PIC simulations of microinstabilities and waves at near-Sun solar wind perpendicular shocks: Predictions for Parker Solar Probe and Solar Orbiter
Zhongwei Yang, Ying D. Liu, Shuichi Matsukiyo, Quanming Lu, Fan Guo, Mingzhe Liu, Huasheng Xie, Xinliang Gao, Jun Guo
DDraft version August 18, 2020
Typeset using L A TEX default style in AASTeX63
PIC simulations of microinstabilities and waves at near-Sun solar wind perpendicular shocks:Predictions for Parker Solar Probe and Solar Orbiter
Zhongwei Yang ,
1, 2, 3
Ying D. Liu ,
1, 4
Shuichi Matsukiyo , Quanming Lu , Fan Guo , Mingzhe Liu , Huasheng Xie,
8, 9
Xinliang Gao , and Jun Guo State Key Laboratory of Space Weather, National Space Science Center, Chinese Academy of Sciences, Beijing, 100190, People’sRepublic of China ([email protected], [email protected]) CAS Key Laboratory of Geospace Environment, Chinese Academy of Sciences, University of Science and Technology of China, Hefei,230026, People’s Republic of China ([email protected]) Key Laboratory of Earth and Planetary Physics, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, 100029,People’s Republic of China University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China Faculty of Engineering Sciences, Kyushu University, 6-1 Kasuga-Koen, Kasuga, Fukuoka, 816-8580, ([email protected]) Los Alamos National Laboratory, NM 87545, USA ([email protected]) LESIA, Observatoire de Paris, Universit PSL, CNRS, Sorbonne Universit, Universit de Paris, 5 place Jules Janssen, 92195 Meudon,France ([email protected]) Hebei Key Laboratory of Compact Fusion, Langfang 065001, People’s Republic of China ENN Science and Technology Development Co., Ltd., Langfang 065001, People’s Republic of China ([email protected]) CAS Key Laboratory of Geospace Environment, Chinese Academy of Sciences, University of Science and Technology of China, Hefei,230026, People’s Republic of China ([email protected]) College of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao, 266061, People’s Republic of China ([email protected]) (Received May 21, 2020; Revised Jul 28, 2020; Accepted Aug 15, 2020)
Submitted to ApJLABSTRACTMicroinstabilities and waves excited at moderate-Mach-number perpendicular shocks in the near-Sun solar wind are investigated by full particle-in-cell (PIC) simulations. By analyzing the dispersionrelation of fluctuating field components directly issued from the shock simulation, we obtain key findingsconcerning wave excitations at the shock front: (1) at the leading edge of the foot, two types ofelectrostatic (ES) waves are observed. The relative drift of the reflected ions versus the electronstriggers an electron cyclotron drift instability (ECDI) which excites the first ES wave. Because thebulk velocity of gyro-reflected ions shifts to the direction of the shock front, the resulting ES wavepropagates oblique to the shock normal. Immediately, a fraction of incident electrons are acceleratedby this ES wave and a ring-like velocity distribution is generated. They can couple with the hotMaxwellian core and excite the second ES wave around the upper hybrid frequency. (2) from themiddle of the foot all the way to the ramp, electrons can couple with both incident and reflected ions.ES waves excited by ECDI in different directions propagate across each other. Electromagnetic (EM)waves (X mode) emitted toward upstream are observed in both regions. They are probably induced bya small fraction of relativistic electrons. Results shed new insight on the mechanism for the occurrenceof ES wave excitations and possible EM wave emissions at young CME-driven shocks in the near-Sunsolar wind. INTRODUCTIONCollisionless shocks are of fundamental interests in astrophysics and space physics. They have been proposed asprimary mechanisms for energy dissipation (e.g., Richardson et al. 2008; Parks et al. 2012) and particle acceleration(e.g., Zank et al. 2006; Guo & Giacalone 2010). Observational studies suggest that large solar eruptions are oftenaccompanied with shocks driven by corona mass ejections (CMEs). They are rich in various plasma waves (Wilsonet al. 2007; Liu et al. 2018) and usually associated with type II radio bursts (Bale et al. 1999; Liu et al. 2009). a r X i v : . [ phy s i c s . s p ace - ph ] A ug Yang et al.
