Stochastic ion and electron heating on drift instabilities at the bow shock
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Stochastic ion and electron heating on drift instabilities atthe bow shock
Krzysztof Stasiewicz, , ⋆ Department of Physics and Astronomy, University of Zielona G´ora, Poland Space Research Centre, Polish Academy of Sciences, Warsaw, Poland
Accepted 2020 May 11. Received 2020 April 01; in original form 2020 March 05
ABSTRACT
The analysis of the wave content inside a perpendicular bow shock indicates thatheating of ions is related to the Lower-Hybrid-Drift (LHD) instability, and heating ofelectrons to the Electron-Cyclotron-Drift (ECD) instability. Both processes representstochastic acceleration caused by the electric field gradients on the electron gyrora-dius scales, produced by the two instabilities. Stochastic heating is a single particlemechanism where large gradients break adiabatic invariants and expose particles todirect acceleration by the DC- and wave-fields. The acceleration is controlled by func-tion χ = m i q − i B − div( E ), which represents a general diagnostic tool for processes ofenergy transfer between electromagnetic fields and particles, and the measure of the lo-cal charge non-neutrality. The identification was made with multipoint measurementsobtained from the Magnetospheric Multiscale spacecraft (MMS). The source for theLHD instability is the diamagnetic drift of ions, and for the ECD instability the sourceis ExB drift of electrons. The conclusions are supported by laboratory diagnostics ofthe ECD instability in Hall ion thrusters. Key words: acceleration of particles – shock waves – solar wind – turbulence – chaos
Terrestrial bow shock represents a great opportunity forinvestigation of various mechanisms for heating and ac-celeration of particles in collisionless plasmas with im-portant implications for astrophysics. Since its discov-ery by Ness et al. (1964) there has been great dealof research on this collisionless shock wave (Wu et al.1984; Gary 1993; Balikhin & Gedalin 1994; Lemb´ege et al.2003; Lefebvre et al. 2007; Treumann 2009; Burgess et al.2012; Mozer & Sundqvist 2013; Breneman et al. 2013;Wilson III et al. 2014; Cohen et al. 2019), however, there isstill no consensus on how particles are accelerated, and whatexactly are processes that heat bulk plasma.The observational advances afforded by multipointmeasurements in space like Cluster (Escoubet et al.1997), THEMIS (Sibeck & Angelopoulos 2008), and MMS(Burch et al. 2016) opened new possibilities for space plasmaphysics. In this report we provide arguments supported bymeasurements from the MMS mission, that a single, non-resonant, frequency independent mechanism can mediateheating of both ions and electrons at the bow shock. Thismechanism relies on gradients of the electric field that en- ⋆ E-mail: [email protected] sure breaking of the magnetic moment, allowing for efficient,stochastic heating of particles by the present electric fluctu-ations, and even by the DC field.The energisation process is controlled by a dimension-less function defined as χ ( t , r ) = m i q i B ∇ · E ≡ N c N c V Ai (1)for particles with mass m i , charge q i in fields B ( t , r ) and E ( t , r ) . N c is the number density of excess charges, N num-ber density, V Ai = B /( µ Nm i ) , c - speed of light. The equiv-alent formula is obtained with substitution ∇ · E = N c q i / ǫ .Stochastic heating occurs when | χ | > . It is a single parti-cle mechanism where large electric field gradients (or spacecharges) destabilise individual particle motion, rendering thetrajectories chaotic in the sense of a positive Lyapunov ex-ponent for initially nearby states. This condition has beenknown from some time, but the importance of div( E ) hasnot been recognised, and the directional gradient ∂ x E x , or k ⊥ ∇ φ was used in previous analyses or simulations (Cole1976; McChesney et al. 1987; Karney 1979; Balikhin et al.1993; Mishin & Banaszkiewicz 1998; Stasiewicz et al. 2000;Stasiewicz 2007; Vranjes & Poedts 2010; Stasiewicz et al.2013; Yoon & Bellan 2019).The recent MMS mission comprises four spacecraft © K. Stasiewicz flying in formation with spacing ∼
20 km and pro-vides high quality 3-axis measurements of the electricfield (Lindqvist et al. 2016; Ergun et al. 2016; Torbert et al.2016), which we shall use to demonstrate applicabilityof (1) to heating of plasma at the bow shock. We usemagnetic field vectors measured by the Fluxgate Magne-tometer (Russell et al. 2016), the Search-Coil Magnetome-ter (Le Contel et al. 2016), and the number density, velocity,and temperature of both ions and electrons from the FastPlasma Investigation (Pollock et al. 2016).
