23A03
Collisionality Dependence of Shielding Factor of Beam Driven Current
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Mitsuru Honda, Mitsuru Kikuchi and Masafumi Azumi¶+# , >1 3H&: /¶¶Japan Atomic Energy Agency, 801-1, Mukoyama, Naka, Ibaraki 311-0193, JapanAZM TechnoScience Soft, 6-16-3-209, Chitosedai, Setagaya-ku, Tokyo 157-0071, Japan*+$9;E7.- 311-0193 ="8C5( '801-1¶AZM  157-0071 ,D6A06-16-3-209

The collisionality dependence of a shielding factor of a neutral beam driven current, which is generally neglected formost models currently used, is shown to be important especially for present-day tokamak experiments. An addition offriction coefficients involving beam ions to the Matrix Inversion method based on the moment approach readily makes
it possible to estimate the collisionality-dependent shielding factor. The models proposed coincide with the collisionlessmodel in the collisionless limit. They clearly elucidate the strong dependence of the shielding factor on collisionality,indicating that using a collisionless model always overestimates a beam driven current.

1. Introduction
There is of growing importance in accurately
estimating the current driven by neutral beam
injection (NBI) for obtaining a fully
current-driven, steady-state plasma. Althoughtangential NBI typically produces circulating
beam ions carrying the substantial toroidal
current, electrons tend to be dragged by these ionsdue to parallel friction and subsequently cancel
the fast-ion circulating current out effectively.
Ohkawa clarified that the effective beam driven
current could flow in an impure plasma [1].
Furthermore, it was found that trapped electrons
could reduce the induced electron circulating
current that cancels the beam-ion current, due tothe neoclassical effect [2, 3]. In general, the ratioof the beam driven current, jb, to the fast-ion
circulating current, j||f , is called a shielding factor
and is often written by! = jb j||f = 1" F(1" G),(1)where F stems from the impure characteristic of aplasma and usually F = Zb/Zeff, valid for mostexperiments, and G represents the neoclassicallytrapped electron correction. Accurately estimating
the beam driven current essentially relies on
accurately estimating G, i.e. !.
Many models for calculating G have been
proposed for practical predictions. However, they
all assume that the electron response is collisionless.
Essentially, we believe that the models should havethe collisionality dependence, because frequentcollisions hinder trapped electrons from orbiting
their banana motion, reducing G compared tocollisionless cases.

Because the G factor purely comes from theneoclassical trapping of electrons, Kim et al. [4]and Lin-Lin and Hinton [5] independently derived
the same expression of G using the neoclassicalapproach and the latter authors also enunciated thatG is closely related to the bootstrap coefficient L31.
The latter model is called the Lin-Liu model
hereafter, which borrows the analytical expressionof L31 from the Hirshman’s theory in the banana
regime [6]. In contrast, focusing on the neoclassicalproperties of G, we find that the Matrix Inversion(MI) method [7], which is developed for calculating
the bootstrap current and is based on the moment
approach [8], is capable of accurately estimating Gand thus !. In this paper, we elucidate theimportance of the collisionality dependence of theshielding factor using the updated MI method,which has recently incorporated the Shaing’sviscosity model [9], called the MI-S method.
2. Analytical derivation of the shielding factor
based on the moment approach
Hirshman and Sigmar [8] showed that the Gfactor could be derived by the momentum balance
equations for electrons. We now derive theshielding factor from the momentum balance
equations with appropriate friction forces between
thermal species and fast ions in a complete manner.
The electron momentum equations are given by!"
B ! " ! # e = BRei + BRe b ,!"B ! " ! #e = BH ei + BH e b ,

(2)
(3)

where R and H denote the friction forces forparticles and heat, respectively. Just adding thesecond terms on the right-hand side of both
equations, frictions against fast ions, distinguishes
these equations from the usual neoclassical moment
equations. After some algebraic calculation, we
have the shielding factor in the form:! = 1" Z b Z eff (1" L31 ),(µ̂ e " !ee )µ̂ e " (µ̂ e + ! ee )µ̂ e & (4)Z #= 1" b %1" e 3 ee 22 e 1 ee 2 e12 ee2 2 (,Z eff $ (µ̂3 " ! 22 )(µ̂1 " !11 ) " (µ̂ 2 + !12 ) 'where ! and ! represent viscosity and friction
coefficients, respectively. This expression, called
the MI-S analytic model, is essentially identical to
the Lin-Liu model that has been derived from the
Fokker-Planck equation if the viscosities in thebanana regime [6] are substituted. In this sense,using eq. (4) with the collisionality dependent
viscosities that can be calculated by the MI-S
method yields the collisionality dependent !. Of
course, the MI-S method alternatively can estimate! in a more fundamental way, expressed by! !
#k =e, j,b Zk nk [(M " L)"1 ]kb
Bjb
! !
!==.
Bj||fZ b nb [( M " L)"1 ]bb

collisionality is taken into account. This clear
degradation of ! in the high collisionality regime
implies that collisionality has a significant impact
on ! and this tendency would be effective in theedge region and/or in present-day tokamak
experiments, as shown in Fig.3 for example.

Fig.2. Collisionality dependence of ! at #=0.1 for
Zeff=2.0.

(5)Here, the matrices are computed by the MI-Smethod.
3. Numerical results
The following calculations have been doneusing a concentric circular equilibrium calculatedby an equilibrium code. Figure 1 shows thecomparison of the MI-S model in eq. (4) and theMI-S analytic model in eq. (5) against the Start
and Cordey model [10], which consists oftabulated numerical values based on thecollisionless Fokker-Planck calculations.

Fig.1. ! profiles in the collisionless regime.

Clearly, for this collisionless case, very good
agreement has been obtained among them for any
Zeff. As shown in Fig. 2, ! rapidly decreases as theeffective collisionality "*e increases, once

Fig.3. Profiles of ! and "*e for JT-60U #48822 at
t=12.7 s.

Acknowledgments
This work was supported by a Grant-in-Aid forYoung Scientists (B) (No 22760667) from theJapan Society for the Promotion of Science (JSPS).
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