Review of Laboratory Experiments on Alfvén waves and their Relationship
to Space Observations
Walter Gekelman
Physics Department. University of California, Los Angeles, 90095-1696
Hannes Alfvén predicted the existence of a hydrodynamic wave in a
perfectly conducting fluid in 1942. It took six years before this
discovery was accepted and ten years before Alfvén waves were
first observed in the laboratory. Now it is widely recognized that
these waves are ubiquitous in space plasmas and are the means by
which information about changing currents and magnetic fields are
communicated. Alfvén waves have been observed in the solar wind,
are thought to be prevalent in the solar corona, may be responsible
for parallel electric fields in the aurora and could cause particle
acceleration over large distances in interstellar space. They have
also been considered as a candidate for heating thermonuclear
plasmas, and are potentially dangerous to confinement. Alfvén
waves have been difficult to observe in basic laboratory experiments
because of their low frequencies and long wavelengths. In this paper
we will present a review of plasma Alfvén wave experiments
performed in recent years. The quality of the laboratory data have
paralleled advances in plasma sources and diagnostics. In the past
few years the quantum jump in data collection on the Freja and FAST
missions have lead to re-evaluation of the importance of these waves
in the highly structured plasma that was probed. Recent laboratory
experiments have examined, in great detail, shear waves generated
by filamentary currents in both spatially uniform and striated
plasmas. Tone bursts, short pulses and interference effects have
been studied with emphasis on structures of the order of the skin
depth, c/ωpe. These are features of significant interest to the space
community.
In fact, it appears the phenomena observed in
laboratory experiments show striking similarities to what has been
observed in space. A comparison of these results will be given.
I. Introduction
Alfvén waves are ubiquitous in space plasmas. They are the means
whereby magnetized plasmas communicate internal information about
1
changing currents and magnetic fields. Their enormous importance is
becoming more apparent with every new satellite and rocket flight. Space
plasmas are tenuous and collisonless but, at the moment, multipoint
measurements are not possible within them because of the great cost of
individual satellites.
In the seemingly other world of fusion physics, Alfvén waves have
been employed in various heating schemes for both ions and electrons. They
are a source of concern in fusion plasmas in which energetic alpha particles
may occur. These particles can destabilize toroidal Alfvén modes which, in
turn, can ill affect the particle confinement.
Despite their key role in many areas of plasma physics there have been
relatively few “basic “ plasma experiments concerning these waves. The
reason for this is that they occur at low frequencies, generally below or near
the ion cyclotron frequency, ωci making them subject to collisional damping.
Fusion plasmas are collisionless, but hot and dense enough to destroy
internal probes and antennas. The basic MHD dispersion relation for the
shear Alfvén wave is ω = k||V A , where V A =
B2
. At ω = ωci, , λ || ∝ 1/√n.
4πρ
Relatively large densities are required so that waves with reasonable parallel
wavelengths will “fit” into the laboratory device. There is a limit to the
density of a cold plasma in that the Coulomb collision frequency which is
proportional to the density will become comparable to the wave frequency.
Recently plasma sources have been developed in which systematic
studies of the basic properties of these waves are possible, and are progressing.
This work will review how the laboratory studies of these waves have
evolved in the past few decades. This will be preceded with several examples
of spacecraft and astronomical examples of Alfvén waves. It is by no means
an exhaustive survey of space related observations but, from the point of
view of a laboratory experimentalist, underscores the enormous importance
of these waves, and serves as a motivation to study them.
2
II. Alfvén waves in space plasmas.
In 1942 Hannes Alfvén proposed the existence of hydrodynamic waves.
This was met with skepticism since it was believed that any conducting fluid
would immediately short circuit wave electric fields. Alfvén also believed
that any theory of cosmic phenomena must also agree with laboratory
experiments [Fälthammar and Dessler, 1995]. A survey, by Lundquist of some
of the investigations into solar physics and cosmic radiation appears as early
as 1952. Notwithstanding the nonexistence of dark matter, the Universe is
upwards of 99% in the plasma state. The great variety of structure in
astrophysical plasmas, now apparent from the wealth of data produced by the
Hubbell space telescope, suggests plasma physics based explanations. Many
astronomers shy away from this approach and act as if the stars are giant gas
balls. In the opening of a recent publication (1995) V. Jatenco-Periera states
“Although there are many astrophysical problems, few plasma physics
solutions have been suggested”. The paper goes on to discuss a variety of
phenomena such as bipolar jets, stellar clouds, and quasar clouds. The waves
in these scenarios are conjectured to be produced by reconnection which
occurs in highly distorted flux tubes. The canonical picture of this is shown
in figure 1.
3
Figure 1: Illustraton of the instability of magnetic flux tubes. They initially twist and reconnect in the
shaded region with the concurrent emission of Alfvén waves. [adapted from Jatenco-Pierra, 1995]
Flux ropes or braided magnetic fields are twisted by plasma motions
and when oppositely directed magnetic field lines are forced together
magnetic field line reconnection occurs. Reconnection can result in localized
heating and directed particle motion, and as figure 1 illustrates the emission
of Alfvén waves from the reconnection region. Several laboratory
experiments [Gekelman et . al, 1982, Ono et al., 1996] have directly observed
reconnection from measured changes in magnetic field topology and
evidence for it in the form of ions jetting from the reconnection site at speeds
approaching VA. However there is no existing laboratory experiment with a
plasma of sufficient size to fit several Alfvén wavelengths into the area
where particle jetting is usually observed. Scaling this type of experiment to
reflect the solar corona, or any astrophysical object is a great challenge, as yet
unmet.
Enormous amounts of energy are released in solar flares, about 10 32
erg. During the explosive phase of a flare which lasts from 10-30 sec the rate
of expansion of the flare increases from about 10 km/s to greater than 100
km/sec. A long duration flare, on the other hand can last from hours to days.
There are many theories which try to explain the various processes
which go on in such a flare. In an early discussion Piddington (1974)
hypothesizes that helically twisted (shear) Alfvén waves occur within the
subsurface solar magnetic fields. The flare is a complex sequence of events
accompanied by the emergence of flux from the solar surface as illustrated in
Figure 2.
4
Figure 2: Two sunspots labeled N and S have equal amounts of positive and negative flux. Some field
lines are closed and others are open. B 1 is a twisted flux “stand” which may be energized from below to
carry Alfvén waves. B2 is similarly excited and projects out into the low corona. B3 is a field line which
is grossly deformed by a very nonlinear Alfvén wave. [adapted from Piddington, 1974]
Emerging flux tubes such as B1 and B2 are comprised of helically twisted
field lines or “flux ropes” [Babcock, 1961]. Subsurface motions are likely to be
responsible for this twisting. After they emerge they can untwist ( Bϕ ≈ 0 ).
This untwisting proceeds as Alfvén waves migrate along the tubes. Alfvén
waves could play a major role in the dynamics of the flares. It is agreed that
magnetic field line reconnection, or the transfer of magnetic energy to heat
and directed motion must play a major role in the energy release. How the
reconnection actually proceeds is still a subject of research and debate but
Alfvén waves must be an important ingredient in the twisting of the “flux
ropes” in a magnetized plasma. They may be part of the “initial condition”
which triggers the reconnection and are certainly radiated from the
reconnection region in a plasma in which the ions are magnetized.
