MHD in Tokamak Plasmas
Guido Huysmans
Association Euratom/CEA Cadarache, France
with contributions from:
T. Hender
(UKAEA, Culham, UK)
Y. Liu
(Göteborg, Sweden)
H. Lutjens
(Ecole Polytechnique, Paris, France)
S. Sharapov
(UKAEA, UK)
J. Lonnroth
(JET, UK)
S. Saarelma
(UKAEA, UK)
M. Becoulet
(CEA Cadarache, France)
P. Maget
(CEA Cadarache, France)
G.Huysmans
workshop : Principles of MHD
21-24/3/2005
ITER relevant MHD
• Edge Localised Modes (ELMs)
– Can cause large heat loads on plasma facing components
– Need to be controlled
• Neo-classical tearing modes (NTMs)
– Can cause pressure limit well below ideal MHD stability limits
• Resistive wall modes (RWMs)
– The relevant modes (external kink modes) for broad current profiles typical
for advanced scenarios
• Disruptions
– Need to be avoided
• Fast particle modes, TAE modes
– Efficiency of alpha particle heating
• Sawteeth
– Source of seed islands
• …
G.Huysmans
workshop : Principles of MHD
21-24/3/2005
Outline
• Ideal (linear) MHD
– Waves, MHD Spectroscopy
•TAE modes
•Resistive wall modes
– MHD Stability limits
•Global (Internal transport barriers)
•Local (ELMs)
... I wish to deal with a model which:
• respects the main physical conservation
laws
• has a decent mathematical structure
• permits analysis in complicated
geometries
Ideal MHD is the only model so far that
satisfactorily combines these features
J.P. Goedbloed (1983)
• Resistive, extended, non-linear, etc. MHD
– Neoclassical tearing modes
– MHD in steady state plasmas
G.Huysmans
workshop : Principles of MHD
21-24/3/2005
Ideal MHD
• The ideal MHD model is very successful in describing and
predicting important aspects of MHD phenomena in
tokamaks:
– Global MHD stability limits
• Disruptions in advanced tokamak discharges
– Local MHD stability limits
• Edge Localised Modes (ELMs)
– Frequencies (spectrum) of stable global modes (waves)
• TAE modes
Modelling tools : linear MHD codes (CASTOR, MISHKA)
G.Huysmans
workshop : Principles of MHD
21-24/3/2005
MHD Spectroscopy
… through continued improvement of both numerical calculations and experimental
observations we may witness the birth of a, new kind of spectroscopy, properly called
MHD spectroscopy, in the coming decade.
J.P.Goedbloed (1993)
Diagnostics of the q(r)- profile in toroidally rotating plasmas
Neutral Beam Injection on JET drives a significant toroidal plasma rotation
Frequencies of waves with mode number n in laboratory reference frame,
fnlab , and in the plasma, fn0 , are related through the Doppler shift : n frot(r)
fnlab = fn0 + n frot(r)
For TAEs frequency does not depend on n, so that
fn0  Va / (2 q R)
On the other hand, TAE with mode number n is located at
qTAE(r)=(m+1/2)/n
Correspondence between frot(r) and qTAE(r) can be inferred from TAE and
the toroidal rotation profile measurements (charge-exchange)
S. Sharapov
G.Huysmans
workshop : Principles of MHD
21-24/3/2005
MHD Spectroscopy
• Magnetic field perturbation measurements at vessel wall (JET #40369)
– High frequency TAE modes (300-400 kHz)
– TAE antenna signal
– Low frequency MHD instabilities (0 – 80 kHz)
G.Huysmans
workshop : Principles of MHD
21-24/3/2005
MHD Spectroscopy
TAEs observed with
magnetic pick-up coils
Resulting q-profile:
TAE’s
EFIT
Rotation profile
Radius [m]
G.Huysmans
workshop : Principles of MHD
21-24/3/2005
Resistive wall modes (RWMs)
•
•
•
The broad current profile of advanced scenarios leads to a relatively low MHD
stability limit due to global kink mode
External kink stabilised by ideally conducting wall, unstable when wall is resistive
Excitation by external perturbation, plasma response (resonant field
amplification) depends on closeness to stability limit : MHD spectroscopy
MARS code
Y. Liu, Göteborg
G.Huysmans
workshop : Principles of MHD
21-24/3/2005
Resistive Wall Modes
• Excitation of JET resistive wall mode:
– Resonant field amplification (RFA), comparison experiment with
MARS simulations
T. Hender, Y. Liu, IAEA, 2004
G.Huysmans
workshop : Principles of MHD
21-24/3/2005
Disruption Limit
• Peaked pressure profile due to transport barrier
in advanced scenarios causes disruption at a
low MHD stability limit
• Good agreement with predicted ideal MHD
stability limit
• Good agreement with calculated ideal MHD and
observed mode structures (JET)
3.8
2
1.5
N
1
exp. trace
0.5
disruption
0
5.4
Electron temperature contours

