The physics of the aurora is one of the foremost unsolved problems of space physics. The mechanisms responsible for accelerating electrons that precipitate onto the ionosphere are not fully understood. For more than three decades, particle interactions with inertial Alfvén waves have been proposed as a possible means for accelerating electrons and generating auroras. Inertial Alfvén waves have an electric field aligned with the background magnetic field that is expected to cause electron oscillations as well as electron acceleration. Due to the limitations of spacecraft conjunction studies and other multi-spacecraft approaches, it is unlikely that it will ever be possible, through spacecraft observations alone, to confirm definitively these fundamental properties of the inertial Alfvén wave by making simultaneous measurements of both the perturbed electron distribution function and the Alfvén wave responsible for the perturbations. In this laboratory experiment, the suprathermal tails of the reduced electron distribution function parallel to the mean magnetic field are measured with high precision as inertial Alfvén waves simultaneously propagate through the plasma. The results of this experiment identify, for the first time, the oscillations of suprathermal electrons associated with an inertial Alfvén wave. Despite complications due to boundary conditions and the finite size of the experiment, a linear model is produced that replicates the measured response of the electron distribution function. These results verify one of the fundamental properties of the inertial Alfvén wave, and they are also a prerequisite for future attempts to measure the acceleration of electrons by inertial Alfvén waves.

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 RE 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 RE and the FAST spacecraft at 1.05–1.65 RE 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, ωpe 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 wave2,14 at kδe1, where k is the perpendicular Alfvén wavenumber. Parallel electron inertia leads to a dispersive nature for the inertial Alfvén wave governed by

ωkz=±vA1+(kδe)2(1+iνte/ω),
(1)

where νte 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+kx̂.

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

Ẽz=kzδekδe(1+iνte/ω)1+(kδe)2(1+iνte/ω)Ẽx,
(2)

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 r3RE, 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 Ez 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, Ex, and Ey.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 ne and temperature Te.

FIG. 1.

Experimental setup in the LAPD. (a) Elsasser probes E1 and E2 make two dimensional measurements of the perpendicular wave fields B and E in the x-y plane. Whistler probes W1 and W2 are used to measure the electron distribution function parallel to the background magnetic field B0. The swept Langmuir probe, denoted as L above, is used to scan the density and temperature profile. (b) The ASW antenna launches Alfvén waves that travel nearly parallel to B0. The amplitude of the oscillating voltage applied to each grid can be adjusted independently to produce well-defined structure in x̂ that determines k of the Alfvén wave.

FIG. 1.

Experimental setup in the LAPD. (a) Elsasser probes E1 and E2 make two dimensional measurements of the perpendicular wave fields B and E in the x-y plane. Whistler probes W1 and W2 are used to measure the electron distribution function parallel to the background magnetic field B0. The swept Langmuir probe, denoted as L above, is used to scan the density and temperature profile. (b) The ASW antenna launches Alfvén waves that travel nearly parallel to B0. The amplitude of the oscillating voltage applied to each grid can be adjusted independently to produce well-defined structure in x̂ that determines k of the Alfvén wave.

Close modal

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=2×105<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, kδe=0.66.

FIG. 2.

A snapshot in time of the Alfvén wave fields in the x-y plane. (a) Measurements from Elsässer probe E1 show B of the Alfvén wave launched by the ASW antenna. Arrows indicate the magnitude and direction of B; color shows the magnitude of By. The magnetic field is polarized almost exclusively in the ŷ direction. (b) Ez is calculated from the measured perpendicular wave fields. The black X and line in both plots correspond to the location of fixed position and radial scan WWAD measurements described in Section V.

FIG. 2.

A snapshot in time of the Alfvén wave fields in the x-y plane. (a) Measurements from Elsässer probe E1 show B of the Alfvén wave launched by the ASW antenna. Arrows indicate the magnitude and direction of B; color shows the magnitude of By. The magnetic field is polarized almost exclusively in the ŷ direction. (b) Ez is calculated from the measured perpendicular wave fields. The black X and line in both plots correspond to the location of fixed position and radial scan WWAD measurements described in Section V.

