A self-similar solution of expanding cylindrical flux ropes for
any polytropic index value
Hironori Shimazu † and Marek Vandas Applied Research and Standards Division, Communications Research Laboratory, Koganei, Tokyo 184-8795
Japan
†
Department of Earth and Space Sciences, Box 351310, University of Washington, Seattle, WA 98195-1310 USA
Astronomical Institute, Academy of Sciences, Boční II 1401, 141 31 Praha 4, Czech Republic
Abstract. We found a new class of solutions for MHD equations that satisfies the condition that cylindrical flux ropes can
expand self-similarly even when the polytropic index γ is larger than 1. We achieved this by including the effects of elongation
along the symmetry axis as well as radial expansion and assuming that the radial expansion rate is the same as the elongation
rate.
INTRODUCTION
However, coronal mass ejections, or flux ropes, expand in MHD simulations even when γ is larger than
1 [12] [13] [14] [15] [16]. Moreover, theoretical models of [17] and [18] showed expanding flux ropes with
γ 1. Recent comprehensive analysis by [19] has concluded that an observed negative correlation between the
temperature and the density of electrons, on which the
derivation of γ was based, cannot be a measure of γ .
It was shown that single-point measurements cannot be
used to determine γ value [20] [21]. To determine γ value
observationally continues to be a matter for debate [22]
[23].
This paper1 describes a new class of solutions for
MHD equations that satisfies the condition that cylindrical flux ropes can expand self-similarly even when γ
is larger than 1. We achieved this by including effects
of elongation along the symmetry axis, which is not included in the previous solutions. This new class of solutions puts new light to the very discussed problem on
the relation between γ and the expansion. Mainly it emphasizes a role of flux rope elongation for its expansion.
Recently a paper which deals with a similar topic to ours
was submitted by [25].
Interplanetary magnetic flux ropes are structures in
which magnetic field vectors rotate in one direction
through a large angle. The sun often ejects flux ropes
as coronal mass ejections (e.g., [1]). When flux ropes
reach the earth, they often cause a geomagnetic storm
(e.g., [2]). Observationally, they were first identified as
“magnetic clouds" in interplanetary magnetic field data
by [3].
While the flux rope propagates in interplanetary space,
it expands because the ambient pressure decreases with
its distance from the sun. The technique of converting time derivatives to self-similarity expansion parameters in MHD (magnetohydrodynamics) equations
was originally applied by [4] and [5] to the explosion of a supernova. The self-similar approach simplifies
time-dependent problems and makes them analytically
tractable. A class of self-similar solutions of the expanding solar corona was found by [6] in the spherical coordinates when the polytropic index γ is exactly 4 3. The
same approach was used by [7], and a theoretical MHD
model was presented describing the time-dependent expulsion of a three-dimensional coronal mass ejection.
Another class of self-similar solutions was found by
[8] and [9]. The solution showed that cylindrical flux
ropes expand self-similarly only when γ is less than 1
for this class of self-similar solutions. This formulation
was applied to real interplanetary flux ropes [10]. Observations of solar wind electrons in interplanetary magnetic flux ropes were interpreted as having γ less than 1,
and that is why flux ropes expand in interplanetary space
[11].
SOLUTION
The MHD equations are solved as follows:
∂ρ
∂t
1
∇ ρ v 0
This paper is a summary of a theoretical part of our paper [24].
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
737
(1)
ρ
∂v
∂t
ρ v ∇v ∇P
∂ Pρ γ ∂t
1
∇ B B
µ
v ∇Pρ γ 0
(2)
U
∂B
∇ v B
(4)
∂t
where t is time, ρ is mass density, P is pressure, µ is
permeability, v is velocity, and B is the magnetic field.
These MHD equations will be solved in cylindrical coordinates (r, θ , z) moving with a flux rope. The cylindrical
flux rope has its axis along z and an axial symmetry is
assumed, that is, the flux rope has a circular cross section
and the quantities do not depend on θ .
and
where
P KDy 2γ D
f
η f¼ 2
∂f
∂ξ
Bθ
(8)
U
and
η f ¼ 2 0
(13)
f
(23)
P KGy 3γ (24)
y3 3γ D¼ G¼ S1 χ y 1
3
K lny
D¼ G¼ S1 χ y 1
3γ
K
K y3 3γ η ∂ G ∂ ξ 3γ 3
ξ ∂ G ∂ η U
K lny η ∂ G ∂ ξ η f ¼ 2¼ 0
(22)
(γ 1)
(γ 1),
(25)
where D is also defined by (10) in this new solution.
