A Generalized Model for the Proton Expansion in Astrophysical Winds. II. The Associated Set of Transport Equations

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Astrophysical Winds. II. The Associated Set of

Transport Equations

François Leblanc, Daniel Hubert

To cite this version:

A GENERALIZED MODEL FOR THE PROTON EXPANSION IN ASTROPHYSICAL WINDS. II. THE ASSOCIATED SET OF TRANSPORT EQUATIONS

AND DANIEL

FRANCÓOIS LEBLANC HUBERT

Departement de Recherche Spatiale, CNRS URA 264, Observatoire de Paris, 92195 Meudon Cedex, France ; Ñeblanc=megasx.obspm.fr Received 1997 February 21 ; accepted 1998 February 4

ABSTRACT

In Paper I, we presented a new model of the velocity distribution function for protons composing a stellar atmosphere expanding in interstellar space, valid from collisional to collisionless regions. In this paper, the set of generalized transport equations associated with this model and the closure assumptions for higher order velocity moments are provided for 9 and 16 moment approximations. The study of the properties of such a set of transport equations in the collisionless limit is presented and discussed. A comparison with the similar bi-Maxwellian approximation is made using two kinds of analysis, in the context of an application to solar wind expansion. Our model is better adapted to high values of the heat Ñux and thus is able to provide a macroscopic parameter evolution for stellar atmosphere expan-sion in a state far from local equilibrium, as well as for the expanexpan-sion of planetary polar winds.

Subject headings : hydrodynamics È plasmas È solar wind È stars : atmospheres È stars : mass loss

1

.

INTRODUCTION

In a previous article(Leblanc & Hubert1997,hereafterPaper I),we have presented a new model for the proton velocity

distribution function (VDF) in astrophysical winds. This function is a polynomial expansion based on an exact solution of the BGK equation for moderately ionized plasmas, and has been constructed following the generalized Grad method (Grad 1958 ; We have emphasized that the main advantage of this function is that it is close to the observed proton VDF Mintzer 1965).

proÐles in the solar wind. Indeed, our zeroth-order approximation is an asymmetric function that displays a suprathermal tail in the magnetic Ðeld direction. In this paper, we present the transport equations, the moment closure, and the one-dimensional derived system associated with our generalized model of the VDF. This approach has been developed to be well adapted to the description of particle states from collisional to collisionless regions (corresponding to large Knudsen numbers and suprathermal velocity distribution functions with large normalized heat Ñux). Moreover, the moment approach of Grad enables us to derive direct relations between the macroscopic and microscopic descriptions of the particles and between the kinetic e†ects and the classic hydrodynamic parameters.

In transitional collisional regions, the characteristics of the distribution functions that compose the solar wind are now better known from the solar corona to interplanetary space, thanks to numerous probes like Ulysses and SOHO, which

provide new information from the solar corona and solar wind(Feldmanet al.1996).But several very important phenomena

still need to be accurately understood, e.g., the heating and increase of wind speed in the corona, the nonadiabatic evolution of

particles between 0.3 and 1 AU(Marschet al.1982 ; Kohl,Strachan, & Gardner1996),the origin of the proton temperature

anisotropy(Marschet al.1982),the importance of heat Ñux in the corona(Withbroe 1988),and the interaction between the

heavy and light ions(Habbal 1996).Improving the heat-Ñux conduction law is also important for obtaining a good

descrip-tion of the thermal forces in the corona and interplanetary medium, where the classical Spitzer law is no longer valid (Dorelli & Scudder1996).In addition to the wave-particle e†ects in all these processes(Jacques 1977 ; Hu,Esser, & Habbal1997),we need an accurate description of the Coulomb collisional e†ects.

In this paper, we establish the new transport equations associated with our generalized velocity distribution function model. We study the fundamental properties generated by the new closure assumption of the equation system, using two approaches that have already been applied in analyzing the bi-Maxwellian model of Demars & Schunk (Cordier 1994a,

Moroko†, & Nadiga The hyperbolicity criterion of Cordier and the realizability

1994b ; Levermore, 1995 ; Levermore 1995).

criterion of Levermore et al. allow us to underline the improvements to the generalized model in the context of measurements made in the solar wind.

Section 2 is devoted to the presentation of the VDF and the macroscopic parameters used to describe the plasma ; we then present the closure assumption for higher order moments and the transport equations. Section 3 deals with the properties of

the generalized system. In° 4we discuss the main advantages of the generalized multimoment equations compared to the set

of transport equations associated with a bi-Maxwellian function. A conclusion is given in ° 5.

2

.

