I− = −5.9 × 10−3 and I+ = 3.6 × 10−4 for 0.8 ≤ r ≤ 0.9 and I− = −6.1 × 10−3 , I+ = 3.5 × 10−4 for 0.8 ≤ r ≤ 1.0. Obviously, these values of helicity integrals calculated for rather arbitrary chosen belts are less impressive compared with the previous, where the belts were isolated on the basis of snapshots of the helicity distribution. The important is that a link between helicity integrals and depth is still visible here. We have demonstrated that the available observational data concerning solar current helicity give some hints concerning its radial distribution. The active regions clearly associated with the upper layers of the solar convective zone demonstrate a significantly more homogeneous distribution of current helicities than the deeper regions. We interpret this as an observational indication that the structure of the solar activity wave deep inside the Sun is substantially more complicated than near its surface. In contrast to a rather smooth structure of the surface activity wave with the dominant pattern propagating from the middle latitudes to the equator, we expect a more complicated structure of activity waves deep inside the Sun. In particular, the waves with “wrong” polarity deep inside the Sun are expected to be more important compared to the main wave than nearer the surface.

6.2.7 Appendix: Quenching Functions The quenching functions φv (B) and φm (B) appearing in the nonlinear α effect are given by √ 1 [4φm (B) + 3L( 8B)] 7

 √ 3 arctan( 8B) φm (B) = 1− √ 8B 2 8B φv (B) =

(6.29) (6.30)

(see Rogachevskii and Kleeorin, 2000), where L(y) = 1 − 2y 2 + 2y 4 ln(1 + y −2 ). The nonlinear turbulent magnetic diffusion coefficients for the mean poloidal and toroidal magnetic fields, η A (B) and η B (B), and the nonlinear drift velocities of poloidal and toroidal mean magnetic fields, V A (B) and V B (B), are given in dimensionless form by

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6 Magnetic Helicity with Solar Dynamo

η A (B) = A1 (4B) + A2 (4B) 3 η B (B) = A1 (4B) + [2 A2 (4B) − A3 (4B)] 2 (B)  V2 (B) V A (B) = V1 (B) + (er + cot θ eθ ) + V(ρ) (B) 2 r V3 (B) V B (B) = (er + cot θ eθ ) + V(ρ) (B) r

(6.31) (6.32) (6.33) (6.34)

where 3 A3 (4B) − 2 A2 (4B) 2 1 V2 (B) = A2 (4B) 2 3 V3 (B) = [A2 (4B) − A3 (4B)] 2 1 V(ρ) (B) = ρ [−5A2 (4B) + 3A3 (4B)], 2 V1 (B) =

Rogachevskii and Kleeorin (2004). The functions Ak (y) are     6 arctan y 1 5 5 A1 (y) = 1+ 2 + L(y) − 2 5 y 7y 14 7y     2 15 15 6 arctan y 1 + 2 − L(y) − 2 A2 (y) = − 5 y 7y 7 7y   2 arctan y 2 (y + 3) − 3 . A3 (y) = − 2 y y The nonlinear quenching of the turbulent magnetic diffusion of the magnetic helicity is given by κ(B) =

  1 1 1 + A1 (4B) + A2 (4B) . 2 2

(6.35)

6.3 Current Helicity of Active Regions as a Tracer of Large-Scale Solar Magnetic Helicity For comparison, we also studied a more basic 2-D mean-field dynamo model with simple algebraic α quenching only. Using these numerical models we obtained butterfly diagrams both for the small-scale current helicity and also for the large-scale

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345

magnetic helicity, and compared them with the butterfly diagram for the current helicity in active regions obtained from observations. This comparison shows that the current helicity of active regions, as estimated by −A · B evaluated at the depth from which the active region arises, resembles the observational data much better than the small-scale current helicity calculated directly from the helicity evolution equation.

6.3.1 The Role of Helicities in Magnetic Field Evolution As the dynamo amplifies the large-scale magnetic field, the large-scale magnetic helicity HM = A·B grows in time (but not monotonically in a cyclic state). We can obtain from Eqs. (6.2) and (6.6) Zhang et al. (2012) ∂ (A · B) = − 2E · B +  · (ϕB + A × E) ∂t =2E · B − 2ηJ · B − ∇ · F M ,

(6.36)

F M = −ϕE − A × E.

(6.37)

where

The evolution of the large-scale magnetic helicity HM can be written in the form ∂ HM + ∇ · FM = 2E · B − 2HC , ∂t

(6.38)

Kleeorin et al. (1995), Blackman and Field (2000a), Brandenburg and Subramanian (2005a), where E = u×b is the mean electromotive force that determines generation and dissipation of the large-scale magnetic field, 2E · B is the source of the large-scale magnetic helicity due to the dynamo generated large-scale magnetic field, and F M is the flux of large-scale magnetic helicity that determines its transport. Since the total magnetic helicity over all scales, HM + Hm integrated over the turbulent fluid, is conserved for very small magnetic diffusivity, the small-scale magnetic helicity changes during the dynamo process, and its evolution is determined by the dynamic equation ∂ Hm + ∇ · F = −2E · B − 2Hc ∂t

(6.39)

Kleeorin et al. (1995), Blackman and Field (2000a), Brandenburg and Subramanian (2005a), where −2E · B is the source of the small-scale magnetic helicity due to the dynamo generated large-scale magnetic field, F is the flux of small-scale magnetic helicity that determines its transport and 2η Hc = Hm /Tm is the dissipation rate of the small-scale magnetic helicity. It follows from Eq. (6.38) and (6.39) that the source

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6 Magnetic Helicity with Solar Dynamo

of the small-scale and the large-scale magnetic helicities is located only in turbulent regions (i.e., in our case, in the solar convective zone). The magnetic part of the α effect is determined by the parameter χc = τ Hc /(12πρ), and for weakly inhomogeneous turbulence χc is proportional to the magnetic helicity: χc = Hm /(18πηT ρ) Kleeorin and Rogachevskii (1999), Brandenburg and Subramanian (2005a), where ρ is the density and ηT is the turbulent magnetic diffusion. It is worth noting that the content discussed here is different from the concepts introduced in Sect. 6.1.2. It is an extension on the conservation of helicities of different scales, although some formulas are similar.

6.3.2 An Estimate for Current Helicity in Active Regions The spatial scale of an active region is much smaller than the solar radius but much larger than the maximum scale of solar turbulent granulation. To estimate the current helicity in an active region, we have to relate the large-scale magnetic field B and its magnetic potential A inside the convective zone as well as the corresponding small-scale quantities inside the convective zone (which determine the small-scale magnetic fluctuations), with the surface magnetic field Bar and its magnetic potential Aar inside active regions, which are the quantities available to observations. We assume that the rise of magnetic tubes is a fast adiabatic process. Let us also assume that the mean magnetic field and the total magnetic helicity vanish at the initial instant, and take into account the magnetic helicity conservation law (as solar plasma is highly conductive, and so we consider the magnetic helicity conservation law to hold at all scales, including that of the whole Sun). If this tube rises rapidly to the surface to produce an active region, the total magnetic helicity in the tube is conserved because the process is rapid. Rising large-scale magnetic field and magnetic potential give the corresponding quantities for active regions, which may thus differ substantially from the corresponding quantities in the surrounding medium. Because the initial total magnetic helicity of the tube, which was almost non-magnetized, was negligible, the magnetic helicity conservation law reads Aar ·Bar  ≈ −A·B,

(6.40)

where the angular brackets denote averaging over the surface occupied by the active region. Now we relate the mean current helicity Bar · (∇ × Bar ) with the magnetic helicity Aar ·Bar . We rewrite it from the first principles with the use of permutation tensors as  2 1 L ar ar ar ar ar , (6.41) B · (∇ × B ) ≈ 2 A ·B  + O 2 L ar R

6.3 Current Helicity of Active Regions as a Tracer of Large-Scale …

347

(see Appendix 6.3.5), where R is the solar radius, and L ar is the spatial scale of an active region. Equations (6.40) and (6.41) yield Bar · (∇ × Bar ) ≈ −

1 A·B. L 2ar

(6.42)

Therefore, the observed current helicity in active regions is expected to be a proxy for −A · B. This idea will be checked using mean-field dynamo numerical modeling and comparison of the numerical results with the observed current helicity in active regions.

6.3.3 Dynamo Models Our approach to compare the dynamo models with observations is as follows. We consider two types of dynamo models. Both types of models are 2-D mean-field models with an axisymmetric magnetic field which depends on radius r and polar angle θ. The third (azimuthal) coordinate is φ, and ∂/∂φ = 0. The dynamo action is based on differential rotation, with a rotation curve that resembles that of the solar convection zone, as known from helioseismological observations, and there is a conventional α-effect. Below we discuss the detailed dynamo models. We use spherical coordinates r, θ, φ and describe an axisymmetric magnetic field by the azimuthal component of magnetic field B, and the component A of the magnetic potential corresponding to the poloidal field. We measure the length in units of the solar radius R , and time in units of a diffusion time based on the solar radius and the reference turbulent magnetic diffusivity√ηT 0 . The magnetic field is measured in units of the equipartition field Beq = u ∗ 4πρ∗ , the vector potential of the poloidal field A in units of R Beq , the density ρ is normalized to its value ρ∗ at the bottom of the convective zone, and the basic scales of the turbulent motions  and turbulent velocity u at the scale  are measured in units of their maximum values through the convective zone. The α-effect is measured in terms of α0 , defined below, and angular velocity in units of the maximum surface value, 0 . In the primitive dynamo model, the α-effect is given by α = αv = χv v

(6.43)

where χv is proportional to the hydrodynamic helicity, Hu , multiplied by the turbulent correlation time τ , and v = (1 + B 2 )−1 is the model for the α quenching nonlinearity. For convenience, we use for most of these computations the code of (Moss and Brooke, 2000)—see also (Moss et al., 2011). This code has the possibility of a modest reduction in the diffusivity, to ηmin , in the innermost part of the computational shell (“tachocline”), below fractional radius 0.7. We define ηr = ηmin /ηT 0 .

