Magnetohydrodynamic Stability of Tokamaks E-Book

Hartmut Zohm

• Herausgeber: Wiley-VCH
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch
• Veröffentlichungsjahr: 2014

Seitenzahl: 390

Beschreibung

This book bridges the gap between general plasma physics lectures and the real world problems in MHD stability. In order to support the understanding of concepts and their implication, it refers to real world problems such as toroidal mode coupling or nonlinear evolution in a conceptual and phenomenological approach. Detailed mathematical treatment will involve classical linear stability analysis and an outline of more recent concepts such as the ballooning formalism. The book is based on lectures that the author has given to Master and PhD students in Fusion Plasma Physics. Due its strong link to experimental results in MHD instabilities, the book is also of use to senior researchers in the field, i.e. experimental physicists and engineers in fusion reactor science. The volume is organized in three parts. It starts with an introduction to the MHD equations, a section on toroidal equilibrium (tokamak and stellarator), and on linear stability analysis. Starting from there, the ideal MHD stability of the tokamak configuration will be treated in the second part which is subdivided into current driven and pressure driven MHD. This includes many examples with reference to experimental results for important MHD instabilities such as kinks and their transformation to RWMs, infernal modes, peeling modes, ballooning modes and their relation to ELMs. Finally the coverage is completed by a chapter on resistive stability explaining reconnection and island formation. Again, examples from recent tokamak MHD such as sawteeth, CTMs, NTMs and their relation to disruptions are extensively discussed.

Leseprobe

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Guide

Cover

Table of Contents

Preface

Chapter 1: The MHD Equations

List of Illustrations

Figure 1.1

Figure 1.2

Figure 1.3

Figure 2.1

Figure 2.2

Figure 2.3

Figure 2.4

Figure 2.5

Figure 2.6

Figure 2.7

Figure 2.8

Figure 2.9

Figure 2.10

Figure 2.11

Figure 2.12

Figure 2.13

Figure 2.14

Figure 3.1

Figure 3.2

Figure 3.3

Figure 4.1

Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figure 5.5

Figure 5.6

Figure 6.1

Figure 6.2

Figure 6.3

Figure 6.4

Figure 6.5

Figure 6.6

Figure 6.7

Figure 6.8

Figure 6.9

Figure 7.1

Figure 7.2

Figure 7.3

Figure 7.4

Figure 7.5

Figure 7.6

Figure 7.7

Figure 7.8

Figure 7.9

Figure 7.10

Figure 7.11

Figure 7.12

Figure 8.1

Figure 8.2

Figure 8.3

Figure 8.4

Figure 8.5

Figure 8.6

Figure 8.7

Figure 9.1

Figure 9.2

Figure 9.3

Figure 9.4

Figure 9.5

Figure 9.6

Figure 9.7

Figure 9.8

Figure 9.9

Figure 10.1

Figure 10.2

Figure 10.3

Figure 10.4

Figure 10.5

Figure 10.6

Figure 10.7

Figure 11.1

Figure 11.2

Figure 11.3

Figure 11.4

Figure 11.5

Figure 11.6

Figure 12.1

Figure 12.2

Figure 12.3

Figure 12.4

Figure 12.5

Figure 12.6

Figure 12.7

Figure 13.1

Figure 13.2

Figure 13.3

Figure 13.4

Figure 13.5

Figure 13.6

Figure 13.7

Figure 13.8

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Hartmut Zohm

Magnetohydrodynamic Stability of Tokamaks

Author

Hartmut Zohm

MPI für Plasmaphysik

Garching

Germany

[email protected]

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Preface

This textbook is based on a University course on MHD stability of tokamak plasmas that I have been developing since 1996. MHD stability of tokamaks is an evolving field, and while a lot of specialist's knowledge exists, I found it very important for students to see that there is a solid theoreticalfoundation in common with other areas of plasma physics from which the understanding of tokamak MHD stability is developed. The book should hence serve to bridge the gap between a basic plasma physics course and forefront research in MHD stability of tokamaks. This means that there are some elements of review of the field's present status in it that will certainly develop over the coming years, but I found it important to point out where we have reached basic understanding and where there are still open ends at the time of writing the book.

At this point, I want to thank all the individuals who have contributed directly or indirectly to this book. First I would like to thank Karl Lackner for all the enlightening physics discussions and especially for always having time for me whenever I entered his office. Some of the most insightful physics arguments presented in this book actually originate from these discussions. Then, I want to thank the colleagues who have worked with me on MHD stability on ASDEX Upgrade for the last 20 years or so, namely Marc Maraschek, Anja Gude, Sibylle Günter and Valentin Igochine, trying to figure out where the experiment knows about the theory. It is also a pleasure to acknowledge the very helpful discussions with Per Helander on stellarator physics and Steve Sabbagh on RWM stability. In preparing the book, Hans-Peter Zehrfeld helped me through all the troubles of editing, formatting and organizing the text. He also contributed greatly to Chapter 2. A special thanks goes to Sina Fietz for her help with the figures and to Emanuele Poli for the thorough proofreading. Last but not least, a major part of the text was written during a stay at University of Wisconsin, Madison, and I want to thank Cary Forest and Chris Hegna for their hospitality and the very useful discussions about the topics of this book.

