Fast Solar Wind Driven by Parametric Decay Instability and Alfvén Wave Turbulence (Springer Theses) 9811610290, 9789811610295

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Springer Theses Recognizing Outstanding Ph.D. Research

Munehito Shoda

Fast Solar Wind Driven by Parametric Decay Instability and Alfvén Wave Turbulence

Springer Theses Recognizing Outstanding Ph.D. Research

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Munehito Shoda

Fast Solar Wind Driven by Parametric Decay Instability and Alfvén Wave Turbulence Doctoral Thesis accepted by The University of Tokyo, Tokyo, Japan

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Author Dr. Munehito Shoda Department of Earth and Planetary Science, School of Science The University of Tokyo Tokyo, Japan

Supervisor Dr. Takaaki Yokoyama Department of Earth and Planetary Science, School of Science The University of Tokyo Tokyo, Japan

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-981-16-1029-5 ISBN 978-981-16-1030-1 (eBook) https://doi.org/10.1007/978-981-16-1030-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 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 translation, 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

Supervisor’s Foreword

This thesis by Dr. Munehito Shoda describes the studies on the fundamental physics of the heating and acceleration in the fast solar wind that is the super-sonic plasma outflow from the Sun. It has been a mystery how the fast wind can be accelerated to the speed beyond that of the classical thermally-driven wind. The Alfvén wave turbulence has been believed to play a key role, however, the maintenance of which remained unresolved. The author focused on the interplay between compressional and non-compressional magnetohydrodynamic waves and proposed a novel theoretical framework, i.e., the parametric-decay-instability-driven turbulence model. Based on this model, the proposed interplay process and consequent reproduction of a fast wind were, for the first time in my best knowledge, successfully demonstrated in a self-consistent three-dimensional simulation. This work yielded essential theoretical predictions for observations by in-operation and future space missions. Tokyo, Japan November 2020

Dr. Takaaki Yokoyama

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Abstract

A continuous injection of energy and momentum is required to sustain the high-temperature corona and high-speed solar wind. A widely accepted idea is that the origin of such energy and momentum lies in the convective motion of the solar surface (photosphere) and the magnetic field transports them to upper atmosphere via Alfvén wave, which is called Alfvén-wave modeling or Alfvén wave scenario of the (fast) solar wind. Although the general framework of Alfvén-wave modeling is, at least theoretically, of little doubt, the detailed physics inside is yet controversial. Specifically, the thermalization mechanism of Alfvén wave in the corona and solar wind is unclear. Bearing in mind that the (magnetic) Reynolds number is quite large, turbulence and resultant cascading are required to obtain adequate heating. A standard model, called reflection-driven Alfvén wave turbulence model, assumes that inward Alfvén waves generated by a partial reflection in the solar wind collide with outward Alfvén waves, triggering Alfvén wave turbulence to heat the solar wind. One problem that recently arose regarding this model is that the heating rate is insufficient. Therefore, we need to revisit and modify the standard model, incorporating new physics. The standard model is based on reduced MHD equations in which compressional waves are ruled out. We assume that the shortage of heating is attributed to neglecting compressional waves. An overall motivation of this thesis is to update the standard model including compressional waves. Permission of compressibility is crucial in the solar wind, where plasma beta is low, because parametric decay instability works with a comparable time scale to Alfvén wave turbulence. For this purpose, we have performed two different simulations: one-dimensional and three-dimensional simulations. In Chaps. 2 and 3, we report the results of one-dimensional simulations. To consider the turbulent heating in one-dimensional geometry, we introduce a phenomenological turbulence model of Alfvén wave turbulence into compressible MHD equations. The turbulence model is set so that it becomes equivalent with commonly used Alfvén wave turbulence model in the limit of incompressibility. In the limit of infinite turn over time of turbulence, this model is identical to the vii

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Abstract

previous work without turbulence. We have found that the dominant heating process in the solar wind is correlation-length dependent; with small correlation length most of the heating is by turbulence and with large correlation-length shock heating is dominant. With the realistic correlation length, the turbulence is slightly stronger than shock. Using parameter survey, we have revealed several conclusions. First, the solar wind velocity is strongly affected by correlation length while the magnitude of energy injection determines the mass-loss rate. This result is consistent with previous works. Second, the observed large density fluctuation in the solar wind acceleration region is explained by parametric decay instability. Specifically, the largest-density-fluctuation region is the same as the largest-growth-rate region of parametric decay instability. Third, the cross-helicity evolution is explained based on linear reflection. These indicate that parametric decay instability plays a crucial role in the solar wind acceleration region but not in the distant solar wind. In Chap. 4, we show the results of the three-dimensional simulation, the motivations of which are to validate the 1D model and to predict the data of Parker Solar Probe. Our simulation is the first-ever three-dimensional self-consistent simulation of the solar wind acceleration. Large density fluctuation is generated by parametric decay instability in the wind acceleration region consistently with 1D simulation. The turbulence in the solar wind is characterized by an imbalanced MHD turbulence in which the power spectra of outward and inward waves have different power indices. As a prediction of Parker Solar Probe, we have shown that the positive correlation between density and parallel-velocity fluctuations would be observed. As a conclusion of this thesis, we propose an updated standard model of the solar wind acceleration: the PDI-driven Alfvén wave turbulence model. Alfvén wave reflection, the source of turbulence in the solar wind, is triggered not only by simple linear refection but also by the parametric decay instability. The compressible waves play a crucial role in the solar wind turbulence and therefore are never ignorable in the simulation.

Parts of this thesis have been published in the following journal articles: M. Shoda, T. Yokoyama and T. K. Suzuki, 2018: A Self-consistent Model of the Coronal Heating and Solar Wind Acceleration Including Compressible and Incompressible Heating Processes, The Astrophysical Journal, 853, 190, https://doi. org/10.3847/1538-4357/aaa3e1 M. Shoda, T. Yokoyama and T. K. Suzuki, 2018: Frequency-dependent Alfvén-wave Propagation in the Solar Wind: Onset and Suppression of Parametric Decay Instability, The Astrophysical Journal, 860, 17, https://doi.org/10. 3847/1538-4357/aac218 M. Shoda, T. K. Suzuki, M. Asgari-Targhi and T. Yokoyama, 2019: Three-dimensional Simulation of the Fast Solar Wind Driven by Compressible Magnetohydrodynamic Turbulence, The Astrophysical Journal, 880L, 2, https:// doi.org/10.3847/2041-8213/ab2b45

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Acknowledgments

It was really lucky for me that I had a chance to take a class by Takaaki Yokoyama, my supervisor. Without him, I would never have studied astrophysics. All of my work I have accomplished are thanks to his continuous encouragement and endless patience. I cannot thank him enough. My colleagues of Yokoyama lab also helped me through various discussions. It was my great honor to be a member of our laboratory. I would also like to show my gratitude to all the members in Space Science Group in the Department of Earth and Planetary Science, School of Science, The University of Tokyo. Seminars in our group are always enjoyable and stimulating, and sometimes tough. All the questions and comments helped me improve my work. During my visit in US, I was supported by Dr. Mahboubeh Asgari-Targhi and her family. I would like to express my appreciation to her. Thanks to her, I experienced and learned a lot of valuable things, including how I should behave when we lose our passport abroad and have to go back to Japan in three days! I also learned the importance of collaboration and discussion with others from her. I owe my growth as a researcher to her. I have been supported by the Research Fellowship for Young Scientists of the Japan Society for the Promotion of Science (JSPS) and Leading Graduate Course for Frontiers of Mathematical Sciences and Physics (FMSP). Nearly all the numerical calculations are carried out on PC cluster, Cray XC30 and Cray XC50 at Center for Computational Astrophysics, National Astronomical Observatory of Japan.

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2 1D Model with Compressional and Non-compressional Nonlinear Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Phenomenology of Quasi-2D Turbulence . . . . . . . . . . . 2.2.2 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Simulation Domain and Boundary Condition . . . . . . . . .

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1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Observational Fundamentals . . . . . . . . . . . . . . . . . . . 1.1.1 Flux-Tube Expansion Factor . . . . . . . . . . . . . . 1.1.2 Property of Turbulence . . . . . . . . . . . . . . . . . 1.1.3 Thermodynamical Properties . . . . . . . . . . . . . 1.1.4 Summary of Observational Aspects . . . . . . . . 1.2 Thermally Driven Wind Model . . . . . . . . . . . . . . . . . 1.2.1 Hydrostatic Solar Atmosphere with Thermal Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Transonic Solution . . . . . . . . . . . . . . . . . . . . . 1.2.3 Generalizations and Limitation . . . . . . . . . . . . 1.3 Wave/Turbulence-Driven Model . . . . . . . . . . . . . . . . 1.3.1 Observational Signatures of Alfvén Waves . . . 1.3.2 Alfvén Wave as a Source of Momentum . . . . . 1.3.3 Numerical Simulations Based on Alfvén Wave Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Theoretical Aspects of Solar Wind Turbulence . . . . . . 1.4.1 Quasi-2D Alfvén-Wave Turbulence . . . . . . . . 1.4.2 Heating-Rate Problem . . . . . . . . . . . . . . . . . . 1.4.3 Parametric Decay Instability . . . . . . . . . . . . . . 1.5 Summary and Motivation of This Thesis . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.3 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Quasi-steady States with Various Correlation Lengths 2.3.2 Density Fluctuation and Cross Helicity . . . . . . . . . . . 2.3.3 Heating Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Dependence on the Injection Amplitude . . . . . . . . . . 2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Heating Mechanism in the Solar Wind . . . . . . . . . . . 2.4.2 Parameter Dependence of the Solar/Stellar Wind . . . . 2.4.3 Photospheric Correlation Length . . . . . . . . . . . . . . . . 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Onset and Suppression of Parametric Decay Instability . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Simulation Domain and Boundary Condition . . . . . . 3.3 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Monochromatic Wave Injection . . . . . . . . . . . . . . . 3.3.2 Growth Rate of PDI in the Expanding/Accelerating Solar Wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Broadband Wave Injection . . . . . . . . . . . . . . . . . . . 3.4 Discussion and Summary . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3D Simulations of the Fast Solar Wind . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Motivation from Theoretical Point of View . . . . . . . 4.1.2 Motivation from Observational Point of View . . . . . 4.1.3 Purpose of 3D Simulation . . . . . . . . . . . . . . . . . . . 4.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Local Spherical Coordinate . . . . . . . . . . . . . . . . . . 4.2.2 Basic Equations and Numerical Solver . . . . . . . . . . 4.2.3 Numerical Settings . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Averaged 1D Structure in the Quasi-Steady State . . 4.3.2 Density Fluctuation and Enhanced Wave Reflection . 4.3.3 Origin of Density Fluctuation: Parametric Decay Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Two-Dimensional Slice: rh-Plane . . . . . . . . . . . . . . 4.3.5 Two-Dimensional Slice: h/-Plane . . . . . . . . . . . . . . 4.3.6 Anisotropy of Turbulence . . . . . . . . . . . . . . . . . . . . 4.3.7 Plasma Heating Mechanism . . . . . . . . . . . . . . . . . . 4.3.8 Prediction for Parker Solar Probe . . . . . . . . . . . . . .

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4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Plasma Heating Scenario . . . . . . . . . . . . . . . . . 4.4.2 Parametric Decay Instability in an Open System 4.4.3 Masking Effect of Line-of-Sight Super Position . 4.4.4 Power Spectrum with Respect to Perpendicular Wave Number . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 The Bottom Boundary Condition . . . . . . . . . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Summary and Discussion . . . . . . . . . . . . . . . . . . . . 5.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . 5.1.1 1D Simulation . . . . . . . . . . . . . . . . . . . . 5.1.2 3D Simulation . . . . . . . . . . . . . . . . . . . . 5.2 General Discussion . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Heating Mechanism in the Solar Wind . . 5.2.2 Turbulence in the Solar Wind . . . . . . . . . 5.3 Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Chromosphere and Transition Region . . . 5.3.2 Toward a Better Modeling of Turbulence References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

General Introduction

Abstract The presence of the supersonic outflow from the Sun, the solar wind, is first predicted by Parker (1958). Since its first observational detection (Neugebauer and Snyder 1962, 1966), the solar wind has been extensively studied from various aspects, including heating and acceleration mechanism, evolution of turbulence, and space weather forecasting. How the solar wind is heated and accelerated is one of the most important problems in astrophysics. Because the magnetic energy is much larger than the thermal and kinetic energies in the corona, the magnetic field is likely to play a role in the solar wind acceleration. The solar wind model based on magnetic energy deposition to the wind is classified into two: wave/turbulence-driven model and reconnection/loop-opening model. Both theoretical and observational arguments support the wave/turbulence-driven model as a model of the fast solar wind, although the conventional Alfvén-wave turbulence turned out to be insufficient to heat the solar wind. This “heating-rate” problem might be solved once the plasma compression is taken into account. Keywords Solar wind · Alfvén wave · Turbulence

1.1 Observational Fundamentals Throughout this thesis, we discuss the theory of the fast solar wind, a relatively fast (500 km s−1 ) component of the solar wind. Interestingly, the fast and slow (500 km s−1 ) solar winds are found to exhibit different properties besides velocity. Observed differences between fast and slow solar winds indicate that the two have different origins. Specifically, the fast solar wind is known to blow out from polar coronal holes or large equatorial coronal holes, while the source of the slow solar wind remains unknown. In the first section, we summarize the observational fundamentals of the solar wind, with especial interest on the difference between the fast and slow solar winds.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. Shoda, Fast Solar Wind Driven by Parametric Decay Instability and Alfvén Wave Turbulence, Springer Theses, https://doi.org/10.1007/978-981-16-1030-1_1

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1 General Introduction

1.1.1 Flux-Tube Expansion Factor The structure of the solar wind is highly influenced by the (large-scale) magnetism of the Sun. In the solar minimum, a clear bimodal structure is found in the latitudinal distribution, with high-speed wind in the high latitude and low-speed wind in the low latitude (McComas et al. 2008). Dipole-dominated solar magnetic field in the solar minimum appears to generate such a clear structure. In the solar maximum, on the other hand, the structure of the solar wind is far more complicated: the fast and slow solar winds have a randomly mixed distribution in the latitudinal direction, corresponding to the complex nature of solar magnetic field (McComas et al. 2008). The connection between the solar wind velocity and local magnetic-field structure is seen in a more quantitative manner once the expansion factor f exp is obtained. The expansion factor of a flux tube is defined by the degree of super-radial expansion from the coronal base. Mathematical definition of f exp is given by f exp =

2 Br, R , 2 Br,ref rref

(1.1)

where Br, is the radial magnetic field in the coronal base. rref is a reference distance above which the magnetic field is (nearly) radial and Br,ref is the radial magnetic field at r = rref . If all the coronal magnetic field lines are open to the interplanetary space, the fields expand radially without super-radial expansion. Actual coronal field lines are not always open, but roughly 90% of the coronal unsigned magnetic flux come from closed field lines. The open field lines then super-radially expand to compensate the space above the closed loops. If we use Potential Field Source Surface method (Schatten et al. 1969) with source surface of 2.5R , one obtain typically f exp ∼ 10. Coronal holes, relatively dark regions in the corona, are found to be the source region of the fast solar wind (Cranmer 2009). In light of small expansion factor of coronal hole, Wang and Sheeley (1990) investigated the relation between the expansion factor and solar wind velocity and found that wind speed is anti-correlated with the expansion factor. That is, the fast solar wind is associated with small f exp and vice versa. Since the expansion factor is possible to estimate from the photospheric magnetic field, the finding by Wang and Sheeley (1990) opened a way to predict the solar wind velocity from the solar photospheric observation. The specific relation between vr and f max are given by several works (Wang and Sheeley 1990; Arge and Pizzo 2000; Fujiki et al. 2015).

1.1.2 Property of Turbulence Fluctuation in the solar wind is known to be turbulent with a broadband power spectrum (Coleman 1968; Belcher and Davis 1971; Podesta et al. 2007). In the fast solar wind, fluctuations are Alfvénic in that velocity and magnetic field are correlated in the sense of outward Alfvén waves, that is,

1.1 Observational Fundamentals

3

δ B/B = −δv/v A ,

(1.2)

where δv and δ B are fluctuations in velocity and magnetic field and B and v A are averaged magnetic field and Alfvén velocity. For this reason, fluctuations in the fast solar wind are interpreted as turbulent Alfvén waves. Turbulent Alfvén wave in not a simple remnant of coronal turbulence advected into the interplanetary space. Measurement of high-order structure function shows that Alfvén waves in the solar wind undergo cascading, which is often found to compensate the adiabatic cooling by the wind expansion (Carbone et al. 2009). Unlike fast solar wind, fluctuations in slow solar wind are not always Alfvénic. Instead, the slow solar wind is often found to exhibit much larger density fluctuation than the fast solar wind. Turbulence in the slow solar wind is thus not a simple superposition of Alfvén waves that is unlikely to generate large-amplitude density fluctuation. We note, however, that some slow solar winds exhibit similar property to fast solar wind in terms of fluctuation and are called Alfvénic slow solar wind. The Alfvénic fluctuations exhibit 1/ f power spectrum, the origin of which is still under debate. In the fast solar wind with large Alfvénic correlation, the power spectrum of magnetic field is described by a broken power law with lower-frequency range proportional to f −1 and higher-frequency range proportional to f −5/3 . A similar power spectrum is found also in the Alfvénic slow solar wind. The spectral break point frequency is found to increase with radial distance. Meanwhile in the slow solar wind with less Alfvénicity, the whole frequency range is fitted by a single power law of ∝ f −5/3 . These differences in the fast and slow solar winds indicate different roles of waves and turbulence in driving the two winds. Specifically, in driving the fast solar wind, Alfvén waves and turbulence are likely to be important.

