PHYSICS OF PLASMAS 18, 055708 (2011)
Resonant drive and nonlinear suppression
of gradient-driven instabilities via
a)
interaction with shear Alfvén waves
D. W. Auerbach,b) T. A. Carter, S. Vincena, and P. Popovich
Department of Physics and Astronomy, University of California, Los Angeles, California 90095-1547, USA
(Received 7 December 2010; accepted 1 March 2011; published online 29 April 2011)
The nonlinear interaction of shear Alfvén waves and gradient-driven instabilities on pressure
gradients in the Large Plasma Device (LAPD) [Gekelman et al., Rev. Sci. Instrum. 62, 2875 (1991)]
at UCLA is explored. Nonlinear response at the beat frequency between two shear Alfvén waves is
shown to resonantly drive unstable modes as well as otherwise damped modes. Resonantly driving
the damped modes is shown to suppress the originally unstable mode, leaving only the beat-driven
response with an overall reduction in fluctuation amplitude. A threshold is observed in the
suppression behavior, requiring that the driven damped mode power be of order 10% of the power in
the saturated unstable mode. The interaction is also observed to be dependent on the parallel
wavenumber of the driven beat wave; efficient coupling and suppression is only observed for
co-propagating beat waves with small parallel wavenumber, consistent with the parallel wavenumber
C 2011 American Institute of Physics. [doi:10.1063/1.3574506]
of the gradient-driven modes. V
I. INTRODUCTION
In magnetized laboratory plasmas, cross-magnetic-field
gradients can drive an assortment of instabilities, including
drift waves1 and Kelvin–Hemholtz (KH) instabilities.2,3
These instabilities and the associated turbulence are
observed to contribute to loss of heat, particles, and momentum across the magnetic field.4 Methods to control gradientdriven instabilities (GDIs) and the resulting transport are
thus desirable to improve confinement, especially in fusionrelated devices such as tokamaks.
Control of drift wave instabilities has been recently
demonstrated in linear plasma devices. In the MIRABELLE
device, an in-plasma electrode array was used to excite parallel currents and directly drive drift waves.5 In those experiments, a spectrum of unstable drift waves were synchronized
to the single driven frequency, reducing the overall fluctuation amplitude. Those studies were extended in the VINETA
device, where direct drift wave excitation and synchronization was achieved using parallel currents driven by electrode
arrays and saddle coils.6 Synchronization in VINETA can
lead to a reduction in cross-field particle transport associated
with drift waves.7
This article expands on previously reported8 laboratory
experiments using nonlinearly driven beat waves driven by
shear Alfvén waves (SAWs) to interact with and control
GDIs. In these experiments, two SAWs at slightly differing
frequency are launched along a magnetic-field-aligned density depletion on which a coherent unstable mode is
observed. When the SAW beat frequency matches the instability frequency, resonant drive of the mode is observed.
However, when the beat frequency is detuned from the instability, resonant driving of nearby stable modes is observed,
along with the suppression of the original unstable mode. The
a)
Paper GI3 1, Bull. Am. Phys. Soc. 55, 108 (2010).
Invited speaker.
b)
1070-664X/2011/18(5)/055708/7/$30.00
ability to control instabilities using waves represents an important extension of the synchronization work on MIRABELLE
and VINETA, potentially enabling the application to fusion
devices where electrodes and saddle coils may not be practical.
High power radio frequency (RF) heating is a tool that is
widely used in current fusion experiments and is expected to
be important for next-step fusion experiments such as ITER.9
Beat waves driven by, e.g., ion-cyclotron range of frequencies
(ICRF) waves could be used to interact with and affect drifttype instabilities.
This article is organized as follows. In Sec. II, the experimental apparatus is described along with techniques for producing density depletions and SAW beat waves. In Sec. III,
the experiments on the resonant drive and control of GDIs
using SAW beat waves are discussed. The resonant drive of
the GDI and other nearby stable modes is shown. The nonlinearity of the suppression of the unstable mode is explored,
and the dependence on a spatial match in kk is demonstrated.
Finally, in Sec. IV conclusions and next steps are discussed.
