PREPARED FOR THE U.S. DEPARTMENT OF ENERGY,
UNDER CONTRACT DE-AC02-76CH03073
PRINCETON PLASMA PHYSICS LABORATORY
PRINCETON UNIVERSITY, PRINCETON, NEW JERSEY
PPPL-3750 PPPL-3750
UC-70
Measurement of the Transverse Spitzer Resistivity
during Collisional Magnetic Reconnection
by
F. Trintchouk, M. Yamada, H. Ji, R.M. Kulsrud, and T.A. Carter
September 2002
PPPL Reports Disclaimer
This report was prepared as an account of work sponsored by an
agency of the United States Government. Neither the United States
Government nor any agency thereof, nor any of their employees, makes any
warranty, express or implied, or assumes any legal liability or
responsibility for the accuracy, completeness, or usefulness of any
information, apparatus, product, or process disclosed, or represents that its
use would not infringe privately owned rights. Reference herein to any
specific commercial product, process, or service by trade name, trademark,
manufacturer, or otherwise, does not necessarily constitute or imply its
endorsement, recommendation, or favoring by the United States
Government or any agency thereof. The views and opinions of authors
expressed herein do not necessarily state or reflect those of the United
States Government or any agency thereof.
Availability
This report is posted on the U.S. Department of Energy’s Princeton
Plasma Physics Laboratory Publications and Reports web site in Fiscal
Year 2002. The home page for PPPL Reports and Publications is:
http://www.pppl.gov/pub_report/
DOE and DOE Contractors can obtain copies of this report from:
U.S. Department of Energy
Office of Scientific and Technical Information
DOE Technical Information Services (DTIS)
P.O. Box 62
Oak Ridge, TN 37831
Telephone: (865) 576-8401
Fax: (865) 576-5728
Email: reports@adonis.osti.gov
This report is available to the general public from:
National Technical Information Service
U.S. Department of Commerce
5285 Port Royal Road
Springfield, VA 22161
Telephone: 1-800-553-6847 or
(703) 605-6000
Fax: (703) 321-8547
Internet: http://www.ntis.gov/ordering.htm
Measurement of the transverse Spitzer resistivity during
collisional magnetic reconnection
F. Trintchouk
∗
, M. Yamada, H. Ji, R. M. Kulsrud and T. A. Carter
Princeton Plasma Physics Laboratory, P.O. Box 451, Princeton, New Jersey 08543
(September 15, 2002)
Abstract
Measurement of the transverse resistivity was carried out in a reconnecting
current sheet where the mean free path for the Coulomb collision is smaller
than the thickness of the sheet. In a collisional neutral sheet without a guide
field, the transverse resistivity is directly related to the reconnection rate. A
remarkable agreement is found between the measured resistivity and the clas-
sical value derived by L. Spitzer. In his calculation the transverse resistivity
for the electrons is higher than the parallel resistivity by a factor of 1.96. The
measured values have verified this theory to within 30% errors.
52.25.Fi, 52.25.Xz
Typeset using REVT
E
X
1
Coulomb collisions among charged particle species were historically the first mechanism
of transport in plasmas to be described by a quantitative theory [1]. In magnetically con-
fined plasma devices this “classical” mechanism is often strongly modified by particle orbit
effects, or is completely dominated by turbulent transport. Nevertheless, the classical value
of electrical resistivity, among other transport coefficients, is universally used as an impor-
tant reference value and the lower bound whenever transport or dissipation phenomena are
discussed.
For plasmas where Coulomb collisions dominate all other dissipation processes, including
wave and turbulence effects the resistivity is determined by the collisional drag on electrons
moving against the background of ions. If a strong magnetic field is applied perpendicular
to the electric field direction, the current is not due to direct acceleration of electrons by
the electric field but is diamagnetic in origin. The transverse or cross-field resistivity was
calculated by L. Spitzer [1] as the rate of momentum transfer from electrons to ions through
collisions:
η
⊥
= 1.03 × 10
−4
T
−3/2
e
Z ln Λ (Ohm · m) (1)
where the electron temperature T
e
is in electron-volts and ln Λ is the Coulomb logarithm.
This resistivity arises from e − i collisions as the electrons drift with respect to the ions and
is determined by the electron velocity. What drives the current or which species carries it
in the laboratory frame is immaterial to Spitzer’s definition. This transverse resistivity is
approximately twice as large as the resistivity without the presence of the magnetic field since
the transverse electron distribution function is not as distorted as in unmagnetized plasma
where the current is carried by energetic electrons experiencing less frequent collisions. We
report the first quantitative measurement of the Spitzer’s transverse resistivity within the
accuracy of 30%, in a highly collisional magnetically reconnecting plasma.
The classical parallel resistivity is equal to that of an unmagnetized plasma. Its value
has been verified experimentally with precision in cylindrical Q-machines by driving current
on the axis of symmetry [2], and later in tokamaks [3]. To the authors’ knowledge no
2
measurements of comparable quality have been reported for the transverse resistivity. The
difficulty of such measurement is due to the E ×B drift resulting from the application of the
transverse electric field. In a typical plasma that incorporates the unrestrained E × B flow
the two terms on the left hand side of Ohm’s law E + v × B = ηj very nearly cancel each
other. Attempting to experimentally evaluate Ohm’s law fails to produce a reliable estimate
of η since it involves subtracting two nearly identical numbers. One way to make the
measurement feasible is to suppress the cross-field flow, which can be achieved by designing
an equilibrium with a magnetic null between regions with oppositely directed fields. Plasma
volumes with oppositely directed fields are brought into contact and the E × B motions are
stagnated against each other. This situation occurs in the field-reversed configuration (FRC).
The flux confinement time observed in FRCs created in θ-pinch machines has so far provided
the only experimental estimates of η
⊥
. The transport in these devices has been found to
be highly anomalous [4], with the resistivities implied by the flux decay generally exceeding
the classical values by a factor from several times to several orders of magnitude. Various
MHD and micro-instabilities were found to correlate with the anomaly factor [5]. These
instabilities, rather then collisions are thought to be the dominant dissipation mechanism in
these experiments.
This letter presents resistivity measurements in the neutral sheet of Magnetic Reconnec-
tion Experiment (MRX), where an electric field is applied perpendicular to the reconnecting
magnetic field. Oppositely directed magnetic field lines merge through the neutral sheet, as
a pressure gradient is created perpendicularly to both E and B (as in FRCs) and a neutral
sheet current is induced. If the plasma were perfectly conductive, this sheet current would
prevent the reconnection of the field lines. If the plasma is collisional, a resistance force is
created against the electric field, the sheet current dissipates and reconnection takes place.
We measured the Spitzer’s transverse resistivity in a highly collisional region of a recon-
necting plasma where the mean free path for Coulomb collisions is smaller than the neutral
sheet thickness.
Fig. 1 (a) shows a poloidal cross-section view of the MRX vacuum vessel. The plasma
3
