Plasma Theory: An Advanced Guide for Graduate Students 303144485X, 9783031444852

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Vladimir Rozhansky

Plasma Theory

An Advanced Guide for Graduate Students

Plasma Theory

Vladimir Rozhansky

Plasma Theory An Advanced Guide for Graduate Students

Vladimir Rozhansky Peter the Great St. Petersburg Polytechnic University St. Petersburg, Russia

ISBN 978-3-031-44486-9 ISBN 978-3-031-44485-2 https://doi.org/10.1007/978-3-031-44486-9

(eBook)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Introduction

Lectures on plasma theory are part of a program for students of the Plasma Physics department of Peter the Great St. Petersburg Polytechnic University. This is an advanced course, and students are supposed to be familiar with the basics of plasma physics in the framework of introductory courses such as “Introduction to Plasma Physics” and “Elementary Processes in Plasma.” Lectures on plasma theory are planned for two semesters. In the first semester, students study kinetic theory and transport processes in plasma, while the second semester is devoted to plasma dynamics, including MHD theory, equilibrium, and stability. More advanced problems, such as neoclassical theory, stochastization of the magnetic field lines, and edge plasma physics, are also considered. Waves in plasma are not included in this course and should be studied separately. Only low-frequency waves and instabilities that are closely connected with the dynamics and transport of plasma, such as MHD and drift waves, are considered. The distinctive feature of this course compared to most courses on plasma physics is that phenomena in both low- and high-temperature plasmas are considered simultaneously so that the theories of slightly ionized and fully ionized plasmas are presented. Therefore, this course might be useful for a wide audience of students and specialists working in different areas, such as nuclear fusion, gas discharge physics and low-temperature plasma applications, space, and astrophysics. At the end of the chapters, one can find examples of using theoretical expressions obtained in the chapter in various applications, experiments, and numerical simulations. For further reading, see [1–15]. The author is grateful to Dr. E. Kaveeva and Dr. I. Senichenkov for useful discussions on different problems considered in this book and to I. Rozhanskaia for help in manuscript preparation for publishing.

v

Contents

1

Plasma Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Collision Operator for Coulomb Collisions . . . . . . . . . . . . . . . . 1.2.1 General Expression for a Flow in the Velocity Space Caused by Collisions . . . . . . . . . . . . . . . . . . . . 1.2.2 Deceleration and Diffusion of Test Particles Cloud in the Velocity Space . . . . . . . . . . . . . . . . . . . . 1.2.3 Momentum and Energy Loss of the Test Particles . . . . 1.2.4 Landau Collision Operator . . . . . . . . . . . . . . . . . . . . . 1.3 Relativistic Collision Operator . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Runaway Electrons in Fully Ionized Plasma . . . . . . . . . . . . . . . 1.6 Distribution Function of Electrons in Slightly Ionized Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.8 1.9 1.10 2

6 8 12 14 15 17 20 25

Approximation f 0 , f 1 . . . . . . . . . . . . . . . . . . . . . . . . Distribution Function in the Electric Field . . . . . . . . . . Impact of Electron-Electron Collisions . . . . . . . . . . . . .

26 30 32

1.6.4 General Expression for f 1 . . . . . . . . . . . . . . . . . . . . . Transport Coefficients for Electrons in Slightly Ionized Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Drift Kinetic Equation in a Stationary Electric and Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gyrokinetic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pellet Ablation in a Tokamak . . . . . . . . . . . . . . . . . . . . . . . . . .

32

1.6.1 1.6.2 1.6.3 1.7

→

1 1 6

→

Transport Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Momentum Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Transport Coefficients in Fully Ionized Plasma. Method of Chapman and Enskog . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 35 38 42 47 47 51

vii

viii

Contents

2.3 2.4

2.5 2.6 2.7 2.8 3

4

5

Summary of the Results for the Fully Ionized Plasma . . . . . . . . Transport Coefficients in Fully Ionized Plasma. Qualitative Considerations . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Friction Caused by the Relative Mean Velocity and Thermal Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Spitzer Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Heat Flux: Conductive and Convective Parts . . . . . . . . 2.4.4 Collisional Heat Production . . . . . . . . . . . . . . . . . . . . 2.4.5 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equation for Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscosity in the BGK Approximation . . . . . . . . . . . . . . . . . . . . Thermal Force for Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . First Ionization Potential Effect and Impurity Retention in a Tokamak Edge . . . . . . . . . . . . . . . . . . . . . . . . .

Quasineutral Plasma and Sheath Structure . . . . . . . . . . . . . . . . . . . 3.1 Quasineutrality Maintenance . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Collisionless Sheath at the Material Surfaces . . . . . . . . . . . . . . . 3.2.1 Electrons in a Capacitor with a Reflecting Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Particle and Energy Fluxes to the Material Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Current-Voltage Characteristics of the Sheath. Floating Potential . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Sheath Structure. Bohm Criterion . . . . . . . . . . . . . . . . 3.3 Impact of Electron Emission. Double Sheath . . . . . . . . . . . . . . . 3.4 Sheath in Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Thermoelectric Current Between Two Electrodes . . . . . . . . . . . Diffusion in Partially Ionized Unmagnetized Plasma . . . . . . . . . . . . 4.1 Ambipolar Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Examples of Solutions of the Ambipolar Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Decay of Initial Perturbation in Infinite Plasma . . . . . . . 4.2.2 Positive Column of Glow Discharge . . . . . . . . . . . . . . 4.2.3 Diffusive Decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Diffusive Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Diffusion of Slightly Ionized Multispecies Plasma . . . . . . . . . . . 4.4 Diffusion in the Ionosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . Diffusion of Partially Ionized Magnetized Plasma . . . . . . . . . . . . . . 5.1 Diffusion and Mobility in a Magnetic Field . . . . . . . . . . . . . . . 5.2 One-Dimensional Diffusion in Magnetized Plasma . . . . . . . . . . 5.2.1 Diffusion Across a Magnetic Field . . . . . . . . . . . . . . . 5.2.2 1D Diffusion at an Arbitrary Angle with a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56 59 59 61 62 64 65 66 68 70 71 75 75 78 79 81 83 83 86 89 91 95 95 98 98 99 101 101 105 109 111 111 115 115 116

Contents

5.3 5.4 5.5 5.6 5.7 6

7

ix

Diffusion of Perturbation in Unbounded Plasma . . . . . . . . . . . Diffusion in Plasma Restricted by Dielectric Walls . . . . . . . . . Diffusion in a Cylinder with Conducting Walls . . . . . . . . . . . . Diffusive Probe in Magnetic Field . . . . . . . . . . . . . . . . . . . . . Experiments in Laboratory Plasma . . . . . . . . . . . . . . . . . . . . .

. . . . .

120 126 130 132 134

Partially Ionized Plasma with Current . . . . . . . . . . . . . . . . . . . . . . 6.1 Plasma with Net Current in the Absence of a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Small Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Nonlinear Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Magnetized Plasma with Current . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 One-Dimensional Evolution . . . . . . . . . . . . . . . . . . . . 6.2.2 Multidimensional Evolution of Small Perturbation in Unbounded Plasma . . . . . . . . . . . . . . . 6.2.3 Effect of Conductivity Recovery in a Weak Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Plasma Clouds in the Ionosphere . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Redistribution of Metal Ions in the Polar Ionosphere. Sporadic Layers . . . . . . . . . . . . . . . . . . . . 6.3.2 Active Experiments with Barium Clouds . . . . . . . . . . .

137

Transport in Strongly Ionized Plasma Across a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Classical Diffusion of Fully Ionized Plasma Across a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Transport of Impurities in Fully Ionized Plasma Across a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Partially Ionized Magnetized Plasma with an Inhomogeneous Neutral Component . . . . . . . . . . . . . . . . . . . . . 7.4 Penetration of Neutral Particles into Hot Tokamak Plasma . . . . .

137 139 141 143 143 144 145 147 147 148 149 149 155 157 160

8

Drift Waves and Turbulent Transport . . . . . . . . . . . . . . . . . . . . . . 8.1 Drift Waves in Inhomogeneous Plasma . . . . . . . . . . . . . . . . . . 8.2 Drift-Dissipative Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Universal Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Fluid Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Kinetic Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Instabilities Caused by the Temperature Gradient . . . . . . . . . . . 8.5 Turbulent Transport Caused by Random Electric Fields . . . . . . . 8.6 Effect of Magnetic Shear on Plasma Instabilities . . . . . . . . . . . .

163 163 166 169 169 173 176 178 181

9

Dynamics of Fully Ionized Plasma in the Absence of a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 9.1 Ion Acoustic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 9.2 Nonlinear Dynamics. Self-Similar Solutions . . . . . . . . . . . . . . . 192

x

Contents

9.3 9.4 9.5

Simple Nonlinear Waves. Overturn . . . . . . . . . . . . . . . . . . . . . 195 Nonlinear Ion Acoustic Waves with Dispersion . . . . . . . . . . . . . 197 Plasma Expansion During Pellet Injection . . . . . . . . . . . . . . . . . 201

10

Magnetohydrodynamics (MHD) . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Magnetohydrodynamic Equations . . . . . . . . . . . . . . . . . . . . . . 10.2 Magnetic Field Frozen in and Skin Effect . . . . . . . . . . . . . . . . . 10.3 MHD Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Nonlinear MHD Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Magnetosonic Waves with Dispersion . . . . . . . . . . . . . . . . . . . 10.6 Alfven Masers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205 205 209 212 219 222 226

11

Dynamics of Plasma Blobs and Jets in a Magnetic Field . . . . . . . . . 11.1 Plasma Motion Across Magnetic Field in Vacuum . . . . . . . . . . 11.2 Deceleration of the Plasma Jet by Ambient Plasma . . . . . . . . . . 11.3 Edge Localized Modes and Filaments . . . . . . . . . . . . . . . . . . . .

229 229 234 238

12

Plasma Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 On the Possibility of Equilibrium in the Absence of a Vacuum Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Equilibrium of a Pinch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 Magnetic Flux Surface Functions . . . . . . . . . . . . . . . . . . . . . . . 12.4 Grad-Shafranov Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Integral Equilibrium in a Tokamak . . . . . . . . . . . . . . . . . . . . . . 12.6 Plasma Equilibrium in a Tokamak with Circular Cross-Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.7 Coordinates for Arbitrary Flux Surfaces . . . . . . . . . . . . . . . . . . 12.8 Force-Free Equilibrium and Pinch with Canonical Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9 2D Modeling of the Tokamak Edge . . . . . . . . . . . . . . . . . . . . .

241

13

Transport Phenomena in Tokamaks . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Fluid Regime (Pfirsch-Schlueter Regime) . . . . . . . . . . . . . . . . . 13.1.1 Qualitative Estimates . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.2 Heat Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.3 Plasma Flows on the Flux Surface, Density, Temperature and Potential Perturbations . . . . . 13.1.4 Particle Fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Radial Electric Field, Poloidal and Toroidal Rotation . . . . . . . . . 13.3 Neoclassical Transport in Collisionless Regimes . . . . . . . . . . . . 13.3.1 Particle Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Ware Drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.3 Estimation of Transport Coefficients in the Plateau Regime . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.4 Estimation of Transport Coefficients in the Banana Regime . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Distribution Function in the Collisionless Regimes . . . . . . . . . .

242 243 245 249 252 256 260 263 267 269 271 271 274 276 279 281 285 285 289 291 293 294

Contents

13.4.1 Plateau Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4.2 Banana Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle and Heat Balance Equations . . . . . . . . . . . . . . . . . . . . . Transport Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

294 298 301 304

Instabilities in Magnetized Plasma . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Rayleigh-Taylor Instability in Fluids . . . . . . . . . . . . . . . . . . . . 14.2 Flute Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Dissipative Modifications of Flute Instability . . . . . . . . . . . . . . 14.3.1 RT Instability in Partially Ionized Plasma . . . . . . . . . . 14.3.2 Flute Instability in Plasma Contacting Metal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.3 Gravitational-Dissipative Flute Instability . . . . . . . . . . . 14.4 Energy Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Kink Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 Tearing Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.7 Geodesic Acoustic Mode and Zonal Flows . . . . . . . . . . . . . . . . 14.8 Equatorial Plasma Bubbles . . . . . . . . . . . . . . . . . . . . . . . . . . .

307 307 310 315 315

13.5 13.6 14

15

16

xi

Magnetic Islands and Stochastic Magnetic Field . . . . . . . . . . . . . . . 15.1 Magnetic Islands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 Stochastic Instability and Magnetic Field Line Diffusion . . . . . . 15.3 Transport in Stochastic Magnetic Field . . . . . . . . . . . . . . . . . . . 15.4 Resonant Magnetic Perturbations in Tokamak . . . . . . . . . . . . . . 15.5 Simulation of Resonant Magnetic Perturbations Effects with Codes and Examples of Experimental Results . . . . . . . . . .

316 318 318 321 326 331 333 335 335 339 344 346 350

Improved Confinement Regime (H-Mode) . . . . . . . . . . . . . . . . . . . . 353 16.1 16.2 16.3

→

→

E × B Drift Shear and Transport Barriers . . . . . . . . . . . . . . . . . 353 Transition from Low to High Confinement Regime (L-H Transition) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 L-H Transition Power Threshold . . . . . . . . . . . . . . . . . . . . . . . 358

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

Chapter 1

Plasma Kinetics

1.1

Boltzmann Equation

The plasma state is generally described by a set of distribution functions → → f α r , V , t for all the plasma components and their quantum states. The distribution function can be denoted as a particle density in a six-dimensional coordinate and → → → → velocity phase space, while the quantity dnα r , V , t = f α d r d V is the number of particles in an infinitesimal element of a phase volume. Subscript α here represents different particles, neutral or ionized ones, as well as different quantum states of atoms, molecules, or ions. Classical ideal nonrelativistic plasma is considered below. The variation in the number of particles in the six-dimensional phase space in the absence of collisions is caused by a flow of a “phase liquid” to the neighboring regions of the phase space and a change in the number of particles over time. It is controlled by a six-dimensional continuity equation 6

∂f α ∂ ðf α x_ i Þ = 0: þ ∂t ∂x i i=1 In the phase space, six “coordinates” xi in the Cartesian coordinate system are three space coordinates ri and three components of the velocity Vi, and “velocities” x_ i correspondingly consist of three components of the velocity Vi and three components of the acceleration V_ i . Therefore, the continuity equation in the phase space has the form 3

3

∂f α ∂ ∂ ðf α V i Þ þ f α V_ i = 0: þ ∂t ∂r ∂V i i i=1 i=1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Rozhansky, Plasma Theory, https://doi.org/10.1007/978-3-031-44486-9_1

1

2

1

Plasma Kinetics

The first term on the l.h.s. represents temporal variation of the distribution function, the second term corresponds to the divergence of a flux in the real space, and the third term corresponds to the flux divergence in the velocity space. The acceleration V_ i is produced by forces applied to a particle. In the plasma →_

V =

→ → Zαe → E þ V×B mα

→

þ g, →

→

where Zα is the charge number of a particle, mα is the particle mass, E and B are → electric and magnetic fields, respectively, and mα g is the gravitational force. The space coordinates and velocities are the independent variables; hence, ∂Vi/ ∂ri = 0. Additionally, since the Lorentz force is perpendicular to the velocity of a particle, we have ∂V_ i =∂V i = 0. Therefore, the continuity equation is reduced to the form ∂f α → ∂f α →_ ∂f α þ V → þ V → = 0, ∂t ∂r ∂V

ð1:1Þ

or → → ∂f α → ∂f α Z α e → E þ V×B þV →þ mα ∂t ∂r

∂f α →

∂V

→

þ g

∂f α →

∂V

= 0:

ð1:2Þ

This equation is known as the Vlasov equation. The Vlasov equation can be rewritten in any generalized coordinates qi and momentum pi. The Vlasov equation in the form of Eq. (1.1) in the general case is derived from the continuity equation in the phase space using a relation ∂q_ i =∂qi þ ∂p_ i =∂pi = 0, which follows from the Hamilton equations q_ i = ∂H=∂pi ; p_ i = - ∂H=∂qi . The left-hand side of the Vlasov equation is equal to a full derivative dfα/dt. Therefore, in the stationary case, according to Liouville’s theorem, the distribution function is constant along the phase-space trajectories of the system. An important statement that follows from Liouville’s theorem is that the stationary distribution function in a collisionless case should be a function of integrals of motion. This important notice provides an opportunity to find the distribution function in various collisionless problems. Accounting for collisions changes Eqs. (1.1) and (1.2), since the distribution function is not constant along the trajectories even in the stationary case. In the process of collisions, the velocities of the particles change as well as their quantum states (we shall not consider the change in the particle positions in the process of collisions assuming that the plasma is an ideal gas). In the presence of collisions, the kinetic equation is given by

1.1

Boltzmann Equation

3

→ → df α ∂f α → ∂f α Z α e → Eþ V×B þV →þ  dt mα ∂t ∂r

∂f α →

∂V

→

þ g

∂f α

= St α ,

→

∂V

ð1:3Þ

where a collision operator on the r.h.s. Stα is responsible for the change in the distribution function during collisions. Equation (1.3) is known as a Boltzmann equation. The collision operator is a sum over all species St α = β

St α β f α , f β :

ð1:4Þ

Each summand corresponds to collisions of species α with all species in the plasma, including particles α. We shall consider elastic collisions when the quantum states of particles remain the same in the process of collisions and the energy and the momentum of the particles are conserved. We assume that before the collision, two species α and β → → have velocities V α and V β , while after the collision, they change their velocities to →0

→0

the values V α and V β correspondingly without changing the spatial coordinate. →

After each collision, the particle with the velocity V α escapes from the infinitesimal → volume in the velocity space d V α . The full number of such escapes caused by the collisions of the particles of α species with the particles of species β with →

→

→0

→0

→

V α , V β → V α , V β per second in the infinitesimal velocity space volume d V α for →

a fixed value of V α is given by the expression →

→

→

dQα-β = d r d V α

→

fα V α fβ V β

→

→

→

dσα β =dΩ V α - V β dΩd V β :

Vβ Ω

→

Here, dσαβ/dΩ is a differential cross-section of scattering to the solid angle Ω. In → addition to losses in the infinitesimal velocity space volume d V α , there is also a →0

→0

→

→

source caused by the collisions V α , V β → V α , V β that transfer particles with →0

→0

→

velocities V α , V β to the velocity space volume d V α : →0

→

dQþ αβ = d r

→0

fα V α fβ V β →0 Vβ

→0

→0 Vα

→0

→0

→0

→0

dσ α β =dΩ V α - V β dΩd V α d V β :

Ω →0

Velocities V α and V β under the integral are not independent but connected by the conservation laws. Indeed, in the process of collision of the particles with velocities →0

→0

→

V α , V β , the particle of species α obtains the velocity V α . Let us change variables in →

→

the integral and integrate over the velocities V α , V β using the conservation of the

4

1 Plasma Kinetics →

→0

→

→0

relative velocity V α - V β = V α - V β . As is known from classical mechanics, →0

→0

→

→

the Jacobian of this transformation is equal to unity: d V α d V β = d V α d V β . Finally, →

→

since integration over d V α is carried out in the vicinity of a chosen value V α , one obtains →

→0

→

dQþ αβ = d r dV α

→0

fα V α fβ V β

→

→

→

dσα β =dΩ V α - V β dΩd V β :

Vβ Ω

→

→

→

Combining sources and sinks in the volume d r d V α , we find →

→

dQþ α β - dQα β = St αβ d r d V α ,

where the collision operator is f 0α f 0β - f α f β

St αβ f α, f β = →

Vβ

→0

→

→

→

dσαβ =dΩ V α - V β dΩd V β :

ð1:5Þ

Ω →0

Here, f 0α  f 0α V α , f 0β  f 0β V β . Equation (1.5) is known as the Boltzmann collision operator. Boltzmann kinetic equation (1.3) is an integro-differential equation that contains all distribution functions of the particles in the system. Therefore, a system of coupled equations for all distribution functions is to be solved. In the process of deriving the collision operator, we assumed that during the collision, the particle coordinate remains unchanged. This is justified provided that the potential energy is significantly smaller than their average kinetic energy, which is of the order of their temperature T. For the charged particles, the average potential energy of the Coulomb interaction is ZαZβe2/4πε0hrαβi, and the average distance between the charged particles hrαβi is of the order of n-1/3, where n is the plasma density. Therefore, the criterion of ideal plasma used in the derivation has the form Z α Z β e2 n1=3 =ð4πε0 TÞ V , we have u ≈ - V , and, therefore, -

∂ ∂V j

U jk

→0 fα ∂ V k f 0β d V = Tβ ∂V j

V 0j V 0k f α →0 δjk V k f 0β d V : 0 3 0 V ðV Þ T β

In the second term, the contribution from components with j ≠ k turns to zero (in the →r reference frame where the mean velocity u is absent); therefore, the remaining integral is given by ∂ ∂V j

2

0 2 0 f α f 0β ðV Þ - V j V Tβ j ðV 0 Þ3

1=2

→0 → 4 mβ ∂ dV = - p n fαV : 3=2 β → 3 2π T β ∂V

The second term in the Landau collision integral Eq. (1.36) is simplified analogously. Finally, the kinetic equation with the simplified collision operator has the form df α ∂ = ναβ → dt ∂V

→

V fα þ

T β ∂f α , mα ∂ → V

ð1:46Þ

where a collision frequency ναβ is defined as p ναβ =

1=2

2mβ ΛZ 2α Z 2β e4 nβ 3=2

12π3=2 ε20 mα T β

:

ð1:47Þ

Equation (1.46) is known as the Fokker-Planck equation. In homogeneous plasmas in the absence of external forces, this equation describes the process of relaxation of a distribution function fα to the Maxwellian distribution function with the temperature Tβ fM α = nα

mα 2πT β

3=2

exp -

mα V 2 2T β

-1 with the characteristic time scale τ = ναβ . Indeed, this Maxwellian distribution M function f α turns to zero the r.h.s. of Eq. (1.46). The Fokker-Planck equation is a linear equation and can be solved analytically. Let us demonstrate how the Fokker-Planck equation can be used to calculate the mobility of impurities – the coefficient that connects the applied electric field and

1.4

Fokker-Planck Equation

19

mean (fluid) velocity of impurities. In the stationary homogeneous plasma in the electric field, the Fokker-Planck equation has the form →

Z α e E ∂f α ∂ = ναβ → mα ∂ → V ∂V

→

V fα þ

T β ∂f α → : mα ∂ V

ð1:48Þ

→

Let us multiply Eq. (1.48) by V and integrate over velocities. After integrating by part, one obtains that the first term on the r.h.s. is proportional to the particle flux → nα u α , while the second term turns to zero. Integral on the l.h.s. is equal → to - Z α nα e E =mα . Finally, →

→

u α = bα E , b α =

Zαe : mα ναβ

ð1:49Þ

Thus, the mobility is given by Eq. (1.49) with the numerical coefficient equal to unity. The distribution function can also be obtained from Eq. (1.48). In relatively small electric fields when the fluid velocity of impurities is much smaller than the thermal velocity, the solution can be sought in the form of a sum of the Maxwellian distribution plus small correction: f = f M + f 1. In the linear approximation, the distribution function on the l.h.s. of Eq. (1.48) can be taken as the Maxwellian one 1 fα =fM α since the electric field is small, while on the r.h.s. f α = f α can be kept since the Maxwellian distribution function turns to zero the r.h.s. Then, it is easy to obtain f 1α =

mα Tβ

→ → u αV

ð1:50Þ

fM α ,

where the mean velocity is given by Eq. (1.49). In other words, the distribution function is given by expansion of a shifted Maxwellian distribution

f α = nα ≈ nα

mα 2πT β mα 2πT β

→

3=2

exp

-

3=2

exp -

→

mα V - u α

2

2T β mα V 2 2T β

1þ

ð1:51Þ mα → → u V : Tβ α

It is, however, worthwhile to note that in the general case, the distribution function in the electric field does have such a simple form.

20

1.5

1

Plasma Kinetics

Runaway Electrons in Fully Ionized Plasma

Let us analyze distribution function of electrons in the homogeneous plasma without a magnetic field in the weak electric field. E Ec, electrons with velocities V > V0 = c(Ec/E)1/2 are accelerated since their friction force is smaller than the electric field. Since the Coulomb cross-section is inversely proportional to the cube of the scattering angle and, therefore, inversely proportional to the cube of the transferred velocity, such electrons are born as a result 4 of scattering of fast electrons with frequency νe = 4πεe2 mn2 V 3 . The number of such 0

e

1.6

Distribution Function of Electrons in Slightly Ionized Plasma

25

electrons born per second by an order of magnitude can be estimated as ( fr is a 1D distribution function of fast electrons that is independent of the velocity) c

νe f r dV = V0

fr e4 n c2 1 : m2e 8πε20 c2 V 20

Assuming that the distribution function of fast electrons is constant, so that their density is nr = cfr, one obtains an estimate for the number of fast electrons born per time unit n_ r = nr νe ðcÞðE=Ec - 1Þ=ð2ΛÞ:

ð1:65Þ

Here, the factor (E/Ec - 1) demonstrates that the avalanche effect has a threshold character.

1.6

Distribution Function of Electrons in Slightly Ionized Plasma

As in the previous section, we shall consider the electrons to be the test particles. Then, the collision operator could be linearized, and the kinetic equation could be significantly simplified. Slightly ionized plasma is defined as the plasma where the following inequality is fulfilled: νee νii is assumed. The friction force consists of two parts →

→

→

→u

→T

R ei  R = R

þR :

ð2:34Þ

The first part of the friction force is connected with the relative velocity of electrons and ions: →

→u

R

→

→

→

= - nme νei 0:51 u k þ u ⊥ ,

→

→

u = u e - u i:

ð2:35Þ

The electron-ion collision frequency in accordance with Eq. (2.22) p νei =

2

ne4 Λ

3=2 12π3=2 ε20 m1=2 e Te

ð2:36Þ

,

where the Coulomb logarithm is given by Λ = 23:4 - 1:15lgn þ 3:45lgT e , T e < 50eV; : Λ = 25:3 - 1:15lgn þ 2:3lgT e , T e > 50eV: The density in the expression for the Coulomb logarithm should be inserted in CGS units, while the temperature is in eV. The second part is known as the thermal force and depends on the electron temperature gradient: →T

R = - 0:71n∇k T e -

3 nνei 2 ωce

→

B × ∇T e B

:

ð2:37Þ

2.3

Summary of the Results for the Fully Ionized Plasma

57

Ion heat flux →

q i = - κik ∇k T i - κi⊥ ∇⊥ T i þ

→

5 nT i 2 eB

B × ∇T i B

ð2:38Þ

,

where the heat conductivities along and across the magnetic field are: κik =

3:9nT i 2nT i νii , κi⊥ = , mi νii mi ω2ci

ð2:39Þ

with the ion-ion collision frequency defined as νii =

ne4 Λ 1 : 2 3=2 1=2 12π ε0 mi T 3=2 i

ð2:40Þ

The electron heat flux is a sum of two contributions: →

→ T

→

→u

qe= qe þ qe :

ð2:41Þ

The flux caused by heat conductivity has the form →T qe

5 nT e = - κek ∇k T e - κe⊥ ∇⊥ T e 2 eB

→

B × ∇T e B

,

ð2:42Þ

where κek =

3:16nT e 2nT e νei , κe⊥ = : me νei me ω2ce

ð2:43Þ

The second part of the electron heat flux is caused by the relative velocities of electrons and ions: →

→u qe

→

B → 3 nT e = 0:71nT e u k ν ×u 2 ωce ei B →

:

ð2:44Þ

Sources and sinks in the heat balance equations correspond to heat transfer between electrons and ions and Joule heating. The latter exists only in the electron equation: Qi = Q Δ =

3me nν ðT - T i Þ, mi ei e

→→

Qe = - R u - QΔ :

ð2:45Þ

58

2 Transport Equations

The viscosity tensor consists of two parts: $

→

$u

π=π

→

$q

þπ :

ð2:46Þ

The first part depends on the mean velocities [9], and the second part depends on the heat fluxes [10]. In the absence of a magnetic field, the first part is: →

$u π jk

= - η0 W jk = - η0 →

∂uj ∂uk 2 → þ - δjk ∇  u , 3 ∂xk ∂xj

ð2:47Þ

→

In the strong magnetic field ( z k B ) →

πzzu = - η0 W zz , → η η πxxu = - 0 W xx þ W yy - 1 W xx - W yy - η3 W xy , 2 2 → η0 η1 u πyy = W xx þ W yy W yy - W xx þ η3 W xy , 2 2 → → η πxyu = πyxu = - η1 W xy þ 3 W xx - W yy , 2 → → πxzu = πzxu = - η2 W xz - η4 W yz , →

ð2:48Þ

→

πyzu = πxyu = - η2 W yz þ η4 W xz : Ion viscosity coefficients are ηi0 = 0:96nT i =νii , nT ν ηi1 = 0:3 i2 ii , ηi2 = 4ηi1 , ωci nT i ηi3 = , ηi4 = 2ηi3 : 2ωci

ð2:49Þ

For electrons, the viscosity is usually small with respect to the ion viscosity. The → second part becomes important when the conductive heat flux q i is of the order of → the convective heat flux nT i u i , and the corresponding expressions can be found in [10]. Viscosity heat production is mostly determined by the main ions, and its main part is Qvis = - πjk

∂V j η0 ∂V x ∂V y ∂V z = þ -2 3 ∂x ∂xk ∂y ∂z

2

:

ð2:50Þ

Numerical coefficients for Z > 1 and intermediate values of the magnetic field can be found in [9, 10].

2.4

Transport Coefficients in Fully Ionized Plasma. Qualitative Considerations

2.4

Transport Coefficients in Fully Ionized Plasma. Qualitative Considerations

59

Let us consider qualitative mechanisms responsible for transport coefficients.

2.4.1

Friction Caused by the Relative Mean Velocity and Thermal Force →

→u

An estimate for the part of the friction force associated with the relative velocity R can be easily obtained as a change of the momentum caused by relative mean velocity of the electrons with respect to the ions multiplied by electron mass → → me u e - u i and by particle density n during the time between collisions νei- 1 . For the motion across the magnetic field, the electron distribution function is close to → the Maxwellian one shifted by u since the rotation over the Larmor radius reduces the anisotropy of the distribution function. Corrections contain a small factor νei/ωce. Hence, the perpendicular part of the friction force coincides with that obtained in the quasihydrodynamic approximation given by Eq. (2.21). In the parallel direction, due to distortion of the distribution function, the numerical coefficient and friction force are almost twice as small as those in the perpendicular direction. The second part of the friction force, which is proportional to the gradient of the electron temperature, is known as the thermal force. This force is connected with the velocity dependence of the electron-ion collision frequency for Coulomb collisions. Let us consider electrons in the absence of mean velocity that collide with ions at rest in the plasma without a magnetic field. Electron chaotic fluxes from the mean free path distance through the unit area situated at z = z0, Fig. 2.1, have different directions and are given by an estimate Γ+~Γ-~nVT with VT being the thermal velocity. Due to the momentum loss of electrons during collisions, two forces proportional to the corresponding fluxes arise R+~R-~menVTνei, which are applied to the ions. In the plasma with a homogeneous electron temperature, these forces balance each other so that the net force is zero. However, in the presence of a temperature gradient, electrons that cross the area come from regions with different electron temperatures. The temperature difference can be estimated as δTe = (dTe/ dz)λe, where λe is the mean free path. Hence, R+ ≠ R- and unbalanced force arises. If, for example, the electron temperature increases from left to right, then since the Coulomb collision frequency decreases with increasing velocity, the force R+ is smaller than R-. As a result, we have an unbalanced force extracting ions into the hot region. This force can be estimated as

60

2

Transport Equations

Fig. 2.1

Fig. 2.2

RT  me

dT dνei δT nV  n e : dT e e T dz

ð2:51Þ

According to Newton’s law, the same force with the opposite sign is applied to electrons. Across the magnetic field, electrons rotate over the Larmor circle, and, therefore, an unbalanced force arises in the direction perpendicular to the temperature gradient (Fig. 2.2). Since electrons bring their momentum from a distance of the order of ρce, the temperature difference should be estimated as δTe = (dTe/dz)ρce. As in the absence of the magnetic field, Eq. (2.51), the force applied to ions is →

B ν × ∇T e , R  n ei ωce B T

while the same force with the opposite sign is applied to electrons.

ð2:52Þ

2.4

Transport Coefficients in Fully Ionized Plasma. Qualitative Considerations

2.4.2

61

Spitzer Conductivity

In homogeneous plasma in the absence of a magnetic field, the electric force is balanced by electron-ion friction. As a result, the current is flowing, and the density is proportional to the electric field: →

→

j = σk E :

ð2:53Þ

Here, the Spitzer conductivity is defined as σk =

ne2 : 0:51me νei

ð2:54Þ

Since the Coulomb collision frequency is proportional to the electron density, the Spitzer conductivity is independent of the density and is determined by the electron temperature, σk  T 3=2 e . It is also worthwhile to note that in the fully ionized plasma, mobility is not defined (in contrast to the slightly ionized plasma); it is possible to obtain only the relative velocity of electrons with respect to the ions from Eq. (2.53) but not the velocities itself. This fact is connected with the absence of the special reference frame, which in slightly ionized plasma is bound to neutral gas. In the literature, a quantity σ⊥ = ne2/(meνei) known as perpendicular conductivity is often used, which, however, does not determine the real physical conductivity across the magnetic field. Indeed, a homogeneous electric field applied across the magnetic field could be turned to zero by the Lorentz transformation to the reference → → frame moving with E × B =B2 velocity. Hence, the homogeneous electric field in the fully ionized plasma does not cause current in the perpendicular direction. In this sense, the perpendicular conductivity in fully ionized plasma is zero, as is the mobility in the direction of the electric field. In the plasma with Z > 1, a numerical coefficient in the expression for friction force in Eq. (2.35) differs from 0.51. Therefore, the expression for Spitzer conductivity has the following form: σk =

ne e 2 u с me νei

ð2:55Þ

with p νei =

2

ni Z 2 e 4 Λ

3=2 12π3=2 ε20 m1=2 e Te

:

ð2:56Þ

62

2 Transport Equations

The numerical coefficient is [9]: сu = 0:51 for Z = 1, сu = 0:44 for Z = 2, сu = 0:4 for Z = 3, сu = 0:38 for Z = 4, сu = 0:3 for Z = 1: If plasma is a mixture of different species, it is convenient to introduce Zeff according to

Z eff =

j

nj Z 2j nj Z j

=

j

nj Z 2j ne

:

ð2:57Þ

j

Since the friction force is a sum of friction forces due to collisions of electrons with different species, assuming that all ions have velocities smaller than the electron velocity so that the current is determined by the electron velocity, the Spitzer conductivity can be rewritten as σk =

e 2 ne f Z eff , me νei Z eff

ð2:58Þ

where the collision frequency of electrons is given by Eq. (2.36), and a function f is approximated according to [10] f Z eff =

4:95Z eff : 1 þ 1:54Z eff

ð2:59Þ

For Zeff = 1, expression (2.54) is recovered. Note that in the literature, a simpler approximation for Spritzer conductivity with f = 1 is often used, which differs significantly from the correct approximation given by Eq. (2.59).

2.4.3

Heat Flux: Conductive and Convective Parts

Similar to the friction force, the heat flux is a sum of two contributions. The first conductive part is a linear function of the temperature gradient and is caused by heat conductivity. It has the form →T qα

= - κα ∇T α = -

3 nχ ∇T α : 2 α

ð2:60Þ

Here, κα is the heat conductivity tensor, and χα is the heat diffusivity tensor. An estimate for the tensor components along and across the magnetic field is obtained similar to the case of slightly ionized plasma. The diagonal components of the heat

2.4

Transport Coefficients in Fully Ionized Plasma. Qualitative Considerations

63

diffusivity tensor are estimated as the square of the random walk step multiplied by the collision frequency. Along the magnetic field (in the absence of it) χek  λ2e νei ,

χik  λ2i νii :

ð2:61Þ

Since the electron and ion mean-free paths for comparable temperatures are of the same order, we have χek =χik  mi =me , i.e., the parallel electron heat diffusivity is much larger than the ion heat diffusivity. Across the strong magnetic field, particles are shifted at a distance of the order of Larmor radius, and, therefore, χe⊥  ρ2ce νei ,

χi⊥  ρ2ci νii :

ð2:62Þ

The perpendicular heat diffusivity coefficients for the comparable temperatures have an inverse relation in contrast to the parallel ones -χi⊥ =χe⊥  mi =me . The estimate for the heat fluxes along the temperature gradients can be easily obtained as the difference in the heat fluxes brought from the mean free path or Larmor radius distance. Along the magnetic field qαk  qαkþ - qαk -  nV T δT α  - nV T λα

dT α , dz

while across the field qα⊥  qα⊥þ - qα⊥ -  nρcα ναα δT α  - nρ2cα ναα

dT α : dx

In the magnetized plasma, there are also off-diagonal Hall heat fluxes that correspond to off-diagonal components of the heat conductivity tensor. The corresponding flux is shown in Fig. 2.2 for electrons. Along the у-axis arises an unbalanced heat flux through the unit area. A one-sided flux is given by an estimate nVTT. Its unbalanced part is of the order of nVTρcdT/dx. As a result, for electrons →T q eΛ

 -

nT e → B × ∇T e : eB2

ð2:63Þ

A similar Hall heat flux for ions has the opposite sign. The flux equation (2.63) is directly perpendicular to the temperature gradient along isotherms, and, therefore, for a constant density, its divergence is equal to zero, i.e., this flux does not lead to a change in the temperature. In the inhomogeneous magnetic field, the divergence of the Hall flux equation (2.63) is not zero. Physically, its divergence coincides with the divergence of the convective flux associated with the drift of the particles in the inhomogeneous magnetic field.

64

2

Transport Equations

Fig. 2.3

For electrons, in addition to the conductive flux, there is also a convective flux →

→u

associated with the relative velocity, q e in Eq. (2.41). Similar to the thermal force, its origin is connected with the velocity dependence of the electron collision frequency. Distortion of the electron distribution function with respect to the shifted Maxwellian function in the absence of a magnetic field leads to a larger contribution →

→

→u

→

of fast electrons to the mean electron velocity u . As a result, heat flux q e  nT e u arises. The mechanism for the similar convective heat flux across the magnetic field is illustrated in Fig. 2.3. The friction force associated with the electron velocity decelerates electrons rotating over the Larmor radius and crossing the area y = 0 in the positive у-direction and accelerates those moving in the negative у-direction at y = 0. The work acting on electrons meνeiuxρce results in the fact that the energy of electrons crossing the area in the downwards direction is smaller than the energy of electrons moving upwards. Multiplying the difference in energies meνeiuxρce by the particle flux nVT, one obtains an estimate for the heat flux in the у-direction →

→u q eΛ

2.4.4

→

B → nT  e νei ×u : ωce B

ð2:64Þ

Collisional Heat Production

During the collision of the light particle with the heavy particle, the fraction ~me/mi of the energy is transported from one particle to another one. Estimating the energy transported from electrons to ions as (3/2)(me/mi)n(Te - Ti) during the time scale νei- 1 , one obtains the heat exchange rate between species

2.4

Transport Coefficients in Fully Ionized Plasma. Qualitative Considerations

65

me nν ðT - T i Þ: mi ei e

QΔ 

This term enters the heat balance equations for electrons and ions with different signs. →→ The term - R u enters only the heat balance equation for electrons and consists →

→u →

of two parts. The summand - R can be rewritten in the form

u represents the Joule heating of electrons and

→

→u →

-R

u=

→T →

The second contribution - R u corresponding heating is reversible.

2.4.5

j2k j2 þ ⊥: σk σ⊥

ð2:65Þ

could be positive or negative, and the

Viscosity

Viscosity is associated with the transport of momentum. Let us consider plasma without a magnetic field in the absence of temperature gradients. Consider, for example, the case when the mean velocity in the y-direction uy is changing in the x-direction. Similar to the heat flux, the difference in one-sided momentum fluxes mnVTuy from the mean free path distance is estimated as πxy  πþ xy - πxy  mnV T δuy  - mnV T λ

duy : dx

After introducing the viscosity coefficient, we have πxy = - η0

duy nT , η0  : dx ν

ð2:66Þ

For ions, the ion-ion collision frequency should be substituted in Eq. (2.66); therefore, ηi0  nmi χik :

ð2:67Þ

Substituting Eq. (2.66) into the momentum balance equation for ions, Eq. (2.4) yields 2

∂ uy ∂uy =D 2 : ∂t ∂x

66

2

Transport Equations

This is the diffusion equation with the diffusion coefficient D = Ti/(miνii). Thus, the viscosity represents the diffusion of momentum in the direction of the velocity gradient. Analogous transport of other velocity components takes place. In the general case, the viscosity tensor is a linear function of the rate-off-strain tensor given by Eq. (2.46): πjk = - η0Wjk. Tensor Wjk is a symmetrical tensor with a zero → → → trace Wjj = 0. This tensor vanishes when plasma rotates as a whole u = ω × r or →

→

stretches as a whole u  r . Viscosity components that are proportional to the heat → → flux have the same order of magnitude if q α  nT α u α . In magnetized plasma, the viscosity tensor has a more complicated character. Components of the viscosity tensor that contain coefficients η1 and η2 correspond to the reduction of the random walk step across the magnetic field to the Larmor radius value. Components with coefficients η3 and η4 are known as gyroviscosity terms; they are independent of the collision frequency and are inversely proportional to the gyrofrequency. Gyroviscosity terms correspond to momentum flow in the Hall direction analogously to the terms, Eq. (2.63), in the heat flow. $ Note also that viscous forces ∇  π are small with respect to the pressure gradient ∇p by factors λ/Lk and ρc/L⊥ along and across the magnetic field, respectively.

2.5

Equation for Entropy

Entropy per single particle, for example, for electrons, is given by se =

3 ln T e - ln n þ const: 2

ð2:68Þ

The equation for the entropy of electrons can be evaluated by combining the particle balance equation with the heat balance equation. It has the form →

q ∂nse → þ ∇  se n u e þ e Te ∂t

þ

QΔ = θe : Te

ð2:69Þ

T e θe = - q e ∇ ln T e - R u -

1 π W : 2 ejk ejk

ð2:70Þ

Here, entropy production per unit volume is →

→→

The l.h.s. of this equation describes entropy variation in time, entropy transport in space, and entropy transfer between electrons and ions. Substituting the expressions for friction, heat flux, and viscosity from Sect. 2.3, one obtains the entropy production in the form

2.5

Equation for Entropy

T e θe =

67

κek ∇k T e Te

2

þ

j2k j2 κe⊥ 1 ð∇⊥ T e Þ2 þ þ ⊥ þ ηe0 W ejk W ejk : Te σk σ⊥ 2

ð2:71Þ

Therefore, in this form, it is clear that the entropy production is essentially positive. For ions, analogously, →

q Q ∂nsi → þ ∇  s i n u i þ i - Δ = θi , Ti Ti ∂t

T i θi =

κik ∇k T i Ti

2

þ

κi⊥ 1 ð∇⊥ T i Þ2 þ ηi0 W ijk W ijk : 2 Ti

ð2:72Þ

ð2:73Þ

The equation for the total entropy sn = sen + sin is of the form →

→

q q ∂ns → → þ ∇  se n u e þ si n u i þ e þ i = θe þ θi þ θei , Te Ti ∂t 2

θei = QΔ

ðT e - T i Þ 1 1 3me nν : = mi ei T e T i Ti Te

ð2:74Þ

If the time scales for heat transport, heat exchange, and heat production are large in comparison with the density and temperature variation time scales, so dissipative processes can be neglected, the process can be treated as adiabatic. In this situation, the equation for electron or ion entropy reads ∂nsα → þ ∇  nsα u α = 0: ∂t

ð2:75Þ

This equation has the same form as the particle continuity equation, and, therefore, quantities nsα and n change identically so that their ratio (entropy per particle) is conserved. Combining Eq. (2.75) with the particle balance equation, one obtains ∂sα þ ∂t

→

u α ∇ sα = 0 :

ð2:76Þ

Therefore, entropy per particle is conserved, and, hence, conserved is the quantity T 3=2 α =n = const :

ð2:77Þ

68

2

Transport Equations

The thermodynamic “fluxes” and “forces” discussed in Sect. 2.1 and, connected by Eq. (2.13), are called conjugate if j m xm :

θ= m

→

→ $

According to Eqs. (2.71), (2.73), and (2.74), “fluxes” q α , R , π αjk , QΔ, and “forces” → ∇Tα/Tα, u , 1/2Wαjk, (Te - Ti)/(TeTi) are the conjugate ones. For transport coefficients Lmn, which connect “fluxes” and “forces” according to Eq. (2.13), the following relation is fulfilled: →

→

Lmn B = Lnm - B

:

This relation represents the general principle of Onsager symmetry of transport coefficients.

2.6

Viscosity in the BGK Approximation

The viscosity tensor can be derived in a simplified form using the so-called BGK (Bhatnagar, Gross, and Krook) form of the collision operator ð2:78Þ

St ðf α Þ = - να f α - f 0α ,

where f 0α is the Maxwellian distribution function and ν is some collision frequency representing relaxation to the Maxwellian distribution function. Let us take the → → → → V - uα V - u α moment of the kinetic equation to obtain j

k

∂M 0jk ∂uαj 0 ∂u ∂ V l M 0jk þ M 0ljk þ M þ αk M 0lj þ ∂t ∂xl ∂xl lk ∂xl eBl 0 0 ε M þ εkml M mj = - να παjk mα jml mk

ð2:79Þ

Here, moments are defined as M 0αj,k...n =

→ →

*

V j - uαj ðV k - uαk Þ . . . ðV n - uαn Þf α r , V , t dV ,

and εjml is the Levi–Civita completely antisymmetric unit tensor, εjml = 1 for an even permutation of the numbers 123, εjml = - 1 for odd permutation, and εjml = 0 if any two indices coincide.

2.6

Viscosity in the BGK Approximation

69

As discussed in Sect. 2.2 in the fluid approximation, there are two “large” terms where the deviation from the Maxwellian distribution function is important-term with the magnetic field and the collision operator. In the other “small” terms, the distribution function can be considered as the Maxwellian one. After substituting the Maxwellian distribution function, we find ∂uαj ∂uαk ∂pα ∂ ðuαl pα Þ δjk þ p þ þ ∂t ∂xl ∂xk ∂xj eBl = ε M 0 þ εkml M 0mj : mα jml mk

þ να παjk

ð2:80Þ

The l.h.s. of this equation can be simplified using Eq. (2.11) in the zero approximation, where dissipative terms are neglected: ∂pα ∂ ∂ 2 ðuαl pα Þ = - pα u þ 3 ∂xl αl ∂t ∂xl The r.h.s. of Eq. (2.80) in the matrix form with the z-axis along the magnetic field is 2παxy

παyy - παxx

παyz

παyy - παxx παyz

- 2παxy - παxz

- παxz 0

2παxy

παyy - παxx

παyz

παyy - παxx

- 2παxy

- παxz

παyz

- παxz

0

eB eBl ε M 0 þ εkml M 0mj = mα mα jml mk Finally, Eq. (2.80) can be simplified to →

$

pα W α þ να π α eB = mα

:

ð2:81Þ

Resolving this algebraic system leads to the viscosity tensor equation (2.48) with the simplified form of viscosity coefficients (for a strong magnetic field) ηα0 = nT α =να , nT ν ηi1 = 0:25 α2 α , ηα2 = 4ηα1 , ωcα nT α , ηα4 = 2ηα3 : ηi3 = 2ωcα Viscosity coefficients ηi3 and ηα4 coincide with those obtained by Braginsky, Eq. (2.49), other coefficients are very similar.

70

2.7

2

Transport Equations

Thermal Force for Impurities

In the plasma with impurities of charge Z and species I, there are thermal forces due to collisions with the electrons and the main ions. In the absence of a magnetic field, the thermal force on impurity ions from electrons can be estimated in the following way. Let us consider the case of test impurities with nI Z 2 > mi can be treated analogously (for the main ions, we assume Z = 1) RTiI 

∂ ∂T ðni mi νiI V Ti Þλii  nI Z 2 i : ∂z ∂z

ð2:85Þ

This part of the thermal force is directed towards the regions of higher temperature of the main ions.

2.8

First Ionization Potential Effect and Impurity Retention in a Tokamak Edge

71

The resulting thermal force on impurities is RTI = αnI Z 2

∂T e ∂T þ βnI Z 2 i , ∂z ∂z

ð2:86Þ

where α and β are numerical coefficients calculated from kinetic theory.

2.8

First Ionization Potential Effect and Impurity Retention in a Tokamak Edge

In recent years, understanding the mechanism of impurity retention/leakage has become a more pressing issue, as impurity seeding into the edge plasma has been found to be a reliable way to obtain divertor detachment. This allows for reducing the target heat loads down to an acceptable level through impurity radiation, and attaining such a regime with impurity seeding and radiative divertor operation is considered to be essential for the operation of future thermonuclear devices. To obtain this highly radiated regime, gas (Ne, Ar, N, and others) is seeded into the edge plasma, and after ionization, it should stay mainly in the divertor region. On the other hand, the thermal force, Eqs. (2.85) and (2.86), extracts impurities upwards towards the separatrix and further into the main plasma, where the radiation of impurities might lead to unacceptable energy loss in the main plasma. To understand in detail the process of impurity retention/leakage in the divertor, one should analyze in detail the parallel momentum balance for ionized impurities and the behavior of both neutral and ionized components at the edge plasma. Edge tokamak plasma is described by 2D modeling, where equations for ionized components are solved on the basis of transport equations discussed in this chapter and neutral particles are followed by the Monte-Carlo kinetic code. The typical simulation region and computational mesh are shown in Fig. 2.4. Here, x is the poloidal coordinate, the full magnetic field has a projection on it, and y is directed perpendicular to the flux surfaces. The flows of the main deuterium ions in the divertor region are governed by the recycling process. Neutral particles are reflected from divertor plates and move into the divertor plasma. After ionization, most of the ions return back to the divertor plates along magnetic field lines, and, hence, along x, driven by the pressure gradient (Fig. 2.5). Their poloidal velocity is of the order of a poloidal sound speed 1=2 bx cs = ðBx =BÞðT e þ T i Þ1=2 =mi ; for further details, see Chap. 9. Part of the deuterium ions that ionize at larger distance from the plates move in the opposite direction towards upstream also driven by the pressure gradient. Ions that move towards the plate recombine at the plate and return back as neutrals.

72

2

Transport Equations

Fig. 2.4 Computational meshes for plasma (violet quadrangular cells) and neutrals (dark-yellow triangular cells) used in the simulations. Shown are the in-vessel structures and locations of gas puff/seeding and pumping

To understand impurity behavior, let us consider the stationary impurity force balance in the direction of the magnetic field →

→

∇ mI nI u I ukI = - ∇k ðnI T I Þ - ZenI ∇k φ þ RIku þ RTIk :

ð2:87Þ

→

Here, a part of the friction force RIu is caused mainly by ion-impurity collisions due → to different velocities, RIu = mi ni νiI uki - ukI , and thermal force RTI is given by Eq. (2.86). Parallel derivatives here can be expressed through poloidal derivatives as |∇k| = bx∂/∂x. Since collisions between impurities are sufficiently strong (the corresponding criterion is νiI >> bxcs/L with L being the poloidal spatial scale), the main contribution to the parallel momentum balance equation (2.86) is produced by the last two terms, so that →

RIku þ RTIk ≈ 0:

ð2:88Þ

In other words, the friction force for impurity ions is close to zero. This equation determines the parallel (poloidal) velocity of impurities: (uIk = bxuIx)

2.8

First Ionization Potential Effect and Impurity Retention in a Tokamak Edge

73

Fig. 2.5 Illustration of leakage/retention mechanism. Zones of maximal ionization are shown schematically by ellipses for deuterium (green ellipses), nitrogen (blue ellipses), and neon (red ellipses). Dotted lines schematically represent the stagnation points of poloidal flow (the difference in their location between species exists in reality, but is not shown here). Solid circles show points of ionization and arrows of corresponding color represent the poloidal flow directions

uIx = uix þ αnI Z 2 bx

∂T e ∂T i þ βnI Z 2 bx =ðbx mi ni νiI Þ: ∂x ∂x

ð2:89Þ

One can see that impurity ions are strongly coupled to the main ions. The thermal force changes their velocity with respect to the main ions and is important in the regions with a strong temperature gradient, i.e., in the region where ionization takes place. Since temperatures rise from the plate towards upstream, thermal force extracts impurities away from the divertor. Now, one can understand the first ionization potential (FIP) effect. Impurities as neutrals are reflected at the plates and move away from the plates. Impurity with a first ionization potential smaller than that of deuterium (N, e.g.) is mainly ionized below the stagnation point of the main ion poloidal flow (closer to the plates than the deuterium ions). Therefore, after ionization, they are dragged to the plates by the main ions. Only a small fraction of them is ionized above the stagnation point and is dragged away upstream (Fig. 2.5). Impurities with ionization potentials higher than that of deuterium (Ne, for example) are mainly ionized above the stagnation point

74

2

Transport Equations

Fig. 2.6 Nitrogen and neon net density distribution in a tokamak. Modeling by the SOLPS-ITER transport code

and are dragged upstream by the main ion parallel flow. As a result, elements with low first ionization potential are better retained in the divertor region than those with high ionization potential. Thermal force can only shift the stagnation boundary for impurity poloidal flow with respect to that for deuterium. Examples of simulations of the impurity density distribution are shown in Fig. 2.6. N with a lower first ionization potential is indeed better retained in the divertor region than Ne.

Chapter 3

Quasineutral Plasma and Sheath Structure

3.1

Quasineutrality Maintenance

Plasma is a quasineutral medium where the densities of the charged particles multiplied by their charge for positive and negative charges coincide with high accuracy. The charge separation in plasma can be estimated from the Poisson equation, which for Z = 1 reads →

∇
E =

e ðn - ne Þ: ε0 i

ð3:1Þ

Indeed, estimating the potential in plasma as T/e, from the Poisson equation, one obtains jni - ne j 

ε 0 j φj ε 0 T  2, eL L2

where L = |∇ ln n|-1 is a typical spatial density scale. Dividing |ni - ne| by ion or electron density, we obtain a condition to be satisfied in the quasineutral plasma jni - ne j r 2d  2 ≪ 1, n L

ð3:2Þ

where the Debye radius is defined as r D = ε0 T=ne2 . Hence, the plasma can be considered a quasineutral medium provided that the Debye radius is much smaller than the characteristic scale of the problem. This condition is violated at the plasma boundaries where scale L decreases since the plasma density is reduced. Everywhere else in plasma, the paradigm of quasineutrality consists of the following. The densities of the charge species are considered to be equal, ni = ne, and the selfconsistent electric field, providing that quasineutrality takes place, is to be found © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Rozhansky, Plasma Theory, https://doi.org/10.1007/978-3-031-44486-9_3

75

76

3 Quasineutral Plasma and Sheath Structure

from the plasma equations with ni = ne. The space charge can be calculated afterwards from the Poisson equation if necessary. In some situations, for example, when a strong current is flowing through the plasma, the potential in the plasma can significantly exceed the value T/e. In this case, the value T/e in the quasineutrality condition, Eq. (3.2), should be replaced by a corresponding value of the plasma potential so that the Debye radius should be replaced by the corresponding effective Debye radius. Now let us consider a process of maintaining quasineutrality when the quasineutrality does not hold from the very beginning, i.e., nonquasineutral perturbations of electrons and ions exist at t = 0. Let us consider the evolution of small perturbations δne and δni on a uniform quasineutral background ne = ni = n0 in a slightly ionized plasma without a magnetic field and in the absence of a plasma current as an example. Let us start with a particle balance for electrons and ions with the electron flux given by Eq. (1.95) and similar for ions: ∂ne þ ∇
ð- De ∇ne þ ne be ∇φÞ = I - R, ∂t

ð3:3Þ

∂ni þ ∇
ð- Di ∇ni - ni bi ∇φÞ = I - R: ∂t

ð3:4Þ

A constant temperature is assumed, so thermal diffusion is absent. Terms on the r.h.s. represent sources and sinks of the charged particles. In the absence of a magnetic field, mobility and diffusion tensors are reduced to scalars, and according to the Einstein relation Dα/bα = Tα/e. Systems (3.3) and (3.4) should be supplemented by Poisson equation (3.1). The linearized system of Eqs. (3.1), (3.3), and (3.4) has the form: ∂δne = ∇
ðDe ∇δne - n0 be ∇φÞ, ∂t ∂δni = ∇
ðDi ∇δni þ n0 bi ∇φÞ, ∂t e Δφ = ðδne - δni Þ: ε0

ð3:5Þ

The solution is sought as the Fourier integral 1 ð2πÞ3 1 → φ r ,t = ð2πÞ3 →

δnα r , t =

→→

→

δnα →k exp i k r d k , →→

→

φ →k exp i k r d k :

3.1

Quasineutrality Maintenance

77

After substituting into Eq. (3.5), we have for the Fourier components ∂δn

→

ek

∂t ∂δn →

= - De k2 δn

→

ek

þ be n0 k 2 φ → , k

ð3:6Þ

= - Di k δn → - bi n0 k φ → : ik k ∂t - k2 φ → = e δn → - δn → =ε0 : 2

ik

k

2

ek

ik

Eliminating the potential from the third equation, one can seek the solution of the remaining equation system for δne →k and δni →k assuming that the time dependence is given by exp(-iωt). Then, we obtain the algebraic system - iωδn

→

ek

= - De k2 δn

→

ek

þ be n0 e δn → - δn ik

- iωδn → = - Di k δn → - bi n0 e δn → - δn 2

ik

ik

ik

→

ek

=ε0 ,

→

ek

=ε0 :

The algebraic system has a nontrivial solution provided its determinant is equal to zero. Solving the corresponding quadratic equation for ω, one obtains two roots: ω1 = - iDe =r 2d ;

ω2 = - ik 2 Di ð1 þ T e =T i Þ,

ð3:7Þ

where the Debye radius is defined as rd = (ε0Te/ne2)1/2. A general solution for the Fourier components is δn → = C 1 expð - iω1 t Þ þ C 2 expð - iω2 t Þ, ek δn → = - ðbi C1 =be Þexpð - iω1 t Þ þ C 2 expð - iω2 t Þ:

ð3:8Þ

ik

During the derivation of Eq. (3.8), the inequalities De ≫ Di and krd ≪ 1 were used. Coefficients C1, C2 are determined from the initial conditions C 1 = δn0 → - δn0→ , ek

ik

C 2 = δn0→ - bi C 1 =be δn0 → ≈ δn0→ , ik

where δn0

→

ek

ik

= δne,i →k ðt = 0Þ.

e,i k ω1- 1

Since ≪ ω2- 1 , evolution is characterized by two different time scales. The first stage is determined by the time scale ω1- 1 , which corresponds to the diffusion of electrons at a distance of the order of the Debye radius. At this fast stage, the ion density perturbation remains practically constant, while the electron density perturbation is strongly redistributed. At the end of this stage, the quasineutrality is established so that δne = δni, as follows from Eq. (3.8). Therefore, quasineutrality maintenance occurs at the so-called Maxwellian time scale

78

3

Quasineutral Plasma and Sheath Structure

τM = r 2d =De = ε0 =σe ,

ð3:9Þ

where σe = enbe is the electron conductivity. The Maxwellian time does not depend on the spatial scale of the perturbation. The second root ω2 corresponds to the joint quasineutral evolution of electrons and ions at a longer time scale. This process of ambipolar diffusion will be considered in the next chapter. It is worth noting that the process of maintaining quasineutrality takes place due to the mobility of electrons in a self-consistent electric field and not by their diffusion. Indeed, neglecting diffusive terms in Eq. (3.6), subtracting the second equation from the first one and substituting potential from the Poisson equation, one finds for the Fourier components ∂ δne →k - δni →k ∂t

= - De r d- 2 δne →k - δni →k :

ð3:10Þ

This equation describes relaxation to the quasineutral state at the Maxwellian time scale. The quasineutrality maintenance occurs at the Maxwellian time scale in the slightly ionized plasma with high collision frequency when τMνeN ≫ 1. In the opposite case, oscillation at plasma frequency takes place, which decays with the -1 , which in this situation is the time scale for establishing time scale νeN quasineutrality.

