Non-local Ohm’s law during collisions of magnetic flux ropes

W. Gekelman,1 T. DeHaas,1 P. Pribyl,1 S. Vincena,1 B. Van Compernolle,1 and R. Sydora2

1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA2Department of Physics, University of Alberta, Edmonton, Alberta T6G2J1, Canada

(Received 20 March 2017; accepted 6 June 2017; published online 27 June 2017)

Two kink unstable magnetic flux ropes are produced in a carefully diagnosed laboratory experiment.

Using probes, the time varying magnetic field, plasma potential, plasma flow, temperature, and

density were measured at over 42 000 spatial locations. These were used to derive all the terms in

Ohm’s law to calculate the plasma resistivity. The resistivity calculated by this method was negative

in some spatial regions and times. Ohm’s law was shown to be non-local. Instead, the Kubo resistiv-

ity at the flux rope kink frequency was calculated using the fluctuation dissipation theorem. The

resistivity parallel to the magnetic field was as large as 40 times the classical value and peaked where

magnetic field line reconnection occurred as well as in the regions of large flux rope current.

Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4990054]

Flux ropes are magnetic structures with helical magnetic

fields and currents and are found in astrophysical plasmas

throughout space.1 Flux ropes routinely occur within the

Earth’s magnetopause boundary layer and also within the

magnetotail. The surface of the sun is littered with arched flux

ropes, which are visible as arched filaments when viewed in

ultra violet or X ray emissions. It is common in nature for flux

ropes to have one or more companions. When two or more

flux ropes are close to one another, they can interact often in

complex ways and collide. During a collision, a small compo-

nent of the magnetic field from each rope can be anti-parallel,

and magnetic field line reconnection can be triggered. When

reconnection is two-dimensional, a magnetic null can exist

with field lines piling into it and reconnecting. This occurs in

the magnetotail2 and has been observed in numerous labora-

tory experiments.3–8 Generally, in nature, the magnetic fields

are three-dimensional and there are no nulls as in the 2D case,

but reconnection happens nevertheless.9 Arched magnetic

flux ropes with helical fields have been studied in the labora-

tory,10 but here, we consider a simpler geometry, with two

side-by-side flux ropes generated from parallel currents in a

background magnetic field. The ropes are kink unstable and

periodically smash into one another, and each time they do

reconnection occurs.11 Ohm’s law links the local plasma

resistivity to the total electric field and current. In this work,

we measure all the quantities necessary to calculate every

term in Ohm’s law and examine the resulting resistivity.

These experiments were performed in the Large Plasma

Device (LaPD) at UCLA.12 The plasma is produced by a DC

discharge between a Barium Oxide coated cathode13 and a

mesh anode. The bulk of the plasma carries no net current

and is therefore quiescent. The background plasma parame-

ters are as follows:

dn

n� 3%; 0:5 � Te � 5 eV ; 1010 � ne � 2� 1012cm�3;

Ti ’ 1eV:

The parallel Lundquist number is �1.8 � 105, which is

based on Spitzer resistivity and an 11 m rope length. The

background plasma is pulsed at 1 Hz with a typical plasma

duration of 15 ms. A schematic of the machine indicating

how the background plasma and flux ropes are generated is

shown in Fig. 1.

The background plasma is generated by an oxide

coated cathode schematically shown on the left of Fig. 1.

A transistor switch14 in series with a capacitor bank is

used to pulse a voltage between the negatively biased cath-

ode and an anode located 50 cm away. The background

plasma is 18 m in length and 60 cm in diameter. A second,

high emissivity cathode15 (LaB6) is inserted in the far side

of the machine. The flux rope cathode is masked with a

carbon sheet into which two circular holes are cut 3.5 cm

in radius and 1 cm apart edge-to edge so it can only emit

from the unmasked areas. The cathode mask is 32 cm from

the second cathode. A second transistor switch, capacitor

bank, and charging supply are used for the ropes. The ref-

erence probe was used to calculate the temporal autocorre-

lation time for the magnetic field. We can reliably

reconstruct the data for 4 ms, about twice the time in our

datasets.

