Assessing implicit large eddy simulation for two-dimensional flow

Numerical models of the atmosphere cannot resolve all relevant scales; the effects of unresolved scales on resolved scales must be represented by a subgrid model or parametrization. When the unresolved scales are similar in character to the resolved scales (as in three-dimensional or layerwise two-dimensional turbulence) the problem is essentially one of large eddy simulation. In this situation, one approach to subgrid modelling is implicit large eddy simulation (ILES), where the truncation errors of the numerical model attempt to act as the subgrid model. ILES has been shown to have some success for three-dimensional turbulence, but the validity of the approach has not previously been examined for two-dimensional or layerwise two-dimensional flow, which is the regime relevant to weather and climate modelling. Two-dimensional turbulence differs qualitatively from three-dimensional turbulence in several ways, most notably in having upscale energy and downscale enstrophy transfers. The question is of practical importance since many atmospheric models in effect use the ILES approach, for example through the use of a semi-Lagrangian advection scheme. In this paper a number of candidate numerical schemes are tested to determine whether their truncation errors can approximate the subgrid terms of the barotropic vorticity equation. Results show that some schemes can implicitly model the effects of the subgrid term associated with the stretching and thinning of vorticity filaments to unresolvable scales; the subgrid term is then diffusive and is associated with the downscale enstrophy transfer. Conservation of vorticity, by using a flux form scheme rather than advective form for advection of vorticity, was found to improve performance of a candidate ILES scheme. Some effects of the subgrid terms could not be captured by any of the schemes tested, whether using an implicit or a simple explicit subgrid model: none of the schemes tested is able to capture the upscale transfer of energy from unresolved to resolved scales. Copyright (c) 2011 Royal Meteorological Society and British Crown Copyright, the Met Office