Eet of Plasma Subsoni Toroidal Flows Indued
by Alfven Waves on Transport Proesses in the
Edge of Elongated Tokamaks
V. S.Tsypin, J. H. F. Severo, I. C. Nasimento,
R. M.O. Galv~ao,and Yu. K.Kuznetsov
Instituteof Physis,UniversityofS~aoPaulo,
RuadoMat~ao,TravessaR, 187,05508-900, S~aoPaulo,Brasil
Reeivedon4May,2000
There is a renewed interest in using Alfven waves (AW) in tokamak plasmas. Previously, AW
were atively explored mostlyfor urrent drive and plasma heating in tokamaks. Presently, the
possibility oftheanomalousandneolassial transportsuppressionbyAWintokamakplasmasis
beingvividlydisussed. AWanalsoinduepoloidalandtoroidalplasmarotation.Toroidalplasma
rotation anreah thesubsoni level. These ows ansubstantiallyaet neolassial transport
both in ollisional and weakly ollisional plasmas. In this paper, the eet of plasma subsoni
toroidalows indued by Alfvenwaves ontransportproesses inthe edgeof elongated tokamak
is investigated. The dependene of poloidal plasma rotation and ion heat ondutivity on the
elongationparameterandtheratioofinduedtoroidalveloity tothesonispeedareanalytially
obtained.
I Introdution
Itiswell-knownthat Alfvenwavesareoneofthebasi
methodsofurrentdrive(AWCD)andplasmaheating
(AWPH) in tokamaks. 1 5
These methodswerelearly
demonstratedinnumerousexperiments 6 8
andwillbe
anew applied in forthoming experiments in TCABR
(Tokamak Chauage Alfven Brasil) tokamak. 9;10
The
omparatively reent appliations of AW in
toka-maks are reation of transport barriers and
suppres-sion ofanomalous and neolassialtransport by these
waves. 11 14
Themainideaistoindueashearedradial
eletrield that antakeapart theturbulentplasma
eddies and thus suppress anomalous transport. 15 17
Suppression of neolassial transport in weakly
olli-sionaltokamak plasmasis onneted with ion banana
orbitsqueezingbytheshearedradialeletrield. 18;19
Alfven waves an be used to eÆiently indue the
stronglyshearedradialeletrield 11 14
thusahieving
suppression of anomalous and neolassial transport.
The rst experimental evidene that AW an be
use-ful to reate plasma rotation, and, onsequently, the
radial eletri eld, was obtained in the Phaedrus-T
tokamak. 20
Besides these very attrative topis, there is also
theadditionalphysialmehanismwhereAW anplay
asubstantialrole,i.e.,theionneolassialtransportin
of this mehanism was emphasized in the papers by
Rogister, 21 23
where theexperimental resultsin
mod-ern tokamaks were arefully analyzed. Rogister
on-siders the transport proesses in tokamak edge
plas-mas to be in the H-mode (the so-alled high regime),
when the harateristi length of plasma marosopi
parametersL
n
isoftheorderoftheionLarmorradius,
i = v
Ti M
i =e
i B
p
, alulated on the poloidal
mag-neti eld B
p , i.e., L
n
i
. Heree
i
andM
i
are the
ionhargeandmass,respetively,B
p
isthe"physial"
omponent of the magneti eld, to dier it from
o-variantorontravariantomponentsthatappearbelow,
and v
Ti =
p
2T
i =M
i
is theionthermalveloity.
How-ever,therearesomerelevantproblemsthatneedtobe
investigated in ollisional tokamak plasmas, in the
L-mode(thelowregime),whentheonditionL
n >
i is
fullled.
Theionneolassialtransportintokamakollisional
plasmas 24
wasoriginallyinvestigatedbyShafranov. He
used thewell-known approahbyPrshand Shluter
toalulatetheionheatondutivityintokamak
plas-maswithirularmagnetisurfaes. Theseresultswere
latergeneralizedforarbitrarygeometryofthemagneti
eld. 25
The next step in this diretion was taken by
Mikhailovskii and Tsypin, 26
who onsideredthe eet
of soni toroidal ows on transport proesses in
hang-valueof theratio oftoroidalveloityU
i
to sound
ve-loity 2
s = (T
e +T
i )=M
i
, and the importane of the
ollisional parameter b = q 2
R 2
= 2
i
were brought up
in this paper. Here, q is the inverse rotational
trans-form, R is the tokamak major radius,
i = v
Ti =
i is
the ion mean free path, T
e and T
i
are the eletron
and ion temperature, respetively, and
i
is the
ion-ion ollision frequeny. 27
This study waslater
ontin-ued todeterminealsotheionheat ondutivitytaking
into aount of large values of the ollisional
param-eter b, 1 < b < M
i =M
e ,
28;29
external fores, 30
and
elongation. 31
In this paper we arry on this
investi-gation studying the eet of plasma subsoni toroidal
owsindued byAlfven waveson poloidal plasma
ro-tation andionheat ondutivityin ollisionalplasmas
of elongatedtokamaks,withsmoothprolesof
maro-sopiplasmaparameters,L
n >
i .
