ARTEMIS Observations of Plasma Waves in Laminar and Perturbed Interplanetary Shocks
L. Davis, C.A. Cattell, L.B. Wilson III, Z.A. Cohen, A.W. Breneman, E.L.M. Hanson
ARTEMIS Observations of Plasma Waves in Laminar and Perturbed Interplanetary Shocks
Lance Davis , C. A. Cattell , L. B. Wilson III , Z. A. Cohen , A. W. Breneman , E. L. M. Hanson School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota, USA NASA Goddard Space Flight Center, Greenbelt, Maryland, USA. Space Sciences Laboratory, UC Berkeley
Key Points: ● The two laminar shocks with burst data in the ramp have large amplitude waves to provide the needed resistivity for energy dissipation ●
Ion acoustic waves and waves in the ion acoustic frequency range with rapid temporal frequency change are the most common wave modes ●
Large amplitude electrostatic waves occur more frequently downstream of perturbed shocks, but amplitudes are comparable between shock types avis et al. , [2019] 1
Abstract
The ‘Acceleration, Reconnection, Turbulence and Electrodynamics of the Moon's Interaction with the Sun’ (ARTEMIS) mission provides a unique opportunity to study the structure of interplanetary shocks and the associated generation of plasma waves with frequencies between ~50-8000 Hz due to its long duration electric and magnetic field burst waveform captures. We compare wave properties and occurrence rates at 11 quasi-perpendicular interplanetary shocks with burst data within 10 minutes (~3200 proton gyroradii upstream, ~1900 downstream) of the shock ramp. A perturbed shock is defined as possessing a large amplitude whistler precursor in the quasi-static magnetic field with an amplitude greater than ⅓ the difference between the upstream and downstream average magnetic field magnitudes; laminar shocks lack these large precursors and have a smooth, step function-like transition. In addition to wave modes previously observed, including ion acoustic, whistler, and electrostatic solitary waves, waves in the ion acoustic frequency range that show rapid temporal frequency change are common. The ramp region of two laminar shocks contained a wide range of large amplitude wave modes. The ion acoustic waves observed in the ramp of laminar shocks could provide resistivities necessary for the energy dissipation. The wave occurrence rates for laminar shocks are higher in the transition region, especially the ramp, than downstream. One perturbed shock exhibited no waves with amplitudes ≳ Early theoretical work on quasi-perpendicular ( θ Bn ≥ 45°, where θ Bn is the angle between the upstream magnetic field and shock normal unit vector) collisionless shocks used the structure of the quasi-static magnetic field to classify the shocks as “laminar”, “quasi-laminar”, “quasi-turbulent”, and “turbulent” based on the upstream average fast mode Mach number M f and plasma beta β [see Greenstadt , 1985;
Mellott , 1985, and references therein]. Until recently [
Wilson III et al. , 2017], laminar shocks were usually thought to be low Mach number ( M f ≲ β ≲ Galeev and Karpman , 1963;
Sagdeev , 1966;
Tidman and Krall , 1971;
Biskamp , 1973;
Greenstadt et al. , 1975;
Mellott and Greenstadt , 1984;
Mellott , 1985;
Farris et al. , 1993]. These have step function-like ramps, with a sharp magnetic field change between the upstream and downstream regions. Note that laminar shocks may still exhibit upstream or downstream electromagnetic fluctuations [
Gary and Mellott , 1985]. Turbulent shocks are expected to occur for high Mach number ( M f ≳
3) and/or high beta ( β ≳
1) [
Sagdeev , 1966;
Kennel and Sagdeev , 1967a,b;
Coroniti , 1970b;
Formisano and Hedgecock , 1973a,b;
Formisano et al. , 1975;
Wilson III et al. , 2012]. Turbulent shocks generally possess some feature in the ramp region resulting in a non-laminar structure. Such a non-laminar avis et al. , [2019] 2 structure could be generated when a whistler precursor has sufficient amplitude, with respect to the amplitude of the shock ramp, to cause perturbations in the ramp structure [
Wilson III et al. , 2017]. This division between laminar and turbulent has recently been questioned by
Wilson III et al. [2017], who looked at the structure of low Mach number ( M f ≤ 3), low beta ( β up ≤ 1), quasi-perpendicular ( θ Bn ≥ 45°) shocks and found that ~78% of these shocks had clear whistler precursors. When the maximum peak-to-peak amplitudes of these precursors ẟB precursor was compared with the difference between the upstream and downstream average magnetic field magnitudes ΔB , the average (median) value of ẟB precursor /ΔB was ~2.2 (~1.1). With such large normalized amplitudes, doubts have been raised about the traditional classification of these shocks as “laminar.” They also note that whistler precursors may be even more common, but could not be resolved without higher (> 11 samples/s) quasi-static magnetic field sampling rates. Whistler precursors have also been observed to be common occurrence over a range of plasma conditions ( M f between 1.2-2.6, β up between 1-16) [ Russell et al.,
Ramírez Vélez et al. , 2012;
Kajdič et al. , 2012]. Whistler precursors are a manifestation of dispersive radiation [
Tidman and Northrop , 1968;
Fairfield et al. , 1974;
Mellott and Greenstadt , 1984], one mechanism through which collisionless shocks may transform bulk flow kinetic energy into other forms of energy. Several other mechanisms have been proposed, including wave-particle interactions [
Sagdeev, 1966; Coroniti , 1970a;
Gary , 1981], particle reflection [
Edmiston and Kennel , 1984;
Kennel et al. , 1985;
Kennel , 1987;
Bale et al. , 2005;
Su et al. , 2012], and macroscopic quasi-static field effects [
Scudder et al. , 1986a,b,c]. At low M f , theory suggests that dispersive radiation and wave-particles interactions dominate [ Kennel et al. , 1985]. Because this study focuses on low Mach number shocks ( M f ≲ Tidman and Northrop , 1968;
Fairfield et al. , 1974] are often observed and can dissipate energy through several processes, including generation of higher frequency waves by creating electron temperature anisotropies or current-driven instabilities [
Gary , 1981;
Hull et al. , 2012], acceleration of halo electrons and thermal ions [
Wilson III et al. , 2012;
Chen et al. , 2018], and deflection and modulation of core particles [
Goncharov et al. , 2014]. Wave-particle interactions dissipate energy through anomalous resistivity, shorthand for the energy and momentum exchange between the electromagnetic fields and particles [
Sagdeev, 1966; Coroniti , 1970a;
Gary , 1981;
Papadopoulos , 1985;
Breneman et al. , 2013;
Wilson III et al. , 2007, 2010, 2012, 2014a,b].
Goodrich et al . [2018] showed that ion acoustic waves may be indicators or facilitators of momentum transfer between reflected and incident ion populations, thus linking these waves to the transformation of bulk flow kinetic energy. In addition, work by
Wang et al. [2020] shows how reflected ions interacting with incident ions generate Debye-scale electrostatic fluctuations, starting as ion acoustic waves that trap ions and decay into electrostatic solitary waves.
