Nonlinear evolution of a weakly unstable wave in a free shear flow with a weak parallel magnetic field

Abstract

A study is made of the nonlinear spatial evolution of an externally excited instability wave in a mixing layer of nearly perfectly conducting fluid with a large Reynolds number in a weak parallel magnetic field.It is shown that the evolution pattern bears a resemblance to that of disturbances in a weakly stratified shear flow with the Prandtl number less than unity which was studied in our earlier publication (Shukhman & Churilov 1997): a weak magnetic field, like a weak stratification when Pr<1, has a stabilizing effect on the nonlinear development of disturbances and in the case when the linear growth rate of the wave is not too large leads either to the instability saturation in the viscous critical layer regime or to the establishment of a unsteady nonlinear critical layer regime where the wave amplitude oscillates without exceeding a certain maximum value. In this case the regime of the quasi-steady nonlinear critical layer is not attained evolutionarily. When the linear growth rate is large enough the magnetic field has no dynamical effect on evolution and the quasi-steady nonlinear critical layer regime with the well-known power-law growth of amplitude (A∝x2/3) is eventually attained.Also, the critical layer structure and the evolution behaviour in the case of a strong difference of dissipation coefficients (i.e. ordinary viscosity and magnetic viscosity) are considered.