3.2

Collisionless Sheath at the Material Surfaces

Near the material surfaces, the quasineutrality condition is violated, and the fluid approximation simultaneously becomes invalid since the typical spatial scale of the density variation L = |∇ ln n|-1 tends to zero. Here, a sheath of space charge is formed with a width of the order of Debye radius. Let us analyze a sheath structure in the absence of a magnetic field for the case when both the electron and ion mean-free paths are larger than the sheath width. Such a sheath is known as a collisionless sheath. Let for definiteness consider a surface that is biased negatively with respect to the plasma. Near the surface at a distance of the order of the Debye radius, a sheath of positive space charge is located (Fig. 3.1). Further away from the surface at the plasma side of the sheath, a so-called presheath is situated. Its width is of the order of the mean-free path of ions, and the preasheath separates the quasineutral plasma and the sheath. In the preasheath, ions are freely accelerated towards the surface by the electric field, so the fluid description is not applicable here; however, this region is still a quasineutral one, while further in the plasma, the fluid description becomes applicable. At the sheath edge, ions and electrons move in completely different ways. Ions are accelerated in the sheath; therefore, all ions that move towards the surface reach

3.2

Collisionless Sheath at the Material Surfaces

79

Fig. 3.1

Fig. 3.2

it. In contrast, a significant fraction of electrons is reflected back in the sheath, and only electrons with large energy can reach the surface. To understand how the electron distribution function is formed, we consider below an electron distribution function in a capacitor with an electric field that decelerates electrons.

3.2.1

Electrons in a Capacitor with a Reflecting Electric Field

Consider a capacitor with an electric field that decelerates electrons and a potential difference Δφ (Δφ is a positive quantity). The flux of electrons is directed from right to left (Fig. 3.2), and their distribution function is assumed to be Maxwellian for positive velocities Vx. Part of the electrons is reflected back by the decelerating electric field, while the other part, which has sufficient energy to overcome the potential difference Δφ, appears at the left from the capacitor. In the stationary case

80

3 Quasineutral Plasma and Sheath Structure

and in the absence of collisions, the distribution function is conserved along the particle trajectories in accordance with Liouville’s theorem. Therefore, in the stationary case, the distribution function should be a function of the integral of motion. Since the full energy E = meV2/2 - eφ is conserved, the distribution function is a function of the full energy. To the right of the capacitor, we assume the distribution function to be Maxwellian for electrons moving from right to left with positive velocities (the potential here is set to zero φ = 0) f þ ð V x ≥ 0Þ  f M = A

m1=2 e ð2πT e Þ1=2

exp -

mV 2x : 2T e

ð3:11Þ

Hence, f = f M(E). To overcome the reflecting potential Δφ, electrons should have a velocity larger than V 0 = 2eΔφ=me , while electrons with a velocity V0 after passing through the capacitor are decelerated to zero velocity. Therefore, to the left of the capacitor for electrons with positive velocities f - ðV x ≥ 0Þ = A exp -

m1=2 eΔφ mV x 2 e exp : 1=2 T e ð2πT e Þ 2T e

ð3:12Þ

There are no electrons with negative velocities in this region: f - ðV x < 0Þ = 0:

ð3:13Þ

In contrast, to the right of the capacitor, there are reflected electrons with kinetic energies smaller than eΔφ, while electrons with larger energies flowing to the right are absent; they are passed through the capacitor. Therefore,

f þ ð V x < 0Þ =

A

m1=2 e ð2πT e Þ

1=2

0,

exp -

mV x 2 , 2T e

jV x j ≤ V 0

:

ð3:14Þ

jV x j > V 0

Finally, inside the capacitor m1=2 me V x 2 - eφðxÞ e exp , f in ðV x ≥ 0Þ = A 2T e ð2πT e Þ1=2 m1=2 me V x 2 - eφðxÞ e f in ðV x < 0Þ = A exp , 2T e ð2πT e Þ1=2 f in ðV x < 0Þ = 0, jV x j > V 0 , V 0 =

2e ðΔφ þ φÞ: me

jV x j ≤ V 0 ,

ð3:15Þ

3.2

Collisionless Sheath at the Material Surfaces

81

Note that even to the right of the capacitor, the electron density does not coincide with А; in particular, it is smaller than А. Indeed, integration of Eqs. (3.12) and (3.14) over velocities yields nþ =

A 1 þ erf 2

ðeΔφ=T e Þ ,

ð3:16Þ

where the error function is defined as z

2 erf ðzÞ = p π

exp - t 2 dt: 0

The electron density inside the capacitor is nin =

A 1 þ erf 2

eðφ þ ΔφÞ=T e

exp

eφ : Te

ð3:17Þ

If the potential drop Δφ > Te/e is large enough, almost all electrons are reflected back, and the error function in Eq. (3.16) is close to unity, so that n+ ≈ A. In this situation, the electron density decreases inside the capacitor starting from the right plate according to the Boltzmann law nin = nþ exp

eφ : Te

ð3:18Þ

At the left plate, the density, Eq. (3.17), does not follow the Boltzmann law due to the absence of a sufficient number of reflecting electrons. In particular, to the left of the capacitor in accordance with Eq. (3.12) n- =

3.2.2

A eΔφ exp : 2 Te

Particle and Energy Fluxes to the Material Surfaces

One can easily calculate particle and energy fluxes to the surface coming from the sheath edge. Since the electron particle flux is conserved in the sheath, it can be calculated using the Maxwellian distribution function, Eq. (3.11), for the positive velocities at the sheath edge (at the right plate of the capacitor). Only electrons with energies larger than eΔφ contribute to the flux:

82

3

Quasineutral Plasma and Sheath Structure

1

Γe = V0

A V x f þ dV x = p 2π

Te eΔφ exp : me Te

ð3:19Þ

The same expression can be obtained by integrating with the distribution function at the surface, Eq. (3.15), but then one should integrate over all positive velocities. If the potential drop in the sheath is significantly larger than Te/e, so that the error function in Eq. (3.16) is close to unity, then the flux of electrons is (ns  n+) n Γe = p s 2π

Te eΔφ exp : me Te

ð3:20Þ

The ion flux to the wall can be easily obtained assuming that at the sheath edge, all ions have the same velocity u0 directed to the surface. This is justified, as shown below, if Te ≫ Ti and ions gain energy equal to Te/2 due to the acceleration in the presheath, so that their velocity at the sheath edge should be u0 = (Te/mi)1/2. The ion flux at the sheath edge coincides with their flux at the surface since all ions are accelerated in the sheath and reach the surface. Hence, Γ i = ns u0 :

ð3:21Þ

While calculating an energy flux, one has to distinguish the flux to the surface and the flux leaving the plasma, these fluxes do not coincide. The easiest way to calculate the electron energy flux to the surface is to use the distribution function, Eq. (3.15), at the surface. One has to use a 3D distribution function. For Δφ ≫ Te/e 1

qe = 0

me V 2 þ 2eΔφ ns m3=2 me V 2 e exp Vx 2 2T e ð2πT e Þ3=2

dV x dV y dV z

ð3:22Þ

= 2T e Γe : While calculating the energy flux of ions to the surface, it is necessary to take into account the ion energy gain in the sheath eΔφ. With an account of energy gain in the presheath Te/2, one obtains qi = ðeΔφ þ T e =2ÞΓi :

ð3:23Þ

The energy fluxes given by Eqs. (3.22) and (3.23) cause heating of the surface. To obtain the energy flux of electrons at the sheath edge leaving the plasma, qes, one has to integrate the energy flux from V0 to infinity over velocities to obtain qes = ðeΔφ þ 2T e ÞΓe :

ð3:24Þ

3.2

Collisionless Sheath at the Material Surfaces

83

The ion energy flux leaving the plasma, qis, is (assuming the small ion temperature) qis =

1 T Γ: 2 e i

ð3:25Þ

The energy fluxes are not conserved in the sheath, the energy is redistributed between the charged particles due to the work performed by the electric field. It is also worth noting that with the account of finite ion temperature, their energy flux increases considerably.

3.2.3

Current-Voltage Characteristics of the Sheath. Floating Potential

For many plasma physics problems, it is necessary to know the connection between the potential drop in the sheath and the current to the surface. This relation, known as the current-voltage characteristic of the sheath, is used as a boundary condition for the potential distribution inside the plasma volume. The collisionless sheath currentvoltage characteristic can be obtained from Eqs. (3.20) and (3.21). For eΔφ ≫ Te, the current density to the surface is j = ens

Te mi

1=2

1 Te -p m e 2π

1=2

exp -

eΔφ Te

:

ð3:26Þ

In the absence of current, the surface is biased negatively with respect to the plasma, and the potential drop is called the floating potential. The isolated surface in plasma is, therefore, biased to the floating potential (by absolute value). In accordance with Eq. (3.26), the floating potential (plasma potential with respect to the surface in the absence of current) is φfl =

mi Te ln e 2πme

1=2

:

ð3:27Þ

The floating potential is of the order of few Te/e (for hydrogen plasma, e.g., of the order of 3Te/e).

3.2.4

Sheath Structure. Bohm Criterion

Let us now find the potential distribution inside the sheath for the negatively biased surface. The potential distribution is described by the Poisson equation

84

3

Quasineutral Plasma and Sheath Structure

d2 φ = eðne - ni Þ=ε0 : dx2

ð3:28Þ

For simplicity, we assume a sharp boundary between the plasma and the sheath located at x = 0, and, at this boundary, we assume quasineutrality ne = ni = ns and zero electric field. The electron density in the sheath decreases in accordance with the Boltzmann distribution, Eq. (3.18). Since the ion flux is conserved in the sheath, the ion density in the sheath can be obtained from the flux conservation condition (φ(x = 0) = 0 is assumed): Γi = ns u0 = ni ðxÞ u20 -

2eφ mi

1=2

:

ð3:29Þ

The ion density decreases deeper in the sheath since the ions are accelerated by the electric field. The electron density, however, should decrease even faster to provide a positive space charge in the sheath (Fig. 3.1). The Poisson equation (3.28) has the form d2 φ eφ u0 = ε0 ens exp Te 2 dx2 u0 - 2eφ=mi

1=2

:

ð3:30Þ

In the dimensionless variables Φ= -

eφ , Te

ξ=

x , rd

rd =

ε0 T e ns e 2

one has d2 Φ 1 = - expð- ΦÞ þ , dξ2 ð1 þ Φ=Φ0 Þ1=2

ð3:31Þ

where Φ0 = mi u20 =2T e . Let us study the character of the solution for small values of Φ. Expanding the r.h.s. of Eq. (3.31), we find that it is equal to Φ 1 - T e =mi u20 . The space charge value, which is proportional to the r.h.s. of Eq. (3.31), tends to zero at the sheath boundary, while its sign depends on the u0 value. If u0 < (Te/mi)1/2, which corresponds to a negative space charge, the solution of Poisson equation (3.31) has an oscillating character, so it is impossible to construct a solution that corresponds to the negative potential drop in the sheath and, hence, a positive space charge. In contrast, if u0 ≥ (Te/mi)1/2, the monotonic potential profile exists corresponding to the potential profile as shown in Fig. 3.1. A condition

3.2

Collisionless Sheath at the Material Surfaces

u0 ≥ ðT e =mi Þ1=2

85

ð3:32Þ

is known as the Bohm criterion. The separation into plasma and sheath is, of course, conventional. In reality, at a considerable distance from the surface, the charge separation, and the electric field strength smoothly increase towards the surface; this region is often referred to as the presheath. The characteristic presheath scales equal the mean-free path of attracting particles (for negatively biased surface to ion mean-free path). This means that this problem demands rigorous kinetic treatment. The value u0 depends on the problem and type of collision. Rigorous analysis gives the quantity of u0 close to u0 = ðT e =mi Þ1=2 :

ð3:33Þ

We shall demonstrate how this value can be obtained from the simplified fluid approach. Let us consider one-dimensional stationary plasma flow towards the surface. Summing up momentum balance equations of electrons and ions neglecting electron inertia and electron-neutral collisions, we have in plasma mi nui

d ð pe þ p i Þ dui - nmi νiN ui : =dx dx

ð3:34Þ

Combining this equation with the particle balance equation in the absence of sources and sinks yields 1-

c2s dui = - νiN : u2i dx

ð3:35Þ

Here, cs = (dp/dρ)1/2 is the sound speed in the plasma, p = pe + pi, ρ = nmi. From Eq. (3.35), one can see that ions are accelerated up to the sound speed at the sheath edge, where acceleration turns to infinity (the density gradient here also turns to infinity as follows from momentum balance equation (3.34). In reality, of course, acceleration remains finite and takes place at the last ion mean free path. In the case considered Te ≫ Ti, the sound speed value cs coincides with u0 given by Eq. (3.33). Equation (3.31) is analogous to the Newton equation of motion. After multiplying both sides of the equation by dΦ/dξ and integration, one obtains 1 dΦ 2 dξ

2

þ W ðΦÞ = 0:

ð3:36Þ

Here, the “potential energy” known as the Sagdeev potential is defined as W ðΦÞ = 1 - expð- ΦÞ - 2Φ0 ðΦ=Φ0 þ 1Þ1=2 þ 2Φ0 : The solution of Eq. (3.36) for Φ0 = 1/2 is

ð3:37Þ

86

3 Φ

1 ξ= p 2

0

Quasineutral Plasma and Sheath Structure

dΦ0 0 1=2

ð1 þ 2Φ Þ

0

- 2 þ expð- Φ Þ

:

ð3:38Þ

For a large potential drop in the sheath, assuming Φ ≫ 1, the potential in the sheath is determined by the Child-Langmuir law ξ=

25=4 3=4 Φ : 3

ð3:39Þ

Inserting the full potential drop and sheath width into Eq. (3.39), in the initial coordinates, we find the relation between them: Lsh =

eΔφ 25=4 r 3 d Te

3=4

:

ð3:40Þ

Hence, the sheath width is of the order of the Debye radius and increases with the size of the sheath potential drop. Analysis for the case of a positively biased surface can be done analogously.

3.3

Impact of Electron Emission. Double Sheath

The structure of the collisionless sheath changes in the presence of additional electron flux from the surface, caused by thermal electron emission or secondary electron emission. We assume that the surface emits electrons with a Maxwellian distribution with temperature Tec, which is much smaller than the temperature of electrons in plasma Teh. Let us restrict ourselves by considering a surface at floating potential. The emitted electrons partially neutralize the positive space charge in the sheath, thus reducing the potential drop in the sheath and increasing the flux of hot electrons from the plasma to the surface. For a small flux of cold electrons Γec, the zero current condition reads Γeh - Γec = Γi ,

ð3:41Þ

where the fluxes Γeh and Γi are defined according to Eqs. (3.20) and (3.21). Then, Eq. (3.41) determines the potential drop as a function of emitting flux Γec. A further increase in the cold electron flux leads to the formation of a negative space charge near the surface, which leads to partial reflection of the cold electron flux. The so-called virtual cathode arises, and a double sheath is formed with a potential minimum close to the electrode (Fig. 3.3). The negative space charge and the negative potential drop restrict the flux of cold electrons so that only part of their

3.3

Impact of Electron Emission. Double Sheath

87

Fig. 3.3

flux Γec = Γ0ec is injected through the double layer into the plasma, while the rest are reflected back to the surface. The potential difference between the surface and potential minimum Δφmin is connected with the flux of cold electrons reaching the plasma by the relation

Γðec0Þ = Γec exp - eΔφðmin Þ =T ec , Therefore, the value Δφmin should be of the order of few Tec/e. Since Tec ≪ Teh, it is possible to neglect Δφmin with respect to the potential drop Δφ between the plasma and the surface. Inside the double layer, the potential profile is determined by the Poisson equation d 2 φ=d x2 = eðneh þ nec - ni Þ=ε0 :

ð3:42Þ

Charged particle densities on the r.h.s. of Eq. (3.42) are the functions of the potential. The ion density is still given by Eq. (3.29), where ns is the ion density at the plasma side of the double sheath. In contrast, the density of hot electrons should be described in a more accurate way than the Boltzmann distribution since the potential drop in the double layer is significantly smaller than the floating potential. We shall take the distribution function given by Eqs. (3.15) and (3.16). According to Eq. (3.17) neh ðΦÞ =

1 A expð- ΦÞ 1 þ erf 2

Φðmin Þ - Φ

:

ð3:43Þ

where Φ = - eφ/Teh. The cold electrons are accelerated inside the double sheath, so neglecting their initial energy, one obtains

88

3

nec ðΦÞ = nec Φðmin Þ exp

Quasineutral Plasma and Sheath Structure

T eh Φðmin Þ - Φ T ec

1 - erf

T eh Φðmin Þ - Φ T ec

T ec , T eh Φðmin Þ - Φ

1 ≈ p nec Φðmin Þ π

ð3:44Þ where nec(Φmin) is the cold electron density at the potential minimum. It is connected with the flux of cold electrons: Γðec0Þ

p 2 = p nec Φðmin Þ π

T ec : me

ð3:45Þ

Now, we can integrate the Poisson equation assuming a zero electric field at both boundaries of the double sheath – at the plasma side of the sheath and at the potential minimum: Φðmin Þ

ðneh þ nec - ni Þ dΦ = 0:

ð3:46Þ

0

Combining this equation with Eqs. (3.43) and (3.44) and the expression for the ion density, Eq. (3.29), with an account of the zero current condition, Eq. (3.41) and the quasineutrality constraint nehs + necs = ns at the plasma side of the sheath, one obtains an equation for the plasma drop in the double sheath for a floating surface: exp - ΔΦ fl =

a=

ΔΦ fl ð1 - aÞ ; a 1þp πΔΦfl

1 þ erf

mi u20 T eh

1þ

ð3:47Þ

2T eh ΔΦ fl - 1 , mi u20

Here, ΔΦfl = eΔφfl/Teh. We neglected the terms containing a small factor (me/mi)1/2. For Teh ≫ Ti accounting for the Bohm criterion mi u20 = T eh , we find eΔφfl =T eh = ΔΦfl = 0:95:

ð3:48Þ

We see that the potential drop in the double sheath that is formed for the case of strong electron emission is significantly reduced with respect to the floating potential Eq. (3.47).

3.4

Sheath in Magnetic Field

89

In general, a double sheath structure can be formed not only near the emitting surface but also in many other situations, particularly at the boundary of plasmas with different parameters, so their role in plasma physics is quite fundamental.

3.4

Sheath in Magnetic Field

The sheath structure for the magnetic field normal or slightly inclined to the surface does not change significantly with respect to the unmagnetized plasma. However, for a magnetic field strongly inclined or parallel to the surface, the sheath structure changes significantly. The reason consists of the following. If the Debye radius is smaller than the charged particle Larmor radii, a so-called magnetic presheath is formed in front of the surface, as shown in Fig. 3.4, with a width of the order of Larmor radius. For negatively biased surface, it is ion Larmor radius. Let us consider the general situation of the inclined magnetic field forming an angle Θ with the surface. At distances of the order of the ion Larmor radius, ions reach the negatively biased surface at a time scale of the order of the inverse cyclotron frequency ωci- 1 . Their distribution functions strongly differ depending on whether their orbits intersect or do not intersect the surface. At the orbits intersecting the surface, the ion density is small since the ions escape to the surface with the thermal velocity, while these orbits are filled due to slower ion flow from the bulk plasma along the magnetic field or by rare collisions with other ions. Closer to the surface, the number of intersecting orbits is larger, and the density at these orbits is correspondingly smaller. At the surface, practically all orbits intersect, and the density ns here can be estimated by equalizing the thermal ion flux ns(T/mi)1/2 and normal component of the ion flux from the bulk plasma n(T/mi)1/2 sin Θ. As a result, the density at the sheath edge ns~n sin Θ should be significantly smaller than that at

Fig. 3.4

90

3 Quasineutral Plasma and Sheath Structure

the outer side of the magnetic presheath. The density drop in the magnetic presheath is especially large for small angles Θ when the magnetic field line is almost parallel to the surface. Electrons in the magnetic presheath are trapped by the electric field to prevent their fast escape along the magnetic field lines, so the potential profile here corresponds to the Boltzmann distribution for electrons. The potential drop in the presheath is, therefore, quite large, it is of the order of Δφps = (Te/e) ln n/ns = (Te/e) ln (sin-1Θ). The detailed simulation of the density profile inside the magnetic presheath, however, is rather complicated since the kinetic approach is required here. On the other hand, one can easily obtain the current-voltage characteristics of the sheath and the magnetic presheath by taking the sum of the potential drops in the sheath and presheath so that the potential drop is calculated from the outer presheath edge to the surface. The normal flux of electrons to the surface is a projection of their parallel flux along the magnetic field: n sin Θ Γe = p 2π

Te eΔφ exp : me Te

ð3:49Þ

The ion flux is conserved and coincides with the normal component of ion flux at the outer edge of the magnetic presheath: Γi = n sin Θ

Te : mi

ð3:50Þ

The ion velocity in Eq. (3.50) corresponds to the Bohm criterion, which can be derived as without the magnetic field. The current-voltage characteristic, which generalizes Eq. (3.26), is thus given by j = en sin Θ

Te mi

1=2

1 Te -p 2π me

1=2

exp -

eΔφ Te

:

ð3:51Þ

For very small angles when the magnetic field is almost parallel to the surface, however, this expression is not applicable. At small angles, the orbits that intersect the surface are filled not by the parallel flux from the bulk plasma but by the collisions of ions from nonintersecting orbits with neutrals or with ions. The orbits intersecting the surface remain almost empty since the time scale of escape to the surface is of the order of the inverse cyclotron frequency ωci- 1 , while the time scale of filling these orbits is νi- 1 . In the stationary case, the ion flux to the surface is equal to the ion flux in the velocity space from nonintersecting to intersecting orbits. The latter can be estimated as

3.5

Thermoelectric Current Between Two Electrodes

91

Γi ≈ nρci νi :

ð3:52Þ

The ion collision frequency here is either the ion-neutral collision frequency or ion-ion collision frequency. The exact expression for the ion flux can be found in [7] and requires kinetic derivation. Therefore, the current-voltage characteristic equation (3.51) can be used for angles Θ > νi/ωci. In the general case, one should use the ion flux to the surface Γi = n sin Θ

3.5

Te þ nρci νi : mi

ð3:53Þ

Thermoelectric Current Between Two Electrodes

The current voltage characteristic of the sheath, equation (3.26), can be used to calculate the thermoelectric current between two electrodes adjacent to the plasma with different electron temperatures. Suppose we have two connected grounded electrodes with the same (zero) potential with different electron temperatures Tþ es , T es at the sheath entrances, as shown in Figs. 3.5–3.6. Let us assume that the plasma between electrodes has zero potential drop and that the magnetic field is normal to the surfaces or absent. The potentials at the sheath entrances in the general case are not equal to the floating potentials given by Eq. (3.27). Indeed, if the electron temperature at the right electrode marked by + is larger than the temperature at the left electrode marked by -, then φþ fl > φfl and the potential difference is applied to plasma to produce infinite current for infinite plasma conductivity. Therefore, for uniform plasma, sheath potential drops in both sheaths must coincide, so that

Fig. 3.5 Net current through the divertor plates for various seeding rates

92

3

Quasineutral Plasma and Sheath Structure

jth

– Tes


a; ð4:37Þ

;

b < a:

The expression for the saturation current could be obtained using electrostatic analogy. The potential profile created by a solitary conductive body in an electrostatic field is also described by the Laplace equation with a constant potential at the surface. The ion saturation current is proportional to the normal component of the density gradient integrated over the probe surface. A similar integral of the potential gradient in the electrostatic problem over the surface of the conductor is proportional to the conductor charge, i.e., to its capacity С. Similarly, the ion saturation current to the probe is proportional to its capacity I sat i = 4πen0 Da C:

ð4:38Þ

The expression for the capacity of the conductor having the form of an ellipsoid of rotation with semiaxes a, a, b can be found, for example, In [11]: p

C=

b2 - a2 ; Archðb=aÞ p a 2 - b2 ; arccosðb=aÞ a;

b>a a>b

:

ð4:39Þ

a=b

For the strong positively biased probes, ions are trapped in the plasma and do not flow to the probe, while the electron saturation current is calculated analogously I sat e = 4πen0 De ð1 þ T i =T e ÞC:

ð4:40Þ

The electron saturation current is significantly larger than the ion saturation current since De ≫ Di. The typical current-voltage (I–V) characteristic of the probe is presented in Fig. 4.2. In the intermediate part of the I–V characteristic, there are two typical potentials. One is the floating potential φf when the current to the probe is zero. The other is the plasma potential (which we assume to be zero) when there is no electric field both in sat the plasma and in the sheath. Since I sat e ≫ I i , a considerable part of the I–V characteristic corresponds to the negative probe potentials. The ion current to the probe for a large negative φ is equal to the ion saturation current, and the ion flux to sat the probe is Γi(a) = Dan0/a. The electron current to the probe is I þ I sat i . For I ≪ I e ,

104

4

Diffusion in Partially Ionized Unmagnetized Plasma

Fig. 4.2 Typical current-voltage characteristic of a diffusive probe

the electric field in the plasma and the main part of the sheath corresponds to the Boltzmann distribution for electrons. Employing the electron flux to the surface for trapped electrons Eq. (3.20) and adding the potential drop in plasma Te/e ln (n0/ns), we obtain for the spherical probe ðsat Þ

I þ Ii eφ = ln Te I ðesatÞ

- ln

r 2d ð1

3a2 : þ T i =T e Þ2

ð4:41Þ

where rd is the Debye radius with electron temperature. This expression determines the I–V characteristic for large negative probe potentials |φ| ≫ Te/e. The floating potential φfl is negative and can be calculated by substituting I = 0 in Eq. (4.41): eφfl Da 3a2 = ln : - ln 2 Te De ð1 þ T i =T e Þ r d ð1 þ T i =T e Þ2

ð4:42Þ

For the probe at plasma potential φ = 0, there is no electric field in the plasma, and the sheath vanishes. The density profile still obeys Eq. (4.35), while the current to the spherical probe is given by the difference between two unipolar diffusive fluxes: I 0 = 4πeðDe - Di Þn0 a:

ð4:43Þ

For positive probe potentials, the I–V characteristics can be obtained in a way similar to Eq. (4.41).

4.3

4.3

Diffusion of Slightly Ionized Multispecies Plasma

105

Diffusion of Slightly Ionized Multispecies Plasma

The ambipolar diffusion equation (4.2) for the pure plasma, which was investigated in the previous section, has several striking peculiarities. First, it is formulated solely in terms of the plasma density. The self-consistent electric field that is always present in the inhomogeneous plasma remains “hidden” in the ambipolar diffusion coefficient Eq. (4.3). The self-consistent problem of calculating the plasma density and potential profiles is reduced to two separate problems. The plasma density profile can be found from Eq. (4.2) with the zero boundary condition independent of the potential profile. The latter can be found, if necessary, from Eq. (4.9). In other words, the evolution of the density profile does not depend on the presence of dc current through the plasma inhomogeneity. The rate of the diffusion process is determined by less mobile particles, i.e., by ions. The electrons are hindered by the electric field, which drags the ions. This drag force enhances ion diffusion and is responsible for the factor (1 + Te/Ti) in the expression for the ambipolar diffusion coefficient, Eq. (4.3). The second remarkable property of the ambipolar diffusion process consists of the fact that despite the nonlinearity of the initial problem, the equation for density is linear (in the absence of sources). These properties, and the striking simplicity of the ambipolar diffusion equation, stimulated numerous attempts to apply such an approach to more complicated situations – to multispecies plasmas, to plasmas in magnetic field, etc. These attempts have not attained serious success over several decades. The reason is that such an approach is principally wrong. It follows from the fact that the ambipolar diffusion problem corresponds to a degenerate case. This means that such a reduction of the nonlinear problem to a system of two linear equations for plasma density and potential is possible only for the case of pure plasma with scalar constant diffusion and mobility coefficients. In this situation, several essentially nonlinear effects strictly compensate for each other. However, in the general case, such compensation is impossible, and the problem cannot be reduced to the ambipolar diffusion equation and remains essentially nonlinear and current dependent. Even in the case of diffusion in currentless plasma, the electric field is determined by the density gradients of all charged particles. Hence, the problem becomes essentially nonlinear since every partial field-driven flux is proportional to the product of the density of a given species and the field strength, which is determined by a linear combination of the density gradients of all species. To understand these nonlinear mechanisms, we consider the simplest 1D example, when N test ions are injected at t = 0 into a small region of infinite and uniform ambient weakly ionized plasma with density n0 and Te = Ti. We neglect the ionization–recombination processes and assume that the mobility and diffusion coefficients of the injected and ambient ions coincide. The only distinction with the ambipolar problem equation (4.2) consists of the fact that we shall now distinguish between the injected and the ambient particles. The evolution process for the injected n1 and for the ambient n2 ion densities is described by the equations

106

4

Diffusion in Partially Ionized Unmagnetized Plasma

∂n1 þ ∇ . ð - D1 ∇n1 - n1 b1 ∇φÞ = 0, ∂t : ∂n2 þ ∇ . ð - D2 ∇n2 - n2 b2 ∇φÞ = 0: ∂t

ð4:44Þ

The quasineutrality condition is satisfied, so n = n1 + n2. Since D1 = D2 (b1 = b2) for the electron profile n(x, T ), Eq. (4.2) is valid, with the ambipolar diffusion coefficient Da = 2D1. Its solution for the case of the point initial condition δn(x, t = 0) = n1(x, t - 0) = Nδ(x) is δnðx, t Þ = p

N x2 : exp 4Da t 4πDa t

ð4:45Þ

The potential profile φ = (Te/e) ln n/n0 corresponds to the Boltzmann distribution for electrons. The total Gaussian profile δn with an effective width of the order of (4Dat)1/2 is, nevertheless, formed in a rather complex way by the diffusive and fielddriven fluxes of the injected and ambient particles. In the linear case, n1 ≪ n0, the electric field perturbation is small, and its influence on the motion of the injected ions is negligible. Their profile is thus determined by the unipolar ion diffusion: n1 ðx, t Þ = p

x2 N : exp 4D1 t 4πD1 t

ð4:46Þ

Perturbation of the ambient ion density is caused by the electric field and diffusion. It can be found as the difference between Eqs. (4.45) and (4.46). The ambient ions are extracted by the electric field from the neighboring zone x ≤ (4D1t)1/2, where the injected ions dominate and are piled up outside of it (Fig. 4.3a). If n1 > n0, the problem becomes nonlinear. The net profile equation (4.45) consists mainly of the injected ions. Their diffusion is practically ambipolar: the field here simply doubles their diffusive flux. The ambient ions are almost absent in the neighboring zone, as they are dragged out by the electric field. In Fig. 4.3b, an example of the numerical solution for such a situation is displayed. As the second example of plasma separation into regions with different ion compositions, we shall consider the problem of diffusive decay of a cylindrical positive column of weakly ionized plasma that consists of electrons and two ion species – positive ions with density n+(x, t) and negative ions with density n-(x, t). The quasineutrality condition demands n = n+ - n-. Expressing the radial electric field from the zero-current condition at the tube wall Γe þ Γn - = Γnþ , we obtain the radial flux of the negative ions

4.3

Diffusion of Slightly Ionized Multispecies Plasma

107

Fig. 4.3 (a) Small perturbation. Density profiles of injected ions, Eq. (4.36), (1) and perturbation of ambient ions (2) for equal mobilities and point initial perturbation. The curve 3 is the perturbation of electron density described by ambipolar diffusion, Eq. (4.45). The spatial coordinate ξ is measured in units of (4D1t)1/2. (b) Numerical simulation for strong disturbance: 1 – injected ions, 2 – ambient ions, and 3 – electron density. The densities are in units of n0

108

4

Γn - = ≈

Diffusion in Partially Ionized Unmagnetized Plasma

n - ∇n De bn - - bnþ Dnþ - n∇n - Dn - be þ Dn - bnþ - 2n - ∇n - Dn - bnþ bn þ nþ þ bn - n - þ be n De bn - n - ∇n - Dn - be n∇n - - 2n - Dn - bnþ ∇n : bn þ nþ þ bn - n - þ be n ð4:47Þ

Imposing the zero boundary condition on all three partial densities and neglecting all the plasmachemical processes, we find that the problem of plasma decay has a solution that corresponds to the fundamental diffusive mode, Eq. (4.33). For the case of cylindrical geometry, it means that all the densities are proportional to the Bessel function J0(ζ1r/a) and to time-dependent factors N-(t), N+(t), N(t): 2N - N þ Dn - bnþ N D dN ζ1 dN = - n= dt NDe dt N - bn - þ N þ bnþ þ Nbe a

2

:

ð4:48Þ

One can see that the decay time for electrons is De =Dn - smaller than that for negative ions. The radial field is Er = -

De N þ Dn - N - - Dnþ N þ d ðln J 0 ðζ1 r=aÞÞ: Nbe þ N þ bnþ þ N - bn - dr

ð4:49Þ

At Te = Ti, the electric field, Eq. (4.49), corresponds to the Boltzmann law both for electrons and for negative ions until the Bessel partial profiles satisfy the condition N ≫ N þ bnþ =be :

ð4:50Þ

Since the relaxation of the partial temperatures usually occurs considerably faster than that of the particle densities, the plasma is isothermal during the main part of the decay, Te = Ti = TN. In this case, the flux Γn - , according to Eq. (4.47), is small, and the decay process at this stage is determined by electron-ion ambipolar diffusion when the radial positive ion flux is practically equal to the electron flux. During this stage, plasma is continuously enriched by the negative ions, and when the relation (4.50) is violated, the field, Eq. (4.49), cannot hinder electrons. The rate of electron loss increases drastically up to a value that corresponds to the electron-free unipolar diffusion, and the whole tube volume is filled with the ion-ion plasma. The second stage of the decay process is determined by ion-ion ambipolar diffusion.

4.4

Diffusion in the Ionosphere

109

Fig. 4.4 Temporal dependence of the square of the cloud half-width: × – longitudinal scale of the ionized barium cloud; o – longitudinal scale of the neutral barium cloud; Δ – transverse scale of the ion barium cloud. The faster longitudinal expansion of the ionized cloud is connected with the factor (1 + Te/Ti) in the diffusion coefficient Da. The decrease of Λk at the later stage is apparently connected with the influence of ambient brightness

4.4

Diffusion in the Ionosphere

In the ionosphere at heights of 200–300 km, we have an example of strongly magnetized partially ionized plasma. The evolution of plasma inhomogeneities is controlled by complicated 3D processes of diffusion and mobility. Understanding complicated plasma evolution is very helpful for analyzing so-called active experiments in space. In some of these experiments with barium clouds, neutral barium vapor is injected into the ionosphere from the spacecraft. The evolution of a neutral barium cloud has a diffusive character in the framework moving with the velocity of the ionospheric wind of the neutral component. Simultaneously, photoionization of the cloud takes place, and a second ionized Ba+ cloud is formed. The evolution of an ionized cloud is different from that of the neutral cloud since ionospheric plasma is magnetized. Experiments with barium clouds are performed at dawn or sunset so that sunlight on the one hand can produce photoionization and, on the other hand, makes observation of the clouds from the Earth possible, which can be done even with pure eye. Since the density in the clouds is very large, the evolution of Ba+ clouds along the Earth’s magnetic field is controlled by 1D ambipolar diffusion (see Sect. 4.1). The global evolution of the plasma cloud is discussed in Sect. 6.3. Figure 4.4 plots the parallel dimensions of two clouds (halfwidth Λk) versus time for the first Soviet Union experiment “Spolokh.” One can see that the square dimensions of the clouds increase linearly with time, which is typical for diffusive processes. One would expect, in accordance with the results of Sect. 4.2.1,

110

4

Diffusion in Partially Ionized Unmagnetized Plasma

Λ2k = 4Da t

ð4:51Þ

Λ2k = 4Di t

ð4:52Þ

for plasma clouds, and

for neutral clouds. Ion and neutral diffusion coefficients are identical. Indeed, in Fig. 4.4, the slope of the curve for ions is larger than that for neutrals since Da/Di = (1 + Te/Ti). Using Eqs. (4.51) and (4.52), one can obtain information on ionospheric parameters at the height of the experiment, neutral density, and temperature ratio.

Chapter 5

Diffusion of Partially Ionized Magnetized Plasma

5.1

Diffusion and Mobility in a Magnetic Field

The diffusion of pure plasma with constant temperatures of electrons and ions is governed by particle balance equations similar to Eqs. (3.3) and (3.4): ∂n - ∇
De ∇n - be n∇φ = I - R; ∂t ∂n - ∇
Di ∇n þ bi n∇φ = I - R: ∂t

ð5:1Þ

Mobility and diffusion tensors are taken in the approximation of elementary theory (1.103) and (1.104). The dependence of the diagonal (perpendicular) components on the magnetic field is shown in Fig. 5.1. The following notations are used: xe =

ωce ; veN

xi =

ωci : viN

Since xe ≫ xi, when the magnetic field is rising from B = 0 at first, the value xe = 1 is reached. For larger magnetic fields xe ≫ 1, the electron diffusion coefficient De⊥ = Dek = 1 þ x2e starts decreasing with the magnetic field as B-2. For such a magnetic field, electrons are considered magnetized. On the other hand, since xi ≪ xe, at xe = 1, the perpendicular ion diffusion coefficient remains practically equal to the parallel ion diffusion coefficient, and ions are not magnetized. When the magnetic field increases further at xexi = 1 (for equal temperatures of electrons and ions), the perpendicular diffusion coefficients of electrons and ions become equal. However, we still have xi ≪ 1, so ions remain unmagnetized. For larger magnetic fields, when xe xi ≫ 1

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Rozhansky, Plasma Theory, https://doi.org/10.1007/978-3-031-44486-9_5

ð5:2Þ

111

112

5

Diffusion of Partially Ionized Magnetized Plasma

Fig. 5.1 Dependence of the perpendicular diffusion coefficients on magnetic field. (1) Electron and (2) ion diffusion coefficient

ion perpendicular diffusion coefficient exceeds that of electrons (the same is true for the mobility coefficients due to the Einstein relation). Plasma with inequality (5.2) fulfilled is called magnetized plasma. Finally, for larger magnetic fields with xi ≫ 1, the ions become magnetized, and their perpendicular diffusion coefficient decreases with the magnetic field as B-2. Let us discuss the physical meaning of perpendicular transport diffusion and mobility coefficients for a strong magnetic field when both electrons and ions are magnetized. In the diffusion tensor (α = i, e)

Dα =

Dα⊥

± DαΛ

0

∓ DαΛ 0

Dα⊥ 0

0 Dαk

ð5:3Þ

the diagonal perpendicular diffusion coefficients in the strong magnetic field xi ≫ 1 are Dα⊥ =

T α vαN  ρ2cα vαN : mα ω2cα

ð5:4Þ

Hence, the perpendicular diffusion coefficients of the charged particles can be estimated as the square of the random walk step during collisions with neutrals (Larmor radius) multiplied by the collision frequency. Nondiagonal components of the tensor, Eq. (5.3), for magnetized ions do not depend on the collision frequency: DαΛ =

Tα eB

ð5:5Þ

5.1

Diffusion and Mobility in a Magnetic Field

113

Fig. 5.2 Hall (diamagnetic) fluxes of electrons and ions, and diamagnetic current

They are connected with the flux known as the Hall or diamagnetic flux, which is directed perpendicular to the density gradient (Fig. 5.2). The flux value can be estimated as the thermal velocity of the charged particles multiplied by the density difference at the Larmor radius scale. For example, if the density gradient is parallel to the x-axis, the flux in the y direction is given by Γαy = ∓ DαΛ

dn dn  V Tα ρcα : dx dx

ð5:6Þ

In the homogeneous magnetic field, these fluxes do not cause density redistribution since they are directed perpendicular to the density gradient along equidensities. It is easy to demonstrate that 0

∇
Dα ∇n = ∇
Dα ∇n ,

ð5:7Þ

0

where diagonal tensor Dα is defined as 0 Dα

=

Dα⊥ 0

0 Dα⊥

0 0

0

0

Dαk

:

Diffusive Hall fluxes in a uniform magnetic field, therefore, do not contribute to the equation system, Eq. (5.1). The difference in electron and ion Hall fluxes leads to the formation of a diamagnetic current. For magnetized electrons and ions, one obtains

114

5 →

→

j

Λ

Diffusion of Partially Ionized Magnetized Plasma

→

= e Γ iΛ - Γ eΛ =

ðT e þ T i Þ → B × ∇n B2

ð5:8Þ

The diamagnetic current, Eq. (5.8), reduces the vacuum magnetic field. Using the → → Maxwellian equation ∇ × B = μ0 j , we find that the magnetic field perturbation is of the order of 2μ nðT e þ T i Þ δB :  β= 0 B B2

ð5:9Þ

Below, we shall assume that β ≪ 1 and thus shall neglect magnetic field perturbations produced by plasma currents. Nondiagonal components of the mobility tensor in the strong magnetic field xi ≫ 1 coincide for electrons and ions: bαΛ =

1 : B

ð5:10Þ →

→

→

The corresponding Hall flux caused by the electric field Γ αΛ = n E × B =B2 is just →

→

E × B drift of electrons and ions with the same velocity. Note that this is correct only for magnetized ions; in contrast, for xi ≪ 1, the ion Hall flux is significantly smaller than that for electrons. Diagonal perpendicular components of the mobility tensor, similar to the diffusion tensor, in a strong magnetic field are proportional to the collision frequency bα⊥ =

evαN : mα ω2cα

ð5:11Þ

The corresponding fluxes in the direction of the electric field can be interpreted as drift produced by friction with neutrals (Fig. 5.3). Indeed, an electric field parallel to the y-axis causes joint drift of electrons and ions along the x-axis with equal velocities uex = uix = E/B. Due to collisions with neutrals, a friction force in the xdirection arises, acting to decelerate charged particles: RαN = - mαnvαNuαx. In turn, these friction forces produce the drift of the particles along the electric field with velocity uαy = ± RαN/(neB). As a result, we have a flux along the electric field Γαy = ± nbα⊥E with mobility, Eq. (5.11). Since the ion-neutral friction force is much larger than the electron-neutral force, the ion flux along the electric field is much larger than that of electrons provided that the plasma is magnetized, Eq. (5.2). For the mobilities, therefore, we have bi⊥ ≫ be⊥. Everywhere below, with one exception, we shall consider the case of magnetized plasma.

5.2

One-Dimensional Diffusion in Magnetized Plasma

115

Fig. 5.3 Flux along the electric field as drift caused by friction with neutrals

5.2 5.2.1

One-Dimensional Diffusion in Magnetized Plasma Diffusion Across a Magnetic Field

One-dimensional (1D) diffusion across a strong magnetic field is very similar to diffusion in the absence of a magnetic field; only the role of charged particles is inversed since in magnetized plasma, ion mobility and diffusion coefficients are much larger than electron mobility and diffusion coefficients. Similar to the case without a magnetic field, multiplying the first equation in Eq. (5.1) by bi⊥, and the second one by be⊥ after summation, one obtains the equation of ambipolar diffusion: ∂n - Da⊥ Δn = I - R: ∂t

ð5:12Þ

Here, the coefficient of ambipolar diffusion is Da⊥ =

De⊥ bi⊥ þ Di⊥ be⊥ T ≈ De⊥ 1 þ i : be⊥ þ bi⊥ Te

ð5:13Þ

In the absence of current through the plasma, the potential corresponds to the Boltzmann distribution for ions: φd = -

Di⊥ - De⊥ T ln n þ const ≈ - i ln n þ const: bi⊥ þ be⊥ e

ð5:14Þ

Hence, in contrast to the situation without a magnetic field, positive plasma perturbation is biased negatively, so that the electric field confines ions and accelerates electrons to keep their fluxes equal.

116

5.2.2

5

Diffusion of Partially Ionized Magnetized Plasma

1D Diffusion at an Arbitrary Angle with a Magnetic Field

Here, we shall study the situation when the plasma density depends only on the coordinate ζ when the density gradient forms an angle β with a magnetic field, as shown in Fig. 5.4. It is convenient to introduce a new coordinate system (x, η, ζ) instead of the initial coordinate system (x, y, z), where the z-axis is parallel to the magnetic field. A new coordinate system is obtained from the initial coordinate system by rotation over the x-axis; therefore, the diffusion and mobility tensors in the new system have the standard form 0

0

D = ADA ;

0

0

b = AbA ,

ð5:15Þ

where ^= A

1 0 0

0 cos β sin β

0 - sin β cos β 0

is the operator of rotation over the x-axis, and A is the inversed operator. In the new coordinate system, the diffusive tensor has the form

jη

z, B η ζ β 0

y

x

→

→

Fig. 5.4 1D diffusion ∇n k ζ for μe ≪ μ ≪ μi, particle fluxes are shown by arrows

5.2

One-Dimensional Diffusion in Magnetized Plasma

0

D =

117

D⊥ - Dxy cos β

Dxy cos β D⊥ cos 2 β þ Dk sin 2 β

- Dxy sin β Dk - D⊥ cos β sin β

Dxy sin β

Dk - D⊥ cos β sin β

Dk cos β þ D⊥ sin β 2

ð5:16Þ

2

Here, the quantitates Dk, D⊥, and Dxy correspond to the components of the tensor Eq. (5.3). The analogous form has the mobility tensor. In the new coordinates initial equation system, Eq. (5.1) has the form ∂n ∂n ∂ ∂φ ± bαζζ n Dαζζ = I - R, ∂ζ ∂t ∂ζ ∂ζ

ð5:17Þ

where α = i, e. Similar to previous cases, after eliminating the electric field, one obtains ∂n ∂ ∂n = I - R, D μ2 ∂t ∂ζ ∂ζ

ð5:18Þ

where μ = cos β, Deζζ biζζ þ Diζζ beζζ biζζ þ beζζ = Dek μ2 þ De⊥ ð1 - μ2 Þ bik μ2 þ bi⊥ ð1 - μ2 Þ þ Dik μ2 þ Di⊥ ð1 - μ2 Þ bek μ2 þ be⊥ ð1 - μ2 Þ -1 : × bek þ bik μ2 þ ðbe⊥ þ bi⊥ Þð1 - μ2 Þ

Dðμ2 Þ =

ð5:19Þ

The potential profile can be found by equalizing the electron and ion fluxes in the ζ direction: φ=

Dek - Dik μ2 þ ðDe⊥ - Di⊥ Þð1 - μ2 Þ ln n þ const: bek þ bik μ2 þ ðbe⊥ þ bi⊥ Þð1 - μ2 Þ

ð5:20Þ

Equation (5.18) has the form of the ambipolar diffusion equation. However, this equation cannot be obtained by equalizing the full fluxes of the charged particles since Γek ≠ Γik, Γe⊥ ≠ Γi⊥ coincides only the ζ projections of the fluxes. In other words, even in the absence of current in the ζ direction, there is still current flowing along the η-axis. Let us discuss the physical mechanism that controls the formation of particle fluxes. If the density gradient is almost perpendicular to the magnetic field, then it satisfies inequality μ ≪ μ0, with μ0 = bi⊥ =bek

1=2

,

Equation (5.19) can be simplified. In this case, we have

ð5:21Þ

118

5

Diffusion of Partially Ionized Magnetized Plasma

D μ2 = De μ2 = ð1 þ T i =T e Þ Dek μ2 þ De⊥ 1 - μ2 :

ð5:22Þ

The diffusion coefficient, Eq. (5.22), is (1 + Ti/Te) times larger than the unipolar coefficient Deζζ. In particular, for μ = 0, when the density gradient is perpendicular to the magnetic field, the coefficient, Eq. (5.22), coincides with the coefficient of ambipolar diffusion, Eq. (5.13). For μ ≪ μ0, the potential given by Eq. (5.20) is negative since electrons are less mobile than ions. The electric field accelerates electrons, while the potential corresponds to the Boltzmann distribution of ions. As seen from Eq. (5.22), there is a critical angle μe = be⊥ =bek

1=2

:

ð5:23Þ

For μ < μe, electrons diffuse mainly across the magnetic field, and the diffusion coefficient, Eq. (5.22), practically coincides with the coefficient of ambipolar diffusion across the magnetic field, Eq. (5.13). On the other hand, at angles μe ≪ μ ≪ μ0, → electrons move mainly along the magnetic field B , and the diffusion coefficient D(μ2) exceeds the perpendicular coefficient of ambipolar diffusion. Ions in both cases diffuse mainly across the magnetic field since their unipolar coefficient Diζζ ≈ Di⊥. As a result, the fluxes of electrons and ions along and across the magnetic field are essentially different. For large angles μ > μ0 between the density gradient and direction normal to the magnetic field diffusion coefficient, Eq. (5.19) is reduced to D μ2 = Di μ2 = ð1 þ T e =T i Þ Dik μ2 þ Di⊥ 1 - μ2 :

ð5:24Þ

We see that for these angles, diffusion is controlled by ions. The potential for Te~Ti corresponds to the Boltzmann distribution for electrons. Now, electrons move → mainly along the magnetic field B , while the direction of the flux depends on the value of μ with respect to a parameter μi = bi⊥ =bik

1=2

:

ð5:25Þ

For μ < μi, the ion flux is mainly caused by the perpendicular motion of ions, while in the opposite case, ions move mainly along the magnetic field. At a special angle μ0 = (Di⊥/Dek)1/2 = μ0(Ti/Te)1/2, the unipolar diffusion coefficients of electrons and ions become equal Deζζ = Diζζ = Di⊥, diffusive fluxes in the direction of the density gradient are also equal to each other, and, therefore, an electric field is absent. At this angle, although the density gradient is almost → perpendicular to B , coefficient D(μ2) is equal to Di⊥. It is much larger than the ambipolar perpendicular diffusion coefficient given by Eq. (5.13). The dependence D(μ2) is shown in Fig. 5.5. Since in the two limiting cases μ = 0, 1, the initial equation system is reduced to the single equation of ambipolar diffusion, one would suppose that diffusion at an

5.2

One-Dimensional Diffusion in Magnetized Plasma

119

Fig. 5.5 Function D(μ2) (solid line) for Te = Ti, xe = 103, xi = 10. Dotted line – different approximations: 1 – Eq. (5.22), 2 – Eq. (5.24), 3 – Eq. (5.27)

arbitrary angle with a magnetic field could be described by a simple equation of anisotropic diffusion

∂n - ∇k
Dak ∇k n - ∇⊥
ðDa⊥ ∇⊥ nÞ = I - R, ∂t

ð5:26Þ

with the coefficients defined by Eqs. (4.3) and (5.13). However, Eq. (5.26) can be obtained by simultaneously assuming Γek = Γik and Γe⊥ = Γi⊥, which requires a nonpotential electric field. Hence, Eq. (5.26) is incorrect. For the 1D problem from Eq. (5.26), it follows that Da μ2 = Dak μ2 þ Da⊥ 1 - μ2 :

ð5:27Þ

This expression strongly differs from the correct Eq. (5.19), especially for μ~μ0, Fig. 5.5. The current along the equidensities can be calculated using jη = en - Diηζ - Deηζ

∂n ∂φ - n biηζ þ beηζ : ∂ζ ∂ζ

ð5:28Þ

Substituting potential equation (5.20), we find jη =

enDek ð1 þ T i =T e Þbi⊥ sin β cos β ∂n : ∂ζ bek cos 2 β þ bi⊥ sin 2 β

The current is zero at β = 0, π/2.

ð5:29Þ

120

5.3

5

Diffusion of Partially Ionized Magnetized Plasma

Diffusion of Perturbation in Unbounded Plasma

Consider diffusion of small perturbation δn on the constant background n0 in a magnetized unbounded plasma. Let us assume for simplicity Te = Ti. The basic equations in the plasma with constant temperatures of the charged particles are given by Eq. (5.1) in the absence of sources and sinks ∂n - ∇
De ∇n - be n∇φ = 0, ∂t ∂n - ∇
Di ∇n þ bi n∇φ = 0: ∂t

ð5:30Þ

The boundary conditions for density and potential are: →

→

n r → 1 = n0 ,

φ r → 1 = 0:

Let us choose the initial perturbation in the form of the delta-function →

→

δn r , 0 = Nδ r : The linearized Eq. (5.30) has the form: ∂δn - ∇
De ∇δn - be n0 ∇φ = 0, ∂t ∂δn - ∇
Di ∇δn þ bi n0 ∇φ = 0: ∂t

ð5:31Þ

The solution is sought as the Fourier integral →

δn r ; t = →

φ r ;t =

→→

1 3

ð2πÞ 1

ð2πÞ3

→

δn → exp i k r d k , k

→→

→

φ → exp i k r d k :

:

k

After substitution into Eq. (5.31), one obtains the equation system for the Fourier harmonics:

5.3

Diffusion of Perturbation in Unbounded Plasma

∂n →k

→

∂t ∂n →k ∂t

121

→

→

→

→

→

→

→

→

→

= - k De k δn →k þ n0 k be k φ →k ,

:

ð5:32Þ

= - k Di k δn →k - n0 k bi k φ →k :

Its solution is →

→

k D k - k D k δn →k φ →k = → e → → i→ , k b k þ k b k n0 e

i

→

:

ð5:33Þ

δn →k ðt Þ = δn →k ð0Þ exp - D k t , where →

D k

→

=

→→

→

2 k De k k Di k

→

→

→

→

k De k þ k Di k

= k 2 D μ2 :

ð5:34Þ →

Here, μ = cos β is the cosine of the angle between the vector k and the magnetic →→ field: μ = k B =ðkBÞ. The diffusion coefficient D(μ2) here coincides with that given by Eq. (5.19): D μ2 =

2 Dek μ2 þ De⊥ ð1 - μ2 Þ Dik μ2 þ Di⊥ ð1 - μ2 Þ : Dek þ Dik μ2 þ ðDi⊥ þ De⊥ Þð1 - μ2 Þ

ð5:35Þ

For the density perturbation, we have →

δn r , t =

N ð2πÞ3

→→

→

exp i k r - D μ2 k2 t d k :

ð5:36Þ

→

To calculate this integral, let us divide k -space into three regions. In the first → region, the vector k is almost perpendicular to the magnetic field: μ ≪ μ0. In this region, as has already been discussed in the previous section, the diffusion coefficient Eq. (5.35) has a simple form of Eq. (5.22): D μ2 = De μ2 = 2 Dek μ2 þ De⊥ 1 - μ2

= 2 Dek k2z þ De⊥ k 2⊥ =k 2 :

In the integral Eq. (5.36) let us switch to Cartesian coordinates. Neglecting small contributions, one can extend integration to infinity. Let us then introduce new coordinates:

122

5

z = z Dek =De⊥

1=2

,

Diffusion of Partially Ionized Magnetized Plasma

kz = kz Dek =De⊥

1=2

:

In the new variables, the integral equation (5.36) is reduced to integral equation (4.19), which can be easily calculated. Hence, the contribution from region μ ≪ μ0 in → the k -space is (in the initial coordinates) ne =

N exp - z2 =8Dek t - ðx2 þ y2 Þ=8De⊥ t : p 1=2 16 2π3=2 t 3=2 Dek De⊥

ð5:37Þ

→

The second contribution in the k -space gives region 1 > μ ≫ μ0. In this region, the diffusion coefficient, Eq. (5.35), can be approximated as D μ2 = Di μ2 = 2 Dik μ2 þ Di⊥ 1 - μ2

= 2 Dik k2z þ Di⊥ k2⊥ =k2 :

→

In the integral over k , the integration region can be extended to zero (neglecting small corrections). Introducing, analogously, new coordinates z = z Dik =Di⊥

1=2

,

kz = kz Dik =Di⊥

1=2

,

and calculating the integral, we find that the contribution from region 1 > μ ≫ μ0 is ni =

N exp - z2 =8Dik t - ðx2 þ y2 Þ=8Di⊥ t : p 1=2 16 2π3=2 t 3=2 Dik Di⊥

ð5:38Þ

Finally, the contribution from region μ~μ0 can be estimated, and it is of the order of n0h, where h=

16

p

N 1=2 2π3=2 t 3=2 Dek Di⊥

:

ð5:39Þ

In the center of inhomogeneity, this contribution is a small correction. Therefore, the density perturbation may be expressed as a sum →

δn r , t = ne þ ni þ Oðn0 hÞ: The potential is calculated analogously and is given by an integral

ð5:40Þ

5.3

Diffusion of Perturbation in Unbounded Plasma

123

→

→

→

Fig. 5.6 Equidensities δn r , t =δnð0, t Þ. Parameters xe = 30, xi = 0.3; z B ; distances in units of (8Di||t)1/2. Depletion regions are dashed

→

φ r ,t =

N ð2πÞ3

→→ → Dek μ2 - Di⊥ ð1 - μ2 Þ 1 exp i k r - D μ2 k2 t d k : ð5:41Þ bek μ2 þ bi⊥ ð1 - μ2 Þ n0

In region μ ≪ μ0, the multiplier in front of the exponent equals -T/e, and in region 1 > μ ≫ μ0, it equals T/e. Integration in Eq. (5.41) yields →

φ r ,t = -

T ne T n i T þ þ O n0 h : e n0 e n0 e

ð5:42Þ

According to Eq. (5.40), the density perturbation is approximately the sum of two Gaussian profiles – “electron” ne and “ion” ni, and lines of constant density represent the superposition of two ellipsoids (Fig. 5.6). The potential is shown in Fig. 5.7. The physical mechanisms responsible for the unusual behavior of the density profile are as follows. Electrons, whose mobility along the magnetic field is the → largest, start diffusing along B . Correspondingly, the potential along the magnetic field becomes negative. Across the magnetic field, the situation is inversed-ions → mobility across B in the magnetized plasma is larger than that of electrons, and, therefore, the potential here is positive. As a result, the potential of a quadrupole configuration arises, as shown in Fig. 5.7, with the saddle point at the origin. Since φ(r → 1) = 0, a potential minimum arises at z = ± z0 along the magnetic field and a → → potential maximum across B at ρ = ρ0 (ρ = ðx2 þ y2 ). The φð r → 1Þ = 0 →

electric field is not able to confine electrons along B everywhere, as well as ions across the magnetic field. As a result, electrons and ions diffuse practically

124

5

Diffusion of Partially Ionized Magnetized Plasma

Fig. 5.7 Equipotentials. Potential in units of Tδn(0,t)/en0, other notations as in Fig. 5.6 →

independently along and across B , correspondingly forming two separate ellipsoids. Particles with other charges cannot diffuse to these ellipsoids from the origin due to their low mobility. Hence, electrons and ions from the ambient plasma should come to “ion” and “electron” ellipsoids correspondingly to compensate space charge to maintain quasineutrality. Their fluxes are driven by an electric field. Therefore, in the ambient plasma depletion regions with reduced ambient plasma density should be formed (dashed regions in Fig. 5.6). The dimensions of the depletion regions are → → (8Dekt)1/2 along B and (8Di⊥t)1/2 across B . Ions are extracted from this region → across B by the electric field towards the electron “ellipsoid,” and electrons – along → B to the ion “ellipsoid.” These fluxes form eddy “short circuiting” currents in the ambient plasma, which are responsible for fast, practically unipolar diffusion of the perturbation. Angle μ0 with respect to the magnetic field, at which depletion regions are formed, can be estimated by equating the characteristic time scales for electrons to → → move along B driven by the electric field and for ions across B . From the condition z2/(bekφ)~ρ2/(bi⊥φ), one obtains μ0 = ρ/z = (Di⊥/Dek)1/2. The potential has a quadrupole character, and, hence, at large distances, it should decrease as r-3. Since far from ellipsoids particle motion is mainly determined by fluxes driven by the electric field, divergence of the fluxes there decreases with distance as r-5, while density perturbations decrease as δn~t/r5.

