VOLUME 88, NUMBER 7 PHYSICAL REVIEW LETTERS 18F
EBRUARY 2002
Exponen ially G owing Tea ing Modes in Rijnhuizen Tokamak P ojec Plasmas
F. Salzedas,* F. C. Schülle , A. A. M. Oomens, and he RTP Team
FOM-Ins i uu oo Plasma ysica “Rijnhuizen,” Associa ion Eu a om-FOM, T ila e al Eu egio Clus e ,
P.O. Box 1207, 3430 BE Nieuwegein, The Ne he lands
(Recei ed 14 Ma ch 2001; published 5 Feb ua y 2002)
The local measu emen o he island wid h w, a ound he esonan su ace, allowed a di ec es o
he ex ended Ru he o d model [P. H. Ru he o d, PPPL Repo -2277 (1985)], desc ibing he e olu ion
o adia ion-induced ea ing modes p io o dis up ions o okamak plasmas. I is ound ha his model
accoun s e y well o he obse ed exponen ial g ow h and suppo s adia ion losses as being he main
d i ing mechanism. The model implies ha he e ec i e pe pendicula elec on hea conduc i i y in he
island is smalle han he global one. Compa ison o he local measu emen s o wwi h he magne ic
pe u bed ield ˜
Bshowed ha w~˜
B1兾2was alid o wid hs up o 18% o he mino adius.
DOI: 10.1103/PhysRe Le .88.075002 PACS numbe s: 52.55.Tn, 52.35.Py
The iden i ica ion o he main d i ing mechanism o he
obse ed MHD (magne ohyd odynamics) ins abili ies in a
okamak plasma is o conside able in e es . Such knowl-
edge o he na u e o he ins abili y can be e y help ul
in he choice o a s abiliza ion p ocedu e needed o a oid
deg ada ion o con inemen o plasma dis up ion. Dis-
up ions a e he mos dange ous ins abili ies in okamak
plasmas, and he p ecu so o densi y limi dis up ions is
o en domina ed by an m兾n苷2兾1MHD ins abili y [1]
(whe e mand na e he poloidal and o oidal Fou ie mode
numbe s, espec i ely). In his Le e , we s udy he expo-
nen ial inc ease o an m兾n苷2兾1 ea ing mode obse ed
p io o a densi y limi dis up ion [see Fig. 1(a)] in RTP
(Rijnhuizen Tokamak P ojec , ci cula limi e , mino a-
dius a苷0.16 m and majo adius R0苷0.72 m). Simila
exponen ial g ow h is obse ed also in o he okamaks [2].
Compa ison wi h he ex ended Ru he o d [3] model sup-
po s adia i e ene gy losses as he main d i ing mecha-
nism o his ea ing mode. To a oid con usion wi h he
well-known neoclassical ea ing modes, we, hence o wa d
denomina e hese ins abili ies as adia i e induced ea ing
mode o RTM. The s udy o RTM is also e y impo an
because u u e la ge okamaks such as ITER a e expec ed
o wo k close o 100% edge adia ion o educe he mal
load on he di e o .
In he plasmas s udied he e, he plasma cu en IP苷
100 kA, he sa e y ac o qa艐4, he a io o he plasma
ene gy o he magne ic ield ene gy, b,was low, and a
he ime he RTM is obse ed collisionali y is high. So
neoclassical e ec s a e dis ega ded. The illing gas was
He and he densi y was amped up using Ne gas injec-
ion. Bo h noble gases a e no abso bed by he essel wall.
This allowed a be e and ep oducible con ol o he elec-
on densi y han wi h hyd ogenic plasmas. Mo eo e , he
densi y a dis up ion could be kep below he cu o den-
si y o he elec on cyclo on emission (ECE) he e odyne
adiome e , used o measu e he ime e olu ion o he elec-
on empe a u e a 20 adial posi ions along a ho izon al
cho d. Elec on empe a u e and densi y we e measu ed a
h ee ime poin s wi h a high spa ial esolu ion Thomson
sca e ing (TS) sys em. Saw ee h a e always obse ed in
hese discha ges [Fig. 1(b)].