Previous investigations reveal that there are two main mechanisms for high frequency electromagnetic wave emissions(e.g., O or X modes) at collisionless shocks. One is synchrotron maser instability (SMI) which usually refers to anelectromagnetic wave emission mechanism at relativistic magnetized shocks (Hoshino & Arons 1991; Plotnikov & Sironi2019). This mechanism is in favor of a positive slope in a electron velocity distribution function (VDF) perpendicularto the ambient magnetic field, such as due to a loss cone or ring velocity distribution, which can become a free energysource for wave emissions. The essentially the same mechanism may operate as long as weakly relativistic anisotropicelectrons exist (even down to a few keV ) in the case of auroral electrons (Wu & Lee 1979). So it may indeed bea potential candidate for radio emissions at non-relativistic CME-driven shocks associated with enhanced electronintensities from < keV to about 200 keV (Liu et al. 2008). The other is the nonlinear three-wave interaction (Pulupaet al. 2010; Gao et al. 2017). First, suprathermal and nonthermal electrons are produced in the shock. Then, theseenergetic electrons stimulate the growth of high-frequency electrostatic (ES) waves, such as Langmuir waves. TheseES waves interact with each other in the nonuniform background solar wind or rippling shock front to produce theobserved EM emissions due to the nonlinear wave interactions (Umeda 2010; Ganse et al. 2012). However, it is stillunclear which mechanism and what wave mode plays a major role at CME-driven shocks near the Sun.Electrostatic waves, such as (1) Langmuir waves and their higher harmonics (Bale et al. 1999; Thejappa & MacDowall2019) and (2) electron cyclotron harmonic waves (Bernstein waves) (Wilson et al. 2010; Goodrich et al. 2018), havebeen widely observed at shocks by spacecraft (ISEE 1, WIND, STEREO, CLUSTER, MMS etc.) around 1 au. Theyare commonly believed to accelerate electrons and provide possible free energy sources of EM emissions at shocks,magnetopause, and solar flares (Graham et al. 2018; Hork´y et al. 2018; Henri et al. 2019). Despite this, there is a lackof in situ observations accurately of determining shock microstructures and associated plasma waves excited in thenear-Sun solar wind. Bale et al. (2016) extracted some solar wind parameters in the near-Sun conditions based on datafrom HELIOS and models, which give us some inspiration. The study of wave properties at shocks under the near-Sunsolar wind condition could help understand ES wave excitations and EM wave emissions at young CME-driven shocksobserved by Parker Solar Probe (PSP) or Solar Orbiter near the perihelion at later encounters. Near the Sun, weexpect a strong magnetization and relatively low values of plasma β (Bale et al. 2016) that may affect characteristicsof waves excited at the shock front. This is our motivation to do the work.Umeda et al. (2012a) directly extract the field components of the wave from a 2-D shock simulation (ambient B in-plane case) with a relatively small box along the shock front ( ∼ c/ω pi , where c/ω pi is the ion inertial length).They identify whistler waves excited by modified two-stream instability (MTSI) (Matsukiyo & Scholer 2003) by usinga dispersion relation analysis of fluctuating electromagnetic field components δB and δE . Based on this method, wedirectly extract high resolution fluctuating electromagnetic field components from 2-D large scale shock simulations(ambient B out-of-plane case, including the complete particle gyro-motion perpendicular to the background B ), andcheck the wave modes from the leading edge of the foot to the ramp. Furthermore, observed linear waves are confirmedby a linear theory tool (BO, a new version of PDRK) (Xie 2019).This paper is organized as follows. We present a description of the simulation model and setup in Section 2. Theshock front wave analysis is presented in Section 3, and we conclude with a summary in Section 4 and discuss theimplications of our results for Parker Solar Probe and Solar Orbiter. SIMULATION MODELWe carry out shock simulations using an open source electromagnetic particle-in-cell (PIC) code