Advertisement

Isotropic Turbulence with Coupled Microstructures. II: Quantum Turbulence

  • Pierre Sagaut
  • Claude Cambon
Chapter

Abstract

This chapter is devoted to Quantum turbulence in superfluid helium and Bose-Einstein condensates. Basic equations and theories related to the Gross-Pitaevski equation, the vortex filament model and the two-fluid hydrodynamic model are discussed. The links between the different theories are emphasized. Main physical mechanisms and models at both hydrodynamic and quantum scales (pseudo-Kolmogorov cascade, Kelvin waves and Kelvin wave cascade, vortex reconnection, dissipative mechanisms, mutual friction effects ...) are detailed.

References

  1. Adachi, H., Tsubota, M.: Numerical studies of counterflow turbulence. Velocity distribution of vortices. J. Low Temp. Phys. 158, 422–427 (2010)ADSCrossRefGoogle Scholar
  2. Babuin, S., Stammeier, M., Varga, E., Rotter, M., Skrbek, L.: Quantum turbulence of bellows-driven \(^4\)He superflow: steady state. Phys. Rev. B 86, 134515 (2012)ADSCrossRefGoogle Scholar
  3. Baggaley, A.W.: The sensitivity of the vortex filament method to different reconnection models. J. Low Temp. Phys. 168, 18–30 (2012)ADSCrossRefGoogle Scholar
  4. Baggaley, A.W., Barenghi, C.F.: Spectrum of turbulent Kelvin-wave cascade in superfluid helium. Phys. Rev. B 83, 134509 (2011)ADSCrossRefGoogle Scholar
  5. Baggaley, A.W., Barenghi, C.F., Sergeev, Y.A.: Quasiclassical and ultraquantum decay of superfluid turbulence. Phys. Rev. B 85, 060501(R) (2012)ADSCrossRefGoogle Scholar
  6. Baggaley, A.W., Barenghi, C.F., Sergeev, Y.A.: Three-dimensional inverse energy transfer induced by vortex reconnections. Phys. Rev. E 89, 013002 (2014)ADSCrossRefGoogle Scholar
  7. Baggaley, A.W., Laurie, J.: Kelvin-wave cascade in the vortex filament model. Phys. Rev. B 89, 014504 (2014)ADSCrossRefGoogle Scholar
  8. Baggaley, A.W., Sherwin, L.K., Barenghi, C.F., Sergeev, Y.A.: Thermally and mechanically driven quantum turbulence. Phys. Rev. B 86, 104501 (2012)ADSCrossRefGoogle Scholar
  9. Barenghi, C.F.: Is the Reynolds number infinite in superfluid turbulence? Physica D 237, 2195–2202 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. Barenghi, C.F., Donnelly, R.J., Vinen, W.F.: Friction on quantized vortices in helium II. A review. J. Low Temp. Phys. 52, 189–247 (1983)ADSCrossRefGoogle Scholar
  11. Barenghi, C.F., Donnelly, R.J., Vinen, W.F.: Thermal excitation of waves on quantized vortices. Phys. Fluids 28, 498–504 (1985)ADSCrossRefGoogle Scholar
  12. Barenghi, C.F., Samuels, D.C., Kivotides, D.: Superfluid vortex reconnections. J. Low Temp. Phys. 126, 271–279 (2002)ADSCrossRefGoogle Scholar
  13. Barenghi, C.F., Samuels, D.C., Kivotides, D.: Scaling laws of vortex reconnections. J. Low Temp. Phys. 136, 281–293 (2004)ADSCrossRefGoogle Scholar
  14. Barenghi, C.F., Skrbek, L., Sreenivasan, K.R.: Introduction to quantum turbulence. Proc. Natl. Acad. Sci. USA 111, 4647–4652 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. Berloff, N.G.: Interactions of vortices with rarefaction solitary waves in a Bose–Einstein condensate and their role in the decay of superfluid turbulence. Phys. Rev. A 69, 053601 (2004)ADSCrossRefGoogle Scholar
