PHYSICS OF PLASMAS 20, 012111 (2013)
n wave resonator
Investigation of an ion-ion hybrid Alfve
S. T. Vincena, W. A. Farmer, J. E. Maggs, and G. J. Morales
Physics and Astronomy Department, University of California, Los Angeles, Los Angeles,
California 90095, USA
(Received 4 June 2012; accepted 27 December 2012; published online 16 January 2013)
A theoretical and experimental investigation is made of a wave resonator based on the concept of
wave reflection along the confinement magnetic field at a spatial location where the wave
frequency matches the local value of the ion-ion hybrid frequency. Such a situation can be realized
by shear Alfven waves in a magnetized plasma with two ion species because this mode has zero
parallel group velocity and experiences a cut-off at the ion-ion hybrid frequency. Since the ion-ion
hybrid frequency is proportional to the magnetic field, it is expected that a magnetic well
configuration in a two-ion plasma can result in an Alfven wave resonator. Such a concept has been
proposed in various space plasma studies and could have relevance to mirror and tokamak fusion
devices. This study demonstrates such a resonator in a controlled laboratory experiment using a
Hþ-Heþ mixture. The resonator response is investigated by launching monochromatic waves and
impulses from a magnetic loop antenna. The observed frequency spectra are found to agree with
C 2013 American Institute of Physics.
predictions of a theoretical model of trapped eigenmodes. V
[http://dx.doi.org/10.1063/1.4775777]
I. INTRODUCTION
1–4
It has long been established
that in a magnetized
plasma, the presence of two ion species allows the perpendicular component of the cold-plasma dielectric coefficient
e? to vanish at a frequency commonly referred to as the ionion-hybrid frequency
xii ¼
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u 2 2
uxp1 X2 þ x2p2 X21
t
x2p1 þ x2p2
;
(1)
where xpj and Xj , respectively, refer to the ion plasma frequency and ion cyclotron frequency of species j. The vanishing of e? causes a collective resonance for waves that
propagate across the magnetic field, such as the compressional or fast Alfven mode. This resonance is perceived to
provide an effective heating channel and is extensively studied5–9 in the context of heating magnetically confined plasmas to fusion temperatures. In contrast to this resonant
behavior, for shear Alfven waves, which propagate predominantly along the confinement magnetic field, the condition
e? ¼ 0 results in a wave cut-off where wave reflection takes
place. This property can be readily seen from the approximate (obtained by dropping the off-diagonal terms in the
dielectric tensor) dispersion relation for shear waves with
large perpendicular wave number k? ,
pffiffiffiffiffi
2
kk ¼ k0 e? ½1 k?
=k02 ek 1=2 ;
(2)
where kk is the parallel wave number, k0 ¼ x=c with c the
speed of light, x the wave frequency, and ek the dielectric
coefficient parallel to the ambient magnetic field. In this context, large k? implies that the transverse scale-length of the
shear wave is shorter than the smallest ion skin-depth, c=xpj ,
in the system. This parameter regime is of significant interest
1070-664X/2013/20(1)/012111/12/$30.00
because it allows the modes to interact strongly with both
electrons10–13 and ions.14–17 An additional consequence of
this parameter regime is that the compressional mode is
evanescent, so that there is no direct coupling, or crossover,
between the shear Alfven wave and the fast mode for frequencies below the largest Xj in the system. For the experimental parameters of this work, to be presented later,
numerical solutions of the full dispersion relation overlap
with those predicted by Eq. (2) except in the narrow regions
where the wave frequency is approximately equal to the fundamental or harmonics of the heavier ion species.
The wave cut-off at the ion-ion hybrid frequency implies
that, in a plasma with two ion species, the shear Alfven wave
exhibits a frequency propagation gap in which the modes are
evanescent over the range X1 x xii , where X1 refers to
the ion cyclotron frequency of the heavier species. Shear
modes can propagate in the two adjacent bands x < X1 and
xii < x < X2 , where X2 refers to the ion cyclotron frequency of the lighter species. Experimental evidence for
such propagation bands has been obtained in tokamak studies18,19 related to the coupling properties of Alfven wave
antennas. Those earlier studies referred to the higher frequency band as “ion-ion hybrid waves.” A more recent
experiment in the Alcator C-Mod tokamak20 has identified
that the compressional mode can excite, via mode conversion, a small-scale wave that propagates away from the ionion hybrid layer, but in this case, the mode has been labeled
as the “electromagnetic cyclotron wave.”
The important role played by the wave cut-off at x ¼ xii
has long been recognized in space plasma studies. Young
et al.21 noted that the significant differences in ULF wave polarization and spectra measured by the GEOS 1 and 2 satellites
could be qualitatively explained by including the effects of a
helium minority ion species in the proton-dominated wave dispersion relation. Motivated by these observations, Rauch and
Roux15 used a raytracing analysis to confirm the conclusions of
20, 012111-1
C 2013 American Institute of Physics
V
012111-2
Vincena et al.
Young et al., and pointed out that multiple reflections of ULF
waves could take place at the locations where x ¼ xii in the
earth’s dipole field. This concept was later supported by Perraut
et al.,22 who compared magnetic field data taken by GEOS 1
and 2 to simultaneous measurements by ground-based magnetometers at the magnetic footprints of the satellites. The results
were consistent with waves being reflected at the ion-ion hybrid
layer in the magnetosphere. Contemporary with those studies,
Temerin and Lysak23 identified that narrow-banded waves were
generated by the auroral electron beam in a limited spatial
region determined by the local value of xii for a mix of HþHeþ ions.
Recently, the reflection properties for various species
concentration scenarios in the geoplasma have been investigated by Hu et al.24 using a hybrid simulation. In other theoretical studies, it has been recognized that the wave
reflection can result in a detached ion-ion hybrid resonator in
the equatorial region of planetary magnetospheres.
Guglielmi et al.25 have calculated the resonator spectrum
and compared it to satellite observations. Their conclusion
was that the resonator concept is “plausible, but we have not
yet confirmed it experimentally.” Significant insight into the
ion-ion hybrid resonator concept has been provided by the
theoretical study by Mithaiwala et al.26 This work concluded
that waves excited by the controlled release of gas in the
Earth’s magnetosphere could be trapped in the ion-ion
hybrid resonator formed by typical abundances of Hþ-Heþ
in the inner radiation belts.
