Calulations of Alfven Wave Heating
in TCABR Tokamak
A.G. Elmov, R.M.O. Galv~ao,
InstituteofPhysis,
UniversityofS~aoPaulo,05315-970,SP, Brazil
S.A. Galkin,A.A. Ivanov, and S.Yu. Medvedev
KeldyshInstituteof AppliedMathematis,RAS,Mosow,Russia
Reeivedon3July,2001
AtwodimensionalodeALTOK,whihisdesignedforalulatingplasmaheatingdueto
radiofre-quenyeldsintheAlfvenandinIonCylotronRangesofFrequeniesinaxisymmetritokamaks,
isusedtoanalyzeAlfvenwaveabsorptioninmultispeiesplasmasinTCABR(TokamakChauage
Alfven Brasilien) [Nul.Fusion30, 503 (1996)℄. A good agreement between the results obtained
with ALTOK ode alulations and with atwo dimensional kineti ode [Phys. Plas., 6(1999)
2437℄isshownforAlfvenwavedissipationinhydrogenplasmas. TheglobalAlfvenwaveresonane
ofthem=0modeisfoundtobethebestandidatetoexplainsomeheatingregimesinTCABR.
I Introdution
ThemethodofAlfvenwave(AW)heatingisbased
on the mode onversion eet[1℄ (named a loal AW
resonane(LAR)[2,3℄)oftheradiofrequeny(rf)eld,
indued by an external antenna, into a kineti AW
that eetively dissipates on eletrons. The analysis
of Alfven wave absorption in tokamaks with two
di-mensional (2D) numerial odes has its originon the
MHD plasmamodeldevelopedin Ref[4℄. Lately, this
model wasextendedto ionylotronrangeof
frequen-ies (ICRF) byinludingionylotron resonanewith
ollisional dissipation (see Ref [5℄). However, the
ef-fetsof wave dissipation oneletronswasmodeledby
anartiialdampingtooveromethelogarithmi
diver-genein LAR.Somekineti2D-odesfor AW heating
weredevelopedfortheaxis-symmetritokamakplasma
geometry [6,7,8℄. However,nonloal eetsin the
ki-netidieletri tensor produed diÆulties in the
pro-edure for solving theMaxwell equations numerially,
espeiallyintheviinityoftheionylotronresonane.
Reently,theMHD plasmamodelwasextended to
amulti-uidplasmamodelwhihinludeseletronand
two speies ion uids [9℄. This two-dimensional
nu-merialode(namedALTOK)an help to resolvethe
problem of numerial solution at the loal AW
reso-nanebeauseitinludesnaturallytheeletron-ion
ol-lision dissipationandthe eletroninertia in the
paral-lel omponent of the dieletri tensor. In this ase, a
slow quasi-eletrostati Alfven wave (SQAW) [10, 11℄
mayappearat the mode onversionmagneti surfae.
This model issuÆientfor analysisof Alfvenand fast
magneto-soniwavesinICRF.
Here, using the ALTOK ode, we present
rele-vant results of numerial alulations related to the
AlfvenwaveheatingexperimentsinTCABR [12℄,
tak-ing into aount arbon and oxygen impurities, and
omparesomeresultswith theorrespondingones
ob-tainedthroughthekinetiode[7℄.
Thepaperisorganizedasfollows. InSetionII,we
briey desribethe toroidalmulti-uid plasma model.
In Setion III,we present theresult of alulationsof
Alfven wave dissipation in the TCABR tokamak.
Fi-nally,weanalyzesomeheatingregimesinTCABRand
summarizethemainresultsofouralulations.
II Plasma model
The standard plasma model inludes Maxwell
equationsandthedieletritensor,whihisalulated,
in ourase, from multi-uid MHD equations (see, for
example[13℄). Assumingharmonidependeneintime
and intoroidalangle 'forthewavemagnetield B,
E= X n E n e in' i!t
; B= X n B n e in' i!t
; j= X n j n e in' i!t ; (1)
theMaxwellequationsanbepresentedasfollows:
rrE ! 2 2 ^ I 4i ! ^ E= 4i! 2 j ext : (2) d wherej ext
isanexternaldrivingurrentintheantenna
andgeneralizedOhm'slawj=E^ isusedwith^being
thesuseptibilitytensorinaloalbasesofoordinates
e k = B 0 jB 0 j ; e N = r
jr j ; e ? =e k e N ; (3) 4i ! ^ = 0 N N? 0 N? ? 0 0 0 k 1 A ; (4)
where ispoloidal magnetiuxandtensoromponentsare
N = X ! 2 p (! 2 B ! 2 ) ; ? = X ! 2 p (! 2 B ! 2 ) ; (5) N? = X ! B ! 2 p !(! 2 B ! 2 ) ; k = X ! 2 p ! 2 ! 2 p;e
!(!+i
e )
; (6)
d
Intheequations
e
is theeletron-ionollision
fre-queny, !
p
, and !
B
are the plasma and ylotron
frequenies, respetively, for speies . The wave
ab-sorption powerdensityisgivenbyW =R e(jE
).
