A subgrid‐scale eddy parameterization is developed, which makes use of an explicit eddy kinetic energy budget and can be applied at both “non‐eddying” and “eddy‐permitting” resolutions. The subgrid‐scale eddies exchange energy with the resolved flow in both directions via a parameterization of baroclinic instability (based on the established formulation of Gent and McWilliams) and biharmonic and negative Laplacian viscosity terms. This formulation represents the turbulent cascades of energy and enstrophy consistent with our current understanding of the turbulent eddy energy cycle. At the same time, the approach is simple and general enough to be readily implemented in ocean climate models, without adding significant computational cost. The closure has been implemented in the Modular Ocean Model Version 6 and tested in the “Neverworld” configuration, which employs an idealized analytically defined topography designed as a testbed for mesoscale eddy parameterizations. The parameterization performs well over a range of resolutions, seamlessly connecting non‐eddying and eddy‐resolving regimes.
In this paper large-eddy simulations (LES) of forced stratified turbulence using two common subgrid scale (SGS) models, the Kraichnan and Smagorinsky models, are studied. As found in previous studies using regular and hyper-viscosity, vorticity contours show elongated horizontal motions, which are layered in the vertical direction, along with intermittent Kelvin-Helmholtz (KH) instabilities. Increased stratification causes the layer thickness to collapse towards the dissipation scale, ultimately suppressing these instabilities. The vertical energy spectra are relatively flat out to a local maximum, which varies with the buoyancy frequency N. The horizontal energy spectra depend on the grid spacing ∆; if the resolution is fine enough, the horizontal spectrum shows an approximately −5/3 slope along with a bump at the buoyancy wavenumber k b = N/u rms , where u rms is the root-mean-square (r.m.s.) velocity. Our results show that there is a critical value of the grid spacing ∆, below which dynamics of stratified turbulence are well-captured in LES. This critical ∆ depends on the buoyancy scale L b and varies with different SGS models: the Kraichnan model requires ∆ < 0.47L b , while the Smagorinsky model requires ∆ < 0.17L b . In other words, the Smagorinsky model is significantly more costly than the Kraichnan approach, as it requires three times the resolution to adequately capture stratified turbulence.
The dynamic Smagorinsky model for large eddy simulation (LES) of stratified turbulence is studied in this paper. A maximum grid spacing criterion of ∆/L b < 0.24 is found in order to capture several of the key characteristics of stratified turbulence, where ∆ is the filter scale and L b is the buoyancy scale. These results show that the dynamic Smagorinsky model needs a grid spacing approximately twice as large as the regular Smagorinsky model to reproduce similar results. This improvement on the regular Smagorinsky eddy viscosity approach increases the accuracy of results at small resolved scales while decreasing the computational costs because it allows larger ∆. In addition, the eddy dissipation spectra in LES of stratified turbulence present anisotropic features, taking energy out of large horizontal but small vertical scales. This trend is not seen in the non-stratified cases, where the subgrid-scale energy transfer is isotropic. Statistics of the dynamic Smagorinsky coefficient c s are investigated; its distribution is peaked around zero, and its standard deviations decrease slightly with increasing stratification. In line with previous findings for unstratified turbulence, regions of increased shear favour smaller c s values; in stratified turbulence, the spatial distribution of the shear, and hence c s , is dominated by a layerwise pancake structure. These results show that the dynamic Smagorinsky model presents a promising approach for LES when isotropic buoyancy-scale resolving grids are employed.
General circulation models use subgrid-scale (SGS) parameterizations to represent the effects of unresolved mesoscale eddies on large-scale motions. Most of the current SGS parameterizations are based on a theoretical understanding of transient eddies, where the mean flow is a temporal average. In this work, we use a spatial filtering analysis to better understand the scale-dependent characteristics of the SGS fluxes. Specifically, we apply the filtering approach to diagnose SGS eddy volume fluxes and eddy velocity scales in a hierarchy of model configurations from a flat bottom channel to an idealized Southern Hemisphere. Importantly, SGS volume fluxes include significant contributions from standing meanders; unlike for transient eddies, the vertically integrated SGS volume flux does not necessarily integrate to 0. To accommodate net vertically integrated eddy fluxes, we define a SGS eddy diffusivity based on planetary potential vorticity (PV) diffusion. We diagnose the transient and standing contributions to SGS fluxes and associated effective diffusivities. In the presence of bottom topography or continental barriers the standing component of the PV diffusivity becomes dominant at large filter scales in the westerly wind region, while the transient component remains dominant in the easterly wind region. Our results suggest that the diagnosed PV diffusivity can be parameterized using mixing length theory based on a priori estimates of SGS velocity and length scales.
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