5.3

Diffusion of Perturbation in Unbounded Plasma

125

Let us calculate the density perturbation and potential far from the ellipsoids. In →

→

→

→

the condition of equal divergences of the fluxes ∇
Γ e = ∇
Γ i , which follows from Eq. (5.30), one can neglect small terms of the order of De⊥/Di⊥ and Dik/Dek. Let us introduce a new variable ζ and a new function Ψ according to ζ = z bi⊥ =bek

1=2

,

Ψ = φ - ðT=eÞ lnðn=n0 Þ:

ð5:43Þ

Then, the condition of equal divergences of the fluxes is reduced to the form ∇
ðn∇ΨÞ = -

2T Δ n: e ⊥

ð5:44Þ

This Eq. (5.44) is the Poisson equation. The net “charge” is Q=

ε0 T Δ ndζdxdy = 0: en0 ⊥

ð5:45Þ

The “dipole moment” is also zero due to the symmetry of the problem. “Quadrupole moment” is given by the integral M=

2ζ2 - x2 - y2

8Tμ0 N ε0 T Δ ndζdxdy = , en0 en0 ⊥

ð5:46Þ

and the quadrupole potential is Ψ=M

2ζ2 - x2 - y2 16πε0 ζ2 þ x2 þ y2

5

:

ð5:47Þ

At large distances φ = Ψ, in the initial coordinates outside ellipsoids, we have →

φ r

=

2 2 2 NT μ0 2μ0 cos α - sin α , 2πen0 r 3 μ2 cos 2 α þ sin 2 α 5=2 0

ð5:48Þ

where α is the angle with the z-axis. Density perturbation could be obtained, for example, from the ion equation, where only perpendicular mobility should be taken into account, ∂n = ∇⊥
ðbi⊥ n∇⊥ ΨÞ = bi⊥ n0 Δ⊥ Ψ: ∂t Substituting potential Eq. (5.48), one finds

ð5:49Þ

126

5 →

Diffusion of Partially Ionized Magnetized Plasma

δn r , t = ×

NTDi⊥ t πμ40 r 5

12 cos 2 α - 36 cos 2 α sin 2 α=μ20 þ ð9=2Þ sin 4 α=μ40 cos 2 α þ sin 2 α=μ20

9=2

:

ð5:50Þ

This expression corresponds to the density perturbation outside ellipsoids. One can see that at angles α~μ0, the density perturbation is negative, so depletion regions are formed in the ambient plasma. When the density perturbation becomes stronger, the depletion regions become deeper, and, at some critical density, the short-circuiting mechanism by the ambient plasma becomes ineffective. In the limiting case of a very large initial density perturbation, when the depth of the depletion region becomes of the order of n0, the main part of electrons and ions of the initial perturbation diffuse together, and the evolution of the main part is governed by the equation of anisotropic ambipolar diffusion Eq. (5.26). Only a small fraction of the ionized particles diffuse due to the unipolar mechanism, forming “electron” and “ion” ellipsoids. For further details of nonlinear evolution, see [7].

5.4

Diffusion in Plasma Restricted by Dielectric Walls

Let us analyze plasma decay in a cylinder with dielectric walls of length L and radius a with the z-axis parallel to the magnetic field. The initial perturbation is supposed to have azimuthal symmetry, with arbitrary dependence on r and z. The boundary conditions at the dielectric surfaces are: nðz = 0, LÞ = 0, nðr = aÞ = 0, Γek ðz = 0, LÞ = Γik ðz = 0, LÞ, Γe⊥ ðr = aÞ = Γi⊥ ðr = aÞ:

ð5:51Þ

It is instructive to introduce four characteristic time scales for the diffusion of electrons and ions along and across the magnetic field ταk =

L2 , π2 Dαk

τα⊥ =

a2 : ð2:4Þ2 Dα⊥

ð5:52Þ

In the magnetized plasma, τe⊥ ≫ τi⊥ and τek ≪ τik. Let us separate diffusion processes into fast and slow. We shall consider processes to be fast if they occur at time scales τf = max (τek, τi⊥). At such time scales, → → it is possible to neglect ion diffusion along B and electron diffusion across B , so the initial equations are reduced to

5.4

Diffusion in Plasma Restricted by Dielectric Walls

∂n = ∂t ∂n = ∂t

∂ ∂n ∂φ Dek - nbek , ∂z ∂z ∂z ∂φ ∂n 1∂ : þ bi⊥ n r Di⊥ r ∂r ∂r ∂r

127

ð5:53Þ

At small time scales of the order of τf, the particles with small diffusion coefficients → → in the corresponding direction, i.e., ions along B and electrons across B , to not have enough time to escape from plasma to the walls. Since the current to dielectric walls should be zero, and Γik(z = 0, L ) = 0, Γe⊥(r = a) = 0 at τ~τf, we have for τ~τf: Γek ðz = 0, LÞ = Γik ðz = 0, LÞ = 0,

Γe⊥ ðr = aÞ = Γi⊥ ðr = aÞ = 0:

ð5:54Þ

Let us seek the solution of Eq. (5.53) for the case when (the opposite case could be considered analogously) τek ≪ τi⊥ :

ð5:55Þ

When this inequality is satisfied, each term on the r.h.s. of the electron Eq. (5.53) is larger than the terms in the ion equation. Since the r.h.s. of these two equations are equal, the terms on the r.h.s. of the electron equation must practically compensate for each other. The corresponding potential is given by the Boltzmann distribution for electrons: φ=

Te ln n þ Ψðr Þ, e

ð5:56Þ

where Ψ(r) is an arbitrary function of the radial coordinate. Let us now integrate Eq. (5.53) over z from 0 to L along the cylinder length. Since the electron flux at the cylinder ends is zero, Eq. (5.54), we find ∂N 1 1 ∂ ∂N 1 ∂Ψ ∂N 1 , r ð1 þ T e =T i ÞDi⊥ = 0, = þ bi⊥ N 1 r ∂r ∂t ∂t ∂r ∂r where N 1 =

L

ð5:57Þ

ndz. For time scales τ~τf, the particles do not leave the plasma volume

0

to the cylinder ends, and, therefore, according to Eq. (5.57), the quantity N1 = const(t) is conserved. Hence, it can be calculated using the initial density profile L

N 1 ðr Þ =

nðr, z, t = 0Þdz: 0

128

5

Diffusion of Partially Ionized Magnetized Plasma

The r.h.s. of ion equation of Eq. (5.57) is zero, the ion flux to the sidewalls is also zero. Hence, Ψ= -

Te þ Ti ln N 1 þ const: e

ð5:58Þ

Substituting potential Eqs. (5.56) and (5.58) into the ion equation of Eq. (5.53), we have ðT þ T i Þ ∂ ln N 1 ∂n 1 ∂ ∂n = r ð1 þ T e =T i ÞDi⊥ : - bi⊥ e n r ∂r e ∂t ∂r ∂r

ð5:59Þ

This is the equation for the density evolution at a time scale of the order of τ~τi⊥. During this evolution, the density profile changes over time due to particle redistribution in the plasma volume without escaping to the walls. At the end of this fast stage, the profile, which corresponds to Γi⊥ = 0, is established. According to Eq. (5.59) at the end of this fast stage n = N 1 ðr Þf ðzÞ: The arbitrary function f(z) can be found by integrating this expression over the radius and introducing a new integral N2(z), which is also conserved, according to a

N 2 ðzÞ =

2πnrdr =

0 a

L

2πnðr, z, t = 0Þrdr, 0 L

a

2πnrdzdr =

N= 0

a

0

2πnðr, z, t = 0Þrdzdr: 0

0

As a result, at the end of the fast stage, the following density profile is established: n=

N 1 ðr ÞN 2 ðzÞ : N

ð5:60Þ

The potential, which corresponds to Eq. (5.60) can be obtained from Eqs. (5.56), (5.58): φ=

Te T ln N 2 - i ln N 1 þ const: e e

ð5:61Þ

Therefore, at time scales τ~τf, the profiles that are given by Eqs. (5.60) and (5.61) are formed due to the redistribution of ions and electrons inside the plasma volume, while integrals N1, N2, and N remain constant. An example of such redistribution is

5.4

Diffusion in Plasma Restricted by Dielectric Walls

129

Fig. 5.8 Scheme of short-circuiting of the electron and ion fluxes in a dielectric cylinder at time scales τ~τf

shown in Fig. 5.8. The short-circuiting currents that flow during plasma redistribution are similar to those analyzed in the previous section. The case τek ≫ τi⊥ could be analyzed in a similar way, and after plasma redistribution at time scale τ~τf, the profiles Eqs. (5.60) and (5.61) are established. For larger time scales t > τf, particle escape to the walls should be taken into account, and the full equation system (5.50) should be solved. However, as the initial condition, the density distribution Eq. (5.60) could be taken. This is an important simplification since this profile is a product of two functions depending on r and z correspondingly. Hence, a solution for density could also be sought in the form of the product of two functions: n = n1(r)n2(z), while the potential could be sought as a sum φ = φ1(r) + φ2(z). The latter at t = 0 coincides with Eq. (5.61). Using the separation of variables method, one obtains n=

Aj Bk J 0 ðζk r=aÞ sinðπjz=LÞ exp - t=τjk , j, k = 1

where 1 = ð1 þ T e =T i ÞDik ðπj=LÞ2 þ ð1 þ T i =T e ÞDe⊥ ðζk =aÞ2 : τjk

ð5:62Þ

130

5

Diffusion of Partially Ionized Magnetized Plasma

The density profile Eq. (5.62) coincides with the solution of the equation of anisotropic ambipolar diffusion ∂n = ∇k
Dak ∇k n þ ∇⊥
ðDa⊥ ∇⊥ nÞ: ∂t At large times τ > τf, ionized particles escape from the volume with equal fluxes, and the decay time is determined by the smallest of two large time scales τs = min (τik, τi⊥). Finally, we see that plasma decay in the dielectric tube occurs in two stages. During the first one, at a fast time scale of the order of τf, the profile Eq. (5.60) is formed due to electron diffusion along the magnetic field and ion diffusion across the field without escape to the dielectric walls. During this stage, short-circuiting currents flow in the plasma. At the second slow stage with the time scale τs joint ambipolar diffusion of electrons and ions takes place, and the density profile is described by Eq. (5.61).

5.5

Diffusion in a Cylinder with Conducting Walls

In a cylinder with conducting walls, short-circuiting currents can flow inside the walls of the conducting vessel. In this case, ionized particles can escape to the walls at a time scale τf = max (τek, τi⊥), i.e., much faster than in the tube with dielectric walls. This phenomenon is known as the Simon effect. Let us analyze plasma decay in such a vessel, and for simplicity, we shall assume up-down and azimuthal symmetry and consider a “short” cylinder with τek ≪ τi⊥. In contrast to the previous case, boundary conditions should be imposed on the potential – φ = 0 at the surfaces and not on the fluxes. Due to condition τek ≪ τi⊥, the potential of the end wall is strongly negative with respect to the plasma, and if the sheath adjacent to this wall can be treated as collisionless, the potential difference between the wall and a point in the plasma (close enough to the end that the problem can be assumed 1D along the magnetic field), according to Eq. (3.20), is T φðr, z, t Þ = - e ln e

p

2πΓek ðr, z = 0, L, t Þ n

T e =me

:

ð5:63Þ

Here, we used the fact that the potential at the sheath edge and the potential in plasma at the same magnetic field line are connected by the Boltzmann relation for electrons: φ/φs = (Te/e) ln n/ns. In a “short” device (τe|| < τi⊥), we can again, as in the case of a dielectric device, seek the potential in the form of Eq. (5.56). Hence, Eq. (5.63) holds nearly throughout the whole plasma volume, except the immediate vicinity of the sidewall. We also assume, as in the previous section, that both slow times τi|| and τe⊥ exceed the two fast times τe|| and τi⊥, so that the small coefficients De⊥ and Di|| can be neglected in the transport equations. In this approximation, the electron flux to the end walls is related to the change in the number of particles in column N1(r,t) by

5.5

Diffusion in a Cylinder with Conducting Walls

131

∂N 1 = - 2Γej j ðr, z = 0, t Þ: ∂t

ð5:64Þ

Substituting Eq. (5.64) into Eq. (5.63), we find eφðr, z, t Þ = - T e ln

π=2j∂N 1 =∂t j n

T e =me

:

ð5:65Þ

Substitution of the potential into the second Eq. (5.53) leads to 2

n∂ N 1 =∂r∂t ∂n T 1 ∂ ∂n T 1 ∂ = 1þ e rDi⊥ - e rDi⊥ T i r ∂r T i r ∂r ∂t ∂N 1 =∂t ∂r

:

ð5:66Þ

There is a small region δ adjacent to the sidewall, where electron motion cannot be considered 1D, and Eq. (5.66) is not correct. Near the sidewall, the electrons are trapped both along and across the magnetic field. The potential with respect to the end wall is given by Eq. (5.63), while the potential with respect to the sidewall is described by an analogous equation provided that the electron gyroradius exceeds the sheath width (see Sect. 3.4): near the sidewall, the perpendicular electron flux can be expressed through the potential as Γe⊥  nρce νeN expð- eφ=T e Þ

ð5:67Þ

Combining Eq. (5.67) with Eq. (5.63), we obtain ρce νeN Γe⊥ ðr = a, z, t Þ  T e =me Γek ðr = a - δ, z = 0, t Þ The ratio Γe⊥/Γe|| is thus very small, and only in the thin layer of width δ near the sidewall, where the divergences of parallel and perpendicular fluxes are comparable, does the electron motion become two-dimensional. Therefore, the width of this layer is δ  LΓe⊥ =Γek  LνeN =ωce :

ð5:68Þ

Equation (5.66) has the particular solution nðr, z, t Þ = AJ 0 ðζ1 r=aÞf ðzÞ expð- t=τi⊥ Þ,

ð5:69Þ

which corresponds to the decay with the “short-circuiting” τi⊥, where τi⊥ is defined by Eq. (5.52). Due to the proper account for the potential drop in the sheath adjacent to the end walls, this decay time differs by the factor (1 + Te/Ti) from the original value derived by Simon from qualitative consideration. The difference becomes

132

5 Diffusion of Partially Ionized Magnetized Plasma

particularly important in the case Te ≫ Ti, which is typical for devices with gas discharge plasmas. The potential drop between a point in the plasma and the end wall is determined by Eq. (5.65). For the profile Eq. (5.69), the radial dependencies of the density and particle escape rate in the column, ∂N1/∂t, coincide, and, as a result, the potential is independent of the radius. In other words, the radial electric field is absent almost in the whole volume of the tube, with the exception of a thin layer of the order of δ, Eq. (5.68), near the sidewalls. By applying voltage between the end and sidewalls, it is possible to control the decay time in the range from τf to τs; for further details, see [7].

5.6

Diffusive Probe in Magnetic Field

In the magnetized plasma, the ionized particles are collected by the probe in a way different from that analyzed in Sect. 4.2. First, in a strong magnetic field, the motion of charged particles in the vicinity of a small electrode (probe) can be described by the transport equations even when their mean free paths exceed the probe dimensions. We shall show below that the criterion of applicability of the electron description in terms of diffusion and mobility is ρce ≪ a,

ð5:70Þ

where ρce is the electron gyroradius and a is the probe size perpendicular to the magnetic field. The reason is that in the strongly magnetized plasma for a positively biased probe, the plasma density is perturbed in a region with the scale lk = aωce/ veN = axe along the magnetic field. Hence, the condition λeN ≪ lk is equivalent to inequality (5.70). For magnetized ions, the fluid description is valid if their gyroradius is smaller than the probe size. However, in the opposite situation, part of the results obtained below remains valid for the positively biased probe. Therefore, the fluid approach in magnetized plasma can be used even in plasma with relatively rare collisions. Density and potential profiles in the fluid approximation are described by the equation system →

→

→

→

→

→

∇
De ∇n - be n∇φ = 0; ∇
Di ∇n þ bi n∇φ = 0: →

ð5:71Þ

→

with the boundary conditions n r → 1 = n0 ; φ r → 1 = 0. For electron saturation current, potential distribution coincides with the Boltzmann distribution

5.6

Diffusive Probe in Magnetic Field

133

for ions φ = - (Ti/e) ln n/n0, which corresponds to the absence of the ion current to the probe. The electron equation in Eq. (5.71) then has the form 2

De⊥ ∂ ∂n ∂ n r þ Dej j 2 = 0: r ∂r ∂r ∂z

ð5:72Þ

By coordinate transformation z′ = z(De⊥/Dek)1/2, this equation is reduced to the Laplace equation Δn = 0:

ð5:73Þ

The probe, which has the form of an ellipsoid of rotation with semiaxes a, a, b, in the new coordinates is transformed into a disk with semiaxes a, a, b/xe. The problem is, therefore, reduced to one considered in Chap. 4 for an unmagnetized probe. The solution is sought in ellipsoidal coordinates, in which the initial variables have the form z2 x2 þ y2 þ 2 = 1: 2 a þ ξ b þ ξ x2e

ð5:74Þ

The plasma density is given by

→

n r

 n0 ð 1 - n Þ = n0 1 e

arctg

1 - γ2e γ2e þξ=a2

arctg

1 - γ2e γ2e

,

ð5:75Þ

where γe = b/axe. From Eq. (5.75), one can see that the longitudinal scale of the depletion region is lk = aωce/veN = axe. For b ≪ axe, this longitudinal scale is much larger than the longitudinal probe size. The electron saturation current in the new coordinates coincides with Eq. (4.40), while the capacity of a disk with small longitudinal semiaxis b/xe is C = 2a/π. When returning to the initial coordinates, one must take into account that the saturation current is also transformed. Finally, I sat e = 8en0 a xe ð1 þ T i =T e ÞDe⊥ :

ð5:76Þ

The large longitudinal scale of the perturbed region in plasma, from which the electron saturation current is collected, leads to a strong restriction on the probe’s perpendicular size. Diagnostics should not change the global plasma distribution, and this is true if lk ≪ L, where L is the typical plasma scale along the magnetic field. This condition is not always satisfied in magnetized plasma.

134

5

Diffusion of Partially Ionized Magnetized Plasma

Fig. 5.9 Dependence of the transverse ambipolar diffusion coefficient in a helium plasma on the magnetic field; sharp increase at B = Bc is due to transition to the turbulent state. Dashed line corresponds to the classical theory, a = 2 cm, L = 75–80 cm

5.7

Experiments in Laboratory Plasma

Diffusion in magnetized plasma has been studied in many laboratory experiments. The first example is related to diffusion perpendicular to the magnetic field in dielectric tubes. Classical ambipolar diffusion has been observed, e.g., in decaying plasma where instabilities have been stabilized by a metal electrode perpendicular to the magnetic field. In Fig. 5.9, the perpendicular ambipolar diffusion coefficient, which was calculated from the measured plasma lifetimes, is displayed versus magnetic field strength. The result is in reasonable agreement with the classical values given by Eq. (5.13), the ambipolar diffusion coefficient is inversely proportional to the squared magnetic field strength. The second example is connected with the density gradient inclined to the magnetic field considered in Sect. 5.2.2. If the magnetic field is parallel to the side surface of the dielectric tube, the particle lifetime in decaying plasma is governed by the perpendicular ambipolar diffusion coefficient. For small inclination angles, the decay time dramatically decreases (Fig. 5.10), in accordance with Eq. (5.19). The plasma decay in a cylinder with conducting surfaces is shown in Fig. 5.11. When the end and sidewalls are disconnected, the time of plasma decay is the same as in the dielectric tube, Eq. (5.62). The acceleration of the decay process can be initiated at any arbitrary moment by short-circuiting of the end and the sidewalls. Before short-circuiting, the slow ambipolar decay process, as described in the preceding section, took place, and after the short-circuiting of the walls, the regime of fast diffusion was observed. The decay rate after short-circuiting increased drastically by more than two orders of magnitude.

5.7

Experiments in Laboratory Plasma

135

Fig. 5.10 Effect of inclination of dielectric tube on the particle lifetime. Experiments in He in a tube of length 90 cm, B = 0.1T. Angle between the tube axis and the magnetic field is α = Δx/L. Different types of dots correspond to the inclination in two perpendicular directions. Solid line – calculations by Eq. (5.19) for L → 1 with d replaced by πa/ζ

Fig. 5.11 Decay of helium plasma in a metallic chamber; a = 2 cm, p = 13.3 Pa, B = 0.2T. The end electrode was simulated by a cylinder in a weak field. Arrow indicates moment when the end and side electrodes are connected; x – ambipolar decay, o – short circuited walls

Chapter 6

Partially Ionized Plasma with Current

6.1

Plasma with Net Current in the Absence of a Magnetic Field

As shown in Chap. 4, in pure plasma, which consists of electrons and one species of ions, the net current through the plasma does not change the density evolution in the absence of a magnetic field. The equation system, Eq. (4.1), for pure plasma is reduced to the equation for ambipolar diffusion equation (4.2) as in the absence of the current, while the current reveals itself only in the additional electric field, Eq. (4.14), which is proportional to the current. The situation becomes essentially different when two or more ion species are present in plasma. Then, as we shall see, the convection of different quasineutral perturbations takes place. Let us consider the evolution of multispecies plasma with р ionized particles: p - 1 ion species and electrons. The basic equations are ∂nα ∂n ∂ Dα α - Z α nα bα E = 0; ∂t ∂x ∂x ∂ne ∂ ∂n De e þ be ne E = 0; ∂t ∂x ∂x

ð6:1Þ

p-1

ne =

Z α nα , α=1

where subscript α corresponds to ion species. The boundary conditions for Eq. (6.1) are:

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Rozhansky, Plasma Theory, https://doi.org/10.1007/978-3-031-44486-9_6

137

138

6

nα ð x → ± 1 Þ = ne ðx → ± 1 Þ = E ð x → ± 1Þ =

Partially Ionized Plasma with Current

n0α ; n0 ; E 0 ≠ 0:

ð6:2Þ

The last condition implies that a finite electric field E0 exists at infinity, which is responsible for the flow of current j. Subtracting the sum of ion particle balance equations from the electron balance → equation in Eq. (6.1), one obtains the current continuity equation ∇  j = 0. In the 1D case, this condition corresponds to the current conservation p-1

j = eE0 be n0 þ α=1

Z 2α bα nαð0Þ = eE be ne þ

∂n þe De e ∂x

p-1 α=1

Z 2α bα nα ð6:3Þ

p-1

∂n Dα Z α α : ∂x α=1

Evaluating the electric field from this equation and substituting it into ion balance equation (6.1), we obtain the equation system for ion densities p-1

∂nα ∂ n0 Z β be þ Z β bβ þ E0 ∂t ∂x β = 1 β

p-1 β=1

p-1

þ

β=1 ∂n ∂ - D α α - Z α bα nα p - 1 ∂x ∂x β=1

Z α bα nα be þ Z β bβ Z β nβ

De - Dβ Z β

∂nβ ∂x

ð6:4Þ = 0:

be þ Z β bβ Z β nβ

In the pure plasma with ions of one species p = 2 with Z = 1 second term on the l.h.s. turns to zero since the sum in the denominator is equal to (be + bi)n1 and cancels with the nominator, so Eq. (6.4) is reduced to ambipolar diffusion equation (4.2). In contrast, in multispecies plasma with several species of ions, this term is quite important, and the ambient electric field E0 to a large extent determines the evolution of the inhomogeneity. Hence, the situation in pure plasma should be considered a degenerated case. Let us analyze the solution of Eq. (6.3) in a so-called drift approximation, when the electric field associated with the current is strong enough, the following inequality is satisfied: E0 ≫ T=ðeLÞ,

ð6:5Þ

where L is the spatial scale of the inhomogeneity. In this situation, diffusive processes play a modest role, and the corresponding term (third one on the l.h.s. of

6.1

Plasma with Net Current in the Absence of a Magnetic Field

139

Eq. (6.4)) can be neglected. For simplicity, let us restrict ourselves to two species of ions, 1 and 2, with Z1, 2 = 1. Then, in the drift approximation, the equation system (6.4) with account of b1, 2 ≪ be is reduced to ∂n1 ∂ n1 = 0, þ b1 E 0 n0 ∂t ∂x n1 þ n2 ∂ n2 ∂n2 = 0: þ b2 E 0 n0 ∂x n1 þ n2 ∂t

6.1.1

ð6:6Þ

Small Perturbations

Let us find the solution of Eq. (6.6) in the drift approximation for small perturbations of ion densities nα = δnα þ n0α ,

δnα ≪ n0α :

The solution of linearized Eq. (6.6) is sought as the Fourier integral þ1

1 δn1,2 ðx, t Þ = 2π

δn1,2k expðikx - iωt Þdk: -1

After substitution into linearized Eq. (6.6), one obtains an algebraic system of equations for δn1k and δn2k. Since the determinant of this algebraic system should be zero, two roots for the frequency are obtained ω1 = 0,

b1 n02 þ b2 n01 ω2 = V: = E0 k n01 þ n02

ð6:7Þ

The corresponding Fourier transforms are b2 n01 ½δn2k ð0Þ þ δn1k ð0Þb2 =b1  þ b2 n01 þ - δn2k ð0Þ þ δn1k ð0Þn02 =n01 expð - ikVt Þ ; b1 n02 ½δn1k ð0Þ þ δn2k ð0Þb1 =b2  δn2k ðt Þ = b1 n02 þ b2 n01 þ - δn1k ð0Þ þ δn2k ð0Þn01 =n02 expð - ikVt Þ : δn1k ðt Þ =

b1 n02

ð6:8Þ

140

6

Partially Ionized Plasma with Current

Fig. 6.1 Small perturbations in current carrying plasma. Injected ions of species 1 with Gaussian profile of width а are injected into plasma with ambient ions of species 2. (а) Initial moment, (b) moment t = 5a/(b1E0). Solid line – injected ions, dash-dotted line – ambient ions, dotted line – electron density. Mobility ratio: b1 = 2b2 ≪ be

Density perturbations contain moving contribution and zero-velocity parts. The stable contribution corresponds to zero frequency ω1, while the moving part contains the density of two species moving with velocity V. This phenomenon of density perturbation propagation in the electric field is known as ambipolar mobility, while velocity V is known as ambipolar mobility velocity. Let us consider a special case when initially background plasma consists of the ions of species 2, so that n01 = 0, and ions of species 1 are injected into the background plasma at t = 0, so that δn2(x, t = 0) = 0. Then, in the limit n01 → 0 in Eq. (6.8), one finds δn1k ðt Þ = δn1k ð0Þ expð- ikb1 E0 t Þ, δn2k ðt Þ = δn1k ð0Þ

b2 ½1 - expð- ikb1 E 0 t Þ: b1

ð6:9Þ

In the initial coordinates δn1 ðx:t Þ = δn1 ðx - b1 E 0 t Þ, b δn2 ðx, t Þ = 2 ðδn1 ðx, t = 0Þ - δn1 ðx, t ÞÞ: b1

ð6:10Þ

The corresponding profiles are shown in Fig. 6.1. We see that at t = 0, injected ions of species 1 are localized in the place of injection, while the perturbation of background ions of species 2 is absent. The perturbation of injected ions moves with the constant velocity b1E0. In other words, injected ions move in the unperturbed electric field. A small perturbation of the electric field has an impact only on the background ions. According to Eq. (6.1), at the place where ions of species 1 were injected, the peak of the background ions is formed with an amplitude larger or smaller than that of injected ions, depending on the ratio of species mobilities b1/b2. The second negative perturbation of the

6.1

Plasma with Net Current in the Absence of a Magnetic Field

141

background ions of the same amplitude accompanies the peak of injected ions. It is interesting to note that in the special case b1 = b2, the moving perturbation of electron density disappears since positive and negative perturbations of injected and background ions compensate for each other. The initial perturbation of the electron density simply stays at rest, which corresponds to the solution for the case of pure plasma, which is described by the ambipolar diffusion equation. In the drift approximation, this implies the absence of any evolution. Hence, even if the mobilities of ions coincide, a moving signal exists that can be observed, provided the injected and background ions differ by some other physical feature. In the general case in multispecies plasma, the arbitrary initial perturbation splits into р - 1 signals, with one of them having zero velocity. From the solution obtained above, we can conclude that the fact that in the pure current-carrying plasma, the density evolution is described by the same ambipolar diffusion equation as in the currentless plasma results from the exact compensation of several rather complicated disturbances of ambient and injected particle densities. In plasmas of more complex composition, in addition to the fixed density disturbance, several modes of propagating weakly damping signals exist. In each of these modes, the densities of all the plasma components are perturbed.

6.1.2

Nonlinear Evolution

We shall start the nonlinear analysis with the simplest case b2 = 0 when the mobility of one species of positive ions, in particular ambient ions, is negligible. The wellknown situation of this kind corresponds to an n-type semiconductor, where posið0Þ tively charged fixed donor ions with spatially uniform density n2 are present. Some situations in gaseous and dusty plasmas can also be described by this approximation. Assuming n2 = n02 = const and n01 = 0, in the drift approximation from Eq. (6.6), we obtain the equation for injected ions ∂n1 ∂n þ V ðn1 Þ 1 = 0, ∂t ∂x

ð6:11Þ

where V(n1) is a nonlinear ambipolar drift velocity: V ðn1 Þ =

b1 E0 n02

2

n1 þ n02

2

:

ð6:12Þ

Equation (6.11) describes propagation in a purely nonlinear wave, and its solution is n1 = n1 ðx - V ðn1 Þt Þ:

ð6:13Þ

142

6

Partially Ionized Plasma with Current

Fig. 6.2 Evolution of nonlinear initial Gaussian profile of injected ions in plasma with ambient ions with zero mobility. Dimensionless density is n1 = n1 =n02 , time is t = tb1 E 0 =a, initial perturbation amplitude nmax 1 ðt = 0Þ = 2. (а) Initial profile and (b) t = 15 after shock formation

In other words, each point of the initial profile of injected ions with density n1 propagates with its own velocity V(n1). According to Eq. (6.12), parts of the profile with larger densities move slower than those with smaller densities; hence, the front of the inhomogeneity becomes shallower with time, while the rear steepens (Fig. 6.2). In the evolution process, a multivalued density profile is formed, which is the consequence of the drift approximation used. This phenomenon is similar to the wave overturn in the standard hydrodynamics. To obtain a physically meaningful solution, it is necessary to keep diffusion terms in the initial equation system. Indeed, in the process of evolution, steepening of the profile takes place, and a sharp density gradient – diffusion shock – is formed, where Eq. (6.5) becomes invalid. Typical spatial scale of this diffusion shock lT =

T eE

ð6:14Þ

corresponds to diffusive and convective terms in Eq. (6.4) being of the same order. Shock position, as in hydrodynamics, could be obtained from the “area rule”. At the multivalued profile, the vertical line is chosen so that the area below the multivalued solution and the area below the profile with shock should coincide (Fig. 6.2). The total number of particles is conserved and remains the same for both profiles, while the gradual part of the new profile with shock satisfies the initial equations since inequality (6.5) is fulfilled. Shock velocity propagation is also determined by particle conservation: W=

Γ1 nþ 1 - Γ1 n1 , nþ 1 - n1

ð6:15Þ

where Γ1 = n1b1E is the particle flux, and quantities nþ 1 and n1 correspond to the density on the right and left sides of the shock, respectively. The shock structure, as in standard hydrodynamics, could be obtained by transfer to the reference frame moving with the velocity W and reduction of the partial derivative equations to the ordinary derivative equations.

6.2 Magnetized Plasma with Current

6.2

143

Magnetized Plasma with Current

6.2.1

One-Dimensional Evolution

In the magnetized plasma, the phenomenon of ambipolar mobility reveals itself even in pure plasma with one species of ions. Let us consider plasma evolution in the same geometry as in Sect. 5.2.2 when plasma density depends only on coordinate ζ, forming an angle β with the magnetic field (Fig. 5.4). In contrast to Sect. 5.2, we → now assume the existence of an arbitrary ambient electric field E 0 . In the coordinates (x, η, ζ), the initial equations, analogous to those considered in Sect. 5.2, have the form ∂n ∂ ∂n ∂φ ∂φ ∂φ = I - R, Dαζζ ± bαζζ n ± bαζx n ± bαζη n ∂t ∂ζ ∂ζ ∂ζ ∂x ∂η

ð6:16Þ

where α = i, e, and the components of the diffusion and mobility tensors are given by Eq. (5.16). The boundary condition for the potential is more general: →

→

∇φ r → 1 = - E 0 :

ð6:17Þ

Note that the components of the electric field Eη and Ex are independent of coordinate ζ and coincide with the corresponding ambient values: Eη = E0η and Ex = E0x. This follows from the electrostatic character of the electric field, so that ∂Eη/ ∂ζ = ∂Eζ/∂η = 0 and ∂Ex/∂ζ = ∂Eζ/∂x = 0. The electric field in the ζ direction can be eliminated similar to Sect. 5.2. Multiplying equations for ions and electrons by beζζ and biζζ correspondingly, after summing them up, we obtain ∂n ∂ ∂n ∂n D μ2 þ V ζ μ2 = I - R, ∂t ∂ζ ∂ζ ∂ζ

ð6:18Þ

where Vζ(μ2) is the ζ component of a vector →

→

V μ

2

→

bek μ2 þ be⊥ ð1 - μ2 Þ bi E 0 - bik μ2 þ bi⊥ ð1 - μ2 Þ be E 0 = : bek þ bik μ2 þ ðbi⊥ þ be⊥ Þð1 - μ2 Þ

ð6:19Þ

According to Eq. (6.18), in addition to the diffusion plasma, perturbation has a convective velocity – ambipolar drift velocity. According to Eq. (6.19), if the electric field belongs to the plane (η, ζ), the ambipolar drift velocity is zero for the ζ-axis parallel or perpendicular to the magnetic field but remains finite for an arbitrary angle between the ζ-axis and magnetic field direction. The ambipolar velocity caused by → → the Ex component for magnetized ions corresponds to E × B drift.

144

6

Partially Ionized Plasma with Current

The electric field in the ζ-direction is obtained by subtraction of the ion equation from the electron equation and integration over ζ. The quantity Eζ contains a → diffusive part and a part linear to the ambient field E 0 .

6.2.2

Multidimensional Evolution of Small Perturbation in Unbounded Plasma

The evolution of small perturbation δn at a constant ambient background with → density n0 in the presence of the ambient electric field E 0 qualitatively differs from the diffusive evolution considered in Sect. 5.3. Initial equations for perturbed → density and potential in the presence of the ambient electric field E 0 are given by → ∂n - ∇  De ∇n - be n∇φ þ be n E 0 = 0, ∂t → ∂n - ∇  Di ∇n þ bi n∇φ - bi n E 0 = 0 ∂t

ð6:20Þ

→

with the same boundary conditions as without E 0 : →

n r → 1 = n0 ,

→

φ r → 1 = 0:

Solution of the linearized equation system, Eq. (6.20), with the initial perturbation in → → the form of delta-function δn r , 0 = Nδ r with the Fourier method leads to the expression →

δn r , t = →

N ð2πÞ3

→

exp i k

→

→

→

r - V μ2 t - D μ2 k 2 t d k ,

ð6:21Þ

where velocity V ðμ2 Þ is defined according to Eq. (6.19). This integral was analyzed in detail in [7]. In particular, it was demonstrated that the initial perturbation is separated with time into several moving perturbations. An example of such an evolution when the electric field is perpendicular to the magnetic field is presented in Fig. 6.3. The initial perturbation here is separated into two blobs moving with velocities close to the velocities of electrons and ions in the ambient electric field (Fig. 6.3a). Polarization of the blobs also leads to their deformation, especially for electron blob along the magnetic field. As in the process of diffusion, quasineutrality is maintained by the short-circuiting currents in the ambient plasma and the formation of depletion regions that move together with the blobs. Strong perturbation, when the density of the ambient plasma is not sufficient to provide quasineutrality,

6.2

Magnetized Plasma with Current

145

→

→

Fig. 6.3 Evolution of density perturbation for E 0 ⊥ B . (a) Small perturbation and (b) strong perturbation. Here OO1 and OO2 are the direction of blobs motion, the depletion regions are dashed

mainly remains at rest while its peripheral parts drift (see Fig. 6.3b). As a result, nonlinear deformation of the blob occurs.

6.2.3

Effect of Conductivity Recovery in a Weak Magnetic Field

Here, we shall consider conductivity in a weak magnetic field xexi ≪ 1 when plasma remains unmagnetized. On the other hand, the magnetic field is assumed to be strong enough so that electrons become magnetized, xe ≫ 1. In such plasma, the perpendicular electron mobility is significantly smaller than the parallel electron mobility be⊥ ≪ bek, but in contrast to the situation considered previously, the perpendicular ion mobility is small with respect to the electron mobility, bi⊥ ≪ be⊥. This case is typical, for example, for the plasma of MHD generators or low ionosphere plasma. Let us consider plasma flow in a channel restricted by two infinite dielectric → surfaces. An electric field E 0 is applied in the x-direction parallel to the surfaces, as shown in Fig. 6.4. The boundary condition at the dielectric surfaces corresponds to the absence of the current in the y-direction: jy = 0. Note that the initial electric field → E 0 causes not only current in the х-direction but also the Hall current of electrons in the у-direction. Therefore, the plasma is polarized, as shown in Fig. 6.4, and an additional electric field in the y-direction, Ey, arises. The ion fluxes in both the х- and у-directions are negligibly small and do not contribute to the current. For simplicity, we shall also neglect the diffusive electric field of the order of T/eL with respect to

146

6

Partially Ionized Plasma with Current

Fig. 6.4 Unmagnetized plasma polarization in a channel with dielectric surfaces

the field E0. Then, the polarization field in the у-direction, Ey, can be found from the condition of zero current normal to the surface jy = - e - be^ nE0 - be⊥ nE y = 0:

ð6:22Þ

The field Ey obtained from Eq. (6.22) is negative: E y = - E 0 beΛ =be⊥ :

ð6:23Þ

Since electrons are magnetized, the ratio beΛ/be⊥ ≫ 1, and the polarization field in the y-direction is much larger than the initial field E0. This polarization field Ey → → causes E y × B drift of electrons in the х-direction. The latter together with the small current caused by the field E0 determines the net current along x: jx = - enbeΛ E y þ enbe⊥ E 0 = enE 0 b2eΛ =be⊥ þ be⊥ = enbek E0 :

ð6:24Þ

The effective conductivity is the coefficient, which connects the current density with the applied electric field and coincides with the conductivity in the absence of a magnetic field σk = ebekn, jx = σeff E 0 = σk E0 :

ð6:25Þ

In other words, conductivity across a magnetic field is “recovered” and coincides with the conductivity in the absence of a magnetic field. The main contribution to this effect of conductivity recovery is due to the polarization electric field, which by factor xe exceeds the initial electric field and produces the Hall current of electrons. This result is of course a direct consequence of the chosen boundary condition at the material surfaces, jy = 0. In the opposite limiting case of the two short-circuiting highly conductive surfaces, the current in the у-direction can flow freely, and the effect of conductivity recovery is absent. The polarization electric field in this case does not exist, and the current along the х-axis is simply jx = enbe⊥E0. In addition to

6.3

Plasma Clouds in the Ionosphere

147

these two limiting cases of short-circuited, jy = 0, and disconnected, Ey = 0, surfaces, there are also possible intermediate cases with finite resistance between the plates. The dependence of the effective conductivity on the resistance value is easily calculated analogously to the limiting cases.

6.3

Plasma Clouds in the Ionosphere

Ionospheric plasma provides many examples of density evolution in the presence of currents and global electric fields.

6.3.1

Redistribution of Metal Ions in the Polar Ionosphere. Sporadic Layers

The first example is the formation of plasma layers in the Polar ionosphere. The traditional explanation of the formation of sporadic layers in the inhomogeneous ionosphere plasma consists of the so-called wind shear model. In the reference frame that moves with the neutral gas across the magnetic field, due to the Lorenz transformation, an electric field arises. The inhomogeneous Hall fluxes in these crossed fields result in the piling up of the plasma density. In the lower part of the polar ionosphere, narrow layers of high plasma density with sharp gradients are often observed. To explain these layers in terms of the wind shear theory, it is necessary to assume that the neutral wind changes its direction on a very short spatial scale. On the other hand, the lifetime of these layers is very large, reaching several hours. This contradicts the fact that the characteristic ionization and recombination times of the main ionospheric ions (NO+, O2+) are of the order of 10–20 s. Therefore, these sporadic layers probably consist of metal ions, n1 which are characterized by very long recombination times. Furthermore, in the Polar ionosphere, a considerable current parallel to the Earth’s magnetic field flows. This current in the multispecies ionospheric plasma can cause the formation of shocks analyzed in this chapter in Sect. 6.1. Indeed, the ambient ion density n2 = n02 can be considered constant due to fast recombination. The injected ions of species 1 can be associated with metal ions of meteor origin with zero recombination. Their density may significantly exceed the background density. Then, the situation is similar to that presented in Fig. 6.2 and is described by Eq. (6.11). The shock arises at the rear side of a density hump (if the current outflows from the ionosphere, the shock is formed at the lower side of an inhomogeneity). For typical ionospheric parameters, the values of lT in Eq. (6.14) are of the order of several hundred meters and L is of the order of several kilometers. These numbers are within the experimentally observed range.

148

6.3.2

6

Partially Ionized Plasma with Current

Active Experiments with Barium Clouds

Drift and deformation of strong plasma inhomogeneities were studied in many experiments where barium (and some other) clouds were injected into the ionosphere. In these experiments, the density of injected Ba+ ions was rather high with respect to the ambient plasma density. Therefore, the main fraction of the injected ions for a long time remained at rest in the reference frame of the neutral wind (the main cloud). Simultaneously, part of the injected ions outflowed in the perpendicular → → E × B direction, creating a tail. The motion of this tail is possible due to the shortcircuiting effect (see Sect. 6.2.2). Only one of the two separate plasma clouds, where barium ions are located, can be seen by spectroscopy measurements, and the second blob consists of ambient ions and remains invisible. A photo of the Ba+ cloud is presented in Fig. 6.5. One can see the shock on the rear side of the cloud and the → → outflow of Ba+ in the E × B direction. The cloud is separated into striations due to → → specific instability. Small-density Ba+ clouds move in the E × B direction without significant deformation, and they can be used to measure the electric field in the ionosphere.

Fig. 6.5 Photo of the barium cloud evolution in the “Spolokh” experiment made from the different points of observation (a) and (b). Plasma is separated into striations stretched along the Earth magnetic field. In (b), the line of sight is perpendicular to the outflow direction

Chapter 7

Transport in Strongly Ionized Plasma Across a Magnetic Field

7.1

Classical Diffusion of Fully Ionized Plasma Across a Magnetic Field

First, let us demonstrate that evolution along the magnetic field (or in the absence of a magnetic field) in fully ionized plasma (in contrast to slightly ionized plasma) is not of diffusive character. To show this, let us sum two momentum balance equations for electrons and single-charged ions: →

→ → due = - ∇pe - en E þ R ei ; dt → → → dui mi n = - ∇pi þ en E þ R ie : dt

me n

ð7:1Þ

Both electric forces and friction are canceled during summation, and neglecting electron inertia, one has →

mi n

dui = - ∇p, dt

ð7:2Þ

where p = pe + pi is the net pressure. This Eq. (7.2) describes density evolution with the velocity of the order of a sound speed. Indeed, estimating the inertia term as mi nu2i =L, where L is the characteristic spatial scale, assuming |∇p|~p/L, we obtain ui~cs = [(Te + Ti/)mi]1/2 with cs being the sound speed. The dynamics of such plasma evolution are considered in Chap. 9. In contrast, the evolution of fully ionized plasma across a magnetic field has a diffusive character, similar to the case of partially ionized plasma. The pressure gradient across the magnetic field is balanced by the Lorentz force, and the radial velocity is significantly smaller than the sound speed so that the inertia term in the momentum balance equation can be neglected. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Rozhansky, Plasma Theory, https://doi.org/10.1007/978-3-031-44486-9_7

149

150

7

Transport in Strongly Ionized Plasma Across a Magnetic Field

We consider diffusion in an infinite cylinder placed in a strong homogeneous magnetic field parallel to the z-axis. The plasma density and temperatures depend only on the radial coordinate. Neglecting viscosity and inertia terms in the momentum balance equations, we have (see Sect. 2.3) →

→

→

→

→

3 nνei B × ∇T e = 0, 2 ωce B

→

→

3 nνei B þ × ∇T e = 0: 2 ωce B

- ∇pe þ en∇φ - en u e × B - nme νei u e - u i →

→

- ∇pi - en∇φ þ en u i × B þ nme νei u e - u i

→

ð7:3Þ The sum of these equations gives us the equilibrium equation →

→

∇p = j × B :

ð7:4Þ

According to Eq. (7.4), the pressure gradient is balanced by the Lorentz force, which is caused by the diamagnetic current flowing in the azimuthal direction. The mechanism of diamagnetic current formation is discussed in Chap. 5. Substituting → → the current from the Maxwellian equation ∇ × B = μ0 j , after integrating over radius, one obtains pþ

B2 B2 = const = 0 , 2μ0 2μ0

ð7:5Þ

where B0 is the vacuum magnetic field. As follows from Eq. (7.5), such equilibrium is possible if β = 2μ0 p=B2 < 1:

ð7:6Þ

Note that for β < < 1, the vacuum magnetic field is only slightly perturbed by the magnetic field of diamagnetic currents. Radial projections of the momentum balance equations (7.3) read dpe dφ þ en - enueϑ B = 0, dr dr dp dφ - i - en þ enuiϑ B = 0: dr dr

-

ð7:7Þ

Note that the radial component of the friction force is absent; it is zero since the radial velocities of electrons and ions must coincide; otherwise, a radial nonzero current should flow, which is restricted by the quasineutrality constraint. The thermal

7.1

Classical Diffusion of Fully Ionized Plasma Across a Magnetic Field

151

force also does not contribute to the radial momentum balance; only the poloidal component of the thermal force exists. The azimuthal velocities according to Eq. (7.7) are: 1 dpe 1 dφ þ = upe þ V 0 , eBn dr B dr 1 dpi 1 dφ uiϑ = þ = upi þ V 0 : eBn dr B dr

ueϑ = -

ð7:8Þ

These velocities consist of two contributions. The first part upe, upi represents the diamagnetic flux of electrons and ions associated with rotation over the Larmor radius in inhomogeneous plasma. The origin of diamagnetic fluxes is discussed in → → Sect. 5.2. The second contribution corresponds to E × B drifts in the radial electric field with velocity V0. The azimuthal projections of the momentum balance equations have the form 3 nνei dT e = 0, 2 ωce dr 3 nνei dT e = 0: - enuir B þ nme νei ðueϑ - uiϑ Þ þ 2 ωce dr

enuer B - nme νei ðueϑ - uiϑ Þ -

ð7:9Þ

Substituting the difference in the azimuthal velocities (ueϑ - uiϑ) obtained from Eq. (7.8) into Eq. (7.9), one finds uer = uir = ur = -

me νei dp 3 νei 1 dT e þ : e2 B2 n dr 2 ωce eB dr

ð7:10Þ

In this approximation, electrons and ions have equal radial velocities that are independent of the radial electric field value. This situation is quite different from the case of partially ionized plasma, where ion and electron radial fluxes are essentially different. Sometimes this fact is known as a statement that in fully ionized plasma transport is “automatically ambipolar.” The radial velocity can be interpreted as a particle drift caused by the azimuthal friction force (Fig. 7.1). Indeed, the azimuthal friction force associated with the particle’s diamagnetic velocities with opposite directions and the azimuthal thermal force is applied both to electrons and ions. These forces cause radial drift of ions with velocity uri = Rieϑ/(eBn) and drift for electrons with velocity ure = - Reiϑ/(eBn). Since according to the third Newton law Reiϑ = - Rieϑ and the signs of particle charges are also different, we have uer = uir. Note that for plasma consisting of ions with charge number Z and density ni (n = Zni), the radial electron velocity remains the same, given by Eq. (7.10), while the radial velocity of ions is smaller uri = Rieϑ/(ZeBni). Their radial flux is Z times smaller than that of electrons:

152

7

Transport in Strongly Ionized Plasma Across a Magnetic Field

Fig. 7.1 Fluid velocities in a cylinder of fully ionized plasma oriented along magnetic field

ni uir = nuer =Z: The radial current, however, is still zero since each ion carries charge Ze. In the approximation described above, the radial electric field remains arbitrary. To find the radial electric field, one has to go to the next approximation for the momentum balance equations and take into account corrections connected with the ion viscosity. Indeed, since the azimuthal velocity of ions given by Eq. (7.8) depends on the radius, there is an azimuthal viscous force applied to ions. This force is proportional to the radial gradient of the azimuthal velocity ∂uiϑ/∂r. In the absence of an ion temperature gradient, this force, in accordance with Eq. (2.48), is given by $

- ∇ π

ϑ

=

1 d d ðru Þ : η dr 1 r dr iϑ

ð7:11Þ

A similar electron viscosity is very small and can be neglected. The additional viscous force applied to ions causes additional ion radial flux, which is absent for electrons, and hence produces radial current jr = -

dpi 1 d 1 d η1 r Er 2 dr endr r dr B

:

ð7:12Þ

Since the radial current must be zero in nonuniform plasma, the following radial electric field is established: Er =

T i d ln n : e dr

ð7:13Þ

This field corresponds to the Boltzmann distribution for ions. In the plasma with an inhomogeneous ion temperature, the additional term in Eq. (7.13) arises, which is caused by the viscosity associated with the azimuthal heat flux. The radial velocity in Eq. (7.10) can be rewritten as

7.1

Classical Diffusion of Fully Ionized Plasma Across a Magnetic Field

ur = - D

dn D 1 dT e dT i þ , ndr T e þ T i 2 dr dr

153

ð7:14Þ

where diffusion coefficient D is defined as D=

me νei ðT e þ T i Þ : e2 B2

ð7:15Þ

The first term on the r.h.s. of Eq. (7.14) corresponds to diffusive flux, while the second one represents thermodiffusion. The diffusion coefficient D corresponds to a simple estimate D  ρ2ce νei

ð7:16Þ

is of the order of square of the electron Larmor radius multiplied by the electron–ion collision frequency. Note that the diffusion coefficient is of the order of the electron heat conductivity coefficient χe and is (mi/me)1/2 times smaller than the ion heat conductivity coefficient, which can be estimated as χi  ρ2ci νii . The ion–ion collisions do not lead to particle diffusion because during the collision process, the leading centers of the two ion orbits shift in the opposite direction without a center mass shift. A special feature of fully ionized plasma is the absence of conductivity perpendicular to the magnetic field, in contrast to the partially ionized plasma case. Indeed, → → the electric field causes only E × B drift for both electrons and ions with the same velocity, and, therefore, there is no friction between them, in contrast to the case of partially ionized plasma. In partially ionized plasma, friction with neutral particles causes additional drift of particles, mainly ions, in the direction of the electric field, which results in current, while in fully ionized plasma, such drift is absent. The perpendicular current in the direction of the electric field can flow only in a nonhomogeneous electric field; in this case, it is produced by viscosity as the current given by Eq. (7.12). Classical diffusion in magnetized fully ionized plasma has a specific feature: the diffusion coefficient is proportional to plasma density because the electron-ion collision frequency is proportional to plasma density. As a result, the corresponding diffusion equation becomes nonlinear. In plasma with a constant temperature, the diffusion equation has the form p ∂n 2π e2 m1=2 2 e ðT e þ T i ÞΛ : - εΔn = 0, D = 2εn; ε = 2 ∂t B2 T 3=2 3ð4πε0 Þ e

ð7:17Þ

We shall demonstrate the specific character of diffusion in fully ionized plasma by constructing a self-similar solution of Eq. (7.17). As an example, let us consider a one-dimensional (1D) case. A self-similar solution can be sought in the form

154

7

Transport in Strongly Ionized Plasma Across a Magnetic Field

nðx, t Þ = t γ nðξÞ;

ξ = x=t δ :

ð7:18Þ

Transformation to the new variables ξ, t yields ∂n dn = γt γ - 1 n - δt γ - 1 ξ , dξ ∂t 2 2 2 2 d n d n = t 2γ - 2δ 2 : 2 dx dξ Equation (7.17) is reduced to an ordinary differential equation if the powers of t in all terms of Eq. (7.17) coincide, which is satisfied for γ = 2δ - 1. One more relation should be obtained from the net particle conservation condition ndx = const = t δþγ

ndξ,

which is satisfied if δ = - γ. From these two equalities, one finds δ = - γ, nðx, t Þ =

1 n x=t 1=3 : t 1=3

ð7:19Þ

Function nðξÞ satisfies the ordinary differential equation ε

d 2 n2 ξ dn n þ þ = 0: dξ2 3 dξ 3

ð7:20Þ

Its solution is nð ξÞ =

1 C - ξ2 : 12ε

In the initial variables, it is convenient to rewrite the solution, replacing the integration constant as a ratio of two new constants: nðx, t Þ =

2=3

a2 2=3

12εt 1=3 t 0

1-

x2 t 0 : a2 t 2=3

ð7:21aÞ

According to the obtained solution, the plasma density becomes zero at a finite distance, and, at the moment t = t0, the density becomes zero at x = a. This fact is a consequence of the dependence of the diffusion coefficient on density – it turns to zero with density. The typical spatial scale of the profile rises with time as t1/3, and the density is reduced correspondingly as t-1/3. The solution in the cylindrical geometry can be obtained in a similar way and has the form

7.2

Transport of Impurities in Fully Ionized Plasma Across a Magnetic Field

nðx, t Þ =

7.2

155

1=2

a2 1=2

16εt 1=2 t 0

1-

r2 t0 : a2 t 1=2

ð7:21bÞ

Transport of Impurities in Fully Ionized Plasma Across a Magnetic Field

Consider the transport of heavy impurities with mass mI, density nI, and charge number Z in the cylindrical geometry, the same as in the previous section. The density of the main ions with Z = 1 is ni, nI < < ni, and subscript i also corresponds to other parameters of the main ions. The main friction force applied to impurities is the friction force with the main ions, while the friction with electrons is (me/mi)1/2 times smaller and, therefore, can be neglected. Since we assume mi < < mI, the friction associated with the difference in velocities and thermal force has the same form as in the case of electron-ion collisions (see Sect. 2.3): →

→u R iI

→

→

= - ni mi νiI u i - u I , →

3 νiI B × ∇T i , n 2 i ωcI B p 4 2 2e Z ΛnI : νiI = p 3=2 12π3=2 ε20 mi T i →T R iI

=-

ð7:22Þ

Azimuthal fluxes of impurity ions are obtained in the same way as for main ions: uIϑ =

c dpI c dφ þ = upI þ V 0 : eZBnI dr B dr

ð7:23Þ

Diamagnetic fluxes of the main ions and impurity ions are shown in Fig. 7.2. For not very steep profile of impurities when d ln ni 1 d ln nI >> dr Z dr

ð7:24Þ

diamagnetic velocity of impurity ions is significantly smaller than that of main ions. Therefore, in plasma with constant ion temperatures, the direction of the azimuthal friction force applied to impurities coincides with the main ion diamagnetic velocity. This force causes the drift of impurity ions to the core of a plasma cylinder. In the general case of an arbitrary ion temperature profile, the azimuthal momentum balance with account of Eqs. (7.23) and (7.8) has the form

156

7

Transport in Strongly Ionized Plasma Across a Magnetic Field

Fig. 7.2 Fluid velocities of the main ions and impurities

eZnI 3 ni νiI dT i u B þ ni mi νiI ðuiϑ - uIϑ Þ þ =0 c Ir 2 ωci dr

ð7:25Þ

Substituting azimuthal velocities into this equation, one obtains the radial flux of impurity ions ΓIr  nI uIr =

1 dnI 1 dT i ni mi νiI c2 T i 1 dni : 2 2 dr Zn dr 2 dr n Ze B i I

ð7:26Þ

Since the collision frequency is proportional to the impurity density, the flux Eq. (7.26) contains the diffusive part (second term on the r.h.s.) and convective part (first and third terms on the r.h.s.). For the condition given by Eq. (7.24), practically all radial flux is convective and directed towards the core of the plasma pinch. In the absence of sources and sinks, the core will be enriched by impurity ions until the outwards diffusive flux compensates for the inwards convective flux so that the net flux turns to zero and a stationary profile is established. It corresponds to equality 1 dni 1 dnI 1 dT i = 0, ni dr ZnI dr 2 dr and is rather peaked: - Z=2

nI = AnZi T i

:

ð7:27Þ

In other words, impurities are gathered in the center of the plasma. Note that the radial flux of impurity ions does not produce radial current and thus does not lead to an additional radial electric field. Indeed, since azimuthal friction forces RiIϑ and RIiϑ differ only by the sign, we have

7.3

Partially Ionized Magnetized Plasma with an Inhomogeneous Neutral Component

ZΓIr = - Γir :

157

ð7:28Þ

The counter flux of the main ions caused by impurities is Z times larger than that of impurities, so the resulting radial current equals zero, and the presence of impurities does not change the radial electric field.

