Weakly nonlinear waves in magnetized plasma with a slightly non-Maxwellian electron distribution. Part 2. Stability of cnoidal waves

摘要

We determine the growth rate of linear instabilities resulting fromlong-wavelength transverse perturbations applied to periodic nonlinear wavesolutions to the Schamel-Korteweg-de Vries-Zakharov-Kuznetsov (SKdVZK) equationwhich governs weakly nonlinear waves in a strongly magnetized cold-ion plasmawhose electron distribution is given by two Maxwellians at slightly differenttemperatures. To obtain the growth rate it is necessary to evaluate non-trivialintegrals whose number is kept to minimum by using recursion relations. It isshown that a key instance of one such relation cannot be used for classes ofsolution whose minimum value is zero, and an additional integral must beevaluated explicitly instead. The SKdVZK equation contains two nonlinear termswhose ratio $b$ increases as the electron distribution becomes increasinglyflat-topped. As $b$ and hence the deviation from electron isothermalityincreases, it is found that for cnoidal wave solutions that travel faster thanlong-wavelength linear waves, there is a more pronounced variation of thegrowth rate with the angle $\theta$ at which the perturbation is applied.Solutions whose minimum value is zero and travel slower than long-wavelengthlinear waves are found, at first order, to be stable to perpendicularperturbations and have a relatively narrow range of $\theta$ for which thefirst-order growth rate is not zero.