On O+ ion heating by BBELF waves at low altitude: Test particle simulations
JJOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1002/?DRAFT,
On O + ion heating by BBELF waves at low altitude: Testparticle simulations Yangyang Shen
1, 2 , David J. Knudsen Abstract.
We investigate mechanisms of wave-particle heating of ionospheric O + ions resultingfrom broadband extremely low frequency (BBELF) waves using numerical test particlesimulations that take into account ion-neutral collisions, in order to explain observationsfrom the Enhanced Polar Outflow Probe (e-POP) satellite at low altitudes ( ∼
400 km)[
Shen et al. , 2018]. We argue that in order to reproduce ion temperatures observed ate-POP altitudes, the most effective ion heating mechanism is through cyclotron accel-eration by short-scale electrostatic ion cyclotron (EIC) waves with perpendicular wave-lengths λ ⊥ ≤
200 m. The interplay between finite perpendicular wavelengths, wave am-plitudes, and ion-neutral collision frequencies collectively determine the ionospheric ionheating limit, which begins to decrease sharply with decreasing altitude below approx-imately 500 km, where the ratio ν c f ci becomes larger than 10 − , ν c and f ci denoting theO + –O collision frequency and ion cyclotron frequency. We derive, both numerically andanalytically, the ion gyroradius limit from heating by an EIC wave at half the cyclotronfrequency. The limit is 0.28 λ ⊥ . The ion gyroradius limit from an EIC wave can be sur-passed either through adding waves with different λ ⊥ , or through stochastic “breakout”,meaning ions diffuse in energy beyond the gyroradius limit due to stochastic heating fromlarge-amplitude waves. Our two-dimensional simulations indicate that small-scale ( <
1. Introduction
The importance of broadband extremely low-frequency(BBELF) waves for transverse ionospheric ion heating andion outflow in the high-latitude auroral region has been wellestablished [
Kintner et al. , 1996;
Bonnell et al. , 1996;
Knud-sen et al. , 1998;
Lynch et al. , 2002]. Based on observa-tions from sounding rockets and satellites, a number of wavemodes may exist in the broadband emission from several Hzup to several kHz, including Alfv´en waves below the ion cy-clotron frequency [
Stasiewicz et al. , 2000a;
Chaston et al. ,2004], ion acoustic waves [
Wahlund et al. , 1998], electrostaticwaves near the ion cyclotron frequency (EIC waves) [
Kint-ner et al. , 1996;
Bonnell et al. , 1996], and electromagneticion cyclotron (EMIC) waves [
Erlandson et al. , 1994].Although it is agreed upon that perpendicular electricfields, either left-hand circularly polarized or linearly polar-ized, are essential for ion acceleration [
Lysak et al. , 1980;
Chang et al. , 1986;
Chaston et al. , 2004], the ion heat-ing mechanism, dominant wave modes, and perpendicularwavelengths that are responsible for ion heating are stillnot well known [
Paschmann et al. , 2003]. Until recently,BBELF heating was thought to occur well above the iono-sphere where collisional effects are negligible [e.g.
Wu et al. ,1999;
Zeng et al. , 2006].
Shen et al. [2018] reported multi-ple examples of BBELF-induced heating at altitudes as lowas 350 km and at spatial scales as narrow as 2 km. These Department of Physics and Astronomy, University ofCalgary, Calgary, Alberta, Canada. Now at Department of Earth, Planetary, and SpaceSciences, University of California, Los Angeles, California,USACopyright 2020 by the American Geophysical Union.0148-0227/20/$9.00 observations call for a more comprehensive and realistic ionheating model in the BBELF wave environment, includingthe effects of both ionospheric ion-neutral collisions and fi-nite perpendicular wavelengths. There has been no suchmodel in the literature yet.Previous studies have suggested different mechanisms ofion heating from auroral plasma waves. Ions can be res-onantly accelerated by lower-hybrid waves when the per-pendicular wave phase velocity is comparable with the ionvelocity, leading to a heated tail [
Lynch et al. , 1996;
Tsuru-tani and Lakhina , 1997]. On the other hand, particles in thebulk of the ion population can be accelerated simultaneouslyby long-wavelength (much larger than the ion gyroradius ρ i )ion cyclotron waves through quasi-linear cyclotron resonance[ Chang et al. , 1986].
Ball and Andr´e [1991] used test parti-cle simulations to study ion heating by BBELF waves andfound that the heating rate increases as the wave frequencyapproaches the ion cyclotron frequency.A similar, but nonlinear, acceleration mechanism is thecoherent ion trapping by electrostatic ion cyclotron waves[
Lysak et al. , 1980;
Ram et al. , 1998].
Lysak et al. [1980]found that, in a single electrostatic ion cyclotron wave, ionswill be trapped in an effective potential well. The energy, orequivalently the gyroradius, of the ion is limited by the firstzeros of the Bessel function of the first kind, governed by theparameter k ⊥ ρ i , where k ⊥ is the perpendicular wavenumber. Lysak et al. [1980] also showed that the presence of addi-tional waves with different wavenumbers can detrap the ion,and the gyroradius barrier can be surpassed.
Lysak [1986]extended this theory to harmonics of EIC waves, in whichcase the upper limits of the ion gyroradius are specified bythe first zeros of J n ( k ⊥ ρ i ), where n is the harmonic inte-ger order. In this paper, we extend these previous resultsof gyroradius limits to EIC waves with the frequency of ω c ,which has not been reported yet. We shall provide consis-tent analytical and numerical calculations for the gyroradiuslimits. a r X i v : . [ phy s i c s . s p ace - ph ] J u l - 2 SHEN ET AL.: ION HEATING BY BBELF WAVES
Another important type of acceleration mechanism isstochastic ion heating, in which ion energy increases dueto random kicks in phase space, increasing temperature.Stochastic ion heating has been studied in detail for EICwaves and lower-hybrid waves [
Karney , 1978;
Papadopou-los et al. , 1980;
Lysak , 1986]. The onset condition is: E ⊥ B (cid:39) (cid:0) Ω i ω (cid:1) ωk ⊥ , where E ⊥ is the amplitude of perpen-dicular wave electric field, B the magnitude of magneticfield, Ω i the ion cyclotron frequency, and ω the wave angu-lar frequency.Stochastic ion heating due to large-amplitude Alfv´enwaves has also been investigated by several subsequent stud-ies [ McChesney et al. , 1987;
Bailey et al. , 1995;
Stasiewiczet al. , 2000b;
Chen et al. , 2001;
Johnson and Cheng , 2001;
Chaston et al. , 2004].
Chaston et al. [2004] found that, fortransverse wave amplitude E ⊥ satisfying E ⊥ B < Ω i k ⊥ , ion mo-tion in the wave field is coherent and the ion may becometrapped by Alfv´en waves in a manner similar to that pro-posed for EIC waves. The ion may be accelerated up to agyrodiameter roughly equivalent to the perpendicular wave-length of the wave. When E ⊥ B > Ω i k ⊥ , the ion can be stochasti-cally accelerated, and the gyroradius limit can be surpassed.In addition, in the presence of a large spatial gradient of elec-tric fields, significant ion acceleration and orbit chaotizationmay take place [ Cole , 1976;
Stasiewicz et al. , 2000b]. How-ever, amplitudes of Alfv´en or EIC waves observed in thelow-altitude ionosphere are generally insufficient to initiatestochastic ion heating, according to the onset condition ofthe stochastic behavior [
Karney , 1978;
Bonnell et al. , 1996;
Chaston et al. , 2004].Ion-neutral collisions, due to large DC electric fields inthe ionosphere, may generate a significant amount of O + ion heating and sometimes lead to anisotropic and toroidalion distributions [ St-Maurice and Schunk , 1979;
Loranc andSt. Maurice , 1994;
Wilson , 1994]. However, collisions canalso put a limit on the ion heating in the ionosphere byrestricting the achievable speed relative to neutral gases[
Schunk and Nagy , 2009]. Few models of ionospheric ionheating by plasma waves take into account both the ef-fects of finite perpendicular wavelengths and ion-neutral col-lisions. Using a multispecies particle simulation,
Providakesand Seyler [1990] showed that collisional current-driven elec-trostatic ion cyclotron waves may be unstable in the bottom-side ionosphere ( <
300 km) and that they are able to gener-ate transverse bulk acceleration of the heavy ions to energiesof more than a few keV in this region. However, the requiredcritical field-aligned current for the cyclotron instabilities isat least 50 µ A/m , which is seen rarely at best, and is sig-nificantly higher than values reported by Shen et al. [2018].
