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Isotropic Turbulence with Coupled Microstructures. II: Quantum Turbulence

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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.

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Notes

  1. 1.

    It must be kept in mind that experiments related to quantum turbulence are all performed in closed configurations, which leads to the use of a potential to account for the confinement appartus.

  2. 2.

    The Magnus force arises when a body with non-zero circulation moves into a flow. The circulation induces an asymmetry in the velocity and pressure distribution along the body surface, yielding a non-zero force.

  3. 3.

    The original HVBK model was not design for turbulence but equations used to describe quantum turbulence are formally the same.

  4. 4.

    This number denotes the number of Kelvin waves with wave number k.

  5. 5.

    It is important noticing that the most advanced theories for reconnection deals with the case of Bose–Einstein condensates, which are often extrapolated for liquid helium for which no such theory exists and only indirect measurements of vortex line length are possible.

  6. 6.

    The shortcoming relation \( E(k) = E_{KW}(k)\) is misleading and should be avoided.

  7. 7.

    As a matter of fact, most existing results dealing on Kelvin wave cascade have been obtained on very simplified configurations with one or very few vortex filaments.

  8. 8.

    At low temperatures, the quantized vortex core radius is about 77 nm in \(^3\)He-B, to be compared to 0.1 nm in \(^4\)He.

  9. 9.

    As quoted by Nemirovskii (2013), it is also important to note that the idea underlying the pile-up of kinetic energy at scales about the inter-vortex distance \(\ell \) is that the Kelvin wave energy at such scales is much smaller than the one given by the Kolmogorov spectrum, leading to a mismatch between the two spectra and the associated cascade rate. This corresponds to the flawed equality \(E(k) = E_{KW}(k)\), as discussed in Sect. 6.4.2.3. As a matter of fact, if the correct kinetic energy, i.e. the one of the velocity fluctuations induced by quantized vortices and Kelvin waves is taken into account, the mismatch and the need for a bottlneck disappear.

  10. 10.

    It is worth noting that this model for the destruction term corresponds to the type I decay discussed in Sect. 6.4.1.1. Another version of Vinen’s equation must be formulated for the type II régime, in which counterflow velocity is small with respect to the self-induced velocity and the destruction term scales as \(L_0^{5/2}\).

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Authors and Affiliations

  1. Laboratoire de Mécanique, Modélisation et Procédés Propres, UMR CNRS 7340, Ecole Centrale de Marseille, Aix-Marseille Université, Marseille, France

    Pierre Sagaut

  2. Laboratoire de Mécanique des Fluides et d’Acoustique, UMR CNRS 5509, Ecole Centrale de Lyon, Écully, France

    Claude Cambon

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Sagaut, P., Cambon, C. (2018). Isotropic Turbulence with Coupled Microstructures. II: Quantum Turbulence. In: Homogeneous Turbulence Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-319-73162-9_6

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