named EPOCH(Arber et al. 2015) with a normalization which is the same as that used in our previous works (Yang et al. 2016,2018; Lembege et al. 2020) to simulate the waves excited at moderate-Mach-number, perpendicular shocks. In thispaper, collisionless shocks were generated by the so-called injection method (Matsukiyo & Scholer 2012; Yang et al.2015), in which particles are injected from the one side of the simulation boundary (at X = 0) at super-Alfv´enic speed V inj = 6 V A in the + X direction and specularly reflected at the other side of the simulation boundary ( X = L x ). Theshock propagates in the − X direction in the present downstream rest frame. The periodic boundary condition is appliedin the Y direction. The number of grid cell is n x × n y = 25 , × , X = ∆ y = 0 . c/ω pi .The box size along the shock surface is 5 c/ω pi . The ion-to-electron mass ratio m i /m e is 100 and the particle numberper cell is 50. In the 2-D simulation, the ambient magnetic field B is along Z and strictly perpendicular to the X − Y simulation plane (i.e., the shock normal angle θ Bn ) is nearly 90 ◦ . This B configuration is similar to that inprevious simulations (Amano & Hoshino 2009; Matsumoto et al. 2013). Based on fitting methods similar to Bale et al.(2016), plasma parameters in the near-Sun solar wind (at about 10 R s ) can be estimated by using PSP data observed hock waves near the Sun R s . Such as the magnetic field B ∼ nT , the protondensity N p ∼ cm − , the proton temperature T p ∼ . eV , and the Alfv´enic velocity V A ∼ km/s . T e /T i = 3is adopted based on previous observations from HELIOS (Liu et al. 2005) and PSP (Maksimovic et al. 2020). Thespeed of fast CME-driven shocks at 10 Rs can often exceed 1500 ∼ km/s (Zhao et al. 2019), and a fraction ofthem have extremely high speeds ( ∼ km/s ) (Liu et al. 2013, 2019). Their corresponding average M A is 5 ∼ ∼
15. Such Mach numbers are much larger than that observed at interplanetary(IP) shocks at 1 au (Wilson et al. 2007; Liu et al. 2018). In this simulation, the Alfv´enic Mach number of the shock isabout 7 ∼
9. Plasma beta values: β e = 0 . β i = 0 . ω pe / Ω ce ∼ .
8, on the order of 10 for the near-Sun solar wind conditions. In addition, weexamined wave properties at 2-D shocks with similar setups for different Mach numbers: M A = 5 ∼ >
10 forslower and faster shocks, respectively. A 3-D shock simulation is also carried out for comparing the ES wave propertyin higher dimensions. In this paper, we focus on the moderate-Mach-number 2-D shock with universal significance.The other cases will be discussed in section 4. SIMULATION RESULTS: WAVE ANALYSIS IN 2-D SHOCK SIMULATIONSFigure 1a shows an overview of the time-evolving shock magnetic field B z averaged along Y in the simulation. Theshock becomes mature and has reached a fully evolved state after t > . − ci in this case. In order to study waveproperties at the shock front with a relatively high resolution in κ − ω space, the electromagnetic field components aresampled in a long period: t = 3 . ∼ . − ci , and a large box including the whole shock front. Figure 1b-d representthe snapshots of the fluctuations of δE x , δE y and δB z at a typical time t = 3 . − ci within the sampling period. Thezoomed B z profile consisting typical shock structures: upstream, foot, ramp, overshoot, and downstream is shown inFigure 1d for reference. The shock ramp is located at about X = 56 . δE x ( x, y, t ) as anexample to show how the data is sampled for the wave analysis. Figure 1e-f illustrate a slice of the sampled shockprofile δE x at Y = L y / X − Y − t space are converted to the shock ramp rest frame. A 3-D Fourier transform isapplied after Hanning windowing to compensate for the nonperiodicity of the data, in both X and t directions (Figure1g-h). Similar process are also carried out for other fluctuation components δE y and δB z .Figure 2a-d shows corresponding phase space plots ( X − V x,y ) of particles at t = 3 . − ci . The solar wind is comingfrom the left hand side and the shock ramp is marked by a black vertical line as in Figure 1b-d. In region A (Figure2a-b), some electrons are trapped and accelerated by the excited