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DENSITY LIMITS IN TOROIDAL PLASMAS
MARTIN GREENWALD
MIT - PLASMA SCIENCE & FUSION CENTER
Presented at43rd Annual Meeting of the APS Division of Plasma Physics
Long Beach, CAOctober 29, 2001
ACKNOWLEDGEMENTS
• All people who have worked in this area over the years
• Note particularly people who contributed figures and ideas for this talk
• Not enough time to show all deserving work
• Mistakes in interpretation are my own
R. Bartiromo
C. Boswell
K. Borrass
L. Giannone
Y. Kamada
S. Kaye
B. LaBombard
B. Lipschultz
A. Mahdavi
V. Mertens
J. Ongena
T. Osborne
W. Suttrop
J. Terry
F. Wagner
S. You
S. Zweben
T. Petrie
J. Rapp
L. Marrelli
B. Rogers
G. Saibene
A. Staebler
OPERATIONAL LIMITS
• Magnetic confinement devices
don't operate at arbitrary plasma
parameters
• There are well established,
distinct limits on plasma
pressure, current, and density
• Understanding these limits and
their implications has always
been an active area of research
DENSITY LIMITS - AN IMPORTANT ISSUE FOR MAGNETIC FUSION
• 2DTR n vσ∝
• Plasma pressure limited by MHD
stability
• At fixed pressure, there is an
optimum temperature ➙ optimum
density
• No guarantee that this density
is achievable in any given device
• Critical issue for conventional tokamak reactor
DENSITY LIMITS - THE PHYSICS PROBLEM
• What physics can limit the density?
−−−− Ideal MHD only cares about pressure (and current) not density
✸ Temperature profile influences current profile
✸ (Resistive MHD could be a factor at low temperatures)
−−−− Radiation cooling ( )2RAD e Z eP n f R T∝
−−−− Neutral shielding: fueling limits
−−−− Density or collisionality dependent transport ➙ edge cooling
• No widely accepted first principles theory available
• Not even agreement on critical physics
OUTLINE OF TALK
• Experimental observations including
−−−− Tokamak
−−−− Stellarators
−−−− Reversed Field Pinches (RFP)
−−−− Spheromaks and FRCs
• Physics basis for density limit
−−−− Neutrals
−−−− Radiation models
−−−− Role of transport physics
• Summary and Discussion
IN TOKAMAKS, LIMIT ULTIMATELY MANIFESTS ITSELF AS DISRUPTION
• General agreement on final scenario
• Current profile shrinkage ➙ MHD
instability ➙ disruption
• Critical questions involve the
evolution to the point where the
current profile collapses
• What is the essential physics of the
bifurcation or catastrophe
• "Hard" terminations also seen at
times in reversed field pinches
"SOFT LIMITS" SEEN IN OTHER DEVICES
• In Stellarator, clear evidence of thermal collapse -
plasma can recover if density is lowered
• No coupling from Te profile through resistivity and
current profile to MHD stability
• Physics is not so clearly confined to edge
• RFPs have quenches as well as fast terminations
• Spheromak and FRC don't have density limit data
operation at "optimized" density.
