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

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