Numerical simulations to understand the dynamics, energetics, and mixing of breaking internal gravity waves

Breaking internal gravity waves are an important component of transport and mixing of heat, dissolved tracers and biological matter in natural water bodies, and much effort has been made in the past few decades to parameterize the breaking and mixing dynamics in large-scale hydrodynamic models.  In this talk I will present results from three-dimensional numerical simulations (both LES and DNS) of laboratory-scale breaking internal gravity waves, both at and away from boundaries.  

I will focus on the breaking dynamics of internal gravity waves propagating along a finite-thickness interface separating two miscible fluids of different densities. The dynamics are fundamentally different from surface gravity waves in that the density difference between the two layers is no more than a few percent, which is typical of internal gravity waves in natural water bodies. Because they are miscible, mixing of the two fluids is of interest in addition to the breaking dynamics.  Owing to the small density difference between the two layers, internal gravity waves can break due to either a shear instability or a convective (overturning) instability that occurs when the fluid velocity within the wave exceeds the wave speed. I will use simulation results to show that the type of instability depends on the thickness of the interface relative to the wavelength.  Although internal wave breaking away from boundaries is more likely to result from a shear instability, breaking on slopes arises both from shear and convective instabilities. Regardless of the primary instability, simulation results show that the secondary instability is always convective, which is then followed by turbulence and mixing.

To quantify the turbulent energetics in breaking internal gravity waves, I will discuss the mixing efficiency, which is the ratio of the amount of energy lost to mixing the density field (since this amounts to an effective increase in the potential energy) to the total amount of energy lost during breaking.  I will discuss both the bulk mixing efficiency, or the domain-integrated mixing efficiency, and the local mixing efficiency, which provides a distribution of instantaneous mixing in space. The mixing efficiency will be related to the dynamics of the turbulence during different stages of wave breaking, particularly for internal wave breaking on slopes. The overall mixing efficiency will then be discussed in terms of the bulk wave properties, including wave steepness and nondimensional interface thickness.  Results show that the mixing efficiency in breaking internal waves is typically higher than the efficiency of mixing in a stratified shear layer due to the presence of convective instabilities. Overall, breaking internal waves are found to be most efficient at mixing when the nondimensional interface thickness is within a given range: interfaces that are too thin are strongly stratified which acts to damp turbulence and limit mixing, while interfaces that are too thick have little fluid to mix, and so are inefficient. The results have important ramifications for turbulence models that seek to parameterize the eddy-diffusivity as a function of the mixing efficiency.