Neolassial Ion Transport in the Edge
of Axially-Symmetri Arbitrary Cross-Setion
Tokamak with Plasma Subsoni Toroidal Flows
J.H.F. Severo, V.S. Tsypin, I.C. Nasimento,R.M.O. Galv~ao,
M. Tendler 1
, and A. N. Fagundes
Institute ofPhysis,UniversityofS~aoPaulo,
Rua doMat~ao,TravessaR,187, 05508-900S~aoPaulo,Brasil
1
TheAlfvenLaboratory, EURATOM-NulearFusionResearh,
RoyalInstituteof Tehnology,10044 Stokholm,Sweden
Reeivedon26June,2001
Generalmetris oflarge aspet ratiotokamaks is usedinthe paper. Generalexpressions for the
neolassialpoloidalplasmarotationV
i
andradialionheatux
Ti
areobtained.Theirdependene
onthesquaredMahnumber=U 2
i =
2
s
isanalyzed(hereUiistheiontoroidalveloityandsis
thesoundveloity,respetively). Someinterestingpeuliaritiesofthisdependeneareemphasized.
I Introdution
Thetheoretialandexperimentalstudyofthe
neolas-sial iontransportin edgetokamak plasmasare
nowa-daysof renewedinterest. 1 4
Inpartiular, these
stud-ies are important in the so-alled H-regimes in large
toroidal failities, haraterized by sharp gradients in
the radialprolesof marosopiplasmaquantities in
theregionofthetransportbarriers. 1;2
Plasmasofsmall
tokamaks,whih play animportant role in
investigat-ingdierentphysialphenomenainfusionresearh,are
mainly in L-regime. Nevertheless, some problems
re-gardingtheneolassialiontransportinthesemahines
are not yet well understood and additionaleorts are
needed. 3;4
Oneoftheseproblemsisaddressedinthis
pa-per, namelythe neolassial iontransport in the edge
of an axisymmetri plasma olumn of arbitrary
ross-setion and with subsoni toroidal ows. These ows
anbeinduedbyparallelneutralbeaminjetionorby
radio frequenywaves.
II Starting equations
The main neolassialquantities observed
experimen-tallyare the ion radialheat ux and the ion poloidal
veloity. 1;2
Untilnowthereisnosatisfatorytheoretial
explanationforthedampingrate
oftoroidalrotation,
whihisusuallysupposedtobeapproximatelyequalto
theenergyonnementtime
E ,
E .
5
WefollowRefs. 6and7,wherethemagnetisurfae
averagedionradialheatuxforollisionalplasmaswas
rstobtained. We use thetoroidal oordinates r;;,
wherer isthemagnetisurfaelabel,and and are
poloidal and toroidal angles, respetively, assume
ax-ial symmetry, i.e., = = 0, large aspet ratio, and
smooth prolesof themarosopi plasma quantities.
Themagnetisurfaeaverage
h:::i= Z
2
0 ( :::)
p
gd Z
2
0 p
gd (1)
oftheradialontravariantomponentq r
i
oftheionheat
ux(seeRef. [8℄),
q r
i =
2p
i
i
M
i !
2
i g
11 T
i
r +g
12
e
T
i
!
5
2 p
i g
33 B
M
i !
i B
p
g
e
T
i
; (2)
leadsto
Ti =hq
r
i i=
2
M
i
p
i
i g
11
! 2
T
i
r
5
2
e
i
g
33 B
Z
2
d p
i
B 2
e
T
i
,
Z
2
p
Here, T
i
is the ion temperature, e
T
i =T
i hT
i iis the
osillatorypart(perturbation)of theiontemperature,
p
i =nT
i
, n=h ni+en isthe plasma densityand ne is
theosillatorypartofit, !
i =e
i B=M
i
istheion
y-lotronfrequeny,B andB
arethemagnitudeandthe
-ontravariantomponentofthemagnetieldB,
re-spetively,and g 11
and g 12
aretheontravariant
om-ponents of the metri tensor bg , whose determinant is
denoted g.
In obtaining Eq. (3), we used also the expressions
fortheontravariantomponentsofthemagnetiled
B=f0; 0
=2 p
g;
0
=2 p
gg; (4)
whereandarepoloidalandtoroidalmagnetiled
uxes,respetively,and
( g
33 =
p
g)==0: (5)
The last equation follows from the ondition j r
0, where j r
is the r-ontravariant omponent of the
plasmaurrentj. Theinequality e
A=A hAi hAi
was also employed, where A stands for the plasma
marosopiquantities.