1.1.3 Thermodynamical Properties Thermodynamical properties of the fast and slow solar winds are also found to be different, which give a crucial hint as to where each type of solar wind originates from. First, the proton temperature is correlated with the wind velocity. We note that, if the in-situ heating in the distant solar wind is absent, efficient thermal conduction of electron works in such a way that electron is always hotter than proton (Hartle and Sturrock 1968). In the fast solar wind, however, proton temperature measured at 1 au is usually larger than electron (Newbury et al. 1998; Cranmer et al. 2017). That proton is hotter than electron in the fast solar wind indicates the presence of extended proton heating. Compositional observation of the solar wind enables the estimation of sourceregion temperature based on the frozen-in property of the ionization state. In the solar wind, collision between ion and electron is insufficient enough to deviate the ionization degree from the Saha–Boltzmann equilibrium with local temperature. The

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1 General Introduction

ionization degree of ion is thus conserved soon after the solar wind becomes collisionless. In other words, the ionization state of ion indicates the electron temperature of the “source region”, where ions are still collisional. We find a clear anti-correlation between solar wind velocity and froze-in temperature, which shows the fact that the fast wind originates from cool corona and vice versa (Cranmer et al. 2017). This anti-correlation is a crucial contradiction to the classical theory of the solar wind (Parker 1958) and demands essential modification of the model. As well as the frozen-in temperature, the compositional difference with respect to the first ionization potential (FIP) is important. It is well known that the elements with low FIP is enhanced in the (core of the) streamer (Raymond et al. 1997). The similar enhancement of the low-FIP element is often observed in the slow solar wind but not in the fast solar wind (von Steiger et al. 2000). This compositional difference can be an evidence that the two types of the solar wind are accelerated by different mechanisms.

1.1.4 Summary of Observational Aspects The observational aspects of the fast and slow solar winds are summarized as follows. 1. The solar wind velocity is found to anti-correlate with the expansion factor of the flux-tube from which the wind blows out. That is, the fast solar wind originates from the small-expansion region such as core of large coronal hole, while the origin of the slow solar wind is likely to be magnetically more complex such as coronal-hole boundary, edge of active region, etc. 2. Fluctuation and turbulence in the solar wind show different properties in fast and slow solar winds. The fast solar wind usually exhibits large Alfvénicity in which fluctuations of velocity and magnetic field are connected in an Alfvénic manner. The Alfvénic fluctuation also shows 1/ f spectrum, indicating energy storage in the Alfvénic fluctuations. The slow solar wind is less Alfvénic and exhibits f −5/3 spectrum in magnetic energy. This difference indicates different roles of turbulence in the fast and slow solar winds. Specifically, Alfvén waves and turbulence are likely to be crucial in the fast wind acceleration. 3. Thermodynamical properties such as proton temperature and compositional difference are also found to be different in the two types of wind. The fast solar wind blows out from cool corona and undergoes little FIP effect, and vice versa for the slow solar wind. Different source region and acceleration mechanism of the fast and slow solar winds are inferred from this observation. Bearing these observational aspects in mind, we move on to the theoretical arguments of the solar wind from below.

1.2 Thermally Driven Wind Model

5

1.2 Thermally Driven Wind Model Close connection between magnetic field structure and solar wind velocity indicates the magnetic origin of the solar wind energy. In fact, in light of the low-beta nature of the solar corona, the solar wind energy should be fed by magnetic field, either directly or indirectly. Even though, a classical theory of the solar wind that is based on thermal expansion of the corona is worth noting. The theory was first described by Parker (1958), and was generalized in various aspects to yield consistent properties with observations.

1.2.1 Hydrostatic Solar Atmosphere with Thermal Conduction Before discussing the dynamical solution, we first discuss the property of hydrostatic equilibrium, which turns out to be unrealizable in the presence of thermal conduction. The hydrostatic equilibrium is described by the following force balance between outward gas pressure and inward gravity. By simply assuming that the solar atmosphere is composed of fully-ionized hydrogen, the force balance equation is given by G M d (2nk B T ) = −nm p 2 , dr r

(1.3)

where n is the number density of hydrogen, k B is the Boltzmann constant, T is the hydrogen temperature, m p is the hydrogen (proton) mass, G is the gravitational constant, and M is the solar mass. Under given temperature T (r ), the above equation yields a solution in terms of pressure as    m p G M dr . p = p exp − 2 2k B r r T (r )

(1.4)

The asymptotic behavior of this solution is worth noting. Suppose, for instance, an isothermal atmosphere (T (r ) = T ), the pressure at sufficiently large r is given by   m p G M , p(r ) ∼ p exp − 2k B Tr

(1.5)

which means the existence of finite pressure at infinitely far region. Since the actual interstellar pressure is negligibly small, the pressure needs to asymptotically approach zero at r → ∞. In terms of temperature, from Eq. (1.4), limr →∞ p = 0 is equivalent with T decaying faster than 1/r . Thus, a hydrostatic equilibrium requires sufficiently fast decrease of temperature.

6

1 General Introduction

Chapman and Zirin (1957) showed, however, that T decreases slower than 1/r in the presence of thermal conduction. According to Spitzer and Härm (1953), the thermal conductive flux in a fully-ionized plasma Fcnd is written in terms of T by Fcnd = −κ0 T 5/2

d T, dr

(1.6)

where κ0 ≈ 10−6 in cgs unit. The stationary condition requires the heating-cooling balance that is satisfied by  d  4πr 2 Fcnd = 0, dr

(1.7)

the solution of which is given by under the vanishing boundary condition limr →∞ T = 0 by T = T



r R

−2/7

.

(1.8)

The hydrostatic solar corona is therefore not sustained. In terms of energy, the outward conductive energy flux forces plasma to move outward.

1.2.2 Transonic Solution The observational indication of anti-Sunward plasma outflow (Biermann 1957), as well as the above theoretical argument, motivated the dynamical model of the solar atmosphere (Parker 1958). The basic equations of the stationary expanding, isothermal atmosphere are  d  ρvr r 2 = 0, dr

v

1 dp G M dv =− − , dr ρ dr r2

p = a 2 ρ,

(1.9)

where a denotes the speed of sound. After some algebra, these equations are reduced to the following ordinary differential equation:  v−

a2 v



dv 2a 2 = 2 (r − rc ) , dr r

(1.10)

where rc = G M /2a 2 is called critical height. The qualitative behavior of the above differential equation is described as follows. Since a meaningful solution should satisfy a subsonic flow speed in the vicinity of the Sun, we assume v < a at r < rc . One of the two possible solutions are what is always subsonic (v < a). In such a case, the sign of dv/dr is positive in r < rc and negative in r > rc , which yields a

1.2 Thermally Driven Wind Model

7

Fig. 1.1 Typical solutions of the isothermal wind equation (1.10) with subsonic basal speed. Higher and lower lines indicate the transonic and subsonic solutions, respectively

maximum at r = rc . The other possible solution requires v = a at r = rc , that is, the wind speed reaches the speed of sound at the critical height. Such a solution yields always positive dv/dr and experience the transition from subsonic to supersonic solutions. The existence of this transonic solution is the most pioneering insight in Parker (1958). Figure 1.1 shows the typical solutions of the isothermal wind with two lines indicating the transonic and subsonic solutions. Although we assumed for simplicity and isothermal atmosphere here, the transonic solution is found to exist as long as the polytropic index γ (= d ln p/d ln ρ) is smaller than the adiabatic value 5/3 (Parker 1965). The observational fact that the solar wind is always supersonic means that the transonic solution is what the nature favors. The reason for this is that the transonic solution is stable with respect to the fluctuation in the distant region, which is later shown by Velli (1994).

1.2.3 Generalizations and Limitation Since the Parker’s solar wind model is found to be correct, a number of attempts are made to generalize the model to explain the observed properties of the solar wind. Several important generalizations are described as below. 1. Sturrock and Hartle (1966) and Hartle and Sturrock (1968) solved the energy equations of ion and electron separately to show that the conductive wind model cannot account for the ion temperature in the solar wind. The density of the solar wind is enough to allow ions and electrons thermodynamically independent due to small frequency of collision. Since the heat flux of electron is much larger than that of ion, if they behave independently, thermal conduction cannot sustain high temperature of ion. Comparable temperature of proton and electron observed in the solar wind indicates that protons are selectively heated by some mechanism at play in the solar wind.

8

1 General Introduction

2. Super-radial expansion of flux tube needs to be considered. As already reviewed in Sect. 1.1, the super-radial expansion factor of flux tube is known to be anticorrelated to the wind velocity. It is therefore worth discussing whether a generalization of Parker’s model can account for such an anti-correlation. Kopp and Holzer (1976) generalized Parker’s wind model considering the super-radial fluxtube expansion and found that the location of the critical point is affected by the shape of the flux tube. Multiple critical points are also found in some range of parameters. Since the heat and momentum injection above (below) the critical point leads to fast (slow) wind, this generalization shows that the super-radial expansion can indirectly control the wind velocity. 3. Collisionless heat transport needs to be considered in the distant region. The widely-used Spitzer–Härm type thermal conduction (Spitzer and Härm 1953) assumes that the mean free path of electron is sufficiently smaller than the scale length of background temperature. In fact, electron observation near 1 au shows the deviation of heat flux from Spitzer-Härm value (Salem et al. 2003; Bale et al. 2013). An alternative can be the free-streaming flux (Perkins 1973; Hollweg 1974, 1976) that takes a form of qcnd =

3 αFS pvr , 4

(1.11)

where αFS is an order-unity free parameter, p is the pressure and vr is the solar wind velocity. Several solar wind models utilize a bridging law between Spitzer– Härm flux and free-streaming flux to account for the temperature profile (Hollweg 1986; Cranmer et al. 2007; Lionello et al. 2014). 4. A wide range of parameter survey of thermally-driven solar wind model failed in explaining the fast solar wind with realistic coronal density (Durney 1972; Holzer and Leer 1980). This motivated Leer and Holzer (1980) to add external momentum and energy injection terms. With this model, the fast solar wind is reproduced by heat/momentum addition in the supersonic region. The model of Leer and Holzer (1980) is further extended by including the chromosphere that regulates the coronal mass by energy balance or evaporation (Hammer 1982; Withbroe 1988; Hansteen and Leer 1995). In spite of the successful explanation of the origin of the supersonic flow, it turned out that the Parker’s model needed substantial modification to explain the fast solar wind; fast/slow solar wind is found to blow out from relatively cool/hot corona which is contradictory to Parker’s prediction. Besides, the origin of coronal heat should also be explained to understand the ultimate origin of the solar wind acceleration. These requirements led to the self-consistent modelling of the solar wind that solves the coronal heating and wind acceleration simultaneously without phenomenological additional terms. Since the magnetic energy dominates in the corona, how the magnetic energy generated or stored in the corona can be converted to the solar wind energy is a key to solve the problem. Theoretical models are classified into two depending on how the magnetic energy is fed to the wind: wave/turbulence-driven model and

1.2 Thermally Driven Wind Model

9

reconnection/loop-opening model (Cranmer 2012). The fast solar wind, the target of this thesis, is believed to be driven by wave and turbulence since magnetic reconnection is found to supply insufficient energy flux in coronal hole (Cranmer and van Ballegooijen 2010; Lionello et al. 2016). In what follows, we concentrate on the theory of wave/turbulence-driven model.

1.3 Wave/Turbulence-Driven Model Solar surface is filled with dynamic granular motion. Once the convective motion is coupled with magnetic field that is open to the interplanetary space, Poynting flux is transported to the wind in the form of field-line oscillation (Alfvén wave). The energy flux contained in the surface granular motion (∼109 erg cm−2 s−1 ) is known to be much larger than that is required to power the solar wind (∼106 erg cm−2 s−1 ). Convection-driven Alfvén wave is therefore a promising candidate that heats and accelerates the solar wind. Solar wind models based on Alfvén-wave heating and acceleration are called wave/turbulence-driven model. This section is dedicated to overview several fundamentals related to the wave/turbulence-driven scenario.

1.3.1 Observational Signatures of Alfvén Waves Various observations report direct or indirect evidence of Alfvén waves in the solar atmosphere and solar wind. In the vicinity of the solar surface, imaging of transverse motion is widely used to detect Alfvén(ic) waves and oscillations in the solar atmosphere. On the photosphere, tracking of granular motions (Matsumoto and Kitai 2010) or bright-point motions (Chitta et al. 2012) are used to measure the transverse velocity, both of which yield velocity amplitude of approximately 1–2 km s−1 . The energy flux of the photospheric transverse motion is in the order of 109 erg cm−2 s−1 , which is much larger than the energy flux required for the solar wind acceleration (∼106 erg cm−2 s−1 , Withbroe and Noyes 1977). The detection of Alfvén waves becomes much difficult in the higher atmosphere because of the superposition effect along the line of sight. When multiple waves exist with comparable contributions to observation, the observed quantity involve a mixture of contributing waves. To avoid this problem, an isolated wave guide such as jet-like structure is required for wave diagnostics. Such observation suffers from the limitation that waves with longer period than the lifetime of jet is in principle not detectable. To overcome such limitation, De Pontieu et al. (2007) use MonteCarlo simulation together with Hinode/SOT observations to obtain the amplitude and period of the chromospheric Alfvén waves as 10–25 km s−1 and 100–500 s, respectively. Such waves have sufficiently large power to drive the solar wind. A similar method is used to diagnose the coronal waves by SDO/AIA (McIntosh et al. 2011), obtaining the similar result with De Pontieu et al. (2007).

10

1 General Introduction

When we have an isolated wave guide, by investigating the oscillation of it, a direct analysis becomes available. Okamoto and De Pontieu (2011) perform a statistical analysis of spicule oscillations to show that the high-frequency waves with period ∼40–50 s are dominantly observed. Srivastava et al. (2017) also observes these high-frequency oscillations and shows that these high-frequency waves can potentially heat the corona. These high-frequency waves are possibly generated by mode conversion (Shoda and Yokoyama 2018). Meanwhile an imaging analysis of coronal plume oscillation indicates that the wave amplitude is not sufficiently large to drive the solar wind (Thurgood et al. 2014). Although it is widely accepted that upward Alfvén waves exist from the chromosphere to the coronal base, the properties of them are still on debate. Indirect methods are used to detect waves in the tenuous corona and solar wind where imaging is no longer available. One of such methods are to measure the nonthermal broadening of spectral lines. When incoherent wave motions exist in the line-of-sight, because of the super position of random Doppler shift, the spectral line becomes broader than is expected from thermal broadening. Such wave/turbulenceinduced line broadening is called non-thermal line broadening and is used to diagnose the turbulence intensity. In the vicinity of the Sun, SOHO/SUMER (Banerjee et al. 1998; Teriaca et al. 2003) and Hinode/EIS (Banerjee et al. 2009; Hahn and Savin 2013) are widely used to observe the line profiles in the corona. In the farer region, the spectral lines are observed by SOHO/UVCS (Esser et al. 1999). The non-thermal line broadening tends to increase with radial distance, indicating the upward wave propagation. In the interplanetary space beyond 0.3 au from the Sun, the velocity fluctuation is observed by in-situ measurements, mainly by Helios (inner heliosphere) and Ulysses (outer heliosphere) (Bavassano et al. 1982, 2000). The radial evolution of Alfvén waves given by these spacecrafts is consistent with simple linear propagation of Alfvén waves (Verdini and Velli 2007).

1.3.2 Alfvén Wave as a Source of Momentum Alfvén waves have a potential to heat the plasma, and the idea of Alfvén wave heating of the solar atmosphere dates back to Alfvén (1947) and Osterbrock (1961). As well as heat, Alfvén waves can also be a source of momentum through wavepressure acceleration (Alazraki and Couturier 1971; Belcher 1971; Jacques 1977; Heinemann and Olbert 1980). The fact that Alfvén waves can push plasma can be understood as follows. Suppose the energy flux of Alfvén wave is conserved during propagation along a flux tube, the magnetic fluctuation of Alfvén wave δ B satisfies δ B2 v A S = const., 4π

(1.12)

1.3 Wave/Turbulence-Driven Model

11

where v A is the Alfvén speed and S is the cross section of flux tube. Using the magnetic flux conservation (B S = const.), we obtain δ B2 = pAW ∝ ρ 1/2 , 8π

(1.13)

where pAW is the Alfvén wave pressure. Thus the acceleration rate by Alfvén wave pressure (aAW ) is given by aAW = −

1 d 1 d pAW ∝ − 1/2 ln ρ. ρ dr ρ dr

(1.14)

Given that the density drops off exponentially with length scale H , the acceleration rate satisfies aAW ∝

1 −1/2 ρ . H

(1.15)

This indicates that the wave acceleration works efficiently in the distant region where ρ is low. Thus Alfvén wave is a promising candidate for the momentum injection in the supersonic region, which is essential in making the fast solar wind.

1.3.3 Numerical Simulations Based on Alfvén Wave Modeling Based on the background above, several theoretical models explains the solar wind acceleration based on Alfvén wave heating and acceleration. Velli et al. (1989) proposes an idea that unidirectional Alfvén waves can drive turbulence in the inhomogeneous solar wind. This idea is extended to coronal heating by Matthaeus et al. (1999) and now is called reflection-driven Alfvén wave turbulence model. A number of theoretical works based on Alfvén wave turbulence model shows that the observed Alfvén waves have potential to explain the observed solar wind via turbulent heating (Cranmer et al. 2007; Verdini et al. 2010; Chandran and Hollweg 2009; Chandran et al. 2011; Usmanov et al. 2011; Lionello et al. 2014; van der Holst et al. 2014; Usmanov et al. 2018). Meanwhile, some models explain the fast solar wind without turbulence (Suzuki and Inutsuka 2005, 2006) based on mode conversion to slow mode waves (Hollweg et al. 1982; Kudoh and Shibata 1999; Matsumoto and Shibata 2010). A common conclusion of these models is that, as long as Alfvén wave dissipates with appropriate rate in the solar wind, the fast solar wind is well explained. For more precise understanding, not only the averaged parameter such as wind velocity or density but also the fluctuations and turbulence inside should also be explained.

