Equation for self-consistent superfluid vortex line dynamics

Turbulence in helium II takes the form of a disordered tangle of quantised vortex line. The existing equation of vortex dynamics used by Schwarz and others to model the evolution of the vortex tangle does not distinguish between the large scale normal fluid velocity and the local variation of the normal fluid velocity introduced by the presence of quantised vortex lines. We derive a new vortex dynamics equation which allows the local normal fluid velocity to be determined by a modified Navier-Stokes equation. Together, the two equations form a self-consistent model to determine the coupled evolution of the normal fluid and quantised vortices.