Breizman, B.N.

Lisak, Mietek

Nyqvist, Robert

Sharapov, S.E.

Chalmers Publication Library

Fast Ion Driven Alfve´n Eigenmodes within the q = 1 Radius
R. Nyqvist1, B. N. Breizman2, M. Lisak1, S. E. Sharapov3
1EURATOM/VR Association, Chalmers University of Technology,
SE-412 96 Gothenburg, Sweden
2Institute for Fusion Studies, University of Texas at Austin,
Austin, Texas 78712, USA
3EURATOM/UKAEA Fusion Association, Culham Science center,
OX14 3DB, Abingdon, UK
I. Introduction
In JET discharges with high-power ICRH and a central safety factor lower than unity, monster
sawtooth crashes are preceeded by so called tornado modes [1], see figure 1 (left), observed
experimentally using both external Mirnov coils and interferometry diagnostics. These modes,
with frequencies in the toroidal Alfve´n eigenmode (TAE) range, appear with decreasing toroidal
mode numbers one by one and they exhibit a peculiar downward sweep in frequency, initially
deviating from the usual TAE scaling fTAE ∝ B/
√
n. Moreover, the tornado modes appear
simultaneaously with a decrease in the intensity of γ-rays emitted from hot ions in the plasma
center, implying that the tornado modes are responsible for a reduction of the hot ion population
inside the q = 1 magnetic surface, and that this redistribution may in fact be the cause of the
monster sawtooth crashes.
O-mode interferometry reveals another interesting feature of the magnetic spectrum on JET:
The tornado modes are often preceeded by a set of core-localized modes in the range 50 − 200
kHz. These modes behave similar to Alfve´n cascades (ACs) in reversed-shear plasmas [2], in
that their frequencies sweep up more or less linearly in time, seemingly from the same initial
frequency but with different sweeping rates. Similar modes have been observed on the Alcator
C-Mod tokamak using phase constrast imaging diagnostics [3].
s
Figure 1: Left: Magnetic spectrum of the plasma center during shot 66203 on JET, measured by O-mode
interferometry diagnostics. Both tornado modes and cascading modes are clearly visible. Right: Mode structure
of (a) cascading mode with q0 = 0.95 and (b) tornado mode with q0 = 0.91 obtained numerically using MISHKA-1.
Two important features of tokamak plasmas undergoing monster sawtooth oscillations is
the smallness of the magnetic shear inside the q = 1 radius and the proximity to the magnetic
axis. When the shear s = r/q dq/dr satisfies the condition s < ǫ, with the inserve aspect ratio
ǫ = r/R0, so called multiple low shear TAEs (MLSTAEs) may be supported by the tokamak
geometry itself. Moreover, a flat q-profile implies broader radial mode structures, with the result
that resonating particles interact with the waves during longer times.
It is of high importance to clarify what kind of Alfve´n eigenmodes may exist within the q = 1
radius, how these eigenmodes are connected to the behavior of the central safety factor profile
and how they affect hot ions inside the q = 1 surface. The aim of the present EPS contribution
is to identify the observed modes and their connection to the safety factor. Calculations of the
transport of fast ions through the q = 1 surface during monster sawtooth crashes in the presence
of the modes identified in this work will be the aim of future investigations.
II. Mode Identification
In order to explain the two types of modes observed, two different paths were explored. First, a
kinetic Alfve´n wave, residing in a potential well at the magnetic axis, was considered analytically.
Second, a slightly reversed low-shear equilibrium was created and studied numerically to find
the possible modes within ideal MHD theory.
A. Kinetic Modes
Kinetic modes close to the magnetic were first predicted theoretically by Rosenbluth and Ruther-
ford in 1975 [4]. By examination of a narrow region well off the magnetic axis, where d/dr ≫ 1/r,
they used WKB theory to solve a generalization of the MHD equation for the radial dependence
of the mode amplitude (including finite ion Larmor radius effects and finite parallel electron
conductivity), and finally obtain a fast varying ”kinetic” Alfve´n wave. A similar approach was
used by Konovalov et al. [5] to explain certain modes with sweeping down frequencies which are
sometimes observed following ACs in reversed-shear plasmas.
In the close proximity of the magnetic axis, the ordering d/dr ∼ 1/r and the high values of
Te and Ti suggests that non-ideal effects may play an important role. For large mode numbers,
m2 ≫ 1, the non-ideal differential equation for the mode amplitude ξ is given by (cf. [4])
ω2
7
4
ρ2i
d4ξ
dr4
− m
2
r2
(
ω2 − ω2A
)
ξ = 0 , (1)
with the additional restriction that the solutions are regular at r = 0 and everywhere finite.
Here, ω is the mode frequency, ρi is the ion Larmor radius, m is the poloidal mode number and
ωA = k‖vA is the Alfve´n frequency, k‖ and the Alfve´n velocity vA being given respectively by
k‖ =
1
R0
(
m
q
− n
)
, vA =
B
µ0ρ
.
Expanding k2‖ to second order in r around the magnetic axis and looking for modes with fre-