Theoretical studies by Moiseenko and Tennfors,27 motivated by analogies to low frequency toroidal Alfven eigenmodes (TAE), have considered the role of vanishing e? in
tokamak geometry. The authors identified a new class of
high-frequency toroidal eigenmodes that they named TLE,
but did not explore the possibility of an ion-ion hybrid resonator. In magnetic mirror geometry, the ion-ion hybrid resonator is a candidate for the type of mode structure that could
facilitate the proposed alpha-channeling concept.28,29
Motivated by the broad interest in this topic, the present
manuscript reports the investigation of an ion-ion hybrid
Alfven wave resonator in the controlled laboratory environment provided by the upgraded large plasma device (LAPD)
operated by the Basic Plasma Science Facility (BaPSF) at
the university of California, Los Angeles (UCLA). A recent
study made in this device30 documented the theoretically
predicted effects produced by two ion species on the propagation of shear Alfven waves of small transverse scale in a
uniform plasma. The study also demonstrated that such
waves experience reflection when the condition x ¼ xii is
encountered in a region of increasing magnetic field (i.e., a
magnetic step). The ion-ion resonator concept was recently
verified in LAPD31 in a preliminary experimental investigation. The current study provides a theoretical framework for
interpreting the earlier results and presents new experimental
data and techniques.
The manuscript is organized as follows. Section II outlines
a theoretical description of trapped eigenmodes of relevance to
the experiment and documents their properties. Section III
describes the experimental set-up and Sec. IV reports the experimental observations. Conclusions are given in Sec. V.
Phys. Plasmas 20, 012111 (2013)
II. THEORY
This section presents a mathematical model based on
the essential elements required to describe the properties of
an ion-ion hybrid Alfven wave resonator in the laboratory. A
key simplifying assumption consists of neglecting contributions from the coupling between the shear and compressional
modes, i.e., the component of the wave magnetic field parallel to the confinement field is ignored. This approximation is
appropriate when the largest transverse scale encountered in
the system is smaller than the smallest ion skin depth, c=xpj .
In this parameter regime, typical of conditions in the LAPD
experiment, the compressional mode is evanescent over the
frequency interval for which the resonator behavior is possible. Accordingly, only the modes with shear polarization are
retained. It is also assumed that the plasma response is
adequately represented by the cold plasma dielectric tensor.
This implies that kinetic features such as electron Landau
damping and Bernstein modes are not included. Such effects
could play a role in the resonator behavior but their consideration is deferred to a future more comprehensive treatment.
The only spatial non-uniformity retained in the model is that
the axial component (in the z-direction) of the confinement
~0 B0 ðzÞ^z .
magnetic field varies with the coordinate z, i.e., B
This paraxial approximation is reasonable for the magnetic
well established in the LAPD experiment. Furthermore,
the density of all plasma particles (ions and electrons) is
assumed uniform. However, in the experiment, there are
radial density gradients that cause the radial eigenfunctions
to differ somewhat from the ideal Bessel functions characteristic of a uniform plasma cylinder model.
With the approximations previously described, the radial
and axial components of the complex amplitude electric field
ðEr ; Ez Þ of a shear Alfven wave at frequency x are governed
by the following coupled equations:
@ @Ez @Er
k02 e? Er ¼ 0;
@z
@z @r
(3)
1@
@Er @Ez
4pik0
k02 ek Ez ¼
jz ;
r
@z
@r
c
r @r
(4)
where jz represents the parallel component of the current
density of an antenna operating at frequency x. A cylindrical
coordinate system ðr; z; hÞ is used with the origin at the center of the plasma column. In Eqs. (3) and (4), it is assumed
further that the field pattern excited by the antenna is azimuthally symmetric.
Noting that the only spatial nonuniformity in the
model enters through the z-dependence of the ion gyrofrequency appearing in e? , Eqs. (3) and (4) can be decoupled
using a suitable radial Bessel function representation. In
describing the fields driven by an antenna of small radial
dimension, it is useful to use a continuous Fourier-Bessel
transform32,33
Ez ðr; zÞ ¼
1
ð
0
J0 ðk? rÞE~z ðk? ; zÞk? dk? ;
(5)
012111-3
Vincena et al.
Er ðr; zÞ ¼
1
ð
Phys. Plasmas 20, 012111 (2013)
J1 ðk? rÞE~r ðk? ; zÞk? dk? ;
(6)
J0 ðk? rÞj~z ðk? ; zÞk? dk? :
(7)
0
jz ðr; zÞ ¼
1
ð
0
The corresponding azimuthal component of the wave
magnetic field, which is readily detectable with B-dot magnetic loop probes, is given by
Bh ðr; zÞ ¼
1
ð
J1 ðk? rÞB~h ðk? ; zÞk? dk? :
(8)
0
With these transformations, the decoupled equations
consist of an axial differential equation for the BesselFourier amplitude of the radial electric field
@ 2 E~r e? 2
4pik? @ j~z
;
ðk? k02 ek ÞE~r ¼
2
@z
ek
k0 cek @z
(9)
and two algebraic expressions for the remaining field amplitudes
1
4pik0 ~
@ E~r
E~z ¼ 2
j
;
(10)
k
?
c z
@z
k? k02 ek
B~h ¼
1
4pk? ~
@ E~r
j
:
þ
ik
e
0
k
2 k2 e
c z
@z
k?
0 k
(11)
The next step consists of solving Eq. (9) for E~r . To simplify
the treatment of shear waves trapped within a magnetic well, the
shape of the well is approximated by three spatial regions in
which exact analytic solutions can be obtained. The low magnetic field portion of the well (bottom of the well) is approximated as a region of uniform magnetic field which is connected,
on both sides, by a magnetic field that increases linearly with
position, i.e., a magnetic ramp. The ramp eventually connects to
another region of uniform magnetic field of larger strength (top
of the well). The piecewise fits are shown as the red dashes in
Fig. 1(a). They are superimposed on the black continuous curve
that corresponds to the actual strength of the magnetic field used
in the experiments reported in Sec. IV. The bottom of the well
has total length of 2l1 while the overall length of the well is 2l2.
Figure 1(a) also illustrates the position and axial extent of the
antenna used in the experiment. It is symmetrically located at
the center of the well (i.e., r ¼ 0 and z ¼ 0) with axial extent
2a. The antenna is modeled as a uniform current channel of current density j0 and radius R. The effect of the radial feeds, necessary in an experiment, is ignored.