Krasnoselskikh et al. [2013] provides a review of the quasi-perpendicular bow shock and the inferred dissipation mechanisms based on Cluster observations. Some of the types of waves observed near shocks include: magnetosonic whistler mode waves [
Fairfield et al. , 1974;
Coroniti avis et al. , [2019] 3 et al. , 1982], solitary waves [
Bale et al. , 1998], ion acoustic waves [
Fredricks et al. , 1968;
Rodriguez and Gurnett , 1975;
Gary et al. , 1975;
Gurnett et al. , 1979a,b;
Formisano and Torbert , 1982;
Fuselier and Gurnett , 1984;
Balikhin et al. , 2005;
Hull et al ., 2006], Langmuir waves [
Filbert and Kellogg,
Kellogg , 2003 and references therein;
Pulupa and Bale , 2008], and electron cyclotron harmonic waves and waves associated with the electron cyclotron drift instability [
Wilson III et al. , 2010, 2014a,b;
Breneman et al. , 2013;
Goodrich et al. , 2018]. Because
Wilson III et al. [2017] showed that whistler precursors perturbing the ramp region are a common occurrence for low Mach number shocks, it is important to examine whether there is a difference in the transformation of the bulk flow kinetic energy into other forms between laminar and perturbed shocks. Since most methods of energy transformation result in processes involving wave-particle interactions, one approach to this question is analyzing waves observed near shocks. This gives rise to several questions about the possible role of higher frequency waves: Are there differences between laminar and perturbed interplanetary shocks in the wave modes and amplitudes observed near/within the ramp region? Are there differences in the number and/or duration of wave packets observed? To answer these questions, we analyze ARTEMIS long duration electric and magnetic wave burst captures to examine waves at frequencies of ~50-8000 Hz near the ramps of 11 interplanetary (IP) shocks. We briefly review waves in this frequency range. Whistler mode waves are electromagnetic, right-hand polarized, and occur in the frequency range between the ion cyclotron and electron cyclotron frequencies [e
Beinroth and Neubauer , 1981;
Lin et al. , 1998;
Ramírez Vélez et al. , 2012]. They have been observed both upstream and downstream of IP shocks [
Fairfield , 1974;
Coroniti et al. , 1982;
Russell et al. , Ramírez Vélez et al. , 2012;
Kajdič et al. , 2012;
Wilson III et al. , 2013]. In the solar wind, whistlers have been observed in two different frequency bands, a lower and a higher. Lower frequency whistlers, with frequencies from the ion cyclotron frequency up to the lower hybrid frequency, are often whistler precursors, discussed above [
Fairfield et al. , 1974;
Hoppe et al. , 1982, 1983;
Russell and Hoppe , 1983;
Russell et al. , Ramírez Vélez et al. , 2012;
Kajdič et al. , 2012]. The higher frequency band of whistlers are most commonly observed near ~0.1-0.3 f ce [ Breneman et al. , 2010;
Hull et al. , 2012;
Wilson III et al. , 2013;
Giagkiozis et al ., 2018], with frequencies up to the electron cyclotron frequency. These higher frequency, narrowband whistlers can have large amplitudes (some > 40 mV/m, in comparison to most whistler amplitudes near ~5 mV/m), and are usually observed at stream interaction regions (~88% of the time) and sometimes at IP shocks (~17%) [
Breneman et al. , 2010]. The whistler mode waves observed in the electric and magnetic field burst data we use in this study are of the higher frequency band. Ion acoustic (IA) waves are electrostatic, are linearly or elliptically polarized, have a rest frame frequency up to the ion plasma frequency, f pi , (typically observed at ~1-10 kHz in the solar wind), and travel parallel or obliquely to the ambient magnetic field [ Gurnett et al. , 1979a,b;
Fuselier and Gurnett , 1984;
Akimoto and Winske , 1985;
Hess et al. , 1998;
Balikhin et al. , 2005]. These waves are thought to be generated by ion-ion or electron-ion drifts [
Gary et al. , 1975;
Formisano and Torbert , 1982;
Fuselier and Gurnett,
Goodrich et al. , 2018, 2019]. A avis et al. , [2019] 4 number of studies [
Gurnett et al. , 1979a,b;
Thomsen et al. , 1985;
Hess et al. , 1998;
Wilson III et al. , 2007, 2014a,b;
Chen et al. , 2018;
Goodrich et al. , 2018, 2019] have concluded that ion acoustic waves are important in dissipating energy in low Mach number shocks. Some waves observed in this study displayed characteristics similar to ion acoustic waves. They are electrostatic, have a peak spacecraft-frame frequency between ~1-10 kHz, are linearly or elliptically polarized, and have the peak power primarily parallel to the ambient magnetic field. However, they also show clear frequency change over time. The observed frequency change, which lasts through the ~10-100 ms duration of a given wave packet, is not easily explained by a change due to Doppler shift. The frequency change is not associated with either a rotation or change in magnitude of the quasi-static magnetic field, sampled at 128 samples/s. The change might be due to changes in density; this cannot be ruled out due to the low sampling rate of density measurements. For the purposes of this paper, these waves are referred to as time-dependent frequency electrostatic waves (TFES). Similar wave behavior has also been observed in a STEREO study of IP shocks [
Cohen et al. , 2019b]. Electron cyclotron harmonic waves (ECH) and waves associated with the electron cyclotron drift instability (ECDI) have some characteristics similar to ion acoustic waves [
Matsumoto and Usui , 1997;
Usui et al. , 1999;
Wilson III et al. , 2010;
Breneman et al.
Goodrich et al ., 2018]. The characteristic that distinguishes ECH waves or waves associated with the ECDI from ion acoustic waves is the presence of integer or half-integer harmonics of the electron cyclotron frequency [
Wilson III et al. , 2010;
Breneman et al.
Goodrich et al ., 2018]. Other features, such as an asymmetric oscillation about a mean or comma-shaped hodograms [
Wilson III et al. , 2010;
Breneman et al.
Goodrich et al ., 2018], have also been used to distinguish these modes. However in the absence of clear harmonics, as discussed by
Breneman et al. [2013], it can be difficult to distinguish between these modes. Due to the typically short wavelength ( ≲ Breneman et al.
Giagkiozis et al. , 2011] are available to make this distinction when harmonics are not present, this type of analysis was beyond the scope of this study. Thus any waveform, including ECH waves and waves associated with the ECDI, exhibiting the characteristics of an ion acoustic mode (i.e., linearly or elliptically polarized, f ~ 1-10 kHz in the spacecraft-frame, mostly parallel or oblique to the magnetic field) was identified as an ion acoustic mode. If the characteristics of a TFES mode (i.e., same as ion acoustic, but with clear frequency change over time) were exhibited, the waveform was identified as a TFES mode. Electrostatic solitary waves are nonlinear bipolar pulses in the electric field which are parallel to the ambient magnetic field and are often associated with electron beams [ Ergun et al ., 1998;
Cattell et al ., 2005;
Franz et al ., 2005]. Solitary waves have been observed at Earth’s bow shock [
Bale et al. , 1998], IP shocks near 1 AU [
Wilson III et al. , 2007] and near 8.7 AU [
Williams et al. , 2005], and as a common occurrence in the lunar wake [
Hutchinson and Malaspina , 2018]. They have also been observed within the magnetosphere, where they potentially act as a mechanism for energy dissipation and particle energization, in the auroral zone [
Ergun et al. , avis et al. , [2019] 5 1998], magnetotail and plasma sheet boundary layer [ Matsumoto et al. , 1994;
Andersson et al. , 2009], reconnection regions [
Cattell et al. , 2002, 2005], and the radiation belt [
Mozer et al. , 2013]. In section 2, we describe the instrumentation and methodology for the identification and selection of IP shocks and the wave modes observed. In section 3, we present one example of a laminar shock and one of a perturbed shock. In section 4, we describe the statistics for the 11 shocks and an analysis of the wave modes observed between different shock types. Discussion and conclusions are presented in section 5 and 6, respectively.
The two ARTEMIS spacecraft entered lunar orbit late in 2011. When the spacecraft are not in Earth’s magnetosphere or in the lunar wake, they directly measure the solar wind. On board each spacecraft are instruments to measure the magnetic and electric fields, plasma velocity, temperature, and density, and particle distributions [
Angelopoulos , 2011]. The instruments that measure the fields are the Electric Field Instrument (EFI), the Flux Gate Magnetometer (FGM), and the Search Coil Magnetometer (SCM). The EFI measures the 3D electric field and waves [
Bonnell et al. , 2008]. The SCM measures the 3D magnetic field fluctuations and waves [
Roux et al. , 2008]. This study uses the wave burst mode captures from the EFI and SCM, which have a duration on the order of ~10 s [
McFadden et al. , 2008] and have a sampling frequency of 16384 and 1024 samples/s, respectively [
Bonnell et al. , 2008;
Roux et al. , 2008]. The FGM measures the 3D quasi-static magnetic fields [
Auster et al. , 2008]. This study used FGM data from either the fast survey or particle burst modes, which have a sampling rate of 4 and 128 samples/s, respectively [
Auster et al. , 2008]. For the events in panel E of Figure 1 and panels B, E, F, G in Figure 2, the particle burst FGM data contained oscillations on the order of a spin period. To remove this artificial effect, a data cleaning algorithm was applied. This algorithm removes periodic noise signals from time-series data for the "target" frequencies: a fundamental frequency set to correspond to the spin period of the spacecraft (~4.3 s) and its harmonics up to the Nyquist frequency. This algorithm is a form of spectral pre-whitening [
MacDonald, avis et al. , [2019] 6 The ElectroStatic Analyzer (ESA) measures the electron and ion distributions over the energy range of several eV to 30 keV for electrons and several eV to 25 keV for ions [
McFadden et al. , 2008]. Three modes are available: full, reduced, and burst packets. These modes are available when the spacecraft is in slow or fast mode, fast mode, or burst mode, respectively. A full packet has a low time resolution (~1 sample/min) but high angular resolution (~2°); a reduced packet has higher time resolution (~15 samples/min) but lower angular resolution (~22°); burst packets have both the higher time resolution and angular resolution [
McFadden et al ., 2008]. This study used reduced packets or burst packets for the higher time resolution. ARTEMIS is the only mission primarily in the solar wind with high resolution particle measurements as well as both the quasi-static and 3D wave electric and magnetic field measurements. For example, STEREO lacks a search coil and Wind cannot transmit three components of both field types for the same time interval. ARTEMIS also provides the longest waveform burst captures (~10 s) commonly taken in the solar wind, enabling observations of the evolution and structure of the entire ramp. In comparison, the longest burst captures on STEREO are 2.1 s; for Wind, the burst captures are typically 17 ms, with the longest being 1.13 s. Note that Cluster and MMS have both made burst measurements throughout the ramps of the quasi-perpendicular bow shock [
Balikhin et al. , 2005;
Krasnoselskikh et al. , 2013;
Goodrich et al. , 2018, 2019]. ARTEMIS has a large database (>130) of observed IP shocks. While other missions with similar or higher quality instruments, such as MMS [
Cohen et al. , 2019a;
Hanson et al. , 2019], have made observations of IP shocks, the number of IP shocks these missions have observed is small in comparison to ARTEMIS. Thus ARTEMIS provides a unique opportunity to study the structure of IP shocks and the associated wave generation and particle energization.
