Equatorial source of oblique electromagnetic ion cyclotron waves: peculiarities in the ion distribution function

Abstract

Electromagnetic ion cyclotron (EMIC) waves are important for Earth’s inner magnetosphere as they can effectively drive relativistic electron losses to the atmosphere and energetic (ring current) ion scattering and isotropization. EMIC waves are generated by transversely anisotropic ion populations around the equatorial source region, and for typical magnetospheric conditions this almost always produces field-aligned waves. For many specific occasions, however, oblique EMIC waves are observed, and such obliquity has been commonly attributed to the wave off-equatorial propagation in curved dipole magnetic fields. In this study, we report that very oblique EMIC waves can be directly generated at the equatorial source region. Using THEMIS spacecraft observations at the dawn flank, we show that such oblique wave generation is possible in the presence of a field-aligned thermal ion population, likely of ionospheric origin, which can reduce Landau damping of oblique EMIC waves and cyclotron generation of field-aligned waves. This generation mechanism underlines the importance of magnetosphere-ionosphere coupling processes in controlling wave characteristics in the inner magnetosphere.

\draftfalse\journalname

JGR: Space Physics

Department of Physics, University of Texas at Dallas, Richardson, TX, USA Department of Earth, Planetary, and Space Sciences, University of California, Los Angeles, USA

\correspondingauthor

David S. Tonoiandavid.tonoian@utdallas.edu

{keypoints}

We report observations of very oblique EMIC waves around their equatorial source region

Oblique EMIC wave generation is associated with field-aligned thermal ion streams and hot, transversely anisotropic ions

The presence of field-aligned thermal ion streams attributes oblique EMIC wave generation to its possible ionospheric source

1 Introduction

Electromagnetic ion cyclotron (EMIC) waves is a natural emission generated by transversely anisotropic ion (proton) populations in the Earth’s inner magnetosphere [Cornwall \BOthers. (\APACyear1970), Cornwall \BBA Schulz (\APACyear1971), Min \BOthers. (\APACyear2015), Yue \BOthers. (\APACyear2019)]. These waves are responsible for resonant scattering of relativistic electrons and energetic (ring current) ions [<]see review¿[and references therein]Usanova&Mann16, which makes this wave mode principally important for the inner magnetosphere dynamics.

Transversely anisotropic ion populations, either injected from the plasma sheet or formed due to the magnetosphere compression by solar wind transients [Jun \BOthers. (\APACyear2019), Jun \BOthers. (\APACyear2021), Kim \BOthers. (\APACyear2021)], usually generate field-aligned EMIC waves [Chen \BOthers. (\APACyear2010), Chen \BOthers. (\APACyear2011)], whereas generation of oblique waves is often suppressed by wave Landau damping due to suprathermal ions [<]see discussions in¿SoriaSantacruz13. Thus, oblique EMIC waves have been mostly detected off the equator [Liu \BOthers. (\APACyear2013)], because wave propagation in highly inhomogeneous magnetic field and plasma will result in wavevector deviation from the field-aligned direction [Rauch \BBA Roux (\APACyear1982), Thorne \BBA Horne (\APACyear1997), Fraser \BBA Nguyen (\APACyear2001)]. It has been shown that oblique EMIC waves can scatter particles via several resonant mechanisms not available to field-aligned waves [<]see review by¿Usanova21. First, wave obliquity modifies the efficiency of relativistic electron scattering [<]see¿Khazanov&Gamayunov07,Lee18:emic,Hanzelka23. Second, the Landau resonance between oblique EMIC waves and cold ions allows EMIC waves to serve as a transmitter of energy between hot and cold ion populations [Omura \BOthers. (\APACyear1985), Kitamura \BOthers. (\APACyear2018), Ma \BOthers. (\APACyear2019)]. Third, oblique EMIC waves can resonate with cold plasmasphere electrons [<]in linear and nonlinear regimes, see¿Wang19:emic and can accelerate and/or precipitate them to form the stable red aurora arcs [Cornwall \BOthers. (\APACyear1971), Thorne \BBA Horne (\APACyear1992)]. Fourth, oblique EMIC waves may scatter energetic ( $\sim 100$ keV) electrons via the bounce resonance [Q. Wang \BOthers. (\APACyear2018), Blum \BOthers. (\APACyear2019)]. All these mechanisms imply the importance of understanding possible sources of oblique EMIC waves: can such waves be generated around the equator or only off-equator as the field-aligned waves propagate away from their equatorial source?