One o he di icul ies in he analysis o he beha io o
ea ing modes is o ela e he measu ed pe u bed poloidal
magne ic ield ˜
Bu共 c兲[Fig. 1(a)], whe e cis he adius
o he pickup coils, wi h he unknown pe u bed adial
magne ic ield ˜
B 共 s兲a he mode esonan su ace, s, ha
is he quan i y whose dynamics is p edic ed by heo y [4].
The usual p ocedu e consis s [5,6] o app oxima ing he
o oidal plasma by a cylind ical one and hen o use he
FIG. 1. (a) Ampli ude o ˜
Bu. (b) ECE Tein he cen e (chan-
nel 10) and in he icini y o he q苷2su ace (channels 6
and 5) [see Fig. 2(a)]. (c) D 兾acalcula ed wi h (4). The
solid ci cles indica e he island wid h es ima ed om TS [see
Figs. 2(c)–2(e)]. The g ay line is he i o Eq. (10). (d) D 兾a
plo ed agains ˜
Bu. The dashed (solid) g ay line ep esen s
Eq. (6) [Eq. (5)]. Symbol 䉭(¶) e e s o LFS (HFS).
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VOLUME 88, NUMBER 7 PHYSICAL REVIEW LETTERS 18F
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cu en p o ile as ound om one o he me hods desc ibed,
o example, in [7], o sol e he equa ion
d2c
d 211
dc
d 2µm2
1m0
djz0
d
Bu0共12n
mq兲∂c苷0, (1)
using ˜
Bu共 c兲as a bounda y condi ion, o ind ˜
B 共 s兲, whe e
cis a scala po en ial ha is ela ed o ˜
Bby
˜
B苷=c 3 ˆz苷21
≠c
≠u ˆ 1≠c
≠ ˆ
u.(2)
The zcomponen o he equilib ium plasma cu en is jz0,
and Bu0is he poloidal componen o he equilib ium mag-
ne ic ield.
In he case ha he plasma cu en ou side he esonan
su ace can be neglec ed and he wall ac s as a pe ec
conduc o , Eq. (1) has an explici solu ion. Then, using
(2), he mcomponen o ˜
Bua cis ela ed o he m
componen o ˜
B a sby
˜
Bum共 c兲苷˜
B m共 s兲µ s
c∂共m11兲11共 c
w兲2m
12共 s
w兲2m,(3)
whe e wis he adius o he essel wall.
In he me hods which use (1) and (3) wi h measu emen s
o ex e nal pickup coils, he mode e olu ion is desc ibed
ia he ampli ude o he pe u bed magne ic ield. In wha
ollows, ano he me hod o measu e he spa ial ampli ude
o he pe u ba ions, di ec ly and locally a ound he eso-
nan su ace, will be used. Basically, i elies on he as-
sump ion ha close o and ou side o he sepa a ix o a
ea ing mode he elec on empe a u e is a lux unc ion
[8]. In his case, a change Dwo he island wid h will
cause a p opo ional displacemen D o he lux su aces
in he neighbo hood o he sepa a ix. The displacemen
D is gi en by
D 苷
DT
dT0
d
,(4)
whe e T0is he empe a u e p o ile, measu ed adially
along he Xpoin and DTis he di e ence in empe a-
u e be ween he wo poin s a he wo angles ha co e-
spond wi h he Xpoin and he Opoin [9]. I is also
implici ly assumed ha he plasma is incomp essible. Fig-
u e 1(c) shows D 兾a, de i ed bo h om he low ield side
(LFS), channel 6 o he adiome e , and on he high ield
side (HFS), channel 16. The posi ion o hese channels is
shown in Fig. 2(a) and a pic o ial illus a ion o D 共LFS兲
is shown in Fig. 2( ). I is expec ed ha he smalle al-
ues o D ha e lowe accu acy since he island sepa a ix
OXO
OX O
m/n=2/1 mode
chn.6
∆
∆T
chn.
FIG. 2. (a) ECE Te共 , 兲iso he ms. The channels posi ion is indica ed a he igh . A ske ch o he m苷2mode sepa a ix is
illus a ed by he do ed lines a ound he g ay a eas. (b) TS Tep o iles a 1,
2, and 3. (c) Zoom in on (b). Fo cla i y, only one
ypical e o ba is shown. (d) and (e) a e he same as be o e o ne共 兲. The dashed line indica es he calcula ed adial posi ion o he
q苷2su ace. ( ) Illus a ion o (4). G ay (black) line, ECE Te共 兲passing h ough he O(X) poin . (g) j共 兲and q共 兲calcula ed
om Te共 1兲.