THE GENERALIZED TRANSPORT EQUATION SYSTEM

2.1. T he Velocity Distribution Function

To describe each species in a gas mixture, we use a separate VDF,f that is a solution of the Boltzmann equation,

s(r,¿s, t),

where r is the position vector,¿ is the species velocity, and t is the time. The distribution function can be viewed as a

s fs

probability density in r, ¿ phase space. For most Ñow situations, mathematical difficulties do not allow us to obtain

s

number of low-order velocity moments of the species distribution function, which are deÐned as SC s pT \ 1 m s

P

V f s(r,¿, t)Csp dcs,

where p is the order of the velocity moment,c is the random velocity, is the species average drift velocity, is a

s\ ¿s[ us us Csp

product of p terms of components(c and V is the velocity space.

s1, cs2, cs3),

The Ðrst step in this type of approach consists of choosing an a priori form forf that is, the type of dependence of the

s,

microscopic descriptionf as a function of the velocity moments. Indeed, to close the transport equations of the generalized

s

state variables, such an approximate expression is needed. As inDemars & Schunk(1979),we construct this function as a

polynomial expansion based on a zeroth-order function(Paper I).In a 16 moment approximation, the macroscopic

param-eters used to describe the plasma Ñows derived from the velocity moments are :

u s\ S¿sT Drift velocity T sA\ (ms/kB)ScsA2 T Parallel temperature T sM\ (ms/2kB)ScsM2 T Perpendicular temperature q s A\ n

smsScsA2 csT Heat-Ñux vector for parallel energy

q

s

M\ 1

2nsmsScsM2 csT Heat-Ñux vector for perpendicular energy

P s\ nsmsScscsT Pressure tensor s s\ Ps[ psMI[ (psA[ psM)e3e3 Stress tensor l s A\ n

smsScsA2 cscsT Higher order pressure tensor related to parallel energy

l s

M\ n

smsScsM2 cscsT Higher order pressure tensor related to perpendicular energy

Q

s\ nsmsScscscsT Heat-Ñux tensor, where_{BoltzmannÏs constant, and}ns is the number density of the species s,_{o is the mass density} ms is the mass,kBis s\ nsms p sA\ nskBTsA Parallel energy p sM\ nskBTsM Perpendicular energy T

s\ 13(TsA] 2TsM)\ (ms/3kB)Scs2T Absolute temperature of species s

q

s\ 1_{The unknowns used in a 16 moment approximation are}2(qsA] 2qsM)\ 12osScs2 csT Heat-Ñux vector. _{n and The higher order velocity moments} s, us, TsA, TsM, ss, qsA, qsM.

and are expressed starting from the approximate VDF as a function of these parameters. We choose the same

l s A, l s M, Q s

convention as inDemars& Schunk(1979).The subscripts p and o denote quantities related to the parallel and perpendicular

directions to the magnetic Ðeld, respectively. Thus, for a vectorA\ (A and (a

1, A2, A3)T, AM\ (A1, A2, 0)T AA\ (0, 0, A3)T superscript T on a tensor means the elements of the tensor should be transposed). But when the symbols p and o are

superscript, the vector AM is deÐned as (A and is deÐned as (0, 0, vector AM being related to the

1 M, A 2 M, A 3 M)T A A M A 3 M)T,

perpendicular thermal energy (see the deÐnition ofq and The unit dyadic I is deÐned as

s A, q s M , l s A, l s M). I\ e 1e1] e2e2] e3e3

where(e is an orthogonal set of unit vectors, with aligned along the direction of the straight magnetic Ðeld lines

1, e2, e3) e3

[corresponding to the letter z in the position vector r\ (x, y, z)T].

To obtain the 16 moment set of transport equations, we multiply BoltzmannÏs equation by m

s, mscs, mscsA2 , mscsM2 /2, mscscs,

and and integrate over velocity space. We then obtain the continuity, momentum, parallel energy,

perpen-m

scsA2 cs, mscsM2 cs,

dicular energy, pressure tensor, parallel and perpendicular heat-Ñux equations, respectively, for species s. To close such a

system, if we neglect the collisional terms, we only have to express higher order momentsl and as a function of the

s A, l

s

M, Q

s

lower order velocity moments. But if we do not neglect the collisional terms, we also need an a priori form of the distribution function to determine the collisional terms for the non-Maxwellian interacting potential. To this end, we use the VDF f

sG A , deÐned inPaper Ias f sG A \ f sG 0 M1 ] / sN , (1) and f sG 0 (r, c sA, csM, t)\ ns m s 4k BnTsMDs* exp

A

[ ms 2k BTsM c sM 2 [csA] Ds* D s * ] 1 E s *

B

_erfc

C

_E s *1@2

A

1 E s *[ c sA] Ds* 2D s *

BD

_{, (2)}

wherec and have been already deÐned ; erfc is the complementary error function & Stegun

sA, csM, TsM (Abramovitz 1964). Ds*

andE are deÐned in The term is a polynomial development determined using the rules of orthogonalization

and a s2\

C

6

A

q s3 A 2o s

B

4@3 [ 2 b sA 2 [ bsA

A

q s3 A o s

B

2

D

~1@2 , b s2\ [as2 q s3 A o s b sA, cs2\ [ a s2 b sA . (5)

The deÐnition of our VDF imposes the condition that the heat Ñux is limited by the inequality q s3 A ¹ 2o

swsA3 \ 2(PsAVsthA)

where is the thermal energy of ions, and is the magnetic ÐeldÈaligned thermal speed.

