Neoclassical electron transport in tokamaks with neutral-beam ...

Neoclassical electron transport in tokamaks with neutral-beam injection 

P. Helander and R. J. Akers 

Euratom/UKAEA Fusion Association, Culham Science Centre, Abingdon, Oxfordshire OX14 3DB, 

United Kingdom 

�Received 29 November 2004; accepted 7 January 2005; published online 11 March 2005� 

The collisional interaction between neutral-beam ions and bulk plasma electrons leads to convective 

transport of particles and energy similar to the well-known Ware pinch. These transport fluxes are 

calculated, and it is found that the particle flux is outward when the neutral beams are in the same 

direction as the plasma current and inward otherwise, while the opposite holds for the electron heat 

transport. This effectively shifts the neutral-beam fueling profile approximately one fast-ion banana 

width outward during coinjection and inward during counterinjection, and could help to explain why 

very different plasma behavior is sometimes observed when the direction of the plasma current is 

reversed. �DOI: 10.1063/1.1864074� 

I. INTRODUCTION 

Over the years, it has been reported from a number of 

tokamaks that plasma behavior can change dramatically if 

the direction of the neutral-beam injection �NBI� is reversed 

with respect to the plasma current, even if the amount of 

current driven by the beams is low. Most recently this phenomenon 

has been observed in the Mega-Ampère Spherical 

Tokamak �MAST�, 1 where reversing the beams has an even 

greater effect than the transition from low �L� to high �H� 

mode confinement. 1 The density profile is much more peaked 

in plasmas with counterinjection, while the temperature profile 

is flatter. Similar behavior had earlier been observed in 

the impurity study experiment ISX-B, 2 the axially symmetric 

divertor experiment, ASDEX, 3 and the Japan Atomic Energy 

Research Institute Fusion Torus, JFT-2M, 4 but does not seem 

prevalent in experiments with higher plasma temperature. 

These observations provide the motivation for the 

present paper, where we reexamine the effect of fast ions on 

neoclassical electron transport. Three decades ago, Connor 

and Cordey 5 discovered that friction between NBI ions and 

bulk plasma electrons gives rise to a convective electron 

flux, qualitatively similar to the Ware pinch but in the opposite 

direction �for cocurrent NBI�. A more complete calculation 

of fast-ion driven transport was outlined by Kim, Callen, 

and Hamnén, 6 who also obtained an accompanying heat flux. 

Unfortunately these results appear to have been largely forgotten. 

In the present paper, we show that the transport coefficients 

describing electron transport driven by fast-ion 

friction are actually identical to other coefficients in the neoclassical 

transport matrix and can therefore easily be accounted 

for in existing transport codes. The kinetic equation 

describing the effect of fast-ion friction on electron transport 

is of the same form as that for ordinary neoclassical diffusion 

in the banana regime, and the corresponding transport coefficients 

are therefore the same. We also show that the net 

effect of the beam-driven pinch is to shift the effective particle 

source of the NBI by approximately one poloidal gyroradius 

outward in the case of coinjection and inward in the 

case of counterinjection. It appears likely that this could help 

PHYSICS OF PLASMAS 12, 042503 �2005� 

1070-664X/2005/12�4�/042503/5/$22.50 12, 042503-1 

to explain some of the experimental observations mentioned 

above. 

II. KINETIC EQUATION 

We consider an axisymmetric tokamak plasma with a 

number of different ion species in a magnetic field B 

=I��+ �� , where � is the toroidal angle, � is the 

poloidal flux, and I��=RB �, with R denoting the major radius. 

The angle � is measured in the direction of the plasma 

current so that � increases with minor radius. For simplicity, 

we assume that the plasma rotates slowly enough that the 

densities of all species are flux functions and standard neoclassical 

theory applies. The drift kinetic equation for electrons 

in a constant magnetic field is approximately 7 

� f e 

� f e 

�t + �v� + vd� · � fe − ev�E� �U = Ce�f e�, �1� 

where v � denotes the parallel velocity, v d is the drift velocity, 

E �=−�B/B�·�A/�t is the parallel component of the inductive 

electric field, and C e is the electron collision operator. The 

gradient is taken at constant magnetic moment and energy 

U=m ev 2 /2−e�, where � is the electrostatic potential. Expanding 

f e= f e0+ f e1+¯ in the smallness of the gyroradius in 

the usual way �assuming small �/�t and E �� gives 8–10 

v�� f e0 = C e�f e0� �2� 

in lowest order, implying a Maxwellian f e0��. In next order 

v�� f e1 + �v d · � �� f e0 

�� + ev�E� fe0 = Ce�f e1�, �3� 

where the cross-field drift is v d· ��=Iv�� v�/� e� and the 

gyrofrequency � e=−eB/m e. Hence, 

T e 

v��g + ev�E� fe0 = Ce�F + g�, �4� 

Te where f e1=F+g and 

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042503-2 P. Helander and R. J. Akers Phys. Plasmas 12, 042503 �2005� 

