Balanced solutions for an ellipsoidal vortex in a rotating stratified flow
Balanced solutions for an ellipsoidal vortex in a rotating stratified flow
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旋转分层流中椭球涡的平衡解
DOI：
10.1017/jfm.2016.462
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发表时间：
2016
影响因子：
3.7
通讯作者：
McKiver W
中科院分区：
文献类型：
--
作者：
McKiver W
本文考虑旋转分层流体中具有均匀位涡的椭球涡在有限Rossby数下的运动，给出了涡的演化。在准地转近似下，通过反演位涡来确定流矩阵，以通过泊松方程获得流函数，该方程具有依赖于椭圆积分的已知解析解。在这里，我们表明，高阶平衡的解决方案，高达二阶的Rossby数，也可以计算分析。然而，在这种情况下，有一个矢量势，需要解决三个泊松方程的每个组成部分。这些方程的源项与椭球内的空间坐标无关，只依赖于在准地转阶的椭圆积分。与准地转情况不同，这些源项一般不会在椭球外消失，并且对空间坐标有非常复杂的依赖性。在椭球的轴与坐标轴对齐的特殊情况下，我们能够导出这些源项，并获得三个泊松方程的完整解析解。然而，如果考虑齐次的情况下，其中的外部源项被忽略，可以得到一个近似的解决方案，具有紧凑的矩阵形式类似的领先阶准地转的情况。这种近似的解决方案被证明是高度准确的任意取向的椭球体的一般情况下，验证通过比较的解决方案与从数值模拟的椭球体使用一个准确的非线性平衡模型，即使在温和的Rossby数。
We consider the motion of a single ellipsoidal vortex with uniform potential vorticity in a rotating stratified fluid at finite Rossby number to provide the vortex evolution. Under the quasi-geostrophic approximation, the flow matrix is determined by inverting the potential vorticity to obtain the streamfunction via Poisson’s equation, which has a known analytical solution depending on elliptic integrals. Here we show that higher-order balanced solutions, up to second order in the Rossby number, can also be calculated analytically. However, in this case there is a vector potential that requires the solution of three Poisson equations for each of its components. The source terms for these equations are independent of spatial coordinates within the ellipsoid, depending only on the elliptic integrals solved at the leading, quasi-geostrophic order. Unlike the quasi-geostrophic case, these source terms do not in general vanish outside the ellipsoid and have an inordinately complicated dependence on spatial coordinates. In the special case of an ellipsoid whose axes are aligned with the coordinate axes, we are able to derive these source terms and obtain the full analytical solution to the three Poisson equations. However, if one considers the homogeneous case, whereby the outer source terms are neglected, one can obtain an approximate solution having a compact matrix form analogous to the leading-order quasi-geostrophic case. This approximate solution proves to be highly accurate for the general case of an arbitrarily oriented ellipsoid, as verified through comparisons of the solutions with solutions obtained from numerical simulations of an ellipsoid using an accurate nonlinear balance model, even at moderate Rossby numbers.
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