Scaling regimes of 2d turbulence with power-law stirring: theories versus numericalexperiments

Abstract

We inquire about the statistical properties of the pair formed by the Navier–Stokesequation for an incompressible velocity field and the advection–diffusion equation for ascalar field transported in the same flow in two dimensions (2d). The system is in a regimeof fully developed turbulence stirred by forcing fields with Gaussian statistics, white noisein time and self-similar in space. In this setting and if the stirring is concentratedat small spatial scales, as if due to thermal fluctuations, it is possible to carryout a first-principles ultraviolet renormalization group analysis of the scalingbehavior of the model. Kraichnan’s phenomenological theory of two-dimensionalturbulence upholds the existence of an inertial range characterized by inverse energytransfer at scales larger than the stirring one. For our model Kraichnan’s theory,however, implies scaling predictions radically discordant from the renormalizationgroup results. We perform accurate numerical experiments to assess the actualstatistical properties of 2d turbulence with power-law stirring. Our results clearlyindicate that an adapted version of Kraichnan’s theory is consistent with theobserved phenomenology. We also provide some theoretical scenarios to accountfor the discrepancy between renormalization group analysis and the observedphenomenology.