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 Alfven waves dispersive and circularly polarized. We introduce the generalized Elsasser 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 Alfven, whistler and ion cyclotron wave turbulence in the presence of a strong uniform magnetic field B-o(e) over cap (parallel to). Hall MHD turbulence is characterized by anisotropies of different strength: (i) for wavenumbers kd(i) >> 1 (d(i) is the ion inertial length) nonlinear transfers are essentially in the direction perpendicular (perpendicular to) to B-o; (ii) for kd(i) << 1 nonlinear transfers are exclusively in the perpendicular direction; (iii) for kd(i) similar to 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 analvze 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(perpendicular to), k(parallel to)) similar to k(perpendicular to)(-5/2)\k(parallel to)\(-1/2) and E(k(perpendicular to), k(parallel to)) similar to k(perpendicular to)(-2). An anisotropic phenomenology is developed to describe continuously the different sealing laws of the energy spectrum; one predicts E(k(perpendicular to), k(parallel to)) similar to k(perpendicular to)(-2)\k parallel to\(-1/2)(1 + k(perpendicular to)(2) d(i)(2))(-1/4). Non-local interactions between Alfven, 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.