Scaling laws for parameterisations of subgrid eddy–eddy interactions in simulations of oceanic circulations

• Subgrid parameterisations are developed for oceanic flows.
• The parameterisation coefficients are determined from high resolution simulations.
• Simulations using these parameterisations produce resolution independent results.
• Parameterisations are scale selective matrices, represented by simple scaling laws.
• These scaling laws can be implemented into both spectral and grid point codes.

Deterministic and stochastic parameterisations with self-similar scaling laws are developed to represent the subgrid eddy–eddy interactions in the large eddy simulation (LES) of oceanic flows simulated with a quasi-geostrophic model. Specifically the parameterisations are developed for flows broadly representative of the mean Antarctic Circumpolar Current. The stochastic parameterisation consists of a drain eddy viscosity and stochastic backscatter noise, whilst the deterministic parameterisation is governed solely by a net eddy viscosity representing the net effect of the drain and backscatter. All of the eddy viscosities are calculated self consistently from the statistics of high resolution benchmark numerical simulations. The subgrid interactions at each vertical level are represented by eddy viscosity scaling laws in potential vorticity space, and the coupling between levels is represented by eddy viscosity scaling laws in baroclinic vorticity space (related to the temperature). The eddy viscosities are dependent on the LES resolution, enstrophy flux, Rossby radius, and the extent of the energy containing range. LESs with the subgrid parameterisation coefficients defined by the scaling laws are also shown to reproduce the kinetic energy spectra of the benchmark simulation. These scaling laws make the subgrid parameterisations more widely applicable as they remove the need to generate the eddy viscosities from higher resolution simulations, and have implications for more complex ocean models.