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PHYSICS OF PLASMAS VOLUME 8, NUMBER 5 MAY 2001 

Collisionless magnetic reconnection in a toroidal cusp* 

J. Egedal, †,a) A. Fasoli, D. Tarkowski, and A. Scarabosio 

Massachusetts Institute of Technology, Plasma Science and Fusion Center, Cambridge, Massachusetts 02139 

Received 26 October 2000; accepted 21 December 2001 

Fast collisionless magnetic reconnection is driven and observed in a toroidal magnetic cusp. For low 

values of the toroidal guide magnetic field, reconnection occurs without the formation of a 

macroscopic current channel. In the absence of a current channel, the measured plasma potential and 

poloidal flows are consistent with predictions based upon single particle theory. Only for large 

values of the toroidal magnetic field compared to cusp poloidal field is a current channel present. 

Also in this case single particle orbit theory provides a good description of the anomalous resistivity 

observed experimentally. © 2001 American Institute of Physics. DOI: 10.1063/1.1355316 

I. INTRODUCTION 

Magnetic reconnection 1 has been observed to cause global 

changes in plasma properties and magnetic field topology 

in a variety of plasmas in very different environments. For 

example, reconnection appears to be the direct cause of the 

heating of the solar corona and solar flares. 2 In fusion devices 

such as tokamaks, reconnection is associated with fast 

relaxation processes that influence the plasma confinement. 3 

Recent satellite observations 4 confirmed that magnetic reconnection 

also occurs at the earth’s geomagnetic tail as a consequence 

of the interaction between the solar wind and the 

earth’s magnetic field creating the auroral phenomena. 5 

During magnetic reconnection, field lines in opposed directions 

cross link, locally forming a magnetic cusp configuration. 

Especially in low collisional plasmas, the reconnection 

process is observed on time scales much shorter than 

expected from theoretical considerations based upon resistive 

magnetohydrodynamic MHD models. 6 

A number of laboratory experiments 7–12 have been designed 

to investigate the mechanisms causing the short 

time scales characteristic of reconnection events. Amongst 

these, the Versatile Toroidal Facility VTF experiment 12 has 

been recently developed to study the plasma dynamics in a 

magnetic cusp configuration and the response to driven magnetic 

reconnection. Two special features make the VTF experiment 

unique: The plasma production stage is separate 

from the reconnection drive and the mean free path of the 

electrons is larger than the dimensions of the device. The 

experimental observations of this experiment become therefore 

important to understand reconnection phenomena in low 

collisional plasmas in space and magnetic fusion devices. 

In this paper we document recent experimental observations 

of magnetic reconnection in the VTF experiment addressing 

the following questions. 

1 Is a macroscopic current sheet necessary for collisionless 

reconnection? 

*Paper VI1 2, Bull. Am. Phys. Soc. 45, 322 2000. 

† Invited speaker. 

a Electronic mail: jegedal@psfc.mit.edu 

2 What is the poloidal drift speed of the plasma during 

reconnection? 

3 How does reconnection affect the plasma potential and 

electron temperature? 

4 Can features of the reconnection process, namely the fast 

time scales, be explained by characteristics of single particle 

orbits? 

The paper is organized as follows: In Sec. II the experimental 

setup is described, together with typical VTF plasma 

profiles. The drive system for the magnetic reconnection and 

the response of the plasma are described in Sec. III. Section 

IV presents the measurements of the poloidal flow velocity. 

In Sec. V particle orbits in the cusp configuration are discussed, 

accounting for the observed features of the plasma 

during the reconnection process. In Sec. VI we provide examples 

of current sheets measured in profiles with a strong 

guide field. Finally, the paper is concluded in Sec. VII. 

II. EXPERIMENTAL SETUP AND PLASMA 

PROPERTIES 

The experiments are performed on the Versatile Toroidal 

Facility VTF at the MIT Plasma Science and Fusion 

Center. 12 The poloidal cross section of the toroidal device is 

shown in Fig. 1. The dashed lines represent contours of constant 

poloidal flux, which coincide with the poloidal projection 

of magnetic field lines. This magnetic cusp configuration 

is produced by poloidal field coils installed outside the stainless 

steel vacuum vessel. The solid lines represent contours 

of constant modulus of the poloidal magnetic field. Plasmas 

are produced by applying 2.45 GHz rf heating delivered by a 

50 kW Varian klystron amplifier via a ceramic window and a 

rectangular cross-section horn antenna. For the maximal operational 

cusp field full cusp the break down of the injected 

gas normally argon at 110 5 Torr occurs on the ECRHring 

electron cyclotron resonance heating shown in Fig. 1. 

