Modeling electron heat transport during magnetic field buildup in SSPX Page: 4 of 17
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The following model, in cylindrical coordinates and calibrated to CTX , is applied to the
0 r3~ 10 F /OT\1-
B21 10 2
0B 2 + 1r [rP(r)] = -7j2 ,(2)
at 2 r ar
where Eq. (1) is the temperature diffusion equation and Eq. (2) the magnetic energy dif-
fusion equation. Here all quantities are expressed in MKS units except for T, which is
in keV. In the equations, n(r) is the electron density (in units of 1020 m-3), T(r, t) the
electron temperature, B(r, t) the magnetic field, and P(r) = d(vAB2/ ) the Poynting
vector (representing helicity injection that drives the turbulent magnetic field b) where
vA = 1.63 X 106(B/vii) [m/a] is the Alfven speed. In addition, gj2 is the Ohmic heating
where 77 = (4 x 10-8)T~a [W/(m3A2)] is the Spitzer resistivity and j = (XB/p) is the
current density, obtained from Maxwell's equations and the Taylor state , assumed in
this model. The diffusion coefficient in Eq. (1) is
X = XGB + Xmag
XGB = B2a ' (3)
Xmag = veLc . (4)
XGB is an approximate Gyrobohm diffusion coefficient arising from electrostatic turbulence,
and Xmag is the Rechester-Rosenbluth coefficient arising from magnetic turbulence . In
Eqs. (3) and (4) a is the minor radius, Ve = 1.88 x 10' VT [m/s] the electron thermal
speed, and L, = 3.2a from fitting previous results to CTX data [1, 41. Manipulating Eq.
(2) to get B in terms of other quantities ,1 we arrive at
1iwe have modeled the spheromak as a cylinder with length (21ra) and radius a.
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Hua, D.D.; Hooper, E.B. & Fowler, T.K. Modeling electron heat transport during magnetic field buildup in SSPX, report, October 1, 1997; California. (digital.library.unt.edu/ark:/67531/metadc691671/m1/4/: accessed November 12, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.