The next step in the chain of events which starts on the sun is the
generation of the solar wind. Data from the Helios 2 satellite which was
analyzed by Hollweg et al. (1982) are consistent with “Alfvén waves making a
significant contribution to high-speed streams”. The measurements were
5
made when the satellite was being eclipsed by the sun. A linearly polarized S
band microwave source was located on the spacecraft and its Faraday rotation
(which measured the fluctuations in magnetic field and density) and
frequency shift (which is related to density fluctuations) were both monitored
on the Earth. A careful analysis indicated that if the Faraday rotation
fluctuations were due solely to density fluctuations the radial variation of the
coronal magnetic field strength could not be predicted accurately. It turned
out the most of the observed fluctuations ( ≈ 96% ) are associated with Alfvén
waves. These waves were conjectured [ Hollweg, 1981] to be associated with
the twisting of magnetic flux tubes at the top of the convection zone (as in the
previous example). The group concludes that coronal turbulence is “most
easily thought of as consisting of long, radially aligned filaments”, and the
Alfvén energy flux could substantially contribute to the solar wind
momentum. A calculation by Parker (1991) indicated that resonant
absorption and thermal conduction both dissipate Alfvén waves in the solar
corona, and could provide the heat input and wave pressure necessary to
accelerate the solar wind to the fast stream speeds (800 km/sec) observed at
the Earth. Since then the Faraday rotation measurements have been refined
[Sakuri and Spangler, 1995] with the result that the amplitudes have been
revised downwards. The jury’s decision on how the solar wind is created
and accelerated is not yet in; at present there is no quantitative model for the
heating and acceleration of the solar wind.
The solar wind contains a complicated mix of Alfvénic fluctuations
and velocity streams. Satellites have magnetometers which measure the
wave magnetic field and particle detectors which measure the velocity of the
solar wind. Coleman (1966) saw propagating MHD waves in the solar wind
satellite data and interpreted them as Alfvénic turbulence. Data analysis is
complicated because the waves are embedded in a plasma which, in the case
of Mariner data [Belcher and Davis, 1971], is convected past the spacecraft at
400 km/sec which is roughly 8 times the wave propagation speed. In spite of
6
this it was determined that the waves are Alfvénic (there is a close correlation
between velocity and density fluctuations) and for the most part are
propagating outward from the sun. Embedded in the streams are sharp
discontinuities. These fluctuations have been conjectured to be “spaghettilike or tubelike” [McCracken and Ness, 1966, Mariani et al., 1973, Thieme et
al., 1990]. They seem to be filamentary with long correlation lengths along
the background magnetic field (> 1.5X106 km) and short ones (≈ 105 km) across
the field [Matthaeus et al., 1990] Interestingly laboratory studies of magnetic
field fluctuations in fusion confinement devices, which support Alfvén
waves [Zweben at al., 1979], have observed magnetic structures with long
correlation lengths along the background magnetic field, and correlation
lengths of the order of the colissionless skin depth, or ion gyroradius across
the field.
The topic of Alfvénic structure is thoroughly reviewed by Tu and
Marsch (1995). A second review paper by Goldstein and Roberts (1995)
addresses MHD turbulence in the solar wind and discusses several, as yet,
unsolved issues. The origin of Alfvénic fluctuations in the solar wind is
basically not yet understood, however it has been established that turbulent
and nonlinear processes, as well as structures within the plasma play a key
role in the behavior of the solar wind as it expands past the Earth.
There are numerous reports of signals from spacecraft that have been
interpreted as encounters with Alfvén waves. Another early work [Hayward
and Dungey, 1982 ] describes a crossing of the plasma sheet boundary layer by
ISEE1 and ISEE2 satellites. waves. Figure 3 shows a schematic of the plasma
sheet and a hypothesized source of Alfvén waves in the far tail. In the insert
at the bottom of figure 3, one sees a sharp drop and subsequent increase in By
as the plasma sheet boundary is crossed. After the initial crossing (which lasts
about a minute) the data becomes complicated (during the following 6
minutes) and is interpreted as motions of the boundary layer over the
spacecraft. Hodograms of the initial crossing show that only the direction of B
7
changes, not its magnitude. This leads the authors to interpret the signal as
being due to a shear Alfvén wave associated with the boundary layer current.
Measurements from the two satellites were used to estimate the thickness of
the current sheet associated with the wave to be 1 km thick. The spatial scale
is of order of the electron inertial length, δ = c/ωpe.
Figure 3: Upper: Schematic diagram showing the path of the wave relative to the Earth and the source
region. Lower: Thirty minutes of magnetic data from ISEE 1 from the time period 10:30-11:00 U.T, 27
March, 1978. The figure shows the plasma sheet entry at about 10:45. [adopted from Hayward and
Dungey, 1982]
8
Advances in satellite instrument design have allowed measurement of
electric and magnetic fields with greater temporal resolution. An early
[Chmyrev et al., 1988] example is magnetic fields measured on the ICB -1300
satellite. The scale is now in seconds rather than in minutes as in the
previous example. Data, reproduced in figure 4a show strong magnetic and
electric field variations on short temporal scales.
Figure 4 a) Variations in the horizontal component of the electric field vector in the frequency range 0.120.8 Hz. b) The distribution of Electric field as contour maps and surfaces for a two dimensional vortex
9
chain. c) The distribution and topographic lines of the density in the two-dimensional vortex chains.
[adopted from Chmyrev et al, 1988]
Observations of these disturbances occurred on both the polar and
equatorward edge of the auroral oval. In both cases the magnetic
perturbations were transverse and in the nanotesla range, and with a very
small B z component (Here the z axis is parallel to the local background
magnetic field.). Hodograms of the electric field show that it rotates and this
leads to an interpretation that the spacecraft was traveling through vortex
chains which were both left and right hand polarized. The measured ratio of
E⊥ /B⊥ was approximately the Alfvén speed. It was also observed that the
electron flux within the “vortex” region exceeds the background by two orders
of magnitude. This was interpreted [Chmyrev et al., 1988] as particles trapped
within an Alfvén vortex and traveling together with it, from a generation
region in the magnetosphere to the observation region in the ionosphere. A
kinetic theory of nonlinear drift Alfvén waves was developed in the
collisionless and low beta limit for analysis of this case. The solutions were
vortex tubes propagating transverse to the background magnetic field which
could be solitary , dipolar tubes as well as vortex chains. Density and electric
fields for this solution are shown in figure 4b and c respectively. The
perpendicular scale size of these filaments are of order Cs/ωci when β > me/Mi
and c/ωpe when β< m e/Mi. The authors conclude that “the process of
transition from quiet uniform auroral arcs to active rayed forms can thus be
explained by means of the theory for drift-Alfvén waves”. When the plasma
inhomogeneity exceeds a certain threshold value of the energy of the vortex
tubes becomes negative. This results in “an explosive ‘condensation’ of the
plasma into vortex filaments.” Time will tell if this is the correct explanation;
what is significant here is the recognition of the importance of structure in
the Aurora and the involvement of Alfvén waves.
Another observation by Cerisier et al. (1987) utilizing the Aureol-3
satellite points to the importance of structure in the Aurora and its
10
connection to Alfvén waves. Data from a crossing of such a structure is
shown in figure 5b. A theory was proposed using a source function which
modeled a channel of current, of infinite length in the y direction, and
channel width ∆x. The geometry for the analysis is shown in figure 5a. The
steady state currents which would be necessary to generate the magnetic field
observed by the satellite have never been observed and Cerisier et al.
hypothesize the current to be associated with an Alfvén wave. A Fourier
analysis using the cold plasma dielectric was used to predict the magnetic field
near the channel. (It might be argued that a the problem is better suited to a
Laplace transform analysis.) The theory predicts that the current layer widths
must be of order of the electron inertial length which in the topside
ionosphere is less than 1 kilometer. The waveform in figure 5 is well
reproduced by the theory assuming the current crossing that is shown.
