Linear theory and measurements of electron oscillations in an inertial Alfvén wave

Although the production of accelerated electrons that generate auroras is not rigorously understood, Alfvén waves are a likely acceleration mechanism. Alfvén waves are measured ubiquitously in Earth's magnetosphere, and they play a key role in the coupling of the magnetosphere-ionosphere system.¹ Alfvén waves can be launched by a sheared plasma flow perpendicular to Earth's magnetic field, or by a sudden dynamic change in convection or resistivity in some region of the magnetosphere,² as can occur when magnetic storms cause shifts in magnetospheric boundaries or when magnetotail reconnection occurs.³ In situ data shows that Alfvén waves and bursts of field-aligned suprathermal electrons occur in conjunction in auroral regions,¹ and Alfvén waves are associated with a significant fraction of electrons precipitating into the ionosphere.⁴ Theoretical progress over several decades has shown viable mechanisms by which Alfvén waves can produce auroral electrons. Early studies found that dispersive Alfvén waves in the magnetosphere are capable of accelerating electrons to auroral energies.^3,5 Kinetic calculations including dispersive Alfvén waves have reproduced the kinetic signatures of accelerated electrons observed in the upper magnetosphere.⁶ While these in situ and theoretical studies are strongly suggestive that Alfvén waves play a key role in accelerating electrons and generating auroras, a controlled test of the acceleration process has not yet been performed.

In the past three decades, the observational study of Alfvénic electron acceleration in the auroral zone has entered a new era with measurements from the FAST, Freja, Polar, DMSP, Geotail, and Cluster missions. Polar measurements of downward Alfvénic Poynting flux at 4–7 R_E in the plasma sheet boundary layer are well correlated with the luminosity of magnetically conjugate auroral structures in a statistical study of 40 plasma sheet boundary layer crossings.⁷ A study of the conjunctions of the Polar spacecraft at 4–7 R_E and the FAST spacecraft at 1.05–1.65 R_E demonstrated statistically that the Alfvénic Poynting flux dominated over electron energy flux at Polar orbits but the electron energy flux was greater than the Alfvénic Poynting flux at the FAST altitudes.^8,9 This evidence supports the picture that Alfvén waves are losing energy via wave-particle interactions to accelerate electrons as they propagate toward the ionosphere. A statistical survey correlating particle fluxes and Alfvén wave fields of more than 5000 polar orbits from the FAST satellite shows that Alfvén waves may be responsible for 31% of all electron precipitation onto the ionosphere.⁴ At magnetic local noon and midnight, Alfvénic activity may account for as much as 50% of electron precipitation. Finally, Schriver et al.¹⁰ used seven FAST-Polar conjunction events to show that, during geomagnetically active times, Polar measured large-amplitude Alfvén waves in the plasma sheet boundary layer, FAST measured field-aligned electron acceleration events, and the Polar UVI imager recorded strong auroral luminosity at the magnetically conjugate point in the ionosphere. They concluded that Alfvén waves are important drivers of auroral acceleration, in addition to quasi-static, field-aligned potentials and the earthward flow of energetic plasma beams from the magnetotail.

Although such conjunction studies provide a powerful tool to explore the evolution of Alfvén waves between two points along approximately the same magnetic flux tube, perfect conjunctions (measurements on the same magnetic field line by both spacecraft) are impossible to achieve. All conjunction studies are subject to uncertainties from the motion and mapping of geomagnetic field lines, time of flight delays for Alfvén waves propagating between points of measurement, and different orbital speeds of the spacecraft.¹ A definitive measurement of electron acceleration by inertial Alfvén waves requires the simultaneous measurement of accelerated electrons and the inertial Alfvén waves responsible for the acceleration. No such verification of this fundamental mechanism in auroral physics has ever been achieved, either observationally or experimentally. Additionally, the space-time ambiguity associated with measurements from a moving platform makes the unambiguous identification of Alfvén waves with a single spacecraft difficult.

Laboratory experiments may be the most direct way to overcome the limitations of conjunction studies. In particular, the controlled and reproducible plasma of the Large Plasma Device (LAPD) at UCLA¹¹ provides the unique capability to verify definitively the acceleration of electrons by inertial Alfvén waves. However, laboratory investigations have not yet been successful because Alfvén wave amplitudes achieved so far have been typically small and interaction lengths are relatively short compared to the magnetosphere. This makes the wave-particle interaction that presumably produces accelerated electrons difficult to observe with traditional diagnostic methods. We address this difficulty using a novel Whistler Wave Absorption Diagnostic (WWAD) to measure suprathermal electrons with high precision.¹² 

The experiments presented here, performed on the LAPD, successfully isolate the linear effect of inertial Alfvén waves on the electron distribution. These measurements verify one of the most basic features of inertial Alfvén waves: the oscillation of electrons participating in the current of the Alfvén wave itself. The linear theory verified by these results will be required for analyzing measurements of the nonlinear electron response containing acceleration. The primary results are given in Figs. 9 and 10. These results show that the amplitude, phase, and perpendicular spatial structure of the measured electron response agree with linear theory. Further analysis considers the importance of non-Alfvénic terms included in the model. While preliminary results were given in a letter by Schroeder et al.,¹³ this longer paper includes a more thorough discussion of the experiment, theory, and analysis, and also includes several more detailed results.

