The magnetohydrodynamic kink instability is observed and identified
experimentally as a poloidal flux amplification mechanism for coaxial gun
spheromak formation. Plasmas in this experiment fall into three distinct
regimes which depend on the peak gun current to magnetic flux ratio, with (I)
low values resulting in a straight plasma column with helical magnetic field,
(II) intermediate values leading to kinking of the column axis, and (III) high
values leading immediately to a detached plasma. Onset of column kinking agrees
quantitatively with the Kruskal-Shafranov limit, and the kink acts as a dynamo
which converts toroidal to poloidal flux. Regime II clearly leads to both
poloidal flux amplification and the development of a spheromak configuration.Comment: accepted for publication in Physical Review Letter

B. Vrs̆nak

J. P. Freidberg

L. Lindberg

M. Tokman

P. M. Bellan

P. M. Bellan

S. C. Hsu

T. G. Cowling

English

arXiv

Crossref

Experimental Identification of the Kink Instability as a Poloidal Flux Amplification Mechanism for Coaxial Gun Spheromak Formation

The magnetohydrodynamic kink instability is observed and identified experimentally as a poloidal flux amplification mechanism for coaxial gun spheromak formation. Plasmas in this experiment fall into three distinct regimes which depend on the peak gun current to magnetic flux ratio, with (I) low values resulting in a straight plasma column with helical magnetic field, (II) intermediate values leading to kinking of the column axis, and (III) high values leading immediately to a detached plasma. Onset of column kinking agrees quantitatively with the Kruskal-Shafranov limit, and the kink acts as a dynamo which converts toroidal to poloidal flux. Regime II clearly leads to both poloidal flux amplification and the development of a spheromak configuration

Hsu, S. C.

Bellan, P. M.

