A weakly nonlinear theory of the spatial evolution of disturbances in a shear flow with a parallel magnetic field

Abstract

A study is made of the spatial downstream evolution of a weakly unstable disturbance excited by an external source of a frequency ω close to the frequency of a marginally stable (neutral) mode, in a mixing layer of conducting, nearly inviscid fluid with a uniform parallel magnetic field. The nonlinear dynamics of such a disturbance is governed largely by two critical layers, i.e., by two narrow regions on the flow profile vx=u(y) near critical levels y=yj (j=±1), on which the resonance condition u(y±1)=c±cA is satisfied (c being the wave’s phase velocity corresponding to the neutral mode, and cA is the Alfven velocity). A nonlinear integrodifferential evolution equation is derived for a complex amplitude of the disturbance, and its solutions are analyzed (numerically and analytically) with different relationships between problem parameters. It is shown that the nonlinearity can play both a stabilizing and destabilizing role depending on the magnitude of the magnetic field and on the degree of supercriticality of the wave. With not too large a magnetic field (cA<cA*), the future of a disturbance depends only on the relationship between the reciprocal of the Reynolds number ν and the wave’s linear spatial growth rate γL (provided that the magnetic Prandtl number is of order unity). With a sufficiently small growth rate (γL≲ν1/3), the critical layer regime throughout the evolution remains a viscous one, and the wave amplitude, upon reaching a very smooth maximum, goes very slowly to zero. At larger values of γL(γL≳ν1/3), when the nonlinearity threshold is attained, a peculiar kind of self-maintaining unsteady regime of critical layers is established, and the amplitude grows explosively, |A|∝(x0−x)−2 (and ultimately reaches values where weakly nonlinear theory becomes invalid). With a sufficiently large magnetic field (cA>cA*, but cA<12Δu, where Δu is the velocity difference), the nonlinearity leads to an “explosion” with any relationship between γL and ν1/3.