12

1 General Introduction

1.4 Theoretical Aspects of Solar Wind Turbulence 1.4.1 Quasi-2D Alfvén-Wave Turbulence Gravitational stratification yields varying Alfvén speed (∝ ρ −1/2 ) in radial distance. This results in the partial reflection of Alfvén waves (Ferraro and Plumpton 1958; Heinemann and Olbert 1980; An et al. 1990; Velli 1993; Hollweg and Isenberg 2007). Decreasing trend of cross helicity with r in the polar heliosphere directly shows the presence of reflection in the solar wind (Bavassano et al. 1982, 2000). Recent ground-based observation of the corona also found a signature of backwardpropagating Alfvén waves (Morton et al. 2015). The Alfvén wave reflection is a key to understand the nonlinear dynamics of Alfvén waves in the corona and solar wind. It is known that a collision of bi-directional Alfvén waves drives nonlinear interaction and resultant cascading (e.g. Priest 2014). In the solar atmosphere, the anti-Sunward Alfén waves are generated by the photospheric magneto-convection and Sunward waves are by reflection. Thus, Alfvénwave reflection drives turbulence and resultant plasma heating. This framework of model is called reflection-driven Alfvén wave turbulence model and is proposed to explain the coronal and solar-wind heating (Velli et al. 1989; Matthaeus et al. 1999). Because the background magnetic field is in general larger than the fluctuating components, the driven turbulence is likely to be quasi-2D one (Shebalin et al. 1983), in which cascading perpendicular to the mean field is preferred. For this reason, the reflection-driven turbulence is often called quasi-2D Alfvén-wave turbulence. A goal of the theoretical model is to understand the property of turbulence as a function of radial distance. Specifically, once the heating (cascading) rate and turbulent pressure (Reynolds and Maxwell stress) are obtained, the large-scale structure of the solar wind is directly calculated. This is, however, a non-trivial task. To obtain the structure of the solar wind, one needs to solve the wave propagation up to 10R , while the required resolution to solve turbulence is many-orders-of-magnitude smaller. It is therefore difficult to deal with the fine turbulent structure and large-scale solar wind simultaneously. A widely-used approach is to model the wave-driven solar wind is to appropriately model the large-scale property of turbulence without solving fine structures, that is, Reynolds-average modeling. For this purpose, several works are dedicated to find an appropriate formulation of turbulent heating rate. Pioneering studies on this strategy were performed by Oughton et al. (2001) and Dmitruk et al. (2002), who solved the wave propagation and its turbulent evolution in the solar corona in the framework of reduced-MHD approximation (where any compressional waves are ignored). Figure 1.2 shows the horizontal profiles of field-aligned electric current (that is enhanced by Alfvén wave turbulence) in the upper corona (top panel) and at the coronal base (bottom panel). As a result of turbulence, finer-scale structures are observed in the upper layer. In the quasi-steady state, the averaged plasma heating rate is the same as averaged cascading rate that is determined by energy containing scale. Using a turbu-

1.4 Theoretical Aspects of Solar Wind Turbulence

13

Fig. 1.2 Horizontal profiles of the electric current along the mean magnetic field in the upper corona (top panel) and at the coronal base (bottom panel). Figure reproduced from Dmitruk et al. (2002) under permission of the American Astronomical Society

lence model by Hossain et al. (1995), Dmitruk et al. (2002) compare the heating rate directly calculated from simulation and phenomenologically obtained from turbulence model. Figure 1.3 compares the directly calculated heating rate (thin line) and phenomenologically estimated heating rate (thick line). The two lines are in agreement with each other, validating the phenomenological description. The phenomenological model of Dmitruk et al. (2002) have been widely used. Several works study the wave propagation with phenomenological turbulent dissipation to show that the turbulent heating is sufficiently large to sustain the solar wind (Cranmer and van Ballegooijen 2005; Verdini and Velli 2007; Chandran and Hollweg 2009). These works are extended to self-consistent modeling that solves the feedback from the waves to background (Cranmer et al. 2007; Verdini et al. 2010; Lionello et al. 2014).

14

1 General Introduction

Fig. 1.3 Plasma heating rates per unit mass directly calculated from simulation (thin line) and estimated from turbulence model (thick line) versus radial distance. Figure reproduced from Dmitruk et al. (2002) under permission of the American Astronomical Society

1.4.2 Heating-Rate Problem Simulation by Dmitruk et al. (2002) only focused on the limited range of radial distance with low spatial resolution. Recently, the capability of computer has increased and it enabled a direct reduced-MHD simulation with large spatial extension and high spatial resolution. Interestingly, the direct numerical simulations yielded a much smaller heating rate compared with the phenomenological model. As a result, the plasma heating rate by quasi-2D turbulence turned out to be much smaller than required to sustain the fast solar wind (Perez and Chandran 2013; van Ballegooijen and Asgari-Targhi 2016, 2017). Figure 1.4 shows the heating rates calculated from three-dimensional reduced-MHD simulation (black-solid line), calculated from the phenomenological model (blue line), and required to sustain the fast solar wind (black-dashed line) as functions of radial distance. There exists a roughly one-orderof-magnitude difference between the required and calculated heating rates, indicating the incapability of heating by reflection-driven turbulence. The phenomenological model, on the other hand, yields a sufficient heating rate by overestimating the actual value. van Ballegooijen and Asgari-Targhi (2017) argues that the inverse cascading that is ignored in the previous models works to reduce the net cascading rate. To summarize, the reflection-driven quasi-2D turbulence turned out to be too weak to sustain the fast solar wind. We thus need to update the model with some mechanism that have been ignored in the literature.

1.4.3 Parametric Decay Instability One solution to compensate the gap between required and simulated heating rates is to introduce density fluctuation. This is because the density fluctuation enhances the wave reflection to promote Alfvén wave turbulence. van Ballegooijen and AsgariTarghi (2016) show that density fluctuation with relative amplitude of 10% can drastically enhance the heating rate to be sufficiently large. Figure 1.5 shows the

1.4 Theoretical Aspects of Solar Wind Turbulence

15

Fig. 1.4 Heating rates required to sustain the solar wind (black dashed line) and obtained from 3D reduced MHD calculation (black solid line). Shown by blue solid line is the heating rate given by the phenomenological model. Figure reproduced from van Ballegooijen and Asgari-Targhi (2016) under permission of the American Astronomical Society

Fig. 1.5 Same as Fig. 1.4 but with density fluctuation. Figure reproduced from van Ballegooijen and Asgari-Targhi (2016) under permission of the American Astronomical Society

heating rates with 10% density fluctuation uniformly distributed from the coronal base to the solar wind. Alfvén wave turbulence in this case generates sufficiently large heating rate to sustain the fast solar wind. Several observations show that large (∼10%) density fluctuation does exist in the solar wind. Figure 1.6 is a compilation of various observations of density fluctuation in the solar wind. Blue rectangle near r = 4R (Coles and Harmon 1989; Spangler 2002; Harmon and Coles 2005) and black circles (Miyamoto et al. 2014) are obtained

16

1 General Introduction

Fig. 1.6 Compilation of the density fluctuation observation in the solar wind. Red and green rectangles approximate the density fluctuation reported in Marsch and Tu (1990). Blue rectangle near r = 4R (Coles and Harmon 1989; Spangler 2002; Harmon and Coles 2005) and black circles (Miyamoto et al. 2014) are from radio-wave observations

from radio-wave observations, while red and green rectangles approximate the density fluctuation obtained by in-situ measurements (Marsch and Tu 1990). Not shown in this figure, but recent observation of the coronal intensity fluctuation also shows the large (10% of mean value) density fluctuation in r > 1.2R (Hahn et al. 2018). The presence of non-negligible density fluctuation is of little doubt. Furthermore, recent simulation indicates that density fluctuation has a potential to enhance the heating of the solar wind. The origin of the density fluctuation is, however, yet unclear. We aim to explain the origin, magnitude and role of the density fluctuation in the solar wind. Numerical simulations based on compressible MHD equations reveal that the density fluctuation is a natural consequence of Alfvén wave propagation in the solar wind. Suzuki and Inutsuka (2005) perform 1D compressible MHD simulation of the fast solar wind to show that the density is highly fluctuating in the wind acceleration region. This is later confirmed by 2D compressible MHD simulation by Matsumoto and Suzuki (2012) as seen in Fig. 1.7. In the solar wind acceleration region (2.5R  r  10R ), density fluctuation becomes as large as a few tens percent. According to time-space diagram in Suzuki and Inutsuka (2006), the density fluctuation comes from slow mode (acoustic) waves. Since acoustic waves of photospheric origin dissipates quickly due to steepening, the slow mode waves in the solar wind should be in-situ generated waves. The parametric decay instability (hereafter PDI) can be a source of slow mode waves in the solar wind. Via PDI, Alfvén waves resonantly collapses into forward acoustic waves and backward Alfvén waves. The growth rate of PDI is higher in lower-beta plasma, and thus the corona and solar wind are preferential regions for PDI growth. Recent theoretical (Tenerani and Velli 2013; Réville et al. 2018) and observational (Bowen et al. 2018) studies indicate that PDI works in the solar wind. The shortage of heating in the standard model is possibly solved by introducing PDI for two reasons. First, by generating slow shock waves, PDI directly enhance

1.4 Theoretical Aspects of Solar Wind Turbulence

17

Fig. 1.7 A snapshot of density fluctuation in the meridional plane with horizontal axis radial distance in the solar wind. Color contours indicate the magnitude of density fluctuation and white lines are magnetic field lines. Figure reproduced from Matsumoto and Suzuki (2012) under permission of the American Astronomical Society

heating rate by adding shock wave heating. Second, by generating density fluctuation, Alfvén wave reflection is promoted. The enhanced reflection amplifies the turbulence, enhancing the heating rate. PDI and turbulence work with comparable timescales. The timescale of PDI (τPDI ) is estimated by the growth rate given by Sagdeev and Galeev (1969): τPDI ≈ 1/γPDI , γPDI =

1 δ B −1/4 β ω0 , 2 B0

(1.16)

where B0 and δ B are the mean and fluctuation parts of magnetic field, β = a 2 /v 2A is the (conventional) plasma beta, and ω0 is the (typical) frequency of Alfvén wave. τPDI is given by  τPDI ≈ 103 s

δ B/B0 10−1

−1 

β 10−2

1/4 

f0 10−3 Hz

−1

,

(1.17)

where we use the wave frequency f 0 = ω0 /2π instead of the wave angular frequency ω0 . Specifically, for the typical solar wind parameters (δ B/B0 = 0.1, β = 0.01), τPDI is approximated as τPDI ≈ τ0 ,

(1.18)

where τ0 = 1/ f 0 is the typical period of the Alfvén waves. Therefore in the solar wind, PDI grows with the timescale of injected wave period. Since the observed wave periods in the solar atmosphere and solar wind (102 − 104 s) are smaller than the Alfvén crossing time from the coronal base to wind acceleration region (104 s), PDI is expected to grow inside the wind acceleration region. Meanwhile, the timescale of Alfvén wave turbulence τAWT is defined as the timescale of outward-wave cascading, which is given as τAWT ≈

2λcorr , z−

(1.19)

18

1 General Introduction

√ where λcorr is the correlation length and z − = v⊥ + B⊥ / 4πρ is the downward Elsässer variable. We assume that the correlation length is proportional to the radius of flux tube: λcorr = λcorr, (B /B)1/2 .

(1.20)

Using this, τAWT is evaluated as  τAWT ≈ 103 s

λcorr, 102 km



B 103 G

1/2 

B

−1/2 

10−1 G

z− 20 km s−1

−1 . (1.21)

Comparing Eqs. (1.17) and (1.21), the timescales of the PDI and Alfvén wave turbulence turn out to be comparable.

1.5 Summary and Motivation of This Thesis The heating and acceleration of the solar wind is one of the most important problems in astrophysics. The fast component of the solar wind, called fast solar wind, is found to exhibit large-amplitude Alfvén waves that are likely to play an essential role in the heating and acceleration. The solar wind model based on Alfvén-wave heating and acceleration, the wave/turbulence-driven model, is now regarded as a promising model of the fast wind. Indeed, the observation in the solar atmosphere indicates the presence of Alfvén wave that is strong enough to power the solar wind. It turned out, from recent numerical simulation of the wave/turbulence-driven solar wind model, that the heating rate from Alfvén-wave turbulence is one-order-ofmagnitude smaller than the required value. This “heating-rate problem” requires us to modify the solar wind turbulence model beyond quasi-2D Alfvén-wave turbulence. One essential ingredient that is ignored in the standard model is the compressional waves. Although the compressional waves are found to be much less energetic than Alfvén waves in the solar wind, it can play an important role as the turbulence driver: enhanced reflection by density fluctuation promotes turbulent heating in the solar wind. Specifically, Alfvén wave itself can be a source of compressional wave generation via parametric decay instability. The overall motivation throughout this thesis is to model the solar wind turbulence with parametric decay instability and quasi-2D cascading process.

References Alazraki G, Couturier P (1971) A&A 13:380 Alfvén H (1947) MNRAS 107:211 An C-H, Suess ST, Moore RL, Musielak ZE (1990) ApJ 350:309

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19

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1 General Introduction

Matsumoto T, Suzuki TK (2012) ApJ 749:8 Matthaeus WH, Zank GP, Oughton S, Mullan DJ, Dmitruk P (1999) ApJ 523:L93 McComas DJ, Ebert RW, Elliott HA, Goldstein BE, Gosling JT, Schwadron NA, Skoug RM (2008) Geophys Res Lett 35:L18103 McIntosh SW, de Pontieu B, Carlsson M, Hansteen V, Boerner P, Goossens M (2011) Nature 475:477 Miyamoto M et al (2014) ApJ 797:51 Morton RJ, Tomczyk S, Pinto R (2015) Nat Commun 6:7813 Neugebauer M, Snyder CW (1962) Science 138:1095 Neugebauer M, Snyder CW (1966) J Geophys Res 71:4469 Newbury JA, Russell CT, Phillips JL, Gary SP (1998) J Geophys Res 103:9553 Okamoto TJ, De Pontieu B (2011) ApJ 736:L24 Osterbrock DE (1961) ApJ 134:347 Oughton S, Matthaeus WH, Dmitruk P, Milano LJ, Zank GP, Mullan DJ (2001) ApJ 551:565 Parker EN (1958) ApJ 128:664 Parker EN (1965) Space Sci Rev 4:666 Perez JC, Chandran BDG (2013) ApJ 776:124 Perkins F (1973) ApJ 179:637 Podesta JJ, Roberts DA, Goldstein ML (2007) ApJ 664:543 Priest E (2014) Magnetohydrodynamics of the sun Raymond JC et al (1997) Sol Phys 175:645 Réville V, Tenerani A, Velli M (2018) ApJ 866:38 Sagdeev RZ, Galeev AA (1969) Nonlinear plasma theory Salem C, Hubert D, Lacombe C, Bale SD, Mangeney A, Larson DE, Lin RP (2003) ApJ 585:1147 Schatten KH, Wilcox JM, Ness NF (1969) Sol Phys 6:442 Shebalin JV, Matthaeus WH, Montgomery D (1983) J Plasma Phys 29:525 Shoda M, Yokoyama T (2018) ApJ 854:9 Spangler SR (2002) ApJ 576:997 Spitzer L, Härm R (1953) Phys Rev 89:977 Srivastava AK et al (2017) Sci Rep 7:43147 Sturrock PA, Hartle RE (1966) Phys Rev Lett 16:628 Suzuki TK, Inutsuka S-I (2005) ApJ 632:L49 Suzuki TK, Inutsuka S-I (2006) J Geophys Res (Space Phys) 111:6101 Tenerani A, Velli M (2013) J Geophys Res (Space Phys) 118:7507 Teriaca L, Poletto G, Romoli M, Biesecker DA (2003) ApJ 588:566 Thurgood JO, Morton RJ, McLaughlin JA (2014) ApJ 790:L2 Usmanov AV, Matthaeus WH, Breech BA, Goldstein ML (2011) ApJ 727:84 Usmanov AV, Matthaeus WH, Goldstein ML, Chhiber R (2018) ApJ 865:25 van Ballegooijen AA, Asgari-Targhi M (2016) ApJ 821:106 van Ballegooijen AA, Asgari-Targhi M (2017) ApJ 835:10 van der Holst B, Sokolov IV, Meng X, Jin M, Manchester WB IV, Tóth G, Gombosi TI (2014) ApJ 782:81 Velli M (1993) A&A 270:304 Velli M (1994) ApJ 432:L55 Velli M, Grappin R, Mangeney A (1989) Phys Rev Lett 63:1807 Verdini A, Velli M (2007) ApJ 662:669 Verdini A, Velli M, Matthaeus WH, Oughton S, Dmitruk P (2010) ApJ 708:L116 von Steiger R et al (2000) J Geophys Res 105:27217 Wang Y-M, Sheeley NR Jr (1990) ApJ 355:726 Withbroe GL (1988) ApJ 325:442 Withbroe GL, Noyes RW (1977) ARA&A 15:363

Chapter 2

1D Model with Compressional and Non-compressional Nonlinear Effects

Abstract Alfvén waves are believed to play a central role in heating and accelerating the fast solar wind. It is, however, still controversial how the energy of Alfvén waves is converted to heat. Large Reynolds number in the corona and solar wind indicates that simple viscous and resistive dissipation cannot account for the heating, and thus, nonlinear effects and resultant cascading must be at play. In this chapter, we propose a new solar wind model including compressional heating process and non-compressional quasi-2D Alfvén wave turbulence, which have been discussed independently in the literature. The quasi-2D turbulence is found to feed sufficient heat to the solar wind, possibly owing to the enhanced wave reflection by compressional waves. The interplay between compressional process and non-compressional turbulence is thus crucial in the fast solar wind. Keywords Turbulence phenomenology · Correlation length · Heating mechanism of the solar wind

2.1 Introduction Broadband magnetic energy spectra observed in the solar wind indicates that plasma is heated by nonlinear energy cascading (Carbone et al. 2009). Specifically in the fast solar wind, in which the Alfvénicity is large, nonlinear Alfvén waves are believed to drive the energy cascading. In other words, Alfvén-wave energy should be converted to plasma thermal energy. The Alfvén-wave heating is believed to occur everywhere in the open-field region from above the coronal base to the distant interplanetary space. In modeling the fast solar wind, the amount of Alfvén-wave thermalization (dissipation) rate as a function of radial distance is crucial. The heat and momentum deposition beyond the sonic point is necessary to sustain the fast solar wind (Leer and Holzer 1980), and therefore, a fraction of Alfvén waves needs to propagate across the subsonic region to accelerate the fast solar wind. Understanding how Alfvén waves dissipate is thus crucial to understand the origin of the fast solar wind.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. Shoda, Fast Solar Wind Driven by Parametric Decay Instability and Alfvén Wave Turbulence, Springer Theses, https://doi.org/10.1007/978-981-16-1030-1_2

21

22

2 1D Model with Compressional and Non-compressional …

As already discussed in the previous chapter, two nonlinear processes are proposed as dissipation mechanisms of Alfvén waves: interaction with compressional wave and quasi-2D turbulence. Turbulent spectra and small amplitude of compressional waves observed in the solar wind support the quasi-2D turbulence scenario. However, the quasi-2D turbulence is found to yield smaller-than-required heating rate from recent direct numerical simulations (Perez and Chandran 2013; van Ballegooijen and Asgari-Targhi 2016), and needs some modification. Specifically, the role of compressional waves (density fluctuation) should be considered in the quasi-2D turbulence model. The density fluctuation possibly plays a non-negligible role in the solar wind energetics. By introducing density fluctuation, due to the enhanced reflection of Alfvén waves, the heating rate of Alfvén wave turbulence becomes sufficiently large (van Ballegooijen and Asgari-Targhi 2016). In this regard, the density fluctuation is essential, although the origin of it is unknown. Meanwhile, observations of density fluctuation indicate that the magnitude of density fluctuation becomes largest (as large as a few 10% of the mean value) in the wind acceleration region (Miyamoto et al. 2014). Fast mode propagation cannot account for such a large density fluctuation (Cranmer and van Ballegooijen 2012), and therefore, the large density fluctuation in the wind acceleration region is a mystery. The radial distribution of compressional waves needs to be discussed in detail. Given this background, we aim to model the solar wind including both compressional waves and quasi-2D turbulence. Based on 1D compressible MHD equations in which the coupling between Alfvén wave compressional waves are automatically solved, we introduce additional terms that correspond to quasi-2D turbulent dissipation. Using this model, we aim to investigate following three issues. 1. What is the dominant heating mechanism in the solar wind? How does it vary with the nature of turbulence on the photosphere? 2. Is the observed behavior of density fluctuation explained by this model? 3. How do the wind properties depend on the photospheric boundary condition? Are there any significant differences from previous model? The reminder of this chapter is organized as follows. Simulation methods are described with an especial focus on the implementation of turbulence phenomenology. Simulation results and several analyses on them are given in the following section. Finally, we discuss and summarize the one-dimensional simulation results.