Close modal

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

ẼxB̃y=vA1+k2δe2(1+iνte/ω).
(3)

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/Ex0.003 for this experiment, it is not possible to measure Ez directly. Instead, Ez is calculated from measurements of the perpendicular wave fields. Combining Eq. (2) with Faraday's law gives

Ẽz=(ω+iνte)δe2(kxB̃ykyB̃x).
(4)

Since the Alfvén wave launched by the ASW antenna is polarized so that ByBx, 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 Ez is shown in Fig. 2(b). To facilitate the development of theory in Section IV, the Ez waveform in Fig. 2(b) is modeled as a standing wave in x̂ and a propagating wave in ẑ

Ez=Ez0ei(kzzωt)cos(kx).
(5)

The most notable difference between B and Ez in Fig. 2 is the quarter wavelength shift in x̂ so that the nodes of B are located where there are antinodes of Ez. Since Ez overlaps spatially with the parallel current Jz, the physical significance of this shift between Ez 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

fe1t+v·x(fe1)eme(E1+v×B1)·v(fe0)eme(v×B0)·v(fe1)=(dfedt)coll,
(6)

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 (vx, vy) of the electron by X=x+vy/Ωce and Y=yvx/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=tan1(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 ωΩce, 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)

f̂e1t+vzf̂e1zemeEzf̂e0vz=(df̂edt)coll.
(7)

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)=dYρdρf̂e(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)eiωt. With this assumption and after integration over Y and ρ, Eq. (7) becomes

iωge1+vzge1zemeEzge0vz=(dgedt)coll,
(8)

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 vz to experience a collision. Using this concept, the time rate of change is (dge/dt)coll=(gege0)/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

ν(vz)=nee44πϵ02me2vz3lnND,
(9)

where ϵ0 is the permittivity of free space and ND 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

(dgedt)coll=ge1ν.
(10)

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 νte.

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

ge1=ge1f+ge1k+ge1h,
(11)

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 theory22Jz=inee2Ez/[me(ω+iνte)]. To this end, we specify the term ge1f(vz) to be included in our solution which has the form

ge1f=ieme(ω+iν)Ez(x,z)ge0vz.
(12)

This is referred to as the fluid term, since its first moment produces the fluid current Jz of the Alfvén wave in the limit that the Coulomb collision rate ν is replaced with the thermal collision rate νte. 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

i(ω+iν)ge1k+vzge1kzekzvzme(ω+iν)Ezge0vz=0.
(13)

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

ge1k=iekzvzme(ω+iν)(ω+iνkzvz)ge0vz×[1ei(ω+iνkzvz)(zz0)/vz]×Ez0eikzzcos(kx).
(14)

Combining Eqs. (12) and (14) gives

ge1f+ge1k=ieme(ω+iνkzvz)ge0vz×[1kzvzω+iνei(ω+iνkzvz)(zz0)/vz]×Ez0eikzzcos(kx).
(15)

The location of the boundary z0 depends on the sign of vz. 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 Ez from Eq. (5) has been written explicitly in the last line of Eqs. (14) and (15) to emphasize that the perpendicular structure of Ez, modeled as cos(kx), 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 zz0 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

ge1h=h(vz)ei(ω+iν)(zz0)/vzcos(kx).
(16)

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(zz0)/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

δvz=eVsmevz,
(17)

where Vs is the voltage oscillation across the antenna sheath. Since the sheath region surrounding the ASW antenna is not diagnosed, the amplitude of Vs is not known. Additionally, it is not known if Vs 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

h(vz)=emevzVs0eiϕsge0vz,
(18)

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

ge1h=eVs0mevzge0vzei(ω+iν)(zz0)/vz+iϕscos(kx).
(19)

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(kx). 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

FIG. 3.

The WWAD is used to measure the suprathermal portion of ge(vz) with high precision. The diagnostic is a chirped microwave interferometer. The voltage controlled oscillator (VCO) drives whistler mode waves that are transmitted through L = 32 cm of plasma. The WWAD output is used to calculate ge(vz).

