Download Three-dimensional electron magnetohydrodynamic reconnection. I

PHYSICS OF PLASMAS
VOLUME 10, NUMBER 7
JULY 2003
Three-dimensional electron magnetohydrodynamic reconnection.
I. Fields, currents, and flows
R. L. Stenzel, M. C. Griskey, J. M. Urrutia, and K. D. Strohmaier
Department of Physics and Astronomy, University of California, Los Angeles, California 90095-1547
共Received 28 August 2002; accepted 7 April 2003兲
In a large laboratory plasma, reconnection of three-dimensional 共3-D兲 magnetic fields is studied in
the parameter regime of electron magnetohydrodynamics. A reversed magnetic field topology with
two 3-D null points and a two-dimensional 共2-D兲 null line is established, and its free relaxation is
studied experimentally. Major new findings include the absence of tilting instabilities in an
unbounded plasma, relaxation times fast compared to classical diffusion times, dominance of field
line annihilation at the 2-D current sheet versus reconnection at 3-D null points, conversion of
magnetic energy into electron thermal energy, and excitation of various microinstabilities. This first
of four companion papers focuses on the magnetic field topology and dynamics. © 2003 American
Institute of Physics. 关DOI: 10.1063/1.1578998兴
of microinstabilities 共ion sound, whistlers, plasma waves兲
have been observed but their role in the dissipation process is
still under investigation.
The material is presented in four companion papers. The
present, Part I, describes the experimental setup, the magnetic field topology, currents and electric fields, reconnection, tearing, annihilation and convection of magnetic fields.
The second paper, Part II,13 deals with the stability of a
freely relaxing FRC to translation, rotation, and precession.
The third paper, Part III,14 deals with the energy flow from
the external source into stored magnetic energy and its release into the particles. The fourth paper, Part IV,15 describes
microinstabilities and their effect on the reconnection rate
and electron heating.
The present paper is organized as follows. In Sec. II, the
plasma device and the measurement techniques are described. The experimental results, divided into various subsections, are presented in Sec. III. The conclusion, Sec. IV,
points out the new findings and implications to other related
research.
I. INTRODUCTION
Magnetic reconnection describes dynamic processes in
plasmas that produce changes in field topologies, associated
plasma transport, and energization of particles at the expense
of magnetic field energy. These processes play an important
role in space and laboratory plasma physics. In spite of a
long history of research,1 reconnection remains at the forefront of interest in many areas.2 The topic is complicated
because of coupled, multiscale processes with different physics near magnetic null points. For example, an outer region
described by single-fluid magnetohydrodynamics 共MHD兲 is
coupled to an intermediate region of electron magnetohydrodynamics 共EMHD兲,3 which in turn is coupled to the unmagnetized region of the null point. The field of reconnection has
evolved from steady-state pictures for two-dimensional
共2-D兲 geometries4 to three-dimensional 共3-D兲, timedependent reconnection models5,6 supported by many numerical simulations including Hall effects.7,8 In space and
fusion plasmas, it is often difficult to observationally resolve
both global and microscopic processes simultaneously.
Hence, dedicated laboratory experiments have been performed aimed at studying reconnection processes under controlled conditions.9,10 The present work describes a new experiment that focuses on reconnection in the EMHD
parameter regime. Compared to earlier work in this parameter regime,11 the present experiment has no boundary effects, reconnection is both driven 共turn-on phase兲 and nondriven 共spontaneous, turn-off phase兲 and the field topology is
3-D. Major findings include 共i兲 a topology change from a
field-reversed configuration 共FRC兲 into a uniform field on a
time scale fast compared to classical diffusion times, but
slower than whistler transit times, 共ii兲 no disruptive tilting
instability of a freely relaxing EMHD FRC, 共iii兲 conversion
of magnetic field energy into electron thermal energy, and
共iv兲 energy conversion which does not take place by reconnection at 3-D null points, but by field line annihilation in a
toroidal magnetic neutral/electron current sheet.12 A variety
1070-664X/2003/10(7)/2780/14/$20.00
II. EXPERIMENTAL ARRANGEMENT
The experiments are performed in a large laboratory
plasma device schematically shown in Fig. 1共a兲. A 1 m diam
⫻2.5 m long plasma column of density n e ⱗ 1012 cm⫺3 ,
electron temperature kT e ⯝3 eV, ion temperature kT i ⭐0.3
eV, argon gas pressure p n ⯝0.4 mTorr 共plus 1% hydrogen for
cathode activation兲 is produced in a uniform axial magnetic
field B0 ⯝⫺5 G ẑ 共away from the cathode兲 with a pulsed dc
discharge (V dis⯝50 V, I dis⯝600 A, t pulse⯝5 ms, t rep⯝1 s兲 by
a 1 m diam oxide-coated cathode. In the early, current-free
afterglow plasma 关 t a⬇115 ␮ s, n/( 兩 ⳵ n/ ⳵ t 兩 )⬇1 ms兴, pulsed
currents 共100 A, t pulse⯝47 ␮ s, t rise⯝5 ␮ s, t fall⭐3 ␮ s兲 are
applied to two insulated magnetic loops forming a Helmholtz
coil 共30 cm diam, 15 cm axial spacing, 4 turns each兲. The
pulse circuit consists of a charged, floating capacitor 共300
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Phys. Plasmas, Vol. 10, No. 7, July 2003
Fields, currents, and flows
2781
FIG. 2. Calculated field lines in vacuum for a strong Helmholtz coil field
共25 G兲 in a weak uniform field 共5 G兲 which is 共a兲 opposing, and 共b兲 aligned
with the coil field.
FIG. 1. 共a兲 Experimental setup with basic plasma parameters. The poloidal
magnetic field lines produced by the Helmholtz coil are shown. 共b兲 Waveform of the applied current to the Helmholtz coil.
␮ F兲, several high-power field effect transistors 共1200 V, 100
A兲 and Rogowski loops to measure the pulsed current. The
voltage applied to the coil is measured with a shunt resistor.
The waveform of the applied coil current, shown in Fig. 1共b兲,
has a slow rise time governed by the coil inductance (L coil
⬇15 ␮ H兲 and circuit resistance (R circuit⬇2 ⍀). Its pulse
length 共47 ␮ s兲 is sufficient for the magnetic field to penetrate
into the plasma and to establish the vacuum-like field topology 共as evidenced by a zero induced voltage on the coil at
t⯝40 ␮ s兲. The abrupt turn-off is made feasible by using fast
field-effect transistors. The reason for the fast switch-off is to
study the free relaxation of the stored poloidal magnetic field
inside the plasma shown schematically in Fig. 1共a兲. Since
there are no conducting boundaries nearby, the relaxation can
proceed self-consistently as in free space.
The Helmholtz coil magnetic field employed in these
experiments is typically stronger (B H⫽25 G兲 than the uniform field (B 0 ⫽5 G兲, resulting in a net field as shown in Fig.
2共a兲. The field in vacuum, calculated from Biot-Savart’s law
and experimentally verified, has two cusp-type 3-D null
points on axis, two O-type null lines and one X-type null
line, all azimuthal 共or toroidal兲, at nearly the radius of the
coil. Under identical conditions, the Helmholtz coil field can
be reversed with a relay switch resulting in a net field as
shown in Fig. 2共b兲. In such case, there are no 3-D null points
on axis but two X- and two O-type null lines, all toroidal, in
the vicinity of the individual coils. Comparison between
these two configurations is made to point out the difference
in the relaxation mechanism and electron energization with
and without 3-D null points. Other configurations such as
two opposing coil fields, a single coil, and coil fields oblique
to B0 have also been investigated, but results are preliminary.
