Coupling of damped and growing modes in unstable shear flow

Abstract

Analysis of the saturation of the Kelvin-Helmholtz (KH) instability is undertaken to determine the extent to which the conjugate linearly stable mode plays a role. For a piecewise-continuous mean flow profile with constant shear in a fixed layer, it is shown that the stable mode is nonlinearly excited, providing an injection-scale sink of the fluctuation energy similar to what has been found for gyroradius-scale drift-wave turbulence. Quantitative evaluation of the contribution of the stable mode to the energy balance at the onset of saturation shows that nonlinear energy transfer to the stable mode is as significant as energy transfer to small scales in balancing energy injected into the spectrum by the instability. The effect of the stable mode on momentum transport is quantified by expressing the Reynolds stress in terms of stable and unstable mode amplitudes at saturation, from which it is found that the stable mode can produce a sizable reduction in the momentum flux.

The impact of magnetic fields on momentum transport and saturation of shear-flow instability by stable modes

“…( 1) and ( 2), which then informs how much energy they remove from fluctuations 25 . As with previous studies of stable modes in shear-flow turbulence 17,27 , we are specifically interested in the dissipationless modes of this system, so eigenmodes are calculated with viscosity and resistivity neglected. While eigenmodes could be calculated with dissipation included, such modes would mix together the physical effects of conservative energy transfer between the shear flow and fluctuations -a primary focus of this paper -and non-conservative dissipation at large scales.…”

“…(1) in Ref. 17 , and describe how fluctuations interact linearly with the background flow and field, and nonlinearly with one another. When the fluctuations are small enough that the nonlinearities can be neglected, Fourier transforming Eqs.…”

“…Following Refs. 17,27 , when describing the eigenmodes of this system, we label the most unstable mode at each k x as j = 1 and its conjugate stable mode as j = 2. Real-space contours corresponding to φ j and ψ j for j = 1 and j = 2 at M A = 40, k x = 0.4 are shown in Fig.…”

“…Particularly in systems with fixed background gradients, saturation might instead occur when the injection of energy by the instability is balanced by the nonlinear transfer of energy to small, dissipative scales. A third, distinct saturation mechanism involves the transfer of energy to large-scale, linearly stable (damped) modes 16,17 .…”

“…16,22 were applied to a hydrodynamic, KH-unstable system in Ref. 17 , showing that stable modes are important in saturating the KH instability, and that when they are excited they can significantly affect turbulent momentum transport (see also Ref. 26 , where the methods were extended to systems with eigenmodes that cannot be derived in closed form, like the system considered here, and thus the calculation must be done numerically).…”