Extending Square Conservation to Arbitrarily Structured C-grids with Shallow Water Equations

Abstract. The square conservation theory is widely used on latitude–longitude grids, but it is rarely implemented on quasi-uniform grids, given the difficulty involved in constructing anti-symmetrical spatial discrete operators on these grids. Increasingly more models are developed on quasi-uniform grids, such as arbitrarily structured C-grids. Thuburn–Ringler–Skamarock–Klemp (TRiSK) is a shallow water dynamic core on an arbitrarily structured C-grid. The spatial discrete operator of TRiSK is able to naturally maintain the conservation properties of total mass, total absolute vorticity and instantaneous total energy. The first 2 integral invariants are entirely conserved during integration, but the total energy dissipates when using the dissipative temporal integration schemes, i.e., Runge-Kutta. The method of strictly conserving the total energy simultaneously uses both an anti-symmetrical spatial discrete operator and square conservative temporal integration scheme. In this study, we demonstrate that square conservation is equivalent to energy conservation in both a continuous shallow water system and a discrete shallow water system of TRiSK, attempting to extend the square conservation theory to the TRiSK framework. To overcome the challenge of constructing an anti-symmetrical spatial discrete operator, we unify the unit of evolution variables of shallow water equations by Institute of Atmospheric Physics (IAP) transformation, expressing the temporal trend of the evolution variable by using the original operators of TRiSK. Using the square conservative Runge-Kutta scheme, the total energy is completely conserved, and there is no influence on the properties of conserving total mass and total absolute vorticity. In the standard shallow water numerical test, the square conservative scheme not only helps maintain total conservation of the three integral invariants but also creates less simulation error norms.