At this location the magnetic field strength is 87.5 mT and 

the electron cyclotron frequency is matched by the heating 

frequency. In addition to the cusp field, a toroidal field 0– 

200 mT may be added using 18 toroidal field coils not 

shown. By selecting appropriate values of the cusp field B c 

1070-664X/2001/8(5)/1935/9/$18.00 

1935 

© 2001 American Institute of Physics 

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1936 Phys. Plasmas, Vol. 8, No. 5, May 2001 Egedal et al. 

FIG. 1. Poloidal cross section of VTF. The solid contour 

lines represent the poloidal magnetic field strength. 

The dashed contour lines correspond to constant levels 

of the poloidal magnetic flux, , which coincide with 

magnetic field lines. 

and the toroidal field B , it is possible to modify the shape 

and location of the ECRH-ring. Highly reproducible target 

plasmas are produced during shots lasting up to 100 ms and 

taken typically every 2 minutes. 

The snapshots in time of the steady state density profiles 

shown in Fig. 2 were measured by a 16-tip Langmuir probe 

moved though the plasma 29 plasma shots per profile. For 

each magnetic configuration the location of the ECRH-ring 

is indicated by the red lines. The plasma is confined partly by 

mirror effects 12 and partly electrostatically. 

The profile in Fig. 2A was obtained in the absence of 

toroidal magnetic field. In this case the magnetic moment is 

not conserved for the particles on field lines passing through 

the region of the absolute magnetic null. As a consequence, 

the measured density is reduced along the separatrix. In Fig. 

2B a small toroidal component (B 30 mT is added, improving 

the confinement of the particles on the separatrix. 

The 1/R dependence of the toroidal magnetic field 

strength makes it possible to place the ECRH-ring to the 

right of the X point. The density profiles obtained in two 

such examples are given in Figs. 2C and 2D. 

The profiles in Fig. 3 were obtained by sweeping the 

bias on the multiple Langmuir probe tips. Electron temperatures 

in the range of 20 eV are typical for the VTF plasmas. 

The plasma potential is, for most of the possible configurations, 

in the range of 60 to 100 V. 

III. RECONNECTION DRIVE SYSTEM AND RESPONSE 

OF THE PLASMA 

For driving magnetic reconnection VTF is equipped with 

a 1 mH ohmic coil system fed through a semiconductor 

switch by a1mFcapacitor. 12 The capacitor is charged to 3 

kV. As shown in Fig. 4, the LC resonance circuit produces 

sinusoidal current pulses inducing electric field pulses of 10 

V/m in the center of the vacuum vessel. 

The density contours in Fig. 5A represent another example 

where the plasma is produced entirely to the right of 

the X line. According to ideal MHD it should not be possible 

for this plasma to cross the separatrix into the quadrants to 

the left of the X line. However, as seen in Figs. 5B and 

5C, during the electric field pulses the plasma drifts unhindered 

across the separatrix. The direction of the drifts in the 

plasma agrees with the EB drift. The measurements described 

in the next section show that the drift velocity is 

given by E /B cusp . 

During the reconnection experiments, the time evolution 

of the magnetic flux over the plasma cross section is measured 

by a multiple magnetic probe. This movable probe has 

30 pick-up coils each with a total area of 0.5 cm 2 for each 

of the three direction in space, providing a spatial resolution 

of 1 cm in the radial direction. Due to the reproducibility of 

the VTF plasma the resolution in the vertical direction can be 

made arbitrarily fine by moving the probe in between plasma 

shots. The signals are integrated ( int 0.5 ms and amplified 

gain100 before digitization. 

