J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)144–147
000–000

Study of Finite-Orbit-Width Effect on Neoclassical Transportin Tokamak Core Region
SATAKE Shinsuke and OKAMOTO Masao
National Institute for Fusion Science, Toki 509-5292, Japan(Received: 9 December 2003 / Accepted: 4 February 2004)

Abstract
Neoclassical transport simulation using the δ f Monte-Carlo method is carried out to investigate the finite-orbitwidth (FOW) effect on the transport near the magnetic axis. The time evolution of the radial electric field to maintainthe ambipolarity of the flux is calculated simultaneously. It is found that, in the near-axis region, the ion heat flux
decreases from the value predicted by the standard neoclassical theory both in the banana and plateau regimes. Though
the radial transport shows a strong dependence on the FOW effect, the ambipolar electric field profile at the steadystate is similar to that calculated in the small-orbit-width limit approximation.
Keywords:
neoclassical transport, finite-orbit-width effect, δ f method, radial electric field, potato orbit

1. Introduction

radial fluxes. In order to study neoclassical transport withconsistent Er , one has to solve the time development of it
according to the equation below

Neoclassical transport theory has been studied in analytic
approaches [1,2] in which small-orbit-width (SOW)approximation is used. In the analysis, the orbit width ofparticle drift motion is assumed to be negligibly small, though
the finiteness of banana orbit does cause the neoclassicalenhancement of particle and thermal transport. Recently,
however, the finite-orbit-width (FOW) effect on theneoclassical transport has attracted attention. Especially in theregion near the magnetic axis, where relatively wide-width
potato particles dominate the transport process, transport
theory in the SOW approximation breaks its assumption.
There have been several attempts to investigate the transportphenomena in the near-axis region by using a direct
simulation in Monte-Carlo method [3,4], or by an analytic
way considering the existence of non-standard potato orbit
around there [5,6]. Recently, we presented a new transport
theory [7] to study this problem by using Lagrangian
description of drift kinetic equation [8]. In this approach, inwhich the finiteness of real orbit widths is appropriately
considered, it is shown that the ion heat conductivity χ i in
the near-axis region becomes much lower than that ispredicted from the standard theory. In our previous study, theradial electric field has been neglected. However, it is shown[9] that the radial electric field is important in neoclassicaltransport with the FOW effect. The intrinsic ambipolarity is
not valid if the FOW effect is included, and the radial electricfield develops so that it retains Γi (Er) = Γe (Er). And once asheared Er profile is established, it modifies banana widthknown as the orbit-squeezing effect, which in turn affects the⎛ c 2 ⎞ ∂Er (r , t )= − Z i e 〈 Γ ineo (r , Er , t )〉 ,⎜1 + 2 ⎟ ε 0v
∂t
⎝
A⎠

(1)

where 〈...〉 means a flux-surface average, and the electron flux
is neglected. It is important to investigate the formation of
the Er profile to give a neoclassical steady state, but aconsistent treatment of this problem is difficult in an analyticway.
In this paper, we investigate the FOW effect onneoclassical transport with radial electric field by using aMonte-Carlo simulation called the δ f method. In Sec. 2, thebasic formulation of the δ f method is reviewed. In Sec. 3,
simulation results are shown in the banana and plateau
regimes. The SOW-limit model calculation using the δ fmethod, which corresponds to conventional theory, is alsoshown. The summary of this paper is given in Sec. 4.

2. Formulation of δ f method
The time development of gyro-averaged distributionfunction of a species a is described by the drift-kineticequation as follows∂ f (x, , µ , t ) + (v + v ) ⋅∇ f + ∂ f aKK ∂K||da∂t a= Σ C ( fa , fb ),b(2)©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: satake@nifs.ac.jp1144

