Magnetic Fluctuations in the Magnetosheath: Weak Turbulence Theory.

The magnetic fluctuations in the magnetosheath and in the vicinity of the magnetopause are of primary importance for magnetospheric physics since they are likely to be the cause of the transfers from the solar wind to the magnetosphere. Several experimental characteristics of these fluctuations strongly suggest that non-linear effects are at work and that turbulence is observed. The first one concerns the shape of the magnetic power spectrum: a power law in w -a (with a of the order of 3) exists between the ion gyrofrequency (¯ 1 Hz) and the lower hybrid frequency (¯ 100 Hz), much evoking the "inertial range" known in hydrodynamic turbulence. Furthermore, the polarization of the magnetic fluctuations suggests that, in this frequency range, an "intermediate mode" is excited in addition to the well-known fast magnetosonic mode. The non-linear coupling could force this mode, so overcoming its linear kinetic damping. We propose the first attempt to interpret the fluctuation properties in the frame of the weak turbulence theory and to quantify the suggested scenario. The moderate amplitudes as well as the apparent phase incoherence of the fluctuations are indeed strong arguments in favor of this theory. In order to describe the plasma behavior above the ion gyrofrequency, we must go beyond the Ideal-MHD system, at least to the Hall-MHD one. In this frame, before looking for the "kinetic equation of waves" and equilibrium spectra one has first to derive all the coupling coefficients thanks to Hamiltonian mechanics. Starting from the Lagrangian of a particle in an electromagnetic field, we demonstrate the existence of an Hamiltonian description for the bi-fluid system. Then, we seek how to reduce it to describe the Hall-MHD system. A perturbation method should then allow to determine the linear solutions (modes) of the system and all the non linear coupling coefficients between these modes. By the way, the energy transfers from the low frequency Alfven fluctuations toward the high frequency modes should be estimated by deriving the "kinetic equation of waves". The general principles of the method have been elaborated by Zakharov and Kuznetsov (1997).