These magnetic measurements reproduce the magnetic 

fields obtained in the absence of plasma. In turn, these measurements 

agree with the fields calculated on the basis of the 

currents applied in the VTF magnetic coils system. Hence, 

we see no evidence of currents in the plasma. Taking into 

account the sensitivity of the magnetic probes, it can be concluded 

that the total toroidal current induced in the plasma is 

less than 5 A. If classical Spitzer resistivity were valid for the 

experimental configuration, a current of the order of 10 kA 

should have been observed. 

Significant plasma currents are only observed in profiles 

where the maximal poloidal magnetic field in the cross section 

is less than 2% of the toroidal field see Sec. VI. 


Phys. Plasmas, Vol. 8, No. 5, May 2001 

Collisionless magnetic reconnection in a toroidal cusp 

1937 

FIG. 2. Color Contours of electron density in various 

magnetic configurations. 

IV. MEASUREMENT OF THE POLOIDAL DRIFT SPEED 

Flows with directions consistent with the EB drift 

have been measured in VTF during reconnection using a 

Mach probe. 12 However, the numerical value of the flow 

speed was not accurately obtained using this method. In order 

to accurately measure the drift speed of the plasma during 

the reconnection process, experiments were conducted 

where the rf heating is pulsed at 2 kHz with a duty cycle of 

12%. Using a sampling rate of 20 kHz for the acquisition of 

the ion saturation current signals measured by the Langmuir 

probe, the evolution of the decaying plasma density profile 

is monitored with good time resolution during the time intervals 

where no rf heating is applied. Figure 6 shows the time 

history of the rf heating A, the electric field induced B, 

and the resulting ion saturation current on one of the Langmuir 

probe tips C. 

The evolution of the plasma density is obtained throughout 

the poloidal cross section as a function of time. It is 

found that the plasma density is at all times nearly constant 

along the magnetic field lines. We may therefore illustrate 

the experimental results by plotting contours of the density 

as function of the magnetic flux and t. The contours 

FIG. 3. Color Contours of density, temperature, pressure 

and plasma potential in a magnetic configuration 

with full cusp and B 80 mT. 



shown in Fig. 6D are obtained by evaluating the density 

along the poloidal location indicated by the dashed red line 

in Fig. 5D. The data used in creating the diagram in Fig. 

6D corresponds to data samples obtained at constant time 

interval t0.5 ms after the rf heating pulses. 

Similarly, the data used in creating the diagrams in Figs. 

7 was obtained with t0.3, 0.5 ms. The value of l 0 

B /B cusp for the magnetic configuration of Fig. 6 and 

Figs. 7A and 7B is 0.3 m, whereas l 0 0.5 m for the 

magnetic configurations forming the basis for the data shown 

in Figs. 7C and 7D. In the diagram of Fig. 6D and the 

four diagrams in Fig. 7 the locations of the plasma boundaries 

are well defined and coincide with the location of the 

black dashed lines. These lines obtained through (t) 

tRE (t) represent the theoretical location of the plasma 

boundary assuming that the plasma is frozen into the poloidal 

magnetic flux. Hence, we find experimentally that the 

plasma is frozen into the poloidal magnetic flux. Due to the 

absence of a detectable toroidal current and an associate diffusion 

region, the poloidal magnetic field reconnect unhindered 

and arbitrarily fast. The rate of reconnection in the 

VTF experiment is solely determined by the operator by imposing 

a given value of the induced toroidal electric field. 

The poloidal drift speed, expressed in terms of the flux 

variable, d/dtRE , translates into a drift speed in real 

space given by v d E /B cusp , independent of the toroidal 

magnetic field. In the presence of a toroidal magnetic field, 

this drift speed is much larger than usual EB drift speed 

given by E /B inferred on the basis of the induced electric 

field where B is the total magnetic field. 

FIG. 4. Loop voltage induced by the ohmic pulse on a copper wire at the 

center of the vessel. 

V. INTERPRETATION OF EXPERIMENTAL RESULTS 

ON THE BASIS OF SINGLE PARTICLE ORBITS 

The mean free path between collisions for the electrons 

in VTF is in the order of 5 m. The thermal speed of the 

electrons is typically 110 6 m/s, so during a particle confinement 

time, p 0.2 ms, the electrons execute hundreds of 

complete particle bounce orbits in the poloidal cross sections. 

It is therefore natural to seek the explanation for the 

observed plasma behavior in the properties of the particle 

orbits. 