  16. Boffetta, G., Celani, A., Dezzani, D., Laurie, J., Nazarenko, S.: Modeling Kelvin wave cascades in superfluid helium. J. Low Temp. Phys. 156, 193–214 (2009)ADSCrossRefGoogle Scholar
  17. Boué, L., Dasgupta, R., Laurie, J., L’vov, V., Nazarenko, S., Procaccia, I.: Exact solution for the energy spectrum of Kelvin-wave turbulence in superfluids. Phys. Rev. B 84, 064516 (2011)ADSCrossRefGoogle Scholar
  18. Briard, A., Gomez, T., Sagaut, P., Memari, S.: Passive scalar decay laws in isotropic turbulence: prandtl number effects. J. Fluid Mech. 784, 274–303 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. Briard, A., Gomez, T.: Passive scalar convective-diffusive subrange for low Prandtl numbers in isotropic turbulence. Phys. Rev. E 101, 011001(R) (2015)ADSCrossRefGoogle Scholar
  20. Fujimoto, K., Tsubota, M.: Bogoliubov-wave turbulence in Bose–Einstein condensates. Phys. Rev. A 91, 053620 (2015)ADSCrossRefGoogle Scholar
  21. Gao, J., Guo, W., L’vov, V.S., Pomyalov, A., Skrbek, L., Varga, E., Vinen, W.F.: Decay of counterflow turbulence in superfluid \(^4\)He. JETP Lett. 103, 648–652 (2016)ADSCrossRefGoogle Scholar
  22. Golov, A.I., Walmsley, P.M.: Homogeneous turbulence in superfluid \(^4\)He in the low-temperature limit: experimental progress. J. Low Temp. Phys. 156, 51–70 (2009)ADSCrossRefGoogle Scholar
  23. Hänninen, R.: Dissipation enhancement from a single vortex reconnection in superfluid helium. Phys. Rev. B 88, 054511 (2013)ADSCrossRefGoogle Scholar
  24. Idowu, O.C., Kivotides, D., Barenghi, C.F., Samuels, D.C.: Equation for self-consistent superfluid vortex line dynamics. J. Low Temp. Phys. 120, 269–280 (2000)ADSCrossRefGoogle Scholar
  25. Jou, D., Mongiovi, M.S., Sciacca, M.: Hydrodynamic equations of anisotropic, polarized and inhomogeneous superfluid vortex tangles. Physica D 240, 249–258 (2011)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. Kivotides, D., Vassilicos, J.C., Samuels, D.C., Barenghi, C.F.: Kelvin-wave cascade in superfluid turbulence. Phys. Rev. Lett. 86, 3080–3083 (2001)ADSCrossRefGoogle Scholar
  27. Kobayashi, M., Tsubota, M.: Kolmogorov spectrum of superfluid turbulence: numerical analysis of the Gross–Pitaevskii equation with small-scale dissipation. Phys. Rev. Lett. 94, 065302 (2006)ADSCrossRefGoogle Scholar
  28. Kobayashi, M., Tsubota, M.: Thermal dissipation in quantum turbulence. Phys. Rev. Lett. 97, 145301 (2006)ADSCrossRefGoogle Scholar
  29. Kobayashi, M., Tsubota, M.: Dissipation of Gross–Pitaevskii turbulence coupled with thermal excitations. J. Low Temp. Phys. 148, 275–279 (2007)ADSCrossRefGoogle Scholar
  30. Koundaurova, L., L’vov, V., Pomyalov, A., Procaccia, I.: Structure of a quantum vortex tangle in \(^4\)He counterflow turbulence. Phys. Rev. B 89, 014502 (2014)ADSCrossRefGoogle Scholar
  31. Koundaurova, L., Nemirovskii, S.K.: Numerical study of decay of vortex tangles in superfluid helium at zero temperature. Phys. Rev. B 86, 134506 (2012)ADSCrossRefGoogle Scholar
  32. Kozik, E.V., Svistunov, B.V.: Kelvin-wave cascade and decay of superfluid turbulence. Phys. Rev. Lett. 92, 035301 (2004)ADSCrossRefGoogle Scholar