(Paper I), w

sA\ (kBTsA/ms)1@2, PsA VsthA

One of the advantages of this deÐnition is thatq is directly implied in the deÐnition of via Another advantage is that if

s3 A f s 0 D s *.

we imposeD then the zeroth-order function is the bi-Maxwellian function used by & Schunk The

s

*\ 0, f

s

0 Demars (1979).

Goldberger, & Low model corresponds to the set of equations obtained with and Our model of

Chew, (1956) D

s

*\ 0 /

s\ 0.

the VDF is the Grad solution(Grad 1958) when we impose the condition thatT and in our zeroth-order

sA\ TsM qs3A \ 0

functionf and construct the polynomial development following the rules of Mintzer.

s 0

2.2. T he 16 Moment Transport Equations

The method of constructing the transport equations has been explained byBurgers (1969)andTanenbaum (1967)and used

byDemars& Schunk(1979)for a 16 moment bi-Maxwellian approximation. The notations are deÐned as follows :D

s/Dt\

is the convective derivative, is the charge of species s, c is the speed of light, B is the magnetic Ðeld, E is the

L/Lt] u

s$ es

electric Ðeld, G is the acceleration due to gravity, L/Lt is the time derivative, and $ is the coordinate space gradient. For the meaning of the mathematical notations see Appendix A. The expressions of the higher order velocity moments as a function of

the lower order velocity moments are obtained from the expression off (see eqs. Below, we only present the new

sG

A [1]È[5]).

truncation assumptions compared to theDemars& Schunk(1979)model, that is, the expressions of the tensorsl and

sA lsM: l sA\ p sA o s [p sM(I[ e3e3)] ss[ ssÆ e3e3[ e3e3Æ ss]] q s3 M q s3 A p sA (I[ e 3e3) ] o s c s4 p sA (e 3e3Æ ss] ssÆ e3e3)] as22 bs4(qsA e3] e3qsA)] (oscs4[ as22 bs4qs3A )e3e3 (6) and l sM\ p sM o s [4p sM(I[ e3e3)] 2psAe3e3] 6ss[ 2ss Æ e3e3[ 2e3e3Æ ss]] 2q s3 M q s3 A p sA e 3e3, (7) where b s4\ 6 q s3 A o s

C

1 b sA ] 2

A

qs3A 2o s

B

2@3 [ b sA

A

q s3 A 2o s

B

4@3

D

, c s4\ 6

A

q s3 A 2o s

B

4@3 ]

A

3 b sA

B

2 [ a s2 2 b s4 q s3 A o s . (8)

The stress tensor equation is obtained by subtracting e times the parallel energy equation and times the

3e3 (I[ e3e3)

Parallel heat Ñux : D sqsA Dt ] qsAÆ $us] qsA($ Æ us)[ e s m sc (q s A Â B)]kB m s $ M(TsApsM)] k B m s [$Æ (T sAss)Æ (I [ e3e3)[ $A(TsAss)] ] $ M

A

q s3 M q s3 A p sA

B

_{] $ Æ}

A

c s4ss p sA

B

_{Æ e} 3e3] $A Æ

A

c s4ss p sA

B

_{] [$ Æ (a} s2 2 b

s4qsA)]e3] e3T Æ [$(as22 bs4qsA)] [ 2$ A(as22 bs4qs3A )] $A

G

os

C

6

A

q s3 A 2o s

B

4@3 ] 3 b sA 2

DH

]

C

Dsus Dt [ G [ e s m s

A

_E]1 cus B

BD

_{Æ [p} sAI] 2e3e3Æ (ss] psAI)] ] 2[q s AÆ $ MusA] qsA($A Æ us)] ($MusA)Æ qsM]\ dq s A dt , (14)

Perpendicular heat Ñux : D sqsM Dt ] qsM Æ $us] qsM($ Æ us)[ e s m sc (q s M Â B)]kB m s [2$ M(TsMpsM)] $A(TsMpsA) [ $ A Æ (TsMss)] $ Æ (TsMss)Æ (3I [ e3e3)]] $A

A

q s3 M q s3 A p sA

B

_]1 2qsMÆ $MusM ]1 2[($MusM)Æ qsM] ($MÆ us)(I] e3e3)Æ qsM]] qsMÆ $AusM] ($AusM)Æ qsA ]

C

Dsus Dt [ G [ e s m s

A

_E]1 cus] B

BD

_{Æ [p} sMI] (I [ e3e3)Æ (ss] psMI)]\ dq s M dt . (15)

We verify the results presented in this subsection in comparison to those presented byDemars & Schunk(1979)by setting

equal to zero, as in the previous subsection. The new terms that correspond to the second, third, and fourth lines of D

s *

and the third term of the second line in change into gradients involving only the stress tensor and

equation (14) equation (15)

the parallel energy. The heat-Ñux gradients and the nonlinear heat-Ñux terms in equations(14)and(15)are then new terms,

di†ering from previous moment approach models. The right-hand side of each equation (see eqs.[9]È[15]) is the velocity

moment of BoltzmannÏs collision integral. These quantities will be evaluated in a forthcoming paper for typical interparticle force and particle populations.

