On the Closure Problem of the Coarse-Grained Hydrodynamics of Turbulent Superfluids
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On the Closure Problem of the Coarse‑Grained Hydrodynamics of Turbulent Superfluids Sergey K. Nemirovskii1 Received: 18 December 2019 / Accepted: 20 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 L(r, t)–the vortex line density (VLD) of the vortex tangle. The question of how to treat the quantity 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 L(r, t) via coarsegrained vorticity ∇ × 𝐯s 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 L(r, t) is proportional to the squared relative velocity between normal and superfluid components ∝ (vn − vs )2 . One more variant of the closure procedure, discussed in the paper, is based on the method in which the vortex line density 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. Keywords Quantum turbulence · Coarse-grained hydrodynamics · Closure problem
The study on the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) approach was carried out under state contract with IT SB RAS (No. 17-117022850027-5), the study on the Hydrodynamics of Superfluid Turbulence (HST) method was financially supported by RFBR Russian Science Foundation (Project No. 18-08-00576) * Sergey K. Nemirovskii [email protected] 1
Institute of Thermophysics, Lavrentyev ave., 1, Novosibirsk, Russia 630090
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Journal of Low Temperature Physics
1 Introduction The concept of quantum turbulence began with the well-known experiment by Gorter and Mellink (GM) on heat exchange in superfluid helium and Feynman’s theory, which explains the GM experiment on the basis of the dynamics of a set of chaotic quantized vortices, or vortex tangle [1, 2]. Since then, the concepts of quantum turbulence as a chaotic tangle of vortex filaments have expanded and appeared in many fields of physics, ranging from superfluids and cold atoms to heavy ions, and neutron stars (see e.g., INT Program INT-19-1a, URL http:// www.int.washington.edu/PROGRAMS/19-1a/). The experimental studies of superfluid turbulence are mainly based on hydrodynamic methods. These include, for example
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