  33. Kozik, E.V., Svistunov, B.V.: Scale-separation scheme for simulating superfluid turbulence: Kelvin-wave cascade. Phys. Rev. Lett. 94, 025301 (2005a)ADSCrossRefGoogle Scholar
  34. Kozik, E.V., Svistunov, B.V.: Vortex-phonon interaction. Phys. Rev. B 72, 172505 (2005b)ADSCrossRefGoogle Scholar
  35. Kozik, E.V., Svistunov, B.V.: Kolmogorov and Kelvin-wave cascades of superfluid turbulence at T = 0: what lies between. Phys. Rev. B 77, 060502(R) (2008a)ADSCrossRefGoogle Scholar
  36. Kozik, E.V., Svistunov, B.V.: Scanning superfluid-turbulence cascade by its low-temperature cutoff. Phys. Rev. Lett. 100, 195302 (2008b)ADSCrossRefGoogle Scholar
  37. Kozik, E.V., Svistunov, B.V.: Theory of decay of superfluid turbulence in the low-temperature limit. J. Low Temp. Phys. 156, 215–267 (2009)ADSCrossRefGoogle Scholar
  38. Landau, L.: The theory of superfluidity of helium II. J. Phys. – USSR 5, 71–90 (1941)Google Scholar
  39. Laurie, J., L’vov, V.S., Nazarenko, S., Rudenko, O.: Interaction of Kelvin waves and nonlocality of energy transfer in superfluids. Phys. Rev. B 81, 104526 (2010)ADSCrossRefGoogle Scholar
  40. Leadbeater, M., Samuels, D.C., Barenghi, C.F., Adams, C.S.: Decay of superfluid turbulence via Kelvin-wave radiation. Phys. Rev. A 67, 015601 (2004)ADSCrossRefGoogle Scholar
  41. Leadbeater, M., Winiecki, T., Samuels, D.C., Barenghi, C.F., Adams, C.S.: Sound emission due to superfluid vortex reconnections. Phys. Rev. Lett. 86, 1410–1413 (2001)ADSCrossRefGoogle Scholar
  42. Lipniacki, T.: Dynamics of superfluid \(^4\)HE: two-scale approach. Eur. J. Mech. B/Fluids 25, 435–458 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  43. L’vov, V.S., Nazarenko, S.: Spectrum of Kelvin-wave turbulence superfluid. JETP Lett. 91, 428–434 (2010)ADSCrossRefGoogle Scholar
  44. L’vov, V.S., Nazarenko, S.V., Rudenko, O.: Bottleneck crossover between classical and quantum superfluid turbulence. Phys. Rev. B 76, 024520 (2007)ADSCrossRefGoogle Scholar
  45. L’vov, V.S., Nazarenko, S.V., Rudenko, O.: Gradual eddy-wave crossover in superfluid turbulence. J. Low Temp. Phys. 153, 140–161 (2008)ADSCrossRefGoogle Scholar
  46. L’vov, V.S., Nazarenko, S.V., Skrbek, L.: Energy spectra of developed turbulence in helium superfluids. J. Low Temp. Phys. 145, 125–142 (2006)ADSCrossRefGoogle Scholar
  47. L’vov, V.S., Nazarenko, S., Volovik, G.E.: Energy spectra of developed super uid turbulence. JETP Lett. 80, 479–483 (2004)ADSCrossRefGoogle Scholar
  48. Nemirovskii, S.K.: Quantum turbulence: theoretical and numerical problems. Phys. Rep. 524, 85–202 (2013)ADSMathSciNetCrossRefGoogle Scholar
  49. Nore, C., Abid, M., Brachet, M.E.: Kolmogorov turbulence in low temperature superflows. Phys. Rev. Lett. 78, 3896–3899 (1997)ADSCrossRefGoogle Scholar
  50. Ogawa, S., Tsubota, M., Hattori, Y.: Study of reconnection and acoustic emission of quantized vortices in superfluid by the numerical analysis of the Gross–Pitaevskii equation. J. Phys. Soc. Jpn. 71, 813–821 (2002)ADSCrossRefzbMATHGoogle Scholar
  51. Proment, D., Nazarenko, S., Onorato, M.: Quantum turbulence cascades in the Gross–Pitaevskii model. Phys. Rev. A 80, 051603(R) (2009)ADSCrossRefGoogle Scholar
