Neoclassical Tearing Modes
O. Sauter1, H. Zohm2
1CRPP-EPFL,
Lausanne, Switzerland
2Max-Planck-Institut für Plasmaphysik, Garching, Germany
Physics of ITER
DPG Advanced Physics School
22 -26 Sept, 2014, Bad Honnef, Germany
With contributions in particular from:
D. Humphreys, R. La Haye, M. Maraschek, E. Poli, M. Reich
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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Outline
• Magnetic islands
• Classical tearing mode -> Rutherford equation
• Neoclassical tearing mode: Modified Rutherford equation (MRE)
• Metastable properties (large "hysteresis" between onset/offset)
• Plasma performance and NTMs
• Strategies for NTM control in ITER:
¾ Will there be more than one mode at the same time?
¾ Stabilization
¾ Preemption
¾ Avoidance
¾ Real-time control of NTMs:
¾ Very complex yet simple
¾ New "robust" control being tested in Europe
• Conclusions
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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Outline
• Magnetic islands
• Classical tearing mode -> Rutherford equation
• Neoclassical tearing mode: Modified Rutherford equation (MRE)
• Metastable properties (large "hysteresis" between onset/offset)
•
NTMS
lead
to
a
"soft"
beta
limit
as
opposed
to
"hard"
• Plasma performance and NTMs
beta limits
leading
• Strategies
for NTM
controltoindisruption
ITER:
¾• NTMs
Will there
be more
than
mode at thedegradation
same time?
mainly
lead
to one
performance
¾• BUT
Stabilization
if you push "too hard", NTMs can lead to
¾ Preemption
disruption
¾ Avoidance
¾ Real-time control of NTMs:
¾ Very complex yet simple
¾ New "robust" control being tested in Europe
• Conclusions
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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, E. Poli
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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The one fluid MHD equations
∂ρ
r
= −∇ ⋅ ( nv )
∂t
equation of continuity
r
r r
r⎞
⎛ ∂v r
ρ ⎜ + (v ⋅ ∇)v ⎟ = −∇p + j × B =0 force equation
⎝ ∂t
⎠
static equilibrium
r r r
r
E + v × B = η j =0 if η=0
=> frozen-in B lines
Ohm’s law
d⎛ p ⎞
⎜ γ ⎟=0
dt ⎝ ρ ⎠
equation of state
+ Maxwell‘s equations for E and B
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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MHD: consequences of Ohm's law
Consider equilibrium Ohm's law...
r
r r 1 r
E = −v × B + j
σ
...and analyse how magnetic field can change:
r
r
r
1
∂B
r r
= −∇ × E = ∇ × (v × B ) −
∇ × (∇ × B )
μ 0σ
∂t
r
r
∂B
1
r r
⇒
= ∇ × (v × B ) +
ΔB
μ 0σ
∂t
Typical time scale of resistive MHD:
τ R = μ 0σL2
Since σ is large for a hot plasma, τR is slow (~ sec for 0.5 m) – irrelevant?
