Neutral particle and radiation effects on Pfirsch–Schlüter fluxes near the
edge
Peter J. Catto, P. Helander, J. W. Connor, and R. D. Hazeltine
Citation: Phys. Plasmas 5, 3961 (1998); doi: 10.1063/1.873115
View online: http://dx.doi.org/10.1063/1.873115
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Published by the American Institute of Physics.
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PHYSICS OF PLASMAS
VOLUME 5, NUMBER 11
NOVEMBER 1998
Neutral particle and radiation effects on Pfirsch–Schlüter fluxes
near the edge
Peter J. Catto
Plasma Science and Fusion Center, Massachusetts Institute of Technology and Lodestar Research
Corporation, 167 Albany Street, NW16-236, Cambridge, Massachusetts 02139
P. Helander and J. W. Connor
UKAEA Fusion (Euratom—UKAEA Fusion Association), Culham Science Centre, Abingdon, Oxfordshire,
OX14 3DB, United Kingdom
R. D. Hazeltine
Institute for Fusion Studies, The University of Texas at Austin, Austin, Texas 78712
~Received 26 May 1998; accepted 4 August 1998!
The edge plasma of a tokamak is affected by atomic physics processes and can have density and
temperature variations along the magnetic field that strongly modify edge transport. A closed system
of equations in the Pfirsch–Schlüter regime is presented that can be solved for the radial and
poloidal variation of the plasma density, electron and ion temperatures, and the electrostatic
potential in the presence of neutrals and a poloidally asymmetric energy radiation sink due to
inelastic electron collisions. Neutrals have a large diffusivity so their viscosity and heat flux can
become important even when their density is not high, in which case the neutral viscosity alters the
electrostatic potential at the edge by introducing strong radial variation. The strong parallel gradient
in the electron temperature that can arise in the presence of a localized radiation sink drives a
convective flow of particles and heat across the field. This plasma transport mechanism can balance
the neutral influx and is particularly strong if multifaceted asymmetric radiation from the edge
~MARFE! occurs, since the electron temperature then varies substantially over the flux surface.
© 1998 American Institute of Physics. @S1070-664X~98!01211-7#
by Rognlien and Ryutov,4 which treats the magnetic field as
constant, enhances classical transport to model anomalous
effects, ignores neutrals and radiation, and, like Hinton and
Kim, works in the large flow limit of Braginskii.5 The large
flow Braginskii expressions for the ion viscosity do not reduce to the small flow form of Hazeltine2 because of the
different orderings employed.
The interaction of neutrals with ions via charge exchange influences the electric field by introducing a flux surface averaged neutral toroidal angular momentum flux that
can compete with or even dominate over that of the ions. In
the absence of neutrals the radial electric field is found to be
the square of the inverse aspect ratio smaller than the radial
temperature gradient.2 As a result, neutral effects on the electric field are expected to be more pronounced in conventional
than spherical tokamaks. The neutrals can be responsible for
strong radial variation of the electric field and thereby be
responsible for large shear in the E3B, poloidal, and parallel
ion flows, which may have an influence on turbulence.6 The
neutrals also introduce a heat flux that can compete with the
radial ion heat flux and thereby enhance heat transport
losses.
To substantially simplify the algebra and illustrate the
effects of neutrals in the most transparent way possible, we
ignore elastic ion–neutral interactions and assume the charge
exchange rate constant is speed independent. Since elastic
ion–neutral collisions are thought to be weaker than charge
exchange collisions for energies greater than about 10 eV,7
I. INTRODUCTION
Tokamak performance appears to be sensitive to the
edge plasma region just inside the last closed flux surface. In
the work presented here we investigate how its behavior can
be affected by interactions with neutral particles and poloidal
asymmetry in the radiation energy loss due to inelastic electron collisions. These processes are usually unimportant in
the plasma core and are therefore normally neglected in
plasma transport equations. However, because their diffusivity is large, neutrals can enhance energy and momentum loss
and be responsible for the radial variation of the electrostatic
potential, even though the neutral density is typically much
smaller than the plasma density. In addition to directly modifying the energy balance, poloidally asymmetric radiation
loss also creates strong poloidal variation in the electron temperature and thereby drives strong convective fluxes. Convective electron fluxes are particularly strong in the presence
of multifaceted asymmetric radiation from the edge
~MARFE!.1 To illustrate the effects of neutrals and radiation
loss we generalize the conventional Pfirsch–Schlüter regime
treatment of tokamak transport2 to include charge exchange
in the short mean-free path limit and an electron energy sink,
both of which may involve significant poloidal asymmetries.
Our model differs from the collisional model of Hinton and
Kim3 by retaining neutrals and a radiation loss term, by considering the weak plasma flow limit, and by neglecting
anomalous effects and the region beyond the separatrix.
