On the kinetic instabilities of uniform magnetized plasmas with generalized loss-cone distribution functions

A general proof is given that in uniform magnetized plasmas described by generalized loss-cone distribution functions (loss-cone index l, thermal velocity αǁ, and perpendicular spread α⊥), electromagnetic, electrostatic, or coupled-mode instabilities are insensitive to the separate values of l and (α⊥/αǁ); they depend rather, on the effective thermal anisotropy Aeff ≡ (T≚/Tǁ)eff − 1, where (T⊥/Tǁ)eff ≡ (l+i) (α⊥2/αǁ2). In the case of parallel propagation this statement is limited only by the linearization assumption; in the oblique propagation case, the additional condition λ⊥/rL ≫ 1 is required (λ⊥ = l/k⊥, where k⊥ is the wave vector perpendicular to the external magnetic field, and rL is the Larmor radius). Thus, dispersion relations and their solutions obtained by using simple bi-Maxwellian distribution functions can be used directly for the complex case of generalized loss-cone distribution functions by simply replacing the anisotropy factor, A = α⊥2/αǁ2 − 1, by Aeff defined above. This result explains earlier conclusions that the growth rate of the whistler instability is independent of the explicit value of the loss-cone index l, for a given thermal anisotropy.