Figure 5: a) Model of a plane current sheet flowing along the z axis and with a Gaussian shape in x. The
satellite crosses the sheet in the xy plane at an angle ϕ with x; this angle is determined from the ratio of the
x and y components of the magnetic field. b) Magnetic pulse observed by the AUREOL-3 satellite on June
8, 1982 [adopted from Cerisier et al, 1987]. Here Bx is along the satellite orbit, that is in the xsat
direction.
11
Evidence of electric fields parallel to the local magnetic field are well
documented [Borovsky, 1993] in the auroral zone and are associated with
auroral arcs [Mcfadden et al., 1990]. There are many mechanisms that could
give rise to these fields and they may be, in fact due to several of them. One
candidate is the parallel electric field of obliquely propagating Alfvén waves
[Hui and Seyler, 1992, Cerisier et al., 1987, Boehm et al., 1990a] another
possibility is electrostatic shocks [Boehm et al., 1990b]. The former has been
observed [Mauk and McIlwain, 1993], on the ATS-6 geostationary satellite,
which utilized ion and electron energy analyzers. For spacecraft traveling in
the auroral ionosphere the Debye length and cyclotron radii are of the order
of, or smaller than the particle detectors, which are usually mounted on
booms. At great distances from the earth these scale lengths are much larger
than the detectors. In either case detectors do not perturb the ambient plasma
and the particle trajectories are hardly curved with respect to the entrance
apertures of the instruments. This is generally not the case in laboratory
plasmas. In a plasma of density ≈ 1012 /cm3 , an electron temperature of 10 eV,
and in a kilogauss magnetic field, the Debye length is 23 microns and the
electron cyclotron radius, Rce, is 100 microns. Unless nanofabrication
techniques are applied (we will discuss this later) the response of the energy
analyzers will strongly vary with direction, and they will perturb the plasma.
Other techniques such as laser induced fluorescence [Stern and Johnson, 1975,
Stern, 1985, Mc Williams and Sheenan, 1986, Anderagg et al., 1997] are non
perturbative, but are expensive, and dependent upon finding an appropriate
optical transition. On the other hand spacecraft charging does not allow for
measurement of the lower energy particles which in some instances may be
the bulk of the distribution.
An example of the high quality of spacecraft particle measurements is
illustrated in figure 6 [Mauk and McIlwain, 1993].
12
Figure 6 a) Ion and electron data associated with a propagating Alfvén/ ion cyclotron wave. b) Wave
modulated electron distribution function. [adopted from Mauk and McIlwain, 1993]
Ion and electron data both parallel and perpendicular to the ambient field
show a strong modulation at about 1 Hz. This is not due to spacecraft
rotation, which is slow on this time scale, but is a result of modulation of
13
particle fluxes as a result of the presence of Alfvén/ion cyclotron waves. The
ions collected by two detectors perpendicular to B0 are shown in the top two
panels of figure 6a. Ions moving roughly parallel to B0 , the ambient field,
(which is in the North-South direction) are shown in the third panel. The
analyzers were configured to collect all ions of energy ≈ 30 eV. The ion flux to
the detectors is modulated by an EXB0 drift. The electric field is that of the
wave for which E|| << E⊥ . The effect of the wave is also clearly visible on the
electron distribution as seen in the lower two inserts of figure 6a. Here the
NS signals are much larger than the perpendicular ones and the detector
directly responds to electron motions along B0 . It was possible to extract the
electron distribution function during a wave period, this is shown in 6b. The
“peak” and “valley” distributions are acquired during the corresponding
intervals of the wave therefore the parallel electric field may be deduced from
the shift of these curves. The data from both detectors is combined to yield:
E||/E⊥ = 4.9X10-4 with E|| = 0.93 µV/m. Alfvén waves that do not propagate
strictly along the magnetic field have a parallel electric field and that is what
has been measured here. They are termed “kinetic” Alfvén waves but this is
due to the fact that they do not follow the pure MHD dispersion relation. We
will address the terminology issue later in this paper.
What emerges from these satellite observations is the importance of
Alfvén waves, the parallel electric fields of these waves and their association
with small scale density striations in the auroral ionosphere. This is again
illustrated in a recent satellite mission that had the temporal resolution to
resolve small scale structures. Figure 7a is data [Louran et al., 1994] from an
auroral region crossing at 58 to 69o, and which spans a time interval of less
than one second. Part of this is magnified in 7b. The data indicate that when
a spike in the electric field (E⊥ ) and dip in the magnetic field (B⊥ ) occur, their
ratio is ∆E/∆B = 5X10 6 m/s ≈ V A. The duration of the event is 60-80 ms which
corresponds to 200-800 m or a distance of order of the electron inertial length,
14
δ= c/ωpe. A deep (δn/n ≈ 0.7) density depression is simultaneously observed.
The authors conjecture that a highly nonlinear Alfvén wave, which they dub
a SKAW (Solitary Kinetic Alfvén W aves), is responsible for the density
depletion. This is believed to be a large amplitude Alfvén wave, only several
cycles in duration, and possessing an electric field parallel to the ambient
magnetic field. Since such a wave cannot be described by “pure” MHD theory
the “kinetic” descriptor became part of its acronym. This is misleading as will
be shown below.
Figure 7 a) An example of a “SKAW” observed on orbit 2033 of the Freja satellite. b) Magnification of
figure a showing 0.1 sec of the electric field, magnetic field and density fluctuations [adapted from Louarn et
al, 1994]
A second example [Volwerk et al., 1996] of Freja data showing the
perpendicular wave magnetic field and the parallel electric measurement are
shown in figure 8. This illustrates an instance in which the detectors
15
registered a short lived, therefore spatially narrow event. The low frequency
turbulence was dominated by strong electromagnetic spikes of duration .02 to
.1 second. This corresponds to a spatial widths of 100 to 500 meters as in the
above example. Figure 8a is a hodogram of the perpendicular magnetic field
at four different frequencies obtained by Fourier analysis of the data. The
bottom trace is the parallel electric field. The higher frequency magnetic
perturbations are clearly associated with spikes in the electric field. The
analysis indicates the SKAWs are embedded in complex current systems
which are laminar on large scales (km). The SKAWS are located on the edges
of these fluctuations and most likely correspond to tubular currents which
can propagate up or down along the field lines.
Figure 8b illustrates a sheet and solitary current and how hodograms of
the magnetic field, measured by a passing spacecraft would appear. We
contrast this with a hodogram taken in a laboratory measurement [Gekelman
et al., 1997b] of two shear Alfvén waves propagating side by side, and out of
phase by 180 degrees.
16
Figure 8 a) Hodograms of the perpendicular magnetic field, B⊥ . The four hodograms correspond to different
cutoff frequencies of the filter which are shown on the right. The parallel electric field is shown below. b)
Three different possibilities for the spacecraft to “cross” the current system. On top of each hodogram is the
trajectory through the current structure. [adopted from Volwerk et al, 1996]
The isosurfaces of constant current are derived from three dimensional
magnetic field measurements (∇ XH = j). The upper figure (9a) is a view
along the background magnetic field and the lower an orthogonal view. The
abscissa in 9b corresponds to a distance of three meters. The diagram has been
stretched in the perpendicular direction; the current density isosurfaces are
several centimeters wide. The hodograms are strikingly similar in the
spacecraft and laboratory, but a note of caution must be struck. In space it is
not possible to distinguish between traversal of a oscillating structure or one
at rest in which the fields are frozen. In space there is no way to ascertain the
time evolution of the density striations and make a positive statement that
they are caused by the wave. In the Freja work this is suggested only because
the waves were intense. In the laboratory experiment the waves also
propagated in narrow density striations, but these were caused by other means
17
which will be discussed later on. What has emerged in the past few years is
that laboratory experiments can be designed to study the same type of
structures seen in space [ Maggs et al. , 1997].