The time-varying magnetic fields produced by the pulsed
current in the coil and the induced currents in the plasma are
measured with a single vector magnetic probe, recording
(B H,x ,B H,y ,B H,z ) versus time at a given position with digital
oscilloscopes 共100 MHz, 8 bit兲. By repeating the highly reproducible discharges and moving the probe to many positions in a 3-D volume, the vector field BH(r,t) may be obtained. The total magnetic field is obtained by adding B 0
⫽⫺5 G to the measured B H,z component, B(r,t)⫽BH(r,t)
⫹B0 . Typically, at a given position, an ensemble average
over 10 repeated shots is formed, which increases the digitization accuracy from 8 bit to approximately 11 bit. The spatial resolution of the B-field measurement is given by the
probe size 共0.75 cm兲, and is sufficient to resolve magnetic
structures on the electron inertial scale length (c/ ␻ pe ⬇1
cm兲. The temporal resolution is on the order of the electron
cyclotron period (2 ␲ / ␻ ce ⬇70 ns兲, and improved by temporal deconvolution using measured waveforms in vacuum and
the coil current waveform. Integration of the probe signal
(⬀ ⳵ B/ ⳵ t) is performed numerically, and the absolute probe
calibration is done with the known field inside the Helmholtz
coil. From the stored traces of BH(r,t), the spatial field topology can be displayed at any instant of time. The current
density is calculated from Ampère’s law, J⫽ⵜ⫻B/ ␮ 0 共the
displacement current is neglected because the magnetic field
evolves on time scales long compared to ␻ ⫺1
pe ). The measurement accuracy is confirmed by checking that ⵜ•B⯝0
when 兩 B 兩 / 兩 ⵜB 兩 is greater than the probe spacing.
Plasma parameters are obtained from a small Langmuir
probe ( ␲ r 2 ⯝2.6 mm2 ). Density fluctuations are detected
with a cylindrical probe 共0.5 mm diameter, 2 mm length兲
connected to a 50 ⍀ coaxial cable. Microwave emissions
near the plasma frequency are done with a superheterodyne
receiver. Light emission from the plasma is detected with a
photomultiplier tube and resolved with a McPherson scanning monochromator. The diagnostic tools for microwave
and light measurements will be described in more detail in
Part IV.
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Stenzel et al.
Phys. Plasmas, Vol. 10, No. 7, July 2003
FIG. 3. Measured vector fields (B y ,B z )(0,y,z,t ⬘ ) in plasma at different times after switch-off of the coil current (t ⬘ ⫽0). 共a兲 Initial field topology when
BH 储 ⫺B0 ; 共b兲 topology when I coil⫽0; 共c兲 field-reversed configuration at t ⬘ ⫽8 ␮ s; 共d兲–共f兲 corresponding data for BH 储 B0 . The steady state field configuration (t ⬘ ⫽0) rapidly relaxes to the uniform background field (B0 ⬇5 G兲.
III. EXPERIMENTAL RESULTS
A. Magnetic field topologies
The change in the magnetic field topology during the
free relaxation of the field induced in plasma by the Helmholtz coil is displayed in three snapshots of measured vector
fields (B y ,B z ) in the central y–z plane (x⫽0) in Figs. 3共a兲–
3共c兲 for BH 储 ⫺B0 and Figs. 3共d兲–3共f兲 for BH 储 B0 . For the
former case, as the coil current starts to decrease (t ⬘ ⫽0), the
field topology is as in vacuum 关Fig. 3共a兲兴: an FRC. After the
coil current has vanished 关 t ⬘ ⯝3 ␮ s, Fig. 3共b兲兴, the field near
the individual coils has decayed, but no significant change
has taken place in the center of the Helmholtz coil. Because
the bulk field is now similar to the vacuum field, this implies
that plasma currents comparable to the coil current (I coil⫻8
turns ⬇800 A兲 have been induced. Since the field near the
individual coils is lower than before, the plasma currents
must be distributed over a region (⯝4 cm兲 larger than the
thin Helmholtz coil windings. As this is an EMHD plasma,
the currents are carried by electrons. It is shown in the following that the current is predominantly a Hall current,
driven by inductive and space-charge electric fields associated with the decaying magnetic fields. There are no conducting boundaries with induced currents nearby. The openloop coils have no influence on the distribution of plasma
currents. The relaxation of the magnetic field proceeds selfconsistently as in free space. The currents are able to redistribute themselves freely and are not localized to a thin current sheet in the magnetic neutral layer where EMHD breaks
down.
Observation shows that from t ⬘ ⫽3 ␮ s to t ⬘ ⫽8 ␮ s, the
FRC contracts radially and expands axially leading to the
configuration shown in Fig. 3共c兲. The axial elongation and
radial contraction are explained in the following as a result
of magnetic field lines being frozen into electron fluid flow.
The toroidal component of the magnetic field forms during
the axial expansion. Although many aspects of the magnetic
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Phys. Plasmas, Vol. 10, No. 7, July 2003
evolution are explained by EMHD, the rapid disappearance
of flux near the wire indicates a breakdown of EMHD. Since
the electron fluid is frozen to the magnetic flux in EMHD,
the disappearance of the flux would require the electrons to
flow into the wire violating ⵜ•J⫽0.
For t ⬘ ⬎8 ␮ s, the FRC maintains a similar structure to
Fig. 3共c兲 but decays in amplitude and shrinks axially, eventually relaxing to the uniform background field. The decay
time scale ( ␶ ⬇15 ␮ s兲 is long compared to a whistler transit
time across the magnetic structure ( ␶ w ⬇3 ␮ s for v whistler
⬇10 cm/ ␮ s and L B⬇30 cm兲 but short compared to the classical diffusion time scale ( ␶ diff⫽ ␮ 0 ␴ SpitzerL B2⬇1.5 ms for
␴ Spitzer⬇150 ⍀ ⫺1 cm⫺1 at kT e ⯝4 eV兲.16 The 15 ␮ s time
scale corresponds to a conductivity of ␴ * ⫽15 ␮ s/( ␮ 0 L B2)
⯝1.3 ⍀ ⫺1 cm⫺1 . This anomalously low conductivity is
thought to be due to ion turbulence observed during the decay of the FRC as shown in Part IV. The magnetic moment
of the FRC oscillates slightly around the z axis but does not
exhibit a major tilting instability. The oscillation is due to the
FRC magnetic field being frozen to toroidal Hall electron
flows17 and is discussed in Part II.13
For purpose of comparison, Figs. 3共d兲–3共f兲 show the topology of the total field for the case when the Helmholtz coil
field is along the uniform field, BH 储 B0 . Again, the initial
field is as in vacuum 关 t ⬘ ⯝0, Fig. 3共d兲兴. In contrast to the
case where BH 储 ⫺B0 , the field in the center of the Helmholtz coil begins to decay as soon as the coil current begins
its switch-off (0⭐t ⬘ ⭐3 ␮ s兲. After the coil current has vanished 关 t ⬘ ⯝3 ␮ s, Fig. 3共e兲兴, little magnetic field energy is left
to relax. Thus, the relaxation proceeds much faster than in
the case of the FRC and an almost uniform field is restored at
t ⬘ ⯝8 ␮ s 关Fig. 3共f兲兴. As shown in Part III,14 the magnetic
energy is axially convected away from the Helmholtz coil at
the whistler speed when BH 储 B0 . This case is similar to the
well-documented formation of a whistler wave packet by a
dipole loop antenna.18 The propagation is predicted by
EMHD theory, but just like the BH 储 ⫺B0 case, EMHD
breaks down near the wire as evidenced by the rapid decay
of the field in that region. It should be noted that EMHD
does not predict propagation in the BH 储 ⫺B0 case because
the two 3-D null points on the axis of the Helmholtz coil
关 z⫽⫾28 cm in Fig. 3共a兲兴 stagnate the wave propagation
causing dissipation to dominate. Much less electron heating
and its associated light emission is observed in the propagating case 共see Parts III and IV兲.