We have made the same development for a 9 moment approximation, for which we used the macroscopic parameters n s, us,

and (see Appendix B) as independent variables. The advantage of such a model is its capacity to provide large

T

sA, TsM, qs

temperature anisotropy and large heat-Ñux values in a simple way. But, obviously, the stress terms are not taken into account,

and in a 16 moment approximation neither is the heat-Ñux anisotropy. The deÐnition off is very close to the deÐnition

sG 0 given for the 16 moment approximation, the only di†erence being the relation

P

V c s 2 c s3fsG0 (r, cs, t)dcs\

P

V c s 2 c s3fsE(r, cs, t)dcs

instead of an equivalent relation, wherec is replaced by for a 16 moment approximation. The exact solution is The

s 2 c sA 2 f s E.

equations for the 9 moment approximation (see eqs.[9]and[B5]È[B8]),the deÐnition of the derived polynomial expansion

(seeeq. [B1]),and the expressions of the higher order velocity moments as a function of the lower order moments (see eqs.

are given in Appendix B. [B2]È[B4])

3

.

PROPERTIES OF THE GENERALIZED TRANSPORT EQUATIONS

3.1. T he Hyperbolicity Criterion

The hyperbolicity criterion is a necessary condition for a linearized system of transport equations to be well posed, being the condition for the existence of a stable and unique solution of a linearized problem associated with Cauchy boundary

conditions. Cordier(1994a, 1994b) andCordier & Girard(1996) have described a mathematical method for studying the

hyperbolicity of a system of transport equations. These works have been applied to systems derived from a polynomial expansion function based on a bi-Maxwellian function for 8, 10, 13, and 16 independent moments. In this section we discuss the hyperbolicity criterion relative to the set of transport equations (9)È(15).

The Ðrst step is to project equations(9)È(15)onto the direction of the magnetic Ðeld. This amounts to supposing that the

motion of the particles is conÐned along the magnetic Ðeld lines, which allows us to reduce the number of independent moments to consider (6 in the case of a 16 moment approximation). We note that this assumption of gyrotropic-dominated

plasmas is often made in studies of the polar and solar winds(Palmadesso, Ganguli, & Mitchell1988 ; Demars & Schunk

If we write the velocity in the form where and are, respectively, the components of parallel and

1991). ¿ (¿

perpendicular to the magnetic Ðeld, we then haveu and the state variables depend only on z : t), t),

sM\ S¿sMT\ 0, ns(z, us3(z,

t), t), t), and t). Furthermore, the pressure tensor can be written in the form

p sA(z, psM(z, qs3A (z, qs3M (z, P\

1

p sM 0 0 0 p sM 0 0 0 p sA

2

,

which implies that the stress tensor s s\ 0.

We also neglect the collisional terms, and consequently the dissipative e†ects that would have generated a less restrictive limit. But this corresponds to the situation observed in the solar wind for large heliocentric distances. The one-dimensional system projected along the magnetic Ðeld is given by

Ln s Lt ] us3 Ln s Lz] ns Lu s3 Lz \ 0 , (16) Lu s3 Lt ] us3 Lu s3 Lz ] k BTsA o s Ln s Lz] k B m s LT sA Lz \ 0 (17) LT sA Lt ] us3 LT sA Lz ] 2TsA Lu s3 Lz ] 1 n skB Lq s3 A Lz \ 0 , (18) LT sM Lt ] us3 LT sM Lz ] 1 n skB Lq s3 M Lz \ 0 , (19) Lq s3 A Lt [ m s 21@3

A

q s3 A o s

B

4@3 Ln_s Lz] 4qs3A Lu s3 Lz ] 3n skB2 TsA m s LT sA Lz ]

C

_u s3] 25@3

A

q s3 A o s

B

1@3

D

Lq s3 A Lz \ 0 , (20) and Lq s3 M Lt [ q s3 A q s3 M n s 2 k BTsA Ln s Lz] 2qs3M Lu s3 Lz [ q s3 A q s3 M n skBTsA2 LT sA Lz ] n skB2 TsA m s LT sM Lz ] q s3 M m skBTsA Lq s3 A Lz ]

A

_u s3] q s3 A n skBTsA

B

Lq s3 M Lz \ 0 . (21)

As explained byCordier (1994a)in a simple species case, the one-dimensional system (see eqs.[16]È[21])has a schematic

form of

LU

Lt ] A(U)

