A three-fluid, 16-moment gyrotropic bi-Maxwellian
fast solar
wind: the effect of He
Lorraine Allen and Xing Li
Department of Physics, University of Wales, Aberystwyth, UK
Abstract. We present a 16-moment, three-fluid description of the solar wind consisting of electrons, protons, and alpha
particles. We assume gyrotropic flow (transport across the magnetic field is neglected) which reduces the 16-moment set of
transport equations to a six-moment set yielding the density, velocity, temperatures parallel and perpendicular to the magnetic
field, and parallel heat conductive fluxes from the parallel and perpendicular directions for each particles species. The model
incorporates the effects of Coulomb collisions. It allows for non-radial divergence of the magnetic field and heating and
momentum addition to the particles. We investigate the influence of the heat conductive flux in shaping the temperature
anisotropy.
INTRODUCTION
nuclei account for 20% of the solar wind mass density
and contribute a non-negligible portion of the flow’s momentum and energy flux at 1 AU [16]. If ion cyclotron
resonance heats and drives the flow in the solar wind acceleration region, the alphas are in queue immediately
before the protons to be heated, if at all, by the waves
and their behavior can serve as a probe of the waves. Because of their lack of emission, alpha particles are not
observable in the inner corona, but Helios observations
in interplanetary space show that the core distribution of
T at 0.3 AU while the
the alphas is anisotropic with T
protons, with an implied ratio of T T
1 4 in the inner corona, are anisotropic with T
T until close to 1
AU [11]. Our goal is to determine if the heat conductive
flux of the alpha particles plays a significant role in governing the temperature anisotropy of the alphas and how
it can be treated in models.
SUMER/ and UVCS/SoHO observations of minor ions
in the inner corona imply that particles with greater
mass-to-charge ratios are rapidly accelerated and preferentially heated in the direction perpendicular to the magnetic field in the fast solar wind [7, 19, 18]. To adequately
model the thermal behavior of minor ions, it is necessary
to know how to treat their heat conductive flux which
is usually assumed negligible but may be significant in
shaping the temperature profile.
In the collision dominated case, the classical value for
the heat flux [17] can be recovered from the dominant
terms in the heat conductive flux equation which is derived by taking velocity moments of the Boltzmann equation. However, since the solar wind is collision dominated only a short distance above the solar base, the classical Spitzer-Harm values of the heat conductive flux are
not valid descriptions in the solar wind. Helios observations show that the heat conductive flux of the protons in interplanetary space is well below that obtained
from the classical expression [16, 11]. Gyrotropic 16moment fluid models, incorporating the heat conductive
flux equations for protons and electrons, yield a proton
heat flux which is lower than the classical value and
which influences the behavior of the proton temperature
anisotropy [14, 9, 10].
Little attention has been paid to minor ions and specifically to alpha particles. On account of the empirical evidence that minor ions are interacting with waves that may
heat and accelerate the solar wind, more exploration of
their behavior is needed. With an empirical abundance of
3-5 % that of the protons in interplanetary space, helium
THE MODEL
The 16-moment fluid equations for protons, electrons,
and alphas are obtained by expanding the distribution
function of each of the fluids about a bi-Maxwellian and
taking velocity moments of the Boltzmann equation. We
assume the gyrotropic limit in which the off-diagonal
terms of the pressure tensor are neglected and thermal
transport occurs only in the direction parallel to the magnetic field. This is valid in the strong magnetic field region close to the Sun and may also be justified in interplanetary space where observations have shown the presence of field-aligned distribution functions and that the
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
339
solar wind heat flux is parallel to the magnetic field [11].
We derive our equations for one-dimensional flow from
the formulation in [1] as
∂ ni
∂t
∂ vi
∂t
∂v
vi i
∂r
1 ∂ ni kT i
n i mi
∂r
1 ∂ n i vi a
a ∂r
0
T 1 da
a dr
mi
e
s
i
c
j
j
j
j
j
j
j
j
j
j
j
j
c
j
j
j
j
j
j
j
j
j
j
j
j
c
j
j
j
j
j
j
j
j
j
j
j
j
2
j
j
ext
j
j0
i
!
! ! ! ! !
!
