Plasma instabilities

This course is given in PPB seminar room every Thursday 9.00-13.00 during the Autumn semester 2023. 

Jonathan Graves will be giving seven or seven lectures related to MHD.

Stephan Brunner will teach three lectures related to the kinetic description of microinstabilities, as well as another set of three lectures related to basic non-linear effects in plasmas.

Here you will find the course notes related to the lectures given by Stephan Brunner on microinstabilities and non-linear effects.

Also available here is information related to the exam (questions, instructions and schedule). The exam will take place on Friday, January 28, 2022.

MHD Lecture 1: Introduction and basic derivation of the ideal MHD model.  Axisymmetric equilibria are investigated, and the Grad-Shafranov equation is derived.  Some essentials on metrics are also provided in notes pages.  Exercises are written for those who have and have not taken Plasma Confinement.  All questions should be reviewed when solutions are provided.

Further reading:

MHD equations: Freidberg Ideal MHD, chapter 2; Wesson, Tokamaks 3rd edition, chapter 2.20

Frozen in theorem: Wesson, Tokamaks 3rd edition, chapter 2.21

Grad Shafranov Equation: Freidberg Ideal MHD, pages 108-112; Wesson, Tokamaks 3rd edition, chapter 3.3

[R, Z, phi] right handed coordinate system, transformations to conventional [R, phi, Z] cylindrical system, and related vector calculus identities, and discussion on flux coordinates: Advanced MHD, by Goedbloed (see pages on flux coordinates in a torus and especially the appendix)

MHD Lecture 2: Introduction to flux coordinates, straight field line coordinates, Fourier expansion of magnetic equilibira, expansion of Grad-Shafranov equation, Shafranov shift and penetration of shaping.

Further reading

Flux coordinates and magnetic field representation: W D. D'haeseleer et al, Flux Coordinates and
Magnetic Field Structure A Guide to a Fundamental Tool of Plasma Theory (I don't have this book, haven't used it, but it was mentioned in the class.  Should order some copies for the library)

Shafranov shift and Vertical field: Freidberg, Ideal MHD, pages 120-124; Wesson, Tokamaks 3rd edition, sections 3.6 - 3.8

Equilibrium representation with inverse aspect ratio and Fourier expansion: there isn't much in textbooks.  There are journal articles which could be pointed out, but these lecture notes have more detail than you will find elsewhere

MHD Lecture 3: Linear ideal MHD force operator, Approaches for solving MHD equations, Convenient form for delta W, Compressibility and comments on kinetic MHD, Inverse aspect ratio expansion of stability equations, Field line bending stabilisation, Internal modes, introduction to toroidal effects including internal kink mode

Further reading

MHD Lecture 4: External kink modes, inclusion of inertia for resolving singularity at rational surfaces, expansion of equations with respect to local layer variable, ideal internal kink mode dispersion relation and layer width, resistive MHD equations and the constant-psi approximation for tearing mode calculations.

Further Reading

For external kink modes the best reference is Wesson, Tokamaks (e.g. 3rd edition). This was one of his main research areas in real life.  For a special case on external kink growth rates, see White, Theory of Confined Plasmas.

For calculation of internal kink inertia effects, expansion with layer variable and growth rate calculation (and layer width), see chapter 4.10 in  White, Theory of Confined Plasmas.  But there is more detail in the course notes.  I have a number of publications on the topic, with extensions to include collisionless kinetic effects, should you like deeper understanding.

For resistive MHD and constant-psi approximation, see chapter 7.8 of Hazeltine and Meiss, Plasma Confinement.  They go into more detail on asymptotic matching and boundary layer theory than we have time for in this course.  It is a subject of its own, very interesting, and very important, should you be interested in it.  Note that White has written a specific book on boundary layer theory with many applications, which you can borrow from me should you want to.

Microinstabilities Lecture 1 

Microinstabilities are small scale instabilities in inhomogeneous magnetized plasmas that drive turbulence and associated transport of particles, heat and momentum.  This turbulent transport is a limiting factor to the confinement properties of a magnetic fusion reactor.

In this first week on the topic of microinstabilities, we will start with a review of various types of drifts in a magnetized plasma, both individual particles drifts as well as average diamagnetic drifts.  We will then pursue with the derivation of a local kinetic dispersion relation for electrostatic waves in a magnetized inhomogeneous plasma. To this end, the linearized Vlasov equation will be solved by integration along the unperturbed trajectories.