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6 Magnetic Helicity with Solar Dynamo

We also used this primitive formulation of alpha-quenching in the model used in the following when producing Fig. 6.11. In the latter case, the diffusivity is everywhere uniform. At the surface r = 1 the field is matched to a vacuum external field, and “overshoot” boundary conditions are used at the lower boundary. In the dynamo model with magnetic helicity evolution, the total α-effect is given by m c χ (6.44) α = αv + αm = χv v + ρ(z) with αv = α0 sin2 θ cos θ v . The magnetic part of the α-effect is based on the idea of magnetic helicity conservation and the link between current and magnetic helicities. Here χv and χc are proportional to the hydrodynamic and current helicities multiplied by the turbulent correlation time, and v and m are quenching functions. The analytical form of the quenching functions v (B) and m (B) can be seen in Sects. 6.2.4 and 6.2.7 in more detail. The density profile is chosen in the form: ρ(z) = exp[−a tan(0.45π z)]

(6.45)

where z = 1 − μ(1 − r ) and μ = (1 − R0 /R )−1 . Here a ≈ 0.3 corresponds to a tenfold change of the density in the solar convective zone, a ≈ 1 by a factor of about 103 . The meridional circulation (single cell in each hemisphere, poleward at the surface) is determined by ∂[r (r, θ)] 1 sin θ r ρ(r ) ∂r ∂(r, θ) 1 = sin θ r ρ(r ) ∂θ

VθM = −

(6.46)

VrM

(6.47)

where (r, θ) = Rv sin2 θ cos θ f (r )ρ, f (r ) = 2(r − rb )2 (r − 1)/(1 − rb )2 , rb is the base of the computational shell. This is normalized so that the max of VθM at the surface is unity. We introduce a coefficient Rv = R U0 /ηT 0 , where U0 the maximum surface speed. Buoyancy is implemented by the introduction of a purely vertical velocity VB = γ Bφ2 r˜ (Moss et al., 1999). We justify the apparent non-conservation of mass by adopting the argument of K. H. Rädler, presented as a private communication in Moss et al. (1990), that the return velocity will be in the form of a more-or-less uniform “rain”. In some ways, the process represents pumping by a “fountain flow”. As a result, the regular velocity V B appears in the governing equations for the largescale magnetic field and magnetic and current helicities but not the equation for density. From the viewpoint of probability theory, in the first case, V B is a mean quantity taken under the condition that an elementary volume is magnetized so it does not vanish while in the second case this means is taken without any condition

6.3 Current Helicity of Active Regions as a Tracer of Large-Scale …

349

and vanishes. In our opinion, this idea can also be constructive for other problems with magnetic helicity advective fluxes, e.g., Shukurov et al. (2006). Please see the discussion in the 6.2.3 section for numerically computed boundary conditions.

6.3.4 Simulated Butterfly Diagrams for Current Helicity We performed an extensive numerical investigation of the models in a parametric range which is considered to be adequate for solar dynamos. We estimate the values of the governing parameters for different depths of the convective zone, using models of the solar convective zone, e.g., Baker and Temesvary (1966), Spruit (1974)— more modern treatments make little difference to these estimates. In the upper part of the convective zone, say at depth (measured from the top) h ∗ = 2 × 107 cm, the parameters are Rm = 105 , u = 9.4 × 104 cm s−1 ,  = 2.6 × 107 cm, ρ = 4.5 × 10−7 g cm−3 , the turbulent diffusivity ηT = 0.8 × 1012 cm2 s−1 ; the equipartition mean magnetic field is Beq = 220 G and T = 5 × 10−3 . At depth h ∗ = 109 cm these values are Rm = 3 × 107 , u = 104 cm s−1 ,  = 2.8 × 108 cm, ρ = 5 × 10−4 g cm−3 , ηT = 0.9 × 1012 cm2 s−1 ; the equipartition mean magnetic field is Beq = 800 G and T ∼ 150. At the bottom of the convective zone, say at depth h ∗ = 2 × 1010 cm, Rm = 2 × 109 , u = 2 × 103 cm s−1 ,  = 8 × 109 cm, ρ = 2 × 10−1 g cm−3 , ηT = 5.3 × 1012 cm2 s−1 . Here the equipartition means magnetic field Beq = 3000 G and T ≈ 107 . If we average the parameter T over the depth of the convective zone, we obtain T ≈ 5, see Kleeorin et al. (2003). We start with the results for the primitive model. Figure 6.9 presents the current helicity butterfly diagrams overlaid on those for the toroidal field. We estimate this quantity based on the idea that the observed current helicity in active regions is expected to trace −A · B, and so we plot in this section −A · B. We see that the plots successfully represent the main feature of the observed helicity patterns. The pattern presented in Fig. 6.9 is quite typical for the model. Of course, one can choose a set of dynamo governing parameters that is less similar to the observations. For example, one can concentrate magnetic fields in the deep layer of the convective zone (say, in the overshoot layer) by choosing a reduction ηr = 0.1 in the nominal “overshoot layer”, instead of ηr = 0.5 (our standard case), as used in Fig. 6.9. This tends to make the helicity wave in the overshoot layer look more like a standing wave but keeps the main features of the surface diagram (Fig. 6.10). The highly anharmonic standing patterns of the butterfly diagram that were discussed as a possible option for some stars, see Baliunas et al. (2006), look however irrelevant for the solar case. We produced the same type of plots for models based on helicity conservation Fig. 6.11. We also present in Fig. 6.12 the small-scale current helicity χc . Here the migration diagrams are presented for the middle radius of the dynamo region: nearer the surface χc displays only relatively weak vacillatory behavior. We see that the small-scale current helicity is strongly concentrated in middle latitudes and helicity

6 Magnetic Helicity with Solar Dynamo

100

100

80

80

60

60

40

40 Latitude, degrees

Latitude, degrees

350

20 0 -20

20 0 -20

-40

-40

-60

-60

-80

-80

-100

-100 0

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0

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Time, non dim units

0.4

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0.8

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100

100

80

80

60

60

40

40 Latitude, degrees

Latitude, degrees

Fig. 6.9 Current helicity of active regions estimated as −A · B [see Eq. (6.42)], overlaid on the toroidal field for the primitive dynamo model: left panel–deep layer, right–surface layer. Cα = −6.5, Cω = 6 · 104 , Rv = 0 (i.e., no meridional circulation), no buoyancy. The diffusivity constant ηr = 0.5 and the bottom of the computational region is at r0 = 0.64. The color palette is hereafter chosen as follows: yellow is positive, red is negative, and green is zero. From Zhang et al. (2012)

20 0 -20

20 0 -20

-40

-40

-60

-60

-80

-80

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-100 0

0.2

0.4

0.6

Time, non dim units

0.8

1

0

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0.4

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0.8

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Time, non dim units

Fig. 6.10 Current helicity of active regions estimated as −A · B [see Eq. (6.42)], overlaid on the toroidal field for the primitive dynamo model with enhanced dynamo activity in the overshoot layer: left panel–deep layer; right–surface layer. The values of Cα , Cω , Rv , and r0 are the same as in the previous figure but diffusivity contrast ηr = 0.1. From Zhang et al. (2012)

oscillations which are available in the model are almost invisible on the background of the intensive belt of constant helicity in middle latitudes. We doubt that such oscillations would be observable. We stress that, if this model produces any traveling helicity pattern, it is situated in the deep layers only. The pattern usually is much more similar to that presented in the right-hand panel of Fig. 6.10 rather than to a traveling wave such as presented in Fig. 6.9. Summarizing, we conclude that the current helicity of the magnetic field in active regions is expected to have the opposite sign to A·B, evaluated at the depth at which the active region originates. Thus, the models presented here are consistent with the interpretation that the mechanism responsible for the sign of the observed helicity operates near the solar surface, cf. e.g., Kosovichev (2012).

100

100

80

80

60

60

40

40 Latitude, degrees

Latitude, degrees

6.3 Current Helicity of Active Regions as a Tracer of Large-Scale …

20 0 -20

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0.6 0.4 Time, non dim units

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351

0.8

1

Fig. 6.11 Current helicity of active regions estimated as −A · B [see Eq. (6.42)], overlaid on toroidal field contours for the dynamo model based on helicity balance, near the middle radius of the dynamo region (r = 0.84, left panel) and near the surface (r = 0.96, right panel) for Cα = −5, Cω = 6 · 104 , Rv = 10 (i.e., with meridional circulation), and buoyancy parameter γ = 1. From Zhang et al. (2012) 100 80 60

Latitude, degrees

40 20 0 -20 -40 -60 -80 -100 0

0.2

0.4

0.6

0.8

1

Time, non dim units

Fig. 6.12 Small-scale current helicity χc overlaid on toroidal field contours for the dynamo model based on helicity balance, near the middle radius of the dynamo region. Cα = −5, Cω = 6 · 104 , Rv = 10 (i.e., with meridional circulation), buoyancy parameter γ = 1. The plots are for fractional radius 0.84. From Zhang et al. (2012)

6.3.5 Appendix: Current Helicity Versus Magnetic Helicity Here we relate the mean current helicity Bar ·curl Bar  with the magnetic helicity Aar ·Bar . First, we rewrite the mean current helicity from first principles with the use of permutation tensors as y

(6.48) Bar · ( × Bar ) = εmpq εmi j lim ∇ xp ∇i Aqar (x)B arj (y) x→y

x  ar = lim (∇ · ∇ y ) Aar (x) · Bar (y) − ∇px ∇qy Aar q (x)Bp (y) , x→y

352

6 Magnetic Helicity with Solar Dynamo

where εi jn is the fully anti-symmetric Levi-Civita tensor, R is the solar radius, L ar is the spatial scale of an active region and r = x − y.1 The use of the full tensor notation and limits are needed here to separate the large-scale and small-scale variables and to obtain a simple final answer in scalar form when there is the separation of scales. Since r = x − y is a small-scale variable, R = (x + y)/2 is a large-scale variable, the derivatives ∇ xp ≡

