准地转方程的非线性滤波稳定化
Nonlinear filtering stabilizations for the quasi-geostrophic equations
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- Clemson University（克莱姆森大学）
- University of Houston（休斯顿大学）
- Technical University of Munich（慕尼黑工业大学）
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中文总结 AI 辅助
针对准地转方程粗网格模拟，提出演化-滤波-松弛（EFR）算法及非线性Bardina正则化，显著提升精度与稳定性，其中EFR在地中海案例中综合性能最佳。
中文摘要 AI 辅助
海洋流动的数值模拟通常需要精细的计算网格来解析芒克尺度，导致高昂的计算成本。基于滤波的大涡模拟（LES）通过模拟未解析尺度的影响，提供了一种放宽网格尺寸要求的方法。为实施这一策略，我们提出了一种名为“演化-滤波-松弛”（EFR）的三步算法，该算法需要（i）求解一个准地转方程（QGE）问题，（ii）对位势涡度场应用一个利用指示函数的非线性亥姆霍兹滤波器，以及（iii）一个最终的松弛步骤。我们证明，EFR算法可以解释为具有额外耗散的扰动QGE问题的分裂格式，并为松弛参数提供了实用的选择。为了比较，我们还研究了QGE的非线性Bardina正则化。在一个经典基准上的数值结果表明，EFR方法和非线性Bardina正则化都显著提高了无LES模型的粗网格模拟的精度和稳定性。此外，在涉及更真实几何形状（地中海）的测试案例中，采用基于去卷积的指示函数的EFR方法在精度、稳定性和计算效率之间取得了最佳平衡。
英文摘要
Numerical simulations of ocean flows typically require fine computational meshes to resolve the Munk scale, leading to high computational costs. Filtering-based large eddy simulation (LES) provides a way to relax the mesh size requirement by modeling the effects of the unresolved scales. For the implementation of this strategy, we propose a three-step algorithm called Evolve-Filter-Relax (EFR) that requires (i) the solution of a QGE problem, (ii) a nonlinear Helmholtz filter for the potential vorticity field leveraging an indicator function, and (iii) a final relaxation step. We show that the EFR algorithm can be interpreted as a splitting scheme for a perturbed QGE problem with additional dissipation and provide a practical choice for the relaxation parameter. For comparison, we also investigate a nonlinear Bardina regularization of the QGE. Numerical results on a classical benchmark show that both the EFR approach and the nonlinear Bardina regularization significantly improve the accuracy and stability of coarse mesh simulations with no LES model. Additionally, the EFR method with a deconvolution-based indicator function delivered the best balance between accuracy, stability, and computational efficiency in a test case involving a more realistic geometry (Mediterranean Sea).