An initial list of IP shock events observed by ARTEMIS between November 2011 and June 2017 was compiled by finding discontinuities in the magnetic field and the ion velocity and density. Only times when ARTEMIS was outside of both the magnetosphere and the lunar wake were considered. If the discontinuity fulfilled the following criteria [
Lumme et al. , 2017] on the ratio of downstream to upstream magnetic field, ion density and ion temperature, as well as the change in the velocity magnitude, within the uncertainties (relevant uncertainties listed in Table 1), avis et al. , [2019] 7 𝐵 𝑑𝑜𝑤𝑛 /𝐵 𝑢𝑝 ≥ 1.2 𝑁 𝑑𝑜𝑤𝑛 /𝑁 𝑢𝑝 ≥ 1.2 𝑇 𝑖,𝑑𝑜𝑤𝑛 /𝑇 𝑖,𝑢𝑝 ≥ 1/1.2 | 𝛥 𝑉| = |𝑉 𝑑𝑜𝑤𝑛 − 𝑉 𝑢𝑝 | ≥ 20 𝑘𝑚/𝑠 it was considered a shock and added to the initial event list. A list of IP shocks observed by the ARTEMIS spacecraft between June 2008 and December 2011 (available at http://themis.ssl.berkeley.edu/data/themis/events/ ) was also used. Only quasi-perpendicular ( 𝜃 𝐵𝑛 ≥ 45 ∘ ) shocks with wave burst data within 10 minutes, either upstream or downstream, of the ramp were selected for this study. The 10 minute interval was chosen because some whistler precursors (e.g. Panel A, Figure 1) can last this long. Other studies have found that precursors, while typically only extending within ~30,000 km, could extend up to 100,000 km [ Kajdič et al. , 2012;
Ramírez Vélez et al. , 2012]. The 10 minute interval corresponds to an average (median) distance of 214,000 (220,000) km and to an average (median) of 3330 (3200) proton gyroradii ( ⍴ gi ) upstream and 1930 (1160) ⍴ gi downstream. The specific distances and number of proton gyroradii are different for each event and depend on local plasma parameters. Note that roughly 56% of the burst capture time analyzed in this study occurred within 500 ⍴ gi of the ramp and 80% occurred within 1500 ⍴ gi . Due to telemetry limitations, of our initial list of >130 shocks, only eleven shocks fulfilled these requirements. For two of the shocks, both spacecraft had burst data available, and the observations from each spacecraft were considered individually, yielding the 13 observations in the event list in Table 1. Of the 13 events, 6 had burst captures overlapping with the transition region, three laminar and three perturbed. Three events, two laminar and one perturbed, had burst captures covering the entire ramp region. Although 13 events does not provide a statistically significant database to reach conclusions for all low Mach number, quasi-perpendicular IP shocks, it is sufficient to fulfill the goal of this study to explore the differences in wave activity in the range of ~50-8000 Hz between laminar and perturbed shocks. To minimize the impact secondary ion populations, such as gyrating and field-aligned ion beams reflected by the shock, have on the ion moment calculations, the core of the solar wind was isolated by utilizing a velocity mask when these secondary ion populations were present, most often in the upstream region [ Wilson III et al. , 2014a, appendix C, and references therein]. From the solar wind core distribution, the ion velocity, density, and temperature moments were recalculated. This recalculation, which was needed because the ESA instrument was originally intended for measuring the slow and hot electron and ion distributions within the magnetosphere, not the fast and cold distributions in the solar wind, was made with the same methodology as described by
Wilson III et al. [2014a]. These recalculated moments, as well as the background magnetic field and electron temperature, were used in the Rankine-Hugoniot (RH) equations to estimate the shock normal and other shock parameters [
Koval and Szabo , 2008], similar to other studies [
Wilson III et al ., 2014a,b, 2016;
Kanekal et al. , 2016]. If there are large amplitude fluctuations near the ramp region, the fluctuations could locally perturb the shock normal and modify the RH equations by affecting the incident flow prior to crossing the shock [
Scholer and avis et al. , [2019] 8
Belcher , 1971]. To avoid this issue, intervals, typically 2-3 minutes, for the upstream and downstream inputs to the RH equations were taken from quiet, undisturbed regions ~1-10 minutes from the shock ramp. The ion velocity, density, and temperature were taken from the corrected ion moments which had a resolution of ~15 samples/min. Uncertainties in the results, which can be significant, are listed in Table 1. For four events, analysis found no stable shock solution with M f > 1.0 within the uncertainties for any of the standard methods for identifying shocks [ Abraham-Shrauner and Yun , 1976]. Two of these events, and one satisfying M f > 1.0, did not pass the criteria that |ΔV| > 20 km/s within the uncertainties after the ion moment corrections were made. This is likely due to inaccuracies because the ARTEMIS detectors were not optimized for solar wind measurements. For these cases, Wind and ACE observations of these shocks (located at L1, whereas ARTEMIS is at 1 AU) were utilized to determine shock parameters and all events passed the above shock criteria. For the five events where there was no stable solution of the RH equations with M f > 1.0 and |ΔV| > 20 km/s within the uncertainties using the ARTEMIS data, the parameters calculated from Wind were used and are listed in Table 1. While large spacecraft separation perpendicular to the flow of the solar wind has been shown to be associated with sometimes large angular deviations of the calculated shock normal [ Szabo , 2005;
Koval and Szabo , 2010], it is not expected that the global structure of shocks evolve dramatically when the spacecraft separation is primarily parallel to the flow of the solar wind, as is the case for Wind and ARTEMIS, especially over a scale of only ~0.01 AU. Thus, we have used Wind-based shock parameters as an estimate. A number of other studies have used Wind similarly and found similar shock parameters between Wind and ARTEMIS [
Möstl et al. , 2012;
Kanekal et al. , 2016;
Oliveira and Samsonov , 2018;
Pope et al ., 2019]. We do note that shock features, such as ripples along the shock surface [
Terasawa et al. , 2005;