Under the assumptions of ideal MHD, Alfvén waves do not produce an electric field parallel to the background magnetic field and are not capable of accelerating electrons in this direction. However, when the scale size perpendicular to the background magnetic field becomes comparable to the electron skin depth $δe=c/ωpe$⁠, the Alfvén wave becomes dispersive and generates a parallel electric field that alters the parallel electron motion.^3,5,14 In this equation, ω_pe is the electron plasma frequency. In the limit $β<me/mi$⁠, the electron thermal velocity is slower than the Alfvén speed $vte<vA$⁠, and the Alfvén wave transitions to the inertial Alfvén wave^2,14 at $k⊥δe≳1$⁠, where $k⊥$ is the perpendicular Alfvén wavenumber. Parallel electron inertia leads to a dispersive nature for the inertial Alfvén wave governed by

where ν_te is the thermal electron collision rate,¹⁵ and the parallel direction is defined by the mean magnetic field $B0=B0ẑ$⁠. We have chosen coordinates so that wave structure across $B0$ is in the $x̂$ direction so that $k=kzẑ+k⊥x̂$⁠.

The collisionally-modified dispersion relation in Eq. (1), discussed by Thuecks et al.,¹⁶ is derived by multiplying electron mass in the electron momentum equation by the complex factor $(1+iνte/ω)$⁠. The modified dispersion relation has been experimentally verified using inertial Alfvén waves in the LAPD.^16,17 Because experiment and theory agree on the collisional changes to the inertial Alfvén wave in the LAPD, we are able to identify the similarities and differences between inertial Alfvén waves in this experiment and in the magnetosphere. The effects of collisions will be revisited in the discussion of inertial Alfvén wave measurements (Section III B) and the kinetic electron behavior (Section IV B).

The electric field aligned with $B0$ is given by

where a tilde is used to indicate quantities that have been Fourier transformed in space and time. In the auroral acceleration zone, at a geocentric radius $r ≲ 3RE$⁠, the cold plasma of primarily ionospheric origin and strong magnetic field of the Earth lead to plasma conditions in the inertial regime, with $β<me/mi$ or $vte<vA$⁠. Therefore, the physics of the inertial Alfvén wave, with its associated parallel electric field, is relevant to the problem of auroral electron acceleration.

Using an Alfvén wave model that calculates particle distributions, Kletzing¹⁸ showed that inertial Alfvén waves can accelerate electrons to velocities near twice the Alfvén speed. These electrons start out moving slower than the wave, are overtaken by it, and are then accelerated out of the front of the wave moving at a velocity greater than the wave speed in a manner analogous to a single-bounce Fermi acceleration process. The electrons are observed in the lower ionosphere as a burst that arrives before the electric and magnetic field signature of the wave itself. This resonant process affects electrons near the phase speed of the inertial Alfvén wave, and since $vA>vte$⁠, the affected electrons are suprathermal. The work by Kletzing was extended to include a realistic variation of plasma density and magnetic field along an auroral field line and demonstrated that an Alfvén wave pulse can produce time-dispersed bursts of electrons like those observed by sounding rockets and satellites.⁶ 

The results from Kletzing¹⁸ also show a non-resonant linear effect of the electromagnetic fields of the Alfvén wave on the electrons. In this linear interaction, the parallel electric field E_z of the inertial Alfvén wave causes oscillations of the electron distribution. These electron oscillations constitute the parallel current of the inertial Alfvén wave. Even though we are ultimately interested in testing the resonant nonlinear acceleration of electrons, measurements of the distribution function will contain the electron response to all orders. Therefore, before an experimental test of the nonlinear behavior can be carried out, we must be able to identify and separate the linear response of the electron distribution function. The work presented here takes the necessary step of measuring and modeling this linear response for inertial Alfvén waves in the LAPD.

The goal of this experiment is to launch inertial Alfvén waves and simultaneously measure their effect on the electron distribution function. The experiment presented here is carried out at UCLA's Large Plasma Device (LAPD).¹¹ The LAPD consists of a 16.5 m linear plasma formed as neutral fill gas is ionized by electrons from a cathode-anode source located at one end of the experiment. Solenoidal coils wrap around the device and produce within the plasma a uniform axial background magnetic field $B0=B0ẑ$⁠. The plasma discharge, or shot, lasts for approximately 10 ms and is repeated every second.

As shown in Fig. 1(a), an Alfvén wave antenna and five probes are used in this experiment. Elsässer probes E1 and E2 each have integrated inductive coils and double probes to make spatially coincident measurements of $∂Bx/∂t, ∂By/∂t$⁠, E_x, and E_y.¹⁹ Whistler probes W1 and W2 are a part of the WWAD, described in Section V, which measures the reduced electron distribution parallel to $B0$⁠. The swept Langmuir probe records the electron density n_e and temperature T_e.

For the experiments presented in this paper, the fill gas is H₂, and the background magnetic field is $B0=1.80±0.01$ kG. The electron density is $ne=(1.0±0.1)×1012$ cm⁻³, and the electron temperature is $Te=2.1±0.2$ eV. The electron density is calibrated to a nearby line-integrated measurement of density from a microwave interferometer. In these conditions, the Alfvén speed is $vA=3.9×106$ m/s, the electron thermal speed is $vte=6.1×105$ m/s, and the electron thermal collision frequency is $νte=1.0×107$ s⁻¹. The electron skin depth is $δe=0.53$ cm. Using previous interferometer measurements in a similar plasma, the ion temperature is estimated to be 1.25 eV, although no ion temperature measurement was available when our experiments were performed. Since $vte/vA=0.15$⁠, or equivalently $βi=2×10−5<me/mi$⁠, inertial Alfvén waves can be produced in these conditions.

