Wave turbulence in incompressible Hall magnetohydrodynamics

We investigate the steepening of the magnetic fluctuation power law spectra observed in the inner Solar wind for frequencies higher than 0.5 Hz. This high frequency part of the spectrum may be attributed to dispersive nonlinear processes. In that context, the long-time behavior of weakly interacting waves is examined in the framework of three-dimensional incompressible Hall magnetohydrodynamic (MHD) turbulence. The Hall term added to the standard MHD equations makes the Alfvén waves dispersive and circularly polarized. We introduce the generalized Elsässer variables and, using a complex helicity decomposition, we derive for three-wave interaction processes the general wave kinetic equations; they describe the nonlinear dynamics of Alfvén, whistler and ion cyclotron wave turbulence in the presence of a strong uniform magnetic field B_0 \(e}_{Vert) . Hall MHD turbulence is characterized by anisotropies of different strength: (i) for wavenumbers textit{kd}_i {≫} 1 (d_i is the ion inertial length) nonlinear transfers are essentially in the direction perpendicular (⊥) to B_0; (ii) for textit{kd}_i {≪} 1 nonlinear transfers are exclusively in the perpendicular direction; (iii) for textit{kd}_i ∼ 1, a moderate anisotropy is predicted. We show that electron and standard MHD turbulence can be seen as two frequency limits of the present theory but the standard MHD limit is singular; additionally, we analyze in detail the ion MHD turbulence limit. Exact power law solutions of the master wave kinetic equations are given in the small- and large-scale limits for which we have, respectively, the total energy spectra E(k_{⊥},k_{Vert}) ∼ k_{⊥}(-5/2) |k_{Vert}|(-1/2) and E(k_{⊥},k_{Vert}) ∼ k_{⊥}(-2) . An anisotropic phenomenology is developed to describe continuously the different scaling laws of the energy spectrum; one predicts E(k_{⊥},k_{Vert}) ∼ k_{⊥}(-2) |k_{Vert}|(-1/2) (1+k_{⊥}(2d_) i(2)(-1/4)) . Non-local interactions between Alfvén, whistler and ion cyclotron waves are investigated; a non-trivial dynamics exists only when a discrepancy from the equipartition between the large-scale kinetic and magnetic energies happens.