F =− Iv� 

� e 

� fe0 . �5� 

�� 

The linearized collision operator is 

C e�f e1� = C ee�f e1� + � a 

� 

ea�L�f D e1� + mev� Va�f e0�, �6� 

Te where the first term on the right describes electron-electron 

collisions and L is the pitch-angle scattering operator. The 

sum is taken over all ion species a and 

2 4 

ea naZae ln � 

�D = 2 2 3 4��0mav is the electron scattering rate in collisions with those ions. 

Their mass is denoted by m a, their charge by Z ae, and their 

parallel velocity by V a�. It is assumed that the thermal speed 

of the electrons exceeds that of all ion species, including the 

beam ions. For all thermal species, it follows from standard 

neoclassical theory that V a� is of the form 8–10 

�7� 

Va� =− I d� I dpa − 

B d� naeB d� + ua��B, �8� 

where p a��=n a��T a�� is the pressure of species a and 

u a��B � is its poloidal velocity. The beam ions �subscript b� 

are different because of their high energy and wide orbits, 

and are therefore regarded as a separate species from the 

majority bulk ions �subscript i�. 

The presence of beam ions affects the electron collision 

operator �6� in two ways: directly through the beam-electron 

friction force �i.e., the term containing V b�� and indirectly by 

changing the friction exerted on the electrons by the other 

ion species �i.e., by changing the other V a��. The first of these 

effects is more important than the second. This is readily 

established by estimating the change of V i� caused by collisions 

between beam and bulk ions as 

�Vi� � �ii Vb�, �9� 

�ib where �ii is the bulk ion collision time and �ib is the bulkbeam 

ion collision time. �This estimate follows either from 

consulting Ref. 11 or from balancing poloidal flow damping, 

which occurs on the time scale �ii, with the drive from beamion 

friction, i.e., �Vi�/� ii�V b�/� ib.� It now follows that the 

corresponding change in the electron collision operator �6� is 

smaller than that resulting from beam-electron friction, i.e., 

ei 

� �Vi� D 

eb � 

� Vb� D �ii ni �� ib nb v 3 

Ti 

� 1, �10� 

Vb�� where v Ti=�2T i/m i� 1/2 is the bulk ion thermal speed. 

The kinetic equation �4� can be written as 

Iv� 

�e� d ln pe e d� 

− 

d� 

v��g − Ce��g� =−Ce�� d� Te +� U 

− 

Te 5 

2� d ln Te�f e0� 

d� 

+ � a 

� D ea m ev� 

T e 

Va�f e0 − ev�E� fe0, �11� 

Te ea 

where Ce�=C ee+� DL and �D=� a�D . The driving terms appearing 

on the right are the electron pressure and temperature 

gradients, the ion flow velocities Va�, and the electric field. 

Since the equation is linear and the driving terms appear 

additively, we need only consider the contribution from the 

beam ions. 12 For reasons that will become clear, we also 

include the driving term from the electron pressure gradient, 

i.e., we consider the equation 

v�� g − C e ��g� = � DIv� 

� e � 

d ln p e 

d� − Z bj bB 

p eIZ eff�f e0, �12� 

where j b=Z ben bV b� is the beam-ion current, 

Zeff = 1 

� ne a 

2 

naZa, �13� 

eb 2 

and we have used �D =�nbZ b/neZ 

eff��D and 

C e �� 

Iv� 

� e 

III. TRANSPORT 

d ln pe d� fe0� =−�D Iv� 

� e 

d ln p e 

d� f e0. �14� 

Once Eq. �12� has been solved for g, the neoclassical 

cross-field transport is obtained from the flux-friction 

relation 8–10 

�n eV e · � �� = I�� m ev� 

eB C e�F + g�d 3 v − n eE � 

B �, �15� 

where angular brackets denote flux-surface averages and the 

collision operator is again given by Eq. �6�. Here, 

I 

eb 

�Cee�F� + �DL�F� + �D B mev� Vb�f e0�d3 v� 

e�� m ev� 

=− meI2Zeff e2 �B 

�ee −2� dpe d� + meIZb e2�ee� jb �16� 

B�, 

if only the pressure gradient term in F is retained as discussed 

above. The cross-field particle flux can thus be written 

as 

�n eV e · � �� = I�� m ev� 

eB C e ��g�d3 v − n eE � 

B � 

− peI2 −2 

Zeff��e � 

me�ee + m eIZ b 

e 2 � ee �� 

j b 

T e 

� � ln p e 

B� −� 1 

�� − Zb/Zeff �jbB�� peI B2��j bB��, �17� 

where the electron collision time is defined in the usual way, 


042503-3 Neoclassical electron transport in tokamaks… Phys. Plasmas 12, 042503 �2005� 

