On the von Kármán-Howarth equations for Hall MHD flows

The von Kármán-Howarth equations are derived for three-dimensional (3D) Hall magnetohydrodynamics (MHD) in the case of an homogeneous and isotropic turbulence. From these equations, we derive exact scaling laws for the third-order correlation tensors. We show how these relations are compatible with previous heuristic and numerical results. These multi-scale laws provide a relevant tool to investigate the nonlinear nature of the high frequency magnetic field fluctuations in the solar wind or, more generally, in any plasma where the Hall effect is important. Turbulence remains one of the last great unsolved problem in classical physics which has evaded physical understanding and systematic description for many decades. For that reason, any exact results appear almost as a miracle. In his third 1941 turbulence paper, Kolmogorov found that an exact and nontrivial relation may be derived from Navier-Stokes equations – which can be seen as the archetype equations for describing turbulence – for the third-order longitudinal structure function (Kolmogorov, 1941). Because of the rarity of such results, the Kolmogorov’s four-fifths law is considered as one of the most important results in turbulence (Frisch, 1995). The derivation of the Kolmogorov’s law uses earlier exact results found by von Kármán and Howarth in 1938 (von Kármán and Howarth, 1938): it is the well-known von Kármán-Howarth (vKH) equation that describes the dynamical evolution of the second-order correlation tensors. Very few extensions of such results (vKH equations and four-fifths law) to other fluids have been made; it concerns scalar passively advected (Yaglom, 1949), such as the temperature or a pollutant in the atmosphere, and astrophysical magnetized fluid described in the framework of MHD (Chandrasekhar, 1950; Politano and Pouquet, 1998; Politano, Gomez and Pouquet, 2003). The addition in the analysis of the magnetic field and its coupling with the velocity field renders the problem more difficult and, in practice, we are dealing with a couple of equations.