Scaling regimes of 2d turbulence with power law stirring: theories versus numerical experiments

TL;DR

This paper compares theoretical predictions and numerical experiments on 2D turbulence with power-law stirring, revealing that an adapted Kraichnan's theory aligns better with observed phenomena than renormalization group results.

Contribution

The study provides the first-principles renormalization group analysis of 2D turbulence with power-law stirring and compares it with numerical experiments, highlighting discrepancies with traditional theories.

Findings

Renormalization group analysis predicts scaling behavior that disagrees with numerical results.

An adapted version of Kraichnan's phenomenological theory matches the numerical observations.

Discrepancies between RG analysis and observed phenomenology are discussed with potential theoretical scenarios.

Abstract

We inquire the statistical properties of the pair formed by the Navier-Stokes equation for an incompressible velocity field and the advection-diffusion equation for a scalar field transported in the same flow in two dimensions (2d). The system is in a regime of fully developed turbulence stirred by forcing fields with Gaussian statistics, white-noise in time and self-similar in space. In this setting and if the stirring is concentrated at small spatial scales as if due to thermal fluctuations, it is possible to carry out a first-principle ultra-violet renormalization group analysis of the scaling behavior of the model. Kraichnan's phenomenological theory of two dimensional turbulence upholds the existence of an inertial range characterized by inverse energy transfer at scales larger than the stirring one. For our model Kraichnan's theory, however, implies scaling predictions radically discordant from the renormalization group results. We perform accurate numerical experiments to assess the actual statistical properties of 2d-turbulence with power-law stirring. Our results clearly indicate that an adapted version of Kraichnan's theory is consistent with the observed phenomenology. We also provide some theoretical scenarios to account for the discrepancy between renormalization group analysis and the observed phenomenology.