论文标题
涡旋对和偶极子上的偶极
Vortex pairs and dipoles on closed surfaces
论文作者
论文摘要
我们在封闭的Riemannian表面上设置了点涡流系统的通用运动方程，从而使涡度总和不为零，因此必须存在反涡流。仔细考虑了与涡流动力学结合的全球循环动力学。 将Kimura的猜想重点放在涡旋对的研究中。这说明涡旋对沿偶极极限沿测量曲线移动，并且先前使用高斯大地测量坐标给了S.〜Boatto和J.〜Koiller的证明。在本文中，我们通过遵循略有不同的途径来得出相同的结论，直接导致带有重新测量时间变量的地球方程。 在最后一部分中，我们解释了如何将平面域中的涡流运动视为封闭表面上涡流运动的特殊情况，在两个附录中，我们提供了一些关于仿射和投射连接的必要背景。
We set up general equations of motion for point vortex systems on closed Riemannian surfaces, allowing for the case that the sum of vorticities is not zero and there hence must be counter-vorticity present. The dynamics of global circulations which is coupled to the dynamics of the vortices is carefully taken into account. Much emphasis is put to the study of vortex pairs, having the Kimura conjecture in focus. This says that vortex pairs move, in the dipole limit, along geodesic curves, and proofs for it have previously been given by S.~Boatto and J.~Koiller by using Gaussian geodesic coordinates. In the present paper we reach the same conclusion by following a slightly different route, leading directly to the geodesic equation with a reparametrized time variable. In a final section we explain how vortex motion in planar domains can be seen as a special case of vortex motion on closed surfaces, and in two appendices we give some necessary background on affine and projective connections.
