Quasiperiodicity and Chaos in the Nonlinear Evolution of the Kelvin-Helmholtz Instability of Supersonic Anisotropic Tangential Velocity Discontinuities
A nonlinear stability analysis using a multiple-scales perturbation procedure is performed for the instability of two layers of immiscible, strongly anisotropic, magnetized, inviscid, arbitrarily compressible fluids in relative motion. Such configurations are of relevance in a variety of astrophysical and space configurations. 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, which is, to the best of our knowledge, 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 quasiperiodic and chaotic wave envelopes, unlike in earlier physical models governed by nonlinear Klein-Gordon equations. In addition, numerical diagnostics reveal the onset of chaos as a consequence of modulation of the external magnetic field.
S. Roy Choudhury and Kevin G. Brown, Quasiperiodicity and Chaos in the Nonlinear Evolution of the Kelvin-Helmholtz Instability of Supersonic Anisotropic Tangential Velocity Discontinuities, Quarterly of Applied Mathematics, LXI, 1, 41-72, (2003).