Caltech Authors - Main

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Experimental Identification of the Kink Instability as a Poloidal Flux Amplification
Mechanism for Coaxial Gun Spheromak Formation
S. C. Hsu∗ and P. M. Bellan
California Institute of Technology, Pasadena, CA 91125
(Dated: April 28, 2003; accepted by Phys. Rev. Lett.)
The magnetohydrodynamic kink instability is observed and identified experimentally as a poloidal
flux amplification mechanism for coaxial gun spheromak formation. Plasmas in this experiment fall
into three distinct regimes which depend on the peak gun current to magnetic flux ratio, with
(I) low values resulting in a straight plasma column with helical magnetic field, (II) intermediate
values leading to kinking of the column axis, and (III) high values leading immediately to a detached
plasma. Onset of column kinking agrees quantitatively with the Kruskal-Shafranov limit, and the
kink acts as a dynamo which converts toroidal to poloidal flux. Regime II clearly leads to both
poloidal flux amplification and the development of a spheromak configuration.
PACS numbers: 52.55.Ip,52.35.Py,52.30.Cv
The spheromak [1, 2] is a simply-connected plasma
configuration in which the magnetic fields are largely de-
termined by dynamo-driven plasma currents. Because
of its topological simplicity and ease of formation, the
spheromak is of interest as a magnetic fusion confine-
ment scheme [3]. Spheromak formation has traditionally
been explained by Taylor’s hypothesis [4] that a turbulent
magnetohydrodynamic (MHD) system relaxes to a state
of minimum magnetic energy subject to the constraint
of constant magnetic helicity. While this hypothesis has
successfully explained the existence and many equilib-
rium properties of spheromaks, it says nothing about the
actual 3D dynamics underlying the relaxation process,
e.g., for coaxial gun spheromaks, the dynamo mecha-
nism which converts injected toroidal flux into required
poloidal flux [5]. The dynamics must be 3D because
Cowling’s theorem [6] shows that purely axisymmetric
processes cannot accomplish this. This Letter experi-
mentally identifies the MHD kink instability [7] as a 3D
mechanism which converts toroidal to poloidal flux in a
coaxial gun system, thereby leading to spheromak for-
mation. This mechanism should also be of fundamen-
tal importance to coaxial helicity injection in spherical
tori [8], relaxation in reversed-field pinches [4], solar coro-
nal plasma instabilities [9], and astrophysical jets [10].
Plasmas in this experiment fall into three regimes de-
pending on peak λgun = µ0Igun/ψgun (where Igun is the
gun current and ψgun is the bias poloidal magnetic flux
intercepting the inner gun electrode), with (I) low values
resulting in a straight plasma column with helical mag-
netic field along the symmetry axis, (II) intermediate val-
ues leading to kinking of the column axis, and (III) high
values leading immediately to a detached plasma with
Btor ≫ Bpol. Onset of column kinking agrees quantita-
tively with the Kruskal-Shafranov limit [7], and the kink
acts as a dynamo which converts toroidal to poloidal flux.
Regime II clearly leads to both poloidal flux amplification
∗Now at: Los Alamos National Laboratory, Los Alamos, NM 87545
and magnetic field profiles consistent with spheromak for-
mation. These results are qualitatively consistent with
recent time-dependent resistive MHD numerical simula-
tions of electrostatically driven spheromaks [11].
Early coaxial gun experiments [12] demonstrated
poloidal flux amplification, and it was postulated that
this was due to an observed helical instability. More re-
cently, toroidal mode number n = 1 central column in-
stabilities in coaxial gun experiments were reported by
several spheromak research groups [13, 14, 15, 16, 17].
Typically, the observations were based on edge magnetic
measurements, and the mode was studied in the context
of relaxation during sustainment. Two hypotheses were
proposed to explain the mode: (1) development of a q = 1
surface in the closed-flux region [13, 15] or equivalently
a magnetic axis kink, and (2) current-driven instability
of the central column [16, 17] or equivalently a geometric
axis kink. Additionally, the mode was shown to couple
power from the central column to the spheromak [14].
The present work offers a significant new result in that a
nonlinear n = 1 geometric axis kink is directly observed
to produce the dynamo and poloidal flux amplification
leading to spheromak formation.
Building on a prior Caltech spheromak formation ex-