SATAKE S. and OKAMOTO M., Study of Finite-Orbit-Width Effect on Neoclassical Transport in Tokamak Core Region

where K = mv 2/2, µ = mv⊥2 / 2B, and C( fa , fb) is the FokkerPlank collision term. We consider here only the ion transport
and the ion-electron collision is omitted. The δ f method isbased on the separation of the distribution function in theform f = f0 + δ f. Then eq. (2) is expanded order by order byassuming δ f /f0 ~ vd /v|| ~ δ << 1. The lowest order is∂f0+ v || ⋅ ∇ f 0 = C( f 0 , f 0 ).
∂t

where r̂ is unit vector normal to the flux surface. Note herethat the FOW effect in this formulation arises from the “vd ·”terms in the lhs of eq. (5). Neglecting these terms results in
the reduction to the SOW-limit neoclassical transport [1,2]where these terms are treated as higher-order ones and thus
excluded from eq. (5). In the next section, we compare theSOW-limit calculation with the FOW case to investigate how
the finiteness of particle orbit affects radial transport. In theSOW-limit case, only the parallel motion of marker is solved,and eq. (8) is replaced by p· = 0. The equation for w· isunchanged since eq. (7) arises from the rhs of eq. (5).

(3)

The time-independent solution of eq. (3) is given by a local
Maxwellian

f0 = fM = nm
2πTexp – KT

3/2(4)

3. Simulation results
Simulations are carried out in two collisional regimes;
in banana and plateau regimes. Estimation of collisionality inthe near-magnetic axis region is given as follows [4,7]with n and T being a flux-surface function. The next orderequation becomes∂ + (v + v ) ⋅ ∇ + e v ⋅ E ∂ – C (, f ) δ f||ddrM∂t∂K= – vd ⋅ ∇ –e Er
T… bananaν* (r = rp) << 1
3/2 …
1 << ν* (r = rp) << (R0 /rp)plateauf M + C ( f M , δ f ), (5)where rp = 2(2q2ρ2R0)1/3 is the fattest potato width passing
through the magnetic axis, and ν* = νceff/ωb is the fraction of
the effective collision frequency to the bounce frequency [1].
We use a simple configuration of circular cross-section with
R0 = 4 m, a = 1 m, and q = 2. Profiles of n and T have anexponential slope. The collisionality is controlled by changing
n0, T0, and B0 so that the potato width rp is unchanged (0.25m) in both cases. Initial Er -profile is given from HintonHazeltine’s relation [1] between u|| and Er in a steady state,
where we set u|| = 0 initially. Both in the two collisionality,
we carry out a normal δf simulation including the FOW effectand the SOW-limit case in which the radial drift motion ofmarkers are neglected.

·where K = ev·E is used, and only the radial electric field isconsidered here. In the δf method, only the δ f part is solved
according to eq. (5). The advantage of this method is thereduction of statistical noise, which is about |δ f / f0|2 times
lower than that in a full-f calculation.
The linearized collision term is treated in a Monte-Carloway. The test-particle collision operator CTP (δ f ) = C(δ f , fM)
is implemented by random kicks in the velocity space [10].
The field-particle operator C( fM,δ f ) is approximated by alinear operator P fM which satisfies conservation lows ofcollision term
3d v M (C TP + P f M ) = 0(6)3.1 Collisionless caseIn this case, we choose ν* (rp) ~– 0.12. Since ν* << 1, a
potato particles can complete its orbit before it is scatteredby collisions. Then the finiteness of potato orbit is expected
to affect neoclassical transport around the magnetic axis. The
numerical result of the ion heat flux is shown in Fig. 1.

for M = 1, v||, and K. In the simulation, δ f is expressed bysimulation markers of which distribution is g = Σi δ(x – xi )δ(v– vi ), and two weights w and p for each marker which satisfy
the relations wg = δ f , pg = fM . Each marker moves so as tosatisfy Dg/Dt = 0, where D/Dt is the lhs of eq. (5). Using
these relations in eq. (5) yields the evolution of two weights
as followseEp– vd ⋅ ∇ – r + P fM ,TfM

(7)

p=e Erpv ⋅ ∇–fM .
fM dT(8)3qneo (W/m2)

w=+4[×10 ]The self-consistent radial electric field is obtained fromeq. (1). The neoclassical particle and heat fluxes are evaluatedasrΓneo=rqneo=d v vd ⋅ rwg ,3d v vd ⋅ rKwg ,32100(9)ChangHinton
Simulation (FOW)
Simulation (SOW)

rp0.5

r/aFig. 1 Ion heat flux in ν* ~– 0.12 case.