Burleigh [2018] used an anisotropic fluid model to study ionupflows in the ionosphere from drivers of both quasilinearwave-ion heating and collisions. But this fluid model sim-plifies wave-ion heating processes and does not incorporatefinite wavelength effects.In this paper, we shall explore, using test particle sim-ulations, whether and how cyclotron and stochastic ionheating from Alfv´en (planar or non-planar) or EIC waveswith different perpendicular wavelengths, along with ion-neutral collisions, might contribute to ionospheric O + ionheating to observed levels, as exemplified by those foundfrom the Enhanced Polar Outflow Probe (e-POP) at 410km altitude [ Shen et al. , 2018]. The test particle simula-tions we use are admittedly simple but allow us to under-stand the microphysical (kinetic) processes in some detail.In the following, we shall first examine two different ionheating mechanisms—stochastic acceleration and coherenttrapping—and then investigate the role of BBELF wavesalong with ion-neutral collisions in explaining O + ion heat-ing under the constraints of e-POP observations.
2. Stochastic ion acceleration by a monochromaticelectrostatic wave
We assume a coherent monochromatic electrostatic wavein a uniform background magnetic field (cid:126)B = B ˆ z : (cid:126)E = E cos( kx − ωt )ˆ x (1)where B is the magnitude of magnetic field, E the ampli-tude of the electrostatic wave, k the wavenumber, and ω thewave angular frequency.We use the Hamiltonian method to describe the chargedparticle’s motion perpendicular to the background magneticfield. The Hamiltonian reads: H = T kinetic + V potential = 12 m [( P y + qB x ) + P x ] − (cid:18) qE k (cid:19) sin( kx − ωt )(2)where P x = mv x and P y = mv y are the ion momentumterms. Following previous similar studies [ Karney , 1978;
Bailey et al. , 1995], we can normalize the physical quanti-ties as t (cid:48) = Ω i t , x (cid:48) = kx , P y (cid:48) = P y km Ω i , P x (cid:48) = P x km Ω i , α = E B k Ω i , and ν = ω Ω i , where the primed variables indi-cate normalized quantities. Ω i = qB m is the ion cyclotronfrequency, α the normalized wave potential, and ν the nor-malized wave frequency. The Hamiltonian takes this newform: H = 12 ( P y + x ) + 12 P x − α sin( x − νt ) (3)where we have dropped the prime symbol. The equations ofmotion for the particle can be found as:˙ x = ∂H∂P x = P x (4)˙ P x = − ∂H∂x = − P y − x + α cos( x − νt ) (5)˙ y = ∂H∂P y = P y + x (6)˙ P y = − ∂H∂y = 0 (7)where we can identify that P y is a constant of motion; there-fore, the dynamics in terms of x and P x are exclusivelydetermined by each other. The symmetric equations canbe conveniently integrated numerically using the 4th-ordersymplectic integrator [ Forest and Ruth , 1990]. Such an in-tegrator is phase-space area conserving and more suitablefor long-term integration. It also takes less computing timethan the classic Runge-Kutta scheme. The orbit of a parti-cle in phase space is often represented by a Poincar´e surfaceof section plot, constructed by marking the particle’s tra-jectory when it passes a constant plane in phase space, e.g.,when the wave phase equals 2 π in our case. We use 30 testparticles that are uniformly distributed in velocity space toexplore their trajectories under the influence of the waveelectric field. The accuracy of the 4th order symplectic in-tegrator has been tested on a simple harmonic oscillator sothat the relative trajectory error is less than 0.01% for 10million integration steps. A stepsize of 0.01 (equivalent to Ω i − ) is chosen as we numerically calculate the particle’smotion over two million steps and construct the Poincar´eplots.In the following, we define the off-resonance case when ν = 0 . ν = 1. Linearlypolarized waves having a frequency much less than the ioncyclotron frequency ( ν = 0 .
1) may represent Alfv´en wavesin the sheet-like current structures in the auroral region. Inthe case described here, when the perpendicular wave elec-tric field and wave vector are approximately parallel, Alfv´en
HEN ET AL.: ION HEATING BY BBELF WAVES
X - 3 waves are in the shear mode. Throughout the paper, theterm “Alfv´en wave” therefore refers to a linearly polarizedwave with a frequency much lower than the ion cyclotronfrequency. We ignore the magnetic perturbation, which isjustifiable because magnetic perturbations at low altitudes( ∼
400 km) are insignificant compared with the backgroundmagnetic field. The on-resonance case corresponds to anEIC wave. Both cases are investigated for different wavepotentials or amplitudes α , in order to understand the ionstochastic motion in the Alfv´enic and ion cyclotron regimes. Figure 1.
Poincar´e surface of section plots for bothAlfv´enic off-resonance (Figure 1a, 1b) and electrostaticion cyclotron (EIC) wave on-resonance (Figure 1c, 1d,1e, 1f) cases for different values of wave amplitude α . ForAlfv´en waves, there is no net ion energy increase when thewave amplitude is small ( α = 0 .
1) (Figure 1a). However,when α approaches 0.8, the ions can undergo stochasticacceleration as shown in Figure 1b. For the on-resonancecases, when α gradually increases from 0.05 to 2.0, weobserve that stochastic motion emerges from the innercircle and expands into outer circles. The largest circleis specified by J ( kρ i ) = 0.Figure 1 shows Poincar´e surface of section plots for bothoff- (Figure 1a, 1b) and on-resonance (Figure 1c, 1d, 1e, 1f)cases when the wave amplitude α is set to different values.In Alfv´en waves, there is no net ion energy increase whenthe wave amplitude is small ( α = 0 . α approaches 0.8, or equivalently, E B k Ω i (cid:39) .
8, the ion canundergo stochastic acceleration. Figure 1b shows such aphonomenon, where ions start to randomly occupy the ve-locity space within a specific circle, which represents theenergy limit that ions can gain. The stochastic onset condi-tion for Alfv´en wave ion heating is consistent with
Chastonet al. [2004]. Within the stochastic region, ions have access to all the available phase space region, or equivalently, ionscan be accelerated to any energy level accessible.An ion trajectory that follows a circle in phase spacemeans that its energy does not increase. However, for EICwaves, the ion’s energy increases and returns back to itsoriginal level cyclically (Figure 1c) if the ion stays in thesystem for a sufficiently long time. In this case, the ion istrapped within wave potential wells. The energy limit is de-termined by harmonic potential structures of the EIC wave,which can be represented by expansion of the wave poten-tial in terms of the Bessel function J n ( kρ i ) of the first kind[ Lysak et al. , 1980;
Gibelli et al. , 2010]. Similar to Figure 1b,when α gradually increases from 0.05 to 2.0, we observe thatstochastic ion motion emerges from the inner circle and ex-pands into outer circles. The largest circle is specified by J ( kρ i ) = 0 [ Lysak et al. , 1980]. The stochastic onset condi-tion for EIC waves was numerically determined by
Karney [1978] as E ⊥ B (cid:39) . (cid:0) Ω i ω (cid:1) ωk ⊥ . The numerical constant wefound instead is 0.4, which is similar to 0.25 found earlierusing different scenarios and parameters. The formula de-rived from Karney [1978] concerns electrostatic waves withfrequencies much larger than the ion cyclotron frequency.
