Exploring the Equivalence between Two-dimensional Classical and Quantum Turbulence through Velocity Circulation Statistics

We study the statistics of velocity circulation in two-dimensional classical and quantum turbulence. We perform numerical simulations of the incompressible Navier-Stokes and the Gross-Pitaevskii (GP) equations for the direct and inverse cascades. Our GP simulations display clear energy spectra compatible with the double cascade theory of two-dimensional classical turbulence. In the inverse cascade, we found that circulation intermittency in quantum turbulence is the same as in classical turbulence. We compare GP data to Navier-Stokes simulations and experimental data from [Zhu et al. Phys. Rev. Lett. 130, 214001(2023)]. In the direct cascade, for nearly incompressible GP-flows, classical and quantum turbulence circulation displays the same self-similar scaling. When compressible effects become important, quasi-shocks generate quantum vortices and the equivalence of quantum and classical turbulence only holds for low-order moments. Our results establish the boundaries of the equivalence between two-dimensional classical and quantum turbulence.