7.3

Partially Ionized Magnetized Plasma with an Inhomogeneous Neutral Component

Partially ionized plasma with inhomogeneous neutral density in a magnetic field is a more complicated object. In contrast to Chap. 5, we shall consider neutral gas to be strongly inhomogeneous. Neutral gas driven by the pressure gradient tries to expand freely with a velocity of the order of the sound speed. However, in the presence of a rather small fraction of ionized particles (small ionization degree), such expansion in the collisional plasma could be stopped due to the interaction of the perpendicular current with the magnetic field. Let us analyze the diffusion of a slightly ionized plasma pinch oriented along the magnetic field. Two conditions are assumed to be satisfied. First, β = 2μ0 p=B2 < < 1:

ð7:29Þ

Here, in contrast to Sect. 7.1, p = pN + pe + pi is the net pressure, which is defined as the sum of the pressures of electrons, ions, and neutral particles. Second, the mean free pass of neutral particles with respect to ion-neutral collisions λNi is supposed to be smaller than the plasma density scale L: λNi < < L:

ð7:30Þ

We shall demonstrate that neutral gas is confined due to the interaction of magnetic field with a small fraction of ionized particles provided that these two conditions are satisfied, and the evolution of neutral particles has a diffusive character. The momentum balance for neutral and ionized particles has the following form provided that electron inertial and viscous forces are neglected: →

→

→

→

→

0 = - ∇pe - en E þRei þReN - en ue × B ; → → → → → du → mi n i = - ∇pi þ en E þRie þ RiN þ en ui × B ; dt → → → du mN nN N = - ∇pN þ RNi þ RNe : dt

ð7:31Þ

158

7

Transport in Strongly Ionized Plasma Across a Magnetic Field

The sum of these three equations reads →

mN n N

→

duN dui þ mi n = - ∇p þ dt dt

→

→

j ×B :

ð7:32Þ

If ion and neutral inertia terms are neglected, then →

→

∇p = j × B :

ð7:33Þ

Substituting the current from the Maxwellian equation, we obtain an equation similar to Eq. (7.5) pþ

B2 B2 = const = 0 , 2μ0 2μ0

ð7:34Þ

but with full pressure p = pN + pe + pi, which includes the pressure of the neutral component. This equation can be satisfied provided that inequality (7.29) is fulfilled. In plasma with a small ionization degree restriction given by Eq. (7.29) is much more severe than that in fully ionized plasma, Eq. (7.6). The mechanism of establishing equilibrium could be analyzed using momentum balance Eq. (7.31). We shall take into account only the largest friction force between ions and neutral particles. Homogeneous temperatures are assumed, so thermal force is neglected. The radial force balance for neutral particles in the absence of inertia reads dpN = RNir = - mN nN νNi uNr , dr

ð7:35Þ

where the ion-neutral collision frequency νNi = nhVσiNi is proportional to the plasma density. Here, the radial ion velocity is set to zero since the radial current is absent in the cylindrical plasma. From Eq. (7.35), the radial velocity of neutral particles is derived: uNr = -

dpN 1 : nnN mN hVσiNi dr

ð7:36Þ

If condition Eq. (7.30) is satisfied, this velocity is smaller than the sound speed, and the transport of neutral particles has a diffusive character. According to the azimuthal momentum balance or ions in the absence of their radial velocity, the azimuthal friction force should be zero RiNϑ = 0, and, therefore, the azimuthal velocities of ions and neutrals must coincide: uNϑ = uiϑ :

ð7:37Þ

7.3

Partially Ionized Magnetized Plasma with an Inhomogeneous Neutral Component

159

Radial momentum balance for ions yields -

dpi þ enE r þ enuiϑ B þ RiNr = 0: dr

ð7:38Þ

Since the radial friction between ions and neutrals is balanced by the pressure gradient of neutral particles, Eq. (7.35), from Eq. (7.38), we have Er þ uiϑ B =

dp : endr

ð7:39Þ

This equation remains correct for any azimuthal velocity of ions and the same for neutrals. In the absence of special forces that could produce rotation of neutral gas and plasma in the azimuthal direction, these velocities should turn to zero due to azimuthal viscosity forces. As a result, a radial electric field is established Er =

dp , endr

ð7:40Þ

which is directed from the periphery to the core. This field is much larger than the field Eq. (7.13) for slightly ionized plasma if pN > > pi. The large radial electric field → → Eq. (7.40) causes E × B drift of electrons in the azimuthal direction, which creates an azimuthal current while ions are at rest. The account of electron–neutral friction leads to ambipolar classical diffusion of electrons and ions, as described above in Sect. 7.1. In the general two-dimensional (2D) case, the velocities of neutrals are obtained from the momentum balance for neutral particles →

→

∇pN = - mN nN νNi u i - u N :

ð7:41Þ

Ion and electron velocities obeyed equations, which are generalizations of the equations of Chap. 1 for moving neutrals →

→

→

→

→

n u i - u N = - Di ∇n þ bi n E þ u N × B →

→

→

→

→

n u e - u N = - De ∇n - be n E þ u N × B

,

:

ð7:42Þ

The latter could be derived by shifting to the reference frame moving with the velocity of neutral particles and using the Lorentz transformation for the electric field.

160

7.4

7

Transport in Strongly Ionized Plasma Across a Magnetic Field

Penetration of Neutral Particles into Hot Tokamak Plasma

Penetration of neutral particles into the hot tokamak plasma may have a diffusive character if the mean free path of neutral particles is smaller than the typical spatial scale of plasma density variation. In the edge plasma, neutral particles flow from the divertor plates due to the recycling process and then move into the plasma. In the collisional case, their motion is controlled by charge-exchange collisions with ions. They are also ionized by plasma electrons. Therefore, in the steady state, their penetration is described by the equation: →

∇  nN u N = - nN νion ,

ð7:43Þ

where νion = nhVσiion is the ionization frequency, and the velocity of neutral particles is given by the 2D analog of Eq. (7.36), →

uN= -

1 ∇pN : nmN hVσiNi

ð7:44Þ

Substituting Eq. (7.44) into Eq. (7.43), one obtains an estimate for the penetration depth of neutral particles into the plasma: m1=2 1 e : Lpen = - p n σNi σion m1=2 i

ð7:45Þ

Here is taken into account the fact that a velocity in ionization frequency is the electron thermal velocity, while the charge-exchange frequency is determined by the ion thermal velocity.

Fig. 7.3 Radial particle flux as a function of tokamak major radius in ASDEX-Upgrade tokamak

7.4

Penetration of Neutral Particles into Hot Tokamak Plasma

161

In large tokamak reactors, the penetration depth is smaller than the distance from divertor plates to the separatrix, so neutral particles practically do not penetrate to the core. In contrast, for smaller tokamaks, the penetration depth can be comparable to the distance to the separatrix, and neutral particles can cross the separatrix, providing an ionization source nNνion inside it. Due to this ionization source, the flux of ionized particles from the tokamak core rises towards the separatrix, leading to a strong impact on the radial density profile. To obtain quantitative results of this impact, one must run 2D transport code outside the separatrix, such as SOLPS-ITER, and 1D code inside the separatrix. An example of the corresponding particle flux in the core plasma obtained with such a combined simulation is presented in Fig. 7.3 for the ASDEX-Upgrade tokamak.

Chapter 8

Drift Waves and Turbulent Transport

8.1

Drift Waves in Inhomogeneous Plasma

In inhomogeneous magnetized plasma, there are special waves generated by the gradient of plasma parameters that can propagate across the magnetic field. Consider fully ionized inhomogeneous plasma with a density gradient parallel to the х-axis, n0(x), Fig. 8.1. The homogeneous magnetic field is perpendicular to the density gradient and is parallel to the z-axis. The temperatures of electrons and ions are assumed to be constant Te, i = const(x). We consider density perturbation, which is periodic in y and z: n1 = AðxÞ exp - iωt þ ik y y þ ik z z :

ð8:1Þ

The problem will be considered in the local approximation, and we shall assume that the density scale of unperturbed plasma in the x-direction satisfies the condition kyLx ≫ 1. The spatial scale for perturbation in the x-direction is also large, so if one performs Fourier transformation for x, then for typical wave vectors, a condition should be satisfied: kx ≪ ky. On the other hand, the spatial scale of perturbation should not exceed that of the unperturbed plasma: kx- 1 < Lx . Along the magnetic field, the wavelength is supposed to be large, so that kz ≪ ky. Wavelengths in the yand z-directions should be larger than the ion Larmor radius and mean-free path, respectively, so that the fluid description should be applicable. Parallel momentum balance for electrons in the first approximation, when electron inertia and friction with ions are neglected, is reduced to balance between the electron pressure gradient and the electric force. For a constant electron temperature, this corresponds to the Boltzmann distribution for electrons

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Rozhansky, Plasma Theory, https://doi.org/10.1007/978-3-031-44486-9_8

163

164

8

Drift Waves and Turbulent Transport

Fig. 8.1 Isodensities in the drift wave

φ1 =

n þ n1 Te T n1 = e ln 0 e n0 e n0

ð8:2Þ

A positive potential perturbation corresponds to a positive density perturbation, while a negative density perturbation is biased negatively (Fig. 8.1). The ion pressure gradient and the electric field cause ion motion along the magnetic field. However, let us assume that the longitudinal shift of ions during time scale ω-1 is significantly smaller than k z- 1 ; the corresponding criterion is presented below. Therefore, ion motion along the magnetic field is neglected. Perpendicular velocities of ions and electrons are the sum of contributions from → → E × B and diamagnetic drifts provided inertia terms are neglected →1 →1 u e⊥

→

E ×B

=

B →1

→1 u i⊥

þ

2

T e ½∇n1 × B , en0 B2

→

E ×B

=

-

2

B

ð8:3Þ

T i ½∇n1 × B : en0 B2

Since diamagnetic fluxes in the homogeneous magnetic field are divergence free → → (their divergence is zero), only E × B drifts contribute to particle balance equations. Their velocities are → →

→

→

u ed = u id = u d =

→

E×B B2

:

ð8:4Þ

Drifts caused by the electrostatic electric field in the homogeneous magnetic field are incompressible

8.1

Drift Waves in Inhomogeneous Plasma

165 →

∇⊥ 

→

E×B B2

= 0:

ð8:5Þ

This statement can be easily checked directly for the electrostatic electric field. Considering this fact, the particle balance equation is given by ∂n → þ u d ∇⊥ n = 0: ∂t

ð8:6Þ

After linearization, one finds - iωn1 þ u1dx

dn0 = 0: dx

ð8:7Þ

According to Eq. (8.4), u1dx = E 1y =B = - ik y φ1 =B. Substituting potential Eq. (8.2) into Eq. (8.7), we obtain the dispersion relation ω = ωd = - k y

T e d ln n0 : eB dx

ð8:8Þ

The frequency ωd is known as the drift frequency, and the wave is called the drift wave. Since the obtained frequency is a real number, the density perturbation oscillates with drift frequency and propagates along the y-axis with a phase velocity V py =

ωd T d ln n0 =- e : ky eB dx

ð8:9Þ

The latter coincides with the diamagnetic velocity of electrons. The phase velocity in the z-direction is Vpz = ωd/kz. Particles in the drift wave oscillate in the x-direction due to the drifts caused by Ey, which on the inhomogeneous background n0(x) leads to the propagation of the drift way along the y-axis. The frequency of the drift wave is real because the electric field E 1y and particle drift u1dx = E 1y =B are shifted in phase by π/2 with respect to the density perturbation. Hence, density changes in the knots, which reveals density propagation along the y-axis. The above derivation is valid provided the following inequality is satisfied: ω Te þ Ti > ω d , 1=2

;

ωs < < ωd :

ð8:30Þ

In the most interesting low collisionality case when ωs ≫ ωd, the increment of the instability is significantly smaller than the real part of frequency ωd. When the collision frequency is reduced to the low limit so that the electron mean-free pass becomes comparable with the wavelength along the magnetic field kz(Te/me)1/2/ νei~1, the ratio of the imaginary to the real part is very small: ωd =ωs  ωd = kz ðT e =me Þ1=2 k2y ρ2ci < < 1: Therefore, we demonstrated that with the account of collisions, the drift wave considered in the previous section becomes unstable with the increment given by Eq. (8.30). The physical mechanism of wave growth consists of the following. In the absence of a friction force, the potential perturbation has the same phase as the density perturbation, while the electric field and particle drift along x are shifted by π/ 2 with respect to the density perturbation. Hence, the density rises in the knots of the wave, and the frequency remains real. If friction is taken into account, a shift between density and potential perturbation arises, and the phase shift between the perturbed electric field and density perturbations differs from π/2. As a result, in the regions where perturbation is maximal, density rise takes place due to perpendicular particle drift. The drift for the inhomogeneous density profile reveals itself as instability. It is also worth mentioning that in the laboratory reference frame, the Doppler shift kyV0 associated with plasma motion caused by the ambient electric field is added to the real part of the frequency.

8.3 8.3.1

Universal Instability Fluid Ions

In low collisional plasma with the mean-free path exceeding k z- 1 , the fluid approach considered in the previous section is not applicable. Nevertheless, in such low collisional plasma drift, waves are also unstable. Here, Landau damping plays a

170

8 Drift Waves and Turbulent Transport

role of friction force, and the corresponding instability is known as universal instability. Ions are still assumed to be fluid kyρci < 1. We shall use for analysis the drift kinetic equation for electrons derived in Chap. 1. In the homogeneous → → magnetic field, guiding centers move across the magnetic field due to E × B drift and along the magnetic field with parallel velocity. Therefore, the drift kinetic Eq. (1.112) for electrons is reduced to →

→

E×B eE ∂f ∂f ∂f ∇⊥ f - z = St : þ Vz þ 2 me ∂V z ∂t ∂z B

ð8:31Þ

One has to keep the collisional integral on the r.h.s. this equation despite collisions being rare. We seek a solution in the form f = f0 + f 1, where f0 is an unperturbed distribution function that corresponds to the unperturbed density profile n0(x), and f 1~ exp (iωt + ikyy + ikzz) is a small correction. After linearizing the electron drift kinetic equation, one obtains - iωf 1 þ ik z V z f 1 þ →1

E 1y ∂f 0 eE1z ∂f 0 = St 1 : B ∂x me ∂V z

ð8:32Þ

→

Here, E = - i k φ1 is the wave electric field. For the collision integral, the simplest model representation may be chosen: St1 = - νf 1. The real collisional integral is given by the Landau form in Eq. (1.36), but as demonstrated below, the result is insensitive to the collisional frequency value since ν → 0, so one can use a simplified form of the collision integral. From Eq. (8.32), correction to the distribution function can be found: f1 =

ky ∂f 0 ek z ∂f 0 φ1 : þ ω - k z V z þ iν B ∂x me ∂V z

ð8:33Þ

The distribution function has real and imaginary parts. The multiplayer in Eq. (8.33) can be rewritten after multiplying both the nominator and the denominator by complex conjugate values: iðν þ γÞ Ω - kz V z 1 : = ω - kz V z þ iν ðΩ - k z V z Þ2 þ ðγ þ νÞ2 ðΩ - k z V z Þ2 þ ðν þ γÞ2 Here, ω = Ω + iγ. Because the universal instability increment γ is much smaller than the real part Ω, the resonance corresponds to Ω = kzVz. Two summands demonstrate different behaviors in the vicinity of this resonance point Ω = kzVz. The first real term turns to zero in the resonance point, and for ν → 0 and γ → 0 is antisymmetric with respect to Vz = Ω/kz. The second term for ν → 0 and γ → 0 is small everywhere except the resonance point, where it goes to infinity. These features can be used for

8.3

Universal Instability

171

the calculation of macroscopic plasma parameters. The real part of the density perturbation for ν → 0 and γ → 0 equals the principal-value integral → k y ∂f 0 ek z ∂f 0 φ1 dV z d V ⊥ : þ Ω - kz V z B ∂x me ∂V z

Re n1 = V:P:

ð8:34Þ

Since the frequency real part is of the order of the drift frequency, we can, in accordance with Eq. (8.10), consider Ω ≪ |kzVz| and Ω - kzVz~kzVz. Hence, Re n1 = V:P:

→ e ∂f 0 φ1 k y ∂f 0 dV z d V ⊥ : me ∂V z V z k z B ∂x

ð8:35Þ

Choosing the Maxwellian distribution function for f0, we find that the first term on the r.h.s. of Eq. (8.35) becomes zero since it is an even function of Vz. In the second term ∂f0/∂Vz = - meVzf0/Te, singularity Vz disappears, and the integral is easily calculated: Re n1 =

eφ1 n : Te 0

ð8:36Þ

Therefore, the real part of the density perturbation corresponds to the Boltzmann distribution for electrons. While calculating the imaginary part, we shall take into account that the integrand has a sharp maximum in the vicinity of the resonance point. Therefore, all functions except the resonant multiplier can be taken at Vz = Ω/kz: Im n1 =

-

k y ∂f 0 ekz ∂f 0 νþγ φ1 þ 2 2 B ∂x me ∂V z ðΩ - k z V z Þ þ ðν þ γÞ

→

V z = Ω=k z

dV z d V ⊥ :

ð8:37Þ Since þ1

-1

νþγ π dV z = , 2 2 k j zj ð Ω - k z V z Þ þ ðγ þ ν Þ

the imaginary part is Im n1 =

eφ1 Te

p me π p ð ω - ωd Þ n 0 : 2 jk z j T e

ð8:38Þ

The resonant multiplier in Eq. (8.33) for ν → 0 and γ → 0 is convenient to write in the form

172

8

Drift Waves and Turbulent Transport

1 1 =P - iπδðω - kz V z Þ: ω - kz V z ω - kz V z

ð8:39Þ

This implies that integrals with such multipliers should be calculated in the following way: with the first term as the principal value and with the second term as the integral with δ-function. Finally, combining Eqs. (8.36) and (8.38), we have n1 =

p me π p ð ω - ωd Þ n0 : 2 jk z j T e

eφ1 1þi Te

ð8:40Þ

This result is independent of the collision frequency value, which justifies the choice of the model collision integral. Collisional imaginary correction to the wave frequency in Eq. (8.40) is caused by friction of the resonant particles with the wave known as Landau damping. To derive the dispersion equation, one must also analyze the continuity equation for ions, which connects the ion density and potential perturbations. Since parallel ion motion can be neglected according to inequality (8.10), which is assumed to be satisfied, the equation for ions coincides with Eq. (8.24) obtained in the previous section in the fluid approximation. Hence, the equation system is a pair of Eqs. (8.24) and (8.40): ω n1 -

n1 þ n0

- ωd þ ω

k2y mi T e eφ1 = 0: Te e2 B2

p me π p ð ω - ωd Þ n 0 = 0 : 2 jk z j T e

eφ1 1þi Te

The determinant of the system should be zero, which gives us the dispersion relation (for increments much smaller than the real frequency) Ω=

ωd 1þ

k2y mi T e e2 B2

;

γ = ωd jk z j

p

k2y mi T e πωd < < 1: 2 2T e =me e2 B2 1 þ ky m2 i T2 e e B

ð8:41Þ

The increment of the universal instability reaches its maximum at kyρci~1 and is equal to γ=ωd  ωd = kz 2T e =me < < 1. The increment strongly decreases for smaller ky. In summary, in plasma with rare collisions, the drift wave is unstable with a small increment compared to the drift frequency. At the edge of applicability for kzλei~1, increment Eq. (8.41) turns into the first Eq. (8.30).

8.3

Universal Instability

8.3.2

173

Kinetic Ions

This instability also exists in a more general situation for kyρci ≥ 1. When the perpendicular wavelength is comparable to or smaller than the ion Larmor radius, one must use a kinetic equation to calculate the ion density perturbation. The linearized kinetic equation for ions has the form ∂f 1 e → → ∂f 1 ∂f 1 e → 1 ∂f 0 þV → þ V×B → = - m E → : m ∂t ∂r ∂V ∂V

ð8:42Þ

Here, f 1 is a correction to the main ion distribution function f0. Note that the small → → → 1 second-order term me E ∂f→ is omitted and me V × B ∂f→0 = 0. This equation can be ∂V

∂V

integrated along unperturbed particle trajectories. Let at the moment t0 particles have → → coordinate r 0 and velocity V 0 . Then, →

r

τ

→

→

→

→

→

V 0 r 0 , V 0, τ0 dτ0 ,

r 0, V 0, τ = r 0 þ t0

→

V

→

→

ð8:43Þ

τ

→

r 0, V 0τ = V 0 þ

→

e m

→

→

→

r 0 , V 0 τ0 × B dτ0 :

V

t0

Since ∂f ∂τ

→

→

r 0, V 0

=

∂f ∂t

→ →

r ,V

þ

∂f → ∂r

→

→

t, V

∂r ∂τ

→

→

∂f

→

þ

∂V → jt,r ∂τ ∂V

=

e → → V×B , m

r 0, V 0

→

→

r 0, V 0

ð8:44Þ

and →

∂r ∂τ

→

→

r 0, V 0

→

∂V ∂τ

→

= V,

→

→

r 0, V 0

ð8:45Þ

the kinetic Eq. (8.42) is reduced to ∂f 1 ∂τ

→

→

r 0, V 0

=-

e → 1 ∂f 0 E → : m ∂V

ð8:46Þ

Correction to the distribution function is connected with the perturbed electric field; →1

in the absence of E , the distribution function is conserved along trajectories. Let us assume that perturbation is switched on at t = - 1 from the zero level and then rises over time due to instability. Then, according to Eq. (8.46),

174

8 t

e f =m

∂f 0

→1

1

E

Drift Waves and Turbulent Transport

→

∂V

-1

dt 0 :

ð8:47Þ

The unperturbed distribution function should be a function of motion integrals; → hence, f 0 V = F ε⊥ = mV ⊥ 2 =2, V z , so ∂f 0 →

∂V

→

= mV ⊥

∂F → ∂F þ ez : ∂ε⊥ ∂V z

ð8:48Þ →→

Potential perturbation φ1 is proportional to exp - iωt þ i k r . Then, from Eq. (8.47), one obtains f1 =

eφ1 ∂F ∂F ∂F - ðω - kz V z Þm þ kz Þ I, m m ∂ε⊥ ∂ε⊥ ∂V z

ð8:49Þ

where 1 0

→

τ

exp iωt - i k

I = -i 0

→

V

→

→

r 0 , V 0 , τ0 dτ0 dt 0 :

ð8:50Þ

τ - t0

Here, integration by parts is used with account of the relation dφ1 → = V ∇φ1 - iðω - kz V z Þφ1 : dt

ð8:51Þ

In a uniform magnetic field, the ion velocity is V x ðt Þ = V ⊥ cos½α0 - ωc ðt - t 0 Þ, V y ðt Þ = V ⊥ sin½α0 - ωc ðt - t 0 Þ,

ð8:52Þ

V z = const, α0 = arctg V y ðt 0 Þ=V x ðt 0 Þ : Integral in the exponent in Eq. (8.50) is easily calculated: →

τ

k

→

V

→

→

r 0 , V 0 , τ0 dτ0 = k z V z t 0-ðζ sinðα0 - ωc τ - ΨÞ-sin½α0 -ωc ðτ - t 0 Þ-ΨÞÞ,

τ - t0

ð8:53Þ

8.3

Universal Instability

175

where ζ = k ⊥ V ⊥ =ωc , Ψ = arctg k y =k x : Part of the exponent in Eq. (8.50) can be expanded using the generating function for the Bessel functions exp½ - iζ sinðα0 - ωc ðτ - t 0 Þ - ΨÞ = n=1 n= -1

ζn J n exp½ - in½α0 - ωc ðτ - t 0 Þ - Ψ:

ð8:54Þ

After integration over t′, one finds I = exp½iðk⊥ V ⊥ =ωc Þ sinðα - ΨÞ

n=1 n= -1

ζn J n exp½ - in½α - Ψ:

ð8:55Þ

Here, α = arctgVy/Vx, ζn = (ω - nωc - kzVz)-1. Now, integrating correction to distribution functions Eqs. (8.49) and (8.55) over velocities both for electrons and ions for Maxwellian unperturbed distribution functions, from the quasineutrality condition n1e = n1i = n1 , one obtains the dispersion relation p 2þi π

ω W ðjkz jV T e ÞI 0 ðk⊥ ρce Þ2 exp - ðk⊥ ρce Þ2 jkz jV T e p ω þi π W ðjk z jV T i ÞI 0 ðk⊥ ρci Þ2 exp - ðk⊥ ρci Þ2 = 0 jkz jV T i

Here, ω 1 from Eq. (8.41), one obtains p ω ωd -i π =0 2þ p d jkz jV T e ω 2πk⊥ ρci

ð8:57Þ

176

8

Drift Waves and Turbulent Transport

Maximal increment is reached at p kz =

π ωd , 2 V Te

ð8:58Þ

and V γmax = Ω = pti , 8 πL

ð8:59Þ

where L = |d ln n/dx|-1 is the density spatial scale length. Therefore, in the kinetic limit, the frequency and the increment become independent of the wave vector.

8.4

Instabilities Caused by the Temperature Gradient

The instabilities considered in the previous section have increments significantly smaller than the drift frequency. More dangerous are instabilities caused by the temperature gradients. Therefore, let us consider the instability caused by the ion temperature gradient, known as ηi instability, which is to a large extent responsible for turbulent transport in fusion devices. For simplicity, we shall consider a situation with only the ion temperature gradient, while the density gradient and the electron temperature gradient are absent: dT 0i =dx ≠ 0, dn0 =dx = dT 0e =dx = 0. Let us demonstrate that in this situation, ηi instability exists with a frequency in the range 1=2

kz ðT i þ T e Þ1=2 =mi

< < jωj < < ky e - 1 B - 1 jdT i =dxj:

We shall analyze it in the fluid approximation kyρci ≪ 1 in the collisionless plasma. In this type of wave, perpendicular particle drift caused by a perturbed electric field at an inhomogeneous temperature background leads to ion temperature perturbations. Parallel pressure gradient caused by ion temperature perturbations drives ions along the magnetic field. Their parallel motion leads to the density perturbation and, due to the Boltzmann distribution of electrons, causes potential perturbation. The equation for ion temperature we shall take in the form of Eq. (2.12), where dissipative terms with heat conductivity, viscosity, and heat exchange are neglected (the corresponding time scales are supposed to be large with respect to ω-1): 3 ∂ n þ 2 ∂t

→

u i∇

→

T i þ nT i ∇  u i = 0:

The linearized particle, momentum, and ion heat balance equations are

ð8:60Þ

8.4

Instabilities Caused by the Temperature Gradient

177

- iωn1 þ ik z n0 u1iz = 0, - iωn0 mi u1iz = - ik z n0 T 1i þ n1 T 0i - ik z en0 φ1 ,

ð8:61Þ

cφ dT 0i - iωn0 T 1i - ik y n0 þ ik z n0 T 0i u1iz = 0: B dx 1

With an account of Boltzmann distribution for electrons n1 eφ1 = 0 n0 Te

ð8:62Þ 1=2

one obtains a closed equation system. Assuming k z ðT i þ T e Þ1=2 =mi < < jωj, from the condition of its zero determinant, we find the dispersion relation ω3 = - k2z

T e 1 d ln T i k : mi y eB dx

ð8:63Þ

This equation has one real root and two imaginary roots, and one of the imaginary roots corresponds to the increment p γ=

3 2 T e 1 d ln T i kz ky mi eB dx 2

1=3

:

ð8:64Þ

More detailed analysis for plasma with a density gradient gives the following criterion for ηi instability (for the same sign of the density gradient as the ion temperature one): ηi =

d ln T i > 2: d ln n

ð8:65Þ

In the kinetic limit k⊥ρci ≫ 1, a similar ηi instability also exists with the increment of the order of that given by Eq. (8.59). A similar instability caused by an electron temperature gradient is known as ηe instability. In contrast to ηi instability, it is short-scaled, kyρci ≫ 1, and high frequency, jωj > > kz ðT i þ T e Þ1=2 =m1=2 e . Ions in this wave have a Boltzmann distribution since their Larmor radius exceeds the perpendicular wavelength: eφ1 n1 =- 0 n0 Ti

ð8:66Þ

For electrons, however, kyρce ≪ 1, and their perpendicular motion is just drift in the perturbed electric field. The linearized equation system for electrons coincides with that for ions Eq. (8.43) with the replacement of subscripts «i» with «е». The increment of ηe instability is analogously

178

8

p γ=

Drift Waves and Turbulent Transport

3 2 Ti 1 d ln T e kz k 2 me y eB dx

1=3

:

ð8:67Þ

Similar to ions, the criterion of instability is ηe > 2:

8.5

ð8:68Þ

Turbulent Transport Caused by Random Electric Fields

As a result of the nonlinear evolution of drift-type instability with various wave vectors, plasma transforms into a turbulent state with turbulent transport of particles, energy, and momentum across the magnetic field. We shall consider this process by taking turbulent diffusion produced by universal instability as an example. The increment of this instability in the fluid limit is much smaller than the real part of the frequency, which is close to the drift velocity. In such situations, it is possible to use the so-called quasilinear approximation. In plasma, many waves with different wave vectors are excited, but due to a small increment, the main impact is produced on the average distribution function. The evolution of the latter leads to saturation of the instabilities at a small level before the interaction of different waves becomes important. As in the linear case, we consider the kinetic equation for electrons in the drift approximation →

→

E×B ∂f ∂f eE ∂f ∇⊥ f - z = St: þ Vz þ 2 me ∂V z ∂t ∂z B

ð8:69Þ

The distribution function is sought in the form →→

f = f 0 þ f 1; f 1 = →

f *k 0 exp - iω →k t þ i k r

k

→→

= →

f *k exp - iΩ →k t þ i k r ,

ð8:70Þ

k

where amplitudes f →k are slow functions of time, varying with a time scale smaller than the time scale ω →- 1 due to a small increment. The function f0 is averaged over a k

8.5

Turbulent Transport Caused by Random Electric Fields

179

time scale larger than ω →- 1 and over a spatial scale larger than the wavelengths. For k

potential, analogously →→

φ1 = →

φ*k 0 exp - iω →k t þ i k r

→→

= →

k

φ*k exp - iΩ →k t þ i k r :

ð8:71Þ

k

Summation in Eqs. (8.70) and (8.71) includes positive and negative values of wave →

vectors k . Since the potential is real, φ - →k = φ→ . Nonlinear terms

→

B2

k

eE z ∂f me ∂V z

→

E×B

∇⊥ f and

are neglected in the linearized equation; in other words, nonlinear interaction between different modes is neglected due to their small amplitudes. In the linear approximation, similar to Eqs. (8.32) and (8.33), f →k =

φ →k ky ∂f 0 ek z ∂f 0 : þ Ω →k - k z V z þ iγ →k þ iν B ∂x me ∂V z

ð8:72Þ

The equation for the slowly changing distribution function f0 can be found from the averaged Eq. (8.69): →1

While

averaging

→

E ×B

∂f 0 þ ∂t

∇⊥ f 1

B2 in

Eq.

-

(8.73)

→

→ →0

→0

eE 1z ∂f 1 = hSt i: me ∂V z the

sums

of

A →k B → 0 exp i k r - iΩ →k t þ γ →k t exp i k r - iΩ → 0 t þ γ → 0 t k

k

k k

→0

k

ð8:73Þ the

type

one must take

→

into account that only terms with k = - k remain finite, while other terms vanish. Additionally, according to Eq. (8.41), Ω →k = - Ω - →k and γ →k = γ - →k . Hence, ∂f 0 ∂t

→

k

2 2 ∂ k ω ∂ ky φ →k πδ Ω →k - k z V z - z ce ky ∂V z ∂x B2

∂ kω ∂ f - z ce k y ∂V z 0 ∂x

= hSt i: ð8:74Þ This equation describes the evolution of the averaged distribution function. Potential perturbations grow with nonlinear increment γ →k , which depends on the distribution function f0. The nonlinear increment differs from the linear increment in Eq. (8.41), calculated for the Maxwellian distribution function. Keeping the arbitrary averaged distribution function in the equations of Sect. 8.3, one obtains

180

8

γ →k ðt Þ =

Ω →k πT e en0

-

Drift Waves and Turbulent Transport

→ ky ∂f 0 ek z ∂f 0 δ Ω →k - k z V z dV z d V ⊥ : þ B ∂x me ∂V z

ð8:75Þ

The average distribution function f0 is controlled by the balance of the diffusion in the velocity space for resonant Particles and the collisional integral. Diffusion in the velocity space makes the distribution function more gradual in an attempt to form a plateau, while collisions lead to its Maxwellization. Solving Eq. (8.75), it is possible to find f0 and, hence, nonlinear increment γ →k . Integration of Eq. (8.75) over velocities with account of Eq. (8.74) leads to the diffusion equation ∂ ∂n0 ∂n0 D =0 ∂t ∂x ∂x

ð8:76Þ

With the diffusion coefficient k2y φ2→

D=

k

→

k

2

B

γ →k Ω2→

:

ð8:77Þ

k

Ion contributions to diffusion produce nonresonant particles. The simplest way to → → calculate ion turbulent flux is to average E × B drifts in the direction of the density gradient: Γx = n

cE y ∂n = -D 0: B ∂x

ð8:78Þ

During averaging, density perturbations shifted in phase with respect to potential perturbations contributing to the flux. Density perturbations can be found using the drift kinetic equation for ions, similar to Eq. (8.72). From Eq. (8.78), it follows that the ion diffusion coefficient in Eq. (8.78) coincides with that for electrons Eq. (8.77). For other types of instabilities with larger increments, the quasilinear approach is not applicable since nonlinear interaction between different modes becomes important. The latter determines the wave spectrum. Nevertheless, the quasilinear diffusion coefficient can be used for rough upper estimates of transport coefficients. Let us assume that modes with the largest wavelengths give the main contribution to the transport and keep only one main term in the sum Eq. (8.77). The amplitude of density perturbations can be estimated as k y n →k  dn0 =dx. At such a level, the density gradient of the perturbation becomes comparable to the average density gradient, and large-scale perturbations are the source of smaller-scale instability. Energy is transported from the larger scales to the smaller scales, while the main large-scale mode saturates. Substituting this estimate into (8.77), keeping only one term in the sum, with account of n →k =n0 ≈ eφ →k =T e and Ω →k  ωd , one finds an estimate for the diffusion coefficient

8.6

Effect of Magnetic Shear on Plasma Instabilities

181

D  k y- 2 γ →k :

ð8:79Þ

The same estimate can be obtained assuming that the increment for the main largescale mode γ →k is balanced by damping produced by turbulent diffusion Dk 2y , which is caused by the whole spectrum of oscillation. The largest possible diffusion coefficient for drift waves is reached for the wavelength of the order of density scale: kyL~1, where L = |d ln n0/dx|-1 and increment of the order of drift frequency. Hence, for γ →k  Ω →k  ωd , we have D  DB =

Te : eB

ð8:80Þ

This diffusion coefficient is known as the Bohm diffusion coefficient and is an upper estimate for drift turbulence. In general, to obtain the turbulent diffusive flux, knowledge of the turbulent spectrum is required to account for the nonlinear interaction between different modes.

8.6

Effect of Magnetic Shear on Plasma Instabilities

Magnetic shear (variation of magnetic field in space) generally suppresses plasma instabilities. Let us analyze its impact using drift instability as an example. First, we define shear in the slab geometry. Consider a magnetic field →

→

→

B = Bz ðxÞ e z þ By ðxÞ e y : →

ð8:81Þ →

→

Such a magnetic field is produced by the current j = jy ðxÞ e y þ jz ðxÞ e x , which is connected with the magnetic field by Maxwell equations -∂Bz/∂x = μ0jy and ∂By/ ∂x = μ0jz. All perturbations in wave Ψ still have the form Ψ = AðxÞ exp - iωt þ ik y y þ ik z z : Since the magnetic field turns in space, from the →* k B = kk B = k z Bz þ k y By , one obtains the parallel wave vector k k ð xÞ =

k z Bz þ ky By : B

ð8:82Þ scalar

product

ð8:83Þ

Let the parallel wavelength tends to infinity at some value x0, where kk(x0) = 0. This is possible when wave vectors kz and ky have different signs. Expanding the value kk in the vicinity of x0 assuming for simplicity that only By is a function of x with By ≪ Bz and Bz/B ≈ 1, we have

182

8

k k ð xÞ = k y

Drift Waves and Turbulent Transport

d By =B ðx - x0 Þ: dx

ð8:84Þ

Let us introduce typical plasma scale length L along the x-axis, for example, L = | d ln n0/dx|-1) and define magnetic shear as Θ=L

d By =B : dx

ð8:85Þ

Then, k k ðxÞ = ky Θðx0 Þ

x - x0 : L

ð8:86Þ

The magnetic shear is the geometric characteristic of the magnetic field; on the other hand, the parallel wavelength depends on magnetic shear. In the cylindrical geometry, magnetic shear can be introduced analogously. Let perturbation have the form Ψ = Aðr Þ expð- iωt þ imθ þ ik z zÞ, →

→

ð8:87Þ

→

B = Bz e z þ Bθ ðr Þ e θ , and only the azimuthal component of the magnetic field Bθ depends on the radius. The azimuthal wavelength is connected with the azimuthal wavenumber according to kθ = m/r. Correspondingly, kk ðr Þ =

kz Bz þ ðm=r ÞBθ : B

ð8:88Þ

In the vicinity of the resonant flux surface with radius r0, where kk(r0) = 0, the parallel wave vector is kk ðr Þ = kθ Θðr 0 Þ

r - r0 , r0

ð8:89Þ

where magnetic shear is defined as Θ = r2

dðBθ =rBÞ : dr

ð8:90Þ

In the general case, when both components depend on the spatial coordinate,

8.6

Effect of Magnetic Shear on Plasma Instabilities

183

x - x0 ; L k ⊥ = k y Bz =B - kz By =B;

k k ðxÞ = k⊥ ðx0 ÞΘðx0 Þ B2 d By =Bz Θ = L z2 dx B

ð8:91Þ

for the slab geometry, and kk ðxÞ = k⊥ ðr 0 ÞΘðr 0 Þ

r - r0 ; r0

m B =B - k z Bθ =B; r z B2 dðBθ =rBz Þ Θ = r 2 z2 dr B k⊥ =

ð8:92Þ

for the cylindrical geometry. In the toroidal geometry, the definition of magnetic shear will be introduced later. Magnetic shear in general leads to stabilization of the oscillations. Indeed, the parallel wave vector strongly increases with distance from the resonant flux surface, → and the wavelength along B decreases. At some distance δ from the resonant flux when the wave surface, the parallel wave vector reaches the maximal value kmax k increment turns to zero. For a universal or ηi instability maximal value of k max k , according to inequality (8.10), = ωd = kmax k

Te þ Ti : mi

ð8:93Þ

For larger values of the parallel wave vector, perpendicular ion drift in the drift wave is replaced by their parallel motion, and a drift wave cannot exist. According to Eq. (8.86) δ=

kmax k L ky Θ

:

ð8:94Þ

The quantity δ is a region of drift wave localization in the vicinity of a resonant flux surface. Outside of this region, the drift wave attenuates. Substituting the value of Eq. (8.93) and the expression for drift frequency Eq. (8.8), assuming Te~Ti, we kmax k have δ

ρci : Θ

ð8:95Þ

This estimate is justified for sufficiently large magnetic shear when δ ≪ L, in other words, for Θ > ρci =L.

184

8

Drift Waves and Turbulent Transport

Therefore, for large magnetic shear, a new characteristic scale δ arises in the direction of the initial inhomogeneity x instead of L. As a result, the local approximation, discussed in the previous sections when the wave amplitude is supposed to be a gradual function of x, is valid only for short wavelengths kyδ > 1. The maximal wave vector in the y direction is thus determined by the condition k min y δ = 1:

ð8:96Þ

Drift waves with larger wavelengths are not generated in plasma with large shear. Restriction of the maximal perpendicular wavelength leads to a reduction in the turbulent transport coefficients. Let us use Eq. (8.79) to estimate the ion heat Eq. (8.96) and δ conductivity coefficient caused by ηi instability. Substituting k min y Eq. (8.95), assuming the increment to be of the order of the drift velocity, one finds χi 

T e ρci : eB ΘL

ð8:97Þ

The ion heat conductivity coefficient is thus δ/L times smaller than the Bohm coefficient. In contrast to the Bohm coefficient, it is inversely proportional to B2 and depends on the plasma characteristic scale L. This dependence is known as Gyro-Bohm scaling. It is important to mention that for this type of instability, the diffusion coefficient should be much smaller than the ion heat conductivity coefficient due to the Boltzmann distribution of electrons in the wave so that the averaged → → E × B drift vanishes. For ηe instability, the maximal value of the parallel wave vector is = ωd = k max k

Te : me

ð8:98Þ

Hence, the localization region for this instability is much smaller: δ

ρce : Θ

ð8:99Þ

The corresponding estimation for electron heat conductivity is, therefore, smaller than that for ions: χe 

T e ρce : eB ΘL

ð8:100Þ

The strong magnetic shear local approach considered previously in this chapter is no longer valid. Indeed, the typical scale of the wave amplitude variation in the xdirection becomes comparable with the wavelength in the y-direction, and, hence, the electric fields and polarization current along x and y are of the same order. Even

8.6

Effect of Magnetic Shear on Plasma Instabilities

185

for smaller perpendicular wavelengths, the divergence of the parallel ion flux changes from the small one for kk → 0 up to values when the wave starts to dampen at x~δ. The rigorous approach implies that the solution is sought in the form of Eq. (8.1), but the equations are reduced to ordinary differential equations with respect to x. Let us demonstrate this approach for collisionless drift waves in the absence of temperature gradients, which corresponds to universal instability in the absence of magnetic shear. In the particle balance for ions in addition to Eq. (8.23), we shall take into account the polarization drift in the x-direction and their parallel velocity: - iωn1 þ in0 k k ðxÞu1ik -

ik y φ1 dn0 m m d 2 φ1 - iωk 2y n0 i2 φ1 þ iωn0 i2 = 0: B dx eB eB dx2

ð8:101Þ

It is taken into account that the wave amplitude spatial scale in the x-direction is much smaller than L. Here, kk(x) changes linearly in the vicinity of the resonance surface (8.86). Parallel ion velocity could be found from the parallel momentum balance equation for ions (Te ≫ Ti) - iωn0 mi u1ik = - ik k ðxÞT e n1 :

ð8:102Þ

One more relation between density and potential perturbations is obtained from electron Eq. (8.40). Neglecting the imaginary part of Eq. (8.40), which is associated with Landau damping, we have n1 = (eφ1/Te)n0 – Boltzmann distribution for electrons. Combining Eqs. (8.101) and (8.102) with an account of this relation, one obtains 2

ρ2ci

k2y T e Θ ðx - x0 Þ2 d 2 φ1 ω þ - 1 þ k2y ρ2ci - d ω mi ω2 L2 dx2

φ1 = 0,

ð8:103Þ

where the Larmor radius is defined as ρci = (Te/mi)1/2(mi/eB). In the new variables ξ=

x - x 0 ωd Θ ρci ω

1=2

expðiπ=4Þ

ð8:104Þ

Eq. (8.103) is reduced to the equation of the quantum oscillator d 2 φ1 þ 2E - ξ2 φ1 = 0, dx2 with energy

ð8:105Þ

186

8

E=

Drift Waves and Turbulent Transport

i ω ω 1 þ k 2y ρ2ci - d : ω 2 ωd Θ

ð8:106Þ

The eigenvalues for the oscillator are 1 E = n þ , n = 0, 1, 2 . . . : 2

ð8:107Þ

Therefore, from Eqs. (8.106)–(8.107), one obtains the dispersion relation. When n is not too large, the frequency is Re ω = Ω = ωd = 1 þ k2y ρ2ci ; Imω = γ = - ωd Θð2n þ 1Þ:

ð8:108Þ

Eigenfunctions are given by Hermite polynomials Hn: φ1 = H n ðξÞ exp - ξ2 =2 :

ð8:109Þ

In accordance with Eq. (8.90), mode slightly damps. At x → 1, eigenfunctions oscillate φ1 ðx → 1Þ = exp - i 1=2

ωd Θ x 2 ω 2ρ2ci

ð8:110Þ

with the typical scale ρci =Θ . Due to the small imaginary correction, the frequency eigenfunctions diverge at infinity. To make them converge, it is necessary to take into account an additional term in the electron equation, which is responsible for universal drift instability. The corresponding analysis leads to an ordinary differential equation that can be solved only numerically. The imaginary part of the eigenvalues remains negative, so drift waves in the slab geometry with magnetic shear are damped. Drift waves caused by the temperature gradient remain unstable in the presence of magnetic shear for kyδ > 1. Note, however, that other factors, such as an inhomogeneous magnetic field, can make drift waves unstable even in the presence of magnetic shear.

Chapter 9

Dynamics of Fully Ionized Plasma in the Absence of a Magnetic Field

9.1

Ion Acoustic Waves

In plasma, ion acoustic or sound waves are a type of oscillation of electrons and ions similar to acoustic waves traveling in neutral gas. However, in contrast to neutral gas, electrons and ions can interact with the electrostatic field that arises in plasma. In the fluid approximation, fully ionized plasma without impurities for Z = 1 is described by the equation system considered in Chap. 2: ∂n → þ ∇  n u α = 0, ∂t → → → duα $ mα n = - ∇pα - ∇  π α ± en E þ R α , dt ∂uαj 3 dT α → → þ ∇  q α = Qα : þ nα T α ∇  u α þ παjk n 2 dt ∂xk

ð9:1Þ

Summing up two momentum balance equations for electrons and ions, neglecting the electron inertia term and ion viscosity, one obtains the net momentum balance equation, where the electric field and friction force are absent: →

nmi

dui = - ∇p: dt

ð9:2Þ

The latter forces cancel each other due to different signs in the partial momentum balance. Here, p = pe + pi is the total pressure. The viscosity term can be neglected with respect to the pressure gradient provided that the mean-free path is smaller than the typical scale L of plasma, which is the criterion of applicability of the fluid approach. As mentioned in Chap. 7, it is possible to estimate the typical velocity in inhomogeneous fully ionized plasma from Eq. (9.2). This typical velocity is of the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Rozhansky, Plasma Theory, https://doi.org/10.1007/978-3-031-44486-9_9

187

188

9 Dynamics of Fully Ionized Plasma in the Absence of a Magnetic Field

order of ui~[(Te + T )i/mi]1/2, and its exact value depends on the problem type. We shall first consider the evolution of small perturbations in unbounded homogeneous plasma with density n0 in the isothermal case. Let all variables be functions on z only. The density and ion fluid velocity are sought in the form: n = n 0 þ n1 ,

ui = u1i , n1  exp ð- iωt þ ikzÞ, u1i  exp ð- iωt þ ikzÞ:

ð9:3Þ

Linearizing the ion particle balance equation and net momentum balance Eq. (9.2), one obtains - iωn1 þ iku1i n0 = 0, - iωmi n0 u1i þ ik ðT e þ T i Þn1 = 0:

ð9:4Þ

This equation system has a nontrivial solution when its determinant equals zero, which leads to the dispersion relation ω = ± kcs ,

ð9:5Þ

where for plasma with constant temperatures cs =

Te þ Ti : mi

ð9:6Þ

Here, the quantity cs is known as the ion sound speed. A more general definition of the ion sound speed is cs =

dp , dρ

ð9:7Þ

where ρ = nmi. For Te, i = const, this definition coincides with Eq. (9.6). We see that in accordance with Eq. (9.5), density perturbation propagates with constant phase velocity cs, and sign ± corresponds to the propagation in one or opposite direction. Dispersion, i.e., dependence of phase velocity on wavelength, is absent for the ion acoustic wave. According to continuity equations for electrons and ions, in the absence of plasma current, ue = ui = u, i.e., electrons and ions move together. Hence, the electron-ion friction force is zero, and from the electron momentum balance, we obtain the Boltzmann distribution for electrons φ1 =

Te T n1 : lnðn=n0 Þ ≈ e e e n0

ð9:8Þ

The positive density perturbation has positive potential, while negative perturbation is biased negatively. The electric field that corresponds to this Boltzmann potential keeps electrons coupled to ions so that their velocities in the acoustic wave coincide.

9.1

Ion Acoustic Waves

189

If the time scale typical for ion acoustic waves ω-1 = (kcs)-1 is smaller than all dissipative time scales – a heat conductivity time scale (k2χe, i)-1 and a heat exchange time scale (2meνei/mi)-1 – then ion acoustic waves follow an adiabatic law. In this situation, entropy per volume s and particle temperatures change with density as Te, i~n2/3 and total pressure p~n5/3. In the general case system, Eqs. (9.4) is given by - iωn1 þ iku1i n0 = 0, dp - iωi mi n0 þ ik n1 = 0: dn

ð9:9Þ

The dispersion law is still given by Eq. (9.5), while the adiabatic sound speed is cs =

dp j = dρ s = const

5ð T e þ T i Þ : 3mi

ð9:10Þ

Finally, a situation is possible when the time scale ω-1 is small with respect to the ion heat conductivity time scale and the heat exchange time scale but is large with respect to the electron heat conductivity time scale. Such a situation may exist due to the high electron heat conductivity with respect to that of ions. In this case, the ion sound speed is isothermal for electrons and adiabatic for ions, and the sound speed is cs =

T e þ ð5=3ÞT i : mi

ð9:11Þ

Since dispersion for ion acoustic waves is absent, i.e., waves with different wavelengths propagate with the same velocity, and any perturbation of arbitrary shape, which can be decomposed into Fourier harmonics, propagates with the same velocity. Hence, the general solution for linearized Eqs. (9.1) and (9.2) has the form n1 = f 1 ðz - cs t Þ þ f 2 ðz þ cs t Þ:

ð9:12Þ

This becomes quite clear if linearized equations are reduced to the form ∂u1 ∂n1 þ n0 i = 0, ∂t ∂z ∂u1i ∂n1 n0 = - c2s : ∂t ∂z

ð9:13Þ

Taking the time derivative in the first equation and the spatial derivative in the second equation and subtracting one from another, one comes to the wave equation

190

9

Dynamics of Fully Ionized Plasma in the Absence of a Magnetic Field

Fig. 9.1 Splitting of rectangular perturbation into two moving ones of half amplitude

2

2

∂ n1 1 ∂ n1 - 2 = 0, 2 cs ∂t 2 ∂z

ð9:14Þ

in which the solution coincides with Eq. (9.12). As an example, let us consider the evolution of rectangular density perturbation, n1 = A for -a ≤ z ≤ a and n1 = 0 outside of this interval, as shown in Fig. 9.1. The initial fluid velocity is set to zero. Such perturbation can be considered as a sum of two perturbations of half amplitudes, which in accordance with Eq. (9.12), propagate with the velocities ±cs. In the course of time, the initial perturbation splits into two perturbations moving in different directions (Fig. 9.1). Ion acoustic waves can also propagate in the collisionless plasma if the electron temperature exceeds the ion temperature, Te > > Ti. This case can be considered in the following way. According to the momentum balance for electrons, the electric force is balanced by the electron pressure gradient. Therefore, the electric force exceeds the ion pressure gradient and viscosity in the momentum balance equation for ions, so both the ion pressure gradient and the ion viscosity can be neglected. As

9.1

Ion Acoustic Waves

191

a result, replacing electric force with electron pressure gradient in the ion momentum balance equation, one ends up with Eq. (9.2) with only the electron pressure gradient →

nmi

dui = - ∇pe : dt

ð9:15Þ

The sound velocity coincides with Eq. (9.6). For comparable temperatures, the situation becomes more complicated, ion acoustic waves strongly dampen due to collisionless Landau damping since the sound speed is of the order of ion thermal velocity, and kinetic analysis is needed. When the perturbation wavelength becomes comparable with the Debye radius, special consideration is required since the quasineutrality is violated and the Poisson equation should be solved. Let us analyze this situation for collisionless plasma with Te > > Ti. The initial equation system in the general case of an arbitrary wavelength is ∂ni → þ ∇  ni u i = 0, ∂t ∂ne → þ ∇  ne u e = 0, ∂t → * dui mi ni = eni E , dt

ð9:16Þ

→

0 = - T e ∇ne - ene E , →

∇  E = e ðni - ne Þ=ε0 : From the electron momentum balance equation, one obtains the Boltzmann distribution for electrons φ = (Te/e) ln (ne/n0), where n0 is an unperturbed density for both → electrons and ions. From the ion momentum balance, we have mi ni ddtu i = - T e ∇ne . The linearized system of particle balance for ions, momentum balance for electrons and ions and Poisson equation in the 1D case is, therefore, - iωn1i þ ikn0 u1i = 0, eφ1 =T e = n1e =n0 , - iωmi n0 u1i = - ikT e n1e , k2 φ1 = e n1i - n1e =ε0 :

ð9:17Þ

After eliminating the potential and ion velocity, one obtains an equation system for two unknowns n1e , n1i . From the condition of zero determinant, one obtains a dispersion relation

192

9

Dynamics of Fully Ionized Plasma in the Absence of a Magnetic Field

ω2 =

k2 c2s , 1 þ k2 r 2d

ð9:18Þ

where the Debye radius is defined as r 2d = T e ε0 =ðn0 e2 Þ. One can see that in the general case, ion acoustic waves have dispersion since the phase velocity depends on the wavelength. For a large wavelength, Eq. (9.18) turns into Eq. (9.5), while for a small wave length krd > > 1, the wave frequency becomes independent of the wavelength and coincides with the ion plasma frequency: ω = ωpi =

n 0 e2 : m i ε0

ð9:19Þ

In multicomponent plasma with p different ion species, there are p roots for phase velocities not counting the direction of propagation. For the Te > > Ti case, p-1 modes propagate with small velocities of the of the order of ion thermal velocities, and one fast mode propagates with a velocity proportional to Te1/2. The general expression for this fast mode can be easily obtained similarly to the above derivation, and the sound speed for the fast mode for a large wavelength is cs =

2 T e Z I mp =mI , mp hZ I i

ð9:20Þ

where mp is the proton mass, mI the ion mass, ZI the charge number, and hX I i =

X I nI = I

nk . In the practically important case for the fusion application k

of a deuterium–tritium mixture (half deuterium, half tritium) for Te = Ti, isothermal electrons and adiabatic ions, two sound speeds are c1s =

9.2

1:1T e , c2s = mp

1:8T e , mp

ð9:21Þ

Nonlinear Dynamics. Self-Similar Solutions

In this section, we shall consider nonlinear evolution in the fully ionized plasma in the situation when the initial sizes of an inhomogeneity are unimportant. Then, the solutions of different problems could be obtained using a self-similar approach when plasma density and ion fluid velocity are sought as a function of one variable, which is a combination of spatial coordinates and time. As an example, let us consider 1D plasma expansion into a vacuum (ambient plasma of small density). At z = 0, a

9.2

Nonlinear Dynamics. Self-Similar Solutions

193

_ ðzÞ. Plasma from the source spreads in source of ionized particles is located: I = Nδ both the positive and negative directions of z. Near the source, a stationary density profile is established, and the plasma flux can be obtained by integrating the particle balance equation over the region in the source neighborhood: _ Let s first consider isothermal case Te = const, Te > > Ti. 2Γðz = 0Þ = Idz = N. Such a situation can be realized due to the large electron heat conductivity, so that a large parallel heat flux compensates for the work spent on plasma expansion. In the equations (subscript for fluid velocity is dropped) ∂n ∂ðnuÞ = 0; þ ∂z ∂t ∂u ∂u ∂p nmi þu =∂t ∂z ∂z

ð9:22Þ

we introduce a new variable ζ = z/t. Both plasma density and ion fluid velocity are supposed to be functions of ζ only: n = n(ζ), u = u(ζ). Derivatives are transformed in the following way: ∂ ∂ζ d 1 d = =- ζ ; t dζ ∂t ∂t dζ

∂ ∂ζ d 1 d = = : t dζ ∂z ∂z dζ

ð9:23Þ

In the new variables,the equation system has the form dn d ðnuÞ þ = 0; dζ dζ dn du du = - c2s : -ζ þ u dζ dζ dζ

-ζ n

ð9:24Þ

Here, the sound speed is defined according to Eq. (9.7). From the first equation, the derivative du/dζ can be obtained as a function of the density derivative. After substitution of the latter into the second equation, one obtains (the sign corresponds to expansion into the positive direction of the z-axis) u = cs þ ζ:

ð9:25Þ

In the isothermal case, the sound speed is constant; hence, du/dζ = 1, and from the second Eq. (9.24), we have the density dn n þ = 0: dζ cs Its solution is

ð9:26Þ

194

9

Dynamics of Fully Ionized Plasma in the Absence of a Magnetic Field

n = A exp -

ζ : cs

ð9:27Þ

_ In the initial variables with an account of the condition nuðz = 0Þ = Γðz = 0Þ = N=2, we have for z > 0 n=

Γðz = 0Þ z exp , cs cs t

z u = cs þ : t

ð9:28Þ

Hence, plasma density decreases exponentially with a spatial scale cst, while ion velocity increases linearly with distance. The same is true for z < 0. In the adiabatic limit, sound speed Eq. (9.7) depends on the density as n1/3 ( p~n5/3). The fluid velocity is still given by Eq. (9.25) From the second Eq. (9.24), one finds a general relation between u and n: u= -

cs

dn : n

ð9:29Þ

Integrating Eq. (9.29) with an account of Eq. (9.25) yields z 4cs ðn0 Þt 3z u = cs ð n0 Þ þ : 4t

3

n = n0 1 -

ð9:30Þ

Here, n0cs(n0) = Γ(z = 0). In the adiabatic limit, the plasma density turns to zero at a finite distance from the source, and the fluid velocity remains finite. Another problem that can be analyzed using the obtained solution is the problem of initial discontinuity decay. Let us consider the initial plasma density discontinuity at t = 0 nðz ≤ 0Þ = n0 ,

nðz > 0Þ = 0:

ð9:31Þ

Let us restrict ourselves to the isothermal case. Then, the solution can be constructed using Eqs. (9.25)–(9.27). The solution that satisfies initial condition Eq. (9.31) is z -1 , cs t u = 0,

n = n0 exp n = n0 ,

z u = cs þ , t

z > - cs t; z ≤ - cs t:

ð9:32Þ

The density profile is shown in Fig. 9.2. The plasma from the region -cst < z < 0 moves to the right, while a wave of rarefaction propagates to the left with the sound speed. It is interesting that the density at z = 0 remains constant and equals

9.3

Simple Nonlinear Waves. Overturn

195

Fig. 9.2 Evolution of initial density shock (dotted line). In time plasma (solid line) flows to the right, while wave of rarefaction moves to the left

n(z = 0) = n0 exp (-1). The solution in the adiabatic case can be constructed analogously.

