functional tests

Electromagnetic ion-ion instabilities :

We consider a two-species three-components plasma consisting of Maxwellian electrons and two ion components represented by drifting Maxwellian distribution, a dense core and a less dense beam. Focusing on parallel instabilities (k x Bo = 0), there are 3 instabilities (see, Theory of space plasma microinstabilities, by S. Peter Gary, Cambridge University Press, 1993) :

Right-hand resonant mode

For this mode, only the particle of the beam are resonant (we hence should be able to see loops of trapped particles). With a uniform magnetic field B=1.0 and a 3 component plasma :

• the core plasma : density : 1.0, temperature : 0.1 (so beta is 0.2)
• the beam plasma : density : 0.01, bulk velocity : 5.0, temperature = 0.1 (cold beam)
• the electron : density : 1.0, temperature : 0.1

With these values, the most unstable wave number should be around 0.19, the growth rate should be 0.09 and the real frequency should be 0.19. For positive real frequency, wave number and bulk velocity, the mode is right hand circularly polarized, and propagating in the direction of the beam.

With these values, the Left-hand Resonant mode is stable, as the beam is too cold ; hence, no particle can resonate as for a k>0 left hand mode, the frequency (and then the particle velocity) are negative.

The non-resonant mode can exist (with a growth rate 3.7 time smaller than the one of the RhR mode) but is less important as it needs a large beam density and/or a large beam temperature in order to favour this firehose-like mode.

These values are the one coming from Gary, "Electromagnetic ion beam instabilities - Hot beams at interplanetary shocks", ApJ, 1985