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.
Similar content being viewed by others
References
C.J. Gorter, J.H. Mellink, On the irreversible processes in liquid helium ii. Physica 15(3–4), 285–304 (1949)
R. P. Feynman. Progress in Low Temperature Physics, Vol. 1, p. 17. North-Holland, Amsterdam (1955)
I.M. Khalatnikov, An Introduction to the Theory of Superfluidity (Benjamin, New York/Amsterdam, 1965)
W. Frank Vinen, D. Shoenberg, Mutual friction in a heat current in liquid helium ii iii. theory of the mutual friction. Proc. R. Soc. Lond. A 242(1231), 493–515 (1957)
S.K. Nemirovskii, V.V. Lebedev, The hydrodynamic of superfluid turbulence. Sov. Phys. JETP 57, 1009 (1983)
S.K. Nemirovskii, W. Fiszdon, Chaotic quantized vortices and hydrodynamic processes in superfluid helium. Rev. Mod. Phys. 67(1), 37–84 (1995)
K.W. Schwarz, Three-dimensional vortex dynamics in superfluid \(he4\): homogeneous superfluid turbulence. Phys. Rev. B 38(4), 2398–2417 (1988)
D. Jou, M.S. Mongiovi, M. Sciacca, Hydrodynamics equations of anisotropic, polarized and inhomogeneous superfluid vortex tangles. Physica D 240, 249–258 (2011)
E.B. Sonin, Dynamics of Quantised Vortices in Superfluids (Cambridge University Press, Cambridge, 2016)
A.W. Baggaley, J. Laurie, C.F. Barenghi, Vortex-density fluctuations, energy spectra, and vortical regions in superfluid turbulence. Phys. Rev. Lett. 109, 205304 (2012)
A.W. Baggaley, C.F. Barenghi, A. Shukurov, Y.A. Sergeev, Coherent vortex structures in quantum turbulence. EPL (Europhys. Lett.) 98(2), 26002 (2012)
N. Sasa, T. Kano, M. Machida, V.S. L’vov, O. Rudenko, M. Tsubota, Energy spectra of quantum turbulence: large-scale simulation and modeling. Phys. Rev. B 84, 054525 (2011)
D.C. Samuels, Response of superfluid vortex filaments to concentrated normal-fluid vorticity. Phys. Rev. B 47(2), 1107–1110 (1993)
G.E. Volovik, On developed superfluid turbulence. J. Low Temp. Phys. 136(5), 309–327 (2004)
D. Kivotides, Spreading of superfluid vorticity clouds in normal-fluid turbulence. J. Fluid Mech. 668, 58–75 (2011)
D. Kivotides, The impact of kinematic simulations on quantum turbulence theory, in New Approaches in Modeling Multiphase Flows and Dispersion in Turbulence, Fractal Methods and Synthetic Turbulence, volume 18 of ERCOFTAC Series, ed. by F.C.G.A. Nicolleau, C. Cambon, J.-M. Redondo, J.C. Vassilicos, M. Reeks, A.F. Nowakowski (Springer, Netherlands, 2012), pp. 1–8
D. Kivotides, Energy spectra of finite temperature superfluid helium-4 turbulence. Phys. Fluids (1994-present) 26(10), 105105 (2014)
D. Kivotides, Mutual-friction induced instability of normal-fluid vortex tubes in superfluid helium-4. Phys. Lett. A 382(22), 1481–1485 (2018)
E. Kozik, B. Svistunov, Theory of decay of superfluid turbulence in low-temperature limit. J. Low Temp. Phys. 156, 215–267 (2009)
S.K. Nemirovskii, Energy spectrum of the quantum vortices configurations. Low Temp. Phys. 41(6), 608–614 (2015)
P.-E. Roche, P. Diribarne, T. Didelot, O. Francais, L. Rousseau, H. Willaime, Vortex density spectrum of quantum turbulence. EPL (Europhys. Lett.) 77(6), 66002 (2007)
D.I. Bradley, S.N. Fisher, A.M. Guénault, R.P. Haley, S. O’Sullivan, G.R. Pickett, V. Tsepelin, Fluctuations and correlations of pure quantum turbulence in superfluid \(^{3}{{ \rm He}}-{{\rm B}}\). Phys. Rev. Lett. 101, 065302 (2008)
S.K. Nemirovskii, Fluctuations of the vortex line density in turbulent flows of quantum fluids. Phys. Rev. B 86, 224505 (2012)
M. Kursa, K. Bajer, T. Lipniacki, Cascade of vortex loops initiated by a single reconnection of quantum vortices. Phys. Rev. B 83, 014515 (2011)
R.M. Kerr, Vortex stretching as a mechanism for quantum kinetic energy decay. Phys. Rev. Lett. 106, 224501 (2011)
S.Z. Alamri, A.J. Youd, C.F. Barenghi, Reconnection of superfluid vortex bundles. Phys. Rev. Lett. 101(21), 215302 (2008)
A.W. Baggaley, The importance of vortex bundles in quantum turbulence at absolute zero. Phys. Fluids 24(5), 055109 (2012)
M.V. Melander, F. Hussain, Cross-linking of two antiparallel vortex tubes. Phys. Fluids A 1, 633 (1989)
N.J. Zabusky, M.V. Melander, Three-dimensional vortex tube reconnection: Morphology for orthogonally-offset tubes. Physica D 37(1–3), 555–562 (1989)