Hartmut Zohm

Garching, August 2014

1The MHD Equations

1.1 Derivation of the MHD Equations

In this book, we will treat the description of equilibrium and stability properties of magnetically confined fusion plasmas in the framework of a fluid theory, the so-called Magnetohydrodynamic (MHD) theory. In this chapter, we are going to derive the MHD equations and discuss some of their basic properties and the limitations for application of MHD to the description of fusion plasmas. The derivation follows the treatment given in [1]. For a more in-depth discussion of the MHD equations, the reader is referred to [2]. Non-linear aspects of MHD are treated in [3]. A good overview of general tokamak physics can be found in [4].

1.1.1 Multispecies MHD Equations

As a magnetized plasma is a many-body system, its description cannot be done by solving individual equations of motion that would typically be a set of, say, equations1 that are all coupled through the electromagnetic interaction. Hence, some kind of mean field theory is needed.

Starting point of our derivation is the kinetic equation known from statistical physics. It describes the many-body system in terms of a distribution function in six-dimensional space , where

1.1

is the probability to find a particle of species at with velocity at time . Here, and are independent variables that, in the sense of classical mechanics, fully describe the system.

The basic assumption of kinetic theory is that fields and forces are macroscopic in the sense that they have already been averaged over a volume containing many particles (say, a Debye-sphere2) and the microscopic fields and forces at the exact particle location can be expressed through a collision term giving rise to a change of along the particle trajectories in six-dimensional space. We note that this has reduced the microscopic problem of the interactions to the proper choice of the collision term.

Evaluating the total change of along the trajectories and keeping in mind that along these, and , where is the force acting on the particle and its mass, the kinetic equation can be expressed as

1.2

where we have assumed that the only relevant force is the Lorentz force and hence explicitly neglected gravity (which is a good approximation for magnetically confined fusion plasmas, but generally not true in Astrophysical applications). According to the above-mentioned description of mean field theory, the fields and will have to be calculated from Maxwells equations using the charge density and current resulting from appropriate averaging over the distribution function in velocity space as will be described in the following.

The kinetic equation is used to describe phenomena that arise from not being a Maxwellian, which is the particle distribution in thermodynamic equilibrium to which the system will relax through the action of collisions. In fusion plasmas, this frequently occurs as the mean free path is often large compared to the system length as is for example the case for turbulence dynamics in a tokamak along field lines. Another important example is when the relevant timescales are short compared to the collision time, such as in RF (radio frequency) wave heating and current drive that can occur by Landau damping rather than collisional dissipation. Here, a description using the Vlasov or Fokker–Planck equation is needed.

However, in situations where is close to Maxwellian, one can average the kinetic equation over velocity space to obtain hydrodynamic equations in configuration space. When doing so, one encounters so-called moments of . The kth moment, which is related to the velocity average of , is given by

1.3

These moments are related to the hydrodynamic quantities used to describe the plasma in configuration space. For the zeroth moment, we obtain

1.4

which is the number density in real space. The first moment of is related to the fluid velocity in the centre of mass frame by

1.5

For the second moment, it is of advantage to separate the particle velocity into the fluid velocity and the random thermal motion according to

1.6

It is easy to show that , as expected for thermal motion, as

1.7

However, the quadratic average is non-zero, representing the thermal energy via

1.8

where is the Boltzmann constant and we have used the definition of the thermal energy density and its relation to the pressure for an ideal plasma. We note that this definition relies on the previous assumption that is close to Maxwellian. More generally, the second moment is defined as a tensor of rank 2, the pressure tensor

1.9

where denotes the dyadic product. The non-diagonal terms of this tensor are related to viscosity, whereas from Eq. 1.8, it is clear that the trace of is equal to , that is for an isotropic system, the diagonal elements of are just equal to the scalar pressure. Therefore, the pressure tensor is also often written as

1.10

where is the unit tensor and the anisotropic part of .

We now integrate the kinetic equation (Eq. 1.2) over velocity space3 to obtain

1.11

which is the equation of continuity for species . Here, we have assumed that the velocity space average of the collision term is zero, meaning that the total number of particles is conserved for each species. Should this not be the case (e.g. by ionization or fusion), the right-hand side would consist of a source term.