(3+1)-dimensional compressible fluid as a (4+1)-dimensional Chern-Simons system

Abstract

A fluid described by an Abelian Chern-Simons action principle in 4+1 dimensions is considered. Letting 3+1 dimensions correspond to the usual space and time, and assuming the fields to be independent of the fifth coordinate, the free theory provides an interpretation as a system of advection equations, where the advecting velocity field is defined as the null vector of the field strength tensor (curvature). The free theory possesses a number of conservation laws which turn out to be prototypical forms of helicity and entropy conservation. Coupling the Chern-Simons field to an external source, a new conserved charge density is obtained which has the form of the Rossby-Ertel's potential vorticity (PV). Finally, by identifying the external current with the Chern-Simons field in a gauge-invariant setting, based on non-relativistic ideas, a self-interacting action principle is obtained whose Euler-Lagrange equations correspond precisely to a classical dissipationless compressible (3+1)-dimensional fluid endowed with thermodynamics, with only one extra condition: a constraint on the initial profile of the PV. After analysing this constraint of the "Chern-Simons fluid formulation", we investigate the helicity conservation of general fluids, going beyond classical analyses of barotropic fluids and no-cross boundary conditions for vorticity (Moffatt 1969). A new fluid helicity invariant for barotropic fluids under generic boundary conditions is obtained and the role of baroclinity in the helicity production is clarified. Inside a region bounded by an isentropic surface, the theory's constraint on the PV gives an integral formula for the mass, and for the evolution of fluid helicity in the baroclinic case. Finally, for an ideal gas exact, steady solutions of the equations of motion are found in a rotating scenario, showing that Ferrel-cell like patterns are produced in a rotating planet.