Rudresh Veerkhare, Hang Yin, Albert Chern
We introduce a simulation method for point vortices on surfaces with arbitrary topology. We show that incorporating the dynamics of the harmonic components on a non-simply-connected surface is equivalent to introducing a new fluid invariant that is conserved over time. This invariant admits an elegant formulation in terms of the divisor class of point vortices, viewed as a divisor on a Riemann surface. To clarify this connection, we establish a nontrivial bridge between fluid dynamics and the theory of divisors in algebraic geometry. By exploiting this invariant, our simulation method becomes a simple modification of the vortex particle-on-mesh method, which extends its applicability to surfaces with arbitrary topology.