Neugebauer and Giacalone , 2005], may evolve from L1 to 1 AU, thus the Wind-based parameters provide only an estimate to the local shock parameters at the time of the ARTEMIS observations. Table 1 gives the date, satellite (using the THEMIS labels), shock classification, the ratios of the average downstream (subscript d) to upstream (subscript u) magnetic field, ion density, ion and electron temperature, the upstream plasma beta, difference between upstream and downstream ion velocities, fast mode Mach number, θ Bn , and the ratio of the fast mode Mach number to the first critical Mach number [ Edmiston and Kennel , 1984] for each event. For perturbed shocks, the ratio of the peak amplitude of the whistler fluctuation to the transition jump ( δB / ΔB ) is shown. The top (bottom) section of the table lists the shock parameters calculated from ARTEMIS (Wind) observations. Events with burst data overlapping with the transition region are denoted by an asterisk ( * ); double asterisks (**) denote a burst capture overlapped with the ramp region, and by extension, the transition region as well; supercritical shocks are denoted by a dagger ( † ). Note that all events satisfy the conditions of a quasi-perpendicular shock within the listed uncertainties of the fast mode Mach number and θ Bn . A more complete list of both ARTEMIS and Wind calculated parameters, including all uncertainties, for these events is available in a public repository ( https://doi.org/10.5281/zenodo.3475588 ). avis et al. , [2019] 9 ARTEMIS-Based Shock Parameters Date Probe ShockType 𝛿𝐵 𝑝𝑟𝑒 𝛥𝐵 𝐵 𝑑 𝐵 𝑢 𝑁 𝑑 𝑁 𝑢 𝑇 𝑖,𝑑 𝑇 𝑖,𝑢 𝑇 𝑒,𝑑 𝑇 𝑒,𝑢 β u |𝛥𝑉| (km/s) M f 𝜃 𝐵𝑛 𝑀 𝑓 𝑀 𝑐𝑟 † B L - 2.67 4.63 2.16 1.51 6.82 58.6±2.7 1.9±0.3 86°±6° 1.8±0.3 2015-06-24 † C P 0.9 2.20 1.94 5.52 1.53 0.34 155±6.4 3.2±1.9 56°±7° 1.4±0.9 2016-04-14** † C P 0.7 1.72 1.73 5.15 1.03 1.07 10.9±11.8 2.2±0.7 71°±6° 1.3±0.4 2017-02-24** B L - 1.81 2.39 3.74 1.47 0.39 30.6±21.2 1.5±0.5 76°±12° 0.7±0.2 2017-02-24* C L - 1.96 2.12 4.66 1.33 0.46 41.2±22.2 1.6±0.4 72°±8° 0.8±0.2 2017-05-20 C P 0.6 2.24 3.81 6.48 1.41 0.35 90.2±20.4 1.4±0.3 52°±17° 0.6±0.1 Wind-Based Shock Parameters 2012-02-20 C L - 1.97 2.57 1.80 1.22 0.55 65.7±14.0 1.9±0.6 74°±23° 0.9±0.3 2012-12-16 C L - 1.42 1.41 1.55 0.99 0.34 25.6±7.3 1.2±0.6 72°±5° 0.5±0.3 2013-05-18 C L - 1.70 1.77 1.58 1.22 0.19 59.5±4.7 1.5±0.2 76°±6° 0.6±0.1 2015-06-27* C P 0.7 1.79 3.58 2.63 1.52 0.06 26.7±15.8 1.0±0.4 51°±2° 0.4±0.1 2017-05-20* † B P 1.8 2.27 2.57 2.18 1.53 0.37 338±11.7 2.1±0.7 74°±12° 1.0±0.3
Table 1: Shock parameters for the 13 events in this study. The top and bottom sections of the table list the shock parameters calculated from ARTEMIS and Wind observations, respectively. Dates with asterisks ( * ) denote a burst capture overlapped with the transition region; double asterisks (**) denote a burst capture overlapped with the ramp region; daggers ( † ) denote supercritical shocks. Subscript u (d) represents the average upstream (downstream) value. Underlined values are from Wind calculations. The classification of each shock as laminar or perturbed utilized a definition similar to that of
Wilson III et al. [2017], where a normalized amplitude between the whistler precursor and the magnetic field jump greater than 10% was considered non-laminar. This definition has been modified to be stricter (10% was changed to 33%), avis et al. , [2019] 10 𝛿 𝐵 𝑝𝑟𝑒 𝛥𝐵 ≳ , 𝛥𝐵 = (𝐵 𝑢𝑝 − 𝐵 𝑑𝑜𝑤𝑛 ) , where B up and B down are the average upstream and downstream magnetic field magnitudes, respectively, and δB pre is the maximum peak-to-peak amplitude of the observed whistler precursor. We refer to an event as perturbed if it had a precursor satisfying this condition. Of the 13 events in this study, 6 were laminar and 7 were perturbed. The ramp region, the region of the largest change in the magnetic field, for each event was found through visual inspection. Any potential foot, overshoot, or precursor was excluded from the ramp region. The ramp regions found in this study resemble the ramp regions found in other studies [ Hobara et al. , 2010;
Mazelle et al. , 2010]. The transition region is the region from the undisturbed upstream to the nominal downstream [
Schwartz and Burgess , 1991;
Wilson III , 2016], including any whistler precursor, foot, or overshoot. Note that the ramp region is contained within the transition region. avis et al. , [2019] 11 Figure 1: (left) The quasi-static magnetic field magnitude and GSE components near the transition region (shaded in grey) for the 7 perturbed events. The magnitude of the jump from the upstream to the downstream field is labeled in black. The number of samples per second is listed in the lower right. The black oval near the time-axis denotes the zoom-in region to the right. (right) Zoom-in of the magnetic field near the ramp (shaded in yellow). Maximum amplitude of the whistler precursor is labeled in black. avis et al. , [2019] 12 Figure 2: (left) The quasi-static magnetic field magnitude and GSE components near the transition region (shaded in grey) for the 6 laminar events. The magnitude of the jump from the upstream to the downstream field is labeled in black. The number of samples per second is listed in the lower right. The black oval near the time-axis denotes the zoom-in region to the right. (right) Zoom-in of the magnetic field near the ramp (shaded in yellow). avis et al. , [2019] 13 Figure 1 shows the magnetic field magnitude and GSE components for each perturbed event; Figure 2 shows the same quantities for each laminar event. The transition region is shaded grey in the left column; the ramp region is shaded in yellow in the right column. The difference between the upstream and downstream average magnetic field magnitude is shown for each event. Figure 1 also shows the maximum peak-to-peak amplitude of the whistler precursor for each event. The sampling rate of the magnetic field is listed in samples/s (sps); note that events in panels A and G of Figure 1 and panel D of Figure 2 had a sampling rate of 4 samples/s. Panel D of Figure 2 shows several under resolved fluctuations near the start of the ramp region, but, due to the low sample rate, it is not possible to determine the amplitude of this fluctuation or if it is a whistler precursor. For this reason, the event was classified as laminar. Because both
Ramírez Vélez et al. [2012] and
Wilson III et al. [2017] found no relation between whistler precursors, a feature associated with perturbed shocks, and θ Bn , M f , or any other shock parameter, we have focused this comparative study of plasma waves on laminar and perturbed shocks instead of θ Bn and M f . All events had M f ≲
3; there was no significant difference between the average M f for laminar and perturbed shocks (average of ~1.62 for laminar, ~1.64 for perturbed). The average θ Bn (~76° for laminar; ~65° for perturbed) differed by ~11° between the shock types. All six laminar events had a θ Bn range from 72-86°; three of seven perturbed events had θ Bn < 70°. Potential bias in our data set based on M f or θ Bn is explored in Section 4. The average downstream to upstream electron temperatures did not vary significantly between shock types (average of 1.29 for laminar, 1.33 for perturbed). However, the average downstream to upstream ion temperature ratio for perturbed shocks was roughly twice that for laminar shocks (average of 2.58 for laminar, 4.65 for perturbed). Properties of the whistler precursors are given in Table 2. Note that the amplitude ratio uses maximum peak-to-peak amplitude of the precursor. The listed frequency is the frequency at peak power found by using Morlet wavelets transforms [ Morlet et al. , 1982;
Morlet , 1982].
