Theoretical and Computational Fluid Dynamics Manuscript No. (will Be Inserted by the Editor) Nonlinear Dynamics of Hydrostatic Internal Gravity Waves
Theoretical and Computational Fluid Dynamics Manuscript No. (will Be Inserted by the Editor) Nonlinear Dynamics of Hydrostatic Internal Gravity Waves
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理论与计算流体动力学手稿编号（将由编辑插入）静水重力内波的非线性动力学
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通讯作者：
B. Khouider
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作者：
S. Stechmann;A. Majda;B. Khouider
层状静水流体具有不同相速度和垂直剖面的线性重力内波。本文推导了一组简化的偏微分方程组(PDE)来描述不同垂直剖面波的非线性动力学。这些方程是通过将完全非线性方程投影到两个重力波的垂直模上而得到的，由此得到的方程在这里被称为双模浅水方程(2MSWE)。2MSWE的非线性的一个关键方面是，它们允许背景风切变和传播的波之间的相互作用。这在热带大气中很重要，在热带大气中，水平传播的重力波与风切变相互作用，并由于对流而产生源项。结果表明，2MSWE具有非线性内孔解，并研究了不同背景风切变下非线性波的行为。当考虑背景切变时，向东和向西传播的波之间存在不对称性。这可能是热带对流大范围组织的一个重要影响，因为对流通常不是各向同性的，而是以波的形式在大尺度上组织的。对于Madden-Julian振荡西风暴发位相的背景切变，给出了这种不对称性的理想化解释；传播的非线性重力波使有组织对流的位势增加到现有对流以西，这与实际观测定性一致。这里的想法也应该对其他物理应用程序有用。此外，2MSWE还具有一些有趣的数学性质：它们是一个能量守恒的非保守偏微分方程组，它们是条件双曲的，它们在整个状态空间既不是真正的非线性也不是线性退化的。理论和数值计算被用来说明这些特征，这些特征在数值格式的设计中是重要的。以简单性和计算量最小为主要设计原则，设计了一种数值方法。数值试验表明，当超音性丧失时，不会引入突变效应，并且该格式可以表示传播不连续而不会引入虚假振荡。
Stratified hydrostatic fluids have linear internal gravity waves with different phase speeds and vertical profiles. Here a simplified set of partial differential equations (PDE) is derived to represent the nonlinear dynamics of waves with different vertical profiles. The equations are derived by projecting the full nonlinear equations onto the vertical modes of two gravity waves, and the resulting equations are thus referred to here as the two-mode shallow water equations (2MSWE). A key aspect of the nonlinearities of the 2MSWE is that they allow for interactions between a background wind shear and propagating waves. This is important in the tropical atmosphere where horizontally propagating gravity waves interact together with wind shear and have source terms due to convection. It is shown here that the 2MSWE have nonlinear internal bore solutions, and the behavior of the nonlinear waves is investigated for different background wind shears. When a background shear is included, there is an asymmetry between the east-and westward propagating waves. This could be an important effect for the large-scale organization of tropical convection, since the convection is often not isotropic but organized on large scales by waves. An idealized illustration of this asymmetry is given for a background shear from the westerly wind burst phase of the Madden–Julian oscillation; the potential for organized convection is increased to the west of the existing convection by the propagating nonlinear gravity waves, which agrees qualitatively with actual observations. The ideas here should be useful for other physical applications as well. Moreover, the 2MSWE have several interesting mathematical properties: they are a system of non-conservative PDE with a conserved energy, they are conditionally hyperbolic, and they are neither genuinely nonlinear nor linearly degenerate over all of state space. Theory and numerics are developed to illustrate these features, and these features are important in designing the numerical scheme. A numerical method is designed with simplicity and minimal computational cost as the main design principles. Numerical tests demonstrate that no catastrophic effects are introduced when hyperbol-icity is lost, and the scheme can represent propagating discontinuities without introducing spurious oscillations.