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The zeroth law is one of the oldest conjecture in turbulence that is stillunproven. Here, we consider weak solutions of one-dimensional compressiblemagnetohydrodynamics and demonstrate that the lack of smoothness of the fieldsintroduces a new dissipative term, named inertial dissipation, into theexpression of energy conservation that is neither viscous nor resistive innature. We propose exact solutions assuming that the kinematic viscosity andthe magnetic diffusivity are equal, and we demonstrate that the associatedinertial dissipation is positive and equal on average to the mean viscousdissipation rate in the limit of small viscosity, proving the conjecture of thezeroth law of turbulence and the existence of an anomalous dissipation. As anillustration, we evaluate the shock heating produced by discontinuitiesdetected 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 thevalue obtained from the turbulent fluctuations. This suggests thatcollisionless shocks can be a dominant source of heating in the outer solarwind.

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