Ageneraltwodimensionalgridwithnon-orthogonal
quadrangular ells is used to disretize Eq.(2). For
theaxis-symmetri equilibrium,aquasiradial-annulus
grid, adapted to magneti surfaes, is applied. The
equilibrium odePOLAR-2D[14℄ isemployedto solve
theGrad-Shafranovequationfor andobtainthegrid
adapted to magneti surfaes. However an arbitrary
grid anbealso usedif theequilibrium magnetield
isdiretlyspeied.
III Alfven Wave Heating
Here,using the2Dkineti[7℄and MHDodes[9℄,
we present relevant results of numerial alulations
disharge 4893 in Ref [12℄) and ompare them with
oldTCA results[15℄. The simulationshave been
ar-riedoutassumingairular ross-setiontokamak
ge-ometry with the following parameters: minor radius
a=0:18m; major radiusR
0
=0:615m; antenna
sur-faeradiusb=0:2m;wallradiusd=0:23m,toroidal
magnetield B
0
=1:1T, plasmaurrentI
p
=54kA,
for theohmi stage,and I
p
= 57kA, for the rf stage.
Thekinetiodealulationsarearriedoutassuming
atemperatureprole given by T
=T 0 (1 r 2 =a 2 ) 2 ;
with=e;i, T
e0
=500eV andT
i0
=160eV,
respe-tively. Wehosethevaluesofthesafetyfatortobein
theinterval1:1 q
0
1:6 at the magneti axis, and
4:4q
0
6:4,attheplasmaboundary. These
parame-tersareonrmedwithASTRAtransportode[16℄
al-ulations (see disussion below). Finally, the eletron
densityproleisassumedtobeofparaboliform with
amaximumentraldensityn
0
=3:210 19
m 3
anda
pedestal value n
a
=110 18
m 3
. The impurity and
an-wavenumbers. Therealstrutureof newTCABR
an-tennasystemistakenintoaountthroughoeÆients
alulatedin Ref.[17℄. Inouralulations,weonsider
only oneantennamode that orrespondsto the main
omponentgiven bytheFourieranalysisof theatual
antennasystem.
Beause of small pressure orretions and to
sim-plify alulations, we use a fore-free equilibrium in
theALTOKodealulationswiththetoroidalurrent
density rj
' = (1
2
)
1
and the plasma density is
n=n
0 (1
)
3
;where
isnormalizedpoloidal
mag-netiuxfuntion,sothat
=0attheaxisand
=1
at the plasma boundary. Tohave the plasma density
andurrentprolesadjustedtotheprolesusedinthe
kinetiode,weuse
1
=1:6;
2
=0:95and
3 =0:7.
InFig.1, weshowthat there is reasonableoinidene
between the orresponding plasmaproles of the
AL-TOKandkinetiodeforthepoloidal angle=2.
Figure1. Density ne andtoroidal urrent prolerj' over
theradiusatthepoloidalangle=2oftokamakrossetion;
thesolidlinesaretheALTOKodeproles,thedottedlines
markprolesusedinthekinetiode.
Here we onsider wave dissipation in the Alfven
wave ontinuum. Generally, in the ylindrial model,
theequation that governsthe Alfven waveontinuum
anbewrittenin theform,
! 2
? (r)=k
2
k (r)
2
;
? =
X
!
2
p
(! 2
!
2
) (7)
Forhydrogenplasmas,theequationfortheAlfvenwave
ontinuumanbesimpliedto
!
A =
k
k
A
q
1+k 2
k
2
A =!
2
;i ;
A =
B
0
p
4n
e m
i ; k
k =
(nq+m)
R
0 q
(8)
In Fig.2, we show the M = 1;N = 4-antenna
impedaneforpurehydrogenplasmasandplasmaswith
arbon impurity (apital letters are used for antenna
vauum modesand lowerase are used for modes
a-tually exited in the plasma). We an observe spikes
of the impedane related to the m = 0 and m = 1
globalwaveresonanessituatedbelowAlfvenwave
on-tinuum. Theorrespondingwavedissipatedpower,
al-ulated with the ALTOK ode for frequenies f =
4:2MHz and 4:6MHz, is presented in Fig.3.
Usu-(8) for hydrogen plasmas does not depend on the q
prole. In this ase, the dissipated wave power
pre-sentedin Fig.3aisstrongly peakedin the plasmaore
and it an be eetively used for plasma ore
heat-ing. Note that our alulations onrm the global
Alfven wave resonane that was fond experimentally
in old TCA deuterium plasmas [15℄ with the
param-eters: B
0
= 1:2T,n
0
= 2:6 10 19
m 3
, q
0
= 1:1,
M= 1;N = 2;f =2:5MHz.