∂ ∂ 1 ∂ ∂ y = + = −∇ p + , ∂x p ∂r p 2 ∂Rp ∂Rp

y

∇p ≡

∂ ∂ 1 ∂ ∂ =− + = −∇ xp + . ∂ yp ∂r p 2 ∂Rp ∂Rp

This implies that

∇x · ∇y = −

∂2 1 ∂2 − 2 4 ∂R2 ∂r

 y

y

y

∇ xp ∇q = ∇ p ∇qx − ∇ p

,

∂ ∂ ∂2 − ∇qx + . ∂ Rq ∂Rp ∂ R p ∂ Rq

y

ar We take into account that div Bar = 0 [i.e., ∇ p B ar p (y) = 0] and div A = 0 [i.e., x ar ∇q Aq (x) = 0]. We also take into account that the characteristic scale of an active region is small compared with the thickness of the convection zone or the radius of the Sun. This implies that

∇ xp

∇qy Aqar (x)B ar p (y)

 2 ∂2 L ar ar ar , = Aq (x)B p (y) ∼ O 2 ∂ R p ∂ Rq R

and therefore this term vanishes. This yields  B · ( × B ) = − ar

ar

  2 ∂2 L ar ar ar , A ·B  +O 2 ∂r p ∂r p R r→0

(6.49)

Now we take into account that the second derivative of the correlation function   ∂2 Aar ·Bar  ∂r p ∂r p r→0 should be negative since as r → 0 the correlation function has a maximum. Thus, we finally obtain Bar · ( × Bar ) ≈

 2 1 L ar ar ar . A ·B  + O 2 L 2ar R

(6.50)

Similar calculations relating to the current helicity and the magnetic helicity in kspace can be found in Appendix C of Kleeorin and Rogachevskii (1999).

1

It is noticed ( × B) · B = ( × ( × A) · B = [( · A)] · B − [ · (A)] · B.

6.4 Reversal Helicity and Solar Dynamo Possibility

353

6.4 Reversal Helicity and Solar Dynamo Possibility 6.4.1 Possibilities on Reversal Helicity in Subatmosphere The formation of reversal magnetic helicity in a series of active regions relative to the hemispheric helicity rule presented in Sects. (5.2.1), (5.4), and (5.7.2) is a notable question. It can be proposed that the transferred reversal magnetic helicity in solar active regions comes from two possibilities: (a) Local generation of magnetic helicity from the subatmosphere. This is a normal case in the mirror-symmetric dynamo. (b) The transequatorial dynamo wave from the different subhemisphere. For analyzing the magnetic helicity in both hemispheres, two different topological twist patterns of magnetic lines of force are shown in Fig. 6.13. Figure 6.13a shows a typical schematic pattern of the twisted magnetic field generated in the subatmosphere. The twisted magnetic lines of force in the subatmosphere take the mirror symmetry relative to the solar equator. It is morphologically consistent with the normal models of the solar dynamo, even if some authors proposed the reversal mirror-symmetrical models for studying the helicity formation process at the different phases of the solar cycle (Choudhuri et al., 2004; Xu et al., 2009). Figure 6.13b shows another possibility in the generation of magnetic helicity in the solar subatmosphere. The magnetic lines of force twist in the same handedness in the subatmosphere of both hemispheres. These phenomena have been presented in the 5.4 section from the observational perspective. As one believes that the twist component of magnetic field transfers in the order B ) (Alfvén, 1942), the transequatorial time of magnetic of Alfvén speed (V A = √ μ0 ρ

a

b

Hc 0

357

1 1 + i γ = −λκ2 − iκV + (κ|D|α) 2 √ . 2

(6.62)

As compare D < 0 , we can obtain 1

γ = −λκ2 − iκV ± (κ|D|α) 2 thus as Re γ > 0

−1 + i √ , 2

(6.63)

1 1 − i γ = −λκ2 − iκV + (κ|D|α) 2 √ . 2

(6.64)

For maximum Re γ, then − λκ2 + and

1 2

1 2

1

(κ|D|α) 2 = Max,

(6.65)

dγ 1 1 1 = −2λκ + 3 κ− 2 (|D|α) 2 = 0. dκ 22

We can find

(6.66)

1

κ=

(|D|α) 3 5

2

23 λ3

.

(6.67)

1 1 − i γ = −λκ2 − iκV + (κ|D|α) 2 √ 2   21   2 1 1 1 (|D|α) 3 (|D|α) 3 1−i (|D|α) 3 −i 5 2 V + |D|α = −λ √ 5 2 5 2 2 23 λ3 23 λ3 23 λ3 2

=−

(|D|α) 3 10

4

2 3 λ3

1

−i

(|D|α) 3 4

2

23 λ3

(6.68)

2

V+

(|D|α) 3 4

1

23 λ3

(1 − i).

As set λ = 1, then 2

γ=−

+

10

23

(|D|α) 3

2

=−

3(|D|α) 3 10

23



2

(|D|α) 3

−i

−i 4 23  1 (|D|α) 3 5

23

1

(|D|α) 3 5

23

2

V+ 2

V+

(|D|α) 3 4

23

(|D|α) 3



4

23

 .

(6.69)

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6 Magnetic Helicity with Solar Dynamo

As set b = 1, then a=

α . γ + λκ2 + iκV

(6.70)

The magnetic helicity density can be written in the form A · B = ab · exp(2γt + 2iκθ) α · exp(2γt + 2iκθ) = γ + λκ2 + iκV 1

=

22 α 1 2

· exp(2γt + 2iκθ)

(6.71)

(1 − i)(κ|D|α) α 1 iπ =( ) 2 · exp(2γt + 2iκθ + ), κ|D| 4 where

and

6.4.2.2

(1 + i)2 = 1 + 2i − 1 = 2i,

(6.72)

√ π π 2 1 π (1 + i) = cos( ) + i sin( ) = ei 4 . √ (1 + i) = 2 4 4 2

(6.73)

Method 2

We can assume the form of a traveling wave in Eqs. (6.53) and (6.54) in the form A = a1 sin(ωt + κθ) + a1 cos(ωt + κθ) B = b1 sin(ωt + κθ)

(6.74a) (6.74b)

Under the assumption of the mirror dissymmetry of magnetic helicity h m = AB with the solar cycle, the solution and the corresponding picture of magnetic helicity have been presented by (Xu et al., 2009). The solution of Eqs. (6.53) and (6.54) shows the mirror asymmetrical component of magnetic helicity relative to the equator with opposite sign. As α and λ have been assumed as the forms α = α0 (1 − B 2 ),

B2 =

b1 2 and λ = 1, 2

(6.75)

6.4 Reversal Helicity and Solar Dynamo Possibility

359

Eqs. (6.53) and (6.54) can be written in the form ⎧ b1 2 ⎪ ⎪ ωa )b1 sin(ωt + κθ) cos(ωt + κθ) − ωa sin(ωt + κθ) = α (1 − ⎪ 1 1 0 ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎨ −κ2 [a1 sin(ωt + κθ) + a1 sin(ωt + κθ)] ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

−[V κa1 cos(ωt + κθ) − V κa1 sin(ωt + κθ)]

(6.76)

ωb1 cos(ωt + κθ) = Dκ[a1 cos(ωt + κθ) − a1 sin(ωt + κθ)] −κ2 b1 sin(ωt + κθ) − V b1 κcos(ωt + κθ).

From Eq. (6.76), we can obtain that ωa1 = −κ2 a1 − V κa1 ,

(6.77a)

2

b1 )b1 − κ2 a1 + V κa1 , 2 ωb1 = Dκa1 − V κb1 ,

−ωa1 = α0 (1 −

0 = Dκa1 + κ b1 . 2

(6.77b) (6.77c) (6.77d)

From Eqs. (6.77a), (6.77b) and (6.77d), we also can obtain a1 = −

a1 ω + Vκ ω + Vκ a1 or =− , 2 κ a1 κ2

(6.78a)

2

−ωa1 = α0 (1 −

D 2 a1 D )(− a1 ) − κ2 a1 + V κa1 . 2κ2 κ

(6.78b)

From Eqs. (6.77c) and (6.77d), we also can obtain − thus a1 = −

ω a1 = κa1 − V a1 , κ

ω + Vκ a1 ω + Vκ a1 or =− . 2 κ a1 κ2

(6.79)

(6.80)

As compared Eqs. (6.78a) and (6.80), we can obtain a1 a1 a1 = or = ±1. a1 a1 a1

(6.81)

From Eq. (6.78b), we can obtain −ω =−

α0 D (2κ2 − D 2 a1 2 ) + ω + 2V κ, 2κ3

(6.82)

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6 Magnetic Helicity with Solar Dynamo

then

4(ω + V κ)κ3 = 2κ2 − D 2 a1 2 , α0 D

and a1 2 = Thus

1 4(ω + V κ)κ3 ]. [2κ2 − 2 D α0 D

a1 b1 ω + Vκ D = a1 a1 2 2κ2 κ ω + Vκ = [α0 D − 2κ(ω + V κ)]. κα0 D 2

(6.83)

(6.84)

(6.85)

The magnetic helicity density can be written in the form A · B = [a1 sin(ωt + κθ) + a1 cos(ωt + κθ)] · b1 sin(ωt + κθ)  √ κ π  ) = [α D − 2κ(ω + V κ)] 1 − 2sin(2ωt + 2κθ + 0 α0 D 2 4

(6.86)

As the magnetic helicity density in the solar convection zone can be separated into osi the normal helicity density h nor m and an oscillate component h m in both hemispheres, the form of magnetic helicity density can be written in the form osi h m = h nor m + h m = A · B + h osi sin(ωo t + κo θ + φo ).