II Magneti eld and metri
To solve our problem, we need to hoose the
oordi-natesystemtoarryoutthealulationsforatoroidal
plasma olumn with ellipti ross-setion. As the
al-ulationsofthemetrishavebeenrepeatedlypresented
inmanypreviouspubliations,wehereshowonlymain
stepstoobtainit. Initially,wetaketheoordinates 0
,
00
andtobeattahedtothegeometrialenterofthe
magneti surfaes,where 0
istheradialoordinatein
thetokamakross-setionand 00
and arethepoloidal
andtoroidalangles,respetively. Thelengthelementin
these oordinatesis
dl 2
=d 02
+ 02
d 002
+(R 0
os 00
) 2
d 2
: (1)
The nextstepis the "enirling"ofthe magneti
sur-faes bymeansofthetransformation
os 0
=exp(=2) 0
os 00
;
sin 0
=exp( =2) 0
sin 00
; (2)
where= p
l
1 l
2
; =ln(l
2 =l
1 ), andl
2 andl
1
are
semi-major and semiminor axes, respetively, of the
mag-neti surfae. Then, wehoose newoordinatesto be
attahed tothegeometri enterofthetokamak
ross-setion,andstraightenthemagnetilinesoffore:
os 0
=ros[+Æ(;r)℄+(r);
sin 0
=rsin[+Æ(;r)℄; (3)
where (r) is the shiftof thegeometri enter of the
magneti surfae, determined from equilibrium
ondi-tions.
The straightening parameter Æ an be found from
theonditionthattheradialomponentoftheurrent
density vanishes, j r
= 0, by taking into aount the
expressionforthemagnetieldwiththestraightlines
offore,
~
B
0
=f0;
0
=2 p
g;
0
=2 p
gg; (4)
(here and are the poloidal and toroidal magneti
uxes,respetively,andg isthemetritensor
determi-nant),andafter ndingthemetritensoromponents
g
ik
, afterthesubstitution ofEqs. (2)and (3)into Eq.
(1),
( g
33 =
p
g)==0: (5)
Asaresult,theparameterÆisequalto
Æ= sin("
+ 0
exp(=2)): (6)
Then, we nd the metri ovariant tensor
ompo-nentsandtheirdeterminant
g
11
=osh sinhos2+2 0
exp( =2)os; (7)
g
22 =r
2
(osh+sinhos2)[1 2os("
+ 0
exp(=2))℄; (8)
g
12 =g
21
=rfsinhsin2 sin[osh("
+r 00
exp(=2))+ 0
exp( =2)℄g; (9)
g
33 =R
2
(1 2"
os); p
g=rR (1 2"
os); (10)
d
where the parameter "
is "
= "exp( =2); and
"=r=Risthetokamakinverseaspetratio. Theangle
dependeneofthemagnetieldanbeobtainedfrom
theexpressionB
0 =
p
g
ik B
i
B k
;wehave
B
0 =B
s
1+"
os+(A"
=2)os 2
; (11)
where
A= "
q 2
[exp(2) 1℄= l
1
R q 2
l 2
2
l 2
1 1
; (12)
and q = 0
= 0
is the inverse rotational transform
de-nedin generalgeometry.
WealsoneedaomponentoftheCristoel'ssymbol
i
= g im
m;k l = 1 2 g mk x l + g ml x k g k l x m ; (13) namely, ="
sin: (14)
Thenon-orthogonaloordinatesarex i
=(r;;) .
III Starting equations
AswementionedintheIntrodution,ourmain goalis
toalulatethesurfaeaveragedionheatux
Ti and
theionpoloidalrotationveloityU
i
asfuntionsofthe
ratio =U 2
i =
2
s
(the squaredMah number) andthe
elongationparameter A. Theion heat ux is dened
bytheexpression
Ti hq
r
i
i; (15)
where h:::i= Z 2 0 (:::) p gd= Z 2 0 p gd (16)
andtheheatux radialontravariantomponentq r i is equalto q r i = 2p i i g 11 M i ! 2 i T i r 5 2 p i g 33 h M i ! i p g T i : (17) Here, p i
is the ion isotropi pressure, !
i = e
i B=M
i
is theionylotron frequeny, g 11
is theontravariant
omponentofthemetritensor,andh
=B
=B isthe
-ontravariantomponentoftheunitvetorh=B=B.
