Proc. 12th Intl. Reflectometry Workshop - IRW12 (Jülich) 18-20 May 2015
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Structure and properties of geodesic acoustic mode oscillations
in ASDEX Upgrade
P. Simon1,2,3 , G. D. Conway2 , U. Stroth2,4 , P. Manz2,4 , A. Biancalani2 and the ASDEX
Upgrade2 Team
1
Institut für Grenzflächenverfahrenstechnik und Plasmatechnologie,
Universität Stuttgart, 70569 Stuttgart, Germany
2
Max-Planck-Institut für Plasmaphysik, 85748 Garching, Germany
3
Institut Jean Lamour, Université de Lorraine, 54011 Nancy, France
4
Physik-Department E28, Technische Universität München, 85747 Garching, Germany
e-mail: [email protected]
1. Introduction
Doppler reflectometry has previously been used on ASDEX Upgrade (AUG) to study the
properties of zonal flows and geodesic acoustic modes (GAMs) [1]. This report presents
recent results from studies on the behaviour of GAM properties under changing plasma
geometry. The scaling of the GAM frequency and amplitude with variation in plasma
elongation, safety factor and other parameters is compared to heuristic and theory-based
models. Additionally, results are presented on the radial structure of the GAM.
2. ASDEX Upgrade Doppler reflectometer setup
Three Doppler reflectometry systems are in operation on ASDEX Upgrade (AUG): two
fixed antenna V-band systems (50 − 75 GHz, O- and X-mode) [2, 3] and one W-band
system (75−108 GHz) with an adjustable tilt angle on the tokamak low-field-side [4]. The
20 MHz sampling rate of the receiver in-phase and quadrature signals, using a 12 bit ADC,
allows the investigation of velocity fluctuations and radial electric field perturbations
with high temporal and spatial resolution. The perpendicular wavenumber k⊥ as well
as the measurement location are provided by the beam tracing code torbeam [5] for
each probing frequency, using a fitted density profile and high resolution equilibrium
reconstruction. The data presented in this work was obtained with the V-band systems.
3. Signal analysis
The detection of GAMs from the raw Doppler reflectometer signal is a multi-step process.
First, the Doppler shift frequency fD , which is proportional to the perpendicular velocity
v⊥ , is calculated from the IQ signal. It is necessary to have a high-resolution time series
fD (t) to measure GAMs. Therefore, fD is calculated from a small window of the complete
IQ signal, which slides forward in time. An average power spectrum is then calculated
from fD (t). GAMs appear as distinct peaks in this frequency spectrum, usually in the
range of 5 − 25 kHz for AUG discharges.
The Doppler shift is obtained from the raw signal by calculating the power spectrum
Sf (f ) with a Fast Fourier Transform (FFT) algorithm and computing a weighted mean:
P
P
fD = f · Sf (f )/ Sf (f ). The original data window (10-15 ms at 20 MHz sampling
rate) is typically separated into sub-windows of 256 points (with 50 % overlap) for this
calculation. The power spectrum is again calculated via an FFT from the fD (t) time
series.
Proc. 12th Intl. Reflectometry Workshop - IRW12 (Jülich) 18-20 May 2015
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4. GAM frequency scaling
GAMs can reduce the turbulence correlation length and provide important contributions
to the effective shearing rate, as long as their frequency is below the inverse turbulence
decorrelation time.
Winsor’s theoretical prediction for circular plasmas, based on a fluid model,
is a scaling
with the ion sound speed
q
cs =
(Te + γi Ti )/m and major radius
R0 : ωGAM = Gcs /R0 , with a weak dependence on safety
factor q in the scale fac√
tor G = 2 + q −2 [6]. For non-circular
plasmas with a boundary elongation κb >
1, deviations from this scaling were detected, especially in the plasma edge. At
ASDEX Upgrade, a heuristic scaling law
for edge GAMs (ρpol > 0.95), including
the inverse aspect ratio ǫ0 = 0.3, was
found by Conway et al. [7]: ωGAM =
Fig. 1: Experimental fGAM against Conway
4πcs /R0 ((1 + κb )−1 − ǫ0 )
scaling law (fscale ).
