Moist thermodynamics is a fundamental driver of atmospheric dynamics across all scales, making accurate modeling of these processes essential for reliable weather forecasts and climate change projections. However, atmospheric models often make a variety of inconsistent approximations in representing moist thermodynamics. These inconsistencies can introduce spurious sources and sinks of energy, potentially compromising the integrity of the models. Here, we present a thermodynamically consistent and structure preserving formulation of the moist compressible Euler equations. When discretised with a summation by parts method, our spatial discretisation conserves: mass, water, entropy, and energy. These properties are achieved by discretising a skew symmetric form of the moist compressible Euler equations, using entropy as a prognostic variable, and the summation-by-parts property of discrete derivative operators. Additionally, we derive a discontinuous Galerkin spectral element method with energy and tracer variance stable numerical fluxes, and experimentally verify our theoretical results through numerical simulations.

翻译：湿热力学是驱动所有尺度大气动力学的基本机制，因此对这些过程的精确建模对于可靠的天气预报和气候变化预测至关重要。然而，大气模型在表示湿热力学时常常采用各种不一致的近似。这些不一致性可能引入虚假的能量源与汇，从而可能损害模型的完整性。本文提出了一种热力学一致且结构保持的湿可压缩欧拉方程形式。当采用分部求和方法进行离散时，我们的空间离散方案能够守恒：质量、水物质、熵以及能量。这些性质是通过离散湿可压缩欧拉方程的斜对称形式、使用熵作为预报变量，以及利用离散导数算子的分部求和性质来实现的。此外，我们推导了一种具有能量示踪剂方差稳定数值通量的间断伽辽金谱元方法，并通过数值模拟实验验证了我们的理论结果。