3.7 Vlasov–Boltzmann equation and fluid theory

The standard fluid MHD theory is described in any text book (e.g., Montgomery and TidmanJump To The Next Citation Point, 1964Jump To The Next Citation Point or StixJump To The Next Citation Point, 1992Jump To The Next Citation Point) of plasma physics and will not be discussed here, however the basic fluid equations are derived from the moments of the Vlasov/Boltzmann kinetic equation below. Here we concentrate instead on the assumptions underlying classical transport theory of plasma with emphasis on basic kinetic concepts and perturbation analysis. Transport theory in a plasma is based on approximate solutions of Equation (9View Equation) for weakly non-uniform media. In our discussion of space plasmas, we will closely follow the article of Dum (1990Jump To The Next Citation Point). The complete and most detailed description of a plasma is in terms of the particle velocity distribution function (VDF), denoted by f = f(x,t,w ), the evolution of which in phase space is generally described by the kinetic Vlasov–Boltzmann Equation (9View Equation), in which we now omit the index. It can also be written in the non-standard form:
df- ∂f- ∂f-- -q ′ du- -∂f- ∂u- -∂f- dt + w ⋅∂x + (w × Ω ) ⋅∂w + (m E − dt ) ⋅∂w − w ⋅∂x ⋅∂w = 𝒞f. (10 )
Here a single dot means the scalar product of vectors, and a double dot the dyadic contraction of tensors. The Coulomb collisions and/or wave-particle interactions are included in this equation via the collision term that involves the acceleration, A, and the diffusion tensor, 𝒟. We introduced the relative or random velocity w = v − u(x, t), with respect to the mean (or bulk) velocity u (x,t), and v is the particle’s velocity in the inertial frame. We also defined the vector gyrofrequency, Ω = qB ∕mc, with q being the charge and m the mass of any particle species. The electric field in the moving frame is denoted as ′ E and according to the Lorentz transformation given by:
1 E ′ = E + -u × B. (11 ) c
As usually, the advective derivative (time change in the moving frame) is given by
-d ∂-- -∂- dt = ∂t + u ⋅∂x . (12 )
The collision operator, which is here understood to describe either binary Coulomb collisions or wave-particle interactions, reads as follows:
( ) -∂- 1- ∂-- 𝒞f = − ∂v ⋅ A − 2𝒟 ⋅ ∂v f. (13 )
It can be written as a velocity divergence of a flux density associated with friction and diffusion, see, e.g., the textbooks of Melrose and McPhedran (1991Jump To The Next Citation Point) and Montgomery and Tidman (1964).

If for Coulomb collisions the so-called Rosenbluth potentials,

∫ f (v ′) ∫ Hj(v ) = d3v′---j-----, Gj (v) = d3v′fj(v ′) | v − v ′ |, (14 ) | v − v′ |
are exploited (Rosenbluth et al., 1957), the related frictional acceleration and diffusion terms can concisely be written as:
( ) ( ) 2 A (v) = Γ 1 + mi- -∂-H (v) = − 1 + mi- ν (v )v, 𝒟 (v) = Γ -∂---G (v). (15 ) i ij mj ∂v j mj ij i ij∂v ∂v j
Only here we used two indices to indicate the two species involved in the binary collision. The gamma factor is defined as Γ ij = 4πe2ie2j∕m2i ln Λ, with the Coulomb logarithm being given via the plasma parameter (i.e., number of particles in the Debye sphere) through Λ = 12πn λ3D, with the total particle density n = ∑ nj j. The Debye shielding length is given by
−2 ∑ −2 λD = λj , (16 ) j
with the individual species Debye length defined as λj = vj∕ωj. As one can see in Equation (15View Equation), the collision frequency is defined as the negative velocity gradient of the Rosenbluth potential. Hernandez and Marsch (1985) evaluated the collisional times scale for temperature and velocity exchange between drifting bi-Maxwellians, and Livi and Marsch (1986) and Marsch and Livi (1985a,b) calculated the collisional relaxation process and the associated rates for non-thermal solar wind VDFs and self-similar and kappa VFDs. This issue is also addressed in a previous review by Marsch (1991aJump To The Next Citation Point). In the case of ion-electron collisions with electron thermal speed obeying ve >> vi, we have the collision frequency −3 νei = 2Γ einive, showing the well-known strong speed dependence of the collisional friction on the relative speed of the colliding partners.

In multi-fluid theory, the dynamic equations for each species are separately obtained (we suppress an index labelling the species) by taking the zeroth, first, second, etc., moment of Equation (10View Equation). The zeroth order moment gives the continuity equation for the density, n, which reads:

dn ∂ --- = − n--- ⋅ u. (17 ) dt ∂x
Note that collisions do not change the particle number, i.e. < 𝒞f >= 0, which is obvious from the form of Equation (13View Equation). Here the angular brackets indicate velocity space integration. Similarly, the momentum equation can be written:
d ∂ 1 nm -- u = − ---⋅ 𝒫 + nq [E + -u × B ] + R. (18 ) dt ∂x c
The couplings between the particles appear through the collisional (or wave-particle) momentum transfer rate, which is given by the first moment of the collision operator, R = m < w 𝒞f >. By taking the other moments of the velocity distribution function, one obtains the zero mean random velocity, the pressure (or stress) tensor, and the heat flux vector:
1 2 < w >= 0,𝒫 = nm < ww >, q = nm < w --w > . (19 ) 2
The thermal pressure p (and kinetic temperature T), and the thermal stress tensor Π are derived as follows:
1- p = nkBT = 3T r𝒫,Π = 𝒫 − ℐp. (20 )
Here Tr denotes the trace, ℐ is the unit tensor, and kB is Boltzmann’s constant. By definition the trace of Π vanishes. In terms of these quantities the internal energy or temperature equation is written as:
3 dT dn ∂u ∂ kB (2n dt-− T -dt) = − Π : ∂x-− ∂x-⋅ q + Q. (21 )
The left side contains the adiabatic changes (of the entropy) owing to advection, and the right side the dissipation terms related with velocity shear (viscosity), heat conduction and collisional friction (Ohmic heating) at a volumetric rate Q, with m- 2 Q = 2 < w 𝒞f >.

In principle, one can continue this scheme, by taking ever higher moments of (10View Equation), which will lead to an infinite chain. For practical purposes, this chain must be terminated, and a way of closure be found. Transport theory is expected to provide this closure, yielding transport coefficients that link the unknown moments q, Π, R and Q with the fluid variables n, T, and u and their spatio-temporal variations. Transport relations providing closure are explicitly given in the review of Dum (1990Jump To The Next Citation Point). The general closure problem of the Vlasov–Boltzmann equation was recently revisited by Chust and Belmont (2006), with emphasis on the behaviour of a collisionless plasma. Yet, the potential relevance of this work for the solar wind remains to be demonstrated.


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