The plasma in the ropes is highly ionized with a temper-

ature of 12 eV. The electrical systems, which generate the

ropes, and background plasma are independent of one

another. Diagnostics include Langmuir probes (n,Te), Mach

probes (flow), and three-axis magnetic pickup probes16,17

(~B). A specialized emissive probe18 measured the plasma

potential, Vp(t), at each of the 42 150 spatial locations. The

time series was digitized at 16 000 timesteps. Subsequently,

Vp was corrected for the electron temperature,19 which was

also measured at each location. The gradient of the plasma

potential was used to determine the space charge contribu-

tion to the electric field.

Ohm’s law has long been used to estimate the plasma

resistivity in the analysis of spacecraft data, in the interpreta-

tion of laboratory experiments and in computer simulations.

Often, one or more terms are dropped using symmetry argu-

ments or assumptions on the relative size of terms. The justi-

fication for this is often questionable.

1070-664X/2017/24(7)/070701/6/$30.00 Published by AIP Publishing.24, 070701-1

PHYSICS OF PLASMAS 24, 070701 (2017)

Ohm’s law for electrons is in SI units

me

ne2

d~J

dt¼ ~E þ~u � ~B þ 1

ner � P

$� 1

ne~J � ~B � gk~Jk � g?~J?:

(1)

Here, P$

is the pressure tensor, n is the plasma density, ~J is

the current (in this experiment, it is carried by the electrons),

e is the absolute value of the electron charge, and ~B is the

total magnetic field. The resistivity, g, is different along and

across the magnetic field. In the steady state and when there

is no magnetic field or pressure, Ohm’s law reduces to the

familiar ~E ¼ g~J . For the first time to our knowledge, all the

terms in the generalized Ohm’s law, Eq. (1), to determine

the resistivity, were measured as a function of space and

time in a flux rope experiment. The goal was to derive the

parallel resistivity from the data. When the dot product of

Eq. (1) with the local magnetic field (the background field

and that of the ropes) is taken, one arrives at

~E � b̂ þ 1

nerP � b̂ ¼ gkJk with b̂ ¼

~B

j~Bj: (2)

In Eq. (2), b̂ is a unit vector along the local value of the mag-

netic field due to the external magnets and plasma current

and P is the scalar pressure. The time varying current term in

Eq. (1) is negligible for our experimental parameters.

The perpendicular magnetic field can be as large as

30 G or about 1/10 of the background field, B0z¼ 330 G. The

flux ropes rotate about each other in addition to being line-

arly kink unstable. As the rope rotation starts at a slightly dif-

ferent time for each experimental shot, a conditional

averaging technique20 was used to define the experimental

start time, which differs from the time the rope currents are

switched on.

The resulting magnetic field lines writhe about themselves

and twist about one another as previously observed.21,22,24,25

When the kink motion drives the ropes into one another, there

are bursts of reconnection and a Quasi Separatrix Layer

(QSL)23 forms. QSLs24,25 are regions in which the magnetic

field connectivity changes rapidly but continuously across a

narrow spatial region. Reconnection results in an induced elec-

tric field which, in principle, can drive reverse currents in the

plasma. The three-dimensional plasma currents are evaluated

from the magnetic field usingr� ~B ¼ l0~J .

