# Cascades and Dissipative Anomalies in Relativistic Fluid Turbulence

## TL;DR

This paper develops a first-principles theory of relativistic fluid turbulence at high Reynolds and Péclet numbers, revealing dissipative anomalies and mechanisms similar to non-relativistic turbulence, with implications for relativistic and classical fluid dynamics.

## Contribution

It introduces a non-perturbative, renormalization-group inspired approach to relativistic turbulence, identifying mechanisms for dissipative anomalies and extending Onsager's theory to relativistic fluids.

## Key findings

- Dissipative anomalies arise from local cascade and pressure-work defect mechanisms.
- Derived 4/5th-law-type relations characterizing singularities and structure-function scaling.
- Lorentz covariance of fluxes is broken by regularization but restored in the limit.

## Abstract

We develop first-principles theory of relativistic fluid turbulence at high Reynolds and P\'eclet numbers. We follow an exact approach pioneered by Onsager, which we explain as a non-perturbative application of the principle of renormalization-group invariance. We obtain results very similar to those for non-relativistic turbulence, with hydrodynamic fields in the inertial-range described as distributional or "coarse-grained" solutions of the relativistic Euler equations. These solutions do not, however, satisfy the naive conservation-laws of smooth Euler solutions but are afflicted with dissipative anomalies in the balance equations of internal energy and entropy. The anomalies are shown to be possible by exactly two mechanisms, local cascade and pressure-work defect. We derive "4/5th-law"-type expressions for the anomalies, which allow us to characterize the singularities (structure-function scaling exponents) required for their non-vanishing. We also investigate the Lorentz covariance of the inertial-range fluxes, which we find is broken by our coarse-graining regularization but which is restored in the limit that the regularization is removed, similar to relativistic lattice quantum field theory. In the formal limit as speed of light goes to infinity, we recover the results of previous non-relativistic theory. In particular, anomalous heat input to relativistic internal energy coincides in that limit with anomalous dissipation of non-relativistic kinetic energy.