Transcript ITERR&D - University of California, Irvine

Computer simulations of fast frequency
sweeping mode in JT-60U and fishbone
instability
Y. Todo (NIFS)
Y. Shiozaki (Graduate Univ. Advanced Studies)
K. Shinohara, M. Takechi, M. Ishikawa (JAERI)
October 6-8, 2003
8th IAEA Technical Meeting on Energetic Particles in Magnetic
Confinement Systems
General Atomics, San Diego
Fast Frequency Sweeping Mode observed
in the JT-60U plasma with NNB injection
K. Shinohara et al., Nucl.
Fusion 41, 603 (2001).
Frequency sweeping takes
place both upward and
downward by 10-20kHz in
1-5 ms.
Particle-MHD hybrid simulation
[Y. Todo and T. Sato, Phys. Plasmas 5, 1321 (1998)]
1. Plasma is divided into “energetic ions”
“MHD fluid”.
+
2. Electromagnetic fields are given by MHD
equations.
3. Energetic ions are described by the driftkinetic equation.
MHD Equations: current coupling model

   (v)
t

1
1
enh
v  v  v   p 
(  B  jh )  B 
E //
t

0 

 [(  v)      v]
B
   E
t
E  v  B    B
p
   ( pv)  ( 1) p  v
t
( 1)v  [(  v)      v]

