Expe imen al Valida ion o a Filamen T anspo Model
in Tu bulen Magne ized Plasmas
D. Ca ale o,1P. Manz,2L. Aho-Man ila,3G. Bi kenmeie ,1,2 M. B ix,4M. G o h,5H. W. Mülle ,1,6 U. S o h,1,2
N. Vianello,7,8 E. Wol um,1ASDEX Upg ade eam,1JET Con ibu o s,*and EURO usion MST1 Team†
1Max-Planck-Ins i u e o Plasma Physics, Bol zmanns aße 2, Ga ching, Ge many
2Physik-Depa men E28, Technische Uni e si ä München, James-F anck-S aße 1, Ga ching, Ge many
3VTT Technical Resea ch Cen e o Finland, Helsinki, Finland
4EURO usion Conso ium, JET, Culham Science Cen e, Abingdon, OX14 3DB, Uni ed Kingdom
5Aal o Uni e si y, Espoo, Finland
6Ins i u e o Ma e ials Chemis y and Resea ch, Uni e si y o Vienna, Wh inge s asse 42, A-1090 Vienna, Aus ia
7Conso zio RFX, C.so S a i Uni i 4,I-35127 Pado a, I aly
8Ecole Poly echnique Fédé ale de Lausanne (EPFL), Swiss Plasma Cen e (SPC), CH-1015 Lausanne, Swi ze land
(Recei ed 28 May 2015; published 17 No embe 2015)
In a wide a ie y o na u al and labo a o y magne ized plasmas, ilamen s appea as a esul o
in e change ins abili y. These con ec i e s uc u es subs an ially enhance anspo in he di ec ion
pe pendicula o he magne ic ield. Acco ding o ilamen models, hei p opaga ion may ollow di e en
egimes depending on he pa allel closu e o cha ge conse a ion. This is o pa amoun impo ance in
magne ic usion plasmas, as high collisionali y in he sc ape-o laye may igge a egime ansi ion
leading o s ongly enhanced pe pendicula pa icle luxes. This wo k epo s o he i s ime on an
expe imen al e i ica ion o his p ocess, linking enhanced anspo wi h a egime ansi ion as p edic ed
by models. Based on hese esul s, a no el scaling o global pe pendicula pa icle anspo in eac o
ele an okamaks such as ASDEX-Upg ade and JET is ound, leading o impo an implica ions o
nex gene a ion usion de ices.
DOI: 10.1103/PhysRe Le .115.215002 PACS numbe s: 52.35.Ra, 52.35.Py, 52.55.Fa, 94.30.cq
In e change ins abili ies a e a ubiqui ous phenomenon in
magne ized plasmas whe e a densi y g adien coexis s wi h a
pa allel o ce densi y o he same sign [1]. This ins abili y
leads o he o ma ion o elonga ed s uc u es known as
ilamen s, lux ubes o p ominences ea u ing a dipole
elec ic pola iza ion in he pe pendicula plane. The esul ing
E×Bd i p opels he ilamen , g ea ly enhancing con ec-
i e pe pendicula anspo . Examples can be ound in
dispa a e con ex s, including as ophysical plasmas such
as acc e ion discs [2] and plane a y magne osphe es [3],
whe e he des abilizing o ce is ypically o he cen i ugal
ype, and labo a o y plasmas [4,5], whe e he o ce densi y
ypically comes om magne ic p essu e g adien s. This issue
is pa icula ly ele an o magne ically con ined usion
plasmas, as i de e mines he p opaga ion o ilamen a y
s uc u es, which ha e become ecognized as he dominan
adial anspo mechanism in he egion be ween he closed
magne ic ield lines and he wall, known as he sc ape-o
laye (SOL) [6–9]. The e o e, ilamen a y anspo
s ongly in luences he pa allel o pe pendicula a io o
he pa icle and hea luxes on o plasma acing componen s,
hus de e mining hei du abili y and he spu e ing o
impu i ies om he main wall.
