4.1 Transport theory in collisional plasma

The conventional way to solar wind modelling of course is the fluid approach, following the early work of Parker (1963). Here we shall review possible ways to remedy the shortcomings of the standard fluid description, e.g., by considering multi-species fluids either, or for a single species, by accounting for higher-order moments of the VDF. The advances made in this respect in modelling the fast solar wind were briefly reviewed by Hansteen et al. (1999). The intention here is not to discuss the existing fluid models of the solar wind themselves, but from the kinetic perspective to identify the weak and questionable points of transport theory in the solar wind and corona, and then to indicate possible physical remedies.

The basic assumption of classical transport theory is that collisions are strong and only permit very small deviations from collisional equilibrium. As a consequence, the particle VDFs can be expanded about a local Maxwellian (LTE, local thermal equilibrium), in which a dependence upon space and time occurs only through the fluid moments,

F (x,t,w ) = F (n(x,t),u(x, t),T (x,t),w ). (22 ) 0 M
This approach following Chapman, Enskog, Cowling, Grad and others (see DumJump To The Next Citation Point, 1990 for further references), typically leads to polynomial expansions in the velocity variable w (we suppress the variables x and t for simplicity) of the form:
f (w ) = F0 (w ) + w ⋅ F1(w ) + ww : ℱ2 (w ) + ...., (23 )
where the higher order terms are proportional to the spatial gradients of the first few moments, such that the linear corrections scale like F ∼ 𝒯 ,𝒩 ,R 1, and the quadratic ones relating to the velocity shear as ℱ2 ∼ 𝒰. The deviation from the local Maxwellian describe heat conduction, particle diffusion, electrical resistivity, and viscosity, where the symbols mean:
𝒯 = ∇T (x,t),𝒩 = ∇n (x,t), (24 )
( ) 1 ∂u [∂u ]T 1 𝒰 = -- --- + --- − -ℐ∇ ⋅ u. (25 ) 2 ∂x ∂x 3
The superscript T denotes the transposed tensor. Taking the appropriate moments of the VDF (23View Equation), yields the sought for transport relations, such that Π ∼ 𝒰 and q ∼ 𝒯, or a drift velocity Δu ∼ 𝒩.

The transport coefficients are calculated by means of a kinetic perturbation theory, where the effects of collisions are included in an approximate solution of the Boltzmann equation, an approach leading to a series expansion that can in lowest order be expressed as:

f (w ) = f0(w ) + 𝜖f1(w ) + 𝜖2f2(w).... (26 )
It is important to note here, that to ensure rapid convergence of such series one requires a smallness parameter, 𝜖, to exist, so that the higher order terms in Equation (26View Equation) can safely be neglected. The small parameter usually is the ratio, 𝜖 = λc ∕L, of the collisional free path, λc, over the gradient scale, L, of the fluid parameters.

The collision term is characterised by the mean time between collisions, τc, and scales as follows: 𝒞f ∼ f∕τc. Therefore, to guarantee small non-uniformity, or to obtain weak deviations from LTE, the following inequalities must be fulfilled:

( ) −1 d- dt ≫ τc,L ≫ λc. (27 )
The lowest-order (of order 𝜖0) uniform and stationary solution of the basic Equation (10View Equation) must therefore obey:
∂f0 w × Ω ⋅∂w--= 𝒞f0. (28 )
If the background magnetic field is only weak, the left hand side of Equation (28View Equation) may also be omitted, and then one simply has, 𝒞f0 = 0, a result which directly leads to a Maxwellian that is known to annihilate the collision operator (13View Equation). Expansions such as in Equation (23View Equation), even when going to much higher order in w or in the cosine of the pitch angle, do in the solar wind hardly converge (see Dum et al.Jump To The Next Citation Point, 1980Jump To The Next Citation Point and MarschJump To The Next Citation Point, 1991a). Consequently, a non-perturbative kinetic treatment suggests itself, since using the classical transport coefficients of Braginskii (1965Jump To The Next Citation Point) in fluid equations for space plasmas becomes questionable, and will often lead to spurious results.

As in the fast solar wind usually N < 1, with the number of collisions being N = (τc-d)−1 dt, Coulomb collisions certainly require a kinetic treatment. However, as shown in numerical model calculations for the solar wind by Livi and Marsch (1987Jump To The Next Citation Point), only very few collisions may already suffice to remove the otherwise extreme exospheric anisotropies in the VDFs. In the slow wind one has N > 5 for only about 10% of the time, but N > 1 for about 30 – 40% of the time. When seeking the lowest order solution of Equation (28View Equation) in the strongly magnetised corona, one obtains a gyrotropic distribution, from which strongly anisotropic transport coefficients may result, which can differ substantially along and transverse to the magnetic field.

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