A review of a deceptively simple topic is presented, i.e., the excitation of whistler modes by antennas. It includes the knowledge of antennas and of the waves and their coupling. This review will show how the research in the last few decades has advanced and become a refined and complex topic which covers nonlinear effects, instabilities, nonuniform fields, whistler modes with orbital angular momentum, wave field topologies, etc. This review is mainly focused on experimental work in laboratory plasmas, but the findings will be related to research on whistler waves in space plasmas, helicon plasma sources in the laboratory, and significant findings of other research groups. This review starts with antenna properties such as radiation patterns, radiation efficiencies, and the topology of the emitted wave packets. Next, the propagation of whistler modes in highly nonuniform ambient magnetic fields will be presented. Even in the linear regime, new phenomena have been discovered such as the reflection of whistler modes from strong magnetic field gradients or the eigenmodes of waves on circular magnetic field lines. Important nonlinear effects of whistler modes are presented. Whistler instabilities will be briefly reviewed although this is a broad topic by itself. Examples of magnetic reconnection in the Hall parameter regime will be shown. This review will also discuss the advances of wave diagnostics in dedicated laboratory plasmas, the difficulties of diagnostics in high power laboratory plasmas, and the limiting multipoint diagnostics in space plasmas.

Whistler waves have been observed already a century ago and have named so because of their whistling tones coming from the ionosphere.1 Intensive theoretical work explained the physics of whistler waves and their dispersion, reflection, and trapping,2,3 all based on plane wave theory. Whistler modes have also been observed in solid state samples and coined them “helicons” due to their helical field lines.4,5 Such low-frequency whistler modes have been found in dense plasma columns, and a theory of helicon eigenmodes in bound plasma columns has been developed.6 Both helicon modes and plane waves are ideal structures which cannot be realized, but they form the basis for physical models. Helicon waves were found to ionize gas efficiently7 and to produce dense plasmas for plasma processing8 and plasma thruster application.9–11 Each community of researchers has different names for whistlers such as helicons, “whistler waves” (plane waves in space, excited by lightning), and general “whistler modes” for any other type of whistler wave packet. Since helicons can be produced in unbound plasmas,12 the definition should be generalized as waves with orbital angular momentum. Such waves have been well developed for electromagnetic “vortex” waves from radio waves to lasers with many useful applications.13,14

The excitation of whistler modes is usually done with electric and magnetic dipole antennas. Since most of the wave energy resides in the wave magnetic field of whistlers, it is advantageous to use magnetic loop antennas. The radiation efficiency of electric dipoles is much weaker since the electric field is shielded by the sheath and the dipole current is much smaller than a loop current.15 High voltages present breakdown effects in the nonlinear regime. Waves can also be excited by modulated16 or unmodulated electron beams via wave-particle resonance.17 The beam excitation is also limited to the linear regime. In space plasmas, whistler modes are usually excited by lightning, instabilities, ionospheric modifications, and rare active wave injections,18 none of which are as versatile as a simple loop antenna.19 

Wave diagnostics has advanced to the level of complete resolution of the Bwave(x,y,z,t) vector components in three-dimensional space and time. A single movable probe resolves the wave topology from repeated measurements in a pulsed laboratory plasma. The perturbation from a single probe is minimal, and the plasma is large compared to the wavelength, uniform, and quiescent. Multichannel digital oscilloscopes have replaced analog interferometer diagnostics.

This review will start with a description of an experimental device used to obtain many of the results described. Several sections and subsections describe the various findings. A new feature is to present space-time dependencies with clickable video clips. This review ends with a conclusion and suggestions of future research.

Although each research environment for studying whistler modes has somewhat different setups, it is still useful to describe one plasma device which produced most of the results described further below. The present plasma source consists of a large, pulsed dc discharge (5 ms on, 1 s off) with a density of ne10111012 cm−3, an electron temperature of kTe ≃ 2 eV, a neutral pressure of pn = 0.4 mTorr Ar, and a uniform axial magnetic field of B0 = 3–10 G in a large vacuum chamber (1.5 m diam, 2.5 m length), shown schematically in Fig. 1(a).

FIG. 1.

(a) Schematic of the experimental setup with basic parameters. (b) Photograph of the discharge plasma with an inserted loop antenna (4 cm diam.) used to excite basic whistler modes. A very small triple loop on a telescopic shaft serves to measure the wave magnetic fields in 3D space and time. Reproduced with permission Phys. Plasmas 25, 032111 (2018). Copyright 2018 AIP Publishing. (c) Photograph of the chamber interior without plasma with a multitude of antennas used to study whistler modes described in this review. (1) Linear wire with return current forming a large semicircular loop, used to create nonuniform ambient fields with null points. (2) Circular antenna array. (3) Electric dipole antenna. (4) Elongated magnetic loop antenna. (5) 5 cm diam m = 1 loop. (6) 15 cm diam m = 0 loop to create cusp and mirror background fields. (7) 4 cm diam loop, rotatable from the m = 0 to m = 1 configuration. Reproduced with permission from Phys. Plasmas 25, 032111 (2018). Copyright 2018 AIP Publishing.

FIG. 1.

(a) Schematic of the experimental setup with basic parameters. (b) Photograph of the discharge plasma with an inserted loop antenna (4 cm diam.) used to excite basic whistler modes. A very small triple loop on a telescopic shaft serves to measure the wave magnetic fields in 3D space and time. Reproduced with permission Phys. Plasmas 25, 032111 (2018). Copyright 2018 AIP Publishing. (c) Photograph of the chamber interior without plasma with a multitude of antennas used to study whistler modes described in this review. (1) Linear wire with return current forming a large semicircular loop, used to create nonuniform ambient fields with null points. (2) Circular antenna array. (3) Electric dipole antenna. (4) Elongated magnetic loop antenna. (5) 5 cm diam m = 1 loop. (6) 15 cm diam m = 0 loop to create cusp and mirror background fields. (7) 4 cm diam loop, rotatable from the m = 0 to m = 1 configuration. Reproduced with permission from Phys. Plasmas 25, 032111 (2018). Copyright 2018 AIP Publishing.

Close modal

The discharge uses a 1 m diam oxide-coated hot cathode, a photograph of which is shown in Fig. 1(b). The afterglow plasma is used since it provides a range of plasma parameters and provides a Maxwellian electron distribution function free of instabilities. Whistler modes are excited with magnetic loop antennas (4–20 cm diam, typically 2–4 turns, insulated from the plasma). The axis of the dipole is oriented either parallel to B0 which creates an m = 0 helicon mode or perpendicular to B0 which excites an m = +1 helicon mode.20 A variety of other electrodes and antennas, used in many experiments on whistler modes described further below, are shown in Fig. 1(c) and briefly explained in the figure caption. The antenna is driven by 5 MHz rf bursts (20 rf periods duration, 5 μs repetition time) whose transient state reveals the group velocity, while the continuous wave (cw) yields the phase velocity. The loop current can vary (Iloop = 1…100 A), which covers the linear regime to the highly nonlinear regime. The high currents are produced with a switched L-C circuit with an inductor L and a capacitor C. When studying linear waves, the field amplitude of B <0.1 G is applied. Conversely, nonlinear waves are excited with Brf>B05 G.

The wave magnetic field is received by a small magnetic probe with three orthogonal loops (6 mm diam) which can be moved together in three orthogonal directions. The spatial field distribution is obtained by moving the probe through orthogonal planes and acquiring at each position an average over 10 shots, which improves the digital resolution and reduces spurious noise. The fields of the antenna in vacuum can also be measured and subtracted from the fields in plasma, which is useful to distinguish shielding currents and wave currents in the antenna near zone. Plasma parameters are measured using Langmuir probes which are also attached to the telescopic probe. All signals are acquired using a 4-channel digital oscilloscope with an amplitude resolution of 8 bit and a time resolution of 10 ns.

In order to produce nonuniform background magnetic fields, we apply a current through a large circular loop (16 cm diam, 4 turns) with the dipole moment parallel to B0. It yields a mirror or cusp field topology depending on the direction of the dipole moment. Two dimensional nonuniform fields B0(y, z) with magnetic X and O points are created with a line current along the x-axis. The loop and line currents (Imax ≃ 600 A-turns) have a slowly varying sinusoidal waveform of one period (T ≃ 300 μs) which is long compared to the rf period (0.2 μs) or the rf burst length (5 μs) but short enough to avoid creating density perturbations. A string of phase-locked rf bursts is created throughout the slowly varying ambient magnetic fields. By recording all bursts, the wave propagation in many different field configurations is obtained within a single shot.

Figure 1(c) shows pictures of the 16 cm diam loop and the straight wire carrying the line current which will create 2D nonuniform ambient fields. The line current is closed by a semicircular return current near the bottom along the chamber wall. The field of the resultant large half loop is calculated from Biot-Savart's law. Likewise, the 3D field of the 16 cm loop is also calculated and superimposed on the uniform axial field.

Besides, the single loop antenna waves have also been excited by different antennas, for example, a circular array of loop antennas. The dipole moments of adjacent loops have been reversed so as to create a multipole of azimuthal mode number m = ±8. The antenna excites a helicon with a high mode number for the study of whistlers near the cyclotron resonance, often called Trivelpiece-Gould (TG) modes.21 

The excitation of whistler modes depends, among other criteria, on the size of the antenna relative to the wavelength. A very small antenna couples best to short wavelength whistler modes, which propagate near the resonance cone. At this angle, the phase and group velocities are nearly orthogonal. The latter determines the wave amplitude, and hence, the radiation pattern is the group velocity cone satisfying sinθ=ω/ωc, where ω is the signal frequency and ωc is the electron cyclotron frequency. Since it is easier to vary the wavelength instead of changing the antenna length, the resonance cone pattern is best seen at low densities or long whistler wavelengths.

Figure 2 shows a comparison of loop radiation patterns in vacuum, late afterglow, and early afterglow. In vacuum, the free space wavelength is λ = c/f =60 m, and the field is essentially the magnetic field of a loop without wave propagation effects. In a low density afterglow plasma, the loop produces a resonance cone pattern. The cone field does not propagate, but its sign oscillates. As the density is increased, the whistler wavelength decreases and the antenna launches propagating waves with oblique phase fronts. The phase and amplitude of the wave propagate axially, which implies that the group velocity is not oblique but nearly field aligned. The transition from resonance cones to whistler modes has been studied earlier.22,23

FIG. 2.

Basic antenna fields excited in vacuum and plasma. (a) Loop field Bz(x=0,y,z) in vacuum. (b) Resonance cones observed in low density plasmas (ωpωc). (c) Low frequency (ω0.3ωc) whistler modes excited in a dense plasma (ωpωc). Reproduced with permission from Phys. Plasmas 25, 032111 (2018). Copyright 2018 AIP Publishing.

FIG. 2.

Basic antenna fields excited in vacuum and plasma. (a) Loop field Bz(x=0,y,z) in vacuum. (b) Resonance cones observed in low density plasmas (ωpωc). (c) Low frequency (ω0.3ωc) whistler modes excited in a dense plasma (ωpωc). Reproduced with permission from Phys. Plasmas 25, 032111 (2018). Copyright 2018 AIP Publishing.

Close modal

Figure 3 summarizes the field topology of linear m = 0 helicon modes in a uniform axial magnetic field. The loop antenna in vacuum excites a dipole-like field (By, Bz)(x = 0,y,z) in plasma. In an unbound plasma, the wave propagates axially and radially outward, which results in V-shaped phase contours. (a) The Bz component has mirror images with respect to the antenna plane. (b) The Bx(y,z) component is an odd function with respect to y and z. It is one component of a toroidal field (Bx, By)(x,y) which links around Bz to form a vortex. (c) The field lines are radial at the ends of the vortex, circular in the middle of the vortex, and straight on the z-axis. (d) The radial phase propagation changes the spherical phase surface into a conical surface. The m = 0 mode has no phase rotation around B0. In helicon plasma devices, the diameter of the plasma column is that of or less than the antenna diameter. The radial wave propagation would not be noticeable. This leads to the impression of a radial eigenmode. In unbound plasmas, it is a wave packet but neither a theoretical helicon nor an ideal plane wave which are not realizable. Since two components vanish on the axis (Bx = By = 0), the m = 0 whistler mode is linearly polarized in the z-direction.

FIG. 3.

Basic properties of an m = 0 whistler mode. (a) Contour plot of the parallel field component Bz(x=0,y,z) at an instant of time during cw conditions. The inclined V-shaped contours are due to radial propagation. (b) Contour plot of the perpendicular field component Bx(x=0,y,z), which vanishes on the z-axis and changes sign across the axis. (c) Isosurfaces of Bz(x,y,z) imply conical phase surfaces. Wave magnetic field lines are approximately parallel to the phase surface. The m = 0 whistler mode has no azimuthal phase variation. When the plasma column is radially confined to the antenna dimensions, no radial propagation would be visible. The radial amplitude variation is assumed to be a standing wave, and the phase propagation would be paraxial, Bz=f(r)expi(kzωt). Such a column mode is frequently called an m = 0 helicon mode. Reproduced with permission from Phys. Plasmas 22, 092111 (2015). Copyright 2015 AIP Publishing.117 

FIG. 3.

Basic properties of an m = 0 whistler mode. (a) Contour plot of the parallel field component Bz(x=0,y,z) at an instant of time during cw conditions. The inclined V-shaped contours are due to radial propagation. (b) Contour plot of the perpendicular field component Bx(x=0,y,z), which vanishes on the z-axis and changes sign across the axis. (c) Isosurfaces of Bz(x,y,z) imply conical phase surfaces. Wave magnetic field lines are approximately parallel to the phase surface. The m = 0 whistler mode has no azimuthal phase variation. When the plasma column is radially confined to the antenna dimensions, no radial propagation would be visible. The radial amplitude variation is assumed to be a standing wave, and the phase propagation would be paraxial, Bz=f(r)expi(kzωt). Such a column mode is frequently called an m = 0 helicon mode. Reproduced with permission from Phys. Plasmas 22, 092111 (2015). Copyright 2015 AIP Publishing.117 

Close modal

The basic properties of an m = 1 helicon mode are shown in Fig. 4. It shows a snapshot of contours for the axial field component Bz(x,y) and a vector field of the perpendicular components (Bx, By)(x,y). The wave field resembles the dipolar field of the antenna whose dipole moment is perpendicular to B0. The major difference between vacuum and plasma fields is that the dipole field in plasma rotates around B0 in the same right-handed sense as plane whistlers do. Click on the video clip [see Fig. 4 (Multimedia view)] to observe the field rotation of an m = 1 helicon mode. The field rotation is the salient new feature of helicons, not the radial eigenmodes. The rotation creates an orbital angular momentum which produces a transverse Doppler shift and new wave-particle and wave-wave interactions.

FIG. 4.

Basic properties of an m = 1 whistler mode. The field components are displayed by contours of the parallel component Bz(x, y) and a vector field of the perpendicular components (BxBy), all at a fixed time under cw conditions. The wave field resembles the dipole field of the loop, but the wave field rotates in time or space. The time variation can be seen by clicking on m1 helicon rotation. The spiraling phase gives rise to the name helicon. Reproduced with permission from Phys. Plasmas 22, 092111 (2015). Copyright 2015 AIP Publishing.117 Multimedia view: https://doi.org/10.1063/1.5097852.1

FIG. 4.

Basic properties of an m = 1 whistler mode. The field components are displayed by contours of the parallel component Bz(x, y) and a vector field of the perpendicular components (BxBy), all at a fixed time under cw conditions. The wave field resembles the dipole field of the loop, but the wave field rotates in time or space. The time variation can be seen by clicking on m1 helicon rotation. The spiraling phase gives rise to the name helicon. Reproduced with permission from Phys. Plasmas 22, 092111 (2015). Copyright 2015 AIP Publishing.117 Multimedia view: https://doi.org/10.1063/1.5097852.1

Close modal

Helicon theory predicts the field propagation in the form Bzf(krr)expi(mϕ+kzzωt), where kr and kz are the radial and axial wave number components, respectively. In a plane z = const, the contours of Bz rotate in time in the +ϕ direction with one rotation per rf period, which identifies it as an m = +1 helicon mode. In unbound plasmas, the radial propagation leads to outward spiraling contours in an x-y plane. Except for the outward radial propagation, the fields in unbound plasmas have very similar radial profiles as predicted by helicon theory, BzJ0(krr) for m = 0 and BzJ1(krr) for m = 1 (see Fig. 2 in Ref. 24). Since neither boundaries nor density gradients exist, the antenna determines the Bessel function profiles. In helicon devices, the radial profile could be one of the three possibilities: boundaries, density profiles, and antenna properties. There are also frequent observations of radial phase shifts,23,25,26 which contradicts the theoretical prediction of “paraxial” wave propagation, f(krr)expi(mϕ+kzzωt). Thus, the widely accepted helicon eigenmode theory is only a very approximate description which does not include the antenna-wave coupling.

An axial plot of Bz(z) shows a strong amplitude oscillation which is sometimes interpreted as a wave reflection,27 whereas it is caused by the field rotation (see Figs. 4 and 5).

FIG. 5.

Phase fronts of an m = 1 helicon mode, displayed as 3D isosurfaces of the axial field component Bz(x,y,z). The snapshot in time shows two helical phase fronts in space. Along the positive Bz surface, the field lines spiral in the direction of +B0, while for Bz < 0, the lines point along −B0 and provide the field line closure. The time variation is shown in a video clip m1 helical phase surface. Note that the spirals translate axially in time but rotate in a transverse plane when the helix passes through the plane, consistent with Bz=f(r)expi[mϕ+kzωt]. Reproduced with permission from Phys. Plasmas 25, 082109 (2018). Copyright 2018 AIP Publishing. Multimedia view: https://doi.org/10.1063/1.5097852.2

FIG. 5.

Phase fronts of an m = 1 helicon mode, displayed as 3D isosurfaces of the axial field component Bz(x,y,z). The snapshot in time shows two helical phase fronts in space. Along the positive Bz surface, the field lines spiral in the direction of +B0, while for Bz < 0, the lines point along −B0 and provide the field line closure. The time variation is shown in a video clip m1 helical phase surface. Note that the spirals translate axially in time but rotate in a transverse plane when the helix passes through the plane, consistent with Bz=f(r)expi[mϕ+kzωt]. Reproduced with permission from Phys. Plasmas 25, 082109 (2018). Copyright 2018 AIP Publishing. Multimedia view: https://doi.org/10.1063/1.5097852.2

Close modal

At a fixed time, the phase surface in 3D space is helical, mϕ+kzz= const., an example of which is shown in Fig. 5. Since Bz has two polarities, there are two helices, both of which are left-handed for parallel propagation (kz > 0). The twist is reversed for the wave on the opposite side of the antenna (kz < 0). By choosing relatively large Bz contours, the radial propagation can be neglected and the observed contours agree by chance with helicon theory. But in unbound plasmas, the radial propagation enters into the phase (krr+mϕ+kzz) such that paraxial propagation breaks down and there are no radial eigenmodes.