Basic models o ilamen s in usion li e a u e (see, e.g.,
K asheninniko [5]) desc ibe how elonga ed s uc u es
p opaga e as he esul o equilib a ion o plasma pola i-
za ion caused by an e ec i e g a i y o ce: aking a educed
MHD app oach on a magne ized s a iona y plasma unde
some a bi a y o ce densi y F, he cha ge conse a ion
equa ion can be exp essed as
∇·d
d nmi
B2∇⊥ϕ¼1
Bb·∇×Fþ∇∥J∥;ð1Þ
whe e n,ϕ,B,J, and mis and o densi y, po en ial,
magne ic ield, cu en densi y, and ion mass, d=d ¼
∂=∂ þðb∇ϕÞ=B ·∇and b¼B=B. Pola iza ion [le -
hand side o Eq. (1)] is hus he esul o an equilib ium
be ween he d i e ( i s e m on he igh -hand side) and he
pa allel closu e e m (second e m on he igh -hand side)
accoun ing o he cu en pa allel o B. Figu e 1shows his
o a SOL con ex , whe e F¼2nmic2
s=Re s ands o he
e ec o cu a u e and ∇B(csis he sound speed and e
indica es he adial di ec ion). This can be easily gene al-
ized o many o he o cing mechanisms such as he
cen i ugal o ce in Keple ian sys ems, g a i y, e c.
[3,10]. Equa ion (1) yields di e en solu ions depending
on he pa allel closu e used o J∥: he usual hypo hesis in
Published by he Ame ican Physical Socie y unde he e ms o
he C ea i e Commons A ibu ion 3.0 License. Fu he dis i-
bu ion o his wo k mus main ain a ibu ion o he au ho (s) and
he published a icle’s i le, jou nal ci a ion, and DOI.
PRL 115, 215002 (2015) PHYSICAL REVIEW LETTERS week ending
20 NOVEMBER 2015
0031-9007=15=115(21)=215002(5) 215002-1 Published by he Ame ican Physical Socie y
he SOL is ha he ilamen ex ends along he ield line o
he a ge pla es, and J∥is hus limi ed by he shea h [7].
In his egime, known as “shea h limi ed”(SL), he
pola iza ion e m is neglec ed and ∇∥J∥¼ð2=L∥Þncseð1−
exp ½−eðϕ−ϕ Þ=TeÞ is canceled by he d i e e m. He e,
ϕ ,e,Te, and L∥a e he loa ing po en ial, elec on cha ge,
elec on empe a u e, and pa allel connec ion leng h. By
sol ing Eq. (1), he pe pendicula eloci y o he ilamen ,
b, is hen ound o scale as
b¼2cs
L∥
Rρs
δb2
∝
1
δ2
b
;ð2Þ
whe e ρsand Ra e he ion gy o adius and okamak majo
adius, and δbis he ilamen pe pendicula size. Howe e ,
se e al phenomena, such as high SOL collisionali y, la ge
X-poin magne ic shea , elec omagne ic e ec s, e c., may
in alida e he wall connec ion hypo hesis, e ec i ely discon-
nec ing he ilamen om he wall [7].In hiscase,∇∥J∥→0
and pola iza ion cancels he d i e e m, leading o a di e en
egime, known as he “ine ial egime”(IN), in which
b¼csffiffiffiffiffiffiffiffiffiffi
~
pe
δb
R
∝δ1=2
b;ð3Þ
whe e ~
peis he p essu e in he ilamen no malized o he
backg ound alue [11].These basic models ha e been e ined
by he inclusion o ealis ic SOL elemen s such as ho ions
[12]andla ge luc ua ionampli udes(“ ull- ”app oxima ion)
[13], leading o imp o ed e sions o Eqs. (2) and (3):
SL
b¼csð1þτiÞL∥
R
~
n
¯
nþ~
nρs
δb2
;
IN
b¼csffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1þτiÞ~
n
¯
nþ~
n
δb
R
;ð4Þ
whe e τi¼Ti=Te,and¯
nis he backg ound densi y. No e
ha he iso he mal limi has been assumed o app oxima e
