  We derive the vorticity equation for an incompressible fluid on a
2-dimensional surface with arbitrary topology embedded in 3-dimensional
Euclidean space by using a tailored Clebsch parametrization of the flow. 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 the
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 2-dimensional, and the
corresponding diffusive equilibria minimize dissipation under the constraint of
curvature energy.
