Constantin A, Johnson R S
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, UK.
Proc Math Phys Eng Sci. 2017 Apr;473(2200):20170063. doi: 10.1098/rspa.2017.0063. Epub 2017 Apr 12.
Starting from the Euler equation expressed in a rotating frame in spherical coordinates, coupled with the equation of mass conservation and the appropriate boundary conditions, a thin-layer (i.e. shallow water) asymptotic approximation is developed. The analysis is driven by a single, overarching assumption based on the smallness of one parameter: the ratio of the average depth of the oceans to the radius of the Earth. Consistent with this, the magnitude of the vertical velocity component through the layer is necessarily much smaller than the horizontal components along the layer. A choice of the size of this speed ratio is made, which corresponds, roughly, to the observational data for gyres; thus the problem is characterized by, and reduced to an analysis based on, a single small parameter. The nonlinear leading-order problem retains all the rotational contributions of the moving frame, describing motion in a thin spherical shell. There are many solutions of this system, corresponding to different vorticities, all described by a novel vorticity equation: this couples the vorticity generated by the spin of the Earth with the underlying vorticity due to the movement of the oceans. Some explicit solutions are obtained, which exhibit gyre-like flows of any size; indeed, the technique developed here allows for many different choices of the flow field and of any suitable free-surface profile. We comment briefly on the next order problem, which provides the structure through the layer. Some observations about the new vorticity equation are given, and a brief indication of how these results can be extended is offered.
从球坐标下旋转坐标系中表示的欧拉方程出发，结合质量守恒方程和适当的边界条件，推导出了薄层（即浅水）渐近近似。该分析由一个基于单一参数微小性的总体假设驱动：海洋平均深度与地球半径之比。与此一致的是，穿过该层的垂直速度分量的大小必然远小于沿该层的水平分量。选择了这个速度比的大小，它大致对应于环流的观测数据；因此，该问题由一个单一的小参数表征并简化为基于该参数的分析。非线性主导阶问题保留了运动坐标系的所有旋转贡献，描述了薄球壳中的运动。该系统有许多解，对应于不同的涡度，所有这些解都由一个新的涡度方程描述：该方程将地球自转产生的涡度与海洋运动产生的潜在涡度耦合在一起。得到了一些显式解，它们表现出任意大小的类似环流的流动；实际上，这里开发的技术允许对流场和任何合适的自由表面轮廓进行许多不同的选择。我们简要评论了下一阶问题，它给出了穿过该层的结构。给出了关于新涡度方程的一些观察结果，并简要指出了如何扩展这些结果。
