The Astrophysical Journal, 563:L183–L186, 2001 December 20
䉷 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A.
EXPERIMENTAL DEMONSTRATION OF HOW STRAPPING FIELDS CAN INHIBIT
SOLAR PROMINENCE ERUPTIONS
J. Freddy Hansen and Paul M. Bellan
Department of Applied Physics, California Institute of Technology, MS 128-95, Pasadena, CA 91125
Received 2001 June 4; accepted 2001 November 16; published 2001 December 12
ABSTRACT
It has been conjectured that the eruption of a solar prominence can be inhibited if a much larger scale, arched
magnetic field straddles the prominence and effectively straps it down. We have demonstrated this effect in a
laboratory experiment where a vacuum strapping field acts on a scaled simulation of a solar prominence. The
required magnitude of the strapping field is in good agreement with a theoretical model that takes into account
the full three-dimensional magnetic topology.
Subject headings: methods: laboratory — MHD — plasmas — Sun: flares — Sun: magnetic fields —
Sun: prominences
two small blue cylinders in Fig. 1) creates an arched vacuum
magnetic field that protrudes into the vacuum chamber. This
vacuum field is 0.6 T at the footpoints, which are 8 cm apart.
Second, 10⫺8 kg of hydrogen gas is injected through orifices
at the vacuum field footpoints into the region where the prominence will form. Third, a 59 mF capacitor bank charged to
3 kV is connected across the footpoints, causing the hydrogen
gas cloud to break down and form a highly conducting plasma
(resistance of 0.02–0.1 Q). After breakdown, the discharging
capacitor ramps up a current through the plasma; this current
reaches a maximum of ∼40 kA at a time of 8 ms after the
breakdown. The plasma has a measured density of between
1019 and 1020 m⫺3 and a temperature of ∼5 eV, corresponding
to a low b (10⫺4 to 10⫺3) and a high Lundquist number
(103–104). The combination of low b, high Lundquist number,
and experimental time larger than the Alfvén time (∼0.05 ms
from a calculated Alfvén velocity of 2 # 10 6 m s⫺1 and assuming a typical length scale of 0.1 m) but shorter than the
resistive diffusive time (∼300 ms) produces a reasonable simulation of the solar parameter regime (Tandberg-Hanssen 1995;
Bellan & Hansen 1998).
The main diagnostic of the experiment consists of a gated,
intensified CCD camera, with a shutter speed of 10 ns. The
camera takes visual light snapshots of the prominence at prescribed times during its evolution. Figure 2 shows such a snapshot of a typical laboratory prominence. Application of various
filters to the camera has shown that the visual light plasma emission is mainly Ha. Although only one image is obtained per
discharge, the high reproducibility of the experiment allows visualization of the prominence evolution in time by assembling a
sequence of images obtained at successively delayed times from
multiple discharges. Figure 3 (left column) shows a typical temporal evolution sequence of the laboratory prominence. The
prominence initially consists of a smooth, arched current channel
that simply follows the applied vacuum magnetic field. However,
as the current increases, the minor radius of the prominence
decreases owing to the pinch effect, and on a finer scale, filaments
making up the channel start twisting around each other (not
visible at the camera resolution in Fig. 3). Magnetic hoop forces
(Shafranov 1966; Miyamoto 1976; Bateman 1978) cause the
major radius to expand, and the superposition of the current selfmagnetic field onto the original vacuum field causes the prominence to develop a helical shape. When viewing the prominence
at a right angle to the line between its footpoints, the helical
1. BACKGROUND
Solar prominences are large, arched plasma structures protruding from the surface of the Sun (Tandberg-Hanssen 1995).
The typical length scale of a solar prominence is 107–108 m.
The ratio of hydrodynamic to magnetic pressure, b p
2m 0 p/B 2, is small, on the order of 10⫺3 to 10⫺1, so that magnetic
forces dominate hydrodynamic forces. The ratio of the diffusive
timescale to the convective timescale (i.e., the Lundquist number, a measure of how well a magnetic field is frozen to the
plasma) is large, reportedly as high as 1014 (Mikić, Barnes, &
Schnack 1988; Mikić & Linker 1994). Being magnetically
dominated, a prominence equilibrium is in a “force-free” state
(Nakagawa 1971), where J p aB so that the magnetic force
J ⴛ B vanishes and where the scalar a p a(r) may be nonuniform.
Prominences occasionally lose equilibrium and erupt, spawning magnetic clouds that travel outward from the Sun while
continuously expanding (Burlaga 1988; Rust 1994). Magnetic
clouds intercepting Earth’s magnetosphere have been observed
to be in force-free states (Burlaga 1988; Rust 1994). Certain
laboratory devices, e.g., reversed-field theta pinches and spheromaks, are also characterized by J p a(r)B force-free states
(Bellan 2000) and so have physics similar to prominences but
with different boundary conditions. We have taken advantage of
this similarity to create laboratory simulations of prominences
by combining spheromak technology with boundary conditions
appropriate to prominences (Bellan & Hansen 1998).
