A global three dimensional hybrid simulation of the
interaction between a weakly magnetized obstacle and the
solar wind
P. Trávníček , P. Hellinger and D. Schriver†
Institute of Atmospheric Physics, AS CR, Boční II. čp. 1401, 14131 Prague 4, Czech Republic.
†
Institute of Geophysics and Planetary Physics, UCLA, Los Angeles, 90095-1567, U.S.A.
Abstract. We study an interaction between the solar wind flow and a conductive obstacle with a weak dipole magnetic field
using a three dimensional implementation of the hybrid code. We show that the hybrid approach is capable of describing most
of the structures formed due to the interaction between the solar wind and a magnetized planet like the bow shock, proton
foreshock, magnetopause, magnetosheath, northern and southern cusps and the current sheath.
INTRODUCTION
mainly describe qualitative results of the overall features
of the interaction between a weakly magnetized obstacle
and the solar wind.
We concentrate mainly on transition regions. One of
these is a collisionless shock wave that results from the
interaction between the "supersonic" flow and the planet.
The shock wave resulting from the interaction with a
small obstacle usually has a multiple, "shocklet" structure. Observations of this shocklet structure in front of
Mars have been discussed by Dubinin et al. [6] (see also
[7]), who referred to the detailed numerical analysis by
Omidi and Winske [8] (see also [9]). Dubinin et al. [6]
studied the role of O + heavy ions in the process of formation of this structure. Shimazu [5] reported the formation of the shocklet structure in front of an unmagnetized
obstacle in a 3-D global hybrid simulations for a pure
electron-proton plasma. Shimazu [5] proposed that the
formation of the shocklet structure is related to the finite Larmor radius of protons and appears for sufficently
small obstacles. We compare our simulation results with
these results and predictions.
The interaction between a collisionless solar wind flow
and an obstacle with a typical scale comparable to ion
scales is beyond the applicability of standard magnetohydrodynamic (MHD) models. To include ion kinetic effects one can use a hybrid scheme, where electrons are
considered as a massless fluid while ions are treated kinetically by the second order particle in cell scheme. Hybrid codes were applied successfully in studies of the interaction between the solar wind and unmagnetized obstacles such as Venus, Mars, and comets [1, 2, 3, 4, 5].
In this paper we are interested in the interaction between the solar wind plasma and a magnetized obstacle.
In a hybrid model the interaction between the solar wind
flow and a magnetized obstacle is determined by three
parameters: the radius of the obstacle R, the velocity of
the solar wind flow v sw , and the dipole moment M. Table 1 gives an overview of different parameters for different planets of the Solar system. We present results of
the three-dimensional (3-D) global simulation of a small
planet with radius R = 8:5 L in (where Lin is proton inertial length) with a weak dipole magnetic field given
by dipole moment M = 1:5 10 4 Bsw L3in =µ0 (where Bsw
is the mean value of the solar wind magnetic field) embedded into a solar wind plasma flow with the velocity
vsw = 3:0 vA (vA is the Alfvén speed). Table 1 shows that
appart from unmagnetized planets Venus and Mars, Mercury is the most likely candidate for the description by
global numerical experiments based on a kinetic model,
however, the scales in our numerical experiment do not
match any of the real parameters yet, since we are limited by available computing resources. In this paper we
THE MODEL
We consider a 3-D numerical model of a small planet
with a weak dipole magnetic field in a solar wind flow
with a bulk speed v sw = 3:0 vA along the x+ axis. The
initial ambient magnetic field ~B (x; y; z) = ~B (cosϕ cosψ ,
sinϕ cosψ , sinψ ) has been defined by setting ϕ = 30 o ,
ψ = 0 (no north-south component). The dipole field of
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
485
the planet is given by the standard formula
of a complicated structure of transition regions that are
similar to those observed around magnetized planets.
The nine panels of Figures 1–3 show the main results of
the simulation at time t = 10 Ω pi1 . The overall structure
of the interaction is already formed, but the simulation
has not yet reached its stationary state. Later on, at t 20 Ω pi1 , boundary effects start to affect the flow.