The general behavior of E~r can be understood by considering its behavior in the three regions of the model magnetic
field profile. At the bottom of the well, E~r corresponds to a
wave with parallel wave number k1 given by Eq. (2). Along
the ramp portions, for a wave frequency smaller than the
value of xii at the top of the well, E~r behaves as a trapped
mode that encounters a cut-off at positions z ¼ 6z0 , where
eR? ðz ¼ 6z0 Þ ¼ 0, and eR? refers to the real part of e? . Beyond
FIG. 1. (a) Experimental (solid black) and model (dashed red) magnetic field profiles of the resonator. The axial extent of the experimental and model antenna is
shown as a horizontal magenta line. (b) Real part of the perpendicular element of
the dielectric tensor corresponding to the curves in (a). Note the change in sign.
the cutoff point, E~r decays exponentially, as the mode is
evanescent. Accordingly, in the ramp region of the magnetic
well, the perpendicular dielectric is represented as the first
term in a Taylor expansion, i.e.,
e? @eR?
ðz0 zÞ þ ieI? ; z > 0;
@z
(12)
with an equivalent expression for z < 0. In Eq. (12), eI? refers
to a constant imaginary part of e? . It is introduced here to provide an ad hoc dissipation channel that prevents the eigenmodes driven by the antenna from attaining unrealistically large
amplitude levels. While beyond the scope of the present paper, eI? could be considered as modeling dissipation due to ion
heating or mode conversion. The inclusion of just collisional
dissipation due to electrons, entering through the imaginary
part of the parallel dielectric coefficient, eIk , is too weak to
limit significantly the resonantly driven response.
With the approximation in Eq. (12), Eq. (9) becomes an
Airy differential equation describing the wave reflection
occurring within the magnetic ramp. It is understood that the
derivative of eR? is evaluated at 6z0 , as is appropriate. Nonlinear phenomena, such as particle heating and flow generation, as well as mode coupling at the reflection point remain
topics for future study. Figure 1(b) illustrates the quality of
the approximation (red dashes) when superimposed on the
continuous value (black curve) of eR? computed for the
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Vincena et al.
Phys. Plasmas 20, 012111 (2013)
experimental conditions described in Sec. III. The magenta
line illustrates the axial location and extent of the antenna.
The source term in Eq. (9) is only present in the uniform
region at the bottom of the well, i.e.,
@ j~z Rj0
¼
J1 ðk? RÞ½dðz aÞ dðz þ aÞ:
@z
k?
(13)
8
>
>
<
~ ? ; zÞ involves seven unknown
The total solution for Eðk
constants Ci that are determined by matching functional
values at the edges of each region of the well, and a jump
condition at the end of the antenna determined by Eq. (13).
Their self-consistent evaluation yields the driven-modes
trapped within the resonator. For z > 0, the solution takes
the form
C1 eik1 z þ C2 eik1 z
C3 eik1 ðzaÞ þ C4 eik1 ðzaÞ
~ ? ; zÞ ¼
Eðk
C Aiðaðz z0 Þ þ ilÞ þ C6 Biðaðz z0 Þ þ ilÞ
>
>
: 5
C7 ek2 ðzl2 Þ
where
a¼
@eR? 2
ðk k02 eRk Þ=eRk
@z ?
1=3
(15)
and
l¼
eI?
a
@eR?
@z
:
(16)
In Eq. (14), k1 is the parallel wave number obtained
from Eq. (2) for the bottom of the well and k2 is the corresponding evanescence coefficient in the region at the top the
well. As usual, Ai and Bi refer to the two linearly independent solutions of the Airy equation. The piecewise solutions
in Eq. (14) are supplemented with the appropriate parity condition for an eigenmode about the point z ¼ 0, i.e., either
symmetric or anti-symmetric.
The solution given in Eq. (14) represents modes confined
within the magnetic well that are driven by an antenna whose
frequency x can be continuously varied. For each chosen
value of x, there is a solution. However, as the frequency
approaches a resonant frequency, the mode amplitude
becomes unrealistically large unless a dissipation mechanism
is present. In exploring the free eigenmodes of the resonator,
i.e., not continuously driven by an antenna, a slight variation
of the previously described scheme is implemented. Instead of
using a continuous Bessel-Fourier transform, as in Eqs. (5)–
(8), a discrete radial Bessel representation is employed, i.e.,
X
(17)
E~z;v ðzÞJ0 ðk0v rÞ;
Ez ðr; zÞ ¼
a>z
l1 > z > a
l2 > z > l1
z > l2
ðantenna regionÞ
ðbottom of wellÞ
;
ðrampÞ
ðtop of wellÞ
(14)
In this case, there is no need to match the solutions at the end
of the antenna. Also, in solving numerically for the pure (i.e.,
not driven) eigenfrequencies of the resonator, the collision frequency is neglected when evaluating ek , and l is set to zero.
The continuous red curve in Fig. 2 displays the frequency dependence of the calculated amplitude of the
antenna-driven, wave magnetic field at an axial position
z ¼ 0 cm, and radial location r ¼ 5 cm, for a plasma consisting of a 60% Hþ-40% Heþ ion mixture. In generating the
red curve, a value of l ¼ 0:2 is used in order to obtain a
continuous spectrum with damping comparable to that
observed in the experiment. The frequency axis is scaled to
the Hþ cyclotron frequency at the bottom of the well. The
horizontal arrows indicate the range of possible frequencies
that can be trapped within the resonator configuration of Fig.
1. It is seen that within the expected frequency range two
amplitude peaks are present; they correspond to trapped
eigenmodes. The axial dependence of the amplitude of the
peak located at x=XH ¼ 0:55 in Fig. 2 is illustrated in
Fig. 3; it corresponds to an axially symmetric Bh component.
v
Er ðr; zÞ ¼
X
E~r;v ðzÞJ1 ðk0v rÞ;
(18)
B~h;v ðzÞJ1 ðk0v rÞ:
(19)
v
Bh ðr; zÞ ¼
X
v
This method, again, decouples Eqs. (3) and (4) into a system
analogous to Eqs. (9)–(11), but with k? replaced by k0v
where J0 ðk0v Rp Þ ¼ 0 and Rp represents the plasma radius.
