The Dynamics of a High Mach Number Quasi-Perpendicular Shock: MMS Observations
H. Madanian, M.I. Desai, S.J. Schwartz, L.B. Wilson III, S.A. Fuselier, J.L. Burch, O. Le Contel, D.L. Turner, K. Ogasawara, A.L. Brosius, C.T. Russell, R.E. Ergun, N. Ahmadi, D.J. Gershman, P.-A. Lindqvist
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The Dynamics of a High Mach Number Quasi-Perpendicular Shock: MMS Observations
H. Madanian , M.I. Desai,
1, 2
S.J. Schwartz, L.B. Wilson III, S.A. Fuselier,
1, 2
J.L. Burch, O. Le Contel, D.L. Turner, K. Ogasawara, A.L. Brosius,
4, 7
C.T. Russell, R.E. Ergun, N. Ahmadi, D.J. Gershman, andP.-A. Lindqvist Southwest Research Institute, 6220 Culebra Rd, San Antonio, TX 78238, USA University of Texas at San Antonio, San Antonio, TX 78249, USA Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO 80303, USA NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA Laboratoire de Physique des Plasmas, CNRS, Ecole Polytechnique, Sorbonne Universit´e, Universit´e Paris-Saclay, Observatoire de Paris,Paris, France Johns Hopkins University Applied Physics Laboratory, Laurel, MD 20723, USA Pennsylvania State University, University Park, PA 16802, USA University of California, Los Angeles, CA 90095, USA KTH Royal Institute of Technology, Stockholm 10044, Sweden (Accepted November 16, 2020)
Submitted to The Astrophysical JournalABSTRACTShock parameters at Earth’s bow shock in rare instances can approach the Mach numbers predicted atsupernova remnants. We present our analysis of a high Alfv´en Mach number ( M A = 27) shock utilizingmultipoint measurements from the Magnetospheric Multiscale (MMS) spacecraft during a crossing ofEarth’s quasi-perpendicular bow shock. We find that the shock dynamics are mostly driven by reflectedions, perturbations that they generate, and nonlinear amplification of the perturbations. Our analysesshow that reflected ions create modest magnetic enhancements upstream of the shock which evolve in anonlinear manner as they traverse the shock foot. They can transform into proto-shocks that propagateat small angles to the magnetic field and towards the bow shock. The nonstationary bow shock showssignatures of both reformation and surface ripples. Our observations indicate that although shockreformation occurs, the main shock layer never disappears. These observations are at high plasma β ,a parameter regime which has not been well explored by numerical models. Keywords: nonlinear amplification — nonstationary — reformation — rippling INTRODUCTIONThe physics of collisionless shocks have been extensively investigated over the past several decades through theoreticalmodels, numerical simulations, in-situ observations, and laboratory experiments (Bykov et al. 2019; Parks et al. 2017;Balogh & Treumann 2013; Schaeffer et al. 2017; Burgess & Scholer 2015; Bell 2014; Treumann 2009; Lembege et al.2004; Gedalin 1997, and references therein). Collisionless shocks in space plasmas are characterized by several keyparameters including Mach number, which specifies the flow speed relative to the phase speed of a characteristic wavemode in the background plasma. Plasma β , or the ratio of the thermal to magnetic pressures, of the incident flow is alsoan important parameter that can affect the growth rate of various plasma instabilities. A third important parameter isthe angle ( θ Bn ) that the upstream magnetic field makes with the shock normal. Shocks exhibit significantly differentcharacteristics based on θ Bn . At quasi-perpendicular shocks ( θ Bn > ◦ ), the guiding center of reflected chargedparticles is driven towards the shock by the solar wind so their trajectories are constrained to about one gyroradius Corresponding author: Hadi [email protected] a r X i v : . [ phy s i c s . p l a s m - ph ] N ov Madanian et al. distance upstream. In contrast, at quasi-parallel shocks ( θ Bn < ◦ ), particles can gyrate and stream along themagnetic field line farther upstream. These differences in trajectories cause different particle distributions, plasmainstabilities, and shock structures (Burgess et al. 2006; Bale et al. 2005). The angle θ Bn = 45 ◦ is conveniently chosento separate these regimes, though in some relativistic shocks the quasi-perpendicular regime begins at much smallerangles ( θ Bn > ◦ ) (Bykov & Treumann 2011). For the rest of the paper we shall focus on the quasi-perpendicularregime.In supercritical shocks, wave-particle coupling within the transition layer is insufficient to dissipate the incidentkinetic energy and to sustain a stable shock layer. The energy balance at these shocks is attained by radiatingdispersive whistler waves (Fairfield 1974; Tidman & Northrop 1968; Wilson III et al. 2012; Wilson III et al. 2017) andreflecting a portion of the solar wind ions upstream (Paschmann et al. 1982; Schwartz et al. 1983). At higher Machnumbers, unbalanced nonlinear growth of the shock layer also leads to a nonstationary shock front. We will considertwo forms of nonstationarity: 1) shock front reformation, and 2) surface ripples. Two common models that describethe reformation