CLASIFICACIÓN DE DESPIDOS:

The stability codes can determine growth or damping rates for a variety of distributions and electromagnetic waves. The distribution may either be specified numerically on a Cartesian or polar grid in velocity, in which case the distribution is interpolated using a cubic spline in order to provide the distribution function and its derivatives for use in the integral (5.4). Alternatively it may be of a specified analytic form chosen through an input flag. In this section we mainly describe its use to determine the growth or damping rates of whistler waves. Its use to determine growth rates of numerical distributions is limited because it is numerically problematic to take derivatives of distributions specified on grids, especially if the distributions are not smooth or if we are interested in regions where the distribution function is small. We have already mentioned figures (5.2a) to (5.2c) which were calculated using numerical data. In these examples however the distributions were smooth and there was little difficulty in taking derivatives. Unfortunately the same could not be said for the distributions displayed in the last chapter which we would have liked to have tested for stability in order to make a more direct comparison of semi-analytically derived growth rates and those derived from fully numerical means.

In the last section we have given a detailed account of the way in which Yjq» the integral in (5.5a), is calculated. We have said very little about the actual calculation of the growth rate given by (5.6). Now equations (5.2) to (5.4) provide an implicit equation for n|;^, which is solved using NAG routine C05AJF. However we make a number of approximations in the process.

First we consider the evaluation of (5.4a), We note first that in cylindrical coordinates (pj_,p||,0) the 0 integration is trivial since the integrand is independent of 0. Secondly it is important to appreciate that the most significant effect a distortion to a Maxwellian distribution has is to alter the absorption properties rather than to alter the basic polarisation and dispersion of the waves. Hence it is typical in numerical work to assume the distribution is Maxwellian for the purposes of calculating (5.4a), so the double integral can be evaluated in

terms of Shkarofsky functions using the code of Owen (1984) if the calculation is performed relativistically, or using the plasma dispersion function otherwise. We adopt this approach. We calculate the integral in (5.4a) nonrelativistically for the purpose of obtaining n|j^ by solving (5.2), and for the purpose of calculating the denominator in (5.6). The reason for the nonrelativistic approximation in the former case is that many evaluations of the LHS of (5.2) are needed to calculate n;;^ implicitly, and the plasma dispersion function is far easier to calculate than the Shkarofsky functions. In the latter case it is because the plasma dispersion function is relatively easy to differentiate with respect to (0, whereas the Shkarofsky functions are not. However we use the relativistic approach to evaluate the numerator of (5.6).

The routine proceeds by calculating the growth rates for the instabilities for a number of given values of the whistler frequency . Optionally the maximum growth rate and the whistler frequency at which it occurs may be found. Then these maximum growth rates can be found as a function of the ECRH frequency and a maximisation over the ECRH frequency can be performed using a cmde search procedure.

The calculation of the growth rates of the O mode is similar to those of whistlers, but is more complicated and involves either four or eight integrals depending on the approximations being used. Details are given in Appendix three.

The actual results are calculated by using numerical integration to calculate the integral in (5.5a). In the case of numerically specified distributions it is essential to use numerical integration, but when we use the distribution predicted by the adiabatic model and the initial distribution is Maxwellian this is not necessary as we have an analytic formula (5.13) which is exact. Although the numerical integration is still performed, the formula is still of use in order to check the coding.

The amount of CPU time required for the numerical integrations is negligible in the case that the distribution is specified analytically. If the distribution is specified numerically it can take a minute or so of CPU time to set up the spline coefficients, but the overall timing is still relatively short.

Despite the apparent simplicity of the code, it is really not possible to make many comparisons of the results with completely analytic results because of the difficulty of evaluating analytically the factors in (5.6) other than

We have already fully explained in the previous section how the distribution function is obtained in the simplified case that the propagation is perpendicular to the field. We have described how the limits of the numerical integration are determined by checking that the conditions (1) and (2) are satisfied on (Tlmim^max)- The only difference between the analysis in section 5.4 and the code is that in the latter case we also determine numerically Trot^transit check that condition (3) is satisfied on ('Hmin^'nmax) (which is generally the case). In the case of oblique propagation the approach is similar but requires greater use of the numerical solution of complicated equations to calculate quantities given implicitly which had explicit values in the perpendicular case.