Linear and nonlinear Kelvin-Helmholtz instabilities of high velocity magnetized shear layers with generalized polytrope laws

The linear and nonlinear Kelvin-Helmholtz instability of tangential velocity discontinuities in high velocity magnetized plasmas with isotropic or anisotropic pressure is investigated. This problem is important in various geophysical, astrophysical and space configurations where there are spatially varying flow speeds.; A new analytical technique applied to the magnetohydrodynamic equations with generalized polytrope laws (for the pressure parallel and perpendicular to the magnetic field) yields the complete structure of the unstable, standing waves in the (inverse plasma beta, Mach number) plane for modes at arbitrary angle to the flow and the magnetic field. The stable regions in the (inverse plasma beta, propagation angle) plane are mapped out via a level curve analysis, thus elucidating the stabilizing effects of both the magnetic field and the compressibility. For polytrope indices corresponding to the double adiabatic and magnetohydrodynamic equations, the results reduce to those obtained earlier using these models. Detailed numerical results are presented for other cases not considered earlier, including the cases of isothermal and mixed waves. Also, for modes propagating along or opposite to the magnetic field direction and at general angles to the flow, a criterion is derived for the absence of standing wave instability—in the isotropic MHD case, this condition corresponds to (plasma beta) ≤1.; We also study the initial-value problem using Laplace transforms to investigate whether the linear instability acts to transform the translational shear flow into rotating vortices. Singularity analysis of the resulting equations using Fuchs-Frobenius theory yields the large-time asymptotic solutions. The instability is found to remain, within the linear theory, of the translationally convective shear type. No onset of rotational or vortex motion, i.e., formation of “coherent structures” occurs.; Finally, we comprehensively analyze the nonlinear evolution and chaos in the instability employing reductive perturbation techniques based on multiple scale expansions, together with a Melnikov function formulation to investigate the possible occurrence of transverse homoclinic orbits. For modes near the critical point of the linear neutral curve, the nonlinear evolution of the amplitude of the linear fields on the slow first-order scales is shown to be governed by a complicated nonlinear Klein-Gordon equation. The nonlinear coefficient turns out to be complex, unlike previously considered cases, and leads to completely different dynamics from that reported earlier. Both the spatially dependent and space-independent versions of this equation are considered to obtain the regimes of physical parameter space where the linearly unstable solutions either evolve to final permanent envelope wave patterns resembling the ensembles of interacting vortices observed empirically, or are disrupted via nonlinear modulation instability. In particular, the complex nonlinearity allows the existence of quasi-periodic and chaotic wave envelopes, unlike in earlier physical models governed by nonlinear Klein-Gordon equations. In addition, a Melnikov function formulation reveals the onset of chaos as a consequence of modulation of the external magnetic field.