The 1 Hz repetition of the discharge facilitates combining data from many shots; however, this raises the question of repeatability. The discharge is believed to be repeatable since probe data shows that shot-to-shot fluctuations are small and random. Automated drive systems move the probes to different locations in the perpendicular x-y plane to collect data during subsequent shots, allowing composite measurements that span the x-y plane.

The Arbitrary Spatial Waveform (ASW) antenna,¹⁶ shown schematically in Fig. 1(b), is used to launch Alfvén waves. The antenna is made of 48 bare copper grid pieces that are immersed in the plasma. Each grid piece extends 30.5 cm in $ŷ$⁠, and the grids are evenly spaced along 30.1 cm in $x̂$⁠. The voltage applied to the grid pieces draws current from the plasma that flows along $B0$⁠. An oscillating voltage is applied to the grids with a frequency much less than the ion cyclotron frequency, and the oscillating current produced in the plasma excites the Alfvén wave mode. All grids are driven at the same frequency, but the voltage amplitude for each grid can be individually adjusted. By tuning the voltage amplitudes of the grid pieces, a pattern is produced in $x̂$ that sets the $k⊥$ of the Alfvén wave as seen in Fig. 2(a). Because of the uniformity of the antenna in $ŷ$⁠, the waves launched by the ASW antenna are effectively two dimensional. Additionally, by tuning the grid pieces so that there is spectral purity in $x̂$⁠, the comparison of theory and experiment is further simplified. For this experiment, the antenna is driven at $125±1$ kHz and is tuned to have a wave pattern composed of $k⊥=±1.24$ cm⁻¹ with an experimental uncertainty of ±0.08 cm⁻¹. For this wave pattern and electron density, $k⊥δe=0.66$⁠.

Elsässer probes E1 and E2, shown in Fig. 1(a), are used to measure the wave fields launched by the ASW antenna. The intensity and polarization of $B⊥$ is shown in Fig. 2(a) using measurements from Elsässer probe E1. This figure shows that $B⊥$ is polarized almost exclusively in $ŷ$⁠, and it shows the wave pattern in $x̂$ has good spectral purity. The wave amplitude is 35 ± 5 mG.

Elsässer probe measurements are also used to verify the Alfvénic behavior of the wave launched by the ASW antenna. This is done by comparing the predicted and measured values of the phase speed, damping, and wave admittance. Because of the low electron temperature, the plasma is collisional for thermal electrons, $νte/ω=13$⁠, where ω is the Alfvén wave frequency. Therefore, the wave field measurements are compared with the collisionally-modified dispersion relation for the inertial Alfvén wave given in Eq. (1). This dispersion relation predicts the Alfvén wave phase speed and damping consistent with the wave field measurements. The predicted phase speed is given by $vph=ω/kzr$⁠, where $kzr$ is the real part of the parallel wave number, and the predicted value is $vph=2.1× 106$ m/s, or $kzr=0.37$ m⁻¹ for the 125 kHz wave used here. The experimental phase speed is found by correlating the arrival of phase fronts between probes E1 and E2, and is found to be $vph=(2.2±0.1)×106$ m/s, or $kzr=0.36±0.015$ m⁻¹. The theoretical value of damping is $kzi=0.29$ m⁻¹, where $kzi$ is the imaginary part of the parallel wave number. The measured damping is $kzi=0.29±0.02$ m⁻¹.

The predicted Alfvén wave admittance comes from Faraday's law and the collisionally-modified two fluid dielectric tensor that was used to produce the dispersion relation

The theoretical value is $|Ẽx|/|B̃y|=9.4×106$ m/s. The experimental value is $|Ẽx|/|B̃y|=(9.8±0.5)×106$ m/s. Given the agreement between the theoretical and measured values of phase speed, damping, and wave admittance, we conclude that the ASW antenna is launching an inertial Alfvén wave.

Since the parallel electric field is small, $Ez/Ex≈0.003$ for this experiment, it is not possible to measure E_z directly. Instead, E_z is calculated from measurements of the perpendicular wave fields. Combining Eq. (2) with Faraday's law gives

Since the Alfvén wave launched by the ASW antenna is polarized so that $By≫Bx$⁠, most of the contribution to $Ẽz$ in this calculation comes from $B̃y$⁠. An inverse Fourier transform is performed on the calculated $Ẽz$⁠, and the resulting E_z is shown in Fig. 2(b). To facilitate the development of theory in Section IV, the E_z waveform in Fig. 2(b) is modeled as a standing wave in $x̂$ and a propagating wave in $ẑ$

The most notable difference between $B⊥$ and E_z in Fig. 2 is the quarter wavelength shift in $x̂$ so that the nodes of $B⊥$ are located where there are antinodes of E_z. Since E_z overlaps spatially with the parallel current J_z, the physical significance of this shift between E_z and $B⊥$ is that there are current channels at the nodes of $B⊥$ as predicted by theory.