We show in Fig. 1 the time-frequency spectrogram of χ forthe bow shock encountered by MMS on 2017-12-12T03:33:30at position (8.9, 11.8, 4.9) R E GSE. The shock had alfv´enicand sound Mach numbers M A =8, and M S =5, and wasstrictly perpendicular. The timeseries for χ was derived from4–point measurements using a general method for comput-ing gradients in space developed for Cluster (Harvey 1998).Frequencies lower than 1 Hz have been removed from theanalysis to ensure that calibration offsets, or satellite spineffects (0.05 Hz, and multiples) would not affect computa-tions.The spectrum of χ is similar to the usual power spec-trum of the electric field, but weighted with B − so it empha-sises the heating region in the foot/ramp of the shock, in-stead of the peak, where the power of E maximises, but thereis no heating. It is sensitive to local charges related to elec-trostatic fields generated in plasma by waves and instabili-ties. Ideally, it should show distribution of electric charges,and time-domain variations at long wavelengths should van-ish by subtraction. However, the spacecraft separation of10–20 km does not make it possible to compute correctlygradients on scales ∼ f lh = f pp /( + f pe / f ce ) − / , where f pp , f pe are proton, electron plasma frequencies, f ce electron gyrofre-quency. The f lh , f ce plots represent the density and mag-netic field variations across the shock, which should helpthe reader to position the foot and the ramp of the shock.The proton gyrofrequency f cp = 0.1–0.4 Hz, is just below thebottom line of the picture.The power of χ correlates well with regions of ion andelectron heating, which is not true for the power spectrumof the electric field. A well known fact is that heating of ionsis not co-located with heating of electrons, as seen also inthe picture. Evident in the picture is a turbulent cascadethat appears to transfer energy from lower frequency modesat the bottom to higher frequencies in localised striationsextending to 4096 Hz, the upper frequency of measurements.Waves above f lh are identified further as LHD and above f ce as ECD.We now turn to a different technique of signal analy-sis – namely, multiresolution frequency decomposition usingorthogonal wavelets (Mallat 1999). This technique differsfrom pass-band filtering. Orthogonal decomposition is ex-act; the signal is divided into discrete frequency dyads thatform − n f N hierarchy starting from the Nyquist frequency f N . Orthogonality means that time integral of any pair of Figure 1.
Power spectrum of χ across the perpendicular bowshock. Over-plotted are: the lower-hybrid frequency f lh (black),temperatures, T e ⊥ (magenta), T i ⊥ (red), both in eV, and electroncyclotron frequency f ce . At the bottom up to f lh there are obliquewhistler waves, followed by striations of Lower-Hybrid-Drift wavesin the range f lh − f ce that extend as Electron-Cyclotron-Driftwaves above f ce . the frequency dyads is zero. The decomposed electric field isshown in Fig. 2 grouped into three layers with perpendicularand parallel components shown separately. In order to savespace, plots show only halves of the waveforms for the par-allel and for one of the perpendicular components. Similardecomposition for the magnetic field is shown in Fig. 3.Noticeable in these plots is the electrostatic characterof waves in panel 2a and electromagnetic character of wavesin panels 2c, 3c. Also interesting is the large parallel electricfield, as well as the compressional magnetic field present inall wave modes.Frequency range between f cp and f ce is occupiedby whistler and lower-hybrid waves which belong tothe same branch of the dispersion equation but dif-fer by the propagation direction and polarisation prop-erties. LH waves propagate close to perpendicular direc-tion and are mostly electrostatic, while whistlers have k-vectors in a wide angular range and become purely elec-tromagnetic in the parallel direction. These two wavetypes can change their mode by conversion on the den-sity gradients and striations (Rosenberg & Gekelman 2001;Eliasson & Papadopoulos 2008; Camporeale et al. 2012)that exist at the bow shock.The analysis of the propagation direction for individ-ual frequency dyads in the range 2–16 Hz shows that theypropagate 120–140 degrees to the magnetic field, which im-plies that these are oblique whistler modes. The propagationdirection is established both from the Poynting vector direc-tion and the minimum variance of the wave magnetic field.At 32 Hz and above, the propagation direction becomesperpendicular, the magnetic component diminishes rapidly MNRAS000
