Energy Spectrum of the Superfluid Velocity Made by Quantized Vortices in Two-Dimensional Quantum Turbulence
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We discuss the configurations of vortices in two-dimensional quantum turbulence, studying energy spectrum of superfluid velocity and correlation functions with the distance between two vortices. We apply the above method to quantum turbulence described by Gross-Pitaevskii equation in Bose-Einstein condensates. We make two-dimensional quantum turbulence from many dark solitons through the dynamical instability. A dark soliton is unstable and decays into vortices in two- and three-dimensional systems. In our work, we propose a method of discriminating between the uncorrelatedturbulence and the correlatedturbulence. We decompose the energy spectrum into two terms, namely the self-energy spectrum Eself(k) made by individual vortices and the interactive energy spectrum Eint(k) made by interference of two vortices. The uncorrelatedturbulence is defined as turbulence with Eint(k)≪Eself(k), while the correlatedturbulence is turbulence where Eint(k) is not much smaller than Eself(k). Our simulations show that in the decay of dark solitons, the vortices created consist of correlated pairs of opposite circulation vortices, leading to the correlatedturbulence.
KeywordsBose-Einstein condensate Quantum turbulence Quantum vortices
Turbulence is one of the most challenging problems in fluid dynamics. The circulation of vortices in classical turbulence (CT) has an arbitrary value and the core of a vortex is not well-defined due to the kinematic viscous diffusion. In contrast, quantum turbulence (QT) is composed of quantized vortices. A quantized vortex is a well-defined topological defect having a definite circulation κ and a thin core of the order of the coherence length ξ.
Recently, the internal structure of QT consisting of quantized vortices has been discussed [1, 2, 3, 4]. In three-dimensional QT of superfluid helium, two kinds of the vortex line density L decaying as t−3/2 and t−1, were observed in experiments and simulations in the previous works [5, 6, 7, 8]. The turbulence decaying as L∝t−3/2 is called the semi-classicalturbulence, which may be thought of as a specific case of the correlatedturbulence. On the other hand, the turbulence decaying as L∝t−1 is called the randomturbulence, which may be referred to as the uncorrelatedturbulence. These kinds of decay are related to the correlation between quantized vortices. In the semi-classicalturbulence, turbulent energy is concentrated on scales ≫l∼L−1/2 where l is the mean distance of vortices. It exhibits a Kolmogorov spectrum, leading to the vortex line length decay L∝t−3/2. In the randomturbulence, the total energy is mainly determined on the scale <l, the decay of the vortex line length obeys t−1. The energy can be delivered from larger ∼l to very short ≪l by kelvin wave cascade [9, 10].
In this paper, we propose a method of deciding whether a sample of turbulence is correlated or uncorrelated by focusing on the histograms of the distance between two vortices. Section 2 describes the analysis of the energy spectrum of the point vortex model. We display the relation between the energy spectrum and the configurations of vortices, which is useful to understand the internal structure of QT. In Sect. 3, we introduce the formation of two-dimensional (2D) QT from many dark solitons in a uniform system by calculating the Gross-Pitaevskii equation (GPE) in order to simulate the dynamics after the solitons are made by the interference of four Bose-Einstein condensates (BECs) [11, 12]. In Sect. 4, we examine whether 2D QT made from dark solitons is correlated or uncorrelated by calculating the energy spectrum of the point vortex model and the histograms.
2 The Analysis of Energy Spectrum of the Point Vortex Model in 2D QT
In order to only consider the contribution of vortices to QT, we address the energy spectrum of the point vortex model, neglecting compressible effects such as sound waves and the density profile of the vortex core. This model is applicable when the mean intervortex distance is much larger than the coherence length ξ. It describes the dynamics of classical vortices in Euler equation . The point vortex model has been constructed as discrete vorticity in CT, so it is applicable to QT in which all vortices have exactly the same circulation κ and the thin core.
h+(l)≃h−(l) (The uncorrelatedturbulence)In randomly distributed vortices, since vortices are uniformly distributed, the correlation functions are h±(l) = constant, regardless of the signs. Thus, Eint(k) is negligible and E(k)≃Eself(k)∝k−1. We refer to this kind of turbulence to the uncorrelatedturbulence. Figure 1(a) shows a sample of the configuration with random vortices. The energy spectrum (Eq. (2)) is obtained by the point vortex model as shown in Fig. 1(b), which is consistent with k−1.
h+(l)≠h−(l) (The correlatedturbulence)
In this case, Eq. (3) E(k) does not show E(k)∝k−1 in low wave number, because h+(l)≠h−(l) may lead to Eint(k)≠0. We refer to this kind of turbulence to the correlatedturbulence.
3 Two-Dimensional Quantum Turbulence of the Gross-Pitaevskii Equation
We show a simulation of the dimensionless GPE. As a method of making QT, this paper shows that 2D QT is created from many dark solitons through the dynamical instability [18, 19], which may occur in experiments when four BECs interfere.
4 The Energy Spectrum of the Point Vortex Model and Correlation Functions
In order to determine whether 2D QT made by dark solitons is correlated or uncorrelated, we investigate the energy spectrum of the point vortex model and the correlation functions h±(l).
One finds that a peak of the energy spectra (Fig. 3(b)) has shifted to the left as the vortices decay and the spectrum at τ=312.5 obeys E(k)∝k−1 in the lower wave number too, which means QT having a lot of opposite sign pairs gradually changes to the uncorrelatedturbulence.
We discussed the configurations of vortices by calculating the energy spectra of superfluid velocity made by vortices and the correlation functions of the distance between two vortices. In this paper, it was found that 2D QT from dark solitons has a lot of opposite sign pairs which we call vortex pairs.
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