Conservation laws, fluxes, and symmetries: lessons from a perturbative approach for self-organized turbulence
Some turbulent flows self-organize into large-scale structures, rather than breaking up into ever-smaller scales. Underpinning this phenomenon is the existence of two sign-definite quantities which are conserved by the dynamics. Two-dimensional turbulence is a prime example, where large-scale mean flows, termed condensates, spontaneously emerge. We review a perturbative theoretical framework for the statistical description of such inhomogeneous turbulence, offering new perspectives on the role of the two conserved quantities. We illustrate the universal properties of the theory, comparing results from two-dimensional Navier-Stokes to those from the large-scale-quasi-geostrophic equation. These two models are limiting cases of the shallow water quasi-geostrophic equation, the former exhibiting long-range fluid element interactions, while the latter has local interactions. We then demonstrate these theoretical ideas in two new settings: first, in rotating three-dimensional turbulence, where two-dimensional condensates are known to form. Considering jet-type condensates, we derive the mean-flow profile and discuss a surprising symmetry breaking. Second, we vary the Rossby deformation radius in the shallow water quasi-geostrophic equation. We obtain novel domain-spanning condensates in all tested regimes and show that they follow two-dimensional Navier-Stokes for deformation radii above the forcing scale, and the large-scale quasi-geostrophic equation for those below, demonstrating the power of these asymptotic models.