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首页> 外文期刊>Communications in mathematical sciences >MINIMAX VARIATIONAL PRINCIPLE FOR STEADY BALANCED SOLUTIONS OF THE ROTATING SHALLOW WATER EQUATIONS
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MINIMAX VARIATIONAL PRINCIPLE FOR STEADY BALANCED SOLUTIONS OF THE ROTATING SHALLOW WATER EQUATIONS

机译：旋转浅水方程组稳定平衡解的MINIMAX变分原理

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摘要

Two well-known variational principles for geophysical flows are combined into a single minimax principle that characterizes distinguished steady solutions of the rotating shallow water ( RSW) equations. On the one hand, in the limit of small Rossby number ?, in which the dynamics becomes quasi-geostrophic and closes terms of the potential vorticity field Q, steady coherent states are characterized as minimizers of (generalized) enstrophy A at a given value of total energy H. On the other hand, for small amplitude motions at finite c balanced states resulting from geostrophic adjustment are characterized as minimizers of the total energy subject to a given potential vorticity Q. Moreover, the organization into a coherent state through potential vorticity mixing occurs on a slow time scale relative to the fast time scale of adjustment through inertia-gravity wave radiation. These two complementary principles suggest a variational characterization of steady balanced states for the RSW equations at finite f. Namely, the functional A-I-0N, where 0 < 0 is a parameter, is first maximized over all RSW fields with given Q, and then minimized over all Q. Any such minimax critical point of A± ON is an exact steady solution of the RSW equations, which represents a physically relevant equilibrium state at finite Rossby number. This minimax principle is implemented numerically for zonal shear flows, and branches of solutions are computed to first-order in E. The results quantify the breakdown of quasi-geostrophy and the asymmetry between cyclonic and anticyclonic structures. In addition, the O(?)-correction is computed for a model of the zonally-averaged winds in Jupiter's weather layer.
机译：地球物理流的两种众所周知的变分原理被组合为一个极小极大值原理，该极小值原理描述了旋转浅水（RSW）方程的显着稳态解。一方面，在小的Rossby数β的极限中，其中动力学变为准地转并封闭了潜在涡度场Q的项，稳态相干态被表征为在给定值α下（广义）涡旋A的极小值。另一方面，对于有限c处的小振幅运动，由地转调节产生的平衡态的特征是在给定势涡Q的作用下总能量的极小值。此外，通过势涡混合，组织成为相干态相对于通过惯性重力波辐射进行快速调整的时间尺度而言，这种情况发生在较慢的时间尺度上。这两个互补原理表明RSW方程在有限f处的稳态平衡状态具有变化特征。即，功能AI-0N（其中0 <0是一个参数）首先在具有给定Q的所有RSW字段上最大化，然后在所有Q上最小化。A±ON的任何此类minimax临界点都是RSW方程，表示在有限Rossby数下的物理相关平衡状态。此minimax原理是通过数值方式实现的，用于区域剪切流，并在E中将溶液的分支计算为一阶。结果量化了准大地层的破裂以及旋风和反旋风结构之间的不对称性。另外，针对木星天气层中纬向平均风的模型计算O（？）校正。

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