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arXiv:2111.11659

Reply to "Comment on 'Theoretical analysis of quantum turbulence using the Onsager ideal turbulence theory'"

Tomohiro Tanogami

Published Nov 23, 2021• 1 citation
arXivHTMLPDF

Abstract

We refute the criticism expressed in a Comment by Krstulovic, L'vov, and Nazarenko [arXiv:2107.10598] on our paper [Phys. Rev. E 103, 023106 (2021)]. We first show that quantization of circulation is not ignored in our analysis. Then, we propose a more sophisticated analysis to avoid a subtle problem with the regularity of the velocity field. We thus defend the main results of our paper, which predicts the double-cascade scenario where the quantum stress cascade follows the Richardson cascade. We also provide a conjecture on the relation between the Kelvin-wave cascade and the quantum stress cascade.

DOI
10.1103/physreve.105.027102

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Comment on “Theoretical analysis of quantum turbulence using the Onsager ideal turbulence theory”

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In a recent paper, Tanogami [Phys. Rev. E 103, 023106 (2021)2470-004510.1103/PhysRevE.103.023106] proposes a scenario for quantum turbulence where the energy spectrum at scales smaller than the intervortex distance is dominated by a quantum stress cascade, in opposition to Kelvin-wave cascade predictions. The purpose of the present Comment is to highlight some physical issues in the derivation of the quantum stress cascade, in particular to stress that quantization of circulation has been ignored.

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Theoretical analysis of quantum turbulence using the Onsager ideal turbulence theory

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We investigate three-dimensional quantum turbulence as described by the Gross-Pitaevskii model using the analytical method exploited in the Onsager "ideal turbulence" theory. We derive the scale independence of the scale-to-scale kinetic energy flux and establish a double-cascade scenario: At scales much larger than the mean intervortex ℓ_{i}, the Richardson cascade becomes dominant, whereas at scales much smaller than ℓ_{i}, another type of cascade is induced by quantum stress. We then evaluate the corresponding velocity power spectrum using a phenomenological argument. The relation between this cascade, which we call quantum stress cascade, and the Kelvin-wave cascade is also discussed.

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We comment on the paper by Sonin [Phys. Rev. B 85, 104516 (2012)] with most statements of which we disagree. We use this option to shed light on some important issues of a theory of Kelvin-wave turbulence, touched on in Sonin's paper, in particular, on the relation between the Vinen spectrum of strong and the L'vov-Nazarenko spectrum of weak turbulence of Kelvin waves. We also discuss the role of explicit calculation of the Kelvin-wave interaction Hamiltonian and ``symmetry arguments'' that have to resolve a contradiction between the Kozik-Svistunov and the L'vov-Nazarenko spectrum of weak turbulence of Kelvin waves.

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Both the Kelvin wave and the Kolmogorov turbulence interpretations presented in the PRL, [v. 103, 084501 (2009) by J. Yepez, G. Vahala, L.Vahala and M. Soe, arXiv:0905.0159] are misleading, and much more theoretical analysis needs to be done for the interpretation of the important numerical results obtained by the authors. A way to do this is suggested.

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Theoretical Analysis of the Energy Cascade in Quantum Turbulence: From Richardson–Kolmogorov Eddies to Kelvin Waves

Abishek Mahalingam

Abstract Quantum turbulence — the chaotic, non-equilibrium state of a superfluid threaded by a dense tangle of quantized vortex lines — provides a uniquely clean testbed for the universal statistical mechanics of turbulent energy transfer. We develop a theoretical account of the energy cascade across all relevant length scales: from macroscopic, quasi-classical Richardson–Kolmogorov eddies, through the dimensionally frustrated L’vov–Nazarenko–Rudenko (LNR) bottleneck at the mean inter-vortex spacing, down to the weakly nonlinear Kelvin-wave cascade on individual filaments, and finally to the zero-temperature dissipation cutoff. Starting from the Gross–Pitaevskii equation and its Madelung hydrodynamic decomposition, we separate the compressible and incompressible components of the flow to isolate the vortical and acoustic energy channels. Vortex reconnection is identified as the microscopic mechanism coupling these regimes, and the competing predictions of the Vortex Filament Model and the Gross–Pitaevskii equation for reconnection dynamics are compared. We review the theoretical controversy between the Kozik–Svistunov six-wave cascade and the L’vov–Nazarenko four-wave cascade, together with the associated tilt-symmetry debate, and present the finite-temperature two-fluid (HVBK) description, showing how mutual friction locks the normal and superfluid components at large scales while damping Kelvin waves at small scales. PACS numbers: 67.25.dk, 47.37.+q, 03.75.Kk, 47.32.C−