FIG. 3.

The WWAD is used to measure the suprathermal portion of ge(vz) with high precision. The diagnostic is a chirped microwave interferometer. The voltage controlled oscillator (VCO) drives whistler mode waves that are transmitted through L = 32 cm of plasma. The WWAD output is used to calculate ge(vz).

Close modal

Since the electron plasma frequency fpe = 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

vz=2π(fw|fce|)/kwzr,
(20)

where vz is the resonant electron velocity, fw is the whistler wave frequency, and kwzr 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 fw, electrons present that have the corresponding velocity vz 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, kwzr < 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 AT is the transmitted wave amplitude, and AR is the received wave amplitude. Using the assumption that the wave damps exponentially like AR/AT=ekwziL, 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 kwzi. As discussed by Thuecks, Skiff, and Kletzing12 and Stix,26 the warm plasma relationship between kwzi and ge(vz) is

ge(vz)=c2kwzr24π4fpe2fwkwzi.
(21)

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 fpe calculated from the electron density ne; the whistler driving frequency fw known from WWAD calibrations; the imaginary part of the parallel whistler wavenumber kwzi measured by the WWAD; and the real part of the parallel whistler wavenumber kwzr 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 vzvA, and since vA6.7vte 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.

FIG. 4.

Overview of WWAD data collection. The decreasing whistler wave frequency, shown on the top left, causes the resonant electron velocity to increase. Resonant velocity is shown here in energy units. During this same window of time, the whistler wave transmission fraction AR/AT is recorded by the WWAD mixer output. The transmission fraction is used to calculate ge(vz), which is plotted against resonant electron energy on the right.

FIG. 4.

Overview of WWAD data collection. The decreasing whistler wave frequency, shown on the top left, causes the resonant electron velocity to increase. Resonant velocity is shown here in energy units. During this same window of time, the whistler wave transmission fraction AR/AT is recorded by the WWAD mixer output. The transmission fraction is used to calculate ge(vz), which is plotted against resonant electron energy on the right.

Close modal

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 vz 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 vz and the Alfvén wave is at a different phase ϕ2. Using these two measurements, ge(vz) at the velocity vz 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 Ez 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 Ez. On the horizontal axis, electron velocity vz is converted to energy units, E=mevz2/2, but the sign of vz 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 vz 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.

FIG. 5.

Measurements of ge(vz) were made using the WWAD at a single position in the perpendicular x-y plane. (a) This composite measurement of ge(vz) is generated by binning 64 phase-shifted data sets, and the result resolves ge(vz) in electron energy and Alfvén wave phase. Vertical black dashed lines indicate the low-energy limit of the diagnostic, below which the whistler signal is completely absorbed by resonant electrons. (b) The background distribution ge0(vz) is removed to show perturbations δge(vz)=ge(vz)ge0(vz). These perturbations contain a periodic structure along the vertical axis that matches the frequency of the Alfvén wave, which indicates that ge(vz) is modified by the Alfvén wave.

FIG. 5.

Measurements of ge(vz) were made using the WWAD at a single position in the perpendicular x-y plane. (a) This composite measurement of ge(vz) is generated by binning 64 phase-shifted data sets, and the result resolves ge(vz) in electron energy and Alfvén wave phase. Vertical black dashed lines indicate the low-energy limit of the diagnostic, below which the whistler signal is completely absorbed by resonant electrons. (b) The background distribution ge0(vz) is removed to show perturbations δge(vz)=ge(vz)ge0(vz). These perturbations contain a periodic structure along the vertical axis that matches the frequency of the Alfvén wave, which indicates that ge(vz) is modified by the Alfvén wave.

Close modal

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 Ez 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.

FIG. 6.

Measurements of ge(vz) were repeated at eight positions in x̂ spanning one perpendicular Alfvén wavelength. The background distribution ge0(vz) has been removed to show δge(vz). At two locations, x=0.86 cm and x = 1.70 cm, δge(vz) is significantly diminished, and these are the approximate locations of nodes in the perpendicular structure of ge1(vz). These regularly-spaced nodes indicate periodic structure of ge1(vz) in x̂ with a scale size matching the perpendicular Alfvén wavelength.

