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

摘要

A study is made of the spatial downstream evolution of a weakly unstable disturbance excited by an external source of a frequency omega 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 upsilon(x)=u(y) near critical levels y=y(j) (j=+/-1), on which the resonance condition u(y(+/-1))=c+/-c(A), 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 (c(A)c(A)*, but c(A)