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VOLUME 88, NUMBER 7 PHYSICAL REVIEW LETTERS 18F
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is u he away om he ECE channels. On he o he ex-
eme, i he sepa a ix c osses he ECE channel hen he
de ini ion (4) is no applicable.
A cha ac e is ic o m苷2islands in okamaks is isible
in he ECE measu emen s, as shown in Fig. 2(a), namely
ha he de o ma ion o iso he ms is symme ic in ela-
ion o he esonan su ace on he HFS [see illus a ion
in Fig. 2(a)]. On he LFS i is mo e di icul o localize
he Xpoin , since he island g ows asymme ically in e-
spec o he esonan su ace, i.e., much mo e owa ds he
cen e o he plasma han o he edge. In his case i is ex-
pec ed ha on he LFS he s ongly asymme ic island leads
o D ⯝w, while on he HFS a symme ic island gi es
D ⯝w兾2. Figu e 1(c) con i ms his expec a ion wi h
D 共LFS兲⯝2D 共HFS兲. Mo eo e , he es ima ed island
wid h om he TS [10], measu ed along a e ical cho d,
comes e y close o he alues o D measu ed on he LFS
[see Figs. 1(c) and 2(c)–2(e)].
In Fig. 1(d) D is plo ed agains ˜
Bu共 c兲, o alues up o
18% o he mino adius, which a e measu ed jus 100 ms
be o e he onse o he dis up ion. Wi hou any assump ion
on he plasma equilib ium, he bes i s o he expe imen al
da a gi e he ollowing o he LFS and HFS:
D 共LFS兲苷共1.04 60.08兲p˜
Bu共 c兲,(5)
D 共HFS兲苷共0.56 60.05兲p˜
Bu共 c兲.(6)
We can compa e he expe imen al ela ion (5) wi h he
well-known exp ession o he island wid h ha ela es he
spa ial and magne ic mode ampli udes:
w苷4s sq
mBuq0p˜
B 共 s兲.(7)
Because o he adia i e con ac ion o he cu en p o-
ile, he cu en o . scan be neglec ed so we can use
(3) o ob ain ˜
B o m苷2. Using he alues om Table I
ha we e calcula ed om TS Te共 1兲[see Figs. 2(b) and
2(g)], he ollowing is ob ained:
w苷共1.14 60.25兲p˜
Bu共 c兲.(8)
The ag eemen be ween (8) and he expe imen ally de-
i ed (5) is e y good. This suppo s he calcula ion o w
in he cylind ical app oxima ion used un il now.
Expe imen al e idence indica es ha he equilib ium pa-
ame e s, Bu共 s兲, s, and q0共 s兲, do no change no ice-
ably in ime du ing he island exponen ial g ow h. Fo
s his can be seen unambiguously a he HFS in Fig. 2(a).
Rela i ely o Bu共 s兲and q0共 s兲, since he cu en di usion
ime is ypically 10 20 ms, i is e y unlikely ha hese
TABLE I. Equilib ium pa ame e s a 苷 scalcula ed om
Te共 1兲in Fig. 2(b), assuming jz~Te共 1兲3兾2.
sTejq
0Buh
(m) (eV) (MA m22)(m
21)(T) (1026Vm)
0.113 127 0.211 32.7 0.165 4.12
pa ame e s change signi ican ly du ing he 0.6 ms o he
mode exponen ial g ow h. So, assuming ha du ing his
pe iod he equilib ium pa ame e s do no show no iceable
changes, Fig. 1(d) p o ides a di ec demons a ion ha (7)
is s ill alid up o island sizes o he o de o 18% o he
mino adius. This ea u e should be emphasized because
he use in he li e a u e (e.g., [5,6]) o (7) o island sizes
o his magni ude is based only on he magne ic measu e-
men s pe o med ou side he plasma, which a e ex apo-
la ed inwa ds up o s.