• Doesn't preclude (or require) a common cause
0
24
48067 B=2.5T48078 B=1.25T
Wdi
a (k
J)
b)
20
1019 m
-3 )
WPLASMA
1.25T 2.5T
Te
ne
(Giannone 2000)
0
ne (
1.5
Te
(keV
)
0
0
1000
0 1 2
Pra
d (k
W)
Time (s)
PRAD
DENSITY LIMIT FIRST CHARAC D BY EMPIRICAL SCALING
• First motivated by
observation that impure
plasmas disrupted at
lower densities
• Murakami limit (1976)
0/T OhmicB R j P∝ ≈
• Hugill plot ~ 1978
• Leading dependence is
with plasma current density
• 2P
LIMB I
nqR a
∝ ≈ (Note absence o
TERIZE
f significant power scaling)
(Axon 1980)
SCALING REFINED BY INCLUSION OF DATA FROM SHAPED TOKAMAKS
• Greenwald limit: 2
PG
In
aπ=
(with n: 1020/m3, IP: MA, a: m)
• Identical to Hugill for circular
plasmas
• Differs significantly for shaped
plasmas
RECENT DATA WITH VERY DIFFERENT PLASMA SHAPE IS ROUGHLY FIT BY
EMPIRICAL LA
• Original data had wide
range in BT, IP, PIN
• Aspect ratio only 3-5
• Elongation only up to
1.5
• Spherical tokamaks
aspect ratio ~ 1.5
elongation ~ 2
W
(Kaye 2001)
IMPURITIES ARE IMPORTANT …. BUT ONLY UP TO A POINT
• Below around ZEFF ~ 2.5, drops out ( EZ
1 2 3 4 5 6Zeff
0.0
2.0
4.0
6.0Li
ne A
vera
ged
Den
sity
[1019
m−
3 ]P = 1400kW
600kW, 990kW, 1
ald limit
P =
Greenw
2i i
FFe
n Z
n≡ ∑ )
7 8 9 10
400kW
Critical densities ofradiative collapse
(Rapp et al, 2000)
DENSITY LIMIT IN TOKAMAKS DOES NOT DEPEND STRONGLY ON INPUT
POWER
x x
xx
x x
+
+
+
+
+
12
10
DIVERTORIp = 0.5–1.9 MABT = 0.8–2.1 Ta ∼ 0.63 mb/a ∼ 1.8Pbeam ≤ 7 MW
Ip(MA)0.51.01.51.91.01.0
BT(T)2.02.02.02.01.71.0
8
6
4
2
00 2 4
Pinput (MW)
n e,d
et/I p
(1019
m–3
/MA)
6 8 10 12
• Power dependence in low
confinement mode (L-mode)
varies from P0 - P0.25
• Role of neutral beam fueling
in power dependence is
unc
(Petrie 1993)
ertain
AT HIGH DENSITIES, HIGH CONFINEMENT (H-MODE) DISCHARGES
DEGRADE THEN REVE -MODE
0
2
4
6
8
0 0.2
P sep
10
L → H
Greenwald Limit• H-mode plasmas have
edge "transport barrier"
➙ Pedestal in Te, ne profile
• Confinement
degradation can set in as
low as 0.3nG
• Threshold power
diverges as the limit is
approached
• H/L transit
RT TO L
0.4 0.6 0.8 1
H-mode
L-mode[MW]
n [10 m ]e20 -3
Marfe Limit threshold
1.2
(Mertens 1997)ion at 0.8-1.0nG
STRONG SHAPING DOES ALLOW FOR BETTER CONFINEMENT IN H-MODE
AS THE DENSITY IS RAISED TOWARD THE LIMIT.
• Increase in confinement at high triangularity at o improved pedestal
stability
ELMy H-mode dW/dt~0
n (r/a~0.7) /nGW
H-f
acto
r
Ip=0. -4.5MA,Bt=1-4.4T, Rp=3 3.5m,
0
1
2
3
4
0 0 2 0.4 0.6 0.8 1
δ < 0.2δ > 0.2
(Kamada 1996) (Ongena 2001)
.2
.0
.8
.6
.40 0.8 1.0 1.2
H98(y,2)
n / nGW
JG01
.230
-35c
Ope e III ELms
δ = 0.46δ = 0.38
ELMy H-Mode
ITER
_
1.0
0.5
e
4-
.
.4 0.6n symbols: Typ
= 0.23 δ= 0.14 δ
tributed t
DETERIORATION IN H-MODE CONF T IS CORRELATED WITH DROP
IN EDGE T ATUR
• CORE EDGEH and T T∇ ∝
• Constant edge pressure implies τE
independent of density
T
/n
c
PE
De
G
0.00
0.04
0.08
0.12
• D
c
b(Hughes 1997)
E
p =
n /n
p ∝ η-0.53
PEDe G
0 0.4 0.6 0.8 1.0
e
on
y
(Osborne 2000)
0.2
INEMEN
EMPER
terioration in edge
finement can be offset
internal transport barrier
SO…THE TRICK FOR EXCEEDING THE EMPIRICAL LIMIT - PEAKED
DENSITY PROFILES
• All indications are that limit is due to edge
• Particles in core apparently don’t drive density limit
0.0 0.2 0.4 0.6 0.8 1.0 1.2Normalized ρ
0
5
10
15
20
25
ne
(x10
19 m
–3)
98893.450098889.4000 x 3.2 (no gas)
98893.2000 x 1.3 • Density profiles not stiff
• Peaked by core fueling,
edge pumping, transport
modification
(Mahdavi 1997)
GOOD CONFINEMENT WITH DENSITY IN S OF nG IS CORRELATED TO
PEAKED DENSIT ILE
• Widely seen (Alcator C, TFTR, DIII-D,
JET, ASDEX, ASDEX-Upgrade,
TEXTOR…)
• Also seen in stellarators (Heliotron E, LHD
• Edge density apparently never exceeds
empirical limit
• Combination of density peaking and stron
plasma shaping open window for high den
Line Average Density
EXCES
Y PROF
2
×1020
)
g
sity operation
1
01.5
05
Input Power (MW)
0
1400 1600 1800 2000Time (ms)
2200 2400 2600 2800
Greenwald “Limit”
Radiative Power (MW)
τ EITER
93–H
τ E
w, p Not Subtractedcore
rad
•
w, p Subtractedcore
rad
• H93 = 1
(Mahdavi 1997)
DENSITY LIMITS IN REVERSED FIELD PINCH
• erating space
terized historically
•
•
•
2 102 0
with fast terminationwithout fast termination
(Bartiromo 2000)
RFP opcharac
by I/N
GnI
N n∝
Same scaling !