Note that this Letter focuses on the mechanism of the erupting prominence, not on a possible associated coronal mass
ejection (CME). A three-part CME (halo, void region, and
underlying prominence) is often, but not always, present in a
prominence eruption. However, it is not known whether a CME
is required for a prominence to erupt or, vice versa, whether
an erupting prominence is required for a CME.
2. EXPERIMENTAL SETUP
Our laboratory prominences are produced by a magnetized
plasma gun mounted nearly flush with the end dome of a large
(2 m long, 1.4 m diameter) vacuum chamber (Fig. 1). The
metal end dome simulates the flux-conserving surface of the
Sun, and since all other walls are far away, the configuration
approximates an infinite half-space. The plasma gun makes a
prominence in three steps. First, an electromagnet (shown as
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SOLAR PROMINENCE ERUPTIONS
Vol. 563
Fig. 2.—Example of a laboratory prominence
Fig. 1.—Experimental setup. Insert shows enlargement of magnetized
plasma gun. The prominence (red) forms between the two D-shaped electrodes.
shape makes the prominence look as if it has a sharp, cusplike
central dip. Experiments with arched vacuum magnetic fields of
lower strength produce prominences with several such dips. One
dip always appears, and owing to symmetry, this dip is in the
middle of the prominence. (From symmetry arguments alone,
the opposite of a dip, namely, a bulge, could appear in the middle,
but this has never been observed in our experiment and is presumed to be energetically unfavorable.)
The new result that we report here is the demonstration of
control over the prominence expansion by an externally produced “strapping” magnetic field, which simulates the largescale vacuum arcade fields believed to straddle actual solar
prominences and inhibit their eruption (Chen 1996; Antiochos
1998; Amari & Luciani 1999). The strapping field in our experiment is generated by a pair of large coils, located on either
side of the plasma gun, just behind the vacuum chamber end
dome (Fig. 1). If one follows a line that is perpendicular to the
electrode plane and goes through the center of the solar gun,
the strapping field starts at approximately zero field strength
in the electrode plane and then rises to a peak amplitude some
distance (8 cm in our case) above the electrode plane. Like the
prominence vacuum magnetic field, the strapping field operates
on a much slower timescale (milliseconds) than the current
ramp-up time (microseconds) and can thus be regarded as effectively constant.
point of the prominence, i.e., the longest distance from the
origin to any plasma-vacuum boundary, was measured in a
similar way. For times early in the evolution, before the prominence takes on a helical shape and develops a dip, h p y. We
also measured the filament radius, b. Each distance, measured
in pixels, was converted to actual length distance by scaling
to known reference distances in the image.
Since both our experiment and many actual prominences (see
the photos in Vrsnak, Ruzdjak, & Rompolt 1991; TandbergHanssen 1995) have an approximately semicircular geometry,
we assume semicircular geometry when we construct a model
for the prominence. Complicating factors exist that would have
to be taken into account if one were to model higher aspect
ratio (not semicircular) prominences. In particular, these high
aspect ratio prominences often display connecting filamentary
structures—“barbs”—to the photosphere that increase the topological complexity. However, each segment of such a complex
prominence would be more accurately treated as a semicircle
than as a line current (see the photos in Tandberg-Hanssen 1995).
For each value of the strapping magnetic field, the observed
expansion velocity dh/dt is nearly constant. It is interesting to
note that constant velocities are also observed for solar prominences (Kahler et al. 1988; Tandberg-Hanssen 1995). The expansion velocity is plotted in Figure 4 (bottom panel) as a function of strapping field strength (peak strapping field strength on
a line that is perpendicular to the electrode plane and goes
through the origin). It is clear that the expansion velocity is
3. RESULTS
The experimental effect of the strapping field is to alter the
prominence shape and to reduce the prominence expansion rate
or even inhibit expansion altogether. Prominence expansion
was quantified as follows. First, a reference point (origin) was
defined in each image as the midpoint between the two orifices
through which gas is injected. Next, the central, cusplike dip
was identified and the distance h (indicated in Fig. 2) from the
origin to the upper plasma-vacuum boundary (as apparent with
a 5%–95% brightness scaling of the image) at the dip was
determined. The distance y from the origin to the “highest”
Fig. 3.—Evolution sequence of a laboratory prominence for various strapping field strengths.
No. 2, 2001
HANSEN & BELLAN
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Fig. 5.—Schematic of the prominence, the electrode plane, and “image”
prominence. In the calculation, the envelope has been chosen to touch y and
h ⫺ 2b. To clarify the sense of the plasma writhing, the envelope has a slightly
reduced minor radius in the schematic.
1. The energy associated with the toroidal magnetic flux
(relative to the vacuum field energy) is
Wtor p
Fig. 4.—Prominence expansion velocity (bottom) and peak toroidal current
(top) as a function of strapping field strength.
reduced with increasing strapping field strength. The rate reduction is approximately proportional to the strength of the applied strapping field. Note that for sufficiently large strapping
fields, 103 mT, the expansion is completely halted. Furthermore,
if a negative strapping field is applied (i.e., a strapping field of
reverse polarity), the expansion velocity exceeds that obtained
with no applied strapping field at all.
