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Modelling ICCD Experiments for Sawtooth
Control in JET
Cite as: AIP Conference Proceedings 871, 350 (2006); https://doi.org/10.1063/1.2404569
Published Online: 30 November 2006
J. P. Graves, W. A. Cooper, S. Coda, L.-G. Eriksson, T. Johnson, and JET-EFDA Contributors
AIP Conference Proceedings 871, 350 (2006); https://doi.org/10.1063/1.2404569
© 2006 American Institute of Physics.
871, 350
Modelling ICCD Experiments for Sawtooth
Control in JET
J. P. Graves∗ , W. A. Cooper∗ , S. Coda∗ , L.-G. Eriksson† , T. Johnson∗∗ and
JET-EFDA Contributors‡
∗
Centre de Recherches en Physique des Plasmas, Association EURATOM-Confederation Suisse,
EPFL, 1015 Lausanne, Switzerland ([email protected])
†
Association EURATOM-CEA, CEA/DSM/DRFC, CEA-Cadarache, F-13108 St. Paul lez Durance,
France
∗∗
Association Euratom-VR, KTH, SE-100 44 Stockholm, Sweden
‡
Proc. 20th IAEA Conference, Vilamoura, Portugal, 2004.
Keywords: MHD instabilities in tokamaks and fast particles
PACS: 52.35.Py, 52.55.Fa, 52.55.Tn
INTRODUCTION
The motivation of this work is to characterise a model for the distribution function of
ICRH populations which can easily be employed in stability codes. The model described
here has already been incorporated in the 3D equilibrium code VMEC [1], and 3D
fluid stability code TERPSICHORE [2]. Furthermore, the model lends itself to relatively
straightforward semi-analytical calculations of macroscopic stability in tokamaks, and in
particular the stability of the internal kink mode in the presence of ICRH ions. Despite
the apparent simplicity of the model, it will be seen that the three radially dependent
parameters in the distribution function enable the recovery of the salient features of the
SELFO [3] simulations. In this work an attempt is made to parameterise the distribution
function for SELFO simulations of the important JET discharge 58934 [4], shown here
in Fig. 1. This discharge demonstrates that an off-axis ion cyclotron resonant surface,
with phasing to enable ion cyclotron current drive (ICCD), can destabilise (shorten
period of) sawteeth even when the sawteeth are initially stabilised by trapped energetic
RF ions in the core. Hence, in the latter part of the discharge two resonant surfaces coexist. It is the sum of these two populations that ultimately require modelling in order to
ascertain the internal kink mode stability.
MODEL ICRF DISTRIBUTION FUNCTION
The distribution of particles F depend only the constants of the particle motion: energy
E = mv2 /2, magnetic moment μ = mv2⊥ /B and flux surface label r. The distribution is
written in terms of a bi-Maxwellian in v and v⊥ :
m 3/2
μBc
|E − μBc |
nc (r)
exp −
F=
−
.
1/2
2π
T⊥ (r)
T (r)
T⊥ (r)T (r)
CP871, Theory of Fusion Plasmas: Joint Varenna-Lausanne International Workshop,
edited by J. W. Connor, O. Sauter, and E. Sindoni
© 2006 American Institute of Physics 978-0-7354-0376-5/06/$23.00
350
FIGURE 1. Pulse 58934 in JET [4], plotting the electron temperature, sawtooth period, sawtooth
inversion radius Rinv and first harmonic H cyclotron resonance layer Rres (H) for +90◦ and −90◦ phasings,
and heating power for the two antennas.
Here in general the critical field strength Bc can in principle also depend on the flux
label r, but a constant Bc is usually more appropriate. In the above nc is the local
density evaluated where B = Bc on the flux surface r. Taking the zero’th moment of
the distribution function yields the variation of the density with respect to B:
n(r, B) = nc NB
where
T⊥a
for B > Bc and
T⊥
1/2 T⊥a T⊥b − T⊥a T⊥
Bc − B 1/2
+
for B < Bc , and
NB =
T⊥
T⊥
T
Bc
−1
−1
Bc
Bc
Bc T⊥
Bc T⊥
and T⊥b = T⊥
.
T⊥a = T⊥
+
−
1−
1−
B
T
B
B
T
B
NB =
Denoting the flux surface average with angular brackets we can identify
2π
−1
2π
2
nc = n G(r) and G = (2π)
dθ
dφ NB
0
0
where G assumes the role of normalising a distribution function defined in terms of n,
and θ and φ are the poloidal and toroidal angles respectively.
351
Now, taking second moments of the distribution function it can be shown that
P = nc T H and P⊥ = nc T⊥ H⊥ ,
where for B > Bc :
H =
while for B < Bc :
H =
H⊥ =
T⊥a
T⊥
T⊥a
T⊥
2
T⊥a
T⊥
+
T⊥
T
+
T⊥
T
1/2 and H⊥ =
3/2 Bc −B
Bc
Bc −B
Bc
1/2 T⊥a
T⊥
3/2 2
,
T⊥b −T⊥a
T⊥
T⊥b −T⊥a
2T⊥
B
Bc
+
2 −T 2
T⊥b
⊥a
T⊥2
.