Figure 9: Experimental data of the parallel currents of two Alfvén waves propagating side by side. The
isosurfaces are of current density |jz | = 45 mA/cm2. The upper curve is an end on view, the lower one a
size view. Also shown are the hodograms which would be measured by a “spacecraft” in the laboratory
device, that moves through the currents rapidly compared with the wave period.
III Alfvén waves in Laboratory Experiments, early measurements.
Alfvén waves were not observed immediately after their existence was
predicted. As pointed out earlier the long wavelength, low frequency
18
characteristics of these waves make them difficult to study. A plasma source
in which the ions are highly magnetized is also required. Further
requirements for basic small amplitude studies are that the plasma be
uniform, quiescent and fairly collisionless. In the early 1950’s when
experiments began there were no such devices.
In cold plasmas, at frequencies below the ion cyclotron frequency, the
Alfvén wave dispersion relation has two branches, the compressional wave
and the shear wave. The compressional wave is isotropic and characterized
by fluctuations in both the magnetic field strength and plasma density. The
shear wave is highly anisotropic, propagating along the ambient magnetic
field direction and, to first order, is characterized by fluctuations in the
direction, but not magnitude, of the magnetic field.
Early experiments explored the compressional wave which was
invariably bounded in the direction orthogonal to the background magnetic
field by the experimental device. Data analysis involved bounded plasma
theory and Bessel function dependencies in the radial field profiles. Finally
vacuum and plasma diagnostic techniques were not as elaborate as they are
today. Nevertheless these early experiments established that Alfvén waves
exist, their dispersion was verified and considering the tools at the
experimentalist’s disposal surprisingly good work was done two decades ago.
19
Figure 10: a) Oscillations of electron current to probe in He at a pressure of 0.02 mm , B0 = 410 Gauss.
[from Bostick and Levine, 1952] b) Received (top) and driving (bottom) wave forms. [from Wilcox et al,
1960 ]
The first observation of an Alfvén wave was in mercury [Lundquist,
1949 a,b]. One of the first experiments in which the waves were glimpsed in
an ionized gas, was done by Bostick and Levine (1952). In this very simple
experiment [Figure 10a] a gas in a toroidal tube was ionized with a pulse
applied to a coil wrapped around it. There is a background toroidal magnetic
field of 400 Gauss. A floating Langmuir probe whose waveform is shown in
fig 10a recorded oscillations of order 104 Hz. The oscillations were interpreted
as a standing MHD wave, and satisfied the relationship f =
20
B0
4πrn . ( L is the
2L
circumference of the torus, r the plasma radius). No waves were seen when
the toroidal magnetic field was zero.
In another experiment [Jephcott, 1959], very similar to Bostick’s, an
antenna wound around a torus was used to ionize a gas (Id = 10 kA, t d = 200
µsec). A background toroidal magnetic field of 3 - 14 kG was present. Two
external coils were used to produce an oscillating magnetic field transverse to
the background field. Magnetic probes picked up 10 G oscillations which
exhibited delay times in accord with the Alfvén propagation speed. The
experiment was highly collisional and no wave structure was measured.
In yet another early experiment by Sawyer et al. (1959) hydrodynamic
waves were observed in a linear discharge 60 cm long and 15 cm in diameter.
There was an axial magnetic field of 500 G. The plasma was produced with a
200 kilojoule capacitor bank in relatively high (.1 - 10 micron*) pressure
deuterium. Magnetic loop probes 3 mm in diameter, electrostatically
shielded, and then set in quartz tubes were used to measure the azimuthal
radial or axial component of the magnetic field. A self generated helical wave
was measured and detailed plots of current distribution were presented. The
wave magnetic field was proportional to 1/√ρ, and the wave seemed to travel
at the Alfvén speed. The resulting wave was deduced to be a mix of
hydrodynamic waves traveling in the axial and circumferential directions.
Although the wave physics is quite complex this is one of the first
experiments where detailed measurements were made.
A better experiment in a linear device was performed by Wilcox et al.
(1960). The plasma was produced by a high pressure ( 100 micron fill, n0
≈ 3.5X1015 cm-3) discharge between the ends using a switch and a capacitor
bank. The experimental setup is similar to that shown in figure 11a. The
wave was launched by applying rf from a ringing capacitor between the
cathode of the plasma source and the wall of the device after the plasma was
*
Note: 1 micron = 10-3 mm mercury, 760 mm of mercury is atmospheric pressure
21
formed. The wave phase velocity was measured with a magnetic pickup
probe as a function of the background magnetic field.
Some data are shown in figure 10b. The time delay between the
sinusoid applied and the received signal is clearly visible and the phase
velocity agrees with the Alfvén speed. The velocity was also measured as a
function of magnetic field which further established the waves to be Alfvénic.
Wave attenuation was also measured and this roughly agreed with theory.
As time elapsed laboratory experiments on Alfvén waves, and detailed
comparison with theory improved. This is illustrated in an experiment
carried out in a linear device filled with argon [Jephcott and Stocker ,1962].
The experimental setup is illustrated in figure 11a and the results in figure
11b,c.
22
Figure 11: a) Diagram of the experimental apparatus. b) Magnetic probe signals at the two locations shown
in a. c) Wave phase velocity as a function of axial magnetic field in Ne and Ar. Solid lines are from
calculation by Woods. [adapted from Jephcott and Stocker, 1962]
The linear device has a background magnetic field in the z direction.
The wave was launched by applying an oscillating waveform, generated in a
LC circuit, to a ball electrode. The circuit closes on the wall of the cylindrical
vacuum vessel. Figure 11b shows the magnetic signal detected by two
magnetic coil probes, 5 mm in diameter and 40 cm apart. The time delay
difference of the waveform between the probes is a measure of the phase
velocity of the waves. Figure 11c shows the wave phase velocity in two gases
23
as a function of magnetic field. The solid curve is the theoretical prediction.
The plasma was well diagnosed. Double Langmuir probes were used to
calculate the density at the position of the magnetic probes, and the electron
temperature (n ≈ 1X1015 cm -3, T e≈ 2.3 eV). The ion temperature was estimated
from Doppler broadening ( Ti ≈ 0.2 eV) The conductivity was estimated from
electric probe measurements. The plasma was collisional; the ion neutral
collision frequency was much greater than the wave frequency ( νei ≈ 4.5X106
Hz, fwave = 125 kHz). The magnetic probes could be moved radially and the
azimuthal component of the wave magnetic field, Bθ(r), was measured. The
data were compared to predictions of a theory of bounded Alfvén waves by
Woods (1962) which included the effect of neutral collisions. The comparison
was excellent.
