Abstract
The breaking of internal gravity waves in the abyssal ocean is thought to be responsible for much of the mixing necessary to close oceanic buoyancy budgets. The exact mechanism by which these waves break down into turbulence remains an active area of research and can have significant implications on the mixing efficiency. Recent evidence has suggested that both shear instabilities and convective instabilities play a significant role in the breaking of an internal gravity wave in a high Richardson number mean shear flow. We perform a systematic analysis of the stability of a configuration of an internal gravity wave superimposed on a background shear flow first considered by Howland et al. (J. Fluid Mech., vol. 921, 2021, A24), using direct–adjoint looping to find the perturbation giving maximal energy growth on this evolving flow. We find that three-dimensional, convective mechanisms produce greater energy growth than their two-dimensional counterparts. In particular, we find close agreement with the direct numerical simulations of Howland et al. (J. Fluid Mech., 2021, in press), which demonstrated a clear three-dimensional mechanism causing breakdown to turbulence. The results are shown to hold at realistic Prandtl numbers. At low mean Richardson numbers, two-dimensional, shear-driven mechanisms produce greater energy growth.
Highlights
Buoyancy budgets of the global oceans suggest that turbulence, on scales too small to simulate directly in computational models, is an important element both to dissipate energy and to close the budget (Wunsch & Ferrari 2004; Gregg et al 2018)
Motivated by the observation in Alford & Pinkel (2000) of coinciding shear and large amplitude waves, Howland, Taylor & Caulfield (2021), denoted HTC21, numerically simulated the idealised flow arising from a superposition of a plane internal wave in a uniform density gradient, and a simple sinusoidal shear profile
The linear optimal energy growth analysis we have employed has provided an explanation of the early phase of the simulations described by Howland et al (2021, referred to as HTC21)
Summary
Buoyancy budgets of the global oceans suggest that turbulence, on scales too small to simulate directly in computational models, is an important element both to dissipate energy and to close the budget (Wunsch & Ferrari 2004; Gregg et al 2018). Motivated by the observation in Alford & Pinkel (2000) of coinciding shear and large amplitude waves, Howland, Taylor & Caulfield (2021), denoted HTC21, numerically simulated the idealised flow arising from a superposition of a plane internal wave in a uniform density gradient, and a simple sinusoidal shear profile They observed a clear three-dimensional, convective-like structure, with a definite spanwise wavelength, very different from the primarily two-dimensional Kelvin–Helmholtz billows and other shear instabilities often thought to dominate breakdown to turbulence. We wish to determine the relative importance of shear-driven and convective growth mechanisms in the amplification of these optimal perturbations as they grow on the evolving background flow To address these twin aims, the remaining three sections of the paper are as follows: § 2 gives the precise flow we are considering, and gives the derivation and implementation details of the DAL algorithm.
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