The Limitation of Bessel Functions for ICME Modeling
C. T. Russell and T. Mulligan
IGPP and ESS, University of California, Los Angeles, CA 90095, USA
Abstract. Most inversions of the structure of magnetic ropes in ICMEs have assumed that the rope can be
approximated as a force-free structure in the Taylor state in which the current is not only parallel to the
magnetic field but everywhere has the same constant of proportionality to the field strength. The solution of
this problem is a magnetic field that is describable by Bessel functions: Jo for the axial component and J1 for
the poloidal component. The Taylor state approximation has a maximum twist that is exceeded by about half
the observed flux ropes. Moreover, many flux ropes are not force-free. The vast majority of non-force-free
ropes have an excess of pressure pushing outward. Thus these ropes are either expanding or are balanced by
non-magnetic forces. Thus while the Bessel function approach may be useful for determining the orientation
of rope axes, its limited ability to correctly measure twist and its inability to assess any magnetic force
imbalance mitigate against its usefulness in studies of ICME genesis and evolution.
B  1
(2)
 + ∇ • (B B) = 0
−∇ 
 2µ  µ
INTRODUCTION
2
Most work on inverting the structure of ICMEs
and especially on their most simple form, the
magnetic cloud, has been performed using a Bessel
function [1,2]. The justification for this is that in a
Tokamak the plasma often relaxes to a state in which
the parallel current per unit magnetic flux is constant
across the diameter of the machine. This so-called
Taylor state [3] is thought to occur due to
reconnection and turbulence in the plasma. While
often true in a fusion plasma, this need not be true in
a space plasma. In this paper we first review the
physics of force-free magnetic flux ropes and show
the solution for the Taylor state. We then show the
greater flexibility afforded by a non-force-free and
non-Taylor-state solution. We apply this solution to
flux ropes encountered by Pioneer Venus at 0.72 AU
and evaluate the applicability of the Taylor state and
force-free assumptions.
o
Thus in a rope the outward force of the magnetic
pressure is balanced by the inward force of the
twisted magnetic field.
A force-free rope has no current perpendicular to B.
Thus we may write
∇ x B = µoJ = αB
(3)
There is no constraint on the functional form of α. It
may vary with distance from the center of the rope. If
it is constant, then we take the curl of (3) to obtain
∇(∇•B) - ∇2B = α(∇ x B)
∇2B = -α(∇ x B)
(4)
The solutions of this second order equation are the
zeroth and first-order Bessel functions [4]
PHYSICS OF FORCE-FREE
FLUX ROPE
Ba = Bo Jo (αr)
Bp = Bo J1 (αr)
Br = 0
In a force-free flux rope there is no net magnetic
stress
JxB=0
o
(5)
If α is constant, the rope is said to be in the Taylor
state
(1)
We can rewrite (1) as the sum of a magnetic pressure
gradient force and the divergence of the magnetic
stress tensor.
In the Taylor state
J  α  is constant.
= 
B  µο 
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
125
Magnetic Flux Rope with Bessel function field and currents
Twist: 1.34 µA/Wb
1.0
Jo(R)
0.6
0.4
0.2
r=0.00
r=0.61
r=1.22
r=1.84
12
Axis of Rope
Field Strength
0.8
8
4
J1(R)
0
0
4
-0.2
0
0.4
0.8
1.2
1.6
2.0
2.4
4
0
Y Radius
2.8
0
-4
Bessel Function Argument R
FIGURE 1. Magnetic field of a Taylor state rope.
are determined, there is very little flexibility in fitting
the field variation. If the field has very little twist,
one can assume that the Bessel function solution has
a limited range, e.g. 0 to 0.4 in Figure 1. If it is very
twisted around the axis, then the fit can be extended
out to 2.4. However, the functional variation with
rope radius is not variable. Another way to visualize
this is given in Figure 2 where field lines in a Taylorstate, Bessel function flux rope are drawn. The
solutions available to the modeler consist only of
choosing the twist of the outer most field line. Once
chosen there is a fixed path from the central (straight)
field line to the most twisted one.
TAYLOR-STATE FORCE-FREE
ROPE: THE BESSEL FUNCTION
SOLUTION
Figure 1 shows the J0 and J1 Bessel functions that
represent the variations of the magnetic field with R
in a Taylor-state, force-free magnetic flux rope. For a
typical flux rope seen in the vicinity of the Earth or
Venus, the α-parameter would be about 1.34 µA/Wb.
The field strength would be about 20 nT in the center
of the rope and the diameter of the rope would be
about 0.2 AU. Since there is only one Bessel function
solution, once the central field strength and the radius
n=3
n=2
n=1
1.0
Magnetic Flux Ropes with Exponential Field
x=0.00
x=0.33
x=0.67
x=1.00
a=0.5
a=0.5
0.6
1
0.4
1
2
2
x=0.00
x=0.33
x=0.33
x=0.67
x=0.67
x=1.00
x=1.00
6
Axis of Rope
6
Axis of Rope
Poloidal Field
y=1-exp(-(x/a)n)
a=0.5; n=2
a=0.5; n=1
a=0.5
0.8
X Radius
-4
FIGURE 2. A Taylor state flux rope.
4
2
4
2
0.2
2
Y Radius
2
0
-2
X Radius
0.5
0.2
0.5
0
4
2
0
0
0.2
0.4
0.6
0.8
1.0 0
0.2
0.4
0.6
0.8
1.0 0
0.2
0.4
0.6
0.8
1.0
X
FIGURE 3. Poloidal and axial fields in a non-force-free
solution for varying exponents and scales.