Limit is quantitatively
identical (!!)
0 100
5 101 9
1 102 0
1.5 102 0
0 100 5 105 1 106 1.5 106 2 106
ne [
m-3
]
Greenwald limit
Ip/ ( a2) [Am- 2]π
RADIATED POWER PROBABLY NOT CRUCIAL FOR RFP LIMIT
• both soft (quench-like)
•
•
0.5
(Marrelli 1998)
RFP has
and hard (disruption-like) density
limits
Radiated power increases at
high density (low I/N), but
Radiated fraction is never
very high
0
0.1
0.2
0.3
0.4
1 2 3 4 5 6 7
1/I dI/dt less than -2 s-1
1/I dI/dt between -2 s-1 and 2 s -1
γ =
P r
ad /
P o
hm
I/N (10 -14 A m)
STELLA S REACH SIMILAR DENSITIES BUT SHOW DIFFERENT
DEPENDENCES
• Different
• Shaping:
• Scaling w
• For mach
stellarato
RATOR
scaling with power, size
B/qR vs I/a2 scaling
ith ι = 1/q
ines with similar size and fields,
r will reach about twice the density
of a tokamak
Ptot [MW]
2.01.51.0.5
5
10
15
20
n ∝ P0.5 W7–AS
n ∝ P0.25 ASDEX
ne,
max
[10
19 m
-3]
t = 2.1 T, 1/qa = 0.34
Power dependenceof density limit
(Staebler 1992)
(Wagner 1997)
00.00
ASDEX: B
W7-AS: Bt = 1.28 T, ι (a) = 0.33both with boronized walls
SCALING FOR STELLARATOR DE IMIT
• Variation in results
• Consensus: ( )0.5/CRITn BP V∝ (Sudo
1990, Giannone 2000)
• But note evidence for stronger B and
weaker size scaling
• Preliminary results from LHD (Large
Helical device) support scaling
• Results generally consistent with
radiation/power balance models
0.4
1
n (1
02
0 m-3
)
4
NSITY L
0.4 1
npred (1020 m-3)4
(Giannone 2000)
"DENSITY LIMIT" IN SPHEROMAK AND FIELD REVERSED
CONFIGURATION (FRC)
• Spheromak and FRC don't
have density limit data
• "Optimized" discharges
obtained by scanning fill
pressure
• Turns out to be quite close to
empirical scaling.
• Significant?
(Jarboe 1985)
PHYSICS MODELS FOR THE DENSITY LIMIT
• NEUTRALS - FUELING AND POWER BALANCE
• RADIATION MODELS - POWER BALANCE
• ROLE OF TRANSPORT PHYSICS
GLOBAL SCALING BY ITSELF IS AN INSUFFICIENT FOUNDATION FOR
PREDICTING THE PERFORMANCE OF FUTURE MACHINES
• Scaling does an OK job, may need small corrections for aspect ratio, power, etc,
but
• Covariance in data, may hide dependences (IP and PIN for example)
• Misses important local physics - density profiles
• Need verified, first principles model
Big questions
• Where does the catastrophe come from?
• How do we compute the density limit?