Experiments on Alfvén waves have not been restricted to ionized
gases. Because the idealized waves are magnetohydrodynamic they also exist
in conducting fluids and certain solid state plasmas. Liquid sodium has a
high conductivity ( σ = 1/ η ) , of order 106 greater than a basic plasma
laboratory plasma such as in the Large Plasma Device, the LAPD [Gekelman et
al., 1991]. The Lundquist number, µ0 LV A/η, which is one measure of how
deeply into the MHD regime one is, is of order 50 for sodium but is greater
than 1010 near a star. This, of course, depends on the scale length, L, for the
phenomenon of concern, which is enormous in space. Recently the
importance of structures embedded within space plasmas has become
apparent. They are present in different situations for different reasons but
seem to be ubiquitous. For example at altitudes below approximately 5000
km, the high latitude ionospheric plasma is found to contain short scalelength, field aligned irregularities or striations [Herman, 1966; Dyson, 1969,
Clark and Raitt; 1976, Kelly and Mozer, 1972; Fejer and Kelley , 1980]. Density
depletions and associated lower hybrid wave activity and fast transverse ions
were observed in rocket experiments [Seyler, 1994]. Striations have also been
24
observed near the magnetic equator [Kelley and Mozer, 1972]. It is not clear
how these structures form. Auroral arcs have structure on several spatial
scales and are generally much narrower than theory predicts [Borovsky, 1993].
Density striations in the auoral ionosphere are characterized by scale lengths
across the magnetic field which are much smaller than the scale length along
the magnetic field. Laboratory experiments have been able to successfully
scale relevant parameter ratios associated with these structures and model
Density and temperature fluctuations have been observed in the solar wind
[Klein et al. , 1993] and correlated with temperature fluctuations. In some
cases the density fluctuations exceeded fifty percent. Finally structure has
been observed beyond 35 AU of scale size of a few hundredths of an AU by
instruments on the Voyager 2 [Burluga et al, 1994].
It is the structure that determines the relevant value of L, not some
other gross dimension, and in many cases it could turn out that very large
Lundquist numbers may not be a necessary condition in laboratory
experiments or computer simulations.
An early experiment in liquid sodium was carried out by Lehnert
(1954.) A copper disk in the liquid was set into oscillation ( f = 30 Hz) and
generated torsional Alfvén waves which propagated along an imposed
magnetic field. When the waves hit the surface of the liquid they gave rise to
a potential difference at two locations which was measured by probes. The
potentials were predicted by a theoretical model and compared favorably with
the experimental results.
Another non-gaseous laboratory experiment on Alfvén waves, which
produced “textbook quality” results was done in a solid state bismuth plasma
[Hess and Hinsch, 1973, also see references within Hasagawa and Uberoi,
1982]. Bi was chosen because the density of holes and electrons is equal and
opposite and “low”, where low is ≈ 1017 cm-3. The sample was cooled to 4.2
degrees K, the Alfvén wave frequency was 5.41 GHz (B = 1T, ωp > ωc). The
MHD cold plasma dispersion relation was verified by plotting the wave data
25
acquired on wave normal surface diagrams [Stix, 1992]. Since the crystal is
anisotropic and wave propagation is normal to the crystal surface a number of
samples were cut from pure Bi at known angles to the principal crystal axis.
Phase velocity diagrams were measured and are reproduced in figure 12.
Agreement with cold plasma theory was excellent. The conditions are very
different from laboratory plasmas, the Alfvén measured velocity was, VA =(25.8)X108 cm/sec (this depends on angle of propagation), but the wavelength
was small, λ = 370 µ.
Figure 12: Phase velocity diagrams for a background magnetic field of 10 kG. [ from Hess and Hinsch,
1973] The magnetic field in each case is along one of the principal axis of the crystal, q is the direction of
the Alfvén wave vector. The x-y plane is the trigonal plane of the crystal.
IV Alfvén waves in Laboratory Experiments, the modern era.
From the 1980’s to the present key basic science experiments on Alfvén
waves have been done by groups in Australia, Japan, and in the United States.
26
In these experiments some fundamental properties of the shear Alfvén wave
have been explored and their direct relevance to the spacecraft measurements
previously discussed has emerged. At the same time a great deal of research
on Alfvén wave heating of thermonuclear plasmas occurred. The greater
part of the waves studied were bounded cavity modes launched by antennas
close to the chamber wall. Reviewing heating experiments is a formidable
task and does not properly fit into the scope of this paper.
A arcjet plasma source [Amagishi et al., 1981] which produced a 2 meter
long, 15 cm diameter (full width at half maximum ≈ 6 cm ) helium plasma (n
≈ 5X1014 /cm 3 , T e=T i = 4 eV, B = 2.5 kG ), [Amagishi and Tsushima, 1984]. At
these densities [Amagishi, 1990] the wavelength is of order 40 cm ( f = 390
kHz, f ci = 950 kHz) but the plasma is not completely ionized (70%) and the
collision rate, νei, is higher than the wave frequency by a factor of ≈ 10 4 . A Stix
coil (1958) which is a series of loops surrounding the plasma, spaced a half
wavelength apart, which carry azimuthal rf currents, which are out of phase
by 180 o in alternate loops, was used to launch the wave.
The antenna is
energized through a pulsed LC circuit; waves were detected by small ( 5 mm
diameter, 100 turns) magnetic probes. In one of the groups early experiments
[Tsushima et al., 1982] a peak in the amplitude of B θ was observed at a
position several centimeters off the device axis.
The location of the peak
matched the radial position where ω2 /k||2 V A2 = 1 - ω2 /ωci2 ; ω is the applied
angular frequency, VA α
1
, and n= n(r). Historically this has been called the
n
field line “resonance” position and this is, to this author’s knowledge the first
observation of this in a laboratory plasma.
27
Figure 13: Schematic of the experimental apparatus, the MPD arcjet. [from Amagishi and Tsushima,
1984] b) Wave packets of Bz with m = -1 propagating along B0 at r = 4 cm (the center of the plasma
column is at r = 0) as a function of axial distance from the exciter. The wave packets consist mainly of
MHD surface waves of m = -1. c) Interfered wave packets of Bθ propagating along B) at r = 4 cm. The
slopes give 2.6X107 cm/s for the m = -1 SAW and 1.4X107 cm/s for the m = -1 MHD surface wave. [ from
Amagishi et al , 1988]
28
A propagation diagram [Amagishi et al., 1988] showing the axial
component, B z (r = 4), of the wave field, as a position of axial distance from the
exciter is shown in figure 13b. The half width at half maximum of the plasma
is 3 cm, therefore this data was acquired on the steep density gradient on the
edge of the column. The temporal delay at larger distances from the exciter is
clearly visible as well as the collisional damping. The amplitude at 40 cm or 1
wavelength from the exciter has dropped by 75%. Data taken at Bθ(r=4) (figure
13c) have contributions due to both the surface fast wave and the shear wave
(labeled SAW).
An analysis of the SAW showed it to satisfy the correct
dispersion relation. Finally at the center of the plasma, (r=0), only the SAW is
observed which is consistent with the other mode existing only on the
plasma surface.
A great deal of recent work on Alfvén waves has been done at several
institutions in Australia. Cross has written a book (1988) on the subject from
an experimentalist’s viewpoint. Experiments on the shear [Cross and Lehane,
1967a] and compressional [Cross and Lehane, 1967b] waves in both linear and
toroidal [Cross et al., 1982] devices have been done over the past 30 years.
Several results of a study [Cross, 1983], in a linear device, on both the shear
and compressional waves are shown in figure 14. The plasmas (Ar, H) were
12 cm in diameter and 2.6 m long with density ≈ 1015 cm-3, T e ≈ T i = 2. eV, B 0z =
7-8 kG. Several types of launching structures were employed, waves were
detected with 8 mm diameter single turn magnetic dipole loops.