As exercise you will further study diamagnetic drifts, in particular derive these drifts using either a kinetic or a fluid-like description (in fact already addressed in MHD lecture 1).

This first lecture on microinstabilities will cover pages 1-13 of the corresponding lecture notes.

Assumed prior knowledge: Familar with the basic concepts of a kinetic description of a (unmagnetized) plasma, in particular the concept of a particle distribution function and its evolution equation, i.e. the Vlasov / Fokker-Planck equation.

If you need to catch up on these concepts, a useful reference is the book by S. Ichimaru, "Basic Principles of Plasma Physics, A Statistical Approach", part of the "Fontier in Physics" series. A limited number of copies are available at the SPC library.

Microinstabilities Lecture 2

After having solved the linearized Vlasov equation in the previous lecture, we will finalize the derivation of the local kinetic dispersion relation for electrostatic instabilities in inhomogeneous magnetized plasmas. This dispersion relation will then be applied for studying various types of waves and instabilities in magnetized plasmas, with particular focus to kinetic effects such as wave-particle resonances and finite Larmor radius effects. We will start by considering  the interchange instability mechanism, arising in a magnetized plasma in presence of an external force opposing a density gradient.

As exercise, we will apply the dispersion relation derived in the lecture for first studying electrostatic waves in a homogeneous magnetized plasma, in particular Electron Plasma Waves (EPS) and Ion Acoustic Waves (IAW). To analytically solve the kinetic dispersion relation, you will make use of the so-called resonant approximation method.

This second lecture on microinstabilities will cover pages 14-20 of the corresponding lecture notes.

Assumed prior knowledge: Derivation of linear kinetic dispersion relation of EPWs and IAWs in homogeneous, unmagnetized plasmas. 

If you need to catch up on these concepts, a useful reference is again the book by S. Ichimaru, "Basic Principles of Plasma Physics, A Statistical Approach".

Microinstabilities Lecture 3

In this last week addressing microinstabilities, we will further pursue the study of various instability mechanisms using the local kinetic dispersion relation derived in the two previous lectures. We will in particular discuss the actual drift wave and the associated instability resulting from resonant wave-particle interaction. We will then pursue with the study of the so-called Ion Temperature Gradient Instability (ITG), which plays a major role driving turbulent transport in magnetic fusion devices at ion scales. We will finish this chapter by also addressing the Electron Temperature Gradient (ETG) instability, which is the analog of the ITG at electron scales. 

As exercise you will derive the dispersion relation of the ITG instability in the frame of a two-fluid model. 

This third lecture on microinstabilities will cover pages 20-34 of the corresponding lecture notes.

Click here to view the recording for this lecture.

MHD Lecture 5: Different types of resistive instabilities are introduced.  It is explained that the constant-psi approximation can be used for tearing modes, and this in turn allows both linear and non-linear treatments to be developed analytically.  The linear growth rate is obtained in terms of Delta', a quantity that can be calculated in terms of the global q-profile. The linear layer width can also be calculated.  The island width can be calculated in terms of the perturbed radial magnetic field fluctuation on the rational surface, and the equation for the helical field can be defined.  Finally, the Rutherford and modified Rutherford equations are derived.  This allows the width of the island to be solved for neoclassical tearing modes, including the effects of bootstrap current and auxiliary current drive.

Further reading

Linear tearing modes are developed in chapter 5 of Toroidally confined plasmas by White.  Theory of confined plasmas, Hazeltine, describes asymptotic matching and boundary layer physics in detail, and also explains the constant-psi approximation. There is a treatment of linear resistive internal kink mode (out of scope of this course) in chapter 5 of Toroidally confined plasmas by White.  Chapter 6 of Tokamaks, Wesson, also provides a very good derivation of resistive internal kink modes.

The helical field line equation, the island width and Rutherford equation is derived with limited detail in Chapter 7 of Tokamaks, Wesson.  Non-linear kink modes are also described in detail in chapter 6 of Toroidally confined plasmas by White.  The latter (R.B. White) was an architect of non-linear tearing modes, and his book contains topics beyond the scope of this course, including non-linear stabilisation mechanisms.