FIG. 6.

Measurements of ge(vz) were repeated at eight positions in x̂ spanning one perpendicular Alfvén wavelength. The background distribution ge0(vz) has been removed to show δge(vz). At two locations, x=0.86 cm and x = 1.70 cm, δge(vz) is significantly diminished, and these are the approximate locations of nodes in the perpendicular structure of ge1(vz). These regularly-spaced nodes indicate periodic structure of ge1(vz) in x̂ with a scale size matching the perpendicular Alfvén wavelength.

Close modal

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 ϕEz3π/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 vz. 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.

FIG. 7.

A DFT is applied over ϕEz to isolate oscillations of ge(vz) at the Alfvén wave frequency. (a) To demonstrate this technique, measurements at a fixed energy E = 29 eV are taken from the full data set. (b) The slice of data at fixed energy is replotted. The fundamental mode over ϕEz seen here is the measured ge1(vz) at this energy. (c) The measured ge1(vz) is isolated using a DFT over ϕEz. The zero frequency mode (average) is the measured ge0(vz), and the first order mode is ge1(vz). Power in higher frequencies is the result of random and systematic errors. (d) This technique is extended to data at each energy to isolate ge0(vz) and ge1(vz). Higher frequency modes generally have lower amplitude than ge1(vz) and are interpreted as the amplitude of random fluctuations and systematic uncertainties.

FIG. 7.

A DFT is applied over ϕEz to isolate oscillations of ge(vz) at the Alfvén wave frequency. (a) To demonstrate this technique, measurements at a fixed energy E = 29 eV are taken from the full data set. (b) The slice of data at fixed energy is replotted. The fundamental mode over ϕEz seen here is the measured ge1(vz) at this energy. (c) The measured ge1(vz) is isolated using a DFT over ϕEz. The zero frequency mode (average) is the measured ge0(vz), and the first order mode is ge1(vz). Power in higher frequencies is the result of random and systematic errors. (d) This technique is extended to data at each energy to isolate ge0(vz) and ge1(vz). Higher frequency modes generally have lower amplitude than ge1(vz) and are interpreted as the amplitude of random fluctuations and systematic uncertainties.

Close modal

The phase of ge1(vz) relative to Ez 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 Ez, so a complex output indicates a phase shift between ge1(vz) and Ez. 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 Ez.

The measured ge0(vz), extracted by applying a DFT to the data of Fig. 5(a), is shown in Fig. 8. The asymmetry between vz<0 and vz>0 is the result of fast primary electrons from the cathode source with vz<0. Two-temperature distributions are used to fit the data. For ge0(vz>0), the best-fitting combination is 98.6% of a 2.5 eV Maxwellian and 1.4% of a 17 eV Maxwellian. For ge0(vz<0), the result is dramatically different corresponding to the asymmetry of Fig. 8. The best-fitting ge0(vz<0) contains 54% of a 21 eV Maxwellian, and 46% of a 132 eV Maxwellian. Since measurements only span the suprathermal tails, these parameters do not necessarily pertain to the bulk plasma. The derivatives of these best-fit functions ge0/vz are used to evaluate the model ge1(vz) in Eqs. (15) and (19).

FIG. 8.

The background distribution function ge0(vz) is extracted from the full set of ge(vz) measurements in Fig. 5(a). Smooth functions are fitted to the measurements, and the derivaties of these functions are used to evaluate the model ge1(vz). The asymmetry seen here is a result of primary electrons from the cathode with vz<0. The uncertainty of measurements is shown in gray.

FIG. 8.

The background distribution function ge0(vz) is extracted from the full set of ge(vz) measurements in Fig. 5(a). Smooth functions are fitted to the measurements, and the derivaties of these functions are used to evaluate the model ge1(vz). The asymmetry seen here is a result of primary electrons from the cathode with vz<0. The uncertainty of measurements is shown in gray.

