The Absolute Statistical Equili-Brium And Exact Solutions Of The Incompressible Hall Magnetohydrodynamics

MHD turbulence is an important nonlinear plasma physics phenomenon.The study of MHD turbulence is of great significance to understand magnetic confinement fusion plasma physics,particle heating and acceleration of the solar wind and magnetic reconnection.In this paper,we intend to use the approximation model of two-fluid(incompressible Hall MHD model)for our study.The incompressible Hall MHD model can grasp some two-fluid effects and the small-scale plasma physics compared with MHD model,and has been widely used to study the fast magnetic reconnection,dynamo effects and solar wind turbulence.Here,we will focus on the incompressible Hall MHD model in the following aspects:By using Fourier helical decomposition,we obtained the dynamic equations of the helical components of fluctuating velocity and magnetic field in the incompressible ideal Hall MHD model.Meanwhile,we proved that the energy,generalized helicity and magnetic helicity are conserved in the wavenumber space.Furthermore,the left-handed and right-handed statistical equilibrium spectra of the energy,generalized helicity and magnetic helicity are obtained through Gibbs ensemble methods.We find that the magnetic helicity dominates the inverse cascade,and the generalized helicity dominates the forward cascade of Hall MHD turbulence.Moreover,which helical sector plays the major role in the turbulence cascade depends on the value of the magnetic helicity and the generalized helicity.In addition,we investigated the exact solutions of the incompressible Hall MHD equations.The ideal Hall MHD equations submit themselves to the circularly polarized waves.The fluctuating velocity and magnetic field are always parallel or anti-parallel,and the correlation coefficient between them are wave number depended.Based on the solutions of the ideal situation,we derived the exact solutions of the incompressible dissipative Hall MHD equations.If the magnetic Prandtl number Prm?1,and the kinematic Reynolds number and the magnetic Reynolds number are both very large,the fluctuating velocity and magnetic field are damped circular polarized waves.Meanwhile,a phase difference will occur between them.The phase difference is relevant to the Reynolds numbers and the wave number,and it makes the fluctuating velocity and magnetic field no longer parallel or anti-parallel like that in the ideal Hall MHD.It is important to understand the non-Alfvenic correlation and selective decay process of space plasma.Finally,the travelling wave solutions of the incompressible ideal Hall MHD equations are obtained.We find that the external magnetic field is unable to enter the superconducting plasma completely(the self-generated magnetic field by the plasma can cancel out the external magnetic field).But the travelling waves can propagate in the plasma,and the direction of propagation is arbitrary.Because that the plasma itself can generate a magnetic field along the direction of propagation.The amplitudes of the travelling waves are proportional to this magnetic field.The previous absolute statistical equilibrium theory does not consider the energy of the magnetic field along the direction of propagation.However,under the consideration of the energy of this magnetic field,we obtained the ensemble average of the total energy and the generalized helicity.The innovations of this paper are as follows:1.By using Fourier helical decomposition,we investigated the absolute statistical equilibrium of the incompressible ideal Hall MHD model,and indicated that the magnetic helicity and the generalized helicity dominate the inverse cascade and forward cascade properties of Hall MHD turbulence,respectively.2.We obtained the circularized wave solutions of the incompressible ideal and dissipative Hall MHD.Especially,when the magnetic Prandtl number is not equal to one,the fluctuating velocity and magnetic field are no longer parallel or anti-parallel.3.Through the analysis of the travelling wave solutions in the incompressible ideal Hall MHD,we find that the external magnetic field is unable to enter the superconducting plasma completely.But the travelling wave can propagate in the plasma,and the magnetic field along the direction of propagation is generated by the plasma itself.In addition,under the consideration of the energy of this magnetic field,we obtained the ensemble average of the total energy and the generalized helicity.