Since the wave vector k is normal to the phase surface, the wave magnetic field lines must follow the phase surface and form helices too (see Fig. 2 in Ref. 28). Along the positive Bz surface, the field lines expand in the +z-direction; for Bz < 0, they return to the antenna. They close across B0 at the front and end of the rf burst. Note that in m = 0 helicons, the field lines closed within each half-wavelength vortex, but in rotating helicons (m0), they spiral throughout the wave packet.

The space-time evolution of a phase surface depends on 4 variables, Bz(x, y, z, t) = const, which can only be displayed in a video clip [see Fig. 5 (Multimedia view)] propagation of 3D phase surfaces. In contrast to a cork screw, the phase helix does not rotate but translates axially. When the left-handed helix passes through a plane z = const, the contours in the x-y plane rotate in the +ϕ direction. The wave on the opposite side of the antenna has a right-handed twist, travels in the −z-direction, which results in a temporal rotation in an x-y plane with the +ϕ direction, and thus remains in an m = +1 mode.

Figure 6 shows radial amplitude profiles of Bz(krr) for (a) m = 0 and (b) m = 1 modes in bound and unbound plasmas and for the vacuum field of the antenna. Helicon theory predicts Bessel function dependences of order given by the mode number m. The radial wave number kr is determined by the radius of the reflecting boundaries. The helicon eigenmode theory does not consider the antenna properties. The measured profile of the antenna vacuum field actually matches well the Bessel function profiles and yields a radial wavenumber of the antenna. The antenna field imposes the same Bessel function profiles on the waves in the uniform unbound plasma. Thus, the antenna properties may also determine the waves in smaller plasma columns. More examples will be shown that phased antennas excite negative helicon modes, which should not propagate according to eigenmode theory.

FIG. 6.

Radial profile of Bz(krr) of the antenna field, the measured helicon modes with (a) m = 0 and (b) m = 1 topology in an unbound plasma, and the predicted Bessel function profiles for bound helicon eigenmodes due to boundary reflections. The conclusion is that in the absence of boundaries, the antenna determines the radial wave properties. The agreement with helicon theory is coincidental. Radial phase shifts are often observed in helicon plasmas which is inconsistent with the predicted radial standing waves or paraxial wave propagation (see Fig. 7 in Ref. 25 and Fig. 7 in Ref. 23 and Fig. 7 in Ref. 26). Reproduced with permission from Phys. Plasmas 23, 052112 (2016). Copyright 2016 AIP Publishing.

FIG. 6.

Radial profile of Bz(krr) of the antenna field, the measured helicon modes with (a) m = 0 and (b) m = 1 topology in an unbound plasma, and the predicted Bessel function profiles for bound helicon eigenmodes due to boundary reflections. The conclusion is that in the absence of boundaries, the antenna determines the radial wave properties. The agreement with helicon theory is coincidental. Radial phase shifts are often observed in helicon plasmas which is inconsistent with the predicted radial standing waves or paraxial wave propagation (see Fig. 7 in Ref. 25 and Fig. 7 in Ref. 23 and Fig. 7 in Ref. 26). Reproduced with permission from Phys. Plasmas 23, 052112 (2016). Copyright 2016 AIP Publishing.

Close modal

A single m = 1 loop antenna has an azimuthal field dependence which oscillates but does not rotate, similar to an azimuthal standing wave. The standing wave can be decomposed into opposing m = +1 and m = −1 modes. An obvious question is why only an m = +1 helicon mode is excited. If an m = −1 mode was excited, the wave field should have no rotation like the antenna field. The preferential excitation of the m = +1 mode is due to the fact that the dominant antenna field is transverse to B0 and in the center of the loop. It matches the field of the m = +1 mode on the axis which resembles even a quasi plane-parallel whistler with right-hand circular polarization. No m = −1 mode with left-handed polarization can be excited with this antenna.

However, a different antenna can excite the m = −1 mode by coupling to the axial magnetic field ±Bz off-axis. Two m = 0 loops with opposite dipole fields along B0 are placed in juxtaposition (offset across B0), forming a quadrupole field. The strongest field components of the loops are the two opposing dipolar Bz components. There is also a transverse field B which has different signs on either side and vanishes in the middle of the antenna. The single loop has one strong transverse field in the middle of the loop, which couples best to the m = +1 helicon mode. It also has Bz components which are odd in z and hence vanish in the center of the loop. Thus, the single loop couples poorly via the Bz component, while the quadrupole couples poorly via the B components.

Figure 7 shows a snapshot of the Bz isosurfaces which exhibit an azimuthal standing wave. The left plane is 15 cm away from the antenna, where the antenna field is negligible and the observed field is a propagating helicon mode. The wave retains the standing wave of the antenna. Thus, the wave is controlled by the oscillating Bz components of the antenna. The single loop is not controlled by the antenna Bz component, otherwise should also produce a helicon with azimuthal standing waves. The axially propagating helicon mode with standing azimuthal oscillations implies that m = +1 and m = −1 modes propagate equally well.

FIG. 7.

Helicon modes excited by a quadrupole antenna (two opposing m = 0 loops). The snapshot of Bz(x,y) contours and isosurfaces and a selected wave field line show azimuthal standing waves which oscillate in space and time but do not rotate in ϕ. This property also holds for higher m-modes. The time dependence of azimuthal standing waves for m = ±4 helicons can be seen in a video clip phase oscillation without rotation. The decomposition of the azimuthal standing wave into opposing rotating fields implies that m = +1 and m = −1 modes propagate axially equally well. A single m = 1 loop does not excite m = −1 helicons. This result shows that antennas determine helicon modes, and boundaries do not. Reproduced with permission from Phys. Plasmas 22, 092111 (2015). Copyright 2015 AIP Publishing.117 Multimedia view: https://doi.org/10.1063/1.5097852.3

FIG. 7.

Helicon modes excited by a quadrupole antenna (two opposing m = 0 loops). The snapshot of Bz(x,y) contours and isosurfaces and a selected wave field line show azimuthal standing waves which oscillate in space and time but do not rotate in ϕ. This property also holds for higher m-modes. The time dependence of azimuthal standing waves for m = ±4 helicons can be seen in a video clip phase oscillation without rotation. The decomposition of the azimuthal standing wave into opposing rotating fields implies that m = +1 and m = −1 modes propagate axially equally well. A single m = 1 loop does not excite m = −1 helicons. This result shows that antennas determine helicon modes, and boundaries do not. Reproduced with permission from Phys. Plasmas 22, 092111 (2015). Copyright 2015 AIP Publishing.117 Multimedia view: https://doi.org/10.1063/1.5097852.3

Close modal

Circular arrays with more elements can be phased such that the antenna controls the sign and magnitude of the helicon mode number. For example, a video clip [see Fig. 12 (Multimedia view)] shows the rotation of an m = −4 helicon mode, which exhibits cw phase rotation and ccw vector rotations. The wave field near the axis is weak, and the polarization is elliptical to linear. Circular antenna arrays with up to 16 loops have been shown to produce propagating m = ±8 helicons, confirmed by both superposition and measurements on inserted array antennas.29 The high mode number yields high azimuthal wave numbers (kϕ= m/r), which can result in wave-electron resonances like Landau resonance (m > 0) and Doppler shifted cyclotron resonance (m < 0). This transverse Doppler shift involves perpendicular electron velocities such as produced by trapped energetic electrons undergoing mirror reflections.

The present results clarify a remaining “puzzle” about negative helicon modes in bound plasmas.10 

A single loop antenna can only excite m = 0 and m = 1 helicons. Their vacuum fields are linearly polarized and do not exhibit a phase rotation. The oscillating m = 1 antenna field can be decomposed into two oppositely rotating fields, one of which (m = +1) is excited because its field rotation matches the circular polarization of whistler modes, which is not the case for the m = −1 mode.

Rotating the antenna field is accomplished by two orthogonal m = 1 loops with a 90° phase shift. When the rotation matches that of an m = +1 helicon, the wave excitation is enhanced. When the antenna field rotates in the −ϕ direction, no m = −1 mode is excited efficiently.30,31 This does not mean that the plasma does not support an m = −1 mode, but the reason lies in the antenna coupling. The rotating field can be decomposed into two linearly polarized antenna fields. These excite m = 1 modes with a Δϕ=90° phase shift whose superposition cancels for the left rotation (m = −1) and adds for the right rotation (m = +1).

In order to impose a specific m-mode, one has to use phased arrays. Constructing large arrays is resource intensive and possibly perturbs the plasma, but there are simpler solutions to obtain the antenna properties. The linear wave superposition is a valid approach known since the Huygens principle32 for creating wave fronts from wavelets. In order to confirm the superposition principle for plasmas, the waves from two spatially separated loops have been measured from (i) each antenna separately and (ii) both together. The superposition from case (i) agrees well with case (ii), confirming that the linear wave superposition is valid in plasmas of uniform density and ambient magnetic field (see Fig. 10 in Ref. 33).

If the superposition holds for two antennas, it also applies for an array of many loops. When closely spaced, the fringing fields of the loops average out and the axial dipole field becomes constant along the array. Figure 8 shows two arrays along the y-axis perpendicular to B0, (a) without and (b) with the phase shift and all loops with the same amplitude and x-direction. The array without the phase shift excites a so-called “quasi” parallel plane whistler, because an “ideal” plane wave has a constant amplitude and phase to infinity, which obviously does not exist in reality. Beyond the radial limits of the array, the field propagates again obliquely like from a single loop and the field amplitude drops off rapidly. At the top of the array, the wing indicates upward phase propagation, but the energy flows nearly along B0. The phased array excites quasi plane waves but with oblique k=(ky,kz), where ky is imposed by the phase shift of the antenna.

FIG. 8.

A linear antenna array is formed by loop antennas in the y-direction. The signal of one loop is measured, and the displaced antennas signals are superimposed, which is valid in a uniform plasma for linear fields. (a) When all loops have the same amplitude and phase, the antenna array produces a plane wave propagating along ±B0. It is an approximate plane wave, which does not hold beyond the ends of the linear array and outside the x = 0 plane. (b) When the signal of each loop is phase shifted, the antenna imposes a wave number component ky which results in an oblique plane wave with k=(ky,kz). Note that beyond the antenna array, the wave amplitude decays rapidly in the ±y direction, implying a group velocity nearly along ±B0. The wave amplitude decreases as the propagation angle θ approached cosθ=ω/ωc. A video clip shows the propagation from a phased linear array producing a quasi oblique plane wave. Reproduced with permission from Phys. Plasmas 22, 072110 (2015) Copyright 2015 AIP Publishing.118 Multimedia view: https://doi.org/10.1063/1.5097852.4

FIG. 8.

A linear antenna array is formed by loop antennas in the y-direction. The signal of one loop is measured, and the displaced antennas signals are superimposed, which is valid in a uniform plasma for linear fields. (a) When all loops have the same amplitude and phase, the antenna array produces a plane wave propagating along ±B0. It is an approximate plane wave, which does not hold beyond the ends of the linear array and outside the x = 0 plane. (b) When the signal of each loop is phase shifted, the antenna imposes a wave number component ky which results in an oblique plane wave with k=(ky,kz). Note that beyond the antenna array, the wave amplitude decays rapidly in the ±y direction, implying a group velocity nearly along ±B0. The wave amplitude decreases as the propagation angle θ approached cosθ=ω/ωc. A video clip shows the propagation from a phased linear array producing a quasi oblique plane wave. Reproduced with permission from Phys. Plasmas 22, 072110 (2015) Copyright 2015 AIP Publishing.118 Multimedia view: https://doi.org/10.1063/1.5097852.4

Close modal

The axial propagation can be seen in the video clip [see Fig. 8 (Multimedia view)] quasi oblique plane waves excited by a phased array. It is worth pointing out that the wave amplitude is constant along the length of the array but drops off rapidly at both ends because the group velocity is nearly field aligned, whereas the phase velocity is near the Gendrin angle, θG=arccos(2ω/ωc)=arccos10/14=44°. The highly field aligned energy flow attenuates the radially outward propagating wave and the proposed radially inward reflected wave. This prevents the formation of standing waves, which are the essence of the radial eigenmode theory and the resultant dispersion relation. A better approach is to Fourier transform the helicon wavepacket and to compare the plane wave modes with the 3D refractive index surface which has been done earlier (see Fig. 10 in Ref. 34).

Some basic properties of oblique plane waves can be recalled from Fig. 9 which displays contours of Bx(y,z) and a vector field (By, Bz)(y,z) in the x = 0 midplane of the phased array antenna. The wave propagates normal to the constant contours, e.g., Bx = 0. At this phase front, the magnetic field lines are parallel to the phase front. At the crest of the Bx contour, (By,Bz)0, i.e., the field points in the x-direction. Following along k, the field rotates circularly around k, left-handed in space and right-handed in time on a phase surface. The phase surface of this plane waves does not rotate and hence is not a helicon mode. When the array consists of m = 0 loops, the near-zone field has a traveling Bx component which couples to the wave Bx component. Likewise, when the array consists of m = 1 loops, its field is a traveling By field. It couples equally well to the wave which also has a By component.

FIG. 9.

Field lines of oblique plane waves, displayed by contours of the transverse field component Bx(x=0,y,z) and a vector field of the in-plane components (By, Bz). The propagation direction or wave vector k is normal to the phase fronts. The field rotates left handed along k, not along B0. The wave vector rotates right handed when the wave propagates through a constant phase plane. The field lines close outside the x = 0 plane.

FIG. 9.

Field lines of oblique plane waves, displayed by contours of the transverse field component Bx(x=0,y,z) and a vector field of the in-plane components (By, Bz). The propagation direction or wave vector k is normal to the phase fronts. The field rotates left handed along k, not along B0. The wave vector rotates right handed when the wave propagates through a constant phase plane. The field lines close outside the x = 0 plane.

Close modal

With the quasi plane waves excited by a linear array, one can compare the refractive index curve, n vs n, where n=kc/ω is the refractive index. The result shows reasonable agreement with plane wave theory even though the present plane wave applies only to two dimensions, n=(ny,nz) (see Fig. 6 in Ref. 35). It is also worth pointing out that the single loop wave packet contains k-modes near the resonance cone; otherwise, the superposition would not produce a plane wave near the resonance cone.

The next step to obtain better plane waves employs two-dimensional antenna arrays. The antenna surface has to have a constant amplitude and phase to excite a parallel plane wave. This cannot be achieved with a large wire loop. Since helicon modes are less well understood than plane waves, a large antenna array of circular cross section has been investigated.

Figure 10 shows the principle of the array with rings of (a) m = 1 and (b) m = 0 loops. Starting with an array of m = 1 loops, all with an equal amplitude and phase, the superposition produces in vacuum a nearly constant circular field Bϕ(x,y), while the loop fringing fields (Br, Bz) nearly average out. However, the wave field in plasma differs since it rotates Bϕ into (Br, Bz). Figure 10(c) shows a snapshot of the wave amplitude |B|(x,y,z = 18 cm). The array launches a beam of constant amplitude over the antenna cross-section, but it is not a plane wave. Figure 10(d) shows contours of Bz and vectors of (Bx, By) in the midplane of the wave packet. The in-plane field B=(Bx,By) is nearly circular, which links with the axial field Bz to form a vortex topology of an oblate vortex. At a later time (e), the end of the propagating vortex passes through the x-y plane. The vectors of B=(Bx,By) are nearly radial, the contours of |B| decrease with radius, and B0. Thus, at the ends of the half-wavelength vortices, the axial field lines diverge radially inward or outward, which forms a 3D null point with a radial fan between two vortices (see the selected stream lines). The topology resembles that of a broad m = 0 mode but is excited by m = 1 loops. The phase does not rotate (m = 0), but the vector polarization changes from linear on the axis to right-hand elliptical and circular with the increasing radius.

FIG. 10.

Waves excited by a circular 2D antenna array. It can consist of (a) m = 0 loops or (b) m = 1 loops. The excited helicon mode number does not depend on the single loop orientation but on the phasing of the entire array. (c) An antenna array of m = 0 loops with a constant amplitude and phase excites a whistler beam of nearly constant amplitude within the cross section of the array. This cannot be achieved by a single m = 0 loop of large diameter. (d) Contours of the parallel field component Bz(x,y,z=18 cm) and vector fields of the perpendicular components (Bx, By) in the center of an m = 0 helicon mode. (e) Contours of B(x,y,z = 18 cm) and vectors of B between two oblate vortices, showing the topology of a radial fan or cusp 3D null point. The radial amplitude profile is determined by the antenna and not by reflections from boundaries or density gradients. Reproduced with permission from Phys. Plasmas 23, 052112 (2016). Copyright 2016 AIP Publishing.

FIG. 10.

Waves excited by a circular 2D antenna array. It can consist of (a) m = 0 loops or (b) m = 1 loops. The excited helicon mode number does not depend on the single loop orientation but on the phasing of the entire array. (c) An antenna array of m = 0 loops with a constant amplitude and phase excites a whistler beam of nearly constant amplitude within the cross section of the array. This cannot be achieved by a single m = 0 loop of large diameter. (d) Contours of the parallel field component Bz(x,y,z=18 cm) and vector fields of the perpendicular components (Bx, By) in the center of an m = 0 helicon mode. (e) Contours of B(x,y,z = 18 cm) and vectors of B between two oblate vortices, showing the topology of a radial fan or cusp 3D null point. The radial amplitude profile is determined by the antenna and not by reflections from boundaries or density gradients. Reproduced with permission from Phys. Plasmas 23, 052112 (2016). Copyright 2016 AIP Publishing.

Close modal

When the circular array of m = 0 loops is phased “radially,” the wave will also acquire a radial wavenumber component kr. Figure 11 shows a snapshot of the axial field component Bz(x,y,z = 18 cm) and the transverse component (Bx, By). The radial wavelength (λr10 cm) is comparable to the axial wavelength. The propagation angle θ=arctan(k/k)45° which is close to the Gendrin angle θG=arccos(2ω/ωc)=arccos(10/14)=44°. For plane oblique whistler waves, the Gendrin mode has oblique phase velocity and parallel group velocity. The present case is a cylindrical Gendrin mode which also exhibits a highly uniform amplitude profile over the antenna cross section. The outward phase propagation is shown in the video clip [see Fig. 11 (Multimedia view)] of the Gendrin helicon beam. For inward propagation, phase focusing does not produce an amplitude enhancement on the axis.

FIG. 11.

Radial phasing of a circular array produces whistler modes which propagate radially and axially. The nearly axial group velocity prevents significant radial energy transport, e.g., no amplitude focusing for radially inward propagation. When the propagation angle is chosen by cosθ=2ω/ωc, the mode is a circular Gendrin mode with a field aligned group velocity. The radially outward propagation is best seen in a video clip of the m = 0 Gendrin helicon mode. Reproduced with permission from Phys. Plasmas 23, 052112 (2016). Copyright 2016 AIP Publishing. Multimedia view: https://doi.org/10.1063/1.5097852.5

FIG. 11.