2.2 Method To take into account the turbulent heating in 1D simulations, we utilize the phenomenological model of turbulence. Additional terms corresponding to turbulent dissipation are introduced to the conventional 1D compressible MHD equations. Since this procedure is not trivial, we first explain how to implement the turbulence model. The basic equations and numerical settings are then described.

2.2 Method

23

2.2.1 Phenomenology of Quasi-2D Turbulence In one-dimensional simulation, the dynamics of turbulence is not directly solved. Even though, the effect of turbulence is approximately incorporated phenomenologically. Specifically, additional source terms that correspond to the phenomenological description of turbulence are included in the one-dimensional MHD equations The implementation of phenomenology is discussed in this section. Let us review the phenomenological turbulence model developed in the reducedMHD framework. The original reduced MHD equations that describe the Alfvénwave propagation in the solar wind are (Zhou and Matthaeus 1990; Verdini and Velli 2007) ∂ ± z + (u ± v A ) · ∇ z ± ∂t     − z ± (u ∓ v A ) · ∇ ln ρ 1/4 + z ∓ (u ∓ v A ) · ∇ ln ρ 1/4 A1/2 = − z ∓ · ∇ z ± , (2.1) √ where z ± = v ⊥ ∓ B ⊥ / 4πρ are Elsässer variables. u and v A denote the bulk and Alfvén velocities, respectively. A denotes the cross section of flux tube that satisfies the divergence-free condition ( ABr = const.). The third and fourth terms in the left hand side of Eq. (2.1) correspond to the WKB (amplification) and reflection terms, respectively (Heinemann and Olbert 1980). The right hand side represents the nonlinear interaction between forward (z + ) and backward (z − ) Alfvén waves. The widely-used phenomenological model simply approximates this nonlinear term by (Hossain et al. 1995; Matthaeus et al. 1999; Dmitruk et al. 2002; van Ballegooijen and Asgari-Targhi 2017) 

 z∓ z ∓ · ∇ z ± ≈ cd rms z ± , 2λ

(2.2)

∓ is the root-mean-squared value of z ∓ where cd is an order-unity parameter, z rms and λ is the perpendicular correlation length. This approximation yields quasi-linear equations for z ± :

∂ ± z + (u ± v A ) · ∇ z ± ∂t

  z∓ − z ± (u ∓ v A ) · ∇ ln ρ 1/4 + z ∓ (u ∓ v A ) · ∇ ln ρ 1/4 A1/2 = −cd rms z ± . (2.3) 2λ

The choice of the coefficient cd is not trivial. Several studies simply assume cd = 1, while a recent 3D direct numerical simulation indicates that cd = 0.1 is possibly more appropriate (van Ballegooijen and Asgari-Targhi 2017). For this reason, we use cd = 0.1 in this study.

24

2 1D Model with Compressional and Non-compressional …

Using the phenomenological approximation Eq. (2.2), we aim to derive a model that incorporates compressional effects and quasi-2D turbulence. The equation of motion in the transverse velocity and the induction equation of the transverse magnetic field are expressed in the one-dimensional coordinate co-expanding with the background magnetic flux tube by,   Br ∂ √ d √ Av ⊥ = A B⊥ , dt 4π ∂r  ∂ 1 ∂ √ B⊥ = √ A (v ⊥ Br − vr B ⊥ ) , ∂t A ∂r

ρ

(2.4) (2.5)

where A denotes the cross section of the flux tube. Using the mass conservation and the solenoidal condition, ρ ∂ d ρ+ (Avr ) = 0, dt A ∂r

∂ (ABr ) = 0, ∂r

(2.6)

we obtain the conservation laws of ρv ⊥ and B ⊥ :

   1 ∂  3/2 3/2 = 0, ρv ⊥ A Br B ⊥ A + ρvr v ⊥ − ∂t 4π 

∂  ∂ B ⊥ A1/2 + (vr B ⊥ − v ⊥ Br ) A1/2 = 0. ∂t ∂r

(2.7) (2.8)

These are a part of conventional one-dimensional compressible MHD equations. To incorporate the phenomenological dissipation of Alfvén wave turbulence, we add source terms in the right hand side. 

   ∂  ρ 1 ρv ⊥ A3/2 + ρvr v ⊥ − Br B ⊥ A3/2 = −ηˆ 1 · ρv ⊥ A3/2 − ηˆ 2 · B ⊥ A3/2 , ∂t 4π 4π

(2.9)

   ∂  ∂  B ⊥ A1/2 + (vr B ⊥ − v ⊥ Br ) A1/2 = −ηˆ 1 · B ⊥ A1/2 − ηˆ 2 · 4πρv ⊥ A1/2 , (2.10) ∂t ∂r

where ηˆ 1 and ηˆ 2 are diagonal matrices: ⎞ ⎛ cd   − |z + 0 x | + |z x | ηˆ 1 = ⎝ 4λ ⎠, cd  + |z y | + |z − | 0 y 4λ ⎞ ⎛ cd   − |z + | − |z | 0 x x ηˆ 2 = ⎝ 4λ ⎠, cd  + |z y | − |z − 0 y| 4λ

(2.11)

(2.12)

2.2 Method

25

√ where λ is the correlation length, z ± = v ⊥ ∓ B ⊥ / 4πρ are Elsässer variables, and x, y denote the perpendicular components. ηˆ 1 and ηˆ 2 are physically interpreted as turbulent dissipation and dynamical alignment, Note that ηˆ 2 arises only    respectively.  when the turbulence is imbalanced (z +  = z − ) and the dynamical alignment proceeds (Dobrowolny et al. 1980; Stribling and Matthaeus 1991; Hossain et al. 1995; Beresnyak and Lazarian 2008; Chandran 2008). Let us show that Eqs. (2.9) and (2.10) are consistent with the approximated reduced MHD equation (2.3). To rule out the compressible waves that do not appear in the reduced MHD equations, we assume ∂ρ/∂t = 0 and ∂vr /∂t = 0. This is not to assume the incompressibility; we still take into account the wind acceleration that comes from the large-scale (mean-field) velocity divergence while the small-scale (wave) divergence/compression is ignored. Under this assumption, Eqs. (2.9) and (2.10) are written as     ∂ ∂ ∂ ∂ ∂ v ⊥ + u v ⊥ − v A b⊥ + uv ⊥ ln A1/2 − v A b⊥ ln ρ 1/2 A1/2 = −ηˆ 1 · v ⊥ − ηˆ 2 · b⊥ , ∂t ∂r ∂r ∂r ∂r

(2.13)

    ∂ ∂ ∂ ∂ ∂ b⊥ + u b⊥ − v A v ⊥ + v A v ⊥ ln A1/2 − ub⊥ ln ρ 1/2 A1/2 = −ηˆ 1 · b⊥ − ηˆ 2 · v ⊥ , ∂t ∂r ∂r ∂r ∂r

(2.14)

√ where we use the notation u = vr and v A = Br / 4πρ. In terms of Elsässer variables these equations are written as ∂ ± ∂ z + (u ± v A ) z ± ∂t ∂r     ∂ ∂ ± ln ρ 1/4 + z ∓ (u ∓ v A ) ln ρ 1/4 A1/2 = −ηˆ ∓ · z ± , (2.15) − z (u ∓ v A ) ∂r ∂r where ⎞ ⎛ cd |z ∓ 0 x| ηˆ = ηˆ 1 ∓ ηˆ 2 = ⎝ 2λ cd ∓ ⎠ . |z | 0 2λ y ∓

(2.16)

Equation (2.15) is consistent with Eq. (2.3).

2.2.2 Basic Equations The basic equations are 1.5D (one-dimensional, three-components) MHD equations in an expanding flux tube, which are given by

26

2 1D Model with Compressional and Non-compressional …

 2   ∂  ρr f + ρvr r 2 f = 0, ∂r 



   ∂ B⊥ 2 ρv ⊥ 2 d  2  ρvr r 2 f + ρvr 2 + p + r2 f = p + r f − ρgr 2 f, ∂r 8π 2 dr  

  ∂  ρ ∂ Br B ⊥ ρv ⊥ r 3 f 3/2 + ρvr v ⊥ − r 3 f 3/2 = −ηˆ 1 · ρv ⊥ r 3 f 3/2 − ηˆ 2 · B ⊥ r 3 f 3/2 , ∂t ∂r 4π 4π       ∂  ∂   r f B⊥ + (B ⊥ vr − Br v ⊥ ) r f = −ηˆ 1 · B ⊥ r f − ηˆ 2 · 4πρv ⊥ r f , ∂t ∂r  d  2 r f Br = 0, dr 



  1 1 ∂ B⊥ · v⊥ 2 B2 ∂ B⊥ 2 e + ρv 2 + r2 f + e + p + ρv 2 + vr r 2 f − Br r f ∂t 2 8π ∂r 2 8π 4π ∂ ∂t ∂ ∂t

= r 2 f (−ρgvr + Q rad + Q cnd ) , p e= , γ −1

ρk B T ., p= μm H

(2.17) (2.18) (2.19) (2.20) (2.21)

(2.22)

G M g= , r2

(2.23)

where G is the gravitational constant and M is the solar mass. f is the expansion factor (Schatten et al. 1969; Wang and Sheeley 1990; Arge and Pizzo 2000; Réville and Brun 2017) of the flux tube in consideration. μ is the mean molecular mass in unit of m H (hydrogen mass); if the plasma is composed of hydrogen, μ = 0.5 for the fully-ionized case and μ = 1 for the rarely-ionized case. In what follows, we use an ancillary variable h to denote the altitude from the solar surface: h = r − R ,

(2.24)

where R is the solar radius. h is mainly used to express the location in the chromosphere, the typical thickness is only a few 10−3 R .

2.2.3 Simulation Domain and Boundary Condition We solve the basic equations from the photosphere to 0.5 au. For any variables, the free boundary condition is applied at the top:  ∂  X = 0 ∂r top

(2.25)

where X stands for the primitive variable such as ρ, p, etc. The choice of the top boundary condition does not affect the calculation because the top boundary always lies in the supersonic and super-Alfvénic region where no fluctuations can propagate backward. This is why we need not use the transmitting boundary condition (Thompson 1987; Del Zanna et al. 2001; Suzuki and Inutsuka 2006) for the top boundary.

2.2 Method

27

Note that the transmitting boundary possibly yields a better result when we simulate the dynamical acceleration of the solar wind in which the radial velocity of the top boundary is subsonic in the initial phase. At the bottom, the fixed boundary conditions are applied to the density, temperature, and radial magnetic field, ρbottom = 1.0 × 10−7 g cm−3 , Tbottom = 6.0 × 103 K,

(2.26)

Br,bottom = 1.0 × 10 G, 3

while the free boundary conditions are applied to the radial velocity vr and inward Elsässer variable:   ∂ −  ∂  vr z = 0, = 0. (2.27) ∂r bottom ∂r bottom The boundary condition of the upward Elsässer variables are given with a broadband spectrum: z+ ∝



2π f max

  dω P(ω) sin ωt + φ rand (ω) ,

(2.28)

2π f min

where P is the power spectrum and φ rand is a random phase function of ω. The spectrum is fixed to P(ω) ∝ ω−1 and f min = 10−3 Hz,

f max = 10−2 Hz.

(2.29)

The root-mean-squared amplitude of z + is set to 2dv + , where dv + denotes the velocity amplitude of outward Alfvén wave and is set to be a free parameter. The radial profile of the expansion factor is given as follows. f (r ) ∝

f max,1 exp [(r − r1 ) /σ1 ] + f 1 f max,2 exp [(r − r2 ) /σ2 ] + f 2 · exp [(r − r1 ) /σ1 ] + 1 exp [(r − r2 ) /σ2 ] + 1

(2.30)

where

f n = 1 − f max,n exp (R − rn ) /σn , f max,1 = 120, f max,2 = 4, r1 − R = 1 Mm, r2 − R = 200 Mm, σ1 = 0.25 Mm, σ2 = 350 Mm.

(2.31) (2.32) (2.33) (2.34)

28

2 1D Model with Compressional and Non-compressional …

Once f (r ) is given, the correlation length λ is calculated under the assumption that the correlation length scales with the radius of flux tube:  λ = λ0

r2 f , 2 R f

(2.35)

where λ0 is the photospheric correlation length that varies as a free parameter. To exclude the turbulent dissipation in the chromosphere, we modify the correlation length in a density-dependent manner.  λ = λ0



r2 f ρ max 1, −16 . 2 10 g cm−3 R f

(2.36)

Conductive heating Q cnd is expressed in terms of radial conductive flux qcnd by Q cnd = −

1 r2

 ∂  qcnd r 2 f . f ∂r

(2.37)

We impose the Spitzer–Härm type conductive flux qSH that is quenched in the distant solar wind. ∂ T ∂r min 1,

qSH = −κ0 T 5/2 qcnd = qSH

ρ −22 10 g cm−3

,

(2.38)

In order to deal with the large scale gap in the lower atmosphere and corona, a non-uniform radial mesh that expands in r is used.

2.3 Result 2.3.1 Quasi-steady States with Various Correlation Lengths Figure 2.1 shows the snapshots of primitive variables (density, temperature, radial velocity, transverse velocity) in the quasi-steady states with dv + = 0.5 km s−1 . Four lines correspond to different photospheric correlation lengths: λ0 = 10 km (red), λ0 = 102 km (orange), λ0 = 103 km (green), λ0 = 104 km (blue). We find in every case the formation of cool lower atmosphere (chromosphere) and hot higher atmosphere (corona) separated by a sharp boundary (transition region). There is no difference in the four lines in the chromosphere because we switch off the turbulent dissipation in the lower atmosphere.

2.3 Result

29

Fig. 2.1 Quasi-steady states with fixed wave injection dv + = 0.5 km s−1 and various correlation lengths. Red, orange, green, and blue lines correspond to photospheric correlation length of λ0 = 10 km, λ0 = 102 km, λ0 = 103 km, λ0 = 104 km, respectively. Each panel shows in the descending order the radial profile of density, temperature, radial velocity, and transverse velocity, respectively. Figure reproduced from Shoda et al. (2018) under permission of the American Astronomical Society

The corona and solar wind show several differences. When the correlation length becomes smaller (as the color of line approaches red), the coronal temperature becomes larger and the solar wind becomes heavier and slower. This indicates that the subsonic corona is heated preferentially by turbulent dissipation, and as a result of enhanced heating with smaller correlation length, the solar wind becomes slower because the energy is injected more to the lower atmosphere (Leer and Holzer 1980; Hansteen and Velli 2012). When the correlation length is large, large fluctuations

30

2 1D Model with Compressional and Non-compressional …

appear in the density and radial velocity. This indicates that the compressible waves are more likely to be generated for larger correlation length. Considering the fact that the growth rate of PDI is larger when the plasma beta is smaller, the large density fluctuation is naturally interpreted as faster growth of PDI because of the smaller solar wind density and smaller plasma beta.

2.3.2 Density Fluctuation and Cross Helicity To see the clearer dependences on correlation length, we calculate the time-averaged fractional density fluctuation n and the normalized cross helicity σc . n indicates the amplitude of compressible waves while σc represents the population ratio between outgoing and reflected Alfvén waves. The time-averaged fractional density fluctuation is calculated as  (2.39) n = δρ/ ρ, δρ = (ρ − ρ)2  where X  represents the time-average of X :

X  =

1 τave



t0 +τave

dt X,

(2.40)

t0

where t0 is when the system reaches the quasi-steady state and τave = 1000 min. The averaged cross helicity σc is defined as σc =

+ − − z rms z rms , + + z− z rms rms

(2.41)

where ± z rms =



z ± 2 .

(2.42)

Figure 2.2 shows the radial profile of n (top) and σc (bottom) with various correlation lengths. We use the same color of line as Fig. 2.1 to stand for the corresponding correlation length. The black circles indicate the observational values by Miyamoto et al. (2014) and show the same trend with the theoretical values. The peak of the density fluctuation in 10−3  r/R − 1  10−2 comes from the steepening of slowmode waves in the chromosphere. Another peak around r/R ≈ 10 is positively correlated with correlation length. This result shows that the correlation length of turbulence controls the physical mechanism responsible for the generation of density fluctuation. In the bottom panel, the cross helicity shows monochromatic dependence on correlation length. Specifically when λ0 = 10−2 Mm, the cross helicity in the solar wind is nearly unity because of the dynamical alignment effect.

