Vortex Flows on Surfaces and Their Anomalous Hydrodynamics

Hydrodynamic flows with large numbers of vortices have been a staple of theoretical hydrodynamics since Onsager and Feynman. Treating such vortices as constituents of a new fluid of their own leads to an interesting anomalous hydrodynamics that exhibits curious phenomena such as odd viscosity. Starting with the incompressible Euler equation, we rigorously derive the entire coarse-grained hydrodynamics of the "vortex matter" consisting of many identical discrete point vortices on an arbitrary closed 2D surface. The resulting flow of vortices differs from the original flow of the underlying fluid by a term that can be related to odd viscosity and also leads to a particular interaction between the vortex density and the scalar curvature. We investigate the bulk hydrodynamics of the chiral vortex matter on an arbitrary closed surface, extending the ideas of Khalatnikov; Wiegmann and Abanov. Placing this important example of a chiral medium onto a curved geometry reveals the geometric nature of odd viscosity. The anomalous odd viscosity of the vortex matter is associated with a special interaction of point vortices with curvature.