Moreover, it differs from the collisional model implemented
1070-664X/98/5(11)/3961/8/$15.00
3961
© 1998 American Institute of Physics
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3962
Catto et al.
Phys. Plasmas, Vol. 5, No. 11, November 1998
they are not expected to quantitatively alter our results. Consequently, we may quite reasonably account for the collisionality between the ions and neutrals by simply employing the
deuterium charge exchange cross section s x and using the
estimate s x 56310215 cm2. The constant charge exchange
rate approximation8 does not appreciably alter the transport
coefficients.9
To estimate the size of neutral diffusivity effects we can
compare the radial neutral and ion heat fluxes, qW n
;( v 2i /N i ^ s v & x )N n ] T i / ] r and qW i ;(q 2 r 2i / t i )N i ] T i / ] r,
where N n and N i are the neutral and ion densities, v i
5(T i /M ) 1/2 the ion thermal speed with T i the ion temperature, r i the ion gyroradius, t i the ion–ion collision frequency, ^ s v & x the charge exchange rate constant, q the
safety factor, and r the minor radius. For T i 5100 eV, N i
5331014 cm23, B55 T, and q53, we obtain
^ qW n –“ c &
^ qW i –“ c &
;
N n v 2i /N i ^ s v & x
N i q 2 r 2i / t i
;
Nn
Ni
3104 ,
with c the poloidal flux function and ^¯& denoting a flux
surface average. Here s x 56310215 cm2 is used for the
charge exchange cross section, and the neutral mean-free
path is 1/N i s x ;0.5 cm. The neutral heat flux is large because the neutral diffusivity, v i /N i s x , is extremely large, of
order 53106 cm2/s for the preceding numbers ~while
q 2 r 2i / t i ;400 cm2/s!. Consequently, even at neutral densities
a thousandth of the plasma density a sizable neutral effect
occurs, and the effect is larger at lower N i and higher B and
T i . Moreover, by comparing viscosities instead of heat
fluxes we will find that neutral densities smaller by the
square of an inverse aspect ratio than the preceding estimate
can alter the electrostatic potential dramatically.
Representing the radiation losses by a sink, S, in the
electron energy balance equation is a sensible approximation
since any non-Maxwellian features in the electron distribution function due to inelastic electron collisions are
negligible.10 We illustrate the effects of radiation loss by
assuming that ion–impurity collisions are negligible to simplify the presentation. To estimate the impurity density necessary to make a poloidally asymmetric energy sink result in
stronger poloidal electron temperature variation than the
usual Pfirsch–Schlüter terms, we note that S
;N I E I N e v e s I , where N e and N I are the electron and impurity densities, s I is the excitation cross section for a typical
energy loss E I , and v e 5(T e /m) 1/2 is the electron thermal
speed. The poloidal variation of the electron temperature is
estimated by balancing the sink S with the parallel electron
heat variation, nW –“q i e , where q i e ;N e v e lnW –“T e . Here l is
the Coulomb mean-free path and nW 5BW /B the unit vector in
the direction of the magnetic field BW . Normalizing the poloidal variation of the electron temperature by the poloidal
variation of the magnetic field magnitude gives
nW –“ ln T e qR E I
;
N s qR,
nW –“ ln B
el Te I I
where e is the inverse aspect ratio and within a Pfirsch–
Schlüter treatment this ratio must be small compared to
unity. For a cool, dense edge the poloidal variation due to
radiation losses is much larger than that due to the usual
Pfirsch–Schlüter electron transport,11 which is proportional
to the electron gyroradius r e ,
nW –“ ln T e qR q r e
;
,
nW –“ ln B
el w
with w the characteristic edge scale length, which is of the
order of the penetration depth 1/N i ( s x s z ) 1/2, where s z is the
ionization cross section. For example, taking w52 cm, T e
5100 eV, N e 5331014 cm23, B55 T, q53, R5100 cm,
s I 55310216 cm2, and E I 525 eV, gives qRN e s I E I /T e
;10, while q r e /w;1023 and qR/l;3. Consequently, localized impurity to plasma density ratios N I /N e
;1023 – 1022 are sufficient to give large radiation sink effects and strong poloidal variation in the electron temperature. Very high impurity densities would give poloidal variations comparable to that of the magnetic field, but would
complicate the analysis by requiring us to keep ion–impurity
collisions and to treat the poloidal electron temperature
variation as the same order as that of the poloidal variation of
the magnetic field.
In Sec. II we consider the flux-averaged description for
the plasma density, ion and electron temperatures, and electrostatic potential, and present the Pfirsch–Schlüter fluxes
with the neutral contributions to the heat and toroidal angular
momentum fluxes. The neutral viscosity is free of the aspect
ratio factors that make the radial variation of the electrostatic
potential weak in the conventional Pfirsch–Schlüter
treatment,2 so that in a large aspect ratio tokamak the strongest impact of the neutrals is on the electrostatic potential.