Be o e he exponen ial g ow h s a s, i is obse ed ha ,
on a e age, he ampli ude o ˜
Bug ows algeb aically o
a pe iod o 艐20 ms. Du ing his pe iod [o which only
he las 4ms a e shown in Fig. 1(a)], he g ow h o he
mode is no mono onic bu shows an i egula modula ion
wi h he ampli ude now and hen dec easing p ac ically o
ze o. This modula ion is o en co ela ed wi h a saw oo h
c ash. This complex beha io will no be discussed he e.
Ins ead we will analyze only he exponen ial phase using
he ex ended Ru he o d model [3],
dw
d
苷C1
h
m0
D02C2
˜
PT
xisland
⬜e 具Te典w,(9)
whe e his he plasma esis i i y, C1苷1.22, and C2苷
0.9hjzq兾共Buq0兲wi h all he quan i ies aken a he eso-
nan su ace. ˜
PTis he o al powe densi y pe pa icle in
he island and he mode is des abilized i ˜
PTis nega i e.
xisland
⬜e and 具Te典a e he e ec i e pe pendicula elec on
he mal conduc i i y and he a e age elec on empe a u e
in he island, espec i ely.
F om 165.8 o 166.2 ms, he e olu ion o he ampli ude
o he mode magne ic ield could be well i ed wi h a
quad a ic inc ease wi h ime implying a linea inc ease in
island wid h. Du ing his pe iod, losses by adia ion inside
he island a e s ill no impo an , and so he second e m
o (9) is negligible. The alue o D0can hen be es ima ed
om he linea inc ease o he island wid h. Using he
alue o h om Table I, we ob ain D0苷2.
Be ween 166.2 and 166.8 ms, he inc ease in he mode
ampli ude is e y well desc ibed by an exponen ial i . This
can be explained by a sudden onse o he second e m o
(9) wi h ˜
PT,0. An ab up nega i e alue o ˜
PTis pe -
ec ly possible due o he na u e o adia i e powe losses.
This is an e ec o he in e se p opo ionali y be ween a-
dia ed powe densi y and he elec on empe a u e. I im-
pu i ies ha e been accumula ing inside he island, once he
empe a u e on he Opoin is low enough such ha adia-
i e ecombina ion o hese impu i ies can occu , a small
dec ease o empe a u e will lead o an inc ease in adia-
ion loss. Ene gy in he island is hen dissipa ed as e ,
se ing ab up ly ˜
PT,0. The hea lowing in o he island
can be enough o keep ˜
PTcons an , du ing he exponen ial
g ow h, as bo h ECE and TS indica e.
The high spa ial esolu ion empe a u e and densi y p o-
iles inside he island a e nei he la no mono onic, bu
i egula [Fig. 2(c)]. On bo h 1and 2p o iles, despi e he
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VOLUME 88, NUMBER 7 PHYSICAL REVIEW LETTERS 18F
EBRUARY 2002
maxima and minima, in he island he a e age alue o Te
does no change. This indica es ha he hea low inside
he island is no pu ely di usi e. The ECE [Fig. 2(a)],
which has lowe spa ial esolu ion han TS and so does
no de ec he i egula i y ound by he TS p o iles, also
shows he island g owing wi hou signi ican changes in
具Te典. Seconda y maxima in Teand newe e also obse ed
in TEXTOR [11,12].
F om he da a o Fig. 1(c), i is possible o es ima e
he a io o ˜
PT兾xisland
⬜e . Assuming ha ˜
PTis cons an
du ing he sho ime in e al in which he mode g ows
exponen ially, in eg a ion o (9) gi es
w共 兲苷
a1
a2
1µw02a1
a2∂e2a2 ,(10)
whe e w0苷w共166.2兲苷0.02a,a1苷c1
hD0
m0
苷8, and
a2苷
0.9hjzq
Buq0具Te典
˜
PT
x⬜e
.(11)
The i ing o (10) o he da a o Fig. 1(c) (dashed line)
gi es a2苷22.3 3103s21. F om (11) and Table I i is
ound
˜
PT兾xisland
⬜e 苷21.0 3106eV m22.(12)
Bo h ˜
PTand xisland
⬜e a e unknown, bu i is possible o
es ima e i s ange o alues. A he ime he TS Te共 1兲
p o ile is measu ed, he ene gy con inemen ime is
e苷3.4 ms which gi es a global e ec i e pe pendicu-
la elec on he mal conduc i i y xglobal
⬜e 苷a2兾共4 e兲苷
1.9 m2s21.I xisland
⬜e was he same as xglobal
⬜e , hen
˜
PT苷21.9 3105eV s21pe pa icle which means ha
du ing he 0.6 ms o exponen ial g ow h an a e age o
1140 eV pe pa icle would ha e o be dissipa ed in he
island. Such a alue is e y la ge since du ing he same
pe iod he empe a u e in channel 6 dec eases 215 eV,
while in channel 5 i inc eases only 65 eV [Fig. 1(b)].