Q
2 ! 10 erg cm s σ
02R
At the coronal base the electron density is 5 ! 10
cm and the temperatures are equal to 6 ! 10 K. The
j
c
The subscript i denotes alphas or protons and subscript j
denotes alphas, protons, or electrons. We assume overall
charge neutrality and negligible current so that ne
ΣZi ni and ne ve ΣZi ni vi where Zi is the charge of ion
i. The partial pressures are given by p j n j kT j where
* denotes the parallel or perpendicular direction and
k is Boltzmann’s constant. The mean temperature and
total heat conductive flux of species j are T j T j 3
2T j 3 and q j q j 2 q j respectively, where q j
and q j are the conductive fluxes from the parallel and
perpendicular direction, respectively. We use collision
terms, denoted by subscript c derived in [2] using the
Bhatnagar-Gross-Krook approximation and written out
explicitly in [3]. We do not include the ionization and
recombination terms used in their model.
i
We present in Figure 1 a model solution with the following parameters:
Geometry: fm 5 r 1 31Rs , σ 0 51 Rs
Momentum Addition:
D p0 5 10 15 g cm 2 s 2 σ p 0 6 Rs
D pext
5 10 12 g cm 2 s 2
Dα 0 5 10 16 g cm 2 s 2 σα 0 6 Rs
5 10 12 g cm 2 s 2
Dαext
Heating:
Q p0 6 10 8 erg cm 3 s 1 σ p 0 5 Rs
Q α 0 1 10 8 erg cm 3 s 1 σα 0 5 Rs
Q e0 2 10 7 erg cm 3 s 1 σe 0 2 Rs
j
j
j
Rs r σ j
j0
RESULTS
c
j
2
i
j
j
j
4
s
4
10Rs
2
j
j
j
j
j
j
Rs r
stants. In the absence of knowing what heats and accelerates the solar wind, the functional forms above allow
for a wide range of mechanisms. Eq.8 originates from Q
been written as the negative divergence of an energy flux
density that is dissipated over length scale σ [5, 20, 15].
It is reasonable to assume that a similar form applies to
the momentum addition. The second term in Eq.7 allows
for extended momentum addition to the particles. It was
originally added to counteract the large inertia of the helium ions by preventing the excessive dip in the alpha
particle velocity in the inner corona and by aiding the
helium velocity to exceed the proton velocity in interplanetary space. Since observations indicate and models
have incorporated extended proton heating [12, 13, 6],
such as by the dissipation of waves, there is likely also
extended momentum addition to the particles.
The equations are simultaneously solved using a fully
implicit, time dependent code described in [6].
i
i
ext
i0
c
i
2
j
e
e
e
r σi
j
1 da
δv
k T T a dr m δ t D
δv
GM
(2)
r n m δt ∂ T
∂ T
∂v
1 ∂ aq v
2T
∂t ∂r ∂ r n ka ∂ r
δ T
2
Q
2qn ka da
(3)
δ t dr n k
∂ T
∂ T
v T da
1 ∂ aq v
∂t ∂r a dr n ka ∂ r
q da
δ T
1
Q
(4)
n ka dr n k δ t ∂ q
∂ q
v q da
∂v
v
4q ∂t ∂r ∂r a dr
3n k T ∂ T
δ q
(5)
m ∂ r δ t ∂ q
∂ q
2v q da n k T ∂ T
∂v
v
2q ∂t ∂r ∂r a dr m
∂r
n k T T T da
δ q
(6)
a
da δ t m
D 1 e R (7)
r
Q
(8)
Q e where * denotes the parallel or perpendicular direction
for species j and D σ D Q and σ are conDi0 e Rs
Di
i
(1)
k Ti
Zi 1 ∂ ne kT e
mi n e
∂r
We assume radial flow through a diverging flux tube
with cross-sectional area a r2 f r where f r is from
[8]. We allow for arbitrary momentum and heating to the
species of the form
7
3
1
e0
s
e
8
3
5
solution matches empirical electron density and inferred
temperature profiles in the inner corona and is consistent
with proton flux (n p v p 3 3 108 cm 2 s 1 ), velocity
(v 700 km s 1 ), and Helium abundance (nα n p 048)
at 1 AU. The steep acceleration of the velocity profile in
340
!
FIGURE 1. Model solution for parallel proton parameters (solid line), perpendicular proton parameters (dotted line), parallel
alpha parameters (short-dash line), perpendicular alpha parameters (dash-dot-dot-dot line), parallel electron parameters (long-dash
line), and perpendicular electron parameters (dash-dot line). The heat flux density is multiplied by the cross-sectional area a of the
flux tube at distance R divided by the cross-sectional area ao at the solar base to account for the expansion of the flux tube. The
electron density is fitted to empirical data [4] (solid dots) and [7] (diamonds). The proton temperature is also fitted to empirical data
[7].
the inner corona results in a noticeable dip in T p and T α
below 3Rs The alphas initially lag behind the protons
because of their large inertia. The extended heating to
the alphas is mainly responsible for the dominance of
the alpha velocity over the proton velocity in the outer
corona. Since nα
n p the momentum addition per
particle is significantly higher for the alphas.
