Gyrokinetic stability theory of z–pinches

From the Vlasov-fluid model a set of approximate stability equations describing the stability of the pure z–pinch is derived. The equations are valid for equilibria with small gyroradius compared with the pinch radius, but the perturbation wavenumber k may be of the order of the gyroradius ρi, δ = kρi = 0(1) - so-called gyrokinetic ordering. The equations are used to study the stability of the m = 0 and m = 1 internal modes of the z–pinch. In the limit of zero gyroradius δ → 0 we recover previously obtained results. For δ ≠ 0 we find that increasing δ at first gives a rapidly decreasing growth rate, and for δ ≈ l the growth rate compared with perpendicular MHD is γ/γMHD ≈ 0·09. For larger δ however, the growth rate increases to a quite large value. For the m = O mode we find, provided that drift resonances can be neglected, a stability criterion for δ ≥ 1, which is fulfilled both for the Bennett equilibrium and the constant-current-density equilibrium.