On the origin of the energy dissipation anomaly in (Hall) magnetohydrodynamics

Incompressible Hall magnetohydrodynamics (MHD) may be the subject of energy dissipation anomaly which stems from the lack of smoothness of the velocity and magnetic fields. I derive the exact expression of which appears to be closely connected with the well-known 4/3 exact law of Hall MHD turbulence theory. This remarkable similitude suggests a deeper mathematical property of the fluid equations. In the MHD limit, the expression of differs from the one derived by Gao et al (2013 Acta Math. Sci . 33 865–71) which presents miscalculations. The energy dissipation anomaly can be used to better estimate the local heating in space plasmas where in situ measurements are accessible.