Because thermal electrons are collisional, the inertial Alfvén waves in this experiment are similar but not identical to those in the collisionless conditions of the lower magnetosphere. The collisionally-modified theory, introduced in Eq. (1) and used in this section for the predicted phase speed, damping, and wave admittance, accurately describes our measurements of the inertial Alfvén wave. Due to this agreement between measurements and theory, we conclude that the inertial Alfvén waves of the magnetosphere and the ones in this experiment are both well-described by the collisionally-modified theory with $νte=0$ in the collisionless case. Aside from the introduction of collisional damping, the modified wave properties described in this section are not fundamentally distinct from the properties of inertial Alfvén waves in the collisionless conditions of the magnetosphere.

In infinite space, linear kinetic Alfvén wave theory is relatively simple because Fourier transforms make the problem algebraic. However, the solution produced here is somewhat complicated by the finite dimensions of the experiment since boundary effects must be considered.

The full electron distribution $fe(x,v,t)$ is a function of three spatial dimensions $x=(x,y,z)$ and three velocity dimensions $v=(vx,vy,vz)$⁠. We assume that $fe(x,v,t)$ is predominantly composed of a static uniform background distribution $fe0(v)$ and a small component that varies in space and time so that $fe(x,v,t)≈fe0(v)+fe1(x,v,t)$⁠. If $B0=(0,0,B0)$ and $fe0(v)$ are static, and if $E1=(Ex,0,Ez),B1=(0,By,0)$⁠, and $fe1(x,v,t)$ are small quantities that vary in space and time and are associated with the Alfvén wave generated by the ASW antenna, then the linearized Boltzmann equation is

where the right hand side represents the effect of collisions.

Electron cyclotron motion, included in Eq. (6), occurs at a frequency 10⁴ times greater than the Alfvén wave frequency. Because the timescales of the cyclotron and Alfvénic oscillations are well-separated, we can average the linearized Boltzmann equation over the cyclotron period, and the resulting equation will still accurately describe changes to $fe1(x,v,t)$ on the slower timescale of the Alfvén wave. To perform this average, the linearized Boltzmann equation is transformed from Cartesian coordinates to guiding center coordinates. A magnetized electron traces a circle in the x-y plane perpendicular to $B0=B0ẑ$⁠, and the center of this circle is denoted by the coordinates (X, Y). These coordinates, called guiding center coordinates,^20,21 are related to the instantaneous position (x, y) and velocity (v_x, v_y) of the electron by $X=x+vy/Ωce$ and $Y=y−vx/Ωce$ where $Ωce=−eB0/me$⁠. The radius of the circle ρ and the pitch angle of the perpendicular velocity $ϕ$ are related to the instantaneous velocity of the electron, $ρ=(vx2+vy2)1/2/|Ωce|$ and $ϕ=tan−1(vy/vx)$⁠. The electron distribution, originally expressed as $fe0(vx,vy,vz)+fe1(x,y,z,vx,vy,vz,t)$⁠, can be rewritten in guiding center coordinates as $f̂e0(ρ,ϕ,vz)+ f̂e1(X,Y,z,ρ,ϕ,vz,t)$⁠. This guiding center quantity is independent of gyrophase $ϕ$ for fluctuations with frequencies $ω≪Ωce$⁠, so we transform Eq. (6) to guiding center coordinates and eliminate $ϕ$ dependence in $f̂e$ by averaging over $ϕ$⁠.

This process yields the linearized gyroaveraged kinetic equation for the guiding center distribution function $f̂e(X,Y,z,ρ,ϕ,vz,t)$

Based on the small amount of variation of the wave and plasma column in the $ŷ$ direction, we assume $f̂e$ is independent of Y. Also, since we are only concerned with the effect of the Alfvén wave on electron motion parallel to $B0$⁠, we eliminate the dependence on ρ. Integrating Eq. (7) over Y and ρ produces the linearized gyroaveraged Boltzmann equation for the reduced parallel electron distribution $ĝe(X,z,vz,t)=∫dY∫ρdρf̂e(X,Y,z,ρ,vz,t)$⁠. Like $f̂e, ĝe$ is divided into a static background and a small linear perturbation $ĝe(X,z,vz,t)≈ĝe0(vz)+ĝe1(X,z,vz,t)$⁠. The X dependence is maintained since the Alfvén wave has structure in the $x̂$ direction.

An additional simplification is possible since the wave launched by the ASW antenna is a sinusoidal burst of 20 cycles at a single frequency ω, and only data from within the burst is analyzed. Because of this, we can assume the time evolution of the linear perturbation to the electron distribution is periodic. This allows us to write $ĝe1(X,z,vz,t)=ge1(X,z,vz,ω)e−iωt$⁠. With this assumption and after integration over Y and ρ, Eq. (7) becomes

where $ĝe0$ has been written as $ge0$ for notational simplicity.

The maximum difference between an electron's instantaneous x position and its guiding center coordinate X is the electron cyclotron radius. The fastest 100 eV electrons analyzed in our data have a cyclotron radius of 1.2 mm, which is smaller than the accuracy of our probe placement and much smaller than the 5.1 cm structure of the Alfvén wave in $x̂$⁠. Consequently, electron behavior on the length scale of the cyclotron radius cannot be detected by our measurements, and such fine resolution is not necessary to resolve variations of electron behavior on the scale of the Alfvén wave in $x̂$⁠. To the level of accuracy achieved in our experiments, it is reasonable to assume that $ge1(X,z,vz,ω)≈ge1(x,z,vz,ω)$⁠. The coordinates x and z should be interpreted as the position in laboratory coordinates where measurements of the electron distribution are performed using the WWAD. The three-dimensional $(x,z,vz)$ linear correction $ge1(x,z,vz,ω)$ is often written as $ge1(vz)$ for compactness, and ω is suppressed when possible in our notation since only a single Alfvén wave frequency is used.