The equatorial generation of oblique EMIC waves requires a viable mechanism to suppress the Landau damping, i.e., reduction of the field-aligned gradient in the ion phase space density around the Landau resonant energies (hundreds of eV). For the electron-scale whistler-mode waves, it has been shown that oblique wave generation can be explained by field-aligned electron streams generated by ionosphere outflow [<]see discussions in¿[]Artemyev&Mourenas20:jgr, which largely suppress Landau damping [<]see, e.g.,¿Mourenas15,Li16. Such ionosphere outflows also include field-aligned ionospheric ion populations, which are often detected in the inner magnetosphere and near-Earth plasma sheet [Yue \BOthers. (\APACyear2017), Artemyev \BOthers. (\APACyear2018)]. Therefore, oblique EMIC wave generation may be explained by a combination of near-equatorial, transversely anisotropic hot ions and field-aligned thermal ion streams.

In this study, we investigate near-equatorial observations of very oblique EMIC waves by Time History of Events and Macroscale Interactions during Substorms (THEMIS) spacecraft [Angelopoulos (\APACyear2008)]. Combining measurements of ion distribution functions [McFadden \BOthers. (\APACyear2008)] and linear dispersion solver [Astfalk \BBA Jenko (\APACyear2017)], we reveal the conditions of such wave generation and demonstrate that very oblique EMIC waves are associated with field-aligned ion streams. The rest of this paper is structured as follows: Section 2 describes THEMIS instrumentation, plasma and wave measurements, methods of data analysis, and details of the linear dispersion solver. Section 3 examines the generation of multiple very oblique EMIC wave events by analyzing the observed ion distribution function and comparing this distribution with that during field-aligned EMIC waves. Section 4 summarizes our results and discusses their implication for modelling of particle dynamics in the inner magnetosphere.

2 Dataset and instruments

We combine near-equatorial THEMIS measurements of EMIC waves [<]of 16Hz sampling rate from the flux-gate magnetometer, see¿Auster08:THEMIS, plasma sheet ion distributions [<]at 3s resolution from the electrostatic analyzer (ESA), covering $<25$ keV energy range and full pitch-angle range, see¿McFadden08:THEMIS, total plasma density inferred from the spacecraft potential [<]see¿Bonnell08,Nishimura13:density, and the numerical dispersion solver for electromagnetic waves – Linear Electromagnetic Oscillations in Plasmas with Arbitrary Rotationally-symmetric Distributions [<]LEOPARD, see¿Astfalk&Jenko17. We use the single-spacecraft maximum variance analysis technique [Means (\APACyear1972)] to estimate the wave normal angle, which comes with an ambiguity of parallel versus anti-parallel directions: the wave normal angle from this technique ranges from $0^{\circ}$ to $90^{\circ}$ , rather than $0^{\circ}$ to $180^{\circ}$ .

An important advantage of LEOPARD model is it can accommodate arbitrary gyrotropic distribution functions in a uniform grid of velocities parallel and perpendicular to the background magnetic field, $(v_{\parallel},v_{\perp})$ . Compared to dispersion solvers that require fitting the measurements to prescribed particle distributions (such as Maxwellians), LEOPARD significantly reduces the uncertainties due to fitting and hence can give a more reliable dispersion solution. Therefore, we use ESA ion distributions in $23$ logarithmically spaced energy channels between $5$ and $25$ keV and $8$ linearly spaced pitch-angle channels and interpolate them to a denser $(v_{\parallel},v_{\perp})$ grid, which is then passed to LEOPARD to evaluate the observed EMIC wave dispersion and growth rate. We will combine LEOPARD and THEMIS measurements to reveal properties of specific ion distributions that are responsible for the generation of very oblique EMIC waves, quite atypical type of EMIC emission. Note that although THEMIS ESA provides plasma sheet ( $<30$ keV) electron distributions, this electron population usually does not resonate with EMIC waves or alter the wave dispersion. Thus, we only treat electrons as the cold background plasma (with electron $\beta=10^{-2}$ ) and do not examine their contribution to EMIC wave growth.