II. METHODS
A. The Large Plasma Device
This research was carried out on the upgraded Large
Plasma Device (LAPD) at the Basic Plasma Science Facility
(BAPSF) at UCLA (Ref. 10). The LAPD creates an 18 m
long, 0.6 m diameter cylindrical magnetized plasma. The
plasma is created at 1 Hz using an indirectly heated bariumoxide-coated cathode discharge (discharge length is typically
10–20 ms). Bulk parameters for the helium plasmas used in
these experiments are: ne 1012 cm3, Te 6 eV, and
B ¼ 800 G. Ion temperature in LAPD is typically Ti 1 eV
(Ref. 11), although it was not measured directly in these
experiments.
Time varying temperature, density, and floating potential are determined using triple Langmuir probes. The temperature and density measurements have been quantitatively
18, 055708-1
C 2011 American Institute of Physics
V
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Phys. Plasmas 18, 055708 (2011)
A plasma gradient is induced in the LAPD (see Fig. 1)
by selectively blocking the high energy primary electrons
that form the plasma. This is done by inserting a small disk
into the anode–cathode region [Figs. 1(a)–1(c)]. The result is
a cylindrical lower density region of plasma (dn=n 50%)
embedded in the bulk plasma and axially aligned with the
background magnetic field [Figs. 1(d) and 1(e)]. Similar conditions in the LAPD have been well documented to generate
GDIs, including drift-Alfvén waves.14,15
Figure 2 shows details of the depletion and associated
GDI fluctuations observed for this article. The
pffiffiffiffiffimeasurements
shown are of ion saturation current (Isat / ne Te ) and pressure
Pe, with Pe calculated from Te and Isat. Figure 2(a) plots pressure contours of the induced depletions, indicating a 6 cm
wide depletion with central pressure about 50% lower than the
background. The spatially averaged spectrum of observed Isat
fluctuations is shown in Fig. 2(e), demonstrating the narrowband fluctuations at 6 kHz and harmonics. The 6 kHz fluctuations are spatially localized to the pressure gradient region, as
shown in Fig. 2(b). Cross-correlation measurements shown in
Fig. 2(d) of the spontaneous fluctuations indicate an m ¼ 1
azimuthal mode number for the dominant mode, with propagation in the electron diamagnetic direction.
The measured frequency, direction, and speed of propagation of these measured modes agree with linear predictions
for drift waves. However, measurements of the relative phase
hx between density and potential fluctuations [shown in Figs.
2(c) and 2(f) indicating the spatial distribution of hx for 6 kHz
and the spatially averaged hx spectrum, respectively] indicate
that hx for the coherent mode is approximately 180 , which is
inconsistent with the 0 < hx < 90 predictions and observations of saturated, single mode drift waves.16 This measured
FIG. 1. Schematic of the drift wave/Alfvén wave interaction experiment.
Top: (a) grid anode, (b) cathode, (c) primary e blocking disk, (d) plasma
boundary, (e) density depletion, (f) shear Alfvén wave antenna. Bottom portion shows the relative scales of the antenna, the depletion, and the relative
uniformity of the drive Alfvén waves over the depletion region.
FIG. 2. (Color online) (a) Contours of measured plasma pressure in the
depletion. (b) Contours of Isat fluctuation power, with pressure contours
overlaid in white. (c) Contours of cross-phase (hx ) between Isat and Vf fluctuations. (d) Cross-correlation showing m ¼ 1 structure of mode. (e) Spatially averaged Isat FFT spectrum. (f) Spatially averaged cross-phase (hx ).
verified using microwave interferometry and swept Langmuir probe measurements. Magnetic field fluctuations associated with Alfvén waves and other fluctuations were
measured using magnetic pickup coils, differentially wound
to eliminate electrostatic pickup. The probes are mounted on
motorized, computer controlled probe drives. Two-dimensional average profiles are acquired over many repeatable
discharges by moving a single probe shot-to-shot. Twodimensional cross-correlation functions are achieved using a
second, stationary reference probe separated axially (along
the background field) from the moving probe.
