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 0.6

n 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 op

charac

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) = 0.33

both 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