FIG. 2. Calculated frequency dependence of the azimuthal wave magnetic
field at r ¼ 5 cm and z ¼ 0. The red continuous curve corresponds to the
driven response; ad-hoc damping strength is included. Vertical blue lines
indicate the eigenfrequencies of ideal, undamped, and undriven eigenmodes,
as described in the text. The middle blue line corresponds to an axially symmetric mode and the other two blue lines to anti-symmetric modes.
012111-5
Vincena et al.
Phys. Plasmas 20, 012111 (2013)
III. EXPERIMENTAL SETUP
FIG. 3. Continuous curve is the calculated axial dependence of the amplitude of the azimuthal magnetic field corresponding to the peak on the red
curve, at x/XH ¼ 0.55, in Fig. 2. Red dashes are the magnetic field strength
of the model resonator. The bottom of the well has a field strength of 750 G
and the top of the well 1250 G. The black marks, labeled probes 1 and 2, correspond to the axial locations of B-dot probes used in experimental
measurements.
The other mode at higher frequency ðx=XH ¼ 0:72Þ is also
symmetric (not shown). The peak at x=XH ¼ 0:8 does not
correspond to a resonator eigenmode since it is outside the
range of possible trapped frequencies. It is associated with a
propagating wave that is not confined by the well. The peak
arises from constructive interference between the two ends
of the antenna, which launch the signals. The three vertical
blue lines in Fig. 2 indicate the values of the eigenfrequencies of ideal, undamped, and un-driven eigenmodes. The
driven peak response at x=XH ¼ 0:55 corresponds to the
middle blue line. The second driven peak at x=XH ¼ 0:72
does not have a corresponding blue line; it appears only
when dissipation is included because its frequency falls near
the edge of the trapping band. In general, the inclusion of
damping shifts the resonance downward in frequency, and
this causes the peak at x=XH ¼ 0:72 to appear in the driven
case. The right and left blue-lines correspond to antisymmetric modes that do not appear in the driven case, as
the perfect symmetry assumed in the model prevents the
antenna from coupling to these modes. In addition to the
axial eigenfunction, Fig. 3 shows the magnetic field strength
of the model resonator as red dashes. For this case, the bottom of the well has a field strength of 750 G and the top of
the well 1250 G. The black marks, labeled probes 1 and 2,
correspond to the axial locations of B-dot probes used in experimental measurements described later.
The experiments are conducted in the upgraded Large
Plasma Device34 (LAPD) at the Basic Plasma Science Facility at the University of California, Los Angeles. This is a cylindrical device with an overall length of 2070 cm and a
main chamber diameter of 100 cm, as sketched in Fig. 4. The
plasma is produced using an electrical discharge between a
heated, oxide-coated nickel cathode and a molybdenum
mesh anode on one end of the device. A movable, electrically floating copper mesh of 50% optical transparency is
swung into place (as noted) in order to terminate the plasma
at the end opposite the discharge source with a conducting
boundary to match the boundary at the anode, or is left open
to provide shear Alfven wave absorption via ion cyclotron
resonance in a magnetic beach configuration.35 The main
plasma column, between the conducting mesh boundaries, is
1660 cm in length (more than 5 wavelengths of typical
Alfven waves) and carries no net current. The typical column
diameter is 60 cm, which is more than 300 ion Larmor radii
but does not support propagation of compressional Alfven
waves for the frequencies used in these experiments. Highly
reproducible plasma discharges 15 ms long (more than a
thousand Alfven transit times) are repeated at 1 Hz; this
allows for the collection of large ensemble datasets using
probes with computer-controlled drives.
The plasma electron densities are typically (1–3)
1012 cm3, as measured using a swept Langmuir probe
calibrated with a microwave interferometer. This yields a
typical ratio Xe =xpe 0:3, where Xe and xpe are the electron gyrofrequency and plasma frequency, respectively.
Electron temperatures are 5–8 eV during the main plasma
discharge, while ion temperatures are approximately 1 eV.
This results in a low-beta plasma with b 2 104 . The
relevant length scale for the shear Alfven wave across
the background magnetic field is the electron inertial length
de ¼ c=xpe , while the parallel wavelength scales as the
ion inertial length di ¼ c=xpi . In the present experiment,
ne ¼ 1 1012 cm3, de ¼ 0:53 cm, and for a pure Hþ
plasma, di ¼ 23 cm.
The confining axial magnetic field is produced with a set
of solenoidal electromagnets controlled by 10 separate
power supplies. The device is typically run with a uniform
magnetic field strength of 300 G to 2000 G, but the independent supplies allow for a variety of field configurations. Since
the ion-ion hybrid frequency is proportional to the strength
of the magnetic field, for this study, a magnetic well
FIG. 4. (Upper) Top view of the upgraded
large plasma device showing the vacuum
vessel, magnetic field coils, and access
ports. (Lower) Schematic of the experimental setup. Note the expanded x-scale
for clarity. The radial legs of the antennas
are externally connected (as described in
the text) to an RF amplifier, driven by either single-frequency tone-bursts or a voltage impulse.
012111-6
Vincena et al.
configuration is formed near the center of the device to establish a resonator, as shown with the solid black line in Fig.
1(a). The axial dependence of the confinement magnetic field
consists of a low-field uniform segment that is ramped-up on
two sides to another region of larger uniform field. The ionion hybrid frequency, at which wave reflection occurs and
gives rise to axial wave trapping, is encountered in the ramp
portions of the well where the field strength increases.
Stable and highly reproducible plasmas can be obtained
that consist of two ion species with a concentration ratio that
can be continuously selected. Various gas mixtures can be
produced using fixed-flow settings in independent mass flow
controllers that regulate the partial pressures of the individual gases: hydrogen, helium, neon, and argon. However, for
the results presented here, only hydrogen/helium mixtures
are used. Four turbo-molecular pumps provide a steady-state
pumping speed to produce a constant neutral fill pressure to
within the error of measurement of both an ion gauge and a
commercial quadrupole mass spectrometer. Generally, the
partial pressures of the neutral gases do not reflect the relative concentration of the ion densities. The method for determining the ion concentration ratios is presented in Sec. IV.