process are as follows: Shock reformation can be caused by nonlinear steepening of whistler waves atthe shock ramp via the so-called gradient catastrophe process (Krasnoselskikh et al. 2002; Lobzin et al. 2007; Galeevet al. 1988; Dimmock et al. 2019). This is a 1D approximated model which does not include dissipation effects dueto reflected ions. Reformation can also be mediated by the dynamics of reflected ions in the foot region. Numericalsimulations have shown that accumulation of reflected ions at the upstream edge of the foot gives rise to a new shockfront which replaces the old shock upon formation (Biskamp Welter & Welter 1972; Hada et al. 2003; Lembege &Savoini 1992; Lemb`ege et al. 2009; Scholer et al. 2003; Hellinger et al. 2002). These models predict shock reformationat low β ( ≤
1) plasmas only. Due to numerical constraints, and unrealistic ion to electron mass ratio and plasma tocyclotron frequency ratio, dispersive effects in these models are highly overestimated. The reformation process has aperiod on the order of solar wind proton gyroperiod.When the upstream Mach number is higher than the nonlinear whistler critical Mach number, shock-generatedwhistler waves cannot propagate in the foot, or phase stand in the solar wind as a precursor or as an isolated soliton(Krasnoselskikh et al. 2013; Galeev et al. 1988). In this case, the dynamics of the shock layer are driven by reflected ionsand instabilities they generate. Reflected ions cause significant temperature anisotropy and can initiate instabilitiesthat excite dispersive waves acting to relax the background anisotropy (Winske & Quest 1988). Surface ripples havebeen associated with large amplitude low frequency waves within the shock ramp and overshoot excited by Alv´enIon Cyclotron (AIC) instability (Lowe & Burgess 2003; Yang et al. 2018; Burgess et al. 2016; Moullard et al. 2006;Johlander et al. 2016; Gingell et al. 2017; Johlander et al. 2018; Hanson et al. 2019). The ripple waves propagate alongthe shock surface at speeds comparable to the local Alfv´en speed. Rippling alters the orientation of the magnetic fieldacross the shock most noticeably observed as oscillations in the normal component.Changes in the upstream solar wind conditions can also lead to displacement of the entire bow shock layer. Thisform of nonstationarity typically occurs on timescales larger than the characteristic timescales of the charged particledynamics.Our goal in this study is to provide observational details of the shock layer dynamics at high Alfv´en Mach numbers( M A > β (Winterhalter & Kivelson 1988). Nonstationary shocksthat undergo reformation show even higher amplification rates (Sulaiman et al. 2015), highlighting the importanceof the shock structure dynamics in particle acceleration. Current models of nonstationarity however, are inconsistentand insufficient in describing the shock dynamics at high Mach numbers (Sundberg et al. 2017). We take a somewhatdifferent approach to shock reformation than the classical view of this process. That is, similar upstream disturbancesor accumulation of reflected ions is seen, but we still see ion signatures due to reflection at the pre-existing shock. Therole of ions in driving the shock layer dynamics and generating upstream instabilities is emphasized. DATA AND METHODS2.1.
Data igh Mach number Shocks with MMS β from the OMNI dataset.AC-Coupled magnetic field variations are measured by the search-coil magnetometer (SCM) instrument (Le Contelet al. 2016), while 3D AC-coupled electric field measurements are provided by double probe sensors in the spin plane(Lindqvist et al. 2016), and on the spin axis (Ergun et al. 2016), collectively known as electric field double probe(EDP). The AC-Coupled electric and magnetic field fluctuations are scanned at 8192 samples per second. The EDPdata have been filtered and reprocessed to remove the spacecraft spin effects. Dynamic power spectral densities (PSDs)of electric and magnetic field fluctuations are generated by applying a Fourier transform on SCM and EDP time seriesdata with a 31.2 ms sliding window. 2.2. Timing Method
The MMS mission consists of four spacecraft. Using measurements at four corners of the MMS tetrahedron, thespeed and propagation direction of a plasma structure are calculated from (Russell et al. 1983; Schwartz 1998): δr δr δr ˆ k s | V s | = δt δt δt (1)where δr ij are the inter spacecraft distances, ˆ k s and | V s | are the propagation unit vector and the speed of the structure,and δt ij are the time lags determined by cross correlating the measurements between spacecraft pairs. This method isbased on the assumption of a planar structure, meaning that there can be uncertainties in the results of this methodif the structure substantially changes on timescales shorter than the transition time between spacecraft.2.3. Shock Normal Calculations
To calculate the shock normal vector at the point of the crossing we used a conic section model of Peredo et al.(1995) fit to the crossing location of spacecraft. From this method, we obtain ˆ n = (0 . , − . , − .