29
Figure 14: a) Profiles of Bθ verses r at several axial positions z, from the antenna for the shear wave in a
hydrogen plasma at B 0 = 7 kG, f = 0.5 Mhz. The absolute values of Bθ are shown at each axial position.
b) Waveform of antenna current and dBr/dt at z = 20 cm in argon at B 0 = 7 kG. The first pulse is the fast
Alfvén wave followed by a highly localized ion acoustic pulse. [from Cross, 1983]
A shear wave was launched with a “coaxial cage” antenna which
produced only a Bθ field. Figure 14a shows the amplitude profile of the shear
wave for this case at several axial positions, z, from the exciter ( fci = 10.7 Mhz).
The wave appears to be highly damped as one gets away from the exciter.
This is in part attributed to the wave having a k⊥ , and therefore being
geometrically attenuated, but the plasma was not fully ionized and collisional
damping must have contributed as well. The author concludes that the shear
wave is “localized” along the magnetic field close to the exciter. In a separate
experiment a coaxial exciter inserted axially along the chamber axis launched
a complicated waveform, which is a fast wave, as shown in figure 14b,
followed by a field aligned ion acoustic wave. The wave shows up in the
magnetic signal at r = 0.5 cm which nearly in line with the exciter, but is not
visible at larger radial positions. The ion acoustic pulse propagated at 3.8X105
cm/sec (corresponding to 1.8 eV electrons) in Argon. This is one of the first
clear experimental observations of the coupling between the fast wave and
ion sound.
30
The
examples
mentioned
have
experiments dealing with Alfvén waves.
tracked
the
development
of
They have evolved from work
which saw disturbances moving at the right phase velocity, to experiments
which carefully measured wave dispersion and compared it to theory. None
of the experiments discussed thus far has a compelling relation with the
auroral data presented earlier. This has changed with experiments on guided
Alfvén waves by the Australian group, and more recent experiments on
Alfvén wave cones in the LAPD device.
An experiment similar to previously described one was done by Borg et
al. (1985) in a torodial device. The plasma initially had a higher electron
temperature, Te = 10 eV, but the wave measurements were carried out after
the main Tokomak discharge when the plasma current was zero.
The
electron temperature was probably low at that time; there is no estimate of it
in the paper. The waves were excited with a magnetic dipole antenna and
detected with magnetic pickup loops. An example of the data is shown in
which shows the wave magnetic field as a function of radial position along a
line located 180o in azimuth from the port which contained the wave exciter.
Figure 15: Comparison of experimental and theoretical wave fields. The background magnetic field is 8
kG, f = 550 kHz, Hydrogen plasma. [from Borg et al , 1985]
31
The dashed line connects the measured field shown as dots. The center
of the wave excitation coil is at r = 5 cm, the position at which the wave
amplitude goes to zero.
A Green’s function theory which modeled the
current in the excitation coil was used along with the cold plasma dielectric.
An electron ion collision term, νei,
was used to calculate damping.
The
calculation predicts that the waves move away from the source in hollow
cones which to first order are guided along the background magnetic field, but
slowly spread across it. The solid curves shown in figure 15 are theoretical
predictions of the wave profile, N is the number of times the wave has circled
the torus. Analysis of the data indicated that N was approximately 1 and that
νei ≈ 100 Mhz. The ratio of the collision frequency to the wave frequency, or
damping coefficient was Γ = νei /ω = 100. The results demonstrated that an “...
Alfvén ray in a collisional plasma may be composed of an arbitrary spectrum
of k ⊥ components up to a maximum k ⊥ = 1/δ" .
The role of the collisionless skin depth and the experimental
observation that the shear wave propagates, in non-ideal MHD plasmas, as
narrow field aligned cones which slowly spread across the background
magnetic field directly connects recent laboratory experiments to spacecraft
observations.
Before proceeding it is necessary to clarify some of the
nomenclature which has come about because research in this area has
proceeded in several parallel tracks in different institutions.
The shear Alfvén wave propagates with the wave magnetic field vector
perpendicular to the background field. The dispersion relation in a plasma
where the ion temperature is much smaller than the electron temperature (Ti
<< T e) can be written:
Z ′(ζ )(s 2 (1− ϖ 2 ) − ζ 2 )
32
= k 2⊥ δ 2
(1)
where Z'( ζ ) is the derivative of the plasma dispersion function [Fried and
1
Conte, 1961] { Z(ξ) =
π
e −z 2
∫−∞ dz z − ξ , Im ζ > 0 } with respect to ζ, ζ = ω/k||a, and
+∞
s= v A /a is the ratio of the Alfvén velocity (v A =
B 2 /(4πnm+ ) to the electron
thermal speed, v the, , and ϖ is the normalized angular frequency ( ϖ = ω/Ω+ ).
The average electron thermal speed is denoted by a = (2T e/me)1/2 with T e
the electron temperature, measured in ergs, and m e the electron mass. The
ion thermal velocity a+ is defined similarly with Ti and m + denoting the ion
temperature and mass.
There are two important limiting cases for this dispersion, when the
wave phase velocity is much less ( s2 >> 1) or much greater (s2 << 1) than the
electron thermal velocity. The parameter s2 is related to the electron
plasma beta, β e = (
8πnkT
) as:
B2
s
2
=
v 2A
a2
=
me 1
M+ βe
(2)
For the limiting case s2 << 1 we have the inertial Alfvén wave or
ω2
k||2
=
v 2A (1 − ϖ 2 )
(1 + k2⊥ δ 2 )
( v 2A >> a 2 )
(3)
In the very low frequency limit and for waves which propagate strictly
along the magnetic field (k⊥ = 0) equation 3 becomes the standard MHD
dispersion.
The experiments of Borg et al. (1985) and those which be
discussed, as well as spacecraft data, all confirm that in situations with
localized fluctuating currents standard MHD fails.
This wave also has a
parallel electric field, a forbidden commodity in ideal MHD.
33
k||k⊥ δ 2
E|| =
2
1 + k⊥ δ 2
E⊥
kA k ⊥ δ 2
=
2
(1 − ϖ 2 )1/2 (1 + k ⊥ δ 2)1/2
E⊥
(4)
Here kA is the “Alfvén wavenumber” defined as ω/VA. For large k ⊥δ (short
perpendicular wavelengths) E|| = (k Aδ)E⊥/(1-ϖ 2)1/2. This wave is termed the
inertial shear Alfvén because of its dependence on the electron collisionless
skin depth, or inertial length.
In the auroral ionosphere the parallel
wavelengths can be of order 100 km , the wave electric field will not change
sign over a half wavelength, and therefore these waves are candidates for
observed parallel electric fields.
The FREJA group has incorrectly dubbed
them SKAWS. The plasma is cold at the FREJA measurement ( β e << m e/MI)
locations and the waves detected are of the inertial variety.
In the other limit, for a hot plasma, or small s2 (vA2 << a2), the plasma
electron beta is larger than the mass ratio and the Alfvén speed is much
slower than the electron thermal speed. The dispersion relation is :
ω2
k||2
= v 2A (1 − ϖ 2 + k 2⊥ ρ 2s )
( v2A << a 2 )
(5)
where ρ s is the ion sound gyroradius, ρ s = c s/Ω+, with cs = (T e/M +)1/2 the ion
sound speed. This dispersion relation describes the propagation of the so
called kinetic Alfvén wave.
