Quantitative estimation of effective viscosity in quantum turbulence

Abstract

We study freely decaying quantum turbulence by performing high-resolution numerical simulations of the Gross-Pitaevskii equation (GPE) in the Taylor-Green geometry. We use resolutions ranging from ${1024}^{3}$ to ${4096}^{3}$ grid points. The energy spectrum confirms the presence of both a Kolmogorov scaling range for scales larger than the intervortex scale $\ensuremath{\ell}$, and a second inertial range for scales smaller than $\ensuremath{\ell}$. Vortex line visualizations show the existence of substructures formed by a myriad of small-scale knotted vortices. Next, we study finite-temperature effects in decaying quantum turbulence by using the stochastic Ginzburg-Landau equation to generate thermal states, and then by evolving a combination of these thermal states with the Taylor-Green initial conditions under the GPE. We use finite-temperature GPE simulations to extract mean-free path by measuring the spectral broadening in the Bogoliubov dispersion relation that we obtain from the spatiotemporal spectra, and use it to quantify the effective viscosity as a function of the temperature. Finally, we perform low-Reynolds-number simulations of the Navier-Stokes equations, in order to compare the decay of high-temperature quantum flows with their classical counterparts, and to further calibrate the estimations of the effective viscosity (based on the mean-free-path computations).