Theonventionalrelationbetweenvetorovariantand
ontravariantomponentsB
=g
B
isalsoused
be-low. The physial poloidal omponent of veloity U
i
is dened by the expression U
i = rV
i
, where V
i is
the -ontravariantomponentof the ionmarosopi
veloityV
i .
From Eqs. (15)-(17) we nd the surfaeaveraged
ionheatux,
Ti = 2p i i osh M i ! 2 i T i r 5 4 B s T i "e i R Z 2 0 d n 0 B 2 T i : (18) Here,n 0
isthepartiledensity.
Further, to nd the poloidal rotation veloity U
i
weemploytheone uidmomentumequation
M i n 0 d i V i dt
= rp r+
1
[jB℄+F h
; (19)
where p=p
i +p
e
istheplasmapressure, istheion
visosity, 27
in whihwetakeintoaountonlyparallel
visosity
k
,therefore,
r^ = 3
2
[h(rh)+(hr)h℄
k
+h( hr)
k 1 3 r k ; (20) d and F h
is the radio frequeny fore aeting
ele-tronsandions. Usingtheambipolarityondition,
Z
rjdr= Z 2 0 Z 2 0 j r p
gdd 0; (21)
whih an be obtained from the urrent ontinuity
equationrj0averagedoverthemagnetisurfae,
oneobtainsfromEq. (19)
p
+( r)
+M i n 0 d i V i dt +rF h p 0; (22) i U i F h p : (23) Here, i
istheoeÆientdesribingtheiontoroidalor
anomalous visosity orfrition withneutrals, and F h
p
and F h
p
are "physial" poloidal and toroidal
ompo-nentsoftheradio frequenyfore F h
,respetively.
Theplasmadensitypoloidalperturbationsen
0 inEq.
(18)anbefound from themomentum equation (19),
taking into aount the ion inertial terms onneted
with thesubsoni toroidal ow, U
i <
s
, and theion
andeletrontemperatureperturbations,
e n 0 n 0 e T i + e T e +e k T i +T e " n 0
os: (24)
UsingEq. (24),theexpressionfortheCristoel's
sym-bolEq. (14), andtheparallel omponentofEq. (19),
oneobtainsfromEq. (22),
1 2r Z 2 d 3 2 e k
lnB " h n 0 ~ T e + ~ T i +e k i sin +F h p
Thus,ourbasiequationsfortheproblemunderstudy
areEqs. (18)and(25).
IV Poloidally perturbed
quanti-ties
WeseefromEqs.(18)and(25)that weshouldndthe
perturbed ion andeletron temperaturesand ion
par-allel visosity. To nd the partile temperatures, we
proeedfromtheheattransportequations, 27 3 2 n d T dt +p rV
= rq
+Q
; (26)
where=i;e,theheatexhangeQ
betweenionsand
eletronsisequalto
Q i = Q e = 3M e M i n 0 e (T e T i
); (27)
andpartileheatuxesare
q ik = 3:91 n 0 T i M i i qR T i ; q ek = 3:16 n 0 T e M e e qR T e ; q ? = 5 2 n 0 T e B
[hrT
℄: (28)
Then, weobtainequationsforperturbedquantities fromEqs. (26)-(28)
T i U i r ~n + 1 qR q ik
+rq
i? 3M e n 0 e M i ~ T e ~ T i
=0; (29)
T e U e r ~n + 1 qR q ek
+rq
e? + 3M e n 0 e M i ~ T e ~ T i
=0; (30)
d
The terms with thepoloidal veloities U
i andU
e in
Eqs. (29)and(30)anbeobtainedusingthe
quasista-tionaryontinuityequations
n
j rV
j +V
j rn
j
=0 (31)
andthefrozen-inondition
r[V
i
B℄0: (32)
Asaresult,wehave
~ V i q ~ V i ; ~ V i = U i r ln( n
0 p
g): (33)
Usingequation(33)andsolvingthesystemofEqs.(29)
and(30),wendtheperturbedpartiletemperatures
~ T i =0:51 " bT 0 r i d(b) U i
1+7:6b M e M i 5U Ti 1+ 2 (34) 3:8b M e M i " q j k e i n 0 +U p sin 0:32" bAT 0 r i d 1 (b) U Ti sin2; ~ T i + ~ T e =0:51 " bT 0 r i d(b) U i
1+15:2b M e M i 5U Ti 1+ 2 (35) 7:6b M e M i q j k e i n 0 +U p sin 0:32" bAT 0 r i d 1 (b) U Ti sin2;
where d(b)=1+2:2b p M e =M i , d 1
(b)=1+0:54b p
M
e =M
i
, andb=q 2 R 2 = 2 i
. Wehavealsoused
with T
e0 T
i0 = T
0
. The expression for U
e in Eq.