quencies close to the on-axis Alfve´n frequency ωA0 = ωA (r = 0), equation (1) becomes
Λ2
d4ξ
dr4
− 1
r2
(
∆+ αr2
)
ξ = 0 , (2)
where the mode frequency has been written as ω = ωA0 + δω, δω ≪ ωA, and we have defined
Λ2 ≡ 7ρ
2
i
4m2
, ∆ ≡ 2δω
ωA0
, α ≡ mq
′′
0
R0q20k‖0
.
Note that all quantities denoted by 0 are to be calculated at the magnetic axis, r = 0. If we
transform to the dimensionless quantities
z =
( |α|
Λ2
) 1
4
r , E =
∆
Λ
√
|α| ∼ 1 ,
equation (2) takes on the form
d4ξ
dz4
− 1
z2
(
E ± z2) ξ = 0 , (3)
where the ± in front of z2 represents the sign of α. The existence of kinetic eigenmodes satisfying
equation (3) may very well depend on the sign of α, which is in turn determined by the signs of q′′
0
and the mode numbers m and n. Assuming that q0 decreases in time from q0 = 1, α > 0 implies
that the mode frequency experiences upward sweeping for q′′
0
> 0 and downward sweeping for
q′′
0
< 0, while α < 0 implies that the mode frequency sweeps up for q′′
0
< 0 and down for q′′
0
> 0.
Finally, the frequency shift δω of the kinetic modes may be estimated via the rescaling expression
for ∆, resulting in
δω = E
vAρi
4q0
√
7
R0
∣∣∣∣mk‖0q′′0m
∣∣∣∣ ∼ 1 kHz .
Modes with such small δω are not distinguishable from shear Alfve´n continuum modes, satisfying
ω2 = ω2
A
, by present day diagnostics. Unambiguous observations of these modes would require
knowledge of the signs of q′′
0
and the mode numbers. However, the dependence of δω on q′′
0
suggests the possibility that observations of up-sweeping kinetic modes could offer new diagnostic
oppurtunities for the magnetic shear.
B. Ideal MHD Modes
Further away from the magnetic axis, ǫ2-terms become more important than the non-ideal effects
already considered, and AC-like Alfve´n eigenmodes can be found within ideal MHD theory [6].
In order to study this regime, the equilibrium code HELENA was used to construct a low-shear
equilibrium with a slighlty reversed safety factor profile, see figure 2 (left). Solutions for the
ideal MHD eigenmode equations were then obtained using MISHKA-1, a full geometry ideal
MHD stability code, to scan through the relevant values of q0, ranging from q0 = 1 and down.
Both sweeping up, core-localized ACs and tornado-like TAEs, located somewhat more off
axis and moving radially outwards, were found for all toroidal mode numbers considered. As
can be seen in figure 2 (right), the frequencies of the ACs sweep from 0 at q0 = 1 all the way up
to the TAE frequency range, where the TAEs appear with frequencies that evolve in a manner
quite similar to the experimentally observed spectrum in figure 1 (left). It should be mentioned
that the TAEs develop strong interaction with global AEs as they move outwards, effectively
prohibiting identification of the radial asymmetry observed in [1], see figure 1 (right).
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
Normalized minor radius, r/a
Sa
fe
ty
 fa
ct
or
, q
(r)
0.80.820.840.860.880.90.920.940.960.981
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Safety factor on axis, q0
N
or
m
al
iz
ed
 fr
eq
ue
nc
y
n=3
n=4
n=5
n=6
n=7
n=8
n=9
Figure 2: Left: Zoom in (q (r) < 1) of the slightly reversed safety factor profile of the low-shear equilibrium
created by HELENA. Right: Mode frequencies, with toroidal mode number n, for the modes identified with
MISHKA-1.
III. Concluding Remarks
Two paths were explored in order to explain core-localized, cascading modes and tornado modes,
both observed just before monster sawtooth crashes on JET and Alcator C-Mod. First, kinetic
Alfve´n modes with large mode numbers were treated analytically in the proximity of the mag-
netic axis. Second, core-localized ACs as well as TAEs exhibiting tornado behavior were found
numerically within ideal MHD theory using a slightly reversed, low-shear safety factor profile.
Both types of modes are plausible candidates for explaining the observations, and in order to
draw unambiguous conclusions, measurements of mode localization, mode number sign and mag-
netic shear,s as well as better frequency resolution in the experiments are needed. The modes
identified here will be used in future modelling of fast ion redistribution through the q = 1
surface during monster sawtooth crashes.
Acknowledgements
The authors are grateful to the colleagues who operated the JET tokamak during the experiments
considered in this contribution. The work was partly funded by UK EPSRC and EURATOM.
The views and opinions expressed herein do not necessarily reflect those of the European Com-
mision.
References
[1] P. Sandquist, S. E. Sharapov et al., Phys. Plasmas, 14, 122506 (2007).
[2] S. E. Sharapov et al., Phys. Plasmas 9, 2027 (2002).
[3] E. M. Edlund et al., Phys. Rev. Letters, 102, 165003 (2009).
[4] M. N. Rosenbluth, P. H. Rutherford, Phys. Rev. Letters, 34, 1428 (1975).
[5] S. V. Konovalov, A. B. Mikhailovskii et al., Phys. Plasmas, 11, 4531 (2004).
[6] B. N. Breizman et al., Phys. Plasmas, 10, 3649 (2003).


Fast Ion Driven Alfvén Eigenmodes within the q=1 Radius

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