The equations governing the poloidal variation of the plasma
density, ion and electron temperatures, and electrostatic potential are obtained in Sec. III to complete our ‘‘four-field’’
model. The strong poloidal variation of the electron temperature due to poloidal variation in the radiation sink is the
source of the large particle and electron heat fluxes found in
Sec. II. These sink-driven convective fluxes can easily dominate the usual Pfirsch–Schlüter particle and electron heat
fluxes that are small in the electron gyroradius. A comparison of the poloidally asymmetric sink-driven convective flux
to typical gyro-Bohm transport is given in Sec. IV, along
with estimates for the neutral density, which confirm the
consistency of our orderings. In Sec. V we present a summarizing discussion.
II. FLUX SURFACE AVERAGED DESCRIPTION
The plasma density N e 5N i , electron T e and ion T i temperatures, and electrostatic potential F are determined by the
flux surface averaged equations for the conservation of number, electron and ion energies, and total toroidal angular momentum, and are flux functions to lowest order in the gyroradius. In the presence of neutrals and a radiation sink S to
account for electron energy loss due to inelastic scattering
and ionization with rate constant ^ s v & z ; the four conservation equations involving the flux surface averaged plasma
W –“ c & 5 ^ G
W –“ c & , neutral particle flux
particle flux5 ^ G
e
i
W
5 ^ Gn –“ c & , electron heat flux5 ^ qW e –“ c & , ion heat flux
5 ^ qW i •¹ c & , neutral heat flux5 ^ qW n –“ c & , toroidal ion angu-
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Catto et al.
Phys. Plasmas, Vol. 5, No. 11, November 1998
J i –“ c & , and toroidal neutral anlar momentum flux5 ^ R zW –p
J n –“ c & , are modified to begular momentum flux5 ^ R zW • p
come
1 ]
]Ne
W –“ c & ! 5 ^ s v & ^ N & N ,
1
~ V 8^ G
e
z
n
e
]t
V8 ]c
S
D
F S
] 3
1 ]
5
N eT e 1
V 8 ^ qW e –“ c & 1 ^ GW e –“ c &
]t 2
V8 ]c
2
52 ^ S & 1
S
D
H F
GJ
where f n and f i are the neutral and ion distribution functions,
and the moments of species k are defined by
DG
W 2
qW k 5Q
k
~2!
5 W W
^ ~ Gi 1Gn ! –“ c &
2
52
3mN e ~ T i 2T e !
,
M t ei
and
J n 5M
p
~3!
and
M Ni
]
1 ]
J i1 p
J n ! –“ c & #
@ V 8 ^ R zW –~ p
^ R zW –VW i & 1
]t
V8 ]c
5
1 W
^ J –“ c & ,
c
~4!
where we assume the neutral density to be smaller than the
W is the mean
plasma density. In the preceding equations V
i
velocity of the ions, M denotes the ion and neutral mass, m is
the electron mass, the electron–ion collision time is t ei
1/2 4
53m 1/2T 3/2
e /4(2 p ) e N e ln L, with ln L the Coulomb
logarithm, z is the toroidal angle variable with zW the corresponding unit vector, the radial variable is the poloidal flux
function c, the magnetic field is written as BW 5I“ z
1“ z 3“ c , with I5I( c )5RB T and B5 u BW u , R is the major
radius and B T the toroidal magnetic field, the current density
is JW , and the flux surface average is defined as
^¯&5
E
8
1
V
dq~ ¯ !
,
BW –“ q
with V 8 5 * d q /BW –“ q and q the poloidal angle variable. Using zW –VW i ;(q r i / e w) v i to estimate the Pfirsch–Schlüter flow
and 4 p ^ JW –“ c & 52 ^ ( ] EW / ] t)–“ c & ;(RB p /ew) ] T e / ] t, we
see that the JW 3BW force term on the right side of Eq. ~4! is of
order ( e v A /qc) 2 compared to the time derivative on the left,
where v A is the Alfvén speed. Often, v A /c!1, so the toroidal JW 3BW force on the right of Eq. ~4! can be neglected.
The terms that arise from the neutrals enter Eqs. ~3! and
~4!, while those due to the sink S alter the plasma particle
and electron heat fluxes in Eqs. ~1! and ~2!. We consider
neutral effects first.
A. Neutral and ion contributions to Pfirsch–Schlüter
fluxes
To describe the neutrals we employ the neutral kinetic
equation with charge exchange collisions12 and ionization
retained, namely
] f n / ] t1 vW –“ f n 5 ^ s v & x ~ N n f i 2N i f n ! 2 ^ s v & z N e f n , ~5!
N k VW k 5
d 3v f k ,
p k 5N k T k 5
] 3
1 ]
N iT i 1
V 8 ^ ~ qW i 1qW n ! –“ c &
]t 2
V8 ]c
1
E
N k5
~1!
3mN e ~ T i 2T e !