LU

Lz \ 0 , (22)

where U is the vector(n and A(U) is a 6] 6 matrix.

s, us3, psA, psM, qs3A , qs3M )T

In the case of a perturbation of the formU\ U the solution of the characteristic polynomial of

0] U1exp [i(kz[ ut)],

A(U) is the phase velocities u/k of the ion waves generated by this perturbation. The condition for the system to be hyperbolic is that the phase velocities all be real. In order to determine these phase velocities, we calculate the characteristic polynomial of A(U), which has the form

P(j)\ (X2 ] 1

4v3X [ 1)[X4 ] 2vX3 [ 6X2 ] v(v2 [ 6)X ] 3 ] 18v4] , (23)

where X\ (u and j is a characteristic velocity of A(U). The parameter is deÐned as

s3[ j)/w_{The Ðrst bracket always has two real roots. But the second part of P has four real roots if and only if}sA, v\ (4qs3A /oswsA3 )1@3, wsA w

sA\ (kBTsA/ms)1@2.

ov o \ 2.00. The condition for hyperbolicity can also be written in the form

oq s3

A o \ 2.00o_sw sA

3 . (24)

This study is similar to the work of Palmadesso et al. (1988), who solved the dispersion relation issuing from a

bi-Maxwellian 16 moment approximation for large-scale magnetospheric ionospheric dynamics and obtained unstable wave

modes for certain values of the macroscopic parameters ; speciÐcally, for heat-Ñux values above0.88o In fact, a study of

swsA3 .

the solutions of the dispersion relation allows us to better illustrate the consequences of the nonhyperbolicity of a moment

approach. The Ðrst bracket of the characteristic polynomial (seeeq. [23])always has two roots :

j\ u s3] w sA 2

GA

v 2

B

2 ^

CA

v 2

B

4 ] 4

D

1@2

H

.

et al. explain that these phase velocities are associated with perpendicular temperature and heat-Ñux

Palmadesso (1988)

waves. The second bracket has four real or complex roots, which are represented inFigure 1as a function of thev parameter.

The two phase velocities that are nearly independent ofv are parallel acoustic waves, and the two others are thermal waves,

according to the very similar work ofPalmadessoet al.(1988).When thev parameter is equal to^2, two waves couple and

generate an unstable wave above this value inq et al. emphasize that such an unstable wave is the

s3

A . Palmadesso (1988)

FIG. 1.ÈDependence of the four phase velocities solution of the characteristic polynomial (seeeq. [23])as a function of thev parameter. From o v o [ 2., we have represented the real part of the complex solution.

moment of the model cannot be dissipated in the Ðne structure, as with a kinetic model, because the highest order moments, which are not taken into account as a result of the truncation, play the role of the Ðne structure in a moment approach. Therefore, the highest moment is perturbed by an artiÐcial, undamped wave that can then interact with another real wave, as

shown inFigure 1,and generate an unstable wave.Equation (24)must be understood as a necessary condition for the stability

of the solution in the case of a linearized problem. It is a rough estimate of the highest value of the heat Ñux that can be modeled with the generalized polynomial approach in applications to noncollisional astrophysical winds.

3.2. T he Moment Realizability

In recent papers, several authors(Levermoreet al.1995 ; Levermore 1995 ; Grothet al.1996)have developed new methods

for estimating the domain of validity of the moment closure associated with the polynomial expansion function. These

authors have noted that the Ðrst-order Chapman-Enskog(Chapman& Cowling1970)model, which leads to Navier-Stokes

equations, always has negative values when the Navier-Stokes equations have proved to be valid for Ñows in the continuum regime. Thus, it seems that the positivity of the polynomial expansion function that generates the set of transport equations is an overly restrictive condition. A more realistic criterion is to Ðnd the conditions that will ensure the existence of a positive distribution function that could generate the same set of velocity moments.

According toHamburger (1944),a necessary condition for a set of moments of a function to generate a positive function is

that the matrix M, deÐned asM\ SUUTF_sNT,must be positive deÐnite. U\ (1,c corresponds to the

velocity-si, csicsj, . . .)T

moment space, andF is any VDF whose velocity moments are used in the approximation considered. Obviously, the derived

s N

criterion cannot be used in practice. In order to construct a useful mathematical condition, Levermore et al. have considered the subspace generated by the velocity moments corresponding to the usual hydrodynamic approximations. Indeed,

accord-ing toLevermore (1995),every closure assumption of a polynomial expansion method must recover this level of description.

Thus, we determine the condition of realizability for U\ (1,c that is the velocity moment subspace used in a

si, cs2)T

hydrodynamic approximation. This condition is obviously a restriction of the exact criterion that corresponds to the 16 moment approximation. In fact, the exact condition for the positivity and deÐnition of the matrix M should have been

calculated for U\ (1,c & Simon But in such a case, M would be much too large (with

si, csicsj, csM2 csi, csA2 csi)T (Reed 1975).