The plasma density was measured everywhere in the

volume by collecting ion saturation current to the six faces

of a Mach probe.26 The ion saturation current, Isat, to a probe

is proportional to Isat / nffiffiffiffiffiTe

p. The electron temperature was

determined by sweeping the Langmuir probe I-V characteris-

tics at thousands of spatial locations. The contours of elec-

tron temperature calculated closely matched contours of

constant current on these planes and at the times the probe

was swept. The background temperature (where there was no

current) was 4 eV. Figure 2 shows the scalar plasma pressure,

P ¼ nð~r ; tÞKTeð~r; tÞ at one time (t¼ 5.5865 ms). The ropes

are switched on at t¼ 0, and in the first few meters, the iso-

surface of constant pressure (P¼ 5 J/m3) mirrors that of the

rope magnetic field. Isosurfaces of lower pressure fill more

of the volume. The proper term in Ohm’s law is the diver-

gence of the pressure tensor. In the experiment, the electron-

electron collisional relaxation rate is 5 MHz or about one

thousand times faster than time scales associated with the

flux rope motion. These collisions isotroprise the pressure

tensor. A typical mean free path for collisions between 10 eV

electrons is 21 cm, far smaller than the length of the ropes.

Finally, the large guide field, in comparison to the transverse

flux rope fields, suppresses meandering orbits of electrons in

the transverse plane, which lead to tensor pressures.

Consequently, we estimate that the pressure is well approxi-

mated by its measured scalar value.

The electric field has two components, ~E ¼ �rVp � @~A@t .

The space charge component is often neglected. This is not

the case here. In Fig. 2(b), the axial electrostatic field is neg-

ative, in the same direction as Jjj inside the current channels

and is of order ten times larger than the inductively induced

electric field [represented by the second term, � @~Az

@t , and

shown in Fig. 2(c)]. It is also evident that the two compo-

nents have different spatial morphologies. The induced elec-

tric field has the opposite sign as the electrostatic field in the

region where the currents are large, and this is expected

when reconnection occurs.

FIG. 1. Notional representation of the

LaPD device and flux rope generation.

The components are labeled. Note that

the center section (which has many

more magnets) has been foreshortened

for the image to fit. Magnetic field lines

of the ropes rendered in red and blue

are data placed in the image to scale.

070701-2 Gekelman et al. Phys. Plasmas 24, 070701 (2017)

The parallel and transverse resistivities were evaluated

plane by plane, and they will be discussed in detail in a

much longer paper in the future. Estimating the error in each

measurement of the terms in Eq. (2) and the total error pro-

duced by adding and dividing them, we estimate that the

experimental error in the measurement of gjj is 20%. Here,

we present in Fig. 3 the parallel resistivity at a fixed time on

two transverse planes.

Contours of field aligned Jjj are also shown in Fig. 3.

The temporal appearance of negative resistivity has nothing

to do with reconnection and is not observed in the QSL

region. The field-aligned resistivity was also calculated and

also has negative regions. The volume-averaged resistivity is

always positive. Reverse currents have been seen in the past

in experiments on reconnection.20 There are axial pressure

gradients that can potentially drive reverse currents, but

these are measured and were included in the Ohm’s law eval-

uation. Assuming that there is no dynamo, one may question

whether Ohm’s law can be used at all in these circumstances.

It is possible that the contributions to Ohm’s law are non-

local and Eq. (1) cannot be used. Non-locality occurs when

fields and pressure gradients along a moving field line all

contribute to the resistivity at a remote point. Ohm’s law

requires a point-by-point balance between momentum gain

from the electric field and loss due to scattering. Jacobson

and Moses27 discovered that this loss and gain may be in

global balance but not locally. The criterion for this is gov-

erned by a parameter a

a ¼ 2L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi@B?

B

� �2s

1

Ekr?Ek

� �2

: (3)

Here, Ek is the local electric field parallel to a magnetic field

line and L is the parallel decorrelation length along the field.

The first term, 2L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi@B?

B

� �2r

, is the mean-square cross field

excursion for an electron which scatters 90� traversing L.

The second is the inverse of the characteristic length for vari-

ation of Ejj. Here, we took the smallest length possible, the

electron mean free path in the center of the flux rope. When

a 1, the local version of Ohm’s law is correct. This term is

evaluated and shown in Fig. 4. As the quantities in Eq. (3)

vary in time, alpha was averaged over one rope oscillation

(from 5.104 ms< t< 5.321 ms). Alpha is as high as 100, the

average over the total plasma volume hai¼ 6.

