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.
A linearly unstable, sinusoidal E × B shear flow is examined in the gyrokinetic framework in both the linear and nonlinear regimes. In the linear regime, it is shown that the eigenmode spectrum is nearly identical to hydrodynamic shear flows, with a conjugate stable mode found at every unstable wavenumber. In the nonlinear regime, turbulent saturation of the instability is examined with and without the inclusion of a driving term that prevents nonlinear flattening of the mean flow and a scale-independent radiative damping term that suppresses the excitation of conjugate stable modes. From a variety of analyses, the nonlinear state is found to have a significant component associated with stable modes. The role of these modes is investigated through a simple fluid model that tracks how momentum transport and partial flattening of the mean flow scale with the driving term. From this model, it is shown that, except at high radiative damping, stable modes play an important role in the turbulent state and yield significantly improved quantitative predictions when compared with corresponding models neglecting stable modes.
Stellar evolution models calculate convective boundaries using either the Schwarzschild or Ledoux criterion, but confusion remains regarding which criterion to use. Here we present a 3D hydrodynamical simulation of a convection zone and adjacent radiative zone, including both thermal and compositional buoyancy forces. As expected, regions that are unstable according to the Ledoux criterion are convective. Initially, the radiative zone adjacent to the convection zone is Schwarzschild unstable but Ledoux stable due to a composition gradient. Over many convective overturn timescales, the convection zone grows via entrainment. The convection zone saturates at the size originally predicted by the Schwarzschild criterion, although in this final state the Schwarzschild and Ledoux criteria agree. Therefore, the Schwarzschild criterion should be used to determine the size of stellar convection zones, except possibly during short-lived evolutionary stages in which entrainment persists.
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