Figure 1 shows the experimentally observed GAM frequency in a set of specially designed shape-scan discharges against the
frequency predicted by this model. During these discharges the plasma boundary elongation was varied between 1.1 < κb < 1.7 for limiter and 1.4 < κb < 1.8 for divertor plasmas.
Multiple frequency sweeps of the reflectometers were performed during each discharge in
order to observe GAM behaviour in the radial range 0.90 < ρpol < 1.00 while the plasma
shape was evolving.
In comparison to the fluid-model scaling by
Winsor, the heuristic model yields better
agreement to the present data, but there
are still a number of disagreements: GAMs
closer towards the plasma centre in limiter
discharges are found to be at a lower frequency than predicted, while good agreement is found in the edge region. For
GAMs in divertor discharges the measured
frequency is higher than predicted.
An approach based on a gyrokinetic model
and incorporating the effects of finite orbit
drift width (ODW) was derived by Gao [8].
Fig. 2: Comparison of the experimental re- It includes a larger number of parameters
along with the characteristic cs /R0 scaling:
sults (fGAM ) with Gao scaling law (fscale ).
the temperature ratio τ = Te /Ti , inverse
aspect ratio ǫ, the local plasma elongation κ and its radial derivate sκ , safety factor q
and the Shafranov shift gradient ∆′ . Similar to the heuristic model, a dependence on
(κ2 + 1)−0.5 yields a decrease in GAM frequency for elongated discharges. A comparison
of the experimental data with the scaling by Gao in figure 2 shows that the theoretical
prediction generally seems to give only an estimate of a lower boundary. Non-linear effects
Proc. 12th Intl. Reflectometry Workshop - IRW12 (Jülich) 18-20 May 2015
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may be responsible for upshift the GAM frequency by a significant amount [9]. The Gao
scaling exhibits less scatter between various radial regions or plasma geometries then
Winsor’s scaling or the heuristic model.
5. GAM amplitude behaviour
The GAM amplitude is calculated as the
peak-to-peak velocity by integrating over
the GAM spectral peak [10]:
P
0.5
f2
−1
AGAM = 4πk⊥
,
f1 S (fD ) 4/1.5
where f1 , f2 ≈ fGAM ± 0.6 kHz. The factor
4/1.5 compensates for the Hanning bellwindow that is applied to minimize spectral leakage. The perpendicular wavenumber k⊥ is typically of the order of 10 cm−1 .
As shown in figures 3 and 4, the GAM amplitudes for this set of discharges fall in the
range of 0.2–1.0 km/s. For the edge region
of L-mode plasmas, the mean perpendicFig. 3: GAM amplitude against boundary ular plasma velocity is in the range of 2–
elongation κb , colour-coded for varying local 5 km/s [11]. The dataset is formed from
safety factor q.
values taken at the GAM radial maximum
in each discharge condition.
Figure 3 shows the measured GAM amplitudes against the plasma boundary elongation κb ,
with variations in local safety factor q marked by the colour. In figure 4, GAM amplitudes
are shown against q, here the colour marks variations in elongation κb . The results in
figure 3 are similar to earlier ASDEX Upgrade experiments [10], when only the limiter
configuration is considered. AGAM exhibits an inverse dependence on the elongation κb ,
with only a small observable variation due to local q. High boundary elongation κb is also
generally accompanied by higher q.
For divertor plasmas the effect of the Xpoint and strong shaping appears to be important. Discharges have higher q in general, and low GAM amplitudes are either
found at low q, or at very high κb . In cases
of high q, AGAM is raised. For q in the
range of 3.5–4.5, the impact of κb is seen
again. For low to moderate q (3-4), AGAM
approaches the limiter range.
Figure 4 shows the dependence of AGAM
on q more clearly. GAMs with low amplitude appear for any qlocal , κb or plasma
configuration, although there is a tendency
to higher AGAM at lower κb for a fixed q. In Fig. 4: GAM amplitude against local safety
divertor configuration, there is an observ- factor q, colour-coded for varying boundary
able trend of high AGAM at higher q.
κb .