•Jul 10, 2026•0 citations

Comment on “Superfluid Turbulence from Quantum Kelvin Wave to Classical Kolmogorov Cascades”

Victor S. L’vov, Sergey Nazarenko

A Comment on the Letter by Jeffrey Yepez et al. [Phys Rev. Lett. 103, 084501 (2009)].Received 16 September 2009DOI:https://doi.org/10.1103/PhysRevLett.104.219401©2010 American Physical Society

Physical Review Letters•May 25, 2010•5 citations
We refute the criticism expressed in a Comment by Krstulovic, L'vov, and Nazarenko [arXiv:2107.10598] on our paper [Phys. Rev. E 103, 023106 (2021)]. We first show that quantization of circulation is not ignored in our analysis. Then, we propose a more sophisticated analysis to avoid a subtle problem with the regularity of the velocity field. We thus defend the main results of our paper, which predicts the double-cascade scenario where the quantum stress cascade follows the Richardson cascade. We also provide a conjecture on the relation between the Kelvin-wave cascade and the quantum stress cascade.
In a recent paper, Tanogami [Phys. Rev. E 103, 023106 (2021)2470-004510.1103/PhysRevE.103.023106] proposes a scenario for quantum turbulence where the energy spectrum at scales smaller than the intervortex distance is dominated by a quantum stress cascade, in opposition to Kelvin-wave cascade predictions. The purpose of the present Comment is to highlight some physical issues in the derivation of the quantum stress cascade, in particular to stress that quantization of circulation has been ignored.
We investigate three-dimensional quantum turbulence as described by the Gross-Pitaevskii model using the analytical method exploited in the Onsager "ideal turbulence" theory. We derive the scale independence of the scale-to-scale kinetic energy flux and establish a double-cascade scenario: At scales much larger than the mean intervortex ℓ_{i}, the Richardson cascade becomes dominant, whereas at scales much smaller than ℓ_{i}, another type of cascade is induced by quantum stress. We then evaluate the corresponding velocity power spectrum using a phenomenological argument. The relation between this cascade, which we call quantum stress cascade, and the Kelvin-wave cascade is also discussed.
We comment on the paper by Sonin [Phys. Rev. B 85, 104516 (2012)] with most statements of which we disagree. We use this option to shed light on some important issues of a theory of Kelvin-wave turbulence, touched on in Sonin's paper, in particular, on the relation between the Vinen spectrum of strong and the L'vov-Nazarenko spectrum of weak turbulence of Kelvin waves. We also discuss the role of explicit calculation of the Kelvin-wave interaction Hamiltonian and ``symmetry arguments'' that have to resolve a contradiction between the Kozik-Svistunov and the L'vov-Nazarenko spectrum of weak turbulence of Kelvin waves.
Abstract Quantum turbulence — the chaotic, non-equilibrium state of a superfluid threaded by a dense tangle of quantized vortex lines — provides a uniquely clean testbed for the universal statistical mechanics of turbulent energy transfer. We develop a theoretical account of the energy cascade across all relevant length scales: from macroscopic, quasi-classical Richardson–Kolmogorov eddies, through the dimensionally frustrated L’vov–Nazarenko–Rudenko (LNR) bottleneck at the mean inter-vortex spacing, down to the weakly nonlinear Kelvin-wave cascade on individual filaments, and finally to the zero-temperature dissipation cutoff. Starting from the Gross–Pitaevskii equation and its Madelung hydrodynamic decomposition, we separate the compressible and incompressible components of the flow to isolate the vortical and acoustic energy channels. Vortex reconnection is identified as the microscopic mechanism coupling these regimes, and the competing predictions of the Vortex Filament Model and the Gross–Pitaevskii equation for reconnection dynamics are compared. We review the theoretical controversy between the Kozik–Svistunov six-wave cascade and the L’vov–Nazarenko four-wave cascade, together with the associated tilt-symmetry debate, and present the finite-temperature two-fluid (HVBK) description, showing how mutual friction locks the normal and superfluid components at large scales while damping Kelvin waves at small scales. PACS numbers: 67.25.dk, 47.37.+q, 03.75.Kk, 47.32.C−