2.3 Result

31

Fig. 2.2 Radial profile of the time-averaged fractional density fluctuation n (top) and the normalized cross helicity σc (bottom). Red, orange, green, and blue lines correspond to photospheric correlation length of λ0 = 10 km, λ0 = 102 km, λ0 = 103 km, λ0 = 104 km, respectively. Black circles in the top panel indicate the observation by Miyamoto et al. (2014). Figure reproduced from Shoda et al. (2018) under permission of the American Astronomical Society

2.3.3 Heating Mechanism How the solar wind plasma is heated is worth investigating. We calculate the heating rate via the following analysis. The internal energy equation is written in the following manner (Cranmer et al. 2007): ∂e ∂e e + p ∂  2  + vr + 2 vr r f = Q rad + Q cnd + Q heat , ∂t ∂r r f ∂r

(2.43)

where Q heat denotes the heat deposition by wave. In the quasi-steady state, by averaging the above equation in time we obtain

Q heat  = vr

e+ p ∂  2  ∂e + 2 vr r f  − Q rad  − Q cnd . ∂r r f ∂r

(2.44)

Note that X  represents the time averaged value of X (see Eq. (2.40)) and therefore

∂e  ≈ 0. ∂t

(2.45)

32

2 1D Model with Compressional and Non-compressional …

Fig. 2.3 Plasma heating rate per particle Q/ρ for three different correlation lengths. Left, middle, and right panels correspond to λ0 = 0.1 Mm, λ0 = 1 Mm, and λ0 = 10 Mm, respectively. The blue lines show the total heating rate that is calculated from the energy equation, while the red lines denote the turbulence heating rate given by the analytical expression. Figure reproduced from Shoda et al. (2018) under permission of the American Astronomical Society

The total heating rate Q heat  is calculated in this manner. Meanwhile, the turbulent heating rate is calculated analytically by (Cranmer et al. 2007; Verdini and Velli 2007) Q turb = ρ

 i=x,y

|z i+ |z i− + |z i− |z i+ . 4λ 2

cd

2

(2.46)

Figure 2.3 shows the radial profiles of Q heat / ρ (blue lines) and Q turb / ρ (red lines) for three different λ0 values. Left, middle, and right panels correspond to λ0 = 0.1 Mm, λ0 = 1 Mm, and λ0 = 10 Mm, respectively. Since waves dissipate only by shock and turbulence, the gap between blue and red lines corresponds to the heating by shock wave. Figure 2.3 shows that the correlation length strongly affects the solar wind heating mechanism; when λ0 = 0.1 Mm the solar wind is heated everywhere by turbulence, while the shock heating becomes dominant when λ0 = 10 Mm. Interestingly, although the profile of Q turb / ρ drastically changes with λ0 , the total heating rate Q heat / ρ is almost invariant. Several noticeable features turns out from the profiles of Q heat / ρ and

Q turb / ρ. First the coronal base (just above the transition region) where r/R − 1  0.1 is heated preferentially by shock waves. This probably comes from a technical reason that we cut off the turbulence heating in the coronal base. But not unrealistic because both observation (Tian et al. 2014; Kanoh et al. 2016) and numerical simulation (Kudoh and Shibata 1999; Matsumoto and Shibata 2010; Iijima and Yokoyama 2015) show the existence of chromospheric shock waves that can drastically heat the base of the corona. In the subsonic corona where 0.1  r/R − 1  1, regardless of the correlation length, turbulence is the dominant heating mechanism. This is why the fast solar wind is generated when λ0 is large (and the turbulence heating

2.3 Result

33

is small), and vice versa. In the solar wind the fraction of shock heating increases because the correlation length increases with radial distance and the timescale of turbulent dissipation becomes large.

2.3.4 Dependence on the Injection Amplitude We have also performed a parameter survey on wave amplitude, which strongly affects the mass flux of the solar wind (Suzuki and Inutsuka 2006; Cranmer et al. 2007). Figure 2.4 shows how the solar wind velocity, maximum temperature, and mass-loss rate varies with correlation length and wave amplitude. Three panels show the time-averaged solar wind velocity vr at r = 0.5 au (left), maximum temperature (center), and mass-loss rate in unit of M yr −1 , respectively. Three lines correspond to different injection amplitude of dv + = 0.7 km s−1 (blue), dv + = 0.5 km s−1 (green), dv + = 0.4 km s−1 (red), respectively, and the horizontal axis indicates the photospheric correlation length λ0 . Regardless of the injection amplitude, the solar wind velocity is larger for larger λ0 . The mass-loss rate shows only a slight trend with respect to λ0 and the maximum temperature is almost constant in λ0 . The dependence on dv + is as follows. The solar wind velocity decreases as dv + increases. This seems mysterious because the larger energy injection leads to smaller wind velocity. The reason for this apparent contradiction is the sensitive dependence of the massloss rate on dv + ; the mass flux becomes 3–4 times larger when dv + increases from 0.5 km s−1 to 0.7 km s−1 . Because heavier wind is more difficult to accelerate, the solar wind velocity decreases for larger energy input. The maximum temperature shows a weak dependence on the injected energy, possibly because the conductive cooling of the corona and solar wind is larger for larger temperature.

Fig. 2.4 Dependence of solar wind parameters on the correlation length and the injected wave amplitude on the photosphere. Horizontal axis shows the photospheric correlation length and the different color of line corresponds to different wave amplitude. Three panels show the a termination velocity of the solar wind, b maximum temperature and c mass-loss rate, respectively. Figure reproduced from Shoda et al. (2018) under permission of the American Astronomical Society

34

2 1D Model with Compressional and Non-compressional …

2.4 Discussion 2.4.1 Heating Mechanism in the Solar Wind As Fig. 2.3 shows, the total heating rate per unit mass is almost invariant with respect to λ0 ; if the turbulent heating becomes smaller, the shock heating compensates to maintain the constant total heating rate. The possible reason for this explained as follows. When the turbulent heating becomes smaller, the plasma beta of the solar wind decreases because of the less chromospheric evaporation, which leads to the faster growth of the parametric decay instability. The faster growth of PDI by smaller turbulence heating yields larger shock heating to keep the total heating constant. This self-regulating mechanism is possibly a reason why the model without Alfvén wave turbulence (Suzuki and Inutsuka 2005) and the model without PDI (Cranmer et al. 2007; Verdini et al. 2010) successfully explained the observation independently.

2.4.2 Parameter Dependence of the Solar/Stellar Wind Figure 2.4 is worth comparing with previous works. First Cranmer et al. (2007) and Verdini et al. (2010) showed that the solar wind velocity is positively correlated with the correlation length, that is, large λ0 . large vr . This is consistent with our result which shows the monotonic increase of vr with respect to λ0 . The dependence of wave amplitude is discussed in Suzuki and Inutsuka (2006) and Cranmer et al. (2007), both of whom show a decreasing trend of vr with respect to λ0 . This is also consistent with our work. The mass-loss rate of our work is also consistent with Cranmer et al. (2007) in that M˙ shows slightly decreasing trend in λ0 and sensitive dependence on dv + .

2.4.3 Photospheric Correlation Length The motivation of parameter survey with respect to λ0 is not only to understand the physics, but also to prepare for the observation of the correlation length. The photospheric correlation length is not observationally constrained. The value of λ0 depend on the wave generation mechanism. Provided that most of the Alfvén waves are generated inside the vortex motion of magnetic patches (van Ballegooijen et al. 1998, 2011), λ0 should be typically 102 km (Berger et al. 1995; Berger and Title 2001). Meanwhile if the swaying motion of the flux tube by the granular cell is a dominant wave-generation mechanism (Steiner et al. 1998), λ0 should be the size of granular cell, that is, 103 km. To solve this problem, we need a super-high-resolution observation that resolves the fine-scale flow inside the magnetic elements, which

2.4 Discussion

35

is now impossible. Realistic numerical simulations, such as Moll et al. (2012) and Iijima and Yokoyama (2017) would be a powerful tool to determine the realistic value of λ0 .

2.5 Summary By introducing new terms corresponding to turbulent dissipation of Alfvén wave turbulence into 1D compressible MHD equations, a model of the solar wind including compressional and non-compressional heating processes are described. The reflection-driven Alfvén wave turbulence (or standard) model is modified by the presence of density fluctuation (slow-mode waves) in two senses. First, due to the generation of shock waves, the heating rate is directly enhanced via a new energy dissipation channel. Second, the reflection rate increases due to the large density fluctuation to activate the turbulence in the solar wind. The shortage of heating in the standard model is resolved in this way. The dominant heating mechanism is dependent on the correlation length. With the solar parameter (photospheric correlation length of 1 Mm), turbulence heating is slightly larger than the shock heating in the wind acceleration region. It does not mean that the role of parametric decay instability is ignorable. Due to the large density fluctuation, in spite of modified turbulence model that weakens the turbulent dissipation (van Ballegooijen and Asgari-Targhi 2017), the fast solar wind is reproduced. It indicates that the turbulence heating is supported by parametric decay instability. Consistently with previous works, the mass-loss rate is determined by the amplitude of injected Alfvén waves. The dependence of the mass-loss rate on the photospheric upward Poynting flux is super-linear, possibly because the reflection rate at the transition region is reduced for large energy input.

References Arge CN, Pizzo VJ (2000) J Geophys Res 105:10465 Beresnyak A, Lazarian A (2008) ApJ 682:1070 Berger TE, Title AM (2001) ApJ 553:449 Berger TE, Schrijver CJ, Shine RA, Tarbell TD, Title AM, Scharmer G (1995) ApJ 454:531 Carbone V, Marino R, Sorriso-Valvo L, Noullez A, Bruno R (2009) Phys Rev Lett 103:061102 Chandran BDG (2008) ApJ 685:646 Cranmer SR, van Ballegooijen AA (2012) ApJ 754:92 Cranmer SR, van Ballegooijen AA, Edgar RJ (2007) ApJS 171:520 Del Zanna L, Velli M, Londrillo P (2001) A&A 367:705 Dmitruk P, Matthaeus WH, Milano LJ, Oughton S, Zank GP, Mullan DJ (2002) ApJ 575:571 Dobrowolny M, Mangeney A, Veltri P (1980) Phys Rev Lett 45:144 Hansteen VH, Velli M (2012) Space Sci Rev 172:89 Heinemann M, Olbert S (1980) J Geophys Res 85:1311 Hossain M, Gray PC, Pontius DH Jr, Matthaeus WH, Oughton S (1995) Phys Fluids 7:2886

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2 1D Model with Compressional and Non-compressional …

Iijima H, Yokoyama T (2015) ApJ 812:L30 Iijima H, Yokoyama T (2017) ArXiv e-prints Kanoh R, Shimizu T, Imada S (2016) ApJ 831:24 Kudoh T, Shibata K (1999) ApJ 514:493 Leer E, Holzer TE (1980) J Geophys Res 85:4681 Matsumoto T, Shibata K (2010) ApJ 710:1857 Matsumoto T, Suzuki TK (2012) ApJ 749:8 Matsumoto T, Suzuki TK (2014) MNRAS 440:971 Matthaeus WH, Zank GP, Oughton S, Mullan DJ, Dmitruk P (1999) ApJ 523:L93 Miyamoto M et al (2014) ApJ 797:51 Moll R, Cameron RH, Schüssler M (2012) A&A 541:A68 Perez JC, Chandran BDG (2013) ApJ 776:124 Réville V, Brun AS (2017) ApJ 850:45 Schatten KH, Wilcox JM, Ness NF (1969) Sol Phys 6:442 Shoda M, Yokoyama T, Suzuki TK (2018) ApJ 853:190 Steiner O, Grossmann-Doerth U, Knölker M, Schüssler M (1998) ApJ 495:468 Stribling T, Matthaeus WH (1991) Phys Fluids B 3:1848 Suzuki TK, Inutsuka S-I (2005) ApJ 632:L49 Suzuki TK, Inutsuka S-I (2006) J Geophys Res (Space Phys) 111:6101 Thompson KW (1987) J Comput Phys 68:1 Tian H et al (2014) ApJ 786:137 van Ballegooijen AA, Asgari-Targhi M (2016) ApJ 821:106 van Ballegooijen AA, Asgari-Targhi M (2017) ApJ 835:10 van Ballegooijen AA, Nisenson P, Noyes RW, Löfdahl MG, Stein RF, Nordlund Å, Krishnakumar V (1998) ApJ 509:435 van Ballegooijen AA, Asgari-Targhi M, Cranmer SR, DeLuca EE (2011) ApJ 736:3 Verdini A, Velli M (2007) ApJ 662:669 Verdini A, Velli M, Matthaeus WH, Oughton S, Dmitruk P (2010) ApJ 708:L116 Wang Y-M, Sheeley NR Jr (1990) ApJ 355:726 Zhou Y, Matthaeus WH (1990) J Geophys Res 95:10291

Chapter 3

Onset and Suppression of Parametric Decay Instability

Abstract It is shown that the compressional process plays a significant role in the fast solar wind. However, it remains still unclear how compressional fluctuations are generated in the solar wind. In this chapter, we seek the possibility that compressional fluctuations are generated from Alfvén waves via the parametric decay instability (PDI). The growth rate of PDI is known to be reduced in the expanding system like the solar wind, and thus it is non-trivial whether Alfvén waves of realistic amplitude and frequency are subject to PDI. By conducting a parameter survey in wave frequency, we find that Alfvén waves of period shorter than 1000 s are unstable and generate density fluctuations. The theoretical growth rate of PDI is consistent of observed profile of density fluctuation, which supports the idea that density fluctuation originates from PDI. Keywords Solar wind density fluctuation · Parametric decay instability · Nonlinear Alfvén wave

3.1 Introduction What we have shown in the previous chapter is that density fluctuation is naturally generated in the solar wind acceleration region. Such density fluctuation is essentially important in discussing the Alfvén-wave dynamics because it significantly enhances the reflection rate of Alfvén waves that result in an enhanced turbulent heating. Indeed, although we have applied a rather small parameter of cd = 0.1 to phenomenologically describe the turbulent dissipation, the turbulent heating rate is large enough to sustain the solar wind. However, it is still unclear how the density fluctuation is generated. The radial profile of the fractional density fluctuation δρ/ρ indicates that the density fluctuation is generated or amplified in the wind acceleration region. Amongst several processes, the parametric decay instability (PDI, Galeev and Oraevskii 1963; Sagdeev and Galeev 1969; Goldstein 1978; Derby 1978) is the most promising one that is responsible for the generation of density fluctuation. The growth rate of PDI is known to be large when the energy of Alfvén wave is comparable to or larger than the © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. Shoda, Fast Solar Wind Driven by Parametric Decay Instability and Alfvén Wave Turbulence, Springer Theses, https://doi.org/10.1007/978-981-16-1030-1_3

37

38

3 Onset and Suppression of Parametric Decay Instability

thermal energy of plasma. Then, a resonant interaction between Alfvén and acoustic (slow-mode) wave results in an exponential growth in amplitude of slow-mode wave. Since the slow-mode wave is essentially acoustic in a low-beta plasma, the growth of slow-mode wave means an exponential generation of density fluctuation. In this chapter, we investigate how PDI proceeds in an accelerating expanding solar wind from numerical simulation. It is known that the growth rate of PDI is suppressed by the acceleration and expansion effects of the solar wind (Tenerani and Velli 2013; Del Zanna et al. 2015). It is therefore non-trivial if an Alfvén wave undergoes PDI against these suppression effects. For this purpose, we conduct a series of simulations with a range of driving frequencies of Alfvén wave, using the same basic equations as the previous chapter.

3.2 Method 3.2.1 Basic Equations The basic equations are the same as what are used in the previous chapter: 1.5D (one-dimensional, three-components) MHD equations in an expanding flux tube with phenomenological turbulent dissipation.  ∂  2  ∂  ρr f + ρvr r 2 f = 0, (3.1) ∂t ∂r       ∂  ∂ B⊥ 2 ρv ⊥ 2 d  2  ρvr r 2 f + ρvr 2 + p + r2 f = p + r f − ρgr 2 f, (3.2) ∂t ∂r 8π 2 dr 

  ∂  ρ ∂ Br B ⊥ ρv ⊥ r 3 f 3/2 + ρvr v ⊥ − r 3 f 3/2 = −ηˆ 1 · ρv ⊥ r 3 f 3/2 − ηˆ 2 · B ⊥ r 3 f 3/2 , ∂t ∂r 4π 4π ∂ ∂t d dr ∂ ∂t

 



∂  r f B⊥ + (B ⊥ vr − Br v ⊥ ) r f = −ηˆ 1 · B ⊥ r f − ηˆ 2 · 4πρv ⊥ r f , ∂r   r 2 f Br = 0,       B⊥ · v⊥ 2 1 1 B2 ∂ B⊥ 2 r f e + ρv 2 + r2 f + e + p + ρv 2 + vr r 2 f − Br 2 8π ∂r 2 8π 4π

= r 2 f (−ρgvr + Q rad + Q cnd ) , e=

p , γ−1

p=

ρk B T ., μm H

g=

(3.3) (3.4) (3.5)

(3.6) G M . r2

(3.7)

In this chapter, we the simulation domain extends from the coronal base to 1 au. Due to the high temperature and low density in this domain, the ionization degree is fixed to unity and radiative cooling is ignored. μ = 0.5,

Q rad = 0.

(3.8)

3.2 Method

39

3.2.2 Simulation Domain and Boundary Condition As already noted, we ignore the photosphere and chromosphre for simplicity. The simulation domain extends from the coronal base to 1 au. Like previous chapter, the free boundary condition is applied at the top:  ∂  X = 0, ∂r top

(3.9)

where X stands for the primitive variable such as ρ, p, etc. This choice does not affect the simulation results (see previous chapter). The fixed boundary conditions are used to the density, temperature, and radial magnetic field as ρbottom = 8.5 × 10−16 g cm−3 ,

(3.10)

Tbottom = 4.0 × 10 K,

(3.11)

Br,bottom = 1.0 × 10 G.

