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 weighted averages of skew-symmetric formulations using both flux form and material form advection of thermodynamic variables, which allow for the necessary cancellations required to conserve entropy without the chain rule. We show that such formulations allow for stable simulations of both the thermal shallow water and 3D compressible Euler equations on the sphere using mixed compatible finite elements without entropy damping.

翻译：包含热动力学的大气系统必须在能量和熵两方面保持稳定。虽然能量守恒可以通过保持运动方程哈密顿形式的斜对称结构来实现，但熵守恒通常需要作为哈密顿系统的额外不变量来推导，并通过精确保持链式法则来满足。这一过程极具挑战性，因为在兼容有限元离散中用于表示热力学变量的函数空间通常在单元边界处不连续。本研究通过构建基于加权平均的斜对称公式运动方程，同时采用热力学变量的通量形式和平流形式对流，从而在不依赖链式法则的情况下实现熵守恒所需的抵消项。我们证明，此类公式能够在不引入熵阻尼的情况下，使用混合兼容有限元在球面上对热浅水方程和三维可压缩欧拉方程进行稳定模拟。