The LES model, developed by Deardorff (1980) and Moeng (1984) for the atmospheric boundary layer, dynamically solves for unresolved (subgrid) turbulent kinetic energy (TKE). Several adaptations of this model are currently used to model oceanic boundary layers, including high-latitude deep convection in response to storm forcing (Harcourt et al., 2001), and forced convection in more shallow free-slip ocean boundary layers (e.g. Skyllingstad and Denbo, 1995; Wang et al, 1996).
Following Moeng (1984), the advection terms of the momentum equations are computed prognostically from vorticity components
Terms involving both subgrid and resolved TKE are absorbed into a modified nonhydrostatic pressure . b is the buoyancy force, and the mean acceleration is removed from the right-hand side of the vertical momentum equation to maintain a prescribed mean vertical velocity.
The LES uses a finite series expansion of the equation of state about a potential reference state with density chosen to minimize truncation errors. After removing the horizontally averaged vertical acceleration , buoyancy fluctuations account for mean scalar dependence in the expansion coefficients, thermobaricity, cabbeling and a halobaric effect through , , and , as modifications to the standard approximation of constant thermal and haline expansion coefficients and . The salinity-related components of cabbeling and the dependence of the haline expansion coefficient on mean salinity are small and generally omitted.
The evolution of potential temperature is computed prognostically from
with corresponding equations for salinity and Lagrangian tracers. The velocity components are subject to the condition of incompressibility, which is applied to the momentum equations to solve diagnostically for . Unresolved stresses are modeled in terms of the symmetric resolved strain rate and local nonlinear eddy viscosity ; scalar fluxes in terms of resolved gradients and eddy diffusivity :
Parameterizations eddy viscosity diffusivity, and , depend upon subgrid TKE , its associated dissipation length scale , turbulent eddy Prandtl number and constant . The length scale of unresolved motion is equal to the resolution scale , unless reduced below that by stability or by proximity to a wall boundary. Parameterizations for the dissipation length scale for stably stratified turbulence are discussed below in the description of simulations. Resolution scale is governed either explicitly by a spatial filter, implicitly by the numerical scheme, or by a combination of the two.
The subgrid TKE equation
includes unresolved buoyant production as well as unresolved shear production and transport, and parameterizes dissipation as . When a portion of the turbulent inertial subrange is resolved, constants and of the subgrid parameterization are determined by integrals over the Kolmogorov energy spectrum for unresolved TKE smaller than