~
p≃T~
n. These scaling laws should be conside ed as uppe
bounda ies, as any de ia ion om he pu e in e change
c oss phase be ween elec ic ield and p essu e luc ua ions
educes he eloci y o he blob wi h espec o p edic ions
[14,15]. Al hough hese models ha e been success ully
compa ed wi h expe imen in basic plasmas [16] and
ex ensi e cha ac e iza ion wo k has been made in okamaks
[17–19], a di ec measu emen o he ansi ion be ween he
wo egimes emains o be achie ed in a usion ele an
plasma. Ne e heless, such ansi ion has been in oked o
explain he o ma ion o he SOL densi y shoulde , i.e., a
subs an ial inc ease o he a SOL densi y adial e- olding
leng h, λn≃½∇ log n−1, obse ed in many okamaks when
a ce ain densi y is exceeded du ing low con inemen
(Lmode) ope a ion [20–22]. Recen expe imen s in
ASDEX Upg ade (AUG) [23] linked he inc ease in λn o
an inc ease o he ilamen size and associa ed pe pendicula
pa icle anspo in he ou e midplane Γ⊥; il, which can be
up o 40% o he o al anspo a e he ansi ion acco ding
o compa ison o EMC3-EIRENE simula ions [24].
This ansi ion akes place as he collisionali y inc eases in
he SOL a he onse o di e o de achmen , when s ong
ecycling begins o cool down he plasma in on o he
a ge pla es [25], sugges ing ha he a o emen ioned dis-
connec ion migh play a key ole. My a e al. [26] p edic ed
his p ocess in a wo egion model o he SOL using he
e ec i e collisionali y Λ,
Λ¼L∥=cs
1=νei
Ωi
Ωe
∝nT−3=2
eð5Þ
as he con ol pa ame e , whe e νei is he elec on-ion
collision a e and Ωi=e s ands o he gy o equency o ions
o elec ons. The disconnec ion akes place o Λ>1,
when he cha ac e is ic pa allel anspo ime is longe
han he in e se o ion-elec on collision equency.
Subsequen wo k indica es ha his esul s in enhanced
pe pendicula anspo as he consequence o inc eased
ilamen eloci ies and c ea ion a es [27]. Resul s om
Re . [23] showed ha he ansi ion in AUG coincides wi h
Λ≃1. The ela ion be ween ilamen egime ansi ion and
he shoulde o ma ion has also been shown in JET [28],
whe e independen measu emen s o collisionali y in
he di e o and midplane egions sugges ed ha only he
collisionali y in he di e o egion could accoun o he
disconnec ion p ocess.
The ques ion o whe he he collisionali y in he di e o
Λdi is de e mining he shoulde o ma ion is o g ea
p ac ical impo ance, as nex gene a ion okamaks a e
o eseen o ope a e wi h pa ially de ached, locally colli-
sional di e o s, while emaining ho and collisionless a
he midplane. I shoulde o ma ion is domina ed by a
ilamen ansi ion induced by di e o collisionali y, his
phenomenon will no be educed by a la ge machine size,
as i would acco ding o o he p oposed mechanisms
such as ioniza ion o neu als o igina ed om main wall
ecycling [29]. In such a case, a shoulde will p obably be
o med [28], which would impac subs an ially he main
FIG. 1 (colo online). Tokamak geome y. A SOL ilamen has
been ep esen ed along wi h he ield line connec ing i o he
a ge pla e (in ed).