The range of amplitude ratios (0.4-1.8) and frequencies (0.04-1.55 Hz) is consistent with
Wilson III et al. [2017]. Our range of frequencies are also consistent with the findings of
Ramírez Vélez et al. [2012]. By definition, every perturbed event in this study had an associated whistler precursor. avis et al. , [2019] 14 Date Probe f (Hz) 𝛿𝐵 𝑝𝑟𝑒 𝛥𝐵 f and amplitude with respect to the difference between the upstream and downstream average magnetic field magnitude 𝛿𝐵 𝑝𝑟𝑒 𝛥𝐵 of the whistler precursor are listed. We use the term wave packet to refer to a subinterval of a wave burst capture which contains an identifiable wave mode of sufficient power above the background. Our algorithm examined only the electric field power spectrum to find subintervals within each burst capture with power above 10 -3 mV /m /Hz, which were then marked as potential wave packets. Note that due to the power threshold, wave packets with low ( ≲ A| were calculated using all three field components, |𝐴| =√𝐴 𝑥2 + 𝐴 𝑦2 + 𝐴 𝑧2 where A i is the i th component of the A field measured from peak-to-peak. Wave amplitudes observed in this study are large compared to both the motional electric field (~few mV/m) and amplitudes inferred from other studies using spectral data [ Gurnett et al. , 1979a,b;
Lin et al. , 1998]. All data were rotated into magnetic-field aligned coordinates (FAC), where the z -axis was parallel to the ambient magnetic field, and the x- and y-axes were perpendicular to both the ambient magnetic field and to each other. To identify wave modes, characteristics such as peak frequency, waveform structure, and polarization were used. Figure 3 shows examples of the wave modes observed in this study: whistler, ion acoustic, TFES, and solitary waves. Panels A, B, and C in Figure 3 show the same avis et al. , [2019] 15 time interval observed by THC on 2013-07-09. Figure 3a illustrates the process for identifying wave modes from wave packets. These panels show the electric and magnetic field data containing two wave packets identified by the algorithm, each highlighted in blue, and also the surrounding fields. Initial inspection would suggest there are three waves: two with peak frequency ~1000 Hz corresponding to each wave packet and one, at lower amplitudes, with peak frequency ~250 Hz observed in association with both packets and the adjacent intervals. To study these two different wave modes, a bandpass filter was applied. The lower bandpass (3b), from 100 to 400 Hz, showed activity in both the electric and magnetic field. The magnetic field hodograms showed right-handed, nearly circular polarization. The electric field hodograms did not show clear polarization, likely because of contamination by the second, higher frequency wave not removed by the bandpass filter. The peak frequency of ~0.38 f ce and the right-handed magnetic field polarization identified this wave as a whistler mode wave. The higher bandpass (3c), from 400 to 8000 Hz, coincides roughly with the ion acoustic frequency range of ~1-10 kHz. Hodograms of the electric field showed nearly linear polarization. The highest amplitude component of the electric field was parallel to the magnetic field. The power spectrum of the parallel component of the electric field showed no change in frequency for either wave packet. The peak frequency of ~1 kHz and linear polarization identified these waves as ion acoustic waves. These distinct ion acoustic waves were identified as two waves rather than one because the power dropped below our power threshold for ≳
50% of either adjacent wave packet duration. Although often each wave packet contained only a single wave mode, sometimes, as in this example, two different wave modes were identified in the same time interval. For the interval shown in Figure 3d, observed by THC on 2015-06-24, the power spectrum showed the peak frequency was within the ~1-10 kHz range characteristic of ion acoustic waves, but was also increasing with time over an interval of ~0.04 s, so the wave was identified as a TFES wave mode. On 2015-06-27, THC observed short pulses in the electric field. These pulses, with the largest amplitude component parallel to the magnetic field, shown in Figure 3e, were identified as solitary waves. Also shown in the power spectrum is an ion acoustic mode wave, identified from the peak frequency of ~4500 Hz and polarization (not shown). Note that even with the EFI sampling rate of 16384 samples/s, electrostatic solitary waves were often under resolved, so amplitudes listed for this wave mode may be underestimates. avis et al. , [2019] 16 Figure 3: Examples of observed wave modes. (a) Electric (left) and magnetic (right) field showing two wave packets, highlighted in blue. A whistler wave and two ion acoustic waves are observed. (b) An example whistler wave. The electric (left) and magnetic (right) field from A with a bandpass from 100 to 400 Hz applied are shown. (c) Example ion acoustic waves. The electric field (left) from A with a bandpass from 400 to 8000 Hz applied and its power spectrum (right) are shown. (d) An example TFES wave. The electric field (left) and its power spectrum (right) are shown. (e) Example solitary waves. Each pulse in the electric field (left) and power spectrum (right) corresponds to a solitary wave. Hodograms for time intervals with the black bars in the electric and/or magnetic field in B, C, and D are shown beneath their respective plot. The starting point is denoted with a plus. avis et al. , [2019] 17
Two illustrative events with burst captures in the transition region are discussed in detail. The 2015-06-21 event was a laminar shock with burst data covering the entire transition region, including the ramp, as well as ~20 s of burst data in the downstream region within 30 seconds (15 ⍴ gi ) of the shock ramp. The 2017-05-20 event observed by THB was a perturbed shock with ~10 s of burst data within the transition region, beginning within 1 second (<1 ⍴ gi ) from the ramp region. Four other events had burst data within the transition region, for a total of three for each shock classification. For brevity, only two of these six events are discussed, one for each shock type. An overview of the 2015-06-21 shock observed by THB is shown in Figure 4. The ramp occurred at roughly 16:27:35, as indicated by the jump in the magnetic field (4d) and the ion density, velocity, and temperature (4c), as well as electron energy flux enhancements, particularly at lower (~10-200 eV) energies. The electron pitch angle distributions between ~100 eV to 1 keV (4b.1-2) showed a large enhancement for roughly 20 s (~12 ⍴ gi ) after the shock across all pitch angles, with the largest increase, antiparallel to the magnetic field, lasting only ~5 s (~3 ⍴ gi ). Electrons with these energies were also observed upstream traveling parallel to the magnetic field away from the sun, consistent with strahl. These features were absent in lower energy distributions (4b.3). The parallel component of the electric field (4e) shows three ≳
150 mV/m peaks in the first burst capture. Only the first peak was associated with a wave mode. The other two peaks were deemed artificial after examination of voltage probe signals. Figure 5 shows the background magnetic field near the ramp (5a). One burst capture (5b,d) covered the entire ramp, shaded in yellow, and transition region, shaded in grey, and a second burst was taken 30 seconds (15 ⍴ gi ) downstream (4e). Because the wave burst magnetic field data started ~1 second after the electric field burst and lacked any significant power (>10 -5 nT /Hz) in the magnetic field burst frequency range of ~30-512 Hz, the particle burst (Nyquist frequency 256 Hz) magnetic field data (5b) and its total power spectrum (5c) are shown instead. The 3D electric burst data (5d) and its total power spectrum (5e) contained more and larger amplitude wave packets than any other burst captures in this study. The electron cyclotron frequency and ion plasma frequency are overplotted on the magnetic and electric power spectra, respectively. Figure 6 shows the electric field burst waveform (6a) and the associated power spectrum (6b), wave angle with respect to the local, 3-second-averaged magnetic field direction [ Means et al. , 1972] (6c), and ellipticity [
Samson and Olson , 1980] (6d) within the ramp region. The shaded regions correspond to different wave packets observed in the ramp. All waves were either linearly or elliptically polarized and had a high degree of polarization. Most waves observed in the ramp had large amplitudes (>20 mV/m) in comparison to the majority of wave packets in this study (see section 4). Region 1 contained an ion acoustic wave, with a peak amplitude of ~169 mV/m and broadband power from ~50-3000 Hz. The wave angle for this wave packet was the most parallel avis et al. , [2019] 18 seen in the ramp, roughly 35° with respect to the local magnetic field. Region 2 contained an ion acoustic wave with an amplitude of ~119 mV/m and broadband power from ~2500-6500 Hz. Region 3 contained a TFES wave, decreasing in frequency from 5000 Hz to 1000 Hz in ~0.1 seconds, with an amplitude of ~28 mV/m. In region 4, an ion acoustic wave was observed with an amplitude of ~36 mV/m and peak frequency of ~2000 Hz. Region 5, a zoom-in of which is shown at the bottom of Figure 6, contained a solitary wave with an amplitude ~18 mV/m. Several low amplitude solitary waves occurred in addition to the larger amplitude solitary wave (small black arrows), but because these waves were below our power threshold, they were not included in the statistics. This variability of wave modes and amplitudes observed within the ramp are consistent with the findings of other studies [
Wilson III et al ., 2007, 2014b;
Breneman et al. , 2013;
Goodrich et al. , 2018;
