Proof of the zeroth law of turbulence in one-dimensional compressible magnetohydrodynamics and heating of the sawtooth solar wind

The zeroth law of turbulence is one of the oldest conjecture in turbulence that is still unproven. We consider weak solutions of one-dimensional (1D) compressible magnetohydrodynamics (MHD) 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, on average, positive and equal to the mean viscous dissipation rate in the limit of small viscosity, proving the conjecture of the zeroth law of turbulence.

We show that discontinuities commonly de- tected by Voyager 1 & 2 in the solar wind at 2–10AU can be fitted by the inviscid analytical profiles. We deduce a heating rate of ∼ 10⁻¹⁸ Jm⁻³s⁻¹ , which is significantly higher than the value obtained from the turbulent fluctuations. This suggests that collisionless shocks are a dominant source of heating in the outer solar wind.