(3.12)

5

1

The free boundary conditions are applied to the radial velocity vr and inward Elsässer variable:   ∂ −  ∂  vr  z  = 0, = 0 (case I & II). (3.13) ∂r bottom ∂r bottom Since the wave dynamics is highly affected by the injection frequency, two different ways of wave injection are employed. The first one corresponds to monochromaticwave injection given by + = 2a sin (2π f 0 t) , z θ,bottom

(3.14)

+ z φ,bottom

(3.15)

= 2a cos (2π f 0 t) ,

where a = 32 km s−1 and f 0 is a free parameter. The second one is more realistic with broadband energy spectrum. +

z ∝



2π f max

  dω ω −1 sin ωt + φrand (ω) ,

(3.16)

2π f min

where f max is fixed to 10−2 Hz and f min is changed as a free parameter. The rootmean-square amplitude of outward Elsässer variable is fixed to 32 km s−1 at the lower boundary. The radial profile of the expansion factor is given as follows.

40

3 Onset and Suppression of Parametric Decay Instability

f (r ) ∝

f max,1 exp [(r − r1 ) /σ1 ] + f 1 , exp [(r − r1 ) /σ1 ] + 1

(3.17)

where   f 1 = 1 − f max,1 exp (R − r1 ) /σ1 ,

f max,1 = 10, r1 = 1.3R ,

σ1 = 0.5R (3.18)

The corresponding correlation length λ is given by  λ = λ

r2 f , 2 R f

(3.19)

where λ0 = 1 Mm. Conductive heating Q cnd is connected to the conductive flux qcnd by Q cnd = −

1 r2

 ∂  qcnd r 2 f . f ∂r

(3.20)

In this chapter, a bridging law between the Spitzer-Härm flux qSH and free-streaming flux qFS is employed. The formulation of each flux is given by qSH = −κ0 T 5/2

∂ T, ∂r

qFS =

3 α pvr , 4

(3.21)

where κ0 ≈ 10−6 in cgs-Gaussian unit, and α is a free parameter of order unity. The choice of α is not trivial. α = 4 is often used in the theoretical models (Hollweg 1986; van Ballegooijen and Asgari-Targhi 2016), while α = 1.05 is the best choice to explain the observed temperature (Cranmer et al. 2009). Here we use α = 2 as an intermediate value. Then, the conductive flux are given by qcnd = ξcnd qSH + (1 − ξcnd ) qFS ,

ξcnd

= min 1,

ρ 10−21 g cm−3

.

(3.22)

We apply uniform mesh the size of which is 103 km.

3.3 Result 3.3.1 Monochromatic Wave Injection Figure 3.1 shows the snapshots of quasi-steady states with various injection frequency. Each panel shows the radial profiles of mass density ρ, temperature T , radial

3.3 Result

41

Fig. 3.1 Snapshots of quasi-steady states with various wave injection frequency f 0 (monochromatic). Blue, green, and red lines correspond to f 0 = 10−2.5 Hz, f 0 = 10−3.5 Hz, f 0 = 10−4.5 Hz, respectively. Transparent solid lines and dashed lines in the bottom panel indicate outward and inward Elsässer variables, respectively. Figure reproduced from Shoda et al. (2018) under permission of the American Astronomical Society

velocity vr , and Elsässer variables z ± , respectively. Three lines indicate the injection frequencies of f 0 = 10−2.5 Hz, f 0 = 10−3.5 Hz, and f 0 = 10−4.5 Hz, respectively. In the bottom panel, the transparent solid lines and dashed lines indicate the outward and inward Elsässer amplitudes, respectively. The solar wind property does not monotonically depend on the injection frequency; the case with f 0 = 10−3.5 Hz yields the most tenuous, coolest, fastest solar

42

3 Onset and Suppression of Parametric Decay Instability vr, 1au [km s−1]

1000

(a)

600

10−5

M˙ [10−14 M yr−1 ]

2.5

(b)

800

400 10−6

Tmax [106 K]

2.0

10−4

10−3

10−2

(d)

1.5

2.0

2.5

1.0

1.5

2.0

0.5

1.0

1.5

0.0 10−6

f0 [Hz]

10−5

10−4

10−3

10−2

0.5 10−6

f0 [Hz]

r∗/R

3.0

(c)

10−5

10−4

10−3

10−2

1.0 10−6

f0 [Hz]

10−5

10−4

10−3

10−2

f0 [Hz]

Fig. 3.2 Time-averaged solar wind parameters with different wave injection frequencies f 0 . Each panel shows the a radial velocity at 1 au, b maximum coronal temperature, c mass-loss rate, and d location of critical point, respectively. Figure reproduced from Shoda et al. (2018) under permission of the American Astronomical Society

wind. This trend is more clearly seen in Fig. 3.2, where we show the time-averaged wind parameters versus f 0 . The non-monochromatic dependence on frequency indicates that two different wave heating mechanisms work depending on the wave frequency, and f 0 = 10−3.5 Hz is a specific frequency that both processes do not work effectively. The parametric decay instability and the Alfvén wave turbulence can be such two mechanism. Since the growth rate of PDI is proportional to the wave frequency, the higher-frequency waves can dissipate easily via PDI. Meanwhile the cascading time of Alfvén wave turbulence depends on the amount of reflected wave, and therefore, low frequency waves that experience stronger reflection are preferable for turbulent dissipation. f 0 = 10−2.5 Hz shows quite different behavior compared with the other two cases; large-amplitude density fluctuations in the solar wind acceleration region and the amplitudes of z ± are comparable around r = 1 au. This indicates that the parametric decay instability occurred only when f 0 = 10−2.5 Hz. Note that the PDI yields the density fluctuation and the enhanced reflected Alfvén waves simultaneously. A mysterious result is that the PDI does not evolve when f 0  10−3.5 Hz. The growth time of PDI is roughly estimated as Eq. (1.17), which, when f 0 = 10−4 Hz, yields τPDI,

f 0 =10−4 Hz

≈ 104 s.

(3.23)

The Alfvén-crossing time from the coronal base to 1 au is τAC =

1 au 1 au  3 ≈ 1.5 × 105 s, v A + vr 10 km s−1

(3.24)

which is much larger than the growth time of PDI τPDI, f0 =10−4 Hz . It means that the Alfvén waves with frequency of 10−4 Hz has sufficiently long time for PDI growth before arriving 1 au, which is inconsistent with our numerical simulation. This indicates the strong quenching of PDI in the solar wind where plasma is expanding and accelerating.

3.3 Result

43

3.3.2 Growth Rate of PDI in the Expanding/Accelerating Solar Wind The acceleration and expansion of the solar wind affects the development of waves and turbulence in the solar wind. Several works investigate the role of wind acceleration and expansion on PDI. Tenerani and Velli (2013) conduct 1D MHD simulations of PDI in a accelerating expanding box model to show the suppression of PDI by the diveregence of the wind. Similar result is confirmed in 2D MHD simulations under an expanding box model (Del Zanna et al. 2015). Here we derive a growth rate of PDI including the effects of expansion and acceleration of the solar wind. Following Tenerani and Velli (2013), we evaluate quantitatively how wind acceleration and expansion quenches PDI. The conservation of mass is written as follows.  ∂  ∂  2  ρr f + ρvr r 2 f = 0. ∂t ∂r

(3.25)

Decoupling the mean (time-averaged) and fluctuation parts as X = X + δ X,

(3.26)

the mass conservation is rewritten as   ∂  ∂  ∂  2  δρr f + δρvr r 2 f + ρδvr r 2 f = 0. ∂t ∂r ∂r

(3.27)

Suppose the δρ and δvr satisfy a characteristic relation of slow-mode wave: δρ/ρ = δvr /cs ,

(3.28)

 ∂  ∂  2  δρr f + δρ (vr + cs ) r 2 f = 0. ∂t ∂r

(3.29)

which yields

We assume that the spatial and temporal variation of δρ is approximately expressed as δρ ∝ exp [i (k(r )r − ωt)] .

(3.30)

Here, for simplicity, we assume that k(r ) is always real and any amplification and decay of δρ are attributed to the imaginary part of ω. Then the relation between k(r ) and ω is given as −iω + ik(r ) (cs + vr ) +

d (cs + vr ) d  2  r f + (cs + vr ) = 0. 2 r f dr dr

(3.31)

44

3 Onset and Suppression of Parametric Decay Instability

When the background is nearly homogeneous, we obtain the dispersion relation of outward sound wave: ω = k(r ) (cs + vr ) .

(3.32)

When the background inhomogeneity is non-negligible, ω is expressed as ω = k(r ) (cs + vr ) − iγexp − iγacc ,

(3.33)

where γexp and γacc are factors of expansion and acceleration effects. γexp = (cs + vr )

  d ln r 2 f , dr

γacc =

d (cs + vr ) dr

(3.34)

Equation (3.33) shows the decay of density fluctuation by the wind expansion and acceleration. Since the growth of density fluctuation is a key in PDI, this result indicates that the PDI is possibly suppressed in the solar wind. Another mechanism that suppresses PDI is the Doppler effect, or the boxstretching effect in the accelerating expanding box model (Tenerani and Velli 2017). When we calculate the growth rate of PDI, we need to know the frequency of initial Alfvén wave. Exactly speaking, that frequency is the wave frequency in the comoving frame with the background plasma. In a stationary solar wind, the frequency of Alfvén wave in the laboratory frame does not change, but it changes in the comoving frame in the presence of accelerating flow. To see this effect, let us calculate how the intrinsic frequency changes as a result of accelerating flow. The dispersion relation of outward Alfvén wave is written as ω0 = k(r ) (v A + vr ) ,

(3.35)

where ω0 is the frequency observed from laboratory frame. When we observe from the co-moving frame, since the wavelength never changes (as long as the relativistic effect is negligible), the wave frequency ω˜ should satisfy ω˜ = k(r )v A

(3.36)

Combining these two equations, we obtain ω˜ =

vA ω0 . v A + vr

(3.37)

This equation shows that, in the presence of trans-Alfvénic flow, the intrinsic frequency becomes smaller. Note that the decrease of intrinsic frequency also appears in the wave action conservation (Bretherton and Garrett 1968; Dewar 1970). In the accelerating flow, the decrease in the wave energy density E is accompanied by the

3.3 Result

45

decrease in the intrinsic frequency ω˜ to make the wave action density E/ω˜ constant (in WKB regime). We can now calculate the effective growth rate. First, the normalized growth rate is calculated as the largest growth mode of the Goldstein–Derby dispersion relation given as (Goldstein 1978; Derby 1978)        ω˜ 2 − β k˜ 2 ω˜ − k˜ ω˜ + k˜ − 2 ω˜ + k˜ + 2 = A2 k˜ 2 ω˜ 3 + k˜ ω˜ 2 − 3ω˜ + k˜ , (3.38) where β = a 2 /v 2A , ω˜ = ω/ω0 , k˜ = k/k0 , and η = δ B/B0 is the normalized Alfvén wave amplitude. ω0 and k0 are the frequency and wave number of the parent wave. frequency 2π f 0 but the frequency We should note the ω0 should not be the injection   modified by the Doppler effect 2π f 0 vr / vr,0 + v A,0 . Denoting this growth rate as γGD , the effective growth rate is given as γeff = γGD − γeff − γacc .

(3.39)

Figure 3.3 shows γexp , γacc (top panel) and γGD , γeff (bottom panel) versus heliocentric distance. Color of line corresponds to injection frequency: f 0 = 10−2.5 Hz (blue), f 0 = 10−3.0 Hz (green), f 0 = 10−3.5 Hz (orange), f 0 = 10−4.0 Hz (red). The gap between solid and dashed line corresponds to the degree of growth-rate suppression. Interestingly, waves with f 0 = 10−4 Hz are stabilized by the wind expansion/acceleration effects. The growth rate of f 0 = 10−3.5 Hz waves is positive but sufficiently small that the wave resonance is prevented by the inhomogeneity of background (Tenerani and Velli 2013).

3.3.3 Broadband Wave Injection We next report the results with broadband wave injection focusing on the consistency with observation. For free parameter f min , we apply f min = 10−3 , 10−4 , 10−5 Hz. Regardless of f min , realistic solar wind is reproduced. Figure 3.4 shows the snapshots of quasi-steady states with different f min . Lines indicate the simulation results and symbols represent the observational values. In every simulation, high-temperature corona and fast solar wind are realistically reproduced. The difference between each run is not so significant as the monochromatic-wave case. Interestingly, the radial profile of density fluctuation is similar regardless of f min . Figure 3.5 shows the calculated density fluctuation the solid lines correspond to the simulations with f min = 10−3 Hz (blue), f min = 10−4 Hz (green), and f min = 10−5 Hz (red), while grey symbols correspond to the observational values. Specifically, density fluctuation becomes maximum in the wind acceleration region for every line. This indicates that, as long as Alfvén waves include high-frequency

10−2

f0 = 10−2.5 Hz

(a)

f0 = 10−3.0 Hz f0 = 10−3.5 Hz

10−3

f0 = 10−4.0 Hz

γexp

10−4

γacc 10−5 100

101

102

r/R −2

10

γeff γGD

(b) γGD [Hz], γeff [Hz]

Fig. 3.3 γexp , γacc , γGD and γeff versus heliocentric distance. Top panel corresponds to γexp (dashed line) and γacc (solid line), while bottom panel shows γGD and γeff . Color of line corresponds to injection frequency: f 0 = 10−2.5 Hz (blue), f 0 = 10−3.0 Hz (green), f 0 = 10−3.5 Hz (orange), f 0 = 10−2.5 Hz (red). Figure reproduced from Shoda et al. (2018) under permission of the American Astronomical Society

3 Onset and Suppression of Parametric Decay Instability

γacc [Hz], γexp [Hz]

46

10−3

10−4

10−5 100

101

102

r/R

components that are subject to parametric decay instability, the observed density fluctuation is explained. Another observational constraint, the cross helicity evolution, is shown in Fig. 3.6. The solid and dashed lines indicate the amplitudes of outward and inward Elsässer variables. Blue, green and red lines correspond to f min = 10−3 Hz, f min = 10−4 Hz and f min = 10−5 Hz, respectively, while the gray lines indicate the observational trend by Bavassano et al. (2000). When f min = 10−3 Hz, the outward wave amplitude becomes drastically smaller than the observational value. This inconsistency is possibly because the parametric decay instability leads to an excessive reflection and dissipation of outward Alfvén waves. In other words, some fraction of outward Alfvén waves at the coronal bottom must be stable with respect to parametric decay instability to explain the observation. Indeed, the simulation is consistent with observation when f min = 10−4 Hz or 10−5 Hz. Note that the Alfvén waves are subject to parametric decay instability only when f 0  10−3.5 Hz.

3.3 Result Fig. 3.4 Snapshots in the quasi-steady states with different f min values. Blue, green and red lines indicate f min = 10−3 Hz, 10−4 Hz, 10−5 Hz, respectively. a mass density. Shown by circles and stars are observations by Wilhelm et al. (1998) and by Lamy et al. (1997), respectively. b temperature. Shown by circles and stars are observations by Landi (2008) and by Cranmer (2004, 2009), respectively. c radial velocity. Shown by stars are ion outflow speed by Zangrilli et al. (2002), while the crosses represent the IPS observations (Kojima et al. 2004). d transverse velocity. Figure reproduced from Shoda et al. (2018) under permission of the American Astronomical Society

47

48

3 Onset and Suppression of Parametric Decay Instability

Fig. 3.5 Fractional density fluctuation n f = δρ/ρ versus radial distance. Red, green and blue lines correspond to f min = 10−5 Hz, f min = 10−4 Hz and f min = 10−3 Hz, respectively. Shown by the gray symbols are the observational values. Figure reproduced from Shoda et al. (2018) under permission of the American Astronomical Society

Fig. 3.6 Amplitudes of Elsässer variables ζ ± (not z ± in this figure) as functions of heliocentric distance. Solid and dashed lines indicate the outward and inward Elsässer variables, respectively. Red, green and blue lines correspond to the minimum frequencies of f min = 10−3 Hz (blue), f min = 10−4 Hz (green) and f min = 10−5 Hz, respectively. Also shown by grey lines are the observational trends by Bavassano et al. (2000). Figure reproduced from Shoda et al. (2018) under permission of the American Astronomical Society

3.4 Discussion and Summary In this chapter, we have investigated the frequency-dependent propagation property of Alfvén waves from the corona to 1 au, focusing on how the parametric decay instability proceeds in the solar wind. The frequency-filtering mechanism can operate in the corona and solar wind due to the bimodal behavior of wave dissipation with respect to frequency. The low-

3.4

Discussion and Summary

49

frequency ( f 0  10−4 Hz) waves undergo linear reflection and generate Alfvénic turbulence from the interaction with counter-propagating waves. The high-frequency ( f 0  10−3 Hz) waves dissipate through the PDI. As a result of the efficient heating, dense, hot and relatively slow winds are driven in the cases with f 0  10−4 Hz or f 0  10−3 Hz. In contrast, the intermediate-frequency ( f 0 ≈ 10−3.5 Hz) waves are not subject to these damping mechanisms. As a result, fast and less dense wind emanates from the relatively cool corona in this case. This indicates that the corona and solar wind have a frequency-filtering effect of the Alfvén wave, and as a result, the medium-frequency wave is the easiest to propagate through the corona and solar wind. The observed fact that hour-scale waves dominate the fluctuation in the solar wind could be attributed to this frequency-filtering mechanism (Belcher and Davis 1971). As long as high-frequency waves that are subject to parametric decay instability have a non-negligible power, the observed density fluctuation is explained (Fig. 3.5) Interestingly, the location of maximum density fluctuation is the same as that of maximum growth rate of parametric decay instability. These results support a scenario that the density fluctuation in the wind acceleration comes from parametric decay instability. Meanwhile, when all the injected Alfvén waves are unstable with respect to parametric decay instability, although density fluctuation is accountable, the cross helicity evolution is no longer consistent with observation (Fig. 3.6). To summarize, high-frequency waves ( f 0  10−3.5 Hz) are required to explain the density fluctuation and low-frequency waves ( f 0  10−3.5 Hz) are required to explain the cross helicity. The broadband nature of the Alfvén waves is a key to understand the observational facts.

