Manejo Agronómico

Previous work on the dielectric tensor has involved making further analytic progress by such assumptions as F

Ȉ`uw J p

, making a non-relativistic, or a weakly relativistic, approximation possible. We have opted instead to use numerical integration immediately, thus evaluating the fully relativistic dielectric tensor, in the belief that the complete relativistic correction may prove relevant in a functional tokamak. Certainly for temperatures less than 5keV such effects are negligible, but at the temperatures of around 10-20keV that one might reasonably expect in a working power plant, such effects could prove significant.

The task of evaluating the dielectric tensor is more difficult than one might expect though. If we look at the integrand we are required to evaluate, Fig (2.3.1)-(2.3.2),we can see immediately the integrand is oscillatory, and moreover slowly convergent, making evaluation a difficult, not to say time-consuming, process. Fortunately, the integrand is loosely periodic as is illustrated when we plot it with the function

­°¯

¥ ž . We have plotted these functions against

3

, the variable of integration from (2.15), although we have renamed itž in the graphs.

Figure 2.3.1: The real part of the integrand plotted alongside cos(s) for & 5 ~ &ˆ º 5 ' #" 5 \ š ` \ 8 ¹

Figure 2.3.2: The imaginary part of the integrand plotted alongside cos(s) for

8 & 5 ~ 8 &ˆ º 5 8 ' #" 5 \ š ` \ 8 ¹

Thus it is considerably easier to evaluate ÿ ¦Gy .z F [ ã|{ J (2.19) where ã { 8 { ˜ Ñ J W]\ š K Ñ \£\ t P * FXP J Ò Ñ Y 4 e Ñ 2 m Y 4 ‡ 3 p * Frq s J s t Œ 9 * Frq s J s 4 t u w (2.20)

Though evaluation of this integral can still be time-consuming, the calculation is considerably quicker, and easier.

Our program calculates this sum, stopping when it considers the integral to have converged, the convergence condition we set beingã {

9 ã { 2 ² . &¹2

. Though greater precision could be asked

for, this accuracy is close to the square root of machine precision, generally accepted as the smallest precision worth asking for.

Each integral is evaluated using Romberg’s Method, which uses k successive refinements of the extended trapezoid rule

Ò |} |~ ½ FT J ‡ T€ ¿ ¡ ½ ½ $ûûu ½ { 2 ¡ ½ { à (2.21) where ‚ T 5 T 5 ûû 5 T

{„ƒ is an equally spaced partition of the interval

¿ T 5 T { Ã, € 8 T… 9 T†… 2 , and ½ … 8 ½ F T… J

By refinement, we mean doubling N, so the interval is split into a greater number of points, closer together. We then treat the integral as a function of h, and extrapolate the continuum limit€

8

&

, from our k existing results.

By taking the integral in manageable intervals ofÞ ` ¡ we ensure we are only required to integrate

a smooth analytic function, a task which Romberg’s method is well suited to. We have also experimented with methods using variable step sizes, but such methods have proved slower, and less reliable.

In Figs (2.3.3)-(2.3.4) we show the behaviour of the top left element of the dielectric tensor over a range of\

t

`

\

which includes the first few cyclotron harmonics. This figure shows clearly the peak in the imaginary part around each harmonic. We will go on to show that this element dominates the behaviour of the EBWs, and that these peaks translate into strong damping of the wave, and the sharp edge to the imaginary part on the low field side.

In Figs (2.3.5)-(2.3.6) we show the behaviour of the dielectric tensor element at a higher temper- ature, 20keV. The peaks in the imaginary part of* ||

broaden, corresponding to wave absorption over a broader range of frequencies. In a higher temperature plasma we should expect the wave to be absorbed further from the cold resonance making access to the interior of the plasma more problematic. Of course this is purely heuristic at this stage in thework, but bears remarking upon in passing here. We go on to calculate the dispersion relation, which will confirm this conclusion however.

Figure 2.3.3: s F * || J plotted against \ ` \ t where ~ 8 &ˆ º 5 8 ' 5 \ š ` \ 8 ¹ Figure 2.3.4: ‡ ’ F * || J plotted against \ ` \ t where ~ 8 &ˆ º 5 8 ' #" 5 \ š ` \ 8 ¹

Figure 2.3.5: s F * || J plotted against \ ` \ t where ~ 8 &ˆ º 5 8 ' 5 \ š ` \ 8 ¹ Figure 2.3.6: ‡ ’ F * || J plotted against \ ` \ t where ~ 8 &ˆ º 5 8 ' #" 5 \ š ` \ 8 ¹