Figure 4 indicates that the non-locality condition is satis-

fied and Ohm’s law as expressed in Eq. (1) cannot be used to

evaluate the parallel resistivity.

Since it was determined that Ohm’s law cannot be used,

we adopt another method to determine the global resistivity,

FIG. 2. (a) Two pressure isosurfaces at

t¼ 5.586 ms. Surfaces of lower pres-

sure extend further away (the pressure

is measured from 0.64< dz< 9.6 m).

(b) and (c) Contours of the axial com-

ponent of the space charge and induced

electric field, respectively, in a plane

dz¼ 3.20 m and at the same time. The

color bar is the same for both. The

planes are transverse to the device axis.

Contour lines are expressed in V/m. In

the center of the diagram where the

magnetic islands and QSL are located,

� @~A@t is of order 0.1 V/m while the space

charge field �r/ can be �4.2 V/m. To

guide the eye, color contours of the cur-

rent density are also shown. The color

map is shown.

FIG. 3. Resistivity approximated using

Eq. (2) shown at t¼ 5.232 ms on trans-

verse planes at two axial locations.

The red areas denote negative resistiv-

ity. The average resistivity on the

plane on the left (dz¼ 1.28 m) is the

Spitzer resistivity, and gjj at 7.04 m (b)

is �1.4 times the Spitzer resistivity.

Also shown are contours of the parallel

current, Jjj, to guide the eye.

070701-3 Gekelman et al. Phys. Plasmas 24, 070701 (2017)

which is based on the fluctuation-dissipation theorem.28 The

AC plasma conductivity tensor was derived by Kubo29 and

is given by

r x;~rð Þ ¼ ne2

mv2the

ð ðe�ixtv� t;~r0ð Þvl tþ s;~rð Þdsdt (4)

and we apply this formula under the assumption of non-

equilibrium steady state in which the currents lead to dissipa-

tion.30,31 Here, n is the plasma density, v� is a component of

the velocity (�¼ x, y, and z), and vthe is the local electron

thermal velocity of the bulk plasma at the movable probe

location r. Equation (4) is a double integral first over the

delay time s and then t for the Fourier transform. This is a

correlation function in which the index � pertains to a veloc-

ity component at a fixed point, ~r0, chosen to lie on a field

line in the center of the average current of one of the ropes.

The other index l is that of the velocity field at all locations,

~r , in the x-y plane for a given z location. The velocity in the

integral is the electron drift, which can be obtained from the

current density ~J ¼ ne~v; which is also measured throughout

the volume. We take this to be the “test particle” velocity in

the Kubo formula. This is an approximation and we have

assumed that the current is from a drifting Maxwellian. The

tensor conductivity, both the real and imaginary parts, was

evaluated at 5 kHz, the rope oscillation frequency. Note that

all relevant quantities in this experiment, such as Q, mag-

netic helicity, and flux rope positions, oscillate at this fre-

quency as well.

The Kubo resistivity, 1/rzz (the axis parallel to Boz and

the device axis), on four planes is shown in Fig. 5. In Figs.

5(a) and 5(b), the two current channels are positioned at the

upper left and right hand corner. The resistivity is also large

in regions where the current is small, for example, the edges

of the plane in Fig. 3. Let us focus on the central region

where current is appreciable and reconnection occurs.

In the first 4–5 m of the interaction region, the collisions

of the ropes are most prominent and is most likely that the

bulk of reconnection occurs there. This is reflected in the

larger resistivity in the center of Figs. 5(a) and 5(b). There

are also large resistivities inside the current channels and in

Fig. 5(c), g peaks inside the currents channels, not in the gra-

dient region. In Fig. 5(d), at dz¼ 8.32 m, the currents have

nearly merged. The enhanced resistivity in the reconnection

region and in the center of the current channels can arise

from different processes.