periment [18], the present experiment simplifies, and
makes more accessible, the spheromak formation process
by using a novel coaxial gun [19]. A schematic of the
gun is shown in Fig. 1, along with the cylindrical coor-
dinate system. The gun consists of two concentric, co-
planar copper electrodes: a 20 cm diameter disk (blue)
biased to negative polarity high voltage surrounded by a
50 cm outer diameter annulus (green) connected to vac-
uum chamber ground. The width of the gap between
electrodes is 6 mm. The inner electrode is mounted on
the end of a vacuum re-entrant port; both are electri-
cally insulated from the vacuum chamber by a ceramic
break. The co-planar arrangement of the electrode sur-
faces is geometrically simple, allows the spheromak for-
mation process to be diagnosed with no mechanical ob-
structions, and is feasible to model numerically [20]. Bias
field poloidal flux ψ-contours are also shown in Fig. 1;
2
FIG. 1: Side schematic of coaxial gun, showing bias field coil
and associated ψ-contours, gas feeds, and magnetic probe.
ψgun can be adjusted from 0–7 mWb. Both Igun and
bias field point toward the inner electrode. The rise time
of ψgun is 10 ms, making it essentially stationary on the
much faster time scale of the gun discharge (tens of µs).
The gun is installed at one end of a much larger vacuum
chamber (about 2 m long and 1.5 m diameter), and thus
boundary effects on the spheromak formation process are
minimized. The gun is powered by an ignitron-switched
120 µF, 20 kV capacitor bank. Hydrogen gas is injected
transiently using fast puff valves at eight equally spaced
toroidal positions on each electrode. Due to the Paschen
effect, the optimum path for plasma breakdown is along
the bias field and not at the gap between electrodes. The
capacitor bank is discharged at t = 0, at which time the
bias field and gas puff have already been introduced, and
breakdown occurs at approximately 4 µs.
The main diagnostics are a multiple-frame fast CCD
camera and a 60-channel magnetic probe array. The
camera takes a sequence of time-resolved images in one
plasma shot. The capability to visually follow the global
evolution of a plasma in one plasma shot is a new and
unique aspect of this work. The inter-frame time is typi-
cally 1 or 1.5 µs, and the exposure time of each frame is
10 or 20 ns. False color is applied to the images for view-
ing. The magnetic probe array, shown in Fig. 1, measures
all three components (R, φ, Z) of B at 20 radial positions
with 2 cm radial spacing. Probe Ḃ signals are acquired
using a digital acquisition system and integrated numer-
ically on a computer. For all B measurements in this
paper, the probe is located at Z = 22 cm from the plane
of the gun electrodes. Kink occurrence is independent of
the probe (see Figs. 2 and 3). Propagation of the plasma
in Z past the stationary probe relates temporal informa-
tion to Z-spatial information. This relationship improves
as the propagation past the probe becomes fast compared
to the plasma expansion rate; this was exploited in the
prior Caltech spheromak experiment [18]. The time de-
pendence thus acts as a proxy for the Z-dependence. Igun
is measured with a Rogowski coil surrounding the ce-
ramic break, and gun voltage Vgun is measured using a
high-voltage probe. Typical parameters are: Vgun = 4–
FIG. 2: Image sequence of shot 2472 taken with a DRS
Hadland Imacon 200 CCD camera (each frame originally 1200
by 980 pixels, 10 bits/pixel). The circular gap between outer
and inner electrodes is visible toward right side of each frame.
B-probe is also visible. Kink is fully developed by 13 µs.
FIG. 3: Images of three plasma regimes which depend on
increasing value of λgun, taken with a Cooke Co. HSFC-PRO
CCD camera (each frame originally 1280 by 1024 pixels, 12
bits/pixel): (I) stable column (shot 1210), (II) kinked column
(shot 1247), and (III) detached plasma (shot 1181). B-probe
was not installed for these shots.
6 kV (charge voltage) and 2–2.5 kV (after breakdown),
peak Igun = 70–120 kA, ψgun = 0.5–2 mWb, B ≈ 0.1–
1 kG, n ∼ 1014 cm−3, and Te ∼ Ti ≈ 5− 20 eV.
Figure 2 shows the time evolution of a kinked plasma
(shot 2472). Gun electrodes are on the right side of each
frame, and the Z-axis is oriented horizontally across the
middle of each frame. At 7 µs, bright arches are visi-
ble soon after breakdown, showing that breakdown oc-
curs along the vacuum bias field lines. As Igun increases,
the arches expand and quickly merge together (8.5 µs),
forming a plasma column (10 µs) which begins to kink