05) in the GSEcoordinates. We can also estimate ˆ n from the mixed mode coplanarity method (Schwartz 1998):ˆ n = ± (∆ B × ∆ V ) × ∆ B | (∆ B × ∆ V ) × ∆ B | (2)where ∆ B = B down − B up is the difference between downstream ( B down ) and upstream ( B up ) magnetic field, andsimilarly ∆ V is the difference in the plasma bulk flow velocity. The upstream magnetic field and flow velocity in thepristine solar wind are listed in Table 1 (i.e., B up = B IMF , V up = V SW ). For the downstream values, the magnetic fieldand the ion velocity moment data are averaged between 03:58:52 and 03:59:00 UT, to give B down = ( − . , − . , . V down = ( − , − , −
48) kms -1 . From Equation (2) we obtain ˆ n = (0 . , − . , − . n are in agreement within 3 ◦ , indicating that the shocknormal estimate is reasonably accurate, though we use the normal vector obtained from the model in our analysis. Wedefine the orthogonal shock normal coordinate basis (NCB) with ˆ n , ˆ t , and ˆ t vectors. These basis vectors are relatedsuch that ˆ t = ˆ n × ˆ B up , and ˆ t is perpendicular to ˆ n and ˆ t . ˆ t and ˆ t are tangent vectors to the shock surface, ˆ n and ˆ t form the coplanarity plane, while ˆ t is along the upstream motional electric field. We use the normal incidenceframe (NIF) to study the ion dynamics. In the NIF frame, the solar wind flow is along the shock normal vector, and Madanian et al.
Table 1.
Upstream plasma and shock parameters.Parameter ValueFlow velocity V SW (-464, 31, -14) kms -1 Magnetic field B IMF (-0.80,-1.97,1.55) nTPlasma density n SW -3 Ion temperature T ∗ i,SW β * f ic ω pi λ i = c/ω pi
72 kmElectron cyclotron frequency f ec M A M fm n (0.96, -0.27, -0.05)Shock angle θ Bn ◦ V shock,n -1 Note — * From OMNI dataset. Vector quantities are in the GeocentricSolar Ecliptic (GSE) coordinates, in which + x is toward the Sun and+ z is normal to the ecliptic plane. the the frame’s velocity is obtained from: V NIF = ˆ n × ( V SW-sh × ˆ n ), where V SW-sh is the pristine solar wind velocityin the shock rest frame. For a nonstationarity shock front dynamically modulated by reformation cycles, the timingmethod cannot be applied to measure the shock parameters. The shock speed along the normal vector ( V shock,n )can be calculated based on the traverse time over the shock foot by a specularly reflected solar wind ion (Gosling &Thomsen 1985). In the absence of more accurate methods, this simple approach provides a good rough estimate ofthe shock speed. OBSERVATIONSThe bow shock crossing event is observed by MMS on 28 December 2015 between 03:56:12 and 04:00:15 UT at (10.9,-4.9, -1.1) R E . The upstream IMF is relatively weak but steady during this time. The solar wind beam temperatureis cold, as measured by the width of the ion beam, while the solar wind density is high. The shock Alfv´enic ( M A ) andfast magnetosonic ( M fm ) Mach numbers are recorded at 27 and 15, respectively, and the shock angle is θ Bn = 83 ◦ .Table 1 includes a list of upstream plasma and shock parameters. Figure 1 shows an overview of plasma and field dataacross the shock measured by the MMS spacecraft 1 (MMS-1) from 03:57:15 UT in the solar wind, until 03:59:00 UTwhen the spacecraft is in the magnetosheath. Panel (a) shows the magnetic field component and strength. Differentregions of the shock are annotated on this panel. We will include the magnetic field profile in other time series figuresfor context. The ion energy spectra in panel (b) shows that solar wind protons form a narrow beam with a bulkenergy ∼ ∼ igh Mach number Shocks with MMS -60060 B G SE [ n T ] E ne r g y [ e V ] [ k e V / ( c m s s r k e V ) ] n [ c m - ] | V | [ k m / s ] E ne r g y [ e V ] [ k e V / ( c m s s r k e V ) ] T e [ e V ] Seconds2015 Dec 28 a) a) b) b) c) c) d) d) e) e) f) f) n SW n Refl. n Bulk |V SW ||V Refl. ||V
Bulk | • B x • B y • B z • |B| weak electron acceleration • T e,|| • T e, ⟂ ElectronsIons
MMS-1
DownstreamRamp/overshoot
Foot ••••••
Figure 1.
Overview of the shock crossing event on 28 December 2015: (a) magnetic field magnitude; (b) ion energy fluxspectrogram in logarithmic scale; (c) density of solar wind ions (black), reflected ions (yellow), and bulk plasma (blue); (d)ion velocity moment of solar wind beam, reflected ions, and bulk plasma flow; (e) electron energy flux spectrogram; amd (f)electron temperature parallel (blue) and perpendicular (red) to the local magnetic field. Data in panels (a - f) are from MMS-1spacecraft. R E : Earth radius. occurs at 03:58:09 UT, followed by two closely distanced events at around 03:58:15 UT. During these instances, boxedwith dashed rectangles, we observe significant isotropic electron energization, evident by the increase in both parallel(blue) and perpendicular (red) temperature components in panel (f).The foot region of supercritical quasi-perpendicular shocks typically exhibits a gradual increase in the magnetic fieldstrength caused by reflected ions. In the foot region in Figure 1, we also observe quasi-periodic and isolated enhance-ments in the magnetic field strength ( | B | ) which, particularly closer to the shock, are as strong as the downstreammagnetic field strength. The enhancements are correlated with increases in the density of reflected ions (the yellow linein panel (c)), and with momentary slowdown in the solar wind flow. These quasi-periodic modulations are consistentwith (partial) reformation cycles of the shock (Sulaiman et al. 2015; Sundberg et al. 2017; Madanian et al. 2020).Top two panels in Figure 2 show four MMS spacecraft tetrahedron in NCB coordinates. Magnetic field data fromall four MMS spacecraft for a sub-interval focused on the shock front are shown in panels (c - f). Dashed verticallines are drawn on three reformation cycles upstream of the shock. A specific characteristics of the wave packets (i.e., Madanian et al. sign change in the B x component) is first seen in MMS-2. Moments later, a similar signature appears in MMS-4, thenMMS-1, and finally in MMS-3 data. Using the timing method, we determine the propagation direction of the featureat around 03:58:08 UT ˆ k NCB = ( − . , . , − .
