LA INTELIG ENC IA

In order to determine the new electron distribution function obtained when ECRH is applied to a plasma, one needs to know the change in energy of an electron as it passes through a single beam of ECRH. Without an analytic model we must resort to finding the distribution on a grid and solving the equations of motion at the grid points numerically. This is impractical using the exact equations of motion within the confines of present computing facilities and even use of the averaged equations requires a large amount of computing resources. Calculations along similar lines have been performed by Nevins et al (1987) based on work by Rognlien (1983). Tlie former paper will be discussed in depth later. However in this case the objective was to determine the energy absorption of a beam incident on a Maxwellian plasma, and no numerical plots of the final distributions were produced. However in this thesis, a small number of final distributions are calculated, and displayed in the form of contour plots in Chapter four.

One object of the study of single particle motion in an ECRH beam profile therefore is to try to provide an analytic formula to give the increase in velocity space components after a single pass. O'Brien et al developed a nonrelativistic theory of heating and included the effects of magnetic field inliomogeneity which enabled them to derive a simplified diffusion coefficient for use in a Fokker-Planck equation in the spirit of Fielding (1980) and Cairns and Lashmore-Davies (1986) by finding the change in perpendicular energy

A(V_i_^) = -Vj_K cos(\|fQ) + K^/4 (2.6a)

and

< A(hc2) > =k2/8 (2.6b)

K = m dt

(2.6c)

and where \jtq is the initial phase of the particle with respect to the wave, the integral

is over the time spent in the beam, and and Q depend on time through z.

The explicit value of K therefore depends on the profile and the magnetic field profile as seen by the electron, as well as the initial frequency mismatch. For the simplest possible case where there is exact resonance, no magnetic field

1 “ CO ;

eE|jk^w eE_w

inhomogeneity and a tophat profile, K= — for the O mode and---

for the X mode, where w is the width of the beam, and where we have made the small Larmor radius approximation. More complicated expressions arise in practice and O'Brien et al give an explicit expression for the case where the beam profile is Gaussian and toroidal effects are included,

A relativistic analogue of this analytic form with similar form has been found by Taylor (1987) and Taylor et al (1988):

A(Uj^2) _ cos(vo) + K%/4 (2.7a) where i^co-k„V„-D/7] dt (2.7b) but with the condition

K^/4 « Uj_K

(2.7c).

j

I

In (2.7a) we have included a minus sign not included in the equation in 35

A]^=1-8Tcos(\|;q),

=% +GTsin(\|^), and

Taylor et al so that the angle refers to the angle the electric field initially makes with the gyroangle of the electron. The underlying assumption in the derivation of (2.7a) is. that the phase remains unchanged throughout heating or changes only I

linearly according to \}/=\|/Q+Acot where Aco=Q/y is the initial frequency mismatch.

Neglecting relativistic effects (2.4c) simply reduces to dy/dt = Aco for small electric fields. However with the relativistic effects included, the phase of the particle becomes detuned as the particle is heated or cooled. This reduces the amount of heating and introduces a much greater degree of nonlinearity. The result was originally derived from the Lorentz equation but can be recovered from the averaged equations. For simplicity we consider the X mode at the perpendicular with a tophat profile.

Taking the limit G->«> (ie ignoring magnetic field gradients) of the equations (2.4) and casting them in a dimensionless form we obtain

^ = -ecos(v) and d\(f 1 1 esin(\|f)₊

di Y p A

U a Î2

where A =ÿ— , P=—^ A e harmonic number, 1 in this case, F _{— “ ~TT ’} 2K _{We proceed by calculating corrections to the original}

0^10 10

values of A and y iteratively including successive orders of the small parameter ex each time. Let Aq and be the values of A and v at t = 0. We denote by the first correction calculated by using the value A = Aq (=1) etc. We obtain

^ 4#

A2=l-excos(Vo)+ < * î ^ )

where the last equation has been obtained using a Taylor expansion of c o s(y ) .

Neglecting terms in et higher than (et)^ we recover (2.7a). It is clear that the condition et=2K/Uj^Q«l is necessary from examination of next stage of the iteration which involves the term (l-etcosyo )”^- Essentially this condition is that the absorbed energy is much less than the thermal energy of the electrons.

To understand why this is important we must understand the work of O'Brien et al, and also earlier work on ECRH by Fielding (1980) and Cairns and Lashmore-Davies (1986). They showed that the heating due to ECRH could be understood as a diffusive process and produced a simplified diffusion coefficient, essentially using equation (2.6b) and the approximation

AU^