PRL 115, 215002 (2015) PHYSICAL REVIEW LETTERS week ending
20 NOVEMBER 2015
215002-2
wall pa icle luxes and e osion. In o de o de e mine he
ole o Λdi in shoulde o ma ion, an expe imen was
ca ied ou in AUG: in L-mode discha ges, di e o de ach-
men depends s ongly on inpu powe PIN occu ing a
lowe midplane densi ies o lowe hea ing powe . He e,
Λdi is de ined as in Eq. (5), using nand TeLangmui p obe
measu emen s om he a ge pla es. Also, L∥has been
es ima ed as he connec ion leng h be ween he a ge and
he X-poin heigh a he lux su ace o he measu emen s
L∥≃1
5πRq95,whe eR≃1.65 m is he okamak majo
adius and q95 ≃5is he sa e y ac o [25]. This assump ion
is based on ecip oca ing p obe and spec oscopic mea-
su emen s o he cold egion ex ension in on o he
di e o a he ime o he ansi ion [23,30]. The e o e,
by ealizing a se ies o densi y amps wi h di e en
hea ing powe s (0, 300, and 600 kW o ECH powe , plus
500 kW o Ohmic hea ing in all cases), Λdi co e s a ange
o mo e han 2 o de s o magni ude, c ossing he Λdi ¼1
c i ical poin a di e en alues o edge densi y, ¯
ne.Thisis
shown in he inse o Fig 2, whe e he e olu ion o he
di e o collisionali y is displayed o h ee PIN cases.
Di e en exponen ial g ow h o Λdi can be obse ed,
depending on he PIN (highligh ed by colo ed dashed
lines). The onse o de achmen is highligh ed by colo ed
e ical lines. De ails on he diagnos ic se up and da a
analysis a e analogous o hose desc ibed in Re . [23].
Typical plasma pa ame e s a he SOL o AUG a e lis ed
in Table I.
Two di e en egimes appea app oxima ely a each side
o Λdi ¼1, as can be seen in Fig. 2. Fo low collisionali y,
he size o ilamen s emains a ound 1 cm, and inc eases
loga i mically wi h Λdi . Howe e , o dominan collision-
ali y (Λdi >1), he size o ilamen s inc eases by up o
an o de o magni ude. The da a se shows no dependence
on PIN; hence, he ansi ion is no de e mined by
a¯
ne h eshold, as can be deduced om he di e en
Λdi ðPIN;¯
neÞpa hs seen in Fig. 2.Fu he mo e, he
eloci y o small ilamen s ollows he SL scaling 1=δ2
b
and la ge ones he IN scaling ffiffiffiffiffi
δb
p. This is shown in Fig. 3,
whe e he measu ed a e age ilamen eloci y ⊥and size
δba e no malized wi h csand ρs, espec i ely, as in
Re . [12]. Also, he uppe bounds in he wo egimes
a e ep esen ed by plo ing he espec i e exp essions o
Eq. (4). In o de o do his, he local magne ic ield
(BSOL ¼1.4T), cu a u e adius (RSOL ¼2.15 m), ne
and Te alues a he p obe posi ion ha e been conside ed.
To es ima e ~
n, he iso he mal hypo hesis has been ex ended
o p obe measu emen s, and he ion sa u a ion luc ua ion
le el ~
ji;sa has been used as a p oxy. Compa ison o Fig. 3
wi h Fig. 2 e eals how bo h g oups o da a poin s
co espond o hose ea u ing Λdi <1and Λdi >1:
The ho izon al do ed line c ossing he elbow in Fig. 2,
is ep esen ed e ically in Fig. 3, whe e i sepa a es
he wo scalings. Again, he same beha io is seen o
di e en PIN alues, indica ing he scaling is independen
o his pa ame e (and hus also o ¯
ne). Fo bo h egimes,
cold and ho ions ha e been conside ed (τi¼3,as
p oposed in Re . [12]). Filamen s in he IN egime a e
be e ep esen ed by he cold model. This is consis en
wi h bo h heo e ical analysis [19] and a SOL measu e-
men s ca ied ou wi h ield analyze s [31], which sugges
FIG. 2 (colo online). Size o ilamen s s collisionali y
pa ame e , Λdi . Colo s indica e di e en hea ing powe s. Thin
solid lines p o ided as a guide o he eye o ma k he wo egimes.
