球坐标系下变密度南极绕极流的显式精确解
Explicit and exact solutions concerning the Antarctic Circumpolar Current with variable density in spherical coordinates
摘要：
我们使用球坐标设计一个新的精确解的地球物理流体动力学的控制方程的无粘和不可压缩流体的一般密度分布和强迫项。后者是最重要的模拟现实的流动，也就是说，在某种平均意义上观察到的海洋流动。由于采用了球坐标，我们不需要采用简化控制方程几何形状的近似（例如$f$ -和$β$ -平面类型）。我们的显式解决方案代表了一个稳定的纯方位角分层流与自由表面，-由于强迫项的列入和地球的几何形状的考虑，通过球坐标-使其适合于描述南极绕极流，并使这种流的结构进行深入分析。在后者方面，我们采用功能分析技术来证明，自由表面的变形是定义在一个独特的和隐式的方式通过施加在自由表面的压力。最后，我们的讨论，设定的表面压力的单调性和表面变形的单调性，符合物理预期之间的关系。
英文摘要：
We use spherical coordinates to devise a new exact solution to the governing equations of geophysical fluid dynamics for an inviscid and incompressible fluid with a general density distribution and subjected to forcing terms. The latter are of paramount importance for the modeling of realistic flows-that is, flows that are observed in some averaged sense in the ocean. Owing to the employment of spherical coordinates we do not need to resort to approximations (e.g. of $f$- and $β$-plane type) that simplify the geometry in the governing equations. Our explicit solution represents a steady purely-azimuthal stratified flow with a free surface, that---thanks to the inclusion of forcing terms and the consideration of the Earth's geometry via spherical coordinates---makes it suitable for describing the Antarctic Circumpolar Current and enables an in-depth analysis of the structure of this flow. In line with the latter aspect, we employ functional analytical techniques to prove that the free surface distortion is defined in a unique and implicit way by means of the pressure applied at the free surface. We conclude our discussion by setting out relations between the monotonicity of the surface pressure and the monotonicity of the surface distortion that concur with the physical expectations.
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