Clearly the temperatures at 1 AU are not in agreement
with observations. T p and T α initially dominated by
the heating terms Q p and Q α respectively, peak in
the inner corona, then plummet due to adiabatic cooling
vT
(the a da
dr term). Extended heating is necessary to increase T p α in the outer corona to match observations in
interplanetary space. Because thermal transport is only
in the parallel direction, the decrease in T results in a
large rise in T for both the protons and the alphas. As
noted previously in the literature, this is a serious problem for the protons since empirically T p 3 105K at
1 AU, although ion cyclotron resonance may serve to de-
""
#
$
crease the parallel temperature in the outer corona. Interestingly, a high parallel temperature is less of a concern
regarding the alphas since T α 0 9 2 106K at 1 AU.
For this solution T α is in agreement with observations,
although T α is much farther below its empirical value
( 0 7 1 106 K) at 1 AU than is T p
Within the first 10 Rs the heat flux profiles are in general proportional to the temperature and not to the gradient of the temperature. The alpha parallel and perpendicular heat conductive flux is several orders of magnitude
below that of the protons and especially of the electrons,
as expected, but its role is not insignificant. The creation
of q α (Eq.5) is a delicate balance between competing
terms in each region of the corona. The flux q α peaks
slightly below 2Rs where there is a local maximum in
T α due to the ∂∂ vr term in equation 3. The parallel heat
flux q α does not occur in the perpendicular energy equation (4) and does not dominate in any of the terms in
the parallel energy equation (3). It is not particularly sig-
!
!
341
!
nificant in affecting either temperature profile. The perpendicular heat flux, q α is not influential in any of the
terms in the perpendicular energy equation. Instead, T
is controlled by the heating and collision terms in the in∂T
vT
ner corona and by the a da
dr and v ∂ r terms farther out.
2q
#
#
the parallel heat flux q α does not appear to play an important role in governing the behavior of T α or T α
Future work will include removal of the ad hoc functions in the Eqs. (7) and (8) and the addition of waves to
heat and accelerate the solar wind. These will provide a
more realistic approach to determining the amount and
location of energy addition and the resulting model parameters. This should allow a better comparison of model
solutions with observations, with the hope that the model
will help explain the different thermal behavior of the alphas and protons in interplanetary space and at 1 AU.
#
However, the nka da
dr term in the parallel energy equation
is dominant from 3 9 Rs transporting energy from
the perpendicular into the parallel direction. This causes
the noticeable rise in T α after T α decreases from 2-3 Rs
due to the sharp rise in vα . T α remains high into interplanetary space. Similar behavior exists for the protons.
Thus q α affects T α . The flux q α is negligible
in the first 0 5Rs due to almost perfect balance be∂T
tween the collision and nT ∂ r terms in equation (6).
These two terms decrease and the temperature anisotropy
[T α T α T α ] term becomes dominate at
2 Rs
This generates a large q α value which peaks 2 Rs
near to the peak of T α Two other terms in Eq.6 are
#
2vq
#
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#
rapidly, and
ate a small q
2vq
a
α
#
% and v∂ q ∂ r nearly balance to creterm in interplanetary space.
da
dr
CONCLUSION
In conclusion we find that, because of the large inertia of
the alpha particles, extended momentum addition is very
beneficial in creating an alpha velocity comparable to or
greater than the proton velocity in interplanetary space.
Extended heating in the perpendicular direction is also
necessary for both the protons and the alphas if T at 1
AU is to match observed values. As has been noted in
the literature, cooling is needed in the parallel direction
for the protons because T p is too large in interplanetary
space. However, because of the high empirical value of
T α at 1 AU, the large interplanetary values of T α in this
model are not excessive. Additional heating directly to
the electrons is needed to match the empirical electron
temperature in interplanetary space.
The parallel and perpendicular heat conductive flux
are several orders of magnitude lower than those of the
protons and electrons, but q α is responsible for increasing T α in the region several solar radii above the coronal base. A high positive q α value is generated in this
region predominantly in response to the large temperature anisotropy that develops due to the increase in T α
from the perpendicular heating. The perpendicular heat
flux q α itself does not significantly influence T α and
REFERENCES
also prominent in this region: a da
dr which serves to
decrease q α is slightly larger than v∂ q ∂ r but both
are much less than the anisotropic term. Beyond 4 Rs
2vq da
a dr dominates over the anisotropic term, which falls
342