(6.87)

6.5 Turbulent Cross-Helicity in Mean-Field Solar Dynamo Problem It is normally believed that the cross-helicity parameter that could be observed on the solar surface is a manifestation of solely near-surface physical processes (Rüdiger et al., 2000, 2011). There is the existence of rapid local alignments of turbulent velocity and magnetic field in the presence of spatially non-uniform pressure and kinetic energy of turbulent fluctuations in the solar convection zone (Mason et al., 2006; Boldyrev et al., 2009). It is notable that the locally aligned turbulent velocity and magnetic field structures spatially dominate even in the case when the mean cross-helicity is zero. Some initial attempts to measure the cross-helicity from SOHO/MDI data Scherrer et al. (1995) were made by Zhao et al. (2011) (see Sect. 5.6). The cross-helicity can be a useful quantity for diagnostics of nonlinear turbulent dynamo processes in the solar convection zone, (see, e.g., Kleeorin et al., 2003). Yoshizawa (1990) and Yokoi (1999) considered the cross-helicity as a part of the dynamo mechanism in turbulent astrophysical flows. The role of cross-helicity

6.5 Turbulent Cross-Helicity in Mean-Field Solar Dynamo Problem

361

conservation for this type of dynamo was explored recently by Sur and Brandenburg (2009). On the other hand, the results of numerical simulations Mason et al. (2006), Boldyrev et al. (2009) showed the existence of rapid local alignments of turbulent velocity and magnetic field in the presence of spatially non-uniform pressure and kinetic energy of turbulent fluctuations.

6.5.1 Transformation Symmetry Properties and Cross-Helicity Any pseudo-scalar can be expressed either as a tensor product of a tensor with pseudotensor or as a scalar product of vector and pseudo-vector. The general expression for u · b may have a fairly complicated structure. Pipin et al. (2011d) restrict in studying the effects that can be important for the solar convection zone dynamics and solar dynamo. The contribution to the solar dynamics is given by gradients of the turbu   2 b lent kinetic fluctuations and magnetic energy density, e.g., ∇ ρu2 and ∇ 2μ0 (all quantities are ordinary vectors). Note, that the mean density is sufficient to describe the effects of the thermodynamic stratification in the polytropic atmosphere. This assumption will be used hereafter. Also, we have to take into account the effects of the global rotation (pseudo-vector ) and large-scale shear,  ∇iU j (tensor). This tensor can be decomposed as follows: ∇i U j = 21 εi j p W p + ∇U {i, j} , where,       W = ∇ × U , (pseudo-vector) and a strain tensor, ∇U {i, j} = 21 ∇i U j + ∇ j U i .

We assume that u · b = 0 in the absence of large-scale magnetic field and take into account effects of the large-scale magnetic field, B (pseudo-vector), and its spatial derivatives, ∇i B j . Similar to the large-scale shear flows we decompose the contribution of the non-uniform magnetic field into effectsof large-scale electric current  J = ∇ × B/μ0 (vector) and magnetic strain tensor, ∇B {i, j} = 21 ∇i B j + ∇ j B i (pseudo-tensor). By analogy with Rädler (1980), and also assuming the scale separation {, τc }  {L , T } of the typical small-scale (, τc ) and the large-scale (L , T ) spatial and temporal variations of the velocity and magnetic fields, we write a combination of above effects as follows:

  b2 τc  2 u·b= B · ∇ κ1 ρu ¯ + κ2 ρ 2μ0 (6.88)        + κ3 μ0 W · J + κ4 μ0  · J + κ5 ∇U {i, j} ∇B {i, j} + o , L where κ1−5 are coefficients that need to be determined). Having in mind the transformation symmetry properties of the quantities that are involved in the problem, we, of course, can construct the pseudo-scalar u · b in many other ways. In principle, Eq. (6.88) may include the terms like G · , U ·  (and similar), or the terms like

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6 Magnetic Helicity with Solar Dynamo

U · B, B · J (and similar), or even higher-order combination of the physical quantities. Later, we will see that such terms do not appear in our analysis of the momentum and induction equations of the fluctuating velocity and magnetic fields if we restrict ourselves to the case of the weak large-scale magnetic field B. In this sense, Equation (6.88) may be incomplete and should be considered with cautions in the analysis of the observational data. The physical interpretation of the first term in Eq. (6.88) was given by Rüdiger et al. (2000). Consider the turbulent medium permeated by the large-scale field B. Convective elements rising, with velocity u expand, (∇ · u) > 0, and induce a fluctuating magnetic field, b ≈ −τc B (∇ · u) (sign is anti-correlated with the sign of (∇ · u)). The same is valid for descending and contracting convective elements. In the anelastic approximation Gough (1969), ∇ · (ρu) ≡ 0 and (∇ · u) = − (u · ∇ log ρ). Therefore the sign of u · b is opposite to the sign of B ( because ∇r log ρ < 0). The effects of density fluctuations are not taken into account in this consideration. u2 For weakly compressible (subsonic) convective flows, 2  1 (Cs is the sound CS speed), the contribution of the buoyancy forces to the cross-helicity is proportional  u2  to ∼ κ6 2 τc g · B , where g is the gravity acceleration. This effect may be of the CS same order magnitude as the first term in Eq. (6.88), particularly close to the surface u2 where 2 can be significant. CS

6.5.2 Mean-Field Theory of Cross-Helicity 6.5.2.1

Calculation of u · b by Averaging in Physical Coordinate Space

As pointed out by Yoshizawa et al. (2000) the evolution equation for the cross-helicity is useful for exploring mean-field dynamo properties based on the cross-helicity effects (also see, Sur and Brandenburg, 2009). Pipin et al. (2011d) derived the following evolution equation for u·b after subtracting the equation for the mean-field cross-helicity U · B:        B i u·b C ∇ j Ti j − ∂t u·b = −∇·F − E · 2 + ∇ × U + , ρ τc

 b2 1 Ti j = ρu i u j − bi b j − δi j , μ0 2 C

F = U u·b + B i u i u − B

u2 . 2

(6.89) (6.90) (6.91)

6.5 Turbulent Cross-Helicity in Mean-Field Solar Dynamo Problem

363

Here, following the τ -approximation approach Vainshtein and Kitchatinov (1983), Kleeorin et al. (1996), Blackman and Field (2002), Blackman and Brandenburg (2002), we replace the third-order correlations   of the fluctuating parameters, and the u·b , here τc is a typical time scale of the terms b · f  with a relaxation term − τc turbulent Approximating these complicated contributions with the simple   motions. u·b term − has to be considered as a questionable assumption. It involves additional τc assumptions (see Rädler and Rheinhardt, 2007), e.g., it is assumed that the secondorder correlations in Eq. (6.89) do not vary significantly on the time scale of τc . This assumption is consistent with scale separation between the mean and fluctuating quantities in the mean-field magnetohydrodynamics. In addition, in the turbulent stress tensor Ti j we consider only the contribution of the turbulent kinetic and magnetic pressure as formulated in Eq. (6.88):

Ti j ≈ δi j

b2 ¯ 2 + κ2 κ1 ρu 2μ0

 ,

(6.92)

In a linear approximation (the weak mean-field case), κ1,2 = 1/3 Kleeorin et al. (1996). In general, the turbulent stress tensor contains terms governing the differential rotation and the meridional circulation (Kitchatinov and Ruediger, 1995). We assume that these terms are stationary and do not contribute to the cross-helicity evolution. Therefore, we simplify Eq. (6.89) as:

   2    u·b b 1  B · ∇ ρu ¯ 2+ , (6.93) − ∂t u·b = −E · (2 + W) + 3ρ 2μ0 τc Thus, the major sources of the turbulent cross-helicity are due to the mean electromotive force,the mean  vorticity, 2 + W, and the gradients of the turbulent energy, representation of the mean electromotive where W = ∇ × U . For the simplest   , where the first term represents α-effect (here, force, E ≈ αB − ηT ∇ × B + o L α is pseudo-scalar) and ηT is the turbulent diffusion coefficient, we can write

        1  b2 u·b 2 ∂t u·b = B · ∇ ρ¯ u + , − α B · 2 + W + μ0 ηT 2 + W · J − 3ρ 2μ0 τc 

(6.94) The first two terms of this equation show that the mean cross-helicity is generated in the presence of the large-scale magnetic field that permeates the stratified turbulent medium. Another important contribution represented by the third term is due to the large-scale electric current and the mean vorticity. A similar equation was used by Yoshizawa et al. (2000) for discussing a mean-field dynamo scenario based on the cross-helicity effects. A complete expression for the mean electromotive force was

364

6 Magnetic Helicity with Solar Dynamo

used by (Kuzanyan et al., 2007) who investigated the time-evolution of the crosshelicity produced by the mean-field dynamo in the solar convection zone. To study the distributions of the cross-helicity on time intervals much shorter than the period of the solar cycle, (Pipin et al., 2011d) neglect the time derivative in Eq. (6.94) and find:

 2       b τc  u·b ≈ B · ∇ ρu ¯ 2+ − ατc B · 2 + W + μ0 ηT τc 2 + W · J. 3ρ 2μ0 (6.95) To estimate the magnitude of the cross-helicity in the near-surface layers, we assume that the energy of the fluctuating magnetic fields can be expressed through the kinetic ¯ b2 = ερu2 , and ε = 1 is the energy equipartition energy of convective motions, 2μ0 condition. The convection in the surface layers is short-lived in comparison to the period of the global rotation. For these the α-effect can be estimated as   conditions, 2 (see, Pipin, 2008). Then, the second term following α ≈ ηT τc  (ε + 1) ∇ log ρu ¯ in Eq. (6.95) is much smaller than the first term because of τc   1 in the subsurface layer. Therefore we will neglect this contribution in the study. For the comparison we introduce the density stratification scale parameter, G ≡ ∇ log ρ, ¯ and rewrite Eq. (6.95) as follows u·b ≈ ηT (ε + 1)

        B · G + B · ∇ log u2 + μ0 ηT τc 2 + W · J. (6.96)

If we neglect the effect of fluctuating magnetic field, we find that the contribution of density stratification agrees with Rüdiger et al. (2010), but the contribution of the turbulent intensity stratification is larger by a factor of 2, in our case. The difference can be explained by the approximations  made   in the calculation of Eq. (6.96). In  particular, the contributions of ∇U {i, j} ∇B {i, j} are not included in Eq. (6.96). The main purpose of this approach is to demonstrate the basic physical contributions to the mean cross-helicity in solar conditions. We have to point out that in the derivation of Eq. (6.96) we did not make assumptions that the turbulence inhomogeneity scale is much larger than the typical scale of turbulent flows, . This assumption is used in previous studies Rüdiger et al. (2011), and strictly speaking, their results can be applied only to the case of a weakly stratified medium, G  1.