ES waves (Figure 1b-c) at the leading edge of the foot(Umeda et al. 2009; Amano & Hoshino 2009; Yang et al. 2018). They immediately form a ring-like velocity distributionrelative to the hot drifting Maxwellian core. Figure 2c-d show that a fraction of incident ions are reflected at the ramp,and the others are directly transmitted to the downstream. At the same time, the electron bulk velocity shifts in both X and Y directions to keep the quasi-neutrality. The ion bulk velocity component V iy can be larger than V ix in regionA due to their gyromotion and the shock acceleration along the shock surface. This overall picture is in consistentwith previous simulations (Scholer et al. 2003; Yang et al. 2009). Figure 2e shows Y − averaged number density profilesof electrons (green), incident ions (blue), and reflected ions (red). Here, we only focus on the wave analysis region andquickly separate the reflected ions before the ramp as in Otsuka et al. (2019). From the foot to the ramp, the percentageof reflected ions increases. Figure 2f-h represent the 2-D density profiles of electrons, reflected ions and incident ions.Combined with corresponding field fluctuations (Figure 1b-d), we find that the coupling between reflected ions andelectrons is strong in region A. The coupling between incident ions and electrons along the shock normal becomesnoticeable in region B.Figure 3 shows the 3-D view of the wave dispersion relation diagram of δE x , δE y , and δB z in the κ x − κ y − ω space.Top and bottom panels denote results from regions A and B, respectively. In the simulation, the electron plasmafrequency ω pe and the Debye length λ De as units are measured in the far upstream undisturbed solar wind. By usinga unified tool for plasma waves and instabilities analysis: BO (Xie 2019), corresponding dispersion relations calculatedby the linear theory are shown in Figure 4 to assist in identifying wave modes observed in the simulation. The input Yang et al. plasma parameters for the linear analysis are directly issued from the simulation data, which is averaged over thewhole sampling period in the shock rest frame. More details are shown in Table 1.Firstly, we study the waves in region A (the leading edge of the foot). In region A, three main wave modes areobserved: (1) the first is an ES wave marked by “ES-1” in Figure 3a-b. This wave is rapidly excited by the relativedrift between incident electrons and gyro-reflected ions. As shown in Figure 2, the main bulk velocity component ofreflected ions is in + Y and − X directions at the beginning of the foot. So the reflected ion beam is strongly coupledwith the Doppler-shifted electron cyclotron harmonic braches on the + κ y and − κ x side (Figure 4a-b). This couplingprocess is discussed as electron cyclotron drift instability (ECDI) by Muschietti & Lemb`ege (2013). They assumedthat the reflected ion beam is straight along the shock normal and the gyro-motion is not considered (i.e., only focus onthe X direction). From the simulation above, we realize that the ion bulk velocity in the Y direction can be large, andeven plays a more important role. This is a new point. (2) the second is also an ES wave marked by “ES-2” in Figure3a-b. After the “ES-1” is excited, a fraction of incident electrons are trapped and accelerated by the “ES-1” wave. Onepossible acceleration mechanism of these electrons is the shock surfing acceleration where the electric field or potentialplays an important role (Zank et al. 1996; Hoshino & Shimada 2002; Amano & Hoshino 2009; Matsumoto et al. 2017).The accelerated electrons immediately form a ring-like velocity distribution. This VDF has a weak positive slope inthe velocity space perpendicular to the magnetic field. Free energy is quickly released through the coupling betweenthe ring and the hot Maxwellian core. The growth rate of “ES-2” peaks at about κ = ± . λ − De in the linear theoryaround the Doppler-shifted upper hybrid frequency (Figure 4). It is consistent with the simulation results (purpleregions in Figure 3a-b). (3) the third wave is an electromagnetic mode (marked by “EM-1” in Figure 3c). This highfrequency EM wave is emitted facing the upstream