4.4 Higher-order gyrotropic multi-fluid equations

In previous sections we discussed the assumptions and limitations of transport theory based on Coulomb collisions, and studied the validity and breakdown of the classical (Spitzer and Härm, 1953Jump To The Next Citation PointBraginskii, 1965) theory in the solar corona and wind. For modelling the polar wind in the Earth polar magnetosphere, Demars and Schunk (1979Jump To The Next Citation Point) developed a set of transport equations in terms of a polynomial expansion (up to the first three orders in speed) about a local bi-Maxwellian. This set includes sixteen relevant moments and the corresponding fluid-type differential equations. The momentum and energy exchange collision terms based on the Coulomb operator (13View Equation) were for interpenetrating bi-Maxwellian gases calculated by Barakat and Schunk (1981). This set was used by Lie-Svendsen et al. (2001Jump To The Next Citation Point) to define a 16-moment solar wind model, which was applied from the chromosphere through the corona into the distant solar wind out to 1 AU. The flux conservation of the radial component of the magnetic field, B (r), for one spatial variable, the radius r, implies that
∂(AB ) ------- = 0, (31 ) ∂r
which means that the area of the flux or flow tube scales like: A (r) ∝ 1∕B (r). Subsequently, the index s refers to particle species s, with mass ms, density, ns, parallel and perpendicular temperatures, T s∥ and T s⊥, respectively, heat fluxes, q s∥ and q s⊥. For completeness, and since they have been used by various authors (Olsen and Leer, 1999Jump To The Next Citation PointLi, 1999Jump To The Next Citation PointLie-Svendsen et al., 2001Jump To The Next Citation Point), we explicitly quote these anisotropic, gyrotropic higher-order fluid equations:
∂n 1 ∂ δn ---s+ ----(nsusA ) = --s, (32 ) ∂t A ∂r δt
∂us ∂us kB [ 1 ∂(nsTs∥) 1 ∂A ] es G ⊙M ⊙ 1 δMs ----+ us----= − --- -----------+ -----(Ts∥ − Ts⊥) + ---E − ---2---+ ---------, (33 ) ∂t ∂r ms ns ∂r A ∂r ms r nsms δt
∂T ∂T ∂u 1 [∂q 1 ∂A ] 1 δE ---s∥-+ us---s∥ = − 2Ts∥--s-− ----- --s∥ + -----(qs∥ − 2qs⊥ ) + -------s∥, (34 ) ∂t ∂r ∂r nskB ∂r A ∂r nskB δt
[ ] ∂Ts⊥ ∂Ts⊥ 1 ∂A 1 ∂qs⊥ 1 ∂A 1 δEs ⊥ -∂t-- + us-∂r--= − usTs⊥A--∂r − n-k-- -∂r--+ A-∂r-2qs⊥ + n-k----δt-, (35 ) s B sB
∂q ∂q ( ∂u 1 ∂A ) k2 n [ ∂T ] δQ --s∥-+ us --s∥= − qs∥ 4---s+ us----- − 3-B--s Ts∥ --s∥- + ---s∥, (36 ) ∂t ∂r ∂r A ∂r ms ∂r δt
( ) 2 [ ] ∂qs⊥-+ u ∂qs⊥--= − 2q ∂us-+ u 1-∂A- − kBns- T ∂Ts⊥- + 1-∂A-T (T − T ) + δQs⊥-.(37 ) ∂t s∂r s⊥ ∂r sA ∂r ms s∥ ∂r A ∂r s⊥ s∥ s⊥ δt
The symbol E is the electric field, and the solar gravity force term is in standard notation. The respective rightmost terms refer to the volumetric transfer rates of particle, momentum, internal energy, and heat flow, which all still have to be specified. They are assumed to include internal exchange, among the species and degrees of freedom, and external exchange, for example of momentum transferred to the bulk flow by an Alfvén wave pressure. Without exchanges the set forms a closed system. The equations are still fairly general. However, to evaluate the exchange terms crucial assumptions have to be made about the form of the VDFs of the species involved. A large variety of VDFs will yield the same moments, and therefore the choice is not unique. The standard procedure, as quoted for example in the paper of Barakat and Schunk (1982Jump To The Next Citation Point), assumes for species s in its rest frame a VDF that takes the following convenient form:
[ ( )2 ( )2 ]( ) fs(w∥,w ⊥) = ----ns-----exp − -w∥- − w⊥-- 1 + Φs (w∥,w ⊥;Ts∥,Ts⊥,qs∥,qs⊥ ) , (38 ) π3 ∕2Vs∥Vs2⊥ Vs ∥ Vs⊥
with the thermal speeds: ∘ ----------- Vs∥ = 2kBTs ∥∕ms, and ∘ ----------- Vs⊥ = 2kBTs ⊥∕ms. The correction function, Φs (w ), is a polynomial and depends on the four scalar velocity moments appearing in the stress and heat flux tensor, and its evaluation requires further calculations. Barakat and Schunk (1982) gave examples for various multi-moment approximations, and presented a discussion of the related model VDFs. We recall the discussion in Subsection 4.1, in which it was emphasised that Φs has to be small against unity for the expansion of the VDF to converge and stay positive definite.