3. Coherent ion cyclotron acceleration byEIC waves
To understand ion cyclotron acceleration from EIC waveswith different frequencies and perpendicular wavelengths,we perform another numerical simulation using the classicRunge-Kutta integrator to resolve ions dynamics. This isbecause we aim later to investigate how different frequencyand wavenumber spectra may affect ion bulk heating, asapplicable to the BBELF-heating scenario. The symplecticintegrator has no advantage in this situation since the prob-lem is not symmetric. We shall report two phenomena of ionacceleration in the single wave case, i.e., the ion gyroradiuslimit and the stochastic “breakout”, where at a certain crit-ical point an ion’s energy can diffuse beyond the gyroradiuslimit through stochastic acceleration.The ion dynamics perpendicular to the background mag-netic field are determined by the Lorentz equation: m ∂(cid:126)v∂t = q ( (cid:126)E ( x, t ) + (cid:126)v × (cid:126)B ) (8)where (cid:126)B only has a ˆ z component and (cid:126)E ( x, t ) is the waveelectric field in the ˆ x direction. For a single wave case, (cid:126)E ( x, t ) = E cos( πλ ⊥ x − ωt )ˆ x . The equations of motion inthe perpendicular directions are:˙ v x = qm (cid:126)E ( x, t ) + Ω i v y (9)˙ v y = − Ω i v x (10)where Ω i is the ion gyrofrequency. The equation can benumerically integrated using the 4th-order Runge-Kutta in-tegrator. The accuracy of the integrator has been tested bycomparing numerical results with the analytical solution ofthe E × B drift. A stepsize of 0 . T gyro is chosen to limitthe relative E × B trajectory error to be within 10 − over 10million integration steps.In the presence of a single cyclotron wave, an ion is ac-celerated over the entire gyroorbit; the ion’s velocity has acomponent in the same direction of the wave electric field.Figure 2a, 2b, and 2c show the gyroradius evolution for ω = Ω i , Ω i , and 2Ω i . The ρ i λ ⊥ limits are approximately0.27, 0.6 and 0.83 for the half, fundamental, and double cy-clotron frequency wave respectively. By comparing these - 4 SHEN ET AL.: ION HEATING BY BBELF WAVES three cases, we observe that ion acceleration by higher-order cyclotron harmonics is generally much less effectivethan by the wave in the fundamental mode, as it takes alonger time for ions to reach the gyroradius limit in the for-mer cases. The numerically calculated gyroradius limits forinteger-harmonic cyclotron waves are consistent with thosepredicted by
Lysak [1986]. The limits are determined by thefirst zeros of J n ( kρ i ) for cyclotron harmonics. However, nocalculation has been reported in the literature for the caseof ω = Ω i . In Appendix A, we show an analytical derivationof the ion gyroradius limit with the emphasis on this half cy-clotron frequency. The analytical predictions are consistentwith the numerical results presented. Figure 2.
Time evolution of the ratio ρ i λ ⊥ for EIC waveswith ω = Ω i , Ω i , and 2Ω i , and for the cases with two dif-ferent wavelengths and two frequencies. The ratio limitsare approximately 0.27, 0.6, and 0.83 in Figure 2a, 2band 2c. Figure 2d shows a moderate ion radius limit( ∼ λ ⊥ , each with half of the original wave am-plitude. As a result, the ion gyroradius exceeds the limits setby both perpendicular wavelengths. The ion can be acceler-ated to a gyroradius more than 2.5 times of the maximum λ ⊥ . This is the detrapping effect discussed in Lysak et al. [1980]. Note that waves with multiple frequencies that addup to the cyclotron frequency can also accelerate ions (notshown here), which is consistent with previous studies [e.g.
Temerin and Roth , 1986].One way to break the ion gyroradius limit is to have multi-ple perpendicular wavelengths in the system as shown in Fig-ure 2e. The other way, which we found through test particlesimulations, is to increase the amplitude of the wave electricfield to a degree that ions eventually diffuse out of the per-pendicular wavelength limit due to stochastic ion heating.Figure 3 shows how the stochastic “breakout” initiates whenthe ratio of E B increases from 0 . V phase ( E =10 mV/m) to 0 . V phase ( E =120 mV/m), where V phase is the phasespeed of the EIC wave. When E B approaches 0 . V phase ( E =105 mV/m) of the wave as depicted in Figure 3c, ionsstart to obtain gyroradii larger than 0 . λ ⊥ . This is a repeat-able threshold valid for the EIC wave in the fundamentalmode. Such a threshold has not been reported yet. Al-though stochastic “breakout” only increases the ion energymarginally, resultant detrapping of the ion from the wave po-tential may contribute to ion loss from a spatially-restrictedheating system or a travelling wave. Figure 3.
Stochastic “breakout”—ions diffusing be-yond the gyroradius limit due to stochastic ion heating—initiates when the EIC wave amplitude increases from 10mV/m to 120 mV/m, corresponding to E B = 0 . V phase to 0 . V phase . When E =105 mV/m, as shown in Fig-ure 3c, or equivalently when E B approaches 0 . V phase ofthe wave, the ion gyroradius can be larger than 0 . λ ⊥ .The magnitude of B in the simulation is 40,000 nT.
4. e-POP observations
One of the major objectives of this paper is to investi-gate essential wave properties, effective perpendicular wave-lengths, as well as microscopic heating mechanisms thatare responsible for BBELF wave-ion heating as observedfrom the Enhanced Polar Outflow Probe (e-POP) satellite.The e-POP scientific payload is part of the multipurposeCASSIOPE satellite [
Yau and James , 2015], launched on29 September 2013 into a polar elliptical orbit plane withan inclination of 81 ◦ , a perigee of 325 km, and an apogeeof 1,500 km. In this section, we show the magnetic field,electric field, and ion observations from the wave-ion heat-ing event on 18 May 2015. Further details on this eventcan be found in Shen et al. [2018]. Relevant instrumentsonboard e-POP are described separately in
Knudsen et al. [2015],
Wallis et al. [2015], and
James et al. [2015]. Thefield and particle observations put important constraints onwave electric field amplitudes and ion heating scales, whichserve as baselines for the numerical test particle simulationsin the next sections.Figure 4 presents a summary of the field observationsfrom e-POP at 410 km altitude between 22:29:47.4 and22:30:28.4 UT. Figure 4a and 4b show the magnetic pertur-bations B x and B y in the spacecraft frame, correspondingto the along-track (+ x , to the south) and cross-track (+ y ,to the west) component, respectively. Magnetic fields aremeasured at 160 samples per second (sps) and bandpass-filtered to be within the frequency range of 3-80 Hz. Fig-ure 4c shows wave electric fields measured from the Radio HEN ET AL.: ION HEATING BY BBELF WAVES
X - 5
Figure 4.