In order to describe the field topology in 3-D space, we
show in Fig. 4 the field properties in an orthogonal plane, the
x–y plane in the center of the Helmholtz coil (z⫽0). Figure
4共a兲 displays contours of B z (x,y,t ⬘ ⫽7 ␮ s兲 for the FRC containing a null layer, and Fig. 4共b兲 shows B z (x,y,t ⬘ ⫽3 ␮ s兲
for the case BH 储 B0 without a null layer. For both polarities,
the field shows good azimuthal symmetry. In the FRC case,
the magnetic null layer 共where EMHD breaks down兲 has no
significant influence on the stability of the field distribution.
Note that at t ⬘ ⫽7 ␮ s the field strength at the center of the
FRC is nearly 10 G, which means the field created by the
plasma currents 共which opposes the 5 G background field兲 is
about 15 G. Four microseconds earlier, the field created by
the plasma currents has a magnitude of about 10 G in the
Fields, currents, and flows
2783
FIG. 4. Axial field component B z in the central x–y plane of the Helmholtz
coil (z⫽0). 共a兲 B z (x,y,t ⬘ ⫽7 ␮ s兲 for BH 储 ⫺B0 showing the magnetic null
layer of the FRC. 共b兲 B z (x,y,t ⬘ ⫽3 ␮ s兲 for BH 储 B0 showing a strong net
field on axis. The field has a good azimuthal symmetry for both polarities.
BH 储 B0 case. This is another indication of how the field
strength in the two cases evolves on different time scales.
B. Moving field lines
The concept of moving field lines, often used to demonstrate magnetic reconnection, is meaningful when the field
lines are tied or frozen into the moving fluid.19 This requires
that magnetic field lines be defined and followed as the experiment progresses. Experimentally, the measured vector
magnetic field is used to trace the field lines, and their motion is determined from v⫽⫺J/(ne). However, this method
is not easy to implement and often leads to erroneous results.
For systems with cylindrical symmetry, flux conservation allows an alternate method: magnetic field lines are identical to
contours of constant poloidal flux, ⌽(r,z)⫽ 兰 r0 B(r ⬘ ,z)
•â 2 ␲ r ⬘ dr ⬘ ⫽const where â is identical to a vector parallel
to the inner part of the separatrix. Following such contours as
time advances is then a very simple process.
We showed earlier that the magnetic field configuration
is nearly axially symmetric ( ⳵ / ⳵ ␪ ⬇0). We can therefore calculate ⌽(r,z) once we identify the axis of symmetry. This is
easiest for the steady-state field configuration 关see Figs. 2共a兲
and 2共b兲 of Part II13兴. The symmetry axis is easily identified
as â⫽ẑ. We calculate ⌽(r,z) for both halves of the plane
separately to account for the small differences in symmetry
between the upper and the lower half planes. The calculated
lines of constant poloidal flux are tangential to the vector
field, confirming that they are field lines. The value of the
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Phys. Plasmas, Vol. 10, No. 7, July 2003
FIG. 5. Contours of constant poloidal magnetic flux 共magnetic field lines兲 for BH 储 ⫺B0 in the central y–z plane at different times after turn-on 共a兲–共c兲 and
turn-off 共d兲–共f兲 of the coil current. 共a兲 At early times, field lines are open but distorted near the coils. 共b兲 With increasing coil current (t⫽5.8 ␮ s兲, closed field
lines encircle each coil, developing X-type magnetic nulls lines that converge to the axis and form 3-D null points 共c兲. 共d兲 During the decay of the coil current,
closed field lines around the coils vanish while the rest of the FRC remains relatively unchanged. 共e兲 In the absence of coil currents, the FRC is maintained
by electron currents, which form a long neutral sheet 共f兲.
flux contour has no relation to the local field strength but
identifies a particular field line. The zero contour defines the
separatrix, the boundary that separates open field lines connect to B0 from closed field lines within the FRC. This is
clearly understood as follows: initially, the flux is zero at the
symmetry axis. Since the field lines are in a single direction
above and below the axis, the flux monotonically increases
above/below the axis. If the field lines are closed, the flux
reaches a maximum and decreases until it changes sign. The
integral has accounted for all the field lines enclosed by the
separatrix at that point. Hence, the ⌽(r,z)⫽0 contour defines the separatrix for axially symmetric magnetic field
configurations.
The time evolution of field lines is shown in Fig. 5 for
BH 储 ⫺B0 and in Fig. 6 for BH 储 B0 . As the coil current begins to grow (t⫽1.6 ␮ s兲, Fig. 5共a兲 shows still open field
lines associated with B0 , but distorted near the Helmholtz
coil, which intersects the y–z plane at four points (y⯝⫾8
cm, z⯝⫾8 cm兲. The first magnetic null lines form just in-
side each of the individual coils as the current increases.
These are X-type nulls in the y–z plane with a toroidal separator. With increasing coil current, these null lines move radially toward the z axis. The motion of the null lines causes
the field lines between the coils to become highly stretched
and to reconnect forming an X- and O-type null lines in the
center plane (z⫽0, r⬍r coil) as shown in Fig. 5共b兲 (t⫽5.8
␮ s兲. The O-type null is bounded by a separatrix through the
3 X-type nulls. In vacuum 共not shown兲, two X-type null lines
are also formed in the coil planes but no nulls in the center
plane. O-type nulls cannot form in vacuum since currents
flow only in the coils.
With rising coil current the closed flux around the coils
increases, which pushes the two X nulls in the coil planes
toward the axis. The X null in the mid-plane remains almost
stationary. Although reconnection at the X nulls feeds flux
into the O null it does not grow, presumably due to field line
annihilation. When the field lines trapped within the separatrix near the axis are all reconnected the two X nulls and O
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FIG. 6. Contours of constant poloidal magnetic flux 共magnetic field lines兲
for BH 储 B0 in the central y–z plane slightly after turn-on 共a兲, steady state
共b兲, and after turn-off 共c兲 and 共d兲 of the coil current. 共a兲 At early times field
lines are open but distorted inward near the coils. 共b兲 Field lines are radially
compressed as the coil current increases, leading to formation of null lines at
steady state. 共c兲 After switch-off of the coil currents (t ⬘ ⲏ 3 ␮ s兲, the field
lines straighten out rapidly due to convection of magnetic energy along
⫾B0 . 共d兲 The emission of two whistler wave packets causes field line compression (z⫽⫾25 cm兲 and rarefaction (z⯝0).
null reach the axis. The O null vanishes and the two X nulls
turn into 3-D null points 关 t⬇8.1 ␮ s, Fig. 5共c兲兴. The null
point at z⯝8 cm is characterized by opposing field lines
approaching the point along the z axis 共this axis is generally
referred to as the ‘‘spine’’兲 and field lines flaring out from the
null point along two conical surfaces that intersect at the null
共these surfaces are referred to as the ‘‘fan’’ planes兲. The null
Fields, currents, and flows
2785
point at z⯝⫺8 cm is essentially identical, but with oppositely directed magnetic fields. A detailed classification of
these types of null point structures is given in Ref. 20, where
these nulls are termed ‘‘regular.’’ At this time, there are two
nested separatrix surfaces, one, where ⌽⫽0, separating
closed field lines around both coils from open field lines, and
the second one, separating closed field lines around each coil
from closed field lines encircling both coils. In contrast to the
first separatrix, this second separatrix cannot be assigned any
particular value because it separates closed field lines from
other closed field lines. Further increase in the coil current
pushes the 3-D null points axially outward until the steadystate condition is reached 共see Fig. 2 of Ref. 17兲. The topological changes during the rise of B H are produced by reconnection driven by the Helmholtz coil current. Flux
originating from the coils reconnects at the axially symmetric X line to form common flux around both coils, as discussed in Sec. III C. However, as shown in Sec. III D, no
flux is transferred across the outer separatrix surface and any
additional flux simply increases the volume bounded by the
outer separatrix surface.
