Measurement of the Transverse Spitzer Resistivity during Collisional Magnetic Reconnection Page: 7 of 18
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The second condition is violated in the reconnection neutral sheet. The magnetic field
in MRX has a null surface in the middle of the current sheet and, to the lowest order, varies
linearly with radius away from null surface. The plasma and current density are both peaked
at the magnetic null line, while Spitzer's resistivity calculations assume spatial uniformity.
It is not obvious which value, r, or r, , if any, applies in the case of MRX current sheet. A
detailed kinetic theory of resistivity in non-uniform magnetic field is beyond the scope of
this letter. In the approximation of circular particle orbits, one can write the Fokker-Plank
equation for the species a as
Oft Ofa(Ofa
S-Vf,-GQ +qE--V (2)
where a = i or e, 0 is the gyrophase angle, qe are the charges and Qe, = qaB/rm. z is
taken to be parallel to B, and all quantities are allowed to vary in the x direction only. (x
corresponds to R - Ro in the MRX device coordinates, where Ro is the major radius of the
field null). Then the diamagnetic current is in the y direction. The right hand side of (2) is
the collisional Fokker-Plank term. The zeroth order distribution functions are taken to be
Maxwellian with densities ne, and temperatures Te that depend on x quadratically as
ne = nto + n"o x2/2
T = To + T11 x2 /2
The distributions make the collisional terms and the terms containing Q vanish. The mag-
netic field at the origin is linear so that Q,, = ' x. Using standard Fokker-Plank theory
one finds the correction to the distribution function to the first order in both 1/(r(Q) and
eES/Te, and by taking the first velocity moment of (2) the electron-ion friction force as
RS=pitzr 3 Spitzer (3)
where jeT (0Te/Ox)/(ctsitzcr 4 ) is the diamagnetic current resulting from the electron
temperature gradient. The resulting resistivity is essentially equal to Spitzer's r, with a
correction due to the temperature gradient. In MRX this correction is ignored since the5
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Trintchouk, F.; Yamada, M.; Ji, H.; Kulsrud, R.M. & Carter, T.A. Measurement of the Transverse Spitzer Resistivity during Collisional Magnetic Reconnection, report, September 18, 2000; Princeton, New Jersey. (https://digital.library.unt.edu/ark:/67531/metadc740671/m1/7/: accessed July 18, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.