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Numerical investigation of the regularity of the pressure for the primitive equations of the ocean

机译：海洋原始方程组压力规律性的数值研究

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摘要

In this work we analyze the regularity of the pressure of the primitive equations (PE) of the ocean by numerical simulation and analysis. This model makes use of the hydrostatic pressure assumption. Several authors have shown that the hydrostatic hypothesis limits the regularity of the surface pressure to L_D~(3/2) (ω), where the weight D is the depth of the domain, and ω is the surface domain. Nevertheless, the same L~2(ω) regularity as in the Navier-Stokes case can be proved when the domain has a talus. We address in this paper the question whether this gap of regularity is also observable in the numerical approximation of the primitive equations. Specifically, for the numerical solver of the primitive equations considered, we prove convergence of the surface pressure in L_D~(3/2) (ω) and in L~2(ω′) for subdomains ω′ is contained in ω with talus. This lack of regularity near the border is confirmed by numerical tests, where we observe the formation of an infinite normal derivative in partial deriv ω of the surface pressure as the size of the talus vanishes. Moreover, we show that the L_D~(3/2) (ω) regularity obtained in the theory for domains without talus is not far from being optimal.
机译：在这项工作中，我们通过数值模拟和分析来分析海洋原始方程（PE）压力的规律性。该模型利用了静水压力假设。几位作者表明，流体静力学假设将表面压力的规律性限制为L_D〜（3/2）（ω），其中权重D是域的深度，ω是表面域。然而，当域具有距骨时，可以证明与Navier-Stokes情况相同的L〜2（ω）正则性。我们在本文中讨论的问题是，在原始方程的数值逼近中是否也可以观察到这种规律性差距。具体而言，对于所考虑的原始方程的数值求解器，我们证明了L_D〜（3/2）（ω）和L〜2（ω'）中的表面压力的收敛性，因为子域ω'包含在距骨中。通过数值测试证实了边界附近这种不规则性，在数值测试中，随着距骨尺寸的消失，我们观察到在表面压力的偏导数ω中形成了无限的正态导数。此外，我们证明了在理论上对于没有距骨的区域，L_D〜（3/2）（ω）正则性距离最优值并不遥远。

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