In he inse , e olu ion o collisionali y wi h edge densi y.
TABLE I. Typical plasma pa ame e s a AUG SOL.
Bn
eTeTiρscsνei
1.5 T 1–10 1018 m−315 eV 50 eV 10−4m25 km=s107s−1
0 50 100 150 200 250
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
δb/ρs
Vpe p/cs
SL (τi = 0)
SL (τi = 3)
IN (τi = 0)
IN (τi = 3)
EM
FIG. 3 (colo online). Filamen scaling. Colo code as in
Fig. 2. Black do ed line indica es he egime sepa a ion, as
obse ed in Fig. 2. Red and black lines indica e IN and SL
egimes,asexp essedinEq.(4). Solid o dashed line indica es
τi¼0=τi¼3. Blue solid line indica es he elec omagne ic
egime desc ibed in Re . [12].
PRL 115, 215002 (2015) PHYSICAL REVIEW LETTERS week ending
20 NOVEMBER 2015
215002-3
ha ilamen s cool down a e he ansi ion. Da a poin s
in he lowe le co ne o Fig. 3, co esponding o he
smalles alues o δb, ollow he IN scaling a he han he
SL. This e ec has been p edic ed [32] o SL ilamen s
wi h δb≪δ≃15ρs,whichwould e e oanIN-like
scaling. Las ly, he elec omagne ic scaling p oposed in
Re . [12], in which elec omagne ic e ec s domina e
collisionali y in he ∇∥J∥ e m, ep esen s a wo se i ing
o he da a han he SL and IN ones.
Finally, global pe pendicula pa icle anspo is ound
o be linked o di e o collisionali y. In Fig. 4, he
densi y p o iles emain almos cons an un il Λdi ≃1is
su passed. Fo Λdi >1, he SOL wid h apidly inc eases
by a ac o o 3, indica ing a subs an ial inc ease in Γ⊥.As
be o e, all pa ame e scans in AUG ma ch on o one cu e
when Λdi is used as he o de ing pa ame e . He e, as an
indica ion o he SOL wid h, λnis de i ed om Li-beam
spec oscopy in he i s 25 mm ou side he sepa a ix, as
explained in Re . [23]. These esul s a e independen on
how de achmen is achie ed: I ni ogen is pu ed in he
di e o , local adia ion cools down he egion inducing
de achmen a a lowe ange o densi ies o a gi en PIN.
In he inse in Fig. 2, seeding would cause a shi o he
dashed line o he le compa ed o he unseeded dis-
cha ge a he same PIN. Howe e , he λn alues om hese
discha ges [emp y(solid) pu ple do s indica ing be o e
(a e ) he pu ing] scale exac ly as he unseeded ones,
despi e he di e en ¯
neand PIN ange (see Fig. 4).
Measu emen s in JET yield e y simila esul s: λnand
Λdi a e calcula ed o he discha ges p esen ed
in Re . [28], using equi alen li hium beam and a ge
p obes da a. The esul ing da a, displayed in Fig. 4as
ligh blue s a s, display a ema kable simila i y o he
ones measu ed a AUG.