(36) an be obtained from momentum equations for
eletronsandions. 27
Wealso need to alulate the parallel visosity e
k
in Eqs. (24) and (25). Note, that the Braginskii's
approah 27
forvisosity(the Navier-Stokestypeofthe
visosity)isnotrelevantforproblemstobeunderstudy
in thispaper. Therefore, wetaketheparallel visosity
k
in the form derived in Refs. 26, 32, and 33. This
type of visosity is the so-alled Burnett type of the
visosity. 34
Besides the spatial derivatives of the ion
veloity,italsoinludesthederivativesoftheion
ther-maluxesq
i
. Theseadditionaltermsaresimilartothe
thermal fore terms in theion-eletron frition in the
momentum equation. 27
Thus,wehave, 26;32;33
e
k =
2
3 p
i
i
(0:96 0:59); (37)
where
=3
h(hr)V
i +
2
5p
i
h(hr)q
i 1
3
rV
i +
2
5p
i rq
i
; (38)
= 6
5
h( hr) q
i
+0:27q
ik
+ 1
3
rlnp
i q
i
r q
i
+0:27q
ik
: (39)
UsingEqs.(28)and(33),wesimplifyEqs. (38)and(39)
= 3
r
U
i
ln
p
gn 2=3
B
+ 1
q 2
R 2
U
i g
22
+U
Ti
ln
B
n
; (40)
= 3
r
0:34U
i
lnn+U
Ti
1:36
lnB 0:84
lnn
: (41)
Finally,weobtaintheexpressionfortheparallelvisosityEqs. (37),(40),and(41),employingEqs. (8),(11),and
(24),
e
k =
1:92"
p
i
r
i fU
i
[(1+0:19)sin+0:5Asin2℄+ (42)
+U
Ti
[( 1:83+1:52)sin+0:92Asin2℄g:
d
Thisexpression givesus the possibility to ndthe
ionambipolarpoloidalveloityU
i .
V Ion uxes and their analyses
Now,weanderivethenalexpressionsforplasma
ro-tationveloityandtheradialionheatuxandanalyze
them. The poloidal veloityU
i
anbeobtainedfrom
Eqs. (25),(35),and(42),
U
i =G
u1 (;A)U
Ti +G
u2 (;A)
q j
k
e
i n
0 +U
p
+1:39
i r
2
" 2
M
i n
0 v
2
Ti G
u3 (;A)F
h
; (43)
where
G
u1
(;A)= f
2 (;A)
f
1 (;A)
; G
u2
(;A)=1:35 M
e
M
i
2
b 2
f
1 (;A)
; G
u3
(;A)= d(b)
f
1 (;A)
; (44)
f
1
(;A)=d(b)
1+ 2
3
( 1+0:19)+0:25A 2
+0:18 2
b
1+15:2b M
e
M
i
; (45)
f
2
(;A)=d(b) 1+ 2
3
( 1:83+1:52)+0:46A 2
0:88b
1+
2
To plot funtions G
u1
(;A);G
u2
(;A);G
u3 (;A)
in Fig. 1, wehoose the parameterb =50. One an
observe that funtion G
u1
(;A) (gure 1), as in the
ase of an axially-symmetri tokamak, 30
hanges sign
at
0
2d(b)=b. Thisquantityoinideswithour
pre-viously obtained results. 26;28 31
It is a very
interest-ing and importantresult that inertialfores, aeting
ions,enforeionpoloidalrotationveloityU
i
tohange
its sign in the absene of radio frequeny waves in a
plasma. This eet, whih was originally obtained in
Ref. 26, isalso relevantforweaklyollisional plasmas
but sign hangingtakesplae at theMah numberof
theorderunity. 35
ThemaximumoffuntionG
u1 (;A)
ours atA0and isG
u1
(;0)2:4. Inreasingthe
parameterA, themaximumof thequantityG
u1 (;A)
dereasesandshiftstosupersonivalues,>1.
Never-theless,onlythease<1isofthepratialinterest.
The funtion G
u2
(;A) haraterizes the role of
the parallel urrent and diamagneti drift in the ion
poloidal rotation veloity. This funtion inreases
with the parameter , but stays at a small level, G
u2
(;A)<<1for<1.