9.3

Simple Nonlinear Waves. Overturn

The evolution of one-dimensional nonlinear propagating waves, which are nonlinear analog of ion acoustic waves, is more complicated. Let us seek a partial solution of Eq. (9.22) in the form of a propagating wave. All physical quantities in the wave (plasma density, pressure, and fluid velocity) are supposed to be functions of z - Vt, where a propagation velocity V, in contrast to the linear case, depends on the wave amplitude. Since density, pressure, and fluid velocity propagate together with the same velocity, one physical quantity can be considered a function of the other. Using this fact, the particle balance equation can be transformed to (ρ = min): ∂ρ dðρuÞ ∂ρ þ = 0: dρ ∂z ∂t

ð9:33Þ

Here, the product ρu is considered to be a function of ρ. In the momentum balance equation ∂u ∂u 1 ∂p þu þ =0 ∂t ∂z ρ ∂z

ð9:34Þ

pressure can be considered as a function of u : ∂p/∂z = (dp/du)∂u/∂z. Then, the equation for velocity is reduced to

196

9

Dynamics of Fully Ionized Plasma in the Absence of a Magnetic Field

∂u 1 dp ∂u þ uþ = 0: ρ du ∂z ∂t The quantities V ðρÞ =

dðρuÞ dρ

in Eq. (9.33) and V ðuÞ = u þ 1ρ

ð9:35Þ dp du

in Eq. (9.35)

represent propagation velocities for density and fluid velocity correspondingly. Since these quantities propagate together, two velocities must coincide V(ρ) = V(u), so dðρuÞ 1 dp = uþ : dρ ρ du

ð9:36Þ

Taking into account dp=du = c2s dρ=du with the sound speed defined according to Eq. (9.7), from Eq. (9.36), we find du c = ± s, dρ ρ

ð9:37Þ

and after integration u= ±

cs dρ: ρ

ð9:38Þ

This expression determines u(ρ) since sound speed is a known function of density; Þ see Sect. 9.1. Hence, the propagation velocity V ðρÞ = dðdρu ρ is V ð ρ Þ = V ð uÞ = u ± c s :

ð9:39Þ

One can see that the propagation velocity is different for different points of a density profile and depends on the density. According to Eq. (9.33), one obtains for the density: ρ = ρðz - V ðρÞÞt:

ð9:40Þ

In other words, each point of the initial profile moves with a constant velocity, which, however, depends on the corresponding density. For example, in the case of isothermal plasma, when cs = const, Eqs. (9.38) and (9.39) yield V ðρÞ = cs ln

ρ ± cs , ρ0

ð9:41Þ

where ρ0 is the constant, e.g., constant density at infinity. Points of the profile with a larger density propagate faster than those with a smaller density (Fig. 9.3). The density profile becomes multivalued with time, and nonlinear steepening and overturn take place. A nonphysical multivalued density profile arises because dissipative

9.4

Nonlinear Ion Acoustic Waves with Dispersion

197

Fig. 9.3 Evolution of simple nonlinear wave

terms, in particular ion viscosity, are neglected in the initial equations. However, the real density profile can be approximately obtained from the equations without viscosity using the “area rule.” Let us add a vertical line, as shown in Fig. 9.3, so that the dashed areas should be equal to each other. Then, the density profile, shown by the solid line, is a solution of the initial equations since it consists of a gradual profile and discontinuity. The total number of particles for such a profile is conserved and equals the initial number of particles. At the profile front, a shock is formed, which is similar to the standard shock in gases. As in hydrodynamics, to investigate a shock structure, it is necessary to shift to a reference frame moving with the shock velocity and to take into account the ion viscosity. The density profile in the shock is obtained from ordinary differential equations, and its typical scale is of the order of the ion mean free path. However, for a large mean-free path situation, plasma differs from gaseous case. When the meanfree path exceeds the Debye radius, the shock structure is determined by the quasineutrality violation and not by dissipation. In such situations, the shock is known to be collisionless, and to analyze its structure, kinetic equations should be solved together with the Poisson equation. The density inside such collisionless shock has an oscillating character.

9.4

Nonlinear Ion Acoustic Waves with Dispersion

The propagation of nonlinear density perturbations with the typical scale of the order of the Debye radius has some special features compared to those considered in the previous section. Dispersion, which exists for linear waves, in the nonlinear situation may be some special case that compensates for nonlinear steepening and overturning of the wave. The corresponding solution is known as a soliton. Let us consider an example of collisionless plasma in the limit of cold ions. The initial equations are

198

9

Dynamics of Fully Ionized Plasma in the Absence of a Magnetic Field

∂ni ∂ðni ui Þ = 0, þ ∂z ∂t ∂ϕ ∂ui ∂u , mi ni þ ui i = - ni e ∂z ∂t ∂z ∂n ∂ϕ 0 = - T e e þ ne e , ∂z ∂z 2 ∂ ϕ = - eðni - ne Þ=ε0 : ∂z2

ð9:42Þ

We shall seek a partial solution in the form of localized nonlinear perturbation propagating with a constant velocity, so all quantities are functions of a variable z - Vt : Ψ = Ψ(ζ) = Ψ(z - Vt). In contrast to the case considered in the previous section, the propagation velocity is assumed to be constant for all profile points. The particle balance equation for ions is reduced to -V

dni dðni ui Þ = 0: þ dζ dζ

ð9:43Þ

Integration yields ni ðui - V Þ = const = - n0 V:

ð9:44Þ

The following boundary conditions are supposed to be satisfied: ne(1) = ni(1) = n0 and ui(1) = 0. The momentum balance for ions in the new variables is mi - V

du dui þ ui i dζ dζ

þe

dφ = 0: dζ

ð9:45Þ

After integration: mi

u2i - ui V 2

þ eφ = 0:

ð9:46Þ

The integration constant is chosen to satisfy the boundary condition φ(1) = 0. From the quadratic Eq. (9.43), one finds ui = V ±

V 2 - 2eφ=mi :

ð9:47Þ

Choosing minus in the solution, combining Eq. (9.47) with Eq. (9.44), we obtain the ion density as a function of the potential: ni =

n0 V : V - 2eφ=mi 2

ð9:48Þ

9.4

Nonlinear Ion Acoustic Waves with Dispersion

199

The momentum balance for electrons gives electron density as a function of potential ne = n0 exp

eφ : Te

ð9:49Þ

Finally, after the substitution of Eqs. (9.48) and (9.49) into the Poisson equation, one obtains ε0

d2 φ eφ = en0 exp Te dζ2

V : V - 2eφ=mi 2

ð9:50Þ

This equation coincides with Eq. (3.30), but here, we seek solutions that satisfy condition φ(0) = 0. Let us introduce dimensionless variables Φ=

ζ eφ , ζ = , rd = rd Te

ε0 T e : n0 e 2

In the new variables, the Poisson equation has the form d2 Φ dζ

2

= expðΦÞ -

1 1 - 2Φ=M 2

1=2

:

ð9:51Þ

Here, M = V/cs is the Mach number. Introducing potential energy so that the r.h.s. equals -dU/dΦ, we have U ðΦÞ = - expðΦÞ - M 2 1 - 2Φ=M 2

1=2

þ 1 þ M2:

ð9:52Þ

Constant choice corresponds to U(1) = 0. This potential energy is known as the Sagdeev potential. For small Φ, the potential energy is given by a parabola U ðΦÞ = -

Φ2 Φ2 : þ 2 2M 2

ð9:53Þ

For M > 1, the potential energy at small Φ is negative and decreases with Φ, as shown in Fig. 9.4. For larger values of Φ, the potential energy rises, passes through zero andpbecomes positive. Indeed, when a square root in (9.52) becomes zero at Φ = M= 2, we have U M 2 =2 = 1 þ M 2 - exp M 2 =2 : If M < 1.6, then this value is positive. Hence, for M < 1.6, the potential energy has the form shown in Fig. 9.4. Here, Φmax corresponds to the root of equation U(Φmax) = 0.

200

9

Dynamics of Fully Ionized Plasma in the Absence of a Magnetic Field

Fig. 9.4 Potential energy profile for 1 < M < 1.6

Fig. 9.5 Ion acoustic soliton

Let us seek a solution of Eq. (9.51), corresponding to zero energy level E = 0 in the range 1 < M < 1:6:

ð9:54Þ

Using mechanical analogy (ζ represents time and Φ – coordinate), one can notice that a solution has oscillatory character, and, at Φ → 0, a particle spends infinite time. The potential profile is given by the equation corresponding to the energy conservation 2

1 dΦ 2 dζ

= - U,

ð9:55Þ

and is shown schematically in Fig. 9.5. The potential profile is given by the integral ζ= ±

dΦ - 2 - expðΦÞ - M 2 1 - 2Φ=M 2

: 1=2

ð9:55Þ

þ 1 þ M2

The solution is the localized potential perturbation moving with a constant velocity V. Such a solution is known as a soliton. The corresponding density perturbation is determined by Boltzmann distribution Eq. (9.49). Each part of the profile moves

9.5

Plasma Expansion During Pellet Injection

201

with the same constant velocity, and the profile keeps its shape Eq. (9.55). The propagation velocity of a soliton depends on its amplitude. This relation can be derived from the equality U(Φmax) = 0, so we have M ðΦmax Þ =

½expðΦmax Þ - 12 : 2½expðΦmax Þ - 1 - Φmax 

ð9:56Þ

The typical spatial scale of the soliton is of the order of the Debye radius, since the dimensionless variable ζ is the distance measured in units of the Debye radius, the potential amplitude is of the order of Te/e, and the propagation velocity is supersonic. Therefore, in contrast to quasineutral nonlinear waves, a special partial solution exists in the form of a soliton when nonlinearity is balanced by dispersion. Solitons exist in the range of Mach numbers given by Eq. (9.54). The maximal velocity is M = 1.6, and the corresponding soliton amplitude according to Eq. (9.56) is also maximal and equals Φlim max = 1:3. For larger Mach numbers, soliton solutions do not exist. At M = 1.6 and Φlim max = 1:3, the square root in Eq. (9.52) becomes zero, and for larger Mach numbers, it becomes imaginary. Physically, this means that at large amplitudes, dispersion cannot balance nonlinearity, so nonlinear perturbations should overturn.

9.5

Plasma Expansion During Pellet Injection

During pellet injection into a tokamak, evaporated neutral particles first expand spherically, and then, after ionization by electrons move mainly along the magnetic field, forming a cigar-like structure (see Figs. 1.3 and 1.4). When the pellet stays at a given flux tube, it may be considered a given constant source of particles, and its parallel expansion is similar to that analyzed in Sect.9.2. The only difference with the self-similar solution Eq. (9.28) with constant cloud temperature consists of the fact that in real tokamak plasma, expanding cold plasma is heated by ambient electrons with high energy. To obtain density and velocity parallel profiles in this situation, Eqs. (9.22) should be supplemented by the energy balance equation: 3n

∂T ∂T þu ∂t ∂z

þ nT

∂qek ∂u : =∂z ∂z

ð9:57Þ

We assume Te = Ti due to strong coupling between electrons and ions in cold plasma. To obtain r.h.s. of this equation,principle kinetic treatment is required, since hot electrons have a large mean-free path and the fluid treatment does not correctly describe the heating process. However, in the modeling, one can use an expression for nonlocal heat conductivity, which, in the situation when the electron mean-free path is comparable to the cloud dimension, is equivalent to the kinetic description. According to this approach,

202

9

Dynamics of Fully Ionized Plasma in the Absence of a Magnetic Field 1

qek = -

κek 0

∂T W ðz, z0 Þdz0 , ∂z0

ð9:58Þ

where z

3:16nT e κek = , me νei

1 W ðz, z Þ = exp 2λðz0 Þ 0

p

λ = 32 2λ0 ,

λ0 =

z0 2 2

dz0 0 n z0 0 , nðz0 Þλðz0 Þ

ð4πε0 Þ T : 8πne4 Λ

An example of simulations of Eq. (9.22) and (9.58) is presented in Fig. 9.6. The dotted line corresponds to the self-similar solution Eq. (9.28) for the isothermal

Fig. 9.6 (a) Density; (b) electron temperature; (c) hydrodynamic velocity profiles for t = 25μs. The plasma source is switched on at t = 0 and is situated at z = 0. Dotted line: calculation according to Eq. (9.28). The plasma and pellet parameters correspond to: n0 = 3 × 19m-3, Te0 = 500eV, N_ = 1023 s - 1 . Initially cold plasma had temperature Te0 = 5eV

9.5

Plasma Expansion During Pellet Injection

203

plasma expanding into a vacuum produced by a constant source. We see that the bulk of the ablatant remains fairly cold. A discrepancy between the self-similar solution for the density and the result of the calculation appears at densities that are an order of magnitude smaller than those at z = 0. The elongation of the tail is due to ablatant heating by the ambient electrons. The velocity profile, which should be linear for isothermal plasma expanding into the vacuum, drops off to zero at distances where the ablatant density becomes comparable with the ambient density and the self-similar solution loses its validity.

Chapter 10

Magnetohydrodynamics (MHD)

10.1

Magnetohydrodynamic Equations

The equation system of the transport equations for density and fluid velocities in a magnetic field for pure plasma is as follows: ∂n → þ ∇  n u e = 0, ∂t ∂n → þ ∇  n u i = 0, ∂t →

→

→

→

0 = - ∇pe - en E - ne u e × B þ R ,

ð10:1Þ

→

nmi

→ → → dui → = - ∇pi þ en E þ ne u i × B - R : dt

Electron inertia here is neglected. It is supposed that the pressure is a known function of density and is determined, for example, by the adiabatic law, which replaces energy balance equations. These basic equations are known as two-fluid magnetohydrodynamic (MHD) equations. Let us transform the initial two-fluid MHD equations into a new equation system where electric fields and plasma currents are excluded. After summing up the two momentum balance equations for electrons and ions, the electric field and friction force are canceled, and with an account of the current density definition → → → j = en u i - u e , one obtains →

nmi

dui = - ∇p þ dt

→

→

j ×B :

ð10:2Þ

This equation can be interpreted in the following way. Two forces are applied to the plasma as a whole: the total pressure gradient and the Lorentz force. They cause © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Rozhansky, Plasma Theory, https://doi.org/10.1007/978-3-031-44486-9_10

205

206

10

Magnetohydrodynamics (MHD)

plasma acceleration, which practically coincides with the acceleration of ions since the total momentum is practically the ion momentum due to their large mass. According to the Maxwellian equation →

→

→

∇ × B = μ0 j þ

1 ∂E : c2 ∂t

For slow processes, which are considered in this chapter, displacement current density (the last term on the r.h.s.) can be neglected. Indeed, since drift velocity is connected with electric field, u = E/B, the ratio →

→ EL u2 ∂E 1 = ∇ × B < τs, and the new current profile repeats the old one, while the current density amplitude becomes larger than the initial one. In the general case, when a magnetic field penetrates into a moving plasma, one must use general Eq. (10.14).

212

10

Magnetohydrodynamics (MHD)

Fig. 10.1 Skinned current density profile during fast net current ramp-up

10.3 MHD Waves We shall analyze one-dimensional MHD waves using single-fluid MHD Eq. (10.14) in the collisionless plasma. The solution is sought in the form when all quantities are functions of coordinate z and time t, and, therefore, are functions of each other. Note that the magnetic field direction does not coincide with the z-axis. Equation corresponding to adiabatic approximation we shall replace by equivalent equation for entropy conservation ds/dt = 0. In the equation for the magnetic field, the term containing conductivity is neglected. Then equation system (10.14) in 1D case is: ∂ux B ∂Bx ∂u þ uz x - z = 0, μ0 ρ ∂z ∂t ∂z ∂uy ∂uy B ∂By þ uz - z = 0, μ ∂t ∂z 0 ρ ∂z B ∂Bx By ∂By 1 ∂p ∂u ∂uz þ uz z þ x þ þ = 0, ρ ∂z μ0 ρ ∂z ∂t ∂z μ0 ρ ∂z ∂Bx ∂u ∂u ∂B - Bz x þ Bx z þ uz x = 0, ∂t ∂z ∂z ∂z ∂uy ∂By ∂By ∂uz - Bz þ By þ uz = 0, ∂t ∂z ∂z ∂z ∂Bz = 0, ∂t ∂p ∂u ∂p þ ρc2s z þ uz = 0, ∂t ∂z ∂z ∂s ∂s = 0: þ uz ∂z ∂t

ð10:28Þ

In the ion particle balance equation, we switched to the pressure derivative, using

10.3

MHD Waves

213

∂ρ ∂p dp ∂ρ = c2s = , j dρ s = const ∂t ∂t ∂t

∂ρ ∂p dp ∂ρ = c2s = : js = const ∂z ∂z dρ ∂z

From the equation for the z-component of magnetic field ∂Bz/∂t = 0, it follows that → Bz is the constant in time. On the other hand, from the condition ∇  B = 0, in which the 1D case is reduced to ∂Bz/∂z = 0, we find that component Bz is also constant in space. Hence, the projection of the magnetic field on the z-axis remains constant during MHD wave propagation: Bz = const. The remaining 7 Eqs. (10.28) for seven variables ux, uy, uz, Bx, By, p, s should be solved simultaneously. In the matrix form, Eq. (10.28) can be written as ∂Ψ ∂Ψ þ Z ð ΨÞ = 0, ∂t ∂z

ð10:29Þ

where Ψ is a vector in a 7D space, Ψ = (ux, uy, uz, Bx, By, p, s), and Z(Ψ) is a matrix

Z ð ΨÞ =

-

Bz μ0 ρ

uz

0

0

0

uz

0

0

0

uz

- Bz

0

Bx

Bx μ0 ρ uz

0

- Bz

By

0

uz

0

0

0 0

0 0

ρc2s 0

0 0

0 0

uz 0

0 uz

0

0 Bz μ0 ρ By μ0 ρ 0

-

0

0

0

0

1 ρ 0

0

:

ð10:30Þ

0

The solution is sought in the form of linear waves, i.e., all quantities are supposed to be a sum Ψ = Ψ0 + Ψ1, where vector Ψ1 represents small perturbations. The linearized Eq. (10.29) is: ∂Ψ1 ∂Ψ1 þ Z Ψ0 = 0: ∂t ∂z

ð10:31Þ

For simplicity, one can choose a reference frame where the unperturbed fluid velocity along the z-axis is zero, uz = 0, and the unperturbed y-component of the magnetic field is also zero, By = 0. The last condition can be reached by rotating the coordinate system over the z-axis. Then, the matrix Z(Ψ0) can be simplified:

214

10

Z Ψ0 =

0

0

0

0

0

0

0

0

0

- Bz 0

0 - Bz

0 0

0 0

-

Bz μ0 ρ -

0

Bx 0

Bx μ0 ρ 0 0

ρc2s 0

0 0

Magnetohydrodynamics (MHD)

0

0

0

Bz μ0 ρ

0

0

0 0

1 ρ 0 0

0 0

0 0

0 0

0 0

0

0

:

ð10:32Þ

Vector Ψ1 is supposed to have the form Ψ1 = A exp (ikz - iωt). Then, from Eq. (10.31), one obtains ZA = VA,

ð10:33Þ

where V = ω/k is the phase velocity. Eq. (10.33) represents the classical problem of finding eigenvalues and eigenvectors for matrix Z(Ψ0). The determinant of the matrix (Z - VI) = 0 should be zero, with I being the unit matrix, detðZ - VI Þ = 0:

ð10:34Þ

From Eq. (10.34), seven eigenfrequencies can be obtained. The first two roots of this equation correspond to the Alfven wave: V A = ± cA cos θ:

ð10:35Þ

Here, the Alfven wave velocity cA is defined according to Eq. (10.15), and θ is the angle between the magnetic field and the z-axis. Signs ± correspond to two directions of propagation, i.e., physically the same type of wave. Two second types of phase velocities represent fast magnetosonic waves:

VF = ±

c2A þ c2s þ

c2A þ c2s 2

2

- 4c2A c2s cos 2 θ

1=2

:

ð10:36Þ

:

ð10:37Þ

The phase velocities of the slow magnetosonic wave are

VS = ±

c2A þ c2s -

c2A þ c2s 2

2

- 4c2A c2s cos 2 θ

The last phase velocity corresponds to the entropy wave:

1=2

10.3

MHD Waves

215

Fig. 10.2 Phase diagram for MHD waves

V E = 0:

ð10:38Þ

The dependence of the phase velocity on the angle with a magnetic field is often expressed on a diagram, known as the phase polar (Fig. 10.2). It can be seen that the Alfven wave propagates mainly along the magnetic field, for θ = 0, π phase velocity coincides with the Alfven velocity VA = cA, and, in the perpendicular direction, the phase velocity is equal to zero. Fast magnetosonic waves can propagate both along and across magnetic fields. In the strong magnetic field cA > > cs, its phase velocity coincides with the Alfven velocity independently of the angle with the magnetic field. The third mode is called the slow magnetosonic wave. This wave propagates mainly in the parallel direction, and, in the strong magnetic field, cA > > cs for θ = 0, π, the phase velocity coincides with the ion sonic velocity cs. The phase velocity of the entropy wave is zero. Eigenvectors for these modes for 0 < θ < π/2 are: AA = ½0, cA cos θ, 0, 0, - Bz , 0, 0, AF =

V F c2A sin θ cos θ Bx V 2 , 0, V F , 2 F 2 , 0, ρc2s , 0 , 2 2 VF - VA VA - VF

Bx V 2 V S c2A sin θ cos θ A = , 0, V S , 2 S 2 , 0, ρc2s , 0 , 2 2 VS - VA VA - VS

ð10:39Þ

S

AE = ½0, 0, 0, 0, 0, 0, 1: Eigenvector components show which quantities are exited in the given mode and what are the relations between them. For example, in the Alfven wave, both the velocity and the magnetic field along the y-axis are perturbed while all the other parameters remain constant. Their ratio is u1y =B1y = - c0A cos θ=B0z . Let us analyze physical processes in different modes for simple cases of wave propagation. We start with the Alfven wave propagating in the parallel direction, θ = 0. Assume that there is an electric field directed along x, periodic in space and

216

10 →

Magnetohydrodynamics (MHD)

→

time, E 1x  expð- iωt þ ikzÞ. This field causes E × B drift along the y-axis with velocity u1y = - E1x =B0 . Simultaneously, the polarization current flows in the xdirection: j1x =

n m n0 mi ∂E 1x = - iω 0 2 i E1x : 2 ∂t B0 B0

ð10:40Þ

This current produces magnetic field perturbation according to the Maxwellian equation: j1x = -

1 ik 1 ∂By = - B1y : μ0 μ0 ∂z

ð10:41Þ

The time-dependent magnetic field in its turn produces an electric field -

∂B1y ∂E1x = iωB1y = = ikE 1x : ∂t ∂z

ð10:42Þ

After eliminating the current and electric field from Eqs. (10.40) to (10.42), one finds the dispersion relation for the Alfven wave: 2

ω2 = k2 c0A :

ð10:43Þ

Particle drift in the Alfven wave takes place at a homogeneous background; hence, no density perturbation arises. The only two perturbed quantities are velocity u1y and magnetic field B1y , and their ratio, according to Eqs. (10.40) and (10.41), is u1y =B1y = - c0A =B0 , which is consistent with the general expression for the eigenvector of the Alfven mode Eq. (10.39). The drift velocity is therefore phase shifted at π with respect to magnetic field perturbation, Fig. 10.3. Plasma displacement ξ = u1y dt is phase shifted at π/2. A fast magnetosonic wave propagating perpendicular to the magnetic field in the z-direction, θ = π/2, can be analyzed similarly. The main magnetic field is parallel to the x-axis (Fig. 10.4) and is assumed to be sufficiently strong, cA > > cs. Consider, as in the previous case, an electric field parallel to the y-axis and periodic in time and → → space: E 1y  expð- iωt þ ikzÞ. Plasma E × B drift is directed along the z-axis with velocity u1z = - E 1y =B0 , while the polarization current j1y =

1 n m n0 mi ∂E y = - iω 0 2 i E1y : B20 ∂t B0

The current causes magnetic field perturbation in the x-direction:

ð10:44Þ

10.3

MHD Waves

217

Fig. 10.3 Bend of magnetic field line and plasma shift ξ during Alfven wave propagation

Fig. 10.4 Magnetic field and density perturbations in the fast magnetosonic wave

j1y =

ik 1 ∂B1x = B1x , μ0 μ0 ∂z

ð10:45Þ

which induces an electric field according to -

∂E1y ∂B1x = iωB1x = = - ikE 1y : ∂t ∂z

ð10:46Þ

218

10

Magnetohydrodynamics (MHD)

From Eqs. (10.44) to (10.46), one obtains the dispersion relation, which coincides with Eq. (10.43). Therefore, the fast magnetosonic wave propagates with the Alfven velocity. The perturbation characteristics are illustrated in Fig. 10.4. The magnetic field is perturbed in this wave; however, the magnetic field lines remain straight. It can be seen from Eqs. (10.44) to (10.45), the particle velocity is in the same phase as the magnetic field perturbation, and u1z =B1x = c0A =B0 in accordance with the general expression Eq. (10.39). The density in this wave is also perturbed, as follows from the particle balance - iωn1 þ ikn0 u1z = 0, so that n1 =n0 = B1x =B0 . The last expression corresponds to the magnetic field frozen in Eq. (10.17) and the conservation of the ratio B/n for motion across the magnetic field. It also corresponds to eigenvector Eq. (10.39). Fast magnetosonic waves propagating across a magnetic field resemble sonic waves, where magnetic pressure B2/2μ0 plays a role of plasma pressure. Indeed, the Alfven velocity can be obtained by replacing the pressure in the expression for the sonic velocity with the magnetic pressure. In the limit cA < < cs, a fast magnetosonic wave is transformed into a sonic wave according to Eq. (10.39). A fast magnetosonic wave, which propagates along a magnetic field, is in fact an Alfven wave with different polarization. At θ → 0, the first and fourth terms in the expression for eigenvector AF diverges as θ-1, since V 2A - V 2f → c2A sin 2 θ and Bx = B sin θ. Other terms remain finite and can be neglected. The ratio of the first and fourth terms is u1x =B1x = c0A =B0 , in other words, it corresponds to an Alfven wave with a perturbed x-component of the magnetic field, with plasma also moving along the x-axis. Slow magnetosonic waves propagating in the parallel direction in a strong magnetic field correspond to sonic waves. The entropy wave represents entropy perturbation in plasma with unperturbed pressure. Let us, for example, consider positive density perturbation n1(z) and and T 1i ðzÞ, so that negative temperature perturbations T 1e ðzÞ 1 1 0 0 n1 =n0 = - T e þ T i = T e þ T i . Since pressure perturbation is absent, such density perturbation will stay at rest in the absence of dissipation, according to the 3=2 pressure balance equation. At the same time, entropy proportional to ln T e,i =n remains perturbed. MHD waves decay due to dissipative processes. In addition, there exists a specific collisionless decay mechanism caused by the spatial dependence of the phase velocity. We shall illustrate it for the case of an Alfven wave propagating along → → magnetic field B 0 k z in the inhomogeneous plasma with density depending on the x-coordinate. According to the pressure balance in Eq. (7.5), p0 + B2/2μ0 = const(x). If β < < 1, the magnetic field is only slightly perturbed B ≈ B0, and the dependence of the Alfven wave on the x-coordinate is mainly determined by the inhomogeneous density cA ð x Þ =

B20 =μ0 mi n0 ðxÞ:

ð10:47Þ

10.4

Nonlinear MHD Waves

219

Consider х-localized wave packet of Alfven waves, where the y-component of the magnetic field is perturbed δBy = bðxÞ expð- iωt þ ikzÞ:

ð10:48Þ

Phase velocity ω/k = cA depends on х; hence, waves with the same phase at z = 0 will diverge at phases for large values of z. In the experiment, magnetic field perturbation is always averaged in space; hence, averaged over х magnetic perturbation should turn to zero due to phase divergence. This effect manifests itself as collisionless damping of Alfven waves, in spite of the absence of real collisional damping. Indeed, the magnetic field perturbation averaged over interval 2δ can be defined as xþδ

1 δBy = 2δ

xþδ

1 δBy dx = 2δ x-δ

bðxÞ expð- iωt þ ikzÞdx:

ð10:49Þ

x-δ

Since ω(x) = kcA(x), one can integrate over ω: expðikzÞ δBy = 2δ

ωþ

bðωÞ expð- iωt Þdω,

ð10:50Þ

ω-

where bðωÞ =

bð x ð ω Þ Þ , ω ± = ωðx ± δÞ: dω=dx

Integrating by parts yields

expðikzÞ bðωÞ expð- iωt Þ δBy = 2δ it

ωþ ωþ ω-

1 þ it

ω-

dbðωÞ expð- iωtÞdω : dω ð10:51Þ

Continuing integration by parts, one obtains a series with terms inversely proportional to t. At t → 1, all terms turn to zero, and hδByi also turns to zero, so the average magnetic field perturbation is damped.

10.4

Nonlinear MHD Waves

In the general case, nonlinear waves are described by Eq. (10.29). In the plane (z, t), special curves can be defined,

220

10

Magnetohydrodynamics (MHD)

Fig. 10.5 Characteristic in (t, z) plane

z = ζðt Þ,

ð10:52Þ

which are called characteristics, Fig. 10.5. At such curve, vector Ψ is defined by initial conditions, while its value Ψ outside of characteristics could not be found for known value Ψ at the characteristics. Let us express derivatives ∂Ψ/∂z and ∂Ψ/∂t as functions of the normal derivative to characteristic ∂Ψ/∂n and derivative ∂Ψ/∂s along the characteristic: ∂Ψ = ∂t ∂Ψ = ∂z

∂Ψ 1 þ V ∂s ∂Ψ V þ 2 ∂s 1þV 1

2

∂Ψ , 1 þ V ∂n 1 ∂Ψ : 2 ∂n 1þV V

2

ð10:53Þ

Here, V = dζ/dt slope of the curve z = ζ(t). Substituting Eq. (10.53) into Eq. (10.29) yields ðZ - VI Þ

∂Ψ ∂Ψ = - ðVZ þ I Þ : ∂n ∂s

ð10:54Þ

According to the definition, vector Ψ outside of the characteristics could not be found for known value Ψ, and, therefore, the derivative ∂Ψ/∂n could not be obtained from Eq. (10.54). On the other hand, ∂Ψ/∂s = 0 at the characteristics. Hence, an invertible matrix for matrix Z - VI does not exist, and Z - VI is a degenerate matrix. Therefore, we have the relation detðZ - VI Þ = 0,

ð10:55Þ

which fully coincides with Eq. (10.34). As in the linear case, 7 types of pure waves and 7 phase velocities dζ(t)/dt = V are obtained from Eq. (10.55). The difference consists of the fact that matrix Z(Ψ) Eq. (10.30) depends on Ψ, and, therefore, V = V(Ψ). Since vector Ψ is conserved at the characteristic, the value V is constant there so that the characteristic is a straight line. At the same time, the slopes of the characteristics are different, and the propagation velocity V depends on amplitude.

10.4

Nonlinear MHD Waves

221

Let us consider a nonlinear Alfven wave propagating along a magnetic field as an example of a nonlinear MHD wave. Perturbed components are supposed to be uy and By. According to the general analysis of Sect. 10.3, the magnetic field component Bz remains constant in the 1D wave. The second and fifth Eqs. (10.28) are: ∂uy B ∂By - z = 0, μ0 ρ ∂z ∂t ∂By ∂uy - Bz = 0: ∂t ∂z

ð10:56Þ

One can easily see that the solution of this equation system coincides with the solution of the system ∂By ∂By -V = 0, ∂t ∂z ∂uy ∂uy -V = 0, ∂t ∂z

ð10:57Þ

if V = cA =

B2z : μ0 ρ

ð10:58Þ

From the third equation of system (10.28) By ∂By 1 dp þ =0 μ0 ρ ∂z ρ dz

ð10:59Þ

one obtains pþ

B2y = const: 2μ0

ð10:60Þ

We see that in the regions where the magnetic field and magnetic pressure increase (note that Bz = const), the plasma pressure and, hence, plasma density decrease. The Alfven velocity Eq. (10.58) is proportional to ρ-1/2, and considering adiabatic law p~ρ5/3, the propagation velocity V~p-0.3; therefore, the propagation velocity decreases with pressure, and according to Eq. (10.60) increases with the magnetic field. As a result, regions with a larger magnetic field propagate faster than those with a smaller magnetic field, so the nonlinear Alfven wave overturns. In the second example, we consider a fast magnetosonic wave propagating across a magnetic field Bx. Since for perpendicular motion, the ratio

222

10

b=

Magnetohydrodynamics (MHD)

Bx = const ρ

ð10:61Þ

is conserved, the momentum balance equation along the z-axis can be rewritten in the form ∂u 1 ∂B2x 1 ∂p 1 ∂p ∂uz þ uz z = = , 2μ0 ρ ∂z ρ ∂z ρ ∂z ∂t ∂z

ð10:62Þ

where p = p þ

b2 ρ 2 : 2μ0

Equation (10.62) and particle balance equation coincide with equations for nonlinear sonic waves, Sect. 9.3, where p is replaced by p. The wave propagates with Alfven velocity (for cA > > cs), with respect to plasma velocity V = uz þ c A ≈ cA =

dp = dρ

b2 ρ = μ0

bBx : μ0

ð10:63Þ

Since the propagation velocity is proportional to B1=2 x , the wave overturns.

10.5

Magnetosonic Waves with Dispersion

We shall demonstrate using magnetosonic waves as an example that in magnetized plasma, soliton-type solutions exist, which propagate perpendicular to the magnetic field without deformation. However, in contrast to sonic solitons, which need deviation from quasineutrality, magnetosonic soliton solutions require an account of electron inertia. Let the magnetic field be directed along the z-axis, while a wave propagates along the x-axis with velocity V. We shall seek solutions with V = const that are independent of the wave amplitude, so all quantities are sought in the form Ψ = Ψ(x - Vt). As basic equations, we take two-fluid MHD equations with zero temperatures in the collisionless limit. The particle balance equation for electrons and ions is ∂n ∂ðnuxe,i Þ þ = 0, ∂t ∂x or, since density is n = n(ξ = x - Vt),

ð10:64Þ

10.5

Magnetosonic Waves with Dispersion

-V

223

dn d ðnuxe,i Þ = 0: þ dξ dξ

ð10:65Þ

Integrating Eq. (10.65), assuming n(ξ → 1) = n0 and uxe, i(ξ → 1) = 0, one obtains nðuxe,i - V Þ = - n0 V:

ð10:66Þ

According to Eq. (10.66), the ion and electron velocities in the x-direction coincide: uxe = uxi = u. The momentum balance equations in the new variables are duxe,i ðuxe,i - V Þ = ∓ eE x ∓ euye,i B, dξ duye,i ðuxe,i - V Þ = ∓ eE y ± euxe,i B: me,i dξ me,i

Electric and magnetic fields are linked by the → → ∇ × E = - ∂ B =∂t, which for variable ξ = x - Vt is

ð10:67Þ

Maxwellian equation

dEy dB =V : dξ dξ

ð10:68Þ

Integration with an account of Ey(ξ → 1) = 0 and B(ξ → 1) = B0 yields E y = V ðB - B0 Þ : →

ð10:69Þ →

From the second Maxwellian equation ∇ × B = μ0 j , we have -

dB = μ0 en uyi - uye : dξ

ð10:70Þ

After summing up the x components in the momentum balance for electrons and ions in Eq. (10.67), with account of uxe = uxi = u and mi > > me, one obtains m i ðu - V Þ

du = e uyi - uye B : dξ

ð10:71Þ

Combining this equation with Eq. (10.70) and particle balance Eq. (10.66), we find m i n0 V

du B dB , = dξ μ0 dξ

ð10:72Þ

B2 - B20 : 2μ0 mi n0 V

ð10:73Þ

and after integration: u=

224

10

Magnetohydrodynamics (MHD)

Now, the sum of two y-components of momentum balance Eq. (10.67) yields meuye + miuyi = 0. Hence, uye > > uyi :

ð10:74Þ

In other words, magnetic field perturbation is created by electron current, while ion current in Eq. (10.70) can be neglected. Substituting the electron velocity uye as a function of the magnetic field derivative from Eq. (10.70) into the second Eq. (10.67), with account of Eq. (10.69), Eq. (10.66) for density and Eq. (10.73), we obtain the equation for the magnetic field

-

B2 - B20 B2 - B20 -V = - V B þ VB0 : 2μ0 mi n0 V 2μ0 mi n0 V

δ2 d dB B2 - B20 -V V dξ dξ 2μ0 mi n0 V

ð10:75Þ Here, a scale δ=

me n0 e 2 μ 0

1=2

ð10:76Þ

determines the characteristic perturbation spatial scale. This scale is known as collisionless skin depth. Multiplying both parts of Eq. (10.75) by dB/dξ and integrating, we have - δ2

dB dξ

2

B2 - B20 -V 2μ0 mi n0 V

2

2

=

B2 - B20 - V 2 ðB - B0 Þ2 þ const: ð10:77Þ 4μ0 n0 mi

The constant is chosen to be zero, so that at infinity B = B0 and dB/dξ = 0. Then, from Eq. (10.77), we obtain δ

dB =± dξ

B - B0 2

- B20

B 2μ0 mi n0 V 2

-1

1-

ðB þ B0 Þ2 : 4μ0 mi n0 V 2

ð10:78Þ

Curves in the phase space are shown in Fig. 10.6. A soliton solution corresponds to a rise of the magnetic field from the value B0 up to Bmax (branch dB/dξ > 0) and a further drop of the field from Bmax back to B0 (branch dB/dξ < 0). At the maximum dB/dξ = 0, so according to Eq. (10.78) V=

ðBmax þ B0 Þ2 : 4μ0 n0 mi

ð10:79Þ

10.5

Magnetosonic Waves with Dispersion

225

Fig. 10.6 Curves in the phase space (B, dB/dξ)

Fig. 10.7 Magnetosonic soliton

This relation connects the velocity of the magnetosonic soliton with its amplitude. The soliton shape can be obtained by further integration of Eq. (10.78). An example is shown in schematic Fig. 10.7. The Alfvenic Mach number is introduced as

M=

V V = cA

2μ0 n0 mi : B0

ð10:80Þ

For small perturbations B → B0, the soliton velocity becomes the Alfven velocity and M → 1. For larger amplitudes, the Mach number rises in accordance with Eq. (10.79) until the denominator in Eq. (10.78) becomes zero. The critical value of the magnetic field according to Eqs. (10.78) and(10.79) is Bmax = 3B0 for Mach number M = 2. In this case, the spatial derivative in the maximum dB/dξ → 1, Fig. 10.7. Therefore, magnetic solitons exist in the following range of Alfvenic Mach numbers: 1 ≤ M ≤ 2:

ð10:81Þ

For larger Mach numbers, the effects associated with electron inertia are not able to prevent overturn, and collisionless shock waves arise.

226

10.6

10

Magnetohydrodynamics (MHD)

Alfven Masers

When plasma is restricted by boundaries in the parallel direction, standing Alfven waves are formed. Such an open magnetic trap is formed in the Earth’s magnetosphere with relatively hot collisionless plasma (Fig. 10.8). Alfven waves are reflected from the ionosphere with high perpendicular conductivity caused by ion-neutral collisions. Therefore, the perpendicular inductive electric field and the perpendicular magnetic field at the level of the ionosphere are small and could be put to zero under imposed boundary conditions, so a standing Alfven wave is formed.

Fig. 10.8 Scheme of Alfven maser: (a) magnetosphere flux tube, (b) open magnetic trap in the laboratory

10.6

Alfven Masers

227

Similar magnetic traps exist in the laboratory. Particles in such magnetic traps (magnetic belts in the Earth and planet magnetospheres) form a strongly nonequilibrium distribution function. Indeed, at the ends of a magnetic tube, perpendicular particle velocity increases due to adiabatic invariant conservation, and since full energy is also conserved, particles with small parallel velocities are reflected back to the trap. As a result, loss cones are formed in the velocity space, which becomes a source of various electromagnetic instabilities. Alfven waves in the “resonator” interact with sources of particles, and a maser is created. Similar standing Alfven waves interacting with fast particles are also generated in tokamaks in the vicinity of resonant flux surfaces, where the magnetic field line after a few turns over tokamak returns to its initial position.

Chapter 11

Dynamics of Plasma Blobs and Jets in a Magnetic Field

11.1

Plasma Motion Across Magnetic Field in Vacuum

Consider a plasma blob moving at t = 0 across the magnetic field along the x-axis → with velocity u 0 , as shown in Fig. 11.1. From the momentum balance equation for electrons, it follows that such motion is caused by an induced electric field in the ydirection, and → →

u0= →

→

E0× B0 B2

:

ð11:1Þ

→

In other words, the blob motion is E × B drift produced by blob polarization. To analyze the further motion of the blob, let us consider the sum of the momentum balance Eq. (10.2), electron inertia is neglected. Multiplying both parts of Eq. (10.2) → by B × , we obtain for current density → →

j =

B × ∇p B2

→

þ

→

B × nmi ddtu i B2

:

ð11:2Þ

The first term on the r.h.s. is a diamagnetic current, which is divergent free in a homogeneous magnetic field. The second term corresponds to inertial or polarization current. If the blob as a whole is decelerated, this current is directed downwards. However, this contradicts the quasineutrality condition →

∇  j = 0:

ð11:3Þ

Hence, vertical current should be zero, and, correspondingly, inertia should also turn → to zero. The blob moves along the x-axis with constant velocity u 0 given by Eq. (11.1). Considering the forces applied to each particle, the Lorentz force eu0B, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Rozhansky, Plasma Theory, https://doi.org/10.1007/978-3-031-44486-9_11

229

230

11

Dynamics of Plasma Blobs and Jets in a Magnetic Field

Fig. 11.1 Polarization of a blob moving across the magnetic field

Fig. 11.2 Magnetized plasma blob in the presence of gravitational force

which shifts particles in the vertical direction, is balanced by the opposite vertical → electrostatic force e E 0 . Strictly speaking, this statement is correct for a blob extended in the direction of motion х when the electric field inside the blob is practically homogeneous. For blobs with comparable sizes in the x and → y directions, in addition to motion with velocity u 0 , blob deformation takes place (see below). In the presence of the force of gravity, the magnetized plasma blob obtains additional polarization, as shown in Fig. 11.2. Indeed, ion drift caused by gravity

11.1

Plasma Motion Across Magnetic Field in Vacuum →

231

→

force in the direction mi g × B generates current in the negative direction of the xaxis: jgx = -

nmi g : B

ð11:4Þ

The contribution of electrons can be neglected since their drift is mi/me times smaller than that of ions. Current jgx leads to blob polarization. The polarization electric field should produce current jp in the opposite direction, which should balance current jgx: jpx = - jgx :

ð11:5Þ

In a fully ionized plasma and uniform electric field, such a current is a polarization or inertial current caused by the electric field changing over time: →

nm ∂ E : j p = 2i B ∂t

→

ð11:6Þ →

This expression corresponds to the second term in Eq. (11.2), if one puts u i = →

→

→

→

E × B =B2 and d u i =dt = ∂ u i =∂t. From Eqs. (11.4) to (11.6), one finds that the

electric field in the x-direction grows linearly with time E x = Bgt: →

ð11:7Þ

→

Increasing electric field produces E × B drift downwards in the direction of the gravitational force with the velocity Vy = -

Ex = - gt, B

ð11:8Þ →

→

which corresponds to blob motion with acceleration a = g . As in the previous case, electric field inhomogeneity leads to its additional deformation. Blob motion in a nonuniform magnetic field is similar to that caused by gravitation. Let us consider as an example the magnetic field of a torus (tokamak) B = B0R0/ R, Fig. 11.3. In such a magnetic field, ionized particles drift vertically with the velocity

V jy =

mj

V 2j⊥ 2

þ V 2jk

ej BR

,

ð11:9Þ

where Vk, ⊥ are single particle velocities. The first term here is caused by a force applied to the Larmor orbit trying to push it out to the weaker magnetic field, and the second term is connected with the centrifugal force in the curvilinear magnetic field.

232

11

Dynamics of Plasma Blobs and Jets in a Magnetic Field

Fig. 11.3 Blob acceleration in nonuniform magnetic field

Electrons and ions drift in opposite directions due to different charge signs, so a vertical current arises. To obtain the expression for this current, one should perform averaging over velocities using the Maxwellian distribution function:

jy = e

*

M V iy f M i - V ey f e dV =

2nðT e þ T i Þ : BR

ð11:10Þ

It is worth noting that the divergence of this current caused by orbit leading centers vertical drifts coincides with the divergence of the diamagnetic current, given by the first term in Eq. (11.2). The current (11.10) should be compensated by the opposite current Eq. (11.6). As a result, the vertical electric field increases with time. This → → field produces E × B outwards drift, so the blob is accelerated in the direction opposite to the magnetic field gradient, and the acceleration is g=

2ðT e þ T i Þ : mi R

ð11:11Þ

In the general case, the blob evolution is governed by equation system (11.2) and (11.3). In the first approximation, the electron and ion velocities coincide since the → → * → → → drift is E × B : u e = u i = E × B =B2 . Let us introduce dimensionless variables x = x=a; y = y=a; t = t ðg=aÞ1=2 ; φ = cφ= Ba3=2 g1=2 ,

ð11:12Þ

where a is the characteristic blob size, φ is the electrostatic potential, and acceleration g is defined according to Eq. (11.1). Then, from Eqs. (11.2) and (11.3), one obtains

11.1

Plasma Motion Across Magnetic Field in Vacuum

2,5

233

y

-0,5

x

-5,5 -3,5

4,5

12,5

Fig. 11.4 Lines of constant densities in nonuniform magnetic field, t = 4

∂φ ∂∇φ ∂ ln n þ J ðφ, ΔφÞ þ þ J ðφ, ∇φÞ ∇ ln n þ = 0, ∂y ∂t ∂t

ð11:13Þ

where the operator J is defined as J ðα, βÞ =

∂ α ∂β ∂β ∂α : ∂x ∂y ∂x ∂y

The particle balance equation in the dimensionless variables is reduced to ∂n þ J ðφ, nÞ = 0: ∂t

ð11:14Þ

Here, it is taken into account that in the homogeneous magnetic field → → ∇  E × B =B2 = 0, and corrections caused by derivatives of the magnetic field are small. Eq. (11.14) describes plasma motion along equipotentials. Numerical modeling of the system Eqs. (11.13) and (11.14) demonstrates that the blob moves in the x direction with acceleration given by Eq. (11.11), and its shape resembles a mushroom since its upper and lower parts move in the opposite direction in the dipole electric field (Fig. 11.4).

234

11

Dynamics of Plasma Blobs and Jets in a Magnetic Field

→

Fig. 11.5 Plasma jet moves across magnetic field, at t = 0 the velocity is u 0

11.2

Deceleration of the Plasma Jet by Ambient Plasma

Consider the dynamics of a plasma jet injected into the ambient magnetized plasma. For simplicity, let us assume that the jet is infinite in the x-direction, and, at t = 0, the jet moves with a velocity u0 across the magnetic field, as shown in Fig. 11.5. Plasma density in the jet nI(y, z) is supposed to be much larger than the ambient plasma density ni, and jet ions mI could differ from ambient ions mass mi. As discussed in the previous section, plasma injected in the x-direction is polarized in the y-direction → so that electric field E 0 , which satisfies Eq. (11.1). However, in the ambient plasma, the electric field propagates along the magnetic field as an Alfven wave. At the front of the Alfven wave, the electric field increases, and, therefore, a polarization current in the y-direction is generated. This current balances the polarization current inside the jet in the -y-direction caused by jet deceleration. The scheme of currents is shown in Fig. 11.6. Let us find the jet deceleration law for the case u0 ≪ cA, where cA = B/(2μ0nimi)1/2 is the Alfven velocity of the ambient plasma. If this condition is satisfied, the perturbed magnetic field in the Alfven wave is small: Bx~uxB/cA ≪ B, and the magnetic field can be considered constant. Alfven wave propagation outside the jet is described by the wave equation 2

2

∂ u ∂ ux = c2A 2x ; ∂t 2 ∂z

c2A =

B2 : 2μ0 mi ni

ð11:15Þ

Inside jet analogously 2

nI mI

2

B2 ∂ ux ∂ ux = : 2 2μ0 ∂z2 ∂t

ð11:16Þ

Let us integrate Eq. (11.16) along z from -z0 to z0, where z0 is chosen to satisfy

11.2

Deceleration of the Plasma Jet by Ambient Plasma

235

Fig. 11.6 Scheme of currents in the ambient plasma during jet injection across magnetic field for z > 0. At negative z,the current pattern is symmetrical

ak < z0 ≪ LA = cA t, where ak is the jet characteristic size along the magnetic field. At this distance, plasma velocity ux can be considered independent of z since it changes at a much larger scale LA. Integration yields

2

MI

B2 ∂ux ∂ ux = 2 2μ0 ∂z ∂t

z0

ð11:17Þ

, - z0

where 1

z0

MI =

mI nI dz ≈ - z0

mI nI dz

ð11:18Þ

-1

is the integral mass of injected ions per unit length. Using symmetry, we consider only positive values of z. For z0 → 0, we find 2

MI

∂ ux ∂t 2

z=0

=

B2 ∂ux μ0 ∂z

z=0

:

ð11:19Þ

236

11

Dynamics of Plasma Blobs and Jets in a Magnetic Field

Fig. 11.7 Velocity profile of the ambient plasma at fixed moment

This expression should be considered as a boundary condition for Eq. (11.15), which is the equation for Alfven wave propagation in the ambient plasma. The initial condition for Eq. (11.15) is: ux =

u0 , 0,

z=0 : z>0

ð11:20Þ

Since the solution of wave Eq. (11.15) is ux = ux(z - cAt), considering the initial and boundary conditions, one obtains ux =

u0 exp

ux =

0

B2 ð z - cA t Þ μ0 M I c2A

z ≤ cA t;

:

ð11:21Þ

z ≥ cA t :

Velocity profile given by Eq. (11.21) is shown in Fig. 11.7. One can see that ambient plasma is dragged by the plasma jet in the x-direction at the scale zτ = cAt. The wavefront propagates along the magnetic field with the Alfven velocity. The jet velocity coincides with the ambient plasma velocity z = 0 and exponentially decreases with time: ux ð0, t Þ = u0 expð- t=τA Þ,

τA =

μ 0 M I cA : B2

ð11:22Þ

Characteristic time τA can be interpreted as follows. The net mass of the ambient plasma per unit length dragged during time τA is (with account of positive and negative values of z)

11.2

Deceleration of the Plasma Jet by Ambient Plasma

237

M i ðτA Þ = 2zτ ni mi = M I :

ð11:23Þ

In other words, at the moment τA, the integral mass of the ambient plasma dragged by the jet equals the jet integral mass. At the same time, the jet velocity decreases considerably with respect to the initial velocity. The electric field that causes plasma drift in the x-direction up to distances z = ± cAt, according to Eq. (11.21), decreases exponentially with time. Hence, a negative polarization current flows in the y-direction, as shown in Fig. 11.6. Integral current (for positive values of z) is obtained by integration along the magnetic field

cA t

I = Ic þ I0 = 0

E y ð z = 0Þ ðnI mI þ ni mi Þ ∂E y dz = , 2 μ0 cA ∂t B

ð11:24Þ

where Ey(z = 0) = u0B exp (-t/τA). Net current consists of two equal contributions. The first one Ic is the integral over the jet, during the integration, the electric field Ey is taken at z = 0, and the integral over nI(z)mI gives MI/2. The second contribution I0 is obtained from the integral over ambient plasma with account of Eq. (11.21) and identity Ey = uxB. This current flows in the region 0 < z < cAt. The net negative current I should be compensated by the positive current in the y-direction. Therefore, at the Alfven wavefront, where ambient plasma is accelerated from zero velocity up to value u0, a positive current IF = - I flows. Now let us separate integral currents in a different way. As a first contribution, we shall now take the integral current of the jet Ic = -

E y ð z = 0Þ : 2μ 0 cA

ð11:25Þ

The second contribution from the ambient plasma is the sum of two currents, including the current at the Alfven wavefront: IW = Ic þ IF =

E y ð z = 0Þ 2μ 0 cA

ð11:26Þ

These two net currents gain balance each other. The second current in the ambient plasma can be rewritten as IW =

W

E y ð z = 0Þ ;

W

=

1 , 2μ0 cA

ð11:27Þ

where the quantity ΣW is known as the Alfven conductivity. In other words, ambient plasma with a propagating Alfven wave is equivalent to a resistor, according to Eq. (11.27) (one has to keep in mind that the same current

238

11

Dynamics of Plasma Blobs and Jets in a Magnetic Field

flows at negative values of z). The paradigm of Alfven conductivity can be used in various problems of plasma physics. All solutions discussed above are applicable in low collisional plasma when the skin time scale is sufficiently large. The corresponding condition is τs = 2μ0 σa2⊥ ≫ τA ,

ð11:28Þ

with a⊥ being the perpendicular jet size. In the opposite case, the electric field is electrostatic, and spreading of the polarization in the ambient plasma has a diffusive character with diffusion coefficient D  c2A τs .

11.3

Edge Localized Modes and Filaments

In edge tokamak plasma with a strong density gradient, special edge localized modes (ELMs) can develop. This phenomenon is typical for high confinement regimes, Chap.16, when the pressure gradient near the separatrix exceeds a critical value, determined by the threshold for some of the instabilities, such as peeling-ballooning and kinetic ballooning. As a result of the nonlinear development of the instability, plasma is split into several blobs or filaments extended in the magnetic field direction. Photographs of such filaments for the MAST tokamak are shown in Fig. 11.8. After formation, filaments move radially across the separatrix and further away towards the major radius, i.e., in the direction opposite to the magnetic field gradient. The perpendicular size of a single filament is of the order of a few cm, while its parallel dimension is controlled by plasma expansion along the magnetic field line analyzed in Chap.9.

Fig. 11.8 Filament structure during ELMs

11.3

Edge Localized Modes and Filaments

239

y

x

----

∇B

E jy -

j||

++++

j||+ jw

Fig. 11.9 Currents short-circuiting scheme through divertor targets

Outside of the separatrix, the plasma blob is polarized due to the ∇B driven vertical current given by Eq. (11.10). In contrast to the situation in vacuum considered above in Sect.11.1, this vertical current could be short-circuited by parallel currents in the ambient plasma, which in turn are closed through divertor targets (Fig. 11.9). The resulting polarization is smaller than that in vacuum, so the compensating polarization current can be neglected. Parallel currents required to close the ∇B driven vertical current could be obtained from the current continuity → equation ∇  j = 0 by integrating it over the parallel filament dimension. From Eq. (11.10), we obtain jk± =

2nðT e þ T i Þ lk , BR l⊥

ð11:29Þ

where lk and l⊥ are the parallel and perpendicular half-widths of the filament (the perpendicular pressure gradient is replaced here by p/l⊥). This is the absolute value of the pair of currents, from the blob to the target and from the target to the blob (Fig. 11.9). Since this current should flow through the sheath to the target, the potential at the sheath edge (and at the filament for infinite parallel conductivity of the ambient plasma) should be larger than the floating potential for current to the target and smaller than the floating potential for current from the target. According to the current-voltage characteristic of the sheath in Eq. (3.26), one obtains

240

11

jþ k = ens jk- = - ens

T es þ T is mi

1=2

T es þ T is mi

Dynamics of Plasma Blobs and Jets in a Magnetic Field

1 T es -p 2π me 1=2

1=2

T es 1 -p 2π me

exp 1=2

eφþ T es

eφ exp T es

: ð11:30Þ :

Here, subscript “s” corresponds to plasma parameters at the sheath edge. It is assumed that the parallel current does not exceed the saturation ion current given by the first term on the r.h.s., otherwise short-circuiting through the target is not possible. Eqs. (11.29) and (11.30) determine the potentials at the sheath edge, and the same potentials are assumed at the blob level. The vertical electric field of the blob is E y = ðφþ - φ - Þ=l⊥ ,

ð11:31Þ

which produces drift in the direction of the major radius ux = E y =B = ðφþ - φ - Þ=Bl⊥ :

ð11:32Þ

For edge tokamak plasma parameters, its value could be in the order of 1 km/s, which is consistent with experimental observations. Very dense filaments that ∇Bdriven vertical current could not be short-circuited should still be accelerated, as described in Sect.11.1.

Chapter 12

Plasma Equilibrium

We shall define equilibrium as a plasma state with flow velocities smaller than sound or Alfven velocities. In such a situation, it is possible to neglect inertia terms in Eq. (10.2) for the total momentum balance with respect to the pressure gradient to obtain →

→

∇p = j × B :

ð12:1Þ

This is the equation for plasma equilibrium. Accordingly, the pressure gradient is balanced by the Lorentz force caused by plasma currents and self-consistent magnetic fields. Plasma equilibrium Eq. (12.1) can be combined with the Maxwellian equations →

j =

→ 1 ∇× B , μ0

→

∇  B = 0:

ð12:2Þ

Substituting the current into Eq. (12.1) yields ∇p = - ∇

→ 1 → B2 þ B∇ B: 2μ0 μ0

ð12:3Þ

The first term on the r.h.s. can be interpreted as the magnetic pressure gradient, and the second can be interpreted as the magnetic tension associated with the curvature of the magnetic field lines. Equations (12.1) and (12.2) or Eq. (12.3) are the basic equations for equilibrium analysis. Let us emphasize once again that plasma flows are still present in plasma.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Rozhansky, Plasma Theory, https://doi.org/10.1007/978-3-031-44486-9_12

241

242

12.1

12

Plasma Equilibrium

On the Possibility of Equilibrium in the Absence of a Vacuum Magnetic Field

Let us demonstrate that equilibrium cannot be obtained by internal currents flowing in plasma in the absence of an external magnetic field produced by currents in the coils. To prove this, let us assume the opposite, so that Eq. (12.3) is supposed to be satisfied with internal currents only. We introduce the magnetic stress tensor πik = pδik þ

B2 1 δ - BB : 2μ0 ik μ0 i k

ð12:4Þ

→

With an account of ∇  B = 0, Eq. (12.3) can be reduced to ∂πik = 0: ∂xk

ð12:5Þ

Let us introduce an integral over an infinite volume, including the region with plasma and the region outside plasma, which after integration by parts is: xi

∂πik dV = ∂xk

xi πik dSk -

∂xi πik dV: ∂xk

ð12:6Þ

In accordance with Eq. (12.5), l.h.s. of this equation is zero. Hence, since ∂xi/ ∂xk = δik, one finds πii dV =

xi πik dSk :

ð12:7Þ

After substitution of the magnetic stress tensor πik in Eq. (12.4), one obtains B2 3p þ dV = 2μ0

B2 pþ 2μ0

→* →

r -

→

rB B μ0

→

dS:

ð12:8Þ

The l.h.s. of this equation is finite and a positive quantity. On the r.h.s., the integral is taken over the surface limiting infinite volume. The plasma pressure at this surface equals zero. If a magnetic field is created by plasma currents, at infinity, it should decay as r-3 (dipole field) or faster. Therefore, the surface integral on the r.h.s. of Eq. (12.8) becomes zero. Therefore, one comes to a contradiction—Eq. (12.8) cannot be satisfied, and hence the assumption that Eqs. (12.3) and (12.5) are incorrect. Finally, one can conclude that to keep plasma in the equilibrium state, it is necessary to create a magnetic field by currents flowing in the coils, where → → mechanical forces compensate for j × B forces.

