We introduce a continuous Galerkin finite element discretization of the non-hydrostatic Boussinesq approximation of the Navier-Stokes equations, suitable for various applications such as coastal ocean dynamics and ice-ocean interactions, among others. In particular, we introduce a consistent modification of the gravity force term which enhances conservation properties for Galerkin methods without strictly enforcing the divergence-free condition. We show that this modification results in a sharp energy estimate, including both kinetic and potential energy. Additionally, we propose a new, symmetric, tensor-based viscosity operator that is especially suitable for modeling turbulence in stratified flow. The viscosity coefficients are constructed using a residual-based shock-capturing method and the method conserves angular momentum and dissipates kinetic energy. We validate our proposed method through numerical tests and use it to model the ocean circulation and basal melting beneath the ice tongue of the Ryder Glacier and the adjacent Sherard Osborn fjord in two dimensions on a fully unstructured mesh. Our results compare favorably with a standard numerical ocean model, showing better resolved turbulent flow features and reduced artificial diffusion.

翻译：本文介绍了一种适用于非静压布辛涅斯克近似的纳维-斯托克斯方程的连续伽辽金有限元离散化方法，该方法适用于多种应用场景，如海岸海洋动力学和冰-海相互作用等。特别地，我们引入了一种对重力项的一致性修正，该修正增强了伽辽金方法的守恒特性，而无需严格强制满足无散条件。我们证明了这一修正能够产生一个包含动能和势能的尖锐能量估计。此外，我们提出了一种新的、基于张量的对称粘性算子，该算子特别适用于模拟分层流中的湍流。粘性系数采用基于残差的激波捕捉方法构建，该方法能够守恒角动量并耗散动能。我们通过数值试验验证了所提出的方法，并将其应用于二维全非结构网格上，模拟了赖德冰川冰舌下方及邻近的谢拉德奥斯本峡湾的海洋环流和基底融化过程。我们的结果与标准数值海洋模型相比表现良好，显示出更好解析的湍流特征和更少的人工扩散。