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
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Reconnection in a hot fusion plasma
Due to high electrical conductivity, magnetic flux is frozen into plasma
⇒ magnetic field lines and plasma move together
A change of magnetic topology is only possible through reconnection
• opposing field lines reconnect and form new topological objects
• requires finite resistivity in the reconnection region
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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Reconnection on ‘rational’ magnetic surfaces
Typical q-profile
Corresponding Bpol
position of q=2
Corresponding Bhel
Bhel = Bpol(1-q/qres)
Helical field (i.e. ‚poloidal‘ field relative
to resonant surface) changes sign:
• reconnection of helical flux can form
new topological objects - islands
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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Reconnection on ‘rational’ magnetic surfaces
Torus has double periodicity
• instabilities with poloidal and toroidal 'quantum numbers'
m=1
m=2
m=3
‚Resonant surfaces‘ prone to instabilites with q = m/n
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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MHD description of tearing mode formation
Δψ +
μ 0 dj ( r ) dr
Bθ (1 − n
m
q ( r ))
ψ =0
j ~ q' => ψ ~ q''
q ' ( ρ ) ~ ∫ jϕ dSϕ + jϕ ( ρ ) ~ I p ( ρ ) + jϕ ( ρ )
ψ ~ w2
ρ
0
Deformation of flux surfaces opens up island of width W
• Tearing Mode equation (∇p = j x B) singular at resonant surface:
implies kink in magnetic flux ψ, jump in B
⇒ current sheet on the resonant surface
2πa
Bp
Bϕ
2πR*q
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
ψψ(r):
(r):helical
helicalmagnetic
magneticflux
flux
j(r):
j(r):current
currentprofile
profile
q(r):
q(r):field
fieldline
linehelicity
helicityprofile
profile
m,n:
m,n:mode
modequantum
quantumnumbers
numbers
NTMs , O. Sauter
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MHD description of "classical" tearing mode formation
"outer"
layer
ψ solution
"outer"
layer
"resistive"
layer
Solution of tearing mode equation can be made continuous, but has a kink
• implied surface current will grow or decay depending on equilibrium j(r)
• the parameter defining stability is Δ‘ = ((dψ/dr)right – (dψ/dr)left) / ψ
• if Δ‘ > 0, tearing mode is linearly unstable – this is related to ∇jresonant surface
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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Tearing Modes – nonlinear growth
Equation for the growth rate of a finite size island of width w:
τ res dW dt = a1 Δ ' + a 2 ∇ p W − a 3 I extern 2
W
• for small ∇p, current gradient (Δ') dominates
⇒ 'classical Tearing Mode', current driven
(most of the time stable except if q profile is "tweaked", which is why resistive
MHD was never a big thing up to end 1990s for tokamak interpretation)
• for larger ∇p, pressure gradient dominates:
⇒ 'Neoclassical Tearing Mode', pressure driven
• adding an externally driven helical current can stabilise
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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Magnetic islands deteriorate performances
degradation
Δτ E
τE
w ρ s3
=4 4
a
Chang et al, 1994, "belt-model"
Local transport stays same outside island
But "short-circuit" across island
Provides accurate simple measure of wsat
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
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Perturbed Bootstrap is driving term=>Neoclassical
B*+δBr
x
δjbs
Bθ = Bθ − B sθ
*
B*
.
Bs~j//
q' s
(ρ − ρ s )Bθ
≈−
qs
, E. Poli
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
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Modified Rutherford Equation (MRE)
Many terms can contribute to total // current within island
dw ρ s
= [ ρ s Δ' ( w) + ρ s Δ bs ' ( w) + ρ s Δ GGJ ' ( w) + ρ s Δ cd ' ( w) +
dt τ R
ρ s Δ pol ' ( w, ω ) + ρ s Δ ech ' ( w) + ρ s Δ wall ' ( w, ω ) + ρ s Δ mn ' ( w, ω ) + ...]
Main terms discussed and compared with experiment on 1st line
"classical" + bootstrap + curvature + polarisation + (EC)CD
Glasser-Greene-Johnson
wall and mode coupling can also be important
in particular, efficient locking of 2/1 mode
Despite limits of MRE, method/coeff. described in Sauter et al PoP 1997 and PPCF
2002 + Ramponi PoP 1999 for Δ'wall describes essentially all exp. Results+prediction
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
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Considering main contributions
MRE ×
1
:
ρ s Δ'
~1 (for wcd~w)
To obtain self-consistent solution
with confinement degradation
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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w structure of steady-state MRE (no CD)
β p (t )
~
f ( w ) = −1 +
w sat ∞ , 0
β p0
w
w 2 + w m2 arg
ITER expected 2/1 parameters
wmarg=3cm
wsat∞0=30cm
t=0 ->βp(t)/βp0 = 1
~
f (w )
w [cm]
βp(t)/βp0=2wmarg/wsat∞0 (=0.2 in this case)
β p , m arg = 2
NTMs are metastable: require seed island
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
w m arg
w sat ∞ , 0
β p ,0
NTMs , O. Sauter
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Typical effects of NTMs
D e c lin e in τ E a t (3 ,2 ) a n d