The analyses using potential described in this paper
directly use the floating potential (Vf) measurements from the
Langmuir triple probes. Studies under similar plasma conditions in the TJ-K experiment12 have shown that fluctuation
measurements dVf and dVp show very similar behavior; however, that work did not directly address the effect of temperature fluctuations. Initial comparisons between results obtained
from dVf and results from a dVp calculated from measured
dVf and dTe (Ref. 13) show no major discrepancies. Additionally, there is no evidence that the currents generated by driven
Alfvén waves affect the potential measurements.
B. Gradient-driven instability on filamentary density
depletion
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Resonant drive and nonlinear suppression of gradient-driven instabilities
Phys. Plasmas 18, 055708 (2011)
180 phasing also indicates that these instabilities are not
causing radial transport due to the potential and density fluctuations, as the radial particle flux Cr is given by
ð
1
jPx ðxÞjkh ðxÞ sinðhx Þdx;
(1)
Cr /
B0
where Px ðxÞ is the cross-power between potential and density fluctuations and kh ðxÞ is the azimuthal wavenumber.17
C. Nonlinear beat wave interaction among Alfvén
waves
A compound antenna consisting of two rectangular current-carrying straps isolated from each other and the
plasma18,19 is inserted into the plasma at the far end of the
depletion, 16 m from the plasma source. The antenna is centered on the depletion with the longer current-carrying straps
parallel to the background magnetic field B0 [see Fig. 1(f)] to
drive two independent dB’s perpendicular to B0. Each strap
of the antenna is driven and controlled independently, giving
rise to two perpendicularly polarized (dB perpendicular to
each other and B0), co-propagating SAWs that propagate
along B0. The waves are both driven near half of the ioncyclotron frequency (fci ¼ 307 kHz at 800 G) with one frequency offset from the second by a variable amount on the
order of a few kHz. The peak wave magnetic field for the
driven Alfvén waves used in this study ranges from 0.1 to 1
G (1 104 . dB=B . 1 103 ). In the absence of the density depletion, the two SAWs interact nonlinearly, driving a
nonresonant quasimode at the beat frequency.20
D. Analytic modeling of instability
As indicated in the experimental 1-D profiles shown in
Fig. 3(a), there are two sources of free energy available: the
pressure gradient and a potential gradient, both due to the existence of the blocking disk. Thus there are two possible instabilities, the drift wave instability, resulting in drift waves with
axial mode number nk 0:5, and the Kelvin–Helmholtz
instability, resulting in “flute-like” modes with nk ¼ 0.
Numerical eigenmode solutions to the linearized Braginskii
two-fluid equations21 were performed, using experimentally
measured density, temperature, and potential profiles. These
results were obtained for nk ¼ 0 and nk ¼ 0:5, and the resulting frequencies and growth rates are shown in Figs. 3(b) and
3(c), respectively. The figure shows frequencies and growth
rates for the five fastest growing modes for both axial mode
numbers and a range of azimuthal mode numbers, indicating
that both types of modes are linearly unstable and have real
frequency approximately equal to the measured frequency.
The predicted phasing hx between density and potential (not
plotted here) indicates that KH-like modes have hx 180
and drift wave modes have hx between 0 and 90 . These
results leave open the possibility that the saturated single
unstable mode observed might be of either type of instability
or a hybrid mode driven by both sources of free energy. For
this article we refer to the modes merely as GDIs, postponing
positive identification of the mode to more detailed theoretical
and experimental investigations.
FIG. 3. (Color online) Analytic predictions for the five fastest growing
modes in (a) the experimentally measured profiles, showing (b) growth rates
and (c) frequencies for flute like (nk ¼ 0, red cross-hatches) and drift wavelike (nk ¼ 0:5, green circles) modes, for a range of mh .
