Fluid Modeling of the VLISM/Solar Wind Interaction With
the 13-Moment Formalism
Ralph L. McNutt, Jr.
The Johns Hopkins University Applied Physics Laboratory
Laurel, MD, U.S.A.
Abstract. The interaction of the solar wind with the Very Local Interstellar Medium is mediated largely by collisional charge
exchange of interstellar neutral atomic hydrogen and protons in the heliosheath. This interaction provides an additional
momentum to the interstellar population over that due to the charged population alone that leads to a decrease in the expected
size of the heliospheric cavity. The interaction is complicated by time variations in the internal solar wind flow, the presence of
both the interstellar and interplanetary magnetic fields, and the large mean free paths for charge exchange. Proper treatment of
the problem calls for a fully six-dimensional, time-dependent kinetic interaction model, yet computational complexities inherent
in such a model have precluded its full implementation. Although the applicability of fluid models to this problem has been
questioned, one can expect them to provide fairly good estimates of the various boundary locations provided that all salient
moments are included. While the ion population can be approximated to first order by a convected Maxwellian, given the
relatively small ion gyroradii, the neutral population acquires significant non-Maxwellian features due to the large mean free
paths for collisions. In this case the lowest-order moment description is the thirteen-moment description of Grad. The thirteen
quantities are the density, temperature, velocity vector, heat flux vector, and five deviator components of the pressure tensor.
Unlike the case of the Navier-Stokes equations, there is no a priori assumption about collisions; the only assumption is how the
hierarchy of fluid equations is to be truncated. To connect the neutral and ion components, moments of the Boltzmann collision
operator must be evaluated for representative distribution functions that give rise to such moments, using the appropriate
collision cross section. New results are reported for the collision operator moments corresponding to the full set of pressuretensor components and the heat flux vector components. All of these can be expressed in closed form in terms of sums of
confluent hypergeometric functions of the first kind (Kummer’s function). This derivation completes the formalism required for
the implementation of a time-dependent, magnetized fluid model of the interaction using the thirteen-moment formalism to
describe the neutral component.
population can be approximated by a convected
Maxwellian distribution, given the relatively small ion
gyroradii. The neutral population acquires significant
non-Maxwellian features due to the large mean free
paths for collisions. To capture these in a fluid context,
we consider the lowest-order moment description that
is the thirteen-moment description of Grad11-13. In this
approach, the thirteen moments are the density,
temperature, velocity vector, heat flux vector, and five
deviator components of the pressure tensor.
INTRODUCTION
Charge exchange of interstellar atomic hydrogen
and protons mediates interaction of the solar wind with
the Very Local Interstellar Medium (VLISM)1-10. The
problem is complicated by time variations in the
internal solar wind flow, interstellar and interplanetary
magnetic fields, and large mean free paths for charge
exchange. There is a need for a fully six-dimensional,
time-dependent kinetic interaction model to “do it