n=1 Stability limit
5.6
5.8
6
6.2
6.4
6.6
6.8
time [s]
1.0
SXR
SXR
0.5
R [m]
Z [m]
0.0
Experimental
-0.5
3.1
G.Huysmans
0.
0.2
time [ms]
0.4
0.6
workshop : Principles of MHD
2.6
2.8
3.0 3.2
R [m]
3.4
3.6
21-24/3/2005
Edge Localised Modes (ELMs)
• Periodic relaxations of the large pressure
gradient at the edge of the plasma (H-mode):
pressure
profile
pedestal
energy
pedestal temperature
centre
edge
High speed video
image from MAST
pedestal density
divertor Da
A. Kirk, PPCF(2005)
G.Huysmans
workshop : Principles of MHD
21-24/3/2005
Ideal MHD Stability Limits
• Ideal MHD stability limits to the pressure gradient due to medium n (~10)
ballooning modes agree well with observed maximum pressure gradient.
Edge Electron temperature
shear
JET #55937
density
0.9
radius
1.0
current
density
G.Huysmans
workshop : Principles of MHD
a
S. Saarelma,
PPCF,2005
21-24/3/2005
Medium n ballooning mode
• Mode structure of ideal MHD ballooning mode (n=10) :
perpendicular velocity
poloidal harmonics
0.7
radius
perpendicular velocity
1.0
S. Saarelma,
PPCF,2005
G.Huysmans
workshop : Principles of MHD
21-24/3/2005
Peeling modes and the separatrix
• Peeling modes are localised external kink modes driven by the edge
current density.
• The stability of peeling modes depends sensitively on q at the edge.
• What happens to peeling modes in the presence of an x-point where q
goes to infinity?
• Limit towards separatrix, ideal MHD:
1.2
0.3
0.8
J1
0.4
Z
0.2
0
0.1
-0.4
-0.8
-1.2
-0.75
0
0.97
0
0.75
R
G.Huysmans
Finite current
gradient
workshop : Principles of MHD
Finite edge
current
0.98
0.99
b
Flux at boundary
1
MISHKA
21-24/3/2005
Influence of separatrix
• Resistive MHD peeling modes are also strongly stabilised by the
approach to the separatrix
• An additional instability (so-called peeling-tearing mode) remains
unstable in the presence of an X-point.
0.05
0.05
growth rate
=0.99
0.04
0.04
0.03
0.03
0.02
0.02
0.01
0.01
=0.998
0
0
1
1.2
1.4
q(0)
1.6
1.8
3
4
peelingq(1)
mode
CASTOR
G.Huysmans
workshop : Principles of MHD
5
peeling-tearing
mode
21-24/3/2005
X-point Geometry
• Non-linear MHD code (JOREK)
– Includes separatrix geometry, open and
closed field lines
– Flux surface aligned finite element grid
– Reduced MHD in toroidal geometry
– Fully implicit time evolution
– under development
G.Huysmans
workshop : Principles of MHD
21-24/3/2005
Peeling modes in X-point Geometry
• Peeling mode stability in X-point geometry agrees with CASTOR
results in the limit to the separatrix:
• Ideal and resistive peeling modes are completely stabilised by the
separatrix
• Resistive peeling-tearing mode remains unstable
0.006
=0.99
Growth
rate(n=1)
0.004
0.002
=0.998
X-point
0
1.4
G.Huysmans
1.6
q(0)
1.8
2
workshop : Principles of MHD
Current perturbation
21-24/3/2005
Non-linear evolution of peeling modes
• Peeling-tearing mode saturates non-linearly
– Line tied boundary conditions
– strong deformation density profile, small perturbation of flux surfaces
-5
10
-6
energy
10
kinetic
-7
10
-8
10
-9
10
magnetic
-10
10
-11
10
JOREK
-12
10
0
5,000
10,000
time
G.Huysmans
workshop : Principles of MHD
21-24/3/2005
Neo-classical Tearing modes
•NTMs are non-linear instabilities :
An island created by another instability
(sawtooth) flattens locally the pressure
profile:
reduction of the bootstrap current inside
the island
further growth of the island
loss of confinement and limit to pressure
Electron temperature profile
Example NTM JET
Heating power
Plasma Energy
MHD
density
SXR
G.Huysmans
Radius [m]
workshop : Principles of MHD
temperature
21-24/3/2005
Neo-classical Tearing Modes (NTMs)
•Tore Supra: MHD mode triggered after monster sawtooth
•Tearing mode with mode numbers m/n=3/2
•Slow decay on resistive time scale ~1sec.
G.Huysmans
workshop : Principles of MHD
21-24/3/2005
Neo-Classical Tearing Modes Simulations
• XTOR code :
– resistive MHD (including transport and bootstrap current) in toroidal
geometry.
– H. Lutjens and J.F. Luciani (Ecole Polytechnique Paris)
Evolution island size
G.Huysmans
workshop : Principles of MHD
21-24/3/2005
NTM Theory versus Simulations
Comparison 0D theory with full numerical simulations (XTOR):
DR
'GGJ  6.35
2
2
 r dw
w