III. DRIVE AND CONTROL OF GRADIENT-DRIVEN
INSTABILITIES
A. Resonant interaction between beat waves
and gradient-driven instabilities
A series of experiments were performed in which the
beat frequency was set near the instability frequency and varied shot-to-shot, with the results shown taken from ensemble
averages of 15–30 shots per data point. These results illustrate the range of interactions between the GDIs and the
beating SAWs. Figure 4(a) shows the measured Isat power
spectrum near the depletion gradient maximum [at approximately x ¼ 2 cm, y ¼ 1 cm (see Fig. 2 axes), 7 m from the
antenna, and 8 m from cathode] as the beat frequency is varied over 0 fbeat 15 kHz. The measured fluctuations at
the beat frequency are observed to be strongest when the
beat frequency is close to the instability frequency (6 kHz),
with two separate resonant peaks centered around 5.5 and
8.25 kHz as shown in Fig. 4(b). The peak rms amplitude of
the beat-response is dIsat =Isat ¼ 15%, which is significantly
stronger than that observed in the absence of the depletion,
where for the same launched SAW parameters, a beat wave
amplitude of dIsat =Isat . 3% is observed.
Clearly indicated in Fig. 4 is the interaction between the
SAW beat wave and the unstable mode at f 6 kHz. As the
beat wave frequency approaches 6 kHz from below, the spontaneous mode begins to decrease in power and is observed to
downshift in frequency, meeting the beat wave where it
appears to resonate near 5.5 kHz. As the beat wave is driven
to higher frequency, the spontaneous mode is almost
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Auerbach et al.
Phys. Plasmas 18, 055708 (2011)
FIG. 5. (Color online) Total rms Isat fluctuations between 0 and 30 kHz for
various antenna currents and beat frequencies, summed over measurements
taken in a 2D plane covering the entire depletion area.
FIG. 4. (Color online) Units normalized to power equivalent of maximum
rms fluctuations of 15%. (a) Measured Isat fluctuation power as a function of
the SAW beat frequency (x axis) and FFT frequency (y axis). (b) Power at the
spontaneous mode frequency (f ¼ 6 6 0:4 kHz) and at the SAW beat frequency (f ¼ fbeat 6 0:4 kHz). (c) Isat fluctuation power spectra at 0.0, 6.0, and
8.25 kHz SAW beat frequency (marked with vertical lines on the 2D multispectrum plot above in part (a). (This figure reprinted from Ref. 8.)
completely absent. Instead, the beat wave resonantly drives
fluctuations around 8 kHz. The instability at 6 kHz returns and
the beat-driven response diminishes when the beat frequency
is increased above the higher frequency resonance.
Figure 4(b) illustrates the shift of fluctuation power from
the instability to the driven fluctuations by showing the
power at the unstable mode frequency (6 6 0:4 kHz) and
power in the beat frequency band (fbeat 6 0:4 kHz) as a function of driven beat frequency, essentially a horizontal and
diagonal line cut of the data in Fig. 4(a). These line cuts, on
a linear scale, illustrate the absence of power in the original
6 kHz instability during the resonant drive of fluctuations
centered around 8.25 kHz. This is made clearer in Fig. 4(c),
which shows the measured power spectrum in the absence of
the SAW beat wave (both antennas set to the same
frequency) and the spectrum in the presence of a 6 kHz and
an 8.25 kHz driven beat wave, illustrating single coherent
modes in all three cases.
Also of note is the reduction in broadband fluctuations
while the unstable mode is suppressed. This can be seen in
Figs. 4(a) and 4(c) with a clear reduction in power around 4
and 12 kHz during the 8.25 kHz drive. Figure 5 further illustrates this by displaying the total rms fluctuations measured
in Isat between 0 and 30 kHz, for various beat frequencies
and various antenna currents (the Alfvén wave magnitude is
proportional to the antenna current). The low-power case
shows a small overall reduction compared to the zero
antenna drive case, presumably due to heating and minor
profile modification from the Alfvén wave drive. More interesting is the frequency-dependent increase and decrease, corresponding with resonantly driving the original instability
and the higher frequency resonance around 8 kHz. During
the peak power input to the system with the beat frequency
set at 8 kHz, the total low-power fluctuations in the depletion
are reduced by about 20%.
