Abrupt Transition between Three-Dimensional and Two-Dimensional Quantum Turbulence

We present numerical evidence of a critical-like transition in an out-of-equilibrium mean-field description of a quantum system. By numerically solving the Gross-Pitaevskii equation we show that quantum turbulence displays an abrupt change between three-dimensional (3D) and two-dimensional (2D) behavior. The transition is observed both in quasi-2D flows in cubic domains (controlled by the amplitude of a 3D perturbation to the flow), as well as in flows in thin domains (controlled by the domain aspect ratio) in a configuration that mimics systems realized in laboratory experiments. In one regime the system displays a transfer of the energy towards smaller scales, while in the other the system displays a transfer of the energy towards larger scales and a coherent self-organization of the quantized vortices.

Figure 1

(a) Spectrum of the incompressible kinetic energy averaged between $t=1$ and 10 for simulations in cubic domains ($N_x=N_y=N_z=512$) and different values of $A_z$. Kolmogorov power laws $\sim k^{-5/3}$ are indicated as a reference by solid black lines. The vertical dashed line indicates the inverse mean intervortex distance. Note the growth of energy and a $\sim k^{-5/3}$ scaling for $k\lesssim 10$ when $A_z$ is small. (b) Total energy fluxes for the same simulations. For small $A_z$ the flux becomes negative for $k\lesssim 10$, and the positive flux for $k>10$ decreases. References for the values of $A_z$ in (a) and (b) are provided in the inset. (c) Same as in (a) for simulations in thin domains, for different values of $\gamma$. (d) Same as in (c) for simulations in thin domains; the inset gives the values of $\gamma$. In all panels, dashed curves highlight simulations for which movies are available in Ref. [18].

Figure 2

Ratio of inverse energy flux to total energy flux as a function of the normalized control parameter $Z$ (either $A_z$ or $\gamma$, normalized by their respective critical values $A_z^c$ or ${\gamma }^c$, see inset). For cubic domains two curves are shown, corresponding to spatial resolutions of ${256}^3$ and ${512}^3$ grid points.

Figure 3

Time evolution of the incompressible kinetic energy (a) in simulations in cubic domains with different perturbation amplitudes $A_z$ (${512}^3$ runs), and (b) in simulations in domains with different aspect ratios $\gamma$. The insets show the total vortex length [18] as a function of time for each case. References are as in Fig. 1; a few labels are provided as guidelines.

Figure 4

(a) Time to reach the maximum length of the vortices as a function of $\gamma /{\gamma }^c$. (b) Normalized maximum total length of the vortices, $l_{\max }/L_x$, as a function of the control parameter. (c) Flow integral length scale normalized by the domain size, $L/L_x$, as a function of the same control parameter.