Power spectrum of χ across the perpendicular bowshock. Over-plotted are: the lower-hybrid frequency f lh (black),temperatures, T e ⊥ (magenta), T i ⊥ (red), both in eV, and electroncyclotron frequency f ce . At the bottom up to f lh there are obliquewhistler waves, followed by striations of Lower-Hybrid-Drift wavesin the range f lh − f ce that extend as Electron-Cyclotron-Driftwaves above f ce . the frequency dyads is zero. The decomposed electric field isshown in Fig. 2 grouped into three layers with perpendicularand parallel components shown separately. In order to savespace, plots show only halves of the waveforms for the par-allel and for one of the perpendicular components. Similardecomposition for the magnetic field is shown in Fig. 3.Noticeable in these plots is the electrostatic characterof waves in panel 2a and electromagnetic character of wavesin panels 2c, 3c. Also interesting is the large parallel electricfield, as well as the compressional magnetic field present inall wave modes.Frequency range between f cp and f ce is occupiedby whistler and lower-hybrid waves which belong tothe same branch of the dispersion equation but dif-fer by the propagation direction and polarisation prop-erties. LH waves propagate close to perpendicular direc-tion and are mostly electrostatic, while whistlers have k-vectors in a wide angular range and become purely elec-tromagnetic in the parallel direction. These two wavetypes can change their mode by conversion on the den-sity gradients and striations (Rosenberg & Gekelman 2001;Eliasson & Papadopoulos 2008; Camporeale et al. 2012)that exist at the bow shock.The analysis of the propagation direction for individ-ual frequency dyads in the range 2–16 Hz shows that theypropagate 120–140 degrees to the magnetic field, which im-plies that these are oblique whistler modes. The propagationdirection is established both from the Poynting vector direc-tion and the minimum variance of the wave magnetic field.At 32 Hz and above, the propagation direction becomesperpendicular, the magnetic component diminishes rapidly MNRAS000 , 1–6 (2020) on and electron heating at the bow shock -100-50050100 m V / m
2 , E || -30-1501530 m V / m
2 , E || -10-50510 m V / m
2 , E || Figure 2.
The measured electric vector, sampled at 8192 s − isdecomposed into 3 bands using orthogonal wavelets (complete sig-nal is exact sum of the components) and transformed to magneticfield-aligned coordinates (FAC). From the bottom: (c) obliquelypropagating whistler waves (2–16 Hz), (b) Lower-Hybrid-Driftwaves (32–256 Hz), and (a) Electron-Cyclotron-Drift waves (512–4000 Hz). To enhance readability we show only halves of thewaveforms for the parallel and for one of the perpendicular com-ponents. -0.4-0.200.20.4 n T
2 , B || -202 n T
2 , B || -10-50510 n T
2 , B || Figure 3.
The measured magnetic signal is decomposed into 3bands and shown in FAC coordinates as in Fig. 2. with increasing frequency, indicating electrostatic, lower-hybrid character. These waves are most likely related to theLHD instability, which is a cross-field current-driven insta-bility that couples LH waves with drift waves generated onthe density gradients (Krall & Liewer 1971; Davidson et al.1977; Daughton 2003). According to theory, there could betwo types of instability. A weaker one – kinetic, should growwhen the scale of the density gradient L n = ( N − |∇ N |) − is in Figure 4. (a) Computed gradient scale of the density L n nor-malised with proton gyroradius r p showing that the region isstrongly unstable for Lower-Hybrid-Drift waves ( L n / r p < | χ | , which correlates well with heating of ions andelectrons. It contains 245,760 data points. (c) Difference E y − E y of the electric field measured by satellites 1 and 2, separated by ∆ y =5 km. the range < L n / r p < ( m p / m e ) / , while the stronger, fluid-type instability occurs when L n / r p < . Here, r p is a protonthermal gyroradius. The instability is caused by the dia-magnetic ion drift V d = T p ( m p ω cp L n ) − = v t p ( r p / L n ) . Themaximum growth rate is at k ⊥ r e ∼
1, i.e. at wavelengths ofa few electron gyroradii. These wavelengths will be Dopplershifted by the prevailing plasma flows V ≈
300 km/s to fre-quencies f = V /( πλ ) ∼ L n of the density gradient at the bow shockthat can be determined directly by the MMS is indeedsmaller than the ion gyroradius, as seen in Fig. 4a. Thisprovides additional support for the interpretation that thebursty waves in the range f lh − f ce of Figures 1 and 2bare mostly LHD waves. They are probably a permanentfeature of quasi-perpendicular bow shocks because of thedensity ramp that drives the instability. Such waves havebeen observed also in other regions of the magnetosphere(Bale et al. 2002; Vaivads et al. 2004; Norgren et al. 2012;Graham et al. 2017).Panel 2a contains electrostatic waves, which have beenclassified by first observers (Rodriguez & Gurnett 1975;Fuselier & Gurnett 1984) as ion-acoustic modes becauseof the large parallel electric field component. Waves inthis frequency range have been analyzed with use ofhigh-time resolution measurements obtained by THEMIS(Mozer & Sundqvist 2013; Wilson III et al. 2014), and bySTEREO and Wind (Breneman et al. 2013), who notedlarge parallel electric fields and identified electron cyclotronharmonics in the spectra. Their conclusion that these areECD waves is supported by the spectrum in Fig. 5 whichshows seven f ce harmonics.The ECD instability (Forslund et al. 1972) is caused bythe perpendicular relative ion-electron drift V d comparableto the thermal electron speed v te . It occurs at the resonance MNRAS , 1–6 (2020)