and is visible in both δE ⊥ and δB z diagrams. In this case, the δE z only has a background noise (not shown here). Hence, the EM mode is an extraordinary electromagnetic mode(X mode). This X mode emission cannot be described in Figure 4 because kinetic relativistic effects are not includedin the present version of our linear solver yet. One possible scenario is the synchrotron maser radiation excited byrelativistic electrons with a ring-type velocity distribution at strongly magnetized plasmas (Hoshino & Arons 1991;Plotnikov & Sironi 2019) under a Doppler-shifted condition. In summary, this EM wave is locally emitted at the shockfront and propagates toward the upstream.Secondly, the waves in region B (i.e., in the middle of the foot) are investigated. Figure 2e shows that the values of n e and B are ≥ n e and B in this region. The local ratios ω pe /ω pe and Ω ce / Ω ce are about 1.414 and 2, respectively. The frequency ratio ω pe / Ω ce is about 5.5 and lower than its upstream value( ω pe / Ω ce ≈ . ω pe amd λ − De as units of ω and κ for region B in bottom panels of Figure 3. Corresponding dispersion relationsfrom the linear theory are shown in Figure 4c-d. In region B, both reflected and incident ion beams couple with theelectron cyclotron harmonic branches in all directions of the simulation plane. The main excited waves are as follows:(1) in Figure 3d, “ES-3” wave modes are excited by ECDI on ± κ x directions. The wave frequency in the ± κ x directionsare comparable to each other as predicted by the linear theory (Figure 4c). (2) in Figure 3e, “ES-4” wave modes havemore harmonic branches in the + κ y direction. This is because the old reflected ions are gyrating back towards theramp and they have a large bulk velocity ( > V A ) in the + Y direction. In contrast, the bulk velocity of incidentions in the − Y direction caused by the deflection ahead of the ramp is relatively low. In summary, the middle of thefoot could be a zoom of ECDI which is triggered in different directions. This is interesting and brand new relative toprevious 1-D simulations (Muschietti & Lemb`ege 2006, 2013) which only discuss ECDI along the x − axis. Our workhas taken a step forward on this basis, and find that the ES waves can be excited along the shock surface as well asalong the shock normal. In addition, we also find some low frequency EM waves (marked by “EM-2” in Figure 3f)associated with the “ES-4” waves. Such low frequency EM wave has a modulating effect on the magnetic field andplasma density profiles (refer Figure 1d and Figure 2f-h). The analysis of such EM wave amplification might requiresa theory considering nonuniform plasmas and gradient background magnetic fields. Hence, it will not be furtherdoscussed in this paper. It is worthy noting that the X mode can exist well in region B due to the local relativisticelectrons, and it has a wave vector towards upstream as in region A. CONCLUSIONS AND DISCUSSIONSThis paper presents PIC simulations of a perpendicular shock in the near-Sun solar wind condition, where themagnetization is relatively high, β is relatively low, and the average shock Mach number is below 10, corresponding to hock waves near the Sun
5a young, fast CME-driven shock propagating in the pristine solar wind. The simulated electromagnetic fluctuationsshow that the shock foot can be segmented into two parts by different wave features.1. At the leading edge of the foot, ES waves are excited by electron cyclotron drift instability in the − X and + Y directions. The instability is triggered by the coupling between the incident electrons and gyro-reflected ions. Thewave vector of this ES wave is oblique to the shock normal, and is mainly along the shock surface. This is because thebulk velocity of reflected ions changes from the − X direction to the + Y direction during the gyro-reflection process.2. In the same region, a fraction of incident electrons can be trapped and accelerated by the above exited ES waves.A secondary instability occurs around the Doppler-shifted upper hybrid frequency due to the coupling between thisaccelerated ring-like electrons and the hot Maxwellian core electrons. In addition, some weak relativistic electronsin the ring VDF could lead to a high frequency EM emission probably induced by SMI. The emitted EM wave is aX mode and propagates upstream. Other potential candidates of X mode emission mechanisms are also need to beconsidered for quasi-perpendicular shocks. For an instance, a combination of wave growth due to electron cyclotronmaser instability and nonlinear wave-coupling processes is suggested for plasmas with ω pe / Ω ce ∼
