Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Saturation of the Neoclassical Tearing Mode...

Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005

Saturation of the Neoclassical Tearing Mode Islands

F. Militello1, M. Ottaviani2, F. Porcelli1, J. Hastie1

1 Burning Plasma Research GroupPolitecnico di Torino

Italy

2 CEA CadaracheFrance


Outline

• NTMs and the generalizations of the Rutherford Equation.

• The asymmetric saturation, mathematical technique, nonlinear solution.

• Resistivity models, self-consistent solution, saturated width relations.

• The Symmetric Model. • The Code and our Results.• Theory and Numerics, do they

agree? • Summary and conclusions.

Theoretical model for Asymmetric saturation (simplified model)

Numerical analysis of the Symmetric NTM


The NTMs

• NTMs may decrease tokamak performance.

• It is important to have a reliable prediction of the size of the saturated NTMs islands.


The Generalized Rutherford Equation

• The nonlinear solution of the model for the island width, w, can be obtained by using an asymptotic matching procedure.

'22.1 dt

dw

After Rutherford (PoP ’73)

x

THE MODEL



• Bootstrap current term, drive for the nonlinear instability when ’<0

22*34.6'22.1

d

b

ww

wCv

dt

dw

Hegna & Callen (‘92)Fitzpatrick (‘95)



• Polarization current term, proportional to the magnetic island poloidal rotation frequency,

3

*

22*34.6'22.1

w

C

ww

wCv

dt

dw p

d

b

x

Smolyakov (’89)Waelbroeck et al (’01,’05)



• Term related to the shape of the equilibrium current density:

bw

w

C

ww

wCv

dt

dw p

d

b 41.034.6'22.13

*

22*

x

Militello & Porcelli (’04)Escande & Ottaviani (’04)

)(2

2

seq xx

dx

Jdb



• Term related to the shape of the equilibrium current density:

bw

w

C

ww

wCv

dt

dw p

d

b 41.034.6'22.13

*

22*

x

Militello & Porcelli (’04)Escande & Ottaviani (’04)

)(2

2

seq xx

dx

Jdb

Valid only for symmetric equilibria. Corrections are required in cylindrical geometry !!!!


The Limit of the Asymmetric Saturation

• Previous investigations on classical asymmetric saturation had two major flaws:

1) limiting model for resistivity,

2) no self-consistency (Ansatz required).

bw

w

C

ww

wCv p

d

b 41.034.6'03

*

22*

We can do better !!


Simplified Model Equations

• Vorticity equation:

• Ohm’s law:

• Energy equation

,, JUt

U

02//// TT

22 J

2U

2/3T JTE

t z )(,


Mathematical technique

• Following Rutherford, we employ an Asymptotic Matching procedure, justified by the smallness of the island width w compared to the macroscopic length, L: w<<L.

xx

M~~

)0(~1

)(

2

2

2

1 1cos'

)0(~log'

s

in rr

wJdd

ww

wrr s /)(

Mout() Min()


Inner Nonlinear Solution• The matching function depends on Jin:

• From the averaged Ohm’s law:

• The flux surface average is:

z

in

EJ

11

dd Metric term !

Resistivity Model !

A.Thyagaraja, Phys. Fluids 24, 1716 (1981)

2

2

2

1 1cos'

)0(~)(

s

inin rr

wJdd

wM


Resistivity Models

• Parallel heat transport is very efficient:

• then, the perpendicular transport acts on scale length of order:

• Below this threshold the perturbation of the temperature are smoothed by perpendicular transport.

• Where the perp. transport is negligible T=T()• Cf. R. Fitzpatrick, Phys. Plasmas 2, 825(1995)

10~ ~4/1

//

cw

02//// TT

//

2/3T


Resistivity Models II1) Small Island case:

2) Non-relaxed Large Island case:

3) Relaxed Large Island case:

Lww c

,Lwwc

,Lwwc

←Core

Edge→


Small Island Case

• The shape of the flux surfaces is defined by Ampere’s law, that can be solved by employing a perturbative technique:

Lww c

1

2

)(')(

12

)(

wrJrJr

EJ

seqseqeq

zin

x


Large Island Cases

• Now, the Ampere’s law is:

• where T() is given by the constrain condition:

Lwwc

2/32 )(2)(

TEE

J zzin

)(')(2)( seqs rTrwdd

dT

Metric term again !