�ee = 3�2��3/2 1/2 3/2 2 

me Te �0 

nee4 . �18� 

ln � 

In Eq. �17�, we have added and subtracted a term proportional 

to �jbB�. This is convenient because the pressure gradient 

appears in combination with such a term in Eq. �21� for 

g below. The neoclassical heat transport is evaluated in a 

similar way from 

�qe · � �� = ITe�� e mev�� B mev 2 

− 

2Te 5 

e�F + g�d 

2�C 3 v�, 

�19� 

and is conveniently written as 

�qe · � �� = ITe�� e mev�� B mev 2 

− 

2Te 5 

e��g�d 2�C 3 v� 

A. Banana regime 

+ 3peTeI2 −2 


2me�ee − 3m eT eIZ b 

2e 2 � ee �� 

j b 

� � ln p e 

B� −� 1 

�� − Zb/Zeff �jbB�� peI B 2��j bB��. �20� 

In the banana regime of low collisionality, Eq. �12� is 

solved by expanding g=g 0+g 1+¯ in the smallness of the 

collision frequency. The lowest-order term vanishes in the 

trapped region of velocity space and is determined by the 

constraint equation 

� B 

Ce��g 0�� =− 

v� 

�DmeI� e d ln pe d� − Zb�jbB��f e0 

peIZeff �21� 

in the passing region. It is clear from this equation that the 

beam-ion contribution is of the same form as that from the 

electron pressure gradient: the effect of the beam ions upon g 

can be expressed by making the replacement 

d ln pe d� → d ln pe d� − Zb�jbB� . �22� 

peIZeff This circumstance has earlier been exploited in an elegant 

calculation of NBI current-drive efficiency 13 and can also be 

used to calculate the cross-field transport given by Eqs. �17� 

and �20�. In these equations, the term containing the electron 

pressure gradient �second term on the right� has also been 

combined with the beam current in the same way as in Eq. 

�22�. It follows that if the neoclassical electron transport in a 

plasma without beams is given by transport laws of the form 

�neVe · � �� =− peI2 −2 


me�ee �l d ln pe 11 

d� + l d ln Te 12 

d� 

a 

+ ��l 11 

a d ln pa d� + l a 

12 

d ln Ta�� d� 

− l13neI �E�B� �B2 , �23� 

� 

�q e · � �� 

T e 

=− peI2 −2 


me�ee �l d ln pe 21 

d� + l d ln Te 22 

d� 

a 

+ ��l 21 

a d ln pa d� + l a 

22 

d ln Ta�� d� 

− l23neI �E�B� �B2 , �24� 

� 

a 

where ljk and ljk are dimensionless coefficients and the sum 

is taken over all ion thermal species a, then the corresponding 

transport laws in a plasma with NBI must be 

�neVe · � �� =− peI2 −2 


me�ee �q e · � �� 

T e 

� l 11� d ln p e 

d� − Z b�j bB� 

p eIZ eff � 

d ln Te + l12 d� + a ��l 11 

a d ln pa d� 

a 

+ l12 d ln Ta�� d� − l13neI �E�B� �B2� + m eIZ b 

e 2 � ee �� 

j b 

=− peI2 −2 


me�ee B� −� 1 

B 2��j bB��, �25� 

� l 21� d ln p e 


p eIZ eff � 

d ln Te + l22 d� + a ��l 21 

a d ln pa d� + l22 a d ln Ta�� d� 

2e2 �� ee jb − l23neI �E�B� �B2� − 3meIZb B� −� 1 

B 2� 

��j bB��. �26� 

Therefore it is not necessary to solve any new kinetic equation 

to calculate the transport caused by the presence of NBI 

ions. Apart from the last terms in Eqs. �25� and �26�, the 

beam-driven transport has the same transport coefficient as 

the ordinary neoclassical diffusion. This conclusion holds regardless 

of the number of different ion species in the plasma 

and their relative densities or collisionalities, as long as the 

electrons are in the banana regime. This is useful since the 

transport coefficients l 11 and l 21 can be looked up in the 

literature and are calculated by numerous codes. They depend 

in a complicated way on the magnetic geometry and the 

densities and temperatures of all the species present in the 

plasma. 