Close modal

Using a DFT over ϕEz to isolate the measured ge1(vz), we are able to compare theoretical and experimental results. For this comparison, we only use single-location measurements of Fig. 5 to examine the amplitude and relative phase of oscillations. Evaluating the model ge1(vz) formed by the sum of Eqs. (15) and (19) requires several quantities. The values for Ez, k, kz, and ω are all based on Elsässer probe measurements of the Alfvén wave. The derivative ge0/vz comes from the smooth functions fitted to the measured ge0(vz) seen in Fig. 8. For vz>0, the boundary z0 is the location of the ASW antenna. For vz<0z0, is the location of the cathode. The two free parameters Vs0 and ϕs of the homogeneous solution (Eq. (19)) are determined using least squares minimization. For vz>0,Vs0=0.84 V and ϕs=0.27 radians. For vz<0, Vs0=0.97 V and ϕs=2.53 radians. A non-zero value of Vs0 for vz<0 indicates the cathode-anode voltage is modulated at the frequency of the Alfvén wave, a process that is not well understood. The free parameters Vs0 and ϕs are scalar values that do not depend on vz. The form of the homogeneous solution as a function of vz is not adjusted in the least squares determination of these quantities.

Fig. 9 shows the direct, quantitative comparison of the measured and modeled ge1(vz). The amplitude of each data point in part (a) of this figure shows how intensely that portion of the reduced parallel electron distribution function oscillates during the inertial Alfvén wave pulse. The phase of the data points in part (b) shows whether the oscillations ge1(vz) lead or lag Ez of the Alfvén wave. For the phase convention used here, a positive phase shift less than π radians relative to ϕEz indicates the oscillations of ge(vz) lag Ez. The measured ge1(vz) lags Ez. Overall, the model agrees with measurements within the error bars. This is believed to be the first demonstration of quantitative agreement between measured oscillations of the suprathermal parallel electron distribution and linear kinetic theory of an inertial Alfvén wave. While the uniqueness of our model cannot be absolutely guaranteed without repeating this experiment at several locations in the parallel ẑ direction, the quantitative agreement of the model with measurements is promising.

FIG. 9.

The linear model accurately describes the amplitude and phase of ge1(vz) measurements taken at a single location. (a) An absolute comparison of the amplitude of the measured and modeled ge1(vz) shows good agreement. The model curve is produced by evaluating Eqs. (15) and (19). (b) Plotted here is the phase of ge1(vz) relative to the phase of Ez. The model is able to replicate the relative phase of the measured ge1(vz). Above E=+50 eV, experimental uncertainty approaches 2π, so that phase data in this range is not meaningful. The uncertainty of measurements is shown in gray.

FIG. 9.

The linear model accurately describes the amplitude and phase of ge1(vz) measurements taken at a single location. (a) An absolute comparison of the amplitude of the measured and modeled ge1(vz) shows good agreement. The model curve is produced by evaluating Eqs. (15) and (19). (b) Plotted here is the phase of ge1(vz) relative to the phase of Ez. The model is able to replicate the relative phase of the measured ge1(vz). Above E=+50 eV, experimental uncertainty approaches 2π, so that phase data in this range is not meaningful. The uncertainty of measurements is shown in gray.

Close modal

Using measurements from multiple locations, it is possible to examine the spatial structure of the measured ge1(vz) and compare with predictions. The idealized Ez waveform in Eq. (5) and the full solution for ge1(vz) in Eqs. (15) and (19) vary like cos(kx), so that the variations in these quantities are predicted to be aligned in x̂. The DFT analysis described in Section VI A is applied to the data from all eight locations in Fig. 6. The intensity of ge1(vz) from each location is mapped in Fig. 10. This figure shows the measured values of By and the calculated values of Ez along the range of the scan taken from Fig. 2. Notably, the measured ge1(vz) is most intense at the antinodes of Ez and is diminished at the nodes of Ez. This confirms the prediction that variations of ge1(vz) and Ez are aligned in x̂.

FIG. 10.