12.2

12.2

Equilibrium of a Pinch

243

Equilibrium of a Pinch

Consider an infinite plasma cylinder in a magnetic field with the cylinder axis parallel to the magnetic field. Current along the cylinder axis can produce an additional azimuthal magnetic field. Such a plasma cylinder is known as a pinch. All pinch plasma parameters are independent of the longitudinal coordinate z and azimuthal angle θ and are functions of the radial coordinate r only. Maxwellian → equation ∇  B = 0 in cylindrical coordinates in the absence of z and θ dependence is reduced to equation r-1d(rBr/dr) = 0, from which it follows Brr = const. Since the magnetic field cannot go to infinity at r = 0, its radial component should be zero: Br = 0. Therefore, only longitudinal and azimuthal components of magnetic fields exist in the pinch, which are correspondingly produced by azimuthal and longitudi→ → nal currents. From the projections of the Maxwellian equation ∇ × B = μ0 j , we have jθ = -

1 dBz , μ0 dr

jz =

1 dðrBθ Þ : μ0 r dr

ð12:9Þ

Here, we use identity →

∇× A

i

=

eijk ∂ðAk hk Þ : hj hk ∂xj

Quantity eijk is the Levi-Civita symbol, and each component changes sign when any two indices are interchanged. It equals ±1 for different indices and zero for coinciding indices. Lame coefficients for cylindrical coordinates are hr = 1, hθ = r, hz = 1. Equilibrium equation for pinch given by the radial balance dp = jθ Bz - jz Bθ : dr

ð12:10Þ

After substituting the current density from Eq. (12.9), one obtains 2 B d ðrBθ Þ dp 1 dBz = 0: þ θ þ μ0 r dr dr 2μ0 dr

ð12:11Þ

The azimuthal magnetic field is connected with the current flowing inside a surface of a given radius: μ I ðr Þ Bθ = 0 , 2πr

r

I ðr Þ =

jz 2πrdr: 0

244

12

Plasma Equilibrium

Therefore, Eq. (12.11) can also be written in the form 2 μ dI 2 dp 1 dBz þ þ 20 2 = 0: dr 2μ0 dr 8π r dr

ð12:12Þ

Equation (12.11) or (12.12) determines pinch equilibrium in the magnetic field. Let us now consider special cases of equilibrium. Pinch without parallel current is called θ-pinch. In the θ-pinch jz = 0 and Bθ = 0, the equilibrium equation after integration over the radius is reduced to pþ

B2z B2 = 0 : 2μ0 2μ0

ð12:13Þ

Here, B0 is the magnetic field outside the plasma. This equation can be satisfied only for small parameter β, β=

2μ0 p < 1: B20

ð12:14Þ

In the opposite case β > 1, there is no equilibrium since the plasma pressure cannot be balanced by the Lorentz force, and the plasma expands radially with a velocity of the order of sound speed as without a magnetic field (see Сhap. 9). In contrast, when the condition Eq. (12.14) is satisfied, equilibrium is established automatically by the diamagnetic current, which arises in the nonuniform plasma due to particle rotation over Larmor orbits. The diamagnetic current reduces the magnetic field inside the plasma column, so Eq. (12.14) is satisfied. In contrast to the θ-pinch, in a z-pinch, a longitudinal magnetic field is absent, Bz = 0, and hence, a diamagnetic current is also absent, jθ = 0. The plasma pressure is balanced by the Lorentz force caused by the interaction of the longitudinal current and the azimuthal magnetic field produced by the current. According to Eq. (12.12) -

μ dI 2 dp = 20 2 : dr 8π r dr

ð12:15Þ

Let us multiply both parts of Eq. (12.15) by πr2 and integrate over r from zero to pinch radius a. On the l.h.s., integration by parts yields a

a

dp πr dr = dr 2

0

2πrpdr: 0

As a result, one obtains the integral equilibrium equation

12.3

Magnetic Flux Surface Functions

245 a

μ 0 I 2 ð aÞ = 8π

2πrpdr = hpiπa2 ,

ð12:16Þ

0

which connects pinch net current and average plasma pressure. Therefore, for equilibrium, the net current in the pinch should be organized in accordance with Eq. (12.15) or (12.16).

12.3

Magnetic Flux Surface Functions

The magnetic flux surface is a key concept for fusion devices. We shall consider here a system of nested toroidal flux surfaces. The magnetic field line, which belongs to the flux surface, either comes to a starting point and closes itself after several turns in the poloidal and toroidal directions or fills ergodically all flux surfaces. In the last case, the magnetic field line can be found in the infinite neighborhood of any given flux surface point. Flux surfaces where the magnetic field line closes itself after several turns are known as resonant flux surfaces. In a tokamak that is toroidally symmetric, all quantities are independent of the toroidal angle. Examples of various flux surfaces are shown in Fig. 12.1. At the edge, magnetic field lines become open and go to material surfaces. The magnetic flux surface that separates regions of different topologies is called the separatrix. Inside the separatrix, nested flux surfaces embrace the field line, which is called the magnetic axis. In addition to toroidally symmetric magnetic configurations, there are more complicated magnetic traps, such as stellarators with helical symmetry. Magnetic islands in tokamaks, which can emerge in the vicinity of rational flux surfaces, also have helical symmetry, and they will be considered later. Meanwhile, we shall restrict ourselves by toroidally symmetric nested flux surfaces. To describe equilibrium, it is convenient to introduce flux surface function quantities depending only on the flux surface and changes from one flux surface to another. The first flux surface function is a toroidal magnetic flux ΨT through the torus cross-section, →

ΨT =

→

Bd S ,

ð12:17Þ

ST

where the integral is taken over the toroidal cross-section, as shown in Fig. 12.2. → Since ∇  B = 0, according to the Gaussian theorem, the magnetic flux through the closed surface is zero. Hence, the magnetic flux through one toroidal cross-section is equal to the flux through another toroidal cross-section because together with the flux surface, they form a closed surface, and the magnetic flux through the flux

246

12

Plasma Equilibrium

Fig. 12.1 Cross-sections of magnetic flux surfaces corresponding to fixed toroidal angle

SP

ST Fig. 12.2 Toroidal and poloidal cross-sections of a torus

surface is zero since the normal to the flux surface component of the magnetic field is absent according to the flux surface definition. Poloidal flux is a magnetic field flux through a surface inside the torus, Fig. 12.2—through the “hole of the bagel”: →

Ψp =

→

Bd S,

ð12:18Þ

Sp

Figure 12.3 shows different variants of poloidal surface choice. The magnetic flux through each of the poloidal cross-sections is the same since two cross-sections with corresponding parts of the flux surfaces form a closed surface with zero magnetic field flux through it. Hence, the poloidal flux is also the flux surface function of the toroidal flux.

12.3

Magnetic Flux Surface Functions

247

Fig. 12.3 Different variants of poloidal surface choice

The total plasma pressure p is the third flux surface function. Indeed, since the → equilibrium inertia terms are neglected, condition B ∇p = 0 is satisfied, which means that pressure should be constant along the magnetic field line and, hence, should be constant over the whole flux surface. Note, however, that partial ion and electron pressures are not flux surface functions. → → From Eq. (12.1) ∇p = j × B , it follows that since ∇p is directed normally to the flux surface, there is no current normal to the flux surface, and current flows only along the flux surface (this statement is valid only provided viscosity is neglected). However, if dissipation and inertia are neglected, current lines belong to the flux surface as magnetic field lines. Since the current density current continuity equation → ∇  j = 0 is valid, the net current through the closed flux surface is zero, similar to the magnetic field flux. Hence, accounting for the current through parts of the magnetic flux surface being zero, one can introduce another two flux surface functions: →

IT =

→

j dS

ð12:19Þ

ST

net toroidal current inside a given flux surface, and →

Ip =

→

j dS

ð12:20Þ

Sp

net poloidal current through the poloidal cross-section. Safety factor q is another quantity that is defined at a given flux surface. For a closed magnetic field line, q is defined as the ratio of the number of turns in the toroidal direction n to the number of turns in the poloidal direction m until it comes to the starting point: q = n/m. For the ergodic magnetic field line, the limit for infinite turns should be taken:

248

12

Plasma Equilibrium

d

d

P

Fig. 12.4 Toroidal and poloidal fluxes between two neighboring flux surfaces. Here φ is a toroidal angle, θ is a poloidal angle, and the third coordinate is the normal to flux surface

q = lim

n

n→1 m

:

ð12:21Þ

By definition, the safety factor q is the flux surface function. Now let us demonstrate that the safety factor equals the derivative q= -

dΨT : dΨp

ð12:22Þ

To prove this, we start with special case q = 1. Consider the volume between two neighboring flux surfaces, which we unfold into a region between two neighboring planes (Fig. 12.4). Toroidal angle φ and poloidal angle θ change in the range (0, 2π), so that the left and right faces of the parallelepiped are glued with each other. The third “radial” coordinate is perpendicular to the planes. Since after one poloidal and toroidal turn the magnetic field line comes to a starting point, it should belong to a dashed surface (not necessary plane), Fig. 12.4. Magnetic flux through this surface is absent, as well as magnetic flux through near and far parts of flux surfaces. Therefore, considering a volume restricted by a dashed diaphragm, near and far parts of flux surfaces, lower and right faces, one can conclude that magnetic flux through the low face equals flux through the right face: dΨT = - dΨp. Minus represents the fact that while moving from inner to outer flux surfaces, toroidal flux rises and poloidal flux decreases. Hence, for q = 1, Eq. (12.22) is satisfied. For q = n/m, the toroidal flux through the end face increases by factor n corresponding to the number of magnetic field line intersections before returning to the starting point, while the poloidal flux increases by factor m in accordance with the number of intersections through the low face. Therefore, Eq. (12.22) remains valid in this situation. Finally, for the ergodic field line in the limit of an infinite number of turns, one also obtains Eq. (12.22).

12.4

12.4

Grad-Shafranov Equation

249

Grad-Shafranov Equation

The shape of flux surfaces for the case of toroidal symmetry can be described by an equilibrium equation where all quantities are flux surface functions. For analysis, we shall consider cylindrical geometry with coordinates (R, φ, z) shown in Fig. 12.5. Coordinate R represents the major radius, and φ is the toroidal angle – all quantities are independent of this angle. The point belonging to the flux surface is, therefore, determined by a pair of coordinates (R, z). The magnetic field has two components: → → → → → the poloidal field B p and the toroidal field B T , B = B p þ B T . The poloidal → → → magnetic field in its turn consists of two components: B p = B z þ B R . The poloidal magnetic flux Ψp by definition is R

2πR0 Bz dR0 :

Ψp = -

ð12:23Þ

0

After taking the derivative, we have the relation Bz = -

1 ∂Ψp : 2πR ∂R

ð12:24Þ

→

From equation ∇  B = 0, which in cylindrical coordinates is given by ∂Bz 1 ∂ ðRBR Þ = 0, þ ∂z R ∂R

Fig. 12.5 Cylindrical coordinate system. R– major radius, φ– toroidal angle

ð12:25Þ

250

12

Plasma Equilibrium

one can obtain the radial component of the magnetic field as a function of the poloidal flux: R

1 BR = R 0

∂Bz 0 0 1 ∂Ψp R dR = : 2πR ∂z ∂z

ð12:26Þ

Analogously, for poloidal current Ip: R

2πR0 jz dR0 :

Ip = -

ð12:27Þ

0

Taking the derivative, similar to Eq. (12.24), one finds jz = -

1 ∂I p : 2πR ∂R

ð12:28Þ

→

Since ∇  j = 0, analogously to Eq. (12.26): R

1 jR = R 0

∂jz 0 0 1 ∂I p R dR = : 2πR ∂z ∂z

ð12:29Þ

The toroidal magnetic field can be derived from the integral Maxwellian equation as a function of the poloidal current: Bφ = -

μ0 I p : 2πR

ð12:30Þ

Toroidal current density jφ =

→ 1 ∇× B μ0

φ

=

1 ∂BR ∂Bz = μ0 ∂z ∂R

2

∂ Ψp ∂ 1 ∂Ψp 1 þR = 2πμ0 R ∂z2 ∂R R ∂R

1 = Δ  Ψp : 2πμ0 R

Here, operator Δ resembles the Laplace operator: Δ =

2

∂ ∂ 1 ∂ þR ∂z2 ∂R R ∂R

:

The radial component of the equilibrium equation reads

ð12:31Þ

12.4

Grad-Shafranov Equation

251

∂p = jφ Bz - jz Bφ : ∂R

ð12:32Þ

Let us substitute the components of the magnetic field and current densities Eqs. (12.24), (12.28), (12.30), and (12.31) to the r.h.s. and use relation ∂p/ ∂R = (dp/dΨp)(∂Ψp/∂R) on the l.h.s. After dividing by ∂Ψp/∂R, one obtains Δ  Ψp = - μ 0 R 2

2 μ2 dI p dp - 02 : dΨp 8π dΨp

ð12:33Þ

This equation is known as the Grad-Shafranov equation. Its solution is a function Ψp(R, z), and equality Ψp(R, z) = const determines the equation for magnetic flux coordinates. The Grad-Shafranov equation can be solved if functions p(Ψp) and Ip(Ψp) are specified. These functions determine the flux surface shape. The outside plasma Grad-Shafranov equation is Δ Ψp = 0:

ð12:34Þ

At the plasma, boundary solutions of Eq. (12.33) Ψpi and Eq. (12.34) Ψpe should coincide as well as their normal derivatives: Ψpi s = Ψpe s ,

∂Ψpi ∂Ψpe = : ∂n s ∂n s

ð12:35Þ

It is convenient to calculate the poloidal flux from the magnetic axis, Ψ = Ψ0p - Ψp ,

ð12:36Þ

where Ψ0p is the poloidal flux corresponding to the magnetic axis. Poloidal flux Ψ turns to zero at the magnetic axis and rises towards the separatrix. It also satisfies the Grad-Shafranov equation. Let us consider a simple example of equilibrium. Let us put dp/dΨ = const and dI2/dΨ = const, with constants chosen in the following way: μ0

dp aΨ = - 40 , dΨ R0

μ20 dI 2 bΨ = - 20 : 8π2 dΨ R0

Then simple solution of this equation has the form R40

2 Ψ 1 a-1 2 = bR20 þ R2 z2 þ R - R20 : Ψ0 2 8

At R = R0 and z = 0 poloidal flux Ψ = 0, this corresponds to the magnetic axis. In the vicinity of the magnetic axis at R ≈ R0

252

12 2 Ψ 1 z2 ða - 1Þ ðR - R0 Þ = ð b þ 1Þ 2 þ : 2 Ψ0 2 R0 R20

Plasma Equilibrium

ð12:37Þ

Therefore, the magnetic flux surfaces in this example have the form of ellipsoids with semiaxes a and b.

12.5

Integral Equilibrium in a Tokamak

For better understanding, let us consider a tokamak with a circular cross-section. Let the last closed flux surface have radius a. We shall also restrict ourselves to the case of a large aspect ratio. a ≪ 1: R

ð12:38Þ

Here, R is the major radius (e.g., of the center of the last closed flux surface). If condition Eq. (12.38) is satisfied, the torus can be in zero approximation considered as a cylinder, and toroidal effects can be considered as small corrections. Let us analyze forces applied to plasma as a whole. In the zero (cylindrical) approximation, forces are directed towards the cylinder center along the minor radius, the so-called minor radius equilibrium. In this approximation, Eq. (12.11) or (12.12). is satisfied. After integrating Eq. (12.12) with weight r2 over a cylinder with radius a using integrating by parts, one finds hpi =

B2iφ B2eφ 2μ0 2μ0

þ

μ0 I 2T ðaÞ : 8π2 a2

ð12:39Þ

Here a

1 h pi = 2 πa

2πrpdr, 0

B2iφ 2μ0

a

1 = 2 πa

2πr

B2iφ dr: 2μ0

ð12:40Þ

0

Magnetic field Biφ is a toroidal magnetic field in plasma that corresponds to magnetic field Bz in a cylinder, and field Beφ is a vacuum magnetic field outside plasma (corresponding to field B0 in Eq. (12.12)), IT is the net toroidal current. Equation (12.39) is an integral equilibrium condition along the minor radius. Since the torus surface element at the outer side is larger than that at the inner side, the forces also arise in the direction of the major radius in contrast to the cylinder case. These forces contain a small parameter a/R. The integral balance of these forces is called equilibrium along the major radius. To calculate the forces along the major radius, we consider the virtual shift of the plasma along the major radius and

12.5

Integral Equilibrium in a Tokamak

253

calculate the work A and potential energy W = - A of the plasma pinch. Then, the force along the major radius would be F = - ∂W/∂R. The potential energy and corresponding force are associated with three different factors: W = W p þ W I þ W B:

ð12:41Þ

The first part of the potential energy Wp is connected with the plasma pressure and difference in surface elements at the outer and inner parts of the torus. When the plasma pinch is shifted along the major radius, its volume increases, and work is performed. The corresponding potential energy is Wp = -

pdV = - 2π2 a2 Rhpi,

ð12:42Þ

∂W p = 2π2 a2 hpi, ∂R

ð12:43Þ

and the force is Fp = -

The second force is caused by the currents interaction flowing in the toroidal direction, or, in other words, by the interaction of the toroidal currents with the poloidal magnetic field: jφBp. The potential energy is given by WI =

LI 2T ðaÞ , 2

ð12:44Þ

with L being the inductivity of the plasma pinch. For torus L = μ0 R½lnð8R=aÞ - 2 þ li =2,

li = B2p =B2p ðaÞ:

ð12:45Þ

Here, Bp(a) is a poloidal magnetic field at the plasma boundary, and averaging is performed similar to Eq. (12.40). When shifting along the major radius, magnetic flux Ψ(a) = ITL is conserved while the current is changing. Therefore, it is more convenient to rewrite the potential in the form WI =

Ψ2 ðaÞ , 2L

ð12:46Þ

The corresponding force FI = -

μ I2 ∂W I jΨ = const = 0 T 2 ∂R

ln

8R l -1 þ i : a 2

ð12:47Þ

254

12

Plasma Equilibrium

The third force is caused by the interaction of the poloidal current with the toroidal magnetic field jpBφ. The corresponding potential energy is B2iφ dV þ 2μ0

WB = Vi

B2eφ dV: 2μ0

ð12:48Þ

Ve

Integrals are taken over plasma volume and outside it correspondingly. After regrouping different terms, one obtains WB = V e þV i

B2eφ dV þ 2μ0

B2iφ B2eφ dV = const 2μ0 2μ0 Vi

B2iφ

þ 2π2 a2 R

2μ0

-

B2eφ 2μ0

:

ð12:49Þ

Here, the first summand is an integral of the vacuum magnetic field over the whole volume and hence is constant. The second integral increases linearly with the major radius. The force in the major radius direction, therefore, equals B2iφ B2eφ ∂W B = 2π2 a2 2μ0 2μ0 ∂R

FB = -

:

ð12:50Þ

The r.h.s. of Eq. (12.50) can be modified using equilibrium over minor radius Eq. (12.39). After the substitution of toroidal magnetic pressure, we have F B = 2π2 a2 hpi -

μ0 I 2T : 4

ð12:51Þ

The sum of three forces Eqs. (12.43), (12.47), and (12.51) yields FR = Fp þ FI þ FB =

μ0 I 2T 8R 3 l 4π2 a2 hpi : ln - þ iþ 2 a 2 2 μ0 I 2T

ð12:52Þ

The force FR accelerates plasma on the major radius direction towards the outer part of the torus. To keep plasma in the equilibrium state, it is necessary to apply additional force in the opposite direction F R = - F R . Such a force is produced by a vertical magnetic field BV, which is created by currents flowing in the special coils. As a result of the interaction of the vertical magnetic field with the toroidal current, the force arises F R = -

jφ BV dV = - 2πRI T BV :

ð12:53Þ

12.5

Integral Equilibrium in a Tokamak

255

Fig. 12.6 X-point formation at tokamak high field side for large βI. The direction of poloidal magnetic field in different regions is marked by arrows

Condition F R = - F R determines the vertical magnetic field: BV =

μ0 I T 8R 3 l 4π2 a2 hpi - þ iþ : ln 4πR a 2 2 μ0 I 2T

ð12:54Þ

The last term in brackets is known as beta poloidal βI =

4π2 a2 hpi 2μ0 hpi = 2 , B p ð aÞ μ0 I 2T

ð12:55Þ

and equals the ratio of the average pressure to the poloidal magnetic field pressure. Plasma equilibrium is violated at large values of βI. Indeed, since a vertical magnetic field is added to the poloidal magnetic field produced by the plasma current, at the torus high field side, their sum turns to zero (Fig. 12.6). The point of zero poloidal magnetic field is called the X-point, and the line separating areas with different topologies that pass through the X-point is called the separatrix. For small values of βI, the vertical magnetic field is small, and the Х-point is located outside the plasma, where the poloidal field created by the toroidal current decreases as r-1 and balances the vertical field. When βI rises, the Х-point is shifted towards plasma and then into plasma, and equilibrium cannot be reached since plasma is reached expands along magnetic field lines. The maximal possible value of βmax I when BV = Hence,

μ0 I T : 2πa

ð12:56Þ

256

12

βmax = I

Plasma Equilibrium

2R 8R 3 l - ln þ - i: a a 2 2

ð12:57Þ

corresponds to maximal βI for tokamaks with circular cross-sections. The βmax I For other shapes, for example, for larger elongation, the values of βmax could be I larger than those given by Eq. (12.57).

12.6

Plasma Equilibrium in a Tokamak with Circular Cross-Sections

Here, we consider equilibrium when the tokamak flux surfaces are close to circles. Let us seek equilibrium with flux surfaces being a system of nested circles with centers shifted with respect to the center of the last closed flux surface, Fig. 12.7. We denote the major radius of the flux surface center of the last closed flux surface as R0, and the major radius of an arbitrary point of the flux surface as R. Minor radius r is defined as the radius of a given flux surface calculated from its center for the last closed flux surfacer = a. The shift of the flux surface center with respect to the center of the last closed flux surface is denoted as Δ(r), and this quantity decreases with r and Δ(a) = 0. This quantity Δ(r) is known as the Shafranov shift. Approximation of small toroidicity is assumed ε=

r ≪ 1: R0

ð12:58Þ

The major radius of the given flux surface center in this approximation is also ≈R0 for the last closed flux surface. The toroidal magnetic field is inversely proportional to the major radius:

Fig. 12.7 Shafranov shift of circle magnetic flux surfaces

12.6

Plasma Equilibrium in a Tokamak with Circular Cross-Sections

257

Fig. 12.8 Surface area between two neighboring flux surfaces

BT =

B0T R0 : R

ð12:59Þ

Here, B0T is a toroidal magnetic field in the center of the last closed flux surface. Considering the geometrical relation R = R0 + r cos θ, Eq. (12.59) reads BT =

B0T ≈ B0T ð1 - ε cos θÞ: ð1 þ ε cos θÞ

ð12:60Þ

The poloidal magnetic field is sought in an analogous form Bp = B0p ð1 - εΛðr Þ cos θÞ:

ð12:61Þ

Functions Λ(r) and Δ(r) should be obtained from the equilibrium equation. These functions are not independent. To find the relation between two functions, consider two neighboring flux surfaces. Let dξ be the distance between them in the normal direction, as shown in Fig. 12.8. Distance dξ depends on the poloidal angle and Shafranov shift dξ = dr þ cos θdΔ = 1 þ cos θ

dΔ dr: dr

ð12:62Þ

Surface element dSp between neighboring flux surfaces, which is crossed by poloidal magnetic flux, is

258

12

dSp = 2πRdξ = 2πR0 ð1 þ ε cos θÞ 1 þ cos θ = 2πR0 1 þ ε þ

Plasma Equilibrium

dΔ dr dr

ð12:63Þ

dΔ cos θ dr: dr

Quadratic in ε terms are neglected here. Multiplying dSp by Bp, one obtains the poloidal flux between two surfaces: dΨp = Bp dSp = B0p ð1 - εΛ cos θÞ2πR0 1 þ ε þ ≈ 2πR0 B0p

dΔ cos θ dr dr

ð12:64Þ

dΔ 1 þ ε - εΛ þ cos θ dr: dr

Since the poloidal flux is a surface function, the quantity dΨp should be poloidally independent, and the coefficient in front of cosθ equals zero. Hence, dΔ = εðΛ - 1Þ: dr

ð12:65Þ

Let us now project equilibriumEq. (12.3) normal to the flux surface: →

→

→

h B∇ B dΔ d B2 : = 1cos θ pþ 2μ0 μ0 dr dr

ð12:66Þ

→

Here, h is the unit vector normal to the flux surface. Identity ∂ ∂ dΔ = 1cos θ , dr ∂ξ ∂r which follows from Eq. (12.62), is also used. The operator on the r.h.s. can be split into four components: →

→

→

→

→

→

→

→

→

→

B ∇ B = B T ∇ B T þ B p ∇ B T þ B T ∇ B p þ B p ∇ B p : ð12:67Þ

The first term on the r.h.s. according to the definition of the curve radius is a vector parallel to the major radius direction, →

→

B T∇ B T = -

B2T → R, R2

→

→

→

h B T∇ B T = -

B2T cos θ: R

ð12:68Þ

12.6

Plasma Equilibrium in a Tokamak with Circular Cross-Sections →

259

→

The second term on the r.h.s. of Eq. (12.67) B p ∇ B T is a vector in the toroidal direction and has no projection on the normal to the flux surface. The third compo→ → nent B T ∇ B p is a poloidal vector and has no projection on the normal to flux surface. The last vector in Eq. (12.67) is directed along the minor radius towards its center, so →

→

B2p : r

→

h B p∇ B p = -

ð12:69Þ

With an account of Eqs. (12.68) and (12.69), Eq. (12.66) is transformed to 2

1-

Bp dΔ d B2 1 B2T =cos θ þ : cos θ pþ 2μ0 r dr dr μ0 R

ð12:70Þ

Let us put all small terms of the order of ε containing cosθ into r.h.s.: d B2 pþ 2μ0 dr

þ

2 B2p B2T dΔ Bp =þ cos θ: μ0 r μ0 R dr μ0 r

ð12:71Þ

In the last term, quantity d( p + B2/2μ0)/dr is replaced by B2p =ðμ0 r Þ, which corresponds to the cylindrical (zero order in ε) approximation. One must keep in mind that on the l.h.s. of Eq. (12.71), in addition to zero-order terms in ε, there are also firstorder terms proportional to ε cos θ. The latter are the result of substitution of toroidal and poloidal magnetic fields in the form of Eqs. (12.60) and (12.61). Therefore, besides the zero-order equation B0 d pþ 8π dr

2

þ

2

B0p 4πr

= 0,

ð12:72Þ

one can obtain a first-order equation from the condition that the coefficient in front of the cosθ term in Eq. (12.71) should be zero. Considering Eqs. (12.65) and (12.72), we have d r B0p dr

2

2μ r 2 dp 2r 0 dΔ = 0 B R0 dr dr R0 p

2

:

ð12:73Þ

After integrating over the minor radius: 2

dΔ 2μ0 ðp - pÞ - Bp r , = 2 dr R0 B0p

ð12:74Þ

260

12

Plasma Equilibrium

where r

1 p= 2 πr

r 0

0

0

2πr pðr Þdr ,

2 Bp

1 = 2 πr

0

2πr0 B0p

2

dr 0 :

0

According to Eq. (12.74), shift of flux surfaces is determined by parameter βI, Eq. (12.55). Derivative dΔ/dr is negative because for pressure, it is decreasing with radius p < p. Hence, inner flux surfaces are more shifted outwards than the outer ones in accordance with Fig. 12.7. The Shafranov shift Δ(r) can be obtained by integrating Eq. (12.74). For a known Shafranov shift, it is easy to calculate the poloidal magnetic field from Eqs. (12.61) and (12.65).

12.7

Coordinates for Arbitrary Flux Surfaces

To describe plasma behavior in fusion devices with flux surfaces of arbitrary shape, special coordinates are introduced. It is convenient to use flux surface coordinates, which determine the position at the flux surface, xi = ða, θ, ζÞ:

ð12:75Þ

Here, for coordinate a, any flux surface function can be chosen to mark a given flux surface. For example, or circular cross-sections small radius can be taken as a. Coordinates θ and ζ correspond to poloidal and toroidal angles, respectively, and change in the interval 0, 2π. Signs for coordinates θ and ζ are chosen so that for circle surfaces and small toroidicity, they coincide with the same directions of the cylindrical coordinates of a pinch, θ and z, Fig. 12.1. If flux surface coordinates are known as functions of Cartesian coordinates, then the covariant and contravariant basis vectors are defined as →i

→

→

ð12:76Þ

dl2 = gik dxi dxk ,

ð12:77Þ

e = ∇xi , e i = ∂ r =∂xi :

In these coordinates, the length element is

→ →

where metric tensor gik = e i e k . Volume element p p → → → dV = gdx1 dx2 dx3 , g= e 1× e 2 e 3= =

detgik :

∇x1 × ∇x2 ∇x3

-1

ð12:78Þ

12.7

Coordinates for Arbitrary Flux Surfaces

261

Additional relations: → →k eie

p e i = g ∇xiþ1 × ∇xiþ2 :

→

= δik ,

ð12:79Þ

Arbitrary vector in arbitrary coordinates: →

→i

→

A = Ai e = Ai e i ;

→→

→ →i

Ai = A e i ;

Ai = A e :

ð12:80Þ

Let us demonstrate that for a magnetic field line, which lies on the magnetic flux surface, the magnetic field can be presented in the form →

B=

1 1 1 ½∇Ψ1 × ∇ζ þ ½∇Ψ2 × ∇θ þ ½∇a × ∇η, 2π 2π 2π

ð12:81Þ

where Ψ1(a) and Ψ2(a) are flux surface functions and η(a, θ, ζ) is a periodic function of θ and ζ. Indeed, a vector with a zero component along ∇a can be written as →

B = x½∇a × ∇θ þ y½∇a × ∇ζ:

ð12:82Þ

→

From equation ∇  B = 0, we have ∂x=∂ζ - ∂y=∂θ = 0:

ð12:83Þ

Hence, quantities x and y can be expressed through one periodic function x = x 0 ð aÞ þ

1 ∂η , 2π ∂θ

y = y0 ð a Þ þ

1 ∂η : 2π ∂ζ

ð12:84Þ

Substituting Eq. (12.84) into Eq. (12.82), one obtains Eq. (12.81), if one denotes x0 ð a Þ =

1 dΨ1 , 2π da

y0 ð aÞ =

1 dΨ2 : 2π da

ð12:85Þ

Function η(a, θ, ζ) must satisfy additional condition – the absence of current normal to flux surface, see Section 12.3, →

j ∇a =

→ → 1 1 ∇ × B ∇a = ∇  B × ∇a = 0: μ0 μ0

ð12:86Þ

It is easy to see that quantities Ψ1 and Ψ2 coincide with poloidal and toroidal magnetic fluxes Ψp and ΨT through the surfaces shown in Fig. 12.2. To demonstrate this, let us integrate Eq. (12.81) over toroidal and poloidal surfaces. Introducing → vector potential A , we have

262

12 →

→

Ψp =

→

Bd S =

→

Sp

→

→

A d l T , ΨT =

Bd S =

→

Plasma Equilibrium

→

Ad l

p

,

ð12:87Þ

ST

where lp, lT contours embracing toroidal and poloidal cross-sections correspondingly. Vector potential, which corresponds to Eq. (12.81), can be written in the form →

A=

1 1 1 Ψ ∇ζ þ Ψ2 ∇θ η∇a þ ∇d, 2π 1 2π 2π

ð12:88Þ

where d is an arbitrary function. After substitution of Eq. (12.88) into Eq. (12.87) with an account of →

∇θd l p = →

∇dd l

p,T

→

→

∇ζd l T = 2π,

∇ad l

p,T

=

→

∇θd l T =

→

∇ζd l p = 0,

= 0, ð12:89Þ

one obtains Ψ1 = Ψp, Ψ2 = ΨT. Hence, the magnetic field depends on the poloidal and toroidal magnetic fluxes according to →

B=

1 1 1 ∇Ψp × ∇ζ þ ½∇ΨT × ∇θ þ ½∇a × ∇η: 2π 2π 2π

ð12:90Þ

In general, angle coordinates could be chosen so that η = 0. Such coordinates are called flux coordinates with straightened magnetic field lines. In such coordinates, (they are not necessarily orthogonal ones) →

B=

1 1 ∇Ψp × ∇ζ þ ½∇ΨT × ∇θ: 2π 2π

ð12:91Þ

The equation for the magnetic field line in these coordinates is dθ →

B ∇θ

→

=

dζ →

B ∇ζ

:

ð12:92Þ

→

Taking the poloidal B ∇θ and toroidal B ∇ζ magnetic fields from Eq. (12.91), we find →

dΨT =da dΨT dζ B ∇ζ = q, === → dΨp dθ =da dΨ p B ∇θ

ð12:93Þ

12.8

Force-Free Equilibrium and Pinch with Canonical Profiles

263

where safety factor q is defined according to Eq. (12.22). Since the safety factor is a surface function, the magnetic field line in the chosen coordinates is a straight line. Such a choice of orthogonal coordinates is natural for tokamaks. Let us denote inverse gradients as ha =

1 1 1 , hθ = , hζ = : j∇aj j∇ζj j∇θj

ð12:94Þ

One can see that hζ = R, where R is the major radius for the flux surface point, since after integrating over loop dlT, one should obtain 2π according to Eq. (12.89). In accordance with Eq. (12.91), physical components of magnetic field Bθ  Bp = -

dΨp =da dΨT =da , Bζ = BT = : 2πha hθ 2πRha

ð12:95Þ

For circular flux surfaces, hθ is just the minor radius r. In the general case, quantities ha, hθ can be specified numerically for a given flux surface shape. Differential operators and integrals for known ha, hθ, and hζ are defined according to general rules. If the poloidal magnetic flux Ψ is calculated from the magnetic axis according to Eq. (12.36), then →

B=-

1 1 ½∇Ψ × ∇ζ þ ½∇ΨT × ∇θ, 2π 2π

ð12:96Þ

dΨ=da dΨT =da , Bζ = BT = : 2πRha 2πha hθ

ð12:97Þ

Bθ  Bp =

12.8

Force-Free Equilibrium and Pinch with Canonical Profiles

In some special situations when the plasma pressure is small with respect to the magnetic field pressure, configurations of force-free equilibrium can exist in plasma. In the equilibrium equation, one can put ∇p = 0. From Eqs. (12.1) and (12.2), we have 1 μ0 According to this equation,

→

→

∇ × B × B = 0:

ð12:98Þ

264

12 →

→

∇× B =μB:

Plasma Equilibrium

ð12:99Þ

In other words, the plasma current should flow along the magnetic field. Below is shown that the quantity μ is independent of the magnetic field. Such a configuration is force-free because current flows along the magnetic field line and the Lorentz force is absent. The current density is proportional to the magnetic field strength. This equation is also known as the Beltrami relation. Force-free magnetic equilibrium is obtained in a toroidal pinch when the toroidal and poloidal magnetic fields are of the same order. Pinch equilibrium in the general case is governed by Eq. (12.11) (or Eq. (12.12), containing two arbitrary functions Bz(r) and Bθ(r). Let us demonstrate that an additional constraint of minimal magnetic field energy with conservation of a specific function known as helicity leads to force-free magnetic equilibrium of type Eq. (12.99). The obtained radial profiles are known as canonical profiles. Let us first define helicity as a volume integral taken over the flux tube volume: →*

K=

A B dV,

ð12:100Þ

→

where A is the vector potential. In plasma with infinite conductivity, this integral is conserved. We shall prove this first for a closed field line. The integral can be rewritten in the form →

K=

→

Ad l

* →

Bd S ,

ð12:101Þ

where dS is the magnetic tube cross-section and dl is the differential length along the magnetic field line. Since the magnetic flux is conserved in the flux tube, it is necessary to prove that d dt

→

→

A d l = 0:

ð12:102Þ

Integral evolves according to d dt

→

→

→ → → → → → ∂A þ V∇ A d l þ A d l ∇ V: ∂t

→

Ad l = →

Here, is used d l =dt =

→

ð12:103Þ

→

l ∇ V . Since the magnetic field is frozen into the plasma

for σ → 1 according to Eq. (10.17), the vector potential satisfies the equation

12.8

Force-Free Equilibrium and Pinch with Canonical Profiles

265

→

→ → ∂A þ V × B = ∇χ, ∂t

-

ð12:104Þ

where χ is an arbitrary scalar function. Substituting Eq. (12.104) into Eq. (12.103), → → → taking into account that V × B d l = 0 and combining the remaining terms, we find →

d dt

→

→

Ad l =

→→

d l ∇ A V þ χ = 0:

ð12:105Þ

To demonstrate that this integral is zero, one has to apply Stokes’ theorem and take →→ into account that ∇ × ∇ A V þ χ = 0. Therefore, helicity is conserved for closed magnetic tubes and infinite conductivity. In the general case with finite collisionality, this is not strictly true since magnetic field lines can reconnect, so the topology is changing. However, there are heuristic arguments that the general structure of the magnetic field does not change significantly during reconnections, and the integral Eq. (12.100) is conserved if integration is performed over the whole plasma volume. Helicity is then redistributed between flux tubes, while the whole sum is conserved. Following this assumption, let us find the magnetic configuration corresponding to the minimum magnetic field energy with restriction of helicity conservation. To do this, one has to seek a minimum B2 dV - λK, 2μ0

L=

ð12:106Þ

where λ is the Lagrange multiplier. From the condition of zero functional variation δL = 0, one obtains →

→ →→ B δ ∇ × A - λδ A B μ0

dV = 0

ð12:107Þ

Using equality →→

→

→

→

→

→

→

→

→

δ A B = Aδ ∇× A þ BδA = A ∇×δA þ BδA

ð12:108Þ

equation (12.107) can be reduced to →

→ B -λA μ0

→

→

→

∇ × δ A - λ B δ A dV = 0:

ð12:109Þ

266

12

Plasma Equilibrium

The first term can be further modified using the expression ∇

→

→

→

→

→

→

a × b = - a ∇× b þ b ∇× a :

Since the volume integral over divergence turns to zero, we have →

∇×

→ B -λA μ0

→

→

→

δ A - λ B δ A dV = 0:

ð12:110Þ

Since variation is arbitrary, the Lagrange equation should be satisfied: →

∇×

→ B -λA μ0

→

- λ B = 0,

ð12:111Þ

or for magnetic field →

∇×

→ B = 2λ B : μ0

ð12:112Þ

Introducing the constant μ = 2μ0λ, we obtain Eq. (12.99) for the field-free configuration. In the projections: 1 drBθ = μBz , r dr dB - z = μBθ : dr

ð12:113Þ

Substitution of the azimuthal magnetic field Bθ from the second equation into the first one leads to the Bessel equation: 1 d dB r z dr r dr

þ μ2 Bz = 0:

ð12:114Þ

Hence, canonical profiles of magnetic field components are given by Bessel functions Bz = B0 J 0 ðμr Þ, Bθ = B0 J 1 ðμr Þ:

ð12:115Þ

The radial profiles of the magnetic field components are shown in Fig. 12.9. The longitudinal magnetic field strongly decreases with radius, while the azimuthal

12.9

2D Modeling of the Tokamak Edge

267

Fig. 12.9 Canonical radial profiles of parallel and azimuthal magnetic field components in a pinchcomponents in a pinch

component increases towards the periphery, and its projection onto the magnetic field line also rises. Profile shapes depend on the parameter μa, where a is the pinch radius. Parameter μa is connected with the ratio of longitudinal current to toroidal magnetic flux:

μa =

μ0 aI : ΨT

ð12:116Þ

The radial profiles become more peaked with the rise of μa, and, for μa = 2.4, the longitudinal magnetic field turns to zero at the plasma boundary, and, for μa > 2.4, the longitudinal magnetic field changes sign at the plasma periphery. Canonical profile Eq. (12.115) and magnetic field reversal are in good agreement with the experimental profiles observed in reversed field pinches.

12.9 2D Modeling of the Tokamak Edge Tokamak edge plasma is a complicated 2D system and, therefore, requires 2D modeling. For such modeling, one needs equilibrium analysis. As a first step, the Grad-Shafranov Eq. (12.33) is solved, and the poloidal flux Ψp(R, z) is calculated. Using this function, a special 2D mesh is generated; see Fig. 12.10 as an example. This mesh is generated as a rectangular mesh; one side coincides with one of the flux surfaces in the x-direction, and the other side is perpendicular to the flux surface in the y-direction. The sizes of each cell are hx, hy correspondingly. In the z-direction, the Lame coefficient is hz = R. Plasma parameters such as density, temperature, velocities, and potential are associated with cell centers, and fluxes are associated

268

12

Plasma Equilibrium

j, V 100 –2.5 50

Z, m

–3 –3.5

0 –4 –4.5 4

5

6

–50

R, m Fig. 12.10 Plasma potential in the 2D modeling of edge tokamak plasma with the SOLPS-ITER transport code

with cell areas. Cell surfaces and volumes differ from cell to cell, and they are used to reduce transport equations in the differential form to their numerical realization. All simulation areas are restricted by magnetic flux at the core side of the plasma, magnetic flux outside the separatrix, and divertor plates, where boundary conditions are imposed. An example of the solution of the transport equations with code SOLPS-ITER is presented in Fig. 12.10.

Chapter 13

Transport Phenomena in Tokamaks

Plasma equilibrium does not imply a full absence of flows over and across flux surfaces; the only requirement is that corresponding velocities should be smaller than the sound or Alfven velocity. In this chapter, such flows are analyzed for a tokamak, and, in particular, particle, energy, and momentum fluxes across flux surfaces are obtained. The fluxes caused by Coulomb collisions are called neoclassical fluxes, and the corresponding transport coefficients are known as neoclassical transport coefficients. Here, the prefix “neo” reflects the fact that fluxes are calculated in complicated geometry; however, the fluxes considered below are purely classical in contrast to turbulent transport, which was analyzed in Chap. 8. To analyze transport for arbitrary flux surfaces, one must use the coordinate system considered in the previous section. However, for simplicity below the simplest variant of circular flux surfaces is considered, and, at the end of the chapter, general expressions valid for arbitrary flux surfaces will be presented. Let us consider the simplest model for a magnetic field, the case of small toroidicity ε ≪ 1 with circular flux surfaces. The model magnetic field is simplified with respect to that considered in Sect. 12.6: BT =

B0T ≈ B0T ð1 - ε cos θÞ, ð1 þ ε cos θÞ

Bp  Bθ = B0p ð1 - ε cos θÞ = ΘBT , B0 =

2 B0T

þ

B0p

ð13:1Þ

2

,

ε = r=R0 :

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Rozhansky, Plasma Theory, https://doi.org/10.1007/978-3-031-44486-9_13

269

270

13

Transport Phenomena in Tokamaks

Fig. 13.1 Coordinate system for circle flux surfaces used for transport analysis

For this model electric field, the ratio Θ = Bp/BT is independent of the poloidal angle θ. Let us assume Θ ≪ 1, so that Θ ≈ B0p =B0 . As a coordinate a, which is the flux surface characteristic, we choose a small radius r. The coordinates θ, ζ correspond to the poloidal and toroidal angles, as shown in Fig. 13.1. The length element in these coordinates is

dl2 = dr 2 þ r 2 dθ2 þ R20 ð1 þ ε cos θÞ2 dζ2 ,

ð13:2Þ

and the components of the metric tensor are grr = 1, gθθ = r 2 , gζζ = R20 ð1 þ ε cos θÞ2 :

ð13:3Þ

Ion Larmor radius ρci is supposed to be small enough, so that ρci =ðrΘÞ ≪ 1:

ð13:4Þ

This inequality is usually fulfilled in tokamaks. The Larmor radius and thermal velocity are defined as ρcj =

2T j =mj ωcj

1=2

;

V Tj =

All macroscopic quantities are sought as series

2T j : mj

ð13:5Þ

13.1

Fluid Regime (Pfirsch-Schlueter Regime)

271

nðr, θÞ = n0 ðr Þ þ n1 ðr, θÞ . . . ; T j ðr, θÞ = T 0j ðr Þ þ T 1j ðr, θÞ . . . ; φðr, θÞ = φ0 ðr Þ þ φ1 ðr, θÞ . . . ;

ð13:6Þ

ujk ðr, θÞ = ujk0 ðr Þ þ ujk1 ðr, θÞ: . . . The first term corresponds to the cylindrical approximation and is independent of the poloidal angle, the second term is linear in ε, and the other terms are higher order in ε. Note that temperatures, density, and potential are not surface functions, in contrast to full pressure, and are functions of the poloidal angle θ. Here, φ0 is the average potential on the flux surface, and uik0 corresponds to the average ion velocity along magnetic field lines.

13.1

Fluid Regime (Pfirsch-Schlueter Regime)

First, we consider the fluid case when the mean-free path of the particles is small with respect to the length of the magnetic field line after one turn in the toroidal direction: λj =

V Tj r < = qR: νj Θ

ð13:7Þ

Here, νi  νii, νe  νei. This regime is typical for small temperatures in small tokamaks or in the separatrix vicinity for larger machines. In this regime, plasma is described by the fluid equations of Chap. 2, and transport studies are relatively simple.

13.1.1

Qualitative Estimates

Here, we present qualitative estimates of particle and heat fluxes caused by toroidicity. Let us start with an estimate of temperature perturbations on the flux surface. Electrons and ions drift vertically in an inhomogeneous magnetic field with the velocities given by Eq. (11.9). This ∇B-driven drift is directed upwards for electrons and downwards for ions for the normal direction of the magnetic field. After averaging over the Maxwellian distribution function, the corresponding velocities are: ugj =

2T j : ej BR

Additionally, the vertical current of guiding centers arises, Eq. (11.10),

ð13:8Þ

272

jg =

13

Transport Phenomena in Tokamaks

2nðT e þ T i Þ : BR

ð13:9Þ

There are vertical convective heat fluxes associated with these vertical drifts: qgj = 5/2Tj(nugj). The divergence of these fluxes in plasma with an inhomogeneous magnetic field is nonzero, which causes an increase in temperature perturbations at the flux surface. As a result, parallel heat flux along magnetic field lines arises to compensate for the divergence of vertical convective heat flux. The corresponding estimate for heat balance reads n0 T 20j κjk T 1j Θ2 :  eB0 R0 r r2

ð13:10Þ

Here, as a typical spatial scale in the radial direction, a minor radius r is taken, and for a scale in the parallel direction, the magnetic field length r/Θ = qR is chosen. From Eq. (13.10) with account of the expression for heat conductivity (κjk~nTj/ (mjνj)), one can estimate the temperature perturbation on the flux surface T 1j qνj ε νj  = : T 0j Θ2 ωcj Θωcj

ð13:11Þ

Ion temperature perturbation is positive at the lower part of the torus and negative at the upper part. For electrons, the upper part is slightly hotter than the lower part, as shown in Fig. 13.2. In the first approximation in ε, the vertical heat flux through the upper and lower parts of the torus is the same but is directed inwards and outwards correspondingly, so the net flux through the flux surface is zero. In the second approximation in ε, since the lower part of the torus is hotter for ions than the upper part, the flux through the lower part is larger than that through the upper part. As a result, unbalanced net ion heat flow, averaged over the flux surface, arises: qj



n0 T 0j nT 0j T  q2 ρ2cj νj : eB0 R0 1j r

ð13:12Þ

This heat flow is q2 larger than the heat flow in a cylinder, (q⊥j = - κj⊥dT0j/dr), see Chap. 2. The electron temperature perturbation is mi =me smaller than the ion temperature perturbation. Therefore, the electron net heat flux through the flux surface is mi =me smaller than that of ions in accordance with Eq. (13.12). Vertical current Eq. (13.9), similar to the vertical heat flux, is closed by a parallel current. The scheme of its closing is shown in Fig. 13.3. The parallel current, which closes the vertical current, is called the Pfirsch-Schlueter current. It has opposite directions on the outer and inner parts of the torus. To estimate it, one has to equalize the divergency of the vertical current and the divergency of the poloidal projection of the parallel current:

13.1

Fluid Regime (Pfirsch-Schlueter Regime)

273

Fig. 13.2 Scheme of ion flux formation through the flux surface

Fig. 13.3 Scheme of current closing: (а) cross section. 1 – Vertical current and 2 – poloidal projection of parallel current; (b) current closing along magnetic field line

p  ΘjPS k =r, BRr

ð13:13Þ

so that jPS k 

qp : Br

ð13:14Þ

The Pfirsch-Schlueter current is q times larger than the diamagnetic current.

274

13.1.2

13

Transport Phenomena in Tokamaks

Heat Conductivity

Now, we calculate the ion temperature perturbation on the flux surface from the ion heat balance equation. Using the expressions for particle fluxes obtained below, one can demonstrate that the terms containing fluid velocities are small with respect to the terms containing heat conductivity, so we neglect convective terms from the beginning. The following inequality is assumed to be satisfied: ρci νj qR < 1, Θr V Ti

ð13:15Þ

which implies that the ion temperature perturbation is small, in accordance with estimate Eq. (13.11). Heat exchange with electrons is also neglected. Then, the stationary heat balance equation for ions is reduced to →

∇  q i = 0,

ð13:16Þ

→

where ion heat flux q i is defined by Eqs. (2.38) and (2.39): 5 nT i q i = - κik ∇k T i - κi⊥ ∇⊥ T i þ 2 eB

→

→

B × ∇T i , B

ð13:17Þ

with parallel and perpendicular heat conductivity coefficients κik =

3:9nT i 2nT i νi , κi⊥ = : m i νi mi ω2ci

ð13:18Þ

Let us neglect the small perpendicular heat conductivity and small perpendicular heat flux associated with the second term on r.h.s. Then, in the chosen coordinates, the heat balance equation is p gςς 1 ∂ p qiθ = 0: p gθθ gςς ∂θ

ð13:19Þ

According to Eq. (13.17), we have the poloidal component of the ion heat flux qiθ =

Θ2 ∂T 1i 5 nT i ð1 þ ε cos θÞ dT i - κik : eB0 dr r ∂θ 2

ð13:20Þ

The first term on the r.h.s. represents the diamagnetic heat flux connected with rotation over the Larmor radius, see Chap. 2. In an inhomogeneous magnetic field, the divergence of it is finite and coincides with the divergence of the vertical guiding center heat flux. The second term represents the poloidal projection of the parallel 1i ). From Eq. (13.19), one obtains heat flux (qik = - κik Θr ∂T ∂θ

13.1

Fluid Regime (Pfirsch-Schlueter Regime)

275

p gςς p qiθ = constðθÞ: gθθ

ð13:21Þ

After linearization of Eqs. (13.20) and (13.21), equalizing the first order in ε terms, containing cosθ, we have κik

n T ε dT 0i Θ2 ∂T 1i cos θ: = 5 0 0i r ∂θ eB0 dr

ð13:22Þ

n0 T 0i εr dT 0i sin θ: eB0 κik Θ2 dr

ð13:23Þ

Integration yields T 1i ðr, θÞ = 5

The perturbation of ion temperature has a maximum at the lower part of the torus and corresponds to estimating Eq. (13.11). If condition Eq. (13.15) is satisfied, the ion temperature perturbation remains small: T1i/T0i < 1. Temperature perturbation T1i Eq. (13.23) can be used to calculate the additional radial heat flux caused by toroidicity averaged over the flux surface. Averaged over the flux surface for quantity f is defined as hhf ii =

1 S

p f gθθ

gζζ dθdζ,

ð13:24Þ

where flux surface area S is given by p

S=

p gθθ gζζ dθdζ:

ð13:25Þ

For the chosen simple model of magnetic field S = 4π2rR0, the surface average is defined according to hhf ii =

1 2π

f ð1 þ ε cos θÞdθ:

ð13:26Þ

The surface-averaged radial ion heat flux is thus 2π

1 qir ð1 þ ε cos θÞdθ = 2π

1 hhqi ii = 2π

5 n0 T 0i ∂T 1i ð1 þ ε cos θÞ2 dθ 2 eB0 r∂θ

0 2π

-

1 2π

κi⊥ 0

∂T 0i ð1 þ ε cos θÞdθ: ∂r ð13:27Þ

276

13

Transport Phenomena in Tokamaks

Here, an additional factor (1 + ε cos θ) appears due to the poloidal dependence of the magnetic field in the denominator. While taking the integral, all terms containing products n1 and T1i and n2 and T2i turn to zero since they contain factor sinθ cos θ. Neglecting small terms in Eq. (13.27), one obtains Shafranov’s heat flux hhqi ii = - 1:6q2 þ 1 κi⊥

dT 0i : dr

ð13:28Þ

Here, it is taken into account that 25n2 T 2i = 1:6: 2e2 B2 κik κi⊥ Hence, the heat flux associated with toroidicity (the first term in Eq. (13.28)) is 1.6q2 times larger than the heat flux in the cylinder (the second term in Eq. (13.28)). The ion flux caused by toroidicity was calculated as the diamagnetic heat flux, corresponding to the last term in Eq. (13.17); however, physically, it corresponds to the vertical guiding center vertical heat flux averaged over the flux surface with the account of temperature perturbation. Note that the main toroidal part of the heat flux in Eq. (13.28) is quadratic in ε.

13.1.3

Plasma Flows on the Flux Surface, Density, Temperature and Potential Perturbations

Plasma flows on the flux surface can be found from the particle balance equations. p gςς 1 ∂ p nujr p gςς ∂r grr

p gςς 1 ∂ p nujθ = 0: þp gςς ∂θ gθθ

ð13:29Þ

Radial particle velocity is obtained from the poloidal projection of the momentum balance equations, where viscosity and inertia terms are neglected. It is thus ujr = -

∂p1j 1 1 ∂φ1 : ej nBT r ∂θ BT r ∂θ

ð13:30Þ

Here, we neglected the perpendicular friction force, which leads to classical radial diffusive flux in a cylinder (see Chap. 7). Since, as demonstrated below, perturbations on the flux surface are rather small, the contribution of radial fluxes Eq. (13.30) to the particle balance equations can be neglected. One can check it by substituting the perturbations obtained below into Eqs. (13.29) to (13.30). For the same reason, we neglect density perturbations in Eq. (13.29) and replace it by

13.1

Fluid Regime (Pfirsch-Schlueter Regime)

277

p gςς 1 ∂ p n u = 0: p gθθ 0 jθ gςς ∂θ

ð13:31Þ

From the radial momentum balance, the poloidal velocity is determined: ujθ = -

E r Bp 1 dpj þ ujT þ : ej nBT dr BT BT

ð13:32Þ

Here, ujT is the toroidal velocity. One can also choose two alternative directions – parallel to the magnetic field and perpendicular to the magnetic field – with the perpendicular direction lying on the flux surface. Then, the radial momentum balance determines the perpendicular velocity component uj⊥ = -

1 dpj Er þ = V 0 þ upj : B ej nB dr →

ð13:33Þ

→

The first term here corresponds to E × B drift caused by the radial electric field, and the second term represents diamagnetic drift. The first and third terms in Eq. (13.32) are projections of the perpendicular velocity on the poloidal direction, and due to the smallness of the poloidal magnetic field, they practically coincide with V0 and upj. Parallel velocity also has a poloidal component (parallel and toroidal velocities almost coincide). Therefore, with good accuracy ujθ = V 0 þ Θujk þ upj :

ð13:34Þ

p gςς p ujθ = constðθÞ: gθθ

ð13:35Þ

From Eq. (13.31), we have

Substitution of Eq. (13.34) yields -

1 dpj Er þ ð1 þ ε cos θÞ2 þ Θujk ð1 þ ε cos θÞ = const ðθÞ: B0 ej nB0 dr

ð13:36Þ

Keeping the first-order terms in ε, one finds parallel velocity ujk ðr, θÞ = - 2ε = uPS jk

Er 1 dp0j þ 0 Bp ej nB0p dr

þ U k ð1 - ε cos θÞ:

cos θ þ U k ð1 - ε cos θÞ

ð13:37Þ

278

13

Transport Phenomena in Tokamaks

Here, velocity U k is an average common parallel velocity of electrons and ions, which is independent of the poloidal angle θ and with sufficient accuracy U k = U T . We shall also neglect the difference in the mean velocities of electrons and ions due to the flow of the average current in a tokamak. The first term uPS jk is a Pfirsch-Schlueter flux. The component of parallel flux, which is proportional to the pressure gradient, closes diamagnetic fluxes. Their divergence in a nonuniform magnetic field is nonzero and coincides with the divergence of the vertical guiding center drifts. A similar term is connected with → → the poloidal E × B drift, whose divergence is also nonzero in a nonuniform magnetic field. The second term in Eq. (13.37) is associated with the average toroidal velocity. Since the surface of the low-field side of the torus is larger than the highfield side surface, the parallel velocity is poloidally perturbed-smaller at the low-field → side to fulfil condition ∇  u j = 0. By subtracting the parallel electron velocity from the ion velocity, after multiplying by the density and electron charge, the expression for the Pfirsch-Schlueter current is obtained jPS k = - 2ε

1 dp cos θ, B0p dr

ð13:38Þ

This is consistent with estimating Eq. (13.14). Here, p = p0e + p0i is the net plasma pressure. Due to the relative velocities of electrons and ions, a parallel friction force arises: → u = 0:51me νe jPS Reik k =e: In addition, a parallel thermal force exists caused by electron temperature perturbation on the flux surface. The electron temperature perturbation T1e can be obtained similar to the ion temperature perturbation from the electron heat balance →

∇  q е = 0:

ð13:39Þ

Here, the electron parallel heat flux contains an additional term proportional to the Pfirsch-Schlueter current, see Chap. 2, Eq. (2.44), 5 nT e q e = - κek ∇k T e - κe⊥ ∇⊥ T e 2 eB

→

→

B × ∇T e B

- 0:71T e jPS k =e:

ð13:40Þ

The electron temperature perturbation is T 1e ðr, θÞ = -

1 dp0 n0 T 0e εr dT sin θ 5 0e þ 1:42 : 2 dr n 0 dr eB0 κek Θ

ð13:41Þ

13.1

Fluid Regime (Pfirsch-Schlueter Regime)

279

It is (mi/me)1/2 smaller than the ion temperature perturbation. The parallel thermal force is calculated using Eq. (13.41): RTeik = - 0:71Θ∂T 1e =r∂θ. As a result, full friction force →

u þ RTeik = Rk = Reik

n20 T 0e ε dT 1 dp0 3:55 0e þ 4:14 cos θ: dr n0 dr eB0 κek Θ

ð13:42Þ

Neglecting inertia and viscosity terms, parallel momentum balance equations for electrons and ions can be written in the form - ΘT 0e

1 ∂n1 1 ∂T 1e 1 ∂φ1 - Θn0 þ en0 Θ þ Rk = 0, r ∂θ r ∂θ r ∂θ

ð13:43Þ

- ΘT 0i

1 ∂n1 1 ∂T 1i 1 ∂φ1 - Θn0 - en0 Θ - Rk = 0: r ∂θ r ∂θ r ∂θ

ð13:44Þ

In these equations, the friction force and electron pressure gradient associated with electron temperature perturbations are (mi/me)1/2 smaller than the other terms. Hence, after neglecting them, one obtains eφ1 n1 T 1i = =: T 0e n0 T 0e þ T 0i

ð13:45Þ

The density perturbation is positive in the upper part of the flux surface, and the poloidal gradient of the full pressure caused by the density gradient balances the pressure gradient caused by ion temperature perturbation. As a result, full pressure p = p0 is not perturbed and is a flux surface function in accordance with the result of the previous chapter. The potential perturbation in the approximation considered corresponds to Boltzmann’s distribution, and the potential perturbation as the density perturbation is positive at the upper part of the torus (Fig. 13.4). The poloidal electric field is negative at the low-field side part of the flux surface and positive at the high-field side, and the drift caused by this electric field is directed inwards at the outer part of the flux surface.