,
M t ei
3963
M
3
E
E
d 3 vvW f k ,
W 51 M
Q
n
2
d 3 vv 2 f k ,
5 W 1
p V 5
2 k k 2
E SWW
d 3v v v 2
E
E
d 3 vv 2 vW f n ,
d 3 v~ M v 2 25T k !vW f k ,
D
1 2J
v I fn ,
3
~6!
with WI the unit dyad.
In the short neutral mean-free path limit, the lowestorder neutral distribution function f 0 may be taken as f 0
[N n f i /N i , where to lowest order f i is the stationary Maxwellian f M . We may then use a Grad moment procedure13 in
Eq. ~5! to write the neutral moments in terms of the ion
moments by adopting the ordering
^ s v & x N i f n ; ^ s v & x N n f i @ vW –“ f n ; ^ s v & z N e f n @ ] f n / ] t.
~7!
Neglecting the time derivative term in the M @vW vW 2( v 2 /3)IJ#
and M vW v 2 /2 moments, we find
N n^ s v & x
2t J W
I “–Q n
pi 1
N i~ ^ s v & x1 ^ s v & z !
3
J n5
p
S E
2 t “• M
W 5
Q
n
D
d 3 vvW vW vW f 0 ,
~8!
S E
N n^ s v & x
W 2 t “– M
Q
N i~ ^ s v & x1 ^ s v & z ! i 2
D
d 3 vv 2 vW vW f 0 ,
~9!
where t [1/N i ( ^ s v & x 1 ^ s v & z )'1/N i ^ s v & x .
Equations ~8!–~9! and the number, momentum, and energy moments of Eq. ~5! provide a complete description of
the neutrals, provided we know the ion distribution function.
To lowest order, conservation of momentum and energy give
relations between the neutral and ion temperatures and mean
velocities:
T n5
2t
^sv&x
W 'T ,
T i2
“–Q
n
i
3N n
^ s v & x1 ^ s v & z
~10!
VW n 5
t
^sv&x
VW i 2
“p n 'VW i .
M Nn
^ s v & x1 ^ s v & z
~11!
and
Using a lowest-order Maxwellian ( f 0 [N n f M /N i ) in Eq.
~9!, we find
N n^ s v & x
W 5qW 2 5 p VW 5
W
Q
Q
n
n
2 n n N i~ ^ s v & x1 ^ s v & z ! i
2
5t
“ ~ N n T 2n ! ,
2M
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~12!
3964
Catto et al.
Phys. Plasmas, Vol. 5, No. 11, November 1998
or, upon using Eq. ~11! and neglecting short mean-free path
corrections, the alternate form,
qW n 52
5tpn
N n^ s v & x
“T n 1
qW .
2M
N i~ ^ s v & x1 ^ s v & z ! i
~13!
Next, we consider the final moment of interest. To
evaluate the last term in Eq. ~8! we need the leading corrections to the Maxwellian that are odd in vW and to simplify the
J k –“ c . To
algebra we note that we need only evaluate R zW –p
evaluate this quantity we extract the required higher-order
terms in the ion distribution function f i from Hazeltine,2 correct the numerical coefficients of temperature gradient
terms,14 and write the result in terms of the ion flow velocity
VW i and ion heat flux qW i given by
S
D
cT i 1 ] p i e ] F
1
nW 3“ c
VW i 5V i nW 1
eB p i ] c T i ] c
~14b!
where nW 5BW /B, nW 3“ c 5InW 2RB zW , I5RB T ,
F
S
G
9
cIT i 1 ] p i e ] F
1
1
V i 5nW –VW i 52
eB p i ] c T i ] c
5 ^ B 2&
1
and
D
^ ~ nW –“ ln B ! 2 & B 2 ] T i
,
20^ ~ nW –“B ! 2 & T i ] c
D
S
5cIp i
B2 ]Ti
12 2
.
q i 5nW –qW i 52
2eB
^B & ]c
~15a!
~15b!
Then, the resulting expression for f i may be written conveniently as
f i 5 f M1
1
S
F
G
d 3 vvW vW vW vW
S
D E
S D
M v2 5
2 f 5
2T i 2 M
Ti
d vv a v b v s v g f M5N i
M
3
d 3 vvW vW vW vW f M ,
2
@ d ab d s g 1 d ag d s b
1 d a s d bg # ,
and
F S E
R zW – “– M
S
D
M v2 7
2 f 50,
2T i 2 M
DG
d 3 vvW vW vW f 0 –“ c
W –zW R1R zW –“W
W –“ c
5“ c –“W
W –zW R ! 2W
W –““ c –zW R2R zW –““ c –W
W
5“ c –“ ~ W
W –zW R ! ,
'“ c –“ ~ W
W [N T @ VW 1(2/5p )qW # and “(R zW )5(“R) zW
where W
n i
i
i i
W
2 z “R. Fortunately, all terms involving ““c may be neglected as small since the radial and/or poloidal variation of
N n , T i , F, zW –VW i , and zW –qW i in the plasma edge is much
stronger than that associated with the poloidal flux function.
