Vorticity equation on surfaces with arbitrary topology embedded in three-dimensional Euclidean space

We derive the vorticity equation for an incompressible fluid on a two-dimensional surface with an arbitrary topology, embedded in three-dimensional Euclidean space and arising from a first integral of the flow, by using a tailored Clebsch parameterization of the velocity field. In the inviscid limit, we identify conserved surface energy and enstrophy and obtain the corresponding noncanonical Hamiltonian structure. We then discuss the formulation of the diffusion operator on the surface by examining two alternatives. In the first case, we follow the standard approach for Navier–Stokes equations on a Riemannian manifold and calculate the diffusion operator by requiring that flows corresponding to Killing fields of the Riemannian metric are not subject to dissipation. For an embedded surface, this leads to a diffusion operator, including derivatives of the stream function across the surface. In the second case, using an analogy with the Poisson equation for the Newtonian gravitational potential in general relativity, we construct a diffusion operator taking into account the Ricci scalar curvature of the surface. The resulting vorticity equation is two-dimensional, and the corresponding diffusive equilibria minimize dissipation under the constraint of curvature energy.