General Criterion for Mirror Instability of a Plasma

A plasma column in an axial magnetic induction B exhibits the well‐recognized ``mirror'' instability if the pressure p⊥ perpendicular to B becomes too high, as follows. A small azimuthally symmetric bulge in the field may trap additional plasma via the usual mirror effect. The increased diamagnetism further weakens the field in the bulge, thus trapping more plasma, and so forth. The criterion for stability against this mode for an azimuthally symmetric velocity distribution function f(v, θ) is $${\displaystyle {\int }_0^{\infty }}\mathrm{dv}{\displaystyle {\int }_0^{\text{π/2}}}{\mathrm{\hspace{0.17em}d\theta \hspace{0.17em}\pi mv}}^4{\sin }^3\theta \tan \theta \frac{\mathrm{\partial f}}{\mathrm{\partial \theta }}<\frac{B_{\mathrm{e}}^2}{{\text{2μ}}_0}\left[\text{1−β+}\frac{{\mathrm{\kappa \rho K}}_0\mathrm{(\kappa \rho )}}{{\text{2K}}_1\mathrm{(\kappa \rho )}}\right]$$ . Here, v is speed, m is mass, θ is colatitude angle, and B_e is the uniform induction outside the plasma; ${\text{β=2μ}}_0{p}_{\perp }{\mathrm{/B}}_{\mathrm{e}}^2$ is (plasma perpendicular pressure/total pressure), κ is axial wavenumber of the unstable mode, and ρ is plasma column radius; K₀ and K₁ are modified Bessel functions of the second kind. The integral is a measure of changing p⊥ if an incipient bulge forms in the plasma column; the κρ term represents stabilizing effect of a finite wavelength perturbation. This criterion is more general than the two‐temperature criteria obtained hitherto.