Wesee from Eq. (3)that, in order to nd
Ti , we
needtoalulateenand e
T
i
. Theequationfor e
T
i follows
fromtheionheattransport equation 8
T
i V
i n~
+
B
B q
ik
+rq
i? +
3M
e n
e
M
i ~
T
i
=0; (6)
whereV
i
isthe-ontravariantomponentoftheionveloityV
i ,and
q
ik
= 3:91 nT
i B
M
i
i B
T
i
; q
i? =
5
2 nT
i
e
i B
2
[BrT
i
℄: (7)
PerturbationsoftheeletrontemperaturearenegletedinEq. (6)(seeexplanationsafterEq. (18)). Tondneone
anemploytheosillatorypartoftheparallelomponentoftheplasmaone-uidmomentum equation
3
2
k
lnB+
p+
k
+M
i n
B
B
d
i V
i
dt
=0; (8)
wherep=p
i +p
e =n(T
i +T
e
)is theplasmapressureand
k
istheparallelionvisosity, 9;10
and
d
i V
i
dt =
V
i
t +(V
i r)V
i :
Hereweusedalsothewell-knownapproximateexpression(see,e.g.,Ref. 3)
r= 3
2
[hrh+( hr)h℄
k
+h(hr)
k 1
3 r
k
;
where is the ion visosity tensor. The veloity omponent V
i
an be found from the poloidal average of the
parallelomponentofthemomentum equation(8)
Z
2
0 d
M
i n
B
B
d
i V
i
dt 3
2
k lnB
=0: (9)
Theparallelionvisosity
k
,enteringEqs. (8)and(9),isdenedby 3;4;9;10
k =
2
3 p
i
i
(0:96 0:59); (10)
where
=3
V
i
ln
p
gn 2=3
B
+ B
2
B 2
V
i g
22
+V
Ti
ln
B
n
; (11)
= 3
0:34V
i
lnn+V
Ti
1:36
lnB 0:84
lnn
; (12)
and
V
Ti =
e h B 2
i D
[BrT
i ℄
E
=
e g
33
p
ghB 2
i
B
T
i
r
Thus,wehaveEq. (3)todeterminetheradialheat
ux
Ti
, andEq. (9) tond the-ontravariant
om-ponent of the plasmaveloity V
i
. To alulate them
weshould solveEqs. (6),(8), and(9).
III Solution of the perturbed
equations
WendfromEq. (8)
B
B
d
i V
i
dt =
1
2 U
2
i lng
33
(14)
and, onsequently,
pe+
k
= h pi
2 lng
33
; (15)
where=M
i U
2
i =(T
i +T
e
)isthesquaredMah
num-ber,pe=en(T
i +T
e )+n
e
T
i
. ToobtainEq. (14),weused
theovariantdierentiationrules
r
i V
k
= V
k
x i
+ k
im V
m
(16)
andimposed thatthemetritensoromponentsg
12 =
g
21
areperiodialfuntions oftheangle .
UsingEqs. (6)and(7),weobtaintheseond order
dierentialequationfortheperturbediontemperature
e
T
i ,
2
e
T
i
2
2:17b r
M
e
M
i e
T
i
=f(); (17)
where
f()= 0:51bT
i
i
5
2 V
Ti V
i
lnn
5V
Ti lnB
; (18)
d
b = B 2
= 2
i B
2
is the ollisionality parameter, and
i =
p
2T
i =M
i =
i
is the ion mean free path. The
order of the parameter B 2
=B 2
for large aspet ratio
tokamaksisapproximatelyq 2
R 2
, whereq isthesafety
fator and R is the torus major radius. Thus we
ap-proximately have b = q 2
R 2
= 2
i
. For the ollisional
plasma the parameter b > 1. We onsider the range
1<b . p
M
i =M
e
in this paper. In this aseonean
omitperturbationsoftheeletrontemperatureinEqs.
(6)and(17). Fortherange1<b.M
i =M
e
,these
per-turbationsshouldbetakenintoaount(see,e.g.,Ref.
3).
As far as the funtion f() and, onsequently, e
T
i
are periodialin andmoreover,proportionaltosin
fortheirularross-setiontokamak 3
andtosinand
sin2forelliptialross-setiontokamak, 4
thesolution
ofEq. (17)hastheform
e
T
i =
1
X
s=1 e
T
is
sins; (19)
where
e
T
is =
Z
2
0
f()sin sd
[(s 2
+2:17b p
M
e =M
i )℄
: (20)
Comparison ofEq. (15)withEqs. (17)-(20) shows
that,tozeroapproximation,oneanuse
lnn
2 lng
33
: (21)
Thisexpressionanbesubstitutedinto Eqs. (3),(11),
(12)and(18). Thesurfaeaveragedparallelomponent
ofthemomentumequation(9),takingintoaountEq.
(8),anbetransformedintotheform
Z
2
0 d
e
k
3
2 lnB
2 lng
33
hni e
T
i
2 lng
33
Usingalsotheidentity B 2 =g 22 B 2 +g 33 B 2 ; (23) weget lnB 1 2 1 q 2 R 2 g 22 lng 33 : (24)
Thusweanexpresseveryperturbedvalueviathe
os-illatorypartsofthemetriomponentsg
22 and g
33 .