6.5.2.2

Calculation of u · b in Fourier Space

Pipin et al. (2011d) calculate correlation u r br , which can be estimated from the lineof-sight observations of velocity and magnetic field in a central part of the solar disk (cf., Zhao et al., 2011). The result of these calculations in the form can be compared with Kleeorin et al. (2003) and Rüdiger et al. (2010):

6.5 Turbulent Cross-Helicity in Mean-Field Solar Dynamo Problem

365

  ηT (6.97) (2 + 3ε) B · G 2      3ηT (ε + 1)  + B · ∇ log u2 + ηT τc (ε + 1) μ0  · J 2     η T τc ηT τc μ0 + (3 + ε) J · W + (23ε + 5) ∇U {i, j} ∇B {i, j} , 4 10

u·b=

where ε is a ratio of the kinetic and magnetic energies of the turbulent pulsations, ηT = u2 τc /3 is turbulent diffusion coefficient, and τc is a typical convection turnover time. The structure of Eq. (6.97) corresponds to Eq. (6.95) though contrary to Eq. (6.96) the contributions of density stratification (G) and turbulent   2 intensity stratification (∇ log u ) are decoupled. For the other coefficients we find T τc κ3 = μ0 ηT4τc (3 + ε), κ4 = μ0 ηT τc (ε + 1) and κ5 = η10 (23ε + 5). Also, we see that the contribution of the stratification effects in Eq. (6.96) agrees but have slightly vs (1 + ε), and that the contributions due to the electric different coefficients (2+3ε) 2 currents are in agreement. Note, that here similarly to Rüdiger et al. (2011) in the derivation of Eq. (6.97) we used the scale separation assumption, G  1 (weakly stratified medium). For the correlation of the radial components we obtain:

      u r br = u r br ρ + u r br J  + u r br J W ,

(6.98)

where   u r br ρ =

ηT G r B r ηT τc (ε + 1) B r ∇r u2 − (2 + ε) + (5 − 2ε) ∇r B r 2 6 5

  u r br J  =

  2εηT τc μ0 4εηT τc r Jr +  · e(r ) × ∇r B 5 5

  ηT τc μ0 ηT τc μ0 u r br J W = (ε − 3) J r W r + (ε + 7) J · W 10 20 η T τc + (163ε + 41) ∇{i U j} ∇{i B j} , 70

(6.99)

(6.100)

(6.101)

where e(r ) is a unit vector in the radial direction. Both Eq. (6.97) and Eq. (6.98) generalize the previous results of Kleeorin et al. (2003) and Rüdiger et al. (2011) by including the effects of the large-scale electric current and velocity shear. In our models, we find that the toroidal component of the large-scale magnetic field is much stronger than the poloidal component. We also discard the effect of meridional circulation in Eq. (6.101). Therefore we define the product of the magnetic and velocity strain by the formula: ∇{i U j} ∇{i B j} = ∇r U φ ∇r B φ + ∇θ U φ ∇θ B φ + ∇r U φ ∇θ B φ + ∇θ U φ ∇r B φ ,

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where ∇r,θ are the covariant derivative components, B φ is the toroidal magnetic field and U φ = r sin θ ( (r, θ) − 0 ) is the large-scale shear flow due to the differential rotation.

6.5.3 The u r br Patterns by Dynamo Models The distributed dynamo model includes the meridional circulation with the geometry of flow which is similar to that used by Bonanno et al. (2002). The maximum velocity of the meridional circulation is fixed to 10 ms−1 . The turbulent generation effects include the α−effect (Parker, 1955) and the  × J effect Rädler (1969). The internal parameters of the solar convection zone are given by Stix (2002). The integration domain is above the tachocline, from 0.71R to 0.972R in radius, and from the pole to the pole in latitude. We use the differential rotation profile given by Antia et al. (1998). contributions to u r br into three parts:  separate   et al.  (2011d)   Pipin u r br ρ , u r br J and u r br J W , and associate the first term of Eq. (6.98) as a contribution of the stratification effects, while the second and third terms represent contributions of the large-scale electric current. In the calculations we assume the equipartition between the kinetic and magnetic energies of fluctuations, ε = 1. Modeling the cross-helicity we found that if we take the surface values for the model quantities in Eq. (6.98) then we obtain that the stratification effects are dominant. This is true for the both types of dynamo models. However, if we integrate the cross-helicity from the surface down to deeper layers, say down to 0.9R , then the contributions of the large-scale current become comparable to the stratification effects and even greater. If we consider the observational problem of finding the u r br , we have to choose suitable spatial and temporal scales for averaging (in the spirit of Zhang et al., 2010b), to distinguish between the processes of the “local” (noted in Mason et al., 2006; Matthaeus et al., 2008) and “global” alignments of velocity and magnetic fields. We can take as a working hypothesis that for larger scales most contribution to the observed cross-helicity comes from deeper layers of the solar convective zone. Kuzanyan et al. (2007) pointed out that the contributions h C /τh and ηh ∇ 2 h C serve to take into account the helicity loss in rough way. The parameter τh is normalized −2 η0 , and ηh = εh η0 where, εh  1, η0 = (u c c ) /3– to the typical diffusion time R typical turbulent diffusivity in CZ. Boundary conditions, for helicity: vanishing of the radial derivatives both at the bottom and at the top. The dynamo problem is treated with the following types of the boundary conditions: superconductor at the bottom and external vacuum.   ∂h C  ∂h C  , = 0. ∂θ θ=0,π ∂r r =.71, .96R

6.5 Turbulent Cross-Helicity in Mean-Field Solar Dynamo Problem

367

One of the purpose of the work to consider the different turbulent effects that may be responsible for generation of the large-scale poloidal magnetic fields. First, we multiply the standard α–effect by factor sin2 θ to make sure that the maximum generation of the poloidal large-scale magnetic field (LSMF) is near equator. Second we add the joint effects due to current helicity and shear to the mean electromotive force (see Pipin, 2007). This will ensures that activity of the toroidal LSMF drifts to equator in course of solar cycle. All this additions can be expressed in the formal way as follows ∂ (h) E˜r = E r + ϕ3 τ 2 h C Br sin θ ∂r ∂ (h) 2 θ θ ˜ E = E + ϕ3 τ h C B sin θ ∂θ r sin θE˜ φ = r sin θE φ − Br η˜ T Cα G sin θ

(6.102) (6.103) 

(a)

(a)

(s)

f 12 ϕ6 cos3 θ − ϕ2 τ sin θ



∂ , ∂θ

(6.104)

(a) (h) (s) where, functions f 12 , ϕ(a) 6 , ϕ3 , ϕ2 were defined in Pipin (2007). Thus, for the α we will use these components of MEMF. Note, that αδ dynamo does not require these tunings. In the αδ dynamo, the interaction between the solar rotational angular vector and the mean electric current is taken into account. Further, the governing parameters of the model are η˜T = Pm u c c and Pm ≤ 1 is the effective Prandtle number; Cα , Cω j ≤ 1 are parameters to control the power of the α and  × J effects. Let us give here two examples of possible dynamo models. As the first example we consider the results for αδ dynamo. The model is discussed with more details in the separate paper by Pipin (2007). On Fig. 6.16 we give Maunder’s-like diagram for evolution of the radial LSMF field, toroidal LSMF, current and cross-helicities. The quantities are taken at the near-surface level. Evolution of LSMF and helicities shown in radial section of SCZ is shown in Fig. 6.17. The second example is for α dynamo with the MEMF tuned as given by Eqs. (6.102, 6.103, 6.104). We use the external boundary vacuum conditions, here, as well. The corresponding diagrams are shown in Fig. 6.18.

6.5.4 Discussion for Cross-Helicity Cross-helicity is an inviscous invariant, and, therefore, our analysis within the solar convective zone should well be relevant for the expected observational data. Recently a numbers of space missions have been launched and further more data on solar magnetic fields and velocity fields are expected within a few forthcoming years. This will get more observational data involved into the theoretical analysis and first of all, as never before, an opportunity to check the theory versus direct observations. We can predict the following properties to be seen upon systematic statistical studies of cross-helicity:

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6 Magnetic Helicity with Solar Dynamo

Fig. 6.16 αδ dynamo. Top, radial magnetic field variations on the surface, middle: variations of current helicity with overlaid isocontours of the toroidal magnetic fields, bottom: variations of the cross-helicity on the near-surface level. From Kuzanyan et al. (2007)

• Hemispheric rule: Cross-Helicity is anti-symmetric over the equator. • Cyclic variation: Cross-Helicity changes sign with 11 yr cycle. Our results show that the turbulent cross-helicity reflects the internal properties of the dynamo processes and allow us to discriminate among different solar dynamo models. We have shown that the spatially temporal patterns of cross-helicity depend on details of the dynamo mechanism. This implies the cross-helicity tracer as a diagnostic tool for solar dynamo models.