(11.5 µs) and then becomes strongly kinked (13 µs). It
will be shown that this sequence leads to a spheromak.
Depending on peak λgun, three distinct plasmas result, as
shown in Fig. 3. Regime I leads to a stable column, II to
a kinked column and then a spheromak, and III to an im-
mediately detached plasma. The transition from regime I
to II (II to III) occurs for peak λgun ≈ 40 (60) m
−1.
In order to show that the helical perturbation is an
MHD kink mode, consider the time evolution of the q(R)
profile (relative to geometric axis), which is shown in
Fig. 4 (shot 2472). The kink mode becomes linearly un-
stable when q(R) = 2πRBZ/LBφ = 1, where L is the
column length; this is the Kruskal-Shafranov limit. Since
the kink involves a shift of the current channel, this re-
quires q = 1 on axis and in its vicinity. As seen in Fig. 4,
3
FIG. 4: Flattening of q(R) profile toward unity (shot 2472)
right before kink onset. Error bars are due to uncertainty in
L. Also shown are Bφ and BZ profiles.
q near the axis is greater than unity at 9.5 µs but flattens
and approaches unity by 11.5 µs, coinciding with onset of
column kinking (Fig. 2). The kink is fully developed at
13 µs and has broken apart by 14.5 µs. The q profile is de-
termined using local BZ and Bφ data from the magnetic
probe array, and L is determined from the CCD images.
In addition, for an ensemble of plasma shots in which a
column forms (regimes I and II), the Kruskal-Shafranov
limit (recast [21] as λgun = 4π/L) is a good predictor [21]
of whether a kink actually develops (Fig. 5). Thus, two
sets of independent data (magnetic probe measurements
of BZ and Bφ, and Rogowski/flux loop measurements
of λgun) both give direct evidence that the helical per-
turbation is an MHD kink mode. It is interesting to
note that the observed kinks always have one axial wave-
length, even though there is no fixed boundary at the
end of the plasma column. One explanation [22] for this
is that there is a strong axial gradient in Bn−1/2, and
thus an effective Alfvén speed discontinuity, which acts
as a rigid boundary. This conjecture is consistent with
the “mushroom cap” on the left side of the kink in Fig. 3.
Next, it is shown that the kink is followed imme-
diately by three signatures of spheromak formation:
(1) appearance of closed ψ-contours [calculated assuming
axisymmetry, ψ(R, t) =
∫ R
0
2πR′BZ(R
′, t) dR′], (2) ψ-
amplification, and (3) magnetic field radial profiles con-
sistent with spheromak formation. It is important to
note that the relationship between closed ψ-contours
and closed flux surfaces becomes ambiguous when ax-
isymmetry is broken. Thus, in the presence of a non-
axisymmetric rotating kink, closed ψ-contours indicate
closed flux surfaces only in a time-averaged way. Shortly
after 13 µs, the kink breaks apart (Fig. 2). This co-
incides with signatures of spheromak formation as ob-
served in the magnetic probe measurements (Fig. 6). At
approximately 13 µs, closed ψ-contours appear and ψmax
is amplified to larger than ψgun (ψ-amplification is due
FIG. 5: Plot of λgun vs. column length L, showing good
quantitative agreement of kink onset with Kruskal-Shafranov
limit.
FIG. 6: Plots (shot 2472) vs. R and time of (top) Bpol vec-
tors and ψ-contours (mWb), and (bottom) ψ along horizontal
dashed line of top panel.
mainly to broadening of the BZ profile after 12 µs). At
15 µs, magnetic field profiles consistent with spheromak
formation are observed (Fig. 7). The measured radial
profiles of BZ and Bφ at 15 µs are compared with Taylor
state solutions [4] in cylindrical geometry, i.e. uniform λ
solutions of
∇×B = λB. (1)
The solutions are BZ ∼ J0(λR) and Bφ ∼ J1(λR), where
J0 and J1 are Bessel functions of order zero and one,
respectively, and the best fit is found for λ ≈ 15 m−1
with a radial offset of 4 cm (displacement of spheromak
off the geometric axis). The slight disagreement between
measured profiles and the Taylor solution is not surpris-
ing since the spheromak is expected to be either (1) still
undergoing relaxation toward the Taylor state or (2) in
a modified relaxed state since it is still being driven by
the gun which has peak λgun ≈ 50 m
−1.
The kink modifies the direction of current-flow from
Z (poloidal) to φ (toroidal), as seen in Fig. 2. Equiva-
lently, it converts toroidal to poloidal flux. This process
4
FIG. 7: Plots showing BZ, Bφ profiles (shot 2472) compared
to Taylor solutions (λ = 15 m−1 and radial offset of 4 cm).
Profiles are plotted for vertical dashed line in Fig. 6 (top).
is paramagnetic, amplifying ψ over the initial applied