The shear Alfven waves are launched using inductively
coupled loop antennas (designated as Antenna 1 and
Antenna 2 in Fig. 4), each consisting of a bare copper rod
(0.48 cm ¼ 0.96de in radius) bent into three legs of a rectangle and placed in the chamber. The axial leg extends along
the confinement magnetic field at the center of the plasma
column, as sketched in Fig. 4. The perpendicular legs extend
out of the chamber via electrically isolated, vacuum feedthroughs. The current path is closed outside of the vacuum
vessel with copper wire, which also forms one side of an isolation transformer, wound on a ferrite core. With this setup,
the antennas are electrically floating with respect to the
plasma. The second side of the transformer is connected to
the output of an RF amplifier. A programmable function generator drives the amplifier input. Monochromatic signals or
pulses can be applied. It is well known36,37 that in cylindrical
plasmas with a vacuum (or low density) gap between the
main plasma and the vacuum vessel, there is the possibility
of exciting radial eigenmodes of the fast (compressional)
Alfven wave; however, previous experiments in LAPD using
similar antennas in plasmas with either a single ion species,38–40 or two ion species,30,31 have not observed such
modes to be excited in the explored range of frequencies.
Wave magnetic fields are measured using magnetic
induction probes (i.e., B-dot loops). These probes are constructed of pairs of oppositely wound induction coils with
0.3 cm diameter loops of 50 turns each. A probe comprises 3
such coil pairs with each orthogonal pair measuring the time
varying magnetic field along a Cartesian direction. The time
series data for each probe are digitized at 12.5 MHz and
recorded with 14-bit resolution. Using the phase-locked experimental setup, the time series from between 2 and 8
plasma discharges are averaged together before storage to
improve the signal-to-noise ratio. After storage, each probe
can be automatically moved (using computer-controlled
stepper motors) to a new spatial location, or the wave frequency can be changed as desired, and the process repeated.
Phys. Plasmas 20, 012111 (2013)
The recorded voltages are proportional to dB/dt, and signals
are numerically integrated to produce B(t). The wave magnetic fields are measured to be small compared to the background magnetic field ðdB=B0 < 104 Þ, so that the radiated
field magnitudes are taken to be linearly proportional to the
amplitude of the antenna current; since the inductance of the
antenna circuit makes the current amplitude a function of
frequency, the magnetic fields presented here have been divided by the measured amplitude of the antenna current as a
function of frequency.
IV. EXPERIMENTAL RESULTS
A. Uniform field
Before presenting data from the magnetic well configuration, it is instructive to review the behavior of shear Alfven
wave propagation in a two-ion plasma with a uniform confining magnetic field. These observations are performed for a
helium-hydrogen plasma at a constant 750 G field. A general
theoretical treatment and specific experimental results for
helium-neon plasmas are given by Vincena et al.30 Figure 5
presents a color contour plot of the measured magnitude of
the y-component of the wave magnetic field radiated from
Antenna 2 (shown in Fig. 4) as a function of frequency and
distance along the line y ¼ 0, where jBh j ¼ jBy j. The data
were acquired with 0.5 cm spatial resolution and 68 frequencies at each location using tone bursts of 20 cycles. The
lower frequency bound is chosen so that the magnetic field
probe (at an axial distance of 528 cm) is at least one parallel
wavelength away from the antenna in the lower propagation
band; the upper bound, x=XHe ¼ 4, is the hydrogen cyclotron frequency. The ion concentration ratio (discussed in
detail later) is nHe =nH ¼ 1, which yields an ion-ion hybrid
FIG. 5. Normalized radiated wave magnetic field amplitude jBy j as a function of frequency and position perpendicular to the uniform confinement
magnetic field (750 G) along the line y ¼ 0. The frequency is scaled to the
helium cyclotron frequency. The upper and lower propagation bands are
both evident and separated by a clear propagation gap. Waves are launched
using Antenna 2 and detected at z ¼ þ528 cm for a Hþ-Heþ plasma of equal
ion densities. The data were acquired with 0.5 cm spatial resolution and 68
frequencies using tone bursts of 20 cycles.
012111-7
Vincena et al.
Phys. Plasmas 20, 012111 (2013)
cutoff frequency of xii =XHe ¼ 2. In this case, the end mesh
shown in Fig. 4 is swung out of the way in order to prevent
reflections, as the antenna launches waves in the positive and
negative z directions. Several features are evident in Fig. 5:
(1) the field is zero on axis, rises in magnitude away from
the axis, reaches a maximum value, and then decays. This is
consistent with the azimuthal magnetic field generated by a
central axial current channel provided by electrons, with the
current closure provided by radial ion polarization current;
(2) the pattern is roughly azimuthally symmetric, with
slightly smaller amplitudes for x < 0, which may be due to
the magnetic field probe (which enters from positive x) perturbing the current channel; (3) the upper ðxii < x < XH Þ
and lower ðx < XHe Þ propagation bands as well as the propagation gap ðXHe x xii Þ are all present. In addition, the
peak magnitudes of the axial (Bz) and radial (Bx) components are, respectively, 8% and 15% those of the By component over the entire frequency range; this is consistent with
the shear (rather than compressional) Alfven mode.
B. Concentration ratio
Since xii determines the lower boundary of the upper
propagation band, a direct measurement of this cutoff frequency also provides a diagnostic for the concentration ratio
of the two ion species. This can easily be seen using a dimensionless reformulation of Eq. (1)
2ii 1Þðl212 x
2ii Þ1 ; 1 x
ii l12 ;
q12 ¼ l12 ðx
(20)
ii xii =X1 , the ratio of the two ion densities is
with x
q12 n1 =n2 , and the mass ratio is l12 m1 =m2 > 1 (i.e.,
“2” labels the light ion). Since the ion masses are known
with great accuracy, the main uncertainty for the density ratio, Dq12 , arises from the uncertainty in determining the
ii . The density ratio
scaled ion-ion hybrid frequency, Dx
ii jDx
ii ,
uncertainty is calculated using Dq12 ffi j@q12 =@ x
ii Dx
ii ðl212 1Þðl212 x
2ii Þ2 . When
which yields Dq12 ¼ 2x
studying the ion-ion resonator, however, it is somewhat
more intuitive to consider the fraction of the lighter ion density to the electron density g2 ¼ n2 =ne since only waves in
the upper propagation band can be trapped in the resonator.