6.6 Estimates of Current Helicity and Tilt of Solar Active Regions and Joy’s Law The tilt angle, current helicity, and twist of solar magnetic fields can be observed in solar active regions. We carried out estimates of these parameters in two ways. Firstly, we can consider the model of turbulent convective cells (such as the approximation of supergranules) which have a loop floating structure toward the surface of the Sun. Their helical properties are attained during the rising process in the rotating stratified convective zone. The other estimate is obtained from a simple mean-field dynamo

6.6 Estimates of Current Helicity and Tilt of Solar Active Regions and Joy’s Law

369

Fig. 6.17 Evolution of LSMF and helicities shown in radial section of SCZ. Time grows from left to right, snapshots are given at 0, 3, 5, and 8 years. The top, the poloidal field lines are overlayed by the gray-scale density plot of the toroidal LSMF, middle–current helicity and bottom–cross-helicity. From Kuzanyan et al. (2007)

model that accounts for magnetic helicity conservation. Both values are shown to be capable to give important contributions to the observable tilt, helicity, and twist Kleeorin et al. (2020). Kuzanyan et al. (2020) Having estimated the tilt, current helicity, and twist from local considerations in active regions we notice that even their sign of them may vary from one active region to another. On the other hand, the averaging of the tilt, current helicity, and twist over all active regions do not depend on the phase of the solar cycle and the sign of the magnetic field (Hale’s polarity law), so from one cycle to another cycle the sign of these quantities holds. This is in good agreement with Joy’s law for tilt as well as Hemispheric Sign Rule for current helicity and twist. Given the considered effects do not vary much with the phase of the solar cycle, we may have the same trend everywhere.

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6 Magnetic Helicity with Solar Dynamo

Fig. 6.18 The α dynamo. Top: radial magnetic field variations on the surface, middle: variations of current helicity with overlaid isocontours of the toroidal magnetic fields, bottom: variations of the cross-helicity on the near-surface level. From Kuzanyan et al. (2007)

6.6.1 The Model Let us consider a simple bipolar active region with the distance L between the opposite polarities. We use a model of a turbulent convective cell of the size of supergranulation with a depth L/2 ≈ 109 cm. This scale is of the order of the density  −1 d log ρ0 (r ) stratification scale in L/2 ≈ Hρ = − where ρ0 is the fluid density dr at the depth H = 109 cm. This scale we associate with the depth of the sunspot formation region. In a classical Rayleigh-Bénard convective roll, horizontal and vertical scales are the same. A superposition of the three convective rolls forms a hexagonal structure (see Chandrasekhar, 1961, e.g., Chap.II, §16, p.48, Fig.7a). This implies that the ratio of horizontal to vertical sizes in the hexagonal structure is about 2. In the Sun and stars, the convection is fully turbulent and is different from classical Rayleigh-Bénard convection. In particular, the convection rolls are formed with their horizontal scales greater than the vertical scales approximately by factor 2. Similar phenomena are observed in Earth’s atmosphere and other natural turbulent convection systems. Large-scale structures like the convection rolls can be isolated from turbulent eddies using scale separation ideas Elperin et al. (2002), Bukai et al. (2009) being applied to solar active regions. This consideration is in agreement

6.6 Estimates of Current Helicity and Tilt of Solar Active Regions and Joy’s Law

371

with observations of super-granulation convection in the Sun. Therefore, the total horizontal extent of this active region is 2L. Let us assume that the active region scales are small compared to the solar radius, and put it at heliographic latitude φ being counted from the solar equator. In terms of observable active regions, we estimate this scale by orders of magnitude as L ∼ 20 − 50 Mm = (2 − 5) × 109 cm. Kuzanyan et al. (2020) consider the momentum equation for u applying the anelastic approximation at the boundary between the solar convective zone and the photosphere ∂u = −∇ ∂t



ptot ρ0

 − gS + Fmag + Fhd + Fvisc + Fccor

(6.105)

which includes the effects of the total pressure, hydrodynamic and magnetic buoyancy, nonlinear local hydrodynamic Coriolis force, viscous forces, and global CoriB2 ρu2 + for the total olis force. The terms appeared in Eq. (6.105) are ptot = 2 8π pressure, where p is the hydrodynamic pressure, ρ density, B is the magnetic field, g the acceleration due to  S the entropy, so that −gS is the buoyancy force;  gravity, ∇ρ0 B2 (B · ∇)B − the non-gradient part of the magnetic force in ρ0 Fmag = 4π ρ0 8π density stratified fluid, where the first term stands for magnetic stress while the second term for magnetic buoyancy; Fhd = u × w local Coriolis force from nonlinear  2 2 local fluid motion, and ρ0 Fvisc = νρ0 ∇ u − ∇(∇ · u) the viscous force, where 3 ν is the molecular viscosity, and ρ0 Fcor = 2ρ0 u ×  the Coriolis force from solar global rotation, w = ∇ × u for the vorticity. In order to eliminate the terms containing gradients of pressure we calculate the curl of that equation (6.105), to obtain the equation for vorticity w, and we are interested in the radial component wr only. We assume for the rough estimate that the contribution from the (∇ × Fmag )r , (∇ × Fhd )r , (∇ × Fvisc )r , (∇ × Fcor )r , can be replaced by a relaxation term as −wτ /τ D , where τ D hasa meaning of the sunspot twisting time (Fig. 6.19). Under these assumptions the radial component of the equation for the vorticity in the spherical coordinates reads wr = [∇ × (u × )]r 2τ D = [(∇ · )u + ( · ∇)u − (∇ · u) − (u · ∇)]r = [( · ∇)u]r − [(∇ · u)]r as [(∇ · )u − (u · ∇)]r = 0 (6.106) θ ∂u r θ u θ ∂u r + − − r ∇ · u = r r ∂θ r  ∂r 1 ∂u r uθ ∂u r + sin θ − sin θ − cos θ∇ · u . =  cos θ ∂r r ∂θ r

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6 Magnetic Helicity with Solar Dynamo

r

θ r

φ θ

φ

Fig. 6.19 the reference system of spherical coordinates

∂u r = The radial derivative of the vertical convective velocity can be estimated as ∂r ur − . Here the negative sign reflects the effect of slow-down velocity in the rising hs flux tubes. Let us estimate sunspot twisting time τ D as the ratio L/va , where Alfvén speed √ is va = Beq / 4π0 . At the upper part of the convective zone typical equipartition value of the mean magnetic field is Beq ∼ 300 G, and density of the solar plasma ρ0 according to estimates of Spruit (1974), is of the order of 4.5 × 10−7 g cm−3 . Thus, Alfvén speed is of the order of va ∼ 1.2 × 105 cm s−1 , and therefore the time scale τ D ∼ (2 − 4) × 104 s ≈ 6 − 12 hours. The duration of the flux tube emerging from the bottom of the convective cell is of the order of τ F = L/2u r . We use the anelastic approximation for convection in the solar convection zone ∇ · (ρu) = 0. The density stratification is in the radial ur d 1 log ρ = ≈ , where Hρ is density stratdirection, so we have ∇ · u = −u r dr Hρ τF ification scale estimate for the radial component of vorticity !r for the motion of bipolar sunspots is obtained r ≈ −

τD τF

  1 4 sin φ + cos φ , ξ

(6.107)

6.6 Estimates of Current Helicity and Tilt of Solar Active Regions and Joy’s Law

373

where instead of co-latitude θ we use the latitude φ = π/2 − θ, then τ F = Hρ /u r and ξ is defined below. The latitudinal derivative of the vertical convective velocity can be ur 1 ∂u r ≈ , where ξ being dependent on the structure of active regions estimated as r ∂θ ξL varies in sign, and in absolute value is around 1-2 due to the hexagonal structure of convection. The sign of ξ can be considered random for super-granulation convection in the Sun. If we consider roll-like convection, it is close to unity while for hexagonal convective cells comprising three rolls, it is close to 2 ( see Elperin et al., 2002; Bukai et al., 2009).

6.6.2

Estimate for Tilt of Sunspots

Given the time of evolution of the active region during the sunspot formation (which contributes to the tilt angle of the opposite polarities), is comparable with the flux tube emerging time τ F , so that we can estimate the tilt as 2πτ D δ = wr τ F = − T

  1 4 sin φ + cos φ , ξ

(6.108)

where the solar siderial rotation period T ≈ 25 d approximately corresponding to Carrington rotation. The coefficient in the front of the brackets in this formula is of order 0.25–0.5. Since the value of sin φ for low latitudes where sunspots mainly occur is comparable with the latter term within the brackets, that is of order 1/8 − 1/4, we can expect the tilt angles of individual active regions to vary a lot and even change signs. This estimate also implies the less variability in tilt angles the higher latitude, which may need to be verified with observations. Given the sign of ξ randomly fluctuates and its value varies with super-granulation, for the averaged tilt δ we may obtain the range (0.25 − 0.5) sin φ, which gives the order up to 15◦ for middle latitudes, and so it fits perfectly well with the observational results of Howard (1991). The observational magnitude of tilt is indeed increasing with departure from the solar equator almost linearly with latitude, or like sin φ, and for middle-range latitudes it is in average 5◦ − 15◦ , see, e.g., Stenflo and Kosovichev (2012), Tlatov et al. (2013) and references therein. Correspondingly, the estimate of twist ϒ for typical tilt angles of order δ ∼ 0.1 − 0.2 (5◦ − 10◦ ) reads ϒ = δ/H B ∼ 10−10 cm−1 = 10−8 m−1 ,

(6.109)

which matches quite well the order of magnitude of the observational results, e.g., Zhang et al. (2002), Zhang et al. (2010b). Notice that the vertical magnetic field energy proportional to Bz2 is of the order of the equipartition magnetic field.