PARAMETERISING THE DISTRIBUTION FUNCTION
Here 3D data files from the SELFO simulations of n, P⊥ and P are used to identify the
three radially dependent parameters of the model distribution function. In particular, in
the following, the LHS of the equations correspond to the parameters in the model, and
the RHS of the equations are the quantities from the SELFO simulations:
nc (r) = n(Rc , Z) , T⊥ (r) =
P⊥ (Rc , Z)
T (r) P⊥ (Rc , Z)
and A(r) ≡ ⊥
=
.
e n(Rc , Z)
T (r)
P (Rc , Z)
Now, since the heating is approximately located on a vertical line through the plasma
cross section we can resolve the minor radius on the LHS of the equations through
r2 = Z 2 + (Rc − R0 )2 with Z defining the distance along a vertical chord R = Rc .
Where there are two resonant surfaces the problem is treated upon assuming the
sum of model distributions. Hence there are now six radially dependent parameters
to resolve namely nc (r), T⊥ (r) and A(r) for the two distributions. The problem has
been treated upon exploitation of simulations with off-axis heating alone. This enabled
identification of the three functions for off-axis heating, and thus when employed in
conjunction with the parameters obtained with on-axis heating alone, provided a first
guess for the distribution function for the combined heating case. The six functions
were then normalised iteratively to provide a best parameter fit of the 3D plots of the
density, parallel pressure and perpendicular pressure. The result of such a procedure for
the case at hand, i.e. JET discharge 58934, is shown in Fig. 2. Plotted are the radial
profiles of nc (r), T⊥ (r) and A(r) for the two coexisting resonant RF populations, and
the resulting flux surfaced averaged density and parallel pressure. The latter two are
compared favourably with the corresponding SELFO data. Finally, Fig. 3 shows the
density and parallel pressure over the entire poloidal cross section. Peaks in the density
and pressure result from the localised deposition of the RF heating, and are again seen
to recover the salient features of the SELFO data.
352
FITTING FUCTIONS FOR SUM OF TWO MODEL DISTRIBUTIONS
800
700
(keV)
600
500
T
n c * 10
−19
0.06
0.04
A
(/m )3
0.08
0.02
400
0
0 0.05 0.1 0.15 0.2 0.25 0.3
0
0.02 0.04 0.06 0.08 0.1
ε
0
0.05 0.1 0.15 0.2 0.25 0.3
ε
ε
Early in Discharge with 1 res. surface
1.2·10
16
14
12
10
8
6
4
2
Later in Discharge with 2 res. surface
18
18
18
1·10
Model and SELFO
comparison of
flux averaged
density
(/m^3)
1·10
17
8·10
17
6·10
17
4·10
17
2·10
17
6·10
17
4·10
17
2·10
0
0
0.2
0.4
0.6
comparison of
flux averaged
density
(/m^3)
17
8·10
0
0.8
0
0.2
0.4
80000
120000
Model and SELFO
comparison of
flux averaged
pressure
(J/m^3)
100000
80000
60000
40000
0.6
0.8
comparison of
flux averaged
pressure
(J/m^3)
60000
40000
20000
20000
0
0
0
0.2
0.4
0.6
0
0.8
0.2
0.4
0.6
0.8
r / 0.9
r / 0.9
FIGURE 2. Plots of nc (r), T⊥ (r) and A(r) for the two distributions to reproduce the SELFO simulations
of the latter part of discharge 58934. Also shown are SELFO and model comparisons of the resulting flux
averaged density and pressure profiles.
SELFO PRESSURE
SELFO DENSITY
MODEL PRESSURE
R−R
0
0/R
-0.1
0
0.1
R−R
0.1
0
0
-0.1
0
0/R
R
R
0
-0.1
0
0
80
60
40
20
0
Z/
0.1
P (kJ/m 3 )
0.1
0.05
Z/
n *10 −19(/m3 )
MODEL DENSITY
-0.1
0
0.1
FIGURE 3. A comparisons between the analytical model and SELFO of the density and perpendicular
pressure surfaces for the parameters in Fig. 2
FAST ION CONTRIBUTION TO STABILITY OF INTERNAL
KINK MODE
Presented here are the equations required to assess the stability of the internal kink mode
for the model distribution function. It is assumed that the fast ion pressure is much
353
smaller than the core pressure, and thus does not significantly modify the equilibrium.