13.1.4

Particle Fluxes

Plasma diffusion in a tokamak is a result of specific convection – radial particle flux is a result of averaging convective fluxes over the flux surface. From Eq. (13.4), one can get the impression that the average particle flux is directed inwards since the outer part of the torus has a larger surface than the inner part. However, this is incorrect, as sine full flux also contains vertical guiding center particle drift

280

13

→

Transport Phenomena in Tokamaks

→

Fig. 13.4 Poloidal electric field and corresponding E × B drifts

(poloidally inhomogeneous fraction of radial diamagnetic flux). In the first approximation, the full flux averaged over the flux surface turns to zero due to Boltzmann’s distribution Eq. (13.45). Hence, to obtain finite particle flux, it is necessary to keep all terms in parallel momentum balance equations. Indeed, according to Eq. (13.30), the radial electron flux is determined by the poloidal electric field and poloidal pressure gradient:

Γe  nuer = -

∂p1e n n ∂φ1 : ej nBT r ∂θ BT r ∂θ

ð13:46Þ

Using the parallel momentum balance for electrons Eq. (13.43), one finds Γe =

Rk : eBp

ð13:47Þ

This relation can also be easily obtained from the toroidal projection of the momentum balance equation of electrons. Here, the Lorentz force eBpΓe is balanced by the toroidal friction force RT ≈ Rk. Substituting Eq. (13.42) into Eq. (13.47), after flux surface averaging, one obtains 2π

1 hhΓe ii = 2π

Rk ð1 þ ε cos θÞdθ eBp

0

:

ð13:48Þ

1 dp0 dT n m ν q2 = - 0 2e 2e 1:3 - 1:12 0e n dr dr e B0 0 Particle flux, similar to heat flux, is q2 times larger than the flux in a cylinder, Eq. (7.10),

13.2

Radial Electric Field, Poloidal and Toroidal Rotation

Γ0 = -

281

n0 me νe 1 dp0 3 dT 0e þ : e2 B20 n0 dr 2 dr

The effective diffusion coefficient Deff = hhΓeii/(-d ln n0/dr) is (mi/me)1/2 times smaller than the ion heat diffusivity coefficient χi = hhqiii/(-3/2n0dTi/dr), and the particle flux is not fully diffusive. The ion flux in the considered approximation coincides with the flux of electrons. Indeed, from Eqs. (13.30) and (13.44), we have Γi = cRk/(eBp) and Γe = Γi since both fluxes are expressed through the same friction force in accordance with Eq. (13.47). To calculate the ambipolar electric field, as in the cylindrical case, it is necessary to take into account ion viscosity terms in the momentum balance equations.

13.2 Radial Electric Field, Poloidal and Toroidal Rotation To analyze the radial electric field and rotation in a tokamak, one has to take into account the viscosity and inertia of ions. For parallel velocities smaller than the sound speed, the viscosity dominates over inertia forces. Considering the viscosity, the total momentum balance equation is given by (subscript i is omitted for brevity) $

- ∇k p - ð∇  π k Þk = 0:

ð13:49Þ

In contrast to the previous section, full pressure is now not a flux surface function. In $ the viscosity tensor, it is sufficient to consider only parallel viscosity π k . This part of the viscosity tensor is connected with the coefficient η0 and is presented in Chap. 2 for a magnetic field parallel to the z-axis. In an arbitrary magnetic field, with an account of the tensorial character of viscosity, it should have the form πkαβ = pk - p⊥

Bα Bβ 1 - δαβ : B B 3

ð13:50Þ

Here, the difference between parallel and perpendicular pressures is connected with the rate-of-strain tensor and derivatives of heat fluxes, →

pk - p⊥ = pk - p⊥

u

→

þ pk - p ⊥

q

:

ð13:51Þ

In accordance with Eqs. (2.47) and (2.48), the first term, which determines compo→

$u

nent π k , equals

282

13 →

pk - p⊥

u

=-

Transport Phenomena in Tokamaks

3 Bj B η W k, 2 0 B jk B

ð13:52Þ

where the rate-of-strain tensor Eq. (2.47) in Cartesian coordinates W jk =

∂uj ∂uk 2 → þ - δjk ∇  u : 3 ∂xk ∂xj

ð13:53Þ

→

The second term pk - p⊥

q

is analogously connected with heat fluxes.

Let us multiply parallel momentum balance Eq. (13.49) by magnetic field B and p p perform volume averaging over the flux surface with weight g = grr gθθ gζζ : →

→

$

- B ∇p - B ∇  π k = 0:

ð13:54Þ

Volume averaged (in contrast to surface averaged Eq. (13.24)) is defined as p f gdθdζ=

hf i =

p

ð13:55Þ

gdθdζ:

After averaging, the first term n in Eq. (13.54) becomes zero since →

→

→

B ∇p = ∇  p B =

→

→

pBd S =

p

→

- p∇  B = ∇  p B gdθdζ = 0 ≠ :

ð13:56Þ

→

Here, equation ∇  B = 0 is used, the volume integral is reduced to surface integrals for neighboring flux surfaces, and the latter is zero since magnetic flux is absent through the flux surface. Hence, after volume averaging, the parallel momentum balance is reduced to →

$

B ∇  π k = 0:

ð13:57Þ

In Cartesian coordinates xk, the average viscosity can be rewritten in the form →

$

B ∇  πk =

∂ Bi πkik ∂xk

- πkik

∂Bi : ∂xk

Substituting Eq. (13.50) into the first term ∂(Biπkik)/∂xk, we find that it is propor* tional to ∇  B and, hence, equals zero. The second term is evaluated as

13.2

Radial Electric Field, Poloidal and Toroidal Rotation

πkik

283

1 Bi Bk ∂Bi B ∂B = p k - p⊥ k : - δik B B 3 B ∂xk ∂xk

∂Bi = p k - p⊥ ∂xk

Therefore, from Eq. (13.57), one obtains →

→

$

B ∇  πk =

p⊥ - pk

B ∇B = 0: B

ð13:58Þ

Let us calculate part of the viscosity tensor associated with nonhomogeneous → velocity. Considering condition ∇  u i = 0, which is satisfied on the flux surface, →

pk - p⊥

u

- 3η0 Θ

=-

B 3 Bj W k = η 2 0 B jk B

∂uik - uiθ ε sin θ þ ΘU k ε sin θ : r∂θ

ð13:59Þ

Here, the last two terms are caused by changes in curvilinear coordinates in space p and are proportional to ∂ gζζ =∂θ. Substituting Eqs. (13.34) and (13.37) for poloidal and parallel velocities, using identity →

$

B ∇  πk

→

u

=-

→

B =B ∇B = Bp ε sin θ=r, one finds

3Θ2 E T ∂n 1 ∂T 0i η B - 0r þ 0i0 þ Uk : þ 0 2 0 ∂r 2R0 Bp enBp eBp ∂r

ð13:60Þ

Accounting for the viscosity caused by parallel heat flux leads to the general expression →

$

B ∇  πk = -

1 ∂T 0i E T ∂n 3Θ2 η B - 0r þ 0i0 þ Uk : þ kT 0 2R20 0 Bp enBp ∂r eBp ∂r

ð13:61Þ

Here, the numerical coefficient kT depends on collisionality and in the fluid approximation considered in this section kT = 2.7 [13]. Condition Eq. (13.57) determines the neoclassical electric field Er =

1 ∂T 0i T 0i ∂n þ B0p U k : þ kT en ∂r e ∂r

ð13:62Þ

Thus, the radial electric field, which turns to zero volume average parallel viscosity, consists of two parts. The two first contributions are proportional to the density and ion temperature gradients; for decreasing radius density and temperature, these terms correspond to a negative (directed from periphery to center) electric field. The last term is caused by the toroidal rotation of plasma. For co-current toroidal rotation

284

13

Transport Phenomena in Tokamaks

(positive in the chosen coordinates), the corresponding contribution to the radial electric field is positive. This last term can be interpreted as an additional electric field B0p U T in the reference frame moving with the velocity U T ≈ U k in accordance with the Lorentz transformation. It is worth noting that the average poloidal velocity according to Eq. (13.32) depends only on the ion temperature gradient uiθ = ð1 - k T Þ

1 ∂T 0i : eB0 ∂r

ð13:63Þ

In contrast, the perpendicular velocity ui⊥ depends on the toroidal rotation velocity according to Eq. (13.33). Toroidal rotation velocity is determined from the toroidal component of the momentum balance equation: $

jr Bp - ∇  π k

ζ

= nmi

dU T : dt

ð13:64Þ

Here, the term on the r.h.s. symbolically represents the radial transport of toroidal momentum. In real situations, radial transport is driven by turbulence since neoclassical flux is rather small. A similar term also enters parallel momentum balance Eq. (13.54); however, it is smaller than the parallel viscosity. In Eq. (13.64), the same term is retained since the toroidal viscosity averaged over the flux surface, as shown below, becomes zero. Note that the account of viscosity, in contrast to that considered in the previous chapter ideal case, leads to the formation of radial current, and lines of current density do not belong to the flux surface. On the other hand, the ambipolarity constraint should be satisfied, which means that the surface-averaged radial current through the flux surface should be zero, hhjr ii = 0:

ð13:65Þ

Ambipolarity condition, according to Eq. (13.64), leads to equation $

-

∇  πk

ζ

Bp

=

T nmi dU dt Bp

:

ð13:66Þ

Let us demonstrate that the l.h.s. of this equation becomes zero. In curvilinear coordinates $

∇  πk

p g 1 ∂ p πkθξ =p g g ∂θ ξ θθ

þp

p πkθζ ∂ gζζ : gθθ gζζ ∂θ

ð13:67Þ

Then, we divide the toroidal viscosity by the poloidal magnetic field and average over the flux surface:

13.3

Neoclassical Transport in Collisionless Regimes

285

$

p g 1 1 ∂ p p πkθξ S Bp g ∂θ Bp gθθ p ∂ gζζ p πkθξ gθθ gζζ dθdζ: p Bp gθθ gζζ ∂θ

∇  πk

þ

1 S

ξ

=

p

gθθ

gζζ dθdζ

ð13:68Þ

p Product Bp grr gζζ is proportional to the poloidal flux between neighboring flux surfaces and is independent of the poloidal coordinate and, hence, can be removed p p from the integral. Using identity g = grr gθθ gζζ , taking the first integral by parts, one finds that the first integral compensates the second one, so that $

∇  πk Bp

ζ

= 0:

ð13:69Þ

Ambipolarity constraint Eq. (13.66) is thus reduced to T nmi dU dt Bp

= 0:

ð13:70Þ

This equation determines the toroidal rotation profile. In the presence of an external force in the toroidal direction, for example, from neutral beam injection, the corresponding force should be added to the r.h.s.

13.3 13.3.1

Neoclassical Transport in Collisionless Regimes Particle Trajectories

In collisionless plasma with a large mean free path λj > qR, the main contribution to radial transport is caused by particles with small parallel velocities. Therefore, let us consider the qualitative character of their motion. In a magnetic field, there are two integrals of motion – full energy E = mjV2/2 - ejφ and magnetic moment μj = mj V 2j⊥ =2B. The particle parallel velocity is a function of energy, and the magnetic momentum is V jk = ±

2 E - ej φ - μ j B : mj

ð13:71Þ

While analyzing the particle motion, it is possible to neglect weak poloidal dependence of the potential. Equations of motion in collisionless plasma are:

286

13

Transport Phenomena in Tokamaks

V 2jk þ μj Bmj dr sin θ, =ωcj R dt V 2jk þ μj Bmj dθ r =cos θ þ ΘV jk þ V 0 : dt ωcj R

ð13:72Þ

Two first terms on the r.h.s. of Eqs. (13.72) are projections of the guiding centers’ vertical drift in the inhomogeneous magnetic field Eq. (11.9) in the radial and poloidal directions, respectively. The contribution ΘVjk is a poloidal projection of → → the parallel velocity, and V0 corresponds to the poloidal E × B drift. By dividing the first equation into the second equation, we can obtain the equation for the trajectory. The characteristics of motion strongly depend on the parallel velocity Vjk. If parallel and perpendicular velocities are of the same order (of the order of thermal velocity, e.g.), then variation of the parallel velocity Eq. (13.71) is small because the magnetic field inhomogeneity is small. The particle completes the full turn over poloidal angle. Such particles are called transit or untrapped particles. The trajectories of transit particles are shown in Fig. 13.5. An ion with positive velocity at the outer midplane (at point О) moves in the counterclockwise direction along the poloidal angle and is shifted radially due to vertical drift. Since the drift Vgi for ions (for the chosen direction of the magnetic field) is directed downwards, the ion in the upper part of its trajectory is shifted inside the flux surface and returns outside of it in the lower part. As a result, the trajectory is closed and lies inside a given flux surface (Fig. 13.5a). The ion trajectory with a negative parallel velocity is shifted outwards of the magnetic surface. For electrons, the vertical drift Vge is directed upwards; hence, the trajectories of an electron with positive velocity at point О are situated outside the flux surface, and the trajectory of an electron with negative velocity is located inside the flux surface (Fig. 13.5b).

Fig. 13.5 Poloidal projections of transit trajectories: (а) ions and (b) electrons. Trajectories 1 correspond to the positive (co-current) direction of parallel velocity Vjk > 0, and trajectories 2 correspond to the negative (counter current) direction of parallel velocity Vjk < 0

13.3

Neoclassical Transport in Collisionless Regimes

287

The average radial deviation from the flux surface can be estimated as Δrj = V gj νbj

-1

:

ð13:73Þ

Here, the bounce frequency νbj =

ΘV jk þ V 0 r

ð13:74Þ

is the frequency (inverse time) for a particle to turn in the poloidal direction. For particles with thermal velocities Δrj = ugj νbj

-1



Tj r  qρcj : eBR ΘV Tj

ð13:75Þ

Here, ugj is the vertical drift velocity for thermal particles in Eq. (13.8). We neglected her contribution from V0~T/eB with respect to the poloidal projection of thermal velocity ΘVTj since condition Eq. (13.4) is satisfied. Let us now discuss the motion of particles with small parallel velocities at the outer midplane. First, assume V0 = 0. According to Eq. (13.71) during motion in the poloidal direction, the parallel velocity of a particle decreases due to the rise of the magnetic field towards high field and energy conservation. For transit particles, the change in parallel velocity is rather small-of the order of εVkj, while for particles with small parallel velocity, their parallel velocity turns to zero, and a stagnation point arises. The trajectory of such particles resembles a banana shape (Fig. 13.6), and particles are called trapped or banana particles. Ions with positive velocity at the

Fig. 13.6 Poloidal projection of trapped particle trajectories: (а) ions and (b) electrons. Trajectories 1 correspond to the positive (co-current) direction of parallel velocity Vjk > 0 at outer midplane, and trajectories 2 correspond to the negative (counter current) direction of parallel velocity Vjk < 0 at outer midplane

288

13

Transport Phenomena in Tokamaks

outer midplane are shifted inside the flux surface (Fig. 13.6a). After reflection from the stagnation point, ions move poloidally towards the low-field side and continue shifting downwards. In the lower part of the trajectory, ions move radially towards the initial flux surface, and, at the reflection point, one more change in parallel velocity takes place. Finally, the poloidal projection of the trajectory has a banana shape. Ions with negative velocity at the equatorial midplane are shifted outside the initial flux surface. Banana trajectories are shown in Fig. 13.6b. Electrons with positive velocity at point О are located outside the flux surface, while electrons with positive velocity are located inside the flux surface. The particle trajectory in space for V0 = 0 is presented in Fig. 13.7a. Let us find the maximal value of the parallel velocity at the outer midplane V max k when the particle remains trapped. For such velocity, the reflection point is situated at the inner midplane. From Eq. (13.71) applied for such particles for the outer and inner midplane, one has 2 E - ej φ - μj Bmin , mj

V max jk = 0=

2 E - ej φ - μj Bmax : mj

ð13:76Þ

Here Bmin, Bmax are the minimal and maximal values at the outer and inner midplanes, respectively. Eliminating full energy yields V max jk =

p 2 μj Bmax - μj Bmin ≈ 2εV j⊥ : mj

ð13:77Þ

For thermal particles, the particle remains trapped if the parallel velocity satisfies the inequality p 0 ≤ V jk ≤ 2εV Tj :

ð13:78Þ

For trapped particles with perpendicular velocities of the order of thermal velocity shift with respect to flux surface and banana width are given by an estimate Δrj = ugj νbj

-1



qρcj Tj r p  p : eBR εΘV Tj ε

ð13:79Þ

The characteristic of untrapped particle motion does not change significantly with → → an account of poloidal E × B drift with the velocity V0, since the latter is smaller than ΘVTj. In contrast, for trapped particles, the situation is different. Due to radial electric field kinetic energy changes with radius, Eq. (13.76) has the form

13.3

Neoclassical Transport in Collisionless Regimes

V max jk = 0=

2 E - ej φout - μj Bmin , mj

2 E - ej φin - μj Bmax , mj

289

ð13:80Þ

where φout, φin are the values of potential at the low- and high-field sides, respectively. Since qρcj φout - φin  V 0 BΔr j  V 0 B p , ε

ð13:81Þ

one can expand the expression under square root in series over V0/ΘVTj. Finally, the particle is trapped at the outer midplane: p 0 ≤ V jk þ ΘV 0 ≤ 2εV Tj :

ð13:82Þ

Hence, the particle whose parallel velocity at the outer midplane satisfies inequality (13.82) has a reflection point, and its trajectory at the (r, θ) plane still has a banana shape. At the same time, the parallel (toroidal) velocity remains finite, of the order of -V0/Θ. Hence particle trajectory is disconnected in the toroidal direction, and toroidal particle precession takes place with the velocity -V0/Θ, see Fig. 13.7b.

13.3.2 Ware Drift For banana particles, there is also a specific effect – the average drift of electrons and ions in the radial direction towards the plasma center. This drift is caused by the toroidal inductive electric field ET, which exists in a tokamak. In the presence of electric field, ET trajectories of trapped particles in the plane (r, θ) become disconnected, as shown in Fig. 13.8. Particles (ions) with positive velocity at the outer midplane are accelerated by the electric field in the upper part of the trajectory and gain additional energy Δmj V 2jk =2. Hence, the position of the reflection point is shifted in the poloidal direction, and the particle is simultaneously shifted inwards in the radial direction due to vertical drift in the nonhomogeneous magnetic field. At the inner part of the trajectory field, ET decelerates particles that now do not reach the reflection point at ET = 0. As a result, the particle is shifted radially with respect to the initial flux surface. The average radial drift velocity can be estimated as follows. The change in the particle radial shift δ(Δr) is connected with the variation in the particle mean parallel velocity

290

13

Transport Phenomena in Tokamaks

Fig. 13.7 Trajectory of a trapped particle. (а) In the absence of radial electric field, trajectory is closed; (b) disconnected trajectory with account of radial electric field

Fig. 13.8 Disconnected trajectory of trapped ion in toroidal electric field

δ Δr j =

ugj r ΔV jk , ΘV jk V jk

ð13:83Þ

where velocity variation ΔVjk is caused by the work of electric field eET: ΔV jk =

eE T r : mj ΘV jk

ð13:84Þ

Flux associated with Ware drift with account of small, of the order of δn=n  fraction of banana particles is hence estimated as

p

ε,

13.3

Neoclassical Transport in Collisionless Regimes

hhΓ ii 

p

291

εnδ Δr j νbj :

Combining with Eqs. (13.84) and (13.85), substituting velocity, we obtain

p

εV Tj as a typical parallel

p E hhΓii  - εn T : Bp

ð13:85Þ

The minus corresponds here to the inwards direction of the Ware drift. Analysis based on the kinetic equation solution shows that the Ware drifts for electrons and ions coincide.

13.3.3

Estimation of Transport Coefficients in the Plateau Regime

The collisionality parameter in this regime satisfies inequality ε3=2
0, Fig. 13.10. The derivative in his region can be obtained from the following estimate: ∂f i Δni Δf i   n fi ∂V ik

p V ik = - V 0 =Θ

εV Ti : fi

ð13:124Þ

Momentum exchange is most effective between trapped and untrapped particles. Hence, the ion distribution function in this region should be similar to the Maxwellian function (at dT0i/dr = 0 coincides with the Maxwellian distribution function). Using this fact, one can replace the derivative by ∂f i ∂V ik

=V ik = - V 0 =Θ

f i mi V ik - U k Ti

V ik = - V 0 =Θ

=

f i mi V þ ΘU k : ΘT i 0 →

ð13:125Þ →

Combining Eq. (13.123) with Eq. (13.124), one obtains the poloidal E × B velocity V0. The corresponding radial electric field is of the order of given by Eq. (13.62), as in other regimes. Therefore, the requirement that the distribution function should be close to the Maxwellian function (more accurately, it should turn to zero ion flux in the velocity space caused by ion-ion collisions) and that it should be conserved along trajectories

300

13

Transport Phenomena in Tokamaks

Fig. 13.11 Electron distribution function in the banana regime

for small parallel velocities determines the radial electric field. The vertical ∇B drift of electrons is directed upwards; hence, in the vicinity of -V0/Θ, more electrons have a negative velocity, and, here, the derivative of their distribution function is negative. On the other hand, for thermal velocities, the electron distribution function should be close to the Maxwellian function bound to the ion distribution function by electron-ion collisions. As a result, the electron distribution function should look as shown in Fig. 13.11. A change in the derivative corresponds to the transition from trapped to untrapped particles. Trapped electrons have a mean velocity of the order with respect to ions (we assume Te~Ti~T ). Hence, with account of their of T0/ΘeBrp fraction  ε from the full electron density, this flux corresponds to the current p j1k  n0 εT 0 =ΘeBr,

ð13:126Þ

which depends on the density gradient as well as on the electron and ion temperature gradients. In addition, there is a second contribution to the current, j2k, which arises due to collisions between trapped and untrapped electrons. The latter produce a drag force and shift the whole body of the electron distribution function with respect to ions. From force balance me νei j2k =e - me νee j1k =e n0 = 0

ð13:127Þ

one obtains j1k~j2k. Two contributions are of the same order and together form a bootstrap current jB, which is a function of density and temperature gradients. By order of magnitude p jB  n0 εT 0 =ΘBr The bootstrap current is directed as the main plasma current.

ð13:128Þ

13.5

Particle and Heat Balance Equations

301

The special character of the distribution function also leads to a reduction in the conductivity current caused by the induced toroidal electric field. Since trapped particles are not involved in the conductivity process, neoclassical conductivity is reduced with respect to the Spitzer conductivity, p = σk 1 - α ε , σNEO k

ð13:129Þ

where α is a numerical coefficient.

13.5 Particle and Heat Balance Equations Below are approximation expressions that can be used practically for all collisionality regimes. The dimensionless collisionality parameter is defined as 1=2

νj =

νj mj qR 1=2

T j ε3=2

:

ð13:130Þ

Particle flux hhΓii = Γ1 þ Γ2 ; p 2n εT m ν d ln T 0e Γ1 = 0 2 2 0e 2 e e K 11 Ae þ K 12 ; dr e B0 Θ d lnðnT 0e Þ 5 d ln T 0e T 0i dlnðnT 0e Þ 1 - k T d ln T 0i Ae = þ ; dr dr T 0e 2 dr 1 þ ν2e ε2 dr p 1 E , Γ2 = - K 13 n0 ε B0 Θ T where K mn = K 0mn

1 1þ

1=2 amn νe

þ bmn νe

þ

ε3 νe c2mn =bmn 1 þ cmn νe ε3=2

for m and n from 1 to 2, K m3 = K 0m3

1 1=2

ð1 þ cm3 νe ε3=2 Þ 1 þ am3 νe þ bm3 νe

Coefficients are given in the table.

:

ð13:131Þ

302

13 K 0mn 1.04 1.2 2.55 2.3 4.19 1.83

mn 11 12 22 13 23 33

Transport Phenomena in Tokamaks

amn 2.01 0.75 0.45 1.02 0.57 0.68

bmn 1.53 0.67 0.43 1.07 0.61 0.32

cmn 0.89 0.56 0.43 1.07 0.61 0.66

Electron heat flux: p 2n0 εT 20e me νe 5 d ln T 0e K 12 Ae þ K 22 hhqe ii þ hhΓiiT 0e = 2 2 2 dr 2 e B0 Θ p 1 E : - K 23 n0 T 0e ε B0 Θ T

ð13:132Þ

Parallel current: p p n0 εT 0e d ln T 0e K 13 Ae þ K 23 jk = - σk E T 1 - K 33 ε : ΘB0 dr

ð13:133Þ

Ion heat flux: 5 hhqi ii þ hhΓiiT 0i 2 p n0 εT 20i mi νi d ln T 0i 5 1 - kT = - 2K 2 : þ T 0i hhΓii 2 2 2 2 dr 1 þ ν2i ε3 e B0 Θ

ð13:134Þ

Here K 2 = 0:66

1 1þ

1=2 1:03νi

þ 0:31νi

þ

1:77ε3 νi : 1 þ 0:74νi ε3=2

ð13:135Þ

Neoclassical radial electric field: Er =

T 0i ∂n 1 ∂T 0i þ kT þ B0p U k , en ∂r e ∂r

ð13:136Þ

where 1=2

kT =

- 0:17 þ 1:05νi þ 3:1ν2i ε3 1=2

1 þ 0:7νi

ð1 þ ν2i ε3 Þ

:

ð13:137Þ

13.5

Particle and Heat Balance Equations

303

Particle balance equation: ∂n0 1 ∂ ðr hhΓiiÞ = I - R, þ r ∂r ∂t

ð13:138Þ

where I and R are particle sources and sinks, respectively. Heat balance equation: 5 3 ∂ðn0 T 0e Þ 1 ∂ þ r hhqe ii þ hhΓiiT 0e = Qe , r ∂r 2 2 ∂t 3 ∂ðn0 T 0i Þ 1 ∂ 5 þ r hhqi ii þ hhΓiiT 0i = Qi : 2 ∂t 2 r ∂r

ð13:139Þ

Sources on the r.h.s. in the absence of additional heating are Qe = jk E T - QΔ - eE r hhΓii, Qi = QΔ þ eE r hhΓii:

ð13:140Þ

Here, the term QΔ represents the heat exchange between electrons and ions: QΔ = 3

me nν ðT - T i0 Þ, mi e e0

ð13:141Þ

and E r is the electric field in the reference frame moving with the toroidal velocity: E r = Er - B0p U k =

T 0i ∂n 1 ∂T 0i þ kT : en ∂r e ∂r

ð13:142Þ

Additional terms on the r.h.s. of the heat balance equations represent the work of the radial electric field, and other work of the poloidal electric field and friction are taken into account during the calculation of radial heat fluxes hhqjii. One should keep in mind that in the real tokamak, turbulent particle and heat fluxes can significantly exceed neoclassical values, especially for particle and electron heat fluxes. In such a situation, Eqs. (13.138) and (13.139) remain the same. Only the work of the electric field could look different, since for neoclassical fluxes, part of the wok of the poloidal electric field and friction was taken into account while integrating over the flux surface. For arbitrary flux surfaces, one should use the surface coordinates described in Sect. 12.7. In these coordinates, the local particle balance has the form 1 ∂ ∂n0 þp g ∂a ∂t

p

g Γ = I - R, ha a

ð13:143Þ

304

13

Transport Phenomena in Tokamaks

p p where g = ha hθ hζ . Let us multiply his equation by g and integrate over poloidal and toroidal coordinates. Assuming constant density at the flux surface, the term on the l.h.s. is transformed to ∂

p gn0 dθdζ ∂ = ½V 0 ðaÞn0 , ∂t ∂t

where V is the volume inside a given flux surface and V′(a) is the volume between neighboring flux surfaces. From the second term, one obtains the net flux through the flux surface. Multiplying and dividing it by V′(a) yield p

→ g Γdθdζ = V 0 ðaÞ Γ ∇a : ha

Here, brackets hi correspond to volume averaging Eq. (13.55). Finally, the averaged particle balance equation is given by → ∂ 0 ∂ ½V ðaÞn0  þ V 0 ðaÞ Γ ∇a ∂t ∂a

= V 0 ðaÞðI - RÞ:

ð13:144Þ

As the flux surface coordinate a, it is possible to choose the equivalent radius of the flux surface ρ, defined via the toroidal magnetic flux, πρ2 B0T = ΨT . Then, Eq. (3.144) takes the form → ∂ ∂ 0 ½V ðρÞn0  þ V 0 ðρÞ Γ ∇ρ ∂ρ ∂t

= V 0 ðρÞðI - RÞ:

ð13:145Þ

For circular flux surfaces ρ = r, V′(r)~r and Eq. (3.145) is reduced to Eq. (13.38). The heat balance equations in the general case are similarly: 3 ∂½V 0 ðρÞn0 T 0e  ∂ V 0 ð ρÞ þ 2 ∂t ∂ρ 3 ∂½V 0 ðρÞn0 T 0i  ∂ þ V 0 ðρ Þ 2 ∂t ∂ρ

→

q e ∇ρ þ

→

q i ∇ρ þ

5 → Γ ∇ρ T 0e 2

5 → Γ ∇ρ T 0i 2

= V 0 ðρÞQe , = V 0 ðρÞQi : ð13:146Þ

The local radial electric field is Ea =

T 0i e

1 d ln n 1 d ln T i B - θ BV k þ kT ha da B ha da

ð13:147Þ

13.6

13.6

Transport Codes

305

Transport Codes

The flux surface averaged transport equations analyzed in the previous section can be used in 1D transport codes for the core region of a tokamak. However, in real tokamaks, in addition to neoclassical transport, strong contributions to particle and heat fluxes are produced by turbulent (anomalous) fluxes. This is taken into account by adding turbulent diffusion, particle convection and turbulent heat conductivities for electrons and ions to the corresponding averaged fluxes. The 1D codes can be separated into two groups. The first group uses some models for turbulent transport coefficients, and such codes are known as “theory-based” codes. Another approach has a phenomenological character; that is, such codes’ turbulent transport coefficients are chosen to obtain a better fit with the experimentally observed density and temperature radial profiles. Codes of the second group are widely used since “theorybased” codes are rather complicated and have restrictions due to limited knowledge of turbulent processes. Example of simulation results with code ASTRA of the second type and comparison with experimental data is presented in Fig. 13.12. The 1D codes cannot be used close to the separatrix and outside of it since the plasma distribution is essentially 2D. In this region, edge plasma simulation can be

Fig. 13.12 Calculated density and temperature profiles, and comparison with experiment for tokamak Globus-M. In the graphs (b) and (d), the data are the same as in (a) and (c) respectively, but the scale is changed to show the details near separatrix

306

13

Transport Phenomena in Tokamaks

performed with codes such as SOLPS-ITER (see Sect. 12.9). However, to perform 2D modelling, particle and heat fluxes from the core should be specified at the core side of the simulation domain as boundary conditions. They could be taken from 1D code describing the core. On the other hand, plasma sources obtained in 2D simulations can be used in 1D code after averaging over the flux surfaces, as well as other averaged plasma parameters obtained in 2D modeling (density, temperatures, potential, etc.). Iterating between 2 codes, after convergence, the simulation gives final profiles in the whole region, as shown in Fig. 13.12.

Chapter 14

Instabilities in Magnetized Plasma

In this chapter, plasma stability in the situation when plasma equilibrium in a magnetic field is achieved is considered. The most dangerous instabilities for plasma confinement are large-scale magnetohydrodynamic (MHD) instabilities, which can be described by MHD equations. There are also other types of instabilities of inhomogeneous plasma with finite conductivity, in particular drift waves, which were analyzed in Chap. 8. The first type of MHD instability is rather similar to instability in fluids, so we start with its consideration.

14.1

Rayleigh-Taylor Instability in Fluids

Consider fluid in the gravitational field, Fig. 14.1. Fluid density ρ0 changes with height, ρ0 = ρ0(z). In equilibrium, the gravity force is balanced by the pressure gradient, -

dp0 - ρ0 g = 0: dz

ð14:1Þ

Equilibrium becomes unstable if fluid density rises with height (heavy fluid on light one). In this situation, Rayleigh-Taylor (RT) instability starts developing. The opposite case is stable. Physically, the equilibrium is unstable to any perturbations: if a parcel of heavier fluid is displaced downwards with an equal volume of lighter fluid displaced upwards, the potential energy of the configuration is lower than the initial state. Thus, the disturbance will grow and lead to a further release of potential energy, as the denser material moves down under the gravitational field, and the less dense material is further displaced upwards. This, for example, is observed when water flows from inverted glass.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Rozhansky, Plasma Theory, https://doi.org/10.1007/978-3-031-44486-9_14

307

308

14

Instabilities in Magnetized Plasma

Fig. 14.1 Heavy fluid on top of light one – Rayleigh-Taylor instability

The evolution of small perturbations is given by Euler equations →

∇  u = 0, ∂ρ → þ u ∇ρ = 0, ∂t → ∂u → → → ρ þ u ∇ u = - ∇p þ ρ g : ∂t

ð14:2Þ

Perturbations are sought in the form of waves: ρ1 = αðzÞ expð - iωt þ ikyÞ, →1

→

u = β ðzÞ expð - iωt þ ikyÞ, p1 = γðzÞ expð - iωt þ ikyÞ:

ð14:3Þ

Amplitudes are unknown functions of coordinate z. The linearized system (14.2) is given by ∂u1z = 0, ∂z ∂ρ - iωρ1 þ u1z 0 = 0, ∂z 1 - iωρ0 uy = - ikp1 ,

iku1y þ

- iωρ0 u1z = -

ð14:4Þ

∂p1 - ρ1 g: ∂z

The velocity y-component u1y from the first equation can be substituted into the third equation, and the pressure perturbation p1 from the third equation is substituted into the last equation. Density perturbation ρ1 from the second equation should also be substituted into the last one. As a result, we have a single differential equation for u1z :

14.1

Rayleigh-Taylor Instability in Fluids

ω2

309

du1 d dρ0 1 u: ρ0 z = k 2 ρ 0 ω 2 þ g dz dz z dz

ð14:5Þ

Here, the partial derivative is replaced by the full derivative since the velocity amplitude depends on z only. For a given wave vector k, Eq. (14.5) corresponds to the Sturm-Liouville problem to seek eigenvalues and eigenfunctions for prescribed boundary conditions. At the boundaries z = 0, L corresponds to a fixed or free boundary: u1z ðz = 0, LÞ = 0;

du1z ðz = 0, LÞ = 0: dz

ð14:6Þ

It is possible to analyze the stability of the equilibrium without solving Eq. (14.5) in the following way. Let us multiply both parts of Eq. (14.5) by complex conjugate  u1z and perform integration over z for the whole region. L.h.s. can be integrated by parts, and the outside integral term turns to zero for any type of boundary condition. For frequency, one obtains g - ω2 =

dρ0 1 2 u dz dz z

:

2

ρ

0

1 1 duz þ u1z k2 dz

2

ð14:7Þ

dz

Since all functions except derivative dρ0/dz are positive, the sign of r.h.s. depends on the sign of derivative dρ0/dz. If density monotonically increases with z and dρ0/ dz > 0 for all z values, then ω2 < 0. In this case, ω = ± iγ, and root ω = iγ corresponds to the increment of instability. In the opposite case, the frequency is real, and the equilibrium is stable. For the short wavelength case, kλ >> 1, where λ is the typical spatial density scale λ = |d ln ρ0/dz|-1, Eq. (14.7) is reduced to g - ω2 =

dρ0 1 2 u dz dz z ρ0 u1z

2

:

ð14:8Þ

dz

In this situation, the increment for RT instability for dρ0/dz > 0 is of the order of γ  g=λ. If eigenfunctions are strongly localized with scales smaller than λ, then ρ0 and dρ0/dz can be taken outside the integral and γ=

g=λ:

ð14:9Þ

310

14

Instabilities in Magnetized Plasma

To find the exact value of the increment and eigenfunctions u1z in the general case, one has to solve Eq. (14.5). We shall demonstrate how the solution is obtained using the example of exponential profile ρ0 = ρ(0) exp (z/λ) for fixed boundary conditions u1z ðz = 0, LÞ = 0. Eq. (14.5) for this case is given by ω2

d2 u1z ω2 du1z g þ - k 2 ω2 þ u1z = 0: λ dz λ dz2

ð14:10Þ

The solution is sought in the form u1z = A expðbzÞ, where the complex number b = b1 + ib2. After substituting into Eq. (14.10), we find ω2 b21 þ 2ib1 b2 - b22 þ

g ω2 ðb þ ib2 Þ - k2 ω2 þ = 0: λ 1 λ

ð14:11Þ

Since at the boundaries the velocity turns to zero, the imaginary part should be b2 = πn/L, where is n an integer. As the frequency should be a real coefficient in front of the imaginary terms, it is zero, and, hence, b1 = - 1/(2λ). The characteristic equation is then given by ω2 b21 - b22 þ

ω2 g b - k2 ω2 þ = 0, λ 1 λ

ð14:12Þ

and increment, we find γ2 = - ω2 =

k2 g 1 λ k2 þ 1 2 þ 4λ

πn 2 L

:

ð14:13Þ

The real eigenfunctions are: u1z = C sin

πnz expð- z=2λÞ cos ky expðγt Þ: L

In particular, for short waves kλ >> 1, γ =

14.2

ð14:14Þ

g=λ.

Flute Instability

In fully ionized magnetized plasma, an analog of RT instability for fluids can develop. It is caused by gravitational force or inhomogeneity or curative of magnetic field. This instability of the RT family is called flute or interchange instability. → Consider plasma in a magnetic field parallel to x-axis, B kx, while gravitational force is directed along z-axis. Equilibrium is described by equation

14.2

Flute Instability

311

Fig. 14.2 Polarization of a perturbation for flute instability

-

dp0 - mi n0 g - j0y B = 0: dz

ð14:15Þ

Let us restrict ourselves to the case of small β when the magnetic field is assumed to be constant and independent of z. The current in the y-direction consists of the sum of the diamagnetic current and current j0g caused by the gravitational force. Current j0g is associated with the drift of ions (electron drift is negligible due to small mass): j0g = - mi gn0 =B. The instability mechanism for short waves that are localized in the z direction in a region much smaller than the plasma spatial scale is illustrated in Fig. 14.2. For periodic density perturbation in the y-direction, perturbed current j1g = - mi n1 g=B arises. To balance this current, an electric field arises, which produces a polarization ∂E 1 current in the opposite direction: j1p = nB0 m2 i ∂ty . The perturbed electric field rises with →

time. For short waves, the condition ∇  j = 0 is equivalent to the condition of zero perturbed current in the y-direction j1g þ j1p = -

1 mi n1 g n0 mi ∂Ey = 0: þ 2 B ∂t B

ð14:16Þ

The mechanism of electric field formation is similar to that discussed in Sect. 11.1. Perturbed electric field causes vertical drift with the velocity u1z = - E 1y =B:

ð14:17Þ

The linearized particle continuity equation considering that drift fluxes are diver→ → gence free (∇  E × B =B2 = 0) has the form

312

14

Instabilities in Magnetized Plasma

∂n1 ∂n þ u1z 0 = 0: ∂t ∂z

ð14:18Þ

For perturbations, Eq. (14.3) from Eq. (14.16), one obtains E1y =

iBg n1 : ω n0

ð14:19Þ

Substituting Eq. (14.17) and (14.19) into Eq. (14.18) with account of ∂n1/∂t = iωn1 yields - ω2 = g

d ln n0 : dz

ð14:20Þ

Hence, for d ln n0/dz > 0, plasma is unstable with increment γ=

ð14:21Þ

g=λ,

where λ = |d ln n0/dz|-1. In the opposite case, the frequency is real, and the plasma is stable. In the general case, flute instability is described by the equation system ∂n → þ ∇  n u e = 0, ∂t ∂n → þ ∇  n u i = 0, ∂t → → → dui → → nmi = - ∇pi þ en E þ en u i × B þ mi n g , dt →

→

ð14:22Þ

→

- ∇pe - en E - en u e × B = 0: From momentum balance equations, perpendicular velocities can be derived: → →

ue=

→ →

ui=

→

E×B B2

→

þ

→

E×B B2

-

∇pe × B enB2

→

∇pi × B enB2

, →

þ

→

mi B × d u i =dt eB2

→

þ

eB2

ð14:23Þ

→

mi g × B

:

Since diamagnetic fluxes are divergence free in a homogeneous magnetic field and drift fluxes are incompressible, the continuity equation is reduced to ∂n → þ u E ∇n = 0, ∂t

ð14:24Þ

14.2

Flute Instability →

→

313 →

→

where u E = E × B =B2 is the drift velocity. Choosing ∇  j = 0 as a second equation, one finds →

∇n

→

mi B × d u E =dt B2

→

þ

→

mi g × B B2

= 0:

ð14:25Þ

The expression for the polarization current velocity can be taken in the first approx→ → imation, i.e., only E × B drift can be substituted. The linearized equation coincides with Eq. (14.18) with account of Eq. (14.17). Linearized Eq. (14.25) is ik -

1 mi n1 g n0 mi ∂Ey þ 2 B ∂t B

þ

∂ n0 mi ∂E 1z = 0: ∂z B2 ∂t

ð14:26Þ

Using Eq. (14.18) for density perturbation, one has one equation for potential ω2

dn d dφ1 n0 = k 2 n 0 ω 2 þ g 0 φ1 : dz dz dz

ð14:27Þ

This equation coincides with Eq. (14.5) for fluid R-T instability, and, hence, stability analyses in plasma are the same as in fluids described in the previous section. Therefore, magnetized plasma is unstable if the gravitational force is directed opposite to the density gradient. In the laboratory plasma, much more important is the “effective gravity” associated with nonuniform and curvilinear magnetic fields (see Chap. 11). In a tokamak, the effective acceleration is given by Eq. (11.11). The instability increment remains of the order of that given by Eq. (14.21), where g is replaced by the effective acceleration g=

2ðT e þ T i Þ : mi R

ð14:28Þ

While developing instability, the wavelength along the magnetic field remains infinite, so perturbations have a flute shape, which gives rise to instability. Magnetic shear is a stabilizing factor for this instability and can suppress it. Indeed, perturbation with kk = 0 at a given z in a sheared magnetic field will have kk ≠ 0 at other z values. Then, positive and negative potential perturbations are “short-circuited” since they belong to the same magnetic field line. In magnetic traps, the situation can be rather complicated – effective acceleration can change sign at different parts of magnetic field lines. In a tokamak, for example, at the low-field side, the sign of acceleration corresponds to an instability increment, while at the high-field side, the signs of effective acceleration and density gradient coincide, which corresponds to stable equilibrium. To analyze stability in such

314

14

Instabilities in Magnetized Plasma

situations, let us consider a magnetic flux tube in plasma with a small β. Such a tube tries to increase its volume and, therefore, to shift to the region where the tube volume can become larger. Magnetic field line curvature should not rise during shift, otherwise magnetic energy would increase. For closed field line, the magnetic tube volume is V=

Sdl =

BS

dl =Ψ B

dl , B

ð14:29Þ

where Ψ is the magnetic flux, which is conserved along the tube and remains constant over time. We shall define quantity U as dl : B

U=

ð14:30Þ

For an infinite magnetic field line belonging to the flux surface, U = nlim →1

1 n

dl : B

ð14:31Þ

where n is the number of toroidal rotations. Since the tube tries to expand and hence to increase U, which is proportional to the tube volume, quantity -U might be considered an analog of the potential energy. Since integral U is a flux surface function similar to full pressure, one may consider p = p(U ). For a small shift of a magnetic flux tube without a change in the field line curvature in accordance with Eq. (14.29), δV/V = δU/U. For the adiabatic process, the pressure variation in the tube is δp = - 5=3pδV=V = - 5=3pδU=U: In the tube neighborhood, the ambient plasma density also changes. Since p = p(U ), we have pðU þ δU Þ = p þ

dp δU: dU

ð14:32Þ

If the ambient plasma pressure rises faster than the tube pressure, the tube returns to initial equilibrium. Hence, plasma would remain stable if 5 p dp >: 3 U dU

ð14:33Þ

Usually, it is possible to neglect r.h.s. of Eq. (14.33), especially in the vicinity of the plasma boundary, and the condition for plasma stability is reduced to

14.3

Dissipative Modifications of Flute Instability

dp > 0: dU

315

ð14:34Þ

In other words, plasma remains stable if inside plasma where pressure is rising, quantity U also has a maximum. Since the magnetic field in this region is smaller than that outside, it is usually said that plasma should stay in the “magnetic pit” for stability. In a tokamak, a “magnetic pit” exists due to the Shafranov shift of the inner flux surfaces towards the low-field side. Hence, tokamak plasma is stable with respect to fluid instabilities. However, ballooning modes can develop in a tokamak. This is a modification of flute instability where perturbation is a function of poloidal angle. On the low-field side, the region of unfavorable curvature instability tends to grow, while on the highfield side, the region of favorable curvature instability is suppressed. Finally, the resulting instability occurs for a large plasma pressure gradient when it exceeds some threshold value.

14.3 14.3.1

Dissipative Modifications of Flute Instability RT Instability in Partially Ionized Plasma

In partially ionized plasma, a dissipative analog of flute (RT) instability exists. It is also caused by gravitational force as well as magnetic field curvature. Let us consider the geometry shown in Fig. 14.2 and perform analysis in local approximation. Plasma is assumed to be magnetized so that xexi >> 1, Chap. 5. In partially ionized plasma, as in fully ionized plasma, one gravitational force causes current in the ydirection. For a periodic perturbation along y, the current perturbation equals j1g = - biΛ mi n1 g:

ð14:35Þ

For magnetized ions xi >> 1, the Hall mobility is biΛ = 1/B, and this current coincides with that in fully ionized plasma. However, in contrast to the fully ionized plasma case, this current is balanced not by the polarization current but by the ion conductivity current provided that the ion-neutral collision frequency is sufficiently large. Instead of Eq. (14.16), we have - biΛ mi n1 g þ en0 bi⊥ E 1y = 0:

ð14:36Þ

Here, the current generated by diffusion en1Di⊥k2 is neglected, which is justified for (in the case of magnetized) ions

316

14

Instabilities in Magnetized Plasma

T i k2 νiN 0, the plasma is unstable with increment γ=

g : νiN λ

ð14:40Þ

RT in partially ionized plasma is responsible, for example, for bubble formation in the equatorial ionosphere.

14.3.2

Flute Instability in Plasma Contacting Metal Surfaces

If fully ionized plasma is restricted by highly conductive surfaces perpendicular to the magnetic field, its polarization is to a large extent short-circuited. However, even in collisionless plasma due to the finite conductivity of the boundary sheath, polarization can survive, and plasma could be unstable. Plasma is supposed to be restricted in the x-direction by conductive surfaces. In the y-direction, the perturbed current is still described by Eq. (14.35), but now it could be short-circuited by parallel currents along the x-axis. Integrating equation * ∇  j = 0 over x between two surfaces, assuming symmetry over x, we have ikI 1g þ 2j1k = 0:

ð14:41Þ

Here L

I 1g

g =mi B

n1 dx = 0

g 1 N mi B

ð14:42Þ

14.3

Dissipative Modifications of Flute Instability

317

is the integral perturbed current, and j1k is the parallel current density at surface x = L. To provide current to the surface, the potential should be perturbed with respect to the floating potential. Potential perturbation corresponds to sheath current-voltage characteristics Eq. (3.26). For floating unperturbed potential, linearized potential perturbation is proportional to the current density j1k = ens cs

eφ1 , Te

ð14:43Þ

where cs = (Te/mi)1/2 is the sound speed velocity and ns is the density at the sheath entrance. Due to high conductivity, potential perturbation is transported along the magnetic field and is independent of x. In addition to potential Eq. (14.43), Boltzmann’s potential φB = (Te/e) ln n exists in the plasma to provide parallel → → momentum balance for electrons. The E × B drift caused by this potential is divergence free and does not contribute to particle balance. Particle balance Eq. (14.18) has the form - iωn1 þ

ickφ1 dn0 = 0: B dz

ð14:44Þ

Let us assume for simplicity that unperturbed density could be factorized n0 = n0x(x) n0y(z), so that |d ln n0/dz|-1 = λ is independent of x. Then, since φ1 = const(x), the perturbation density profile in the x-direction repeats the unperturbed profile n0(x). The combination of Eqs. (14.41)–(14.44), hence, yields ω=

i 2 2 Lx d ln n0 , k ρci g cs dz 2

ð14:45Þ

where L

Lx =

n0 dx=ns , ρci =

ðT e =mi Þ1=2 : ωci

0

For d ln n0/dz > 0, the plasma is unstable with increment γ=

g 2 c , νeff = 2 2 s : νeff λ k ρci Lx

ð14:46Þ

318

14

Instabilities in Magnetized Plasma

14.3.3 Gravitational-Dissipative Flute Instability This modification of flute instability is connected with finite conductivity along the → magnetic field and finite wave length along B . This instability becomes important when flute instability with kk = 0 cannot develop but local conditions for plasma polarization, such as unfavorable magnetic field curvature, exist. The analysis is performed analogously to previous cases, but in the current balance, the parallel conductivity current is taken into account. The resulting frequency satisfies the equation ω2 þ iωs ωg þ ω2g = 0,

ð14:47Þ

where ωs =

k 2x ωci ωce , k 2y 0:51νei

ω2g = -

g : λ

ð14:48Þ

For ωs > ωg ω2g 0 for all possible shifts, plasma equilibrium

14.5

Kink Instability

321

remains stable. Therefore, the energy principle provides an opportunity to analyze the sign of δW to predict plasma stability for MHD perturbations.

14.5

Kink Instability

For flute instability, the source of energy is plasma thermal energy, while for kink instability, magnetic field energy serves as energy storage. Below, several examples of kink instability are presented. Let us consider the plasma cylinder to be infinite in the z direction. The current flows along z, and a parallel magnetic field is absent, so this is the z-pinch considered in Chap. 12. Such pinch can be unstable with respect to radial perturbations ξ = Aðr Þ expð- iωt þ imθ þ ikzÞ:

ð14:67Þ

Here, m = kθr integer number due to periodicity over azimuthal coordinates. The plasma element shift ξ here takes place in the radial direction. For m = 0, the perturbation has azimuthal symmetry and is of the constriction type (Fig. 14.3). In the region of constriction, where the plasma radius decreases, the azimuthal magnetic field increases since the full current I is conserved and Bθ~r-1. Hence, the Ampere force IBθ rises, pinch contracts, and perturbation further develops. Plasma pressure also rises but not as quickly since pressure can equalize along pinch, so it cannot stop contraction.

Fig. 14.3 Constrictions onset in z-pinch

322

14

Instabilities in Magnetized Plasma

To study kink instability with m = 0, one can use the energy principle. Let us assume that pinch is restricted in the radial direction by the metal wall to exclude the vacuum region from consideration. Stability condition δW > 0 Eq. (14.66) is simplified for m = 0. After integrating by parts in Eq. (14.66), one finds δW =

5 0 B20 p þ μ0 3

→

2

∇ ξ

-

→ 4B20 dp0 B20 2 ξ∇  ξ þ ξ2 þ μ0 r dr 2πr r

→

d r > 0: ð14:68Þ

Here →

∇ ξ =

1 d ðrξÞ: r dr

Expression under the integral can be rewritten in the following way: 5 0 B20 p þ μ0 3

δW =

þ

-

→

∇ ξ

4B20 2 2 ξ μ0 r

B2 5 4 p0 þ 0 μ0 3

-

2

4B20 μ0 r

2

5 0 3p

B2

þ μ0

ξ

→

dr

0

ð14:69Þ 0

2B20

dp 2 þ ξ2 þ dr μ0 r r

→

d r > 0:

The first integral is positive, so Eq. (14.69) is satisfied if the second integral is also positive, and, hence, -

10=3 r dp0 , < p0 dr 1 þ 56 β

ð14:70Þ

where β = 2μ0 p0 =B20 . Therefore, inequality (14.70) should be satisfied to stabilize the kink mode m = 0. The longitudinal magnetic field stabilizes mode m = 0 since magnetic field lines during contraction are compressed together with plasma and magnetic pressure increases. However, on the other hand, mode m = 1 can become unstable. Let us consider instability for the m = 1 mode and thin current carrying pinch in the longitudinal magnetic field. The perturbed pinch with m = 1 is shown in Fig. 14.4. Since the current flows along the perturbed pinch, the current’s azimuthal component arises, and the radial Lorentz force jθBz leads to instability onset. The azimuthal current depends both on the radial displacement and on the helix pitch distance along z. The radial momentum balance reads

14.5

Kink Instability

323

Fig. 14.4 Instability of thin current-carrying pinch for m = 1 mode

ρ

d2 ξ = jθ Bz = jk Bz kξ: dt 2

ð14:71Þ

Hence, increment γ=

IkBz πa2 ρ

1=2

:

ð14:72Þ

Expressing current through the azimuthal magnetic field, one obtains γ=

cA ð2ΘkaÞ1=2 , a

ð14:73Þ

where a is the pinch radius, Θ = Bθ/Bz, and cA = Bz/(μ0ρ)1/2 is the Alfven velocity calculated for magnetic field Bz. Therefore, the characteristic time for kink instability onset is determined by the Alfven time, in contrast to flute instability, where it is determined by the sound speed time. To obtain the stability criterion for a cylinder of finite radius with respect to mode m = 1, we consider a simplified model. Let us assume that a helical perturbation is added that does not change the cylinder shape. Two cross-sections separated by a quarter of the wavelength are shown in Fig. 14.5. Assuming that at z = 0 pinch is shifted to the left (dotted line), then at distance z = λ/4, pinch should be shifted downwards. Let us consider the situation when the magnetic field line at distance z = λ/4 turns on an angle larger than π/2. Since the magnetic field is frozen into the plasma, the magnetic field line turns at the same angle with respect to the center of the perturbed pinch (dotted line). At the same time, with respect to the unperturbed pinch, the magnetic field line turns on a larger angle. To provide rotation on a larger angle, the magnetic field at the upper part of the pinch should rise with respect to the equilibrium angle. The magnetic pressure in the upper part also increases, and, therefore, the pinch is further shifted downwards, which leads to instability onset.

324

14

Instabilities in Magnetized Plasma

Fig. 14.5 Displacements of pinch cross-sections separated by λ/4 in the process of kink instability onset for m = 1

Fig. 14.6 Lines of constant phase for radial displacement and magnetic field lines in the plane (z, θ). Situation corresponds to instability onset

In the opposite case, when the magnetic field line turns on an angle smaller than π/2 for length λ/4, the pinch is stable. The safety factor for the cylinder can be defined as qð aÞ =

Bz 2πa Bz ka : = Bθ λ Bθ

ð14:74Þ

Therefore, pinch remains stable for q > 1: This inequality is known as the Kruskal-Shafranov condition.

