On the Closure Problem of the Coarse-Grained Hydrodynamics of Turbulent Superfluids

Abstract

The coarse-grained hydrodynamics of turbulent superfluid fluids describes the fluid flow in terms of two equations for the averaged normal and superfluid velocities. These two equations are coupled via the mutual friction term, which contains the quantity $${\mathcal {L}}(r,t)$$ –the vortex line density (VLD) of the vortex tangle. The question of how to treat the quantity $${\mathcal {L}}(r,t)$$ –the so-called closure procedure is crucial for the correct description of flow of turbulent superfluids. The article provides a critical analysis of several approaches to the closure procedure. The first one, which is usually referred to as HVBK method suggests to express the quantity $${\mathcal {L}}(r,t)$$ via coarse-grained vorticity $$\nabla \times {\mathbf {v}}_{s}$$ using the famous Feynman rule. This method and idea on the vortex bundle structure, which justifies the use of the HVBK approach, are analyzed and discussed in detail. Another approach that has been popular before, but is still used sometimes, is called the Gorter–Mellink relation. This method suggests that the VLD $${\mathcal {L}}(r,t)$$ is proportional to the squared relative velocity between normal and superfluid components $$\propto (v_{n}-v_{s})^{2}$$ . One more variant of the closure procedure, discussed in the paper, is based on the method in which the vortex line density $${\mathcal {L}}(r,t)$$ is not expressed directly via the velocity (and/or vorticity) field, but is an independent equipollent variable, controlled by a separate equation. The latter approach is called as hydrodynamics of superfluid turbulence (HST). The advantages and disadvantages of each method are discussed.