The opposite limit will be treated elsewhere. The ideal growth rate is identified with the
normalised potential energy terms:
γ
π ˆ
ˆ
ˆ
= − (δW
I + δW A + δW k )
τA
s1
ˆ I the ideal MHD, δW
ˆ A the adiabatic contribution due to pressure anisotropy,
with δW
ˆ k the non-adiabatic (kinetic) contribution
and δW
−1/2
ˆ k = − 3ε1
δW
π23/2
2μ0
B20
r1
0
r
dr
r1
3/2 2
1+κ
2
Fk (r),
with κ the elongation and
Fk (r) =
1
dk2
0
F1
2
1+κ
Fq2 Jt
+ 2sF2 − ζ
1
4q2
+ F3
,
and Jt is determined by the model distribution:
1/2
2
5 nc T⊥ A |εc +ε(2k −1)|
1/2
A
(nc T⊥ A ) − 2
1+A|εc +ε(2k2 −1)|
t
J =
,
5/2
(1 + A|εc + ε(2k2 − 1)|)
where εc defines the resonant surface location through the identity Bc = B0 /(1 + εc ).
The following is a reasonable fit of Fq :
Fq = 2E(k2 ) − K(k2 ) + 4[1 − q(r)][E(k2 ) + (k2 − 1)K(k2 )]
where E and K are compete elliptic integrals of the second and first kinds. Furthermore,
F1 = 2E(k2 ) − K(k2 ), F2 = 2E(k2 ) + 2(k2 − 1)K(k2 ),
4
F3 = [(2k2 − 1)E(k2 ) + (1 − k2 )K(k2 )]
3
0 dP 2
and ζ = − 2Rμ
q is ‘ballooning’ parameter, and P the plasma pressure.
B2 dr
ˆ A is the sum of trapped (t) and passing (p) contributions
The anisotropic term δW
given by:
ˆ t,p
δW
A
FAt (r) =
1
dk
0
2
−1/2
3ε
= − 1 3/2
π2
2μ0
B20
2Gt1 + Gt2
r1
0
r
dr
r1
t
5/2
[1 + ε(2k2 − 1)]
J and
354
3/2 FAp (r) =
2
1+κ
1
dk
0
2
FAt,p (r),
2G1p + G2p
5/2
[k2 + ε(2 − k2 )]
J p,
with
Jp =
and
Gt1 ,
Gt1 =
Gt2 =
G1p =
−
G2p =
−
Gt2 ,
G1p ,
G2p
A1/2 |k2 εc +ε(2−k2 )|
A
k5 (nc T⊥ A1/2 ) − 52 nckT2⊥+A|ε
k2 +ε(2−k2 )|
c
(k2 + A|ε
c
5/2
k2 + ε(2 − k2 )|)
are given in Ref. [6]:
(1 − k2 )K(k2 ) + (2k2 − 1)E(k2 ) ,
2ε 2
2
2E(k ) − K(k ) +
(1 − 4k2 )K(k2 ) + (8k2 − 4)E(k2 ) ,
3
2ε 2
2
2
2
)E(k
)
−
2(1
−
k
)K(k
)
(2
−
k
3k2
2
4
2ε
(4k − 12k2 + 8)K(k2 ) + 7k4 + 8k2 − 8 E(k2 ) ,
4
15k
2
2ε 4
2
2
2
2
2
2
2E(k ) + (k − 2)K(k ) −
−
8k
+
8
K(k
)
+
4k
−
8
E(k
)
3k
3k2
2
2ε
4
2
2
4
2
2
−
3k
+
2)K(k
)
−
2(k
−
16k
+
16)E(k
)
.
16(k
15k4
2ε
3
CONCLUSIONS
The work presented here provides a means of assessing the stability of the internal
kink mode with ICRH distributed ions. It will also enable analysis of the impact of
the fast ions on the equilibrium [1, 2]. Calculations of the fast ion contribution to ideal
ˆ k indicate that the internal kink mode is strongly stabilised with the
ˆ A + δW
stability δW
single resonant surface close to the magnetic axes. However, calculations with the two
resonant surfaces reveal that the contribution of the ICRH population to ideal stability
is negligible. For the latter case the net effect of the ICRH population on sawteeth is
thus felt predominantly through the effect of ICCD on the evolution of the safety factor.
Sawtooth control techniques in ITER will employ localised current drive in conditions
where there is strong ideal stabilisation from alpha particles.
ACKNOWLEDGMENTS
This work was partly funded by the Fonds National Suisse de la Recherche Scientifique
and EURATOM.
REFERENCES
1.
2.
3.
4.
5.
6.
W.A. Cooper, J P Graves, et al, Nucl. Fusion 46, 683 (2006).
W.A. Cooper, J P Graves, et al, Fusion Science and Technology 50, 245 (2006).
J. Hedin, et al, Nucl. Fusion 42, 527 (2002).
L-G. Eriksson, et al, Phys. Rev. Lett. 92, 235004 (2004).
J. P. Graves, et al, Phys. Rev. Lett. 84, 1204 (2000).
J. P. Graves,et al, Phys. Plasmas 10, 1034 (2003).
355
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