right”. However, such an approach is computationally
intensive, requiring simplifying assumptions3-5,10, and
time-dependence introduces additional complexities2.
Closure of the Moment Equations
Following Grad, consider distribution functions
that contain non-trivial moments through the fourth
order. These functions will then non-trivially have 15
fourth-order moments and 20 moments of orders 0, 1,
2, and 3 that must satisfy 20 moment equations. We
can then reduce the 10 third-order moments to the
FLUID APPROACH
Fluid models should provide good estimates of
many details1,6-9, e.g., boundary locations, and the ion
CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
© 2003 American Institute of Physics 0-7354-0148-9/03/$20.00
194
components of the heat flow vector, leaving 1 zerothorder moment, 3 first-order moments, 6 second-order
moments, and 3 non-trivial components of the 10
third-order moments. This is the “thirteen-moment
approximation.”
φ1 ≡
(2)
t
t
1
p1 − p1I
2
2
5
:(v1 − u1 )(v1 − u1 ) + 85
2
6 q 1 ⋅ (v1 − u 1 )(| v1 − u1 | − 2 w1 )
n1m1 w1
w1 p1
and the assumed distribution function for the neutral
population is entirely equivalent to the 13-moment
equation set.
For the ions, we assume an isotropic Maxwellian
distribution
Boltzmann Collision Operator
Moments of the Boltzmann collision operator
connect the neutral and ion components in the VLISM
problem, so these components must be evaluated up
through deviator components of pressure tensor and
heat flux vector. We assume a power-law cross
section, which provides a good fit to the chargeexchange cross section. In general, the moments of the
collision operator can then be expressed in closed form
in terms of sums of confluent hypergeometric
functions of the first kind (Kummer’s function). As a
check on the derivations, we can then verify the
general results against results for Maxwell molecules
(molecules that obey a repulsive inverse-fifth-power
force law).
n
f2 = f M, 2 =
2
3/ 2
π
w
e
3
2
−|v 2 −u2 | 2 / w 22
(3)
and we introduce the auxiliary quantities wi via
1
2
mi wi2 ≡ kTi as well as ∆U ≡ u 2 − u1 , κ ≡
| ∆U |
and α ≡ κ ∆U =
1
,
w + w22
2
1
.
w12 + w22
Collision Operator
The collision operator for the moments of any
dynamical quantity φ can be written as
∂ϕ1 
=
 ∂t  collision
=
∫∫∫ ( f ′f ′ − f f
1 2
1 2
dσ
3
)ϕ1
dσ
dΩd 3 v1d 3 v2
dΩ
(4)
∫∫∫ f f (ϕ ′ − ϕ ) dΩ dΩd v d v
1 2
1
1
3
1
2
The cross section integrals are
dσ
∫ dΩ(1− cos θ ) dΩ ≡ σ
2
P
(5)
and
dσ
∫ dΩ(1− cosθ ) dΩ ≡ σ
FIGURE 1. Exact (solid lines) and fit (broken lines) charge
exchange cross sections for protons on atomic hydrogen and
elastic scattering cross section and fit for atomic hydrogen
scattering.
(6)
where
σ M, P = σ M 0, P0 | v 2 − v1 |−ν
Model Equations
(7)
Variation of the ratio of σM/σP14 with ν is
approximated by
For the neutral distribution we assume a
Maxwellian modified to have a non-zero set of
deviator stress components and non-zero heat flux.
2
2
n
f1 = f M,1(1+ φ1) = 3 / 21 3 e −|v1 −u1 | / w1 (1+ φ1)
π w1
M
σ P σ P 0 A2 (ν )
=
=
≈ 0.4108ν + 0.6600
σ M σ M 0 A1(ν )
(1)
(8)
≈ 0.0326ν + 0.3425ν + 0.6797
2
Where the correlation coefficients for the linear and
quadratic cases are 0.9955 and 0.9978, respectively.
where
195
Pressure Tensor and Heat Flux Vector
by
The full pressure-tensor collision operator is then
given by
The pressure tensor and heat flux vector are given
t
and
p 1 ≡ ∫ m1(v1 − u 1 )(v1 − u 1 ) f1d 3 v1
∫
q1 ≡
1
2
 ∂p M 
nnσ
 1,ij 
= µ ν1 1/2 2+Mν 0/ 2
g0 κ
 ∂t  collision
m1(v1 − u 1 ) | v1 − u 1 |2 f1d 3 v1

