Variational asymptotics for rotating shallow water near geostrophy: a transformational approach
Variational asymptotics for rotating shallow water near geostrophy: a transformational approach
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地转附近浅水旋转的变分渐进：一种变革方法
DOI：
10.1017/s0022112005008256
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发表时间：
2006
影响因子：
3.7
通讯作者：
M. Oliver
中科院分区：
文献类型：
--
作者：
M. Oliver
本文介绍了一个统一的变分框架，在这个框架中可以导出近地转浅水的经典平衡模式和几个新的模式。我们的方法是基于一致截断的旋转浅水拉格朗日函数的近恒等变换的渐近展开。模型简化是通过施加退化（在半地转尺度模型）或不可压缩性（在准地转尺度模型）相对于新的坐标。在一阶，我们恢复了经典的半地转和准地转方程，鲑鱼的L_1 $和大规模的半地转方程，以及一个单参数家庭的模式，两者之间的插值。我们确定了该家族中的一个成员，与之前已知的模型不同，它比所有其他一阶模型具有更好的规律性-因此与大规模涡旋运动保持一致。此外，我们明确推导出二阶模型所考虑的所有情况下。虽然这些二阶模型涉及非线性位涡反演，并没有明显的共享良好的性能或他们的一阶对应，我们提供了一个明确的调查二阶模型，并指出了几个途径的探索。
We introduce a unified variational framework in which the classical balance models for nearly geostrophic shallow water as well as several new models can be derived. Our approach is based on consistently truncating an asymptotic expansion of a near-identity transformation of the rotating shallow-water Lagrangian. Model reduction is achieved by imposing either degeneracy (for models in a semi-geostrophic scaling) or incompressibility (for models in a quasi-geostrophic scaling) with respect to the new coordinates. At first order, we recover the classical semi-geostrophic and quasi-geostrophic equations, Salmon's $L_1$ and large-scale semi-geostrophic equations, as well as a one-parameter family of models that interpolate between the two. We identify one member of this family, different from previously known models, that promises better regularity – hence consistency with large-scale vortical motion – than all other first-order models. Moreover, we explicitly derive second-order models for all cases considered. While these second-order models involve nonlinear potential vorticity inversion and do not obviously share the good properties or their first-order counterparts, we offer an explicit survey of second-order models and point out several avenues for exploration.