论文标题
在稳定分层的流体的静静力极限上
On the hydrostatic limit of stably stratified fluids with isopycnal diffusivity
论文作者
论文摘要
本文涉及严格证明在重力影响下连续分层不可压缩的流体的静静力极限。 与以前的研究相比，这项工作的主要区别是没有任何（正则化）粘度贡献添加到流体动力学方程中。仅考虑厚度扩散效应。这项工作中的扩散效应是出于对海洋学的应用，这是由额外的对流术语引起的，其特定形式是由Gent和McWilliams在1990年代提出的，以模拟非疾病分辨率系统中地球涡流相关性的有效贡献。 本文的结果在很大程度上取决于稳定分层的假设。我们为稳定分层的流体建立了静水方程和原始（非静态）方程的良好性，以及它们在消失的浅水参数极限下的收敛性。这些结果以高但有限的Sobolev规律性获得，并仔细说明所涉及的各种参数。 我们分析的一个关键要素是使用等音坐标对系统进行重新制定，从而使我们能够提供细致的能量估计值，而这些能量估计值在原始的Eulerian坐标系统中不太明显。
This article is concerned with rigorously justifying the hydrostatic limit for continuously stratified incompressible fluids under the influence of gravity. The main distinction of this work compared to previous studies is the absence of any (regularizing) viscosity contribution added to the fluid-dynamics equations; only thickness diffusivity effects are considered. Motivated by applications to oceanography, the diffusivity effects in this work arise from an additional advection term, the specific form of which was proposed by Gent and McWilliams in the 1990s to model the effective contributions of geostrophic eddy correlations in non-eddy-resolving systems. The results of this paper heavily rely on the assumption of stable stratification. We establish the well-posedness of the hydrostatic equations and the original (non-hydrostatic) equations for stably stratified fluids, along with their convergence in the limit of vanishing shallow-water parameter. These results are obtained in high but finite Sobolev regularity and carefully account for the various parameters involved. A key element of our analysis is the reformulation of the systems using isopycnal coordinates, enabling us to provide meticulous energy estimates that are not readily apparent in the original Eulerian coordinate system.
