Kolmogorov Spectrum of Superfluid Turbulence: Numerical Analysis of the Gross-Pitaevskii Equation with a Small-Scale Dissipation

Abstract

The energy spectrum of superfluid turbulence is studied numerically by solving the GrossPitaevskii equation. We introduce the dissipation term which works only in the scale smaller than the healing length, to remove short wavelength excitations which may hinder the cascade process of quantized vortices in the inertial range. The obtained energy spectrum is consistent with the Kolmogorov law.PACS numbers: 67.40. Vs, 47.37.+q, 67.40.Hf The physics of quantized vortices in liquid 4 He is one of the most important topics in low temperature physics [1]. Liquid 4 He enters the superfluid state at 2.17 K. Below this temperature, the hydrodynamics is usually described using the two-fluid model in which the system consists of inviscid superfluid and viscous normal fluid. Early experimental works on the subject focused on thermal counterflow in which the normal fluid flowed in the opposite direction to the superfluid flow. This flow is driven by the injected heat current, and it was found that the superflow becomes dissipative when the relative velocity between two fluids exceeds a critical value [6] in superfluid 4 He at temperatures above 1 K, and their results were consistent with the Kolmogorov law. The Kolmogorov law is one of the most important statistical laws [7] of fully developed CT, so these experiments show a similarity between ST and CT. This can be understood using the idea that the superfluid and the normal fluid are likely to be coupled together by the mutual friction between them and thus to behave like a conventional fluid [8]. Since the normal fluid is negligible at very low temperatures, an important question arises: even without the normal fluid, is ST still similar to CT or not?To address this question, we consider the statistical law of CT [7]. The steady state for fully developed turbulence of an incompressible classical fluid follows the Kolmogorov law for the energy spectrum. The energy is injected into the fluid at some large scales in the energycontaining range. This energy is transferred in the inertial range from large to small scales without been dissipated. The inertial range is believed to be sustained by the self-similar Richardson cascade in which large eddies are broken up to smaller ones, having the Kolmogorov lawHere the energy spectrum E(k) is defined as E = dk E(k), where E is the kinetic energy per unit mass and k is the wave number from the Fourier transformation of the velocity field. The energy transferred to smaller scales in the energy-dissipative range is dissipated by the viscosity with the dissipation rate, which is identical with the energy flux ǫ of Eq. (1) in the inertial range. The Kolmogorov constant C is a dimensionless parameter of order unity.In CT, the Richardson cascade is not completely understood, because it is impossible to definitely identify each eddy. In contrast, quantized vortices in superfluid are definite and stable topological defects. A BoseEinstein condensed system yields a macroscopic wave function Φ(x, t) = ρ(x, t)e iφ(x,t) , whose dynamics is g...

Numerical Studies of Quantum Turbulence

“…Then, the incompressible kinetic energy spectrum was found to exhibit the Kolmogorov −5/3 power law in a limited time period. After this study, Parker et al also confirmed the Kolmogorov law in QT in a rotating system [125], and Kobayashi et al introduced a phenomenological dissipation term into the GP equation and obtained the Kolmogorov law [126]. Figures 23 (a) and (b) are the vortex configuration and the incompressible kinetic energy spectrum in Ref.…”

“…Figures 23 (a) and (b) are the vortex configuration and the incompressible kinetic energy spectrum in Ref. [126], where the decaying QT is generated by the initial state with random phase and the Kolmogorov law is confirmed. Furthermore, Kobayashi et al generated the stationary steady state QT using the dissipative GP equation and obtained the Kolmogorov law [127,128].…”

“…Originally, this study focused on superfluid helium, but this result is important for understanding QT in ultracold atomic BECs. After this study, some theoretical works studied QT in ultracold atomic BECs, finding the same Kolmogorov −5/3 law [125,126,127,128,129,130].…”

Quantum turbulence—from superfluid helium to atomic Bose–Einstein condensates

“…Araki et al made a vortex tangle arising from Taylor-Green vortices and obtained an energy spectrum consistent with the Kolmogorov law. The third used the modified GP model by Kobayashi and Tsubota [44,45].…”

Macroscopic Excitations in Confined Bose–Einstein Condensates, Searching for Quantum Turbulence

“…In this work, we limit ourselves to the numerical solutions of the time-reversal invariant GP equation, valid for temperatures as close as possible to absolute zero, but we recognize that in order to make a direct comparison with actual experiments, being these at finite temperature, one must take into account dissipative effects. There are already efforts along these lines 24,25,26,27,28,29,30 including the effects of the thermal cloud that surrounds a confined BEC. True stationary states, of course, are found in those cases…”