59) in GSE coordinates, consistent with the order of observations.The structure is in propagation at an angle of ∼ ◦ with the background magnetic field. The propagation direction ofother cycles are also annotated on the figure. The main shock transition layer (i.e., ramp and overshoot) in this figurebegins at ∼ n , one would expect to see changes in magnetic field data in all three spacecraft at about the sametime. However, data in Figure 2 indicate that the enhancements are generated upstream and their propagation vectorhas an Earth-ward component, and they significantly modulate the shock front. -40040 B G SE [ n T ] MM S - -40040 B G SE [ n T ] MM S - -40040 B G SE [ n T ] MM S - B G SE [ n T ] MM S - c) c) d) d) e) e) f) f) • B x • B y • B z • |B| b)a) !k !" = (−0.28, 0.75, −0.59)!k !" = (−0.1, 0.87, −0.47)!k !" = (−0.41, 0.58, −0.7) Figure 2.
Panels (a) and (b) show the MMS tetrahedron formation in n − t and t − t planes of NCB coordinates, respectively.Panels (c - f) show the magnetic field data from spacecraft 2, 4, 1, and 3, respectively. Dashed vertical lines are drawn on threereformation cycles when a major sign change in B x is observed in MMS-2 data. ˆ k NCB show the propagation direction of eachwave packet in NCB coordinates. Structures are in propagation towards the shock and along the magnetic field.
Shock Reformation
Reformation cycles observed in the foot have a period of ∼ . ∼ . f − ic , consistent with previous observationsSundberg et al. (2017) and Sulaiman et al. (2015). They appear to be a proton/subproton scale effect. Figure 3 shows igh Mach number Shocks with MMS V n distribution spectrogram.The ion data in panels (b - d) are in the NIF moving with a velocity of V NIF = (-34.89, -112.35, -55.8) kms -1 in theshock rest frame. The solar wind beam at V n ∼ −
400 kms -1 is evident in panel (b). Reflected ions with positive V n velocities are observed, as well as another population with a small negative V n component. The latter is identified asreflected ions that have already been turned around by the motional electric field. Panel (c) shows ion distributionsalong ˆ t . Since both solar wind and reflected ions travel at nearly perpendicular angles to B , their projected velocityalong B is zero. The last panel shows the V t distribution of ions. Farther upstream from the shock, reflected ions havehigher V t velocities as they have spent a longer time in the upstream motional electric field. The trace of reflectedion in the velocity space (not shown) farther from the shock more closely falls along the predicted trajectory of aspecularly reflected ion (Paschmann et al. 1982; Madanian et al. 2020).For reformation cycles at the beginning of the interval near the upstream edge of the foot, pileup of reflected ionsare collocated with the magnetic field enhancements. Closer to the shock, solar wind and reflected ions have beeninterrupted by the reformation cycles, as seen in distributions near the last four vertical lines in panel (d). These lastcycles also show markedly higher bulk electron acceleration and we observe solar wind beam compression for the firsttime at 03:58:08 UT. These observations indicate that buildup of reflected ions at the upstream edge of the foot doesnot instantly generate a new shock front but they are observed quasi-periodically in the foot. Closer to the shock,they begin to exhibit shock-like plasma heating and acceleration, and interrupt both the solar wind and reflected ionsfrom the main shock. We refer to these cycles as ”proto-shocks”, as they do exhibit shock-like behavior, but are notstrong or expanded enough to fully replace the main shock.We also note that in Figure 3.b after the last reformation cycle at 03:58:23 UT, unlike previous cycles, the solar windbeam continues to decelerate to V n ∼
0. At this time, we observe strong ion reflection with intensities higher thanpreviously observed, while ion distributions in panels (c) and (d) show a heated solar wind plasma. These featuresare significantly different than those of the upstream reformation cycles, and the measurements at 03:58:25 UT mostlikely represent the first encounter with the main shock layer. In other words, while reformation occurs in the foot,the main shock layer (or the boundary at which the main shock properties are observed) never disappears. The solarwind flow gradually decreases throughout the foot. But as evident in Figure 1.d, solar wind bulk flow speeds areconsistently higher than the downstream bulk plasma flow speed. In addition, reflected ions are constantly presentin the foot except when interrupted by the proto-shocks. These ions also seem to occupy a broad range of positive V n velocities, rather than being a beam of ions. This can be due to simultaneous reflection taking place from tworeflection surfaces, non-specular reflection (Sckopke et al. 1983), or interaction with upstream proto-shocks. The shocksurface may also be rippled, leading to reflection at different directions and causing reflected ions to be at differentstages of their gyration and consequently different energy levels by the time they arrive at the location of the MMSspacecraft (Ofman & Gedalin 2013). 2D particle-in-cell (PIC) simulations have shown that both shock reformationand shock surface rippling can affect the excitation of the electrostatic waves in the foot by modifying the intensity ofreflected ions at different locations across the shock (Hao et al. 2016; Matsukiyo & Scholer 2003).3.2. Surface Ripples