Before solutions for Eq. (8) can be found, the collision term needs to be specified. The simplest collision model is the velocity-dependent Krook operator.¹⁶ This operator was the basis for the collisionally-modified Alfvén wave properties in Section III B.¹⁷ This collision term restores $ge(vz)$ to the background $ge0(vz)$ in the time $τc(vz)$ required for electrons with velocity v_z to experience a collision. Using this concept, the time rate of change is $(dge/dt)coll=−(ge−ge0)/τc$⁠. Deviations from the background distribution are $ge(vz)−ge0(vz)≈ge1(vz)$⁠, and the collision time is the inverse of the Coulomb collision rate $τc(vz)=1/ν(vz)$⁠. The Coulomb collision rate is

where ϵ₀ is the permittivity of free space and N_D is the number of particles in a sphere with the Debye radius. The velocity-dependent collision operator used here is formed by the product of $ge1(vz)$ and the collision frequency

The minus sign indicates that collisions counteract departures from $ge0(vz)$⁠. Since faster electrons have fewer Coulomb collisions, the Krook collision operator is weaker at higher velocities. The velocity-dependent collision rate ν is distinguished in this paper from the thermal collision rate ν_te.

Since electrons in this experiment are collisional, the relevant solution for their kinetic behavior is altered from the kinetic response of collisionless magnetospheric electrons. This is particularly true for thermal electrons in our experiment; since $νte/ω=13$⁠, a 2 eV thermal electron will undergo on average 13 collisions during a single Alfvén wave period. However, the suprathermal electrons measured by the WWAD are significantly less collisional. The lowest energy 15 eV electrons measured by the WWAD (discussed in Section V) have a significantly reduced collision rate so that $ν/ω=0.72$⁠. For 30 eV electrons, $ν/ω=0.26$⁠. These lower collision rates for fast electrons produce modest changes to the suprathermal portion of the solution for $ge1(vz)$ as compared to the Vlasov solution where ν = 0. Consequently, the behavior of suprathermal electrons measured and modeled here is still relevant to the auroral magnetosphere.

While there is one complete solution for $ge1(vz)$⁠, we have chosen to express our particular solution in three parts. This is done for convenience, and to highlight different contributions to the solution. Our solution is the sum

where $ge1f(vz)$ reproduces the parallel current expected from two fluid theory, $ge1k(vz)$ is a kinetic correction, and $ge1h(vz)$ is a homogeneous solution that includes non-Alfvénic effects of the ASW antenna.

One requirement we impose on our particular solution is that it should be consistent with the fluid properties of the Alfvén wave. Consequently $ge1(vz)$ must include the parallel current of the Alfvén wave required by two fluid theory²² $Jz=inee2Ez/[me(ω+iνte)]$⁠. To this end, we specify the term $ge1f(vz)$ to be included in our solution which has the form

This is referred to as the fluid term, since its first moment produces the fluid current J_z of the Alfvén wave in the limit that the Coulomb collision rate ν is replaced with the thermal collision rate ν_te. Although $ge1f(vz)$ by itself does not satisfy the linearized Boltzmann equation (Eq. (8)), we are free to include it as a part of the solution.

An additional term $ge1k(vz)$ is developed so that the combination $ge1f(vz)+ge1k(vz)$ is a solution to the linearized Boltzmann equation. Since this term bridges fluid and linear kinetic theory, $ge1k(vz)$ is referred to as the kinetic correction. Using Eq. (12) and inserting the combination $ge1f(vz)+ge1k(vz)$ into Eq. (8) gives

This equation is solved by direct integration from the boundary z₀ to the location of measurements z.

Integration of Eq. (13) from the boundary of the experiment z₀ to the measurement location z gives

Combining Eqs. (12) and (14) gives

The location of the boundary z₀ depends on the sign of v_z. For electrons with $vz>0$⁠, traveling left-to-right in Fig. 1(a), the boundary z₀ is taken to be the ASW antenna. This placement of the boundary assumes electrons with $vz>0$ are unperturbed in the region z < 0. This assumption is reasonable since the density is significantly lower in this region due to the shadow cast by the ASW antenna, and this likely diminishes the coupling of the Alfvén wave to the plasma behind the antenna. For electrons with $vz<0$⁠, traveling right-to-left in Fig. 1(a), the boundary z₀ is the cathode.

The idealized form of E_z from Eq. (5) has been written explicitly in the last line of Eqs. (14) and (15) to emphasize that the perpendicular structure of E_z, modeled as $cos(k⊥x)$⁠, is present in $ge1(vz)$ as well. If only the first term in the square brackets of Eq. (15) is kept, the solution reduces to the infinite space Fourier transform solution of Eq. (8). The second term in brackets in Eq. (15) corrects the infinite space solution to account for the finite interaction length $z−z0$ of electrons with the Alfvén wave.