As a result, when we gather up the preceding expressions
J i term, we find that
and neglect the N n /N i correction to the p
we may write
K S
K S
J n –“ c & '2 t “ c –“ N n T i R zW –VW i 1
^ R zW –p
52 t “ c –“
2N n W W
R z –Q i
5N i
2N n W
R z –qW i
5N i
DL
DL
~17!
,
W the energy flux defined in Eq. ~6!,
with Q
i
zW –VW i 5 zW –VW n
D
52
~16!
(x 2 )
where f M5N i (M /2p T i ) 3/2 exp(2M v2/2T i ) and L (3/2)
2
4
2
5 @ x 27x 1(35/4) # /2 is a Sonine or generalized Laguerre
polynomial. In writing Eqs. ~15!–~16! we assume that the
neutral density is small enough not to affect the usual
Pfirsch–Schlüter results.
We may then use Eq. ~16! to evaluate the last term of
Eq. ~8! by first noting that upon using “ v f M
52(M /T i ) vW f M to integrate by parts,
E
E
E
d 3 vvW vW vW vW
where the d i j are Kronecker delta functions and we have
used the orthogonality of L (3/2)
to L (3/2)
and L (3/2)
. As a
2
1
0
result, the last term shown in Eq. ~16! does not contribute
and we obtain
M v2 5 2
M W
V i –vW 1
2
qW –vW
Ti
2T i 2 5p i i
8q i v i ~ 3/2!
L
~ M v 2 /2T i ! f M1¯ ,
75p i 2
d 3 vvW vW vW vW L ~23/2! ~ M v 2 /2T i ! f M
5
~14a!
and
5cp i ] T i
nW 3“ c ,
qW i 5q i nW 1
2eB ] c
E
1
cRT i
S
e
F
9
5 ^ B 2&
and
J
z –qW i 52
5cRp i
2e
1 ]pi
pi ]c
1
S
1
e ]F
Ti ]c
D
^ ~ nW –“ ln B ! 2 & B 2T ] T i
20^ ~ nW –“B ! 2 &
12
B 2T
^B &
2
D
]Ti
]c
Ti ]c
.
G
,
~18!
~19!
In expressions ~17!–~19! we may use T i 5T n , and the definitions of moments are as in Eqs. ~6!. Recall, also that B T is
the toroidal magnetic field.
If we also neglect the N n /N i correction to qW i , then Eq.
~13! gives the neutral heat flux to be
^ qW n –“ c & '2
K
L
5tpn
“ c –“T n .
2M
~20!
To complete the neutral description we use Eq. ~11! and
W to obtain the neutral parneglect the N n /N i correction to G
i
ticle flux,
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Catto et al.
Phys. Plasmas, Vol. 5, No. 11, November 1998
^ GW n –“ c & '2
K
L
t
“ c –“ ~ N n T n ! .
M
~21!
The ion fluxes in Eqs. ~3! and ~4! are the standard
Pfirsch–Schlüter results:2,11,14
^ qW i –“ c & 52
8M c 2 I 2 p i
5e t i
2
SK L
1
B
2
2
and
^ R zW –pi –“ c & '2
2
16M 2 c 3 I 4 p i ] T i
25e 3 t i
^ B 22 &
^ B 2&
23
D
^ B 22 &
^ B 2&
1
1
1
^B &
2
D
]Ti
~22!
]c
F SK L
SK L
DG
e ]F
1
]c Ti ]c
B4
47 ] T i
1
50T i ] c
B4
2
^ B 2& 2
,
K L
B4
2
^ B 22 &
^ B 2&
;
e2
B4
S
q ie
B
B
5
K L
B4
23
^ B 22 &
^ B 2&
1
^ B 2& 2
D
52 ~ S2 ^ S & ! ,
2eB 2 ] c
2
E
d q ~ S2 ^ S & !
1L ~ c ! ,
BW –“ q
q
x
where the flux function L is determined by employing the
parallel heat conduction expression
~23!
,
2
2eB 2 ] c
5cI p e ] T e
q i e 52
p e T e t ei
@ k 21~ nW –“ ln p e 1eE i /T e !
m
1 k 22nW –“ ln T e # ,
~26!
where, for Z51, k 2151.4, and k 2254.1 ~see Ref. 5 or 13,
for example!. Using the constraint
^ q i e B & 52 ~ p e T e t ei /m ! k 21e ^ E i B &
to determine L, gives
while
1
5cI p e ] T e
2
where, as usual, poloidal derivatives of density and temperature are neglected compared to derivatives of magnetic field.
