J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)148–151
000–000

Relation between the Radial Electric Field and the Flowin a Tokamak Plasma with ∇T = 0
OKAMOTO Masao1,2, SATAKE Shinsuke1 and NAKAJIMA Noriyoshi121
National Institute for Fusion Science, Gifu 509-5292, JapanThe Graduate University for Advanced Studies, Hayama, Kanagawa 240-0193, Japan(Received: 19 December 2003 / Accepted: 23 February 2004)

Abstract
The toroidal viscosity determines the neoclassical radial electric field in an axisymmetric tokamak plasma as innon-axisymmetric systems. This toroidal viscosity is very small and obtained only when the finite orbit width effect isconsidered. In the case of uniform temperature (∇T = 0), the wave equation is derived to show the oscillatory behaviourof the radial electric field. The oscillatory radial electric field converges a steady state which satisfies the satandard
neoclassical relation between the parallel flow and the radial electric field.
Keywords:
neoclassical radial elelctric field, tokamak, non-axisymmetric system, finite orbit width, toroidal viscocity,
parallel flow

1. Introduction

fourth order particle flux and obtained the radial electric fieldat the steady state. This fourth order particle flux is very small
and attributed to the finite orbit width (FOW) effect.
In the general case of non-zero ∇T, it is difficult to obtainhigher order flux without solving numerically the drift kinetic
equation. To include comprehensively the FOW effect instudying neoclassical transport, the δ f Monte Carlo method
has been developed [4-11]. Since this method includes
accurately both FOW effect and collisional effect which
ensures the momentum and energy conservations, it isapplicable to the neoclassical transport independent of εordering. Recently, Okamoto et al. [11] calculated theneoclassical radial electric field in a tokamak plasma with aflow by using a δ f simultion code which solves
simultaneouslly the drift kinetic equation and the equation for
the time development of the radial electric field. The radial
electric field oscilates with a frequency of the geodesic
acoustic mode and converges to a steady state value.
In the present paper, we derive the wave equation of theoscilatory electric field in the case of ∇T = 0, assuming thatthe radial particle flux is calculated by the δ f simulation. It isshown that the oscilatory electric field converges and theresultant steady state radial electric field and the parallel flow
has the same relation with that of the standard neoclassicaltheory. It is conjectured that the standard neoclassicalrelationship between the radial electric field and the parallel
flow holds even for the plasma with FOW.

It is a very important subject to examine whatmechanism determines the radial electric field in toroidal
systems, because the radial electric field plays an importantrole for the plasma transport. The radial electric field isdetermined by the ambipolar condition : Γe (Er) = Γi (Er),where Γ e and Γ i are electron and ion particle fluxes,respectively, and Er is the radial electric field. The particle
fluxes arise from various physical processes such as
neoclassical transport, anomalous transport, charge exchange,particle loss out of the confinement region.
In the present paper, we consider only neoclassicaltransport to determine the radial electric field in axisymmetrictokamaks. It is well known, in axisymmetric tokamaks, thatthe particle transport is intrinsically ambipolar, which meansthat the electron and ion particle fluxes are independent of
the radial electric field, and thus it is not determined.
However, this intrinsically ambipolar condition holds only inthe lowest order of ε in the drift kinetic equation, where ε =ρp / L << 1 with ρp the poloidal Larmor radius and L thecharacteristic length of the plasma. The standard neoclassicaltheory [1,2] holds in this lowest order of ε and thus in thelimit of the small orbit width (SOW) of the plasma particles.
Therefore, to determine the neoclassical radial electric fieldEr in an axisymmetric tokamak, the fourth order O(ε 4) ofparticle flux must be calculated solving the drift kinetic
equation up to the second order O(ε 2). Assuming the constanttermperarure (∇T = 0), Rosenbluth et al. [3] calculated the©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: okamoto@LHD.nifs.ac.jp1148

Okamoto M. et al., Relation between the Radial Electric Field and the Flow in a Tokamak Plasma with ∇T = 0➞

2. Parallel and toroidal viscosities
The equation for momentum balance is given, at thesteady state, byna e a E + u a × B + Fa – ∇Pa – ∇ ⋅ Πa = 0(1)Taking toroidal component (R2∇φ) gives the particle flux inthe radial direction for species a,Γa ψ ≡ na u a ⋅ ∇ψ = R ∇φ ⋅ na u a × B2Γa ψ = –R ∇φ ⋅2

∇ψ(2)

(3)

It is noted that the relations ∑ Fa = 0 (momentum conservation
during the collision), ∑ ea na = 0 (charge neutrality), and ∑ Πa= 0 (viscosity force balance) hold. Then, ∑ ea Γaψ = 0, whichis consistent with vanishing radial current at the steady state.
In the axisymmetric tokamak, the toroidal viscosity issmall due to axisymmetry, which is on the order of O(ε 4) if ε➞ ➞➞➞is small [3], and parallel force balance B 〈B · Fa〉 = 〈B · ∇ ·Πa〉 dominates the particle diffusion. Then, in the lowest order,
the relation Γaψ = Γiψ holds independently of the radialelectric field Eψ . This is the intrinsic ambipolarity.
The total particle flux in the radial direction is given,omitting the subscript ψ, byΓ = –Γee – Γei + Γii + Γie+ 4π nm c 2