Proc. 12th Intl. Reflectometry Workshop - IRW12 (Jülich) 18-20 May 2015
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Fig. 5: Spectrogram of limiter discharge #29722. Left: low κb , GAM frequency changes
continously. Right: high κb , one locked GAM
6. Radial GAM structure
The radial structure of GAMs at ASDEX Upgrade has been shown to vary depending
on the discharge conditions [7]. Figure 5 illustrates how this effect can even be observed
during a single discharge, as the shape of the plasma is changing. The left graphic shows
a spectrogram of the fD time series over the radial region 0.80 < ρpol < 0.99 in the lowelongation phase (κb = 1.12) of a limiter discharge. A strong GAM peak at 16 kHz is
found at ρpol ≈ 0.96, and GAMs at increasing frequency of up to 23 kHz can be detected
closer to the core. The right figure, taken from the same discharge but at κb = 1.67,
shows a different situation. The GAM is now only detected at one locked frequency
fGAM ≈ 8 kHz in a narrow radial region 0.95 < ρpol < 0.98. The narrower overall radial
region is due to changes in the density profile. This transition from continous GAM (or
multiple GAM plateaus) to single locked GAM structure is observed in all shape-scan
discharges. Typical AUG plasmas in elongated divertor configuration are therefore more
likely to exhibit the locked-frequency structure.
7. Discussion and conclusions
This report gives an overview of recent Doppler reflectometry investigations at ASDEX
Upgrade, based on a series of discharges in which a scan of the plasma shape was perfomed.
The GAM frequency was shown to vary as a function of plasma geometry. Comparisons
with the heuristic model by Conway based on plasma elongation show agreement with
previous AUG experiments. The gyrokinetic model by Gao incorporates more plasma
parameters and gives a lower boundary estimate of the GAM frequency.
The GAM amplitudes are measured to be in the range of 0.2–1.0 km/s. In limiter plasmas
the amplitude decreases with increasing elongation. Low-q GAMs in divertor plasmas
behave similarly, however AGAM increases significantly at high q. The roles of GAM drive
and damping are currently under investigation.
The radial structure of the GAM is shown to change significantly depending on the shape
of the plasma. In the case of low elongation, the GAM is found over a wide radial range
(> 5 cm) with a frequency continuum. At high κb the GAM region shrinks to 1–2 cm and
no variation in GAM frequency is observed.
Proc. 12th Intl. Reflectometry Workshop - IRW12 (Jülich) 18-20 May 2015
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8. Acknowledgements
This work has been carried out within the framework of the EUROfusion Consortium and
has received funding from the Euratom research and training programme 2014-2018 under
grant agreement No 633053. This work was also supported by the European Commission
within the framework of the Erasmus Mundus International Doctoral College in Fusion
Science and Engineering (FUSION-DC). The views and opinions expressed herein do not
necessarily reflect those of the European Commission. This work was also performed
within the framework of the Helmholtz Virtual Institute on Plasma Dynamical Processes
and Turbulence Studies using Advanced Microwave Diagnostics.
9. References
[1]
[2]
[3]
[4]
G. D. Conway et al., Plasma Phys. Control. Fusion 47, 1165 (2005).
S. Klenge, PhD Thesis, University of Stuttgart (2004).
G. D. Conway et al., Plasma Phys. Control. Fusion 46, 951 (2004).
T. Happel et al., Design of a new Doppler Reflectometer Front End for the ASDEX
Upgrade Tokamak (Proc. 10th International Reflectometry Workshop, Padua, Italy,
2011).
[5] E. Poli et al., Comput. Phys. Commun. 136, 90 (2001)
[6] N. Winsor et al., Phys. Fluids 11, 2448 (1968).
[7] G. D. Conway et al., Plasma Phys. Control. Fusion 50, 055009 (2008).
[8] Z. Gao, Plasma Science and Technology 13, 15 (2011)
[9] Z. Qiu et al., Phys. Plasmas 22, 042512 (2015)
[10] G. D. Conway et al., Plasma Phys. Control. Fusion 50, 085005 (2008)
[11] G. D. Conway et al., Plasma Phys. Control. Fusion 46, 951 (2004)