The particle confinement in the VTF device is analyzed 

in Ref. 12, based upon the guiding center approximation. In 

Ref. 13 an analysis was given for the particle orbits in a 

linear magnetic cusp including and electric field along the 

direction of the X line, summarized here below. 

The crucial mechanism for confining the particles in a 

magnetic cusp is the adiabatic invariance of the magnetic 

moment mv 2 /(2B), where v is the particle velocity 

perpendicular to the magnetic field. We assume that a sufficient 

magnetic guide field in the direction of the X line is 

present, such that the magnetic moment is conserved at all 

locations throughout the cross section. In the absence of an 

electric field the particles gyrate around specific field lines 

and are confined to regions of the magnetic configuration 

where BE k . Here E k is the particle kinetic energy, which, 

FIG. 5. Color Density contours measured before, during 

and after an ohmic pulse. The ohmic pulse induces 

EB-flows, the directions of which are illustrated by 

the red arrows in B and C, which perturb the shape 

of the density profile. The red dashed lines in D indicate 

the locations along which the plasma density is 

evaluated in Figs. 6 and 7 as a function of time and the 

magnetic flux variable . 




1939 

FIG. 6. Color Experimental scheme for measuring the 

poloidal flow of the plasma during an ohmic pulse. A 

The rf power is pulsed at 2 kHz with a duty cycle of 

12%. B Time history of induced electric field. C The 

resulting ion saturation current measured on a single 

probe tip. D Density contours as a function of and 

t, using data samples measured t0.5 ms after the rf 

pulses. 

FIG. 7. Color Plasma density measured as a function 

of and t during two ohmic pulses for two different 

plasma configurations: A, B full cusp; C, D half 

cusp. The data is measured at, respectively, t0.3 

ms and t0.5 ms after the rf pulses. The dashed lines 

indicate the theoretical location of the plasma edges 

when the plasma moves across field lines with the 

speed v d E z /B p . 



FIG. 8. Trajectories of an argon ion in a linear magnetic cusp, see text. 

in the absence of an electric field, is a constant of motion. 

Since the strength of the magnetic cusp field varies linearly 

with the distance from the center of the X line, the particles 

are confined to a certain region of a field line. The points 

where BE k are the orbit bounce points. Here the velocity 

component v , parallel to the magnetic field, changes sign 

and the particle is reflected back along the same field line. 

With an electric field E in the direction of X line, the 

particle can gain or lose a certain energy E as it travels 

along a field line. The sign and value of E depend on the 

specific magnetic configuration. The orbit bounce points are 

now shifted to the location where BE k E. For now, 

the exact location of the bounce point is not important. What 

is important is that the particles are trapped even when an 

electric field is present. Apart from deviations due to the E 

B, curvature and B drifts across field lines, the particle 

will mirror back along the path it initially followed. Due to 

this trapping, when the orbit motion is averaged over one 

orbit bounce time, the particle conducts no current in the 

direction of the induced electric field. This situation is similar 

to the trapped particles in tokamaks, which do not contribute 

to the toroidal conductivity. The main difference is 

that in a cusp configuration all particles are trapped, including 

particles on the separatrix. Therefore in the absence of 

collisional detrapping the plasma has no conductivity in the 

direction of the X line. 

Figure 8A shows an example of an orbit of a 15 eV 

single ionized argon ion in a linear magnetic cusp: B 

b 0 (xxˆyŷl 0 ẑ). The orbit is calculated for b 0 0.3 T/m 

and l 0 0.25 m, for a homogeneous electric field E z 10 

V/m imposed along the direction of the X line. Figure 8B 

shows the guiding center representation of the same orbit. 

We note how the argon ion practically follows the magnetic 

field lines trapped between the dashed lines B1 and B2. The 

ion is not conducting any current in the negative z direction, 

rather it experiences a drift in the positive z direction as it 

approaches the separatrix. 

In Ref. 14 it was found that trapped particles in tokamaks 

in the presence of a toroidal electric field, E , experience 

a pinch effect known as the Ware pinch causing the 

trapped particles to drift across the poloidal cross section at 

the speed v d E /B p , where B p is the poloidal magnetic 

field strength. By arguments similar to those applied in the 

paper by Ware we find 13 that all particles in the cusp are 

subject to this pinch effect and drift across the cross section 

at the Ware pinch speed. This is exactly the speed that is 

measured experimentally see Figs. 5 and 6. 

