Abstract Accurate representation of vertical turbulent fluxes is crucial for numerical ocean modeling, both in global and coastal applications. The state-of-the-art approach is to use two-equation turbulence closure models which introduces two dynamic equations to the system. Solving these equations numerically, however, is challenging due to the strict requirement of positivity of the turbulent quantities (e.g., turbulence kinetic energy and its dissipation rate), and the non-linear source terms that may render the numerical system unstable. In this paper, we present a Discontinuous Galerkin (DG) finite element discretization of the Generic Length Scale (GLS) equations designed to be incorporated in a DG coastal ocean model, Thetis. To ensure numerical stability, the function space for turbulent quantities must be chosen carefully. In this work, we propose to use zeroth degree elements for the turbulent quantities and linear discontinuous elements for the tracers and velocity. The spatial discretization is completed with a positivity preserving semi-implicit time integration scheme. We validate the implementation with standard turbulence closure model benchmarks and an idealized estuary simulation. Finally, we use the full three-dimensional model to simulate the Columbia River plume. The results confirm that the coupled model generates realistic vertical mixing, and remains stable under strongly stratified conditions and strong tidal forcing. River plume characteristics are well captured.

中文翻译：

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两方程湍流闭合模型的不连续伽辽金离散化

摘要 垂直湍流通量的准确表示对于全球和沿海应用中的海洋数值模拟至关重要。最先进的方法是使用两方程湍流闭合模型，该模型将两个动力学方程引入系统。然而，由于对湍流量（例如，湍流动能及其耗散率）的正性的严格要求以及可能使数值系统不稳定的非线性源项，以数值方式求解这些方程具有挑战性。在本文中，我们提出了通用长度尺度 (GLS) 方程的不连续伽辽金 (DG) 有限元离散化，旨在纳入 DG 沿海海洋模型 Thetis。为了确保数值稳定性，必须谨慎选择湍流量的函数空间。在这项工作中，我们建议对湍流量使用零度元素，对示踪剂和速度使用线性不连续元素。空间离散化是通过保留正性的半隐式时间积分方案完成的。我们使用标准湍流闭合模型基准和理想化的河口模拟来验证实施。最后，我们使用全三维模型来模拟哥伦比亚河羽流。结果证实，耦合模型产生了真实的垂直混合，并在强烈分层条件和强潮汐强迫下保持稳定。河流羽流特征被很好地捕获。空间离散化是通过保留正性的半隐式时间积分方案完成的。我们使用标准湍流闭合模型基准和理想化的河口模拟来验证实施。最后，我们使用全三维模型来模拟哥伦比亚河羽流。结果证实，耦合模型产生了真实的垂直混合，并在强烈分层条件和强潮汐强迫下保持稳定。河流羽流特征被很好地捕获。空间离散化是通过保留正性的半隐式时间积分方案完成的。我们使用标准湍流闭合模型基准和理想化的河口模拟来验证实施。最后，我们使用全三维模型来模拟哥伦比亚河羽流。结果证实，耦合模型产生了真实的垂直混合，并在强烈分层条件和强潮汐强迫下保持稳定。河流羽流特征被很好地捕获。并在强分层条件和强潮汐强迫下保持稳定。河流羽流特征被很好地捕获。并在强分层条件和强潮汐强迫下保持稳定。河流羽流特征被很好地捕获。