Cohen et al. , 2019b]. These wave modes were also observed outside the ramp region. All waves with amplitudes >20 mV/m occurred within ~5 seconds (~3 ⍴ gi ) of the shock ramp. This proximity to the ramp suggests that the free energy source of these waves is located within or in close proximity to the shock ramp region, consistent with the findings of Wilson III et al. [2007]. Two of the waveforms within ~5 seconds (~3 ⍴ gi ) of the ramp, shown in Figure 7, are TFES waves and highlight the distinction between this wave mode and ion acoustic waves: the characteristic frequency decreasing (7a) or increasing (7b) with time. The shaded time intervals correspond to the waveforms in the right column of Figure 7. The wave period in T1 is less than that in T2, and the wave period in T3 is greater than that in T4. If the change in frequency observed for the wave in A (B) was due to Doppler shift, assuming the wave vector was constant because no change in the direction or magnitude of the quasi-static magnetic field was observed, the velocity of the solar wind would need to have changed by a factor >2 in ~0.1 (~0.2) seconds, which was not observed. We examined whether there were harmonics of the electron cyclotron frequency to test for the ECDI. Although the waves had multiple peaks, the peaks were not separated by either integer or half integer multiples of the electron cyclotron frequency, and did not provide evidence for ECH waves or ECDI driven waves. avis et al. , [2019] 19 Figure 4: An overview of the 2015-06-21 shock event as observed by THB. (a) Electron energy flux from ~10 eV to 30 keV from ESA. (b.1-3) ESA electron pitch angle distributions for 19 eV, 167 eV, and 867 eV. Color bar is in units of log electron flux. (c) The ion density (cm -3 , black), velocity (km/s, red, divided by 8), and temperature (eV, green). (d) Magnetic field in GSE coordinates along with the magnitude (black). (e) Parallel component of the electric field wave burst captures. avis et al. , [2019] 20 Figure 5: The 10 s wave burst capture that covers the entire transition region (grey) and ramp (yellow) for the 2015-06-21 shock event. (a) The background magnetic field in GSE coordinates. (b) The magnetic field waveform in FAC. (c) The total magnetic field power spectrum with the electron cyclotron frequency overlaid. (d) The electric field waveform in FAC. (e) The total electric field power spectrum with the ion plasma frequency overlaid. avis et al. , [2019] 21 Figure 6: Zoom-in of the ramp region. Shaded regions 1-5 are different wave packets. (a) Electric field waveform in FAC. (b) Total electric field power spectrum. (c) Wave angle with respect to the magnetic field. (d) Ellipticity. (5) Zoom-in electric field burst of region 5, showing a solitary wave and nearby, smaller-amplitude solitary waves (black arrows). avis et al. , [2019] 22 Figure 7: The electric field wave burst waveform data and power spectra for two TFES waves. (a) Decreasing frequencies with time can be seen in the power spectrum. (b) Increasing frequencies can be seen. (T1-T4) Electric field waveforms corresponding to the shaded times in (a) and (b). An overview of the 2017-05-20 shock observed by THB is shown in Figure 8 (same format as Figure 4). This event was a reverse shock with a ramp at roughly 09:52:47 (8d) and had a wave burst capture overlapping the transition region, beginning immediately upstream (<1 ⍴ gi ) of the ramp (8e). A clear enhancement was seen for electrons below 100 eV (8a). Upstream of the shock, electron pitch angle distributions (8b.1-3) show a population of electrons, likely strahl, traveling antiparallel, away from the sun, for energies between ~10-500 eV. Another population is shown travelling parallel, towards the sun, for energies between ~500-5000 eV. Downstream of the shock, there was a strong perpendicular enhancement of electrons for energies between ~100-3000 eV which lasted for roughly 8 minutes (~435 ⍴ gi ); this enhancement was absent for energies below 100 eV. This event had a burst capture ~150 s upstream (~135 ⍴ gi ) of the ramp (9c), and one which covered part of the transition region, starting within 1 s upstream (<1 ⍴ gi ) of the ramp (9b). There were also two burst captures ~4 minutes (~217 ⍴ gi ) downstream, but they contained no waves of significant power and are not shown. Note that no burst covered the ~1 s duration of the ramp region itself. Figure 9 shows the magnetic field around the ramp region (shaded in yellow) and the transition region (shaded in grey) (9a). A perturbed shock with a clear whistler precursor is seen. The amplitude and frequency of this precursor were ~8 nT and ~1.6 Hz; the difference in the average upstream and downstream background magnetic fields was ~6.6 nT. Both burst captures and their respective power spectra are shown in Figure 9. This event had fewer and lower amplitude wave packets compared to most other events in this study. Only one wave packet with avis et al. , [2019] 23 sufficient power was observed within the transition region, occurring ~2.5 seconds (~2 ⍴ gi ) upstream of the ramp. The color scale in the power spectra of Figure 9 is the same as in Figure 5, 6, and 7 to highlight the lack of large amplitude (>20 mV/m) waves. Seven wave modes were identified upstream, six ion acoustic waves and one solitary wave. All waveforms had amplitudes between ~2-8 mV/m. Note that the periodic spikes (some denoted by black arrows in 9c), which appear in both burst captures, were only observed by a single voltage probe, V5, indicating that the spikes are artificial. Similar to those shown in Figure 4e, the high power broadband spikes in both burst captures were also determined to be artificial. THC (~17000 km, ~85 ⍴ gi from THB), which observed a similar transition region, took two burst captures ~8 minutes (~980 ⍴ gi ) downstream, containing no waves above the power threshold of our study. Figure 8: An overview of the 2017-05-20 shock event as observed by THB. (a) Electron energy flux from ~10 eV to 30 keV from ESA. (b.1-3) ESA electron pitch angle distributions for 11 eV, 167 eV, and 867 eV. Color bar is in units of log electron flux. (c) The ion density (cm -3 , black), velocity (km/s, red, divided by 40), and temperature (eV, green, divided by 15). (d) Despun background magnetic field in GSE coordinates along with the magnitude (black). (e) Parallel component of the electric field wave burst captures. avis et al. , [2019] 24 Figure 9: Two 10 s electric field wave burst captures near the ramp of the 2017-05-20 shock event. The ramp region is shaded yellow. The transition region is shaded in grey. (a) The background magnetic field. A whistler precursor is clearly evident. (b) The electric field waveform wave burst capture in field aligned coordinates (red is z , green is y , blue is x ). This burst overlaps the transition region. (c) The second upstream electric field burst capture in field aligned coordinates. Arrows above the waveform data highlight the periodic spikes deemed artificial. Each burst capture waveform has its corresponding power spectrum shown below. The number of identified wave packets observed near laminar and perturbed shock ramps is summarized in Table 3. During the 13 events, there were ~290 s of burst capture time, corresponding to 26 individual burst captures within 10 minutes (average of 3330 ⍴ gi upstream, 1930 ⍴ gi downstream) of each ramp; 56% of the burst time occurred within 500 ⍴ gi of the ramp and 80% occurred within 1500 ⍴ gi . To compare the occurrence rate of different wave modes between the four regions and two shock types, we normalized the total number of observed waves in each wave mode by the total burst capture time for a given region and shock type. Table 3 lists these rates for each shock type, shock region, and wave mode. These rates only include waves with sufficient power ( ≳ -3 mV /m /Hz or an amplitude roughly ≳ avis et al. , [2019] 25 given region and shock type. Most of the burst capture time and wave packets occurred downstream of both shock types, with both types having a similar amount of burst capture time. The major difference between the two shock types is in the occurrence rate of the different wave modes. In the downstream region of perturbed shocks, the occurrence rate for ion acoustic waves was ~2.7 times that for laminar shocks; TFES waves had ~2.3 times the occurrence rate. For solitary waves and whistler modes, the statistics are less significant due to the lower number of observations. The occurrence rate for solitary waves was ~7 times greater downstream of perturbed shocks compared to laminar shocks; for whistler waves, the occurrence rate was ~10 times greater. For six events (three laminar and three perturbed), burst captures occurred during the transition region. Both shock types had similar burst capture times in this region, and the occurrence rate for ion acoustic and whistler mode waves was similar for both types. For TFES waves, however, the occurrence rate within the transition region for perturbed shocks was nearly triple that for laminar shocks. This difference may provide a clue to the physical mechanism that results in the rapid frequency change. The average (median) amplitude for waves in the transition region for laminar shocks was ~22 (~12) mV/m; for perturbed shocks, it was ~16 (~10) mV/m. Note that one perturbed shock with burst data in the transition region did not observe any waves with power above 10 -3 mV /m /Hz. In the transition region, waves near laminar shocks on average have higher amplitudes than waves near perturbed shocks. Three events (two laminar and one perturbed) had burst captures covering the entire ramp region. For these two laminar shocks, the occurrence rate of waves in the ramp region was significantly higher than either up- or downstream. The peak amplitudes for both laminar shocks were due to ion acoustic waves, and the amplitudes were, respectively, ~169 mV/m and ~9 mV/m. For the perturbed event, no wave with amplitude ≳ avis et al. , [2019] 26 Shock Type, Region Ion Acoustic TFES Solitary Waves High f Whistlers Burst Time Laminar, Upstream (1) - (0) 3.333 (1) - (0) - (0) 0.3 s Laminar, Transition (3) 0.612 (20) 0.183 (6) 0.153 (5) 0.061 (2) 32.7 s Laminar, Ramp (2) 5.000 (6) 1.667 (2) 3.333 (4) - (0) 1.2 s Laminar, Downstream (5) 0.328 (36) 0.164 (18) 0.018 (2) 0.018 (2) 109.8 s Perturbed, Upstream (1) 0.588 (6) - (0) 0.098 (1) - (0) 10.2 s Perturbed, Transition (3) 0.591 (22) 0.511 (19) 0.081 (3) 0.054 (2) 37.2 s Perturbed, Ramp (1) - (0) - (0) - (0) - (0) 1.3 s Perturbed, Downstream (7) 0.884 (90) 0.373 (38) 0.118 (12) 0.079 (8) 101.8 s Total 174 82 23 14 292.0 s Table 3: Occurrence rate (in units of number of waves/sec) for a given shock type, region, and wave mode. The total number of waves, shown in parentheses next to the rate, was normalized by the total burst capture length for each region. In the left column, the number of events with observations for each shock type and region is shown in parentheses. The number of wave packets observed was not equally distributed among the events in our dataset. Three events, one laminar (THB 2015-06-21) and two perturbed (THC 2015-06-24; THC 2015-06-27), observed significantly more wave packets than other events. When these events are removed, laminar shocks have higher occurrence rates than perturbed shocks for all regions that have burst capture data to compare. No strong dependence of wave occurrence rates based on M f or θ Bn was observed. Any observed differences were more likely due to the limited number of events than any relationship to either M f or θ Bn . Significant numbers of additional events, beyond the scope of this study, would be required to provide statistically significant conclusions on θ Bn or M f dependence. The high frequency whistler mode waves observed (~0.5-3 mV/m; ~50-1000 pT) had smaller electric field amplitudes compared to other wave modes in this study, but also had some of the longest durations (~0.1-10 s; ~8-2000 wave periods). The magnetic field amplitudes are similar to, but in some cases higher, than the range of amplitudes (~10-100 pT, δB/B ~0.1-1%) found in other studies of high frequency whistlers in the solar wind [ Lacombe et al ., 2014;