ROLE OF NEUTRALS DENSITY LIMIT
• Self shielding - limits gas
fueling
• Energy loss via ionization and
charge exchange
• Sets edge gradient length
cause unstable pressure
profile
• Despite this - nG is not
describing a fueling limit -
obviated by core fuelingRelatively small
large reduction i
closed flux surfa
C es ! Open field lines
"IN THE
losed lin
increase in density leads to
n ionization inside last
ce
(LaBombard)
RADIATION POWER BALANCE - EDGE OR SCRAPE-OFF LAYER (SOL)
h Motivation
−−−− Very dirty plasmas don't reach high density
−−−− ( )2RAD e Z eP n f R T∝ - edge cooling
h Choose physical phenomenon to model
−−−− Global thermal collapse
−−−− Radiation condensation
−−−− Poloidal detachment
−−−− Divertor detachment
−−−− Radiation dominated transport ➙ MHD unstable
pressure profiles
h Solve coupled equations for energy, momentum, particle balance
(+ Ad hoc assumption to relate "edge" density to core density)
RADIATIVE CONDENSATION - MARFE THRESHOLD
• MARFE = MARmar wolFE
• At low temperatures ( )
0dR T
dT<
• With insufficient conducted
power, radiative collapse occurs
• At constant pressure T↓ n↑
further increases ( )2e en R T
• In some machines, MARFEs
appear just before the density limi
• So… compute density limit by calc
• However - MARFEs observed fr
t
ulating MARFE threshold
om 0.4 -1.0 nG
(Boswell)
RADIAL STABILITY - POLOIDAL DETACHMENT
• Assumes limit associated with PRAD = PIN (Seen in some machines)
• Plasma is no longer coupled thermally to wall
• Compute radial stability from perturbation analysis for radiating layer at r = ap
• Stability criteria 3
2P
P
a dn
n da− > (Assumes density profile fixed)
• Can get result for scaling law assuming ohmic heating and 2E naτ ∝
• Transport assumption probably not correct
• With other assumptions - don’t get result much like experiments
• (Not universal - PRAD/PIN = 0.3 - 1.0 at limit)
DIVERTOR DETACHMENT - SCRAPEOFF LAYER MODEL
• As density ↑ , Te↓ allows Te
gradient along open field lines
"divertor marfe"
• At sufficiently low temperatures,
neutral collisions dominate
momentum transport
• Leads to drop in plasma
pressure at divertor plate
• Radiation zone moves up to x-
point (x-point marfe)
• Theory uses detachment threshold as crit
• In experiment, detachment occurs from
0
203040500
1
2
3
4
1021
eV
m-3
0
10
eV
ne =1.5x1020ne =1.8x1020ne
L Medium Highde density density
=1.0x1020
-
ow nsity
eria for density limit
0.4-1.0 nG
(LaBombard 1994)
105 15
100 5 15 100 5 15Distance beyond separatrix (mm)
Sh
eath
-L
imit
ed
det
ach
ed - Upstream - Divertor
e Pressure:
- Upstream - Divertor
e- Temperature:
SCRAPE-OFF LAYER/DETACHMENT THEORY
• Analytic theory - divertor two point model - forced in r law form
• Finds critical separatrix density ( )
5/16
11/16
x
SEPq B
nqR
⊥−∝
• Requires assumption - Bohm
transport
• Reasonable agreement with
JET, ASDEX-Upgrade data
• Numerical simulations find limit
diverges for ZEFF ➙ 1 0
0.5
1.0
1.5
2.0
3
n exp/
n calc
to powe
( )10
16 1xwhere x
ββ
−=+
5 7 9
nexp [1019m-3]
(Borrass 1997)
IS THERE MORE PHYSICS INVOLVED?
h Problem with radiation models
−−−− Power and impurity dependence too strong ➙ ( )/ 1LIM IN EFFn P Z∝ −
−−−− Threshold mechanisms show up well below density limit
−−−− Transport assumptions: theories are incomplete at best
h Evidence for increased transport as cause of edge cooling
−−−− Transient transport experiments (Greenwald 1988, Marinak 1993)
−−−− Fluctuation measurements (Brower 1991)
−−−− Detailed probe measurements in edge (LaBombard 2001)
• General observations of edge turbulence at high densities
−−−− Universal result?