Integrating from a convenient angle x to q gives
q ie
1/2 4
where t i 53M 1/2T 3/2
i /4p e N e ln L. For aspect ratios of order unity, the classical contributions,5 which we have neglected for simplicity, could be added to Eqs. ~22! and ~23!.
Hazeltine2 considered the case without neutrals and
noted that for small inverse aspect ratio,
1
BW –“
3965
;
e4
B4
q i e 52B
,
W i –“ c & 50 gave the variation
so that in the steady state ^ R zW –p
of the electrostatic potential to be weak ~order e 2 ! compared
to that of the ion temperature. However, the neutral viscosity, of course, is free of these aspect ratio factors so that in a
large aspect ratio tokamak the strongest impact of the neutrals is on the electrostatic potential! In the steady state when
the neutrals dominate over the ions the radial variation of the
W n –“ c & 50,
electrostatic potential is simply given by ^ R zW –p
21
so that e ] F/ ] c ; ] T i / ] c ;N i ] p i / ] c .
B. Sink and electron contributions to Pfirsch–Schlüter
fluxes
The usual Pfirsch–Schlüter treatment of the electron particle and heat flows is modified by the presence of an energy
sink S due to radiation losses. We consider the edge region
inside the separatrix so the only sink appears in the electron
heat balance equation. Our energy sink treatment corresponds to the weak equilibration limit of Wong,15 except that
we retain induced electric field terms.
When the diamagnetic electron heat flux,
qW 'e 5 ~ 5cp e /2eB 2 ! BW 3“T e ,
~24!
is inserted into the lowest-order electron heat balance equation,
“–qW e 52S,
the resulting equation for the poloidal variation is
~25!
1
2
E
q
x
B
^B &
2
d q ~ S2 ^ S & !
BW –“ q
K
5cI p e
2e
E
B2
S
q
x
B
^B &
2
d q ~ S2 ^ S & !
BW –“ q
2
1
B
D
]Te
]c
2
L
p e t ei k 21eB ^ BE i &
m ^ B 2&
.
~27!
For large, poloidally varying radiation losses, the new terms
involving the sink S can be much larger than the usual
Pfirsch–Schlüter terms.
To find expressions for the radial electron particle and
heat fluxes, we also need the usual forms for the parallel
current:11,13
J i5
ep e t ei
5cI
m
S
@ k 11~ nW –“ ln p e 1eE i /T e ! 1 k 12nW –“ ln T e #
B
^ B 2&
2
1
B
D
]p
]c
1
p e t ei k 11e 2 B ^ BE i &
mT e ^ B 2 &
,
~28!
where the only novelty is that the total pressure p5p e 1p i
1 p n contains the neutral pressure p n , which for our purposes is negligible. In Eq. ~28!, k 1151.9 and k 125 k 21 for
Z51.5,13 From Eqs. ~26!–~28! we can obtain nW –“T e and
nW –“p e 1eN e E i , which allow us to determine the radial electron fluxes:
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3966
Catto et al.
Phys. Plasmas, Vol. 5, No. 11, November 1998
^ qW e –“ c & 5
K L
SKE
L
K E
LD
SK L D
S
D
5cIp e
BW –“T e
2e
B2
52
2
2
3
5cm k 11I
q
2e kt ei
1
x
q
B2
^B &
2
x
d q ~ S2 ^ S & !
BW –“ q
5c 2 mI 2 T e
1
2e 2 kt ei
B2
5 k 11N e ] T e
]c
2
d q ~ S2 ^ S & !
BW –“ q
2
2 k 21
1
^ B 2&
]p
~29!
]c
and
^ GW e –“ c & 5 ^ GW i –“ c &
5
cI
e
52
2
K
1
B2
e kt ei T e
^ B 2&
1cIN e
2
W –“p 1eN BE i ! 2cN ^ R zW –EW &
~B
e
e
e
cm k 12I
1
1
^ B 2&
L
where again w and R are the radial scale length of the edge
region and the major radius, and l5 v i t i is the Coulomb
mean-free path. Inequality ~31! follows because the poloidally varying portion of the ion temperature is small compared to its flux surface averaged value by (q r i /w)(qR/l)
!1, while the poloidal variation of B is of order e. Not
surprisingly, inequality ~31! is more restrictive by only aspect ratio factors than the requirement that the radial ion heat
diffusion time w 2 t i /q 2 r 2i be larger than the parallel ion heat
conduction time (qR) 2 /l v i . If this later condition is not
satisfied, the transport along and across the field occurs on
similar time scales, making the problem two dimensional. In
the absence of a sink, the poloidal variation of the electron
temperature would be smaller than that of the ion temperature ~and plasma density and electrostatic potential! by
re /ri .
The poloidal variation of the ion temperature is found by
equating the usual expressions for the parallel ion heat conduction and its Pfirsch–Schlüter counterpart,
K
B
SKE
q
x
2
E
x
L
LD
^ B 2&
k 22
zW –EW
2
]p
]c
2
2
BT
2
c mI
e 2 kt ei
5 k 12N e ] T e
2
2
]c
,
nW –“T i 5
1
5cI p i
2e
S
B
^B &
2
2
1
B
D
]Ti
,
]c
~32!
