Appendix C: for Chapter 6: Analytical Theory of TAE

С.1 DERIVATION OF ТАЕ EQUATIONS

We start from the main governing equation (4.12) of ideal MHD

and apply the linearization procedure which gives the following expressions:

The last term in (C.3) associated with equilibrium pressure is zero as By substituting all terms from (C.3) into (C.l), we obtain

Two of the terms here give

with the use of the equilibrium equation.

The linearized governing equation for a perturbed plasma takes the following form:

We introduce scalar and vector potentials for the electromagnetic waves of Alfvenic type, so the perturbed fields of the wave are

Shear Alfven waves satisfy the condition of zero parallel perturbed fields, that is, so that only parallel component of the vector potential is involved:

Furthermore, as these waves have no parallel electric field, the scalar and vector potentials are related via

The cross-field perturbed velocity of plasma is simply Using the Amperes law, we find

For simplicity, by considering a small-/} limit (so that 6P —> 0.), we obtain

As the waves considered are extended along equilibrium magnetic field, k_n «: k±, we obtain

This equation describes shear Alfven waves in low-/! plasmas. We reduce this three-dimensional equation to one-dimensional coupled equations by assuming solution in the form

Where c.c. stands for “complex conjugate.” For the equilibrium magnetic field in the form

we obtain

where the metric components should be taken from the flux-type expressions given in Appendix B. Note that the parallel wave vector is given by

The bending energy term takes the following form:

Considering the first term as an example, we obtain the following expression for this term:

We substitute (p{r,&) = (r)exp(-imt?), and use cosz? = [e‘° +e~^ie}j2 to pick the /и-th ele- #11

ment of the Fourier decomposition

to obtain the following expression for the first term:

Express this term via the parallel wave vector to obtain

Representing the other three terms, we arrive at the complete bending energy term:

The inertial term is analysed in a similar manner:

The ш-th Fourier component of this term is

The sum of the bending energy term and the inertial term gives for two coupled harmonics the following equations coupled through toroidicity:

These equations are the starting point equations (6.3) and (6.4) from Chapter 6.

C.2 RADIATIVE DAMPING OF TAE AND KINETIC TAE

For simplicity, we consider modes with high poloidal mode number, m » 1, in a torus with circular

r

magnetic flux surfaces, low magnetic shear, and small inverse aspect ratio. — <к 1. In this limiting case, each eigenmode is highly localised in the vicinity of the centre of TAE-gap and consists of two neighbouring poloidal harmonics (p_m (r), (m- l)-th amplitudes of the electrostatic potential is given by the following coupled fourth-order differential equations [6.11,6.17]:

where e = ^ j^j, the differential operator L„, is defined as and the first-order Larmor radius term is defined as

For solving (C.28), we follow the TAE-ansatz and consider “outer” region away from the TAE-gap and “inner” layer surrounding the TAE-gap. The outer region has only cylindrical part of equations similar to (6.18) and (6.19). In the outer region, we neglect both toroidicity and non-ideal effects and solve it similarly to TAE. In contrast to the outer region, the vicinity of the TAE-gap has both small parameters, the toroidal coupling, and the non-ideal Larmor radius terms, playing an important role as the solutions of (C.28) in the inner layer are highly peaked and the small parameters mentioned are in front of the terms with the high-order derivatives. We keep the terms of (C.28) with high- order derivatives to obtain the following equations for the inner layer [6.11,6.17]:

where the normalized frequency g is given by (6.13), the dimension-less radial variable z is given by (6.16), and the non-ideal parameter A² is given by (6.31). The coupled equations (C.31) can be integrated once to give

Here, we introduced U = ^1, v = . and the integration constants C_m, C,„., have to be chosen

dz dz

I jH

so that at z —> <*> the asymptotes of U, V match onto the “outer” solutions of (6.24) at <к 1. The order of the inner layer equations (C.32) can be further reduced by Fourier transformation (C.32) and solving the problem in the k, -space [6.15-6.17]. Denoting the respective Fourier transforms of t/(z), V(z) with u(k), v(k), we obtain

where

Eqs. (C.33) and (C.34) are to be solved with the boundary conditions u(k) —> <*>, v(k) —> <*> as к —> °°, and the jump conditions at the origin, which result from the ^-functions in (C.33):

Aw'ay from k=0 (that corresponds to the MHD outer region), (C.33) and (C.34) are combined to give two uncoupled second-order differential equations for u(k), v(k):

which in the limit of moderate к, к < A⁴ takes the form of a Schrodinger-type equation

where E = - (l - g¹) is the eigenvalue associated with the mode eigenfrequency, and

is the effective potential for the Schrodinger problem in the k-space.

The most important part of (C.38) is the first (quadratic) term, the sign of which depends on the sign of the normalized frequency g. In accordance with Figure 6.2 (right), we have g <0 for the mode frequencies below the centre of TAE-gap frequency &>„, and g > 0 - for the mode frequencies above ft)₀- Consequently, the quadratic term of (C.38) gives a hill of the potential for the modes below (o₀, and a potential well - for the modes above 0),_h Figure C.l illustrates the form of the potential (C.38) for two reference values, g = ±1. This is how the asymmetry in KAWs between the

FIGURE C.1 Effective potential (C.38) for modes at the bottom tip, g=—1, and at the top tip, g=+l,of the TAE-gap.

frequency regions above and below the central frequency of the TAE-gap to,, = [/(2qR) manifests itself in the Л-space.

The problem of the radiative damping, in terms of the Schrodinger equation approach in the Л-space we developed here, is to investigate what fraction of и tunnels through the potential V(k) due to the non-ideal Larmor radius effects. The usual TAE mode obtained in ideal MHD approach with its eigenvalue (6.14) and (6.25) corresponds to the potential well of the 5(Л)-1'ипсГюп type in (C.33), (C.34) taken in the absence of the non-ideal effects, 2=0. Radiative damping for TAE corresponds to the tunnelling through the potential barrier from Л=0 to the turning points,

After tunnelling, the radiative wave matches the outgoing wave solution

which is the large- Л WKB solution of (C.36). For TAE at the bottom of the TAE-gap, the asymptotic matching procedure gives the value of the radiative damping (6.30).

Near the upper boundary of the gap, g = +l, the non-ideal effects provide a local well. The asymptotic analysis in this case shows an existence of a quasi-stationary discrete spectrum (6.32) of kinetic TAE modes inside this potential well. Frequencies of KTAEs are just above the gap, and these modes have a relatively low radiative damping (6.33).