IV Ion uxes
Let us simplify expressions forperturbed valuesusing
Eqs. (21)and (24). TheFourieromponent e
T
is of the
perturbediontemperature e
T
i
hastheform
e T is = 1:3bT i i d s ( b) h V Ti 1+ 2 + 5 V i i Z 2 0 sins lng 33 d V Ti q 2 R 2 Z 2 0 sins g 22 d ; (25) whered s (b)=s
2 +2:17b p M e =M i
. Theparallel visosity
k
,Eqs. (10)-(12),anbeexpressedintheform
k = 0:96p i i V i
( 1+0:46) lng 33 1 q 2 R 2 g 22 + (26) +1:83V Ti
( 1+0:83) lng 33 1 q 2 R 2 g 22 :
Equation(22)anbetransformedinto
Z 2 0 d k 3 2 + lng 33 3 2q 2 R 2 g 22
+h ni 1 X s=1 e T is Z 2 0
dsins lng
33
=0: (27)
Substituting Eqs. (25)and(26)intoEq. (27),onendsthepoloidalveloityV
i
(tobeomparedwithRef. 12),
V i = 1:83V Ti f 2
(;b;A;D)=f
1
( ;b;A;D); (28)
where
f
1
(;b;A;D)= 1+ 2 3
(1+0:46)A
33 +A
22
(2+1:13)A
23 +0:36 2 bD 33 ; (29) f 2
(;b;A;D)= 1+ 2 3
(1+0:83)A
33 +A
22
(30)
(2+1:5)A
23 0:48b h 1+ 2 D 33 D 23 i ; A 33 = Z 2 0 d lng 33 2 ; (31) A 22 = 1 q 4 R 4 Z 2 0 d g 22 2 ; (32) A 23 = 1 q 2 R 2 Z 2 0 d lng 33 g 22 ; (33) D 33 = 1 X s=1 1 d s (b) Z 2 0
dsins lng 33 2 ; (34) D 23 = 1 q 2 R 2 1 X s=1 1 d s (b) Z 2 0
dsins lng 33 Z 2 0
dsins g
22
: (35)
Equation(3)forthesurfaeaveragedionheatuxanberewrittenasfollows
Substitution ofEqs. (19),(25),(21),and(24)intoEq. (36)resultsin
Ti =hq
r
i i=
2p
i
i
M
i !
2
i T
i
r
g 11
1+0:8q 2
g 2
33
1+
2
2
D
33
(2+)D
23 +D
22
(37)
0:37
1+
2
D
33 D
23
f
2
(;b;A;D)=f
1
(;b;A;D) i.
g 11
h p
gi Z
2
0 p
gd
;
where
D
22 =
1
q 4
R 4
1
X
s=1 1
d
s (b)
Z
2
0
dsins g
22
2
: (38)
d
V Estimates
Let usanalyzethe expressionsforthe ionpoloidal
ve-loity Eq. (28) and the ion heat ux Eq. (37) in a
general ase. WhenthesquaredMah number
van-ishes,=0,oneobtainsfrom Eq. (28)
V
i
= 1:83V
Ti
; (39)
whih agrees with results of Refs. 11, 3, and 2.
Equations (28) and (39) also onrm the
Hazel-tine theorem, 13
whih says that the so-alled residual
plasmapoloidal rotationin tokamaks depends onlyon
the ion temperature gradient and not on gradientsof
othermarosopiplasmaparameters. Estimatesshow
thattheparametersA
23 andD
23
arenegative,(A
23 <0
and D
23
<0). Hene, theparameter f
1
( ;b;A;D)is
positive, f
1
(;b;A;D) > 0, i.e., the denominator in
Eqs. (28) and (37) is positive and has no roots as a
funtion of . A remarkable property of the poloidal
veloity V
i
is the hange of sign at avalue
0 of the
parameter. This resultsfrom thefat of takinginto
aount inertial fores in the starting equations.
As-sumingthatthepoloidalveloityhangessignat<1
(seein detailRef. 11),wendfromEq. (30)
0
2:1(A
33 +A
22 2A
23 )=[b(D
33 D
23
)℄: (40)
Fromtheapproximateequation,
V
i
0:88V
Ti
b(D
33 D
23 )
(A
33 +A
22 2A
23
+0:36 2
bD
33 )
; (41)
onendstheritialquantity
k ,
k 1:67
r
A
33 +A
22 2A
23
bD
33
; (42)
orrespondingtothemaximumofthepoloidalveloity
V
i(max ) ,
V
i(max )
0:73V
Ti p
b(D
33 D
23 )
p
D
33 (A
33 +A
22 2A
23 )
: (43)
For>
k
, the poloidal veloity V
i
dereases slowly
withthegrowthof.