Radial phasing of a circular array produces whistler modes which propagate radially and axially. The nearly axial group velocity prevents significant radial energy transport, e.g., no amplitude focusing for radially inward propagation. When the propagation angle is chosen by cosθ=2ω/ωc, the mode is a circular Gendrin mode with a field aligned group velocity. The radially outward propagation is best seen in a video clip of the m = 0 Gendrin helicon mode. Reproduced with permission from Phys. Plasmas 23, 052112 (2016). Copyright 2016 AIP Publishing. Multimedia view: https://doi.org/10.1063/1.5097852.5

Close modal

When the circular array is phased “azimuthally,” the wave field rotates and propagates axially, i.e., forms a helicon mode. Figure 12 presents a snapshot of the wave fields at z = 15 cm from the antenna where the vacuum field is negligible. The contours of the axial field Bz(x,y) component have four pairs of alternating maxima, which rotate in time as shown by the video clip [see Fig. 12 (Multimedia view)] of the m = −4 helicon rotation. The rotation is in the −ϕ direction, while the vectors rotate in the +ϕ direction. Thus, polarization and phase rotation are not identical in helicons. An m = −1 helicon mode does not imply a left-handed vector rotation. The polarization is always right-handed although locally linear polarization also arises and is readily explained by wave interference.

FIG. 12.

Azimuthal phasing of a circular array produces whistler modes which rotate around B0 and propagate along B0 and hence are generalized helicon modes in unbound plasmas. The sign and magnitude of the phase shift determine the mode number. The example shows the field components Bz(x,y,z=18 cm) and (Bx, By) of an m = −4 helicon beam. The rotation is best seen in a video clip of m = −4 helicon rotation. Note that the phase front rotates clockwise (cw), while the vectors rotate ccw, with the latter being typical for electron whistler modes. Multimedia view: https://doi.org/10.1063/1.5097852.6

FIG. 12.

Azimuthal phasing of a circular array produces whistler modes which rotate around B0 and propagate along B0 and hence are generalized helicon modes in unbound plasmas. The sign and magnitude of the phase shift determine the mode number. The example shows the field components Bz(x,y,z=18 cm) and (Bx, By) of an m = −4 helicon beam. The rotation is best seen in a video clip of m = −4 helicon rotation. Note that the phase front rotates clockwise (cw), while the vectors rotate ccw, with the latter being typical for electron whistler modes. Multimedia view: https://doi.org/10.1063/1.5097852.6

Close modal

Further examples are shown in video clips (see video s1 in the supplementary material) of an m = −1 helicon rotation excited by a large array of m = 1 loops. It shows one pair of opposite Bz(x,y) contours and one pair of dipolar (Bx, By) fields all of which rotate clockwise (−ϕ). The individual vectors rotate counterclockwise (ccw) except in two O-type null points near the center where only Bz remains finite and forms a linear polarization in the z-direction.

The rotation of the (Bx, By) vectors in a higher order (m = −3) helicon mode is shown in the video (see video s2 in the supplementary material) vector field rotation of an m = −3 helicon. There are 6 transverse dipolar fields pointing alternately radially inward and outward. Most of the wave vectors are right-hand polarized with some exceptions near the six null points in the center. The field amplitude has a minimum at the center (see Fig. 9 in Ref. 36), which is not surprising because m-mode helicons are rotating multipoles of order m. In addition, with the decreasing radius, the azimuthal wavelength approaches the electron inertial scale which is the limit for whistler mode propagation.29 

Axial antenna arrays have also been investigated. In its simplest form, an “end-fire” array consists of two loop antennas displaced axially and phase shifted or rotated. The radiation properties are shown in Fig. 13. The contours of B2(y,z) show that the radiation is highly asymmetric. The directionality arises from the axial antenna spacing (λ/2) and a temporal phase delay (θ=90°), which adds the components from both antennas on one side of the antenna and subtracts them on the other side. The polar plot at the bottom shows that Bmax2 is almost 100 times larger on one side compared to the opposite side. Making the array longer did not improve the directionality significantly. These results are of practical value and of value for understanding the radiation from axially long antennas. These are typically λ/2 long and radiate asymmetrically which is sometimes associated with the poor propagation of m = −1 helicons.37 The directionality due to interference of different axial parts of the antenna has not yet been discussed.

FIG. 13.

Directionality of helicon modes of an axial antenna array. Two loops are axially shifted by λ/4 and phase shifted in time by 90°. Both shifts lead to the addition of field components on one side and subtraction on the opposite side. The polar plot of the wave energy density, shown below, indicates a 20 dB increase to the right side compared to the left side at the location of the circle above. Reversing the temporal phase shift switches the direction of preferred radiation. Reproduced with permission from Phys. Plasmas 22, 072110 (2015). Copyright 2015 AIP Publishing.118 

FIG. 13.

Directionality of helicon modes of an axial antenna array. Two loops are axially shifted by λ/4 and phase shifted in time by 90°. Both shifts lead to the addition of field components on one side and subtraction on the opposite side. The polar plot of the wave energy density, shown below, indicates a 20 dB increase to the right side compared to the left side at the location of the circle above. Reversing the temporal phase shift switches the direction of preferred radiation. Reproduced with permission from Phys. Plasmas 22, 072110 (2015). Copyright 2015 AIP Publishing.118 

Close modal

A single small loop antenna excites helicon modes with V-shaped phase contours, resulting from both axial and radially outward wave propagation. A large m = 0 loop with the diameter comparable to the wavelength excites both radially inward and outward waves, i.e., a converging and a diverging cone (see Fig. 3 in Ref. 38 and Fig. 9 in Ref. 39). The converging phase does not imply an amplitude maximum on the axis. The same radiation pattern arises for a large circular array of loops [see Fig. 1(c), item 8]. When the array is phased to produce an m = +1 helicon mode, the contours of Bz(x,y) exhibit two rotating spirals as shown in Fig. 14; one winds outward, and the other inward toward the axis. The rotation is shown in the video clip [see Fig. 14 (Multimedia view)] phase rotation of an m = 1 helicon mode with radially inward and outward propagation. The inner wave propagation can be eliminated by many concentric rings of loops, i.e., a uniform 2D array.

FIG. 14.

Inward and outward radial phase propagation of an m = 1 helicon mode excited by a circular array of a single radius. Displayed are a snapshot of Bz(x,y) contours (phase fronts) and (Bx, By) vectors. The black circle is the location of the circular array of m = 0 loops. The white lines are spirals due to radial outward and inward phase propagation. The radial and azimuthal motions of the phase fronts in time are shown in the video clip of spiral phase fronts of an m = 1 helicon mode excited by a circular antenna array. Reproduced with permission from Phys. Plasmas 23, 052112 (2016). Copyright 2016 AIP Publishing. Multimedia view: https://doi.org/10.1063/1.5097852.7

FIG. 14.

Inward and outward radial phase propagation of an m = 1 helicon mode excited by a circular array of a single radius. Displayed are a snapshot of Bz(x,y) contours (phase fronts) and (Bx, By) vectors. The black circle is the location of the circular array of m = 0 loops. The white lines are spirals due to radial outward and inward phase propagation. The radial and azimuthal motions of the phase fronts in time are shown in the video clip of spiral phase fronts of an m = 1 helicon mode excited by a circular antenna array. Reproduced with permission from Phys. Plasmas 23, 052112 (2016). Copyright 2016 AIP Publishing. Multimedia view: https://doi.org/10.1063/1.5097852.7

Close modal

A hodogram is the curve where the tip of a three-component wave electric or magnetic field vector traces out in time, typically one period for cw waves. In order to satisfy Maxwell's equation ·B(r)=0 for plane waves, one must have k·B=0. This implies that the wave magnetic field lies in the phase front. The hodogram surface is the local phase front, and its normal is parallel to the wave vector k. There is a sign ambiguity that the hodogram normal is parallel or antiparallel to the wave vector k, depending upon the direction of propagation with respect to B0.

Hodograms are a standard tool in space plasma physics to determine the direction of wave propagation.40,41 Hodograms are mainly used to determine the type of polarization. For plane-parallel whistlers, the hodogram rotation is anticlockwise, which produces the right-handed rotation around B0. For plane but oblique whistlers, the field vector rotates around the k-vector but not around B0,42,43 also see Fig. 7(c) in Ref. 35. Single point hodograms are difficult to interpret; unless, one makes assumptions of plane waves. Multipoint measurements are needed to distinguish plane waves from wave packets.

Although not frequently used in laboratory plasmas, hodograms are a powerful diagnostic tool for tracing the flow of phase and energy, shown here for helicon modes. It requires multipoint measurements to trace streamlines through the vector field of k. By expanding this concept to the Poynting vector, the flow of energy can also be obtained.

Figure 15 shows an example of hodogram diagnostic results. A single-point hodogram is shown in Fig. 15(a), taken near the axis of an m = 1 helicon. It has a right-hand circular polarization with respect to the normal n or k which is slightly oblique to B0. Figure 15(b) shows hodograms with linear polarization taken on the axis of an m = 0 helicon mode. Since Bz is the only field component, the polarization is along B0.

FIG. 15.

Examples of phase and energy flow obtained with hodograms. A hodogram is formed by the rotation of the wave magnetic field during one cycle. It can be (a) a circle (circular polarization) whose normal indicates the direction of the phase velocity, (b) an oscillation along a line (linear polarization), and (c) the concept can be extended to other vectors such as the Poynting vector S. In whistler wave packets, elliptical polarization is most common. (d) By performing multipoint hodogram measurements, the streamlines through the vector field of normals (k-vectors) yield the flow of the phase velocity. The out-of-plane normal indicates a phase rotation around the axis of the wave packet. (e) Contours of the axial Poynting vector component and streamlines of the time-averaged in-plane components of S. (f) Vector field of hodogram normals (nx, ny) indicating the phase rotation in an x-y plane normal to B0. (g) Snapshot of the transverse Poynting vector which indicates an angular field momentum. In time, the opposing S-vectors peak where Bz(x,y) has peaks, and both rotate in time in the + ϕ direction, characteristic for an m = +1 helicon mode. (h) 3D isosurface of the perpendicular Poynting vector, showing the helical flow of energy or momentum. The latter has been determined quantitatively. Reproduced with permission from Phys. Plasmas 25, 032112 (2018). Copyright 2018 AIP Publishing.119 

FIG. 15.

Examples of phase and energy flow obtained with hodograms. A hodogram is formed by the rotation of the wave magnetic field during one cycle. It can be (a) a circle (circular polarization) whose normal indicates the direction of the phase velocity, (b) an oscillation along a line (linear polarization), and (c) the concept can be extended to other vectors such as the Poynting vector S. In whistler wave packets, elliptical polarization is most common. (d) By performing multipoint hodogram measurements, the streamlines through the vector field of normals (k-vectors) yield the flow of the phase velocity. The out-of-plane normal indicates a phase rotation around the axis of the wave packet. (e) Contours of the axial Poynting vector component and streamlines of the time-averaged in-plane components of S. (f) Vector field of hodogram normals (nx, ny) indicating the phase rotation in an x-y plane normal to B0. (g) Snapshot of the transverse Poynting vector which indicates an angular field momentum. In time, the opposing S-vectors peak where Bz(x,y) has peaks, and both rotate in time in the + ϕ direction, characteristic for an m = +1 helicon mode. (h) 3D isosurface of the perpendicular Poynting vector, showing the helical flow of energy or momentum. The latter has been determined quantitatively. Reproduced with permission from Phys. Plasmas 25, 032112 (2018). Copyright 2018 AIP Publishing.119 

Close modal

Figure 15(c) presents a hodogram of the Poynting vector S=E×H, where H=B/μ0 and E is the Hall electric field which dominates over the resistive parallel field (Ez0). The vector rotates twice per rf period and has a conical surface for an m = 1 helicon caused by the azimuthal and axial energy flow. The time average vector S yields the parallel energy flow (B2/2μ0=vgroup·S).

Multipoint hodogram measurements yield the flow of the phase velocity. Figure 15(d) displays contours of the hodogram normal component nx and streamlines of the vector field (ny, nz). The odd dependence of nx(y) confirms the helicon phase rotation. The streamlines show no paraxial propagation but a significant radial wave spread without a significant amplitude decay. Figure 15(e) shows contours of the average Poynting vector S(y,z) and streamlines through the vector field (Sy, Sz). The latter shows much less energy spread than the phase spread. Collisional damping and some wave spread account for the energy loss along B0.

The phase and energy rotation of an m = 1 helicon mode are shown in Figs. 15(f) and 15(g), respectively. The snapshot of the wave vectors (kx, ky) shows the phase rotation with some azimuthal variation. The Poynting vector field (Sx,Sy)=E×(Bz)/μ0 has two peaks with opposite directions created by the two opposing peaks in Bz. The off-axial peaks rotate in time and carry an orbital angular field momentum L=r×S/vgroup2. An isosurface of S is shown in Fig. 15(h), which reveals a helical spiral similar to the phase. It is the Poynting vector which determines energy and angular momentum flow, not the wave vector which determines the phase.

Hodograms are useful to quantify the wave polarization. They yield the sense of rotation and the ellipticity ϵ=Bmin/Bmax. Theory predicts that plane-parallel whistlers have a right-hand circular polarization with respect to B0 or k for oblique propagation. Plane waves are a theoretical concept, but all real waves are finite-size wave packets. Linear polarization of whistler wave packets is readily observed, for example, in m = 0 helicon modes. Figures 3(a) and 3(b) show that the perpendicular field components vanish on the z-axis, while the parallel component Bz maximizes on the axis. This yields a linear polarization along B0. In general, a linear polarization arises when two field components vanish, while the third one is nonzero for at least one rf period.

Whistler modes have phase shifts between different components, and hence, interference creates nodes at different locations. An example is shown in Fig. 16, where two m = 0 modes propagate axially against each other. Standing waves are created in the axial direction, while radial waves continue to propagate outward. Interference nodes for the perpendicular components (a) Bx(x,y) and (b) By(x,y) form in the midplane (z15 cm) and adjacent planes spaced ±λ/2 apart. The waves cancel at these planes because the B components have opposite signs on either side of the antennas. However, the axial Bz component [Fig. 16(c)] has the same sign on both sides of the loop antennas, and hence, their signals add in the midplane. With one remaining component, the polarization is linear along the z = 15 cm line. The same conditions also arise along the y = 0 line, where B(y,z) vanishes (d) without or (e) with collisions. Wave damping creates unequal amplitudes and nonzero nodes, and the polarization becomes elliptical.

FIG. 16.

Linear polarization caused by interference. (a) Contours of the perpendicular field component Bx(y,z) of two counter propagating m = 0 helicons which form standing waves in the axial direction. The nodes are indicated by vertical dashed lines. In addition, there are nodes of B on the horizontal z-axis at x = y = 0 for all m = 0 modes irrespective of wave interference. (b) Contours of By(y,z) which has nodes at the same location as the nodes of Bx(y,z) since (Bx, By) is either azimuthal or radial [see Fig. 3(c)]. (c) The axial field component Bz(y,z) is an even function in z and hence forms an antinode (Bmax) in the central vertical line (x = 0, y, z = 0) which creates linear polarization. (d) Hodogram ellipticity ϵ=Bmin/Bmax for a single m = 0 mode which produces linear polarization on the z-axis (x = y = 0) without interference. (e) Ellipticity for interfering waves which adds ϵ0 along the dashed vertical lines. (f) Amplitude modulation Brms=[(B2)1/2B]/B) which peaks where only one field component exists, coinciding with linear polarization [also see Fig. 10(f) in Ref. 31]. Note that single m = 1 modes have no linear polarizations, but interfering m = 1 modes do [see Fig. 6(f) in Ref. 20]. Reproduced with permission from Phys. Plasmas 25, 032111 (2018). Copyright 2018 AIP Publishing.

FIG. 16.

Linear polarization caused by interference. (a) Contours of the perpendicular field component Bx(y,z) of two counter propagating m = 0 helicons which form standing waves in the axial direction. The nodes are indicated by vertical dashed lines. In addition, there are nodes of B on the horizontal z-axis at x = y = 0 for all m = 0 modes irrespective of wave interference. (b) Contours of By(y,z) which has nodes at the same location as the nodes of Bx(y,z) since (Bx, By) is either azimuthal or radial [see Fig. 3(c)]. (c) The axial field component Bz(y,z) is an even function in z and hence forms an antinode (Bmax) in the central vertical line (x = 0, y, z = 0) which creates linear polarization. (d) Hodogram ellipticity ϵ=Bmin/Bmax for a single m = 0 mode which produces linear polarization on the z-axis (x = y = 0) without interference. (e) Ellipticity for interfering waves which adds ϵ0 along the dashed vertical lines. (f) Amplitude modulation Brms=[(B2)1/2B]/B) which peaks where only one field component exists, coinciding with linear polarization [also see Fig. 10(f) in Ref. 31]. Note that single m = 1 modes have no linear polarizations, but interfering m = 1 modes do [see Fig. 6(f) in Ref. 20]. Reproduced with permission from Phys. Plasmas 25, 032111 (2018). Copyright 2018 AIP Publishing.

Close modal

Polarization also affects the wave amplitude vs time. For a circularly polarized wave, the amplitude of a rotating vector remains constant. For a linearly polarized wave, the amplitude oscillates. Figure 16(f) shows contours of the root-mean-square amplitude, which maximize where ϵ minimizes.

On the axis of an m = 1 helicon mode, the parallel field component vanishes, while the perpendicular components maximize, which results in circular polarization. However, interference between m = 1 helicons also produces linear polarization at planes Δz=±λ/2 from the midplane.

Interference commonly arises when waves reflect from boundaries. The boundaries result from steep gradients in the refractive index or conductivity, such as variations in density, magnetic field, and conductivity. Reflections from boundaries have been studied earlier,27,44,45 but the details of the change in helicon rotation, force, and torque transfer to the boundary, group and phase velocity changes, helicity conservation, etc., are as complicated as the near zone of antennas.

Helicity describes field lines with a twist, writhe, linkage, or knots. Its basic definition is H=A·(×A)d3r, where A is a vector field like the vector potential, magnetic field, current density, fluid velocity, etc. Helicity is important for magnetic field topologies,46 fluid mechanics,47 and molecular chemistry.48 

The current density J=nev has a simple relation to the wave magnetic field B, provided that it is small compared to the guide field B0. From Faraday's law B/t=×E and the Hall relation E+v×B=0, one obtains B/t=×(v×B)=B0·J/ne. For plane monochromatic waves /t=ω,/z=kz and defining a Hall conductivity σH=ne/B0 and parallel phase velocity vz=ω/kz, the relation simplifies to J=vzσHB. It predicts that the wave field is force-free, J·B=0, which requires the field lines to be twisted or writhed. The sign of the helicity depends on wave propagation along B0 even when the wave propagation is oblique.

Relevant observations are presented in Fig. 17 which (a) shows contours of the helicity density component JxBx of an m = 0 helicon mode. The sign changes with the propagation direction along B0, the amplitude is proportional to Bx2, and hence, the spatial period is λ/2. Figure 17(b) shows the ratio J/B for an m = 1 helicon mode propagating along B0. The ratio is surprisingly uniform in the region of the wave. It has a value close to theory J/B=vσH even though a helicon is not a plane wave.

