Generalized semi-geostrophic theory on a sphere
Generalized semi-geostrophic theory on a sphere
复制标题
球体的广义半地转理论
DOI：
10.1017/s0022112005003812
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发表时间：
2005
影响因子：
3.7
通讯作者：
M. J. Sewell
中科院分区：
文献类型：
--
作者：
M. Cullen;R. Douglas;I. Roulstone;M. J. Sewell
结果表明，浅水流半地转方程的解可以通过与现有 $f$ 平面解中使用的方法类似的方法在球面几何中找到和分析。地转平衡的稳定状态被识别为能量最小化器，并构建了寻找最小化器的程序，这是位涡反演的一种形式。这定义了 $f$ 平面理论中使用的地转坐标变换的概括。该过程通过计算进行了演示。演化方程在变换后的坐标中采用简单的形式，但正如文献中先前工作所预期的那样，它们不能精确地表达为地转运动。相关的位涡不遵守拉格朗日守恒定律，但它确实遵守通量守恒定律以及相关的环流定理。变换坐标中的流的发散主要是与地转运动自然相关的，附加项来自球体的曲率，以及广义坐标变换中可变科里奥利参数产生的额外“曲率”。这些项是估计的，并且发现对于正常数据来说非常小。该估计在计算中得到验证，确认了通常使用半地转理论得出的局部 $f$ 平面近似的准确性。
It is shown that the solution of the semi-geostrophic equations for shallow-water flow can be found and analysed in spherical geometry by methods similar to those used in the existing $f$-plane solutions. Stable states in geostrophic balance are identified as energy minimizers and a procedure for finding the minimizers is constructed, which is a form of potential vorticity inversion. This defines a generalization of the geostrophic coordinate transformation used in the $f$-plane theory. The procedure is demonstrated in computations. The evolution equations take a simple form in the transformed coordinates, though, as expected from previous work in the literature, they cannot be expressed exactly as geostrophic motion. The associated potential vorticity does not obey a Lagrangian conservation law, but it does obey a flux conservation law, with an associated circulation theorem. The divergence of the flow in the transformed coordinates is primarily that naturally associated with geostrophic motion, with additional terms coming from the curvature of the sphere and extra ‘curvature’ resulting from the variable Coriolis parameter in the generalized coordinate transformation. These terms are estimated, and are found to be very small for normal data. The estimate is verified in computations, confirming the accuracy of the local $f$-plane approximation usually made with semi-geostrophic theory.