Figure 1 shows the number density n as a grey scale
plot. The left panel displays the equatorial-plane cut
n = n(x; y; z = 0), the middle panel displays the first
meridian-plane cut n = n(x; y = 0; z) and the right panel
displays the second meridian-plane cut n = n(x = 0; y; z).
Arrows give the magnetic field vector ~B = ~B (x; y; z)
projected onto the given plane. One can clearly recognize the quasi-perpendicular and quasi-parallel shock
regions on the left panel. The left panel also shows the
anti-clockwise injection of protons from the front behind
the planet around the planetary surface in the equatorial
plane. The middle panel shows the formation of the current sheath where the density n is higher, while the vector
~
B of the magnetic field changes its direction.
Figure 2 shows the magnitude of the magnetic field
j~Bj in the same format as Figure 1. Arrows represent
the magnetic field vector ~B = ~B (x; y; z) projected onto
the given plane. The magnetopause is clearly visible
in the numerical experiment on the middle panel together with the formation of a current sheath. The position of the magnetopause can be approximately predicted using the expression for the balance between dynamic pressure of the solar wind flow and the magnetic
pressure of the
p model dipole magnetic field R MP =R =
[(Beq =Bsw )=( 2 vsw =vA )]1=3 where RMP stands for the
distance of the nose of the magnetosphere from the
center of the planet and B eq denotes the magnitude of
the magnetic field at the planet’s equator B eq = 1:5 104 =8:53 Bsw . This expression provides a rough theoretical estimate for the position of the nose of the magnetosphere R MP; MHD 1:7 R. Closer evaluation of the
B2 in the x direction suggests, that the magnetopause
is formed around R MP; model 1:4 R, however the shock
front is not yet well separated from the magnetopause
as we have not reached stationary state of the simulation. The structure of a magnetosphere begins to appear
as seen by the formation of northern and southern cusp
regions.
The last set of panels shown on Figure 3 describes
flows of protons in the simulation box. The density n is
shown in the the same format as Figure 1. Arrows represent the proton flow velocity ~u p = ~u p (x; y; z) projected
onto the given plane. Figures 1–3 reveal an asymmetry
of the magnetosphere (see all right panels). The shock
observed in the simulation has the shocklet structure (apparent especially in the area of the perpendicular shock,
seen in all of the left panels).
µ0 M 2 sin λ ~er + cos λ ~eλ
3
4π r
M0
B
2 sin λ ~er + cos λ ~eλ
=
3 sw
0
r
where M stands for the magnetic dipole moment of the
planet, and r is the distance measured from the center
of the planet. M and r are in SI units, while M 0 and
r0 are in the dimensionless units used in the simulation
(Bsw L3in =µ0 and Lin respectively). Here also ~e r and ~eλ are
unit vectors in the r and λ directions, with r, λ being
radius and magnetic latitude respectively in a spherical
coordinate system which has its origin in the center of
the obstacle. No gravitational terms are included in the
model. The numerical scheme describing the dynamics
of the plasma is based on a 3-D version of the CAM
algorithm developed by Matthews [10].
The spatial dimensions of the simulation box are
(Lx ; Ly ; Lz ) = (104; 92; 92) L in with dx = dy = dz =
0:8 Lin . The radius of the planet is R = 8:5 L in , which is
roughly ten times smaller than the radius of Mercury. We
use the magnetic dipole strenght B dip given by magnetic
field moment M (see Table 1), M 0 = 1:5 104 Bsw L3in = µ0 .
Initially the simulation box is filled with a homogeneous plasma (32 protons per cell) with β e = β p = 0:5.
The box has open boundaries along the x axis. Solar wind
protons, with a Maxwellian distribution function, are injected at the boundary plane x = 0:33 L x with the bulk
speed vsw = 3 vA . The particles can freely escape the
box at the opposite (x = 0:66 L x ) side. The volume has
periodic boundary conditions in both directions perpendicular to the solar wind plasma flow. The other boundary in the model consists of the surface of the planet.