Figure 4 (top panel) also shows the peak plasma current as
a function of strapping field, which provides an independent
measurement of when the plasma is fully strapped down. As
the plasma expands, its inductance increases. Thus, a plasma
that is strapped down has lower inductance, which in turn
corresponds to a larger peak current (as the energy 12 CV 2 originally stored in the capacitor is converted to inductive energy
1
2
2 LI ). The current measurements of Figure 4 (top panel) indicate that the plasma is fully strapped down at about 73 mT
(since the peak current does not increase for larger values of
the strapping field).
4. DISCUSSION
Figure 5 shows the topology of the prominence (red) and a
fictitious “image” prominence (green) and their writhing around
a fictitious toroidal envelope (light blue). The three-dimensional
image current is a generalization of similar image currents previously used in two-dimensional solar prominence models (e.g.,
van Tend & Kuperus 1978; Martens & Kuin 1989; Lin & Forbes
2000). In the situation where the expansion is halted, the prominence and its image are in an equilibrium resulting from the
balancing of three distinct magnetic forces acting to change the
envelope major radius R. The basis of these three forces can be
seen as follows (Shafranov 1966; Miyamoto 1976; Bateman
1978):
冕
Bf2 ⫺ Bf2 vac
2pR dS,
2m 0
(1)
where dS is the envelope cross-sectional area. A virtual displacement along the major radius direction gives the tension
along the magnetic field lines in the toroidal direction, Bf, tending
to decrease R:
Ftension p ⫺
⭸Wtor
p 2p
⭸R
冕
1
(Bf2 vac ⫺ Bf2 )dS.
2m 0
(2)
2. The energy associated with the toroidal current is
Wpol p
1 2
(LItor ) 2
f2
LItor p
p pol ,
2
2L
2L
(3)
where L is the circuit inductance associated with the toroidal
current. This circuit inductance can be written as L p
m 0 R [ln (8R/a) ⫺ 2 ⫹ li /2], with a the envelope minor radius
and li the internal inductance per unit length (Shafranov 1966;
Miyamoto 1976; Bateman 1978). A virtual displacement, holding
the poloidal flux fpol constant, along the major radius direction
gives the poloidal hoop force associated with the toroidal current,
Itor, tending to increase R:
Fhoop p ⫺
⭸Wpol
f 2 ⭸L
1 2 ⭸L
p pol2
p Itor
.
⭸R
2L ⭸R
2
⭸R
(4)
3. The force due to the interaction of the strapping field,
Bs, and toroidal current, Itor, tending to increase or decrease R
depending on the polarity of the strapping field, is
Fstrapping p
冕
ˆ
Jtor ⴛ Bs 2pR dS p 2pRItor Bs R.
(5)
Thus, the force-balance equation is (Shafranov 1966; Miyamoto
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SOLAR PROMINENCE ERUPTIONS
1976; Bateman 1978)
2pRItor Bs p 2p
冕
1
1 2 ⭸L
2
(Bfvac
⫺ Bf2 )dS ⫹ Itor
.
2m 0
2
⭸R
(6)
The poloidal current can be determined by comparing Figure 2
with Figure 5. The experimental data in Figure 2 shows that the
current channel spirals around the envelope axis twice on its
path from one electrode to the other, or four times around the
envelope axis when we consider the image current (Fig. 5, green)
in the electrode plane. Thus, the poloidal current is Ipol p
NItor, where the number of poloidal turns is N p 4, and so the
associated toroidal magnetic field due to prominence current is
Bf p m 0 NItor /2pR. Integrating Bf over the entire envelope cross
section, pa 2, gives the strapping field required for force balance:
Bs p
1
2m 0 RItor
⫹
冕
(
2
Bfvac
dS ⫺
m 0 N 2Itor a 2
8pR 3
)
m 0 Itor
8R
li
ln
⫺1⫹ .
4pR
a
2
(7)
Vol. 563
Using the measured values of h p 51 Ⳳ 2 mm, y p 62 Ⳳ
2 mm, 2b p 26 Ⳳ 1 mm [giving R p (y ⫹ h ⫺ 2b) /2 p 43
mm and a p (y ⫺ h ⫹ 2b)/2 p 19 mm], Itor p 47.3 Ⳳ 2.0 kA,
and a measured average vacuum toroidal field over the envelope
of Bfvac p 0.12 Ⳳ 0.02 T, we can now calculate the required
strapping field for force balance, provided we estimate the internal inductance li. For this inductance we set li p 12 , corresponding to a uniform distribution of toroidal current over
the cross section of the envelope, a fair assumption considering
that 2b ≈ a (extreme values of li would be 0 or 1). Using these
measured values, the strapping field required for force balance
is calculated to be Bs p 82 mT. Uncertainties are Ⳳ5 mT from
measurement errors and an additional Ⳳ27 mT from extreme
values of li. This calculated value corresponds reasonably well
to the measured 73–103 mT used to strap down the prominence
in the actual experiment.
We conclude that vacuum strapping fields can inhibit the
eruption of solar prominences and that force-balance arguments
of the type presented here give a reasonable quantitative description of strapping field effects on solar prominences.
Supported by US Department of Energy grant DE-FG0397ER54438.
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