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arXiv:0709.2501

Generalized vorticity in transitional quantum turbulence

A. Bershadskii

Published Sep 16, 2007• 1 citation
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Abstract

Transition to condensate state in the degenerate gas of bosons is studied using the Gross-Pitaevskii equation. It is shown that adiabatic invariance of an enstrophy (mean squared vorticity) based on a generalized vorticity $curl (|ψ|^2 {\bf v})/|ψ|^2$, controls dynamics of the transitional turbulence just before formation of the condensate with its tangles of quantized vortices ($|ψ|^2 {\bf v}$ is the weighted velocity field defined on the macroscopic wave function $ψ$). Scaling of the angle-averaged occupation number spectrum, $N_k \sim k^{-1}$, has been obtained for the free developing transitional turbulence for the weak nonlinear (and completely disordered) initial conditions. Results of the three-dimensional numerical simulations have been used to support the theoretical consideration.

DOI
10.48550/arxiv.0709.2501

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Transition to condensate state in the degenerate gas of bosons is studied using the Gross-Pitaevskii equation. It is shown that adiabatic invariance of an enstrophy (mean squared vorticity) based on a generalized vorticity curl(∣ψ∣2v)/∣ψ∣2curl (|ψ|^2 {\bf v})/|ψ|^2curl(∣ψ∣2v)/∣ψ∣2, controls dynamics of the transitional turbulence just before formation of the condensate with its tangles of quantized vortices (∣ψ∣2v|ψ|^2 {\bf v}∣ψ∣2v is the weighted velocity field defined on the macroscopic wave function ψψψ). Scaling of the angle-averaged occupation number spectrum, Nk∼k−1N_k \sim k^{-1}Nk​∼k−1, has been obtained for the free developing transitional turbulence for the weak nonlinear (and completely disordered) initial conditions. Results of the three-dimensional numerical simulations have been used to support the theoretical consideration.
We investigate formation of Bose-Einstein condensates under nonequilibrium conditions using numerical simulations of the three-dimensional Gross-Pitaevskii equation. For this, we set initial random weakly nonlinear excitations and the forcing at high wave numbers and study propagation of the turbulent spectrum toward the low wave numbers. Our primary goal is to compare the results for the evolving spectrum with the previous results obtained for the kinetic equation of weak wave turbulence. We demonstrate existence of a regime for which good agreement with the wave turbulence results is found in terms of the main features of the previously discussed self-similar solution. In particular, we find a reasonable agreement with the low-frequency and the high-frequency power-law asymptotics of the evolving solution, including the anomalous power-law exponent ${x}^{*}\ensuremath{\approx}1.24$ for the three-dimensional wave action spectrum. We also study the regimes of very weak turbulence when the evolution is affected by the discreteness of the Fourier space, and the strong turbulence regime when the emerging condensate modifies the wave dynamics and leads to formation of strongly nonlinear filamentary vortices.
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