MHD Lecture 6: The slides begin with a recap on the magnetic operator and straight field line coordinates.  Then the separation of fast and slow oscillations respectively perpendicular and parallel to the equilibrium field are dealt with using an eikonal approach.  The ballooning problem in axisymmetric toroidal geometry is tackled in the limit of infinite toroidal mode number.  It is shown that the ballooning energy and associated Euler-Lagrange equation can be defined in terms of arbitrary flux coordinates.  The equations are then written in terms of the axisymmetric equilibrium coordinates developed in week 2.  This enables the ballooning problem to be defined in terms of toroidal effects in an analytically tractable and transparent fashion.  The ballooning s-alpha diagram is described, and the ballooning representation of eigenmodes is discussed.  The effects of toroidicity on the local magnetic shear is shown to be the reason for strong ballooning instability in some cases, and for strong stability on other cases.  Finally, the infinite n interchange problem is explored from the secular behaviour of the ballooning equation.  The concepts of average curvature and field line bending stabilisation is discussed.

Further reading:

Most of the material from this lecture can be seen in condensed form in R. B. White, Theory of Toroidally Confined Plasmas, chapters 4.11-4.14.  The notes of this course, and associated exercises, generally delve deeper, and are integrated with the earlier slides and exercises (on equilibrium coordinates, etc). Note however that it might be useful to refer to White to see the inclusion of inertia and compression in some of his development.  The course notes do not consider inertia and compression because investigations and discussions are limited to marginal stability conditions.

Some elementary explanations on ballooning and interchange modes can be found in Wesson, Tokamaks, chapter 6.

MHD Lecture 7: The slides start with an elementary explanation into how fast particles modify the MHD equations.  It relies on the derivation given in week 1, which yields the perpendicular momentum equation in terms of a pressure tensor (i.e. modification of the usual right hand side j^B-grad P).  The lecture is broken two parts, both are dedicated to investigating the effects of fast ions on pressure driven MHD instabilities.  First the fast ions are considered in the collisional limit (anisotropic fluid limit).  Departure from usual MHD stability occurs when the distribution function is anisotropic, meaning P-parallel not equal to P-perp.  A modified bi-Maxwellian is deployed and it is shown that the plasma is more unstable for populations with parallel temperatures smaller than perpendicular temperatures because the effective curvature is weighted most strongly on the low field side of the configuration, the so called region of poor curvature.  Fast ion populations are stabilising for large parallel temperatures (relative to perpendicular temperatures) due the effective curvature being weighted most heavily in the region of good curvature.  Hence under this collisional approximation, one would expect an ICRH distribution to be destabilising to pressure driven instabilities, while a tangential-aligned neutral beam injected population would be stabilising.  Second, we investigate the more realistic scenario where the fast ions are weakly collisional.  It is found that the effects of trapped hot ions on low frequency pressure driven instabilities is cancelled.  Passing particles spend most of their time in the region of good curvature, the associated passing ion effective average curvature is strongly stabilising for all types of fast ion populations, either parallel or perpendicular anisotropic.  Except for the case where there is a vanishing passing ion fraction (e.g. in an extreme case with ICRH applied on the LFS) where the effects of fast ions would be neutral for a low frequency instability.  The effects of fast ion populations on internal kink modes and interchange modes are quantified.  Modified marginal stability conditions and linear growth rates are obtained.  If time would have allowed, a further week on this topic would have investigated resonant instabilities such as fishbones and toroidal Alfven modes, where fast ions can instabilities.  More time would also have been spent on linear solutions of the drift kinetic equations.

Further reading:

Text books provide very little information on the details of fast ion interaction with MHD instabilities.  Having written this lecture it does not seem obvious why that is the case, at least for fast ion stabilisation of low frequency modes.  Once the foundations for MHD instabilities are understood it is possible to develop a fairly concise intuitive and quantitative understanding of the subject.  Roscoe White was one of the architects of the topic, and his book does have a chapter on fishbones and the stabilisation of low frequency modes.  See R. B. White, Theory of Toroidally Confined Plasmas.  Of course there are many academic papers dedicated to the topic, but these are notoriously difficult to get into.  You might find my thesis useful background reading because it was dedicated partly to the topic and starts on elementary physics.   It is added on this moodle.  The first part of chapter 4 on collisionless thermal ions can be ignored for this lecture, but otherwise chapter 3 provides solutions to the drift kinetic equations, and the second half of chapter 4 and chapter 5 deals with anisotropic hot ion populations and their effects on the internal kink mode.

