An energy-conserving restoration scheme for the shallow-water equations

The numerical methods that solve the governing equations in an atmospheric dynamical core are designed to dissipate potential enstrophy and prevent the build-up of kinetic energy at the grid scale. A side-effect of this is the dissipation of total energy which should be conserved. Energy fixers are used in climate models to replace the dissipated energy by modifying the temperature in the thermodynamic equation, and stochastic backscatter schemes have also been developed for use in weather prediction models. Here, we present the first steps towards designing a deterministic energy-conserving restoration scheme that considers the conversion of kinetic energy to heat, replacing kinetic energy lost due to model error, and the backscatter of kinetic energy. The energy-conserving restoration scheme (ECRS) is presented in the context of the shallow-water equations on the sphere. It is designed to be used with any existing shallow-water equation scheme (called the preliminary scheme) which can adequately dissipate potential enstrophy, and in this article we use a semi-implicit semi-Lagrangian (SISL) scheme. For each prognostic variable, a spatial pattern is chosen; this is added to the preliminary scheme solution, and the amount added is calculated to ensure energy conservation. Results from short-term test cases show that ECRS and SISL have very similar error norms. For long-term simulations, ECRS conserves energy to a good approximation whereas SISL dissipates energy.