Propagation of a Toroidal Magnetic Cloud through the
Inner Heliosphere
Eugene Romashets* and Marek Vandas_
*
Institute of Terrestrial Magnetism, Ionosphere, and Radio Wave Propagation of Academy of Sciences, Troitsk,
Moscow Region, 142190, Russia, Email: [email protected]
_
Astronomical Institute, Academy of Sciences, Bocni II 1401, 14131 Praha 4, Czech Republic,
Email:[email protected]
Abstract. An analytical solution for a potential magnetic field with arbitrary intensity around a toroidal magnetic cloud
has been found. The background external field may have a gradient. The solution is used for calculation of magnetic
cloud propagation. Obtained velocity profiles show a good agreement with in situ observations near the Earth’s orbit.
INTRODUCTION
Φ0 = B0 x −
The idea that magnetically isolated bodies, also
called “magnetic clouds”, may exist in inerplanetary
space has been considered before[1-5]. There are many
reasons for these structures to retain common field
lines with the Sun [6,7]. On the other hand, the
possibility for such objects to be isolated from the
solar magnetic field may not excluded. In order to
verify if this hypothesis is consistent with observations
of magnetic clouds, including time delays of their
launches and arrivals to the Earth’s orbit, their
velocities observed by coronographs close to the Sun
and at 1 AU, we calculated dynamics of isolated
bodies of toroidal shape and found their velocity
profiles for 5 R < R < 220 R , where RS is solar
radius.
S
B1 2
x − y2) ,
(
2L
(1)
which yields the magnetic field components,
as follows:
Bx = B0 −
B1
B
x , By = 1 y , Bz = 0
L
L
(2)
Φ0 is a harmonic function, i.e., ∆Φ0 = 0 , which
ensures the condition divB = 0 is fulfilled. A toroid
with the major radius R0 and the minor radius r0 is
inserted into this field. The intrinsic system of the
toroid will be described in the toroidal coordinates µ ,
η , and ϕ :
S
x=
CALCULATION
a sinh µ cos ϕ
cosh µ − cos η
,
y=
a sinh µ sin ϕ
cosh µ − cos η
a sin η
, z = cosh µ − cos η
, (3)
The parameter a is defined as a = R20 − r 20 , and
the surface of the toroid is given by µ = µ 0 , where
cosh µ 0 = R0 / r 0 . The expression for the scalar
potential Φ0 is arranged as a sum of the harmonic
It is assumed that the main (homogeneous)
→
magnetic field B 0 is directed along the x axis. A small
magnetic field gradient B1/L along the x axis is added
to the main field. The scalar magnetic potential of the
resulting field is
functions x , and x 2 − y 2 . These functions can be
expressed as a sum of toroidal harmonics :
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
180
5/2
2
2 a
∑ Q (cosh µ ) cos nη cos 2ϕ .
x − y = cosh µ − cos η
3π
3/ 2
∞
a
1
x = cosh µ − cos η 2
∑ Qn −1 / 2 cosh µ cos nη cos ϕ (4)
n = −∞
∞
2
2
n = −∞
2
n −1 / 2
(
π
)
Qmn −1/ 2 are the Legendre functions of the second
kind. A way to obtain these relationships is described
in [8]. Now we shall look for the potential magnetic
field which (i) has normal components to the surface
of the toroid equal to zero and which (ii) tends to the
undisturbed field (2) at larger distances from the
toroid. That is, we add to the scalar potential (1),
expressed in the toroidal harmonics, additional toroidal
harmonic functions, for which the additional magnetic
field becomes negligible when µ → 0 , and for which
the total magnetic field component Bµ = 0 for µ = µ 0 .
The scalar magnetic potential Φ1 of the total field is
2
F = cosh µ − cos η
1
{
∞
[
0
n=0
n
n −1 / 2
a
− B ∑ε Q
3L
∞
1
n=0
n
[
a
(5)
×
π
(cosh µ ) − a1n P (cosh µ )] cos nη cos ϕ −
(cosh µ ) − a P (cosh µ )] cos nη cos 2ϕ ,
1
1
× B ∑ε Q
3/ 2
n −1 / 2
2
n −1 / 2
2
2
n
n −1 / 2

where the coefficients a1n and a 2n were formally
added and they will be selected in order for the two
above mentioned conditions, (i) and (ii), to be
m
satisfied. Pn−1 / 2 are the Legendre functions of the first
kind, ε n = 1 for n = 0 and ε n = 2 for n>0. Condition
(ii) is fulfilled by the selection of Legendre functions
of the first kind [8]. Condition (ii) implies that Bµ = 0
at µ = µ0 for all η and ϕ .
In Figures 1-3 magnetic field magnitude contours
and field lines for the case R0 / r 0 = 5 are shown.
FIGURE 1. Contours of modified by the insertion of a
toroid magnetic field’s magnitude in xy plane, z = 0 (a),
and z = r 0 (b).
Once the magnetic field around the toroid is found,
the diamagnetic force acting on it along the x axis can
be calculated:
F =−
x
= −a
2
1
∫ ( B + B ) e dS = F + F =
8π
sinh µ
∫ ∫
8π
0
π
2π
−π
0
2
2
η
ϕ
(B + B ) (
2
2
η
ϕ
x ⋅
xη
In Figure 4 one can see radial velocities of toroidal
magnetic cloud as a function of the radial distance
from the Sun. The diamagnetic force given by (6) was
used for calculation. Also gravity and the drag force of
solar wind plasma were taken into account, in a way
similar to that used in [8].
(6)
xϕ
cosh µ cos η − 1) cos ϕ
0
(cosh µ − cos η)
3
dϕdη
0
181
FIGURE 4. Velocity profiles of a toroidal magnetic cloud
for the following initial velocities close to the Sun:100 km s-1
(solid line), 400 km s-1 (dashed line), and 800 km s-1 (dotted
line). The dashed-dotted line shows the ambient solar wind
velocity.
FIGURE 2. Field lines for the case of initially non-uniform
field. B1 = B0 4 , L = 2 R0 .
ACKNOWLEDGEMENTS
This work was supported by EU/INTAS/ESA grant
99-00727 and by AV CR project S1003006.
REFERENCES
1. Klein, L. W., and Burlaga, L. F., J. Geophys. Res., 87,
613-624 (1982).
FIGURE 3. Field lines for the case of initially non-uniform
field. Side view.
2. Ivanov, K. G., and Harshiladze, A. F., Solar. Phys., 98,
379-386 (1985).
CONCLUSIONS
3.
Magnetic field modification caused by the insertion
of toroidal body with super-conductive walls into
current free medium was calculated. This result can be
applied for space and laboratory plasmas.
Ivanov, K. G., Harshiladze, A. F., and Romashets, E. P.,
Solar. Phys., 143, 365-372 (1993).
4. Vandas, M., et al., J. Geophys. Res., 98, 11467-11475
(1993).
5. Vandas, M., et al., J. Geophys. Res., 98, 21061-21069
(1993).
It is shown that the maximum field increase in the
case of a sub-sonic cloud is of the order of 2 times in
magnitude.
6. Kahler, S. W., and Reames, D. W., J. Geophys. Res., 96,
9419-9424 (1991).
The obtained formulas for the diamagnetic force
can be used for calculation of velocity profiles of
toroidal clouds from the Sun to the Earth’s orbit.
7.
Chen, J., and Garren, D. A., Geophys. Res. Lett., 20,
2319-2322 (1993).
8. Romashets, E. P., and Vandas, M., J. Geophys. Res., 106,
10615-10624 (2001).
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