DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Size and amplitude scaling of ELM-wall interaction on JET and ITER W.Fundamenski and O.E.Garcia

DSOL ITPA meeting, Toronto W. Fundamenski 8/11/2006

TF-E

Size and amplitude scaling of ELM-wall interaction on JET and ITER

W.Fundamenski and O.E.Garcia

Euratom/UKAEA Fusion Association, Culham Science Centre, Abingdon, OX14 3DB, UK

This work was funded jointly by the UK Engineering and Physical Sciences Research Council and by the European Communities under the contract of Association between EURATOM and UKAEA. The view and opinions expressed herein do not necessarily reflect those of the European Commission.


TF-EOpen question: Size & amplitude scaling

A.Loarte et al., Phys. Plasma, 11 (2004) 2668T. Eich et al., J. Nucl. Mater., 337-339 (2005) 669

Why do bigger, more intense ELMs deposit a larger fraction of their energy on the wall ?

or, in the context of a filament model, why do larger ELM filaments travel faster ?


TF-E

Origin of plasma motion in non-uniform B-field

Momentum conservation equation

Denoting LHS by F and taking a curl yields a general plasma vorticity equation,

where is the magnetic curvature, eg. in MHD, the two terms on the RHS correspond to flux tube bending (kink) and interchange (ballooning). Term by term, this equation can be recast as a charge conservation or current continuity equation,

which expresses the balance of polarisation, parallel and diamagnetic currents

R.D.Hazeltine and J.D.Meiss, Phys. Rep., 121 (1985) 1, or Plasma Confinement (1992) Addison-Wesley, New York


TF-E

Note that the divergence of the diamagnetic current,

is equal to the divergence of the current due to guiding centre, magnetic drift

Inertia + curvature = interchange motion

In toroidal geometry, grad-B points towards the major axis, which leads to a vertical polarisation of charge and an outward radial ExB drift, as shown for a plasma filament (blob), see left.

This is the origin of the interchange instability and ballooning transport in tokamaks, i.e. turbulent motions in regions of unfavourable magnetic curvature (pointing along the pressure gradient).


TF-ETwo-field interchange model

Invoking the thin layer approximation, one obtains the reduced vorticity equation

where x, y, z are the radial, poloidal and parallel co-ordinates, is the kinematic

viscosity and is the electric drift vorticity.

We complement this with an advection-diffusion equation for a generic

thermodynamic variable ,

where is its collisional diffusivity. We normalise by a characteristic cross-field blob size l, the ideal interchange growth rate 1/and the characteristic variation

N.Bian et al, Phys. Plasmas., Phys. Plasmas 10 (2003) 671


TF-EDimensional values for typical ELM filaments

O.E.Garcia, N.H. Bian, W.Fundamenski., submitted to Phys. Plasmas

Let us set the model parameters at typical large tokamak values

And choose the cross-field filament size and amplitude as

This yields the following values of gyro-radius, sound speed and interchange rate,

such that the ideal velocity, , is equal to 25 km/s.


TF-ENon-dimensional model equations

N.Bian et al, Phys. Plasmas., Phys. Plasmas 10 (2003) 671; O.E.Garcia et al, Phys. Plasmas, 12 (2005) 090701

This gives the non-dimensional model equations

Where andare the non-dimensional diffusivity and viscosity. Their product and ration define the Rayleigh and Prandtl numbers,

Ra is the ratio of boyancy and collisional dissipation, while Pr is the ratio of viscosity and diffusion.


TF-EIdeal (non-dissipative, collisionless) limit

For larger Rayleigh numbers, the viscous term is negligible

Which gives the only dimensionally allowable scaling of the transverse velocity,

Hence, in the ideal (collisionless) limit, the radial Mach number increases as the square root of the cross-field filament size and the relative thermodynamic amplitude, i.e. the perturbation compared to some background value.

In other words, provided dissipation forces are small, we expect larger and more intense perturbations to travel faster (aside from the obvious scaling with the sound speed), in broad agreement with ELM measurements on JET, see below.

O.E.Garcia et al, Phys. Plasmas, 12 (2005) 090701


TF-EInitial conditions and moments

Consider a gaussian filament, initially at rest,

Define the centre-of-mass position, velocity and effective diffusivity as

where the dispersion tensor is given by

O.E.Garcia, N.H. Bian and W.Fundamenski., submitted to Phys. Plasmas


TF-ENumerical simulations of filament motion

Radial distanceRadial distance

density, pressure vorticity

O.E.Garcia, N.H. Bian and W.Fundamenski., subm. to Phys. Plasmas


TF-EPosition, velocity and diffusivites vs. time

O.E.Garcia, N.H. Bian and W.Fundamenski., submi. to Phys. Plasmas

Filament velocity, in units of , increases from 0 to < 1, then

decays gradually with time.


