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 the stable simulation of both the thermal shallow water and 3D compressible Euler equations on the sphere using mixed compatible finite elements without entropy damping.

翻译：结合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强制执行，但熵守恒通常被推导为哈密顿系统的额外不变量，并通过链式法则的精确保持来实现。这尤其具有挑战性，因为在相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。在本文中，我们通过使用热力学变量的通量形式和平流形式构建加权平均的斜对称公式的运动方程来规避这一问题，该方法允许在无需链式法则的情况下实现熵守恒所需的抵消。我们证明，这类公式能够在使用混合相容有限元方法且无熵阻尼的球面上，稳定模拟热浅水方程和三维可压缩欧拉方程。