The measurement process is repeated in an abbreviated form at 8 locations in x̂ spanning one perpendicular Alfvén wavelength. The top panel shows the measured By and the calculated Ez for the same range in x̂. Measurements of By are used to calculate Ez. Errors in By measurements are smaller than the markers. The bottom panel is an intensity map showing variations in the strength of measured ge1(vz) along the scan. The significant result seen here is that the measured ge1(vz) vanishes at the nodes of Ez, as predicted.

FIG. 10.

The measurement process is repeated in an abbreviated form at 8 locations in x̂ spanning one perpendicular Alfvén wavelength. The top panel shows the measured By and the calculated Ez for the same range in x̂. Measurements of By are used to calculate Ez. Errors in By measurements are smaller than the markers. The bottom panel is an intensity map showing variations in the strength of measured ge1(vz) along the scan. The significant result seen here is that the measured ge1(vz) vanishes at the nodes of Ez, as predicted.

Close modal

The model ge1(vz) includes the terms ge1f(vz)+ge1k(vz) that are attributable to the Alfvén wave and an additional homogeneous solution ge1h(vz) used to describe non-Alfvénic perturbations at the same frequency as the Alfvén wave. It is worth considering the relative importance of the Alfvénic and non-Alfvénic terms. While the experimental data cannot answer this question, the model can provide insight.

Fig. 11 shows the amplitude of the Alfvén wave terms and the homogeneous solution. Consider the two halves of this plot separately. For vz>0, where the boundary is the ASW antenna, the Alfvénic and non-Alfvénic contributions are approximately equal in amplitude. Since the ASW antenna functions by applying an oscillating voltage to the plasma, the importance of the homogeneous solution which includes an oscillating voltage boundary condition is expected. For vz<0, the boundary is the cathode. The significance of the homogeneous solution for this half of the data implies the cathode-anode voltage is oscillating at the frequency of the Alfvén wave. The presence of an oscillating voltage at this boundary is not well understood. An expected feature, confirmed by Fig. 11, is that the Alfvén wave terms are more significant for vz>0 than for vz<0. This is expected since vz>0 is the resonant direction and the denominator of the Alfvénic terms in Eq. (15) goes to zero as the velocity approaches resonance.

FIG. 11.

The amplitudes of the Alfvén wave solution and the homogeneous solution are shown here. According to the model, both the homogeneous solution and the Alfvén wave solution are significant for vz>0. For vz<0, the homogeneous solution dwarfs the Alfvén wave terms. The insignificance of the Alfvén wave solution for vz<0 is expected since this is the non-resonant direction.

FIG. 11.

The amplitudes of the Alfvén wave solution and the homogeneous solution are shown here. According to the model, both the homogeneous solution and the Alfvén wave solution are significant for vz>0. For vz<0, the homogeneous solution dwarfs the Alfvén wave terms. The insignificance of the Alfvén wave solution for vz<0 is expected since this is the non-resonant direction.

Close modal

A new antenna has been built that inductively couples to the Alfvén wave instead of electrostatically like the ASW antenna. The electrostatic design of the ASW antenna may be responsible for the prominence of the homogeneous term. The new inductively-coupled antenna is designed to reduce the homogeneous term.

The experimental measurement of ge1(vz) enables additional analysis of assumptions used to produce the homogeneous solution in Eq. (19). The full solution is the sum ge1=ge1f+ge1k+ge1h. Rearranging gives ge1h=ge1ge1fge1k. Using this rearranged form along with data for ge1(vz) and theoretical values for ge1f(vz) and ge1k(vz) gives values for ge1h(vz) as a hybrid of data and theory. The benefit of this hybrid is that ge1h(vz) is evaluated using a different set of assumptions from the ones used in Section IV D to produce the analytical expression for ge1h(vz). The hybrid values require both the assumptions of the measurement and analysis techniques to produce the measured ge1(vz) and the assumptions used to produce the analytical form of the Alfvén wave terms ge1f(vz)+ge1k(vz) (Eq. (15)). Conversely, the analytical form of ge1(vz) assumed there is an oscillating voltage at the boundary producing a non-Alfvénic perturbation in the motion of the suprathermal electrons. Fig. 12 compares the hybrid values for ge1h(vz) with values of the analytical expression for ge1h(vz) (Eq. (19)). There is good agreement, indicating the non-Alfvénic effects of the ASW antenna are sufficiently described by the homogeneous solution developed in Section IV D and justifying the use of this homogeneous solution in the analysis.

