Nonlinear Advection Schemes and Energy Cascade on Semi-Staggered Grids

Abstract

A common problem with nonlinear advection schemes is the false accumulation of energy at the smallest resolvable scales. To keep this process under control, following Arakawa (1966), a number of energy and enstrophy conserving schemes for staggered and semi-staggered grids have been designed. In this paper, it is demonstrated that, in contrast to the staggered grid, the conservation of energy and enstrophy on the semi-staggered gods does not guarantee that the erroneous transport of energy from large to small scales will be effectively restricted. Using a new approach to the application of the Arakawa Jacobian, a scheme for a semi-staggered grid which exactly reflects the Arakawa theory for nondivergent flow is obtained for the first time. This is achieved by conservation of energy and enstrophy as defined on the staggered grid. These two quantities are of higher accuracy and cannot be calculated directly from the dependent variables on the semi-staggered grid. It is further demonstrated that the amount of energy which can be transported toward smaller scales is more restricted than for any other scheme of this type on both staggered and semi-staggered grid. Experiments performed with the proposed scheme and a scheme which conserves energy and enstrophy as defined on the semi-staggered grid reveal visible differences in long-term integrations which are in agreement with the theory and demonstrate the advantages of the new scheme.