FIG. 17.

Helicity of helicon modes. (a) Helicity density component JxBx (y,z) for an m = 0 helicon mode excited by a single loop antenna. The sign depends on the direction of wave propagation along B0. (b) Contours of the ratio J/B(y,z) which is fairly constant in the region of wave propagation. Theory predicts that J/B=σHv, where σH=ne/B0 is the Hall conductivity and v is the phase velocity along B0. (c) Time dependence of the spatially averaged ratio |J|/|B|(t) and the ratios of Jx/Bx and Jy/By. The component ratios are nearly time independent and comparable to the total ratio. With J=×B/μ0B, the wave field is a force-free Beltrami field. Reproduced with permission from Phys. Plasmas 25, 032112 (2018). Copyright 2018 AIP Publishing.119 

FIG. 17.

Helicity of helicon modes. (a) Helicity density component JxBx (y,z) for an m = 0 helicon mode excited by a single loop antenna. The sign depends on the direction of wave propagation along B0. (b) Contours of the ratio J/B(y,z) which is fairly constant in the region of wave propagation. Theory predicts that J/B=σHv, where σH=ne/B0 is the Hall conductivity and v is the phase velocity along B0. (c) Time dependence of the spatially averaged ratio |J|/|B|(t) and the ratios of Jx/Bx and Jy/By. The component ratios are nearly time independent and comparable to the total ratio. With J=×B/μ0B, the wave field is a force-free Beltrami field. Reproduced with permission from Phys. Plasmas 25, 032112 (2018). Copyright 2018 AIP Publishing.119 

Close modal

From 3D measurements, the spatial average of the helicity of field components can be compared with the total |J|/|B|. Figure 17(c) shows the time dependence of the total helicity density and those of transverse components, JxBx and JyBy. The values are comparable and exhibit small temporal oscillations.

Having established that the wave topology is close to a Beltrami field,46 one has to reconsider the topology of the wave field lines. These must be twisted to obtain (×B)B. This is confirmed by plotting field lines and flux ropes in 3D space. Figure 18(a) shows the field at the start of an m = 1 helicon mode in a curved magnetic field B0, an upcoming topic. The new feature is that a bundle of lines in the form of a rope with a triangular cross section exhibits a pronounced writhe. If two adjacent field lines are launched next to each other, they twist around each other. Its right-handed writhing number exceeds the single rotation of the left-handed helicon twist, resulting in a net positive helicity. Figure 18(b) shows that strong writhing is also seen in whistler modes without helicon topology. Thus, whistler mode theories predict JB, but displays of field lines without a twist are inconsistent with real field lines. The problem is that most field lines are plotted in 2D,49 whereas the helicity is only visible in 3D.

FIG. 18.

Field line writhing of force-free whistler modes. A flux tube (rod) with a triangular cross section is writhed, i.e., adjacent field lines in the tube twist around each other. The twist is right-handed, implying positive magnetic helicity density. It is found in (a) helicon modes and (b) oblique whistler modes in both uniform and nonuniform B0 fields. Force-free fields must be twisted, which can only be seen from measurements of 3 components in 3D space. Flux ropes and vortices in MHD plasmas have related topologies. Reproduced with permission from Phys. Plasmas 25, 082109 (2018). Copyright 2018 AIP Publishing.

FIG. 18.

Field line writhing of force-free whistler modes. A flux tube (rod) with a triangular cross section is writhed, i.e., adjacent field lines in the tube twist around each other. The twist is right-handed, implying positive magnetic helicity density. It is found in (a) helicon modes and (b) oblique whistler modes in both uniform and nonuniform B0 fields. Force-free fields must be twisted, which can only be seen from measurements of 3 components in 3D space. Flux ropes and vortices in MHD plasmas have related topologies. Reproduced with permission from Phys. Plasmas 25, 082109 (2018). Copyright 2018 AIP Publishing.

Close modal

The helicity of the magnetic field H=A·Bd3r is predicted to be conserved in ideal MHD plasmas.50 Experiments on electron MHD (EMHD) vortices, i.e., m = 0 helicons,51 have shown that magnetic and current helicities have the same properties. A simple loop antenna possesses no helical fields or currents, and hence, conservation requires that the excited vortices on either side of the antenna have opposite helicities (see Fig. 4 in Ref. 52). Waves injected with knotted or linked currents excite waves directionally.53 Wave reflection reverses the helicity.45 The helical topology of the wave angular momentum has similar properties because the momentum is also a conserved quantity. Since Bz is an odd function in z and y, the instantaneous angular momentum is opposite on either side of the antenna. Likewise, the linear momentum of two oppositely propagating waves cancels. Thus, the total momentum of the two ejected waves carries no net momentum, consistent with the fact that the antenna injects no momentum. The polarization and field rotation are right-hand circular on both sides (m = +1), but the phase spirals have opposite twists on both sides because the axial wave vectors are reversed. The helicon field line spirals have opposite helicities. The writhe of the flux ropes also reverses on both sides of the antenna.

Low-frequency whistler modes have been excited by electric dipoles and magnetic loop antennas. Electric antennas are primarily used for active experiments in space plasmas, while magnetic antennas are used for helicon sources. There are significant differences in the physics and performance of the two types of antennas, and hence, the observations should be briefly described. An experiment has been done to compare the performance of a dipole and an elongated magnetic loop in the same plasma parameter regime and the same rf source,15 described next.

Figure 19 displays the antennas and the excited waves. Snapshots of the parallel field component Bz(x,y) are shown with the antenna schematic super imposed. The magnetic loop has been elongated to the same length as the electric dipole (15 cm). The wave signals are measured one wavelength away from the antenna and thus are not near-zone antenna fields. The magnetic loop has a normal along B0 and hence excites an m = 0 mode or an elongated vortex [Fig. 19(a)]. The electric dipole excites a much weaker field and a different topology that resembles an m = +1 helicon mode [Fig. 19(b)]. The field rotates in time in the +ϕ direction. The electric field cannot excite this topology, but a current loop in the m = 1 configuration could explain it. The antenna current is determined by the antenna voltage and the sheath capacitance. The return current in the plasma forms a current loop, which excites the m = 1 helicon mode. The electric field cannot excite the observed mode, first because of shielding by the sheath and second because an electric field inside the plasma would produce E×B0 Hall currents normal to the dipole and magnetic fields along the dipole with same polarities, which is not observed. Thus, the electric dipole excites whistlers like a magnetic antenna, but with comparatively weak antenna currents. The magnetic loop current is over two orders of magnitude larger than the dipole current. Consequently, the wave magnetic field ratio is similarly different, as shown in Fig. 19(c) by the waveforms of the peak Bz component. The ratio of the radiated power or radiation resistance of magnetic to electric dipoles is 8400. The radiation resistance is obtained from the radiated power, not the real part of the dipole impedance at the center point.

FIG. 19.

Whistler wave excitation by an electric dipole and a magnetic loop. (a) Snapshot of Bz(x,y) contours at one wavelength from the magnetic loop (Δz = 15 cm). The loop is elongated so as to match the lengths of both antennas. The loop normal is aligned with B0 such that the antenna launches an m = 0 helicon mode. (b) Bz contours for the electric dipole exhibit two peaks of opposite polarity which in time rotate in the +ϕ direction. The antenna current is closed by an opposing return current in the plasma which forms a horizontal loop that excites an m = 1 helicon mode. The electric field cannot penetrate into the plasma due to the sheath. (c) Bz(t) waveforms for both antennas, showing that the magnetic loop excites waves with two orders of magnitude larger for the loop than the dipole. The reason is that the loop carries a large current through the wire, while the dipole current is a small displacement current through the sheath. Thus, the magnetic loop excites much stronger whistler modes than an electric dipole. Reproduced with permission from Phys. Plasmas 23, 082120 (2016). Copyright 2016 AIP Publishing.

FIG. 19.

Whistler wave excitation by an electric dipole and a magnetic loop. (a) Snapshot of Bz(x,y) contours at one wavelength from the magnetic loop (Δz = 15 cm). The loop is elongated so as to match the lengths of both antennas. The loop normal is aligned with B0 such that the antenna launches an m = 0 helicon mode. (b) Bz contours for the electric dipole exhibit two peaks of opposite polarity which in time rotate in the +ϕ direction. The antenna current is closed by an opposing return current in the plasma which forms a horizontal loop that excites an m = 1 helicon mode. The electric field cannot penetrate into the plasma due to the sheath. (c) Bz(t) waveforms for both antennas, showing that the magnetic loop excites waves with two orders of magnitude larger for the loop than the dipole. The reason is that the loop carries a large current through the wire, while the dipole current is a small displacement current through the sheath. Thus, the magnetic loop excites much stronger whistler modes than an electric dipole. Reproduced with permission from Phys. Plasmas 23, 082120 (2016). Copyright 2016 AIP Publishing.

Close modal

These results may appear to be obvious since helicon sources use magnetic antennas but no electric dipoles. Loop antennas have not deployed properly on satellites,54 and hence, electric dipoles have been used; their radiation resistances are not only studied extensively55 but also measured in laboratory plasmas;56,57 but no in-situ wave fields can be measured since it requires multipoint measurements.

Electric and magnetic dipoles are not the only types of antennas which can excite whistler modes. A small plane electrode which draws a current pulse creates a propagating current pulse in the plasma. It has been shown that the current propagates in the whistler mode.58 Two electrodes are required to close the current. If one ejects an electron beam, the current penetrates at the wave speed, while the faster beam front is current-neutralized. A model of an electrodynamic tether flown in space has been investigated in the laboratory.59,60 The motion of tethered electrodes across the ambient field creates “whistler wings” which have also been seen on magnetized asteroid Gaspra moving rapidly through the solar wind.61 

Lower hybrid (LH) waves are highly perpendicular whistler modes above the LH frequency, ωlh(ωceωci)1/2. They are nearly electrostatic modes and hence better excited by an electric antenna with large k and small k rather than a magnetic loop. Figure 4 in Ref. 62 shows the antenna and its excited waves. Close to the resonance, the LH waves propagate at an angle θp(me/mi)1/289.8° across B0 and have a group velocity at an angle θg90θp=0.2°. Across B0, LH waves are backward waves.

The propagation of the linear whistler mode in weakly nonuniform magnetic fields has been studied for a long time in space plasmas63 and laboratory plasmas.64 Ray tracing has been used to predict the whistler wave propagation in the magnetosphere.63,65–67 When the magnetic field changes on the scale of the wavelength (|B/B|λ), ray tracing breaks down.68 There are few experiments on this topic38,69 although there are many applications both in space plasmas (whistlers near crustal magnetic fields,70,71 magnetic null point geometries,72–74 shocks,75 and helicon thruster exhaust76) and in laboratory plasmas (helicon devices with nonuniform magnetic fields77 and toroidal helicon sources78–80). The present experiment investigates a number of field configurations and describes new surprising results in the following Secs. VI A–VI C.

An axisymmetric nonuniform magnetic field is produced by a large loop with dipole moment along a uniform axial magnetic field. The result is a single mirror field with mirror ratio Bmax/B010. On the axis, an m = 1 helicon mode is excited with a 5 cm diam loop.

Figure 20 displays contours of the helicity density JxBx of an m = +1 helicon mode propagating from the right side toward the mirror field on the left side. The rf waveform consists of a burst (20 cycles) whose front is shown in Fig. 20(a). The wave has not yet reached the strong mirror point, and no reflection is observed. Figure 20(b) shows a snapshot at the end of the rf burst, when the incident wave has ended. One observes a reflected wave traveling obliquely to the right. The sign of the helicity has been reversed. In the middle of the rf burst, both incident and reflected waves are present as shown in Fig. 20(c). Interference occurs near the sides of the incident wave. At wave nodes, the polarization is nearly linear. The reflection vanishes with the decreasing magnetic field, and hence, it does not arise from the conducting loop. The reflection is interpreted by the strong gradient in the refraction coefficient, which depends on the magnetic field, kc/ωωp/(ωωc)1/2. No density gradients are observed in the reflection region. A video clip [see Fig. 20 (Multimedia view)] shows the continuous time dependence of the reflection of a whistler mode from a strong mirror magnetic field.

FIG. 20.

Propagation of an m = +1 helicon mode into a strong mirror field. Displayed are contours of the helicity density JxBx whose sign determines the direction of propagation along B0 which is against B0 for the incident wave. Three snapshots show the wave burst at three times in the rf burst. (a) There are no reflections until the incident wave reaches the mirror field. (b) There are only the reflected waves when the incident wave vanishes at the end of the rf burst. (c) Both counter propagating waves are seen in cw conditions. Interference creates localized regions of linear polarization. A slow-motion time variation is shown in the video clip reflection of a whistler mode from a strong mirror magnetic field. Reproduced with permission from Phys. Plasmas 25, 082110 (2018). Copyright 2018 AIP Publishing.120 Multimedia view: https://doi.org/10.1063/1.5097852.8

FIG. 20.

Propagation of an m = +1 helicon mode into a strong mirror field. Displayed are contours of the helicity density JxBx whose sign determines the direction of propagation along B0 which is against B0 for the incident wave. Three snapshots show the wave burst at three times in the rf burst. (a) There are no reflections until the incident wave reaches the mirror field. (b) There are only the reflected waves when the incident wave vanishes at the end of the rf burst. (c) Both counter propagating waves are seen in cw conditions. Interference creates localized regions of linear polarization. A slow-motion time variation is shown in the video clip reflection of a whistler mode from a strong mirror magnetic field. Reproduced with permission from Phys. Plasmas 25, 082110 (2018). Copyright 2018 AIP Publishing.120 Multimedia view: https://doi.org/10.1063/1.5097852.8

Close modal

By reversing the current of the large loop, the magnetic field topology changes from a mirror to a cusp field. Figure 21(a) shows the ambient magnetic field lines of the field reversed configuration (FRC) and contours of the wave magnetic field component Bx(y,z). The m = 1 helicon mode propagates against the 3D null point, but does not penetrate into the FRC. However, the notion that a plane wave is absorbed by a null point is demonstrably not correct. The incident wave packet has transverse dimensions exceeding the null point region where cyclotron resonance can absorb the waves (ω>ωc). The outer wave contours continue to propagate along the separatrix around the FRC. A true plane wave has a negligible loss of energy since the transverse dimension of the null point is negligible compared to that of the wave. Conversely, a wave packet smaller than the absorption region of the null point would be completely absorbed.

FIG. 21.

Wave propagation inside and outside the separatrix of an FRC. (a) Waves excited outside the separatrix are absorbed at the 3D null point, but off-axis, the waves continue to propagate around the separatrix. If waves are excited on the axis inside the FRC, they are absorbed at the cusp null point but can continue to propagate off-axis along closed field lines. (b) Field rotation of Bz(x,y) for an m = 1 helicon mode inside and outside the FRC. The sense of rotation reverses due to the reversal of B0, which is best seen in two video clips, a field rotation in the +ϕ direction when kB0 > 0 as in the uniform field outside the FRC and a rotation in the −ϕ direction inside the FRC where kB0 < 0. Thus, both the signs of helicity and orbital angular momentum depend on the propagation direction with respect to the ambient field. Reproduced with permission from Phys. Plasmas 25, 082110 (2018). Copyright 2018 AIP Publishing.120 Multimedia views: https://doi.org/10.1063/1.5097852.9; https://doi.org/10.1063/1.5097852.10

FIG. 21.

Wave propagation inside and outside the separatrix of an FRC. (a) Waves excited outside the separatrix are absorbed at the 3D null point, but off-axis, the waves continue to propagate around the separatrix. If waves are excited on the axis inside the FRC, they are absorbed at the cusp null point but can continue to propagate off-axis along closed field lines. (b) Field rotation of Bz(x,y) for an m = 1 helicon mode inside and outside the FRC. The sense of rotation reverses due to the reversal of B0, which is best seen in two video clips, a field rotation in the +ϕ direction when kB0 > 0 as in the uniform field outside the FRC and a rotation in the −ϕ direction inside the FRC where kB0 < 0. Thus, both the signs of helicity and orbital angular momentum depend on the propagation direction with respect to the ambient field. Reproduced with permission from Phys. Plasmas 25, 082110 (2018). Copyright 2018 AIP Publishing.120 Multimedia views: https://doi.org/10.1063/1.5097852.9; https://doi.org/10.1063/1.5097852.10

Close modal

When the wave is excited inside the FRC, it is again absorbed on the spine, while off-axis, the wave propagates along closed field lines and then across the separatrix into the open field line region. It is worth noting that the FRC reverses the sign of the helicon mode number so that the field rotates ccw around B0 like the right-hand circular rotation of the electrons. Figure 21(b) shows a snapshot of JxBx(x,y) contours, which are positive when the wave propagates along a uniform axial field, i.e., without the FRC.

When the FRC reverses B0, the wave propagates opposite to B0, which reverses the direction of rotation, i.e., the sign of the helicon mode number m and the sign of the helicity density [Fig. 21(c)]. The spatial phase helix mϕ+kzz = const is left-handed for m > 0 (no FRC) and right-handed for m < 0 (inside the FRC).

The wave propagation has also been investigated in ambient fields with X-type magnetic nulls. These arise from the superposition of the field from a line current along the x-direction and a uniform field along the z-direction. The resultant 2D field lines (Bx 0) are shown in Fig. 22(a) in a larger plane covering the null point, closed fields around the line current, and open fields outside the separatrix. The separatrix is the field line passing through the null point. The m = 0 antenna is located between the X and O-points. The data plane is smaller due to limited probe access. Contours of Bx(y,z) are shown in the data plane of Figs. 22(a) and 22(b). The waves propagate on both closed and open field lines which are best followed in a video clip [see Fig. 22 (Multimedia view)] about the propagation of a whistler mode across the separatrix. At the start of the wave burst, the waves follow down the closed field lines within the separatrix. Next, the downward propagation stagnates, and the wave crosses the separatrix and propagates diagonally upward along the open field lines. Their phase front normals are nearly aligned with the open B0 lines, implying parallel wave propagation. The phase fronts are curved because the B0 lines are spreading apart.

FIG. 22.

Wave propagation from an m = 0 loop antenna located inside the separatrix between the X and O-points. Displayed are ambient magnetic field lines in (a) an extended plane to show the field topology for B0. The y-z plane for data taking had to be reduced. It shows in (a) and (b) contours of Bx(y,z) and in (c) contours of JxBx(y,z). The wave propagation is best seen in a video clip reflection of a whistler mode across the separatrix. The antenna launches waves down the closed field lines, stagnates, crosses the separatrix, and travels diagonally upward along the open field lines. The propagation reversal appears similar to a reflection of whistler modes from a strong density gradient. The negative sign of helicity density confirms that the waves on closed field lines are downward, elsewhere upward. The outgoing waves are quasi parallel whistlers whose phase front is curved because the field lines spread outward. There is no evidence for an m = 0 helicon mode. (d) Interference arises at the interface between downward and upward propagating waves. In this region, the contours of the hodogram ellipticities ϵ=Bmin/Bmax show low values, i.e., linear polarization, which is consistent with interference. Reproduced with permission from Phys. Plasmas 25, 082108 (2018). Copyright 2018 AIP Publishing.121 Multimedia view: https://doi.org/10.1063/1.5097852.11

FIG. 22.