FIG. 12.

The model for the homogeneous solution ge1h(vz) produced in Section IV D uses a number of assumptions, and the effectiveness of these assumptions is tested by comparing the model and hybrid values of ge1h(vz). Part (a) shows the amplitude and part (b) shows the phase of ge1h(vs). There is good agreement between the model and the hybrid, indicating that the assumptions used in the derivation of ge1h(vz) are sufficient to describe the non-Alfvénic perturbations originating at the boundary. The uncertainty of the hybrid values is shown in gray.

FIG. 12.

The model for the homogeneous solution ge1h(vz) produced in Section IV D uses a number of assumptions, and the effectiveness of these assumptions is tested by comparing the model and hybrid values of ge1h(vz). Part (a) shows the amplitude and part (b) shows the phase of ge1h(vs). There is good agreement between the model and the hybrid, indicating that the assumptions used in the derivation of ge1h(vz) are sufficient to describe the non-Alfvénic perturbations originating at the boundary. The uncertainty of the hybrid values is shown in gray.

Close modal

This research investigated the oscillations of the reduced parallel electron distribution function ge1(vz) caused by an inertial Alfvén wave. Unlike the Alfvén waves of ideal MHD, inertial Alfvén waves have a parallel electric field that causes oscillations in the parallel electron motion and is expected to produce accelerated electrons. Until this study, neither effect had been definitively verified. The acceleration of electrons by inertial Alfvén waves likely contributes to the generation of a significant fraction of auroras, and the suprathermal portion of ge(vz) is of particular interest since the acceleration affects electrons above the thermal speed. Measurements were carried out in UCLA's Large Plasma Device (LAPD) using the Whistler Wave Absorption Diagnostic (WWAD), an electron cyclotron absorption diagnostic that accurately measures the suprathermal tails of ge(vz). Using a series of phase-shifted data sets, we produced a composite measurement of ge(vz) with sufficient resolution in Alfvén wave phase to isolate oscillations at the Alfvén wave frequency. The model for these oscillations ge1(vz), derived from the linearized gyroaveraged Boltzmann equation, includes an Alfvén wave solution and an additional homogeneous solution that accounts for non-Alfvénic effects generated by the antenna. The model accurately reproduces the amplitude, phase, and spatial structure of the measured oscillations. By measuring and modeling the linear oscillations of ge(vz), this experiment verifies one of the fundamental distinctions between ideal MHD and inertial Alfvén waves. While plasma in the lower magnetosphere is collisionless, in this experiment the inertial Alfvén wave and the electron response ge1(vz) are modified by electron collisions. However, since the suprathermal electrons are significantly less collisional than the thermal electrons, the properties of the suprathermal portions of the measured and modeled ge1(vz) are not fundamentally distinct from the collisionless scenario.

Having successfully measured and modeled the linear oscillations of the reduced electron distribution function, we turn our attention to the nonlinear Alfvén wave-particle physics. Ongoing experiments are testing a new higher-power Alfvén wave antenna and exploring the use of innovative analysis techniques that employ field-particle correlations to directly compute the rate of energy transfer between the parallel electric field and the electrons.27 

This work was supported by the NSF Graduate Research Fellowship under Grant No. 1048957, NSF Grant Nos. ATM 03-17310 and PHY-10033446, NSF CAREER Award No. AGS-1054061, DOE Grant No. DE-SC0014599, and NASA Grant No. NNX10AC91G. The experiment presented here was conducted at the Basic Plasma Science Facility, funded by the U.S. Department of Energy and the National Science Foundation. The data used are available from the corresponding author upon request.

This work is part of a dissertation to be submitted by J. W. R. Schroeder to the Graduate College, University of Iowa, Iowa City, IA, in partial fulfillment of the requirements for the Ph.D. degree in Physics.

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