# Numerical Measurements of Scaling Relations in Two-Dimensional Conformal Fluid Turbulence

## TL;DR

This study numerically investigates relativistic scaling relations in 2D conformal fluid turbulence, confirming their validity and exploring deviations from non-relativistic theories across different Mach numbers.

## Contribution

It extends previous analytical work by providing numerical validation of relativistic scaling relations in 2D conformal turbulence and compares them with non-relativistic limits.

## Key findings

- Relativistic scaling relations are accurately satisfied in inverse-cascade simulations.
- Non-relativistic relations perform worse as Mach number increases.
- Inverse cascade energy spectrum follows a $k^{-2}$ scaling, not Kolmogorov's $k^{-5/3}$.

## Abstract

We present measurements of relativistic scaling relations in $(2+1)$-dimensional conformal fluid turbulence from direct numerical simulations, in the weakly compressible regime. These relations were analytically derived previously in Westernacher-Schneider, Lehner, Oz (2015) for a relativistic fluid; this work is a continuation of that study, providing further analytical insights together with numerical experiments to test the scaling relations and extract other important features characterizing the turbulent behavior. We first explicitly demonstrate that the non-relativistic limit of these scaling relations reduce to known results from the statistical theory of incompressible Navier-Stokes turbulence. In simulations of the inverse-cascade range, we find the relevant relativistic scaling relation is satisfied to a high degree of accuracy. We observe that the non-relativistic versions of this scaling relation underperform the relativistic one in both an absolute and relative sense, with a progressive degradation as the rms Mach number increases from $0.14$ to $0.19$. In the direct-cascade range, the two relevant relativistic scaling relations are satisfied with a lower degree of accuracy in a simulation with rms Mach number $0.11$. We elucidate the poorer agreement with further simulations of an incompressible Navier-Stokes fluid. Finally, as has been observed in the incompressible Navier-Stokes case, we show that the energy spectrum in the inverse-cascade of the conformal fluid exhibits $k^{-2}$ scaling rather than the Kolmogorov/Kraichnan expectation of $k^{-5/3}$, and that it is not necessarily associated with compressive effects. We comment on the implications for a recent calculation of the fractal dimension of a turbulent $(3+1)$-dimensional AdS black brane.