von Kármán-Howarth relationship for helical magnetohydrodynamic flows

We derive an exact equation for homogeneous isotropic magnetohydrodynamic (MHD) turbulent flows with nonzero helicity; this result is of the same nature as the classical von Kármán-Howarth (VKH-HM) formulation for the kinetic energy of turbulent fluids. Helical MHD is relevant to the astrophysical flows such as in the solar corona, or the interstellar medium, and in the dynamo problem. The derivation involves the new writing of the general form of tensors for that case, for either vectors or (pseudo)axial vectors. It is shown that, for general third-order tensors, four generating functions are needed when taking into account the nonmirror invariance of helical fluids, instead of two as in the fully isotropic case. The new equation obtained, denoted by VKH-HM, links the dissipation of magnetic helicity to the third-order correlations involving combinations of the components of the velocity, the magnetic field, and the magnetic potential. Finally, in the long-time and nonresistive limit, this relationship leads to a linear scaling with separation of the third-order tensor, correlating the two normal components of the electromotive force and of the magnetic potential.