3.8
4.0
4.2
4.4
4.6
4.8
5.0
5.2
5.4
0.0
0.5
1.0
1.5
f
c,C+3
Im
pedance (au)
frequency (MHz)
Figure2. TheantennaloadingimpedaneIMN plottedas
a funtion of the generator frequeny f for poloidal and
toroidal wave numbers M = 1;N = 4 of the antenna
spetrum. Theimpedaneurvesarealulatedwiththe
ki-netiode(solidforpurehydrogen,dotedwith0:2impurity
plasmas)anddashedlineorrespondtheMHDode
alula-tionsforplasmawith0:2%-arbonimpurityTheplasma
pa-rametersaren
0
=3:210 19
m 3
;T
e0
=500eV,B
t
=1:1T,
q
0
=1:1,q
a =4:4.
Beause of an unontrollable inrease of the light
impurities,suhasarbonandoxygen,suppliedbythe
hamberwallduring the rf pulse, theglobal wave
fre-queny and Alfven wave ontinuum an be strongly
modied if the Alfvenwavefrequeny is about of the
ion ylotron frequeny. In Fig.4, we show the
ylin-drial Alfven ontinuum (7) modiationin a plasma
with light impurities in shot # 4893. This situation
is also alulated using 2-D odes. For example, in
TCABR hydrogen plasma with 0:2% of three times
ionizedarbon (or fourtimes ionizedoxygen),the
y-lotron frequeny is f = 4:2MHz and it an aet
Alfven wave dissipation. That eet is also
demon-stratedinFig.2wheretheAlfvenwaveontinuumis
de-stroyedforfrequeniesaboutthis ylotron frequeny.
In this ase, the better heating regimes an be our
about f = 4:6MHz that is above the C +3
12
Figure3ab.Distributionofdissipatedwavepowerover
toka-mak ross-setion in the spike of the m = 0-GAW
reso-naneinhydrogenplasmasshownforf =4:2MHz(a)and
f = 4:6MHz with 0:2alulations with M = 1;N = 4
antennaongurationforentralplasmadensityn0=3:2
10 19
m 3
.
IV ASTRA alulations
Theresultsof therfheating disharge# 4893 [12℄
havebeenanalyzedwith theASTRA ode [16℄
. Before
the rf pulse (the referene time is 53ms) the plasma
parameters are: urrent I
p
= 54 kA, loop-voltage
U(a) 2.77V, line-averaged density n
e
= 2:110 13
m 3
,
dia
=0:49(see Fig.4). Assumingthe
Alator-saling oeÆient, the data in Fig.5 an be adjusted
with Z
ef
=5:2. Inthemiddleof therf pulse (the
ref-erenetimeis61.5msatthemaximumoftheurrent),
andonsideringasmalldensitygrowton
e
=2:210 13
m 3
withthesameprole,thedatainFig.4(I
p
=57kA,
U(a) 1.74 V and
p
0:1 for the referene time
61.5 ms) have been adjusted with P
rf
= 60kW and
Z
ef
5:4 with Alator-saling diusion. The result
indiates aneÆientAlfvenwaveheating,asmall
ur-rent drive I
d
1:2kA and urrent drive eÆieny
k =(2en R !)0:02A=W :
-1
0
1
3
4
5
6
7
+3
C
12
+5
O
16
Alfvén continuum
generator frequency
fre
que
nc
y
(R-R
0
)/a
Figure 4. Plot of the ylindrial Alfven ontinuum taken
fromEq(7)anddierentarbonandoxygenimpurity
y-lotron resonanes in hydrogen plasmas for onditions in
Fig.2.
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18
0.0
0.5
1.0
1.5
2.0
2.5
3.0
(b)
n
e
U
oh
U
rf
U
cd
n
e[
13]
cm
-3
, U
(V
)
radius (m)
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18
0.0
0.1
0.2
0.3
0.4
0.5
0.6
(a)
T
e,rf
T
e,oh
P
rf
T
e
(
keV
),
P
rf
(W
/10/
cm
3
)
Figure 5. Distribution of eletron temperature (a), loop
voltage and dissipated wave power is alulated with the
ASTRAodeandshownforohmi(t1=53ms,dottedlines)
andrf(t2 =61:5ms,dashlines)stageinN4893disharge.
Loopvoltageunderrfpowerisshownwithrfheating
(dash-dotline)andtakingintoaount1.2kAurrentdrive(solid
Inonlusion,weansay:
Alfven wave absorption of oupled side-bands
har-monis exited by M = 1 antenna auses a broad
powerdepositionatm=0loalAWresonanethat
sur-passthepowerdeposition ofglobal AW attheplasma
enter;
hoosing generator frequeny properly in m = 0
Alfven wave ontinuum, Alfven wave absorption an
beonentratedintheplasmaore;
the ASTRA ode alulations onrm Alfven wave
heating T
e
(0) 250eV and non-indutive urrent
I
d
1:2kA driven by one module antenna with
60kWdissipatedpowerintheplasma.
toanalyzetheAWdissipationproperly,theeet of
ion ylotron resonanezones introdued by partially
ionizedimpuritiesshould betakenintoaount.
Aknowledgments Authors thankful to
Dr.G.V.Pereverzev for the help with ASTRA
alu-lations. ThisworkissupportedbytheMinistryof
Si-ene and Tehnology/Brazil, through Pronex Projet
andFAPESP(FoundationoftheStateofS~aoPaulofor
theSupport ofResearh).
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