用于热力浅水方程的、具有能量与熵守恒特性并含迎风格式的相容有限元方法
Energy and entropy conserving compatible finite elements with upwinding for the thermal shallow water equations
AI总结：
本文提出一种守恒能量与浮力相关熵的热力浅水方程相容有限元格式，通过中心与迎风通量及变分时间积分，在热地转不稳定性和双涡旋算例中验证了稳定性与守恒性。
摘要：
在本工作中，我们开发了一种新的热力浅水方程相容有限元格式，该格式能够守恒能量以及由浮力相关二次示踪物方差给出的数学熵。我们的方法依赖于重新表述控制方程，以实现热力学变量的间断近似和变分连续时间积分。一个关键创新是引入了中心通量和迎风通量。所提出的半离散系统在使用中心通量时守恒离散熵，在使用迎风通量时单调耗散熵，并且守恒能量。全离散格式在连续层面上体现了熵守恒。一种新的线性化 Jacobian 同时考虑了中心通量和迎风通量，其捕捉浮力大幅变化并长时间模拟热力不稳定流动的能力，在两个不同的瞬态案例研究中得到了验证。第一个案例涉及热地转不稳定性，结果表明引入迎风通量能够抑制虚假振荡，同时成功守恒能量并单调耗散熵。第二个案例是双涡旋，其中受约束的全离散格式被证明在时间上实现了精确的熵守恒。
英文摘要：
In this work, we develop a new compatible finite element formulation of the thermal shallow water equations that conserves energy and mathematical entropies given by buoyancy-related quadratic tracer variances. Our approach relies on restating the governing equations to enable discontinuous approximations of thermodynamic variables and a variational continuous time integration. A key novelty is the inclusion of centred and upwinded fluxes. The proposed semi-discrete system conserves discrete entropy for centred fluxes, monotonically damps entropy for upwinded fluxes, and conserves energy. The fully discrete scheme reflects entropy conservation at the continuous level. The ability of a new linearised Jacobian, which accounts for both centred and upwinded fluxes, to capture large variations in buoyancy and simulate thermally unstable flows for long periods of time is demonstrated for two different transient case studies. The first involves a thermogeostrophic instability where including upwinded fluxes is shown to suppress spurious oscillations while successfully conserving energy and monotonically damping entropy. The second is a double vortex where a constrained fully discrete formulation is shown to achieve exact entropy conservation in time.
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