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Subgrid-scale (SGS) parameterisations of turbulence with self similar scaling laws are developed for large eddy simulations (LESs) of atmospheric and oceanic f ows. The stochastic SGS modelling approach of Frederiksen and Kepert (2006) is used to determine the model coeff cients self-consistently from higher resolution reference direct numerical simulation (DNS). The resulting LES then replicates the statistics of the DNS at the resolved scales. In general the SGS model coeff cients depend on both the zonal and total wavenumbers, making them anisotropic. The f ow f elds are simulated using a two-level quasi-geostrophic model that incorporates the processes of baroclinic instability and the interaction of synoptic-scale structures and inhomogeneous Rossby wave turbulence. Two specif c basic f ows are analysed: an atmospheric f ow with large scale jets in the midlatitudes; and an oceanic f ow representative of the Antarctic Circumpolar Current. Despite the obvious differences, these f ow f elds exhibit similar turbulent properties. In both cases the turbulent energy in the system is injected at the Rossby wavenumber (kR), and there is a constant transfer of enstrophy from the Rossby waves to the small-scale (high wavenumber) structures. The key difference, is that in the present simulations the baroclinically unstable Rossby waves within the atmosphere occur at kR ≈ 14, whilst in the ocean kR ≈ 140. This makes the ocean a more computationally challenging case, as a f ner grid is required to resolve the energy injection. The DNSs presented within capture the energy injection of both cases, as they have a triangular wavenumber truncation of T = 504. This is equivalent to 1536 longitudinal and 768 latitudinal grid points. It is found that for both the atmosphere and ocean, the SGS model coeff cients are approximately isotropic if the LES truncation wavenumber is signif cantly larger than kR. The isotropised prof les decrease in magnitude and become steeper as resolution increases. A unif ed scaling law is determined that represents both the atmosphere and ocean SGS processes, by non-dimensionalising the model coeff cients on the basis of the f nal energy containing wavenumber (kE).

Subgrid parameterisation with scaling laws for atmospheric and oceanic flows