Dynamics of Cohomology in Fluids

We present a topological analysis of the vorticity formulation in describing fluid dynamics.  Despite its widespread use in fluid mechanics, this formulation is insufficient at describing fluid dynamics on a non-simply-connected domain.  What is missing is an equation of motion for fluid's cohomology component, which exhibits fascinating dynamics previously under explored.  Using geometric language, we derive the new equation of motion and establish new conservation laws, as Casimir invariants in Hamiltonian mechanics, for fluids on domains with general topology.  Significantly, we present the first vortex method on curved surfaces with genus and boundaries that is consistent with Euler equation.  The talk also provides an overview of geometric fluid mechanics and implications of our work in geometric mechanics.