Once again the standard MHD dispersion
relation is recovered in the case k ⊥ = 0, and ϖ ≈ 0. Finally the parallel electric
field in this case is
E ||
=
− k||k ⊥ ρ s2
E⊥
1 − ϖ2
=
− k Ak ⊥ ρ s2
E
(1− ϖ 2 )(1− ϖ 2 + k 2⊥ ρ 2s )1/2 ⊥
34
(6)
For k ⊥ ρ s large the parallel electric field is E|| = -(kAρ s)E⊥/(1-ϖ 2).
Here the
relevant scale length is the ion sound gyroradius.
The experiments on shear waves discussed thus far were highly
collisional.
For weakly ionized plasmas electron neutral collisions will
dominate. (νen = p0 Pcv the, p0 is the neutral particle concentration, Pc the
collision probability [Brown, 1959])
plasmas Coulomb collisions (ν ei
For dense cold and highly ionized
4
π ne e lnΛ
2 m1/2
Te3/2
e
=
where ln(Λ) is the
Coulomb logarithm, T e is in ergs, m e in grams ) will dominate. In the LAPD
device at UCLA, which has a 10 meter long, 40 cm diameter quiescent plasma
in an axial magnetic field of up to 2 kG it has been possible to achieve Γ =
νei/ω < 1/30, and study both the inertial and kinetic regimes of these waves.
Shear Alfvén waves propagate in cones strictly in the inertial limit,
and in the limit of large k⊥ . Here the second order differential equation
describing the wave has characteristics
dr
dz
= ±
kA δ
, which define the
(1 − ϖ 2 )1/2
cone angle. No such characteristic exists in the kinetic limit. These waves
have been dubbed Alfvén resonance cones by several authors; this is
misleading. The Alfvén “cones” are a consequence of the group velocity of
the waves.
They move along cones in the inertial case.
There is no
resonance, i.e. infinity, in the electric or magnetic field at the cone angle. This
is in contrast to the resonance cones of Kuehl (1962) and Fisher and Gould
(1971) which is the magnetized plasma response to an oscillating point charge.
In a collisionless plasma the electric field, which is the interference pattern of
many waves with the same ω and differing k, becomes infinite along a cone
angle θ c = tan -1 (-ε⊥ /ε||), where ε is the plasma dielectric. (The cones exist only
if ε⊥ and ε|| have opposite signs). Resonance cones have been studied in a
variety of earlier plasma experiments in both the linear [Ohnuma, 1994] and
nonlinear [Stenzel and Gekelman, 1977, Gekelman and Stenzel, 1977]
35
regimes. Figure 16 shows a schematic of the LAPD with some of its
parameters scaled to basic lengths. Alfvén waves were radiated using small
sources consisting of semi-transparent wire mesh disks with radii on the
order of δ . Assuming azimuthal symmetry, the magnetic field radiated from
these wire mesh antennas has only a component in the azimuthal direction,
r
B = Bθ (r,z ) eˆ θ .
Figure 16: Schematic of the LAPD device showing the plasma source on the left. Several characteristic
length ratios for typical operating parameters are given for a helium plasma. The machine has 128 access
ports, the magnetic field may be tailored with the use of 7 independent power supplies.
The spatial dependence of the radiated magnetic field is given by an
integral expression derived by Morales et al. (1994). The radiation pattern
was derived for the inertial case with, and without collisions. At the same
time experiments [Gekelman et al., 1994] were done in the LAPD which
agreed well with this theory. The radiation patterns for the kinetic case
have been recently calculated and are the subject of a forthcoming
publication [Morales and Maggs, 1997].
This experiment is discussed in detail in a paper by Gekelman et al.
(1997a) performed in a highly ionized He plasma in which collisions with
36
neutrals were not important and the ion-electron collision was low (Γ
= 0.48). Instantaneous values of the perpendicular component of the wave
magnetic field at 441 spatial locations in a plane perpendicular to the
background magnetic field is shown for a wave with frequency, f = 320 kHz,
(ϖ = 0.77, β e = 2.3X10-4) is shown in figure 17.
Figure 17: Measured magnetic field for a wave with ϖ = 0.77 in a plane 1.54 m from the exciter.
The wave burst was 28 cycles long and the data shown was collected
during the burst. The largest vector in the diagram has length of 660 mG
(B⊥ max/B0z = 6.0X10 -4). The axial location of the data plane is 1.0 parallel
wavelengths ( λ || = 1.54 m) from the antenna. The planar data illustrates
that the wave magnetic field exhibits a high degree of azimuthal symmetry
but some departure from symmetry is evident. Wave propagation across
the magnetic field is also evident from the reversal of the field direction
moving radially outward from the antenna. A series of experiments
37
exploring the wave radiation pattern from 0.11< s2 < 4 will be the subject of
a future publication.
If more than one Alfvén wave propagates due to a series of localized,
fluctuating currents, the waves will interfere. This has been explored in the
case of two waves propagating side by side which were both in and out of
phase [Gekelman et al., 1997a]. The interaction of two Alfvén waves was
studied by using two separate disk antennas as wave sources and measuring
the spatial pattern of the radiation. The waves were found to interact linearly
even at the highest current levels that could be produced using the disk
antennas. These currents produce radiated fields as high as 10-3 of the
background magnetic field. To attain large currents it is necessary to apply a
positive DC bias to the Alfvén wave exciters and superimpose the AC wave.
This procedure avoids a rectification of the drawn current since without a DC
bias only the ion saturation current may be drawn from a plasma. Drawing a
large current in a plasma results in the formation of a field aligned density
cavity. In the case of this experiment, the density cavities were deep, δn/n
≈ 60 %. This is reminiscent of the Freja data but in space it is not at all clear
how the density cavities form. The largest wave magnetic fields in the
laboratory experiment were found to occur in the region between the two disk
exciters when the disk exciters are driven 180 degrees out of phase.
Figure 18 is of volumetric data taken at 3600 spatial positions inside a
box in the center of the plasma (plasma diameter 40 cm, box width 20 cm, box
length = 2.8 m, ϖ = 0.5, B 0 = 1.32 kG, n = 2.6X10 12 cm-3, β e = 6.0X10-4). Shown, on
several planes, as field lines is B⊥, the component of the wave field
perpendicular to the background magnetic field ( the field-aligned component
of the wave field is measured to be negligibly small).
38
Figure 18: Volumetric data of the currents and magnetic fields of two shear Alfvén waves, 180o out of
phase and propagating side by side. Streamlines of the magnetic field are shown in several planes. The
axial current, jz = ± 45 mA/cm2, of the waves are shown as color coded isosurfaces. In the insert the vector
magnetic field B⊥(x,y) in a plane 1.26 meter from the exciter, ϖ = 0.50, is shown an a 20X20 cm plane.
The vectors are color coded according to their magnitude, the largest being 0.5 Gauss.
39
An insert at the bottom of the figure shows the magnetic vector field in
one plane. Here the largest wave magnetic field between the exciters is of
order 0.5 G with Bz /B⊥ ≈ 0.06. If a spacecraft traveled vertically upward in this
plane along a line to the left of center it would see a sudden burst of magnetic
field pointing in the negative x direction and little else. Also displayed are
isosurfaces of constant field-aligned wave current jz. The axial wave currents
are largest along field lines threading the disk exciters where the wave
magnetic fields are small.
Since a three dimensional volume data set of the radiated field was
taken the currents associated with the wave can be derived by taking the curl
of the magnetic field ( the displacement current is negligible at Alfvén wave
frequencies).
The data establishes the closure of wave currents across
magnetic field lines. This is clearly visible in figure 19 which shows two
isosurfaces of constant current, j z = 45 mA/cm 2 and streamlines of current
which are drawn as brown tubes.