TURBULENT TRANSPORT IN EDGE INCREASES WITH COLLISIONALITY
h Two regimes observed in scrape-off
layer (SOL)
−−−− Near-SOL: steep gradients
−−−− Far-SOL: flat profiles
h Particle flux and transport
−−−− Near-SOL: cross-field transport low
−−−− Far-SOL: cross-field transport high
h Fluctuation changes character
−−−− Near-SOL: low amplitude, short
correlation times and lengths
−−−− Far-SOL: large amplitude, bursty, long
:899
Te
204060
(eV
)
0
correlation times
Ionization Source-5 0 5 10 15 20
-5 0 5 10 15 20
-5 0 5 10 15 20
-5 0 5 10 15 200.01
0.1
1.0
Density
ρ (mm)
1019
1020
(m-3
)LimiterShadow
Sh
ots
: 10
0060
2017
:899
, 100
0602
021:
597,
100
0517
023
Deff = -Γ⊥ /∇ n
1022
1023
(m-3
s-1)
(m2 s
-1)
1021
1020
(m-2
s-1) Particle Flux
1.11.6 x1020
m-3ne:2.6
(LaBombard 2000)
WE CAN VISUALIZE THE FAR-SOL FLUCTUATIONS
• Images taken with fast CCD
camera
• 4 µsec framing time
• D2 gas puff: image Hα
• Large "blobs" dominate far SOL
• Seen to move poloidally and
radially
• Correlation length, correlation
time, propagation velocity
consistent with probe measurements(Zweben, Terry 2001)
1 cm
TURBULENCE DRIVEN CONVECTION CAN COMPETE WITH PARALLEL
TRANSPORT
• In far SOL, cross-field transport
overwhelms parallel transport
• As density is increased, region of large
fluctuations and transport move inward
toward separatrix
• Parallel transport ~Te7/2 is stable with
respect to temperature perturbations
• Collisionality driven cross-field
transport is unstable
0
10
(eV
)0
0.01
0.1
1.0
(MW
)
00.01
0.1
1.0(M
W)
50
(a)
(b)
(c)
0.1
1.0
(102
0 m
-3) ensity
(d)
D
5 10 15 20
5 10 15 20
5 10 15 20
⊥ Convection
Sh
ots
: 10
0060
2017
:899
, 100
0517
023:
899
LimiterShadow
Electron Temperature
ρ (mm)
nnG
:0.190.43
// Conduction
// Conduction
⊥ Convection
nnG
= 0.19
nnG
= 0.43
(LaBombard 2000)
AS THE DENSITY LIMIT IS APPROACHED, HIGH TRANSPORT REGIME
CROSSES SEPARATRIX AND MOVES INT PLASMA
• Has the potential to explain range
of density limit phenomena
• Once perpendicular transport
dominates, stabilizing influence is
lost
• Threshold condition not known
• Requires that complex transport
physics have the "correct" form
• Where does IP (or BP) dependence
come from?
O MAIN
(LaBombard 2001)
NEED IMPROVED MODELS FOR EDGE TURBULENCE
• Unfortunately, theory and models for edge turbu not understood well
enough yet
• 3D gyro-fluid simulations have found regime of e high transport
• 2 /Rq d drα β= −
• 0 0/D s s nc t L Lα ρ=
2
n n
T
nL L
λ ∝ →
• Region of high transport
consistent with high density, low
temperature
• No quantitative predictions yet
lence are
xtremely
(Rogers, Drake 1998)
EXPERIMENTAL SUPPORT FOR TURBULENCE MODEL
α d
α
0 1 20
1
2
3 ASDEX Upgrade1996 data, divertor IT only
ee
enhancedtransport reduced t
3 4
Type I ELMsType III ELMs
L-ModeMARFE
H-mode:
ransport
(Suttrop 1999)
DISCUSSION - MECHANISMS
• Various models proposed - progress has been made but none are entirely
satisfactory
• Physics strongly coupled - cause and effects hard to untangle
• May need combination of turbulent transport and atomic mechanisms
• Lead to investigation of very different physics
• Need to use self-consistent profiles, transport, power balance etc. for all models
SUMMARY
• Substantial progress has been made in understanding this interesting and
important problem
• It is remarkable that simple empirical laws can capture such complex
physics
• The similarity of the limit across a wide range of confinement devices is
remarkable as well
• By peaking the density profile, it is possible to obviate what is essentially
and edge limit.
• Still, it remains a significant challenge to understand the underlying
physics
Recommended