Abstract
We formulate the equations for point vortex dynamics on a closed two-dimensional Riemannian manifold in the language of affine and other kinds of connections. This can be viewed as a relaxation of standard approaches, using the Riemannian metric directly, to an approach based more on local coordinates provided with a minimal amount of extra structure. The speed of a vortex is then expressed in terms of the difference between an affine connection derived from the coordinate Robin function and the Levi–Civita connection associated with the Riemannian metric. A Hamiltonian formulation of the same dynamics is also given. The relevant Hamiltonian function consists of two main terms. One of the terms is the well-known quadratic form based on a matrix whose entries are Green and Robin functions, while the other term describes the energy contribution from those circulating flows which are not implicit in the Green functions. One main issue of the paper is a detailed analysis of the somewhat intricate exchanges of energy between these two terms of the Hamiltonian. This analysis confirms the mentioned dynamical equations formulated in terms of connections.This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.
Highlights
We study point vortex motion on closed two-dimensional Riemannian manifolds, mainly from the point of view of Riemann surfaces with additional
Since point vortices have infinite energy some kind of renormalization is necessary when passing from the general Euler equations for the fluid, assumed non-viscous and incompressible, to the dynamics for the vortices, often written in Hamiltonian form
The main specific results in the paper are theorems 7.1 and 9.1. The first of these describes the dynamics for a system of point vortices in terms of the difference between two affine connections, one coming from the metric and the other being a connection canonically derived from the Robin function
Summary
We study point vortex motion on closed two-dimensional Riemannian manifolds, mainly from the point of view of Riemann surfaces with additional. Since point vortices have infinite energy some kind of renormalization is necessary when passing from the general Euler equations for the fluid, assumed non-viscous and incompressible, to the dynamics for the vortices, often written in Hamiltonian form. The main specific results in the paper are theorems 7.1 and 9.1 The first of these describes the dynamics for a system of point vortices in terms of the difference between two affine connections, one coming from the metric ( being the same as the classical Levi–Civita connection) and the other being a connection canonically derived from the Robin function. We shall need the Hodge theorem only for 2-forms, in which case it makes the one-point Green function appear naturally. This is essential because, in our applications, V will have the role of being (part of) a stream function, and the conjugate periods of V will enter into the circulations of the flow, which are to be conserved in time by Kelvin’s Law (2.9)
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