Hydrodynamics of two-dimensional compressible fluid with broken parity: Variational principle and free surface dynamics in the absence of dissipation

We consider an isotropic compressible nondissipative fluid with broken parity subject to free surface boundary conditions in two spatial dimensions. The hydrodynamic equations describing the bulk dynamics of the fluid and the free surface boundary conditions depend explicitly on the parity-breaking nondissipative odd viscosity term. We construct an effective action which gives both bulk hydrodynamic equations and free surface boundary conditions. The free surface boundary conditions require an additional boundary term in the action which resembles a 1+1D chiral boson field coupled to the background geometry. We solve the linearized hydrodynamic equations for the deep water case and derive the dispersion of chiral surface waves. We show that in the long-wavelength limit the flow profile exhibits an oscillating vortical boundary layer near the free surface. The layer thickness is controlled by the ratio between the odd viscosity (νo) and the sound velocity (cs), δ∼νo/cs. In the incompressible limit, cs→∞, the vortical boundary layer becomes singular with the vorticity within the layer diverging as ω∼cs. The boundary layer is formed by odd viscosity coupling the divergence of velocity ∇·v to vorticity ∇×v. It results in nontrivial chiral free surface dynamics even in the absence of external forces. The structure of the odd-viscosity-induced boundary layer is very different from the conventional free surface boundary layer associated with dissipative shear viscosity.