Tong et al ., 2019]. High frequency whistlers were seen in only three events (2013-07-09, 2015-06-27, and 2017-02-24). Even though most of the observed whistlers were ≲ δB were as large as 7% of the avis et al. , [2019] 27 background field B . It is possible that other events had whistlers, but either those whistlers did not have high enough amplitude to trigger a wave packet or did not occur simultaneously with another higher amplitude wave mode. One whistler, with an amplitude of ~0.5 mV/m and ~0.15 nT ( δB/B ~0.6%) and a frequency ~200 Hz (0.43 f ce ), was observed ~1 minute (~320 ⍴ gi ) downstream on 2013-07-09 with a duration at least as long as the 10-second wave burst capture. A second whistler, with an amplitude of ~0.5 mV/m and ~0.1 nT ( δB/B ~1%) and a frequency ~160 Hz (0.36 f ce ), occurred ~3 minutes (~960 ⍴ gi ) downstream and also lasted the full duration of the 10-second wave burst capture. Both of these long duration whistlers had a ~10° wave angle and had right-handed (~0.7) ellipticity, consistent with Lacombe et al . [2014]. Figure 10: The amplitude distributions of wave packets near (a) laminar shocks and (b) perturbed shocks. Wave packets containing ion acoustic (blue), TFES (orange), solitary (green), and whistler mode (red) waves are shown separately in each distribution. Figure 10 shows the distribution of the amplitudes of individual wave packets for laminar and perturbed shocks, subdivided into ion acoustic (blue), TFES (orange), solitary (green), and high frequency whistler mode (red) waves. Most wave packets (193/279 ≈ 69%) had amplitudes between 2 and 10 mV/m. The average (median) amplitude of the wave modes observed was 10.2 (6.6) mV/m for ion acoustic, 12.0 (7.5) mV/m for TFES, 15.3 (14.4) mV/m for solitary waves, and 1.1 (0.9) mV/m and 0.42 (0.32) nT for whistler modes. No significant difference in amplitude was found between laminar and perturbed shocks. Similarly, no significant dependence of amplitude on either M f or θ Bn was found. Differences in the duration of ion acoustic and TFES wave modes between shock types were examined using the duration of wave packets normalized by the average upstream ion plasma frequency. The resulting distribution for both wave modes peaked around 5-10 average upstream ion plasma periods with a long tail towards longer durations. There was no difference between laminar and perturbed shocks. For all wave packets analyzed, including whistler and solitary waves, most packets (247/279 ≈ 89%) lasted for less than 100 ms; the median duration was ~31 avis et al. , [2019] 28 ms. These short durations indicate that using filter bank data for identifying wave modes in ramps may be problematic. Typically, the highest sampling filter bank data has a ~1/8 or 1/16 second timescale (roughly 125 or 63 ms) [ Cully et al. , 2008;
Wygant et al. , 2013]. The short duration of a large portion of observed packets (223/279 ≈ 80% had durations less than 60 ms) provides evidence that averaged filter bank data cannot be used to accurately assess wave modes in or near shocks.
Three shocks (two laminar, one perturbed) had burst captures that overlapped the shock ramp. In the ramps of the two laminar shocks, ion acoustic, TFES, and solitary waves were observed. The occurrence rates for these wave modes were 10-20 times greater in the ramp region than downstream. In contrast, no waves with amplitudes ≳ Krasnoselskikh et al. , 2002]. Although the number of shocks is small, the high occurrence rate of waves in both of the observed laminar shock ramps provides support for the idea that laminar shocks could dissipate energy through wave-particle interactions. Given the high occurrence rate of ion acoustic waves in and near the ramp of laminar shocks, the effective collision rate of ion acoustic waves can provide an estimate of the anomalous resistivity due to the observed waves [
Hellinger et al. , 2004;
Petkaki et al. , 2006, 2008;
Yoon and Lui , 2006, 2007], and thus also the energy dissipation of laminar shocks via this mechanism. The maximum amplitude waves for the two laminar shocks with waves within the ramp were ion acoustic modes and were ~169 mV/m and ~9 mV/m, respectively. The shock with the larger amplitude also had the highest ratio of fast mode Mach number to the critical Mach number ( M f /M cr ≈ 1.8) of this study. These waves could provide resistivities 𝜂 𝐼𝐴 = 𝜈/(𝜀 𝑜 𝜔 𝑝𝑒2 ) , where 𝜔 𝑝𝑒 is the electron plasma frequency and ν is the effective collision frequency in the presence of current driven ion acoustic waves [ Watt et al. , 2002], of ~3250 Ωm for the ~169 mV/m wave and ~40 Ωm for the ~9 mV/m wave. Note that these resistivities are lower bound estimates calculated under the assumption T e >> T i . For T e ~ T i , which is the case for many of the events in this study, Vlasov simulations with realistic mass ratios have observed resistivities 1-3 orders of magnitude larger than this lower bound estimate [ Hellinger et al. , 2004;
Petkaki et al. , 2006, 2008;
Yoon and Lui , 2006, 2007]. We compare our resistivities to the findings of
Wilson III et al. [2014b], who found the average (standard deviation) η IA to be 15.9 (107) Ωm and that wave-particle interactions can provide enough energy dissipation for low-to-mid Mach number shocks even with wave energy dissipation efficiencies as low as ~0.01%. The resistivity for the first shock in our study is substantially higher and that for the second shock is above average. For further comparison, we took the ratio of the energy dissipation rates due to wave-particle interactions (-j ·δE, where j is the current density and δE is the fluctuations in the electric field) to the change in kinetic energy density across the shock ramp (
12 𝛥(𝜌𝑈 𝑠ℎ𝑛2 )𝛥𝜏 , where ⍴ is the scalar mass density, U shn is the shock normal speed in the plasma rest frame, and Δ 𝛕 is the shock ramp duration); this is the avis et al. , [2019] 29 same ratio as Equation 3b in Wilson III et al. [2014a]. For the two laminar shocks discussed above, this ratio was 93.4 and 1.8, respectively. For the perturbed shock with burst data in the ramp, this ratio was only 0.07. The large resistivities calculated for our two laminar shocks with burst data in the ramp and the fact that the ratio of energy dissipation to change in kinetic energy is greater than one provide evidence that laminar shocks could, depending on the efficiency of the energy/momentum exchange, dissipate energy solely through wave-particle interactions. Downstream of shocks, where most of the burst capture data was obtained, there is a stark difference in the wave occurrence rate between laminar and perturbed shocks. When normalized by the amount of burst capture time, perturbed shocks were observed with significantly more wave packets downstream than laminar shocks, with the occurrence rate of ion acoustic and TFES waves downstream of perturbed shocks being ~2-3 times that downstream of laminar shocks. Both solitary waves and whistler mode waves were also much more common. These higher occurrence rates are due to two events with significant wave packet generation. The high occurrence rate may be due to increased ion heating, as seen in our perturbed events, and in the study by
Wilson et al. [2012], who also reported gyrating ions downstream of a perturbed shock which could provide free energy for the waves. Further analysis of ion/electron heating or the determination of free energy sources is beyond the scope of this study. Future investigation of sources of free energy near these structures could provide further insight into the difference in occurrence rates. For the other perturbed events, the wave occurrence rates are low in all regions, suggesting that the source of free energy for the two perturbed events with high occurrence rates is not present at all perturbed shocks. No significant difference was found in either the wave amplitudes or wave packet durations between the two shock classifications. Similarly, no amplitude or duration dependence on θ Bn or M f was seen. The average (median) duration of all wave packets was 46 ms (23 ms). A large portion (80%) of wave packets had a duration less than 60 ms. This short duration provides evidence that average filter bank data cannot be used to accurately identify wave modes in or near shocks. Initial examination of ion distributions showed features consistent with reflected ion beams for a portion of shocks in this study; these features were present for an equal number of laminar and perturbed shocks. Further analysis of ion distributions was beyond the scope of this study. Although the small number of events in this study precludes a definite result that applies to all low Mach number, quasi-perpendicular IP shocks, it is sufficient to study the differences in wave activity in the range of ~50-8000 Hz between laminar and perturbed shocks. Recent findings at the high Mach number ( M f ~7), terrestrial bow shock by Goodrich et al . [2018] using MMS observations found similar wave amplitudes and durations to those in this study.
