Physical vs. numerical dispersion in nonhydrostatic ocean modeling
Physical vs. numerical dispersion in nonhydrostatic ocean modeling
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DOI：
10.1016/j.ocemod.2011.07.002
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发表时间：
2011
期刊：
影响因子：
3.2
通讯作者：
S. Vitousek;O. Fringer
中科院分区：
文献类型：
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作者：
S. Vitousek;O. Fringer
内波的许多大尺度模拟都是用海洋模型求解原始（流体静力学）方程来计算的。然而，在某些情况下，内波可以代表非线性和非流体静力学（色散）之间的动态平衡，因此可能需要计算昂贵的非流体静力学模拟得到很好的解决。原始方程的大多数离散化是二阶精度的，从而导致截断误差（三阶导数和更高）中的奇阶项产生的数值色散。这种数值色散模拟由于非水溶性引起的物理色散。在本文中，我们确定的数值色散系数与常见的原始方程的离散。我们比较这个系数的物理色散系数从Boussinesq方程或KdV方程。结果表明，在最低阶，数值色散与物理色散之比为Γ=Kλ2，其中K是一个依赖于控制方程离散的O（1）常数，λ是网格间距比，λ <$Δx/h1，其中Δx是水平网格间距，h1是内界面的深度.除了推导出这种关系，我们验证，它确实在一个非流体静力学海洋模式（SUNTANS）。为了确保物理效应相对于数值效应的相对优势，模拟要求Γ 1。基于这一条件，正确分辨非静力效应所需的水平网格间距为λ<O（1）或Δx<h1。当这个条件不满足时，数值色散取代了物理色散，模型内波在非线性和数值色散之间动态平衡。满足这一条件可能是一个显着的额外的分辨率要求超出了目前的国家的最先进的海洋建模。
Many large-scale simulations of internal waves are computed with ocean models solving the primitive (hydrostatic) equations. Under certain circumstances, however, internal waves can represent a dynamical balance between nonlinearity and nonhydrostasy (dispersion), and thus may require computationally expensive nonhydrostatic simulations to be well-resolved. Most discretizations of the primitive equations are second-order accurate, inducing numerical dispersion generated from odd-order terms in the truncation error (3rd-order derivatives and higher). This numerical dispersion mimics physical dispersion due to nonhydrostasy. In this paper, we determine the numerical dispersion coefficient associated with common discretizations of the primitive equations. We compare this coefficient to the physical dispersion coefficient from the Boussinesq equations or KdV equation. The results show that, to lowest order, the ratio of numerical to physical dispersion is Γ=Kλ2, where K is an O(1) constant dependent on the discretization of the governing equations and λ is the grid leptic ratio, λ≡Δx/h1, where Δx is the horizontal grid spacing and h1is the depth of the internal interface. In addition to deriving this relationship, we verify that it indeed holds in a nonhydrostatic ocean model (SUNTANS). To ensure relative dominance of physical over numerical effects, simulations require Γ≪1. Based on this condition, the horizontal grid spacing required for proper resolution of nonhydrostatic effects is λ<O(1) or Δx<h1. When this condition is not satisfied, numerical dispersion overwhelms physical dispersion, and modeled internal waves exist with a dynamical balance between nonlinearity and numerical dispersion. Satisfaction of this condition may be a significant additional resolution requirement beyond the current state-of-the-art in ocean modeling.