Summary of the field observations frome-POP at 410 km altitude between 22:29:47.4 and22:30:28.4 UT on 18 May 2015. Figure 4a and 4b showmagnetic perturbations B x and B y in the spacecraftframe. The magnetic field time series are sampled at 160per second and bandpass filtered to be within the 3-80Hz range. Figure 4c shows wave electric fields from theRadio Receiver Instrument (RRI) in the cross-track (y)direction. E y electric fields has been lowpass filtered witha cutoff of 80 Hz. Time-series electric fields are under-estimated at frequencies below 7 Hz due to limitation ofthe instrument dynamic range. Note that magnetic fluc-tuations do not always accompany wave electric fields,meaning the waves are sometimes electrostatic. Figure4d displays the calculated AC Poynting flux from B x and E y . Alfv´enic magnetic fluctuations up to 300 nT,perpendicular electric fields up to 8 mV/m, and Poynt-ing uxes up to 0.8 mW/m are observed at near 22:30:08UT. These field fluctuations are colocated with strongO + ion heating, indicated within the red-shaded region.Receiver Instrument (RRI) in the y direction. Electric fieldsare sampled at 62,500 sps and bandpass-filtered from 7 Hzto 80 Hz in the very low-frequency (VLF) mode. Note thatmagnetic fluctuations do not always accompany wave elec-tric fields, meaning the waves are sometimes electrostatic.In fact, Shen et al. [2018] showed that electric fields withinBBELF waves measured by e-POP in this region are mostlylinearly polarized perpendicular to the magnetic field withfrequencies up to 1 kHz (in their Figure 6). One clear sig-nature of O + ion cyclotron waves is present at the exactlocation of observed ion heating in this case, which will beshown in detail later. Figure 4d displays the calculated ACPoynting flux from B x and E y . We observe Alfv´enic mag-netic fluctuations up to 300 nT, perpendicular electric fieldsup to 8 mV/m, and Poynting fluxes up to 0.8 mW/m atnear 22:30:08 UT. These field fluctuations, embedded withinBBELF waves, are also colocated with strong O + ion heat-ing up to a temperature of 4.3 eV as indicated within thered-shaded region [ Shen et al. , 2018].To demonstrate microstructures of ion heating, BBELFwave spectra, Alfv´en wave and cyclotron wave characteris-tics, we present a 1.7-second zoomed-in view of the mea-surements in Figure 5. Figure 5a shows a heated two-dimensional ion enery-angle distribution measured fromthe Suprathermal Electron/Ion Imager (SEI) instrument at 22:30:08.8 UT. The maximum O + ion temperature, repre-sented by the width of the distribution at 55 ◦ pitch angle(the white dashed line), is determined to be approximately4.3 eV, which has been validated through a Monte-Carlocharged particle ray tracing simulation [ Burchill et al. , 2010].The noteworthy feature in the image is that most of the ionsignal lies within the energy of approximately 100 eV, as
Figure 5.
Zoomed-in view of the microstructure of ionheating within 1.7 s, along with the BBELF wave spec-trum and short-time-scale electric fields recovered in thefrequency range of 3–150 Hz, for the event shown in Fig-ure 4. Figure 5a shows a heated two-dimensional ion dis-tribution measured from the Suprathermal Electron/IonImager (SEI) instrument at 22:30:08.75 UT. This plot isadapted from
Shen et al. [2018] in their Figure 4c. Theprojection of the magnetic field direction onto the de-tector is indicated by the black dashed line. Ion energyincreases outward from the centroid of the distribution,which is indicated by the white dot in the image center.The ion energy detectable from SEI goes up to approx-imately 230 eV at the periphery. Figure 5b shows the E y and B x amplitude-frequency spectra averaged overa 0.4-second measurement period inside an ion heatingregion. The black solid curve shows the corrected E y .The black dashed curve displays the fitted function formof E y = 4 . (cid:16) f (cid:17) . . The magnetic perturbation spec-trum is shown in the red line. Figure 5c compares themeasurements of E y and B x within 1.7 s of ion heating.The magnetic field fluctuations are bandpass filtered in3–30 Hz. Note that the time-series E y in Figure 5c havenot been corrected in amplitude as we have done for thespectrum plot in Figure 5b. In Figure 5c, at the timelocation of O + ion heating (dark-red dotted grid line), E y shows fluctuations with a frequency of approximately100 Hz. These waves are linearly polarized perpendicu-lar to the magnetic field and are consistent with O + ioncyclotron waves with a perpendicular wavelength of ap-proximately 150 m. A zoomed-in view of these cyclotronwaves is shown in Figure 5d. It is worth mentioning thatanother ion heating event with a temperature of near 2eV was observed at +1.5 s in Figure 5c, which is notdiscussed here. - 6 SHEN ET AL.: ION HEATING BY BBELF WAVES indicated by the red pixels (not saturated). This is one ofthe observables we use to compare simulations with obser-vations. In addition, statistical observations from e-POPhave shown that ion heating by BBELF waves is associ-ated with ion downflows in the low-altitude (325-730 km)auroral downward current region [
Shen et al. , 2018].
Shenet al. [2018] applied the“pressure cooker” ion heating modelwith down-pointing electric fields in the return current re-gion [
Gorney et al. , 1985] to explain the low-altitude e-POPobservations. Based on these results, we conclude that O + ions can be forced to remain within BBELF heating regionslong enough for their energies to saturate..Figure 5b presents the E y and B x amplitude-frequencyspectra calculated within a 0.4-second measurement periodinside an ion heating region. We can correct the magni-tude measurement of E y below 10 Hz to account for theinstrument filter response. This is plotted as the blacksolid line in Figure 5b. This frequency spectrum up toapproximately 120 Hz is fitted with a power-law function E y = 4 . (cid:16) f (cid:17) . . After taking into account the othercomponent of perpendicular electric field that we do notmeasure, we assume much larger total electric fields as E = 8 (cid:16) f (cid:17) . . This is another important baseline settingfor simulations. The O + cyclotron frequency is about 38 Hzas indicated by the vertical orange line. In Figure 5b, theratios of the corrected E y / B x in the frequency range of 3–10Hz suggest that the electromagnetic fluctuations have phasespeeds of 400–700 km/s. Assuming O + dominated plasmawith an electron density of 10 m − near 400 km altitude,the expected Alfv´en speed is approximately 700 km/s, whichis consistent with the observations.Figure 5c compares the measurements of E y and B x within 1.7 s of the ion heating period. In this case, theelectric field fluctuations are lowpass filtered with a cutoffof 150 Hz. Although time-series electric field measurementsdeteriorate at frequencies lower than 10 Hz, we can see in-phase oscillations of electric field and magnetic field pertur-bations, with an oscillation period of approximately 0.2 s,which is identifiable near +0.5 s and +1.5 s UT on the timeaxis. The fairly detailed correlation between E y and B x , in-ferred Alfv´en speeds, along with the macroscopic Poyntingfluxes as shown in Figure 4d, strongly suggest that Alfv´enwaves are present within the measured BBELF wave spec-trum. Most interestingly, right at the time location of O + ion heating (indicated by the dark-red dotted grid line), E y shows fluctuations with a frequency of approximately 100Hz. These waves are not observed outside the ion heat-ing time intervals (within 0.2 s centered around +0.5 s UTon the time axis). The waves are linearly polarized per-pendicular to the background magnetic field based on two-component electric field ( E y and E z ) hodogram analysis(provided in Supporting Information). These features areconsistent with electrostatic O + ion cyclotron waves withamplitudes of up to 5 mV/m and a perpendicular wave-length of approximately 150 m. Such a wave scale is inaccord with the gyroradius (180 m) of 100 eV O + ions. Azoomed-in view of these cyclotron waves is shown in Fig-ure 5d. Therefore, our observations show that both Alfv´enwaves and EIC waves are present within the BBELF wavespectrum and are associated with ion heating observed bye-POP.Based on previous studies, ion cyclotron waves maybe generated by velocity-shear driven instabilities [ Ganguliet al. , 1994], current-driven instabilities [
Kindel and Kennel ,1971], ion-beam-driven instabilities [
Hendel et al. , 1976], ornonlinear breaking of Alfv´en waves [
Seyler et al. , 1998]. Theexact nature of how these kinetic processes produce EICwaves is beyond the scope of this study.
