关于正压可压缩原始方程的内在静水能量结构

中文摘要 AI 辅助

在可压缩原始方程中，重力不仅仅是一个源项：通过它所强制的静水压平衡，它组织了系统的能量几何。静水压平衡迫使正压焓与位势之和独立于垂直坐标，这是一个内在的首次静水积分，与地球物理流体动力学中的经典蒙哥马利势一致，并且用该积分检验连续性方程可得到总机械能恒等式，无需对垂直速度进行单独估计。本注记的核心是一个增广静水恒等式，通过用Bresch-Desjardins型增广速度检验动量方程而获得：对首次积分再次求导会产生两个结构函数，它们控制其交叉项，此处精确刻画了这些交叉项消失的情形——一个函数当且仅当压力为二次型时恒为零，另一个当且仅当压力为等温时恒为零，没有任何状态方程能使两者同时消失。所得的增广恒等式还分离出一个涉及垂直速度水平梯度的耦合项，该项不受本文所发展的能量结构控制，这是通过直接途径闭合Bresch-Desjardins型熵的一个局限，而非不可能性定理。对于等温状态方程，第二个函数的消失迫使垂直密度剖面为纯指数形式，这正是Ersoy-Ngom-Sy变量替换所依据的分解——蒙哥马利首次积分在该唯一状态方程下的乘法表达式，在此基础上我们给出了Wang、Dou和Jiu存在性定理的详细返回原始变量的论证。

英文摘要

In the compressible primitive equations, gravity is not merely a source term: through the hydrostatic balance it forces, it organizes the system's energy geometry. The hydrostatic balance forces the sum of the barotropic enthalpy and the geopotential to be independent of the vertical coordinate an intrinsic rst hydrostatic integral, coinciding with the classical Montgomery potential of geophysical uid dynamics and testing the continuity equation against it gives the total mechanical energy identity, with no separate estimate on the vertical velocity. The core of this note is an augmented hydrostatic identity, obtained by testing the momentum equation against a BreschDesjardins-type augmented velocity: dierentiating the rst integral again produces two structural functions governing its cross terms, whose vanishing cases are characterized exactly here one vanishes identically if and only if the pressure is quadratic, the other if and only if it is isothermal, no law annihilating both. The resulting augmented identity also isolates a coupling involving the horizontal gradient of the vertical velocity that is not controlled by the energetic structure developed here, a limitation of this direct route to closing a BreschDesjardinstype entropy, not an impossibility theorem. For the isothermal law, the vanishing of the second function forces the vertical density prole to be a pure exponential, exactly the factorization underlying the Ersoy NgomSy change of variables the multiplicative expression of the Montgomery rst integral at that one law on which basis we give a detailed return-to-original-variables argument for the existence theorem of Wang, Dou and Jiu.

发表机构

• Université Assane Seck de Ziguinchor（齐吉尼奥尔阿萨内塞克大学）

机构由 AI 辅助整理，请以论文原文为准。