TF-ERange of Rayleigh and Prandtl numbers


TF-EMaximum radial velocity vs. Ra and Pr

Maximum velocity, in units of , is only weakly dependent on

collisional dissipative effects.


TF-ESheath dissipation: earlier theories

S.I.Krasheninnikov, Phys.Lett. A, 283 (2001) 368; D.A.D’Ippolito et al, Phys. Plasmas 9 (2002) 222

In earlier theories, the interchange term was assumed to be non-linear in and the effect of parallel currents was included via a so-called sheath-dissipative term,

This form allows an analytical solution in the ideal (collisionless) limit

which gives the following transverse velocity. Note the strong, inverse size scaling, and no amplitude dependence,


TF-ESheath dissipation: improved model


In line with the earlier derivations, we also introduce the sheath-dissipative term, but not the strange, non-linearity,

Spectral decomposition (Fourier transform) of this equation

reveals the major difference between viscous and sheath dissipation. The former damps small spatial scales (large k), while the latter damps large spatial scales (small k), which should affect the morphology of plasma filaments.


TF-EEffect of sheath dissipation (Ra = 104, Pr = 1)


TF-E


Maximum radial velocity vs. Ra and (Pr =1)

Maximum velocity, in units of , decreases substantially due to

sheath dissipative effects

Earlier, ideal sheath-dissipative solution


TF-EExpression for ELM energy to wall on JET

Interchange driven amplitude scaling with convective ion losses

combined with moderate-ELM (W/W = 5%, W/Wped=12%) e-folding length, yields

so that fraction of ELM energy to wall can be approximated as

where ped is the pedestal width and SOL is the separatrix-wall gap.

eg. when W/W reduced by a third, then (Wwall/W0) = 10 % for 3 cm gap, see below.

W.Fundamenski et al, PSI 2006; subm. to J.Nucl.Mater


TF-EComparison with JET data presented earlier

Model prediction plotted for a range of SOL = 1 – 5 cm, assuming ped = 3 cm

Comparison with JET data reveals fair agreement, considering scatter in data and approximate nature of the model

y = 1.1828e-4.1699x

R2 = 0.3388

0

0.2

0.4

0.6

0.8

1

1.2

0.05 0.1 0.15 0.2

WELM / Wped

Wd

iv /

WE

LM

5 cm4 cm3 cm2 cm1 cmJET data (2.5 - 4 cm)Expon. (JET data (2.5 - 4 cm))


TF-EELM-wall & limiter interaction on ITER

W.Fundamenski et al., Plasma Phys. Control..Fusion, 48 (2006) 109

Same prescription as used to match JET data (Type-I ELMs, W/W = 5 %)

~ 8 % of ELM energy onto main wall at 5 cm (omp)

~ 1.5 % of ELM energy onto limiter at 15 cm (omp)

ITER 2nd separatrix movable limiter

Nor

mal

ised

ELM

fila

men

t qua

ntiti

es

Normalised time since start of parallel losses

R.Aymar et al., PPCF 44 (2002) 519


TF-EPrediction for ITER: ELM amplitude scaling

Approximate amplitude scaling (in fact, the e-folding length increases with distance)

0.001

0.01

0.1

1

0 0.05 0.1 0.15 0.2

WELM / Wped

Ww

all

/ WE

LM

2.5 cm 5 cm 10 cm15 cm 20 cm

Nominal Type-I ELM size on ITER

ELMs size required from material limits

8 % @ 5 cm

0.6 % @ 5 cm


TF-EConclusions

• JET data indicates that bigger (more intense) ELMs deposit a larger fraction of their energy on the main chamber wall, which suggests that the radial Mach number increases with ELM size

• Two-field interchange model used to study size & amplitude scaling

• It was found that over a wide range of conditions, the radial Mach number is expected to increase as the square root of both ELM size and amplitude,

• This implies that radial e-folding length of ELM filament energy also increases

• Model predictions in fair agreement with JET data

• Preliminary predictions for ITER indicate the added benefit of reducing the ELM size: for small ELMs, W/Wped < 3%, less than 1% of ELM energy deposited on the wall (near 2nd separatrix at upper baffle); contact with main wall is negligible.