Analysis of Eq. (37) showsthat themagneti
sur-fae averagedradialion heat ux
Ti
isan inreasing
funtion of theparameter :The fators that
hara-terize the non-irularity of the plasma ross-setion,
suhaselliptiity, triangularity,reduetheroleof
neo-lassialeetsin
Ti
forallvaluesoftheparameter.
Theseresultsoinidewiththepreviousstudiesofthis
problem,fullledforelliptialandirularross-setion
tokamaks, 3;4
andwedemonstratethemhere.
VI Elliptial tokamak
Intheaseoftheelliptialtokamak wendfrom Eqs.
(28)
U
i =G
u (;b)U
Ti
; (44)
whereU
i =V
i ,U
Ti
=( 1=M
i !
i )T
i
=,= p
l
1 l
2 ,
l
1 and l
2
are the semiminor and semimajor axes of a
tokamakross-setion,respetively,
G
u
(;b)= f
2 (;b)
f
1 (;b)
; (45)
f
1
(;b)=d(b)
1+ 2
3
( 1+0:19)+0:18 2
b; (46)
f
2
(;b)=d(b)
1+ 2
( 1:83+1:52) 0:88b
1+
d(b)=1+2:2b p
M
e =M
i
. ObtainingEq. (44),the
pa-rameter
A= l
1
R q
l 2
2
l 2
1 1
(48)
wasassumedtobesmall, A1.
UsingEq. (37),wederivethemagnetisurfae
av-erageoftheradialionheatuxintheShafranovform, 6
Ti =
2nT
i
i
M
i !
2
i T
i
l
2
1 +l
2
2
2l
1 l
2
1+3:2q 2
l 2
1
(l 2
1 +l
2
2 )
G
T (;b)
; (49)
where
G
T
(;b)=
1+
2
1+ 2
3
f
3 ()
f
1 (;b)
; (50)
f
3 ()=
1+
2
( 1+0:19)+
5
(1:83+1:52): (51)
d
Equations(44)and(49)oinideswiththeproper
equa-tionsofRef. 4. AsfarastheoeÆientl 2
1 = l
2
1 +l
2
2
in
Eq. (49)islessthan1,l 2
1 = l
2
1 +l
2
2
<1,theroleof
neo-lassialeets inthe radialion heatux dereases in
nonirularross-setiontokamaksin omparisonwith
irularross-setionones.
ThequantitiesG
u
(;b)andG
T
(;b)areplottedin
Figs. 1 and 2, respetively. One an see that
fun-tion G
u
(;b) (Fig. 1) hanges sign at
0
2d(b)=b.
Fortheollisional parameterb p
M
i =M
e
, the
quan-tity is equalto
0
0:1. Themaximumof funtion
G
u
(;b) is ahieved when b
m
50,
m
1, and is
G
u (
m ;b
m
) 3. It followsfrom Fig.2 that the
neo-lassial ontribution in the radialion heat ux, as a
funtionof ,isdroppingwithinreasingb.
Figure 1. The dependene of the funtionGu(;b) on
fordierentmagnitudesofthe parameterb: b
1
=10(solid
urve)andb2=50(dashedurve).
Figure 2. The dependeneof the funtionGT(;b) on
for dierentmagnitudes oftheparameterb: b
1
=10(solid
urve)andb2=50(dashedurve).
VII Conlusions
Generalexpressionsforneolassialpoloidalplasma
ro-tation V
i
and radial ion heat ux
Ti
for an
axially-symmetri arbitrary ross-setion tokamak edge with
plasma subsoni toroidal ows are obtained in the
presentpaper. TheirdependeneonthesquaredMah
number is analyzed. It is shown that there is a
re-markablepropertyofthepoloidalveloityV
i
tohange
signat avalue=
0
, whih resultsfrom takinginto
aountinertialforesinthestartingequations. There
also existsaritialvalueof,
k
,whihorresponds
to themaximum of the poloidal veloityV
i(max) . For
>
k
, thepoloidal veloityV
i
isa dereasing
fun-tion of . Analysis of the magneti surfaeaveraged
radial ion heat ux
Ti
shows that this ux is an
in-reasingfuntion of theparameter: Thenonirular
neo-lassial eets in
Ti
for any value of the parameter
. These resultsonrm previousstudiesofthis
prob-lem.
Aknowledgments
Thisworkwassupported bytheResearh Support
Foundationof the Stateof S~ao Paulo(FAPESP),
Na-tional Counil of Sienti and Tehnologial
Devel-opment (CNPq), and Exellene Researh Programs
(PRONEX) RMOG 50/70grant from the Ministryof
Siene andTehnology,Brazil.
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