5. Test particle simulation: Ion heatingfrom BBELF waves
Wave frequency measurements from a single spacecraftcan be complicated to interpret as to whether they are tem- poral or spatial structures. This is especially true as thee-POP satellite moves at a speed of 7.8 km/s, which trans-lates to 7.8 Hz for static structures with a perpendicularwavelength of 1 km. In this section, we examine wave-ionheating from BBELF waves, which are either Alfv´en waves( ν =0.01, different from the earlier simulations to be consis-tent with common observations in the ionosphere) or EICwaves ( ν =1) with varying wavenumber spectra. BBELFwaves contain many wave modes, including EIC waves thatare slightly above the ion gyrofrequency and ion acousticwaves that are below the ion gyrofrequency [ Kintner et al. ,1996;
Bonnell et al. , 1996;
Wahlund et al. , 1998]. Here weterm any electrostatic wave as EIC wave so far as it has a fre-quency near the ion gyrofrequency. We only consider wave-ion heating in the two-dimensional plane perpendicular tothe background magnetic field. Our objective is to find theasymptotic limit of perpendicular ion heating from plasmawaves, which justifies our neglecting the effects of paralleltransport, including the mirror force and others, that tendto remove hot ions from the heating region. Potential paral-lel electric fields in Alfv´en waves, when their perpendicularwavelengths are comparable to the electron inertial length[e.g.
Goertz and Boswell , 1979;
Lysak and Lotko , 1996], areignored in our simulations because ion parallel velocities (onthe order of 1 km/s) are much less than the Alfv´en waveparallel phase velocity (on the order of 1000 km/s) so thations cannot be resonantly accelerated, and their perpendicu-lar temperature is insignificantly affected by parallel electricfields [
Chaston et al. , 2004].In the simulation, there are in total 20000 particles, hav-ing an initial two-dimensional Maxwellian distribution in ve-locity space with a temperature of 0.2 eV, and exhibiting auniformly distribution in the spatial domain x = [0 , y = [ − , . T gyro , to evolve the 20000-particle system numerically.Note that there exists no spatial boundary for an ion’s tra-jectory to evolve in the simulation. The ion temperatureis calculated for the whole ion population and is expressedas T ⊥ = m i k B < ( v x − < v x > ) + ( v y − < v y > ) ) > .Ion temperatures are recorded every 500 steps, correspond-ing to every 5 T gyro . This ensures that we investigate iondynamics at a constant ion gyro-phase, indicating secu-lar variations. The wave electric field is generally in theform E x = (cid:80) k E cos( k x x − ωt + φ rand ), where 40 different k modes are utilized with random phases φ rand . The waveamplitudes are E = 8 (cid:16) f (cid:17) . .In addition, many previous studies suggest that Alfv´enwaves become increasingly non-planar and display two-dimensional (2D) field variations with decreasing scale sizes,especially when λ ⊥ approaches 1 km or less [ Volwerk et al. ,1996;
Chaston et al. , 2004]. To account for this fieldtopology, we perform additional test runs with both k x and k y for Alfv´en waves, with electric fields specified as E x = (cid:80) k x ,k y − k x Φ( ωt + φ rand )cos( k x x )cos( k y y ) and E y = (cid:80) k x ,k y k y Φ( ωt + φ rand )sin( k x x )sin( k y y )[ Chaston et al. , 2004],where k x and k y are equal to bring out the maximum heat-ing rate and k Φ is the field amplitude specified accordingto e-POP observations (the same as E ). Note that the 2Dperpendicular electric field in this equation is curl-free andtherefore the ion flow is incompressible. This is similar to HEN ET AL.: ION HEATING BY BBELF WAVES
X - 7 that used in
Chaston et al. [2004] but excludes variations inthe field line ( z ) direction.Since observations of ion heating from e-POP are in theionosphere, we have to take into account ion-neutral colli-sional effects, which play an important role in limiting ionheating from BBELF waves. Atomic oxygen O + ions in theionosphere will primarily experience O + –O resonant chargeexchange (RCE) collisions and polarization collisions, wherethe ion interacts with the electric dipole it induces in theneutral particle. At altitudes of about 400 km, chemicalreactions between O + ions and N and O neutral gasesare of secondary significance. We only include O + –O RCEcollisions in the simulations, since long-range polarizationcollisions only deflect O + ions that have low relative speedswith neutral particles, and they are less important when T O + >
400 K [
Barakat et al. , 1983;
Wilson , 1994]. Detailson how we simulate O + –O collisions are discussed in Ap-pendix B.Table 1 lists the wave modes, Doppler-shifted frequen-cies, perpendicular wavelengths, durations of numerical in-tegration, and ion heating features in the eight test runs weperformed (see the end of the document). Each test run Figure 6.
Snapshots of ion distribution and temperatureevolution for four test runs: (Run-3) 0.5 Hz planar short-scale shear Alfv´en waves (Figure 6a and 6b), (Run-4) 0.5Hz planar large-scale Alfv´en waves (Figure 6c and 6d);(Run-5) 0.5 Hz planar short-scale Alfv´en waves with in-creased amplitudes to initiate stochastic heating (Fig-ure 6e and 6f); (Run-8) 0.5 Hz two-dimensional Alfv´enwaves with equal k x and k y and with increased ampli-tudes (Figure 6g and 6h), respectively. Snapshots of thevelocity distribution are taken at the end of 1000 gyrope-riods of integration. Snapshots of temperature are takenevery 5 gyroperiods. The red contour (100 eV) in the ve-locity plots represents the observed O + ion energy limitfrom e-POP. has been assigned a number indicated in the first column.Broadband emission can be interpreted either as temporalor Doppler-shifted spatial signals in the spacecraft frame,and both contributions will affect the spectrum we observe[ Stasiewicz et al. , 2000a;
Chaston et al. , 2004]. We assumethat much of the broadband frequency spectrum is due toDoppler shift ( (cid:126)k ⊥ · (cid:126)V ) from a satellite passing through waveswith finite λ ⊥ at a relative speed of 7.8 km/s. The only ex-ception is Run-1 (temporal limit k = 0), where broadbandemissions from 0.1Ω i to 4Ω i are all temporal variations withan infinitely long wavelength. Run-2 (spatial limit ω = 0)treats all fluctuations as stationary spatial structures with-out time oscillations. No heating is observed for these twocases. The movies showing phase space evolution of all thetest runs are in Supporting Information (SI).Figure 6 shows snapshots of the ion distributions (Fig-ure 6a, 6c, 6e, and 6g) and temperature evolution (Fig-ure 6b, 6d, 6f, and 6h) for the test runs of Run-3, Run-4,Run-5, and Run-8. In the case of short-scale shear Alfv´enwaves, we observe no ion acceleration or heating as demon-strated in Figure 6a and 6b. On the other hand, large-scaleAlfv´en waves give rise to only a 0.2 eV ion temperature in-crease through coherent trapping. The periodic oscillationin the ion temperature is due to time oscillation of 0.5 HzAlfv´en waves. Short-scale planar shear Alfv´en waves onlygenerate a small amount of stochastic ion heating up toless than 0.6 eV (Figure 6e and 6f) even as we increase thewave amplitudes tenfold. Ion heating is still insignificant, asshown in Figure 6g and 6h, when we adopt a different fieldtopology with equal k x and k y and with wave amplitudesincreased tenfold (approximately 20 mV/m, 10 mV/m, and5 mV/m for the three wavelengths included).Our result of 2D Alfv´en wave ion heating differs from Chaston et al. [2004], who suggested that non-planarity ofsmall-scale Alfv´en waves can facilitate ion energization up toseveral keV through coherent trapping at low-altitudes andstochastic ion heating at higher altitudes. The differencecan be explained by the fact that electric field amplitudesof Alfv´en waves with a perpendicular wavelength of 800 mobserved from e-POP (Doppler-shifted to 10 Hz in the satel-lite frame) are approximately 2 mV/m, which is two orders
Figure 7.