Rewriting the plasma quasi-neutrality condition ðn1 þ n2 ¼ ne Þ
as q12 ¼ ðg1
2 1Þ along with Eq. (20) yields
g2 ¼
2ii Þ
ðl212 x
;
2
ii Þðl12 1Þ
ðl12 þ x
ii l12 ;
1x
(21)
with corresponding uncertainty
Dg2 ¼
ii Dx
ii l12 ðl12 þ 1Þ
2x
:
2ii Þ2 ðl12 1Þ
ðl12 þ x
Figure 6(a) displays frequency power spectra jBy ðxÞj2
obtained from time-series data acquired at a single location:
x ¼ þ5 cm, z ¼ þ528 cm, for four different concentrations of
hydrogen (gH ¼ g2 ) in plasmas comprising hydrogen and helium ions at a fixed electron density ne ¼ 1 1012 cm3 and
a confining field of 750 G. The cross-field location is chosen
FIG. 6. (a) Fourier power spectra of wave magnetic fields for four values of
the hydrogen ion fraction, gH ¼ nH =ne obtained at x ¼ þ5 cm and
z ¼ þ528 cm. Waves are launched with a current pulse applied to Antenna 2,
as described in the text for a uniform magnetic field of 750 G and Hþ-Heþ
plasmas. Spectra are shown relative to the maximum observed value, and
frequencies are scaled to the helium gyro-frequency. The values for gH are
ii (marked
determined by identifying the scaled ion-ion cutoff frequency x
with a color-coded triangle for each spectrum) and the use of Eq. (21). A second propagation gap is evident at x ¼ 2XHe when the ion-ion hybrid frequency lies below this harmonic, and increases the uncertainty of determining
xii in this region. ((b)-(e)) Measured magnetic field wave amplitude,
jBy ðx; xÞj, along the line y ¼ 0 for the same setup and values of gH given in
(a). For each subplot, the magnitude is scaled to its maximum value.
012111-8
Vincena et al.
to position the probe where the received signal from Fig. 5 is
largest, which avoids the low fields due to the central current
channel and from the radial decay of the radiated pattern.
Here, waves are again launched by Antenna 2, but are generated with a 200 mV input pulse of 300 ns width to the 60 dBgain RF amplifier instead of a tone burst. The resulting
current waveform IðtÞ in the antenna has a peak amplitude of
0.25 A. The consistency of the two methods will be shown
later. The power spectra are computed using fast Fourier
transforms (FFT’s). The pulse method of excitation allows
one to essentially obtain vertical cuts (either amplitude or
power) of Fig. 5 at a chosen spatial location in a matter of
seconds using a digital oscilloscope with an FFT math function. With the frequency axis scaled to the ion cyclotron frequency of the heavier species ðX1 ¼ XHe Þ, the value of the
ii , is found by noting the
scaled ion-ion hybrid frequency, x
lower bound of the upper propagation band in each power
spectrum. These frequencies are identified by the colored triangles on the horizontal axis of Fig. 6(a), and are used in Eq.
(21) to obtain the values of gH shown in the numerical insert
in this figure. It should be noted that, in the present experiment, when xii lies below the second harmonic of helium,
2XHe , a narrow propagation gap is also observed in the vicinity of this harmonic. This causes an increase in the estimate
ii . This additional gap is an effect associated with finite
of Dx
ion temperature, which results in Bernstein mode-like features, as discussed in Ref. 30.
Using these methods, it is possible to rapidly achieve a
desired value of gH (equivalently an ion concentration ratio)
by adjusting the relative fill pressures of the neutral gases
while monitoring the resulting spectra. In addition (as is
done in the present study), a constant, line-integrated electron density can be maintained by monitoring the real-time,
phase-integrated output of a microwave interferometer while
adjusting the absolute fill pressures. Other methods exist for
determining the ion concentration ratio. For plasmas with
low density (ne < 1010 cm3), low temperature (Te < 4 eV
and Ti < 1 eV), and with low ionization fraction (0.1%),
the spectroscopic-Langmuir technique, described by Ono
et al.41 can be used. Using the group velocity of ion acoustic
waves and the electron temperature, as reported by Hala and
Hershkowitz42 in a multi-dipole plasma, also provides an
accurate ion concentration measurement for a two ion species plasma in that environment. Reflectometry, which
exploits the perpendicular resonance and cutoff of externally
launched compressional Alfven waves, has been reported by
Watson et al.43 as a diagnostic for the ion concentration
ratios in the DIII-D tokamak.
The pulsed method of wave excitation may also be
employed as the magnetic field probe is moved to different
locations perpendicular to the background magnetic field.
The resulting amplitude spectra are shown as color contour
plots in Figs. 6(b)–6(e), which correspond to the four values
of gH determined in part (a). The amplitudes have been normalized to the maximum value within each subplot. Note
that Fig. 6(d) from the pulsed excitation may be directly
compared with Fig. 5, which is generated with the same
plasma parameters but using a different excitation method.
As a function of both distance and frequency, the contour
Phys. Plasmas 20, 012111 (2013)
plot from the pulsed excitation exhibits a remarkable fidelity
in the amplitude distribution with the tone-burst result, which
demonstrates the consistency between the two methods.
C. Resonator configuration
The experimental magnetic field profile shown in Fig.
1(a) is produced in the device, while Antenna 2 is removed
and replaced by Antenna 1. This antenna is driven in the
same manner used for the results of Fig. 5—using 20-cycle
sinusoidal wave packets. The same magnetic well configuration is explored for three different concentration ratios of
hydrogen and helium ions, as presented in Fig. 7. Note that,
for the resonator, it is within the upper (hydrogen) propagation band that the trapped eigenmodes are formed. Therefore,
the experimental results are presented with the frequencies
scaled to the hydrogen cyclotron frequency XH ; due to the
axial field variation, this scaling is performed using the value
of XH at z ¼ 0, that is, at the bottom of the well. Figures
7(a)–7(c) display the spatial variation of the ion cyclotron
frequencies and ion-ion hybrid frequencies for: (a)
gH ¼ 0:25, (b) gH ¼ 0:38, and (c) gH ¼ 0:61. Ion concentration ratios are determined using the method described previously, using Antenna 1 and measured with Probe 2 (shown
in Fig. 4). The determination is performed for each case
using uniform magnetic field profiles of 1250 G and 750 G—
the extremum fields for the resonator. The results are consistent between these two field values and are assumed to hold
in the nonuniform profile. In Figures 7(a)–7(c), the lower
propagation bands ðx < XHe Þ are shaded in yellow; the
propagation gaps ðXHe < x < xii Þ are white; the upper
propagation bands are shaded in cyan; and overlaid with a
third color within the upper bands (matching the color of the
subplot to the immediate right (d-f)) are the wave trapping
regions. In the trapping regions, the waves satisfy two criteria: (1) They can be launched by the antenna, i.e., x < XH at
the center of the well (z ¼ 0), and (2) their frequency can
match the reflection condition x ¼ xii ðzÞ at a location within
the plasma column. Note that, for all three cases, tunneling
between potential eigenmodes in the resonator and the lower
propagation bands is prevented since the minimum xii in the
system is chosen to be everywhere greater than the maximum value of XHe . Also note that, for gH ¼ 0:25, the upper
boundary of the trapping region is determined by the x <
XH ðz ¼ 0Þ criterion, while a substantial propagating upperband exists for the gH ¼ 0:61 case.