16M cI
25eB t i
S
B2
12
^ B 2&
D
]Ti
]c
~33!
,
B2
~30!
with k 5 k 11k 222 k 12k 2155.8.
The convective contributions due to the radiation sink S
in the particle and electron heat fluxes of Eqs. ~29! and ~30!
vanish if S is a flux function. Consequently, poloidal variation in S is necessary to drive convective fluxes and, as we
shall see in the next section, is responsible for the strong
poloidal variation of the electron temperature that results in
the convection.
Equations ~17!, ~20!–~23!, ~29!, and ~30! are the fluxes
to be inserted in Eqs. ~1!–~4!, with zW –VW i 5 zW –VW n and zW –qW i
given by Eqs. ~18! and ~19!.
where nW –“ ln Ti;qri /lw, as remarked earlier.
Total parallel pressure balance and the requirement that
the total pressure be a lowest-order flux function then gives
an equation for the poloidal variation of the plasma density,
nW –“p5nW –“ @ N e ~ T i 1T e !# 50,
e
nW –“F2nW –“ ln p e
III. POLOIDAL VARIATION
5
In the edge region just inside the separatrix, strong poloidal variation is observed and expected because of the presence of neutrals and radiation. Within the framework of a
Pfirsch–Schlüter treatment, the poloidal variation of the
plasma density, ion temperature, and potential must be assumed weak compared to that of the magnetic field. As can
be verified a posteriori, for a high aspect ratio ( e 5r/R
!1) tokamak, this assumption requires
~31!
~34!
where the neutral pressure contribution is neglected as small.
The poloidal variations of the electrostatic potential and
the electron temperature follow from Eqs. ~26!–~28!, which
can be combined to obtain nW –“p e 1eN e E i and nW –“T e , or
Te
q r i /w! e l/qR,
32M
nW –“ ln T i 5
to obtain
F K LG SK L
DS
D
^ BE i &
125p i T i t i
d q ~ S2 ^ S & !
BW –“ q
d q ~ S2 ^ S & !
BW –“ q
q
q i i 52
SE
m k 12B
k p e T e t ei
K
3 B2
2
E
q
x
q
x
mcI
eB kt ei
1
d q ~ S2 ^ S & !
2 2
BW –“ q
^B &
LD
DS
d q ~ S2 ^ S & !
BW –“ q
S
12
B2
^ B 2&
1
e
Te
k 22 ] p
pe ]c
@ BE Ii 2 ^ BE Ii & #
2
5 k 12 ] T e
2T e ] c
and
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D
~35!
Catto et al.
Phys. Plasmas, Vol. 5, No. 11, November 1998
nW –“T e 5
SE
m k 11B
k p e t ei
K
S
3 B2
q
x
E
3 12
q
x
d q ~ S2 ^ S & !
1
2 2
WB –“ q
^B &
d q ~ S2 ^ S & !
BW –“ q
B2
^B &
2
DS
k 21 ] p
pe ]c
LD
2
1
mcI
eB k T e t ei
5 k 11 ] T e
2T e ] c
D
,
~36!
where the superscript I on E i denotes that only the induced
electric field enters on the right side of Eq. ~35!. Notice that
in the absence of a poloidally varying energy sink S that the
poloidal variation of the electron temperature is weak and
poloidal variation of the electrostatic potential is therefore a
Maxwell–Boltzmann response to lowest order.
As noted in the Introduction, only poloidal asymmetry in
the radiation sink S and the condition qRN I s I E I /T e
@q r e /w is required to make the radiation sink-driven poloidal variation stronger than the usual Pfirsch–Schlüter poloidal variation of the electron temperature.
Equations ~33!–~36! are the four equations for the poloidal variation of the four unknowns T i , N e , F, and T e . Notice that the neutrals do not influence the poloidal variation
since their density is assumed low compared to that of the
plasma.
larly strong if a MARFE is formed, since the electron temperature then varies substantially over the flux surface,
making F@1.
We can also estimate the size of the neutral heat flux. To
do so, we first estimate the neutral density by assuming that
the plasma edge inside the separatrix is fully recycling. The
convective, poloidally asymmetric radiation sink-driven outward particle flux,
G s ;D BF
To get a feel for the size of the fluxes driven by a poloidally asymmetric radiation sink, S;N I E I N e v e s I , we can
compare it with gyro-Bohm heat transport in the following
way. We use our estimate from the Introduction to define F
as
F[
nW –“ ln T e qR E I
;
N s qR,
nW –“ ln B
el Te I I
and then note from Eq. ~29! that poloidal variation in B is
needed to generate a radial heat flux so that ^ B 22 BW –“T e &
; e T e F/qRB. As a result, the radial electron heat flux driven
by a poloidally asymmetric radiation sink, q s , is of order
q s ;D BF
pe
,
R
where D B[cT e /eB denotes the Bohm diffusion coefficient.
