Frame invariance and conservation in two-dimensional subgrid models

We explore how symmetry and conservation requirements affect the form of subgrid models in large eddy simulations (LES) of two-dimensional incompressible turbulence. We expose two possible strategies that can be adopted in order to preserve these symmetries in the two-dimensional case: a representational approach where the invariance properties are taken as the basic ingredient guiding the formulation of the model and the variational Euler-Poincaré procedure where the invariance properties are a built-in feature of the resulting Euler-α equations. Assuming spatial locality, we find that at leading order in spatial derivatives of the resolved velocity field, the only compatible model is the strain-diffusivity model. Formal analogies between the Euler-α model and the strain-diffusivity model allow us to derive a new "ω-α" model with conservation properties presumably more suited to the two-dimensional situation. In this process, the strain-diffusivity operator appears as a truncated version of both Euler-α and "ω-α" models. The numerical properties of the three considered models are estimated and discussed.