K. Stasiewicz Hz at 2017-12-12T03:33:27.300-7.800 -3 -2 -1 ( m V / m ) / H z Figure 5.
Power spectrum of E ⊥ from the electron heating region.Obliquely propagating whistlers extend from 2 up to f lh ∼
16 Hzfollowed by LHD waves. Clearly seen are Electron-Cyclotron-Driftwaves with 7 harmonics starting from f ce ≈
500 Hz. k ( V d − v t p ) = n ω ce , couples electron Bernstein modes withion-acoustic waves, and produces wavelengths smaller thanthe electron gyroradius (Lashmore-Davies 1971). In the cur-rent theoretical scenario (Muschietti & Lemb´ege 2013) thisinstability is presumably produced by ion beams reflectedfrom the shock and moving perpendicularly against the so-lar wind and stationary electrons. This is a rather unrealisticscenario because two ion populations cannot have perpen-dicular ExB drift in opposite directions at the same time,and different from electrons (there are no other drifts intheir model). This model would put the location of ion-beamdriven ECD instability in front of the shock, around time03:33:15 in our data, while the maximum of ECD waves isat 03:33:27 in the electron heating region. Despite the un-realistic scenario, the mathematics and conclusions of thispaper are correct and valuable.The data presented here suggest another mechanism forthe ECD instability. The LHD electric fields of 20 mV/m in B ∼
15 nT produce ExB drift of 1300 km/s, comparable tothe electron thermal speed. Since this field exists in narrowchannels with a width of a few electron gyro-radii, only elec-trons can participate, making ion-electron drift V d ∼ v te . In-cidentally, this instability attracted attention of researchersworking with Hall ion thrusters, because it affects their per-formance (Ducrocq et al. 2006; Boeuf & Garrigues 2018). Inthese devices, the instability is also caused by the ExB driftof electrons only, because ions have gyroradius larger thanthe drift channels.The theory of ECD instability that links electron cy-clotron harmonics with ion-acoustic waves and wavelengthsat scales below the electron gyroradius provides convinc-ing explanation for waves in panel 2a, which exhibit bothlarge E k and E ⊥ , contain f ce harmonics, and Doppler shiftedstructures smaller than electron gyroradius. We focus now on the fundamental question what kind ofprocesses can increase the ion energy by 400 eV (from 50to 450), and the electron energy by 50 eV (from 20 to 70eV), within a few seconds, or during one proton gyro-period, as seen in Fig. 1. Additional constraint is that heating ofelectrons is isotropic, while ions is mostly perpendicular.The mechanism of stochastic ion heating on LH wavesrelated to condition (1) has been explained by Karney(1979), and on other wave structures by other authors:Cole (1976); McChesney et al. (1987); Balikhin et al.(1993); Mishin & Banaszkiewicz (1998); Stasiewicz et al.(2000); Stasiewicz (2007); Vranjes & Poedts (2010);Stasiewicz et al. (2013); Yoon & Bellan (2019). The essenceis that particles stressed by strong gradients (or equivalentlyby local space charges, div( E )= ρ / ǫ ) loose adiabaticityand can be scattered along the electric field for directacceleration by both DC- and wave-fields. The computed χ (Fig. 4b) shows values greatly exceeding 1, needed tostochastically perturb proton orbits, however, significantion heating is observed when χ > .Formally, the error of χ is equivalent to the error ofthe electric field measurements, i.e., ∼ E y components measuredby satellites 1 and 2. Similar differences are seen on any pairof the satellites. A difference ∆ E y ≈
100 mV/m on scales ofthe electron gyroradius r e ≈ B =15 nT wouldproduce χ ≈ χ well above the re-quired threshold for electron heating, which is reached incomputations shown in Fig. 4b.The presented data imply the following scenario for therapid ion heating seen in Fig. 1. The diamagnetic currentrelated to increasing density gradient at the foot of the shock(Fig. 4a) drives LH waves that couple/convert to obliquewhistlers on density gradients and turn into LHD instabilitywhen the threshold is exceeded ( V d ∼ v t p ). This results inincrease of the wave amplitude, and cascade to shorter scales( kr e ∼ χ > that can perturb incomingsolar wind ions. The ion energisation is probably a one stepprocess (Stasiewicz 2007) (see Fig. 1 in that paper) in whichions gain energy from the DC electric field and not fromwave fields. There is a quasi-DC (below 1 Hz) electric field of E =5 mV/m in this region. Protons stochastically perturbedby χ ∼ E , and acquire large perpendicular velocity duringone gyro-period. To acquire 400 eV required for ion heatingone needs electric potential from E =5 mV/m on a distanceof 80 km. A displacement of ion gyrocenters by less than agyroradius is sufficient to account for ion energisation.Electrons require χ larger by a factor m p / m e to startstochastic heating. Apparently, amplitudes of LHD wavescannot produce gradients strong enough to make χ > ∼