ψgun = 1.7 mWb. The paramagnetism is understood by
realizing that kinks involve perturbations with depen-
dence exp(ik · x) where k ·B = 0. The latter means
kZ = −nBφ/RBZ. Thus, a locus of constant phase of
the kink is given by φ = (Bφ/RBZ)Z, meaning that the
kinked current channel is a right (left) handed helix if
JZBZ > 0 (< 0). This will always lead to amplification
of the original BZ. The additional ψ introduced by the
kink can be estimated by approximating the helix as a
solenoid with current I, turns per length 1/L, and radius
a. The solenoid formula for the field inside the solenoid
is BZ = µ0I/L; the ψ produced by the solenoid is πa
2BZ
and thus depends non-linearly on the kink amplitude a.
Using the measured values a ≈ 5 cm, L ≈ 20 cm, and
I ≈ 60 kA at 13.5 µs, the ψ generated by the kink is
predicted to be approximately 1 mWb, which is within
a factor of 2 of the observed amplification of ψmax over
ψgun. The discrepancy is within the accuracy of a and
L measurements and of the ψ calculation assuming ax-
isymmetry in the presence of the rotating kink. Because
the coaxial gun can only inject toroidal flux, 3D plasma
dynamics must be responsible for ψ exceeding ψgun. The
dynamics are provided by the kink, which amplifies ψgun
by converting toroidal to poloidal flux; thus, in this case,
the kink constitutes the dynamo intrinsic to spheromak
formation.
Regime II clearly leads to all three signatures of sphero-
mak formation: closed ψ-contours, ψ-amplification, and
proper magnetic field profiles. Regime I lacks both closed
ψ-contours and ψ-amplification. Regime III may have
closed ψ-contours; however, Btor ≫ Bpol associated with
high Igun, which could indicate that the plasma is still re-
laxing. This work indicates a close relationship between
the kink threshold (λgun = 4π/L) and the spheromak
formation threshold, which according to a static force-
free treatment is also proportional to an inverse length.
For example, solutions of Eq. (1) for uniform λ indi-
cate that closed ψ-contour equilibria exist for threshold
λ ∼
√
x211/a
2 + π2/h2, where a (h) is the radius (length)
of the flux conserver and x11 is the first root of J1 [2].
Geometric differences among different experiments could
result in different eigenmode spectra of Eq. (1) relative
to the kink stability thresholds.
In summary, the MHD kink instability has been identi-
fied experimentally as a poloidal flux amplification mech-
anism for coaxial gun spheromak formation. An n = 1
central column helical instability was observed during
formation using multiple-frame CCD imaging. Onset of
the perturbation was shown using two independent sets of
data to agree quantitatively with the Kruskal-Shafranov
limit. The kink acts as a dynamo which converts toroidal
to poloidal flux, and it is followed immediately by three
key signatures of spheromak formation: (1) closed ψ-
contours, (2) ψ-amplification, and (3) magnetic field pro-
files similar to the Taylor state.
The authors acknowledge S. Pracko, C. Romero-
Talamás, and D. Felt for technical assistance and
H. Bindslev for asking about kink paramagnetism. Sup-
ported by a US-DoE Fusion Energy Postdoctoral Fellow-
ship and US-DoE Grant DE-FG03-98ER544561.
[1] T. R. Jarboe, Plasma Phys. Control. Fus. 36, 945 (1994).
[2] P. M. Bellan, Spheromaks (Imperial College Press, Lon-
don, 2000).
[3] H. S. McLean et al., Phys. Rev. Lett. 88, 125004 (2002).
[4] J. B. Taylor, Rev. Mod. Phys. 58, 741 (1986).
[5] A. al-Karkhy et al., Phys. Rev. Lett. 70, 1814 (1993).
[6] T. G. Cowling, Mon. Not. R. Astron. Soc. 94, 39 (1934).
[7] J. P. Freidberg, Ideal Magnetohydrodynamics (Plenum
Press, New York, 1987).
[8] R. Raman et al., Phys. Rev. Lett. 90, 075005 (2003).
[9] B. Vrs̆nak, V. Ruz̆djak, and B. Rompolt, Solar
Physics 136, 151 (1991).
[10] H. Li et al., Astrophys. J. 561, 915 (2001).
[11] C. R. Sovinec, J. M. Finn, and D. del-Castillo-Negrete,
Phys. Plasmas 8, 475 (2001).
[12] L. Lindberg and C. T. Jacobsen, Phys. Fluids 7, S44
(1964).
[13] S. O. Knox et al., Phys. Rev. Lett. 56, 842 (1986).
[14] P. K. Browning et al., Phys. Rev. Lett. 68, 1718 (1992).
[15] M. Nagata et al., Phys. Rev. Lett. 71, 4342 (1993).
[16] R. C. Duck et al., Plasma Phys. Control. Fus. 39, 715
(1997).
[17] D. Brennan et al., Phys. Plasmas 6, 4248 (1999).
[18] J. Yee and P. M. Bellan, Phys. Plasmas 7, 3625 (2000).
[19] S. C. Hsu and P. M. Bellan, IEEE Trans. Plasma Sci. 30,
10 (2002).
[20] M. Tokman, S. C. Hsu, and P. M. Bellan, Bull. Amer.
Phys. Soc. 47, 42 (2002).
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61 (2001).