Γi =∇ψB22∂2Φ = 4π e Γi∂t ∂ψ

(6)

dv vd ⋅ ∇ψ f(7)Equation (6) with (7) holds independently of the magnitudeof pressure gradients. That is, eqs.(6) and (7) are unrelated tothe ordering of ε = ρp / L. In the standard neoclassical theory,like-particle collisions produce no particle flux in the SOW
limit, and hence Γi vanishes. However, in the δ f simulation,
Γi remains finite even if the pressure gradient is very small,because the effect of FOW is rigorously taken into accountby solving guiding center equations. If the FOW effect issmall, Γi is small and a small Eψ is generated. If the FOW
effect is large, Γi is large and a large Eψ is generated. The
radial electric field evolves so as to vanish Γi to maintain thecharge neutrality.
In this paper, only the case of ∇T = 0 is considered. For
simplicity, it is assumed that the magnetic surfaces areconcentric and circular. Then, equation (7) becomes(4)

Here, Γee , Γei , Γii , and Γie are radial particle fluxes caused byelectron-electron, electron-ion, ion-ion, and ion-electron
collisions, respectively. In the lowest order, Γei = Γie
independent of Eψ , and Γee is very small compared to Γii ,since the particle fluxes due to self-collisions are attributed
to FOW effects in the axisymmetric tokamak. Γii comes fromthe toroidal viscosity Γii = 〈∇ · (R2∇φ · Πa) 〉, which depends
on Eψ . It is concluded that Γii (Eψ ) = 0 determines Eψ .
In non-axisymmetric tori (stellarator, helical system like➞LHD), the toroidal viscosity 〈Bt · ∇ · Πa〉 dominates theparticle diffusion over the parallel viscosity and Γeψ (Eψ ) =Γiψ (Eψ ) determines Eψ .

21 + c2vA∂E r= – 4π e∂t

3d v f vdr(8)where vA = B/(√4π nm) is the Alfven speed. The radial andpoloidal drift velocities are given byv 2|| + v 2
sin θ
2 Ω 0 R0(9)22E r v || + v–sin θB2 Ω0 R0(10)vdr = –

3. Wave equation
Only ions are treated in the present paper. We consider aplasma in an axisymmetric tokamak and all the variables
depend only on the radial position ψ and the poloidal angle
θ. In the coordinates (ε, µ, ψ, θ), the drift kinetic equation is
given by [1]∂f e ∂Φ ∂f++ v || + vd ⋅ ∇f = C ( f , f )∂t m ∂t ∂ε2

In this equation, Γi = Γii is the ion particle flux. (Γee is verysmall and neglected.) Φ is the electric potential, ψ is thetoroidal flux (the radial electric field is Eψ = – ∂Φ / ∂ψ), andthe second term on the left hand side is attributed to theclassical polarization current. The bracket 〈 …〉 means the flux
average. The ion particle flux Γi is defined by

1 F +n Eaea a2+ 1 ∇ ⋅ R ∇φ ⋅ Πaea➞

where v|| is the parallel velocity, vd is the drift velocity, and C
is the collision operator. This drift kinetic equation is solved
numerically by the δ f method with two weights [5]. However,to solve this equation, the equation for the radial electric fieldis required.
From Poisson equation and the continuity equation, thetime development for the radial electric field is given byvd θ = –where R0 is the major radius and Ω0 = eB0 /mc with B0 themagnetic field at the axis. The parallel flow velocity has aform of(5)2149

Okamoto M. et al., Relation between the Radial Electric Field and the Flow in a Tokamak Plasma with ∇T = 0B0Bv || (r , θ ) = v ||Equation (18) suggests that the radial electric fieldinteracts with moving particles to generate an oscillation witha frequency ωGAM. The solution to eq.(18) is(11)withB = B0 1 – r cos θ .
RE r = E r + Ae(12)∂2 E r∂t

2= – 4π e3dv

∂fv
∂t dr(21)The amplitude A is Landau damped and Er is changed slowly
by magnetic pumping [13]. It is known that the damping ratedepends strongly on the value of the safety factor q. In thelong time limit after the GAM oscillation damps out or after
time averaging, the relation between the radial electric field
and the flux averaged parallel flow becomes

Taking the time derivative of Eq.(8) yieldsε⊥– i ω GAM t(13)with22ε⊥ ≡ 1 + c2 ∼ c2 .
vA vAv || = –(14)From the drift kinetic equation (5)∂fe ∂Φ ∂f – v + v ⋅ ∇f + C .
=– m||d∂t ∂ε∂tv || vzv 2thfM(22)

This is just the relation given by the standard neoclaassicaltheory [1]. Note that the equation (18) or (21) has been
derived without SOW assumption.

(15)

4. Summary

The standard neoclassical theory gives an exact solution
to the drift kinetic equation for the case of ∇T = 0 [1]. Thesolution is a shifted-Maxwellian distribution function fSM
expressed as,

f SM = 1 + 2q R0 v 2th 1 dn e– E .
2r Ω n dr T r

The analysis has been restricted to the plasma with notemperature gradients (∇T = 0). The purpose of this restriction
is to compare the relation of the radial electric field and theparallel flow between the standard neoclassical theory in theSOW limit and the present analysis assuming no SOW limit.
Both relations agree completely. This suggests that the FOW
effect may not affect the standard relation between the radialelectric field and the parallel flow velocity.

(16)

where fM is a Maxwellian with a thermal speed vth given by2nfM =exp e Φ – ε2
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(17)Here, T is the ion temperature. It is noted that the shifted
Maxwellian annihilates off the collision term ; C( fSM , fSM) =0.
Insering Eq.(16) together with Eq.(17) into Eq.(13)yields, up to the order of O(εt = r/R0) and O(〈v||〉/vth),∂2 E r+ ω 2GAM E r = H∂2t(18)where ωGAM is the GAM (Geodesic Acoustic Mode) frequency[12] given byv2
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4 R0(19)H = ω 2GAM Te n1 dn + rm Ω v ||dr eq R0(20)andwhere q is the safety factor, m the ion mass, e the ioniccharge, and Ω is the ion cycrotron frquency. The wave
equation is the same as that in ref. [11] in the small 〈v||〉 limit.
3150

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