20 mV/m canproduce ExB drift of electrons comparable to the thermalvelocity ( V d = V E ∼ v te ), which is needed to start ECD in-stability. This instability will produce stronger fields ∼ χ > and initiate electronheating. The mechanism for the ECD instability at the bow MNRAS , 1–6 (2020) on and electron heating at the bow shock shock is similar to that occurring in Hall thrusters. In bothcases only electrons are drifting, because ions are preventedfrom the ExB drift by large gyroradius compared to the sizeof drift channels.Stochastic heating of electrons occurs in localised burstsand is quenched by increasing B , because of the dependence V E ∝ B − for the instability driver, and χ ∝ B − for thestochastic condition. In most of the region, χ < m p / m e , sothe electrons respond adiabatically, T e ⊥ ∝ B , and a majorpart of the energy increase for electrons can be attributedto the increasing B at the ramp of the shock. Such perpendic-ular adiabatic heating must be accompanied by isotropiza-tion, possibly by E k of waves shown in Figure 2a.It is interesting to note that by expressing div( E ) ∼ E / L E , and V E = E / B we can rewrite the condition χ e > as V E v te > L E r e (2)which implies that to start electron heating we need the ratioof the drift to thermal velocity exceeding the ratio of theelectric gradient scale to the gyroradius. This fits propertiesof ECD waves, and the instability condition.Another implication of equation (1) is the connectionto charge non-neutrality. With typical V A ∼
100 km/s at thebow shock, we need local charge non-neutrality exceeding10 − to start ion heating, and 2 × − to heat electrons. Alsothe presence of E k ∼ E ⊥ in ECD waves in Fig. 2a is a naturalconsequence of space charges, which would produce electricfields in both directions. The observations and the above analysis support the hy-pothesis on the universal character of the relation (1) andits applicability for identifying the heating processes of bothelectrons and ions, independent of the wave mode and thetype of instability.The plasma heating mechanism at quasi-perpendicularbow shocks appears to be a two-stage process. In the firststage, the ion diamagnetic drift on the density gradients, V d ∼ v t p , ignites the LHD instability that produces E ∼ χ ≫ that heats ions. In the second stage,the electric field of LHD waves produces electron ExB drift , V E ∼ v te , which ignites the ECD instability that producesstronger fields, E ∼
100 mV/m, and χ > m p / m e , that heatselectrons. Part of the electron energy increase can be at-tributed to adiabatic heating T e ⊥ ∝ B with isotropization byECD waves.The mechanism of the ECD instability at the bow shockis similar to the experimentally verified ECD instability inion Hall thrusters, i.e., the ExB electron drift. All elementsof the heating scenario described above have experimentalsupport in MMS measurements.Finally, it should be emphasised that this paper presentsfor the first time spectrum of the divergence of the elec-tric field measured in space. Figure 1 is a time-frequencyspectrogram of the electric charge distribution, weightedwith B − , inside the bow shock. Obviously, it is an approx-imation, because gradients on small scales are not well re-solved. The credits should go entirely to the members ofthe FIELDS consortium of the MMS project (Torbert et al. 2016; Lindqvist et al. 2016; Ergun et al. 2016) for designingthe instruments and performing the project. ACKNOWLEDGEMENTS
Special thanks to members of the FIELDS consortium of theMMS mission for making instruments capable of measuringdivergence of the electric field in space. The author is grate-ful to Yuri Khotyaintsev for his invaluable help with thesoftware for data processing. The MMS data are availableto public via https://lasp.colorado.edu/mms/sdc/public/
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