( 1) B  [    B]
0
Equilibrium
 1   2
I M
2 p
R
 2  R
 Rjh, 
R R R z

(2 ) 2 
I  2RB
I
M
K
2
1 K
j h,R  
R z
1 K
j h,z 
R R
 An extended Grad-Shafranov equation for anisotropic
energetic ion pressure is found (equivalent to Eq (9) in
 [Belova et al. Phys. Plasmas 10, 3240 (2003)]). We solve
a stream function K in the zero orbit width limit.
Numerical method
 MHD: finite difference with 4th order accuracy in space
and time.
Number of grid points: (101, 16-48, 101) for (R, , z) coordinates
5
Time steps: (2-7)10
 The df particle simulation is applied to energetic ions.
5
6
Number of particles N: 510 -410
number n=1
 Retaining fluctuations with toroidal mode
5
to reduce numerical noise (N=510 ). Numerical
convergence is checked with a simulation where
toroidal
6
mode number n=0-6 is retained and N= 410 .
Condition (E36379, t=4.0s)
1. R0 = 3.4 m, a = 1.0 m
2. B = 1.2 T
3. DD plasma
4. ne at the plasma center 1.6 1019 m-3
5. Alfvén velocity at the plasma center 4.6 106 m/s
6. Injection energy 346 keV (5.75 106 m/s)
7. Only parallel velocity is considered.
8. Maximum velocity of the slowing down distribution
is assumed to be 80% of the injection velocity.
9. Finite viscosity and resistivity (10-5 or 210-5 R0vA)
are assumed for nonlinear simulation.
Investigation for different energetic ion
pressure profiles
2.0 10 -2
I
h
1.5 10 -2
1.0 10 -2
II
III
5.0 10 -3
IV
0.0 10 0
0
0.2
0.4
0.6
r/a
0.8
1
Red curve is a classical
distribution based on the OFMC
code calculation.
The energetic ion beta value at
the plasma center is 1.9% for the
red curve.
For the other curves central
beta value is reduced.
All the profiles are similar at
r/a>0.6.
0.6
6
0.5
5
0.4
4
Toroidal mode
number n=1
Alfven continuum
0.3
q(0)=1.6
EPM
II III
I
3
q
 A
q-profile and shear Alfvén continuous
spectra
q=2.5 at r/a=0.8
2
0.2
q
0.1
1
0
0
0
0.2
0.4
0.6
0.8
1
r/a
Mode spatial profile depends on energetic ion pressure profile.
They do not peak at the TAE gap at q=2.5.
------> Energetic Particle mode (EPM)
2.0
2.0
1.5
1.5
z/a
z/a
Mode spatial profile (toroidal electric field)
depends on energetic ion pressure profile
1.0
1.0
0.5
0.5
0.0
0.0
2.5
3.0
3.5
R/a
4.0
Profile I: h(0)=1.9%
2.5
3.0
3.5
R/a
4.0
Profile III: h(0)=0.8%
A new kind of EPM
 A
6
0.5
5
0.4
4
Alfven continuum
0.3
EPM
II III
I
3
2
0.2
q
0.1
1
0
0
0
0.2
0.4
0.6
0.8
1
r/a
2. Nonlocal EPM is studied for reversed shear plasma and
trapped energetic ions [H.L. Berk (2001), F. Zonca
(2002)].
3. The EPMs found in this work is similar to the nonlocal
EPM but in monotonic shear plasma and with passing
energetic ions.
4. Theoretical exploration is needed!
q
1. The EPMs are not on the shear
Alfvén continuum. They are
different from the local
(resonant) EPM [L. Chen (1994),
C.Z. Cheng et al. (1995)].
0.6
0.6
6
0.5
5
0.4
4
Alfven continuum
3
0.3
EPM
2
0.2
q
0.1
1
0
0
0
0.2
0.4
0.6
r/a
0.8
1
q
 A
Investigation on different q profiles
with energetic ion pressure profile I
 The safety factor q is
uniformly reduced by
0.2; q=qexp-0.2.
 A gap at q=1.5
(r/a~0.3) appears.
 A local (resonant)
EPM is destabilized.
 Frequency is on the
Alfvén continuum.
2.0
2.0
1.5
1.5
z/a
z/a
Nonlocal EPM and local (resonant) EPM
1.0
1.0
0.5
0.5
0.0
0.0
2.5
3.0
3.5
R/a
4.0
q profile from experiment
Nonlocal EPM
2.5
3.0
3.5
R/a
4.0
Test q profile (q=qexp-0.2)
Local (resonant) EPM
Spiral structure
Nonlocal EPM and local (resonant) EPM (II)
2.5 10 -5
2.0 10 -4
2.0 10 -5
1.5 10 -4
m=2
1.0 10 -4
1.0 10
5.0 10
-5
[a.u.]
[a.u.]
1.5 10 -5
Cosine part
of n=1
electrostatic
potential.
m=2
m=1
5.0 10 -5
0.0 10
m=1
0
m=3
-6
-5.0 10 -5
m=3
0.0 10 0
-1.0 10 -4
-5.0 10 -6
-1.5 10 -4
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
2.5 10 -5
2.0 10 -4
2.0 10 -5
1.5 10 -4
1.5 10
-5
1.0 10
-5
1
0.8
1
m=2
1.0 10 -4
m=3
m=2
0.8
r/a
[a.u.]
Sine part of
n=1
electrostatic
potential.
[a.u.]
r/a
0.6
5.0 10 -5
m=1
0.0 10 0
m=3
5.0 10 -6
-5.0 10 -5
0.0 10 0
-1.0 10 -4
m=1
-5.0 10 -6
-1.5 10 -4
0
0.2
0.4
0.6
0.8
r/a
Nonlocal EPM
1
0
0.2
0.4
0.6
r/a
Local (resonant) EPM
Time evolution of the nonlocal EPM
(energetic ion pressure profile I: classical)
0.0008
0.3
0.90
0.0006
0.0004
0.80
0.0002
d
E m/n=2/1 cos
0.5
0.85
0
-0.0002
0.75
0.70
-0.0004
0.7
0.5
0.3
0.65
-0.0006
-0.0008
0
200
400
 At
600
0
100
200
300
At
400
500
Left: Time evolution of cosine part of the toroidal electric field.
dBmax/B~10-2.
Right: Frequency shifts only downward by 33% (~17kHz) in 6
102 Alfvén time (~0.5ms).
Consider reduced energetic ion pressure
profiles
 In the experiments fast FS modes and abrupt
large events take place with time intervals
much shorter than the slowing down time.
 These activities lead to redistribution and loss
of energetic ions.
classical
distribution
gives
an
 The
overestimate of energetic ion pressure profile.
 We consider reduced pressure profiles whose
shape is the same as the classical distribution.
Time evolution of the nonlocal EPM
(classical distribution is reduced to 2/5)
0.0002
0.3
0.00015
0.3
1.1
5 10-5
1.0
d
E m/n=2/1 cos
0.0001
0
-5 10-5
0.9
-0.0001
-0.00015
0.7
0.5
0.8
-0.0002
0
400
800
At
1200
200
400
At
600
800
Left: Time evolution of cosine part of the toroidal electric field.
dBmax/B~210-3
Right: Frequency shifts upward by 14% (~7kHz) and
downward by 23% (~12kHz) in 103 Alfvén time (~0.8ms). Close
to the experiment.
Radial beta profiles
0.02
0.008
t=0
t=586
0.015
t=0
t=1004
0.007
0.006
h
h
0.005
0.01
0.004
0.003
0.005
0.002
0.001
0
0
0
0.2
0.4
0.6
0.8
1
r/a
Initial: classical pressure
profile.
40% reduction in central
beta. Flattened at r/a<0.4.
0
0.2
0.4
0.6
0.8
1
r/a
Initial: reduced pressure
profile.
15% reduction in central
beta. Flattened only at
r/a<0.2.
2.0
2.0
1.5
1.5
1.5
1.0
1.0
0.5
0.5
0.0
0.0
0.0
3.0
3.5
R/a
4.0
2.5
At=84
3.0
3.5
R/a
4.0
2.5
At=252
2.0
1.5
1.5
1.5
0.5
z/a
2.0
1.0
1.0
0.5
0.0
3.0
3.5
R/a
4.0
At=252
3.5
R/a
4.0
3.5
R/a
4.0
1.0
0.5
0.0
2.5
3.0
At=335
2.0
z/a
z/a
Reduced
pressure
profile
1.0
0.5
2.5
Change
in mode
profile
is small.
z/a
2.0
z/a
Classical
pressure
profile
z/a
Time evolution of mode spatial profile
(toroidal electric field)
0.0
2.5
3.0
3.5
R/a
4.0
At=418
2.5
3.0
At=924
Comparison with hole-clump pair creation
[H.L. Berk et al. Phys. Lett. A 234, 213 (1997); 238, 408(E) (1998).]
0.04
Blue circles: Frequency
upshift and downshift.
d=0.4L
0.035
d/A
0.03
0.025
Red triangles: Only
downshift.
0.02
0.015
0.01
Consistent with the holeclump pair creation.
0.005
0
0
0.02
0.04
0.06
L/A
0.08
0.1
For d/A=0.025, L/A=0.05, the theory predicts frequency
shift d=0.44 L(dt)1/2=0.11 A in 103 Alfvén time. This
corresponds to 22kHz. Simulation result is 10kHz. In good
agreement with a factor of 2.
Summary for fast FS mode
1. A new kind of nonlocal EPM is found near the plasma
center with monotonic magnetic shear and passing
energetic ions.
2. When a classical distribution is taken for the initial
condition, frequency shifts only downward.
3. When a reduced distribution is taken, frequency shifts
both upward and downward. The rate of frequency
sweeping is close to the fast FS modes in the experiments.
4. Frequency shifts both upward and downward are
consistent with the hole-clump pair creation.
5. A great redistribution of energetic ions takes place when
the classical distribution is taken for the initial condition.
Computer Simulation of Fishbone oscillation
(Y. Shiozaki)
GOAL
Clarify self-consistent nonlinear evolution of
precessional fishbone oscillations; spatial profile,
frquency evolution, saturation mechanism, energetic
ion transport
METHOD Particle + MHD hybrid simulation (current coupling)
Bulk plasma : Ideal MHD equation
Fast ion : kinetic equation using delta-f method
BACKGROUND
Fishbone instabilities was first identified during nearperpendicular NBI injection in PDX tokamak.
Initial condition of simulation
1: Major radius
Minor radius
1.43 [m]
0.43 [m]
2: Magnetic field at axis
1.5 [T]
3: Bulk plasma component
Beam plasma component
H+ plasma
D+ plasma
4: Density
51019 [m-3]
5: Velocity distribution of NBI fast ion
isotropic slowing down distribution, injection energy 50keV
6: Spatial profile of fast ion pressure
Gaussian distribution
Pressure gradient scale at q=1 surface
0.15 [m]
Frequency and growth rate at various energetic ion
beta
Case 1
 (1/  A )
Case 2
 (1/  A )