In this particular case we use a conductive obstacle with
all fields defined in the interior. For the particles reaching the surface of the planet we use reflective boundary
condition. The sphere representing the planet is placed
at the origin of the Cartesian coordinate system so that
xmin = 0:33 Lx and xmax = 0:66 Lx . The x axis points
toward the Sun, z + points toward the magnetic north
pole. The length scales in figures 1-3 are expressed in
units of R.
The duration time of the simulation 20 Ω pi1 (Ω pi : the
proton gyro frequency in the solar wind) is comparable
to the characteristic time R=v A = 8:5Ω pi1 . Due to the
short time duration we do not consider the rotation of
the planet.
~
Bdip
=
RESULTS
In this section we summarize the results of the numerical
experiment. The initial conditions lead to the formation
486
TABLE 1. Some parameters of the plasma environment around planets of the Solar system obtained from different sources
determining scales and units used in our numerical model. The first column gives the distance of the planet from Sun in [AU ].
The next two columns list parameters of the solar wind environment of the planet: the proton particle density np and the strength
of the interplanetary magnetic field Bsw . Values of the solar wind velocity vsw are equal to the upstream Alfvén Mach number
MA of the solar wind flow surrounding the given planet. The Earth magnetic dipole moment is MEarth = 8 1022 Am2 .
Name
Mercury
Venus
Earth
Mars
distance
[AU]
0.31
0.47
0.72
1.0
1.5
[cm
np
3]
Bsw
[nT]
Lin
[km]
73
32
14
7
3.1
46
21
10
6
3.4
26.7
40
61
86
129
[km s
vA
1]
118
82
59
50
42
radius RM
[km] [Lin ]
2 439
2 439
6 052
6 378
3 397
91
60
99
74
26
vsw = 300km=s
[vA ]
2.5
3.6
5
6
7
dipole moment M
[MEarth ] [L3in Bsw =µ0 ]
4:7 10
4:7 10
4
4
1.0
-
5.4107
3.5108
2.6109
-
FIGURE 1. Gray scale plots of the density n = n(x; y; z) in three perpendicular planes (left) z = 0, (middle) y = 0, (right) x = 0
through the center of the planet. Regions with higher density of protons are darker. Arrows correspond to the magnetic field vector
~
B = ~B (x; y; z) projected onto the given plane.
FIGURE 2. Gray scale plots of the magnitude of magnetic field j~Bj = B(x; y; z) in three perpendicular planes (left) z = 0, (middle)
y = 0, (right) x = 0. Arrows represent the magnetic field vector ~B = ~B (x; y; z) projected onto the given plane.
FIGURE 3. Gray scale plot of the density n = n(x; y; z) in three perpendicular planes (left) z = 0, (middle) y = 0, (right) x = 0.
Arrows correspond to the proton flow velocity ~up = ~u p (x; y; z) projected onto the given plane.
487
CONCLUDING REMARKS
around the planetary surface. The ring continues to form
until the end of the simulation at t = 20 Ω pi1 . The current
sheath is initially formed in northern hemisphere slowly
moving to the equatorial plane. The current sheath makes
an angle of roughly 10 o with respect to the equatorial
plane. This changes in the latter stage of the simulation
and the current sheath moves down towards the equator.
We have also noticed formation of sites with relatively
negligible particle flow in the magnetotail (vanishing ~u p
on left panel of Figure 3).
Small planets with small magnetospheres relative to
the scales given by the plasma parameters (like Mercury)
provide an opportunity to study the interaction between
the solar wind and a real magnetosphere using a fully kinetic model. The size of the simulation box could easily
be doubled in all three dimensions and further doubled
in two chosen dimensions with respect to sizes we have
used. This makes global three dimensional kinetic simulations of a magnetosphere of small planets possible such
that we can study the influence of all parameters involved
with the formation of the magnetosphere.
Due to limited computing resources we have not
reached a stationary state in our simulation (cf. [9, 5]).
The stationarity, the use of more realistic parameters, and
formation of the shocklet structure will be the subject of
future work.