la rg e d e c lin e a t (2 ,1 )
50536
¾ 3/2 NTMs lead to 10-20%
losses
βn
¾ 2/1 NTMs lead to greater
losses and to disruptions
7
P N B I( 1 0 W )
2 /1 s ta b ilis e d
a t lo w b e ta
Sketch of time evolution
N B n = 2 re d u c e s
s a w te e th
n=1
n=2
P e rtu rb e d ra d ia l
f ie ld c a u s e s s o f t s to p
T im e (s )
Hender, chap 3, NF 2007
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
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Standard Scenario: Sawteeth and Mode Locking
• As discussed in ST session,
crashes after long sawtooth period
can trigger several modes
• Modes can lock rapidly (within
0.4s in this JET case)
JET #58884
2
n=1
n=2
1
0
JET #58884
2
β
N
1
0
SXR
5000
0
10
NBI
5
2/1
0
ICRH
5
4/3
3/2
0.4s
0
14
15
16
17
time [s]
18
19
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
Modes locks
(no disruption)
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NTMs in ITER – predictions for Q=10 scenario
Q=10 operation
point
Full stabilisation
with 7 MW
Full stabilisation
with 20 MW
ITER burn curves in the presence
of ECCD
at q=3/2 and q=2
(O. Sauter and H. Zohm, EPS 2005, also
Plasma Phys. Contr. Fusion 52 (2010))
HH=1.0
Impact on Q in case of continuous stabilisation (worst case):
• Q drops from 10 to 5 for a (2,1) NTM and from 10 to 7 for (3,2) NTM
• with 20 MW needed for stabilisation, Q recovers to 7, with 10 MW to Q > 8
• note: if NTMs occur only occasionally, impact of ECCD on Q is small
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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What are we controlling?
Predicted 2/1 in ITER
30
20
2/1 island width time evolution
3
2
10
1
0
0
0.65
20
40
60
time [s]
80
βp
0.6
Conf. degradation
0.55
0.5
0
20
40
60
time [s]
80
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
1. Island starts
• At large wseed from ST trigger
• At small wseed
• Without trigger from q profile
(Δ'>0)
2. Island grows and rotates or locks
100
• Growth rate depends on beta
• Beta drop depends on ρres, w
• Locking depends on β, q95, and
error field
3. Island saturates or plasma disrupts
• Depends on island size
• On proximity to beta limit
• On w versus a-ρres (on q95)
• On island overlap
100
NTMs , O. Sauter
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How are we controlling?
ITER 2/1 mode
Criteria for ITER:
Add CD on ρ2/1
wcd≤5cm
AND
ηNTMwcd≥5cm
(ηNTM=jcd/jbs≥1)
Compute P required
for various models:
wcd
wmarg
pol type
χ⊥ type
CW or 50%
Sauter, PPCF 2010
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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Why and when do we intervene?
2/1 mode in ITER starting/triggered at t=0
Lock level 2
ΔQ>10%
Lock level 1
Pre-emptive
ECCD stab. on ρres
Start of |ρdep-ρres| - wcd/2<w/2
• Lots of "conditions" and constraints to take into account
• Extra plots are to be added:
• Proximity to disruption, mode freq. vs time (fast diagn. used to predict locked mode)
• Confinement degradation/proximity to H-L transition, etc
• Using real-time control+simulation, these will also be known predictively
• Calculations within few ms and prediction for next 10-20s => opens new "control space"
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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Mode coupling: usually favourable
¾4/3 mode triggered by sawtooth crash is seen to stabilize 3/2 mode
observed in JET and TCV, similar to FIR-NTMs seen on AUG and with XTOR
Bi-coherence confirm
nonlinear coupling
Sauter, PPCF 2002; Raju PPCF 2003
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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Scalings for predictions for ITER
• Scalings of βp,onset hard to use as prediction since
• Couples seed island and bootstrap drive
• Needs similar trigger mechanisms
• Very hard to predict triggered island size
• Use ramp-down experiments to perform scalings of:
• wsat(βp,max) (allows test of bootstrap drive and Δ')
• βp,marg
(includes also effects of stabilising terms)
( stab. terms and provides info size of trigger)
• wmarg
• Then can test the physics against these scalings
• βp,marg scaling: ITER metastable to both 3/2 and 2/1 modes
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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Importance of trigger mechanism
Controlling sawteeth changes significantly βonset
2nd harmonic ICCD
1st harmonic minority ICRH
ΔτST↑
NTM onset
+90: phase: βN,onset≈1
-90: No NTM with βN up to 2
Sauter et al, PRL 2002
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
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βN,marg< L-H threshold
marginal β
q95=2.54 case does not disrupt even with 2/1 mode
Power ramp-down studies
marg
onset
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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NTMs Avoidance technique required in JET high performance:
control of 1st sawteeth
2/1 NTM triggered
leads to end of shot
(soft-stop)
Lack of sawtooth
control actuators is a
difficulty for these
scenarios (high
current and shape)
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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"Ultimate" control of NTMs: ECCD
Why ECCD in ITER? (4 ports dedicated to NTM stab.)