Wave propagation from an m = 0 loop antenna located inside the separatrix between the X and O-points. Displayed are ambient magnetic field lines in (a) an extended plane to show the field topology for B0. The y-z plane for data taking had to be reduced. It shows in (a) and (b) contours of Bx(y,z) and in (c) contours of JxBx(y,z). The wave propagation is best seen in a video clip reflection of a whistler mode across the separatrix. The antenna launches waves down the closed field lines, stagnates, crosses the separatrix, and travels diagonally upward along the open field lines. The propagation reversal appears similar to a reflection of whistler modes from a strong density gradient. The negative sign of helicity density confirms that the waves on closed field lines are downward, elsewhere upward. The outgoing waves are quasi parallel whistlers whose phase front is curved because the field lines spread outward. There is no evidence for an m = 0 helicon mode. (d) Interference arises at the interface between downward and upward propagating waves. In this region, the contours of the hodogram ellipticities ϵ=Bmin/Bmax show low values, i.e., linear polarization, which is consistent with interference. Reproduced with permission from Phys. Plasmas 25, 082108 (2018). Copyright 2018 AIP Publishing.121 Multimedia view: https://doi.org/10.1063/1.5097852.11

Close modal

The reversal of propagation arises in regions of increasing magnetic field strength. The wave reversal can be explained again by the strong gradient in B0 near the line current, similar to the reflection by a mirror field shown in Fig. 20. The propagation direction is obtained from the sign of the helicity density JxBx(y,z) whose contours are shown in Fig. 22(c). Negative helicity indicates downward propagation opposite to the B0 lines. But the best confirmation of the described space-time dependence is given by the video clip [see Fig. 22 (Multimedia view)] reflection of a whistler mode across the separatrix. Interference arises at the boundary between downward and upward propagating waves. The interference creates elliptical to linear polarization shown by the ellipticity contours in Fig. 22(d).

Since field lines travel partially along closed field lines, an attempt has been made to produce trapped whistler eigenmodes on circular B0 lines. Concentric field lines have been achieved with a strong field of the line current and a weak axial field (B0=3 G). There is no field line gradient along B0(y,z), which could reflect waves. The antenna is close to the line current to minimize asymmetries. The antenna vacuum field has been subtracted from the total field measured, and thus, the displayed contours are created by whistler modes.

Figure 23(a) displays the circular B0 lines and contours of the transverse Bx(y,z) component during cw conditions. The evolution of the cw condition is shown in a video clip [see Fig. 23 (Multimedia view)] whistler mode propagation across closed field lines. It shows that the wave field rapidly spreads out along the circular dc field B0 while also expanding across B0.

FIG. 23.

Whistler modes propagating on circular magnetic field lines, excited by an m = 0 loop close to the linear wire at y = z = 0. (a) Snapshots of the wave magnetic field component Bx(y,z). The contours have nulls along the y-axis and peaks of opposite signs along the z-axis, which results in a sinusoidal profile along circular B0 lines. In time, the azimuthal topology remains the same, but the phase fronts propagate radially outward, shown in a video clip whistler mode propagation across closed field lines. The mode can be described by an azimuthal standing wave and a propagating radial wave. The standing wave is due to oppositely propagating waves from the antenna, not due to boundary reflections. (b) The waves from each side of the antenna travel in opposite directions with respect to B0, and the JxBx(y,z) contours have opposite signs on either side of the antenna. The mode is a standing mode along B0 with propagation across B0. (c) Oppositely traveling waves create interference near the z = 0 plane. In this area, the whistler modes are nearly linearly polarized (ϵ ≃ 0). Away from the interface, the waves are circularly polarized (ϵ ≃ 1). Reproduced with permission from Phys. Plasmas 25, 082108 (2018). Copyright 2018 AIP Publishing.121 Multimedia view: https://doi.org/10.1063/1.5097852.12

FIG. 23.

Whistler modes propagating on circular magnetic field lines, excited by an m = 0 loop close to the linear wire at y = z = 0. (a) Snapshots of the wave magnetic field component Bx(y,z). The contours have nulls along the y-axis and peaks of opposite signs along the z-axis, which results in a sinusoidal profile along circular B0 lines. In time, the azimuthal topology remains the same, but the phase fronts propagate radially outward, shown in a video clip whistler mode propagation across closed field lines. The mode can be described by an azimuthal standing wave and a propagating radial wave. The standing wave is due to oppositely propagating waves from the antenna, not due to boundary reflections. (b) The waves from each side of the antenna travel in opposite directions with respect to B0, and the JxBx(y,z) contours have opposite signs on either side of the antenna. The mode is a standing mode along B0 with propagation across B0. (c) Oppositely traveling waves create interference near the z = 0 plane. In this area, the whistler modes are nearly linearly polarized (ϵ ≃ 0). Away from the interface, the waves are circularly polarized (ϵ ≃ 1). Reproduced with permission from Phys. Plasmas 25, 082108 (2018). Copyright 2018 AIP Publishing.121 Multimedia view: https://doi.org/10.1063/1.5097852.12

Close modal

The phase fronts are nearly field aligned, indicating a radial wave vector component kr. The circular contours have opposite signs in opposite hemispheres and hence develop a null at the interface, the z = 0 plane. For a fixed radius, the wave amplitude has a periodic variation along the phase front. In time, the field changes its sign but does not change its parallel sinusoidal amplitude profile. Thus, the parallel field forms a standing wave with wavelength equal to the circumference. The standing wave arises from counter propagating waves traveling down the circular field lines. The antenna has an odd Bx(z) dependence, which determines the nodes above and below the loop. Other components have different locations for their nodes.

Figure 23(b) shows contours of the helicity density JxBx. In the left hemisphere, the negative helicity indicates wave propagation down and against B0, while in the right hemisphere, the waves with positive helicity travel down along B0. The interference of counter propagating waves leads to alternating helicities and linear polarization. The ellipticity shown in Fig. 23(c) indicates linear polarization in the z-plane, where B has nodes, while B has antinodes.

From these observations, one can conclude that the waves are azimuthal standing waves with λ=2πr63 cm at the radial middle (r =10 cm) and perpendicular wavelength λ15 cm. Thus, the waves propagate obliquely at an angle θ=arctan(kr/kϕ)76° which is close to the oblique cyclotron resonance angle θres=arccos(ω/ωc)77° for B08 G. These waves are quantized in the azimuthal direction and propagate radially, which resemble a radial waveguide mode rather than a trapped mode. The azimuthal or parallel quantization number is mθ=± 1. The mode is not created by reflections as in waveguides but by the field topology of the m = 0 antenna. Plane wave theory for propagation near the oblique cyclotron resonance angle predicts that the group velocity cone angle is given by θgroup=arcsin(ω/ωc)90°θphase14°. Plane wave theory does not explain the fast radial transport seen during the wave growth. Helicon mode theory relates resonance cone modes to Trivelpiece-Gould modes21 which are electrostatic boundary modes81 on cylindrical plasma columns. The present highly oblique waves are not T-G modes but magnetic whistler modes in a uniform and unbound plasma.

Whistler modes exhibit many nonlinear phenomena. The threshold for nonlinearity depends on the phenomenon, time and spatial scales, collisionality, etc. In the nearly collisionless and unbound environment of space plasmas, nonlinear wave-particle interactions do not require large wave amplitudes in terms of Bwave/B0, but in small collisional laboratory plasmas, the wave magnetic field may become comparable to the ambient field, which creates strong nonlinearities rapidly. This will be the situation described below. Nonlinear effects arise from the magnetic and electric forces, E+v×B, which can modify density,82–84 temperature,85 and the ambient magnetic field86 in the so-called modulational instabilities.87 Instabilities also arise from wave-particle interactions such as instabilities due to Landau resonance by beams17 and cyclotron resonance by non-Maxwellian electron distributions (Te,>Te,).88 Wave-wave interactions such as parametric instabilities also involve nonlinear effects.89,90

Electron heating produces two types of plasma modifications: For sufficiently long time scales, the heated channel assumes pressure balance (nkTe+B2/2μ0= const), which creates a density trough and duct whistler modes, leading to “thermal” self-focusing. Figure 24 shows an example of whistler filamentation. In a uniform plasma, a small amplitude wave decays axially due to radial wave spread [Fig. 24(a)]. However, a large amplitude wave creates a field aligned channel of high temperature and low density, which ducts the whistler wave and maintains its amplitude [Fig. 24(b)].

FIG. 24.

Comparison of linear and nonlinear whistler waves. (a) Axial interferometer traces of a small amplitude whistler mode whose amplitude decay is mainly due to wave spreading and Coulomb collisions. (b) Interferometer trace of a large amplitude whistler mode which is ducted by a density trough caused by electron heating. Reproduced with permission from Phys. Fluids 19, 865 (1976). Copyright 1976 AIP Publishing.122 

FIG. 24.

Comparison of linear and nonlinear whistler waves. (a) Axial interferometer traces of a small amplitude whistler mode whose amplitude decay is mainly due to wave spreading and Coulomb collisions. (b) Interferometer trace of a large amplitude whistler mode which is ducted by a density trough caused by electron heating. Reproduced with permission from Phys. Fluids 19, 865 (1976). Copyright 1976 AIP Publishing.122 

Close modal

For very short intense rf pulses, the electrons can be heated before the ions begin to move such that no density depression is formed. The whistler wave propagates preferentially in the field-aligned hot plasma channel because the Coulomb collisions are reduced and the wave damping becomes negligible (see Fig. 10 in Ref. 91). The heated flux tube creates diamagnetic electron currents which form a force-free magnetic flux rope (see Fig. 10 in Ref. 92). This nonlinear effect is a modulation of the plasma conductivity which could be of interest to ionospheric modifications by high power radio waves. The intense rf fields create heating, density modifications, instabilities, and wave-wave interactions among different types of waves including whistler modes.93 

The density depression can also arise from the ponderomotive force due to a strong nonuniform electric field, an example of which is shown in Fig. 25. A strong nonuniform electric field is produced by focusing a resonance cone. The latter is produced with a circular antenna which is not a magnetic loop but an electric antenna. The radiation pattern of a point antenna is a diverging cone, and a multitude of point sources on a circle produces a converging cone. When a short intense rf burst is applied, the ponderomotive force at the focus expels electrons, which creates a space charge electric field and accelerates ions away from the focus. The result is little electron heating but a strong acceleration of ions. In time, the focus deteriorates into plasma and field turbulence.94 

FIG. 25.

Density perturbations by the ponderomotive force of a focused resonance cone. (a) Schematic picture showing a circular electric antenna with a short intense rf burst in the whistler mode regime (ω/ωc = 0.08). The converging cone forms a focus of the high rf electric field whose ponderomotive force expels plasma. (b) Perturbed density profile showing a density hole at the focal point which evolves into turbulence. The expelled ions are energized by 2 orders of magnitude, whereas the electron energization is negligible. Reproduced with permission from Phys. Plasmas 20, 108 (1977). Copyright 1977 AIP Publishing.

FIG. 25.

Density perturbations by the ponderomotive force of a focused resonance cone. (a) Schematic picture showing a circular electric antenna with a short intense rf burst in the whistler mode regime (ω/ωc = 0.08). The converging cone forms a focus of the high rf electric field whose ponderomotive force expels plasma. (b) Perturbed density profile showing a density hole at the focal point which evolves into turbulence. The expelled ions are energized by 2 orders of magnitude, whereas the electron energization is negligible. Reproduced with permission from Phys. Plasmas 20, 108 (1977). Copyright 1977 AIP Publishing.

Close modal

Most plasma instabilities have a phase of wave growth followed by saturation created by nonlinear effects. An example is a whistler instability created by an electron beam17 which involves nonlinear wave-particle scattering. Waves grow from fluctuations into broadband VLF hiss which consists of oblique whistler modes whose parallel phase velocity is in resonance with the beam electrons, ω=kzvbeam. The instability properties are best shown with a single frequency test wave launched from a “point” antenna. Figures 26(a) and 22(b) show the wave amplitude along the z-direction on either side of the antenna. Against the beam, the antenna excites a resonance cone, and with the beam, a growing wave is observed. Expanding the measurement to 2D, the phase contours show the resonance cone in the left half plane and oblique waves in the right half plane [Figs. 26(c) and 26(d)]. The wave amplitude is presented by a contour plot and shows the obliquely growing whistler modes along the beam and the decaying resonance cone against the beam [Figs. 26(e) and 26(f)]. The group velocity is nearly orthogonal to the phase velocity. The wave stops growing when the electron beam broadens. The scattering of electrons in velocity space is an example of a nonlinear wave-particle interaction.

FIG. 26.

Whistler mode instability created by a broad electron beam. A point source antenna excites (a) a resonance cone pattern against the beam and (b) a growing wave with the beam. (c) The stable resonance cone diverges, while in (d), the unstable waves have converging phase fronts. (e) and (f) The amplitude contours of stable waves decay, while the unstable waves diverge and show growth along the oblique group velocity vg. The unstable whistler mode propagates near the oblique resonance where group and phase velocities are orthogonal. Reproduced with permission from R. L. Stenzel, J. Geophys. Res. 82, 4805 (1977). Copyright 1977 American Geophysical Union.

FIG. 26.

Whistler mode instability created by a broad electron beam. A point source antenna excites (a) a resonance cone pattern against the beam and (b) a growing wave with the beam. (c) The stable resonance cone diverges, while in (d), the unstable waves have converging phase fronts. (e) and (f) The amplitude contours of stable waves decay, while the unstable waves diverge and show growth along the oblique group velocity vg. The unstable whistler mode propagates near the oblique resonance where group and phase velocities are orthogonal. Reproduced with permission from R. L. Stenzel, J. Geophys. Res. 82, 4805 (1977). Copyright 1977 American Geophysical Union.

Close modal

Since the magnetic field carries most of the whistler wave energy [(B2/μ0)/ϵE2=c2/v21], a strong nonlinearity arises from the J×B term in Hall Ohm's law. The time average of the oscillating product creates a dc magnetic field J(t)×B(t)t, which has been observed as a dc magnetic field near the antenna,95 and the excitation of harmonics, which can propagate as whistler modes, provided ω<ωc.

Wave magnetic fields comparable to the ambient field have been produced with large loop currents. This is accomplished by switching a charged capacitor into the loop inductance, which produces a damped current oscillation. The loop is oriented in the m = 0 mode, which produces an axisymmetric wave field. When the loop field exceeds the ambient field, the excited wave field adds or reverses the ambient field on alternating polarities. Figure 27 shows snapshots of vector fields (By,Bz)total for (a) a field reversal and (b) a field enhancement. The field reversal produces a topology similar to a spheromak or vortex, while the field enhancement produces a mirror field with a twist. Both topologies propagate as whistler modes, which is shown in a video clip [see Fig. 27 (Multimedia view)] of the excitation of whistler mirrors and spheromaks.

FIG. 27.

Whistler modes with the wave magnetic field exceeding the ambient dc guide field (B0 = 5 G). Snapshots of the vector fields of (By, Bz) showing (a) a whistler spheromak and (b) a whistler mirror. In a spheromak, the wave field on the axis reverses the ambient field. In a mirror, it adds to the ambient field. The fields propagate along B0 = 5 G, which is shown in a video clip Emission and propagation of whistler spheromaks and mirrors. First, the whistler mirror is excited (yellow contours) and then the spheromak (blue contours). Spheromaks propagate slower than linear whistler modes, while mirrors propagate faster than small amplitude whistlers. Strong electron heating causes spheromaks to damp faster than linear whistlers or mirrors. Reproduced with permission from Phys. Plasmas 15, 042307 (2008). Copyright 2008 AIP Publishing.123 Multimedia view: https://doi.org/10.1063/1.5097852.13

FIG. 27.

Whistler modes with the wave magnetic field exceeding the ambient dc guide field (B0 = 5 G). Snapshots of the vector fields of (By, Bz) showing (a) a whistler spheromak and (b) a whistler mirror. In a spheromak, the wave field on the axis reverses the ambient field. In a mirror, it adds to the ambient field. The fields propagate along B0 = 5 G, which is shown in a video clip Emission and propagation of whistler spheromaks and mirrors. First, the whistler mirror is excited (yellow contours) and then the spheromak (blue contours). Spheromaks propagate slower than linear whistler modes, while mirrors propagate faster than small amplitude whistlers. Strong electron heating causes spheromaks to damp faster than linear whistlers or mirrors. Reproduced with permission from Phys. Plasmas 15, 042307 (2008). Copyright 2008 AIP Publishing.123 Multimedia view: https://doi.org/10.1063/1.5097852.13

Close modal

Figure 28 shows a snapshot of the 3D total field lines (Btotal=Btotal+Bambient) of a “whistler spheromak.” The field topology is that of a vortex with two cusp null points, a separatrix forming a boundary between inner closed field lines and outer open lines. The field is nearly force free and propagates without deforming its topology. The propagation speed decreases with the increasing wave magnetic field, while the speed of a “whistler mirror” increases with the amplitude. These nonlinear effects can be explained by the modified ambient magnetic field in the frame of the wave packet.

FIG. 28.

Snapshots of the total field lines of a whistler spheromak in 3D space. The spheromak propagates along B0. Its stability arises from the force-free field topology. The strong wave field reverses the ambient field, creating two cusp type magnetic null points on the axis. A toroidal field twists the field lines into a vortex field similar to the wave field of linear m = 0 helicons. Its helicity is negative since the propagation is opposite to B0. The transverse field has a toroidal null line [(By, Bz) = 0], where electrons are freely accelerated leading to strong currents, heating, and instabilities. Reproduced with permission from Phys. Plasmas 15, 042307 (2008). Copyright 2008 AIP Publishing.123 

FIG. 28.

Snapshots of the total field lines of a whistler spheromak in 3D space. The spheromak propagates along B0. Its stability arises from the force-free field topology. The strong wave field reverses the ambient field, creating two cusp type magnetic null points on the axis. A toroidal field twists the field lines into a vortex field similar to the wave field of linear m = 0 helicons. Its helicity is negative since the propagation is opposite to B0. The transverse field has a toroidal null line [(By, Bz) = 0], where electrons are freely accelerated leading to strong currents, heating, and instabilities. Reproduced with permission from Phys. Plasmas 15, 042307 (2008). Copyright 2008 AIP Publishing.123 

Close modal

Another nonlinear effect is the strong damping of a whistler spheromak compared to the whistler mirror. The damping arises from strong electron heating in a region of high current density. The 3D null points are current free and hence do not cause dissipation. But there is a toroidal null line, where (By,Bz)=0, and a toroidal field Bϕ exists, where electrons can be freely accelerated by an inductive electric field Eϕ. Figure 29 shows contours of Jx(y,z) as two whistler spheromaks have been ejected from the loop antenna. The current peaks in the toroidal null line of the in-plane field lines. The electron temperature rises from 2.5 eV to 15 eV, creating light emission and whistler instabilities. In contrast, the whistler mirror is formed by Hall currents which are distributed widely with low current densities. It produces negligible heating and no instabilities and hence has little damping.