The electrostatic design of the ASW antenna produces currents in the plasma by collecting charged particles. This design allows the antenna to fit within the accessibility constraints of the experiment while still exciting Alfvén waves with a tunable $k⊥$ spectrum. Because of its electrostatic design, it is expected that the ASW does not couple exclusively to Alfvén waves, and consequently some of the current injected into the plasma is not converted to the Alfvén wave mode. The result is that there is some non-Alfvénic perturbation to $ge(vz)$ driven at the same frequency as the Alfvén wave. A simple model of these additional kinetic effects is developed here, and the efficacy of the model is examined in the analysis of Section VI.

The homogeneous solution can be used to account for non-Alfvénic effects of the ASW antenna. To model these effects, we consider the perturbation of the ASW antenna in general terms. The oscillating voltage on the antenna grids will cause the voltage across the surrounding sheath to oscillate as well. Electrons traversing the oscillating sheath voltage will be perturbed and propagate ballistically to the point where $ge(vz)$ is measured. A homogeneous solution to Eq. (8) is

The cosine pattern in $x̂$ is imposed by the antenna, so this structure is included in the homogeneous solution as well. The function $h(vz)$ describes the non-Alfvénic perturbation to $ge(vz)$ at the location of the boundary z₀. This boundary condition $h(vz)$ translates to the measurement location z with a ballistic phase shift $ω(z−z0)/vz$ that accounts for the time of flight between z₀ and z.

To find $h(vz)$⁠, we approximate the effect of the ASW antenna. Because the period of voltage oscillations on the ASW antenna is much longer than the amount of time required for a suprathermal electron to pass by the antenna, the oscillating voltage appears approximately static in the reference frame of a passing suprathermal electron. The velocity change $δvz$ of an electron traversing a static voltage is

where V_s is the voltage oscillation across the antenna sheath. Since the sheath region surrounding the ASW antenna is not diagnosed, the amplitude of V_s is not known. Additionally, it is not known if V_s is in phase with the Alfvén wave, so $Vs=Vs0eiϕs$ is allowed to be complex, with an amplitude $Vs0$ and phase $ϕs$⁠. The perturbation to the electron motion $δvz$ alters the distribution function $ge(vz)$ at the antenna, and the change can be approximated as $h(vz)=δvz(∂ge0/∂vz)$⁠. In more complete terms, $h(vz)$ is

where $Vs0$ and $ϕs$ are the unknown amplitude and phase of sheath voltage oscillations. Finally, the homogeneous solution can be expressed as

The full solution $ge1(vz)=ge1f(vz)+ge1k(vz)+ge1h(vz)$ is the sum of Eqs. (15) and (19). Several features are worth identifying. First, the denominator of the Alfvénic contribution $ge1f(vz)+ge1k(vz)$ in Eq. (15) approaches resonance in the direction of wave propagation. Second, the homogeneous solution $ge1h(vz)$ in Eq. (19) has two unknown constants $Vs0$ and $ϕs$⁠. These two constants are the only free parameters in the full solution $ge1(vz)$⁠. Third, the entire solution varies like $cos(k⊥x)$⁠. These features will be revisited in Section VI when this model is compared with measurements of $ge1(vz)$⁠.

The suprathermal portion of the reduced parallel electron distribution function $ge(vz)$ is measured by the Whistler Wave Absorption Diagnostic (WWAD). A series of data sets was recorded using the diagnostic at a single location, and a second series was collected at multiple locations to provide a scan across the perpendicular Alfvén wavelength in $x̂$⁠.

This diagnostic operates by sending a probe wave through a short length of plasma, and the measured damping of the wave is used to calculate $ge(vz)$⁠. A schematic, shown in Fig. 3, includes the addition of power detectors since the diagnostic's original description.¹² The electron cyclotron absorption technique used here can be performed using any wave mode with an electron cyclotron resonance. This technique was first implemented by Kirkwood.²³ Previous electron cyclotron absorption measurements have used X-mode waves on MIT's Versator II to determine the fraction of the parallel current drive carried by suprathermal electrons.^24,25

Since the electron plasma frequency f_pe = 9.0 GHz is greater than the electron cyclotron frequency $|fce|=5.0$ GHz, the LAPD plasma is overdense $fpe>|fce|$⁠. The electron cyclotron frequency, $fce=−eB0/2πme$⁠, is negative as defined here since it includes the sign of the electron charge. In overdense plasmas, only the right-hand circularly polarized whistler mode wave propagates at frequencies just below $|fce|$⁠. Because of this, wave data at these frequencies can be unambiguously interpreted as whistler mode waves. Additionally, the whistler wave has an electron cyclotron resonance. For these reasons, we have chosen to use a small-amplitude whistler wave to probe the reduced parallel electron distribution $ge(vz)$⁠. The whistler wave is transmitted between two 1″ dipole antennas immersed in the plasma and separated by L = 32 cm. These antennas are shown in Fig. 1(a) as W1 and W2. Because of the overdense plasma conditions, these antennas must be inserted into the plasma to excite whistler waves. The whistler antennas are designed with a minimal profile, and our whistler wave propagation model suggests that the antennas make only a $∼1$% localized depression in the background plasma density between the probes.

