Long-Wave Instability in Marginally Stable Shear Flows

Abstract

The reductive perturbation theories used by physicists typically fall into two classes: adiabatic elimination of strongly damped variables in dissipative systems, or averaging over fast oscillations in finite dimensional Hamiltonian systems. Our interest here is a third class which in a peculiar way combines aspects of both methods; this is the development of reduced descriptions for instabilities in nearly ideal fluids and plasmas. Near the ideal limit, the dissipative method is of no use. Moreover, because ideal fluids and plasmas are infinite dimensional Hamiltonian systems, there is a continuum of modal frequencies with no obvious time-scale separation that can be exploited to obtain simplifications. Yet simultaneously there is a peculiar form of (apparent) dissipation—Landau damping in plasmas and Orr-Kelvin decay in fluids. There are the essential physical ingredients which must be retained in any respectable approximation. As a concrete example exemplifying the difficulties outlined above, we derive a reduced model of weakly nonlinear disturbances on marginally stable, high Reynolds number shear flows. Long-wave theory is suggested because shear flows, such as that shown in Fig. 1, first become marginally unstable to disturbances with infinite scale in the streamwise direction [1]. More specifically, flows that have negative vorticity everywhere and contain an inflection point in the velocity profile can become unstable through k ­ 0, where k is the streamwise sxd wave number. Weakly nonlinear descriptions are subtle developments of the linear theory. But for shear flows the underlying linear description presents difficulties associated with the “critical level” singularity where the disturbance travels locally with the speed of the ambient flow [2]. The singularity introduces some difficult mathematics in the expansion; this is symptomatic of the modal continuum. Fortunately, a combination of long-wave theory with matched asymptotic analysis (using a development of the techniques in [3]) saves the day. The expansion opens with a study of the linear stability of a basic state like that in Fig. 1; there is an inflection point at the center of the channel. This is the critical level of the marginally stable mode. In the bulk of the flow, the leading order solution is given by the marginal, linear eigenfunction. However, the modal amplitude, denoted bsx, td is undetermined. This is the first term of the