Coarse-grained Hydrodynamics of turbulent superfluids: HVBK approach and the bundle structure of the vortex tangle

Abstract: In the comment I develop a critical analysis of the use of the HVBK method for the study of three-dimensional turbulent flows of superfluids. The conception of the vortex bundles forming the structure of quantum turbulence is controversial and does not justify the use of the HVBK method. In addition, this conception is counterproductive, because it gives incorrect ideas about the structure of the vortex tangle as a set of bundles containing parallel lines. The only type of dynamics of vortex filaments inside these bundles is possible, namely, Kelvin waves running along the filaments. At the same time, as shown in numerous numerical simulations, a vortex tangle consists of a set of entangled vortex loops of different sizes and having a random walk structure. These loops are subject to large deformations (due to highly nonlinear dynamics), they reconnect with each other and with the wall, split and merge, creating a lot of daughter loops. They also bear Kelvin waves on them, but the latter have little impact. I also propose and discuss an alternative variant of study of three-dimensional turbulent flows, in which the vortex line density $ \mathcal{L}(r,t)$ is not associated with $\nabla \times \mathbf{v}_{s}$, but it is an independent variable described by a separate equation.