3.6 Basics of Vlasov–Boltzmann theory

The coronal expansion and solar wind acceleration are a complex processes, which require a kinetic description if the detailed particle velocity distributions are to be evaluated. The coronal magnetic field guides the outflow of plasma out to the Alfvénic critical surface, where the ram pressure of the wind starts exceeding the magnetic pressure of the coronal field, and where thus the solar wind is ultimately released from the Sun. The subsequent almost spherical expansion and the large-scale inhomogeneity continuously compel the solar wind plasma to attain a variable state of dynamic statistical equilibrium between the particles and electromagnetic field fluctuations. In principle, all these kinetic processes are fully described by the Boltzmann–Vlasov equation for the phase-space distribution for each species, fj(x,t,v ), which is a measure of the number of particles at time t in a volume surrounding position x and with velocities in a certain range around v. The kinetic equation reads:
[ ( ) ] [ ] ∂-+ v ⋅ ∂-+ g + -ej(E + 1v × B ) ⋅-∂- f = d-f (9 ) ∂t ∂x mj c ∂v j dt j c,w
with the interplanetary magnetic field, B (x, t), electric field, E (x,t), and the Sun’s gravitational acceleration, g(x ). Coulomb collisions or wave-particle interactions are also included and described by the Fokker–Planck collision integral or the quasilinear diffusion operator on the right hand side of Equation (9View Equation). Here the particle charge is e j, its mass m j, speed v, and space coordinate x. The speed of light in vacuo is c. In addition to Equation (9View Equation), Maxwell’s equations have to be solved with the self-consistent current density and charge density of the multi-component plasma of the solar corona and wind, which are calculated from the velocity moments of Equation (9View Equation). This complex problem has not been solved for the corona, but solutions of simplified versions of this problem, restricted to a single particle species and special geometries, do exist, and will be discussed in Section 7.
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