3 Oblique EMIC events

We now analyze in details multiple events with very oblique EMIC waves. These are typical events of this wave population and hence their characteristics will be representative of the entire population.

3.1 Detailed analysis of the first event

The first event shows very oblique EMIC waves observed by THEMIS-A spacecraft on February 17, 2020, between 17:50-17:58 UT (Figure 1). During this event, hydrogen-band EMIC waves (of frequencies within $0.3-0.9$ of the proton cyclotron frequency $f_{cH+}$ ) were detected in the dawn flank of the outer edge of the inner magnetosphere (MLT $\sim 07$ , $L\sim 10$ ) with the presence of hot, plasma sheet ions. The EMIC wave normal angle can reach $80^{\circ}$ during this event. The ion beta is $\beta\sim 1.7$ , typical for the inner plasma sheet edge/outer edge of the inner magnetosphere [Yue \BOthers. (\APACyear2017), Artemyev \BOthers. (\APACyear2018)].

Figure 1 shows the field-aligned population of thermal ( $\lesssim 1$ keV) ions and hot ( $>2$ keV), transversely anisotropic ion population. Both the flux and anisotropy of hot ions are large right around the moment of intense, very oblique EMIC waves. Such transversely anisotropic ions are likely responsible for EMIC wave generation [Chen \BOthers. (\APACyear2010), Chen \BOthers. (\APACyear2011), Yue \BOthers. (\APACyear2019)], but instead of more typical field-aligned waves we observe very oblique waves. Therefore, certain features in the ion distribution significantly alter the generation mechanism. Most likely the thermal field-aligned ion population affects wave generation and moves the positive growth rate to high wave normal angles. To verify this assumption, we will combine the measured ion distribution and linear dispersion solver.

Refer to caption — Figure 1: Observation of oblique, hydrogen-band EMIC waves by THEMIS-A spacecraft on February 17, 2020. a) Wave magnetic field power spectrum, b) wave normal angle; top and bottom lines represent hydrogen ion $H^{+}$ and helium ion $He^{+}$ cyclotron frequencies, respectively; vertical dashed lines mark the time interval that is used to average the ion phase space density for subsequent investigations of wave dispersion properties. c-g) Ion pitch-angle distributions for different energy ranges as a function of time.

Both the field-aligned thermal ion population and hot, transversely anisotropic population can be well seen in Figure 2, where we plot the 1-min averaged (around the time of the most intense wave spectrum) ion distribution in $(v_{\parallel},v_{\perp})$ plane. The velocities are normalized to the Alfv\a’en velocity $v_{A}=B/\sqrt{4\pi nm_{p}}$ , where $n$ is the plasma density calculated from spacecraft potential, $m_{p}$ is proton mass. Compared with the isotropic distribution (shown in black dashed curves), observations clearly show a strong transverse anisotropy at $v_{\perp}>0.5v_{A}$ and field-aligned anisotropy at $v_{\perp}<0.25v_{A}$ . This $(v_{\parallel},v_{\perp})$ distribution is then passed to the LEOPARD solver [Astfalk \BBA Jenko (\APACyear2017)] to calculate the wave dispersion relation, where we can determine the frequency–wave normal angle region of positive growth rate and compare this with observations of very oblique EMIC waves. Figure 3 shows results of LEOPARD calculations: the positive growth rate is indeed confined to very oblique wave normal angles, peaking at $74^{\circ}$ , and to wave frequencies around $0.35-0.55f_{cH+}$ (note throughout the paper we also use $\Omega_{ci}=2\pi f_{cH+}$ ). These ranges of wave normal angles and frequencies are quite close to those observed by THEMIS (see Fig.1), which confirms that the ion distribution from Fig. 2 can reliably reproduce the main wave properties. Note that there is no positive growth rate for field-aligned (or small wave normal angle) waves (not shown).

