Stability and mixing of shear layers forced by standing internal waves

Stability and mixing of shear layers forced by standing internal waves Alexis Kaminski and John Taylor Department of Applied Mathematics & Theoretical Physics, University of Cambridge A.K. Kaminski@damtp.cam. ac.uk Abstract Turbulence generated by breaking internal waves is an important source of mixing in the ocean interior, and is often interpreted in terms of the stability of the vertical shear. However, observations and numerical studies suggest that vertical internal wave strain may also play a key role in stratified flow stability. Here we model the effects of wave strain by imposing a spatially- and temporally-periodic standing wave onto a stably-stratified shear flow. To examine the linear stability of this flow, we compute optimal perturbations for a range of target times, wave amplitudes, and stratifications. The standing wave catalyzes perturbation growth in the shear layer, leading to enhanced energy gains. We then perform direct numerical simulations to examine the nonlinear perturbation evolution. We find that in some cases the linear growth is sufficient to trigger nonlinear effects and transition to turbulence. The mixing efficiency varies in time with the background wave, suggesting that the phase of the flow may be an important parameter in describing the mixing. Introduction Vertical mixing in the stably-stratified ocean interior is thought to be a key player in setting the vertical density structure and distributions of tracers which are important for climate and biogeochemical cycles. However, the turbulent eddies responsible for mixing are typically on the order of several meters in size, and are unresolved in ocean circulation models. Parameterizations of the physics leading to mixing are therefore required. Mixing is often assumed to arise in the ocean interior as the result of shear instabilities. The Miles-Howard theorem states that, for parallel, steady, inviscid stratified shear flows, a necessary condition for instability is that the gradient Richardson number be less than 1/4 somewhere in the domain. This concept has been used as the basis of several mixing parameterizations (e.g. Mellor and Yamada, 1982; Price et al., 1986). Theoretical and numerical work has suggested other linear mechanisms by which perturbations may grow and lead to mixing in linearly stable flows (Farrell and Ioannou, 1993; Kaminski et al., 2014). Additionally, and perhaps more relevant to the question of ocean mixing, is the fact that geophysical flows are rarely parallel and steady – assumptions which are key in the Miles-Howard theorem. Oceanic observations of overturns and dissipation suggest that shear and gradient Richardson number alone may not be the entire story in determining regions of enhanced mixing. Increasingly, there is evidence that vertical strain, γ (the normalized distance be- tween adjacent isopycnals), and the vertical strain rate, 1/γ ∂γ/∂t ≈ ∂w/∂z, may also play an important role in mixing events (Alford and Pinkel, 2000; Aucan et al., 2006; Levine and Boyd, 2006). In addition to oceanic observations, there is numerical evidence for mixing induced by vertical strain in stratified flows. For example, in their numerical sim- ulations of stratified turbulence forced by large-scale standing internal waves, Carnevale et al. (2001) observe the development of small-scale vertical “spouts” in regions of strong VIII th Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016