The θ component of (2) is zero under the condition (18).
From the z component of (2), we get
where y is the evolution function of time. The f is the
generating function of η , satisfying
f
(21)
where G is a function of η and ξ .
From the r component of (2), U is expressed as
(11)
(12)
(20)
(10)
f ¼ 0
(18)
η f ¼ 21 2 y 2 Bz 2µ SD1 2 y 2 ρ G¼ η 1y 3 (7)
and
ry 1 0
vz ξ ẏ
(6)
2µ Sχ
and χ , S, and K are positive constants. The center dot
means the derivative by the time t, and the prime is the
derivative by the self-similar parameter η satisfying
η
for the simplest case. A new solution for (1), (3), and (4)
is given by
vr η ẏ
(19)
(9)
1)
(γ 1).
χ S lny (γ
This is consistent with the assumption that the radial
expansion rate is the same as the elongation rate. We will
solve (1) - (4) that satisfy
(5)
1 2Sy 2
1 2Sy 2
In this study we include effects of axial elongation (zdependence) as well as radial expansion. The vθ and Br
are assumed to be 0. An additional self-similar parameter
ξ is introduced by
ξ zy 1 (17)
Before showing the new solution, we will review
the previous solution. This solution assumes selfsimilarity and no dependence on z (one-dimension and
r-dependence only). Following the procedure described
in [9], the solutions for (1), (3), and (4) were
η f ¼ 21 2y 1 Bz 2µ SD1 2 y 2 ρ D¼ η 1 y 2 New solution with elongation
Previous self-similar solution without
elongation
Bθ
y2 2γ
2
K lny
K
(16)
Equation (15) can be regarded as an equation of motion
under potential U. When γ is larger than 1, U takes
a minimum value. Thus, flux ropes do not expand but
oscillate when γ is larger than 1.
(3)
and
vr η ẏ
2γ
χ S
(14)
From the r component of the equation of motion (2), y
satisfies
dU
ÿ (15)
dy
ξ ∂ G ∂ η 1)
(γ
(γ
1).
(26)
For (25) to agree with (26), the conditions
χ1
738
(27)
and
1 ∂G
η ∂η
Features of the new solution
1 ∂G
ξ ∂ξ
(28)
must be satisfied. From (28) we get
G a exp
cη 2
ξ 2 The most important effect of elongation of the axial
length appears in the azimuthal component of the magnetic field:
(29)
Bθ
where a and c are constants. Thus, the pressure distribution is expressed as
P Ka exp cr2
z y y 3γ 2
2
The ρ is given from (23) by
ρ
2ac exp
cr2
z2 y2 y 3 mK
y 3γ 3 2kc
Bθ
(30)
U
K
y3 3γ
3γ 3
K lny
(31)
(32)
1)
(γ 1).
(γ
(33)
This expression shows that U is a monotonically decreasing function of y. Thus, flux ropes obeying this solution
expand for any γ value.
Equation (15) with the given potential (33) can be
integrated yielding
2K y3 3γ c 1 2
1
dy dt 2K3 lny3γ c 1 2
1
where c1 is a constant. In case γ
this equation further, and get
y c1 t
t0 2
(γ 1)
(γ 1),
(34)
5
3, we can integrate
K c1
1 2 excluding elongation
(36)
y 2
including elongation
(37)
The difference of Bθ dependence on y comes from the
volume increase in the axial direction. This is the essential effect of the elongation. It was suggested the lack of a
three-dimensional effect as the reason the previous solution does not expand when γ is greater than 1 [18]. This
statement agrees with our solution.
We consider that a flux rope has a finite volume. The
radius r1 r0 y and the axial length l l 0 y are determined
by the initial radius r 0 and the initial length l 0 (at t t0 ), respectively. r1 is taken for Bz to be zero at the
boundary.