FIG. 29.

Current density contours of whistler spheromaks excited by a loop current. Field lines of the total (wave and background) magnetic field. The snapshot shows the large toroidal current density in the center of the spheromaks. The time evolution of the propagating whistler mirrors and spheromaks is shown in a video clip Emission and propagation of whistler spheromaks. Reproduced with permission from Phys. Plasmas 15, 042308 (2008). Copyright 2008 AIP Publishing.124 Multimedia view: https://doi.org/10.1063/1.5097852.14

FIG. 29.

Current density contours of whistler spheromaks excited by a loop current. Field lines of the total (wave and background) magnetic field. The snapshot shows the large toroidal current density in the center of the spheromaks. The time evolution of the propagating whistler mirrors and spheromaks is shown in a video clip Emission and propagation of whistler spheromaks. Reproduced with permission from Phys. Plasmas 15, 042308 (2008). Copyright 2008 AIP Publishing.124 Multimedia view: https://doi.org/10.1063/1.5097852.14

Close modal

The nonlinear effects of a whistler spheromak are summarized in Fig. 30. The oscillating coil current in Fig. 30(a) shows the timing when spheromaks and mirrors are excited. Only spheromaks produce hot electrons, which cause light emission and whistler instabilities, Fig. 30(b) shows Langmuir probe traces for three different conditions, i.e., the unperturbed plasma, the weak heating from a whistler mirror, and the strong heating by a whistler spheromak. The latter indicates two electron populations with tail energies of 15 eV. The high-frequency instabilities have been further investigated. Figure 30(c) shows a time-of-flight diagram of the wave amplitude δBx(z,t). The contours show that the wave is produced at the spheromak and that they propagate ahead of the spheromak and in the opposite direction of the spheromak. Figure 30(d) shows a spatial hodogram δB(x,y,z), confirming that the oscillations are circularly polarized whistler modes, leaving the spheromak in both directions. The axial component δBz(x,y,t) has two peaks of opposite polarity, which rotate in the ϕ direction and identify the wave as an m = 1 helicon mode [see Fig. 6(a) in Ref. 96]. Thus, the instability source is a dipole across B0, while the spheromak source is a dipole along B0. The 3D topology changes in space and time.

FIG. 30.

Electron heating and whistler instabilities created in a toroidal null line of spheromaks or FRCs. A strong coil current ejects whistler spheromaks and mirrors during current reversals. Only the spheromak creates strong heating and light emission. Higher frequency oscillations are seen during electron energizations. (b) Sampled Langmuir traces show energetic electron tails in spheromaks, but no heating in mirrors. Reproduced with permission from Phys. Plasmas 15, 042309 (2008). Copyright 2008 AIP Publishing.125 (c) Time-of-flight diagram δBx(z,t) showing that the excited whistler modes travel ahead and back from the spheromak, which acts as a moving wave source. (e) Spatial magnetic hodogram shows that the unstable waves consist of right-hand circularly polarized whistler modes propagating axially away from the spheromak. Reproduced with permission from Phys. Plasmas 15, 062109 (2008). Copyright 2008 AIP Publishing.

FIG. 30.

Electron heating and whistler instabilities created in a toroidal null line of spheromaks or FRCs. A strong coil current ejects whistler spheromaks and mirrors during current reversals. Only the spheromak creates strong heating and light emission. Higher frequency oscillations are seen during electron energizations. (b) Sampled Langmuir traces show energetic electron tails in spheromaks, but no heating in mirrors. Reproduced with permission from Phys. Plasmas 15, 042309 (2008). Copyright 2008 AIP Publishing.125 (c) Time-of-flight diagram δBx(z,t) showing that the excited whistler modes travel ahead and back from the spheromak, which acts as a moving wave source. (e) Spatial magnetic hodogram shows that the unstable waves consist of right-hand circularly polarized whistler modes propagating axially away from the spheromak. Reproduced with permission from Phys. Plasmas 15, 062109 (2008). Copyright 2008 AIP Publishing.

Close modal

When the loop is rotated with its dipole moment perpendicular to B0, the total field has two 3D null points near the loop. There is no single toroidal null line on which the current flows and closes (see Figs. 4–7 in Ref. 97). Electron heating occurs near the loop wire. The electron energization also excites higher frequency whistlers.

Whistler spheromaks also exhibit strong wave-wave interactions. When two small amplitude vortices propagate against each other, there is no interaction. The waves pass through each other without changing energy and momentum (see Fig. 4 in Ref. 33). On the other hand, two large amplitude current rings with equal toroidal currents slowly converge, merge, and stop to propagate (see s3 in supplementary material). The helicity of the oppositely propagating spheromaks cancels, and an FRC remains. It is less stable and performs a precession (see s4 supplementary material). Displayed are field lines and toroidal current density contours. The tilt in 2D is a projection of a conical rotation in 3D.

The measure for nonlinearity by the ratio Bwave/B0 becomes meaningless when B0 = 0. A better measure for nonlinearity is whether the wave field can magnetize the electrons, eBwave/mω>1. If this ratio is below unity, the wave is evanescent, but for large ratios, the magnetized electrons can support a complicated whistler mode since it has to propagate in a nonuniform and time-varying magnetic field. Without an axial guide field, the direction of the dipole moment does not matter. This eliminates the possibility that helicon modes can exist since there is no cylindrical geometry and no wave rotation around an axial guide magnetic field.

This regime has been investigated with a large loop field (Bmax30 G) in uniform and unmagnetized plasma. Figure 31 shows [(a) and (b)] schematic field lines and [(c) and (d)] measured field lines and field amplitudes. The snapshots are displayed at a zero crossing of the antenna current [(a) and (c)] and when the coil current rises [(b) and (d)]. At the zero crossing, the field resembles the free space loop field but is maintained by the induced plasma currents. After the zero crossing, the coil current rises with opposite polarity, which creates a field-reversed configuration (FRC). Inside the separatrix, there is an induced toroidal current near the loop, and outside the separatrix, there are “open” field lines with an O-type null carrying a toroidal current, recalling that no field lines are open. With increasing coil current, the toroidal nulls propagate vertically and the 3D cusp nulls expand along the central z-axis. The process continues: The decreasing coil current induces plasma currents, which maintain the expanding FRC, and the current reversal forms a new FRC with opposite polarity, etc. Thus, the stacking of toroidal currents may be called a nonlinear whistler mode propagation in the ±y direction. With the increasing distance, the field decays until the wave is absorbed at its own cyclotron resonance ωc = ω.

FIG. 31.

Magnetic field topologies of a current loop in an unmagnetized plasma. (a) Schematic field lines when the coil current has a zero transition and the field is produced by induced electron currents. (b) Schematic field lines when the coil current grows after a sign reversal. It creates an FRC in the remnant expanded field from the previous half cycle. The dashed rectangle is the area where measurements were possible. (d) Measured field lines and contours of the field amplitude when Icoil = 0. There is a toroidal null line above the loop where the electron current density peaks. (f) The same as (d) but at the time of (b). The FRC is created by the reversed antenna current. The field lines outside the separatrix are remnants from (a,d) which have expanded in (y,z)-directions. Another toroidal current is outside the measurement plane above the coil as in (b). Reproduced with permission from Phys. Plasmas 16, 022103 (2009). Copyright 2009 AIP Publishing.126 

FIG. 31.

Magnetic field topologies of a current loop in an unmagnetized plasma. (a) Schematic field lines when the coil current has a zero transition and the field is produced by induced electron currents. (b) Schematic field lines when the coil current grows after a sign reversal. It creates an FRC in the remnant expanded field from the previous half cycle. The dashed rectangle is the area where measurements were possible. (d) Measured field lines and contours of the field amplitude when Icoil = 0. There is a toroidal null line above the loop where the electron current density peaks. (f) The same as (d) but at the time of (b). The FRC is created by the reversed antenna current. The field lines outside the separatrix are remnants from (a,d) which have expanded in (y,z)-directions. Another toroidal current is outside the measurement plane above the coil as in (b). Reproduced with permission from Phys. Plasmas 16, 022103 (2009). Copyright 2009 AIP Publishing.126 

Close modal

The induced toroidal current near the antenna produces electron heating, light emission, and whistler instabilities. Figure 32(a) shows the waveform of the large coil current (400 A-turns) produced by a switched LRC (L = inductance, R = resistance, C = capacitance) circuit of low frequency (f =0.2 MHz). Figure 32(b) displays the light emission which peaks at each coil current peak when a strong toroidal plasma current flows just outside the loop. Note that light and heating arise every half cycle, while in the case of a guide magnetic field, the FRCs are formed every full cycle. The repeated electron heating causes a rising light background, which is likely to produce ionization with longer rf waveforms. The light is produced by electron-neutral collisions associated with strong electron heating. Non-Maxwellian electron distributions can create whistler instabilities, shown in Figs. 32(c) and 32(d). The large lower frequency amplitudes occur when the toroidal current propagates vertically outward. The time-of-flight diagram of the waves is shown by contours of δJx(y,t). The slope of the contours outside the loop indicates outward propagation in the ±y direction, but inside the loop, the frequency and propagation direction differ.

FIG. 32.

Timing of the field penetration into an unmagnetized plasma. (a) Coil current waveform. (b) Light emission from energized electrons. The light peaks coincide with the coil current peaks. The slowly rising background light is due to a slower electron temperature relaxation than heating. (c) High frequency whistler modes created by strong toroidal currents. The large peaks are associated with outward propagation of the outer toroidal current. (d) Time-of-flight diagram for the wave current δJx(y,t). The inclined contours show the radial outward propagation of the large spikes. Higher frequency oscillations are excited within the coil area. Reproduced with permission from Phys. Plasmas 16, 022103 (2009). Copyright 2009 AIP Publishing.126 

FIG. 32.

Timing of the field penetration into an unmagnetized plasma. (a) Coil current waveform. (b) Light emission from energized electrons. The light peaks coincide with the coil current peaks. The slowly rising background light is due to a slower electron temperature relaxation than heating. (c) High frequency whistler modes created by strong toroidal currents. The large peaks are associated with outward propagation of the outer toroidal current. (d) Time-of-flight diagram for the wave current δJx(y,t). The inclined contours show the radial outward propagation of the large spikes. Higher frequency oscillations are excited within the coil area. Reproduced with permission from Phys. Plasmas 16, 022103 (2009). Copyright 2009 AIP Publishing.126 

Close modal

Thus, in the linear regime, no waves are injected into the plasma, while in the nonlinear regime, the waves penetrate and transfer magnetic energy into electron kinetic energy, as also seen from the strong damping of the LRC circuit in plasma. These observations should be relevant to applications like rf plasma sources without the need for an external magnetic field.

Reconnection deals with the problem of rapid conversion of magnetic energy into plasma kinetic energy.98–101 This topic arose in space plasmas with solar flares and magnetic substorms, where magnetic energy is converted into particle energy. This indicates a breakdown of ideal MHD where magnetic fields are “frozen” into the plasma and cannot change energy and helicity. However, at magnetic null points, ideal MHD must break down and dissipation arises. MHD theories described the null point area as a “dissipation” region. Closer inspection of the dissipation region showed that approaching the null point, first, the ions are demagnetized, then the electrons, and finally, the plasma is fully demagnetized in a very thin layer. The different magnetization creates space charge electric fields and Hall currents. The Hall reconnection regime describes unmagnetized but mobile ions which are accelerated by space charge fields.102 On faster time scales, the ions cannot respond and only the magnetized electrons respond to changing magnetic fields. This is the regime of EMHD.103 Reconnection in the Hall MHD regime has been studied first in laboratory devices,104–106 and this topic is still of present interest.107 It has also been observed in the magnetosphere.108 

In ideal EMHD, electron acceleration does not dissipate magnetic energy since accelerated electrons generate currents and regenerate magnetic fields. Thus, the dissipation problem is still not solved. Laboratory experiments show that fast electron drifts create turbulence, which scatters electrons analogous to collisional effects, resulting in electron heating.

Two examples of magnetic reconnection are described below; one with a linear current sheet of finite length in a parameter regime characterized by Hall MHD. Reconnection is not driven MHD flows but by an external source which induces an electric field and drives a parallel current along the separator. The current is controlled by the electric field and effective resistivity. The inductive electric field is opposed self-consistently by a space charge field such that the net electric field and current satisfy Ohm's law. The plasma potential rises at the electron source (cathode), which provides the enhanced current in the neutral sheet. Electrons are accelerated to nearly the inductive voltage. Thus, the energy conversion occurs at the cathode boundary rather than in the current sheet. This result shows the importance of understanding the global circuit in the reconnection problem.

A second experiment employs circular geometry where a toroidal current sheet closes without boundary effects. The inductive electric field cannot be reduced by a nonrotational space charge field so that the anomalous resistivity limits the current. The reconnection is not driven by an external energy source but by the decaying internal magnetic field, an example of spontaneous reconnection. The magnetic field is annihilated, and electrons are heated. During the decay, the neutral X-type topology changes into an O-type which expands axially as a nonlinear whistler mode, an evidence for EMHD physics.

Figure 33 shows schematic pictures of (a) the linear and (b) the circular current sheet experiments. In both cases, magnetic nulls are produced by two parallel conductors carrying time varying currents produced by an external pulsed power supply. The plasma current is driven by the inductive electric field along the conductors. Hall currents and ion currents are negligible compared to the electron current in the magnetic null region.

FIG. 33.

Geometry of two reconnection experiments, forming linear and toroidal current sheets. (a) In a large plasma column, two sheet conductors with parallel currents Is form an X-type null line on the y-axis (x = z = 0) in vacuum. In plasma, the induced currents develop a double Y-type neutral sheet or O-type nulls. The plate current varies sinusoidally on a time scale of Hall MHD. The axial inductive electric field is opposed by a space charge electric field. (b) Cross-sectional view of the current sheets and field lines in vacuum. Reproduced with permission from R. L. Stenzel and W. Gekelman, J. Geophys. Res. 86, 649 (1981). Copyright 1977 American Geophysical Union.127 (c) Experimental setup for studying EMHD reconnection toroidal current sheets. Two parallel circular line currents form an X-type null line between them. The Helmholtz coils carry a slowly rising large current which is abruptly switched of within 1 μs, and the relaxation of the embedded magnetic field is studied. The magnetic field forms neutral sheet and converts the magnetic energy into electron heat on a time scale two orders of magnitude faster than theory predicts. (d) Picture of the 30 cm diameter Helmholtz coils in a large vacuum chamber free of boundary effects. Reproduced with permission from Phys. Plasmas 10, 2780 (2003). Copyright 2003 AIP Publishing.128 

FIG. 33.

Geometry of two reconnection experiments, forming linear and toroidal current sheets. (a) In a large plasma column, two sheet conductors with parallel currents Is form an X-type null line on the y-axis (x = z = 0) in vacuum. In plasma, the induced currents develop a double Y-type neutral sheet or O-type nulls. The plate current varies sinusoidally on a time scale of Hall MHD. The axial inductive electric field is opposed by a space charge electric field. (b) Cross-sectional view of the current sheets and field lines in vacuum. Reproduced with permission from R. L. Stenzel and W. Gekelman, J. Geophys. Res. 86, 649 (1981). Copyright 1977 American Geophysical Union.127 (c) Experimental setup for studying EMHD reconnection toroidal current sheets. Two parallel circular line currents form an X-type null line between them. The Helmholtz coils carry a slowly rising large current which is abruptly switched of within 1 μs, and the relaxation of the embedded magnetic field is studied. The magnetic field forms neutral sheet and converts the magnetic energy into electron heat on a time scale two orders of magnitude faster than theory predicts. (d) Picture of the 30 cm diameter Helmholtz coils in a large vacuum chamber free of boundary effects. Reproduced with permission from Phys. Plasmas 10, 2780 (2003). Copyright 2003 AIP Publishing.128 

Close modal

Figure 34(a) shows that the electron current modifies the field topology from an X-type to a sheet or (c) into an O-type null when the electric field is reversed. Note that there is also a weak axial guide magnetic field. The electrons experience an E × B vertical flow into the neutral sheet and horizontal jetting out of the sheet. Figure 34(b) shows that the unmagnetized ions follow the leading electrons due to in-plane space charge fields. These are the characteristics of Hall MHD reconnection. For strong plasma currents and weak electrode currents, the total field can exhibit O-type null points [Fig. 34(c)].

FIG. 34.

Characteristic field topologies of the linear reconnection experiment. (a) Magnetic field (Bx, Bz) with an elongated 2D horizontal neutral sheet at y = 0. Note that there is a guide field By = 10 G. (b) Ion drift velocity field, normalized to the sound speed, (vx,vz)/cs. The ions stream vertically into the neutral sheet and are jetted horizontally outward. (c) Instead of the double Y-shaped neutral sheet, the plasma can also form one or more O-type magnetic islands. Reproduced with permission from J. Geophys. Res. 86, 649 (1981). Copyright 1977 American Geophysical Union.127 

FIG. 34.

Characteristic field topologies of the linear reconnection experiment. (a) Magnetic field (Bx, Bz) with an elongated 2D horizontal neutral sheet at y = 0. Note that there is a guide field By = 10 G. (b) Ion drift velocity field, normalized to the sound speed, (vx,vz)/cs. The ions stream vertically into the neutral sheet and are jetted horizontally outward. (c) Instead of the double Y-shaped neutral sheet, the plasma can also form one or more O-type magnetic islands. Reproduced with permission from J. Geophys. Res. 86, 649 (1981). Copyright 1977 American Geophysical Union.127 

Close modal

In the linear reconnection geometry, the electron acceleration along the separator is limited by a space-charge electric field, which opposes the axial inductive electric field so as to create a reduced net field. The space charge field produces a positive voltage drop at the cathode almost as large as the inductive voltage along the current sheets. The electron emission increases with the cathode sheath potential and thus is not controlled by the plasma local resistivity but by a boundary current source where the inductive voltage drop occurs. This finding shows the relevance of considering the global current “circuit” which may also be relevant in solar and space reconnection geometries. Reconnection cannot occur if the current in an X-point cannot be provided or closed throughout the circuit.

The circular reconnection geometry eliminates the issues of current closure and global 3D effects. Toroidal geometries are unlikely to arise in space plasmas but are common in laboratory fusion devices. Without boundary effects, the major problems of reconnection rates, energy conversion, and heating mechanisms remain to be resolved.

The circular current sheet is obtained by bending the two linear current conductors into two parallel coils between which an X-type null line appears [see Figs. 33(c) and 33(d)]. The Helmholtz coil field exceeds the axial guide field and forms a field reversed configuration (FRC). The coil current rises slowly so as to embed the vacuum field into the plasma. At its peak, the current is switched off within <2μs and the relaxation of the embedded field drives the reconnection free of an external energy source. There is no guide magnetic field in the toroidal neutral sheet.