10 in the outer corona(Ni et al. 2020).3. From the middle of the foot all the way to the ramp, the incident ions begin to couple with the electrons. In thisregion, the incident ion beam begin to deflect in the − Y direction when approaching the ramp, and the old gyro-backreflected ions have a large bulk velocity in the + Y direction. Therefore, ES waves can be excited by ECDI in both ± X and ± Y directions. These multi-directional ES waves propagate across each other and form fibrous-like patternin electric field and particle number density profiles. This is brand new and different from early 1-D simulations andtheoretical models of shock front ES wave excitations.4. The long-wavelength low-frequency EM waves associated with ES harmonics in the + Y direction is strengthenas it approaches the ramp. In addition, the high-frequency X mode is also observed in this region.Furthermore, we carried out two additional shock simulations with lower and higher Mach numbers as mentioned inSection 2. Preliminary results indicate that the ECDI is robust and can be observed in all cases. Buneman instabilityappears only at extremely fast CME-driven shocks with a speed > km/s (i.e., M a >
10) (Liu et al. 2019).Normally this is not the case and the shock speed is as slow as that used in this paper. At slower shocks (Ma ≤ B out-of-plane from the 3-D simulation is similar to that observed in this paper. However, ES wavessampled in the B in-plane propagate almost along the shock normal. This is because the main ion bulk velocitycomponents is not included in this cross section. More completely, a 4-D FFT is required for the wave analysis in 3-Dshock cases. This is still infusible under most of the current storage conditions.As we all know, the Sun is entering the solar maximum in the next 5 ∼ . R s to less than 10 R s ) (Bale et al. 2016; Fox et al. 2016; Kasper et al. 2016). Parker Solar Probe and Solar Orbiter arelikely to observe high frequency EM emissions (e.g., X modes) toward upstream accompanied with local energeticelectrons, and electrostatic waves induced by ECDI or BI along the shock normal or along the shock surface at fastCME-driven shocks. ACKNOWLEDGMENTSWe thank the referee for helpful comments and the NASA Parker Solar Probe Mission for use of data. The authors aregrateful to Dr. L. Muschietti from UC Berkeley for helpful discussions on PDRK/BO benchmarks. The computationsare performed by Numerical Forecast Modeling R&D and VR System of State Key Laboratory of Space Weather,and HPC of Chinese Meridian Project. This work is supported by the NSFC (41574140, 41674168, 41774179), theSpecialized Research Fund for State Key Laboratories of China, Youth Innovation Promotion Association of theCAS (2017188), the Open Research Program Key laboratory of Geospace Environment CAS (GE2017-01), the OpenResearch Program Key laboratory of Polar Science, MNR (KP202005), Beijing NSF (1192018), Beijing MunicipalScience and Technology Commission (Z191100004319001, Z191100004319003), Beijing Outstanding Talent TrainingFoundation (2017000097607G049), and the Strategic Priority Research Program of CAS (XDA14040404). S. M.acknowledges partial support by Grant-in-Aid for Scientific Research (C) No.19K03953 and (B) No.17H02966 fromJSPS. J. G. is supported by the Shandong Provincial National Natural Science Foundation(ZR2017MD012). Yang et al.
REFERENCES
