Entropy and energy conservation for thermal atmospheric dynamics using mixed compatible finite elements

Abstract

Atmospheric systems incorporating thermal dynamics must be stable with respect to both energy and entropy. While energy conservation can be enforced via the preservation of the skew-symmetric structure of the Hamiltonian form of the equations of motion, entropy conservation is typically derived as an additional invariant of the Hamiltonian system, and satisfied via the exact preservation of the chain rule. This is particularly challenging since the function spaces used to represent the thermodynamic variables in compatible finite element discretisations are typically discontinuous at element boundaries. In the present work we negate this problem by constructing our equations of motion via a novel formulation which allow for the necessary cancellations required to simultaneously conserve entropy and energy without the chain rule. We show that such formulations allow for stable simulation of turbulent dynamics for both the thermal shallow water and 3D compressible Euler equations on the sphere using mixed compatible finite elements without entropy damping.