16] 16 elements) to give a useful criterion. So we restrain our analysis, following the advice of Levermore (1995).

The matrix corresponding to a 16 moment approximation is then o s 0 0 0 2psM] psA 0 p s11 ps12 ps13 2qs1 M 16\

a

0 ps21 ps22 ps23 2qs2

b

, (25) 0 p s31 ps32 ps33 2qs3 2p sM] psA 2qs1 2qs2 2qs3 rs where r s\ 8p sM 2 o s ]4psMpsA o s ]3psA2 o s ] 6o s

A

q s3 A o s

B

4@3 .

For a symmetric and diagonalizable matrix such asM a condition equivalent to its positivity is that all the determinants of

16,

one-dimensional Ñows(q and the matrix (see is

s1\ qs2\ ps12\ ps13\ ps23\ 0, ps33\ psA, ps11\ ps22\ psM), M16 eq. [25])

easier to analyze. We Ðnd as a condition for the moment realizability in one-dimensional Ñows o s[ 0 , psM[ 0 , psA[ 0 , and 2psM2 ] psA2 ] 3os2

A

q s3 A o s

B

4@3 [ 2o s q s3 2 p sA [ 0 . (26)

This condition is a more general criterion than the hyperbolicity condition. In this case, the limitation on the normalized heat Ñux depends on both temperature and heat-Ñux anisotropies. It is signiÐcant if we compare it to the equivalent conditions determined for a bi-Maxwellian model, which we present in the following section.

4

.

DISCUSSION

Only the equations of the heat-Ñux vectors (eqs.[14]and[15])are di†erent from those in the model ofDemars& Schunk

In our model, and depend on the heat Ñuxes and This property is a direct consequence of the zeroth-order

(1979). l s A l s M q s A q s3 M .

nonequilibrium distribution function bringing a heat Ñux, whereas the bi-Maxwellian approximation ofl and does not

s

A l

s M

take into account the asymmetric character of the microscopic nonequilibrium state. The new expression of the velocity

momentSc is In this expression, the new term is derived from the suprathermal long tail of

sA

4 T Sc

sA 4 T

Max] (qs3A /os)4@3. (qs3A /os)4@3

the microscopic state.

The consequence of this new truncation of the generalized approach is to modify the equations of the heat-Ñux vector related to the direction of the magnetic Ðeld by the introduction of heat-Ñux gradients and nonlinear terms. The one-dimensional system of transport equations is a good illustration of this fact. In contrast to the equivalent equation obtained

byDemars& Schunk(1979),in the equation forq (see the variations of and as a function of the position

s3

M eq. [21]), n

s, qs3A , TsA

z along the magnetic Ðeld are taken into account. In the equation forq the spatial gradient of in the direction z of the

s3

A , n

s

magnetic Ðeld does not appear in the equivalent equation of the bi-Maxwellian approach(Demars& Schunk1979),and the

multiplier to the spatial gradient ofq is modiÐed. In consequence, the equations of the heat-Ñux vector in the direction of the

s3 A

magnetic Ðeld are more strongly coupled to the other equations than with a bi-Maxwellian approximation. The Monte Carlo

simulations (Barghouthy, Barakat, & Schunk 1993) show correlated temperature and heat-Ñux proÐles with larger

tem-perature and heat-Ñux anisotropies than with a bi-Maxwellian model. Moreover, they show large gradients of the heat Ñux in

the transitional regions, which are then better accounted for thanks to the heat-Ñux gradients of equations(20)and(21).On

the other hand, a good estimate of the pressure anisotropy, which depends on a good estimate of the heat Ñuxes, is important to accurately reproduce the important e†ects of the mirror force (Spitzer 1952).

We are able to evaluate the greater accuracy of the truncation of the system using the realizability method that we present

in ° 3.2.If we apply this analysis to the bi-Maxwellian model for one species and for one-dimensional Ñows, we Ðnd the

following conditions : o s[ 0 , psM[ 0 , psA[ 0 , and 2psM2 ] psA2 [ 2os q s3 2 p sA [ 0 . (27)

If we compareequation (27)to the conditions obtained for a similar application deÐned inequation (26),we can establish that

the moment realizability provides a more restrictive condition on the velocity moments for a bi-Maxwellian polynomial expansion than for a generalized model. This is a consequence of the asymmetric character of the observed VDF, which is

better taken into account with the new term(q in

s3 A /o

s)4@3 equation (26).

In order to justify the interest of a greater coupling between the transport equations, we use the hyperbolicity condition as

applied by Cordier(1994a, 1994b) to a 16 moment bi-Maxwellian model. The mathematical description of this method is

presented in° 3.1.For the same assumptions as used previously, Cordier found the condition

oq s3

A o \ 0.91o

swsA3 . (28)

Once again, we note that the generalized model generates a less restrictive condition (see eq. [24]). Indeed, two phase

velocities are associated with an unstable wave in q when for a generalized model. As shown by

s3 A oq s3 A o [ 2.00o_sw sA 3

et al. this limit derives from the same criterion as those for a 16 moment bi-Maxwellian model.