MHD Lecture 8: no physical lecture, but some slides will be provided summarising the seven weeks of MHD lectures

Non-Linear Effects Lecture 1

Plasmas are intrinsically non-linear systems. Non-linear effects obviously lead to the saturation of instabilities, which often involves the coupling to other modes. This spectral spreading may in some cases lead to a state of turbulence. In this introduction to non-linear effects in plasmas, we shall mainly study weak non-linear effects, involving waves whose dispersion properties remain close to the ones defined by the linear dynamics.

Certain non-linear effects can be studied in a fluid-like description, such as the basic mechanism of resonant three wave interaction. Other effects are however of kinetic nature as for example non-linear Landau damping involving the resonant interaction between particles trapped in a finite amplitude plasma wave.

In this first lecture on non-linear effects, we will begin to address the mechanism of resonant three wave interaction. We will study first a simpler, but closely related problem defined by a set of three non-linearly coupled harmonic oscillators. Under certain conditions leading to so-called parametric instability, energy can be efficiently transferred from one oscillator to the two others. It will in particular be shown that for the oscillators to already be effectively coupled requires that their eigenfrequencies verify a so-called frequency matching condition.   

As exercise you will further study the conditions for parametriic instability for the system of three non-linearly coupled harmonic oscillators, in particular deriving the amplitude threshold condition when accounting both for small frequency mismatch and damping.

This first lecture on Non-Linear Effects will cover pages 1-5 as well as 33-37 of the corresponding lecture notes.

Click here to view the recording for this lecture.

Non-Linear Effects Lecture 2

After having studied the related problem of three non-linearly coupled harmonic oscillators in the previous lecture, we will address the mechanism of resonant three wave coupling by considering the relatively simple illustration of Stimulated Raman Scattering (SRS) in a non-magnetized plasma. SRS involves the interaction of (1) an incident laser beam, (2) an Electron Plasma Wave (EPW), and (3) the light wave resulting from the scattering of the incident light on the density perturbations associated to the EPW. 

To study SRS we will thus start deriving appropriate model equations for light waves and EPWs in the frame of a fluid-like description, retaining the essential non-linear coupling terms. Using a similar procedure as for the system of coupled harmonic oscillators, we will then derive equations for the slow time and length scale variation of the envelopes of the three waves involved in SRS. It will be shown that effective coupling of these waves requires that the frequency-wave vector pairs associated to these waves not only verify corresponding dispersion relations, but also that these frequencies and wave vectors verify matching conditions. Conditions for the development of a parametric instability, leading to the efficient transfer of energy from one wave to the two others will also be derived.

Finally, the number of solutions to the combined set of constraints defined by the dispersion relations and the matching conditions on frequencies and wave-vectors will be discussed, both in the case of wave propagation in one and three dimensions.

This second lecture on Non-Linear Effects will cover pages 37-48 of the corresponding lecture notes.

Click here to view the recording for this lecture.

Non-Linear Effects Lecture 3

We will pursue the study of a system of three non-linearly coupled waves. Assuming frequency and wave number matching conditions are verified, allowing effective energy transfer between the waves, we will determine in which proportions energy is transferred from one wave to the two others. To this end one introduces the concepts of action amplitude and action density of a wave. It will be shown how wave action appears as the classical analogue of the number of wave quanta in a quantum mechanical description; this is expressed in form of the so-called Manley-Rowe relations.

During this lecture, we will also carefully study the closely related topic of wave energy in a continuous dispersive media, identifying contributions from the electromagnetic fields as well as kinetic contributions from the medium itself. As exercise you will derive the wave energy related to Transverse Electromagnetic Waves (TEMWs) as well as Electron Plasma Waves (EPWs), providing useful results for identifying the action amplitudes of the three waves involved in Stimulated Raman Scattering (SRS).

Finally, we will derive analytic solutions to the non-linear system of equations for the time evolution of the amplitudes of three resonantly coupled waves. These solutions are expressed in terms of elliptic functions and in particular illustrate the most obvious saturation mechanism of parametric instabilities: pump depletion. Time permitting, we will discuss other possible saturation mechanisms.

This third lecture on Non-Linear Effects will cover pages 48-60 of the corresponding lecture notes.

Click here to view the recording for this lecture (Note: this recording is unfortunately incomplete as it is missing the last 1,5 hrs of the lecture).