  52. Roche, P.E., Barenghi, C.F., Leveque, E.: Quantum turbulence at finite temperature: the two-fluid cascade. EPL 87, 54006 (2009)ADSCrossRefGoogle Scholar
  53. Rorai, C., Skipper, J., Kerr, R.M., Sreenivasan, K.R.: Approach and separation of quantised vortices with balanced cores. J. Fluid Mech. 808, 641–667 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. Sagaut, P.: Large-Eddy Simulation of Incompressible Flows - An Introduction, 3rd edn. Springer, Berlin (2005)zbMATHGoogle Scholar
  55. Samuels, D.C., Barenghi, C.F.: Vortex heating in superfluid helium at low temperature. Phys. Rev. Lett. 81, 4381–4383 (1998)ADSCrossRefGoogle Scholar
  56. Sasa, N., Kano, T., Machida, M.: Large scale numerical simulation of superfluid turbulence. Phys. Rev. B 84, 054525 (2011)ADSCrossRefGoogle Scholar
  57. Sasa, N., Kano, T., Machida, M., L’vov, V.S., Rudenko, O., Tsubota, M.: Energy spectra of quantum turbulence: large-scale simulation and modeling. Prog. Nucl. Sci. Technol. 2, 609–612 (2011)CrossRefGoogle Scholar
  58. Schwarz, K.W.: Three-dimensional vortex dynamics in superfluid \(^4\)He: line-line and line-boundary interactions. Phys. Rev. B 31, 5782–5804 (1985)ADSCrossRefGoogle Scholar
  59. Schwarz, K.W.: Three-dimensional vortex dynamics in superfluid \(^4\)He: homogeneous superfluid turbulence. Phys. Rev. B 38, 2398–2417 (1988)ADSCrossRefGoogle Scholar
  60. Schwarz, K.W.: Anomalous decay of turbulence in \(^4\)He. Phys. Rev. Lett. 66, 1898–1901 (1991)ADSCrossRefGoogle Scholar
  61. Skbrek, L.: Flow phase diagram for the helium superfluids. In: IUTAM Symposium on Elementary Vortices and Coherent Structures: Significance in Turbulence Dynamics. Book Series: Fluid Mechanics and Its Applications, vol. 79, pp. 361–366 (2004)Google Scholar
  62. Skbrek, L.: Energy spectra of quantum turbulence in He II and \(^3\)He-B: a unified view. JETP Lett. 83, 127–131 (2006)ADSCrossRefGoogle Scholar
  63. Skbrek, L.: A simple phenomenological model for effective kinematic viscosity of helium superfluids. J. Low. Temp. Phys. 161, 555–560 (2010)ADSCrossRefGoogle Scholar
  64. Skbrek, L., Niemela, J.J., Sreenivasan, K.R.: Energy spectrum of grid-generated He II turbulence. Phys. Rev. E 64, 067301 (2001)ADSCrossRefGoogle Scholar
  65. Skrbek, L., Sreenivasan, K.R.: Developed quantum turbulence and its decay. Phys. Fluids 24, 011301 (2012)ADSCrossRefGoogle Scholar
  66. Skrbek, L., Stalp, S.R.: On the decay of homogeneous isotropic turbulence. Phys. Fluids 12(8), 1997–2019 (2000)ADSCrossRefzbMATHGoogle Scholar
  67. Swanson, C.E., Wagner, W.T., Donnelly, R.J., Barenghi, C.F.: Calculation of frequency- and velocity-dependent mutual friction parameters in helium II. J. Low Temp. Phys. 66, 263–276 (1987)ADSCrossRefGoogle Scholar
  68. Tchoufag, J., Sagaut, P.: Eddy-damped quasinormal Markovian simulations of superfluid turbulence in helium II. Phys. Fluids 22, 125103 (2010)ADSCrossRefGoogle Scholar
  69. Tisza, L.: The viscosity of liquid helium and the Bose-Einstein statistic. Comptes Rendus hebdomadaires des séances de l’Acad. des Sci. 207, 1186–1189 (1938)Google Scholar