When the coil current is switched off, the field-reversed
topology is maintained by induced electron currents 关Fig.
5共d兲兴. These are not confined to the location of the coils but
flow in an axially elongated layer. The two original O points
merge into a single O-type null. During the relaxation, the
3-D null points slowly move away along the z axis 关Fig.
5共e兲兴, while the O-type null layer moves radially inward 关Fig.
5共f兲兴, thereby elongating the FRC. The elongation is explained in the following as due to the frozen-in condition. As
the 3-D null points migrate outward, the axis of symmetry, â,
is observed to tilt, e.g., by about 6° at t ⬘ ⫽3 ␮ s in Fig. 5共e兲.
The symmetry axis is identified from the unit vectors of
(0,B y ,B z ) at each time step, and the flux integral is performed along a line perpendicular to this axis. The tilting of
the FRC is discussed in detail in Part II.13 As time
progresses, closed field lines disappear without reconnecting.
They annihilate as they converge into the neutral sheet. This
is discussed in more detail in Sec. III E. The 3-D null points
and the null line collapse near the center when the closed
flux has eroded. However, when B⫽0, there are still strong
plasma currents flowing which produce B⫽5 G opposite to
B0 . In the absence of a null layer, these are electron Hall
currents.
When the Helmholtz coil field is reversed (BH 储 B0 ), the
field line motion is as shown in Fig. 6. The initial field perturbation near the coils 关Fig. 6共a兲兴 penetrates rapidly into the
plasma. Closed field lines around the coils reconnect at
X-type null lines that lie outside the coils and, in this case,
the measurement volume 共see Fig. 2兲. These carry small
electron currents such that vacuum-like reconnection allows
fast field penetration within t⫽5 ␮ s. The addition of the coil
field compresses the magnetic field inside the coil in steady
state 关Fig. 6共b兲兴. When the coil current is switched off, the
closed field lines around the coils vanish, and whistler wave
packets are emitted, as explained in Sec. III F. This reduces
the compression and the field lines in the center of the coil
straighten out 关Fig. 6共c兲兴. As the packets propagate, the field
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2786
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Phys. Plasmas, Vol. 10, No. 7, July 2003
FIG. 8. Absence of reconnection at the 3-D null points on axis. Flux contours ⌽(y,z)⫽const 共field lines兲 are shown at four consecutive time steps in
the vicinity of the left 3-D null point during the early phase of the FRC
relaxation. Field lines do not move through the null point but all lines
convect together in the ⫺z direction.
FIG. 7. Driven reconnection at the X-type null point located between the
coils (z⫽0, r⯝r coil). The motion of selected field lines (⌽⫽const) is
shown in four consecutive time steps. 共a兲 Closed and open field lines are
separated by a crossing field line, the separatrix (⌽⫽1 kG cm2 ). 共b兲 Increasing flux forces the former separatrix into the open field line region,
with a different field line (⌽⫽1.5 kG cm2 ) becoming the separatrix, a process that is repeated and 共c兲 and 共d兲.
lines are compressed and develop the opposite curvature 关 z
⫽⫾25 cm, Fig. 6共d兲兴. Thus, magnetic energy is rapidly convected rather than dissipated. This case is similar to the extensively studied generation of a whistler wave packet by a
single loop antenna.18
C. Driven reconnection
During the rise of the coil current, opposing magnetic
field lines move against one another at the X-type null line
between the coils 关 r⯝r coil , z⫽0, Fig. 5共b兲兴. The motion of
selected field lines is shown in Fig. 7 by snapshots of flux
contours, ⌽(y,z)⫽const, at four consecutive time steps in a
y–z plane at x⫽0. At t⫽7.8 ␮ s 关Fig. 7共a兲兴 after turn-on, the
X-type line is clearly displayed. Note that the X line is not as
in vacuum, because during the rise time of the coil current
there is an additional component, B z,p , due to inductively
driven plasma currents. At this selected time, the ⌽⫽1
kG cm2 contour defines the separatrix between the field lines
that close around each coil (⌽⬎1 kG cm2 ) from those that
close around both coils (⌽⬍1 kG cm2 ). At a later time 关 t
⫽8.8 ␮ s, Fig. 7共b兲兴, the field line defined by the ⌽⫽1
kG cm2 contour has reconnected and moved outside the
separatrix now defined by ⌽⫽1.5 kG cm2 . At t⫽9.8 ␮ s
关Fig. 7共c兲兴, the separatrix is now defined by the ⌽⫽2 kG
cm2 contour, which has moved into the null line and reconnected. Reconnection at the X line continues until steady
state (dI coil /dt⫽0) is achieved at t⯝39 ␮ s, when the X-line
location is identical to that of the vacuum case.
For the time range displayed in Fig. 7, the flux at the X
line (r⯝13 cm兲 changes by 2.5 kG cm2 , generating an in-
ductive electric field of E ␪ ⯝⫺60 mV/cm along the X line. If
the electrons were fully unmagnetized in the vicinity of the X
line, this electric field would drive a J ␪ ⯝ ␴ 储 E ␪ ⯝1.5 A/cm2 .
However, the total B field is strong enough to keep the electrons magnetized throughout the region. Calculation of J x ,
equivalent to J ␪ in this plane, from ⵜ⫻B 共not shown兲 indicates that the current is not localized at the X line, and that it
flows across the field lines as a Hall current. Since it is
spread over a large area, the current density at the X line is
only J ␪ ⯝0.5 A/cm2 . This is not surprising since the separatrix field lines intersect at approximately 84°, which implies
a nearly zero current density at the X line. If the field lines
intersected at 90°, Maxwell’s equations (ⵜ•B⫽0) require
that the current density be identically zero. Thus, the plasma
currents reduce the plasma reconnection rate by 10% of the
vacuum rate,21 and the field line dynamics are predominantly
driven by the Helmholtz coil current. Driven reconnection,
however, is not the main focus of the present work as it
previously has been studied by many authors.11,22–26
D. Absence of reconnection at 3-D null points
After the FRC has been established and a steady-state
achieved, the Helmholtz coil is turned off and the relaxation
of the FRC is observed. Because considerable interest has
been shown in reconnection processes near 3-D null points in
space plasma physics,27,28 where three types of reconnection
processes have been identified, we turn our attention to the
dynamics of the magnetic field lines near the cusp-type null
points at (0,0,⫾24 cm兲 that form the axial boundary of the
FRC. Figure 8 displays snapshots of flux contours ⌽(y,z)
⫽const at four consecutive time steps in the vicinity of the
3-D null point located at z⫽⫺24 cm immediately after the
Helmholtz coil current begins its turn-off (t ⬘ ⫽0).
Three key observations are 共1兲 the axis of the FRC exhibits a slight tilt, 共2兲 the null point and separatrix structure
move together in the ⫺z direction, and 共3兲 magnetic field
lines do not move through the null point. The FRC tilts and
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Phys. Plasmas, Vol. 10, No. 7, July 2003
Fields, currents, and flows
2787
eventually processes because toroidal electron drifts cause
the field lines to rotate, as discussed in Part II.13 The motion
of the separatrix structure in the ⫺z direction is due to poloidal electron flows, v z , which carry the frozen-in magnetic
fields axially away from the center of the FRC, but never
toward it. This flow is examined later on in Sec. III G. Because no field lines move through the null point, there is no
flux transfer across the separatrix, hence no reconnection.