As discussed above, the gain in energy of the particle as 

it moves along the B-field line in the direction of the electric 

field causes a shift in the location of the bounce point. This 

explains why the loci of the bounce points B1 and B2 in Fig. 

8B are not symmetric around the lines yx. The 

bounce points are found to be shifted in the positive y direction. 

Similarly, for the electrons one finds that the bounce 

points are shifted in the positive x direction. In a plasma, this 

macroscopic separation of the charges is not possible: An 

electrostatic potential will build up enforcing charge 

quasineutrality. Note that an electrostatic potential in the 

cross section does not influence the Ware pinch drift. Macroscopic 

charge separation is avoided if the particles do not 

experience a change in energy as they move along field 

lines. 13 This requires the electrostatic potential 

0.5E z l 0 logx/y. 

This potential was included when calculating the particle orbit 

shown in Fig. 8C. The argon ion now bounces symmetrically 

around the lines yx, while drifting with the 

magnetic field lines at the Ware pinch speed. 

A direct measurement of the electrostatic potential was 

conducted on VTF during driven reconnection. The data 

shown in Fig. 9 was obtained under conditions identical to 

those used when obtaining the data shown in Figs. 3 and 5. 

The diagrams A–C show the changes in the plasma density, 

electron temperature, and plasma potential obtained by 




1941 

FIG. 9. Color A–C Changes in plasma parameters 

as a response to the induced reconnection electric field 

E . D The theoretical predicted change given in the 

plasma potential. The intervals above the figures correspond 

to the full range 1, 1 in the color code. 

subtracting two profiles for each of these quantities measured 

during periods with opposite sign of the induced electric 

field. It is seen that the changes in density and temperature 

are very modest, implying that no significant heating of the 

electrons is taking place during the magnetic reconnection 

process. We attribute the changes which are present to the 

induced flows combined with the localized ECR-heating of 

the plasma. 

In the area of the dashed field lines the density and temperature 

remain constant under the sign reversal of the reconnecting 

electric field. In this area the measured change in 

plasma potential is consistent with the theoretical change 

given by the equation above. 

Besides the induced electric field E , the total electric 

field E in the plasma includes the contribution from 

the electrostatic potential. The total electric field is perpendicular 

to the magnetic field and the frozen in-law, E 

v EB B0, is fulfilled at all locations where the actual 

electrostatic potential in the plasma agrees with the expression 

of calculated above. Here v EB EB/B 2 is the 

usual EB drift. Note that the poloidal component of v EB 

is indeed the pinch speed E /B cusp , independent of . 

FIG. 10. Color Density contours measured during the 

formation of current layers for four different ratios between 

the toroidal field and the cusp fields: l 0 

B /B cusp 23,17,12 m. The dashed lines represent 

contours of constant poloidal magnetic flux. 



FIG. 11. Color Contours measured current densities 

for the same conditions as in Fig 10. The dashed lines 

represent contours of constant poloidal magnetic flux. 

VI. MEASUREMENTS OF CURRENT SHEETS 

In magnetic configurations where the toroidal magnetic 

field is much stronger than the cusp field, the majority of the 

particles are no longer trapped. In such configurations we do 

indeed find a current sheet. In Figs. 10 and 11 snap shots in 

time of the plasma density and current density at the beginning 

of the formation of the current layers are shown. These 

correspond to the formation of current sheets for three different 

values of the ratio of toroidal field and cusp field: l 0 

B /B cusp 23, 17, 12 m note that the profiles of all 

the previous figures are obtained with l 0 1m. In Fig. 12 

*E / j max is the ratio of the electric field and the current 

density at the center of the current sheet. Here E 

(R,Z) 

(d/dt)/R, where (R,Z) R0,Z0 RB dl is the 

poloidal flux variable, R is the major radius, Z describes the 

vertical position, and the subscript denotes the poloidal 

plane. is calculated on the basis of the currents applied in 

the poloidal field coils and the plasma current profiles 

j (R,Z). The profiles of j (R,Z) are obtained through a 

singular fit to the measured magnetic field profiles. 