In applications of the moment Equations (32View Equation)–(37View Equation) to the modelling of coronal expansion and wind acceleration, the exchange rates on the right sides of the set must be specified. Given model VDFs such as (38View Equation), one can evaluate the self-collision integral (13View Equation), or the wave-particle exchange terms (63View Equation), and thus obtain the rates requested for closure of the set. Examples for Coulomb collisions are given in the 16-moment fluid model with Alfvén-cyclotron wave heating by Li (1999Jump To The Next Citation Point) and Lie-Svendsen et al. (2001Jump To The Next Citation Point). With respect to wave-particle interactions, the heating and acceleration rates after (63View Equation) were, for bi-Maxwellians and power-law wave ESDs, calculated by many authors (Marsch et al., 1982aJump To The Next Citation PointIsenberg, 1984aJump To The Next Citation PointLi and Habbal, 1999Li, 1999Jump To The Next Citation PointMarsch and Tu, 2001aJump To The Next Citation Point).

The standard VDF, resulting from the procedure used by Demars and Schunk (1979) and Li (1999Jump To The Next Citation Point), implies the following third-order-polynomial correction function:

( ( ) ( ) ) w∥ 2qs⊥ ( w ⊥ )2 qs∥ 2 ( w ∥)2 Φs(w ∥,w ⊥) = − 2--- ----2---- 1 − ---- + ----3 1 − -- --- . (39 ) Vs∥ ρsVs⊥Vs∥ Vs⊥ ρsVs∥ 3 Vs∥
What they called improved transport equations for fully ionised gases were recently proposed by Killie et al. (2004) to improve the description of Coulomb collisions, for which the transfer rates for momentum, energy, and heat flux were anew calculated within the eight-moment fluid equations. They found transport coefficients that deviate by less than 20% from the rigorous values (Spitzer and Härm, 1953) obtained from solving the Fokker–Planck equation.

Presently, there is no basic kinetic model of the solar corona and solar wind. The various forms of the multi-moment multi-species fluid equations were used to study different physical process and such characteristics as the dependence of solar wind parameters on variations in the coronal heating function (Olsen et al., 1998), the acceleration of the wind when being based on the above gyrotropic transport equations (Olsen and Leer, 1999Jump To The Next Citation Point), the heating and cooling of protons by turbulence-driven ion-cyclotron waves in fast solar wind (Li et al., 1999), the effect of transition region heating on the generation of the solar wind from coronal holes (Lie-Svendsen et al., 2002), and the coronal energy budget, abundances and temperatures of solar wind minor ions (Lie-Svendsen and Esser, 2005). In their 16-moment model, Lie-Svendsen et al. (2001) also included the heat-flux moment equation explicitly, and integrated the fluid Equations (32View Equation)–(37View Equation) all the way out from the chromosphere, through the corona into the inner heliosphere.


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