To reveal the ion distribution that is responsible for the generation of oblique EMIC waves, we decompose the observed $(v_{\parallel},v_{\perp})$ distribution into three ion populations, each fitted by a bi-maxwellian distribution. Figure 4 shows (a) the observed distribution, (b) the cold ion population with $\beta\sim 10^{-2}$ (this population does not contribute to EMIC wave generation, but is needed to realistically match the total ion density with observations), (c) sum of cold and thermal, with $\beta_{\perp}/\beta_{\parallel}\sim 0.45$ , ion populations (the latter one is parallel anisotropic and used to represent the $<1$ keV field-aligned ion streams), (d) sum of cold, thermal, and hot, with $\beta_{\perp}/\beta_{\parallel}\sim 3.8$ , ion populations (the latter one is transversely anisotropic and provides free energy for EMIC generation). Comparison of panels (a) and (d) in Fig. 4 shows that this three-component fitting agrees well with the observed ion distribution. Using this fitted distribution, we then evaluate the contribution of each ion population to the EMIC wave dispersion and generation. Figure 5 shows the dispersion relations of EMIC waves with $74^{\circ}$ propagation angle from the observed ion distribution and from the fitted distributions in Figs. 4 (b-d). For comparison, we also show the cold plasma dispersion for field-aligned waves [Stix (\APACyear1962)]. Solutions of the wave dispersion for cold-only (blue trace) and cold $+$ thermal (cyan trace) populations are quite similar, apart from the stronger damping in the latter. Introducing the hot transversely anisotropic population (orange trace) results in positive growth rate, which is larger than the growth rate from the observed ion distribution, but occupies a similar frequency range. The main role of the thermal, field-aligned anisotropic population is to provide strong cyclotron damping of low wave normal angle waves: note that the energy of the first cyclotron resonance decreases with wave normal angle decrease and reaches $1$ keV (around the energy of the thermal population) for field-aligned waves. A secondary, yet important, role of the thermal population is in reducing the total ion anisotropy: although the anisotropy of the entire ion distribution, $\beta_{\perp}/\beta_{\parallel}=1.8$ , does not exceed much the threshold for field-aligned EMIC generation [Yue \BOthers. (\APACyear2019)], the anisotropy of the hot population is sufficiently large ( $\beta_{\perp}/\beta_{\parallel}\approx 4$ ) to produce very oblique waves [<]see the analogical mechanism of oblique whistler-mode wave generation by the highly anisotropic electron component in¿Gary11:pop

We also investigate the contribution of different resonances to the oblique EMIC wave dispersion and growth rate. Figure 6 shows the result for the first cyclotron resonance only, $n=1$ (orange), first cyclotron and Landau resonances $|n|\leq 1$ (cyan), and for all resonances up to $|n|=10$ (red). There is barely any difference between wave dispersions for these three cases, i.e., the wave dispersion is mainly provided by the ion population contributing to the first cyclotron resonance. The comparison of wave growth rates shows that the same ion population is responsible for wave generation, whereas Landau damping reduces the magnitude of the growth rate, especially at small wave frequencies.

Figure 7 further confirms the principal role of the first cyclotron resonance in generating the observed oblique EMIC waves, by combining contours of constant phase phase density (from Fig. 2) and resonant conditions for two approximations. The grey shaded region in the left panel shows the cyclotron resonant velocities calculated from the dispersion relation for $74^{\circ}$ wave normal angle and wave frequencies of positive growth rate, $\omega/\Omega_{ci}\in[0.38,0.55]$ . Comparison of constant phase phase density contours (blue traces) and resonance curves (contours of constant energy in the wave rest frame, shown in red) demonstrates that within the grey shaded region the gradient along the resonance curves (seen from the comparison of constant energy curves, shown in dashed black, and phase space density contours, shown in blue) corresponds to an increase in the phase space density, which will drive EMIC wave growth [Lyons \BBA Williams (\APACyear1984)]. Similarly, the same conclusion can be drawn for the resonant velocity range for monochromatic waves, with a frequency corresponding to the peak wave intensity in observations and variations of the resonance condition along magnetic latitudes (right panel): the positive gradient of the ion phase space density along resonance curves within the purple shaded region will amplify the waves as they propagate away from the generation region near the equator.

