Scalar Dissipation Criticality in Compressible Magnetized Turbulence

Abstract

We extend the Obukhov-Corrsin theory of scalar turbulence to compressible flows with spatially variable, anisotropic diffusivity. Combining density and diffusivity into a single positive matrix field, a transport landscape, permits an exact scale-by-scale balance in which the sign-indefinite commutator between filtering and diffusion is eliminated rather than estimated. If the density-weighted third-order velocity and scalar increments scale as $\ell^α$ and $\ell^β$, respectively, we prove that anomalous scalar dissipation is impossible when $α+2β>1$. Remarkably, this threshold is independent of the anisotropy and spatial regularity of the diffusivity, provided it remains uniformly elliptic. Codimension-one shocks in both velocity and scalar have $α=β=1/3$ and therefore lie exactly at the critical threshold. For statistically stationary turbulence we further obtain an exact density-weighted constant-flux relation, providing a compressible analogue of the relation underlying Yaglom's law. The results apply directly to passive-scalar transport in both gases and magnetized plasmas and provide testable diagnostics for simulations of compressible magnetohydrodynamic turbulence. Because the same operator governs anisotropic heat conduction, the results carry over to the electron temperature of a magnetized plasma, and they imply that gradient statistics in such flows must be contracted with the transport landscape rather than with the density.