  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.