P H Y S I C A L R E V I E W L E T T E R S week ending30 MAY 2003VOLUME 90, NUMBER 21Experimental Identification of the Kink Instability as a Poloidal Flux Amplification
Mechanism for Coaxial Gun Spheromak Formation
S. C. Hsu* and P. M. Bellan
California Institute of Technology, Pasadena, California 91125, USA
(Received 24 February 2003; published 30 May 2003)215002-1The magnetohydrodynamic kink instability is observed and identified experimentally as a poloidal
flux amplification mechanism for coaxial gun spheromak formation. Plasmas in this experiment fall
into three distinct regimes which depend on the peak gun current to magnetic flux ratio, with (I) low
values resulting in a straight plasma column with helical magnetic field, (II) intermediate values
leading to kinking of the column axis, and (III) high values leading immediately to a detached plasma.
Onset of column kinking agrees quantitatively with the Kruskal-Shafranov limit, and the kink acts as a
dynamo which converts toroidal to poloidal flux. Regime II clearly leads to both poloidal flux
amplification and the development of a spheromak configuration.
DOI: 10.1103/PhysRevLett.90.215002 PACS numbers: 52.55.Ip, 52.30.Cv, 52.35.Pypoloidal flux. Regime II clearly leads to both poloidal
flux amplification and magnetic field profiles consistent
shown in Fig. 1;  gun can be adjusted from 0–7 mWb.
Both Igun and bias field point toward the inner electrode.The spheromak [1,2] is a simply connected plasma
configuration in which the magnetic fields are largely
determined by dynamo-driven plasma currents. Because
of its topological simplicity and ease of formation, the
spheromak is of interest as a magnetic fusion confinement
scheme [3]. Spheromak formation has traditionally been
explained by Taylor’s hypothesis [4] that a turbulent mag-
netohydrodynamic (MHD) system relaxes to a state of
minimum magnetic energy subject to the constraint of
constant magnetic helicity. While this hypothesis has
successfully explained the existence and many equilib-
rium properties of spheromaks, it says nothing about the
actual 3D dynamics underlying the relaxation process,
e.g., for coaxial gun spheromaks, the dynamo mechanism
which converts injected toroidal flux into required poloi-
dal flux [5]. The dynamics must be 3D because Cowling’s
theorem [6] shows that purely axisymmetric processes
cannot accomplish this. This Letter experimentally iden-
tifies the MHD kink instability [7] as a 3D mechanism
which converts toroidal to poloidal flux in a coaxial gun
system, thereby leading to spheromak formation. This
mechanism should also be of fundamental importance
to coaxial helicity injection in spherical tori [8], relaxa-
tion in reversed-field pinches [4], solar coronal plasma
instabilities [9], and astrophysical jets [10].
Plasmas in this experiment fall into three regimes
depending on peak gun  0Igun= gun (where Igun is
the gun current and  gun is the bias poloidal magnetic
flux intercepting the inner gun electrode), with (I) low
values resulting in a straight plasma column with helical
magnetic field along the symmetry axis, (II) intermediate
values leading to kinking of the column axis, and
(III) high values leading immediately to a detached
plasma with Btor  Bpol. Onset of column kinking agrees
quantitatively with the Kruskal-Shafranov limit [7], and
the kink acts as a dynamo which converts toroidal to0031-9007=03=90(21)=215002(4)$20.00with spheromak formation. These results are qualitatively
consistent with recent resistive MHD numerical simula-
tions of electrostatically driven spheromaks [11].
Early coaxial gun experiments [12] demonstrated po-
loidal flux amplification, and it was postulated that this
was due to an observed helical instability. More recently,
toroidal mode number n  1 instabilities in coaxial gun
experiments were reported by several spheromak research
groups [13–17] in studies of relaxation during sustain-
ment. Two hypotheses were proposed to explain the in-
stability: (1) development of a q  1 surface in the
closed-flux region [13,15] or equivalently a magnetic
axis kink, and (2) current-driven destabilization of the
central column [16,17] or equivalently a geometric axis
kink. Additionally, the instability was shown to couple
power from the central column to the spheromak [14].
This work offers a significant new result in that a non-
linear n  1 geometric axis kink is directly observed to
produce the dynamo and poloidal flux amplification lead-
ing to spheromak formation.
Building on a prior Caltech spheromak formation ex-
periment [18], the present experiment simplifies the
spheromak formation process by using a novel coaxial
gun [19], shown in Fig. 1. The gun consists of two con-
centric, coplanar copper electrodes: a 20 cm diameter
disk (blue) biased to negative polarity high voltage sur-
rounded by a 50 cm outer diameter annulus (green) con-
nected to vacuum chamber ground. The width of the gap
between electrodes is 6 mm. The inner electrode is
mounted on the end of a vacuum reentrant port; both
are electrically insulated from the vacuum chamber by
a ceramic break. The coplanar arrangement of the elec-
trode surfaces is geometrically simple, allows the spher-
omak formation process to be diagnosed with no
mechanical obstructions, and is feasible to model numeri-
cally [20]. Bias field poloidal flux  contours are also 2003 The American Physical Society 215002-1
FIG. 2 (color). Image sequence of shot 2472 (DRS Hadland
Imacon 200 CCD camera). The circular gap between outer and