40
Figure 19: Isosurfaces of current for the same case as in Figure 18. Also shown are streamlines of current
which are seen to close mainly at the quarter wave location where the parallel currents go to zero. Cross
field currents clearly link them and maintain ∇•j = 0.
The length along the magnetic field rendered is roughly 1 m.
The
location at which the isosurfaces disappear is a node in the wave, which
travels along the device at V A = 8.8X107 cm/s.
The wave current which
propagates nearly along the background magnetic field between nodes, crosses
the magnetic field and closes at the nodes.
This is a polarization current
which is carried by the ions.
The issue of striated plasmas can also be addressed in laboratory
plasmas. In the case just described a striation was caused when a DC current
was drawn through the plasma. Such a current can locally heat the electrons,
modify the potential in the current channel and then drive out the ions. This
has been previously observed [Stenzel at al, 1981] in a plasma where the
electrons were magnetized and the ions were not. In the case reported here,
the half width of the striation was roughly two ion gyroradii and a similar
scenario is possible. It is clear, however, that the shear Alfvén waves did not
create the density depression.
41
Figure 20: a) Frequency spectra of the magnetic |δB(ω )|, and density, |δn(ω )| fluctuations for βe ≈ 10 -3.
b) Spatial dependence of the root mean square value of fluctuations in the axial current, δj|| , and the Fourier
amplitude of the magnetic fluctuations at ω /ω ci ≈ 0.9, |δB(ω )|. The plasma density profile is n0. The
current fluctuations and the magnetic Fourier amplitude are plotted using arbitrary units. Only half the
magnetic fluctuation is plotted because of probe interference. [from Maggs and Morales, 1996]
42
The presence of a density striation is enough to create Alfvén waves.
This has been observed in a recent experiment by Maggs and Morales (1996a).
A ten meter long density striation several centimeters in diameter was
created by a circular paddle which interfered with plasma. The density profile
of the striation is shown in figure 20b along with the spatial profiles of
magnetic field and current (measured with a double sided Langmuir probe)
fluctuations. The fluctuations are observed only at the location of the steepest
density gradient. The spectrum of these oscillations is shown in figure 20a.
In the case for β e ≈ 10-3 the magnetic and density fluctuations, which are
eigenmodes of the density cavity accompany each other.
The magnetic
fluctuations are associated with the shear wave since their axial magnetic
field was negligible. A theoretical calculation [Maggs and Morales, 1996b,
Peñano et al., 1997) showed them to be drift Alfvén waves driven by a radial
pressure gradient associated with the striation.
The lowest frequency
eigenmode corresponds to an axial wavelength which is twice the machine
length ( λ/2 = 10 m ). The striation is about a million Debye lengths long, but
the fluctuations do not destroy it. It is possible that the structures observed in
the auroral ionosphere are replete with these waves and that the parallel
electric field of the Alfvén waves is responsible for electron precipitation.
The largest parallel electric fields occur when k⊥ c/ωpe ≈ 1, or at scale length of
order of the electron skin depth. This is the size of the density gradients in
this experiment and scales well with the thickness of auroral arcs [Borovsky,
1993].
V. Summary and Conclusions
What does the future hold?
It is clear that with improved plasma
sources and diagnostics that laboratory experiments can properly scale
interesting phenomena that occur in space. Laboratory experiments allow for
43
acquisition of fully three dimensional, time dependent data sets. The initial
conditions are known, and experiments may be reproduced millions of times.
Laboratory devices can be rapidly reconfigured to perform many experiments,
probes that break can be removed and repaired.
When an interesting
phenomena has been identified by a spacecraft, or rocket, and the basic physics
of it is not well understood, the laboratory is the ideal place to study it.
Spacecraft on the other hand are as essential as laboratory devices. Most of
the key observations made since this area of research opened up were not
predicted beforehand.
The branch of space science happens to have two
laboratories, one up there, the other down here.
The days in which they
operated in parallel, with hardly any interaction are over.
The latest spacecraft such as FREJA [Lundin et al.,, 1994] and FAST (Fast
Auroral
Snapshot
Explorer,
http://sunland.gsfc.nasa.gov/smex/fast/,
http://plasma2.ssl.berkeley.edu/fast/ ) have very high digitization and
telemetry rates. This has directly led to the discovery of fine scale structures
which permeate the aurora, and possibly all of the magnetosphere. Another
satellite mission due to be launched in 2000 is the CLUSTER (1997) mission.
This is the second attempt (the first attempt catastrophically ended when the
Ariane-5 rocket carrying the mission exploded just after takeoff) to deploy
four satellites in a tetrahedral configuration to study the near Earth and solar
wind plasmas and perform coordinated three dimensional measurements.
The purpose of the mission will be to measure small scale plasma structures
in the solar wind, magnetopause, auoral zone, polar cusp and magnetotail.
The next step in satellites will be miniaturization and deployment of
multiple, inexpensive sensors so that multipoint measurements can be made
without prohibitive cost. In laboratory plasmas advances are still being made
in source development, but the next big leap will also come in sensor
technology. Presently magnetic field pickup loops are based on the same
principles as ones used thirty years ago. Probes are easily larger than both
electron gyroradii and the Debye length, therefore they perturb the plasma. It
is time to develop microscopic probes using micro-machining and possibly
44
nanotechnology. An example of this is shown in figure 21 which illustrates
the rough size of the electron and ion gyroradii and Debye length compared to
the size of Freja.
Figure 21: The Boltzman equation and several techniques for the measurement of relevant terms. To do
laboratory plasma physics on the “Boltzman” scale microscopic detectors must be developed such as the
one shown on the bottom right. The smallest plasma scale sizes are shown next to it. The diameter of the
circle is a Debye length. For comparison the Debye length and gyroradii are shown on the left for the Freja
satellite.
The spacecraft is larger than rce or λ D but detectors mounted on these
booms are smaller then these fundamental lengths. The satellite imposes a
minimum perturbation on its environment. Also shown on the lower right
is a 100 µ diameter magnetic field coil with 100 turns connected to a buffer
amplifier [Eyre et al., 1995]. Also shown are the typical electron cyclotron
radius and Debye length in the LAPD device. The scales are similar, in fact
the device could be made smaller. Triplets of magnetic probes, to measure B,
which pop off of the silicon substrate have also been constructed. The top of
figure 21 is the Boltzman equation with references to how various terms in it
are measured. Presently volumetric measurements of certain quantities such
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as B(r,t) and E(r,t) can be routinely performed in the laboratory. Measurement
of the distribution function can be done locally with velocity analyzers (which
are large) or Laser Induced Fluorescence, which is usually done at a small
subset of positions and angles, and is expensive and time consuming.
Microscopic probes, which can be designed to take advantage of effects that
happen at small scales can be deployed by the thousands without disturbing
the plasma. Data sets will then increase to terabyte size and will contain
information on the statistical mechanics of plasmas.
Plasma Physics is a
relatively new science. Its subject matter ranges from the humble discharge
in a fluorescent tube to the structure of galactic jets. Collaborations between
the diverse groups which have studied this field in relative isolation, until
now, are just beginning.
Acknowledgments:
The author would like to acknowledge the collaboration with Jim
Maggs, Steve Vincena, David Leneman and George Morales, in the Alfvén
wave experiments. In addition I would like to thank Joe Borovsky for
suggestions on references for this paper. Finally I would like to acknowledge
the many useful comments and the thorough reading by the two referees.
The work was supported by the Office of Naval Research and by the National
Science Foundation (ATM and AMOP)
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