During this event, sharp magnetic gradient of the ramp, typically present at quasi-perpendicular shocks, is replacedby large amplitude magnetic oscillations. As observed in MMS-4 data shown in Figure 4.a, the normal componentof the magnetic field ( B n ) periodically reverses sign during this period, which can be attributed to a rippled shocksurface (Lowe & Burgess 2003; Johlander et al. 2016). B n data band-pass filtered in the 0.56 - 1.22 Hz frequencyrange from all four spacecraft are shown in panel (b). The selected frequency range covers most of the high amplitudevariations we observe, but does not include the lower frequency variations due to upstream reformation cycles ( ∼ . δt = − . δt = 23 . δt = − . V ph − sc = 136 kms -1 . The wave propagates mostly alongthe shock surface with the wave vector ˆ k NCB = (-0.22, 0.80, -0.54). After correcting for the Doppler effect usingthe locally measured plasma velocity, the wave phase speed in the local plasma rest frame is ∼
41 kms -1 or ∼ . V A ,where V A is the average local Alfv´en speed. The wave’s characteristic wavelength is λ wave = V ph − sc /f sc = 153 kmor 2 . λ i , where λ i is the upstream ion inertial length. These properties are consistent with surface ripples and large Madanian et al. -80-4004080 B NC B [ n T ] n t1 t2 |B| -800-4000400 v n [ k m / s ] -800-4000400 -4000400800 v t [ k m / s ] -4000400800 10 d f [ s k m - ] v t [ k m / s ] Seconds2015 Dec 28 a) a) b) b) c) c) d) d) MMS-2 solar windreflected ionsreturning ions
Figure 3.
The MMS-2 spacecraft observations revealing the fine scale ion dynamics across the shock. Panel (a) shows themagnetic field in the NCB (Section 2). Panels (b - d) show ion velocity distributions in the NIF and along ˆ n , ˆ t , and ˆ t ,respectively. Vertical dashed lines mark the reformation cycles in the foot. Different ion populations are annotated in panel (b). amplitude ion-scale waves generated by AIC instability (Lowe & Burgess 2003; Davidson & Ogden 1975). Waveshave an elliptical polarization, and are far below the lower hybrid frequency. There exist some deviations in theshifted signals, particularly for spacecraft 3, indicating that other processes with similar frequencies could be in play.Nonetheless, the good alignment of the shifted signals verifies that the time lags are properly determined.Similar waves with comparable amplitude are present around the reformation cycle at the beginning of the intervalbetween 03:58:05 and 03:58:05 UT. It seems that the large amplitude waves creating the surface ripples are also presentwithin the incoming proto-shock, which suggests that rippling begins to develop in upstream proto-shocks. Fartherupstream, the wave amplitude becomes very small. It is worth noting that the period of ripple waves is quite differentthan the periodicity of the reformation process, which enables to distinguish the two effects. Numerical simulationsby Gingell et al. (2017) has shown a similar scenario, though with different shock parameters. The authors showedthat surface ripples are modulated by the periodic reformation of the shock front, and transient ripples develop at thenewly formed shock on timescales shorter than the upstream ion gyroperiod. This further signifies our observationsand the feasibility of our interpretations. Similar processes have also been observed for oblique shocks Lefebvre et al.(2009), and have been investigated for astrophysical quasi-parallel shocks through numerical simulations (Caprioli & igh Mach number Shocks with MMS -4004080 B NC B [ n T ] n t1 t2 |B| -40-2002040 B n [ n T ] mms1 mms2 mms3 mms410 20 30 40-40-2002040 B n [ n T ] mms1 mms2 mms3 mms4 Seconds2015 Dec 28 0358: a) a) b) b) c) c) Band-pass filteredShiftedMMS-4 Figure 4.
Analysis of shock ripple properties via the timing method: a) Magnetic field magnitude and components in the NCBfrom spacecraft 4; b) normal component of the magnetic field B n from all four spacecraft, data are band-pass filtered in the(0.56-1.22) Hz frequency range; and c) B n signals from spacecraft 2, 3, and 4 are shifted in time to account for time delays inthe observations with respect to spacecraft 1. The vertical dashed lines specify the time period used for cross correlation andidentification of time lags. Spitkovsky 2014a; Caprioli et al. 2015). Our observations of this process at a nearly perpendicular shock suggest thatsuch shock reformation/generation process is a global phenomenon that can occur for a wide range of shock parameters.3.3.