inner electrodes is visible toward right side of each frame. B
probe is also visible. Kink is fully developed by 13 s.
FIG. 3 (color). Images of three plasma regimes which depend
on increasing value of gun (Cooke Co. HSFC-PRO CCD
camera): (I) stable column (shot 1210), (II) kinked column
(shot 1247), and (III) detached plasma (shot 1181). B probe was
not installed for these shots.
FIG. 1 (color). Schematic of coaxial gun, showing bias field
coil,  contours, gas feeds, and magnetic probe.
P H Y S I C A L R E V I E W L E T T E R S week ending30 MAY 2003VOLUME 90, NUMBER 21The rise time of  gun is 10 ms, making it essentially
stationary on the much faster time scale of the gun
discharge (tens of s). The gun is installed at one end
of a much larger vacuum chamber (about 2 m long and
1.5 m diameter), and thus boundary effects on the spher-
omak formation process are minimized. The gun is pow-
ered by an ignitron-switched 120 F, 20 kV capacitor
bank. Hydrogen gas is injected transiently using fast puff
valves at eight equally spaced toroidal positions on each
electrode. Because of the Paschen effect, the optimum
path for plasma breakdown is along the bias field and not
at the gap between electrodes. The capacitor bank is dis-
charged at t  0, at which time the bias field and gas puff
have already been introduced, and breakdown occurs at
approximately 4 s.
The main diagnostics are a multiple-frame fast charge-
coupled device (CCD) camera and a 60-channel magnetic
probe array. The camera takes a sequence of time-
resolved images in one plasma shot. The interframe
time is typically 1 or 1:5 s, and the exposure time of
each frame is 10 or 20 ns. False color is applied to the
images for viewing. The magnetic probe array, shown in
Fig. 1, measures all three components (R,, Z) ofB at 20
radial positions with 2 cm radial spacing. Probe _B signals
are acquired using a digital acquisition system and inte-
grated numerically. For all B measurements in this paper,
the probe is located at Z  22 cm from the plane of the
gun electrodes. Kink occurrence is independent of the
probe (see Figs. 2 and 3). Propagation of the plasma in Z
past the stationary probe relates temporal information to
Z-spatial information. This relationship improves as the
propagation past the probe becomes fast compared to the
plasma expansion rate; this was exploited in the prior
Caltech spheromak experiment [18]. The time depen-
dence thus acts as a proxy for the Z dependence. Igun is
measured with a Rogowski coil surrounding the ceramic
break, and gun voltage Vgun is measured using a high-
voltage probe. Typical parameters are Vgun  4–6 kV
(charge voltage) and 2–2.5 kV (after breakdown), peak215002-2Igun  70–120 kA,  gun  0:5–2 mWb, B  0:1–1 kG,
n 1014 cm3, and Te  Ti  5–20 eV.
Figure 2 shows the time evolution of a kinked plasma
(shot 2472). Gun electrodes are on the right side of each
frame, and the Z axis is oriented horizontally across the
middle of each frame. At 7 s, bright arches are visible
soon after breakdown, showing that breakdown occurs
along the vacuum bias field lines. As Igun increases, the
arches expand and quickly merge together (8:5 s), form-
ing a plasma column (10 s) which begins to kink
(11:5 s) and then becomes strongly kinked (13 s). It
will be shown that this sequence leads to a spheromak.
Depending on peak gun, three distinct plasmas result, as
shown in Fig. 3. Regime I leads to a stable column, II to a
kinked column and then a spheromak, and III to an
immediately detached plasma. The transition from re-
gime I to II (II to III) occurs for peak gun  40 60 m1.
In order to show that the helical perturbation is a MHD
kink mode, consider the time evolution of the qR profile
(relative to geometric axis), which is shown in Fig. 4 (shot
2472). The kink mode becomes linearly unstable when
qR  2RBZ=LB  1, where L is the column length;
this is the Kruskal-Shafranov limit. Since the kink in-
volves a shift of the current channel, this requires q  1
on axis and in its vicinity. As seen in Fig. 4, q near
the axis is greater than unity at 9:5 s but flattens and215002-2
FIG. 4. Flattening of qR profile toward unity (shot 2472)
right before kink onset. Error bars are due to uncertainty in L.
Also shown are B and BZ profiles.
FIG. 5. Plot of gun vs column length L, showing agreement
with Kruskal-Shafranov limit.
FIG. 6. Plots (shot 2472) vs R and time of (top) Bpol vectors
and  contours (mWb), and (bottom)  along horizontal
dashed line of top panel.
P H Y S I C A L R E V I E W L E T T E R S week ending30 MAY 2003VOLUME 90, NUMBER 21approaches unity by 11:5 s, coinciding with the onset of
column kinking (Fig. 2). The kink is fully developed at
13 s and has broken apart by 14:5 s. The q profile is
determined using local BZ and B data from the magnetic
probe array, and L is determined from the CCD images.
In addition, for an ensemble of plasma shots in which a
column forms (regimes I and II), the Kruskal-Shafranov
limit (recast [21] as gun  4=L) is a good predictor [21]
of whether a kink actually develops (Fig. 5). Thus, two
sets of independent data (magnetic probe measurements
of BZ and B, and Rogowski/flux loop measurements of
gun) both give direct evidence that the helical perturba-
tion is a MHD kink mode. It is interesting to note that the
observed kinks always have one axial wavelength, even
though there is no fixed boundary at the end of the plasma
column. One explanation [22] for this is that there is a
strong axial gradient in Bn1=2, and thus an effective
Alfve´n speed discontinuity, which acts as a rigid bound-