Goodrich et al. [2018] observed a number of ion acoustic waves in the ramp of an oblique bow shock crossing, with amplitudes occasionally exceeding 100 mV/m and durations between 10-100 ms. This is consistent with our findings of ion acoustic mode amplitudes and durations near laminar shocks. In addition to the waves observed in the ramp region, observed reflected ions, which they suggest may provide free energy, may also play a role in energy dissipation for this high Mach number bow shock crossing.
Goodrich et al . [2018] also observed solitary waves with avis et al. , [2019] 30 amplitudes between 100-300 mV/m, much greater than the amplitudes observed near shocks in this study. The variability of wave modes observed in the ramp region of the bow shock by
Goodrich et al . [2018], which includes whistler mode waves, solitary waves, ion acoustic waves, and Bernstein mode waves potentially generated by the EDCI, is similar to the variability of wave modes seen near the lower Mach number IP shocks in this study.
Twelve quasi-perpendicular interplanetary shocks observed by the ARTEMIS spacecraft with electric and magnetic wave burst captures within 10 minutes (average of ~3330 ⍴ gi upstream and ~1930 ⍴ gi downstream, depending on local plasma parameters), either upstream or downstream, of the ramp region were analyzed. Two shocks had burst data observations at both spacecraft, yielding a total of 13 events. The shock structure was classified as either laminar or perturbed. Perturbed shocks were characterized by whistler precursors with an amplitude ẟB on the order of the difference between the upstream and downstream average magnetic field magnitudes ΔB ( ẟB/ΔB ≥ 1/3). This is the first study to examine the differences in waves from ~50-8000 Hz for laminar and perturbed shocks. The burst captures taken near or in the ramp region of these shocks enabled a study of waves in the frequency range of ~50-8000 Hz for the electric field and ~1-500 Hz for the magnetic field; waves with power (amplitude) above 10 -3 mV /m /Hz ( ≳ f ce ) whistler mode waves. 1. The two laminar shocks with burst captures covering the ramp region had a variety of wave modes with amplitudes of up to ~169 mV/m. Based on the observed ion acoustic wave occurrence rate and amplitudes, laminar shocks could potentially dissipate energy primarily through anomalous resistivity due to wave-particle interactions. In contrast, the single perturbed shock with burst data in the ramp contained no waves with amplitudes above 2 mV/m, indicating that dissipation was via other mechanisms, likely dispersion via the whistler precursor. 2.
In the downstream region, two perturbed shocks had 2-3 times the occurrence rate of waves than laminar shocks. This large wave occurrence rate may be associated with ion heating and gyrating ions from large amplitude whistler precursors, as discussed by
Wilson III et al . [2012]. Other perturbed events lacked waves of significant power, suggesting that the source of free energy for the two perturbed shocks with high occurrence rates is not present for all perturbed shocks. 3.
TFES waves (waves with characteristics similar to ion acoustic waves, but with quick changes in frequency with time) were observed ~2-3 times as often in the transition region and downstream of perturbed shocks than laminar shocks. This may provide a clue to the physical mechanism that results in the frequency change. avis et al. , [2019] 31 4.
No significant differences in average wave amplitude or wave packet duration for waves with frequencies between 50 and 8000 Hz was found between laminar and perturbed shocks. We have shown for the first time that laminar shocks contain a variety of large amplitude wave modes in the ramp, and also that the occurrence rate of waves is higher in the transition region, especially the ramp, than downstream. Perturbed shocks lacked waves of significant amplitude in the ramp. The difference in ramp and downstream occurrence rates suggests that the mechanism transforming kinetic bulk flow energy into other forms differs between the two shock types. Our results are consistent with anomalous resistivity due to wave-particle interactions providing energy dissipation within the ramp region for laminar shocks. This is found to not be the primary mechanism for perturbed shocks, with dispersion via a whistler precursor likely providing this change instead.
Acknowledgements
The ARTEMIS data is publicly available from CDAWeb, as well as other sources through the use of the SPEDAS software [
Angelopoulos et al. , 2019]. This research was supported by NASA grants NNX16AF80G, NNX14AK73G, and 80NSSC19K0305. L.B.W. was partially supported by
Wind
MO&DA grants and a Heliophysics Innovation Fund (HIF) grant. The work was also supported by the International Space Science Institute's (ISSI) International Teams programme.
Appendix A: Wave Packet ID Algorithm
In any given burst capture, 0 to ≳
100 wave packets meeting the power criterion could be found. We implemented an algorithm to guarantee that each wave packet was identified with a consistent set of criteria, namely a power threshold in the electric field. The algorithm used the total power in the electric field and marked time-frequency pairs (each has dimensions of ~8 ms by ~66 Hz) when the power exceeded a preset threshold. For this study, the threshold was set to 10 -3 mV /m /Hz; typical background values were ≲ -5 mV /m /Hz. The boundary of a wave packet, which is made up of grouped, adjacent marked pairs, occurred where there were no longer marked pairs adjacent in either the time or frequency domain. The boundary in the time domain provided the time interval of the wave packet. The algorithm can identify waveforms with different characteristic frequencies in the same time interval, allowing identification of multiple wave modes. Low amplitude packets ( ≲ avis et al. , [2019] 32 Figure 11: (a) Electric field data from a wave burst capture. Waveforms inside of black boxes are what were considered wave packets. (b) The total electric field burst power spectrum. Red ovals indicate the rough time and frequency boundaries of each wave packet. Each initial wave packet was then fit with a convex hull to find its peak-to-peak amplitude. For the upper (lower) bound, the maximum (minimum) value in a 1 ms interval was used to fit the bounds. The amplitude was then found by taking the difference between the upper and lower bounds. Figure 12 shows an example of the convex hull surrounding a wave packet. The maximum amplitude and duration of each wave packet, along with the number of time-frequency pairs, was output to a list at the end of the algorithm. avis et al. , [2019] 33 Figure 12: The convex hull (red) fitting for a wave packet. The maximum amplitude (black arrow) and starting and ending points (blue lines) of the wave packet are labeled. Waveforms with power near the threshold, but not consistently above it, would often generate individual or small clumps (<5) of time-frequency pairs. Waveforms with borderline power were removed from the initial list of candidate wave packets by removing wave packets with <5 pairs. The remainder were then carefully sorted such that no two wave packets contained the same waveform in order to avoid double counting. Lastly, clear examples of artificial signals, such as features only observed by a single boom pair in the raw field data, were removed. This final list was used in this study. References
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