 m2
2(kT2 − kT1 )

2
2
δ y
+ 2α e1ie1 j 
y + κw1 [y1,ν − y2,ν ]
 ij 1,ν m1 + m2

 m1 + m2 2,ν

2

1

m1
2  m 2 w2
2

I κw
[y − y1,ν ] −
[w1 y 2,ν + w 22 y1,ν ] +
 2 ijkh 1  m1 + m2 2,ν

m1 + m 2





2κ w12 (kT2 − kT1 ) 1 2 2

 E ijkhα 2  [y 3,ν − y 2,ν ]
+ 2 κ w1 [w1 − w 22 ] +

×

m
+
m




1
2
+  p1,hk − δhk  

p




m
−
m
1

2
e1h e1kδijα 2 [y 2,ν − y1,ν ] 12 w12 1

+


m
+
m
 1

2





 2m2 w12
4
4
4 


e
e
e
α
y
−
2y
+
y
−
κ
w
−
2y
+
y
κ
w
e
+
y
[
]
[
]
3, ν
2,ν 
1
3, ν
2,ν
1,ν
1 

 1i 1 j 1h 1k  4,ν

 m1 + m2



Change to velocities in the center of mass frame
and with respect to the center of mass. Define
mm
m v + m2 v 2 ,
and
g ≡ v 2 − v1,
µ≡ 1 2 ,
G≡ 1 1
m1 + m2
m1 + m2
ˆ ≡ G − u to obtain
G
1
t
∂p1 
ˆ +G
ˆ g d 3Gˆd 3 g
= µ ∫∫ f1 f 2 gσ M gG
 ∂t  collision
t
m
+µ m1 +2m2 ∫∫ f1 f2 gσ P 12 g2 I − 32 gg d 3Gˆ d 3 g
(
)
(
(9)
)
∂q1 
 2 2
= 1µ
f f gσ g Gˆ + g
 ∂t  collision 2 ∫∫ 1 2 M 
[ ] d Gˆ d g
2
m2
m1 + m2
3
3
ˆ (G
ˆ ⋅ g)d 3Gˆd 3 g
+µ ∫∫ f1 f2 gσ M G
+µ
m2
m1 + m2
∫∫
f1 f2 gσ P

 
 2 2(3 + ν2 )κ (kT2 − kT1 )

  [y 2,ν − y1,ν ]κ w2 −
m1 + m2
E ijh

 +


  1
2
2
ν
+ [2 + 2 ][y1,ν − y0,ν ][κw 2 − κw1 ]


  2

 m2 − m1 
8 κ 1/ 2q1,hα 
ν
1
+
+

e1hδij 2 [2 + 2 ][y1,ν − y 0,ν ]
5 n1m1
 m2 + m1 


2
ν
[2 + 2 ][− y 2,ν + 2y1,ν − y 0,ν ]κ w1


e1ie1 je1h 2α 2 

2(3 + ν2 )κ (kT2 − kT1 )
+ [y 3,ν − 2y2,ν + y1,ν ] κw22 −




m1 + m2



(10)
( g Gˆ − [Gˆ ⋅ g]g)d Gˆd g
1
2
2
3
3
2
3
 ∂pP 
nnσ
m2
 1,ij 
= µ ν1 1/2 2+P0
g0 κ ν / 2 m1 + m2
 ∂t  collision
To evaluate, shift the variables once more using
v1 − u1 = G * − κw12 (g− ∆U) and
ˆ ≡ G − u1 = G * + κw12 ∆U + m 2+κm (kT2 − kT1 )g , so that the
G
operators now contain forms like
1
∫dG e
3
1
π 3 / 2κ 3/ 2 w13 w32
*
−

2
α y + 1 δ − 3 e e +
(2 ij 2 1i 1 j )
2,ν
κ




− 3 y I + E 3 α 2 [y − y ] 
ijkh 2
2,ν
1,ν
4 2,ν ijkh


p

× 12 w12  1,hk − δhk  +e1h e1kδijα 2 [3 + ν2 ][y2,ν − y1,ν ] + 
 p1



−3[3 + ν2 ][y 2,ν − y1,ν ]
2


e1ie1 je1h e1kα  21
+ 2 [y 3,ν − y 2,ν ]













[2 + ν2 ][y 0,ν − y1,ν ][12 δij − 32 e1ie1 j ]e1h +
 8 κ1/ 2q1,h
ν

+ 5 n m α{3+ 2 } 3
1
5
1 1
 2 [y 2,ν − y1,ν ][2 E ijh − 2 e1ie1 je1h ]