References Bavassano B, Pietropaolo E, Bruno R (2000) J Geophys Res 105:15959 Belcher JW, Davis L Jr (1971) J Geophys Res 76:3534 Bretherton FP, Garrett CJR (1968) Proc R Soc Lond Ser A 302:529 Cranmer SR, Matthaeus WH, Breech BA, Kasper JC (2009) ApJ 702:1604 Cranmer SR (2004) In: Walsh RW, Ireland J, Danesy D, Fleck B (eds) ESA special publication, vol 575, SOHO 15 coronal heating, p 154 Cranmer SR (2009) Living Rev Solar Phys 6:3 Del Zanna L, Matteini L, Landi S, Verdini A, Velli M (2015) J Plasma Phys 81:325810102 Derby NF Jr (1978) ApJ 224:1013 Dewar RL (1970) Phys Fluids 13:2710 Galeev AA, Oraevskii VN (1963) Sov Phys Dokl 7:988 Goldstein ML (1978) ApJ 219:700 Hollweg JV (1986) J Geophys Res 91:4111 Kojima M, Breen AR, Fujiki K, Hayashi K, Ohmi T, Tokumaru M (2004) J Geophys Res (Space Phys) 109:A04103 Lamy P, Quemerais E, Llebaria A, Bout M, Howard R, Schwenn R, Simnett G (1997) In: Wilson A (ed) ESA special publication, vol 404, Fifth SOHO workshop: the corona and solar wind near minimum activity, p 491 Landi E (2008) ApJ 685:1270

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3 Onset and Suppression of Parametric Decay Instability

Miyamoto M et al (2014) ApJ 797:51 Sagdeev RZ, Galeev AA (1969) Nonlinear plasma theory Shoda M, Yokoyama T, Suzuki TK (2018) ApJ 860:17 Suzuki TK, Inutsuka S-I (2005) ApJ 632:L49 Tenerani A, Velli M (2013) J Geophys Res (Space Phys) 118:7507 Tenerani A, Velli M (2017) ApJ 843:26 van Ballegooijen AA, Asgari-Targhi M (2016) ApJ 821:106 Wilhelm K, Marsch E, Dwivedi BN, Hassler DM, Lemaire P, Gabriel AH, Huber MCE (1998) ApJ 500:1023 Zangrilli L, Poletto G, Nicolosi P, Noci G, Romoli M (2002) ApJ 574:477

Chapter 4

3D Simulations of the Fast Solar Wind

Abstract From a series of one-dimensional simulations, it is indicated that the interplay between parametric decay instability and Alfvén-wave turbulence plays a critical role in heating and accelerating the fast solar wind. To validate this idea, we perform, in a self-consistent manner, the first three-dimensional compressional MHD simulation of the fast solar wind acceleration. Large-amplitude density fluctuation is generated in the solar wind as predicted by previous one-dimensional simulations. Such density fluctuation plays a significant role in wave reflection, an essential process in triggering turbulence. We conclude that the fast solar wind is driven by compressional MHD turbulence initiated by parametric decay instability. Keywords Fast solar wind · Magnetohydrodynamic turbulence · Direct numerical simulation

4.1 Introduction One-dimensional models with phenomenological turbulence model are discussed in the previous chapters. By introducing compressible processes into the standard Alfvén wave turbulence scenario, the observed density fluctuation turns out to be accountable by parametric decay instability. 1D simulation is, however, still unrealistic in that we do not directly deal with turbulence. Another problem arises when we need to compare the observation of turbulence with simulation, because observational characteristics of turbulence such as power spectrum or correlation function are difficult to predict from 1D simulation. To summarize, although 1D simulation is useful in understanding the underlying physics, it has problems from both theoretical and observational aspects; theoretically turbulence phenomenology should be validated and modified using 3D simulation and observationally a direct prediction for in-situ measurement is necessary.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 M. Shoda, Fast Solar Wind Driven by Parametric Decay Instability and Alfvén Wave Turbulence, Springer Theses, https://doi.org/10.1007/978-981-16-1030-1_4

51

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4 3D Simulations of the Fast Solar Wind

Based on this background, we conduct the first, three-dimensional, magnetohydrodynamic simulation of the solar wind acceleration. Although there exists 3D MHD simulation of the coronal loop heating (Dahlburg et al. 2016; Matsumoto 2018), no 3D MHD simulations have ever successfully performed for coronal hole and fast solar wind. One of the difficulties in the 3D simulation of the solar wind is the small timescale of thermal conduction. Based on the assumption that the thermal conduction is Spitzer-Härm type, then the timescale of conduction τcnd is given as 2k B −5/2 ne T γ−1 −5/2    x 2  n e  T −5 , ≈ 4 × 10 s 106 K 1 Mm 104 cc

τcnd = κ−1 0

(4.1) (4.2)

where x is the smallest spatial scale of fluctuation, practically the grid size. Thus the time scale of thermal conduction is in the order of 10−5 s, while the required time to accelerate the solar wind τacc is given as the sound crossing time as  τacc ≈

rout

rin

dr  2 × 104 s, a + vr

(4.3)

where rin and rout are radial distances of inner and outer boundaries, respectively, and a and vr are the sound and radial bulk velocities, respectively. Thus we need approximately 109 time steps for the solar wind to be accelerated, which is numerically too expensive. A recently proposed numerical method called super-time-stepping method (Meyer et al. 2012, 2014) enables us to reduce the numerical cost coming from parabolic terms such as thermal conduction. By employing this method, 3D simulations become available. As an introduction of three-dimensional simulation, we discuss the reason why we need to conduct 3D simulation from both theoretical and observational view points.

4.1.1 Motivation from Theoretical Point of View The theoretical motivations for 3D simulation are to overcome several limitations of 1D simulations. The first limitation is that the phenomenological turbulence model used in Chaps. 2 and 3 would be problematic for application to compressible MHD equations. The model is derived from reduced MHD, in which fast and slow mode waves are excluded. Note that the fast-mode turbulence has different feature than Alfvén wave turbulence, and thus the two modes should be solved independently (Suzuki et al. 2007; Cranmer and van Ballegooijen 2012), while our 1D model ignores the difference between fast mode wave and Alfvén wave in low-beta region. The phenomenology of Alfvén wave turbulence is also worth revisiting. The simplest model of the reduced MHD turbulence assume that inward and outward Elsässer

4.1 Introduction

53

Fig. 4.1 Comparison of the growth rates of parametric decay instability between 1D and 3D settings. The angles between the mean magnetic field and the wave number of density fluctuation are 0◦ (1D setting) and 45◦ (3D setting). Red solid and dashed lines indicate the magnitude of normalized density fluctuation as functions of time. Blue solid and dashed lines are the fitted lines in the growth phase of instability. Figure reproduced from Shoda and Yokoyama (2018) under permission of the American Astronomical Society

variables have the same correlation length (λ+ = λ− = λ) and the correlation length −1/2 expands with the radius of flux tube (λ ∝ Br ). In reality, the correlation lengths of inward and outward Elsässer variables should be different and be solved independently via transport equations (Zank et al. 2017; Shiota et al. 2017; Zank et al. 2018). Besides, in the presence of compressible waves (density fluctuation) with finite perpendicular wave number, the phase mixing begins to work (Heyvaerts and Priest 1983), which could have a significant impact on the correlation length transport. For example, the phase mixing plays a non-negligible role in the early phase of parametric decay instability (Shoda and Yokoyama 2018) and therefore should be taken into account. In 1D models, we limit the propagation direction parallel to the mean field. In the actual corona and solar wind, the direction of wave propagation is possibly oblique to the mean field. In other words, fluctuations can be inhomogeneous perpendicular to the mean magnetic field. For example in the coronal base, taking the horizontal correlation length of 10 Mm, correlation time (period) of 100 s and Alfvén speed of 1 Mm s−1 , then the angle of wave number vector with respect to mean (vertical) magnetic field is k = tan−1



100 Mm 10 Mm



= 84◦ ,

(4.4)

which shows the almost perpendicular nature of coronal-wave propagation. In fact, the density fluctuation near the coronal base is observed to be highly inhomogeneous perpendicular to magnetic field line (Raymond et al. 2014). Structure in the perpendicular direction must be considered. Besides, the oblique wave number affects the growth of parametric decay instability. Figure 4.1 compares the growth of parametric

54

4 3D Simulations of the Fast Solar Wind

decay instability between parallel and oblique propagations of compressible wave. Here the oblique case corresponds to the propagation angle of 45◦ . Dashed (parallel) and solid (oblique) red lines represent the root-mean-square density fluctuation versus time. The dashed and solid blue lines are the fitted exponential function. The oblique propagation case shows 26% smaller growth rate than the parallel propagation case. In other words, the 1D calculation overestimates the growth rate and thus the magnitude of density fluctuation.

4.1.2 Motivation from Observational Point of View While the solar wind is believed to be heated and accelerated by turbulence, until 2017, observation of turbulence had been available beyond 0.3 au from the Sun, which is much farer from the wind acceleration region near 10R . A new spacecraft Parker Solar Probe (hereafter PSP) has been launched and begun to observe the much closer region (Fox et al. 2016). The perihelion of PSP is expected to be as close as 9.86R from the center of the Sun, and thus the direct observation of solar wind turbulence in the wind acceleration region would be available. PSP has four instruments: FIELDS for electromagnetic fields (Bale et al. 2016), ISIS for energetic particles (McComas et al. 2016), WISPR for coronal imaging (Vourlidas et al. 2016) and SWEAP for major particles (Kasper et al. 2016). These instruments are expected to yield essential hints as to how the solar wind is accelerated. By providing synthesized data based on our simulation results, one can make a direct comparison with the observed data with better interpretations. Suppose PSP flies in a quasi-periodically oscillating vortex with length L and period τ , then the time series of PSP data is predicted to have two time scales: τ and L/vPSP where vPSP is the velocity of PSP. In other words, PSP data involve both temporal and spatial variations of turbulence. In the vicinity of the Sun, these two time scales can be similar, and thus it is highly possible that observation cannot distinguish the spatial and temporal variation by itself. Numerical simulation and the direct prediction from it would be helpful to distinguish the two variations.

4.1.3 Purpose of 3D Simulation To summarize the introduction, we have two purposes to perform a 3D simulation. First, to test the results and conclusions of 1D simulations, we discuss how similar the results of 3D simulations are to those of 1D simulations. Specifically, since the parametric decay instability (PDI) is the key in 1D simulations, we investigate whether PDI really works in the three-dimensional setting. Second, we aim to provide theoretical predictions for Parker Solar Probe. The data of Parker Solar Probe would involve both spatial and temporal variations of

4.1 Introduction

55

turbulence in the solar wind, making the physical interpretation of the observed data difficult. For better interpretation, a direct prediction from simulation is required.

4.2 Method The basic equations are ideal MHD equations with gravity and Spitzer-Härm type thermal conduction and are solved in the spherical coordinate. We describe in this section the basic equations and numerical setting in this study.

4.2.1 Local Spherical Coordinate The spherical coordinate is appropriate to follow the radial expansion of magnetic field line and is used in the calculation. Specifically, in order to make the horizontal coordinates (θ and φ) symmetric, we use the local spherical coordinate that is defined as follows. In the usual spherical coordinate, a gradient of scalar X is given by ∇ X = er

∂X 1 ∂X 1 ∂X + eθ + eφ . ∂r r ∂θ r sin θ ∂φ

(4.5)

The local spherical coordinate assumes θ ≈ π/2, which yields the approximated gradient as ∇ X ≈ er

∂X 1 ∂X 1 ∂X + eθ + eφ . ∂r r ∂θ r ∂φ

(4.6)

The accuracy of this approximation depends on the range of θ we consider. Replacing  θ with θ = θ − π/2, sin θ is approximated as 1 2 sin θ ≈ 1 − θ . 2

(4.7) 2

Therefore, the error of our approximation is at most in the order of θ . In the solar wind simulation, the required horizontal extension of the simulation domain at the coronal base is 10–20 Mm (Perez and Chandran 2013). In this case, the maximum error is   2   10 Mm 2 ≈ 10−4 . max θ = R

(4.8)

56

4 3D Simulations of the Fast Solar Wind

4.2.2 Basic Equations and Numerical Solver The basic equations we use are the ideal MHD equations with gravity and thermal conduction given as  1 ∂ 1 ∂  1 ∂ ∂ U+ 2 Fr r 2 + Fθ + F φ = S, ∂t r ∂r r ∂θ r ∂φ

(4.9)

where ⎞ ⎛ ⎞ ρvθ ρvr Br Bθ ⎟ ⎜ ⎜ ⎟ B2 ⎞ ⎛ ⎟ ⎜ ⎜ ⎟ ρvr vθ − ρ ρvr2 − r + pT ⎟ ⎜ ⎜ ⎟ 4π 4π ⎟ ⎜ ⎜ ⎟ 2 ⎜ ρvr ⎟ B B B ⎟ ⎜ ⎜ ⎟ r θ θ 2 ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ρvr vθ − ρvθ − + pT ⎜ ρvθ ⎟ ⎟ ⎜ ⎜ ⎟ 4π 4π ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ B B B B ⎜ ρvφ ⎟ r φ θ φ ⎟ ⎜ ⎜ ⎟, U =⎜ ⎟ , Fr = ⎜ ρvr vφ − ρvθ vφ − ⎟ , Fθ = ⎜ ⎟ ⎜ Br ⎟ 4π 4π ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ 0 vθ Br − vr Bθ ⎜ Bθ ⎟ ⎟ ⎜ ⎜ ⎟ ⎝ B ⎠ ⎟ ⎜ ⎜ ⎟ B − v B v 0 r θ θ r φ ⎟ ⎜ ⎜ ⎟ vr Bφ − vφ Br ⎟ ⎜ ⎜ ⎟ vθ Bφ − vφ Bθ e ⎠ ⎝ ⎝ ⎠ Br Bθ (e + pT )vr − (v · B) (v · B) (e + pT ) vθ − 4π 4π ⎛

⎞ ⎞ ⎛ 0     2 B B ⎟ ⎟ ⎜ ⎜ r φ ⎟ ⎜ ⎜ ρ v 2 + v 2 /r + 2 p + Br /r + ρg ⎟ ρvr vφ − ⎟ ⎟ ⎜ ⎜ θ φ 4π 4π ⎟ ⎟ ⎜ ⎜   Bθ Bφ ⎟ ⎟ ⎜ ⎜ 1 v − ρv θ φ ⎟ ⎟ ⎜ ⎜ Br Bθ − ρvr vθ /r 4π ⎟ ⎟ ⎜ ⎜ 2 4π ⎟ ⎟ ⎜ ⎜ B   φ ⎟ ⎟, ⎜ 2 1 , S = Fφ = ⎜ ρvφ − + pT ⎟ ⎟ ⎜ ⎜ Br Bφ − ρvr vφ /r 4π ⎟ ⎟ ⎜ ⎜ 4π ⎟ ⎟ ⎜ ⎜ v B − v B r r φ φ ⎟ ⎟ ⎜ ⎜ 0 ⎟ ⎟ ⎜ ⎜ vφ Bθ − vθ Bφ ⎟ ⎟ ⎜ ⎜ /r B − v B ) (v r r θ θ ⎟ ⎟ ⎜ ⎜   0 ⎠ ⎠ ⎝ ⎝ vr Bφ − vφ Br /r Bφ (v · B) (e + pT ) vφ − ρgv + Q r cnd 4π ⎛

ρvφ

where e=

1 p B2 + ρv 2 + , γ−1 2 8π

pT = p +

B2 , 8π

g=−

G M . r2

(4.10)

G is the gravitational constant and M is the solar mass. For the specific heat ratio γ, we use the adiabatic value (γ = 5/3). The hyperbolic divergence cleaning method (Dedner et al. 2002) is used to remove the numerically generated ∇ · B. Note that the conserved variable U and corresponding fluxes F r,θ,φ are the same as those in Cartesian coordinates. The differences arise in the geometry factor in F r and the source terms. To numerically solve the ideal MHD equations, we use the HLLD solver (Miyoshi and Kusano 2005) to calculate F r,θ,φ . In the reconstruction procedure, we use 5th-order MP5 method (Suresh and Huynh 1997) for v and B and 2nd-order MUSCL method (van Leer 1979) with minmod limiter (Roe 1986) for ρ, p and

4.2 Method

57

ψ. High-order reconstructions of velocity and magnetic field are to minimize the numerical damping of Alfvén waves. Third-order SSP Runge–Kutta method (Shu and Osher 1988) is used for time integration. We use the same equation of state as van Ballegooijen and Asgari-Targhi (2016); we assume fully ionized hydrogen plasma with 10% contamination of helium. The equation of state is given as p = c1 ρT,

(4.11)

where c1 = 2.3k B / (1.4m H ). Here k B is the Boltzmann constant and m H is the hydrogen mass. The radiative cooling, which is in general not negligible in the coronal base (Matsumoto and Suzuki 2014; van Ballegooijen and Asgari-Targhi 2016), is excluded for simplicity. Instead of solving the radiative cooling, we fix the coronal base temperature and attribute the cooling near the base to thermal conduction. We use the following formula for thermal conduction. ⎛

Q cnd = −∇ · q cnd ,

q cnd

⎞ 0 bˆr2 0 ⎜ ⎟ = −κ0 ξ(r )T 5/2 ⎝ 0 f red bˆθ2 0 ⎠ · ∇T, 0 0 f red bˆφ2

(4.12)

where κ0 = 10−6 in cgs unit and bˆ = (br , bθ , bφ ) = B/ |B| is the unit vector parallel to the local magnetic field. Note the difference from the Spitzer-Härm type conductive flux q SH given by ⎛

q SH

bˆr2 ⎜ = −κ0 T 5/2 bˆ bˆ · ∇T = −κ0 T 5/2 ⎝ bˆr bˆθ bˆr bˆφ

bˆr bˆθ bˆθ2 ˆbθ bˆφ

⎞ bˆr bˆφ ⎟ bˆθ bˆφ ⎠ · ∇T. bˆφ2

(4.13)

The first difference is the factor ξ(r ). This comes from the observational implication that the Spitzer-Härm conductive flux is larger than the observed heat flux. To solve this problem, Hollweg (1974, 1976) proposed an idea that the conductivity switches into another form in the vicinity of wind acceleration region. Following this idea, we quench the conductivity beyond r ≈ 5R via ξ(r ) as  2  rsw ξ(r ) = min 1, 2 , r

(4.14)

where rsw = 5R . ˆ We ignore the non-diagonal compoThe other differences are found in matrix bˆ b. nent of bˆ bˆ for two reasons. First, the non-diagonal component of bˆ bˆ can numerically cause negative temperature (Sharma and Hammett 2007), and second, because the direction of local magnetic field is random, the non-diagonal components vanish

58

4 3D Simulations of the Fast Solar Wind

when time-averaged: ⎛

bˆ 2 ⎜ ˆ rˆ ⎝ br bθ bˆr bˆφ

bˆr bˆθ bˆθ2 ˆbθ bˆφ

⎞ ⎛ bˆ 2 bˆr bˆφ ⎜ r ⎟ bˆθ bˆφ ⎠ ≈ ⎜ ⎝0 bˆφ2 0

⎞ 0 0 ⎟ bˆθ2 0 ⎟ ⎠, 2 ˆ 0 b

(4.15)

φ

where X denotes the time-average of X . Since the thermal conduction affects the wind velocity and mass flux and we are interested in the time-averaged value of them, the non-diagonal components are ignored. In addition, the conductive fluxes in θ and φ components are reduced by the factor of f red , reducing the numerical cost of thermal conduction. The role of thermal conduction is to radially distribute the heat deposited by wave dissipation, and therefore the horizontal thermal conduction can be reduced unless unrealistically large temperature gap is observed. We apply f red = 0.1 in this study. The thermal conduction is numerically solved separately from the ideal MHD equations using super-time-stepping method (Meyer et al. 2012, 2014), an explicit solver of parabolic equation that reduces the numerical cost.

