[论文解读] Mathematical study of degenerate boundary layers: A Large Scale Ocean Circulation Problem

Anne-Laure Dalibard, Laure Saint‐Raymond|arXiv (Cornell University)|Mar 26, 2012

Advanced Mathematical Modeling in Engineering参考文献 16被引用 7

一句话总结

本文对大尺度海洋环流模型中的定态Munk方程进行了严格的渐近分析，解决了由于地转退化而在南北海岸附近出现的退化边界层问题。研究建立了新的北部/南部边界层类，其尺度为$\mathfrak{E}^{1/4}$，大于经典西部/东部边界层（尺度为$\mathfrak{E}^{1/3}$），并证明了在边界层分离和非局部效应存在的情况下仍能实现收敛，消除了先前研究中对强迫项和区域几何形状的限制性假设。

ABSTRACT

This paper is concerned with a complete asymptoticanalysis as $\\mathfrak{E} \ o 0$ of the stationary Munk equation $\\partial\\_x\\psi-\\mathfrak{E} \\Delta^2 \\psi=\ au$ in a domain $\\Omega\\subset \\mathbf{R}^2$, supplemented with boundaryconditions for $\\psi $ and $\\partial\\_n \\psi$. This equation is a simplemodel for the circulation of currents in closed basins, the variables$x$ and $y$ being respectively the longitude and the latitude. A crudeanalysis shows that as $\\mathfrak{E} \ o 0$, the weak limit of $\\psi$ satisfiesthe so-called Sverdrup transport equation inside the domain, namely$\\partial\\_x \\psi^0=\ au$, while boundary layers appear in the vicinity ofthe boundary.These boundary layers, which are the main center of interest of thepresent paper, exhibit several types of peculiar behaviour. First, thesize of the boundary layer on the western and eastern boundary, whichhad already been computed by several authors, becomes formally verylarge as one approaches northern and southern portions of the boudary,i.e. pieces of the boundary on which the normal is vertical. Thisphenomenon is known as geostrophic degeneracy. In order to avoid suchsingular behaviour, previous studies imposed restrictive assumptionson the domain $\\Omega$ and on the forcing term $\ au$. Here, we provethat a superposition of two boundary layers occurs in the vicinity ofsuch points: the classical western or eastern boundary layers, andsome northern or southern boundary layers, whose mathematicalderivation is completely new. The size of northern/southern boundarylayers is much larger than the one of western boundary layers($\\mathfrak{E}^{1/4}$ vs. $\\mathfrak{E}^{1/3}$). We explain in detail how the superpositiontakes place, depending on the geometry of the boundary.Moreover, when the domain $\\Omega$ is not connex in the $x$ direction,$\\psi^0$ is not continuous in $\\Omega$, and singular layers appear inorder to correct its discontinuities. These singular layers areconcentrated in the vicinity of horizontal lines, and thereforepenetrate the interior of the domain $\\Omega$. Hence we exhibit some kindof boundary layer separation. However, we emphasize that we remainable to prove a convergence theorem, so that the singular layerssomehow remain stable, in spite of the separation.Eventually, the effect of boundary layers is non-local in severalaspects. On the first hand, for algebraic reasons, the boundary layerequation is radically different on the west and east parts of theboundary. As a consequence, the Sverdrup equation is endowed with aDirichlet condition on the East boundary, and no condition on the Westboundary. Therefore western and eastern boundary layers have in factan influence on the whole domain $\\Omega$, and not only near theboundary. On the second hand, the northern and southern boundary layerprofiles obey a propagation equation, where the space variable $x$plays the role of time, and are therefore not local.

研究动机与目标

• 分析当$\mathfrak{E} \to 0$时，定态Munk方程在具有复杂边界几何形状的区域中的渐近行为。
• 解决在北部和南部边界段附近由于地转退化导致经典边界层形式上趋于无穷大的问题。
• 构建一个完整的渐近展开，包含叠加的西部/东部与北部/南部边界层，且不假设极点处强迫项为零。
• 证明尽管存在边界层分离与非局部相互作用，近似解仍能收敛。
• 消除先前研究中的限制性假设，这些假设要求在临界边界点附近强迫项或区域几何形状趋于零。

提出的方法

• 在边界附近使用局部边界层坐标系推导多尺度渐近展开，不同边界段采用不同的标度。
• 引入一类新的北部/南部边界层剖面，其由一个传播方程控制，其中$x$作为时间变量。
• 通过匹配渐近展开与截断技术，构建东部、西部、北部和南部边界层的校正项。
• 定义内部奇异层，以校正当区域在$x$方向不连通时Sverdrup解中的不连续性。
• 使用提升过程$\psi^{\mathrm{lift}}$和奇异层$\psi^{\Sigma}$来稳定不连续性，并确保在$H^2$范数下的收敛性。
• 应用曲线坐标变换，并在加权Sobolev范数下进行估计，以控制余项分析中的误差项。

实验结果

研究问题

• RQ1 在北部和南部海岸附近，法向量为竖直方向时，边界层的行为如何，这会导致地转退化？
• RQ2 与经典西部/东部边界层相比，北部/南部边界层的正确标度与结构是什么？
• RQ3 当Sverdrup解$\psi^0$不连续时，如何从数学上描述并控制边界层分离？
• RQ4 为何西部和东部边界层会对整个区域产生非局部影响，这对Sverdrup方程有何影响？
• RQ5 在叠加的、非局部的和奇异的边界层存在的情况下，能否建立收敛定理？

主要发现

• 北部和南部边界层的尺度为$\mathfrak{E}^{1/4}$，显著大于经典西部/东部边界层的尺度$\mathfrak{E}^{1/3}$。
• 在角部区域发生西部/东部与北部/南部边界层的叠加，其中后者在厚度上占主导地位。
• 由于边界层的非局部影响，Sverdrup方程在东部边界具有狄利克雷条件，而在西部边界无条件。
• 当区域在$x$方向不连通时，$L^2$范数下尺度为$\mathfrak{E}^{1/8}$、$H^2$范数下尺度为$\mathfrak{E}^{-3/8}$的内部奇异层可校正$\psi^0$中的不连续性。
• 尽管存在边界层分离，收敛定理仍成立，误差项通过截断与提升技术得到控制。
• 北部/南部边界层的剖面遵循关于$x$的传播方程，使其影响在整个区域内呈现非局部性。