  70. Tsatos, M.C., Tavares, P.E.S., Cidrim, A., Fritsch, A.R., Caracanhas, M.A., dos Santos, F.E.A., Barenghi, C.F., Bagnato, V.S.: Quantum turbulence in trapped atomic Bose–Einstein condensates. Phys. Rep. 622, 1–52 (2016)ADSMathSciNetCrossRefGoogle Scholar
  71. Tsubota, M.: Quantum turbulence - from superfluid helium to atomic Bose–Einstein condensates. J. Phys. Condens. Matter. 21, 164–207 (2009)CrossRefGoogle Scholar
  72. Tsubota, M.: Quantum hydrodynamics. Phys. Rep. 522, 191–238 (2013)ADSMathSciNetCrossRefGoogle Scholar
  73. Tsubota, M., Adachi, H.: Simulation of counterflow turbulence by vortex filaments. Statistics of vortex reconnections. J. Low Temp. Phys. 162, 367–374 (2011)ADSCrossRefGoogle Scholar
  74. Tsubota, M., Fujimoto, K., Yui, S.: Numerical studies of quantum turbulence. J. Low Temp. Phys. 188, 119–189 (2017)ADSCrossRefGoogle Scholar
  75. Vinen, W.F.: Classical character of turbulence in a quantum liquid. Phys. Rev. B 61, 1410–1420 (2000)ADSCrossRefGoogle Scholar
  76. Vinen, W.F.: Decay of superfluid turbulence at a very low temperature: the radiation of sound from a Kelvin wave on a quantized vortex. Phys. Rev. B 64, 134520 (2001)ADSCrossRefGoogle Scholar
  77. Vinen, W.F.: Theory of quantum grid turbulence in superfluid \(^3\)He-B. Phys. Rev. B 71, 024513 (2005)ADSCrossRefGoogle Scholar
  78. Vinen, W.F.: An introduction to quantum turbulence. J. Low Temp. Phys. 145, 7–24 (2006)ADSCrossRefGoogle Scholar
  79. Vinen, W.F.: Quantum turbulence: achievements and challenges. J. Low Temp. Phys. 161, 419–444 (2010)ADSCrossRefGoogle Scholar
  80. Vinen, W.F., Niemela, J.J.: Quantum turbulence. J. Low Temp. Phys. 128, 167–231 (2002)ADSCrossRefGoogle Scholar
  81. Vinen, W.F., Tsubota, M., Mitani, A.: Kelvin-wave cascades in turbulent superfluid \(^4\)He at very low temperatures. J. Low Temp. Phys. 134, 457–462 (2004)ADSCrossRefGoogle Scholar
  82. Volovik, G.E.: Classical and quantum regimes of the superfluid turbulence. JETP Lett. 78, 533–537 (2003)ADSCrossRefGoogle Scholar
  83. Volovik, G.E.: On developed superfluid turbulence. J. Low Temp. Phys. 136, 309–327 (2004)ADSCrossRefGoogle Scholar
  84. Walmsley, P.M., Golov, A.I.: Quantum and quasiclassical types of superfluid turbulence. Phys. Rev. Lett. 100, 245301 (2008)ADSCrossRefGoogle Scholar
  85. Yepez, J., Vahala, G., Vahala, L., Soe, M.: Superfluid turbulence from quantum Kelvin wave to classical Kolmogorov cascades. Phys. Rev. Lett. 103, 084501 (2009)ADSCrossRefGoogle Scholar
  86. Yoshida, K., Arimitsu, T.: Inertial-range structure of Gross–Pitaevskii turbulence within a spectral closure approximation. J. Phys. A: Math. Theor. 46, 335501 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  87. Zuccher, S., Caliari, M., Baggaley, A.W., Barenghi, C.F.: Quantum vortex reconnections. Phys. Fluids 24, 125108 (2012)ADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Laboratoire de Mécanique, Modélisation et Procédés Propres, UMR CNRS 7340, Ecole Centrale de MarseilleAix-Marseille UniversitéMarseilleFrance
  2. 2.Laboratoire de Mécanique des Fluides et d’Acoustique, UMR CNRS 5509Ecole Centrale de LyonÉcullyFrance

Personalised recommendations