Electromagnetic instability related to trapped velocity integration

[David Dickinson]

Attached is a sample input file demonstrating a numerical instability in a relatively simple setup. This shares the characteristics of several previously reported instabilities so could be related. The primary difference/notable point is that here we find the instability is very sensitive to the treatment of the trapped pitch angle velocity space integration.

This instability goes away with

1. Setting fapar / beta = 0.
2. Increasing ky significantly.
3. Setting opt_source = F (this is due to the fact this introduces large dissipation when bakdif /= 0).
4. Increasing ntheta far enough.
5. Switching to new_trap_int = F or nmax = 2 (note nmax=2 should be equivalent to the trapezium rule used by new_trap_int = F albeit calculated with a different, slightly more accurate, method).

Below is a comparison of three cases, new_trap_int = T ; nmax = default, new_trap_int = T ; nmax = 2 and new_trap_int = F

Setting new_trap_int = T ; nmax = 3 delays the onset of instability to around t = 30. Running both new_trap_int = T ; nmax=2 and new_trap_int = F cases for longer shows a much more weakly growing (more physical) instability.

‌

Looking at the velocity space weights we can see that for new_trap_int = T ; nmax = default the trapped weights are highly oscillatory

This remains true for new_trap_int = T ; nmax = default ; ntheta = 128 which is not seen to blow up (indeed it seems to have somewhat lower growth rate than the new_trap_int = F ; ntheta = 32 case, suggesting ntheta=32 may not be fully converged).

‌