Furthermore, because the system is axially symmetric, the
calculation of the flux contours demands that field lines inside the separatrix always be associated with positive flux
contours while those outside are always negative 共see Sec.
III F兲. It is not possible to have a positive flux contour pass
through the separatrix, because there cannot be a positive
contour outside. The situation here is not the same as the
driven reconnection discussed in the previous section where
flux inside and outside the inner separatrix are of one sign.
Since the enclosed flux does erode in time 关i.e., the peak
value of ⌽(y,z) inside the FRC decreases兴, it must be removed by a different process, namely annihilation.
E. Magnetic annihilation at an elongated O-type null
layer
Magnetic reconnection of opposing antiparallel field
lines has been termed field line ‘‘annihilation.’’4,29 This onedimensional limit may approximately apply to a very elongated 2-D X, Y, or O-type neutral sheet. The present FRC
develops an oblate neutral sheet and the field lines are observed to be annihilated in it. Figure 9 displays snapshots of
a single flux contour, ⌽(y,z)⫽600 G cm2 , at five consecutive time steps in the vicinity of the axially symmetric
O-type null layer. Because of the symmetry, only the upper
half plane of the y–z plane needs to be shown. This single
flux contour 共or field line兲 is chosen for clarity and is characteristic of the other adjacent field lines which behave similarly. After the start of the free relaxation 关Fig. 9共a兲, t ⬘ ⫽3
␮ s兴, the closed field line begins to stretch axially and contract radially resulting in the configuration of Fig. 9共b兲, t ⬘
⫽6 ␮ s. By t ⬘ ⫽7 ␮ s, the field line assumes a very elongated
loop 关Fig. 9共c兲兴, with adjacent sides of the loop forming opposing field lines. At t ⬘ ⫽7.5 ␮ s, the opposing lines begin to
pinch inward at z⯝⫾10 cm and form three small islands,
shown in Fig. 9共e兲 at t ⬘ ⫽8 ␮ s, and finally decay. The FRC
topology changes by closed field lines annihilating at the
elongated neutral sheet, not by reconnecting with the open
field lines at the 3-D null points. The field lines evolve in this
way because they are frozen into the electron fluid. The
flows 共as shown in Sec. III G兲 are consistent with the predictions of EMHD theory.
F. Convection of magnetic field
Although the same initial magnetic field strength, B H , is
applied to the plasma for both polarities, BH 储 ⫾B0 , the field
relaxation rates and mechanisms appear to be quite different,
judging from the magnetic topology observed. To investigate
this difference, the poloidal magnetic flux due to magnetic
field lines created by plasma currents alone is calculated. The
contribution of the background field, B0 , and the Helmholtz
FIG. 9. Magnetic field line tearing and annihilation at the elongated O-type
null layer. A single flux contour ⌽(y,z)⫽600 G cm2 is followed in time in
the upper y–z plane. 共a兲,共b兲 The closed field line stretches axially and contracts radially during the free relaxation. 共c兲,共d兲 The elongated field line
moves into the null layer and 共e兲 tears into three small islands, which eventually vanish. All other flux contours behave similarly.
coil field, BH , are ignored because they simply introduce
offsets when added to that of the propagating plasma
currents, ⌽ p (z,t)⫽ 兰 关 B z (r,z,t)⫺B 0 ⫺B H,z (r,z,t) 兴 2 ␲ r dr.
The limit of the integral is forced to be the radial location
where the z component of the plasma magnetic field changes
sign. Choosing this limit prevents counting the return flux
which, if fully included, would lead to a value of zero.
Figure 10 shows the poloidal magnetic flux calculated in
this fashion for both directions of B0 . For the BH 储 ⫺B0 case
关Fig. 10共a兲兴, the contours indicate that the flux is constrained
to decay within the volume bounded by the slowly propagating separatrix. The constraint arises because the relaxing
wave packet, which consists of whistlers, approaches the 3-D
null points with decreasing velocity. The change in velocity
is due to the dependence of the plane wave whistler group
speed on the square root of the magnetic field, v g
⯝2(c/ ␻ pe )( ␻␻ ce ) 1/2. Thus the field stagnates at the null
points. Because the expanding separatrix leaves the measurement volume, it is difficult to determine how much flux does
get transmitted through it. However, it has been shown elsewhere that flux transmission through a stationary separatrix
is extremely inefficient.30 On the other hand, the magnetic
field residing outside the separatrix is free to travel away, and
its reflection from the chamber end wall is observed at 10 ␮ s
⬍t ⬘⬍15 ␮ s.
In contrast, the flux rapidly propagates away, v w⬇18
cm/␮ s, from the center of the Helmholtz coil for the BH 储 B0
case 关Fig. 10共b兲兴. This is consistent with the propagation first
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2788
Stenzel et al.
Phys. Plasmas, Vol. 10, No. 7, July 2003
FIG. 10. Poloidal magnetic flux, ⌽ p (z,t ⬘ ), in the z–t ⬘ plane showing the
difference between convection and dissipation. 共a兲 For BH 储 ⫺B0 , the flux
diffuses axially and decays mainly within the separatrix 共dashed line兲, while
for the other case 共b兲, it propagates away from the end of the Helmholtz coil
(z⯝⫺8 cm兲 and reflects at the chamber wall (z⯝⫺90 cm兲. 共c兲 Flux spatial
decay in the presence of null points (BH 储 ⫺B0 ) is much stronger than for
propagation in the whistler mode along open field lines (BH 储 B0 ).
observed in the net field lines of Fig. 6. For the given parameters 关 B tot⬇20 G, n e ⬇1⫻1012 cm⫺3 , f ⬇1/(⌬t)⫽1/(3 ␮ s兲兴,
the speed is in good agreement with that predicted from the
whistler dispersion relation, v g ⬇30 cm/␮ s. Because the field
seen by the packet decays to the ambient field, the speed of
the reflected wave packet 共10 ␮ s ⬍t ⬘ ⬍15 ␮ s兲 is reduced
accordingly. Additionally, reflection at the end wall disperses
the wave packet changing its frequency spectrum and, therefore, its central group velocity. The magnitude of the contours indicates that the field created by the plasma is smaller
for this case than when BH 储 ⫺B0 . The difference is understood when one examines the response of the plasma while
fulfilling Faraday’s law: in the former case the changing
magnetic field cannot propagate away and thus is maintained
past the end of the coil current, while in the latter the field
carries away the source currents as it travels.
Figure 10共c兲 shows the spatial decay characteristics of
the flux. It is obtained by plotting the peak value of ⌽ p (z,t)
at each z position. To make a comparison between the two
Helmholtz coil polarities easier, each case is normalized to
the peak flux value at z⫽0, ⌽ p (0). For BH 储 ⫺B0 , the flux
decays over a much smaller spatial scale length with an
e-folding distance of 26 cm. The concentrated flux loss results in significant local electron heating and corresponding
light emission that is not observed in the BH 储 B0 case.14 The
BH 储 B0 case has a much longer decay distance of 62 cm,
consistent with collisional damping seen in single-loopantenna generated whistler waves.18
FIG. 11. Schematic of fluid flows and frozen-in field open lines of a relaxing
EMHD FRC. 共a兲 A toroidal electron flow, v tor , in a cylinder of finite axial
length, rotates an axial field line B, equivalent to producing toroidal field
components ⫾B tor where ⳵ v tor / ⳵ z⫽0. 共b兲 The source for the toroidal fields
are two poloidal electron flows, ⫾ v pol , which convect the magnetic field
lines axially and radially elongating and compressing the FRC.
