Decaying superfluid turbulence near an anomalous non-thermal fixed point

Abstract

Anomalously slow coarsening in a dilute two-dimensional (2d) superfluid -- associated with a non-thermal fixed point in the closed system's universal dynamics and driven by three-vortex collisions which initiate vortex-pair annihilations -- is found to exhibit spatial scaling characteristics of Kraichnan-Kolmogorov turbulence. During a universal interval, when the characteristic length scale related to the inter-defect distance grows as $\ell_\text{v}\sim t^{\,β}$, with $β\approx1/5$, moments of the superfluid velocity circulation around an area of extent $r$ scale as predicted by classical turbulence theory, $Γ^{p}(r)\sim r^{4p/3}$. Intermittency corrections seen for higher values of $p$ are found to be consistent with values measured for fully developed, classical turbulence. Hence, we link the decaying quantum turbulence cascade in a closed superfluid to universal dynamics close to a non-thermal fixed point. The decay exponent $β$ clearly deviates from values known for classical systems.

Figures (11)

• Figure 1: Normalized current $|\vb{j}|/\langle |\vb{j}| \rangle$ at four times $t\in[0,10^3,10^4,10^5](2\pi)/\omega_{z}$, with streamlines indicating the flow of $\vb{j}$. The system approaches space-time scaling near the anomalous NTFP within the time interval $t\approx(0.3\dots2)\cdot10^4(2\pi)/\omega_{z}$, corresponding to a decaying turbulent cascade, together with spatial scaling characteristics akin to 2d fully developed turbulence. The insets show the corresponding snapshots of the density normalized to its spatial average, $\rho_2/\langle \rho_2 \rangle$.
• Figure 2: (a) Second moment $\Gamma^2(r)$ of the velocity circulation (\ref{['eq:Circulation']}) about squares of side length $r$ at logarithmically spaced times in the interval $t\in\{10^3\dots10^{5}\}(2\pi)/\omega_z$. The dashed vertical lines indicate the healing length $\xi$ and the inter-defect distances $\ell_\mathrm{v,i}\approx32\,a_\mathrm{ho}$ and $\ell_\mathrm{v,f}\approx100\,a_\mathrm{ho}$ at the earliest and latest times in the interval, respectively. Below the scale set by $\ell_\text{v}(t)$, we recover an approximate scaling with $r^2$, which increases to Kraichnan-Kolmogorov scaling $\sim r^{8/3}$ on larger scales. The upper-left inset shows the local logarithmic derivative of $\Gamma^{2}$ calculated via finite differences over two grid-spacings. It also displays the transition from an approximate $r^2$ towards and $r^{8/3}$ power-law. The lower-right inset shows the decay of the second moment of circulation at $\ell_\mathrm{v,i}$ over time with exponent $\alpha=0.44(1)$. (b) Average inter-defect distance (red) and vortex number (blue). At intermediate times, both follow power-laws with $\beta\approx1/5$ and $\alpha\approx2\beta$, respectively. (c) Comparison of the fitted time-dependent $\beta$ exponent extracted from the vortex number in (b) and the maximal local slope $\lambda_{2,\text{max}}$ from (a). The system approaches universal dynamics, indicated by $\beta\approx1/5$, at the highlighted intermediate times, which is accompanied by the buildup of a transient 2d turbulent Kraichnan-Kolmogorov scaling corresponding to a decaying IEC.
• Figure 3: (a) $p$th-order moments of the velocity circulation, normalized at the inter-defect distance $\ell_\mathrm{v}$, at $t=9\cdot10^3(2\pi)/\omega_z$. Between the healing-length $\xi$ and the inter-defect scale $\ell_\mathrm{v}$ we recover scaling as $r^2$. The inertial range is set to the highlighted interval $300<r/a_\mathrm{ho}<3000$. The upper-left inset shows the logarithmic derivatives of the moments using finite difference over two grid-spacings, which approach power-laws within the inertial range, approximately given by $4p/3$ (dashed lines). The lower-right inset shows the averaged scaling exponents within the inertial range vs. $p$, which exhibit intermittent deviations from $4p/3$ for higher moments. The deviations closely follow the linear function $1.14p+0.58$, as predicted for stationary IECs Zhu2023a.PhysRevLett.130.214001Mueller2024.PhysRevLett.132.094002. (b) Higher-order moments of the average inter-defect distance distribution as a function of time, normalized at $t_0=3\cdot10^3(2\pi)/\omega_z$. The upper-left inset displays the logarithmic derivatives, obtained similarly to $\beta$ in Fig. \ref{['fig:fig_2']}c, which approach power-law behavior within the inertial range. In the lower-right inset the extracted scaling exponents closely follow $p/5$, indicating that the inter-defect distribution is nearly Gaussian and does not show 'intermittent' behavior.
• Figure 4: We show ten occupation number spectra $n_k$ logarithmically spaced in time in the interval $t\in[10^4,10^5](2\pi)/\omega_z$ in the lower-left inset. In the main plot, the spectra are rescaled using the scaling exponents $\alpha=0.40\pm0.03$ and $\beta=0.22\pm0.03$ demonstrating self-similar evolution in time. The UV cutoff at the reference time $t_0=10^4(2\pi)/\omega_z$ is set to $k_\mathrm{max}(t_0)=2\cdot10^{-1}/a_\mathrm{ho}$.
• Figure 5: Incompressible energy spectra $E_\mathrm{ic}(k)$ at logarithmically spaced times in the interval $t\in[10^3,10^5](2\pi)/\omega_z$ with the healing scale $k_\xi$ and the initial $k_\mathrm{i}$ and final $k_\mathrm{f}$ inter-defect scales marked by vertical gray-dashed lines. For comparison we further show the expected scaling of Kraichnan-Kolmogorov turbulence $\sim k^{-5/3}$, of the vortex' velocity field $\sim k^{-1}$, and of the vortex core $\sim k^{-3}$.