In this paper we discuss results of a 3-D numerical model
of a small planet (R = 8:5 L in ) with a weak dipole magnetic field (M = 1:5 10 4 Bsw L3in =µ0 ) embeded in a solar
wind plasma flow with bulk speed v sw = 3:0 vA . These
values can be compared with the corresponding parameters describing the plasma environment of the solar system planets in Table 1.
The main results are shown on Figures 1–3: The simulation clearly shows important kinetic effects like the
differences between a turbulent quasi-parallel part and a
more stable quasi-perpendicular part of the bow-shock
region (Figures 1–3, left panels).
The quasi-perpendicular part of the bow-shock has a
shocklet structure. Shimazu [5] suggested that this structure forms due to finite Larmor radius effects of protons. Shimazu [5] used a 3-D hybrid code and showed
that the shocklet bow-shock structure exists for an (unmagnetized) obstacle with a radius R = 1:6 r L (where
rL = vsw Ω pi1 ), but does not exist for an obstacle with
a radius R = 6:4 rL . In our case the effective radius of
the magnetized obstacle is about R 3 r L and the result
is consistent with the predictions and results of Shimazu
[5]. The formation of the shocklet structure deserves further study. There is still an open question on whether
this effect is a transient [8], caused by nonstationarity
of the numerical experiment, or whether it is a consequence of the kinetic processes around a relatively small
obstacle. One may also ask whether the spatial resolution dx = 0:8 Lin (and worse, cf. [5, 4]) is sufficient to resolve the shock structure with a typical shock thickness
of about Lin .
The spatial size of the magnetosphere with respect to
the radius of the obstacle R roughly corresponds to the
proportions of the Hermean magnetosphere. Hence the
magnetic dipole field is hidden well inside the obstacle
forming a very thin magnetosphere. The magnetopause
is more apparent in the northern hemisphere on the middle panels and its position R MP; model 1:4 R is in a good
agreement with predicted theoretical value R MP; MHD 1:7 R. We have noticed the early stage of the formation
of northern and southern polar cusps. The magnetotail is
not yet formed due to the limited time span of the simulation. However we have seen regions with bi-directional
proton flows together with bi-directional magnetic field
in the simulation. This is a sign of the existence of reconnection sites.
The initial bulk speed v sw of the solar wind plasma is
set to 3:0 vA in entire volume of the simulation box. This
flushes out particles initially located behind the planet
and produces a wake with very low particle density. This
wake begins to fill by the anti-clockwise injection of protons in the equatorial plane thus forming a temporary belt
ACKNOWLEDGMENTS
Authors acknowledge support of the grant ESA
PRODEX 14529/00/NL/SFe and NSF International
Grant INT-0010111. Authors thanks to N. Meyer
and F. Pantellini who had useful suggestions to the
manuscript.
REFERENCES
1.
2.
Brecht, S. H., Geophys. Res. Lett., 17, 1243–1246 (1990).
Brecht, S. H., and Ferrante, J. R., J. Geophys. Res., 96,
11209–11220 (1991).
3. Brecht, S. H., Ferrante, J. R., and Luhmann, J. G., J.
Geophys. Res., 98, 1345–1357 (1993).
4. Brecht, S. H., J. Geophys. Res., 102, 4743–4750 (1997).
5. Shimazu, H., J. Geophys. Res., 106, 8333–8342 (2001).
6. Dubinin, E. M., Sauer, K., Baumgärtel, K., and Srivastava,
K., Earth Planets Space, 50, 279–287 (1998).
7. Dubinin, E. M., Sauer, K., Lundin, R., Baumgärtel, K.,
and Bogdanov, A., Geophys. Res. Lett., 23, 785–788
(1996).
8. Omidi, N., and Winske, D., J. Geophys. Res., 92,
13,409–13,426 (1987).
9. Omidi, N., and Winske, D., J. Geophys. Res., 95,
2281–2300 (1990).
10. Matthews, A., J. Comput. Phys., 112, 102–116 (1994).
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