Ion velocity distribution and temperature evo-lution for the test runs (Run-6 and Run-7) of short-scale( λ ⊥ ∼ + –O collisions collisionsare included, ion temperature increases rapidly (within50 gyroperods, corresponding to near 1 s) to a steadystate of 26 eV. - 8 SHEN ET AL.: ION HEATING BY BBELF WAVES of magnitude smaller than 200 mV/m electric fields used by
Chaston et al. [2004] for the same wavelength. It is gener-ally not likely to observe a 200 mV/m electric field at 10 Hzfrom low-altitude ( < Basu et al. , 1988;
Ball and Andr´e , 1991;
Wu et al. , 2020]. As we artificiallyincrease the wave amplitudes of non-planar 2D Alfv´en wavesin our simulation to 200 mV/m ( λ ⊥ ∼
500 m), we observesignificant ion heating up to 20 eV (not shown here), con-sistent with the O + ion temperature obtained by Chastonet al. [2004]. The temperature increase will drop to 1 eVwith an electric field amplitude of 100 mV/m. Therefore,the relatively weak electric fields at low altitudes limit therole of either planar or 2D Alfv´en waves in explaining theion heating we observed through either coherent trapping orstochastic ion heating.The insignificant amount of ion heating from small-scaleAlfv´en waves from our simulations is also in contrast tothe study of
Stasiewicz et al. [2000b], who suggested thatstochastic ion heating from small-scale (of the order of 100m perpendicular wavelength) Alfv´en waves can explain thebulk ion heating observed at Freja and FAST altitudes( > Stasiewicz et al. [2000b] does. Alfv´en waves may play anindirect role in ion heating cases at low altitudes, whichneeds to be investigated in the future. In addition to thetest runs described here, we also run cases that take into ac-count waves with multiple frequencies below the cyclotronfrequency. No significant difference is observed comparedwith the single-frequency Alfv´enic cases.Figure 7 presents results of the other test runs (Run-6and Run-7) for short-scale ( λ ⊥ ∼
70m – 200 m) cyclotronwave ion heating with (Figure 7a and 7b) and without (Fig-ure 7c and 7d) ion-neutral collisions. In the absence of col-lisions, small-scale cyclotron waves with wavelengths of lessthan 200 m effectively heat ions to temperatures of largerthan 50 eV within 200 gyroperiods (Figure 7b), correspond-ing to approximately 5 seconds. Figure 7a illustrates thataccelerated ions obtain energies well above the 100 eV ob-served limit (the red contour line) due to cyclotron heating.However, many ions are trapped at energies less than 100eV, which can be attributed to the finite wavelength effect.When collisions are present, ions are isotropically restrictedwithin the 100 eV limit and the temperature surges to asteady state of 26 eV within 50 gyroperiods (near 1 second).Figure 7c and 7d demonstrate that the effect of collisionsis to limit as well as to heat the system to a steady-stateion temperature. Compared with the other test runs, small-scale cyclotron wave ion heating regulated by ion-neutralcollisions can be most effective in heating O + ions at iono-spheric altitudes where e-POP operates.
6. Test particle simulation: Cyclotron ionheating regulated by E , λ ⊥ and collisions In this section, we use test particle simulations to ex-plore how ion-neutral collisions, together with λ ⊥ and waveamplitudes, affect ion heating from cyclotron waves in theionosphere. For multiple test particle simulations, we spec-ify a single cyclotron wave with either an infinite wavelengthor a wavelength of 200 m, and with varying amplitudes of1, 2, 4, or 6 mV/m. Note that the 200 m wavelength is cho-sen to represent the finite wavelength effect of cyclotron ionheating. We vary the collisional frequency by changing theatomic oxygen O neutral density from 5.7 × cm − , which is calculated from the MSIS00 model [ Hedin , 1991] at 410 kmaltitude, to half and tenth of this value, calculated at an alti-tude of 450 km and 540 km respectively. A linearly polarizedwave can be decomposed into right-handed and left-handedcircularly polarized waves with equal amplitudes and onlyleft-handed polarized waves can interact with left-handedlyrotating ions in the magnetic field through cyclotron res-onance. Thus, only the left-handed polarized power of cy-clotron waves are taken into account for comparison with themodel of quasilinear cyclotron heating according to
Changet al. [1986], who suggested that the ion heating rate fromcyclotron waves with an infinite wavelength can be expressedas: dWdt = q E m i (11)where E is the power spectral density of the left-hand com-ponent of electric fields. The cases of cyclotron heating withan infinite wavelength are termed “temporal” in the follow-ing.Figure 8 presents the simulation results comparing themagnitude of, and the integration time to, a steady-stateion temperature from cyclotron heating with varying wave-lengths, wave powers, and collisional frequencies at altitudesfrom 410 km to 540 km. Figure 8a, 8b and 8c show increas-ing steady-state temperatures, from several eV to tens ofeV for E = 0 . , with decreasing collision fre-quencies by tenfold for a 100-km change in altitude. Ionheating limited by the finite wavelength effect (the blackdots and lines) has a steady-state temperature smaller thanthat of temporal cyclotron heating (red triangles and lines)from wave electric fields with the same power. The finitewavelength effect is not discernible when collision frequen-cies are relatively high and wave powers are relatively weak,but dramatic when collision frequencies are small and wavepowers strong, as shown in Figure 8a and 8c. For tem-poral cyclotron ion heating, the temperature limits rise bymore than ten times, from 4.3 eV to 49 eV when E = 0 . , as the collision frequency is lowered by a factor Figure 8.
Simulation results comparing the magnitudeof (Figure 8a, 8b and 8c), and the integration time to(Figure 8d, 8e and 8f), a steady-state ion temperaturefrom cyclotron heating with varying wavelengths, wavepowers, and collision frequencies at altitudes from 410km to 540 km. Steady-state ion temperatures from thecyclotron wave with a perpendicular wavelength of 200m, and with an infinite wavelength, are indicated by theblack dots and lines, and red triangles and lines, respec-tively. Simulations have been run for 1,000 gyroperiods inall cases. The theoretically calculated steady-state tem-peratures from Equation 12 are displayed as the cyantriangles for left-hand polarized cyclotron waves in thetemporal cyclotron heating model.