Upstream Electromagnetic Perturbations
In this section we examine electrostatic and electromagnetic wave activities upstream of, and within the shock layer.We restrict our discussion to identifying wave features associated with reformation cycles, noting that each structure hasinternal wave characteristics quite different than the others. Characterizing the nature and the generation mechanismof all these waves, although critical, is beyond the scope of this paper.Figure 5 shows the dynamic power spectral densities (PSDs) of electric and magnetic field perturbations. In panel(a) the magnetic field profile is shown as a reference. Panels (b) and (c) show, respectively, the electric and magneticfield PSDs in the 3 − f lh , the blue line in panel (b)) are observed. These perturbations upstream ofthe ramp are correlated with the reflected ion densities and solar wind decelerations discussed in Figure 1. The totalelectric and magnetic wave powers (integrated over the whole frequency range) oscillates with the same period as theshock reformation. The shock transition layer, between 03:58:24 UT to the end of the interval, is distinguished by highpower broadband electric perturbations. The ratios of the electric to magnetic field fluctuations are shown in panel(d). Yellow and red colors correspond to ratios much greater than 1, indicative of electrostatic waves. As expected,strong electrostatic waves are present within the main shock transition layer (Scudder 1995; Bale et al. 1998, 2005;Vasko et al. 2018; Goodrich et al. 2018; Wang et al. 2020; Wilson III et al. 2014). We also see similar electrostaticwaves upstream of the shock layer inside the last three reformation cycles, which distinguishes them from the earlier0 Madanian et al. | B | [ n T ] f [ H z ] -10 -4 [ n T / H z ] f [ H z ] -10 -5 [ m V m - / H z ] f [ H z ] -3 E ( f ) / c . B ( f ) Seconds2015 Dec 28 a) a) b) b) c) c) d) d) f ce f lh MMS-1
E(f) / c.B(f) f ce f ce Figure 5.
Electromagnetic wave activity in the foot region observed by MMS-1: a) Magnetic field profile; b) power spectraldensity (PSD) of magnetic field fluctuations; c) PSD of electric field fluctuations; and d) ratio of electric to magnetic fieldfluctuations in the frequency domain ( E ( f ) /cB ( f ), the conversion factor c is the speed of light). The red solid line on panel (b)shows the electron cyclotron frequency f ce . 0 . f ce and 0 . f ce frequency lines are shown with red dotted lines. The f ce line isrepeated on panels (c) and (d) for reference. The blue line on panel (b) is the lower hybrid frequency f lh . sequences. These cycles, or proto-shocks, also show significant bulk electron heating (Figure 1.f). This is an importantobservation as it reveals that not all cyclic enhancements exhibit shock-like properties, and bursty high frequencyelectrostatic waves are not restricted to the main shock layer.Our wave analysis also shows that the sporadic high frequency magnetosonic waves between 0 . f ce and 0 . f ce (shown with red dotted lines in panel (b)) are circularly right-hand polarized and in propagation quasi-parallel tothe background magnetic field (wave angle < ◦ ). The waves are consistent with a source of electron temperatureanisotropy ( T e, ⊥ /T e, (cid:107) >
1, see Figure 1.f) (Kennel & Petschek 1966; Gary & Wang 1996), which is likely created bymagnetic increases during the reformation process, and associated gyrobetatron effects.At the beginning of the interval in Figure 5.c, isolated high frequency quasi-electrostatic fluctuations near the electroncyclotron frequency ( ∼
100 Hz) are observed. Their generation mechanism could be related to reflected ions. Theyare generated near the upstream edge of the foot, where specularly reflected ions are accelerated along the motionalelectric field and travel almost perpendicular to the magnetic field. The ion gyration trajectory at that point can beconsidered as almost a straight line, hence providing a nonmagnetized fast ion component that destabilizes the plasmaand causes generation of high frequency quasi-electrostatic waves (Muschietti & Lemb`ege 2013; Sundberg et al. 2017;Omidi & Winske 1987). More investigations are required to better identify the nature of the instability.Propagation and evolution of the upstream cyclic enhancements throughout the foot is rather nonlinear. Eachenhancement is accompanied by a burst of low frequency waves, some of which have characteristics consistent with thewhistler wave mode, while some show no particular polarization. To verify this nonlinear pattern, we perform waveanalysis on magnetic field data from all spacecraft, over various frequency ranges and time periods. Three illustrativeexamples are discussed in Figure 6, though many other frequency bands were examined. Panels (a) and (b) show igh Mach number Shocks with MMS -505 B G SE [ n T ] Bx By Bz |B|30 40 50-505 B G SE [ n T ] Bx By Bz |B|
Seconds2015 Dec 28 0357: a) a) b) b) MMS-3MMS-4 -1 0 1B
Int. [nT]-101 B M a x . [ n T ] -1 0 1B Min. [nT]-101 B M a x . [ n T ] -1 0 1B Min. [nT]-101 B I n t. [ n T ] -1 0 1B Int. [nT]-101 B M a x . [ n T ] -1 0 1B Min. [nT]-101 B M a x . [ n T ] -1 0 1B Min. [nT]-101 B I n t. [ n T ] -1 0 1B Int. [nT]-101 B M a x . [ n T ] -1 0 1B Min. [nT]-101 B M a x . [ n T ] -1 0 1B Min. [nT]-101 B I n t. [ n T ] l Max. / l Int. = 2.5 l Max. / l Min. = 4.6 l Int. / l Min. = 1.8 l Max. / l Int. = 1.8 l Max. / l Min. = 835.9 l Int. / l Min. = 457.3 l Max. / l Int. = 1.0 l Max. / l Min. = 1880.9 l Int. / l Min. = 1811.6
Figure 6.