G. Field line twisting and stretching by fluid flows
The observed magnetic field line dynamics can be interpreted as convections due to field lines frozen into electron
flows. In EMHD, the electron flows and corresponding field
line convection are coupled in ways that are not analogous to
MHD. To provide a theoretical backdrop, a qualitative picture of what EMHD predicts for this experiment is presented,
and the experimental results that quantitatively confirm the
EMHD predictions are given. This is an important result because the parameter regime of the plasma gives a calculated
classical electron mean free path that is similar in size to the
magnetic field structures. This would seem to indicate that
fluid theory is not applicable to this experiment. However,
ion-acoustic turbulence, discussed in Part IV, decreases the
electron mean free path permitting fluid theory to apply.
The rise and fall of the coil current induces a plasma
current that is primarily toroidally directed, consistent with
Lenz’s law. Because all currents are carried by electron drifts
in EMHD, the corresponding electron fluid flow speed is
given by v ⫽⫺J/ne. The frozen-in condition of EMHD predicts that this flow sweeps the axial magnetic field in the
toroidal direction. This is shown schematically in Fig. 11共a兲
for an open field line in the FRC case, i.e., a line outside the
separatrix surface. Since the toroidal drift exists only near the
Helmholtz coil, the field lines are only rotated in the center
( 兩 z 兩 ⬍ axial FRC length兲 but not elsewhere. This implies that
near each end of the Helmholtz coil, toroidal field components of opposite signs are generated, one to produce the
twist, the other to undo it. No net twist or helicity, H
⫽ 兰 A"B dV, where A is the magnetic vector potential, is
generated. The toroidal field must be generated by an axially
directed current which must close radially, i.e., a local poloidal current system, (J r , J z ), is formed. This current system
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Phys. Plasmas, Vol. 10, No. 7, July 2003
Fields, currents, and flows
2789
FIG. 12. Measurement of electron flows and magnetic field during the free relaxation of the FRC. 共a兲 Toroidal electron flow, ( v x , v y )⫽⫺ⵜ⫻B/( ␮ 0 ne) in the
central x – y plane. The electron flow inside the null layer (r⯝13 cm兲 is driven by the toroidal inductive electric field E ␪ ⬀⫺ ⳵ B z / ⳵ t, while the cross-field Hall
drift is due to a radial space charge electric fields Er . 共b兲 Poloidal electron flow, ( v y , v z )(0,y,z). The axial flow convects the radial field component outward,
the radial flow convects the axial field components radially inward. This produces a highly elongated FRC. 共c兲 Strong B x (0,y,z) produced by the toroidal twist
at the ends of the FRC, B x ⬍0 for y⬎10 cm, z⯝⫺20 cm and B x ⬎0 for y⬍⫺10 cm, z⯝⫺25 cm. In the center of the FRC (z⯝0), B x is due to a
tilt/precession 共Ref. 13兲.
is centered at the radial position where B tor is formed, outside
the separatrix. The corresponding poloidal flows, vpol , are
sketched in Fig. 11共b兲. The effect of the axial flow is that it
convects the radial fields of the FRC axially outward, while
the radial flow convects the axial field lines radially inwards.
This results in the observed elongation of the FRC shown in
Figs. 5共d兲–5共f兲. Furthermore, the tearing into three islands
共Fig. 9兲 is likely caused by the radial inflows at two axial
locations off center, z⯝⫾10 cm. Because the presented
magnetic field measurement is an average (N⫽10) at each
spatial location, it is clear that the position of the island
formation is reproducible.
The qualitative predictions of EMHD theory shown in
Fig. 11 are supported by laboratory observations. The toroidal electron flow that carries the axial magnetic field lines
in the toroidal direction is calculated from the current density, J⫽ⵜ⫻B/ ␮ 0 . Figure 12共a兲 shows vectors of vtor(x,y)
in the z⫽0 plane. Outside the null layer (r null⯝13 cm兲,
the flow is a Hall electron drift due to a radial space-charge
electric field, vd ⫽Er ⫻zB z /B z2 , parallel to the inductive
electric field E ␪ , while inside the null layer, the flow is
antiparallel to E ␪ . Electron pressure gradient drifts, vdia
⫽ⵜ(nkT e )⫻B/(neB 2 ), cannot account for the observed
drift, because they point in the opposite direction of the drift
for B z ⬎0. The poloidal flow, vpol(y,z), that causes the axial
expansion of the FRC and the island formation of the field
lines is presented in Fig. 12共b兲 in the x⫽0 plane. The flows
are peaked off-axis at the radius of the toroidal current layer.
As predicted, the poloidal flow/current creates the out-ofplane toroidal field component shown in Fig. 12共c兲. Note that
B ␪ ⫽⫺B x for y⬎0 and B ␪ ⫽B x for y⬍0. The contour plot
shows that B x is largest near the outer edges of the y–z plane
(z⬇⫾20 cm, y⬇⫾12 cm兲, i.e., near the ends of the FRC,
and changes sign with y and z. Thus, the electron flow along
the separator creates a similar quadrupole-like structure as
theoretically predicted for a 2-D X-type null point.3,31,32 In
the center of the FRC, the B x component is caused by a
different effect, i.e., a tilt of the FRC.13
The toroidal field modifies the structure of the field lines
that intersect 3-D null points. In the absence of a toroidal
field, the null point is classified as a simple radial cusp-type
null, otherwise it is classified as a spiral null point. Detailed
definitions of these terms and how null points are quantitatively classified are given in Ref. 20. A few characteristic
field lines are traced in Fig. 13 near the null point at z
⯝⫺30 cm. Closed field lines along the spine flare out in a
fan to form part of the separatrix surface. Open field lines
just outside the separatrix 共not shown for clarity兲 have the
same fan lines but oppositely directed spine lines. The twist
of the fan lines is direct evidence of the toroidal field, B ␪ ,
produced by field line rotation in the toroidal electron flow.
Since the twist increases with radius, the fan lines are purely
radially directed near the axis, but deviate from the radial
direction with the toroidal displacement growing radially.
The field line twist/helicity density changes from lefthanded/negative inside the separatrix to right-handed/
positive outside the separatrix. The same holds true for the
other null point at z⯝30 cm, so the entire FRC has no net
FIG. 13. Measured field topology of a 3-D spiral null point of the relaxing
FRC. The field lines are traced from the inner spine to the left fan assuming
⳵ / ⳵ ␪ ⫽0. The twist of the fan lines is a manifestation of B ␪ and v z .
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2790
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Phys. Plasmas, Vol. 10, No. 7, July 2003
FIG. 14. 共a兲 Schematic picture of an axial magnetic field line frozen into the
toroidal electron flow of the decaying field with BH 储 B0 . The field line
rotation creates toroidal field components ⫾B ␪ where ⳵ v tor / ⳵ z⫽0. 共b兲 Measured toroidal field ⫾B x (0,y,z) and poloidal currents Jpol⫽(J y ,J z ) during
the free relaxation of the field shown in Figs. 3共c兲 and 3共d兲.
helicity. During the turn-on of the coil current, the toroidal
and poloidal electron drifts are reversed from those during
current turn-off. The axial inflow now creates a concave
separatrix surface near the 3-D null points such that the FRC
becomes oblate 关Fig. 5共c兲兴 compared to the prolate shape in
steady state 关Fig. 5共d兲兴 or decay 关Fig. 5共f兲兴.
When the Helmholtz coil field is parallel to the uniform
field, the drifts are reversed from that of the FRC. Figure
14共a兲 presents a schematic picture of the field line twist and
Fig. 14共b兲 shows data of the resulting out-of-plane toroidal
field component B ␪ and poloidal currents (J y ,J z )(y,z). The
toroidal electron drift rotates the field lines, a radial drift
compresses the field lines, and an axial drift stretches them.
