Commutation error correction for large eddy simulations of convection driven dynamos

Abstract

An error arises in large eddy simulations when the order of spatial differentiation and filtering operations is commuted on a non-uniform grid. The magnitude of the commutation error is controlled by spatial variations in the width of the filter function, which is conventionally defined in terms of the local grid size. We have previously found large commutation errors in the sub-grid scale (SGS) terms of a numerical dynamo model, particularly near boundaries where the grid size changes most rapidly. In the present study, we propose a correction for the commutation error which is specifically designed for use with the nonlinear gradient SGS model. The commutation error correction for each SGS term can be expressed as the product of the spatial derivative of the second-order moment of the filter function and the second spatial derivative of the SGS term. We test the correction using output from a fully resolved convection-driven dynamo simulation in a rotating plane layer. As an example, we evaluate the SGS heat flux on a coarser grid with and without the commutation error correction. The result is tested using the resolved dynamo solution on a finer grid. We find that the commutation error correction is comparable in magnitude to the SGS heat flux near the boundary. Addition of the correction term significantly improves the modeled SGS heat flux near the boundaries.