energetic (%)
When fast ion beta is 1.0%, the
frequency
 is oscillating between
4kHz and –4kHz
energetic (%)
For relatively high energetic ion
beta values
fishbone mode is

destabilized
This result is similar to [W. Park et al., Phys. Plasmas 6, 1796 (1999)].
Mode Structure of kink mode
sin component of radial velocity
cos component of radial velocity
Case 1
q-profile
Radial velocity of (1,1) mode is
well-known top-hat structure.
vradial
q
r a
Mode Structure of fishbone
Case 2
sin component of radial velocity
cos component of radial velocity
vradial
q
r a
q-profile
Around q=1 surface phase of
the mode profile changes
(spiral structure). Two step
structure is slightly seen.
Higher resolution is needed to
clarify the two step structure.
Temporal evolution of poloidal magnetic field and
frequency of the fishbone mode, h=3%
B
f (kHz)
Tim e[ A ]
Saturation level dB/B~510-3
Tim e[ A ]
Frequency shifts
downward to zero.
Time evolution in each toroidal mode number
MHD energy ( toroidal mode number N=0 )
MHD energy ( toroidal mode number N=1 )
MHD energy ( toroidal mode number N=2 )
Tim e[ A ]
The n=1 mode saturates at
1400 Alfvén time. The n=0
and n=2 modes continues
to grow after the n=1 mode
saturation.
MHD Energy
Transferred Energy (from energetic particles to MHD)
Total energy
Energy transfer from energetic
ions to MHD continues even
after the saturation of the n=1
mode growth.
Tim e[ A ]
Summary for fishbone mode
1. Linear stability and nonlinear evolution of fishbone
mode is investigated. Frequency shifts downward at
saturation. Saturation level is dB/B~510-3.
2. After saturation of the n=1 mode, n=0 and n=2
modes continues to grow. Energy transfer from
energetic ions to MHD fluid keeps a constant level.
3. Future work:
3.1 investigate energetic ion transport.
3.2 investigate generation of nonlinear MHD
modes, interaction between energetic ions and
the nonlinear MHD modes.