The Spitzer resistivity s is calculated using an electron 

temperature of 20 eV. It is seen that the constant Spitzer 

resistivity is insufficient to account for the low values of the 

current observed. Using an approach similar to that of Ref. 

15, the discrete particle enhanced inertial resistivity pei was 

introduced in Ref. 16 to describe the inertial resistivity based 

upon the time each individual particle spends in the region of 

the X line in a linear magnetic cusp. Although the values of 

pei are too low by a factor of about 6 to describe the 

measured values of * it is seen that pei accurately reproduces 

the scaling of * as a function of l 0 . It should be 

noted that pei was obtained on the basis of heuristic arguments 

for a linear cusp geometry. Using the measured geometry 

of the magnetic field a more accurate value of pei can 

be obtained which may fully account for the observed values 

of *. This work is currently underway. The current sheets 

shown in Fig. 11 evolve very differently in time. For the case 

where l 0 23 m, the X point evolves into an O point and the 

total plasma current increases from 1 kA to 10 kA. In the 

other two cases the X points are maintained throughout the 

pulses and the total plasma currents do not increase. 

VII. CONCLUSION 

Collisionless magnetic reconnection is routinely observed 

in the Versatile Toroidal Facility. In the cases where 

the strength of the cusp magnetic field is in the order of or 

larger than the strength of the toroidal magnetic field (l 0 

1 m the following experimental observations have been 

made 

1 Fast collisionless magnetic reconnection occurs without 

the formation of a macroscopic current sheet. 

FIG. 12. Inferred resistivities. A circles; *E / j ; diamonds, pei ; 

triangles, s ; B diamonds, */ pei ; triangles, */ s . 




1943 

2 The poloidal drift speed of the plasma is given by v d 

E /B p , where E is the toroidal reconnection electric 

field and B p is the field strength in the poloidal cross 

section. 

3 During the reconnection process a poloidal electrostatic 

potential is present. The electrostatic potential is such 

that the sum of the reconnection electric field and the 

electrostatic field is perpendicular to the magnetic field 

lines everywhere, except very close to the magnetic X 

line and separatrix. 

4 The temperature of the electrons is observed to be essentially 

constant during the reconnection process. 

We find that the observed behavior of the plasma is consistent 

with effects that can be attributed to the conservation 

of the magnetic moments of the individual particle. The conservation 

of magnetic moment implies that particles follow 

well-defined trajectories trapped in the direction of the X 

line. Hence, in the absence of collisional detrapping, the particles 

cannot conduct any current in this direction. Rather, an 

electric field along the X line causes them to drift across the 

poloidal cross section including the separatrix at the velocity 

E /B cusp . Furthermore, based on the orbit analysis, we 

find that an electrostatic potential is necessary to avoid 

charge separation. Away from the immediate vicinity of the 

separatrix, the theoretical potential is consistent with the 

measured electrostatic potential. In these areas the plasma 

obeys the frozen in-law. The frozen in-law is broken in the 

vicinity of the X line where finite Larmor radius effects or 

other mechanisms eliminates the unphysical discontinuities 

in the theoretically predicted potential. Because the poloidal 

pinch velocity E /B cusp applies to all particles in the plasma, 

the plasma is at all locations frozen into the poloidal magnetic 

flux. 

Only in magnetic configurations in which the toroidal 

magnetic field is much stronger than the cusp magnetic field 

(l 0 10 m do we observe a toroidal current. Our preliminary 

results for these configurations where trapping of particles 

does not apply suggest that the ratio *E / j of the 

toroidal electric field and toroidal current density scales with 

the particle enhanced inertial resistivity pei introduced in 

Ref. 16. However, it should be noted that */ pei 6. Work 

is underway to improve the estimate of pei using the measured 

magnetic field configuration. 

ACKNOWLEDGMENTS 

The authors acknowledge the help and support of Professor 

M. Porkolab, C. Lemmerz, R. Riddols, N. Darlymple, 

E. Fitzgerald, D. Gwinn, and W. Parkin. 

This work was partly funded by Department of Energy, 

Junior Faculty Development Award DE-FG02-00ER54601. 

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