HEN ET AL.: ION HEATING BY BBELF WAVES
X - 9 of ten from 410 km to 540 km. In this case, the parame-ter ν c f ci decreases from about 10 − to 10 − , where ν c is thecollision frequency between neutral O and heated O + ionsin the steady state and f ci is the ion cyclotron frequency inHz. Overall, collisions play a critical role in regulating ionheating limits in the ionosphere. We observe drastic varia-tions in the ion heating limit with even a 100-km change inaltitude for the same wave electric fields.The ion heating limit from temporal cyclotron heatingwith collisions can be theoretically calculated by solving theenergy equation: dWdt = q E m i − ν c m i ¯ v = 0 (12)where ¯ v is the two-dimensional root-mean-square speed inthe steady state. Note that ν c is constant and can be ap-proximated by n ( O ) Q (¯ v )¯ v in a steady-state, where Q (¯ v ) isthe total collision cross-section. This equation implies thatfor each effective resonant O + –O collision taking place, themomenta of the two colliding particles exchange completely.The loss of energy due to collisions is balanced by the addi-tion of energy due to ion heating. After we obtain a steady-state root-mean-square speed ¯ v based on Equation 12, wederive the steady-state ion temperature of a two-dimensionalMaxwellian distribution based on the assumed relation:¯ v = (cid:114) kT i m i (13)We display the theoretically calculated steady-state temper-atures for cyclotron waves with different powers as the cyantriangles in Figure 8a. The simulated (red) and theoret-ically calculated (cyan) temporal cyclotron heating limitsagree well when E is below 2 (mV/m) . When the wavepower becomes larger, the calculated Maxwellian tempera-tures become much greater than, and deviate from, the sim-ulated limits. This is because the simulated ion distributionbecomes increasingly non-Maxwellian and more flat-shapeddue to larger electric fields, rendering Equation 13 incorrect.Figure 8d, 8e and 8f display integration times to reacha steady-state temperature as the collision frequency de-clines. As E and the collision frequency increase, ions reacha steady-state temperature more rapidly, generally in tensof to hundreds of cyclotron periods, corresponding to ap-proximately 1 s to 10 s. Finite wavelength effects heat ionsto a steady state temperature more efficiently when colli-sion frequencies are small (Figure 8e and 8f). In Figure 8d,temporal cyclotron heating accelerates ions more effectivelythan with a wave having a finite wavelength, and collisionsprevail over finite wavelengths in limiting the system to asteady state. Comparing Figure 8d with Figure 8e, as thecollision frequency decreases by half, we observe a cross-overbetween collisional and finite wavelength effects in heatingions to a steady-state temperature. In general, the effectsof λ ⊥ , wave power and collision collectively determine themagnitude and efficiency (time to reach a steady-state tem-perature) of ion heating in the ionosphere.Note that our neglecting of the magnetic mirror force mayoverestimate the ion heating limits only slightly, since withinthe period of 10 s, cold ionospheric ions cannot travel faralong the field line and transfer of energy from perpendic-ular to parallel via the mirror force is quite limited. Forexample, ionospheric O + ions with a parallel temperatureof 1 eV have parallel speeds of the order of 3 km/s. Ona time scale of 10 s, an O + ion flowing upward due to themagnetic mirror force will travel a vertical distance of 30km. The perpendicular energy of the ion with an initialvalue of 10 eV will decrease by only 0.1 eV. In addition, inlight of evidence of a low-altitude “pressure cooker” effect[ Shen et al. , 2018], ions may well be trapped in the heatingregion long enough to reach the heating limits derived hereunder the assumption of negligible field-aligned transport.
7. Summary and Conclusions
In this paper, we have examined the physics of ion heatingfrom Alfv´en waves ( ω (cid:28) Ω i ) and cyclotron waves ( ω = Ω i )using test particle simulations, with an objective to explainhow BBELF waves heat ionospheric ions as observed fromthe e-POP satellite in the low-altitude ( ∼
400 km) iono-sphere. We have investigated the effects of finite perpen-dicular wavelengths, wave amplitudes, and ion neutral col-lisions, on ionospheric ion heating. The results of our testparticle simulations in this paper show the following:1. The numerical onset thresholds of stochastic ion heat-ing from the EIC wave and Alfv´en wave, as illustrated bythe Poincar´e surface of section plots, are consistent withprevious studies [
Karney , 1978;
Lysak , 1986;
Bailey et al. ,1995;
Chaston et al. , 2004]. One important characteristic ofstochastic ion heating is that ions undergoing stochastic ac-celeration can still be limited by the large potential structureof the wave, or similarly by the perpendicular wavelength.2. The ion gyroradius limit for the EIC wave with halfof the cyclotron frequency is 0.28 λ ⊥ , which has been bothnumerically and analytically derived. The ion gyrora-dius limit from a single EIC wave can be surpassed eitherthrough adding waves with different λ ⊥ , or through stochas-tic “breakout”, when the wave electric field amplitude satis-fies E B ≥ . V phase , where V phase is the wave phase velocity.3. In contrast to previous studies focusing on higher al-titudes [ Stasiewicz et al. , 2000b;
Chaston et al. , 2004], oursimulations indicate that both planar and non-planar small-scale ( < λ ⊥ ≤
200 m.4. The interplay between finite perpendicular wave-lengths, wave amplitudes, and ion-neutral collision frequen-cies collectively determine the ionospheric ion heating limit,which begins to decrease sharply with decreasing altitudebelow approximately 500 km altitude, where the ratio ν c f ci becomes significant ( > − ) with decreasing altitude.Our model of ion heating by Alfv´en waves is limited tothe perpendicular plane. Ion dynamics in the direction par-allel to the magnetic field is not included; as a result, themagnetic mirror force and parallel electric fields are not in-cluded. While effects of parallel dynamics on ion heatingare not included, they are expected to be small in the low-altitude (410 km) ionosphere as discussed above. - 10 SHEN ET AL.: ION HEATING BY BBELF WAVES
Appendix A: Ion gyroradius limit from asingle EIC wave
Lysak [1986] has given the derivation of the ion gyrora-dius limit for EIC waves with ω = nω c . Here we present ananalytical derivation of the ion gyroradius limit using thegyro-orbit-averaging method for EIC waves with half of thecyclotron frequency, since this has not been reported else-where in the literature.Following the main text, we assume a single coherent EICwave with a form E = E cos ( k ⊥ x − Ω i t ) in a backgroundmagnetic field (cid:126)B in the ˆ z direction. Since the wave is electro-static in nature, the wave scalar potential may be expressedas Φ = − E k ⊥ sin( k ⊥ x − Ω i t ). The ion can be viewed as beingtrapped by the wave potential, therefore the total energy ofthe wave-ion system is conserved. The maximum of the ion’skinetic energy corresponds to the minimum of the wave po-tential energy, which translates to finding zeros of the waveelectric field. Considering secular increase of the ion’s gyro-radius, we shall obtain a wave electric field that is averagedover one ion gyro-orbit. The electric field is first rewrittenin the reference frame of the guiding center: E = E cos( k ⊥ x + k ⊥ ρ i cos(Ω i t ) − Ω i t ) (A1)where x is guiding center coordinate, ρ i is the ion gyrora-dius. From trigonometric identities, this can be expressedas: E = E { cos( k ⊥ x )cos( Ω i t )cos( k ⊥ ρ i cos(Ω i t ))+ cos( k ⊥ x )sin( Ω i t )sin( k ⊥ ρ i cos(Ω i t )) − sin( k ⊥ x )cos( Ω i t )sin( k ⊥ ρ i cos(Ω i t ))+ sin( k ⊥ x )sin( Ω i t )cos( k ⊥ ρ i cos(Ω i t )) } (A2)This form can be further expanded using the Bessel functionidentities equations cos( z cos θ ) = J ( z ) + 2 ∞ (cid:80) n =1 ( − n J n ( z )cos(2 nθ ),sin( z cos θ ) = − ∞ (cid:80) n =1 ( − n J n − ( z )cos((2 n − θ ). After in-serting these Bessel identities into the equation and inte-grate the electric field within one ion gyro-orbit < E > = π (cid:82) π E d (Ω i t ), we have the wave electric field in the finalform: < E > = E cos( k ⊥ x ) {− π J ( k ⊥ ρ i )+ 435 π J ( k ⊥ ρ i ) − π J ( k ⊥ ρ i ) + · · · } + E sin( k ⊥ x ) { π J ( k ⊥ ρ i ) + 415 π J ( k ⊥ ρ i ) − π J ( k ⊥ ρ i ) + · · · } (A3)The maximum magnitude of the averaged electric field isfound when setting sin( k ⊥ x ) = cos( k ⊥ x ) = √ , in whichcase the wave potential magnitude is at its largest. Themaximum ion kinetic energy, or equivalently the ion gyro-radius limit, is obtained from the first zero of < E > . Sincethe terms after J ( k ⊥ ρ i ) are negligibly small, we can trun-cate the series and find the solution to be k ⊥ ρ i = 1 .