S. Kida, M. Takaoka, F. Hussain, Collision of two vortex rings. J. Fluid Mech. 230, 583–646 (1991)
O.N. Boratav, R.B. Pelz, N.J. Zabusky, Reconnection in orthogonally interacting vortex tubes: Direct numerical simulations and quantifications. Phys. Fluids A 4, 581 (1992)
B.V. Svistunov, Superfluid turbulence in the low-temperature limit. Phys. Rev. B 52(5), 3647–3653 (1995)
E.B. Sonin, Vortex oscillations and hydrodynamics of rotating superfluids. Rev. Mod. Phys. 59(1), 87 (1987)
R.N. Hills, P.H. Roberts, Superfluid mechanics for a high density of vortex lines. Arch. Ration. Mech. Anal. 66(1), 43–71 (1977)
D.D. Holm, Introduction to hvbk Dynamics. In Quantized Vortex Dynamics (Springer, Berlin, 2001)
R.J. Donnelly, Cryogenic fluid dynamics. J. Phys.: Condens. Matter 11(40), 7783–7834 (1999)
S.K. Nemirovskii, Nonuniform quantum turbulence in superfluids. Phys. Rev. B 97, 134511 (2018)
D. Jou, M.S. Mongiovì, Nonequilibrium thermodynamics of unsteady superfluid turbulence in counterflow and rotating situations. Phys. Rev. B 72, 144517 (2005)
L. Kondaurova, V. Efimov, A. Tsoi, Influence of quantum turbulence on the processes of heat transfer and boiling in superfluid helium. J. Low Temp. Phys. 187(1), 80–89 (2017)
L. Boué, V. L’vov, A. Pomyalov, I. Procaccia, Enhancement of intermittency in superfluid turbulence. Phys. Rev. Lett. 110, 014502 (2013)
V.S. Lvov, S.V. Nazarenko, G.E. Volovik, Energy spectra of developed superfluid turbulence. JETP Lett. 80, 479–483 (2004)
J. Bertolaccini, E. Lévêque, P.-E. Roche, Disproportionate entrance length in superfluid flows and the puzzle of counterflow instabilities. Phys. Rev. Fluids 2, 123902 (2017)
D.H. Wacks, C.F. Barenghi, Shell model of superfluid turbulence. Physical Review B 84(18), 184505 (2011)
L. Skrbek, K.R. Sreenivasan, Developed quantum turbulence and its decay. Phys. Fluids 24(1), 011301 (2012)
J. Salort, P.-E. Roche, E. Leveque, Mesoscale equipartition of kinetic energy in quantum turbulence. EPL (Europhys. Lett.) 94(2), 24001 (2011)
R.J. Donnelly, Quantized Vortices in Helium II (Cambridge University Press, Cambridge, 1991)
S.W. Van Sciver, Helium Cryogenics. International Cryogenics Monograph Series (Springer, Berlin, 2013)
W.F. Vinen, Mutual friction in a heat current in liquid helium ii. iii. theory of the mutual friction. Proc. R. Soc. Lond. A 242, 493–515 (1957)
W.F. Vinen, Mutual friction in a heat current in liquid helium ii. i. experiments on steady heat currents. Proc. R. Soc. London A 240, 114 (1957)
W.F. Vinen, Mutual friction in a heat current in liquid helium ii. ii. experiments on transient effects. Proc. R. Soc. Lond. A 240, 128 (1957)
V.B. Eltsov, A.P. Finne, R. Hanninen, J. Kopu, M. Krusius, M. Tsubota, E.V. Thuneberg, Twisted vortex state. Phys. Rev. Lett. 96, 215302 (2006)
V.B. Eltsov, A.I. Golov, R. de Graaf, R. Hanninen, M. Krusius, V.S. L’vov, R.E. Solntsev, Quantum turbulence in a propagating superfluid vortex front. Phys. Rev. Lett. 99, 265301 (2007)
V.B. Eltsov, R. de Graaf, R. Hanninen, M. Krusius, R.E. Solntsev, V.S. L’vov, A.I. Golov, P.M. Walmsley, Chapter 2 turbulent dynamics in rotating helium superfluids, in: Progress in Low Temperature Physics: Quantum Turbulence, volume 16 of Progress in Low Temperature Physics, pp. 45 – 146. Elsevier, Amsterdam (2009)
R.G.K.M. Aarts, A.T.A.M. de Waele, Numerical investigation of the flow properties of he ii. Phys. Rev. B 50(14), 10069–10079 (1994)
M. Tsubota, T. Araki, S.K. Nemirovskii, Dynamics of vortex tangle without mutual friction in superfluid \(4he\). Phys. Rev. B 62(17), 11751–11762 (2000)
N.G. Berloff, B.V. Svistunov, Scenario of strongly nonequilibrated bose-einstein condensation. Phys. Rev. A 66(1), 013603 (2002)
L. Kondaurova, V. Andryuschenko, S. Nemirovskii, Numerical simulations of superfluid turbulence under periodic conditions. J. Low Temp. Phys. 150, 415–419 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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)
Rights and permissions
About this article
Cite this article
Nemirovskii, S.K. On the Closure Problem of the Coarse-Grained Hydrodynamics of Turbulent Superfluids. J Low Temp Phys 201, 254–268 (2020). https://doi.org/10.1007/s10909-020-02483-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10909-020-02483-6