Beyond magne ic usion, hese esul s can be ega ded as
he alida ion o a gene ic model o he p opaga ion o
s uc u es esul ing om in e change ins abili ies in mag-
ne ized plasmas. In his sense, no only SOL ilamen s and
ilamen a y s uc u es obse ed in he Jo ian and Sa u nian
magne osphe es by he Galileo and Cassini spacec a
[33,34] can be desc ibed wi h isomo phical cu a u e o
cen i ugal o ce-d i en in e change ins abili y models
[7,10], bu also hei e ec on densi y adial p o iles is
s ikingly simila o such di e en sys ems: The adial
densi y p o ile o he equa o ial iogenic plasma o us in he
Jo ian magne osphe e is domina ed by he con ec ion o
“isola ed in e changing lux ubes”[3]. In his case, he
pa allel closu e condi ion is de e mined by he ionosphe ic
conduc i i y a high plane a y la i ude ins ead o he
plasma-wall in e ac ion in he di e o egion. The adial
dis ance a e which connec ion be ween he equa o and
he ionosphe e disappea s is es ima ed a ound 20 Jo ian
adii [10]. Galileo measu emen s show how he adial
densi y g adien is s ongly dec eased when c ossing his
disconnec ion h eshold [35]. In pa icula , he cha ac e -
is ic g adien wid h ½∇ log n−1—λnin he SOL case—is
inc eased by a ac o o 3 be ween 20 and 40 Jo ian adii,
which is simila o he inc ease obse ed in Fig. 4 o
Λdi >1. Also, he azimu al wa e numbe o he in e -
change ins abili y—which would be he magne osphe ic
equi alen o 1=δb—is expec ed o dec ease by a ac o o 6
upon disconnec ion [36], which is again consis en wi h he
esul s obse ed in Fig. 2.
Summa izing, he ole o Λdi as a con ol pa ame e
o he ansi ion o bo h ilamen and global
pe pendicula anspo egimes has been demons a ed,
a e sepa a ing i s e ec om o he plasma pa ame e s
such as ¯
neo PIN. The eloci y scaling o ilamen s
changes om 1=δ2
b o ffiffiffiffiffi
δb
pwhen he c i ical alue Λdi ¼1
is su passed. These esul s ep esen s ong e idence o
ilamen p opaga ion being go e ned by he mechanism
p esen ed in Re . [26], in which he in e up ion o he
pa allel ci cui due o collisionali y swi ches ilamen s
om he SL o he IN egime. This change in SOL
u bulence is accompanied by a subs an ial inc ease o he
pe pendicula pa icle lux Γ⊥in he a SOL, measu ed in
wo di e en okamaks. The ema kable simila i y o
AUG and JET esul s suppo s he p edic ion o a shoulde
o ma ion in ITER. This has impo an implica ions o
main wall pa icle luxes and e osion in u u e usion
de ices. Finally, he simila i y be ween ilamen s d i en
by isomo phic ins abili ies in such dispa a e con ex s as
he SOL o usion de ices and plane a y magne osphe es
poin s owa ds a uni e sal ea u e o anspo in mag-
ne ized plasmas.
This wo k has been ca ied ou wi hin he amewo k o
he EURO usion Conso ium and has ecei ed unding
om he Eu a om esea ch and aining p og amme
2014–2018 unde G an Ag eemen No. 633053. The iews
10−2 10−1 100
10
15
20
25
30
35
40
45
50
55
Λdi
λn (mm)
AUG (ohmic)
AUG (300 kW)
AUG (600 kW)
AUG Seeding ( e )
AUG Seeding (gas)
JET
FIG. 4 (colo online). Densi y e- olding leng h scaling wi h
di e o collisionali y. Black, blue, and ed ci cles ep esen PIN
as in Fig. 2. Pu ple ci cles co espond o seeded discha ges
[emp y(solid) mean be o e (a e ) N2pu ing]. Ligh blue s a s
a e da a poin s om JET. S aigh lines ma k he wo egimes as
in Fig. 2.
PRL 115, 215002 (2015) PHYSICAL REVIEW LETTERS week ending
20 NOVEMBER 2015
215002-4
and opinions exp essed he ein do no necessa ily e lec
hose o he Eu opean Commission. The i s au ho
belongs o he “Leyendas U banas”g oup c ea ed by
CSIC in 2014.
*See he Appendix o F. Romanelli e al., P oceedings o he
25 h IAEA Con e ence 2014, S . Pe e sbu g, Russia.
†See h p://www.eu o‑ usionscipub.o g/ms 1
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PRL 115, 215002 (2015) PHYSICAL REVIEW LETTERS week ending
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