The whistler wave is damped by Doppler-shifted electron cyclotron resonance. The resonance condition is

where v_z is the resonant electron velocity, f_w is the whistler wave frequency, and k_wzr is the real component of the parallel whistler mode wavenumber. The parallel wavenumber is generally complex since damping is included as a function of z. The underlying physical process of this diagnostic is that for a whistler wave at frequency f_w, electrons present that have the corresponding velocity v_z given by Eq. (20) are resonant with the wave and absorb energy from the wave. When there are more resonant electrons, the damping is more severe. Therefore, by measuring the damping of the whistler wave, we can determine how many resonant electrons are present. Because $fw<|fce|$⁠, one of the implications of Eq. (20) is that measurements of resonant electrons with $vz>0$ are made using whistler waves traveling in the opposite direction, k_wzr < 0. For $vz<0$ measurements, a second data set is required where the direction of whistler propagation is swapped, $kwzr>0$⁠, using the switch shown in Fig. 3.

The WWAD measures the damping of the whistler wave by outputting a signal proportional to the transmission fraction $AR/AT$⁠, where A_T is the transmitted wave amplitude, and A_R is the received wave amplitude. Using the assumption that the wave damps exponentially like $AR/AT=e−kwziL$⁠, and since $AR/AT$ is measured and the probe separation L is known, we can experimentally determine the imaginary part of the parallel whistler wavenumber k_wzi. As discussed by Thuecks, Skiff, and Kletzing¹² and Stix,²⁶ the warm plasma relationship between k_wzi and $ge(vz)$ is

Since every quantity on the right hand side of this equation is measured or known, $ge(vz)$ can be calculated from experimentally known quantities. These known quantities include: the electron plasma frequency f_pe calculated from the electron density n_e; the whistler driving frequency f_w known from WWAD calibrations; the imaginary part of the parallel whistler wavenumber k_wzi measured by the WWAD; and the real part of the parallel whistler wavenumber k_wzr calculated using warm plasma theory and verified by time-of-flight measurements of the whistler wave.

By scanning $0.75<fw/|fce|<0.95, AR/AT$ is recorded for a range of resonant velocities $3.5<|vz/vte|<9.5$⁠. Because there are so many lower velocity bulk electrons, whistler waves resonant with these electrons are completely absorbed. Consequently, the bulk of $ge(vz)$ is inaccessible to this diagnostic. However, since we are primarily interested in electron dynamics near $vz≈vA$⁠, and since $vA≈6.7vte$ for this experiment, the WWAD is able to measure the necessary portion of $ge(vz)$⁠.

Fig. 4 shows sample WWAD data from an ensemble average of 1024 shots. During a 10 μs window in the shot sequence, the whistler wave frequency is down-chirped. According to the resonance condition in Eq. (20), the resonant velocity (shown in energy units) increases as the whistler wave frequency decreases. The transmission fraction $AR/AT$ is shown for the same window of time. At later times $AR/AT$ approaches unity since there are fewer resonant electrons at higher resonant velocities. Using $AR/AT, ge(vz)$ is calculated using Eq. (21) and plotted against resonant electron energy. As shown here, a sweep of one side of the suprathermal distribution $ge(vz)$ takes 10 μs. Measurements of the opposite half of the distribution function are obtained in a subsequent data set by transmitting the whistler wave in the opposite direction.

In order to test the linear theory, WWAD measurements must resolve changes to $ge(vz)$ as the Alfvén wave cycles through $2π$ of phase. The most direct approach would be to scan $ge(vz)$ several times during the 8 μs period of the Alfvén wave. However, $ge(vz)$ scans require 10 μs, and a faster scan would increase the frequency of the WWAD output beyond the Nyquist frequency of the available digitizers.¹² Unfortunately, this direct approach is not possible for our experimental setup. Multiple phase-shifted data sets are used to overcome this constraint. Consider a single moment in the shot sequence where the WWAD measures $ge(vz)$ at a particular velocity v_z and the Alfvén wave phase is $ϕ1$⁠. In the same moment of a subsequent shot, where only the phase of the Alfvén wave has been adjusted by shifting the time of the ASW antenna trigger, $ge(vz)$ is recorded at the same velocity v_z and the Alfvén wave is at a different phase $ϕ2$⁠. Using these two measurements, $ge(vz)$ at the velocity v_z is now known for two different Alfvén wave phases. This technique was extended to 64 phase shifted data sets, each of 1024 shots, so $ge(vz)$ measurements span Alfvén wave phase in intervals of $2π/64$⁠. The two halves $ge(vz>0)$ and $ge(vz<0)$ were measured separately, each in a series of 64 data sets.

The distribution function was measured at a fixed location in space to determine the amplitude and phase of oscillations $ge1(vz)$ as an inertial Alfvén wave propagated past the point of $ge(vz)$ measurements. The location of these measurements was $x=−2.1$ cm and y = 0 cm. The location is also indicated by the black “X” in Figs. 2(a) and 2(b). This location was chosen to be an antinode of E_z and consequently the position where $ge1(vz)$ is expected to be maximum.