Properties of magnetic fluctuations. Panels (a) and (b) show the magnetic field components and magnitude measuredby MMS-3 and MMS-4, respectively. The first row of hodograms at the bottom correspond to the first time interval labeledwith the yellow box for MMS-3. The second row corresponds to the second purple interval for MMS-4, and third row shows theprincipal components of the magnetic field for the blue interval on MMS-3. Hodograms show the background subtracted, lowpass filtered ( <
20 Hz) data. The background magnetic field is pointing into the B
Max. − B Int. plane (the first column), and theblue dots mark the beginning of each interval. The band-pass frequency range is (0.1 - 4) Hz for the first interval, (1.4 - 1.8) Hzfor the second interval, and (1.9 -2.1) Hz for the third interval. magnetic field data from MMS-3 and MMS-4 between 03:57:24 and 03:57:59 UT. We apply the minimum varianceanalysis on select intervals highlighted with yellow and blue in MMS-3 and purple in MMS-4 time series. Hodogramsat the bottom show band-pass filtered, background subtracted field variances for B
Max. − B Int. in the first column,B
Max. − B Min. in the second column, and for B
Int. − B Min. in the third column. The ratios of corresponding eigenvaluesare annotated on each panel, and the background magnetic field points into the page of the B
Max. − B Int. planes.During the first interval (yellow segment) we find no identifiable wave pattern at any frequency range in MMS-3 data.2
Madanian et al.
This is evident in the first row hodograms which show field variations in the 0.1 - 4 Hz frequency range in that period.For the second interval (purple segment), right-handed elliptically polarized waves in the 1.4 - 1.8 Hz frequency rangeare observed. In the last interval, using MMS-3 data, we observe 2-Hz waves with right-handed circular polarization,the typical signatures of the whistler mode waves. By comparing the wave activity in the middle interval (03:57:40 -03:57:42 UT) in MMS-3 and MMS-4, we notice a small frequency shift in the high amplitude waves. In addition, thelast wave packet around 03:57:57 UT, the amplitude of the magnetic peak in MMS-3 data shows a decrease comparedto that in MMS-4, while the position of the peak has also changed within the cycle. Note the separation betweenMMS-3 and MMS-4 along the magnetic field is about 25 km ( ∼ . λ i ).Overall, we find that close to the shock, waves observed by all spacecraft share the same wave normal angle dis-tribution (irrespective of the 180 ◦ ambiguity). Far from the shock, each spacecraft sees distinct wave characteristicsand the wave distributions appear to switch between relatively high and relatively low wave normal angles, and thisbehavior intensifies for 2-10 Hz waves. Not only polarization, amplitude, and duration of waves change from one cycleto another, waves also substantially evolve during the short travel between spacecraft. The nature of instabilities alsovaries from one cycle to another, showing complex and nonlinear evolution of wave packets as they propagate in thefoot. These variations however, all begin with modest magnetic enhancement in the IMF generated by reflected ions.They transform into proto-shocks as they propagate Earth-ward.These signatures are inconsistent with ultra-low frequency (ULF) waves which have circular polarization and aperiod similar to the upstream ion gyroperiod. The waves are also inconsistent with ion Weibel instability (IWI)which generates linearly polarized waves. Interaction of reflected ions with incoming solar wind electrons or ions cancause foot instabilities that excite waves in the whistler mode branch. Modified Two Stream Instability (MTSI) dueto relative drift between reflected ions and incoming solar wind electrons (fast drift), and incoming solar wind ions andelectrons (slow drift) has been frequently considered (Marcowith et al. 2016; Muschietti & Lemb`ege 2017; Matsukiyo& Scholer 2003; Umeda et al. 2012; Comi¸sel et al. 2011; Wilson III 2016; Hull et al. 2020). This instability however, ifexcited, creates significant ion heating throughout the foot and suppresses the reformation process (Shimada & Hoshino2005; Matsukiyo & Scholer 2006), rather than creating episodic enhancements that we show in the foot. Furthermore,Gary et al. (1987) indicated that (fast drift) MTSI becomes dominant at low electron beta ( β e < . β e more resonant electrons stabilize this instability through increased electron Landau damping. Electron data forthe time period we discussed in this paper show β e ≥ .
2, and therefore fast drift mode MTSI is most likely notsignificant. The slow drift mode of MTSI could be a more viable candidate at high β plasmas. Wave properties around1.6 Hz in the middle interval (purple segment) of Figure 6, indicate that the wave is in propagation towards the ramp(ˆ k GSE = − . , − . , .