A flux rope model for interplanetary magnetic clouds
was constructed assuming that the total magnetic helicity is conserved by [26]. Our model agrees with their
model in the dependence on l of the magnetic field
strength, mass density, radius, and volume. Their results
also showed that a flux rope evolves self-similarly, which
agrees with our assumption.
The dynamics of flux ropes was investigated by [18].
His model starts with an equilibrium state of pressure between a flux rope and the ambient medium. He treated
propagation in interplanetary space by the change of
physical parameters of the ambient medium. In our
model, we do not assume an initial equilibrium as other
self-similar models do not assume. However, our model
can trace a flux rope evolution during propagation in interplanetary space by the decreases of the ambient pressure and density with time.
In our solution the magnetic flux is conserved at all
times. The magnetic energy decreases with increasing
time. In the models of [26] and [18], the magnetic energy
decreases as the flux rope expands. Our solution agrees
with their results.
In our solution the magnetic flux is conserved at all
times. The magnetic energy is expressed as
where k is Boltzmann’s constant and m is the average
mass of the solar wind particles.
Finally, we get
y 1
is modified into
For the density to be positive anywhere and to be zero in
infinitely distant regions, a and c must both be positive.
In this new solution we do not need the condition (14),
under which the density was positive or zero in the
previous solution. The temperature T is expressed as
T
(35)
where t0 is a constant; vr vz 0 everywhere at t t0 .
We took the plus sign assuming that y is a monotonically
increasing function of t. In our solution y 1 is finite at
t t0 , while in the previous solution y 1 is infinitely
large. Thus, an infinitely large magnetic field strength is
suppressed in our solution.
1
2µ
1π
yµ
l
r1
0
0
l0
0
B2θ
r0
0
B2z 2π rdrdz
2µ SD
η f ¼ 2η d η dξ (38)
The magnetic energy decreases with increasing y and
with increasing time. Our solution agrees with the models of [26] and [18], in which the magnetic energy decreases as the flux rope expands.
739
The r component of the Lorentz force is given by
1 µ ∇
B Br χ 1SD¼ y 5 ported by the JSPS overseas research fellowship. M. V.
was supported by project S1003006 from AV ČR.
(39)
The other components are 0. Thus, when χ 1 (27), a
force-free state is maintained at all times. Expansion is
caused by the pressure gradient force. The ambient and
internal pressures control the expansion. This point is
different from the solution of [9], in which the force-free
state occurred only for one specific time.
Our solution is different from the models of [26] and
[18] in maintaining the force-free state. In their models
the force-free state was not maintained. The difference
comes from the internal magnetic field configuration,
which may be caused by the curvature of the cylindrical
axis. If we consider a curved tube as a flux rope like [26]
or [18], it may be difficult to maintain the force-free state.
As with some other self-similar solutions, velocity
grows without limit as r or z increases. Although we
consider finite volume around the origin as a flux rope,
infinitely large velocity in infinitely distant regions (for
large r as well as for large z) seems strange. However,
in our opinion it is significant to show possible expansion
in the medium where γ is greater than 1 in the same
framework of self-similarity as shown in the previous
solution.
The relation between T and ρ is considered. From (32)
T decreases with time when γ is greater than 1. Thus T
and ρ show a positive correlation in this case. If γ is less
than 1, they show a negative correlation. If γ is equal to
1, T is constant. The relation between the correlation and
γ is the same as the previous solution.
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SUMMARY
15.
We found a new class of solutions for two-dimensional
MHD equations that satisfies the condition that a cylindrical flux rope can expand self-similarly for any γ value
by considering the effects of axial elongation as well as
radial expansion. This new class of solutions puts new
light to the very discussed problem on the relation between γ and the expansion. Mainly it emphasizes a role
of flux rope elongation for its expansion. This solution
showed that the flux rope expands maintaining a forcefree state.
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ACKNOWLEDGMENTS
24.
The authors thank Dr. Katsuhide Marubashi (Communications Research Laboratory, Japan), and Dr. Shin-ichi
Watari (Communications Research Laboratory, Japan)
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