• ECCD can drive localised current
• Deposition location is well defined and depends
essentially on launching mirrors
• Mirrors can be oriented in real-time to control NTMs
• Aim:
• Replace missing bootstrap current by driving jCD in Opoint of island
• (Modify jtot to decrease further Δ') (in theory)
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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, E. Poli
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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ECCD Localized at Islands Can Replace Missing
Bootstrap Current and Stabilize NTM
•
ECCD deposition must be
accurately positioned at
n = 2 Mirnov
q=m/n rational surface where
NTM island forms
10
9
(G) 8
7
6
5
R (cm) of 2f e
c
• Alignment accuracy
required in DIII-D ~ 1 cm
136
134
132
130
128
126
2900
DIII-D Experiment
3000
3100
3200
Time (ms)
3300
3400
20
2
j (A/cm )
•
EC total current drive (for 2
15
MW injected) ~30 kA ~2%IP
10
jECCD
jBS
5
•
NTM control achieved at
ASDEX-U, JT-60U, DIII-D,
FT-U
0
0.5
0.6
0.7
ISLAND
ρ=r/a
w ~ 7 cm
from ECE radiometer
D. Humphreys, R. La Haye
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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0.8
NTM Control Requires Achieving and Sustaining Dynamic
Island/ECCD Alignment
Detect Mode Onset
Search&Suppress
Locate ECCD Deposition
Locate Island
OR
Align
Target Lock
No
Detect Island
Suppression
No
Yes
Maintain Alignment
Active Tracking
D. Humphreys, R. La Haye
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
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DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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Main problem: there is always some mismatch between ρec,dep and
ρm/n due to equilibrium reconstruction, launcher inaccuracies,
changes in density profiles, etc
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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New control strategy developed on TCV
•Add a systematic sweep around target position
•Limits systematic errors, use slow growth of tearing modes (resistive timescale)
2/1 stabilization with robust control
•Works for both preemption and stabilization, scales for ITER
•Allows more precise physics studies => better prediction for ITER
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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Simple to add to present feedback control
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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Simple to add to present feedback control
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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ITER scenarios and NTMs
• Standard scenario:
• Monotonic q profile, medium to high shear:
-> quite stable in shaped plasmas
• In H-mode, quite "meta"-stable
• Sawteeth stabilized by fast particles:
-> large seed islands can be triggered at ST crashes
• Should control sawteeth
• Hybrid and advanced scenarios
• Flat or reversed q profile:
-> quite unstable to classical tearing
(no need for seed islands)
• bN quite large -> more "meta-stable"
-> need NTM preemption and stabilization
• ITER plasma rotation is small -> quite prone to mode locking
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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Conclusions
• Basic physics of NTMs well understood
• Modified Rutherford Equation allows us to understand main physics
mechanisms
• Detailed "first principles" calculations should not rely on MRE but on 3D
MHD codes coupled to kinetic codes
• Nevertheless "fitted" MRE can be used for predictive calculations
• In burning plasmas, performance will decide best strategy for NTM
control. Note small modes can have large effect on neutron rate.
• Best strategy depends on scenario, mode onset and available actuators
• 2/1 mode is clearly main mode to avoid/control
• Detrimental effect of multiple modes not expected
• Apart from 2/1 locking, one has time to control NTMs
• Sawtooth control for standard scenario and preemptive ECCD for hybrid
and advanced scenarios seem best at present
• New robust NTM control strategy can do the job on ITER
DPG Advanced Physics School, The Physics of ITER, Bad-Honnef, Germany, Sept. 2014
NTMs , O. Sauter
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