Palmadesso (1988),

These necessary conditions (see eqs.[24], [26], [28], and[27]) consist of a limitation of the normalized heat Ñux by the

free-streaming Ñux, the parallel energy brought by particles whose velocity is the thermal velocity. We propose that these moment equations can accurately describe the nonequilibrium states that may correspond to high values of the heat Ñux, and could therefore describe the particle nonequilibrium states from collisional to collisionless regions. Indeed, the Navier-Stokes equations, which are well adapted to the description of the collisional regions, are included in the generalized model.

The evolution of the normalized heat Ñux in the solar wind from 0.3 to 1 AU measured by the Helios solar probe(Marschet

al.1982)is shown inFigure 2.This Ðgure displays the decrease of the normalized heat Ñux with heliocentric distance. This

decrease could be explained by wave-particle interactions.Figure 2 could also explain the difficulties ofDemars & Schunk

who used a 16 moment bi-Maxwellian model to simulate the solar wind. In that paper, they show that a singular point (1991),

in the parallel heat Ñux radial evolution appears around 0.4 AU. InFigure 2,the value of the normalized heat Ñux largely

exceeds the limit deÐned by equations(27)and(28) (taking into account measurement errors in the normalized heat Ñux ;

et al.

Marsch 1982).

The e†ect of the thermal force, a force that depends on the heat Ñux and appears in the velocity equation, needs to be

accurately determined. According toDorelli& Scudder(1996),the Spitzer law is not valid in the solar corona, and according

FIG. 2.ÈNormalized heat Ñux from 0.3 to 1 AU, obtained from the values of the macroscopic parameters determined byMarsch (1982)andMarschet al. from the Helios measurements. The solid line corresponds to the slow winds (300È400 km s~1) and the dashed line to the fast winds (600È700 km s~1). (1982)

the thermal force depends on more macroscopic parameter evolutions. This model should then be better adapted to model

and so determine the right e†ects of the thermal force in the acceleration processes, and so to explain why inFigure 2 the

normalized heat Ñux is greater for fast winds than for slow winds.

In the polar wind, simulations predict that for all species the heat Ñux greatly exceeds the limiting value that guarantees the hyperbolicity of the system of transport equations associated with a 16 moment bi-Maxwellian approximation (Robineau, Blelly, & Fontanari1996).Although estimated in the collisionless limit, these results show, as for the solar wind, the difficulties in modeling states far from local equilibrium with a bi-Maxwellian approximation.

5

.

CONCLUSION

InPaper I,we analyzed the microscopic properties of a generalized moment approach to stellar atmosphere expansions, in

particular to the solar and polar winds, the properties of which are now better known thanks to several probes (Helios 1 and 2, Interplanetary Monitoring Platform (IMP) 8, Ulysses, and SOHO) or by Monte Carlo simulations. This second paper provides

the associated set of generalized transport equations, constructed using the method ofBurgers (1969)andTanenbaum (1967)

and previously used byDemars & Schunk(1979). We have not given the corresponding collisional terms. Two levels of

approximation are considered. The 16 moment approximation can describe temperature and heat-Ñux anisotropies using the

same independent macroscopic parameters as the 16 moment bi-Maxwellian approximation ofDemars& Schunk(1979) ;the

9 moment approximation is a simpler model, without pressure-tensor or heat-Ñux anisotropy. We also give the higher order velocity moment closure assumptions for both approximations. Well adapted to describing thermalized state and temperature anisotropy, a 16 moment bi-Maxwellian approximation is not able to model large heat-Ñux values. However, the heat-Ñux radial dependence in the solar wind shows the importance of this parameter in describing the internal energetic state of the protons. In order to be able to accurately reproduce the heat-Ñux evolution as measured by probes in the solar and polar winds from collisional to collisionless regions, we have improved the moment approach.

A comparison with the bi-Maxwellian model is provided in order to illustrate this improvement. To this end, we use two di†erent analyses of the system of equations : the hyperbolicity approach, which studies the well-posed nature of the set of transport equations, and a realizability analysis, which evaluates the physical nature of the moment closure. Both analyses show that the description of the heat Ñux in the transitional collisional region is improved.

Although it is mainly applied to the solar wind, the generalized moment approach can also model stellar atmosphere and planetary wind expansions.

We thank S. Cordier, P.-L. Blelly, and A. Mangeney for helpful comments on this paper.