0.65w
c
 '(w)  'GGJ (w)  'boot (w) ( non MHD)
1.22 dt
Rq
' boot  6.35 o J boot,o
2
2
Bo ss
w  1.8w c 
• Reasonable agreement for
small island widths (thresholds)

G.Huysmans
workshop : Principles of MHD
1/ 4
  
wc  2 2 
 // 
w
rsR
; s
nss

• Disagreement on island
saturation size
21-24/3/2005
MHD in Plasmas with Current Drive
• Tore Supra, steady state discharges
LH Power (MW)
Transformer flux (Wb)
Te(0) (keV)
• Discharge duration : > 6 min
• Injected energy
: > 1 GJ
Plasma stable until small
(harmless) MHD instability
sets in at t=258s
Line density (x1019m-2)
Hard-X
75 keV
(a.u.)
Ti(0) (keV)
Neutron (x1010/s)
q
Zeff
G.Huysmans
workshop : Principles of MHD
21-24/3/2005
Oscillations in Tore Supra
• The interaction between the deposition of the driven current, the
temperature, q-profile and the (improved) confinement can lead to an
oscillatory regime:
Quiet
giant oscillations
G.Huysmans
oscillations
MHD
workshop : Principles of MHD
double tearing mode
21-24/3/2005
Tore Supra : MHD regime
• Tore Supra steady state scenario sensitive to MHD instabilities
– reversed q-profile with qmin~2
– Linear tearing mode and later double tearing almost always unstable.
– relevant stability criterion : full reconnection of the double tearing mode
TS - 31503
8
Ip
Te(0)
dB/dt
magnetic axis
MHD regime
6
MA, T/s
1.0
TS 31503
0.2
keV T (keV)
4
0.0
0.5
2
-0.2
0.0
0
0
5
10
15
20
25
t=10 sec., f=2.7 kHz
t=13.6 sec., f=0.8 kHz
2.4
2.5
2.6
2.7
2.8
R (m)
time (s)
P. Maget
G.Huysmans
workshop : Principles of MHD
21-24/3/2005
Current Holes
• A current hole is the absence of toroidal current in the central part of the
plasma.
perturbed current profile
• Current holes are formed with a
strong off-axis current drive.
– Off-axis current tends to drive central
current density to negative values
– Central current density fixed close to
zero by an n=0/m=1 internal kink
instability (2D non-linear MHD
simulations)
• In JET, sawtooth like crashes are
observed in plasmas with a current
hole.
– Crashes not due to n=0/m=1 mode
– n=1 postcursors observed
– 3D simulations of JET current hole
plasmas
n=0/m=1 internal kink mode
JOREK
G.Huysmans
workshop : Principles of MHD
21-24/3/2005
Current Holes
• The reversed q-profile of current hole plasmas can
be unstable to double tearing modes
–in this example, a complete reconnection occurs in
between the outer q=2 surface and the radius of the
current hole
–the current hole survives the crash
–density profile shows a fast collapse
kinetic energy
10
3
-4
total
10
q
-5
2
10
-6
10
-7
n=1
0
G.Huysmans
4,000
time
8,000
12,000
1
0.2
0.4
0.6
0.8
radius
workshop : Principles of MHD
21-24/3/2005
Conclusions
• The relevance of (and interest in) MHD in tokamak fusion
plasmas has grown significantly over the past years.
• Linear Ideal MHD is surprisingly accurate for a number of applications in
tokamak plasmas
• MHD Spectroscopy is now a valuable diagnostic for the q-profile
• Non-linear and extended MHD is becoming more relevant/necesary for
comparison with experiment
• Interaction between MHD and current deposition profile is important
combined MHD transport simulations
G.Huysmans
workshop : Principles of MHD
21-24/3/2005
Discussion
• Challenges (to Physics and Numerics):
– Simulation of complete ELM cycle
• Different ELM types : I, II, III etc.
– Trigger of Neo-classical tearing modes
• FIR NTM regime (NTM interaction)
• Island rotation
–
–
–
–
–
G.Huysmans
Convergence MHD models – Turbulent Transport models
Kinetic effects in reconnection
Sawteeth models
Fast particle physics
Integrated tokamak modelling
workshop : Principles of MHD
21-24/3/2005