To determine whether the interaction between the driven
beat-frequency fluctuations and the GDI was due to Alfvén
wave interactions or simply antenna near field effects, the previously described experiments were repeated with the antenna
frequency fant set above the ion-cyclotron frequency fci, with
the antenna current, and thus the antenna B fields kept constant. Because SAWs do not propagate above fci, any observed
effect in this second case would indicate that the observed
results were not due to Alfvén wave effects. For these experiments where fant > fci and no propagating SAWs were present,
no resonance or suppression was observed, indicating that the
previously described resonance and suppression behaviors are
indeed due to the presence of the two beating Alfvén waves.
B. Mode structure of gradient-driven instabilities and
driven mode
The two peaks in Fig. 4(b) suggest that the SAW beat
wave is resonantly driving separate resonant modes at f ¼ 5.5
kHz and at f ¼ 8.25 kHz. This conjecture is supported by the
spatial structure of the beat wave response found using
cross-correlation measurements. Figure 6 shows the twodimensional cross-phase [row (a)], the spatial distribution of
fluctuation power [row (b)], and the fluctuation power spectrum [row (c)], for four cases: the beat wave in the absence
of the depletion [column (0)], the spontaneous 6 kHz mode
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Resonant drive and nonlinear suppression of gradient-driven instabilities
Phys. Plasmas 18, 055708 (2011)
FIG. 6. (Color online) 2D phase (from cross-correlation), spatial power distribution, and fluctuation spectrum of Isat for the beat wave without the depletion,
the spontaneous mode (no SAWs) and 6 kHz and 8 kHz beat waves. Plot (b0) is on a different scale for clarity.
in the absence of SAWs [column (1)], the 6 kHz mode when
the beat wave is at 6 kHz [column (2)], and the 8 kHz mode
when driven at 8 kHz [column (3)]. For the three cases with
the antenna-driven beat wave, the antenna current is identical. Note that for the two beat wave cases in the presence of
the depletion, the power in the beat wave response is concentrated on the density gradient [Figs. 6(b2) and (b3)], similar
to the spontaneous mode [Fig. 6(b1)]. In contrast to this observation, in the absence of the depletion [Fig. 6(b0); note
the different scale], the beat wave has a much broader spatial
distribution consistent with the pattern of the launched
SAWs. The beat wave power without the depletion is significantly lower, barely above the background fluctuations
[Fig. 6(c0)]. The asymmetry in the mode structures shown
here is assumed to be due to observed and repeatable asymmetries in the background plasma which result in changes in density up to 10% on the length scale of the depletion. Although
the fluctuations are not completely cylindrically symmetric,
they are strongly coherent across the entire depletion region.
The main distinction between the two modes comes in
the observed mode structure, as illustrated by the cross-phase
measurements in row (a) of Fig. 6. The mode structure of the
beat-driven response at 6 kHz [Fig. 6(a2)] is nearly identical
to the spontaneous mode [Fig. 6(a1)] and indicates an m ¼ 1
mode structure. The peak in the beat-driven response at 8 kHz
corresponds to a distinct m ¼ 2 mode, as shown in Fig.
6(a3). These mode numbers were quantitatively verified by
counting zero crossings of the cosine of the cross-correlation
phase in an annular region at the radius of maximum fluctua-
tion strength. The shift from m ¼ 1 to m ¼ 2 character
appears at approximately 7 kHz, at the minimum between the
two resonant peaks in fluctuation at the beat wave frequency,
confirming that these peaks indicate resonant drive of the
m ¼ 1 and m ¼ 2 modes.
All three modes described here (the unstable mode, the
mode driven at 6 kHz, and the mode driven at 8 kHz) are
similar in spatial structure, direction of propagation, amplitude, and frequency, with the only distinction being the m
number of the 8 kHz mode. They are also similar in the phasing hx between potential and density fluctuations, indicating
that all are essentially the same type of fluctuation. The
measured phasing hx ffi 180 is also unchanged, indicating
that the resonant drive of these modes is not affecting transport due to the fluctuations.
