Spontaneous vortex crystal formation in classical rotating flows

Vortex crystals, ordered structures observed in superconductors and rotating superfluids, have also been hypothesized to form in classical fluids, based on numerical simulations and observations of the Jovian polar atmospheres. We perform direct numerical simulations of the Navier-Stokes equations in rotating frames, to investigate the spontaneous emergence of metastable vortex crystals. We analyze the energy spectrum, vortex morphology, and spatio-temporal dynamics to understand their roles in crystal formation and evolution. In addition, we explore domains with varying aspect ratios to examine their impact on the vortex lattice. Our results indicate a relationship between the crystal lifespan and dissipation, and we propose a scaling law linking the rotation rate, domain geometry, and vortex lattice periodicity. Finally, we identify a critical threshold in the control parameter, the Rossby number, suggesting a behavior similar to that found in phase transitions.