The measurement of the plasma resistivity in a current

carrying plasma is difficult. The worst thing one can do is to

FIG. 4. a(x, y, and z). Equation (3)

was evaluated at every point along

field lines, nearly 8 m in length. avaries from zero to over 100 in the

region where the flux ropes are found.

Alpha is averaged over one half cycle

of the rope motion. Also shown are

magnetic field lines of the flux ropes at

one time during the half cycle

(t¼ 5.18 ms).

070701-4 Gekelman et al. Phys. Plasmas 24, 070701 (2017)

divide the external voltage imposed on the system by the cur-

rent through it.32 It is a simple matter to measure the current

but the external voltage has no relation to the internal electric

field due to sheaths, which always develop on the boundaries.

Experiments have tried to correct for the boundary effects,33

but one cannot be sure that the corrections are valid. Some

experiments have used Ohm’s law with assumptions in order

to neglect terms to estimate the resistivity.34–36 It is not clear

if their conclusions are valid. The closest experiment to this

one was reported by Intrator et al.21 The flux ropes were pro-

duced by two plasma guns at the end of a vacuum vessel with

an axial field and there was no background plasma. The mea-

surement relevant quantities were spatially sparse making

integration along field lines impossible. The value of the

global resistivity was estimated from an incomplete Ohm’s

law and was of order 1.3 larger than classical. The Kubo resis-

tivity has been used in computer simulations of a 3D magnetic

null point to calculate the plasma resistivity.37 The simulation

included chaotic orbits at the null point where there was no

exact method to calculate the resistivity.

In the experiment reported here, all the terms in Ohm’s

law have been evaluated in space and time using the most

comprehensive data set on magnetic flux ropes acquired to

date. Ohm’s law yielded a non-physical negative resistivity

at some spatial locations. This was also the case when the

resistivity was evaluated by integrating the electric field and

plasma current along field lines. We examined the possibility

that Ohm’s law was non-local and could not be used to eval-

uate g. This turned out to be the case where in the future

those who use it in experimental or simulation results should

do so very carefully.

The AC Kubo conductivity was derived from the data

and the parallel resistivity evaluated. The AC resistivity is

enhanced in the flux rope currents and in the region between

the current channels at a location where the QSL is large

and reconnection occurs. What is the cause of anomalous

resistivity (that is anything larger than the classical one)?

The culprit is generally thought to be scattering of the elec-

trons by waves or turbulence. There are a number of candi-

dates such as ion acoustic turbulence and Langmuir waves,

basically any phenomenon that has a large fluctuating elec-

tric field. In this experiment, the quantities measured to cal-

culate the terms in Ohm’s law were filtered to exclude

everything above 3 MHz. With high frequency probes

ð2:5MHz � f � 1 GHzÞ, we measured the spectra of electric

and magnetic field fluctuations. The electric field spectra are

exponential from 1 to 10 MHz and have a broad peak at

20 MHz which is �20 dB from that of the rope oscillations.

A second broad peak at 80 MHz (near the lower hybrid fre-

quency) is 15 dB below the first. Based on this, we think

that there are little, if any, high frequency contributions to

the resistivity.

One of the authors (W.G.) would like to thank George

Morales for interesting discussions. We also would like to

thank Zoltan Lucky, Marvin Drandell, and Tai Ly for their

expert technical support. The work was funded in part by a

Grant from the University of California, Office of the

President (12-LR-237124). It was performed at the Basic

Plasma Science Facility, which is funded by DOE (DE-FC02-

07ER54918) and the National Science Foundation (NSF-PHY

1036140).

FIG. 5. Kubo AC resistivity on four planes. It is divided by the local Spitzer resistivity at each spatial location; therefore, 41 means that gKubo ¼ 41gSpitzer .

Since the temperature and thus the Spitzer resistivity oscillate in time, its mean value at each location was used. The Kubo resistivity is always positive.