(10)
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1

SATAKE S. and OKAMOTO M., Study of Finite-Orbit-Width Effect on Neoclassical Transport in Tokamak Core Region

SOW-limit case is similar to the FOW case. Then it seems
that the neoclassical steady state is affected little from thenon-locality of the radial transport. Precise investigation will
be done in the future by introducing a non-zero initial u||profile.

Around the magnetic axis, one can see that qneo drops fromthe expectation by the conventional neoclassical theory [11],in which the existence of potato particles have been neglected.
The FOW effect on qneo appears in the region r < 1.5rp where
potato particles dominate transport process. The sametendency has also been seen in our analytic theory considering
the FOW effect [7], in which the drop begins at r ~ rp. The
result of SOW-limit calculation agrees well with theconventional theory, though we cannot have a reliable value
at r < 0.1 because of an unexpected accumulation ofnumerical noise.
In Fig. 2, the Er and 〈u||〉 profiles in the FOW-case areshown. Radial transport of toroidal angular momentum
develops the u|| profile in time, but its value remains negligible
compared to the thermal velocity. Though the radial transport
is significantly modified, the steady-state Er profile is almost
the same as its initial value which is given by a conventional
theory. It is of interest that the steady-state Er profile in the+3

This case we choose ν* (rp) ~– 10. In this situation, potato
particles are perturbed by collisions before they complete theirfull orbit. Moreover, the main contribution to the radialtransport comes from resonant passing particles. Then it has
been thought that the plateau transport in the near-axis region
is not affected by potato particles [12]. However, as can beseen in Fig. 3, the ion heat flux decreases at the near-axis
region like as in the banana regime case. This is contrastive
to the previous δ f calculation [4] without Er , which shows agood agreement of the resulting χi with the theoretical valueat ν* (rp) ≥ 10. Since the convective heat flux qconv. = 1.5Γi Ti
remains substantially if Er is not solved, the difference ispartly caused by the ambipolar Er in our simulation. It is also
to be noted that the resonant particles, which are characterized
by a small |v|| / v|, have a wide orbit width as potato in thenear-axis region [13]. Therefore, the transport in the plateau
regime should not be simply considered in an analogy withthe transport theory valid in the region away from the axis.
We are planning a more precise analysis on the plateau regime
transport in the near-axis region using δ f simulation.

+3[×10 ][×10 ]

0Er (init.)
Er (stead., FOW)
u|| (stead., FOW)
Er (stead., SOW)

2420

<u||> (m/s)

Er (V/m)

3.2 Collisional case

4. Summary
Neoclassical transport around the magnetic axis has been
investigated by δ f transport simulation with radial electricfield. Both in the banana and plateau regimes, the ion heat
flux is found to decrease from the theoretical value which
neglects potato orbits. We have demonstrated that the δ fmethod can reproduce the conventional neoclassical theory
in the SOW limit by omitting the drift motion of simulationmarkers. Though the radial transport is modified largely by
the existence of potato particles, it is found that the steadystate Er profiles are similar between the FOW- and SOW-limit
simulation results. This work is supported by JSPS Research
Fellowships for Young Scientists, No. 10177.

6802
10.5

r/aFig. 2 Er and 〈u||〉 profiles (ν* ~– 0.12).

+6[×10 ]References

qneo (W/m2)

2

ChangHinton
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[7] S. Satake, M. Okamoto and H. Sugama, Phys. Plasmas
9, 3946 (2002).
[8] I.B. Bernstein and K. Molvig, Phys. Fluids 26, 1488(1983).
[9] W.X. Wang et al., Phys. Rev. Lett. 87, 055002 (2002).

100rp0.51r/aFig. 3 Ion heat flux in ν* ~– 10 case.

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SATAKE S. and OKAMOTO M., Study of Finite-Orbit-Width Effect on Neoclassical Transport in Tokamak Core Region[10] X.Q. Xu and M.N. Rosenbluth, Phys. Fluids B 3, 627(1991).
[11] C.S. Chang and F.L. Hinton, Phys. Fluids 25, 1493
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