Simplified hydrodynamic analysis of superfluid turbulence in He II: Internal dynamics of inhomogeneous vortex tangle

The hydrodynamic theory of superfluid turbulence is presented in a new simplified form. It applies to flow situations frequently encountered in practice, in which the thermohydrodynamic environment of a superfluid turbulent tangle of quantized vortices may be considered, in a first order of approximation, as given. Flow quantities like the mass density, the entropy density, and the drift velocities of mass and elementary excitations act, accordingly, as external parameters with respect to the internal dynamics of the vortex tangle. The internal dynamics is completely specified by a kinematic equation governing the time evolution of the line-length density of the quantized vortices and a dynamic equation involving the impulse density of the vortex tangle. The derivation of these equations starts from a variational principle that is reminiscent of Hamilton's principle in classical mechanics and proceeds, in order to include dissipative effects, by using methods of the thermo- dynamics of irreversible processes. A new quantity called superfluid turbulent pressure is introduced which shows many properties that are familiar from the ordinary pressure in a classical fluid. Two important particu- lar cases are considered in more detail, viz., homogeneous superfluid turbulent flow and flow situations in which the vortex tangle is in permanent internal equilibrium. When diffusion of the vortex-tangle impulse is taken into account and dispersive effects are disregarded, the dynamic equation of the vortex tangle assumes, in the case of internal equilibrium, the form of Burgers' equation with a nonlinear source term. This equation, which is new, may be considered as a natural generalization of Vinen's equation to inhomogeneous superfluid turbulence. Some exact solutions which represent uniformly propagating superfluid turbulence fronts are listed in the Appendix. @S0163-1829~96!01033-8#