Toroidal and poloidal drifts are self-consistently coupled resulting in 3-D field topologies in EMHD. During the current
rise, the axial elongation and radial pinching of the field
perturbation induces the emission of two propagating current
vortices. As the coil current is turned off, a radial electron
outflow is established near the center of the coil due to E
⫻B drifts.14 This outward flow is balanced by inward flows
axially away from the coil. As the drift region expands axially a compression front moves away from the coil center,
causing the field lines to bulge, as observed in Fig. 6共d兲. The
result is a propagating displacement of the magnetic field
lines transversely to the background field, a characteristic of
propagating whistlers and Alfvén waves.19
A quantitative check of the frozen-in concept for both
orientations of B is presented in Fig. 15. The toroidal displacement of the electron fluid in the axial center of the
relaxing FRC (z⫽0) is determined from s v (t)⫽ 兰 v tor dt ⬘
FIG. 15. Comparison of the toroidal displacements s(t) of the electron fluid
and the magnetic field lines to check the frozen-in condition. 共a兲 During the
relaxation of the FRC, the fluid rotation and field line rotation assume the
same values after current switch-off (t ⬘ ⬎3 ␮ s兲. 共b兲 During the turn-on of
the coil current with BH 储 B0 , the growing angular displacements of the fluid
and the field lines agree when the whistler vortex is located within the
measured volume.
⫽⫺ 兰 t0 J tor /(ne) dt ⬘ and plotted versus time in Fig. 15共a兲. At
the chosen radius (r⯝12 cm兲, the fluid rotates by s v ,max
⯝15 cm or ␪ ⯝72°. The rotation is limited because the toroidal current moves in time radially inward. The net field
line rotation at the same radius is determined by s B (t)
z
⫽ 兰 ds⫽ 兰 0max(B tor /B z ) dz ⬘ . After the coil current has vanished (t ⬘ ⬎ 3 ␮ s兲, the field line is displaced by essentially the
same distance as a fluid element. For the case of BH 储 B0 and
during the turn-on of the coil current, the angular displacements of the fluid and the field lines grow in time 关Fig.
15共b兲兴. There is good agreement as long as the propagating
whistler vortex remains within the measurement volume. After the propagating twist B tor leaves the region of observation, the field line displacement cannot be accounted for.
Within these constraints, we conclude that the magnetic field
is frozen into the electron fluid.
H. Neutral sheet currents versus Hall currents
Reconnection is associated with the formation of current
sheets in regions of magnetic neutral sheets. The thickness of
the current sheet is thought to be of order L, the scale length
on which the magnetic Reynolds number is R m ⫽ ␮ 0 ␴ v AL
⯝1, where v A is the Alfvén speed in MHD plasmas. For the
classical conductivity, ␴ Spitzer , this value leads to unphysically small values in collisionless plasmas. Anomalous resistivity, inertial, and Hall effects are thought to broaden the
current sheet but the topic is still under investigation. Recent
laboratory experiments have shown that current sheets in
MHD plasmas assume a thickness comparable to the ion inertial scale, c/ ␻ pi . 24,33,34 Much less is known in EMHD
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Phys. Plasmas, Vol. 10, No. 7, July 2003
FIG. 16. 共a兲 Toroidal electron current density, J x , in the center of the relaxing FRC vs radius y and time t ⬘ . Current layer shrinks radially in time
like the magnetic null layer 共dashed black contour兲, but current does not
exactly peak where B⫽0. 共b兲 Corresponding data for the case BH 储 B0 without null points. Hall current rapidly decays in the central plane (z⫽0) due
to the emission of two whistler vortices in ⫾B0 direction.
plasmas about the current sheet thickness. By analogy to
MHD, one would expect for the present parameters a sheet
thickness of either L⯝( ␮ 0 ␴ v ) ⫺1 ⭐ 1 mm, for v ⫽ v whistler
⯝10 cm/ ␮ s and ␴ ⫽ ␴ Spitzer⫽150 ⍀ ⫺1 cm⫺1 (kT e ⯝4 eV兲,16
or the electron inertial length c/ ␻ pe ⯝1 cm at n e ⯝5⫻1011
cm⫺3 . Neither of these values is observed in the present
experiment.
From the measured magnetic field, the current density is
obtained via Ampère’s law. The dominant component is
given by J x ⫽( ⳵ B z / ⳵ y⫺ ⳵ B y / ⳵ z)/ ␮ 0 . The peak of J x usually occurs at a smaller radius than that for B z ⫽0. Figure 16
presents contour plots of J x versus radius and time for both
polarities of BH . For the FRC 关Fig. 16共a兲兴, the null layer
共dashed black contour兲 initially expands beyond the individual coils at t ⬘ ⯝2 ␮ s. The expansion of the null layer
coincides with the rapid field decay near the coils seen in
Fig. 3. In time, the peak current density and the null layer
move radially inward and converge. For t ⬘ ⬍10 ␮ s, the peak
current density remains inside the null layer during the convergence. Because the peak current flows in the magnetized
region of the plasma 共not in the neutral layer兲 and it is perpendicular to the field lines, the current is predominantly a
Hall current. This is the current mentioned in Sec. III A that
satisfies the force-free condition of EMHD and accounts for
the stability of the FRC structure. Despite this stability, the
FRC’s magnetic moment oscillates around the z axis, causing
the decrease and subsequent increase of J x observed at t ⬘
⯝6 ␮ s in Fig. 16共a兲. The oscillation is discussed in Part II.13
For the opposite polarity (BH 储 B0 ) shown in Fig. 16共b兲,
the toroidal current profile exhibits markedly different behavior. The currents form two whistler wave packets which
propagate away from the Helmholtz coil region in the ⫾B0
direction. It was mentioned in Sec. III A that the whistler
transit time is about ␶ w ⬇3 ␮ s. This time scale is consistent
with the time it takes for the majority of J x to vanish in the
figure.
Fields, currents, and flows
2791
FIG. 17. 共a兲 Radial profiles of the axial field component B z (y) at x⫽z
⫽0, t ⬘ ⫽6 ␮ s for both polarities of BH . Note the field reversal for
BH 储 ⫺B0 and the absence of null layer and faster relaxation for BH 储 B0 .
共b兲 Radial profiles of the toroidal current density, J x (y)⯝ ␮ ⫺1
0 ⳵ B z / ⳵ y. For
the FRC, the current is not localized in the magnetic null layer but flows
adjacent to it as a Hall current across field lines. The widths of the current
‘‘sheets’’ with or without the magnetic neutral layer are comparable, ⌬y
⯝7 cm⯝7c/ ␻ pe .
Although the time evolution of the current profile appears quite different for the two cases, the fact that both
current systems are Hall currents leads to similar spatial profiles in J x . To see this, radial profiles of the dominant magnetic field component B z (y) and J x (y) at x⫽z⫽0 and t ⬘
⫽6 ␮ s are shown in Figs. 17共a兲 and 17共b兲, respectively. In
the FRC case, the magnetic null layer is seen at y⫽⫾10 cm.
For BH 储 B0 , there is no sign change in the magnetic field,
but B H,z has decayed more rapidly due to the vortices convecting fields away from the Helmholtz coil region. Figure
17共b兲 shows the radial profiles of the current density, J x (y),
scaled differently for the two configurations. In both cases,
the current ‘‘sheet’’ has a full width at half maximum of
⌬y⯝7 cm, which is much broader than the electron inertial
length c/ ␻ pe ⯝1 cm or the electron cyclotron radii, r ce ⯝1
cm (kT e ⫽3 eV, B⫽5 G兲. There is no enhancement of the
current in the neutral layer for the FRC case. The bulk current, then, is a Hall current flowing perpendicular to the field
lines.