The resulting wave amplitude spectra jBy ðxÞj (measured
with Probe 1 at z ¼ þ128 cm and x ¼ þ5 cm) are presented
in Figs. 7(d)–7(f). Each of these sub-figures corresponds to
the case to its immediate left (a-c). The waves are launched
using the 20-cycle tone burst technique. However, the data
displayed are obtained after the antenna current is switched
off, so that a comparison to the theoretical predictions of the
un-driven eigenmode case is appropriate. Specifically, the
displayed spectra are obtained from FFTs of a 10-cycle, Hanning-windowed time series, starting immediately after termination of the antenna current. For this linear study, each
spectrum is normalized to its maximum value; although, as
in Fig. 6, the relative noise floor increases with decreasing
012111-9
Vincena et al.
Phys. Plasmas 20, 012111 (2013)
FIG. 7. ((a)-(c)) Axial variation of the upper (cyan) and lower (yellow) propagation bands; the cold-plasma propagation gap (middle, white); and the trapping
regions (color-coded) for three concentrations of hydrogen ðgH ¼ nH =ne Þ in a Hþ-Heþ plasma. ((d)-(f)) Normalized wave magnetic field amplitude spectra
measured at x ¼ þ5 cm and z ¼ þ128 cm as launched in the resonator by Antenna 1. Each of these three subplots corresponds to the case to its immediate left.
The semi-transparent horizontal bars delineate the trapping regions, and the arrows mark the manually identified modes. The broad "peak" in (f) for 0:7 <
x=XH < 1 is a continuum of freely propagating modes—not another resonance.
concentrations of hydrogen. For reference, within each subplot, a semi-transparent horizontal bar is drawn arbitrarily at
the normalized height of 0.5 and covers the frequency range
of potential resonator eigenfrequencies. Within these bands,
individual spectral peaks are manually identified and marked
with vertical arrows to indicate the experimentally observed
eigenfrequencies for each case. The upper propagation band
in Fig. 7(f) is clearly visible between 0:7 < x=XH < 1 and
should not be interpreted as evidence of a fourth eigenfrequency for that case.
The magnetic field spectra may also be used to calculate
the Q-values for the resonator eigenmodes. In order of
increasing frequency, the Q-values for the eigenmodes of
case gH ¼ 0:25 are QðgH ¼ 0:25Þ ¼ [15,12,11], while the
remaining cases yield QðgH ¼ 0:38Þ ¼ [8.6,11,12] and
QðgH ¼ 0:61Þ ¼ [11,13,15]. While there is no clear trend
among the three cases, the overall range Q-values indicates a
resonant cavity with relatively large dissipation. Estimates of
eigenmode damping based on the inclusion of collisions or
Landau damping yield much larger Q-values. At this stage,
the low experimental Q-values are not well understood, but
other possible energy loss channels are discussed in Sec. V.
Figure 8 displays the results of overlaying the observed
resonator eigenfrequencies from the three cases of Figures
7(d)–7(f) with the theoretically predicted un-driven eigenfrequencies and displayed as functions of a continuous variation
in gH . The range of possible eigenfrequencies is bounded on
the lower end by xii at z ¼ 0 (lower red curve) and the continuum of waves propagating above xii at the high-field ends
of the well (upper red curve). Again, for the frequencies
used, the antenna does not launch shear waves above the
hydrogen cyclotron frequency at z ¼ 0 (marked by the orange
horizontal line). The theoretical values of the resonant frequencies are determined by the model described in Sec. II.
012111-10
Vincena et al.
Phys. Plasmas 20, 012111 (2013)
FIG. 8. Comparison of measured eigenfrequencies to theoretical predictions
for three ratios of hydrogen-to-electron densities, gH, in the heliumhydrogen resonator. Frequencies are scaled to the hydrogen cyclotron frequency at the center of the resonator (z ¼ 0). The dashed black lines are the
theoretical predictions and the data points are color-coded to the marked
peaks in the spectra shown in Figs. 6(d)–6(f). The vertical semi-transparent
frequency bands correspond to the horizontal bands of the same color in
Figs. 6(d)–6(f). The solid red and orange lines bound the range of possible
eigenfrequencies, as explained in the text.
The predicted eigenfrequencies of undriven modes are
shown as dashed lines. Both even and odd parity modes are
represented, as the undriven case does not have the strict
symmetry requirement inherent in the idealized driven-case
and; also, the exact form of the source is removed from consideration. For the highest concentration of hydrogen
ðgH ¼ 0:61Þ, both the number and frequency of the observed
spectral peaks match well with the theoretical prediction. For
lower hydrogen concentrations, the agreement is not as
good, which is primarily attributed to the larger width of the
observed peaks in comparison to the predicted frequency
spacing. This feature makes it difficult to discern individual
peaks from broader, merged peaks. This topic is explored
further by examining the low quality factors of the
resonances.