The funtion G
u3
(;A) desribes the role of
ex-ternal fores in induing the ion poloidal rotation.
Thisfuntiondereases withthegrowthoftheplasma
toroidalrotation,and,orrespondingly,sodoestherole
ofexternalfores ontheionpoloidalrotation. The
in-duedradialeletrield,whih isurrentlysupposed
tobethedominantquantityfortransportbarrier
re-ationintokamaks, 12 17
anbefoundusingthe
expres-sionwhihis dened viatheion veloity from theion
motionequation, 27
E
r
B
(U
i h
p +U
pi U
i );
U
pi =
1
M
i n
0 !
i p
i
r
: (47)
UsingEqs. (11),(24),(34),(42),and(43),wederive
themagnetisurfaeaveragedradialionheatuxfrom
Eq. (18). Itisonvenienttowritethenalequationin
theShafranovform, 24;25
Ti =
2nT
i
i
M
i !
2
i T
i
r
osh+1:6q 2
" 2
" 2
G
T1
(;A)+G
T2 (;A)
1
U
Ti
q j
k
e
i n
0 +U
p
(48)
1:39
i r
2
" 2
U
Ti M
i n
0 v
2
Ti G
T3 (;A)F
h
;
where
G
T1
(;A)=
1+
2
f
3 (;A)
f
1 (;A)
; (49)
G
T2
(;A)=0:76 M
e
M
i
b(1+=2)
f
1 (;A)
1+ 2
3
( 1+0:19)+0:25A 2
0:18 2
b
d(b)
; (50)
G
T3
(;A)=
5f
1 (;A)
1+ 7:6bM
e
M
i
1+
2
; (51)
f
3
(;A)=
5
1+7:6b M
e
M
i
1+ 2
3
(1:83+1:52)+0:46A 2
+ (52)
+
1+
2
1+ 2
3
( 1+0:19)+0:25A 2
+1:4
1+
2
M
e
M
i
2
b 2
d(b) :
d
Inasupposition that theollisionalparameterb
satis-es the inequality 1<b < p
M
i =M
e
, one obtainsthe
well-knownresultfortheradialionheat ux. 24;25
TheoeÆientsG
T1
(;A)andG
T3
(;A) are
plot-ted in Fig. 2. The quantityG
T2
(;A)is small under
theonditionbM
i =M
e
,therefore,itisnotpresented
in Fig. 2. The funtions G
T1
(;A) and G
T3 (;A)
haraterize the neolassial ontribution and the
in-uene of external fores, respetively. The funtion
G
T1
(;A) is a growing funtion of the elongation A
fores inreases with the Mah number, nevertheless,
lessintensivelyfor largerelongations [see thefuntion
G
T3
(;A)inFig. 2℄.
VI Conlusion
Wehavestudied theeet ofplasmasubsonitoroidal
owsinduedbyAlfvenwavesonpoloidalrotationand
ionheatondutivityofollisionalplasmasofelongated
uxesaresometimesin asatisfatoryoinidenewith
theexperimentalresults, 36
as opposed to the eletron
Figure. 1 The dependene of the funtions Gu1(;A),
Gu2(;A), and Gu3(;A) on , for dierent magnitudes
oftheparameterA:A1=0;A2=1;A3=2::
Figure. 2Thedependeneof thefuntions GT
1
(;A)and
GT3(;A)on,for dierent magnitudesof theparameter
A:A1=0;A2=1;A3=2::
anomalous transport. This paper is a
generaliza-tion of our previous results for the ase of elongated
tokamaks. 26;28 31
It is shown that a tokamak
elon-gation an substantially aet ion neolassial uxes.
These uxesarepresentedin thepaperasfuntionsof
three quantities to be proportional to 1) the
onven-tionaliontemperaturegradient,2)theparallelurrent
and diamagneti drift, and 3) radio frequeny fores.
The proper oeÆientsare plotted on the Fig. 1and
Fig. 2. One more, the importane of the ollisional
parameterbis emphasizedin thepaper(seealsoRefs.
21-23,26,28-31). Itisalsodemonstratedthatthevery
interestingeetofhangingthesignoftheionpoloidal
rotationveloityin ollisionaltokamakplasmas,whih
dependsonthesquareoftheMahnumber,obtained
previously, 26
also takes plae in elongated tokamaks.
Thepoloidal veloityhasamaximumasafuntion of
of the ionpoloidal rotationis theonsequene of
tak-inginto aountinertialfores aetingions. It isalso
shownthattheradiofrequenyforesaninsomeases
determinetheionneolassialuxesinelongated
toka-maks.
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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