Mathematical justification of the point vortex dynamics in background fields on surfaces as an Euler–Arnold flow

Abstract

The point vortex dynamics in background fields on surfaces is justified as an Euler–Arnold flow in the sense of de Rham currents. We formulate a current-valued solution of the Euler–Arnold equation with a regular-singular decomposition. For the solution, we first prove that, if the singular part of the vorticity is given by a linear combination of delta functions centered at \(q_n(t)\) for \(n=1,\ldots ,N\), \(q_n(t)\) is a solution of the point vortex equation. Conversely, we next prove that, if \(q_n(t)\) is a solution of the point vortex equation for \(n=1,\ldots ,N\), there exists a current-valued solution of the Euler–Arnold equation with a regular-singular decomposition such that the singular part of the vorticity is given by a linear combination of delta functions centered at \(q_n(t)\). As a corollary, we generalize the Bernoulli law to the case where the flow field is a curved surface and where the presence of point vortices is taken into account. From the viewpoint of the application, the mathematical justification is of significance since the point vortex dynamics in the rotational vector field on the unit sphere is regarded as a mathematical model of geophysical flow in order to take effect of the Coriolis force on inviscid flows into consideration.