The induced current is driven by the toroidal inductive electric field created by the decay of the Helmholtz coil field. No space charge field can cancel the rotational inductive field. Figures 35(a) and 35(b) show the field topology just after switch-off of the coil current when the reconnection starts. Within microseconds, the current modifies the X-type null into an elongated magnetic neutral sheet. The axial expansion is evidence for the whistler mode propagation. The toroidal electron flow convects the in-plane fields so as to produce the out-of-plane fields, shown in Fig. 35(c). They form a quadrupole field associated by in-plane currents. The quadrupole field can also be understood as a vortex field of an m = 0 helicon mode which develops right-handed linkage when propagating along B0 and left-handed linkage for propagation in the −z direction. The FRC also has two 3D null points on its axis. They carry a negligible current since there is no inductive electric field on the axis.

FIG. 35.

Magnetic fields of the toroidal reconnection experiment. (a) Contours of constant flux (field lines) just after the switch-off of the Helmholtz coil current. 2D null lines are between the coils, and two 3D null points are on the z-axis. (b) Vector field of the axisymmetric in-plane field (By, Bz), showing the early formation of a toroidal neutral sheet (y±15 cm, −10 < z <10 cm). (c) Contours of the out-of-plane Bx component which exhibits a quadrupole topology. This is created by the convection of (By, Bz) by Jx. Alternatively, Bx is consistent with (Jy, Jz), which is shown by the vector field. A video clip shows the time evolution of the decaying field into an elongated 2D neutral sheet. Reproduced with permission from Phys. Plasmas 10, 2780 (2003). Copyright 2003 AIP Publishing.128 Multimedia view: https://doi.org/10.1063/1.5097852.15

FIG. 35.

Magnetic fields of the toroidal reconnection experiment. (a) Contours of constant flux (field lines) just after the switch-off of the Helmholtz coil current. 2D null lines are between the coils, and two 3D null points are on the z-axis. (b) Vector field of the axisymmetric in-plane field (By, Bz), showing the early formation of a toroidal neutral sheet (y±15 cm, −10 < z <10 cm). (c) Contours of the out-of-plane Bx component which exhibits a quadrupole topology. This is created by the convection of (By, Bz) by Jx. Alternatively, Bx is consistent with (Jy, Jz), which is shown by the vector field. A video clip shows the time evolution of the decaying field into an elongated 2D neutral sheet. Reproduced with permission from Phys. Plasmas 10, 2780 (2003). Copyright 2003 AIP Publishing.128 Multimedia view: https://doi.org/10.1063/1.5097852.15

Close modal

Fluctuations are abundant in reconnection experiments. In the linear reconnection experiment, the major results are summarized in Fig. 36. The electron drift inside the current sheet exceeds the ion sound speed, giving rise to current-driven ion sound turbulence (Te ≫ Ti). It creates a broad frequency spectrum shown in Fig. 36(a). Figure 36(b) confirms from cross-spectral measurements that the turbulence exhibits the dispersion of ion sound waves. Broadband turbulence is known to produce anomalous resistivity, which in turn causes electron heating.

FIG. 36.

Waves and instabilities in the linear reconnection experiment. (a) Frequency and (b) wave number spectra of density fluctuations characterize the noise as ion acoustic turbulence. (c) High frequency magnetic fluctuations in the frequency regime of whistler modes. (d) Cross-correlation measurements (dots) fall close to the refractive index surface of whistler modes confirming whistler turbulence. (e) Microwave emissions during the discharge with a large spike during the reconnection event. The electromagnetic signal is thought to be mode-converted electron plasma waves created by energetic electrons. (f) Electron distribution function during reconnection, showing tail electrons in the neutral sheet. The distribution function is obtained using a 3D velocity analyzer. Reproduced with permission from Phys. Fluids 89, 2715 (1984). Copyright 1984 AIP Publishing.129 

FIG. 36.

Waves and instabilities in the linear reconnection experiment. (a) Frequency and (b) wave number spectra of density fluctuations characterize the noise as ion acoustic turbulence. (c) High frequency magnetic fluctuations in the frequency regime of whistler modes. (d) Cross-correlation measurements (dots) fall close to the refractive index surface of whistler modes confirming whistler turbulence. (e) Microwave emissions during the discharge with a large spike during the reconnection event. The electromagnetic signal is thought to be mode-converted electron plasma waves created by energetic electrons. (f) Electron distribution function during reconnection, showing tail electrons in the neutral sheet. The distribution function is obtained using a 3D velocity analyzer. Reproduced with permission from Phys. Fluids 89, 2715 (1984). Copyright 1984 AIP Publishing.129 

Close modal

Figure 36(c) reveals that magnetic fluctuations arise in the frequency regime of whistler modes. Cross spectral measurements of vector components, e.g., BxBz, are Fourier transformed in k space and compared with the refractive index curve for plane waves, shown in Fig. 36(d). These measurements confirm that the waves are whistler modes, yet do not determine the mechanism and significance of whistler mode turbulence. Their low amplitude (BwhistlerB0) suggests that wave-wave and wave-particle interactions are negligible and hence are not important for electron heating. Experimentally, whistler instabilities are observed from large currents in magnetic null lines. Theory suggests that inverse Landau damping109 and temperature anisotropy110 can produce whistler instabilities.

Figure 36(e) shows yet another frequency regime, i.e., electromagnetic (em) waves near the electron plasma frequency (fp12 MHz). These are observed external to the plasma and interpreted as unstable electron plasma waves which mode-convert on density gradients into em waves. Plasma waves are excited by electron beams which scatter electrons effectively.17 Measurements of the electron distribution inside the current sheet have shown broad energetic electron tails as shown in Fig. 36(f). Both electron plasma waves and ion acoustic waves have short wavelengths (ve/fpe few mm), which cause effective electron scattering.

Returning to the circular reconnection experiment, we observe that the loss of the magnetic field is too fast for ions to move, and thus, the physics can be characterized by EMHD reconnection. The magnetic energy is transferred into electron heating and light emission. The mechanism is again current-driven ion sound turbulence whose evidence is shown in Figs. 37(a) and 37(b). Strong density fluctuations are observed in the current sheet during the FRC decay. The fluctuation spectrum falls into the range of ion acoustic waves (ffpi).

FIG. 37.

Results of the toroidal reconnection experiment. (a) Helmholtz coil current which creates a strong inductive electric field and toroidal current causing current-driven instabilities. (b) Spectrum of the density fluctuations fall into the frequency regime for ion acoustic waves. (c) Langmuir probe traces showing the dramatic temperature rise just after switch-off compared to before. (d) The stored magnetic energy decays with a time constant 300 times shorter than the classical collisional decay time. Current-driven ion sound turbulence is the likely mechanism for rapid field annihilation and strong electron energization. Reproduced with permission from Phys. Plasmas 10, 2810 (2003). Copyright 2003 AIP Publishing.130 

FIG. 37.

Results of the toroidal reconnection experiment. (a) Helmholtz coil current which creates a strong inductive electric field and toroidal current causing current-driven instabilities. (b) Spectrum of the density fluctuations fall into the frequency regime for ion acoustic waves. (c) Langmuir probe traces showing the dramatic temperature rise just after switch-off compared to before. (d) The stored magnetic energy decays with a time constant 300 times shorter than the classical collisional decay time. Current-driven ion sound turbulence is the likely mechanism for rapid field annihilation and strong electron energization. Reproduced with permission from Phys. Plasmas 10, 2810 (2003). Copyright 2003 AIP Publishing.130 

Close modal

The electron temperature rises dramatically in the early current sheet [Fig. 37(c)]. The temperature decreases in time and broadens in space along the toroidal neutral sheet. The loss of the magnetic energy integrated over the entire volume is shown vs time in Fig. 37(d). The energy decay time is over two orders of magnitude smaller than the theoretical value. The fast reconnection or annihilation event arises from large currents, creating strong turbulence, which heats the electrons. This situation arises in neutral sheets where electric fields and currents are parallel. The null point topology (X,Y, or O) is of secondary importance.

This section will summarize significant findings of the research on whistler modes excited by antennas. It concludes by suggesting future work on unresolved problems and possible applications.

The research, performed over several decades on this topic, covers many aspects, such as linear and nonlinear waves, uniform and nonuniform ambient magnetic fields, null point physics such as EMHD reconnection, instabilities, and whistlers with orbital angular momentum. New results were made possible by advances in large plasma devices, wave diagnostics, and data processing.

Probe diagnostics can now record simultaneously the three components of the wave magnetic field with 3D spatial and temporal resolutions. Such data yield the wave current density unambiguously, the flow of the phase from the vector field of hodogram normals, and the flow of the energy from the Poynting vector field.

Electric field probes are more difficult to build when both inductive and space charge fields arise.113 Particle distributions have been measured using directional velocity analyzers with the resolution in 3D velocity space, 3D real space, and time, albeit at a high price in time for both acquiring and evaluating multidimensional data.114 With a single vector magnetic probe, one can obtain multidimensional data in repeated experiments, provided that the plasma is stable and pulsed at a fast rate. In high power plasmas, device probe arrays have to be used which may create perturbations. In space plasmas, single point measurements from one satellite115 have advanced to a cluster of a few satellites,116 which provides limited spatial wave properties for phenomena of scales larger than the satellite spacing.

Turning to laboratory plasma experiments, the following major discoveries were made. The filamentation of whistler modes in density troughs were observed. Narrow field-aligned density depressions can duct whistler modes which contradicted the ducting theories for whistlers in space plasmas. Ducts can be formed by radiation pressure or by electron heating not only in the wave but also in the near zone of antennas. The importance of antennas in the wave excitation has become obvious and stimulated further research on the antenna-wave coupling.

It was found that helicon modes exist in uniform and unbound plasmas like space plasmas. Helicon modes were usually thought to be eigenmodes of a thin cylindrical plasma column with radially reflecting boundaries. However, the characteristic property of helicons is their phase rotation around the straight guide magnetic field. Combined with the parallel propagation, the phase fronts are helical. In an unbound plasma, the antenna determines the perpendicular phase and amplitude profiles. The highly field aligned group velocity prevents the radial energy spread. Antennas are an important component that determines the helicon properties. A simple loop antenna excites m = 0 and m = 1 modes. A phased array excites positive and negative mode numbers, radial inward and outward propagation, approximate plane waves, Gendrin modes, and standing waves. It has been clarified that the rotation of the wave phase and that of the field vectors are generally different. The polarization of ideal plane whistler waves is right-hand circular, while that of wave packets ranges from linear to right-hand circular polarization. Wave interference creates linear polarization.

When the antenna size becomes small compared to the parallel wavelength, the wave packet changes from parallel to oblique waves. In a Maxwellian plasma, the amplitude profile assumes a cone at the group velocity resonance angle. However, when a fast beam penetrates the plasma, a whistler instability arises which creates oblique waves with parallel phase velocities equal to the beam velocity (VLF hiss). A small antenna now excites oblique waves instead of a cone. The unstable oblique waves grow outward, while the phase converges inward.

With a circular ring antenna, a converging resonance cone has been produced. At focus of the cone, the field amplitude is strongly enhanced and exerts a ponderomotive force on the electrons. The electrons are expelled, creating a space charge electric field, which also expels ions. The density modification changes the resonance cone, and the nonlinear interaction creates turbulence with strongly energized ions.

Advancing the diagnostics of wave fields to 3D, it became obvious that their fields are nearly force-free. It explained the stability of very strong wave packets such as whistler spheromaks. It also shows that field lines in 3D space are always twisted as in flux ropes. It has been shown that large amplitude whistlers can propagate in unmagnetized plasmas. The propagation is possible when the large wave magnetic field magnetizes the electrons. Since the wave amplitude decays, it encounters cyclotron resonance and absorption, an example of magnetic field annihilation which is a topic worth to pursue further.

Wave propagation in nonuniform ambient fields arises frequently. A new effect has been observed, i.e., the wave reflection at a strong magnetic field gradient. Reflection arises from a sudden change in the refractive index. Since the magnetic field determines the refractive index of whistlers, it explains the effect.

Wave propagation on circular field lines has been studied. A loop near the origin forms an isotropic radiation pattern across B0, while on straight field lines, the radiation is along B0. The antenna excites waves to both sides, which forms standing waves on circular field lines. The radial propagation across the ambient field requires a perpendicular group velocity component.

Hall reconnection has been studied in two different geometries, linear and circular. The first case shows that a space charge electric field develops self-consistently and opposes the applied inductive field, resulting in a strong net electric field at the electron source. The current is determined by the localized field in front at the cathode and not by the electric field along the separator, showing the importance of the global geometry.

The second reconnection experiment was done in a toroidal field where the inductive electric field cannot be reduced by a nonrotational space charge field. The current is limited by anomalous resistivity produced by current-driven ion sound turbulence which scatters and heats the electrons effectively. The reconnection electric field is due to the decaying magnetic field created by plasma currents, not by external conductor currents. Since the initial magnetic field decays completely. it is an example of spontaneous EMHD magnetic annihilation. The stretching of an initial X-type null line into an elongated double Y neutral sheet is evidence for whistler mode propagation in an EMHD plasma. Anomalous resistivity due to short-wavelength ion sound turbulence is much more effective than that of whistler wave turbulence.

The above new findings are of intrinsic value and possibly useful for applications in space and laboratory plasmas. Active experiments of VLF injection from antennas into the magnetosphere are of interest to remediate trapped energetic electrons which damage the spacecraft electronics. One approach is to inject strong whistler modes for scattering and removal of energetic electrons.111 This problem involves the knowledge of antennas for exciting strong whistlers. A comparison between electric and magnetic dipoles in a laboratory plasma has shown that the loop antennas radiate orders of magnitude better than electric dipoles. The scattering of electrons involves Doppler-shifted cyclotron resonance. The injection of helicon modes offers two types of Doppler shifts, the conventional parallel wave-particle resonance (ωc=ωkv) and the yet “unknown” perpendicular Doppler shift due to the propagation around B0 (ωc=ωkϕvϕ). The scattering of electrons due to the orbital angular momentum of helicons should be tested in a laboratory plasma. Another application of injecting strong VLF/ELF (ELF=Extremely Low Frequency) waves is for communication with submarines and underground mining and earthquake research.112 

Although much has been learned about antenna excitations of whistler modes, there remain still many open questions.

In space plasma research active, wave injection experiments are performed with magnetic loop antennas. Multipoint wave diagnostics also need to be advanced.

In the laboratory, the effort on helicon plasma devices has shifted to applications such as plasma processing and plasma thrusters. The understanding of the helicon physics is far from complete. The wave physics is nonlinear since the wave ionizes the gas, resulting in wave absorption, nonuniformities, and wave reflections. The radial wave reflection from boundaries or density gradients is not clear. The observed noncollisional wave damping has been explained by Landau damping and Trivelpiece-Gould modes, but none has been verified by direct observations. The probe diagnostics is difficult due to high densities, small plasma size, and a strong rf environment. Electric fields have not yet been measured, no 3D wave properties have been taken, and no space-time evolution with rf bursts can be obtained since the plasma depends on ionization. Due to various limitations, the focus been has shifted to engineering problems.

The basic science of whistler modes with orbital angular momentum needs to be pursued. The new perpendicular Doppler shift should create wave-particle and wave-wave interactions which have not yet been studied. The propagation of helicon modes on curved magnetic field lines is another open topic. When helicons propagate on curved field lines, the conservation of angular momentum requires a precession. Propagation on circular field lines does not support helicons which require cylindrical geometry. Nonlinear effects of large amplitude whistler modes on nonuniform background magnetic fields have not yet been studied. The propagation of whistler modes in reconnection fields should be investigated experimentally to compare with theories. Scattering of electrons by whistler modes needs to be demonstrated experimentally.

See the supplementary material for video clips of an m = 1 helicon rotation, vector field rotation of an m = −3 helicon, merging of current rings, and precession of an FRC formed by opposing whistler spheromaks.

The author greatly thanks his collaborator Dr. Manuel Urrutia for assistance in many experiments, particular with digital data processing and visualization. Support from the National Science Foundation and the Department of Energy for grants and from the Air Force for contracts is gratefully acknowledged and specified below for the last 20 years: Nos. NSF PHY 1414411, NSF PHY 9713240, NSF/DOE 20101721, NSF 0076065, PHY 1004284, NSF/PHY 9303821, FA8718-05-C-0072, and FA8721-11-C-0009.