3.2 Comparison of ion distributions during field-aligned and oblique EMIC events

Figure 8(b-e) shows four more examples of THEMIS observed very oblique EMIC waves in the dawn flank. All these events share similar properties of the event discussed in Section 3.1:

•

Waves are proton band EMIC waves with clear maximum of the wave intensity around $[0.3,0.5]$ of local proton cyclotron frequency and with wave normal angles typically exceeding $60^{\circ}$ (left panels).
•

Ion distribution functions (middle panels) observed around the oblique EMIC wave burst include a very anisotropic hot (a few keVs) ion population ( $\beta_{\perp}/\beta_{\parallel}>3$ ) and a field-aligned anisotropic thermal ( $\leq 1$ keV) ion population. The latter can lead to strong cyclotron damping of field-aligned EMIC waves.
•

Combining the measured ion distribution and the linear dispersion solver, LEOPARD, we show positive growth rates for very oblique EMIC waves (right panels), as well as damping for the field-aligned waves (not shown). Therefore, the most important condition for very oblique EMIC wave generation is likely the combination of transversely anisotropic hot ions (providing wave growth via cyclotron resonance) and field-aligned thermal ions (providing cyclotron damping of small wave normal angle waves).

To verify the importance of the thermal field-aligned population for very oblique EMIC wave growth, we further selected a typical field-aligned EMIC wave event observed by THEMIS in the dusk flank. As shown in Figure 8(f), in the absence of field-aligned thermal ions, the field-aligned waves can be generated via cyclotron resonance with transversely anisotropic hot ions. Moreover, in the absence of the field-aligned population, the anisotropy of the entire ion distribution is mainly determined by the anisotropy of hot ions, which does not need to be as large as in events with oblique waves in order to generate field-aligned waves. As a result, in the event from Fig. 8(f) the hot ion anisotropy, $\beta_{\perp}/\beta_{\parallel}=1.3$ , is sufficient to generate field-aligned EMIC waves, but insufficient to generate oblique waves (the growth rate of oblique waves is negative for this event; not shown).

4 Discussion and conclusions

In this study we analyze several events with THEMIS observations of very oblique EMIC waves around the equator. During these events, the observed ion distributions consist of a highly transversely anisotropic ( $\beta_{\perp}/\beta_{\parallel}\sim 4$ , $>2$ keV) hot ion population and a field-aligned anisotropic thermal ion population ( $\beta_{\perp}/\beta_{\parallel}\sim 0.3$ , $<1$ keV). It is this field-aligned thermal ion population that prohibits the generation of field-aligned EMIC waves: the cyclotron resonant energy of waves with small wave normal angles is $\sim 1$ keV, where the transverse anisotropy is insufficient to drive these waves. Therefore, such thermal ( $<1$ keV) field-aligned ion population play a key role in producing very oblique EMIC waves, which are often observed on the dawn flank. The energy range and field-aligned anisotropy of this population suggest that these are likely ionospheric outflow ions [<]see statistics of ion pitch-angle distributions in¿[]Artemyev18:jgr:RBSP&THEMIS,Yue17:ions. Overlapping of such outflow, probably enhanced at the dawn flank due to strong plasma sheet electron precipitation driven by whistler-mode waves [Thorne \BOthers. (\APACyear2010), Ni \BOthers. (\APACyear2016)], and the hot ion population, likely drifted from the dusk flank after being injected from the plasmasheet [<]e.g.,¿Birn97:ion,Gabrielse14,Ukhorskiy18:DF, creates favorable conditions for the generation of very oblique EMIC waves. This further implies that very oblique waves are likely a result of magnetosphere-ionosphere coupling, in contrast to the more typical field-aligned waves generated in the dusk flank due to plasmasheet injections or on the day side due to magnetosphere compression by the solar wind [Yue \BOthers. (\APACyear2019), Jun \BOthers. (\APACyear2019), Jun \BOthers. (\APACyear2021)].