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6.6.3 Estimates for Current Helicity and Twist In Dynamo Models After we have estimated the tilt angles as well as current helicity and twist using local effects for rising flux tubes in solar active regions, we are going to estimate these values from the axially symmetric spherical shell dynamo model, see Sect. 6.2.3. Let us use Eq. (6.42) for the mean current helicity of the active region in the assumption of conservation of total magnetic helicity (see Sect. 6.3) HcA R = B A R · ∇ × B A R  ≈ −

1 B 2 R ˜ A · B = − ∗ 2 A˜ B, 2 L ar L ar

(6.110)

where A = R A˜ B˜ ∗ , B = B˜ B∗ ; B∗ = 10ρ0 ηT /R is the characteristic magnetic field produced by the dynamo mechanism and ηT is the turbulent magnetic diffusion coefficient (see Zhang et al., 2012; Kleeorin et al., 1995), and R ≈ 7 × 1010 cm is the solar radius. The twist of magnetic fields of an active region can be estimated as 1/2

ϒ≡

˜ R A·B A B A R · ∇ × B A R  ≈− 2 2 ≈− , A R 2 (B )  L ARB B˜ L 2A R

(6.111)

where we assumed that (B A R )2  ≈ B2 . Note that the product and ratio of A and B vary within the solar cycle but do not vary from one (odd/even) cycle to another (even/odd). The sign of A · B is mainly negative (3/4 of the period), so the sign of ϒ > 0, i.e., opposite to the one produced by the Coriolis force. According to observations the horizontal size of the active region is about the size of a super-granule, i.e., L A R ∼ (2 − 5)Hρ , where Hρ ∼ 109 cm. So, we estimate L A R ∼ (2 − 5) × 109 cm = 20 − 50 Mm. Using Eq. (6.111), we estimate the twist of the magnetic field in the active region as ˜ R A ≈ −(0.3 − 1) × 10−10 cm−1 , (6.112) ϒ ≈− B˜ L 2A R ˜ B˜ ∼ 6 × 10−3 . (see Appendix B and also the figures of Zhang where typically A/ et al. (2012)). This value is of order 10−2 which is typical for the most kinematic dynamo models of α type. Now, let us take the value of B˜ of order 0.2-0.5 (as it is on in the bottom of the convective zone). If we adopt the value of Beq ∼ 500 − 1000 G as somewhere in the sunspot umbra near or just beneath the photosphere, we estimate the magnitude for the current helicity as Hcar = ϒB2 B˜ 2 ∼ −10−5 G2 cm−1 = −10−3 G2 m−1 .

(6.113)

6.6 Estimates of Current Helicity and Tilt of Solar Active Regions and Joy’s Law

375

This value is comparable with the observational results of Zhang et al. (2010b) giving the order of (1 − 2) × 10−5 G2 cm−1 for current helicity, and (1 − 2) × 10−10 cm−1 for the twist. In the presence of helicity fluxes, some fraction of the dynamo-born helicity along with some (let us assume the same fraction) of helicity due to Coriolis force are removed from the photosphere and injected into the solar corona. Let us denote the fraction of this helicity injection as . Then combining formulae (6.107), (6.110), and (6.113), we can derive the following expression for the remaining total current helicity of an active region Hcar T O T

2πτ D = −

T

  2 Bz 1 1 − (1 − ) 2 A · B. 4 sin φ + cos φ ξ HB L ar

(6.114)

Hereby the ejection of helicity from an active region into the corona is Hcar F L

2πτ D =

T

 2  Bz 1 1 4 sin φ + cos φ − 2 A · B. ξ HB L ar

(6.115)

Note that the sum of the total remaining and the flux parts of helicities in formulae (6.113) and (6.114) is equal to the amount of helicity produced by the dynamo as in the formula (6.110). The particular value of in formula (6.114) is not known and it can be estimated from the comparison of theoretical dynamo models and the observational data for tilt, vertical magnetic field, twist, and current helicity in solar active regions. The main contribution in the total helicity for large active regions is probably due to Coriolis force while for smaller ones from dynamo generation mechanism. The distinction between the two can be determined by the latitude and phase of the solar cycle. Please see Sects. 3.4 and 5.2.2 for the statistical relationship between the observed magnetic twist and tilt angles in solar active regions. It is more complex than the theoretical expectation. This question requires further investigation with the use of calibrated dynamo models and available observational data.

Chapter 7

More Questions

Parker (2009) has pointed out: “Leighton (1969) remarked many years ago that ‘were it not for magnetic fields, the Sun would be as uninteresting as most astronomers seem to think it is;’ this is the stuff that makes science so fascinating. · · · · · · These scientific puzzles, and many others, have been with us for years, and we are beginning to be haunted by Wigner’s dictum: The important problems in physics are rarely solved; they are either forgotten or declared to be uninteresting. The hope is that sufficient attention to the problem may ultimately evade Wigner’s dictum. Consider, then, the origin of the magnetic fields of the Sun, as a place to begin the discussion.” The phenomenon of solar activity closely related to the human living environment originates from the change of solar magnetic field, so the research of solar magnetic field has been strongly attractive and it is the core subject of solar physics. The sun is a unique star that we have to observe its detailed magnetic activities and plays an irreplaceable role in astrophysics and the study of electromagnetic phenomena in nature. In recent years, space solar observation data show us an unprecedented amount of detailed information of the sun in different bands, which makes it possible for us to make unprecedented progress in the study of the physical structures and explosive activities of the sun. It should be noted that when we investigate deeply, it is still not very clear about many of the basic problems, and many seemingly authoritative explanations may have some loopholes. When we feel clear about some of these problems through our efforts, some new and deeper problems appear in front of us. From the point of view of the physical observation and study of the solar magnetic field, the problems that I feel puzzled about are summarized in the following. A senior scientist once told, “if there are 100 solar physicists, there may be 100 physical models of solar flares in their minds”. This may be an exaggeration, but it reflects our lack of understanding of the sun from one side and some misunderstandings. Similarly, the following discussion and questions are only for thinking.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Zhang, Solar Magnetism, https://doi.org/10.1007/978-981-99-1759-4_7

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7 More Questions

7.1 Measurements of Solar Magnetic Fields Because we can’t get close to the sun to measure its magnetic field directly, the information we get from the sun’s magnetic field is usually based on the theoretical framework system based on the spectral Zeeman effect in the magnetic field, and the diagnosis and analysis method of solar polarized light gradually established from it. In addition, the Hanle effect and the polarization component of the radio band also carry the information of the solar magnetic field. How to infer the solar magnetic field from the information of the radiation transfer process of polarized light in the solar atmosphere is of great significance. However, due to the cognitive defects on the basic structure of the solar atmosphere itself and the nonuniqueness of the magnetic field information carried by polarized light, the measurement of the solar magnetic field is still the core topic in magnetic field research. One of the most obvious examples is the difference in the vector magnetograms of the solar active region obtained by different instruments for the observations of solar magnetic fields, which we cannot fully explain up to now to mention some basic questions involved in the measurement. Some of them include: (1) The parameters of spectral lines in the study of the radiative transfer in the solar atmosphere are usually based on a series of classical assumptions, such as local thermodynamical equilibrium. In polarized spectrum, the rationality of various parameters is still worthy of further discussion; (2) The solar atmosphere is a non-uniform and moving plasma. It is little known about the complexity of the radiative transfer process of magnetic sensitive lines in the solar atmosphere. Due to the anisotropy of the magnetic atmosphere, how to determine and correct the influence of magneto-optical effect on the magnetic field measurement is a significant subject; (3) The 180◦ uncertainty in the direction of the transverse magnetic field is an inherent defect for inferring the magnetic field from the solar polarized lights; (4) As the solar upper atmosphere is optically thin and the temperature rises suddenly, the measurement of the magnetic field in the solar chromosphere and transition region will encounter more uncertainties. It is more difficult and important to accurately measure and analyze the weak magnetic field in the coronal atmosphere; (5) It is generally believed that the limitation of the spatial resolution of the solar magnetic field from the observations is on the scale of the photon-free path in the solar atmosphere, that is, the order of 100 km in the photosphere. The details of the solar magnetic field are still a basic topic.

7.2 Basic Configuration of Solar Magnetic Fields Because the magnetic field plays a key role in solar activity, it is of great significance to study the basic structure and evolution of the solar magnetic field. It is normally believed that more accurate measurements of the solar magnetic field are taken in the

7.3 Solar Magnetic Cycles

379

lower solar atmosphere, while the magnetic field structure in the upper atmosphere of the sun is usually based on morphological and theoretical analysis. The accuracy of the magnetic field structure in the upper solar atmosphere inferred by these methods is still questionable. Therefore, it is more difficult to get a consistent conclusion about the distribution of the magnetic field in the upper atmosphere of the sun. In addition, it is difficult to infer the horizontal current in the solar atmosphere exactly, because the vector magnetograms related to the different layers of the sun normally have different magnetic sensitivities and noise levels. It is also related to obtaining the complete magnetic (current) helicity and other basic physical quantities. Therefore, it should be said that our understanding of the basic structure of the solar magnetic field is still not very clear. In the aspect of the analysis and research of the solar magnetic field structure, many topics need to be further studied: (1) The basic structure of the solar magnetic field lines in the recognition of the physical characteristics of the magnetic flux tubes. It is important to know the basic form of the cosmic magnetic field; (2) The internal relationship between sunspots and magnetic field in solar active regions includes the basic form of magnetic emergence and moving magnetic structures; The distribution of solar magnetic shear, current, and helicity, as well as the possible spatial form and evolution, associated with the solar non-potential magnetic energy storage, are determined from the observations; (3) The reconnection and annihilation mechanism of the magnetic fields and the relationship with solar flare-coronal mass ejections, as well as the further confirmation of the possible observational characteristics; (4) There is a question of the possible relationship between the magnetic fields and the chromospheric and coronal heatings.