2
G*2
κw12 w 22
* 2n
(G )
(11)
which can be integrated immediately. The remaining
integrands are then of the form
∫ d ge κ
3
− |g−∆ U | 2 ν +1
g gi1 gi2 gi3 ...gin
(12)
≈
Γ(2+ n + 12 ν )
M(− 12 − 12 ν ,n + 32 ,− α 2 )
Γ(n + 32 )
(13)
[( + n + ν )+ α ]
5
4
1
4
2
1
(1+ν )
2
MAXWELL-MOLECULE LIMIT
These general expressions are complex. We can
reduce the complexity and provide for a cross-check in
the derivation, by going to the limit of Maxwell
molecules for which ν=-1. Going back to the original
definitions, for the moments of the collision operator,
we obtain for the pressure tensor operator
To write out the results, define the unit vector
e 1 ≡ ∆Uˆ ≡ ∆U /∆U and use it and the Kronecker delta to
define the fully symmetric tensors
Ihkji ≡ δhkδij + δ hiδkj + δkiδhj
(14)
Ehkji ≡ e1h e1kδ ij + e1j e1iδ hk + e1h e1iδkj
(15)
+e1k e1jδhi + e1h e1jδ ki + e1k e1iδ hj
Ehkj ≡ δ hke1j +δ hj e1k +δ kj e1h
(18)
The expressions for the heat flux vector are even
more complex and are not given here.
that can be integrated analytically in terms of
yn,ν (α ) ≡
(17)
  3 n1 p 2 3 n2 p1 1
t 

+2
+ 2 n1n 2 ∆U 2  I
t

 2 m

∂p 1P 
m1
m
2
2


= µ (g0σ P 0 )
t
t


 ∂t  collision,ν =−1
m1 + m2  n1p 2
np

−  32
+ 32 2 1 + 32 n1n2 ∆U∆U  
m1

  m2
(16)
196
(19)
t
 M
 2

t
t
2m
∂p1 
= µ (g0σ M 0 )
(n1p 2 − n2p1)+ m + 2m n1n2∆U∆U 
 ∂t collision,ν =−1

 m1 + m2
1
2
(20)
ACKNOWLEDGMENTS
This work was supported in part under NASA
Grant NAG5-11060.
Corresponding results for the heat flux vector are
(21)


 m 2  n1q 2 6m12 + 2m 22 n2 q1

−
8

 m1 + m 2  m 2 (m1 + m 2 )2 m1


2
2
t
t
M




 m 2  n1p 2
 m − m1  n2 p1

 ∂q1 


= 12 µ(g0σ M 0 ) +∆U ⋅ 8
+ 2 2



 ∂t  collision ,ν = −1




m1 + m 2
m1 
 m1 + m 2 m2


2
2
2








m
n
p
−
m
n
p
m
m

2
1
2 1
2
 1 2 + 3 2
 ∆U 2 
+ 4n1n 2
+∆U12






m
+
m
m
m
+
m
m
m
+
m

1
2
2
1
2
1
1
2


2
REFERENCES
1. Pauls, H. L., Zank, G. P., and Williams, L. L., J.
Geophys. Res., 100, 21595 (1995).
2. Liewer, P. C., Karmesin, S. R., and Brackbill, J. U., J
Geophys. Res., 101, 17199-17128 (1996).
(22)

−





+∆U  3m1 + m2 3p1 n − 2m2 3p2 n − 2m2 n n ∆U 2 
2
1
1 2

 m1 + m2 m1

m1 + m2 m2
m1 + m2
4
(n2q1 + n1q2)
m1 + m2
t
t

P

m2
m1 − 3m2
p
4m2
p
1
 ∂q1 
= 2 µ(g0σ P 0 )
∆U ⋅ 1 n2 −
∆U ⋅ 2 n1
+
m1 + m2
m1 + m2
m1
m1 + m2
m2
 ∂t  collision,ν =−1