The di e ence o 150 eV be ween hese wo channels is
almos 1 o de o magni ude smalle han he p e iously
ound alue 1140 eV. Assuming ha only 150 eV we e
dissipa ed in he island, hen ˜
PT艐22.5 3105eV s21
pe pa icle (which co esponds o 艐270 kW). This
alue is in good ag eemen wi h he adia ion losses in he
ou e plasma laye s a compa able empe a u e alues and
is e y close o he minimum elec on cyclo on powe o
90 kW ha was ound necessa y o s abilize he m苷2
RTM in simila discha ges [9,13]). So om (12) i ollows
ha xisland
⬜e 艐0.25 m2s21.
This alue o he e ec i e elec on hea conduc i i y
inside he island is almos 1 o de o magni ude smalle
han he global e ec i e elec on hea conduc i i y o he
plasma. This educ ion in di usi i y inside he island is in
ag eemen wi h p e ious indica ions [11,12,14]. Mo eo e ,
i is co obo a ed by he obse a ion in TS Te共 1, 2兲o he
seconda y maxima in empe a u e and densi y inside he
island, ela i ely o he sepa a ix. The Temaxima a e
mo e p onounced in TS Te共 3兲. Despi e his p o ile was
measu ed immedia ely a e he exponen ial g ow h ˜
Bu共 c兲
does no indica e any s ong change in he island s uc u e.
Only la e a 苷167.2 ms, du ing he dis up ion ˜
Bu共 c兲
shows a sha p spike, indica ing he p obable des uc ion o
he island.
One o he au ho s (F. Salzedas), was suppo ed by he
Po uguese Founda ion o Science and Technology FCT
unde P og ama PRAXIS XXI-G an No. BD/4531/94.
This wo k was pe o med unde he Eu a om-FOM Asso-
cia ion ag eemen , wi h inancial suppo om he Du ch
esea ch o ganiza ion NWO and Eu a om.
*P esen add ess: Cen o de Fusão Nuclea , Associa ion
Eu a om-IST, A . Ro isco Pais, 1049-001 Lisboa,
Po ugal.
Elec onic add ess: [email p o ec ed]
[1] F. C. Schülle , Plasma Phys. Con olled Fusion 37,A135
(1995).
[2] M. Schi enhelm e al., Nucl. Fusion 37,1255 (1997).
[3] P. H. Ru he o d, PPPL Repo -2277 (1985).
[4] P. H. Ru he o d, Phys. Fluids 16,1903 (1973).
[5] Z. Chang e al., Phys. Re . Le . 74,4663 (1995).
[6] B. Ca e as e al., Nucl. Fusion 19,1423 (1979).
[7] H. Sol wisch, Plasma Phys. Con olled Fusion 34,1669
(1992).
[8] R. Fi zpa ick, Phys. Plasmas 2,825 (1995).
[9] F. Salzedas, Ph.D. hesis, Uni e si ei U ech , 2000 (URL:
h p://www.lib a y.uu.nl/digia chie /dip/diss/1940669/in-
houd.h m).
[10] TS has a e y high spa ial esolu ion bu only one poin
in ime is gi en while ECE has a high empo al bu low
spa ial esolu ion. Bo h e ec s cause some e o in he
loca ion o he sepa a ix.
[11] P. C. de V ies e al., Plasma Phys. Con olled Fusion 39,
439 (1997).
[12] P. C. de V ies e al., Nucl. Fusion 37,1641 (1997).
[13] F. Salzedas e al. ( o be published).
[14] B. P. an Milligen e al., Nucl. Fusion 33,1119 (1993).
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