TF-E


Radial distance from mid-pedestal location

Pea

k io

n im

pact

ene

rgy

Ion impact energies on JET and ITER

Predicted peak ELM filament quantities on JET and ITER (moderate Type-I ELMs)

• JET: Ti,max(rlim) ~ 185 eV (ion impact energy ~ 0.6 keV) at 4 cm

• ITER: Ti,max(rlim) ~ 350 eV (ion impact energy > 1 keV) at 5 cm; ~ 100 eV at 15 cm

• Lower bound estimates for moderate (W/W ~ 5 %) Type-I ELMs

JET ITER

nmax (m3) 8.251018 1.21019

Ti,max (eV) 185 350

Te,max (eV) 74 140

n,max (mm) 47 54.5

Ti,max (mm) 41 42.5

Te,max (mm) 25 27.5


TF-E

The change in kinetic energy is related to the compression of the polarisation current,

Compression of diamagnetic current is related to the magnetic curvature,

Hence, plasma motions are amplified when energy flows opposite to the magnetic curvature vector, i.e. in region of bad curvature.

Energy equation for interchange motions


TF-E

,

,

,

232

21

252

21

,

,

aaaaaiie

eie

aaaaaaaaaaaa

aaaaaanaa

nTumQQ

TnTumQSt

nnSt

n

χΓqq

DuΓΓ

)(1 2 tG

dt

dGDu

t nnn

),(),(),(

exp1

),,;,(

0

2

trSttrrGrdtdtrn

t

tD

tur

tDDutrG

n

rt

nnn

nununn

n urt

drrtG

,

,

,expexp),(

Advective-diffusive description of ELM filament

• Conservation equations for mass & energy

• Green’s function = advective-diffusive wave-packet (filament)

• Radial velocity and diffusivity prescribed



TF-E

Sτ

εε

t

Sτ

εε

t

Snt

eie

eie

e

iie

eii

i

nn

1

1

,1

,

,

11,

111,

1,

||

2||

,||

5

3

2

5 ,~

aTnana

aa

sn

L

Mc

L

Parallel loss model of ELM filament evolution


Key elements of parallel loss model

• Temporal evolution of n, Te and Ti in the filament frame of reference

• Above quantities represent averages over the filament volume

• Time and radius related by filament radial velocity, which is not calculated by the model

• Parallel loss treated by convective and conductive removal times

• Acoustic loss of plasma

• Electrons cooled faster than ions


TF-EFilament evolution for nominal JET conditions

Nor

mal

ised

ELM

fila

men

t qua

ntiti

es

Ion-

to-e

lect

ron

tem

pera

ture

rat

io


Normalised time since start of parallel losses

n’

Ti’

Te’

• As expected Te decays faster than Ti, which decays faster than n

• As the initial density is increased, e-i equipartition becomes more effective and the two temperatures converge more quickly


TF-EGood agreement with all dedicated JET data

Experiment Parallel loss model

Limiter probes + TC:

neELM(rlim) ~ 2.41018 m-3

TeELM(rlim) ~ 25-30 eV

n,maxELM

~ 50 mm, Te,maxELM

~ 30 mm

Nearly all power found on the divertor


TeELM(rlim) ~ 30 eV

n,maxELM

~ 47 mm, Te,maxELM

~ 32 mm

Fraction of ELM energy to wall ~ 5 %

Outer gap scan + IR & TC:

WELM

~ 33-35 mm, W,maxELM

~ 22-24 mm WELM

~ 36 mm, W,maxELM

~ 22 mm

RFA measurements of ion energies:

Jsat, Icoll, Te (reproduced by model)

Ti,maxELM(rlim) ~ 100 eV,

Te,maxELM(rlim) ~ 40 eV,


n,maxELM

~ 48 mm, Ti,maxELM

~ 52 mm

Te,maxELM

~ 30 mm, WELM

~ 32 mm

Fraction of ELM energy to wall ~ 15 %

ELM energy deficit based on IR:

~ 30 % for ~ 3 cm gap and W/W ~ 5 % ~ 28 % for 3 cm based on WELM

~ 35 mm

Documents

DSOL ITPA meeting, Toronto W. Fundamenski8/11/2006 TF-E Size and amplitude scaling of ELM-wall interaction on JET and ITER W.Fundamenski and O.E.Garcia