Craig's XY-distribution and the statistics of Lagrangian power in two-dimensional turbulence

TL;DR

This paper analyzes the probability distribution of energy injection in 2D turbulence, revealing a sharply peaked distribution at zero with asymmetric exponential tails, well described by Craig's XY distribution, highlighting the statistical properties of Lagrangian power.

Contribution

The study demonstrates that the power distribution in 2D turbulence can be accurately modeled by Craig's XY distribution, providing analytical expressions for its asymptotic behavior and tail asymmetry.

Findings

Power distribution peaks sharply at zero

Tails are asymmetric and exponential

Distribution is well described by Craig's XY distribution

Abstract

We examine the probability distribution function (pdf) of energy injection rate (power) in numerical simulations of stationary two--dimensional (2D) turbulence in the Lagrangian frame. The simulation is designed to mimic an electromagnetically driven fluid layer, a well-documented system for generating two--dimensional turbulence in the laboratory. In our simulations, the forcing and velocity fields are close to Gaussian. On the other hand, the measured PDF of injected power is very sharply peaked at zero, suggestive of a singularity there, with tails which are exponential but asymmetric. Large positive fluctuations are more probable than large negative fluctuations. It is this asymmetry of the tails, which leads to a net positive mean value for the energy input despite the most probable value being zero. The main features of the power distribution are well described by Craig's XY distribution for the PDF of the product of two correlated normal variables. We show that the power distribution should exhibit a logarithmic singularity at zero and decay exponentially for large absolute values of the power. We calculate the asymptotic behavior and express the asymmetry of the tails in terms of the correlation coefficient of the force and velocity. We compare the measured pdfs with the theoretical calculations and briefly discuss how the power pdf might change with other forcing mechanisms.