Measurements from this location are shown in Fig. 5(a). Values of $ge(vz)$ are binned into a two-dimensional grid by resonant velocity and the phase of the Alfvén wave's parallel electric field E_z. On the horizontal axis, electron velocity v_z is converted to energy units, $E=mevz2/2$⁠, but the sign of v_z is maintained so that the two halves of the distribution function can be distinguished. Consider cuts along each axis separately. A cut parallel to the horizontal axis shows how $ge(vz)$ varies with electron velocity for a given phase with respect to the Alfvén wave. A cut parallel to the vertical axis shows for a fixed v_z how $ge(vz)$ changes as the Alfvén wave advances through $2π$ of phase. The vertical black dashed lines at +15 eV and −20 eV indicate the low energy limit of $ge(vz)$ measurements. At energies closer to zero, there are many more electrons available to resonate with the whistler wave, and consequently the whistler signal is completely absorbed. The difference between these cutoff energies is consistent with an asymmetric distribution function believed to be caused by the cathode-anode source that produces fast electrons with $vz<0$⁠.

As expected, the dominant trend in the measured $ge(vz)$ is the background distribution $ge0(vz)$⁠, which can be seen in Fig. 5(a) as the decreasing number of electrons as one looks left or right from zero on the energy axis. The background $ge0(vz)$ can be separated from this full set of $ge(vz)$ measurements by averaging over $ϕEz$⁠. Since perturbations to $ge(vz)$ caused by the Alfvén wave are periodic in $ϕEz$⁠, averaging over $ϕEz$ removes these perturbations and produces $ge0(vz)$⁠. Fig. 5(b) shows $δge(vz)=ge(vz)−ge0(vz)$ and is the basis for one of the key experimental results of this paper. The essential feature of this plot is the structure along the vertical axis that is periodic over $2π$ of $ϕEz$⁠. This is the first evidence that the data have captured one of the primary distinctions between ideal MHD and inertial Alfvén waves: electrons oscillating in the parallel electric field E_z of the inertial Alfvén wave.

Measurements at multiple locations are used to search for spatial variations in the strength of $ge(vz)$ oscillations. The data shown in Fig. 6 were collected at eight equally spaced locations in $x̂$ spanning one perpendicular wavelength of the Alfvén wave $λx=5.1$ cm. The range of the scan is shown by the black line in Figs. 2(a) and 2(b). Due to time constraints of the experimental run, for this spatial scan only eight phase-shifted data sets were collected at each location, and only $ge(vz>0)$ was measured. Similar to Fig. 5(b), the average trend along the energy axis $ge0(vz)$ has been removed to show smaller variations $δge(vz)$⁠. This figure should be read left to right, starting with the top row. In the first panel at $x=−1.50$ cm, $δge(vz)$ is periodic across $ϕEz$⁠. This is the same feature seen in Fig. 5(b). The periodic structure is diminished at $x=−0.86$ cm and x = 1.70 cm. These positions are the approximate locations of $ge1(vz)$ nodes. The positions of these measurements have an uncertainty of ±0.1 cm.

Additional insight comes from the phase variation of $δge(vz)$ across each node. This feature can be seen in the panels on either side of the node at $x=−0.86$ cm. To the left of the node, at $x=−1.50$ cm, $δge(vz)$ is at maximum near an electron energy of approximately 12 eV and wave phase of $ϕEz≈3π/4$ radians. On the other side of the node, at $x=−0.22$ cm, $δge(vz)$ is at minimum for the same values of electron energy and $ϕEz$⁠. The minimum at these energy-phase coordinates becomes a maximum again after crossing the node at x = 1.70 cm. The even spacing of two nodes across one perpendicular Alfvén wavelength and the sign change of $δge(vz)$ across each node further indicates Alfvénic modifications of $ge(vz)$⁠.

Here we discuss the analysis used to isolate the measured oscillations of $ge(vz)$ at the Alfvén wave frequency.

A discrete Fourier transform (DFT) is applied to the $ϕEz$ dimension of the single position measurements shown in Fig. 5(a). The first Fourier mode is, by definition, the portion of $ge(vz)$ that is once-periodic across the range of this dimension $ϕEz=0$ to $ϕEz=2π$⁠. In other words, this Fourier mode contains the oscillations of $ge(vz)$ at the Alfvén wave frequency.

An example of this process is shown in Fig. 7. A vertical slice of data, indicated in part (a) of the figure, is taken at a fixed energy and replotted in part (b). A DFT is applied to the data in part (b), giving the Fourier amplitudes shown in part (c). The zero frequency mode is the average across the phase axis which is the experimental value of $ge0(vz)$ for this slice of data with velocity v_z. The first mode is the experimental $ge1(vz)$ for this slice of data. We interpret the signal power in higher Fourier components as the amplitude of random fluctuations and systematic uncertainties. The total power in these higher-frequency components is used to calculate the uncertainty of the first Fourier mode. This process is applied to the data at each velocity and the Fourier amplitudes are plotted in part (d) of Fig. 7. Given the resolution in $ϕEz$ of these measurements, this analysis produces Fourier modes up to $ge32$ or 32 times the Alfvén wave frequency. However, only modes $ge0$ through $ge5$ are shown here. Modes 2–5 are representative of the signal amplitude in the higher frequency modes.

The phase of $ge1(vz)$ relative to E_z can also be extracted from the DFT results. A complex DFT output indicates a phase shift between the input signal and the Fourier kernel. In this case, the Fourier kernel is based on the phase of E_z, so a complex output indicates a phase shift between $ge1(vz)$ and E_z. This shift is found by taking the phase angle of $ge1(vz)$ from the DFT output. For the convention used here, a positive phase shift less than π radians indicates that the peak of $ge1(vz)$ is delayed in time with respect to the peak of E_z.