Small Island Relation

• SI Saturation Relation:

sr

ab

Aa

swaw 44.0

22

68.085.4

1log41.0'0 2

srJ

rJa

seq

seq 21

)(

)('

srJ

rJb

seq

seq 21

)(

)(''

)(

)('

s

ss rq

rqrs

Hastie, Militello, Porcelli PRL (2005)


Small Island Relation

• SI Saturation Relation:

• Thyagaraja:

• Pletzer et al.:

sr

ab

Aa

swaw 44.0

22

68.085.4

1log41.0'0 2

-285.4 10~ ew

w

wa1

log41.0' 2

2

1log41.0' 2 Aa

waw


Non-Relaxed Large Island Relation

• NRLI Saturation Relation:

• No log(w) and A contributions!

• The thermal boundary layer around the separatrix (where ≠ ) brings them back (but multiplied by wc).

sr

abaw 09.027.08.0'0 2

Hastie, Militello, Porcelli PRL (2005)


Relaxed Large Island Relation• RLI Saturation Relation:

• Similar to the NRLI relation• log(w) contribution from the outer solution, not from the

inner solution as in the SI case! • The thermal boundary layer around the separatrix (where

≠ ) would introduce additional log(w) and A terms.

sr

ab

waw 09.027.011.1

1log15.1'0 2


The Complete Model

• The full model contains many additional effects. • The Generalized Rutherford Equation is

obtained by using strong physical assumptions.• The effect of rotation is not completely clarified.• With numerical investigations it is possible to

check the assumptions and shed some light on the relevant physics.


The 4-Field Model

UJdt

dU 2,

)(, * x

nCJJ

yvn

dt

dbeq

nDvJy

vdt

dn 22** ,,

vy

vndt

dv 2*,

x

2U

2J

-The model evolves the 4 fields:

Stream functionMagnetic fluxn: Perturbed densityv: Parallel ion velocity

-2D, slab geometry-Symmetric equilibrium-Constant


Symmetric NTM

• Complete model (Small Island Case):

• And symmetric case:

s

p

d

b

r

ab

Aa

swaw

w

C

ww

wCv44.0

22

68.085.4

1log41.034.6'0 2

3

*

22*

bw

w

C

ww

wCv p

d

b 41.034.6'03

*

22*


Simplifying the Model…

• Averaging Ohm’s law:

• J is “almost” a function of

UJdt

dU 2, ~,J ),()( yxPFJ

x

nCyxPyxPJJ beq

),(),(




• The density equation gives:

nDvJy

vdt

dn 22** ,,

yvnJ /,~ *-1

x

nCyxPyxPJJ beq

),(),(

From Ohm’s Law




• The density equation gives:

nDyvnt

n 2*

2* ,/,

Transport Equation

x

nCyxPyxPJJ beq

),(),(


Numerical Matching

• All the terms can be substituted in Min and evaluated numerically.

• The islands rotates at and the Polarization term is small.

),(),( yxPyxPx

nCJJ beq

3

*

22*34.641.0'0

w

C

ww

wCvbw p

d

b

1/ *


Boundary Conditions

• The bootstrap current term must be corrected.

x

Magnetic field – Contour plot

Numerical solution: -spectral code, -double periodicity,


034.641.0'22

*

ds

sbs ww

wCvbw

Cb=1

Cb=2

Cb=1.4


Conclusions• A systematic investigation of

the saturation of the NTMs has been carried out with theoretical and numerical tools.

• New terms describing the asymmetric saturation have been added to the Generalized Rutherford Equation.

• Theoretical models have been compared to the numerical data obtained with solving the complete symmetric model.

Three new saturation relations describing different physical scenarios

Good agreement but the position of the tangent bifurcation is not well predicted

Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Saturation of the Neoclassical Tearing Mode...

Documents

Transcript of Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005 Saturation of the Neoclassical Tearing Mode...