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个方面保持稳定。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强制执行，但熵守恒通常被推导为哈密顿系统的一个额外不变量，并通过精确保持链式法则来实现。由于在相容有限元离散中表示热力学变量的函数空间通常在单元边界处不连续，这一任务尤为困难。在本文中，我们通过构建基于热力学变量的通量形式和平流形式的斜对称公式的加权平均运动方程来解决这一问题，这些公式使得无需链式法则即可实现熵守恒所需的抵消。我们证明，此类公式允许在球面上使用混合相容有限元方法进行热浅水方程和三维可压缩欧拉方程的稳定模拟，且无需熵阻尼。