D. Wavelet analysis
In order to analyze the spectral content of the fluctuating
magnetic fields radiated by a pulsed antenna, a continuous
wavelet transform is used. The measurement is performed in
a Hþ-Heþ plasma with gH ¼ 0:45 and ne ¼ 1:3 1012
cm3. The impulse excitation provides power over the entire
range of modes potentially trapped in the resonator and
allows observation of their decay. A Morlet wavelet of order
24 is employed, and the separation between scales is chosen
to ensure that the resonances can be resolved. The transform
is applied to an xy-plane (transverse to the confinement magnetic field) of data taken with probe 1 for the magnetic configuration shown in Fig. 1(a). A characteristic wavelet
amplitude spectrum is shown in Fig. 9(a), which is obtained
at a radial distance of r ¼ 17.5 cm from the center of the
antenna at axial location, z ¼ 128 cm. The frequencies are
scaled to the hydrogen cyclotron frequency and time is
scaled to the cyclotron period of hydrogen, sH, at z ¼ 0. The
FIG. 9. (a) Color contours of Morlet wavelet amplitude of magnetic field
fluctuations show the response of the Hþ-Heþ resonator after excitation with
a current impulse of width Dt ¼ sH at t ¼ 0. The hydrogen concentration is
gH ¼ 0:45 and the probe is at z ¼ þ128 cm and x ¼ þ17.5 cm. (b) Instantaneous wavelet spectra obtained from vertical cuts in (a) at four different
times. Vertical dashed lines are the predicted resonator eigenfrequencies.
red lines are reference frequencies of interest, and the
trapped modes should fall between the ion-ion hybrid frequencies at 750 G and 1250 G, respectively. The blue contour is a confidence measure, which outlines when the
amplitude of the wavelet transform of the antenna current
has fallen to 10% of its peak value. Although the antenna
pulse only lasts one cyclotron period, it is artificially broadened by the width of the Morlet wavelet. When looking in
the trapped band, it is clear that there are at least two frequencies that last for significantly longer time than the rest
of the signal. These are candidate resonant modes.
To obtain a better understanding of how the peaks decay
in time, a projection is taken from the wavelet amplitude
contour plot at four different times (t=sH ¼ 10, 15, 20, and
25), as illustrated in Fig. 9(b). The vertical dashed lines
shown in Fig. 9(b) are the theoretically predicted values of
the resonator eigenfrequencies for the un-driven case, computed for the relevant experimental parameters. To determine the characteristic decay time of the observed peaks in
the frequency spectrum, a temporal slice of the wavelet
012111-11
Vincena et al.
FIG. 10. Exponential fits to the temporal decay of the wavelet amplitude for
the two prominent spectral peaks (x0 =XH ¼ 0.56 and 0.71) in Fig. 9(b).
Phys. Plasmas 20, 012111 (2013)
amplitude spectrum is extracted for the prominent peaks at
x0 =XH ¼ 0.56 and 0.71. Assuming an exponential decay of
~
the amplitude of the signals, jBðx;
tÞj / expðt=2sÞ, the
power decay time, s, can be determined from a semi-log plot
of the wavelet amplitude as a function of time. This is shown
in Fig. 10, where a fit is made to the exponential decay
region. For the two peaks, x0 =XH ¼ 0.56 and 0.71, the decay
times are s ¼ 4.23 and 3.52 ls respectively. Thus, the quality
factors Q ¼ sx0 of the resonances are Q ¼ 17.3 and 18.0.
These values are slightly higher, but in reasonable agreement
with the range of Q values ð8:6 Q 15Þcomputed using
the spectral widths obtained from the 20-cycle tone burst excitation method.
To test that the resonances identified from the wavelet
analysis exhibit a global structure in the plasma column, frequency dependencies of the wavelet amplitude at t=sH ¼ 25,
similar to those shown in Fig. 9(b), are shown in Fig. 11(a),
for four different locations. The four sample positions are
each located in a different quadrant relative to the antenna,
as indicated by the asterisks in the insert (color coded). The
color-coded curves are normalized to the maximum value
attained within the frequency range displayed. The peaks
near the theoretically predicted resonances (vertical dashed
lines) do not change appreciably with position across the
plasma column. Additionally, to verify that these global
properties are also present in the axial direction, similar displays are made in Fig. 11(b) of data from probes 0, 1, and 2,
whose z-positions are shown in Fig. 4. It is seen that the
prominent peaks in the transverse plane that correspond to
predicted eigenfrequencies also display a global structure
along the axial direction.
V. CONCLUSIONS
FIG. 11. (a) Wavelet spectra of magnetic fluctuations at t=sH ¼ 25 excited
by a current pulse, as in Fig. 9(a). The four curves correspond to different
radial positions in a plane transverse to the confinement magnetic field. The
relative positions sampled are indicated by the color-coded asterisks in
the inserted grid. The grid is 40 40 cm in size. The vertical lines indicate
the theoretically predicted eigenfrequencies. (b) Similar to (a) but sampling
at different axial positions corresponding to the locations of probes 0, 1, and
2, shown in Fig. 4. The radial location is indicated by the black asterisk in
part (a). The results demonstrate that the observed spectral peaks at the predicted eigenfrequencies are global modes.
This theoretical and experimental investigation has
established the fundamental aspects of an Alfven wave resonator based on the concept of wave reflection along the confinement magnetic field at a spatial location where the wave
frequency matches the local value of the ion-ion hybrid frequency. Measurements in a controlled laboratory environment using a Hþ-Heþ mixture are found to be well described
by a cold plasma model that retains only the axial variations
in the confinement magnetic field. Good agreement is found
between the measured frequencies of resonant modes,
excited by a tone burst, and the resonator eigenfrequencies
predicted by the model. The spectral widths of the eigenfrequencies yielded relatively low resonant Q-values for the
resonator, ranging from 8.6 to 15. Using wavelet analysis, it
has been demonstrated that the observed resonant peaks correspond to radial and axial global modes, trapped within the
resonator region, where the magnetic field is lower. The
wavelet technique has also extracted the effective resonant
Q-value for the resonator eigenmodes in the pulsed excitation mode; they attain a value of Q 17, consistent with the
tone-burst results. Determination of the underlying damping
mechanisms warrants a detailed future investigation. Candidate mechanisms include the possible excitation of ion Bernstein modes, transit time acceleration in the reflection region,
and enhanced wave refraction due to a combination of
012111-12
Vincena et al.
magnetic field-line bending and radial density gradients.
However, ray tracing studies by Mithaiwala et al.26 have
identified that higher order dispersive effects cause a radial
focusing of resonator modes for magnetospheric conditions.
In summary, this study has conclusively validated the previously predicted existence of an ion-ion hybrid Alfven wave
resonator in magnetized plasmas with two ion species. These
laboratory results provide confidence and guidance for the
future application of the concept of an ion-ion hybrid resonator to space plasmas and fusion devices.
ACKNOWLEDGMENTS
This study was partially sponsored by ONR under the
MURI Grant No. N000140710789. The experiments were
performed at the Basic Plasma Science Facility (BaPSF) at
UCLA, which is jointly supported by DOE-NSF. The work
of W.A.F. was supported by DOE Grant DE-SC0007791.
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