070701-5 Gekelman et al. Phys. Plasmas 24, 070701 (2017)

1C. T. Russell, E. R. Priest, and L.-C. Lee, Physics of Magnetic Flux Ropes,

AGU Geophysical Monograph Series Vol. 58 (American Geophysical

Union, Washington, DC, 1990); S. Lukin, “Self-organization in magnetic

flux ropes,” Plasma Phys. Controlled Fusion 56, 060301 (2014).2P. D. Henderson, C. J. Owen, I. V. Alexeev, J. Slavin, A. N. Fazakerley,

E. Lucek, and H. Reme, “Cluster observations of flux rope structures in

the near-tail,” Ann. Geophys. 24, 651 (2006).3R. L. Stenzel, W. Gekelman, and N. Wild, “Magnetic field line reconnec-

tion experiments 4. Resistivity heating and energy flow,” J. Geophys. Res.

87, 111, doi:10.1029/JA087iA01p00111 (1982).4W. Gekelman, R. L. Stenzel, and N. Wild, “Magnetic field line reconnec-

tion experiments 3. Ion acceleration, flows and anomalous scattering,”

J. Geophys. Res. 87, 101, doi:10.1029/JA087iA01p00101 (1982).5R. L. Stenzel, W. Gekelman, and N. Wild, “Magnetic field line reconnec-

tion experiments 5. Current disruptions and double layers,” J. Geophys.

Res. 88, 4793, doi:10.1029/JA088iA06p04793 (1983).6W. Gekelman and R. L. Stenzel, “Magnetic field line reconnection experi-

ments, part 6 magnetic turbulence,” J. Geophys. Res. 89, 2715,

doi:10.1029/JA089iA05p02715 (1984).7M. Yamada, H. Ji, S. C. Hsu, T. Carter, R. M. Kulsrud, N. Bretz, F. Jobes,

Y. Ono, and F. Perkins, “Study of driven magnetic reconnection in a labo-

ratory plasma,” Phys. Plasmas 4, 1936 (1997).8M. Yamada, “Review of the recent controlled experiments for the study of

local reconnection physics,” Earth Planets Space 53, 509 (2001).9E. R. Priest and P. D�emoulin, “Three-dimensional magnetic reconnection

without null points,” J. Geophys. Res. 100, 23443, doi:10.1029/

95JA02740 (1995).10S. K. P. Tripathi and W. Gekelman, “Dynamics of an erupting arched

magnetic flux rope in a laboratory plasma experiment,” Sol. Phys. 286,

479 (2013).11W. Gekelman, T. De Haas, W. Daughton, B. Van Compernolle, T.

Intrator, and S. Vincena, “Pulsating magnetic reconnection driven by three

dimensional flux-rope interactions,” Phys. Rev. Lett. 116, 235101 (2016).12W. Gekelman, P. Pribyl, Z. Lucky, M. Drandell, D. Leneman, J. Maggs, S.

Vincena, B. Van Compernolle, S. Tripathi, G. Morales, T. Carter, and Y.

Wang, “The Upgraded LAPD device, a machine for studying frontier basic

plasma physics,” Rev. Sci. Instrum. 87, 025105 (2016).13D. Leneman, W. Gekelman, and J. Maggs, “The plasma source of the large

plasma device at UCLA,” Rev. Sci. Instrum 77, 015108 (2006).14P. Pribyl and W. Gekelman, “24 kA solid state switch for a plasma dis-

charge experiments,” Rev. Sci. Instrum. 75, 669–673 (2004 ).15C. Cooper, W. Gekelman, P. Pribyl, and Z. Lucky, “A new large area lan-

thanum hexaboride plasma source,” Rev. Sci. Instrum. 81, 083503 (2010).16R. H. Loveberg, “Magnetic probes,” in Huddlestone and Leonard,

“Plasma Diagnostic Techniques” (Academic Press, New York, 1965), p.

69.17E. T. Everson, P. Pribyl, C. G. Constantin, A. Zylstra, D. Schaeffer, N. L.

Kugland, and C. Niemann, “Design, construction, and calibration of a

three-axis, high-frequency magnetic probe (B-dot probe) as a diagnostic of

exploding plasmas,” Rev. Sci. Instrum. 80, 113505 (2009).

