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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
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
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
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
The NTMs
• NTMs may decrease tokamak performance.
• It is important to have a reliable prediction of the size of the saturated NTMs islands.
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
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
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
The Generalized Rutherford Equation
• 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)
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
The Generalized Rutherford Equation
• 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)
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
The Generalized Rutherford Equation
• 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
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
The Generalized Rutherford Equation
• 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 !!!!
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
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 !!
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
Simplified Model Equations
• Vorticity equation:
• Ohm’s law:
• Energy equation
,, JUt
U
02//// TT
22 J
2U
2/3T JTE
t z )(,
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
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()
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
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
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
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
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
Resistivity Models II1) Small Island case:
2) Non-relaxed Large Island case:
3) Relaxed Large Island case:
Lww c
,Lwwc
,Lwwc
←Core
Edge→
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
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
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
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 !
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
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)
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
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
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
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)
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
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
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
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.
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
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
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
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*
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
Simplifying the Model…
• Averaging Ohm’s law:
• J is “almost” a function of
UJdt
dU 2, ~,J ),()( yxPFJ
x
nCyxPyxPJJ beq
),(),(
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
Simplifying the Model…
• Averaging Ohm’s law:
• The density equation gives:
nDvJy
vdt
dn 22** ,,
yvnJ /,~ *-1
x
nCyxPyxPJJ beq
),(),(
From Ohm’s Law
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
Simplifying the Model…
• Averaging Ohm’s law:
• The density equation gives:
nDyvnt
n 2*
2* ,/,
Transport Equation
x
nCyxPyxPJJ beq
),(),(
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
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/ *
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
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/ *
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
Boundary Conditions
• The bootstrap current term must be corrected.
x
Magnetic field – Contour plot
Numerical solution: -spectral code, -double periodicity,
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
034.641.0'22
*
ds
sbs ww
wCvbw
Cb=1
Cb=2
Cb=1.4
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
034.641.0'22
*
ds
sbs ww
wCvbw
Cb=1
Cb=2
Cb=1.4
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
034.641.0'22
*
ds
sbs ww
wCvbw
Cb=1
Cb=2
Cb=1.4
Fulvio Militello – EFTC 2005 Aix-en-Provence 28/9/2005
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