8, whichtranslates to the ratio ρ i λ ⊥ = 0 .
28. This analytical solutionis consistent with the numerical result shown in Figure 2a.Using the same procedure and change the wave frequencyto ω = Ω i , 2Ω i , 3Ω i and 4Ω i , the resulting averagedwave electric fields are, respectively, − E sin( k ⊥ x ) J ( k ⊥ ρ i ), − E cos( k ⊥ x ) J ( k ⊥ ρ i ), E sin( k ⊥ x ) J ( k ⊥ ρ i ), and E cos( k ⊥ x ) J ( k ⊥ ρ i ).The first zeros of J n ( k ⊥ ρ i ) correspond to ρ i λ ⊥ =0 . , . , . , . n = 1 , , ,
4. This result is inagreement with
Lysak [1986].
Appendix B: Simulating O–O + collisions We use an approximate method of O + –O collisions, whichis essentially that used by Barakat et al. [1983] and
Wil-son [1994]. The probability that a particle suffers no col-lision in the time interval t overall follows Poisson statis-tics P nc ( t ) = e − ν c t . The probability of collision is therefore P c ( t ) = 1 − e − ν c t . ν c depends on the neutral O density n ( O ), RCE collisional cross section, and the relative speed g between O + and O. Ideally, the overall probability of col-lision during the time dt can be written in a general form as P c ( t ) = 1 − e − dtn ( O ) (cid:82) σ ( g ) gf ( g ) dg , where σ ( g ) is the differen-tial cross-section of RCE collisions and f ( g ) is normalizedso that (cid:82) f ( g ) dg = 1. We adopt the same approximatemethod as that used by Wilson [1994], where the integralin the exponential is replaced by n ( O ) Q ( g ) g ( Q ( g ) is thetotal cross section) and g is found at each time step. Weassume the background neutral O gas is near stationary sothat for each RCE collision the O + velocity is set to 0 witha probability of 0.5. The neutral O density is obtained fromMSIS00 model [ Hedin , 1991] using geophysical parametersrelevant to the ion heating event reported by
Shen et al. [2018]. If ν c is a constant, the relationship between succes-sive collisional time interval dt and P c ( t ) can be simplified as dt = − n ( O ) Q ( g ) g ln(1 − P c ( dt )). To take into account speeddependence of the collision time dt , we use the “null colli-sion” method [ Lin and Bardsley , 1977;
Winkler et al. , 1992].The simulation steps are:1. Specify a very small constant dt , which represents themaximum ν c possible to occur. We use a dt of about 0.03 sin this paper compared with 0.05 s used by Wilson [1994].2. Assume collisions occur at this constant dt . Calculatethe impact parameter b based on b = (cid:113) − ln(1 − r ) πn ( O ) gdt , where r is a psudorandom number between 0 and 1 representing thecollision probability within time dt .3. Calculate a critical impact parameter b cr , which is de-termined by the experimental Q ( g ) measured for RCE colli-sions. We use the same phenomenological RCE model asin Wilson [1994]. b cr is calculated for each g such thatthe total cross section for RCE satisfies Q ( g ) = (10 . − .
95 log g ) × − cm . The probability of charge ex-change to occur during each collision is 0.5.4. Compare b with b cr . If b ≤ b cr , a real charge exchangecollision occurs during this time interval dt and the O + ionvelocity is set to 0; if b > b cr , the assumed collision is “null”and the O + ion velocity is not changed.5. Proceed to the next dt and repeat the above process.Using this approach, we find that each test particle withina heated (4 eV for example) ion population experiences ap-proximately 10 collisions within the integration time of 1000gyroperiods. This magnitude of O + –O collision frequencyis comparable with values estimated by Schunk and Nagy [2009] assuming a reduced temperature of approximately 2eV.
HEN ET AL.: ION HEATING BY BBELF WAVES
X - 11
Acknowledgments.
This work was supported by an EyesHigh Doctoral Recruitment Scholarship from the University ofCalgary, the Natural Sciences and Engineering Research Councilof Canada (NSERC), and the Canadian Space Agency (CSA).e-POP was funded with support from CSA and MDA Cor-poration. In 2017 e-POP has been incorporated as one part(Swarm Echo) of the Swarm constellation funded by the Eu-ropean Space Agency. e-POP data are accessible throughhttp://epop-data.phys.ucalgary.ca/. Simulation data and codesnecessary to reproduce the results are available from PRISMDataverse at University of Calgary’s Data Repository throughhttps://doi.org/10.5683/SP2/PWYYZQ.
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X - 13 T a b l e . P a r a m e t e r s f o r t h e t e s t p a r t i c l e s i m u l a t i o n r un s . N u m b e r W a v e M o d e ω D o pp l e r − s h i f t e d λ ⊥ I n i t i a l k ⊥ ρ i I n t e g r a t i o n T i m e H e a t i n g f e a t u r e R un - T e m p o r a l L i m i t k = . Ω i - Ω i ∞ m T g y r o n o h e a t i n g R un - S p a t i a l L i m i t ω = . Ω i - Ω i m - m . - . T g y r o n o h e a t i n g R un - D A l f v ´ e n . H z . Ω i - Ω i m - m . - . T g y r o n o h e a t i n g R un - D A l f v ´ e n . H z . Ω i - . Ω i m - m . - . T g y r o t r a pp i n g R un - D A l f v ´ e n . H z S t o c h a s t i c . Ω i - Ω i m - m . - . T g y r o s m a ll s t o c h a s t i c h e a t i n g R un - C y c l o t r o n Ω i . Ω i - . Ω i m - m . - . T g y r o s t r o n g c y c l o t r o nh e a t i n g R un - C y c l o t r o n Ω i + C o lli s i o n . Ω i - . Ω i m - m . - . T g y r o li m i t e d c y c l o t r o nh e a t i n g R un - D A l f v ´ e n w i t h E ⊥ . Ω i , . Ω i , . Ω i m , m , m . - . T g y r o nn
X - 13 T a b l e . P a r a m e t e r s f o r t h e t e s t p a r t i c l e s i m u l a t i o n r un s . N u m b e r W a v e M o d e ω D o pp l e r − s h i f t e d λ ⊥ I n i t i a l k ⊥ ρ i I n t e g r a t i o n T i m e H e a t i n g f e a t u r e R un - T e m p o r a l L i m i t k = . Ω i - Ω i ∞ m T g y r o n o h e a t i n g R un - S p a t i a l L i m i t ω = . Ω i - Ω i m - m . - . T g y r o n o h e a t i n g R un - D A l f v ´ e n . H z . Ω i - Ω i m - m . - . T g y r o n o h e a t i n g R un - D A l f v ´ e n . H z . Ω i - . Ω i m - m . - . T g y r o t r a pp i n g R un - D A l f v ´ e n . H z S t o c h a s t i c . Ω i - Ω i m - m . - . T g y r o s m a ll s t o c h a s t i c h e a t i n g R un - C y c l o t r o n Ω i . Ω i - . Ω i m - m . - . T g y r o s t r o n g c y c l o t r o nh e a t i n g R un - C y c l o t r o n Ω i + C o lli s i o n . Ω i - . Ω i m - m . - . T g y r o li m i t e d c y c l o t r o nh e a t i n g R un - D A l f v ´ e n w i t h E ⊥ . Ω i , . Ω i , . Ω i m , m , m . - . T g y r o nn o h e a t i nn