Four-thirds law of energy and magnetic helicity in electron and Hall magnetohydrodynamic fluids

Abstract

In this paper, by exploiting the feature of the Hall term, we establish some local version four-thirds laws for the dissipation rates of energy and magnetic helicity in both the inviscid electron and Hall magnetohydrodynamic equations in the sense of Duchon–Robert type. Four-thirds law involving the dissipation rates of energy for the Hall magnetohydrodynamic equations generalizes corresponding the work of Galtier and new 4/3 laws for the dissipation rates of magnetic helicity in these systems are first observed. To this end, we derive local Kármán–Howarth–Monin type equations of the energy and magnetic helicity in these equations.