22) with V ph − sc = 34 kms -1 and λ wave = 21 . ∼ λ e , where λ e is the upstream electroninertial length. The plasma rest frame frequency of the wave is about 8 Hz ∼ . f lh . Since these characteristics aresomewhat consistent with model predictions for drift mode of MTSI (Muschietti & Lemb`ege 2017), we do not rule outthe possibility of some waves at certain frequencies and during some intervals being generated by the slow drift modeof MTSI. CONCLUSIONS AND DISCUSSIONBy studying the dynamics of a quasi-perpendicular shock at very high Mach numbers, we are able to isolate andinvestigate the effects of ion dynamics on the shock structure. The high upstream Mach number of the upstream flowdoes not allow for shock-generated dispersive whistler waves to propagate throughout the foot and the shock dynamicsare tied to the reflected ion dynamics.We observe signatures of shock reformation in the form of magnetic enhancements that evolve in a nonlinear mannerto form proto-shocks as they traverse the foot. This is a different mechanism than previously suggested by simulations.What we observe here, instead, is that quasi-periodic reformation is indeed initiated by reflected ions through generationof modest magnetic enhancements at the upstream edge of the foot; however, the enhancement do not immediatelyreplace the main shock. As they convect Earth-ward, they transform into proto-shocks through nonlinear amplificationof electric and magnetic fields within the enhancements. This amplification is separate from the compression andamplification that occur at the main shock layer. The proto-shocks possess high frequency electrostatic waves andexhibit significant electron heating, and interfere with both the solar wind flow and reflected ions from the main shock.We show that Alfv´en Ion Cyclotron (AIC) waves within the shock ramp and overshoot form surface ripples with awavelength of 2.1 ion inertial lengths, and an average period of ∼ . igh Mach number Shocks with MMS M nlw ∼ . (cid:28) M fm , M A ), we observe intermittent whistler waves between 0 . f ce and 0 . f ce frequency range (Figure 5.b). At thebeginning of the interval, they coincide with reformation cycles, but later are observed in between cycles. Thesewhistler waves are correlated with the electron temperature anisotropy. The most likely cause of the anisotropy isgyrobetatron effects associated with the increased magnetic fieldat reformation cycles. In Figure 6, we show that somecycles also carry locally generated low frequency whistler waves.Our observations are unique for high plasma β shocks. This regime of shock parameters has been under studied bynumerical simulations. We observe signatures of shock reformation with upstream plasma β ∼
9. Most simulationstudies have indicated that at high β ( > β conditions are normally achievedby increasing the ion temperature, which causes a smooth but extended increase of the magnetic field in the shockfoot. Unrealistic ion to electron mass ratios can also lead to overestimation of dispersive effects, which also work toincrease the ion temperature (Lemb`ege et al. 2009). Imposing single population isotropic Maxwellian distribution foreach particle species is rather unrealistic and definitely affects the instabilities present in the foot region of quasi-perpendicular shocks and foreshock region of quasi-parallel shocks. Nonetheless, models can provide interpretationsfrom a different stand point, and a detailed simulation analysis for these observations is left for a future study.The high solar wind plasma β in the event we discussed here is due to the high plasma density and the very weakIMF strength, both of which also contribute to achieving the high Alfv´en Mach number under the typical solar windspeed. Relatively weak upstream magnetic field is common in interstellar and astrophysical shocks (Donnert et al.2018; Petrukovich et al. 2019), and our observations and interpretations can, to some extent, be applied to thosestructures. ACKNOWLEDGMENTSWe would like to thank D. Sibeck for helpful discussions. All data used in this study are publicly available viathe MMS Science Data Center (https://spdf.gsfc.nasa.gov/pub/data/mms), and NASA/GSFC’s OMNIWeb service(http://cdaweb.sci.gsfc.nasa.gov) for solar wind data. Data access and processing software includes the publicly avail-able SPEDAS package (Angelopoulos et al. 2019). This work was supported in part by the NASA Award Number80NSSC18K1366. L.B.W. acknowledges partial support through an International Space Science Institute (ISSI) team.A.L.B. is supported by NASA grant 80NSSC20M0189. The French LPP involvement for the SCM instrument issupported by CNES and CNRS. REFERENCES Angelopoulos, V., Cruce, P., Drozdov, A., et al. 2019, SpaceScience Reviews, 215, 9, doi: 10.1007/s11214-018-0576-4Bale, S. D., Kellogg, P. J., Larson, D. E., et al. 1998,Geophysical Research Letters, 25, 2929,doi: 10.1029/98GL02111Bale, S. D., Balikhin, M. A., Horbury, T. S., et al. 2005,Space Science Reviews, 118, 161,doi: 10.1007/s11214-005-3827-0Balogh, A., & Treumann, R. A. 2013, Physics ofCollisionless Shocks: Space Plasma Shock Waves(Springer New York), 1–500,doi: 10.1007/978-1-4614-6099-2 Bell, A. R. 2004, Monthly Notices of the RoyalAstronomical Society, 353, 550,doi: 10.1111/j.1365-2966.2004.08097.x—. 2014, Brazilian Journal of Physics, 44, 415,doi: 10.1007/s13538-014-0219-5Bell, A. R., & Lucek, S. G. 2001, Monthly Notices of theRoyal Astronomical Society, 321, 433,doi: 10.1046/j.1365-8711.2001.04063.xBiskamp Welter, D. H., & Welter, H. 1972, Nuclear Fusion,12, 663, doi: 10.1088/0029-5515/12/6/006Burch, J. L., Moore, T. E., Torbert, R. B., & Giles, B. L.2016, Space Science Reviews, 199, 5,doi: 10.1007/s11214-015-0164-9 Madanian et al.