ary. This conjecture is consistent with the ‘‘mushroom
cap’’ on the left side of the kink in Fig. 3.
Next, it is shown that the kink is followed immediately
by three signatures of spheromak formation: (1)
appearance of closed  contours [calculated assuming
axisymmetry,  R; t  RR0 2R0BZR0; tdR0], (2)  am-
plification, and (3) magnetic field radial profiles consis-
tent with spheromak formation. It is important to note
that the relationship between closed  contours
and closed-flux surfaces becomes ambiguous when axi-
symmetry is broken. Thus, in the presence of a nonaxi-
symmetric rotating kink, closed  contours indicate
closed-flux surfaces only in a time-averaged way.
Shortly after 13 s, the kink breaks apart (Fig. 2). This
coincides with signatures of spheromak formation as
observed in the magnetic probe measurements (Fig. 6).
At approximately 13 s, closed  contours appear215002-3and  max is amplified to larger than  gun ( amplification
is due mainly to broadening of the BZ profile after 12 s).
At 15 s, magnetic field profiles consistent with sphero-
mak formation are observed (Fig. 7). The measured radial
profiles of BZ and B at 15 s are compared with Taylor
state solutions [4] in cylindrical geometry, i.e., uniform 
solutions of
r
 B  B: (1)
The solutions are BZ  J0R and B  J1R, where
J0 and J1 are Bessel functions of order zero and one,
respectively, and the best fit is found for   15 m1
with a radial offset of 4 cm (displacement of spheromak
off the geometric axis). The slight disagreement between
measured profiles and the Taylor solution is not surprising
since the spheromak is expected to be either (1) still
undergoing relaxation toward the Taylor state or (2) in a
modified relaxed state since it is still being driven by the
gun which has peak gun  50 m1.
The kink modifies the direction of current flow from Z
(poloidal) to  (toroidal), as seen in Fig. 2. Equivalently,215002-3
FIG. 7. Plots showing BZ, B profiles (shot 2472) compared
to Taylor solutions (  15 m1 and radial offset of 4 cm).
Profiles are plotted for vertical dashed line in Fig. 6 (top).
P H Y S I C A L R E V I E W L E T T E R S week ending30 MAY 2003VOLUME 90, NUMBER 21it converts toroidal to poloidal flux. This process is para-
magnetic, amplifying  over the initial applied  gun 
1:7 mWb. The paramagnetism is understood by realiz-
ing that kinks involve perturbations with dependence
expikx where kB  0. The latter means kZ 
nB=RBZ. Thus, a locus of constant phase of the kink
is given by   B=RBZZ, meaning that the kinked
current channel is a right (left) handed helix if JZBZ > 0
(< 0). This will always lead to amplification of the origi-
nal BZ. The additional  introduced by the kink can be
estimated by approximating the helix as a solenoid with
current I, turns per length 1=L, and radius a. The solenoid
formula for the field inside the solenoid is BZ  0I=L;
the  produced by the solenoid isa2BZ and thus depends
nonlinearly on the kink amplitude a. Using the measured
values a  5 cm, L  20 cm, and I  60 kA at 13:5 s,
the  generated by the kink is predicted to be approxi-
mately 1 mWb, which is within a factor of 2 of the
observed amplification of  max over  gun. The discrep-
ancy is within the accuracy of a and L measurements and
of the  calculation assuming axisymmetry in the pres-
ence of the rotating kink. Because the coaxial gun can
inject only toroidal flux, 3D plasma dynamics must be
responsible for  exceeding  gun. The dynamics are
provided by the kink, which amplifies  gun by converting
toroidal to poloidal flux; thus, in this case, the kink
constitutes the dynamo intrinsic to spheromak formation.
Regime II clearly leads to all three signatures of spher-
omak formation: closed  contours,  amplification, and
proper magnetic field profiles. Regime I lacks both closed
 contours and  amplification. Regime III may have
closed  contours; however, Btor  Bpol associated with
high Igun, which could indicate that the plasma is still
relaxing. This work indicates a close relationship between
the kink threshold (gun  4=L) and the spheromak
formation threshold, which according to a static force-
free treatment is also proportional to an inverse length [2].
Geometric differences among different experiments215002-4could result in different eigenmode spectra of Eq. (1)
relative to kink thresholds.
In summary, the MHD kink instability has been iden-
tified experimentally as a poloidal flux amplification
mechanism for coaxial gun spheromak formation. An
n  1 central column helical instability was observed
during formation using multiple-frame CCD imag-
ing. Onset of the perturbation was shown using two
independent sets of data to agree quantitatively with the
Kruskal-Shafranov limit. The kink acts as a dynamo
which converts toroidal to poloidal flux, and it is followed
immediately by three key signatures of spheromak for-
mation: (1) closed  contours, (2)  amplification, and
(3) magnetic field profiles similar to the Taylor state.
The authors acknowledge S. Pracko, C. Romero-
Talama´s, and D. Felt for technical assistance and
H. Bindslev for asking about kink paramagnetism. This
work was supported by U.S.-DOE Grant No. DE-FG03-
98ER544561 and Fusion Energy Sciences Fellowship
Program.*Current address: P-24 Plasma Physics Group, Los
Alamos National Laboratory, Los Alamos, NM 87545,
USA.
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Experimental Identification of the Kink Instability as a Poloidal Flux Amplification Mechanism for Coaxial Gun Spheromak Formation

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