It is interesting to compare the magnitude of the beamdriven 

particle pinch with the usual Ware pinch, which is 

given by the term proportional to l 13 in Eq. �25�. The electric 

field is related to the Ohmic current density j � by the neoclassical 

Ohm’s law 

�j�B� = �1−l33��ˆ nee2�ee �E�B�, �27� 

Zeffme where �ˆ is the normalized Spitzer resitivity and l 33 the correction 

due to particle trapping. The beam-driven and Ware 


042503-4 P. Helander and R. J. Akers Phys. Plasmas 12, 042503 �2005� 

pinch terms in Eq. �25� can thus be written as 

�neVe · � ��pinch = IZ −2 

eff��e �� 

me�ee l13 − 

�ˆ �1−l33� l11Zb �jbB� Zeff �j �B� 

�B −2 ��B 2 � + Z b 

Zeff� �jb/B� �B−2� − �j bB��. �28� 

Since the dimensionless coefficients in this expression are all 

of order unity, it follows that the beam-driven pinch is in 

general comparable to the Ware pinch if the beam current is 

comparable to the Ohmic current. To make this more explicit, 

consider a standard circular-flux-surface equilibrium 

with inverse aspect ratio ��1 and Z eff=1. The transport coefficients 

are then �ˆ =1.96, l 11=1.53f t, l 13=1.66f t, and l 33 

=1.31f t, where f t=1.46� 1/2 is the trapped-particle fraction. 

To lowest order in � 1/2 we thus obtain 

�neVe · � ��pinch � ftmeR e2 �1.53jb − 0.85j��. �29� 

�ee The comparison can also be made in terms of the beamdriven 

current, which differs from j b by the current-drive 

efficiency, 

�j NBCDB� 

�j bB� 

according to Ref. 13. Thus, 

=1− Zb �1−l31� �30� 

Zeff �neVe · � ��pinch � ftmeR e2 �0.63� 

�ee −1/2jNBCD − 0.85j�� 31� 

and it follows that the beam-driven pinch per unit of NBIdriven 

current is somewhat larger than the Ware pinch per 

unit of Ohmic current. Moreover, the directions are different: 

whereas the Ware pinch is always inward, the beam-driven 

particle pinch is outward with coinjection and inward with 

counterinjection. The friction between coinjected �counterinjected� 

beam ions and bulk electrons cause the latter to move 

radially outward �inward�. Note that this is in the opposite 

direction to the motion of the injected ions upon ionization. 

�A coinjected ion moves inward from where it was ionized; a 

counterinjected one moves outward.� 

Entirely similar conclusions hold for the beam-driven 

heat flux, except that the signs are reversed because l 21 and 

l 31 are negative. The Ware heat flux is outward, whereas the 

analogous beam-driven heat flux is inward with coinjection 

and outward with counterinjection. 

B. Pfirsch–Schlüter regime 

The situation is slighly different for finite electron collisionality. 