3. Baranov, V. B., Izmodenov, V. V., and Malama, Y. G.,
J. Geophys. Res., 103, 9575 (1998).
4. Linde, T. J., Gombosi, T. I., Roc, P. L., Powell, K. G.,
and DeZeeuw, D. L., J. Geophys. Res., 103, 1889-1904
(1998).
These results can finally be compared with the
limiting forms with ν=-1 by taking q2=0, p2 to be
diagonal, and m1=m2=m. The results are
5. Lipatov, A. S., Zank, G. P., and Pauls, H. L., J. Geophys.
Res., 103, 20631 (1998).
t
t
 M
t
(23)
 ∂p1 
= 12 (g0σ M 0 ) n1 p2 I − n2 p1 + n1n2 m∆U∆U
 ∂t  collision ,ν = −1
[(
]
)
t 3

I − 2 ∆U∆U 

(24)
 M
 ∂q1 
= 14 (g0σ M 0 ) −2n2 q1 + ∆U{5n1 p2 + n1n 2 m∆U 2 }
 ∂t  collision ,ν = −1
(25)
t
t

 ∂p1P 
 p t


= 14 m(g0σ P 0 )n1 n2  − 34 w12 1 − I +


 ∂t  collision,ν =−1

 p1
[
[ ∆U
1
2
2
]
]
6. McNutt, R. L., Jr., Lyon, J., and Goodrich, C. C., J.
Geophys. Res., 103, 1905-1912 (1998).
7. McNutt, R. L., Jr., Lyon, J., and Goodrich, C. C., J.
Geophys. Res., 104, 14803-14809 (1999).
8. McNutt, R. L., Jr., Lyon, J., Goodrich, C. C., and
Wiltberger, M., “3D MHD Simulations of the
Heliosphere-VLISM Interaction,” in Solar Wind Nine,
edited by S. R. Habbal et al., AIP Conference
Proceedings 471, New York: American Institute of
Physics, 1999, pp. 823-826.
t
−2n2 q1 − ∆U⋅ p1 n 2
 (26)
∂q1P 
1


= 8 (g0σ P 0 )
2 
 ∂t collision,ν =−1
+∆U(6 p1 n2 − 5p2 n1 − n1 n2 m∆U )
9. McNutt, R. L., Jr., Wiltberger, M., Lyon, J., and
Goodrich, C. C., “A Fluid Approach to the
Heliosphere/VLISM
Problem,”
in
The
Outer
Heliosphere: The Next Frontiers, edited by K. Scherer et
al., COSPAR Colloquia Series, Vol. 11, New York:
Pergamon, 2001, pp. 89-98.
SUMMARY AND CONCLUSIONS
The complete formalism still requires the general
formulation for the heat-flux vector. This approach
does provide an internally consistent fluid model
applicable to Knudsen numbers ~1 or greater, while
voiding the inherent assumptions of Chapman-Enskog
approach, viz. (1) the collisionality assumption is
replaced with assumed distribution function form, (2)
the lowest-order kinetic results should still be
captured, and (3) the approach trades integration over
velocity space for assumed closure in moments
equations. Even the thirteen-moment approach is still
inherently complex except for Maxwell molecules. In
addition, a numerical model is still required to provide
cross checks with kinetic results and explore further
the parameter space for the interaction.
10. Müller, H.-R., Zank, G. P., and Lipatov, A. S., J.
Geophys. Res., 105, 27419-27438 (2000).
11. Grad, H., Commun. Pure and Appl. Math., 2, 331-407
(1949).
12. Grad, H., “Principles in the Kinetic Theory of Gases” in
Encyclopedia of Physics, Vol. XII Thermodynamics of
Gases, edited by S. Flügge, Springer-Verlag, Berlin,
1958, pp. 205-294.
13. Gombosi, T. I., and Rasmussen, C. E., J Geophys. Res.,
96, 7759-7778 (1991).
14. Chapman, S., Manchester Mem., 66, 1-8 (1922).
197