We construct an exact solution modelling the geophysical dynamics of an inviscid and incompressible fluid which possesses a variable density stratification, where the fluid density may vary with both the depth and latitude. Our solution pertains to the large-scale equatorial dynamics of a fluid body with a free surface propagating steadily in a purely azimuthal direction, and is expressed in terms of cylindrical coordinates. Allowing for general fluid stratification greatly complicates the Bernoulli relation—which relates the imposed pressure to the reciprocal fluid distortion at the free-surface—thereby acting as a constraint on the existence of a solution. Employing the implicit function theorem, we establish the existence of solutions and determine that the requisite monotonicity properties hold for the flow solutions we found. Furthermore, since the fluid velocity and pressure are prescribed by explicit formulae in the framework of cylindrical coordinates, our solution is amenable to analysis by the short-wavelength stability approach, which we investigate.

中文翻译：

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圆柱坐标中的分层赤道流

我们构建了一个精确解，模拟具有可变密度分层的无粘性和不可压缩流体的地球物理动力学，其中流体密度可能随深度和纬度而变化。我们的解决方案与自由表面在纯方位角方向上稳定传播的流体的大规模赤道动力学有关，并用柱坐标表示。允许一般的流体分层使伯努利关系变得非常复杂——它将施加的压力与自由表面的相互流体变形联系起来——从而限制了解的存在。利用隐函数定理，我们确定解的存在性，并确定我们找到的流解具有必要的单调性。此外，