Wave turbulence in incompressible Hall magnetohydrodynamics
Abstract
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 longtime behavior of weakly interacting waves is examined in the framework of threedimensional 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 threewave 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 largescale 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)) . Nonlocal interactions between Alfvén, whistler and ion cyclotron waves are investigated; a nontrivial dynamics exists only when a discrepancy from the equipartition between the largescale kinetic and magnetic energies happens.
 Publication:

Journal of Plasma Physics
 Pub Date:
 2006
 DOI:
 10.1017/S0022377806004521
 arXiv:
 arXiv:physics/0608227
 Bibcode:
 2006JPlPh..72..721G
 Keywords:

 Physics  Space Physics;
 Physics  Plasma Physics
 EPrint:
 72 pages, 3 figures