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Atmospheric systems incorporating thermal dynamics must be stable withrespect to both energy and entropy. While energy conservation can be enforcedvia the preservation of the skew-symmetric structure of the Hamiltonian form ofthe equations of motion, entropy conservation is typically derived as anadditional invariant of the Hamiltonian system, and satisfied via the exactpreservation of the chain rule. This is particularly challenging since thefunction spaces used to represent the thermodynamic variables in compatiblefinite element discretisations are typically discontinuous at elementboundaries. In the present work we negate this problem by constructing ourequations of motion via weighted averages of skew-symmetric formulations usingboth flux form and material form advection of thermodynamic variables, whichallow for the necessary cancellations required to conserve entropy without thechain rule. We show that such formulations allow for stable simulations of boththe thermal shallow water and 3D compressible Euler equations on the sphereusing mixed compatible finite elements without entropy damping.

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