ABSTRACT Here we examine the motion of an isolated ellipsoidal vortex in a rotating stratified fluid. We derive an analytical solution to a set of balanced equations at the next order to quasi-geostrophic theory, providing insights into geophysical vortices at finite Rossby number ε. This is achieved through the solution of a set of complicated Poisson equations. Though complicated, the analytical solution give rise to a velocity field that depends linearly on the spatial coordinates inside the vortex, and, thus preserves the ellipsoidal form. From this general solution, we determine a number of equilibria where the vortex rotates steadily about the vertical axis and examine their stability. At the next order to QG, one finds asymmetry in the behaviour of cyclonic and anti-cyclonic vortices, with anti-cyclonic vortices rotating faster and generally more unstable than cyclonic vortices.

中文翻译：

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有限罗斯比数下的平衡椭球涡

摘要 在这里，我们研究了旋转分层流体中孤立椭球涡旋的运动。我们以准地转理论的下一阶推导出一组平衡方程的解析解，提供对有限罗斯比数 ε 的地球物理涡旋的见解。这是通过求解一组复杂的泊松方程来实现的。尽管很复杂，但解析解会产生一个速度场，该速度场与涡旋内部的空间坐标呈线性相关，因此保留了椭球形式。从这个通用解决方案中，我们确定了涡流围绕垂直轴稳定旋转的多个平衡点，并检查了它们的稳定性。在 QG 的下一个阶，人们发现气旋和反气旋涡旋的行为不对称，