4.2.3 Numerical Settings The simulation domain extends from the coronal base (rmin = 1.02R ) to beyond the wind acceleration region (rmax = 20R ). We solve in the range of θmin ≤ θ ≤ θmax and φmin ≤ φ ≤ φmax , where θmax = φmax =

L bottom rad, 2R

θmin = φmin = −

L bottom rad. 2R

(4.16)

Note that the horizontal size of numerical domain at the bottom boundary is set to L bottom = 20 Mm. As auxiliary variables, we introduce x, y and L(r ) as x = r θ,

y = r φ,

L(r ) = r (θmax − θmin ) = r (φmax − φmin ) ,

(4.17)

where x and y represent the displacements in θ and φ directions and L(r ) represents the size of flux tube at radial distance of r . The number of grid points are (6600, 192, 192) in (r, θ, φ) directions. The radial grid size is fixed to 2 Mm independently of r . As the initial condition, we impose the isothermal Parker wind solution (Parker 1958) with basal density of ρ0 = 8.5 × 10−16 g cm−3 and temperature of T = 1.1 × 106 K. The periodic boundary conditions are used for θ and φ directions. The bottom boundary (r = rmin ) conditions are as follows. The fixed boundary condition is used for density, temperature and radial magnetic field as

4.2 Method

59

ρbottom = 8.5 × 10−16 g cm−3 ,

Tbottom = 6.0 × 105 K,

Br,bottom = 2.0 G, (4.18)

while the free boundary condition is used for the radial velocity:  ∂  vr  = 0. ∂r bottom

(4.19)

The horizontal components of the velocity √ and magnetic field are given in terms of ± Elsässer variables (z θ,φ = vθ,φ ∓ Bθ,φ / 4πρ). The upward Elsässer variable is fixed as

+ z θ,φ,bottom =

3  10 3  

      P n x , n y , l exp i πn x θ/θmax + πn y φ/φmax + 2πl f 0 t + θ,φ n x , n y , l ,

n x =1 n y =1 l=1

(4.20)  −1/2 −1/2     where P n x , n y , l ∝ n 2x + n 2y l and θ,φ n x , n y , l is the random phase + function. The amplitude of z θ,φ,bottom is determined so that the root-mean-squared  2 2 + + value of z θ,bottom + z φ,bottom is fixed to 60 km s−1 . The typical frequency f 0 is f 0 = 10−3 Hz.

(4.21)

The free boundary condition is applied to the inward Elsässer variables.  ∂ −  = 0. z θ,φ  ∂r bottom

(4.22)

As for the top boundary, we apply a marginal numerical region that extends beyond 1000R with radially expanding mesh. Since the supersonic and super-Alfvénic outflow is generated in the quasi-steady state, the choice of boundary condition does not affect the calculation result.

4.3 Result 4.3.1 Averaged 1D Structure in the Quasi-Steady State We calculate the basic equations until the system reaches a quasi-steady state. To see the radial variance of the solar wind, we first calculate the temporal and horizontal averages of solar wind parameters. Here we√ directly average the density, temperature, √ radial velocity, Alfvén velocity v A = Br / 4πρ and sound velocity a = p/ρ in

60

4 3D Simulations of the Fast Solar Wind 103 ± [km s−1] v⊥,rms, z⊥,rms

< ρ > [g cm−3 ]

10−14 10−16 −18

10

10−20 −22

100.0

v⊥,rms

101

+ z⊥,rms − z⊥,rms

0

100.5

10

101.0

100.0

100.5

1.2

103

1.0

102

< vr > < vA >

101

101.0

r/R

r/R 104

< T > [106 K]

< vr >, < vA >, < a > [km s−1]

10

102

0.8 0.6

100 100.0

100.5

0.4 100.0

101.0

100.5

101.0

r/R

r/R

Fig. 4.2 Horizontally and temporally averaged variables as functions of r in a quasi-steady state. Left top: mass density. Right top: root-mean-squared transverse velocity (black) and Elsässer variables (blue: outward, red: inward). Left bottom: radial velocity (black), Alfvén velocity (blue) and sound velocity (red). Right bottom: temperature

time and horizontal direction. Note that the isothermal sound speed is used because the thermal conduction is much faster than the time scale of sound wave. We define temporal-horizontal averaging operator as X =

1 τsim L(r )2





τsim 0



xmax

dt

ymax

dx xmin

dy X,

(4.23)

ymin

where t = 0 corresponds to the time the system reached the quasi-steady  state. Meanwhile the rms values are calculated for transverse velocity v⊥ = vθ2 + vφ2  2 2 ± and Elsässer variables z ⊥ = z θ± + z φ± : v⊥,rms = vθ2 + vφ2 1/2 ,

± z ⊥,rms = z θ± + z φ± 1/2 . 2

2

(4.24)

τsim = 3000 s is used for temporal averaging. Note that τsim is larger than the longest period of upward Alfvén waves 1/ f 0 = 1000 s. Figure 4.2 shows the horizontally and temporally averaged variables versus r . The left top panel shows the mass density profile. The right top panel shows the

4.3 Result

61

transverse velocity (black line), upward Elsässer variable (red line) and downward Elsässer variable (blue line). The left bottom panel shows the radial velocity (black line), Alfvén velocity (red line) and sound velocity (blue line). The right bottom panel shows the temperature profile in unit of 106 K. The wind velocity vr and mass-loss rate M˙ w measured at r = 20R is vr = 591 km s−1 ,

M˙ w = ρvr r 2 r =20R = 1.94 × 10−14 M yr −1 ,

(4.25)

the latter of which is comparable with the observed value (2–3 ×10−14 M yr −1 ). The result that the maximum coronal temperature exceeds 106 K and that the solar wind asymptotic velocity approximates 600 km s−1 shows that the coronal heating and fast solar wind acceleration are successfully reproduced from our simulation. This result indicates that a sufficient amount of thermal energy is supplied by Alfvén waves and turbulence. In light of the fact that the reduced MHD modeling cannot and compressible MHD model can explain the required heating rate, the importance of compressibility in the solar wind is clear.

4.3.2 Density Fluctuation and Enhanced Wave Reflection The transverse velocity disturbance is as large as or slightly larger than the sound speed in the solar wind, indicating that Alfvén waves have potential to compress plasma and fluctuate density. Such density fluctuation can enhance the energy cascading rate up to sufficient level to sustain the solar wind (Carbone et al. 2009; van Ballegooijen and Asgari-Targhi 2016). To understand the role of density fluctuation in wave dynamics, we calculate the density fluctuation and wave reflection rate. The magnitude of density fluctuation is defined by the dispersion of density: δρ =



ρ2 − ρ 2 .

(4.26)

The wave reflection rate is derived as follows. We use the linearized equation for radially propagating Alfvén waves in the spherically expanding magnetic field (Heinemann and Olbert 1980) given as 

      ∂  ∂ ∂ ∂ + vr ± V A,r z± = z± + vr ∓ V A,r ln ρ1/4 ∂t ∂r ∂t ∂r      ∂  ∂ + vr ∓ V A,r ln r ρ1/4 , − z∓ ∂t ∂r

(4.27)

where z + and z − denote the upward and downward Alfvén wave amplitudes, respectively. The second term in the right hand side corresponds to the reflection term. Defining the reflection rate ωref as the inverse of the timescale of upward-to-downward wave energy transfer, we get

62

4 3D Simulations of the Fast Solar Wind 10−2 reflection rate [Hz]

δρ/ρ

100

10−1

10−2 100.0

100.5

r/R

101.0

10−3

10−4

10−5 100.0

100.5 r/R

101.0

Fig. 4.3 Radial profiles of the normalized density fluctuation (δρ/ρ , left panel) and reflection rates (ωref and ωref,0 , right panel). In the right panel, solid and dashed lines correspond to ωref (including density fluctuation effect) and to ωref,0 (excluding density fluctuation effect), respectively

  ∂   ln r ρ1/4 . ωref = vr + V A,r ∂r

(4.28)

A classical understanding is that the radial inhomogeneity of the mean solar wind density makes nonzero ωref , triggering reflection and turbulence (Velli et al. 1989; Matthaeus et al. 1999; Cranmer et al. 2007; Verdini et al. 2010). On the contrary, van Ballegooijen and Asgari-Targhi (2016) pointed out the importance of density fluctuation for ωref . To reveal how density fluctuation enhances the reflection rate, we calculate the reflection rate with and without fluctuation as     d ln r ρ 1/4 , ωref,0 = vr + V A,r dr   ∂   ln r ρ1/4 . ωref =  vr + V A,r ∂r

(4.29) (4.30)

Figure 4.3 shows the radial profiles of density fluctuation normalized by mean value (left panel) and reflection rates with (solid line) and without (dashed line) density fluctuation (right panel). Density fluctuation in the order of 10% of mean value is ubiquitously observed in the solar wind. Besides, the density fluctuation is observed to be maximum in the solar wind acceleration region (r/R  10). These behaviors are consistent with previous observations (Carbone et al. 2009; Miyamoto et al. 2014; Hahn et al. 2018). A large gap between solid and dashed lines in the right panel directly means that the density fluctuation is a main driver of Alfvén-wave reflection. As a result, the wave reflection rate (solid line, right panel) becomes spatially correlated with the magnitude of density fluctuation (solid line, left panel). Owing to the existence of density fluctuation, the reflection rate in the solar wind becomes approximately ten times larger. As a result, the plasma heating

4.3 Result 1.0

0.8

2δρ/ρ γeff [mHz]

Fig. 4.4 Solid line: averaged doubled fractional density fluctuation 2δρ/ρ versus radius. Black dashed line: effective growth rate of parametric decay instability γPDI in unit of mHz

63

growth rate of PDI

0.6

0.4

0.2 magnitude of density fluctuation (doubled)

0.0 100.0

100.5

101.0

r/R rate is expected to become ten times larger due to the density fluctuation, which is consistent with a previous RMHD study (van Ballegooijen and Asgari-Targhi 2016). The radial profile of the magnitude of density fluctuation shows several interesting behaviors. Near the coronal base, the magnitude of density fluctuation is as large as 10% of background, which is consistent with recent observation of the intensity fluctuation in the corona (Hahn et al. 2018). We should note, however, that our calculation can overestimate the density fluctuation near the bottom boundary, because the nonlinearity of Alfvén waves can be larger due to relatively smaller coronal magnetic field (2 G). The trend that the magnitude of the density fluctuation increases in the wind acceleration region is consistent with previous one-dimensional simulations and radio-wave observations Miyamoto et al. (2014).

4.3.3 Origin of Density Fluctuation: Parametric Decay Instability Now that we show the generation and role of density fluctuation in the solar wind, we discuss the physical mechanism relevant to the generation of it. Bearing in mind that the parametric decay instability (PDI) is a candidate mechanism to generate the density fluctuation, here we investigate whether the generated density fluctuations have consistent property with the theory of PDI. According to the results in Chap. 3, the magnitude of density fluctuation is expected to be spatially correlated with the growth rate of PDI, which would support the idea that the density fluctuation is generated via PDI. The growth rate of PDI in the accelerating, expanding solar wind is given as

64

4 3D Simulations of the Fast Solar Wind

γPDI = γ˜ GD

v A d f 0 − 2 (vr + a ) /r − (vr + a ) , v A + vr dr

(4.31)

where γ˜ GD is the normalized growth rate calculated from Goldstein–Derby dispersion relation:        ω˜ 2 − β k˜ 2 ω˜ − k˜ ω˜ + k˜ − 2 ω˜ + k˜ + 2 = A2 k˜ 2 ω˜ 3 + k˜ ω˜ 2 − 3ω˜ + k˜ , (4.32) where β = a 2 /v 2A is plasma beta, ω˜ = ω/ω0 and k˜ = k/k0 are the frequency and wave number normalized by parent wave values, and η = δ B/B0 is the normalized wave amplitude. The second and third terms in the right hand side correspond to the suppression by the wind expansion and acceleration, respectively. Figure 4.4 shows the radial profiles of the relative density fluctuation (solid line) and the growth rate of parametric decay instability (solid line). For better visualization, the density fluctuation is doubled and the growth rate is shown in unit of mHz. The trend of the solid and dashed lines are similar. Specifically, the growth rate of PDI increases in 100.5  r/R  101.0 , so does the magnitude of density fluctuation. This results is the first evidence that the origin of density fluctuation is PDI. We also investigate the spatial scale of density fluctuation through auto-correlation function. For any variable X , spatial auto correlation C X (l) is defined as C X (l) = X (x + l, t)X (x, t)

(4.33)

Since the mean magnetic field makes the turbulence anisotropic, we calculate the parallel and perpendicular auto-correlation functions as follows:   C X l = X (x + l  , t)X (x, t) ,

(4.34)

C X (l⊥ ) = X (x + l ⊥ , t)X (x, t) ,

(4.35)

displacement vectors with where l  and l ⊥ denote the parallel and perpendicular   respect to the mean magnetic field and l,⊥ = l ,⊥ . Note that the spatial variation scale of auto-correlation function corresponds to the spatial scale of X . Specifically, the spatial power spectrum of X is defined as the Fourier transformation of C X (l). Figure 4.5 shows the auto-correlation functions of fractional density fluctuation ρ/ρ − 1 (black solid line) and x-component of upward Elsässer variable z + x with respect to parallel (top panel) and perpendicular (bottom panel) displacements, measured at r/R = 10. Both parallel and perpendicular auto-correlation functions show that density fluctuation has smaller scale than upward Elsässer variable. Given that the majority of density fluctuation comes from slow-mode waves, these results show that slow-mode waves have smaller spatial scale than upward Alfvén waves. This is consistent with the theory of PDI in that smaller-scale daughter slow-mode waves are generated from parent Alfvén waves. Therefore, the smaller scale density fluctuation is the second evidence of density fluctuation excitation by PDI.

4.3 Result

65 1.2

parallel auto correlation

Fig. 4.5 Parallel (top panel) and perpendicular (bottom panel) auto-correlation function of fractional density fluctuation ρ/ρ − 1 (black solid line) and x-component of upward Elsässer variable z+ x (red dashed line), measured at r/R = 10 Note that the scale of x axis is different

0.8

0.4

0.0

−0.4

density fluctuation upward Alfven wave

−0.8 −4.0

−2.0

0.0

2.0

4.0

l [102 Mm]

perpendicular auto correlation

1.2

0.8

0.4

0.0

−0.4

density fluctuation upward Alfven wave

−0.8 −1.2

−0.6

0.0

0.6

1.2

l⊥ [102 Mm]

4.3.4 Two-Dimensional Slice: rθ-Plane We next discuss the spatial structure in two-dimensional slices. First we focus on r θ-plane to see the radial evolution of turbulence. Figure 4.6 shows a snapshot of variables in a r θ-plane (φ = φmin ) in the quasi-steady state. Shown in each panel is the relative density fluctuation δρhrz /ρ hrz , temperature T , radial velocity vr , upward Elsässer variable of φ component z φ+ and downward Elsässer variable of φ component z φ− , respectively. Here ρ hrz and δρhrz are defined as ρ hrz =

1 L(r )2





xmax

ymax

dx xmin

δρhrz = ρ − ρ hrz .

dyρ,

(4.36)

ymin

(4.37)

66

4 3D Simulations of the Fast Solar Wind

Fig. 4.6 A snapshot of simulation result on r − θ plane. Displayed from left to right are the density fluctuation (δρhrz /ρ hrz ), temperature (T ), radial velocity (vr ) in unit of, φ-component upward Elsässer variable (z φ+ ) and φ-component downward Elsässer variable (z φ− ), respectively

Highly turbulent structures are observed for each variable except temperature. Note that temperature experiences fast spatial diffusion by thermal conduction that smooths out the fine structures. The magnitude of density fluctuation is observed to increase in the wind acceleration region. This trend is consistent with the onset of parametric decay instability discussed in 1D simulations. What is important is that the density fluctuation is horizontally highly inhomogeneous, which triggers phase mixing in the solar wind (Shoda and Yokoyama 2018). An interesting feature of the solar wind turbulence is that z φ− clearly has smaller horizontal scale than z φ+ . A natural interpretation of this feature is that the cascading timescales of z φ± are different; when the cascading of z φ− proceeds faster than z φ+ , the smaller structure is expected to be more apparent for z φ− .

4.3.5 Two-Dimensional Slice: θφ-Plane For more detailed investigation of the scale difference between and outward and inward Alfvén waves and the radial evolution of it, the horizontal profiles of z ± are discussed here. Figures 4.7 and 4.8 show the horizontal structures of Elsässer variables (z θ± , z φ± ) at r = 3R and r = 20R , respectively. The difference in spatial structures of outward and inward Elsässer variables is observed in both figures. Furthermore, the difference is more significant in the inner region. To see the scale difference more quantitatively, we calculate the spectra of inward and outward Elsässer energies E ± with respect to perpendicular wave number,

4.3 Result

67

Fig. 4.7 Spatial structures of Elsässer variables on the horizontal plane at r = 3R . The four panels correspond to z θ+ (left top), z θ− (right top), z φ+ (left bottom) and z φ− (right bottom), respectively, in unit of km s−1 . The corresponding color bar is shown in the right side of panel

defined as ±

E (k⊥ )r =r0

1 = k⊥



 √

k⊥