The motion of point vortices on closed surfaces
The motion of point vortices on closed surfaces
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DOI：
10.1098/rspa.2014.0890
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发表时间：
2015-04-08
期刊：
影响因子：
--
通讯作者：
Boatto S
中科院分区：
文献类型：
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作者：
Dritschel DG;Boatto S
我们开发了一个数学框架的动力学的一组点涡的一类可微表面共形的单位球。当涡旋环流的和不为零时，需要补偿均匀涡度场来满足高斯条件（即拉普拉斯-贝尔特拉米算子的积分必须为零）。在可变高斯曲率表面上，这导致自诱导涡运动，这是在平面、球体或双曲面上完全不存在的特征。我们推导出明确的运动方程的旋转表面上的旋涡和计算各种表面的解决方案。我们也应用这些方程来研究任何旋转表面上的涡环的线性稳定性。在旋转椭球面上，在扁球面或足够长的球面上，只有两个旋涡是不稳定的。这扩展了已知的平面结果，其中有七个涡是边缘不稳定的（Thomson 1883 A treatise on the motion of vortex rings，pp. 94-108; Dritschel 1985 J. Fluid Mech.157，95-134（doi：10.1017/S 0022112088003088）），以及球体，其中如果足够接近赤道，四个涡流可能是不稳定的（Polvani & Dritschel 1993 J. Fluid Mech.255，35-64（doi：10.1017/S 0022112093002381））。
We develop a mathematical framework for the dynamics of a set of point vortices on a class of differentiable surfaces conformal to the unit sphere. When the sum of the vortex circulations is non-zero, a compensating uniform vorticity field is required to satisfy the Gauss condition (that the integral of the Laplace–Beltrami operator must vanish). On variable Gaussian curvature surfaces, this results in self-induced vortex motion, a feature entirely absent on the plane, the sphere or the hyperboloid. We derive explicit equations of motion for vortices on surfaces of revolution and compute their solutions for a variety of surfaces. We also apply these equations to study the linear stability of a ring of vortices on any surface of revolution. On an ellipsoid of revolution, as few as two vortices can be unstable on oblate surfaces or sufficiently prolate ones. This extends known results for the plane, where seven vortices are marginally unstable (Thomson 1883 A treatise on the motion of vortex rings, pp. 94–108; Dritschel 1985 J. Fluid Mech. 157, 95–134 (doi:10.1017/S0022112088003088)), and the sphere, where four vortices may be unstable if sufficiently close to the equator (Polvani & Dritschel 1993 J. Fluid Mech. 255, 35–64 (doi:10.1017/S0022112093002381)).