Proof of the zeroth law of turbulence in one-dimensional compressible magnetohydrodynamics and shock heating

TL;DR

This paper proves the zeroth law of turbulence in one-dimensional compressible magnetohydrodynamics by demonstrating positive inertial dissipation, with implications for shock heating in the solar wind.

Contribution

It introduces inertial dissipation as a new form of energy loss in turbulence and proves the zeroth law in a specific MHD context with exact solutions.

Findings

Inertial dissipation equals viscous dissipation in the small viscosity limit.

Discontinuities in the solar wind produce significant shock heating.

Collisionless shocks may dominate heating in the outer solar wind.

Abstract

The zeroth law is one of the oldest conjecture in turbulence that is still unproven. Here, we consider weak solutions of one-dimensional compressible magnetohydrodynamics and demonstrate that the lack of smoothness of the fields introduces a new dissipative term, named inertial dissipation, into the expression of energy conservation that is neither viscous nor resistive in nature. We propose exact solutions assuming that the kinematic viscosity and the magnetic diffusivity are equal, and we demonstrate that the associated inertial dissipation is positive and equal on average to the mean viscous dissipation rate in the limit of small viscosity, proving the conjecture of the zeroth law of turbulence and the existence of an anomalous dissipation. As an illustration, we evaluate the shock heating produced by discontinuities detected by Voyager in the solar wind around 5 AU. We deduce a heating rate of $\sim 10^{-18}$J m$^{-3}$ s$^{-1}$, which is significantly higher than the value obtained from the turbulent fluctuations. This suggests that collisionless shocks can be a dominant source of heating in the outer solar wind.