APPENDIX A

MATHEMATICAL OPERATIONS A and B are tensors of dimension 2. The contracted product of two tensors is deÐned as

(AÆ B)

with the summation convention for equal component indices. The double contracted product of two tensors is deÐned as

A : B\ A

ikBki. We also deÐne the products between a tensor A and a vector u by

(uÆ A)

i\ ukAki, (A Â u)

ij\ vilkAljuk,

wherev is equal to[1 or 1 when ilk is an even or odd permutation of 1, 2, 3, respectively, and equal to zero if two indices are

ilk

equal. With two vectors u and¿,this product is deÐned as usual by

(u  v)

k\ vijkuivj. The operations with a gradient $ are deÐned as

($Æ A) k\ LA kl Lx l , ($u) ij\ Lu j Lx i . We have deÐned two operations with subscripts o and p as

$ Mu\

1

u 1/Lx1 Lu 1/Lx2 0 Lu 2/Lx1 Lu 2/Lx2 0 Lu 3/Lx1 Lu 3/Lx2 0

2

, $ Au\

1

0 0 Lu 1/Lx3 0 0 Lu 2/Lx3 0 0 Lu 3/Lx3

2

. APPENDIX B

THE 9 MOMENT TRANSPORT EQUATIONS

The velocity moments used in a 9 moment approximation are the densityn and

s,

u

s\ S¿sT Species drift velocity

T

sA\ (ms/kB)ScsA2 T Parallel temperature T

sM\ (ms/2kB)ScsM2 T Perpendicular temperature

q

s\ 1_{The term}2osScs2 c_fsT_{is the same as for a 16 moment approximation, but with}Heat-Ñux vector for energy. sG 0 D s *\

A

qs3 o s

B

1@3 .

The terms E* andT are deÐned as for a 16 moment approach. The polynomial part of the velocity distribution function is

0 given by / s\ 2

C

cs2 [

A

4 b sM ] 1 b sA

BD

q sÆ csM o sas1A , (B1)

whereb and are deÐned in the same way as for the 16 moment approximation (see and

sA bsM eq. [4]) a s1 A \ 8 b sM 3 ] 2 b sMbsA2 ] 6 b sM

A

q s3 o s

B

4@3 .

In order to close the left-hand side of the transport equations, we need the expression of the velocity momentsP and

s, Qs, With the new expression of the approximation of the velocity distribution function, we obtain

Q s\ 4 b sM 3 a_s1A (Iqs] qsI] e1qse1] e2qse2] e3qse3)[ 4 b sM 3 a_s1A qs3(Ie3] e3I] e1e3e1] e2e3e2] e3e3e3) ] 2

A

1[ 10 b sM 3 a s1 A

B

_(e 3e3qs] qse3e3] e3qse3)]

A

60 b sM 3 a s1 A [ 4

B

_q s3e3e3e3, (B3) l s\ o s 2

A

4 b sM 2 ] 1 b sMbsA

B

_I]o s 2

C

1 b sMbsA ] 6 b sM

A

q s3 o s

B

4@3 ] 3 b sA 2 [ 4 b sM 2

D

_e 3e3 ] 1 b sMas1A

C

₂₄

A

q s3 o s

B

5@3 ]18 b sA q s3 o s ] 8 b sM q s3 o s

D

_(q se3] e3qs[ 2qs3e3e3) . (B4)

With these closure assumptions, we can determine the left-hand side of the transport equations. The continuity equation

(seeeq. [9])is the same as for the 16 moment approximation.

Momentum : o s D sus Dt ] $M(psM)] $A(psA)] e3T Æ

C

_$ A

A

4q s3qs o sbsMas1A

BD

_]

C

_$Æ

A

4q s3qs o sbsMas1A

BD

_e 3 [

C

$ AÆ

A

8q s3qs o sbsMas1A

BD

_e 3[ osG[ nses

A

E] 1 cus B

B

_\dM s dt , (B5) Parallel energy : D spsA Dt ] $M Æ

A

16q s b sM 3 a s1 A

B

_{] $ Æ (2q} s)] 2o s b sA ($Æ u sA)] 8q s3 o sbsMas1A ($ MusA)T : qse3] psA$Æ us\ dp sA dt , (B6) Perpendicular energy : D spsM Dt ] $M Æ

A

8q s b sM 3 a s1 A

B

_{] p} sM$Æ usM] psM$Æ us] 4q s3 o sbsMas1A ($ AusM)T : qse3\ dp sM dt , (B7) Heat Ñux : D sqs Dt ]

C

D sus Dt [ G [ e s m s

A

_E]1 cus B

BD

_Æ

CA

_2p sM] p sA 2

B

_{I] (p} sA[ psM)e3e3 ] 4qs3 o sbsMas1A (q se3] e3qs[ 2qs3e3e3)

D

] $

A

2o s b sM 2 ] o s 2b sMbsA

B

_{] $} A

G

o s 2b sMbsA ]os 2

C

₆

A

q s3 o s

B

4@3 ] 3 b sA 2

D

_[2o s b sM 2

H

]

A

$ A

G

1 a s1 A b sM

C

₂₄

A

q s3 o s

B

5@3 ]18 b sA q s3 o s ] 8 b sM q s3 o s

D

_q sM

HB

T Æ e 3] $ Æ

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C

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D

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