In order to calculate the transport in this case, it is 

convenient to write the solution of Eq. �12� as g=g 0+g 1, 

where g 0 satisfies 

v��g0 − Ce��g 0� = �DIv�� e d ln pe d� − Zb�jbB��f e0. �32� 

peIZeff This is the same equation as Eq. �12� except that a fluxsurface 

average has been taken of the bracket on the righthand 

side, so that g0 is the banana-regime solution considered 

in the preceding section and g1 satisfies 

v��g1 − Ce��g 1� =− �DZbv� �jbB − �jbB��f e0. �33� 

�epeZeff In the Pfirsch–Schlüter regime, the first term on the left can 

be neglected and it follows that the friction associated with 

g 1 is 

� mev�Ce��g 1�d3 v = Zb �jbB − �jbB��. �34� 

�e�ee The accompanying particle flux, 

I 

=− meIZb e�� m ev� 

B C e ��g 1�d 3 v� 

e 2 � ee �� 

jb B� −� 1 

B2� ��j bB��, �35� 

is added to Eq. �25�, where it has the effect of canceling the 

last term. The total flux thus becomes 

�neVe · � �� =− peI2 −2 


me�ee � l 11� d ln p e 


p eIZ eff � 

d ln Te + l12 d� + a ��l 11 


a 

+ l12 d ln Ta�� d� − neI� �E�B� �B2� −� E� B��, 

�36� 

where we have modified the term involving E � to be appropriate 

for the Pfirsch–Schlüter regime. 8–10 The heat flux calculation 

is analogous and gives 

�q e · � �� 

T e 

=− peI2 −2 


me�ee � l 21� d ln p e 


p eIZ eff � 

d ln Te + l22 d� + a ��l 21 


a 

+ l22 d ln Ta�� . �37� 

d� 

IV. DISCUSSION AND CONCLUSIONS 

As we have seen, the friction with beam ions �or any 

population of fast ions with a net parallel flow� drives convective 

cross-field transport of bulk plasma electrons given 

by Eqs. �25� and �26� in the banana regime and Eqs. �36� and 

�37� in the Pfirsch–Schlüter regime. These fluxes are comparable 

to the corresponding Ware fluxes �for similar Ohmic 

and beam currents�, but in the opposite direction. For cocurrent 

NBI the particle flux is outward and will therefore ef- 


042503-5 Neoclassical electron transport in tokamaks… Phys. Plasmas 12, 042503 �2005� 

fectively result in the beam electrons being deposited further 

away from the magnetic axis than one might naively expect. 

In order to estimate the magnitude of this effect, we note 

that the flux-surface averaged electron continuity equation is 8 

�ne �t = Se − 1 � 

V� �� V��neVe · � ��, �38� 

where V�� is the volume of the flux surface labeled by �, 

and Se�� is the particle source strength delivered by the 

beams. Because of the last term on the right, the “average” 

radial position of the source, 

�¯ 0 =� �S edV/� S edV �39� 

�where the volume integral is taken over the entire plasma 

volume� is effectively shifted to � ¯ =� ¯ 0+�� ¯ , with 

�� ¯ =−� � 1 � 

V� �� V��neVe · � ��dV/� SedV =� �neVe · � ��dV/� SedV, �40� 

where we have integrated by parts and ignored a boundary 

term, assuming that the beam density is small at the lastclosed 

flux surface. The slowing-down time for the beam 

ions by friction on the plasma electrons is 

�s = mb�ee 2 , �41� 

meZb and the beam-driven particle pinch can thus be written as 

�n eV e · � �� = l 11 

IZ b 

� b0B 0� s 

�n bV b�B�, �42� 

2 −2 where B0=1/�B � and �b0=Z beB0/m b. For simplicity, we 

have negelected the last term in Eq. �25�, which is small at 

large aspect ratio and vanishes in the Pfirsch–Schlüter regime 

�see Eq. �36��. Hence, 

�� ¯ I�nbVb�B� =� l11 dV/� SedV, �43� 

� b0B 0� s 

where the beam-ion density is of order n b�S b� s, so that 

�� ¯� l11ZbIVb� . �44� 

�b0 It follows that the particle source is effectively shifted �outward 

with coinjection, inward with counterinjection� by 

about a beam-ion banana width, which can be quite a significant 

distance in many experiments �in particular, spherical 

tokamaks�. Insofar as the density profile depends on the fu- 

eling source, one would thus expect the beam-driven pinch 

to have an important effect. 

A similar calculation suggests that the beam-driven heat 

flux given by Eqs. �26� and �37� is less important. It shifts 

the energy deposition profile from the beams by a distance 

equal to the poloidal gyroradius multiplied by T e/E b, where 

E b is the beam energy. Since E b�T e, the heating profile, 

which is already spread out over at least one beam-ion orbit 

width, is not much affected. 

Without performing detailed transport modeling, it is difficult 

to judge whether the beam-driven particle pinch was 

indeed responsible for the density peaking observed during 

counter-NBI in ISX-B, ASDEX, and JFT-2M as mentioned 

in the Introduction. In the case of MAST, such modeling �to 

be published separately� suggests that the Ware pinch is certainly 

very important in shaping the density profile, and that 

the beam-driven pinch, though usually smaller, can sometimes 

be significant in present experiments and can be expected 

to become more important as the beam power is increased 

in planned upgrades of the device. 

ACKNOWLEDGMENTS 

The authors would like to thank Jack Connor and Peter 

Catto for reading the manuscript and Wayne Houlberg for a 

useful discussion. 

This work was funded jointly by EURATOM and the 

UK Engineering and Physical Sciences Research Council. 

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12 The other terms contribute additively to the transport, and are calculated in 

the usual theory. For instance, the terms involving the flow velocities Va� of thermal ions contribute to the transport by terms proportional to the ion 

a 

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13 

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