126
4
2
0
2
0
Y Radius
Distance from Center
0
X Radius
x=0.00
x=0.33
x=0.33
x=0.67
x=0.67
x=1.00
x=1.00
6
Axis of Rope
0.5
6
-2
a=1; n=2
x=0.00
x=0.33
x=0.33
x=0.67
x=0.67
x=1.00
x=1.00
1
Axis of Rope
1
0.4
2
-2
a=1; n=1
1
0.6
0
Y Radius
0
-2
a=2
a=2
a=2
0.8
2
2
1.0
Axial Field
y=exp(-(x/a)n)
0
0
1
0
2
2
-2
-2
0
X Radius
0
Y Radius
2
-2
-2
0
X Radius
FIGURE 4. Different flux ropes with varying n and a.
18
10
16
8
14
Counts
Counts
12
6
4
10
8
6
4
2
2
0
0
2
4
8
16
32
0
-0.5
64
0
0.5
Magnetic Flux [TWb]
1.0
1.5
2.0
2.5
I⊥ /I ||
FIGURE 5. Occurrence rate of different flux content
in PVO ropes.
FIGURE 6. Occurrence rate of ropes with varying
perpendicular to parallel current strengths.
NON-FORCE FREE CYLINDRICALLY SYMMETRIC MODEL
In practice we have used an “expansion” factor δ
to account for asymmetry in the time profile, most
probably due to the expansion of the rope as it moves
across the observer.
Both models also solve for the orientation of the
rope, clock and cone angles, and the impact
parameter, the distance of the satellite from the
central axis of the rope at closest approach.
Figure 3 shows the magnetic field profiles for the
poloidal and axial components of the field for
different exponents, n, and different scale lengths of
the exponentials, a. In this model the poloidal and
axial fields are fit independently and can have
different scale lengths and different exponents. This
flexibility allows the rope to be non-force-free and if
force-free to not be in the Taylor state. Examples of
ropes that can be made are shown in Figure 4.
We express the poloidal and axial fields in terms
of components
Ba = B1 [exp {-(r/σal)m}]
Bp = Bo [1-exp {-(r/σpl)n}]
that increase and decrease from the center of the rope
respectively as exponentials raised to powers n, and
m, each with their own independent field strength,
exponential scale length and exponent [5]. Here l is
the radius of the outside edge of the rope. The two
parameters of the Bessel function fit have been
replaced by six in the exponential fit.
14
2.0
12
1.5
8
I ⊥/I ||
Counts
10
1.0
6
4
0.5
correlation:
0.67567
2
0
0
0
0.2
0.4
0.6
∆ alpha/<alpha>
0.8
1.0
0
2.0
4.0
6.0
8.0
10.0
Twist [µ A/Wb]
FIGURE 7. Distribution of ropes with varying fractional
alpha change.
127
FIGURE 8. Correlation of non-force-free character
with the magnitude of alpha.
net outward force
net inward force
Nov 24, 1979
2.0
Twist
[µA/Wb]
high
8
I⊥ / I||
1.5
6
1.0
4
0.5
2
0
-30
low
-20
-10
0
10
20
30
0
Net Force
FIGURE 9. Ratio of perpendicular to parallel current strength versus net force in arbitrary units.
These ropes all have a nearly force-free character.
The majority of ropes that have a net force, have an
outward force and that net outward force is greater,
the larger is the non-force-free fraction of the current.
The alpha parameter is color coded. Those ropes with
high alpha are generally also not force-free but there
seems to be no correlation of net force and alpha.
APPLICATION TO PIONEER VENUS
DATA
Pioneer Venus spent an entire solar cycle in orbit
about Venus at 0.7 AU from the Sun. Its 24-hour
orbit took it well into the solar wind much of the time
making it very well suited for ICME studies. We
have applied the non-force-free model to the ICMEs
identified during that time. Figure 5 shows the
inferred flux content of these ropes making allowance
for the (separately) inferred average oval crosssection of these ropes. The median flux content is
about 1013 Wb. Figure 6 shows the number of ropes
with varying ratios of integrated perpendicular to
parallel current where the absolute value of the
current has been used in the ratio. Thus these ropes
could have zero net force but have non-force-free
regions within them. About half of the ropes have
significant non-force-free regions. Alpha gives the
parallel current per unit magnetic flux and is constant
in the Taylor state. Most of the ropes have at most
small deviations from the Taylor state. This is
illustrated in Figure 7 that shows the distribution of
the peak to minimum variation in alpha normalized
by the average alpha. Examination of the ropes with
the largest variations in alpha indicate that some of
these may be due to poor fits to the rope structure. If
we plot the quantification of the non-force-free nature
of the ropes I⊥/I‖ versus alpha or twist we obtain the
values shown in Figure 8. Clearly those ropes with
the strongest parallel currents (per unit field strength)
statistically are the ropes with the greatest non-forcefree character. The dashed line shows the alpha value
for a typical flux rope inferred from the Bessel
function fit. Since the Bessel function has no I⊥, only
the values near the base of this line could be
accurately portrayed with this model.
In Figure 9 we study the net force in the flux
ropes. A minority of ropes have net inward force.
SUMMARY
The Bessel function model of a cylindrically
symmetric flux rope is adequate only for force-free
ropes in the Taylor state. Magnetic clouds seen by
Pioneer Venus in the 1980’s were often not force-free
and not in the Taylor state. When ropes had an
imbalance of force it was generally outward. Ropes
are in general not cylindrically symmetric. We need
multiple observations of flux ropes at separations of
0.1 to 0.4 AU to better understand their true
geometry.
ACKNOWLEDGMENTS
This work was supported by the National
Aeronautics and Space Administration under research
grants NAG5-10834 and NAG5-11583.
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