One distinction between the m ¼ 1 and m ¼ 2 modes
is that the m ¼ 1 mode is unstable on the gradient, and the
m ¼ 2 mode is apparently stable. Figure 7 shows time traces
of measured Isat, one single trace with the beat frequency set
at 8 kHz, and one trace averaging 30 shots with conditions
identical to the single time trace. The 6 kHz instability is not
phase-locked to the experimental timing because it is an instability that grows out of the plasma noise. Thus, the 6 kHz fluctuations are mostly washed out in the average trace, leaving
the phase-locked 8 kHz mode behind. In the average time
trace, immediately after the RF antennas begin to drive the
beating Alfvén waves, the resonant growth of the 8 kHz mode
can be seen. After the RF turns off (antenna currents ring
down completely within about 20 ls), the 8 kHz fluctuations
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Phys. Plasmas 18, 055708 (2011)
FIG. 7. (Color online) Isat time traces showing the unstable 6 kHz mode and
the driven 8 kHz mode. The upper trace shows an average of 30 shots, and
the lower trace is a single shot time trace. Data traces are offset for visibility
and are on the same scale, with fluctuations on the order of dIsat =Isat 15%.
continue for several oscillations (about 300 ls), indicating
both that the 8 kHz mode is a normal mode of the system, and
that it is a damped mode. After about four oscillations of the 8
kHz mode it is mostly gone, and the 6 kHz mode begins to
return indicating that the damping constant of the 8 kHz mode
is on the order of the fluctuation frequency.
FIG. 8. (a) Power in the instability band (6:5 6 0:4 kHz, solid line) and
driven beat-response band (8:0 6 0:4 kHz, dashed line) vs SAW drive current for the case of 8.0 kHz beat frequency, along with a reference instability
power (6:5 6 0:4 kHz, dotted line) at zero beat frequency (both antennas at
same frequency). (b) Fluctuation power in the instability band (6:5 6 0:4
kHz) as a function of beat frequency and antenna current. The white contour
is the 50% suppressed level at each power. Reprinted with permission from
D. W. Auerbach, T. A. Carter, S. Vincena, and P. Popovich, Phys. Rev. Lett.
105, 135005 (2010). Copyright 2010, The American Physical Society.
C. Nonlinearities in resonance and suppression
The resonant drive of the GDIs and the associated suppression of the primary instability depends strongly and nonlinearly on the amplitude of the drive Alfvén waves, a
parameter controlled by the current in the Alfvén wave
antennas. Figure 8(a) shows the variation in the 8 kHz
(driven) and 6 kHz (GDI) amplitudes as the SAW antenna
current is increased (for fixed 8 kHz beat frequency). A
threshold is clear for affecting the instability, requiring the
driven fluctuation power to be on the order of 10% of the
power of the original unstable mode. It is also interesting to
note that the driven mode saturates as the instability is completely suppressed. Also shown is the instability power as a
function of antenna current in the case of zero beat frequency. A reduction of the instability power at high SAW
amplitude is observed which can be attributed to profile
modification due to electron heating by the launched SAWs
(Ref. 19). This effect contributes to modification of the instability in the presence of SAWs, but is less than 10% of than
the reduction observed with nonzero beat frequency.
The width in beat frequency over which control of the
instability is observed also varies with the amplitude of the
driven beat wave response. Figure 8(b) shows the fluctuation
power at the unstable mode frequency as a function of beat
wave frequency and antenna current. At higher currents, the
instability (m ¼ 1) begins to be suppressed in favor of the
higher frequency mode driven by the beat wave (m ¼ 2).
The white contour in Fig. 8(b) indicates where the instability
power is at 50% of the “background” power with the antennas driving a zero beat frequency Alfvén wave. The broadening of the range over which the instability is controlled as
the beat wave power is increased is indicated by this contour.
Similar observations are made in VINETA (Ref. 6), where
the effect was compared to “Arnold Tongues” seen in nonlinearly driven unstable oscillators.22,23
D. Dependence of control on spatial structure of beat
wave
The observed suppression and control appears to be dependent on the axial structure of the driven beat wave. Section III B describes the structure of the GDI as having an
angular mode number of m ¼ 1 or m ¼ 2, and an axial
mode number of nk ¼ 0:5 or nk ¼ 0. To examine the effect
of beat wave axial mode number on suppression and control
of the GDI, the beat waves were driven by counter-propagating Alfvén waves created by inserting antennas into the
plasma at z ¼ 0 m and z ¼ 12 m, in the coordinates shown in
Fig. 1. The resulting observations were compared to the copropagating behavior from the single antenna at z ¼ 0 m.