ð14:75Þ

14.5

Kink Instability

325

For mode m ≥ 2, one can use the following arguments. Let us unveil the pinch surface as a rectangle, as shown in Fig. 14.6. The vertical axis corresponds to coordinate z, and the horizontal axis represents azimuth angle θ. Radial perturbations shift the plasma boundary, and a rippled surface is formed (Fig. 14.6). The longitudinal pinch current also shifts radially together with plasma and current displacements, leading to an increase in perturbations of the magnetic field radial component B1r . A time-dependent perturbed radial magnetic field induces an electric field. Due to very high plasma conductivity, this inductive electric field should be practically perpendicular to the magnetic field with an almost zero parallel component. The → → induced electric field in turn produces radial E × B drift of the plasma element. The magnetic field perturbation and displacement are connected according to Eq. (14.52) as B1r  iBz ξkk ≈ iξðkBz - mBθ =aÞ. Here, the sign in the brackets should be negative to correspond to an outwards current shift. In other words, the induced electric field should cause outwards plasma drift, further increasing displacement (Fig. 14.6). In the opposite case, plasma remains stable. Hence, the stability criterion is q > m. For a cylinder of finite length L, the wave vector k = 2πn/L. In particular, for a torus with radius R, we have k = n/R. In this situation, the stability criterion has the form qð aÞ =

m Bz a > : n Bθ R

ð14:76Þ

The condition (14.76) should be satisfied at the plasma boundary. If the current density is constant as a function of radius, the azimuth magnetic field increases linearly with radius, and the q value is independent of radius. In this model case, Eq. (14.76) is satisfied for the whole plasma volume. For any realistic current profile, the safety factor depends on the radius. For the standard current density distribution, when the current density decreases with radius, q(r) rises with radius. Hence, inside the plasma pinch, many flux surfaces exist with radius r = rres, where qðr res Þ =

m : n

ð14:77Þ

The corresponding flux surfaces are known as resonance flux surfaces. At the resonance surface, the kink mode is neutral stable, but at different sides, it is either stable or unstable, since due to dependence q(r) condition q(r) > m/n is satisfied at one side and is not satisfied on the other side. Stability in such situations depends on the q derivative, on the rate of q variation with radius, in other words, on magnetic field shear. The required shear can be estimated for the following reasons. Let us consider a mode localized in the vicinity of the resonance flux surface. Pressure perturbation according to Eq. (14.56) is p1~ - ξdp/dr. Pressure perturbation and magnetic field curvature RS = rBz/Bθ produce a destabilizing force F1  -

dp ξ , dr Rs

ð14:78Þ

326

14

Instabilities in Magnetized Plasma

which tries to accelerate plasma outwards. This force is balanced by the stabilizing force associated with magnetic field deformation, which can be estimated as →

F2 =

→

B∇ B μ0



iBz kk B1r : μ0

ð14:79Þ

Using Eq. (14.52), B1r  iBz ξkk , and expansion kk in the resonance flux surface vicinity Eq. (8.92) k k ðr Þ = k θ Θðr - r res Þ=r res , where Θ is the magnetic shear introduced in Chap. 8, assuming kθ(r - rres)~1, from F1 = F2, we obtain 2

B2z Θ - dp=dr < : B2z =ð2μ0 Þ B2θ r

ð14:80Þ

This inequality is known as the Suydem stability criterion. Therefore, sufficiently strong magnetic shear can suppress plasma instabilities localized in the vicinity of resonance flux surfaces. A similar Mercier criterion was obtained for toroidal geometry.

14.6 Tearing Instability As shown in the previous section, MHD instabilities localized in the vicinity of resonance flux surfaces can be stabilized by magnetic field shear. However, in the vicinity of resonance flux surfaces, another special instability can develop, which is connected with a change in magnetic field topology. This is known as the tearing mode. This instability is connected with finite conductivity and is accompanied by the formation of magnetic islands – regions with different magnetic field topologies near resonance magnetic flux surfaces. Let us analyze the tearing instability for the plasma slab in Fig. 14.7. The unperturbed magnetic field B0y ðxÞ is directed along the y-axis and is a function of the x-coordinate (linear, for example). At x = 0, the magnetic field turns to zero,

Fig. 14.7 Unperturbed magnetic field near neutral layer

14.6

Tearing Instability

327

which corresponds to the neutral layer. Such a magnetic field is created by a current sheet with j0z = ð1=μ0 Þ∂B0y ðxÞ=∂x. Plasma equilibrium is supposed to be satisfied, so that -

∂p0 þ B0y j0z = 0: ∂x

ð14:81Þ

We consider the perturbation periodic in the y-direction and independent of z: → B1x  expð- iωt þ ikyÞ. However, according to equation ∇  B = 0, perturbed are both components of the magnetic field, which are connected by ∂B1x þ ikB1y = 0: ∂x

ð14:82Þ

At some distance from the resonance flux surface, magnetic field lines are perturbed, as shown in Fig. 14.8, without changing topology. The perturbed timedependent magnetic field induces an electric field in the z-direction: ∂E 1z ∂B1x =: ∂t ∂y

ð14:83Þ

Since tearing instability is aperiodic with only an imaginary part of the frequency, the perturbed electric field is shifted in phase by π/2 with respect to the B1x perturbation. Electric field causes plasma drift along the x-axis with the velocity u1x = -

E1z : B0y

ð14:84Þ

The drift velocity is directed towards the neutral layer x = 0 in the regions where magnetic field lines are compressed and away from it in the regions of magnetic field line divergence. At the neutral layer, the velocity Eq. (14.84) becomes infinity when the unperturbed magnetic field goes to zero. Hence, in the vicinity of the neutral layer, it is necessary to take into account finite conductivity, and instead of Eq. (14.84), one should use the electron momentum balance along the z-axis: E1z = - u1x B0y þ ηj1z ,

ð14:85Þ

where resistivity η = σk- 1 = 0:51 me νei =ne2 . Currents in the z-direction flow in a thin layer in the vicinity of x = 0, where the magnetic field is close to zero. They create a magnetic configuration with closed field lines – magnetic islands. Indeed, a positive magnetic field perturbation B1x near x = 0 shifts the magnetic field line to positive values of x. While moving along x, the unperturbed magnetic field increases,

328

14

Instabilities in Magnetized Plasma

Fig. 14.8 Magnetic island formation

and the magnetic field line starts going in the y direction. The phase of the perturbed magnetic field B1x starts to change, and, hence, the magnetic field line returns to the neutral line x = 0 and then shifts further to the region of negative x. Here, the unperturbed magnetic field changes sign and shifts the magnetic field line in the opposite -y direction. The phase of the magnetic field perturbation B1x changes again, and, as a result, the magnetic field line performs a full turn, forming a magnetic island (Fig. 14.8). Far away from the neutral line, magnetic field perturbation B1x leads only to deformation of the magnetic field line without changing topology. A line separating closed- and open-field lines is known as the separatrix. In the vicinity of the neutral line, plasma drift motion in the x-direction caused by E 1z has different signs for positive and negative x (to the neutral line or away from it) due to different signs of the unperturbed magnetic field. Hence, here, the plasma motion along the x-axis should transform into motion along the y-axis. → Plasma motion is assumed to be incompressible (∇  u = 0), so that

∂u1x þ iku1y = 0: ∂x

ð14:86Þ

Linearized Eq. (10.14) for magnetic field perturbation is reduced to 2

∂B1x η0 ∂ B1x = iku1x B0y þ - k 2 B1x : μ0 ∂x2 ∂t

ð14:87Þ

This equation contains a small coefficient η0 in front of a high derivative. Far from the neutral layer, the last term on the r.h.s. is small, but while approaching x = 0, the second derivative becomes large, and the term ∂B1x =∂t is balanced by the second 2 derivative ∂ B1x =∂x2 multiplied by the small coefficient η0. Looking from large 2 distances, the second derivative ∂ B1x =∂x2 has a break, while in reality, the first

14.6

Tearing Instability

329

derivative changes quickly inside a boundary resistive layer with width 2ε. Let us introduce a quantity Δ0 =

dB1 dB1x ðþεÞ - x ð- εÞ =B1x ð0Þ: dx dx

ð14:88Þ

The value of Δ′ is determined by the solution of MHD equations in the outer region with corresponding boundary conditions. Inside the resistive layer, the second derivative can be estimated as ∂ B1x Δ0 B1x  : ε ∂x2 2

ð14:89Þ

Inside the resistive layer, all terms of Eq. (14.87) are of the same order. Introducing increment γ, one obtains 2

γB1x 

c2 η0 ∂ B1x : μ0 ∂x2

ð14:90Þ

c2 η 0 Δ 0 : μ0 ε

ð14:91Þ

From (14.89) and (14.90), we have γ

Another relation between the increment and resistive layer width could be obtained from the momentum balance equation. In the x-direction j1z B0y , the force is balanced by the pressure gradient ip1/ε: j1z B0y  ip1 =ε:

ð14:92Þ

Pressure perturbation accelerates plasma along the layer in the y-direction, and ikp1  ρ0 γu1y 

ρ0 γu1x : kε

ð14:93Þ

We used Eq. (14.86). It is supposed that the resistive layer is thin, so that kε ≪ 1,

ð14:94Þ

and since according to Eq. (14.86), plasma is incompressible, u1y ≫ u1x -plasma is strongly accelerated along the layer. After eliminating the pressure perturbation from Eqs. (14.92) and (14.93), we find

330

14

j1z 

ρ0 u1x γ ðkεÞ2

Instabilities in Magnetized Plasma

:

ð14:95Þ

From the Maxwellian equation, we have 1 ∂ B1x B1x Δ0  , μ0 kε μ0 k ∂x2 2

j1z 

ð14:96Þ

and a combination of Eqs. (14.96) and (14.97) yields γρ0 u1x 

B1x Δ0 kεB0y : μ0

ð14:97Þ

The unperturbed magnetic field can be estimated as 0

B0y  B0y ε:

ð14:98Þ

0

Substituting estimate γB1x  ku1x B0y  ku1x B0y ε in accordance with Eq. (14.87) into Eq. (14.97), after the elimination velocity, one finds

γ2 

B0y

0 2

k 2 ε3 Δ 0

μ0 ρ 0

:

ð14:99Þ

Combining Eqs. (14.99) and (14.91), we find an increment of resistive tearing instability

γ

B0y

0 2=5

k2=5 ðΔ0 Þ

4=5

ðμ0 Þ4=5 ðρ0 Þ1=5

ðη0 Þ

3=5

:

ð14:100Þ

Let us denote the typical scale of plasma parameter variation along x as L and introduce two characteristic times 1=2

τA =

Lðμ0 ρ0 Þ L2 μ0 L = ; τ = , s 1=2 cA η0 B0y

ð14:101Þ

where τA is the Alfven time and τs is the skin time. Then, Eq. (14.100) may be rewritten in terms of τA and τs, giving

14.7

Geodesic Acoustic Mode and Zonal Flows

331

- 2=5 - 3=5 τs :

γ  τA

ð14:102Þ

Therefore, resistive tearing instability grows on a time scale that is intermediate between a short MHD time scale τA and a long skin time scale τs. The resistive layer width may be obtained from (14.99): ε  L

τA τs

2=5

≪ 1:

ð14:103Þ

The tearing instability grows exponentially until the magnetic island width becomes comparable to the resistive layer width ε, after which nonlinear effects become important. In a cylinder or torus, tearing instability can develop around resonance flux surfaces, q = m/n; see the next chapter. In a tokamak, the so-called neoclassical tearing mode can also exist, and the latter is caused by bootstrap current variation due to the formation of magnetic islands.

14.7 Geodesic Acoustic Mode and Zonal Flows In a tokamak-specific oscillation of density, poloidal and parallel velocities and a radial electric field are observed, known as the geodesic acoustic mode (GAM). We shall consider GAM for a tokamak with a simple circular cross section, Sect. 13.1. Let us assume that in addition to the stationary radial electric field given by Eq. (13.62), an oscillating perturbed radial electric field exists. Corresponding perturbed potential G φG 0 ðr Þ = A ðr Þ expð- iωt Þ

ð14:104Þ

is independent of the poloidal coordinate. Amplitude AG(r) is an arbitrary function of radius, so the radial scale of this mode remains unspecified. Perturbed radial electric → → G field E G 0 = - dφ0 =dr causes poloidal E × B drift in addition to stationary drift with G velocity V G 0 = - E 0 =B. Corresponding poloidal flows are known as zonal flows analogously to zonal flows in planet atmospheres. In a nonuniform magnetic field, velocity V G 0 has finite divergence, which produces close parallel plasma fluxes similar to the formation of Pfirsch-Schlueter fluxes (Fig. 14.9). These parallel fluxes, in contrast to the Pfirsch-Schlueter fluxes, periodically change their direction, and their amplitude is proportional to cosθ. The parallel velocity, hence, is a uG ik ðθ Þ = uik cos θ,

ð14:105Þ

332

14

Instabilities in Magnetized Plasma

Fig. 14.9 Poloidal velocities and density perturbations in geodesic acoustic mode a with uik being a periodic function of time. These parallel fluxes are driven by a parallel pressure gradient, so periodic pressure perturbation proportional to sinθ should exist, namely,

a nG 1 = n1 sin θ:

ð14:106Þ

Therefore, the perturbed radial electric field has a poloidal wavenumber m = 0, and the density and parallel velocity perturbations correspond to m = 1. The linearized particle balance equation has the form - iωnG 1 -

n0 Θuak 2ε sin θ - n0 V G 0 sin θ = 0: r r

ð14:107Þ

Here, the second term represents divergence of the poloidal projection of parallel flux, and the third term corresponds to divergence of the poloidal flux associated with the poloidal rotation. The parallel momentum balance is given by - iωmi n0 uaik cos θ = - Θ

Te þ Ti a n1 cos θ: r

ð14:108Þ

We assume that species temperatures remain unperturbed due to large heat conductivity and heat exchange. To determine the radial electric field, one needs the current balance equation. In particular, it is sufficient to put the net current integrated over the flux surface to zero. Due to density perturbations, the vertical ∇B-driven current integrated over the flux surface remains finite and is equal to 2π

hhj∇B ii4π2 rR = - 4π2 rR 0

4π2 rðT e þ T i Þna1 2ðT e þ T i Þ G n1 sin θdθ = : BR B

ð14:109Þ

14.8

Equatorial Plasma Bubbles

333

This current is compensated by the polarization current caused by radial electric field temporal variation: 4π2 rRhhjP ii = - 4π2 rR

mi n0 ∂V G 0 : B ∂t

ð14:110Þ

From the condition of zero radial current hhj∇Bii + hhjPii = 0, one obtains c2s ε nG 1 - iωV G 0 = 0, r n0 where cs = amplitudes

ð14:111Þ

ðT e þ T i Þ=mi . From Eqs. (14.107) and (14.108), we have for na1 Θuk 2ε G þ þ V 0 = 0, n0 r r 2 a Θc n - iωuaik þ s 1 = 0: r n0 a

iω

ð14:112Þ

By setting the determinant of the system (14.111) and (14.112) to zero, we obtain the dispersion relation for frequency, whose solution is ω2 =

c2s 1 2þ 2 : q R2

ð14:113Þ

In the approximation considered, GAM has a real frequency of the order of ion bounce frequency. According to nonlinear analysis, GAM has increment or decrement due to nonlinear interaction with drift ways and can stay in a dynamic balance with them.

14.8

Equatorial Plasma Bubbles

Plasma bubbles that occur in the equatorial ionosphere are plasma depletion regions with respect to the background ionosphere density. They are observed in the equatorial Earth F-region of the ionosphere at heights, where ionospheric plasma density increases with height. Such depletions can be directly measured in situ by satellite probes. Their formation is a result of the nonlinear development of Rayleigh-Taylor instability in the partially ionized plasma considered in Sect. 14.3.1. Indeed, in the equatorial ionosphere, Earth magnetic field lines are parallel to the Earth surface and hence are perpendicular to the density gradient. Below the Fregion maximum, the density gradient is directed upwards and is antiparallel to the gravity force mig. This bottom layer is unstable due to gravitational RT instability

334

14

Instabilities in Magnetized Plasma

Fig. 14.10 Contour plots of density perturbations at 2000 s, 5000 s, and 10,000 s. The plus and minus signs indicate the enhancements and depletions of the electron density relative to the background. The dashed curve represents the electron density profile of the background ionosphere

because it is located below the heavy layer. An increment is given by Eq. (14.40). A perturbation initiated in the bottom layer rises like a bubble and penetrates the top layer (layer above the F-region maximum). Electron density irregularities exist along the trace of the bubble, as well as within and around the rising bubble. Examples of bubble evolution obtained in the simulations are shown in Fig. 14.10.

Chapter 15

Magnetic Islands and Stochastic Magnetic Field

15.1

Magnetic Islands

In the previous chapter, it was demonstrated that tearing instability leads to the formation of magnetic islands. The same configuration can be created by magnetic field perturbations caused by currents in the external coils. We shall study the magnetic island structure in more detail independently of the mechanism of their formation. Let us consider the same magnetic field B0y ðxÞ as in the previous section, Fig. 14.7, i.e., changing sign at x = 0. Near x = 0, magnetic field is a linear function 0

B0y ðxÞ = B0y x:

ð15:1Þ

In the general case, in addition to the magnetic field B0y ðxÞ, there exists a magnetic field component B0z ; here, for simplicity, we shall assume B0y ðxÞ ≪ B0z . Magnetic field perturbations are supposed to be stationary and contain the sum of harmonics B1x =

ky , k z

B →k exp ik y y þ ik z z :

ð15:2Þ →

First, we consider the case B0z = 0 and kz = 0. Due to condition ∇  B = 0, it is possible to introduce a magnetic flux function for the full perturbed magnetic field as Bx = -

∂Ψ , ∂y

By =

∂Ψ : ∂x

ð15:3Þ

For a single harmonic with real amplitude B1x = B →k sin k y y

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Rozhansky, Plasma Theory, https://doi.org/10.1007/978-3-031-44486-9_15

335

336

15

Ψ = Ψ0 þ

Magnetic Islands and Stochastic Magnetic Field

1 0 0 2 B →k cos ky y: B x þ ky 2 y

ð15:4Þ

One must keep in mind that both components of the magnetic field B1x and B1y are perturbed since perturbation B1x is a function of x. Line Ψ = const coincides with the magnetic field line (Fig. 14.9). Indeed, along this line, dΨ = dx∂Ψ/∂x + dy∂Ψ/ ∂y = 0, and, therefore, according to Eq. (15.3), along this line, -Bydx + Bxdy = 0. The latter equation coincides with the equation for the magnetic field line. According to Eq. (15.4), at the X-point at the separatrix with coordinates x = 0, y = 0, the value of the flux function is Ψ = Ψ0 þ

B →k : ky

ð15:5Þ

At the O-point at the separatrix with coordinates x = W/2, y = π/ky we have Ψ = Ψ0 þ

1 0 B 2 y

0

2

W 2

-

B →k : ky

ð15:6Þ

Here, W is the maximal width of the separatrix (island width). From Eqs. (15.5) and (15.6), the island width is determined: 1=2

W =4

B →k ky B0y

:

0

ð15:7Þ

In the presence of field B0z , the magnetic field line goes mainly along the z-axis. For a single harmonic with kz = 0, the magnetic field line projection to the xy plane acts in the same way as for B0z = 0, forming magnetic islands in the vicinity of x = 0. For harmonics B1x = B →k exp ik z z þ ik y y with finite wave vectors ky and kz, plane x = x0 is of special character since the magnetic field perturbation phase remains constant along the unperturbed magnetic field line. The plane x = x0 is called the resonance plane. Phase kzz + kyy = 0 is constant along the straight line z = - (ky/kz) y. The equation for the unperturbed magnetic field line is given by the straight line 0

z = yBz0 = B0y x0 . These two lines coincide when x0 = -

k z B0z ky B0y

0

:

ð15:8Þ

Since wave vectors can have the same signs or different ones, the value x0 could be positive or negative.

15.1

Magnetic Islands

337

Fig. 15.1 Magnetic islands in the vicinity of resonance flux surfaces

Similar to case B0z = 0, the magnetic field line moves together with the unperturbed line, making turns around it. For an observer moving along an unperturbed magnetic field, its end forms an island. Island formation is caused by a change in the magnetic field y-component when the magnetic field line is shifted along x and a further change in the perturbation B1x phase. The projection of the magnetic field line along the unperturbed line on the plane xy forms the structure shown in Fig. 15.1. The island width in the general case is still given by Eq. (15.7). In the presence of several harmonics, a system of magnetic islands arises, and each island is formed in the vicinity of its resonant flux surface in accordance with Eq. (15.8). In a tokamak, magnetic islands are formed in a similar way around resonant flux surfaces. Let, for circular magnetic flux surfaces, us consider radial magnetic field perturbations, namely, B1r =

m, n

Bmn expðimθ - inζÞ:

ð15:9Þ

Resonant magnetic surfaces are defined by Eq. (14.77), qðrres Þ =

m : n

ð15:10Þ

338

15

Magnetic Islands and Stochastic Magnetic Field

The role of the magnetic field B0y changing with x plays poloidal magnetic field B0θ changing with radius. One can introduce a poloidal magnetic field additional to that on the resonant flux surface: 0 0 B0 θ = Bθ ðr Þ - Bθ ðr res Þ:

ð15:11Þ

It is connected to the corresponding flux as  B0 θ = ∇Ψ × ∇ζ:

ð15:12Þ

Here, in contrast to Eq. (12.91), 2π is included in the flux function. The resonant harmonic forms a magnetic island due to the radial poloidal magnetic field dependence in Eq. (15.11), analogous to the slab geometry. The magnetic field line turns in space around the unperturbed magnetic field line. Analogous to Eq. (15.3), the flux function for the net magnetic field is introduced for a single harmonic Eq. (15.9): Ψ = Ψ ðr Þ þ Ψr cosðmθ - nζÞ:

ð15:13Þ

Phase ky + kzz in the slab geometry is now replaced by phase mθ - nζ. The perturbed field B1r is obtained as B1r = -

∂Ψ : r∂ðθ - ζ=qÞ

ð15:14Þ

As in the slab case, the flux function can be expanded in the vicinity of the resonant flux surface Ψ = Ψ ðr res Þ þ

1 d 2 Ψ ðr Þðr - r res Þ2 þ Ψr cosðmθ - nζÞ: 2 dr 2 res

ð15:15Þ

Similar to the slab case, one obtains W = 4r

Bmn q mBθ rq0

1=2

ð15:16Þ

In the general case, radial magnetic field perturbations lead to the formation of magnetic island chains near resonance flux surfaces given by Eq. (15.8) for the slab case and by Eq. (15.10) in a toroidal geometry.

15.2

Stochastic Instability and Magnetic Field Line Diffusion

339

15.2 Stochastic Instability and Magnetic Field Line Diffusion In the presence of several magnetic field harmonics with sufficient amplitudes, the magnetic field can become stochastic. This occurs when the distance between neighboring flux surfaces Δx0 = x0 k y , kz - x00 k 0y , k0z Þ (or Δr res = jr res ðm, nÞ - r 0res ðm0 , n0 Þj in the cylindrical geometry) becomes smaller than the single island width. This condition of stochastization is known as Chirikov criterion

Δx0
Lc, and (c) random walk for l ≫ Lc

340

15

Magnetic Islands and Stochastic Magnetic Field

general theory, motion becomes unstable according to the Kolmogorov-ArnoldMoser theorem. The so-called Kolmogorov length Lc exists, so that for l > Lc, magnetic field lines start to diverge exponentially. This phenomenon is known as stochastic instability. At larger distances, ergodization starts – a random walk of the magnetic field line ends in the perpendicular plane. The flux tube cross-section starts to deform strongly, stretching in one direction and squeezing in the other direction, since the tube cross-section area should be conserved due to magnetic flux conservation (Fig. 15.2b). With a further increase in distance l, the magnetic tube crosssection becomes rather complicated, as shown in Fig. 15.2c. The length of the “sleeves” increases and their width decreases to maintain a constant cross-sectional area. The average square of radial displacement, in particular displacement in the xdirection, at l > Lc is proportional to distance l in accordance with random walk displacement character: ðΔxÞ2 = 2Dst l:

ð15:18Þ

Here, quantity Dst is the magnetic field line stochastic diffusion coefficient. To calculate the stochastic diffusion coefficient in the presence of magnetic field perturbations, one must first analyze the magnetic field line equations for the perturbed magnetic field line. In the slab geometry dx Bx = ; B dl

dy By x = = , dl B a

ð15:19Þ

0

where a = B0z = B0y . Integration of the second equation along the magnetic field line, which at l = 0 corresponds to x = x0 and y = 0, yields x

xl 1 y= 0 þ a a

x0 dl:

ð15:20Þ

x0

For a single harmonic, which satisfies Eq. (15.8), inserting Eq. (15.20) into the perturbation phase, one obtains the first of Eq. (15.19) in the form ky dx B →k = 0 exp i a dl Bz

x

x0 dl :

ð15:21Þ

x0

Now, we demonstrate that Eq. (15.21) is equivalent to the Newton equation for charged particle motion in the field of electrostatic longitudinal waves. For periodic waves, we have

15.2

Stochastic Instability and Magnetic Field Line Diffusion

341

Fig. 15.3 Phase plane V, x

d2 x dV e = = E 0 cosðωt - kxÞ: 2 dt m dt

ð15:22Þ

In the reference frame moving with wave phase velocity, x=

ω t þ x, k

ð15:23Þ

The equation of motion can be written as e dV e = E 0 cos kx = E 0 cos k m dt m

Vdt :

ð15:24Þ

Equation (15.24) is equivalent to Eq. (15.21), and the variables in Eq. (15.21) correspond to variables for the problem of particle motion in the following way: l $ t; x $ V;

B →k B0z

$

ky e $ k: E ; m 0 a

ð15:25Þ

The particle motion character depends on its energy. If the particle kinetic energy in the wave framework is small, the particle is trapped between the wave potential hills. Particles with large kinetic energy remain untrapped with the velocity modulated by wave potential. Various behaviors are illustrated by the phase plane in Fig. 15.3. The boundary velocity corresponds to the separatrix and is found from the condition 2 mV c =2 = 2eφmax , so Vc = 2 and separatrix width

eφmax m

1=2

=2

eE0 mk

1=2

,

ð15:26Þ

342

15

Magnetic Islands and Stochastic Magnetic Field

ΔV = 2V c = 4

eE 0 mk

1=2

:

ð15:27Þ

Equation (15.27) corresponds to the width of the magnetic island given by Eq. (15.7) with account of similarity in Eq. (15.25). If several waves act on charged particles, then in the phase plane, several zones of finite motion arise for particles trapped in each wave with width Eq. (15.27). In the case of islands, the overlapping motion of a single particle becomes stochastic. In this situation, in accordance with ergodic theory, the description of a single particle is equivalent to the description of an ensemble of particles with their distribution function. The corresponding kinetic equation is given by ∂f ∂f eE1 ∂f = 0, þV þ m ∂V ∂t ∂x

ð15:28Þ

where the small electric field is a sum of many harmonics E1 =

E k expð- iωk t þ ikxÞ:

ð15:29Þ

k

In the quasilinear approximation, a solution is sought in the form f = f 0 þ f 1;

f1 =

f k expð- iωk t þ ikxÞ:

ð15:30Þ

k

In the quasilinear approximation, the interaction between different harmonics is neglected, and only their collective impact on distribution function f 0 is taken into account. In the linear approximation, the kinetic equation is reduced to equations for a single harmonic ð- iωk þ ikV Þf k = -

eE k ∂f 0 , m ∂V

ð15:31Þ

so that fk = -i

eEk ∂f 0 1 : m ∂V ωk - kV

ð15:32Þ

Here, in accordance with Eq. (8.39) 1 1 =P - iπδðωk - kV Þ: ωk - kV ωk - kV

ð15:33Þ

15.2

Stochastic Instability and Magnetic Field Line Diffusion

343

The equation for f 0 is obtained with account of quadratic terms in the kinetic Eq. (15.28) after space averaging: eE 1 ∂f 1 ∂f 0 =: m ∂V ∂t

ð15:34Þ

During space averaging, only terms with k = - k′ and the imaginary part of Eq. (15.33) survive (see also Sect. 8.5): ∂ ∂f 0 = ∂t ∂V

πe2 jE k j2 ∂f 0 δðωk - kV Þ : 2 m ∂V

k

ð15:35Þ

This is a diffusion equation in the velocity space with a quasilinear diffusion coefficient πe2 jE k j2 δðωk - kV Þ: m2

D= k

ð15:36Þ

Diffusion in the velocity space establishes a plateau in the region of the distribution function where resonances exist. Using the analogy between the ergodic motion of the charged particle under the impact of many wave harmonics and the motion of the magnetic field line end for overlapping magnetic islands, we introduce the distribution function fB of the field line ends in the perpendicular plane. Similar to Eq. (15.36), in accordance with Eq. (15.25), the equation for the magnetic field line end density also has the form of a diffusion equation: 2

∂f B ∂ = ∂l ∂x

π

B →k 2 B0z

ky , k z

δ kz þ ky

x ∂f B , a ∂x

ð15:37Þ

x : a

ð15:38Þ

where the stochastic diffusivity coefficient is 2

Dst =

π

B →k

ky , k z

2 B0z

δ kz þ ky

Similar to the toroidal geometry with account of identity δ(n/R - mBθ/rBζ) = Rδ (n - m/q), Dst =

m, n

πR

jBmn j2 δðn - m=qÞ: B2

ð15:39Þ

344

15

Magnetic Islands and Stochastic Magnetic Field

The coefficient of stochastic diffusion has dimension of length and determines the mean square of the magnetic field line shift in the perpendicular direction as a function of length along the magnetic field line, in accordance with Eq. (15.18).

15.3 Transport in Stochastic Magnetic Field Magnetic field line stochasticity causes stochastic transport across a magnetic field. It is possible to estimate the diffusion coefficient of a test particle, for example, an electron, in the following way. Let the electron mean free path λmfp be larger than the Kolmogorov length Lc. In such a situation, electrons move freely along the magnetic field line, simultaneously shifting in the perpendicular direction together with it. Perpendicular displacement becomes irreversible only after collision when the electron shifts at a distance of the order of its gyroradius ρce. Initially, the electron was “smeared” over a magnetic tube with radius ρce. At large distances l > Lc, the perpendicular cross-section of a tube becomes similar to that shown in Fig. 15.2, while the perpendicular size of the “sleeves” decreases to less than ρce. Therefore, electrons during rare collisions shift to completely different flux tubes and “forget” their history; as a result, motion becomes irreversible. Since for time t the electron passed distance l = Vkt along the magnetic field line, then in accordance with Eq. (15.18) ðΔxÞ2 = 2Dst V k t,

ð15:40Þ

and test particle diffusion coefficient Dtest = Dst V k :

ð15:41Þ

Replacing the parallel electron velocity by their thermal velocity, one can obtain an estimate of the electron heat conductivity in a stochastic magnetic field as χe = Dst V Te :

ð15:42Þ

This expression is known as the Rechester-Rosenbluth formula. Diffusion in a stochastic magnetic field is significantly smaller – it is controlled by ions with velocities (mi/me)1/2 times smaller than those of electrons. An ambipolar electric field arises in the plasma, which slows down electrons and accelerates ions, and the resulting ambipolar diffusion coefficient can be estimated as D = Dst cs , where cs =

ðT e þ T i Þ=mi is the ion sound velocity.

ð15:43Þ

15.3

Transport in Stochastic Magnetic Field

345

If a stochastic magnetic field is produced by plasma turbulence, stochastic transport coefficients can be estimated in the following way. In the equation for the magnetic field, Eq. (10.14) →

→ → 1 ∂B → = ∇ × u × B - ∇ × σ - 1∇ × B , μ0 ∂t

ð15:44Þ

One can estimate collisionless conductivity as σ

ne2 , me ω

ð15:45Þ

where ω is the typical frequency of the turbulence. The typical spatial scale can be estimated as a scale where the magnetic field line becomes unfrozen, i.e., all three terms in Eq. (15.44) are of the same order. Comparing the first and third terms Bω =

me ωB , μ0 ne2 δ2

ð15:46Þ

one obtains the scale δ=

c , ωpe

ð15:47Þ

where ωpe = ne2 =ε0 me is the electron plasma frequency. Scale δ is known as the collisionless skin layer. For the tokamak, one can assume that the magnetic field line displacement coincides with δ at the Kolmogorov length, which could be estimated as qR. In other words, B →k =B 

c : ωpe qR

ð15:48Þ

Inserting this estimate into Eqs. (15.42) and (15.39), we obtain an estimate for electron heat conductivity in a stochastic magnetic field: χe 

c2 V Te : ω2pe q2 R

ð15:49Þ

This estimate is known as the Okawa formula for electron heat conductivity in the stochastic magnetic field for developed turbulence. This electron heat conductivity coefficient is inversely proportional to the plasma density and is independent of the magnetic field. However, in the real tokamak, such a level of turbulence is probably not reached.

346

15.4

15

Magnetic Islands and Stochastic Magnetic Field

Resonant Magnetic Perturbations in Tokamak

Radial magnetic perturbations are created in a tokamak by special helical external coils with periodicities of a few m and n in the poloidal and toroidal directions. The current in these coils in the toroidal geometry produces resonant magnetic perturbations (RMPs) of the radial magnetic field with many harmonics. In the region near the separatrix (on the core side), where the value of the safety factor is large and, hence, the distances between resonant flux surfaces are small, magnetic islands in the vicinity of resonant flux surfaces overlap, and a stochastic region is established. In this region, electron transport increases, as demonstrated in the previous section. As a result, it is possible to control transport in the edge plasma. Radial particle and heat fluxes associated with RMPs could be calculated in the quasilinear approximation for the collisionless case. The drift kinetic equation for electron guiding centers, Eq. (1.112), reads eE ∂f ∂f E x ∂f ∂f = 0: - x bx þ V k bx þ m ∂t ∂x B ∂y ∂V k

ð15:50Þ

Here, the slab geometry is considered bx = Bx/B, where x represents the minor radius and y corresponds to the poloidal direction. Bx = →

B →k exp ik y y þ ik z z =

k

→

B →k exp ik k l :

ð15:51Þ

k

The radial electric field Ex = - ∂φ0/∂x. Distribution function is sought in the form f = f 0 þ f 1;

f1 = →

f →k exp ik x x þ ik y y :

ð15:52Þ

k

In the linear approximation for a single harmonic, we have f →k = i

b →k ∂f 0 eE0 ∂f 0 Vk : ky V 0 þ kk V k m ∂V k ∂x

ð15:53Þ

Here, V0 = B-1∂φ0/∂x. As before, 1 1 =P ky V 0 þ kk V k kk V k

þ iπδ kk V k :

ð15:54Þ

It is supposed that kyV0 < < kkVTe. Here, we neglect the potential perturbation φ →k :, assuming that it is smeared out by perpendicular viscosity and inertia due to the small distance between ergodic field lines in the perpendicular direction. Radial fluxes can be calculated using definition

15.4

Resonant Magnetic Perturbations in Tokamak

347 →

Γe =

V k bx f 0 þ f 1 d V ,

qe =

→ mV 2 V k bx f 0 þ f 1 d V : 2

:

ð15:55Þ

The radial particle flux Γe is quadratic with respect to the magnetic field perturbation. Substituting Eqs. (15.52)–(15.54) into Eq. (15.55), one obtains Γe = - n

∂ ln T e ∂ ln n e ∂φ0 8T e D þ 0:5 : πme st ∂x T e ∂x ∂x

ð15:56Þ

The electron heat flux is given by qe = - 2nT e

∂ ln T e 8T e ∂ ln n e ∂φ0 D þ 1:5 : πme st ∂x T e ∂x ∂x

ð15:57Þ

Here, the stochastic diffusion coefficient of the magnetic field lines is defined according to Eq. (15.38) or (15.39). Since the ion flux in the stochastic magnetic field is much smaller than the electron flux, Γi  me =mi Γe , the radial electron current flows in the stochastic magnetic field: je = σSt E x - E St x ,

ð15:58Þ

where σSt = e2 n

8 D , πme T e st

ð15:59Þ

and ESt x = -

∂ ln T e T e ∂ ln n þ 1:5 : e ∂x ∂x

ð15:60Þ

In the absence of an ion current, the radial current should turn to zero due to the ambipolarity constraint, and, hence, the radial electric field should coincide with ESt x . This electric field is positive, i.e., it is directed from the core towards the tokamak edge. Ion current in the radial direction can be generated by the ion neoclassical effects described in Chap. 13. The radial ion current becomes nonzero when the radial turbulent transport of toroidal (parallel) momentum is taken into account. In the presence of this turbulent transport neoclassical condition, Eq. (13.57) → $ B ∇  π k = 0 should be replaced by a more general one:

348

15 →

Magnetic Islands and Stochastic Magnetic Field

$

- B ∇  π k = nmi The term nmi

dU k dt

dU k : dt

ð15:61Þ

here represents the radial transport of parallel momentum. The

toroidal projection of the momentum balance equation reads ji By

=

nmi

dU T dt

:

ð15:62Þ

Here, the surface-averaged ion neoclassical toroidal viscosity is zero, see Chap. 13. The radial current in Eq. (15.62) is an ion current that compensates for the radial current of electrons in the stochastic magnetic field in Eq. (15.55) in accordance with the ambipolarity constraint ji þ je = 0:

ð15:63Þ

Ion radial current accelerates plasma in the toroidal direction. Note that there is no contribution from the electron current since the latter flows along magnetic field lines → → and does not produce a j × B force. Neglecting the difference between Uk and UT and between volume and surface averaging, from Eqs. (15.61) and (15.62), one can obtain the radial ion current as a function of parallel viscosity: →

ji = -

$

B ∇  πk BBy

:

ð15:64Þ

After substituting Eq. (13.61), the ion radial current can be expressed as a function of the difference between the radial electric field and the neoclassical value: , ji = σNeo E x - ENeo x

ð15:65Þ

where the neoclassical electric field is defined according to Eq. (13.62) E Neo x =

1 ∂T i T i ∂n þ kT þ By U k : en ∂x e ∂x

ð15:66Þ

Neoclassical conductivity in the Pfirsch-Schluter regime according to Eqs. (15.64), (15.65), and (13.61) σNeo =

3 η : 2B2 R2 0

ð15:67Þ

15.4

Resonant Magnetic Perturbations in Tokamak

349

Ex

x=0

x

2 1

Fig. 15.4 Radial electric field in the outer equatorial midplane at the core side of separatrix. 1 – Without RMPs and 2 – with RMPs, x = 0 corresponds to separatrix position

Here, η0 is the Braginsky viscosity coefficient. In collisionless regimes, it should be replaced by neoclassical viscosity coefficients [13]. Ambipolarity constraint Eq. (15.63) determines the radial electric field Ex =

Neo Neo Ex σst E St x þσ : st Neo σ þσ

ð15:68Þ

In the absence of RMPs, the radial electric field coincides with the neoclassical field and is negative (directed from the edge towards the core), while in the presence of RMPs, the electric field becomes less negative (Fig. 15.4), and, in the extreme case, σSt ≫ σNeo is positive: Ex = E St x . The radial current can be calculated from Eq. (15.58) or (15.65). Toroidal rotation generated by this current is determined by Eq. (15.62). If the turbulent transport of toroidal momentum is strong enough, the resulting toroidal rotation does not contribute to the neoclassical electric field and can be neglected in Eq. (15.66). In the opposite situation, one has to solve the differential equation for toroidal rotation, which follows from Eqs. (15.62), (15.65), and (15.66), and then obtain the current and electric field. For further details, see [14]. Radial current is associated with radial particle flux Γi = ji/e and, hence, leads to density reduction in the core, see schematic Fig. 15.5. This is known as the “pump out” effect in the presence of RMPs. One more important issue concerns the so-called “plasma response.” In the stochastic magnetic field, current flows along magnetic field lines and creates magnetic field perturbations so that the vacuum magnetic field produced by RMPs coils could be significantly reduced by currents in the plasma. Hence, magnetic field perturbations should be calculated self-consistently so that all treatment becomes rather difficult and requires simulations with complicated codes.

350

15

Magnetic Islands and Stochastic Magnetic Field

n

1 2

x=0

x

Fig. 15.5 Plasma density in the outer equatorial midplane at the core side of separatrix. 1 – Without RMPs and 2 – with RMPs, x = 0 corresponds to separatrix position

15.5

Simulation of Resonant Magnetic Perturbations Effects with Codes and Examples of Experimental Results

The impact of RMPs could be simulated by 2D transport codes of the SOLPS-ITER type. The effects of stochastic magnetic fields could be taken into account by introducing an additional ion current perpendicular to the flux surfaces (equal to the electron current) and additional particle and electron heat fluxes associated with stochasticity, which are considered in the previous section. The stochastic diffusion coefficient of magnetic field lines is calculated by a special code for given currents in the helical magnetic coils. Examples of such modeling for MAST tokamak are shown in Figs. 15.6, 15.7, and 15.8. The modeling results are compared with experimental measurements. One can see that the radial electric field at the core side of the separatrix with RMPs switched on becomes less negative than in the shot without RMPs, in accordance with Eq. (15.68). The pump-out effect is also clearly seen both in the experiment and in the modeling. The electron temperature profile does not change significantly. This is explained by two factors acting in opposite directions. Switching on RMPs leads to an increase in stochastic electron heat conductivity, which should result in electron temperature reduction. On the other hand, the density drop due to the pump-out effect causes a temperature rise to maintain the same power flux from the core. The two factors compensate for each other while keeping the electron temperature almost unchanged.

15.5

Simulation of Resonant Magnetic Perturbations Effects with Codes. . .

351

2000

Er (V/m)

0

–2000

σST = 0.0 code σST = 5.0e - 3 code

Icoils = 0 (#21712) Icoils = 1kA (#21714)

–4000

Icoils = 1.4kA (#21713) –0.04

–0.02

0.00

0.02

0.04

r - rLCF.S (m)

Fig. 15.6 Comparison of experimental and simulated radial electric field profiles at the outer midplane for shots with and without RMPs

Fig. 15.7 Electron density at the outer midplane for shots with and without RMPs

352

15

Magnetic Islands and Stochastic Magnetic Field

Fig. 15.8 Electron temperature at the outer midplane for shots with and without RMPs

Chapter 16

Improved Confinement Regime (H-Mode)

→

→

E × B Drift Shear and Transport Barriers

16.1

→

→

Spatial inhomogeneity (shear) of the E × B drift velocity may lead to turbulence suppression and to a reduction in turbulent transport. If it occurs in the localized region, a transport barrier is formed with reduced turbulent transport and strong density and temperature gradients. To understand this effect, we consider a simple slab geometry, as shown in Fig. 16.1. Here, Fig. 16.1a corresponds to the laboratory → → frame, while Fig. 16.1b corresponds to the moving reference frame where the E × B drift velocity V0( y) changes sign at y = 0. Let us first look at the evolution of the paint spot in a river with water flowing in absence of V0( y), paint evolution has a diffusion the x-direction with V0( y). In the p character, so its size increases as Dt (Fig. 16.2a): t

x

2

t

=

″

′

V x t V x t dt dt 0

0

1

t ′

″

=

dt 0

′

V x t ′ V x t ′ - τ dτ = 2Dt = y2 : 0

ð16:1Þ Here, velocities represent the random walk process. In the presence of shear flow, there is an additional displacement along x, which depends on the y-coordinate,

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Rozhansky, Plasma Theory, https://doi.org/10.1007/978-3-031-44486-9_16

353

354

16

Improved Confinement Regime (H-Mode)

y,E V0 ( y) x

a y,E V0 ( y) x

b Fig. 16.1 Shear flow. (a) Laboratory reference frame and (b) moving reference frame

y √ √

x

a y

x t

3/2

b Fig. 16.2 Evolution of paint in the river. (a) Diffusion in the absence of shear flow and (b) stretching of the paint in the moving reference frame

16.1

→

→

355

E × B Drift Shear and Transport Barriers

y

x

a y

x t 3/2

b Fig. 16.3 Vortices in turbulent plasma. (a) Without shear flow and (b) with shear flow t

x

2

t

t 0

=

00

0

V x ðt ÞV x ðt Þdt dt 0

2Dt þ 2Dt

00

dV 0 dy

yð t 0 Þ

= 2Dt þ

0

0 2

t 2 = 2Dt þ 2D

t

dV 0 dy

dV 0 00 dV 0 0 00 yðt Þ dt dt dy dy

0

2

t3 : ð16:2Þ

0 Here, we assume dV dy to be independent of y. One can see that the spot is stretched in the x-direction, and, at large times, its dimension in the x-direction increases as t3/2, as shown in Fig. 16.2b. Since the number of particles in the spot is conserved, the dimension in the y-direction decreases accordingly. A similar effect has shear flow on the vortex in turbulent plasma, as shown in Fig. 16.3. In the presence of shear flow, vortexes become extended in the direction of the flow. One can introduce decorrelation length Lc. For the pure diffusion case, the decorrelation time for typical vortices can be estimated as the time of diffusive damping:

τ1 = L2c =D:

ð16:3Þ

For strong shear jdV 0 =dyj > D=L2c , according to Eq. (16.2), a shorter decorrelation time can be introduced:

356

16

Improved Confinement Regime (H-Mode)

- 1=3 τ2 = L2=3 ðdV 0 =dyÞ - 2=3 : c D

ð16:4Þ

In the situation without shear, the turbulent diffusion coefficient can be roughly estimated from the balance D=L2c  Dk 2⊥  γk , where k⊥ is a typical wave vector, so that D  γk k⊥- 2 . The strong shear turbulence level is determined by estimate 2=3 γk  τ2- 1  k⊥ D1=3 ðdV 0 =dyÞ2=3 . Since increment γk increases with k⊥, the balance is reached at larger wave vectors, i.e., large-scale vortexes are smeared out. As a result, the diffusion coefficient in the region of strong shear decreases, and a transport barrier is formed.

16.2

Transition from Low to High Confinement Regime (L-H Transition)

The L-H transition is observed in a tokamak when the heating power exceeds some threshold. After that, a fast transition from gradual density and temperature profiles to profiles with transport barriers at the edge takes place (Fig. 16.4). The temperature profiles look similar. A region with a steep density gradient (transport barrier) is formed in the separatrix vicinity at the closed flux surfaces, and the barrier width is of the order of few cm at the equatorial midplane. One can understand L-H transition on the basis of radial electric field evolution and poloidal rotation shear. Indeed, the radial electric field is close to the neoclassical field, Eq. (15.66). If we neglect the contribution from toroidal rotation and the ion temperature gradient for simplicity, then the radial electric field would be close to

n`

pedestal

H

L

x=0

x Transport barrier

Fig. 16.4 Density profiles in the L and H modes

16.2

Transition from Low to High Confinement Regime (L-H Transition)

ENeo x ≈

T i ∂n , en ∂x

357

ð16:5Þ

and the shear of the poloidal rotation (in the y-direction) would be ωs =

∂ ENeo ∂ T i ∂n x =B ≈ ∂x ∂x en ∂x



Ti : eBLn

ð16:6Þ

Here, we neglect the weak dependence on the major radius of the tokamak, and Ln is the typical density spatial scale in the separatrix vicinity. For the gradual density profile in the L-mode, Ln is of the order of the tokamak minor radius. When shear Eq. (16.6) remains smaller than the typical increment of drift instability γ, plasma turbulence is not suppressed by poloidal rotation shear, and the corresponding turbulent transport remains at the level typical for the low-confinement regime (Lmode). With increasing heating power, the ion temperature at the separatrix also rises, and the poloidal rotation shear in Eq. (16.6) rises correspondingly. At some threshold value, ωs becomes equal to the typical increment of drift instability γ, and the turbulent level decreases compared to that in the L-mode, while the turbulent transport coefficients are slightly reduced. The reduction in the turbulent diffusion coefficient in turn leads to an increase in the density gradient. Indeed, the density profile near the separatrix is determined by ionization source I and diffusion: ∂n = Γ ð xÞ = - D ∂x

0

I ðx0 Þdx0 :

ð16:7Þ

xðcoreÞ

Particle flux Γ is supposed to be constant in the core. Therefore, while the diffusion coefficient is reduced, the density gradient rises to maintain the same particle flux Γ. The radial electric field, which according to Eq. (16.5), is proportional to the density gradient, becomes larger, the shear of the poloidal rotation further increases, turbulence is further suppressed, the diffusion coefficient is further reduced and the avalanche-type process of density profile steepening is launched. As a result, the transition to a profile with a transport barrier (H-regime) takes place (Fig. 16.4). A question about the factors that determine the final steady state profile in the Hmode requires special treatment. According to the present-day understanding of very steep density profiles, some new modes, known as edge localized modes or ELMs, start to develop, which prevent profiles from further steepening. Since the profile in the core becomes more gradual, general confinement in the H-mode is better than that in the L-mode. Transition into the H-mode can be caused not only by an increase in heating power but also by other factors, such as pellet injection, which leads to density gradient steepening, geodesic acoustic modes, etc.

358

16.3

16

Improved Confinement Regime (H-Mode)

L-H Transition Power Threshold

The critical shear value for the L-H transition in Eq. (16.6) is proportional to the ion temperature in the separatrix vicinity and hence to the power coming from the core to the edge. It is inversely proportional to the toroidal magnetic field. One can also assume that with increasing density, the edge ion temperature decreases and increases with decreasing density. Therefore, if the L-H transition occurs when the shear of the poloidal rotation exceeds a given value of increment of drift instability γ, one would expect that the critical power for the L-H threshold should have the following scaling: P  nB:

ð16:8Þ

This can be checked in the modeling with 2D transport codes. An example of such modeling is shown in Fig. 16.5. The critical poloidal rotation shear was chosen to be ωs = 3  105s-1 at a reference point 2 cm from the separatrix (in the modeling interval, around this value was taken). One can see that indeed a linear dependence of the power threshold on the density and magnetic field Eq. (16.8) is observed. Similar scaling is observed on many tokamaks. Moreover, the absolute value of the power threshold is in good agreement with the experimental value, which justifies the choice ωs = 3  105s-1. Note that for low densities, the power threshold deviates from the simple scaling in Eq. (16.8). This is explained by the fact that power can be spent to heat electrons so that the electron temperature at the edge rises with power, while the ion temperature rises only due to heat transfer from electrons to ions, so at low densities, the ion temperature can increase slower than for the large density case.

Fig. 16.5 Heating power that is necessary to achieve a given value of poloidal rotation shear (ωs = 3  105s-1 at a reference point) for ASDEX Upgrade for different plasma densities, I = 1MA, B = 2T. Simulations with SOLPS-ITER

Bibiliography

1. Landau, Lev D.; Lifshitz, Evgeny M. (1976). Mechanics. Vol. 1 (3rd ed.). ButterworthHeinemann. ISBN 978-0-7506-2896-9 2. Landau, Lev D.; Lifshitz, Evgeny M. (1975). The Classical Theory of Fields. Vol. 2 (4th ed.). Butterworth-Heinemann. ISBN 978-0-7506-2768-9 3. Landau, Lev D.; Lifshitz, Evgeny M. (1980). Statistical Physics. Vol. 5 (3rd ed.). ButterworthHeinemann. ISBN 978-0-7506-3372-7 4. Golant V. E., Zhilinsky A. P., Sakharov I. E. Fundamentals of Plasma Physics. New York, Chichester: Wiley, 1980. 5. Chen F. F. Introduction to Plasma Physics and Controlled Fusion. Third Edition. 2016 Springer. 6. Goldston R. J., Rutherford P. H. Introduction to Plasma Physics. 1995 Taylor & Francis, NY. 7. Rozhansky V. A., Tsendin L. D. Transport Phenomena in Partially Ionized Plasma. 2001 Taylor & Francis NY. 8. Sivukhin D. V. in Reviews of Plasma Physics Vol 1, ed by M.A. Leontovich, Consultants Bureau 1965. 9. Braginskii S.I. in Reviews of Plasma Physics Vol 1, ed by M.A. Leontovich, Consultants Bureau 1965. 10. Zhdanov V. M. Transport Phenomena in Multispecies Plasma. 2002 Taylor & Francis NY. 11. Landau, Lev D.; Lifshitz, Evgeny M.; Pitaevskii, Lev P. (1984). Electrodynamics of Continuous Media. Vol. 8 (2nd ed.). Butterworth-Heinemann. ISBN 978-0-7506-2634-7 12. Fitzpatrik R. Plasma Physics. An Introduction. 2015 Taylor & Francis NY. 13. Helander P., Sigmar D.J. Collisional Transport in Magnetized Plasma. 2005 Cambridge University Press. 14. Rozhansky V in Reviews of Plasma Physics V.24, ed. by V.D. Shafranov, Springer -Verlag 2008. 15. Wesson J., Campbell D J. Tokamaks. 2004 Clarendon Press.

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Rozhansky, Plasma Theory, https://doi.org/10.1007/978-3-031-44486-9

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Index

A Acceleration in nonuniform magnetic field, 206, 313 Alfven masers, 226–227 Alfven wave propagation, 234, 236 Alfven waves, 166, 214–216, 218, 219, 221, 226, 227, 234, 237 Ambipolar diffusion, 78, 95–109, 115, 117, 118, 126, 130, 134, 137, 138, 141, 344 Ambipolarity, 284 Ambipolarity constraint, 284, 285, 347–349 Ambipolar mobility, 140, 143

B Bohm criterion, 83–86, 88, 90 Bootstrap current, 300, 331 Braginsky equations, 69

C Chapman-Enskog method, 51 Chirikov criterion, 339 Classical diffusion, 149–155, 159 Collision integral, 17, 18, 27, 28, 30, 32, 37, 51, 53, 170, 172, 295 Collisionless dissipation, 197 Collisionless sheath, 78–86 Conductivity recovery, 145–147 Convective flux of impurities, 156 Current, 37, 61, 76, 96, 113, 137, 150, 184, 188, 205, 229, 241, 271, 311, 335

D Deceleration by ambient plasma, 234–238 Depletion regions, 102, 123, 124, 126, 133, 144, 145, 333 Diffusion in the ionosphere, 109–110 Diffusive decay, 101, 106 Diffusive shocks, 142 Dispersion, 7, 8, 10, 165, 168, 172, 175, 177, 186, 188, 189, 191, 192, 197–201, 216, 218, 222–225, 333 Dissipative flute instabilities, 318 Distribution function in low collisionality regimes, 294 Distribution functions, 1–5, 7, 17–23, 25–33, 37, 40, 41, 45, 47, 50–56, 59, 64, 68, 69, 79–82, 87, 89, 170, 171, 173–175, 178–180, 227, 232, 271, 294–301, 342, 343, 346 Double sheath, 86–89 Drift waves, 163–166, 169, 172, 181, 183–186, 307 Drift-dissipative instability, 166–169

E Electron temperature gradient mode, 56, 59, 70, 176, 177, 279 Electron-ion collisions, 12, 13, 155, 300 Energy principle, 318–322 Entropy, 66–68, 189, 212, 214, 215, 218 Equatorial bubbles, 316, 333, 334 Equilibrium, 5, 50, 150, 158, 241–245, 247, 249–252, 254–260, 263, 264, 267, 269, 307, 309, 310, 313, 314, 319, 320, 323, 327

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 V. Rozhansky, Plasma Theory, https://doi.org/10.1007/978-3-031-44486-9

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362 F Fast magnetosonic waves, 214–216, 218, 221 Filaments in edge tokamak plasma, 238 Floating potential, 83, 86–88, 91, 92, 103, 104, 239, 317 Flute instability, 310–318, 321, 323 Flux surface functions, 245–249, 260, 261, 279, 281, 314 Flux surfaces, 71, 182, 183, 227, 245–249, 251, 252, 256–263, 267, 269–280, 282–289, 292, 293, 298, 299, 303–306, 314, 315, 325–327, 331, 332, 337–339, 346, 350 Force-free equilibrium, 263–267 Friction, 20, 24, 48–50, 52–56, 59–62, 64, 66, 72, 114, 115, 149–151, 153, 155, 156, 158, 159, 163, 166, 169, 170, 172, 187, 188, 205, 207, 276, 278–281, 303 Frozen in magnetic field, 209–212, 218, 264, 323

G Geodesic acoustic mode (GAM), 331–333 Grad-Shafranov equation, 249–252 Gyrokinetic equation, 38–42

Index M Magnetic islands, 245, 326–328, 331, 335–350 Magnetic presheath, 89, 90 Magnetic shear, 181–186, 313, 326 Magnetized plasma, 56, 63, 66, 112, 114–119, 123, 126, 132–134, 143–147, 157–159, 163, 222, 230, 234, 307–327 Magnetohydrodynamic (MHD) equations, 205 Magnetosonic solitons, 222, 225 Mapping, 339 MHD equations, 205, 208, 222, 307, 318, 329 MHD waves, 212–222 MHD waves with dispersion, 218 Momentum equations, 47–50, 52 Multispecies plasma, 105–108, 137, 138, 141 Mushroom-type shape, 233

N Neoclassical fluxes, 269, 284, 303 Neoclassical transport coefficients, 269

O 1D transport codes, 305 Overturn, 142, 195–197, 201, 221, 222, 225

H H-mode, 353–358

I Integral equilibrium, 244, 252–256 Ion acoustic waves, 187–192, 195, 197–201 Ion temperature gradient mode, 152, 176, 177, 208, 283, 284, 300

K Kinetic equation, 2, 4, 6, 7, 18, 20, 21, 25–27, 35–37, 39, 40, 47–49, 51, 68, 170, 173, 178, 180, 197, 291, 294, 295, 342, 346 Kink mode, 322, 325

L L-H transition threshold, 356–358 L-mode, 357 Localization region, 184

P Parallel viscosity, 281, 283, 284, 348 Partially ionized plasma, 109, 137, 149, 151, 153, 157, 315–316, 333 Pellet injection, 42, 201–203 Perpendicular ambipolar diffusion, 134 Perpendicular viscosity, 346 Pfirsch-Schlueter currents, 272, 273, 278 Pfirsch-Schlueter regime, 271–281, 293, 294 Pinch, 156, 157, 243–245, 253, 260, 263–267, 321–325 Plasma blob, 229, 239 Plasma clouds in the ionosphere, 147–148 Plasma decay in laboratory plasma, 134 Plasma expansion in a magnetic field, 238 Plasma jet, 234–238 Plateau and banana regimes, 291–301 Polarization electric field, 146, 231 Poloidal rotation shear, 356–358 Pump-out effect, 350

Index Q Quasineutrality, 75–78, 84, 88, 95, 97, 106, 124, 144, 150, 175, 191, 197, 222, 229, 296

R Radial electric field, 106, 132, 151, 152, 156, 157, 159, 277, 281–285, 288, 290, 297, 299, 300, 302–304, 331–333, 346–351 Radial profile steepening, 357 Rayleigh-Taylor instability, 307–310, 333 Rechester-Rosenbluth heat conductivity, 344 Resonant magnetic perturbations (RMPs), 346–352

S Self-similar solutions, 192–195 Shafranov shift, 256, 257, 260, 315 Shocks, 142, 147, 148, 197, 225 Short-circuiting through ambient plasma, 124, 126, 144 Skin effect, 209–211 Slow magnetosonic wave, 214, 215, 218 Solitons, 197, 200, 201, 222, 224, 225 Sound speed, 45, 71, 85, 149, 157, 158, 188, 189, 191–194, 196, 209, 244, 281, 317, 323 Spitzer conductivity, 61–62, 301 Splitting of small perturbation, 190 Stochastic magnetic field, 335–350

363 T Tearing mode, 326, 331 Thermal force, 56, 59–60, 64, 70–74, 150, 151, 155, 158, 207, 278, 279 Thermoelectric current, 91–93 Tokamaks, 269–294 Toroidal and poloidal rotation, 281–285 Transport barrier, 353–357 Transport coefficients, 33–35, 50–56, 59–66, 68, 180, 184, 269, 291–294, 305, 345 Transport in stochastic magnetic field, 344–345 Trapped and untrapped particles, 293, 298, 299 Turbulence suppression, 353 Turbulent transport, 176, 178–181, 184, 269, 305, 347, 349 2D transport codes, 350

U Unipolar diffusion, 108, 118, 124 Universal instability, 169–176, 178, 185

V Viscosity, 48–50, 58, 65–66, 68–69, 150, 152, 153, 159, 176, 187, 190, 197, 247, 276, 279, 281–284, 348, 349

Z Zonal flows, 331–333