7.3 Solar Magnetic Cycles The research on the periodicity of solar magnetic activity is the core subject, which involves the basic problem of solar generator mechanisms. Through the statistical analysis of a large number of sunspot active regions, people found the phenomenon of the solar 11-year cycle, and then through the observation of the solar magnetic field, found the 22-year polarity cycle of the solar large-scale magnetic field. A large number of observations of the solar surface magnetic field provide us with an important opportunity and breakthrough point to understand the formation of the solar internal magnetic field. The systematic observation of the vector magnetic field in the solar active regions provides us with the distribution characteristics of the helical solar magnetic field. It is a new window to study and analyze the formation of the solar internal magnetic field. Meanwhile, due to the opacity of the solar atmosphere, even though we can know the general situation of the solar convective region through the diagnosis of helioseismology, we still know little about the magnetic field information inside the sun, so that we cannot completely confirm which parameters play a decisive role in the solar generator process. How to infer the structure and evolution of the solar

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7 More Questions

internal magnetic field from the observed information of the solar surface magnetic field is challenging. Some topics to be further explored include: (1) The statistical distribution and characteristics of non-potential magnetic field and helicity with solar cycles, and their correlation with solar flares and coronal mass ejections; (2) How the solar magnetic field forms and how it interacts with the velocity field; (3) How the magnetic (current) helicity contributes to the process of the solar dynamo; (4) Whether the solar dynamo theory is an important means to effectively explore and predict the changes of solar activity cycles.

7.4 Space Weather Space weather is the concept of changing environmental conditions in near-Earth space or the space from the Sun’s atmosphere to the Earth’s atmosphere. It is distinct from the concept of weather within the Earth’s atmosphere (troposphere and stratosphere). Space weather is the description of changes in the ambient plasma, magnetic fields, radiation, and other matter in space. The magnetic fields are the driving source of space weather from the solar storms that erupted in the solar atmosphere. The term space weather is sometimes used to refer to changes in interplanetary (and occasionally interstellar) space, while we also do not understand the trigger process of solar storms well until now. A basic question is how do the solar magnetic fields extend from the solar surface to the solar-terrestrial space with solar storms.

Postscript

Recalling that, after graduating from university and being assigned to the solar division in the Beijing Astronomical Observatory of the Chinese Academy of Sciences (the predecessor of the National Astronomical Observatories), I first went to the Nanjing Astronomical Instrument Factory of the Chinese Academy of Sciences to participate in the development of the solar magnetic field telescope led by Academician Ai Guoxiang. After that, under the leadership of Academician Ai Guoxiang, I participated in the establishment of the Huairou Solar Observing Station for the observation and operation of the Solar Magnetic Field Telescope, which was at the international leading level at that time. During this process, my research direction on the subject of solar vector magnetic field was determined. I am very grateful for the many academic benefits and enhancements I have gained during this period. Looking back on my research for more than 40 years, my main research is carried out around the observation and theoretical analysis of the magnetic field at Huairou Solar Observation Station of the National Astronomical Observatory, which has become the main thread of this book. I remember that when I first joined the research group of the Solar Magnetic Field Telescope, I followed Academician Ai Guoxiang to study and calculate the formation of FeIλ5324.19Å spectral lines in the solar magnetic atmosphere. Under his patient guidance, I entered the field on the measurements of the solar magnetic field. The procedure I use now for radiation transfer in the solar magnetic atmosphere is based on his work. After this, I studied the radiation transfer of the Hβ spectral line, a unique working spectral line of the Solar Magnetic Field Telescope. I have the honor to first theoretically analyze the properties of the magnetic fields in the solar quiet and active regions, such as spectral line broadening and formation depth. These researches have given me a deep understanding of the measurement theory of the solar magnetic field and provided a very useful foundation for the subsequent analysis of the observing data of the solar magnetic field at Huairou. The successful operation of the Solar Magnetic Field Telescope in Huairou led to the observation of a large number of vector magnetic fields in the solar active regions and also in the quiet region. In-depth analysis of these data became the first © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 H. Zhang, Solar Magnetism, https://doi.org/10.1007/978-981-99-1759-4

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task. During this period, we found interesting phenomena such as the change of the photospheric vector magnetic field in the active region before and after the flares, and the relationship between the evolution of the solar photospheric magnetic field and the current emergence in active regions. Participating in the “The sun never sets” observation of the solar magnetic field jointly conducted with the Big Bear Solar Observatory in the United States to determine the network lifetime and the long-term evolution of the magnetic field in the active regions, etc., which have been attracted great attention in the international solar physics community. With colleagues, we explored the relationship between magnetic shear, current and current helicity characteristics in the solar active regions, current helicity, and kinematic helicity obtained by helioseismic methods, as well as the internal connection between the solar magnetic field and solar flare-coronal mass ejections, etc. In the process of further discussing the observation data of the Huairou vector magnetic fields, the influence of the magneto-optical effect on the observation accuracy is analyzed. The video observation of the solar chromospheric magnetic fields is one of the features that distinguish the Huairou Solar Magnetic Field Telescope from other instruments in the world. It is logical to carry out the observation of the fine structural characteristics of the chromospheric magnetic field in the quiet region of the sun, thus discovering the fiber structure of the chromospheric super-penumbral magnetic field in the solar active regions, and analyzing the reasons for the signal inversion in the strong magnetic field region of the chromosphere sunspot. These studies draw our attention to the solar magnetic field in the upper solar atmosphere. The accumulation of a large number of systematic observations of the solar vector magnetic fields at the Huairou Station makes a possible for the statistical analysis of the magnetic (current) helicity in the solar active regions. We firstly discuss the relationship between the magnetic helicity of solar active regions and the solar activity cycles. A direct finding is the delayed effect of the statistical distribution of the magnetic (current) helicity in the solar active regions to the sunspot numbers and the variation of the magnetic (current) helicity swing with the solar cycles. The observational facts aroused the interest of scientists in Russia and other countries in the above-mentioned magnetic helicity results and the formation mechanism of the internal magnetic field of the Sun, and intrinsically linked the statistical characteristics of the helicity in the solar active regions with the theory of solar turbulence dynamo. Naturally, it also involves the study of the magnetic turbulence spectrum derived from the helicity of the active regions. I feel that in the process of research, some foreign observational data such as the solar magnetic field are also used. Using these data in conjunction with Huairou’s observational data, I feel that they complement each other and have achieved good results in the research works. In the process of the above research, together with colleagues, the development of the full-solar disk vector magnetic field observation was successfully carried out. Fortunately, the instrument is the first in the world to obtain high-resolution full-solar disk vector magnetograms. The instrument is still functioning well so far. As a person who has been engaged in the observing research of solar physics for a long time, I have the honor to participate in the operation and organization of the Huairou Solar

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Observation Station, the promotion of some space projects on solar magnetic field observation, and the organization of multiple total solar eclipses, which have been expanded my knowledge of solar physics research. Looking back on years of research work, I gradually understand part of the content of solar physics. I know that what I have presented in this book is only part of the physics of the Sun. Solar physics is a physical system established based on actual measurements. It covers such a wide range, and the basic knowledge involved covers many fields of physics and is intertwined with each other. When you dig into it, you will find that while we know something about it, there are many issues that are not fully solved or not solved. There are many kinds of modern sciences, which are broad and profound. Even in the field of solar physics, it is impossible to understand or make achievements. As Einstein said, any branch of science has the potential to devour a lifetime of experience. There are a large number of solar physics and related books at home and abroad, covering almost all directions. I feel that combined with my main research topics in recent years and cooperation with my colleagues and friends (and also former graduate students) in China, as well as international collaborators. I am very grateful for their cooperation in my solar physics research, especially Li Wei, Bao Shudong, Zhang Mei, Tian Lirong, Bao Xingming, Liu Yu, Dun Jinping, Su Jiangtao, Liu Jihong, Zhang Yin, Ruan Guiping, Guo Juan, Xie Wenbin, Wang Xiaofan, Chen Jie, Gao Yu, Xu Haiqing, Yang Shangbin, Sun Yingzi, Liu Suo, Li Xiaobo, Zhao Mingyu, Yang Xiao, Wang Dong, and other colleagues (and former graduate students) in the National Astronomical Observatories of the Chinese Academy of Sciences (or who were there); Mao Xinjie of Beijing Normal University; Song Mutao of Purple Mountain Observatory, Lin Jun of Yunnan Astronomical Observatory; Wang Haimin, Li Jing, Wang Tongjiang, Zhao Junwei, A. Pevtsov in the United States; Japanese Scientists T. Sakurai, H. Kurokawa; Greek scientist A. Nindos. In the study of magnetic helicity and solar cycle, Russian scientists D. Sokoloff, K. Kuzanyan, V. Pipin, O. Chumak; J. Büchner in Germany; A. Brandenburg in Sweden; Israeli scientists N., Kleeorin, I. Rogachevskii, and British scientist D. Moss and others. Their fruitful works have made progress and gratifying achievements in our research. In the process of exploring the measurement method of solar magnetic field, we have obtained good cooperation from our colleagues at the National Astronomical Observatory of the Chinese Academy of Sciences, especially Deng Yuanyong, Yang Shimao, Wang Dongguang, Lin Ganghua, Hu Keliang, Lin Jiaben, Wang Guoping, and others. Here, I would like to thank my colleagues at home and abroad for their good cooperation and harmonious academic atmosphere, which enabled me to do some useful work in the field of solar physics. In addition, because this article evolved in the form of a series of research results, and inserts some background knowledge to make it possible to string together, there will inevitably be various problems. Due to the limitation of level and cognition, there may be inappropriate points in the book, and readers are welcome to criticize and correct them. Its background knowledge is mostly quoted from other people’s materials or research results, and the sources are listed. Please be aware that the unit system of the formulas in different materials may vary.

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The achievements in the research work also stem from the strong support of the national major projects, the Chinese Academy of Sciences, and the National Natural Science Foundation of China.

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