1.
H.
Barkhausen
,
Phys. Z.
20
,
401
(
1919
).
2.
E. V.
Appleton
and
J. A.
Ratcliffe
,
Proc. R. Soc. London A
128
,
133
(
1930
).
3.
R. A.
Helliwell
,
Whistlers and Related Ionospheric Phenomena
(
Stanford University Press
,
Stanford, CA
,
1965
).
4.
P.
Aigrain
, in
Proceedings of the International Conference on Semiconductor Physics, Prague, 1960
(
Academic Press
,
New York and London
,
1961
), p.
224
.
5.
C. R.
Legéndy
,
Phys. Rev.
135
,
A1713
(
1964
).
6.
J. P.
Klozenberg
,
B.
McNamara
, and
P. C.
Thonemann
,
J. Fluid Mech.
21
,
545
(
1965
).
7.
R. W.
Boswell
,
Plasma Phys. Controlled Fusion
26
,
1147
(
1984
).
8.
A. J.
Perry
,
D.
Vender
, and
R.
Boswell
,
J. Vac. Sci.
9
,
310
(
1991
).
9.
C.
Charles
,
J. Phys. D: Appl. Phys.
42
,
163001
(
2009
).
10.
F. F.
Chen
,
Plasma Sources Sci. Technol.
24
,
014001
(
2015
).
11.
S.
Shinohara
,
Adv. Phys.: X
3
,
1420424
(
2018
).
12.
R. L.
Stenzel
and
J. M.
Urrutia
,
Phys. Rev. Lett.
114
,
205005
(
2015
).
13.
M.
Padgett
and
L.
Allen
,
Contemp. Phys.
41
,
275
(
2000
).
14.
15.
R. L.
Stenzel
and
J. M.
Urrutia
,
Phys. Plasmas
23
,
082120
(
2016
).
16.
C.
Krafft
,
P.
Thevenet
,
G.
Matthieussent
,
B.
Lundin
,
G.
Belmont
,
B.
Lembege
,
J.
Solomon
,
J.
Lavergnat
, and
T.
Lehner
,
Phys. Rev. Lett.
72
,
649
(
1994
).
17.
R. L.
Stenzel
,
J. Geophys. Res.
82
,
4805
, (
1977
).
18.
A. V.
Streltsov
,
J. J.
Berthelier
,
A. A.
Chernyshov
,
V. L.
Frolov
,
F.
Honary
,
M. J.
Kosch
,
R. P.
McCoy
,
E. V.
Mishin
, and
M. T.
Rietveld
,
Space Sci. Rev.
214
,
118
(
2018
).
19.
I. G.
Kondratev
,
A. V.
Kudrin
, and
T. M.
Zaboronkova
,
Radio Sci.
27
,
315
, (
1992
).
20.
J. M.
Urrutia
and
R. L.
Stenzel
,
Phys. Plasmas
25
,
032111
(
2018
).
21.
K. P.
Shamrai
and
V. B.
Taranov
,
Plasma Sources Sci. Technol.
5
,
474
(
1996
).
22.
R. L.
Stenzel
,
Radio Sci.
11
,
1045
, (
1976
).
23.
A. W.
Degeling
,
G. G.
Borg
, and
R. W.
Boswell
,
Phys. Plasmas
11
,
2144
(
2004
).
24.
J. M.
Urrutia
and
R. L.
Stenzel
,
Phys. Plasmas
23
,
052112
(
2016
).
25.
K.
Niemi
and
M.
Krämer
,
Phys. Plasmas
15
,
073503
(
2008
).
26.
C. M.
Franck
,
O.
Grulke
,
A.
Stark
,
T.
Klinger
,
E. E.
Scime
, and
G.
Bonhomme
,
Plasma Sources Sci. Technol.
14
,
226
(
2005
).
27.
M.
Light
,
I. D.
Sudit
,
F. F.
Chen
, and
D.
Arnush
,
Phys. Plasmas
2
,
4094
(
1995
).
28.
R. L.
Stenzel
and
J. M.
Urrutia
,
Phys. Plasmas
25
,
082109
(
2018
).
29.
R. L.
Stenzel
and
J. M.
Urrutia
,
Phys. Plasmas
23
,
092103
(
2016
).
30.
A. V.
Karavaev
,
N. A.
Gumerov
,
K.
Papadopoulos
,
X.
Shao
,
A. S.
Sharma
,
W.
Gekelman
,
A.
Gigliotti
,
P.
Pribyl
, and
S.
Vincena
,
Phys. Plasmas
17
,
012102
(
2010
).
31.
J. M.
Urrutia
and
R. L.
Stenzel
,
Phys. Plasmas
21
,
122107
(
2014
).
32.
C.
Huygens
,
Lenses Waves
9
,
213
253
(
1690
).
33.
J. M.
Urrutia
,
R. L.
Stenzel
, and
M. C.
Griskey
,
Phys. Plasmas
7
,
519
(
2000
).
34.
C. L.
Rousculp
,
R. L.
Stenzel
, and
J. M.
Urrutia
,
Phys. Plasmas
2
,
4083
(
1995
).
35.
R. L.
Stenzel
and
J. M.
Urrutia
,
Phys. Plasmas
21
,
122108
(
2014
).
36.
R. L.
Stenzel
and
J. M.
Urrutia
,
Phys. Plasmas
22
,
092113
(
2015
).
37.
M.
Light
and
F. F.
Chen
,
Phys. Plasmas
2
,
1084
(
1995
).
38.
M. E.
Gushchin
,
S. V.
Korobkov
,
A. V.
Kostrov
,
A. V.
Strikovsky
,
T. M.
Zaboronkova
,
C.
Krafft
, and
V. A.
Koldanov
,
Adv. Space Res.
42
,
979
(
2008
).
39.
R. L.
Stenzel
and
J. M.
Urrutia
,
Phys. Plasmas
22
,
092112
(
2015
).
40.
V. S.
Sonwalkar
,
X.
Chen
,
J.
Harikumar
,
D.
Carpenter
, and
T.
Bell
,
J. Atmos. Sol.-Terr. Phys.
63
,
1199
(
2001
).
41.
O.
Santolik
,
M.
Parrot
, and
F.
Lefeuvre
,
Radio Sci.
38
,
1010
, (
2003
).
42.
O. P.
Verkhoglyadova
and
B. T.
Tsurutani
,
Ann. Geophys.
27
,
4429
(
2009
).
43.
P. M.
Bellan
,
Phys. Plasmas
20
,
082113
(
2013
).
44.
W.
Amatucci
,
D.
Blackwell
,
E.
Tejero
,
C.
Cothran
,
L.
Rudakov
,
G.
Ganguli
, and
D.
Walker
,
IEEE Trans. Plasma Sci.
39
,
637
(
2011
).
45.
R. L.
Stenzel
,
J. M.
Urrutia
, and
M. C.
Griskey
,
Phys. Plasmas
6
,
4458
(
1999
).
46.
M. A.
Berger
, in
Magnetic Helicity in Space and Laboratory Plasmas
, Geophysical Monograph No. 111, edited by
M. R.
Brown
,
R. C.
Canfield
, and
A. A.
Pevtsov
(
American Geophysical Union
,
Washington, DC
,
1999
), pp.
1
9
.
48.
D.
Zhao
,
T.
van Leeuwen
,
J.
Cheng
, and
B. L.
Feringa
,
Nat. Chem.
9
,
250
(
2017
).
49.
F. F.
Chen
and
R. W.
Boswell
,
IEEE Trans. Plasma Sci.
25
,
1245
(
1997
).
50.
E.
Priest
and
T.
Forbes
,
Magnetic Reconnection: MHD Theory and Applications
(
Cambridge University Press
,
2000
).
51.
R. L.
Stenzel
,
J. M.
Urrutia
, and
M. C.
Griskey
,
Phys. Rev. Lett.
82
,
4006
(
1999
).
52.
R. L.
Stenzel
,
J. M.
Urrutia
, and
M. C.
Griskey
,
Phys. Scr.
T84
,
112
(
2000
).
53.
R. L.
Stenzel
,
J. M.
Urrutia
, and
M. C.
Griskey
,
Phys. Plasmas
6
,
4450
(
1999
).
54.
V. S.
Sonwalkar
,
U. S.
Inan
,
T. F.
Bell
,
R. A.
Helliwell
,
O. A.
Molchanov
, and
J. L.
Green
,
J. Geophys. Res.
99
,
6173
, (
1994
).
55.
T.
Wang
and
T.
Bell
,
J. Geophys. Res.
77
,
1174
, (
1972
).
56.
D. D.
Blackwell
,
D. N.
Walker
,
S. J.
Messer
, and
W. E.
Amatucci
,
Phys. Plasmas
14
,
092106
(
2007
).
57.
P.
Pribyl
,
W.
Gekelman
, and
A.
Gigliotti
,
Radio Sci.
45
,
RS4013
, (
2010
).
58.
J. M.
Urrutia
and
R. L.
Stenzel
,
Phys. Rev. Lett.
62
,
272
(
1989
).
59.
R. L.
Stenzel
and
J.
Urrutia
,
Geophys. Res. Lett.
16
,
361
, (
1989
).
60.
J. M.
Urrutia
and
R. L.
Stenzel
,
Geophys. Res. Lett.
17
,
1589
, (
1990
).
61.
M. G.
Kivelson
,
L. F.
Bargatze
,
K. K.
Khurana
,
D. J.
Southwood
,
R. J.
Walker
, and
P. J.
Coleman
, Jr.
,
Science
261
,
331
(
1993
).
62.
R. L.
Stenzel
and
W.
Gekelman
,
Phys. Rev. A
11
,
2057
(
1975
).
63.
M. G.
Rao
and
H. G.
Booker
,
J. Geophys. Res.
68
,
387
, (
1963
).
64.
M. K.
Paul
and
D.
Bora
,
J. Appl. Phys.
105
,
013305
(
2009
).
65.
66.
H.
Breuillard
,
Y.
Zaliznyak
,
V.
Krasnoselskikh
,
O.
Agapitov
,
A.
Artemyev
, and
G.
Rolland
,
Ann. Geophys.
30
,
1223
(
2012
).
67.
A. V.
Streltsov
,
J. R.
Woodroffe
, and
J. D.
Huba
,
J. Geophys. Res.
117
,
A08302
, (
2012
).
68.
K. G.
Budden
,
The Propagation of Radio Waves
(
Cambridge University Press
,
Cambridge, UK
,
1988
).
69.
A. V.
Kudrin
,
P. V.
Bakharev
,
T. M.
Zaboronkova
, and
C.
Krafft
,
Plasma Phys. Controlled Fusion
53
,
065005
(
2011
).
70.
J. S.
Halekas
,
D. A.
Brain
,
D. L.
Mitchell
, and
R. P.
Lin
,
Geophys. Res. Lett.
33
,
L22104
, (
2006
).
71.
Y.
Tsugawa
,
Y.
Katoh
,
N.
Terada
,
H.
Tsunakawa
,
F.
Takahashi
,
H.
Shibuya
,
H.
Shimizu
, and
M.
Matsushima
,
Earth, Planets Space
67
,
36
(
2015
).
72.
X. H.
Deng
,
M.
Zhou
,
S. Y.
Li
,
W.
Baumjohann
,
M.
Andre
,
N.
Cornilleau
,
O.
Santolik
,
D. I.
Pontin
,
H.
Reme
,
E.
Lucek
,
A. N.
Fazakerley
,
P.
Decreau
,
P.
Daly
,
R.
Nakamura
,
R. X.
Tang
,
Y. H.
Hu
,
Y.
Pang
,
J.
Buechner
,
H.
Zhao
,
A.
Vaivads
,
J. S.
Pickett
,
C. S.
Ng
,
X.
Lin
,
S.
Fu
,
Z. G.
Yuan
,
Z. W.
Su
, and
J. F.
Wang
,
J. Geophys. Res.
114
,
A07216
, (
2009
).
73.
C. M.
Franck
,
O.
Grulke
,
A.
Stark
,
T.
Klinger
,
E. E.
Scime
, and
G.
Bonhomme
,
Geophys. Res. Lett.
41
,
2721
, (
2014
).
74.
Y.
Chen
,
K.
Fujimoto
,
C.
Xiao
, and
H.
Ji
,
J. Geophys. Res. Space Phys.
120
,
6309
6319
, (
2015
).
75.
B. T.
Tsurutani
,
E. J.
Smith
,
M. E.
Burton
,
J. K.
Arballo
,
C.
Galvan
,
X.-Y.
Zhou
,
D. J.
Southwood
,
M. K.
Dougherty
,
K.-H.
Glassmeier
,
F. M.
Neubauer
, and
J. K.
Chao
,
J. Geophys. Res.
106
,
30223
, (
2001
).
76.
A.
Cardinali
,
D.
Melazzi
,
M.
Manente
, and
D.
Pavarin
,
Plasma Sources Sci. Technol.
23
,
015013
(
2014
).
77.
S.
Shinohara
and
A.
Fujii
,
Phys. Plasmas
8
,
3018
(
2001
).
78.
P. K.
Loewenhardtt
,
B. D.
Blackwell
, and
S. M.
Hamberger
,
Plasma Phys. Controlled Fusion
37
,
229
(
1995
).
79.
T.
Shoji
,
H.
Kikuchi
,
Y.
Sakawa
,
C.
Suzuki
,
G.
Matsunaga
, and
K.
Toi
, in
IEEE Conference Record—Abstracts. The 30th International Conference on Plasma Science ICOPS 2003
(IEEE Conference Publications,
2003
), p.
466
.
80.
C. G.
Jin
,
T.
Yu
,
Y.
Zhao
,
Y.
Bo
,
C.
Ye
,
J. S.
Hu
,
L. J.
Zhuge
,
S. B.
Ge
,
X. M.
Wu
,
H. T.
Ji
, and
J. G.
Li
,
IEEE Trans. Plasma Sci.
39
,
3103
(
2011
).
81.
A. W.
Trivelpiece
and
R. W.
Gould
,
J. Appl. Phys.
30
,
1784
(
1959
).
82.
R. L.
Stenzel
,
Geophys. Res. Lett.
3
,
61
, (
1976
).
83.
V. I.
Karpmann
,
R. N.
Kaufman
, and
A. G.
Shagalov
,
Phys. Fluids B
4
,
3087
(
1992
).
84.
A. V.
Kostrov
,
A. V.
Kudrin
,
L. E.
Kurina
,
G. A.
Luchinin
,
A. A.
Shaykin
, and
T. M.
Zaboronkova
,
Phys. Scr.
62
,
51
(
2000
).
85.
H.
Sugai
,
M.
Maruyama
,
M.
Sato
, and
S.
Takeda
,
Phys. Fluids
21
,
690
(
1978
).
86.
R. L.
Stenzel
,
J. M.
Urrutia
, and
K. D.
Strohmaier
,
Phys. Rev. Lett.
96
,
095004
(
2006
).
87.
V. I.
Karpman
and
H.
Washimi
,
J. Plasma Phys.
18
,
173
(
1977
).
88.
S. P.
Gary
and
J.
Wang
,
J. Geophys. Res.
101
,
10749
, (
1996
).
89.
M.
Krämer
,
Y. M.
Aliev
,
A. B.
Altukhov
,
A. D.
Gurchenko
,
E. Z.
Gusakov
, and
K.
Niemi
,
Plasma Phys. Controlled Fusion
49
,
A167
(
2007
).
90.
J. S.
Zhao
,
J. Y.
Lu
, and
D. J.
Wu
,
Astrophys. J.
714
,
138
(
2010
).
91.
J. M.
Urrutia
and
R. L.
Stenzel
,
Phys. Plasmas
3
,
2589
(
1996
).
92.
R. L.
Stenzel
and
J. M.
Urrutia
,
Phys. Plasmas
3
,
2599
(
1996
).
93.
A.
Vartanyan
,
G. M.
Milikh
,
B. E.
Eliasson
,
A.
Sharma
,
C.
Chang
,
M.
Parrot
, and
K.
Papadopoulos
,
J. Geophys. Res.
103
,
23717
, (
2013
).
94.
R. L.
Stenzel
and
W.
Gekelman
,
Phys. Fluids
20
,
108
(
1977
).
95.
R. L.
Stenzel
and
J. M.
Urrutia
,
Phys. Rev. Lett.
81
,
2064
(
1998
).
96.
J. M.
Urrutia
,
R. L.
Stenzel
, and
K. D.
Strohmaier
,
Phys. Plasmas
15
,
062109
(
2008
).
97.
J. M.
Urrutia
,
R. L.
Stenzel
, and
K. D.
Strohmaier
,
Phys. Plasmas
16
,
022102
(
2009
).
98.
E. N.
Parker
,
Phys. Rev.
107
,
830
(
1957
).
99.
S. I.
Syrovatskii
,
Annu. Rev. Astron. Astrophys.
19
,
163
(
1981
).
100.
D.
Biskamp
,
Astrophys. Space Sci.
242
,
165
(
1996
).
101.
E. G.
Zweibel
and
M.
Yamada
,
Annu. Rev. Astron. Astrophys.
47
,
291
(
2009
).
102.
M. E.
Mandt
,
R. E.
Denton
, and
J. F.
Drake
,
Geophys. Res. Lett.
21
,
73
, (
1994
).
103.
A. S.
Kingsep
,
K. V.
Chukbar
, and
V. V.
Yan'kov
, in
Reviews of Plasma Physics
, edited by
B. B.
Kadomtsev
(
Consultants Bureau
,
New York
,
1990
), Vol.
16
, p.
243
.
104.
P. J.
Baum
and
A.
Bratenahl
,
J. Plasma Phys.
18
,
257
(
1977
).
105.
R. L.
Stenzel
and
W.
Gekelman
,
Phys. Rev. Lett.
42
,
1055
(
1979
).
106.
A. G.
Frank
,
Plasma Phys. Controlled Fusion
41
,
A687
(
1999
).
107.
M.
Yamada
,
J. Geophys. Res.
104
,
14529
, (
1999
).
108.
F. S.
Mozer
,
S. D.
Bale
,
T. D.
Phan
, and
J. A.
Osborne
,
Phys. Rev. Lett.
91
,
245002
(
2003
).
109.
R. L.
Tokar
and
D. A.
Gurnett
,
JGR Space Phys.
90
,
105
(
1985
).
110.
S. P.
Gary
,
D.
Winske
, and
M.
Hesse
,
J. Geophys. Res.
105
,
10751
, (
2000
).
111.
W. E.
Amatucci
,
E. M.
Tejero
,
D. D.
Blackwell
,
C. D.
Cothran
, and
C. L.
Enloe
,
Dispatch
4
,
15
(
2014
); available at https://www.dtra.mil/Research/DTRIAC/DTRIAC-Dispatch/.
112.
J. Y.
Liu
,
K.
Wang
,
C. H.
Chen
,
W. H.
Yang
,
Y. H.
Yen
,
Y. I.
Chen
,
K.
Hatorri
,
H. T.
Su
,
R. R.
Hsu
, and
C. H.
Chang
,
J. Geophys. Res. Atmos.
118
,
3760
, (
2013
).
113.
R. L.
Stenzel
,
Rev. Sci. Instrum.
62
,
130
(
1991
).
114.
R. L.
Stenzel
,
W.
Gekelman
,
N.
Wild
,
J. M.
Urrutia
, and
D.
Whelan
,
Rev. Sci. Instrum.
54
,
1302
(
1983
).
115.
D. A.
Gurnett
, in
AGU Monograph 103
, edited by
R.
Pfaff
,
J.
Borovsky
, and
D.
Young
(
American Geophysical Union
,
Washington, DC
,
1998
), Chap. 14, pp.
121
136
.
116.
X. H.
Wei
,
J. B.
Cao
,
G. C.
Zhou
,
O.
Santolik
,
H.
Reme
,
I.
Dandouras
,
N.
Cornilleau-Wehrlin
,
E.
Lucek
,
C. M.
Carr
, and
A.
Fazakerley
,
J. Geophys. Res.
112
,
A10225
, (
2007
).
117.
J. M.
Urrutia
and
R. L.
Stenzel
,
Phys. Plasmas
22
,
092111
(
2015
).
118.
R. L.
Stenzel
and
J. M.
Urrutia
,
Phys. Plasmas
22
,
072110
(
2015
).
119.
R. L.
Stenzel
and
J. M.
Urrutia
,
Phys. Plasmas
25
,
032112
(
2018
).
120.
R. L.
Stenzel
and
J. M.
Urrutia
,
Phys. Plasmas
25
,
082110
(
2018
).
121.
J. M.
Urrutia
and
R. L.
Stenzel
,
Phys. Plasmas
25
,
082108
(
2018
).
122.
R. L.
Stenzel
,
Phys. Fluids
19
,
865
(
1976
).
123.
R. L.
Stenzel
 et al.,
Phys. Plasmas
15
,
042307
(
2008
).
124.
J. M.
Urrutia
,
R. L.
Stenzel
, and
K. D.
Strohmaier
,
Phys. Plasmas
15
,
042308
(
2008
).
125.
K. D.
Strohmaier
 et al.,
Phys. Plasmas
15
,
042309
(
2008
).
126.
R. L.
Stenzel
 et al.,
Phys. Plasmas
16
,
022103
(
2009
).
127.
R. L.
Stenzel
and
W.
Gekelman
,
J. Geophys. Res.
86
,
649
, (
1981
).
128.
R. L.
Stenzel
 et al.,
Phys. Plasmas
10
,
2780
(
2003
).
129.
W.
Gekelman
and
R. L.
Stenzel
,
J. Geo. Res. Space Phys.
89
,
2715
, (
1984
).
130.
R. L.
Stenzel
 et al.,
Phys. Plasmas
10
,
2810
(
2003
).

Supplementary Material