In addition to the possible resonant interactions between oblique EMIC waves and magnetospheric particles, as discussed in the introduction, a new resonant mechanism has been proposed recently that makes oblique EMIC waves potentially more important. It has been shown by \citeAHanzelka23:emic that very oblique and sufficiently intense EMIC waves may resonate with energetic electrons via the so-called fractional (or subharmonic) resonances [Lewak \BBA Chen (\APACyear1969), Smirnov \BBA Frank-Kamenestkiǐ (\APACyear1968)]. In contrast to cyclotron resonances with integer resonant numbers, the factional resonance is a purely nonlinear effect providing electron scattering in resonances with fractional numbers [Terasawa \BBA Matsukiyo (\APACyear2012)], which reduces the electron energy in resonance with EMIC waves to sub-relativistic values [Hanzelka \BOthers. (\APACyear2023\APACexlab\BCnt2)]. THEMIS observations of very oblique EMIC waves and the proposed formation mechanism of these waves suggest that sub-relativistic electron precipitation on the dawn side [<]see statistics of such precipitation in¿[]Tsai23 may be partly driven by EMIC waves, not exclusively by whistler-mode waves.

Figure 9 shows the energies of electrons and ions in resonance with very oblique EMIC waves (we use a wave normal angle of $74^{\circ}$ and plot resonant energies as a function of wave frequency for typical plasma frequencies as from Figs. 8). Note that the electron resonant energies for fractional resonances fall between the regions bounded by the first cyclotron resonance, $n=-1$ , and Landau resonance, $n=0$ [Hanzelka \BOthers. (\APACyear2023\APACexlab\BCnt2)]. Comparing the results for the cold plasma dispersion ( $\beta=10^{-2}$ ) and for the observed hot plasma with $\beta\approx 1.7$ , we can see a higher resonant energy for the hot plasma case due to a decrease of the wave number at a fixed frequency [Silin \BOthers. (\APACyear2011)]. Cyclotron and Landau resonant energies for protons are within $[1,10]$ keV for $\omega/\Omega_{cp}<0.5$ , where most of wave power is observed (see Fig. 8 (a)). This energy range allows oblique EMIC waves to heat thermal ( $\sim 1$ keV) ions and scatter ring current ( $\sim 10$ keV) ions into the loss cone (note that these resonant energies are calculated at the equator, which will increase in the off-equatorial region). The range of resonant energies for electrons, on the other hand, is much wider: from $\sim 1$ eV in Landau resonance to $1$ MeV in cyclotron resonance. The fractional resonances with a resonance number $\in[0,1]$ will fill this gap and allow EMIC waves to also scatter hundreds of keV electrons. This is likely the most interesting and potentially important implication of very oblique waves.

To conclude, using THEMIS observations in the inner magnetosphere, we have investigated the generation mechanism of very oblique EMIC waves. Six typical events of such waves are observed at the dawn flank, in contrast to the more common field-aligned waves observed at the dusk and noon sectors. Very oblique EMIC waves are usually accompanied by ion distributions consisting of two main populations (except for the cold plasma population contributing to the total density). 1) A thermal ion population, at $\leq 1$ keV, with the field-line anisotropy providing the cyclotron damping of field-aligned EMIC waves and potentially reducing the Landau damping of oblique EMIC waves. This population shares the same properties of ionospheric outflow as reported previously for the inner magnetosphere [<]e.g.,¿Yue17:ions. 2) The hot ion population, at $>5$ keV, with a strong transverse anisotropy ( $\beta_{\perp}/\beta_{\parallel}\sim 4$ ) providing the cyclotron resonant growth of oblique EMIC waves with wave normal angles exceeding $60^{\circ}$ . These observations underline the importance of magnetosphere-ionosphere coupling in producing very oblique EMIC waves.

Acknowledgements.

We acknowledge the support of NASA contract NAS5-02099 for the use of data from the THEMIS Mission, specifically K. H.Glassmeier, U. Auster and W. Baumjohann for the use of FGM data (provided under the lead of the Technical University of Braunschweig and with financial support through the German Ministry for Economy and Technology and the German Center for Aviation and Space (DLR) under contract 50 OC 0302). Work of D.S.T., X.-J. Z., and A.V.A. are supported by NSF grant #2329897, and NASA grants #80NSSC20K1270, #80NSSC23K0403, #80NSSC23K0108 .

Open Research

THEMIS data is available at http://themis.ssl.berkeley.edu. Data access and processing was performed using SPEDAS V4.1 and its Python-based implementation, see \citeAAngelopoulos19 and [Grimes \BOthers. (\APACyear2022)].

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