In the FRC case, the current density yields an electron
drift velocity of v d,e ⫽J/n e e⯝107 cm/s, which is small compared to the electron thermal velocity ( v th,e ⯝108 cm/s兲 but
large compared to the ion sound speed 关 c s ⯝(kT e /m i ) 1/2
⯝3⫻105 cm/s兴. With T e ⰇT i , the electron drift excites
current-driven ion sound instabilities. As mentioned previously, the Part IV companion paper15 presents measurements
of the turbulence, where it is proposed as the mechanism for
the anomalous conductivity and the observed decay time that
is much shorter than the classical diffusion time.
The current density, J x 共calculated from the curl of B), is
integrated to find the total toroidal current, I tor⫽ 兰 J x dydz.
Its time dependence is shown in Fig. 18 for both polarities of
BH . The coil current and the plasma currents are both measured. During the quasi steady-state period of I coil (t⯝40
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2792
Stenzel et al.
Phys. Plasmas, Vol. 10, No. 7, July 2003
FIG. 18. Net toroidal current 共coil and plasma兲 vs time t. During turn-on, the
coil current is opposed by induced electron currents, resulting in a slower
current rise than in vacuum. After turn-off of the coil current, plasma currents of comparable strength are induced. Time variations of plasma currents
are slower in the FRC configuration than for BH 储 B0 .
␮ s兲, the measured current agrees well with that in the individual coils. During turn-on (0⬍tⱗ40 ␮ s兲, the coil current
is opposed by induced electron currents, resulting in a slower
net current rise than in vacuum. After turn-off of the coil
current, plasma currents of comparable strength are induced
and decay in time as the field relaxes. Time variations of
plasma currents are slower in the FRC case because the magnetic field is trapped between null points. For BH 储 B0 , the
field propagates out of the volume, which accounts for the
faster current decay. The secondary peak at t⯝57 ␮ s is
caused by the reflection of the wave packet from the chamber
wall, which is about ⌬z⯝45 cm away. This will be discussed
in depth in Part III.14
I. Electric fields
Electric fields in EMHD are both due to induction
and space charges.35 The inductive electric field is due to
the decay of the axial magnetic field, E ␪ ⬀⫺ ⳵ B z / ⳵ t.
We have calculated E ␪ ⫽⫺(d⌽/dt)/(2 ␲ r) from ⌽(x,y)
where
r⫽ 关 (x⫺x c ) 2 ⫹(y
⫽ 兰 r0 dr ⬘ 兰 20 ␲ B z (r ⬘ , ␪ ⬘ ) d ␪ ⬘ ,
2 1/2
⫺y c ) )] and (x c ,y c ) is the center of the FRC, based on
the observed symmetry ( ⳵ / ⳵ ␪ ⯝0). For the FRC case we
find that the toroidal inductive electric field increase radially
and peaks at a larger radius (r⯝15 cm兲 than the neutral sheet
(r⯝10 cm兲 where the flux assumes a maximum. Due to axial
symmetry there are no space charge electric fields or pressure gradients in the toroidal direction. In the neutral sheet
E ␪ drives the toroidal current J ␪ . Outside the neutral sheet
the inductive field produces a radial electron Hall drift, v r
⫽E ␪ /B z , directed into the neutral sheet, which carries
frozen-in field lines into it. The essentially incompressible
electrons (ⵜ•J⯝0) cannot stagnate in the null layer, and
they flow axially outward at the ends of the null layer forming two poloidal current loops (J r ,J z ). A rotational electron
flow and a diverging energy flow 共Poynting’s vector is parallel to the radial E⫻B drift兲 imply that the frozen-in condition is broken near the null layer. The radial electric field
can be calculated from the EMHD force law, ⫺neE
⫹J⫻BÀⵜp e ⫽0, where p e ⫽nkT e is the electron pressure.
It will be shown in Part III14 that there is a radial electron
pressure gradient due to a density maximum on axis. The
electron temperature peaks in the current sheet but ⵜp e remains radially inward. Since J flows in the ␪ direction and B
reverses sign across the neutral sheet, the radial space charge
electric field must change sign. The peak radial electric fields
and pressure gradients at z⫽0 are of order E r ⬇ 兩 ⵜ p e /ne 兩
ⱗ0.4 V/cm.
For purpose of comparison, we have also calculated E ␪
for the case BH 储 B0 . In the absence of a magnetic null the
electrons remain magnetized everywhere. Since BH is now
reversed, the direction of E ␪ is also reversed but not the
radial electron E␪ ⫻Bz drift. The latter gives rise to the poloidal current loop of a whistler vortex. The radial spacecharge electric field points outward and is of similar magnitude as the inductive field. There are no significant radial
pressure gradients. The toroidal and poloidal Hall currents
are found to be of comparable magnitude.
Finally, it should be pointed out that especially in the
case BH 储 ⫺B0 , there must be a space-charge electric field
E z along the nonuniform axial magnetic field B z (z), so as to
balance the magnetic force on the electrons, ⫺ ␮ e ⵜB⫺eE
⯝0, where ␮ e ⫽m e v⬜2 /2B is the electron magnetic moment.
The potential difference between the null points and the center of the FRC should be of order kT e /e⯝4 V. The unmagnetized ions will respond to this electric field, but their displacement and energy gain are small due to the short time
scale involved, ⌬z⫽(eE/2m i )(⌬t) 2 ⯝0.5 cm and eE⌬z
⯝0.25 eV, for E⯝0.5 V/cm, ⌬t⯝10 ␮ s.
IV. CONCLUSIONS
This first part of a series of articles on EMHD reconnection describes the magnetic field topologies of a time-varying
Helmholtz coil field inside a uniformly magnetized background plasma. Of particular interest is the period of free
relaxation after the rapid switch-off of the coil currents. Two
configurations are compared: First, with the dipole moment
opposite to the weaker background field, an EMHD FRC is
produced. Second, with the dipole moment along the background field, a large amplitude whistler vortex is generated.
The relaxation of the fields in these two cases is different.
The FRC fields stagnate and dissipate while the vortex fields
propagate. The motion of magnetic field lines in the presence
of electron fluid flows is discussed. Frozen-in, axisymmetric
field lines are defined by contours of constant poloidal flux.
Driven reconnection near an X-type null line, absence of reconnection near a 3-D spiral null point, and annihilation of
field lines near an elongated O-type null line are deduced
from the motion of field lines near magnetic null points.
Annihilation is responsible for the disappearance of closed
field lines in a freely decaying FRC. Field line annihilation in
O-type nulls implies a breakdown of the frozen-in condition
since the highly incompressible electrons cannot have a divergent flow, just like the field lines. The frozen-in concept
has been verified experimentally in regions of magnetized
electrons. For example, the poloidal field in a relaxing
EMHD FRC is associated with a toroidal electron drift which
twists the frozen-in field lines and creates toroidal fields or
poloidal currents/drifts. These convect the poloidal fields,
produce an elongated FRC and drive electron tearing modes.
The coupling of toroidal and poloidal electron flows produces, in general, 3-D magnetic fields with helicity density
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Phys. Plasmas, Vol. 10, No. 7, July 2003
but not necessarily net helicity. The toroidal electron current
in an FRC is not localized within the magnetic neutral sheet
but broadened by Hall currents outside the null layer. This is
an important difference to 2D numerical simulations which
predict electron current sheets of width c/ ␻ pe . 5 The forcefree nature of EMHD fields may account for the observed
stability of the FRC to tilting.
ACKNOWLEDGMENT
The authors gratefully acknowledge support for this
work from NSF Grant No. 0076065.
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