A gyro-Bohm heat flux q g in the edge is of order
q g ;D B
ripe
,
w2
giving
q s w 2F
.
;
q g r iR
For the parameters listed in the Introduction, r i R/w 2 ;1,
while our results require F!1. However, for stronger radiation losses than our treatment allows, F.1 is permissible
and sink-driven fluxes larger than gyro-Bohm fluxes can occur. In particular, the sink-driven transport should be particu-
Ne
,
R
must equal the inward neutral flux,
G n;
v 2i N n
N i^ s v & xw
,
for a fully recycling edge. Equating these two fluxes gives
the following estimate for the neutral to plasma density ratio:
Nn
Ne
;
N i ^ s v & x wD BF
v 2i R
.
If this estimate is used to eliminate the neutral density in the
ratio of neutral to ion heat flux given in the Introduction we
obtain
^ qW n –“ c &
^ qW i –“ c &
IV. SINK-DRIVEN AND NEUTRAL FLUX ESTIMATES
3967
;F
w t iD B
R
q 2 r 2i
;F
l
w
qR q r i
@
F
e
,
where the Pfirsch–Schlüter validity inequality ~31! is used to
demonstrate the consistency of our orderings. Consequently,
a poloidal radiation sink asymmetry resulting in a poloidal
variation of the electron temperature of F; e can ~i! cause a
sink-driven outward convective plasma particle flux that balances the incoming neutral particle flux; ~ii! result in diffusive neutral and convective sink driven heat fluxes that are
larger than the Pfirsch–Schlüter ion heat flux; and ~iii! cause
the neutrals to determine the radial behavior of the electrostatic potential.
V. DISCUSSION
We have derived a complete system of equations for the
radial and poloidal variation of the plasma density N e 5N i ,
electrostatic potential F, ion temperature T i , and electron
temperature T e in the presence of neutrals and a poloidally
asymmetric radiation sink. Equations ~17!, ~20!–~23!, ~29!,
and ~30! are the radial fluxes to be inserted in Eqs. ~1!–~4!,
with toroidal velocity and heat flows given by Eqs. ~18! and
~19!. Equations ~33!–~36! are the four equations for the poloidal variation of the four unknowns T i , N e , F, and T e .
The neutral density can either be assumed to be specified or
can be found from the neutral continuity equation,
]Nn
1“–~ N n VW n ! 52 ^ s v & z N n N i ,
]t
with the neutral velocity given by Eq. ~11! with p n 5N n T i .
As noted in the Introduction and at the end of Sec. II A,
rather small neutral densities can result in large effects on the
radial variation of the electrostatic potential and introduce
neutral heat and angular momentum fluxes as large as, or
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3968
Catto et al.
Phys. Plasmas, Vol. 5, No. 11, November 1998
larger than, those associated with the usual Pfirsch–Schlüter
ion fluxes. Since the neutrals are localized to the edge, strong
shear in the E3B, poloidal, and parallel flows can result,
which may have an influence on the level of the turbulence.6
The effects of a poloidally asymmetric radiation sink are
also retained in our system of equations. For a collisional
edge, the poloidal variation in the electron temperature due
to radiation losses can easily be much larger than that due to
the usual Pfirsch–Schlüter electron transport and the resulting convective electron heat and particle fluxes can be comparable to gyro-Bohm fluxes. Consequently, edge transport
descriptions retaining only diffusive fluxes are expected to
be incomplete for many of the situations of experimental
interest.
J n we neglected
Note added in proof. When evaluating p
departures of the neutral particle and heat flows from those
of the ions @recall Eqs. ~11! and ~13!# because only the toroidal component of these flows appears in Eq. ~17!. To retain
these departures the leading short mean free path correction
to f 0 must be retained in Eqs. ~8! and ~9!.
ACKNOWLEDGMENTS
The authors wish to thank Sergei Krasheninnikov for
helpful discussions and suggestions that have improved our
estimates. We thank the referee for bringing Ref. 15 to our
attention.
This research was jointly supported by U.S. Department
of Energy Grants No. DE-FG02-91ER-54109 at the Plasma
Science and Fusion Center of the Massachusetts Institute of
Technology and No. DE-FG05-80ET-53088 at the Institute
for Fusion Studies at the University of Texas at Austin, and
by the U.K. Department of Trade and Industry and Euratom.
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~iv! the slash removed in S 01 between Eqs. ~51! and ~52! so it reads as
S 01 5x 3 12x 5 ; ~v! an extra power of mass removed in Eq. ~77!; and ~vi! the
last term in Eq. ~82! should be 2.9 n ] 2 T/ ] t ] r.
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