18M. Martin, J. Bonde, W. Gekelman, and P. Pribyl, “A resistively heated

CeB6 emissive probe,” Rev. Sci. Instrum. 86, 053507 (2015).19F. F. Chen, in Plasma Diagnostic Techniques, edited by R. H.

Huddlestone and S. L. Leonard (Academic Press, New York, 1965), Chap.

4, pp. 113–200.20W. Gekelman, T. DeHaas, B. Van Compernolle, W. Daughton, P. Pribyl,

S. Vincena, and D. Hong, “Experimental study of the dynamics of a thin

current sheet,” Phys. Scr. 91, 54002 (2016).21T. Intrator, X. Sun, G. Lapenta, L. Dorf, and I. Furno, “Experimental onset

threshold and magnetic pressure pile-up for 3D reconnection,” Nature 5,

521 (2009).22B. Van Compernolle and W. Gekelman, “Morphology and dynamics of

three interacting kink-unstable flux ropes in a laboratory magnetoplasma,”

Phys. Plasmas 19, 102102 (2012).23V. S. Titov, T. G. Forbes, E. R. Priest, Z. Mik̂ız, and J. A. Linkler, “Slip

squashing factors as a measure of three-dimensional magnetic

reconnection,” Astrophys. J. 693, 1029, doi:10.1088/0004-637X/693/1/

1029 (2009).24E. Lawrence and W. Gekelman, “Identification of a quasi-separatrix layer

in a reconnecting laboratory magnetoplasma,” Phys. Rev. Lett. 103,

105002 (2009).25W. Gekelman, E. Lawrence, and B. Van Compernolle, “Three-dimen-

sional reconnection involving magnetic flux ropes,” Astrophys. J. 753, 131

(2012).26M. Hudis and L. M. Lidsky, “Directional Langmuir probes,” J. Appl.

Phys. 41, 5011 (1970).27A. Jacobson and R. Moses, “Non local DC electrical conductivity of a

Lorentz plasma in a stochastic magnetic field,” Phys. Rev. A 29, 3335

(1984).28L. Onsager, “Reciprocal relations in irreversible processes,” Phys. Rev.

32, 110 (1931).29H. Kubo, “Fluctuation dissipation theorem,” Rep. Prog. Phys. 29, 255 (1966).30D. J. Evans, E. Cohen, and G. P. Morris, Phys. Rev. Lett. 71, 2401 (1993).31D. J. Evans and D. J. Searles, “The fluctuation theorem,” Adv. Phys.

51(7), 1529–1585 (2002).32A. W. DeSilva and A. D. Rakhel, “Electrical resistivity and thermody-

namic properties of a dense tungsten plasma,” Int. J. Thermophys. 26(V4),

1137 (2005).33D. Roehling, “Electrical resistivity of a partially ionized cesium plasma,”

Adv. Energy Convers. 3, 69 (1963).34P. J. Baum, A. Bratenahl, and G. Kamin, “Current interruption and impul-

sive flux transfer solar flare models,” Astrophys. J. 226, 286 (1978).35A. Kuritsyn, M. Yamada, S. Gerhardt, H. Ji, R. Kulsrud, and Y. Ren,

“Measurements of the parallel and transverse spitzer resistivities during

collisional magnetic reconnection,” Phys. Plasmas 13, 055703 (2006).36M. C. Zarnstorff, K. McGuire, M. G. Bell, B. Gek, D. Johnson, D.

McCune, H. Park, A. Ramsey, and G. Taylor, “Parallel electric resistivity

in the TFTR tokamak,” Phys. Fluids B 2, 1852 (1990).37D. A. St-Onge and R. D. Sydora, “Kubo conductivity tensor for two and

three dimensional magnetic nulls,” Phys. Rev. E 90, 033103 (2014).

070701-6 Gekelman et al. Phys. Plasmas 24, 070701 (2017)