To determine the predicted axial mode structure of the
beat waves, the kinetic Alfvén wave dispersion relation
2 2
(2)
qs
x2 ¼ kk2 v2A 1 þ k?
(where qs is the sound gyroradius) was used along with and
the three-wave matching rules kkbeat ¼ kk2 kk1 (Ref. 20) to
compute the parallel wavelength of the beat wave.
Using the measured plasma parameters and taking
k? 1:25 cm1 based on the typical scale size of the Alfvén
wave pattern of 5 cm, a typical 6 kHz beat wave driven
between two Alfvén waves near 0.5 xci can be shown to
have a wavelength of 2:6 m for counter-propagating
Alfvén waves, and for co-propagating waves, 130 m (longer wavelength than the fundamental nk ¼ 0:5 mode, which
has 34 m wavelength). The wavelength of the co-
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Resonant drive and nonlinear suppression of gradient-driven instabilities
Phys. Plasmas 18, 055708 (2011)
driven damped mode was observed for a finite range of beat
wave frequencies around the driven mode frequency; this
range widened with increased beat wave amplitude. This
coupling of the beat wave to the driven modes was shown to
be dependent on the spatial structure of the beat wave.
These results provide motivation for several follow up
studies. The connection between stable and unstable modes
is being investigated, with the hope of understanding the
mechanism for coupling between the modes and possibly
related damping of the stable modes. These results and the
reported use of ICRF beat waves to drive Alfvén eigenmodes
in JET and ASDEX upgrade,24,25 suggest that ICRF beat
waves might be used to interact with drift wave turbulence in
tokamak plasmas.
ACKNOWLEDGMENTS
FIG. 9. (Color online) Power in the instability (blue, crosses) and driven
beat wave (red, stars) along with a zero antenna drive instability case
(black), showing power coupling and suppression for both co-propagating
and counter-propagating driven Alfvén waves.
propagating beat wave has been directly measured in the absence of the depletion and is consistent with three-wave
matching.20 Thus a co-propagating beat wave is a much better match in parallel wavenumber to the GDI.
Figure 9 shows the measured GDI power and beat wave
power during an experiment in which the beat frequency was
varied shot-to-shot for both co- and counter-propagating
Alfvén waves. The primary instability in this set of experiments has a slightly different frequency than previous cases
due to differences in the background plasma profiles (in part
caused by the presence of the second antenna). The main
instability in this case is an m ¼ 1 mode at f 7 kHz. The
beating Alfvén waves can excite a second (stable) mode at 5
kHz. The 5 kHz mode also has m ¼ 1 character, but has a
distinct spatial distribution of power. In this case driving the
system with a 5 kHz beat wave generated by co-propagating
waves resonantly drives the 5 kHz mode while suppressing
the growth of the 7 kHz mode. The counter-propagating
case, however, produces much less response at the beat frequency and resonance at 5 kHz, and suppression of the primary instability is not observed.
IV. CONCLUSIONS
The data presented here show the ability to control GDIs
with shear Alfvén beat waves. Two co-propagating SAWs
with distinct, controllable frequencies were launched along a
narrow field-aligned density depletion on which coherent
unstable fluctuations were observed. The nonlinear beatresponse driven by the two Alfvén waves was seen to resonantly drive two distinct modes of the system, both the GDI
and a similar stably damped mode of the system near in frequency to the GDI. The amplitude of the original GDI was
reduced or suppressed completely during the drive of the
nearby stable mode. A reduction in broadband fluctuations
was also observed accompanying the suppression of the primary mode. The suppression of the GDI in favor of the
These experiments were performed using the Basic
Plasma Science Facility at UCLA, which is supported by
DOE and NSF. This work was supported by NSF Grant No.
PHY-0547572.
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