# Cascades and Dissipative Anomalies in Compressible Fluid Turbulence

## TL;DR

This paper explores the nature of dissipative anomalies in compressible turbulence, revealing mechanisms like pressure-work defect and negentropy cascades, and provides theoretical expressions linking singularities to turbulent energy and entropy transfer.

## Contribution

It introduces a non-perturbative approach based on Onsager's principles to analyze anomalies in compressible turbulence, including new mechanisms for energy and entropy dissipation.

## Key findings

- Dissipative anomalies arise from cascade and pressure-work defect mechanisms.
- Entropy anomalies involve negative entropy input and small-scale negentropy cascades.
- Derived 4/5th-law expressions relate singularities to cascade processes.

## Abstract

We investigate dissipative anomalies in a turbulent fluid governed by the compressible Navier-Stokes equation. We follow an exact approach pioneered by Onsager, which we explain as a non-perturbative application of the principle of renormalization-group invariance. In the limit of high Reynolds and P\'eclet numbers, the flow realizations are found to be described as distributional or "coarse-grained" solutions of the compressible Euler equations, with standard conservation laws broken by turbulent anomalies. The anomalous dissipation of kinetic energy is shown to be due not only to local cascade, but also to a distinct mechanism called pressure-work defect. Irreversible heating in stationary, planar shocks with an ideal-gas equation of state exemplifies the second mechanism. Entropy conservation anomalies are also found to occur by two mechanisms: an anomalous input of negative entropy (negentropy) by pressure-work and a cascade of negentropy to small scales. We derive "4/5th-law"-type expressions for the anomalies, which allow us to characterize the singularities (structure-function scaling exponents) required to sustain the cascades. We compare our approach with alternative theories and empirical evidence. It is argued that the "Big Power-Law in the Sky" observed in electron density scintillations in the interstellar medium is a manifestation of a forward negentropy cascade, or an inverse cascade of usual thermodynamic entropy.