Amano, T., & Hoshino, M. 2009, ApJ, 690, 244,doi: 10.1088/0004-637X/690/1/244Arber, T. D., Bennett, K., Brady, C. S., et al. 2015, ppcf,57, 113001, doi: 10.1088/0741-3335/57/11/113001Bale, S. D., Reiner, M. J., Bougeret, J.-L., et al. 1999,Geophys. Res. Lett., 26, 1573,doi: 10.1029/1999GL900293Bale, S. D., Goetz, K., Harvey, P. R., et al. 2016, SSRv,204, 49, doi: 10.1007/s11214-016-0244-5Fox, N. J., Velli, M. C., Bale, S. D., et al. 2016, SSRv, 204,7, doi: 10.1007/s11214-015-0211-6Ganse, U., Kilian, P., Vainio, R., & Spanier, F. 2012, SoPh,280, 551, doi: 10.1007/s11207-012-0077-7Gao, X., Lu, Q., & Wang, S. 2017, Geophys. Res. Lett., 44,5269, doi: 10.1002/2017GL073829Goodrich, K. A., Ergun, R., Schwartz, S. J., et al. 2018,J. Geophys. Res., 123, 9430, doi: 10.1029/2018JA025830Graham, D. B., Vaivads, A., Khotyaintsev, Y. V., et al.2018, J. Geophys. Res., 123, 2630,doi: 10.1002/2017JA025034Guo, F., & Giacalone, J. 2010, ApJ, 715, 406,doi: 10.1088/0004-637X/715/1/406Henri, P., Sgattoni, A., Briand, C., Amiranoff, F., &Riconda, C. 2019, Journal of Geophysical Research(Space Physics), 124, 1475, doi: 10.1029/2018JA025707Hork´y, M., Omura, Y., & Santol´ık, O. 2018, Physics ofPlasmas, 25, 042905, doi: 10.1063/1.5025912Hoshino, M., & Arons, J. 1991, Physics of Fluids B, 3, 818,doi: 10.1063/1.859877Hoshino, M., & Shimada, N. 2002, ApJ, 572, 880,doi: 10.1086/340454Kasper, J. C., Abiad, R., Austin, G., et al. 2016, SSRv,204, 131, doi: 10.1007/s11214-015-0206-3Lembege, B., Yang, Z., & Zank, G. P. 2020, TheAstrophysical Journal, 890, 48,doi: 10.3847/1538-4357/ab65c5Liu, M., Liu, Y. D., Yang, Z., Wilson, III, L. B., & Hu, H.2018, ApJL, 859, L4, doi: 10.3847/2041-8213/aac269Liu, Y., Luhmann, J. G., Bale, S. D., & Lin, R. P. 2009,ApJL, 691, L151, doi: 10.1088/0004-637X/691/2/L151Liu, Y., Richardson, J. D., & Belcher, J. W. 2005,Planet. Space Sci., 53, 3, doi: 10.1016/j.pss.2004.09.023Liu, Y., Luhmann, J. G., M¨uller-Mellin, R., et al. 2008,ApJ, 689, 563, doi: 10.1086/592031Liu, Y. D., Luhmann, J. G., Lugaz, N., et al. 2013, ApJ,769, 45, doi: 10.1088/0004-637X/769/1/45Liu, Y. D., Zhu, B., & Zhao, X. 2019, ApJ, 871, 8,doi: 10.3847/1538-4357/aaf425 Maksimovic, M., Bale, S. D., Berˇciˇc, L., et al. 2020, ApJS,246, 62, doi: 10.3847/1538-4365/ab61fcMatsukiyo, S., & Scholer, M. 2003, J. Geophys. Res., 108,1459, doi: 10.1029/2003JA010080—. 2012, J. Geophys. Res., 117, A11105,doi: 10.1029/2012JA017986Matsumoto, Y., Amano, T., & Hoshino, M. 2013, PhRvL,111, 215003, doi: 10.1103/PhysRevLett.111.215003Matsumoto, Y., Amano, T., Kato, T. N., & Hoshino, M.2017, PhRvL, 119, 1,doi: 10.1103/PhysRevLett.119.105101Muschietti, L., & Lemb`ege, B. 2006, ASR, 37, 483,doi: 10.1016/j.asr.2005.03.077—. 2013, J. Geophys. Res., 118, 2267,doi: 10.1002/jgra.50224Ni, S., Chen, Y., Li, C., et al. 2020, ApJL, 891, L25,doi: 10.3847/2041-8213/ab7750Otsuka, F., Matsukiyo, S., & Hada, T. 2019, High EnergyDensity Physics, 33, 100709,doi: 10.1016/j.hedp.2019.100709Parks, G. K., Lee, E., McCarthy, M., et al. 2012, PhRvL,108, 061102, doi: 10.1103/PhysRevLett.108.061102Plotnikov, I., & Sironi, L. 2019, MNRAS, 485, 3816,doi: 10.1093/mnras/stz640Pulupa, M. P., Bale, S. D., & Kasper, J. C. 2010,J. Geophys. Res., 115, A04106,doi: 10.1029/2009JA014680Richardson, J. D., Kasper, J. C., Wang, C., Belcher, J. W.,& Lazarus, A. J. 2008, Nature, 454, 63,doi: 10.1038/nature07024Scholer, M., Shinohara, I., & Matsukiyo, S. 2003,J. Geophys. Res., 108, 1014, doi: 10.1029/2002JA009515Thejappa, G., & MacDowall, R. J. 2019, ApJ, 883, 199,doi: 10.3847/1538-4357/ab3bcfUmeda, T. 2010, J. Geophys. Res., 115, A01204,doi: 10.1029/2009JA014643Umeda, T., Kidani, Y., Matsukiyo, S., & Yamazaki, R.2012a, J. Geophys. Res., 117, A03206,doi: 10.1029/2011JA017182Umeda, T., Matsukiyo, S., Amano, T., & Miyoshi, Y.2012b, Physics of Plasmas, 19, 072107,doi: 10.1063/1.4736848Umeda, T., & Nakamura, T. K. M. 2018, Physics ofPlasmas, 25, 102109, doi: 10.1063/1.5050542Umeda, T., Yamao, M., & Yamazaki, R. 2009, ApJ, 695,574, doi: 10.1088/0004-637X/695/1/574Wilson, L. B., I., Cattell, C., Kellogg, P. J., et al. 2007,PhRvL, 99, 041101, doi: 10.1103/PhysRevLett.99.041101 hock waves near the Sun Wilson, L. B., I., Cattell, C. A., Kellogg, P. J., et al. 2010,J. Geophys. Res., 115, A12104,doi: 10.1029/2010JA015332Wu, C. S., & Lee, L. C. 1979, ApJ, 230, 621,doi: 10.1086/157120Xie, H. S. 2019, Computer Physics Communications, 244,343, doi: 10.1016/j.cpc.2019.06.014Yang, Z., Huang, C., Liu, Y. D., et al. 2016, ApJS, 225, 13,doi: 10.3847/0067-0049/225/1/13Yang, Z., Liu, Y. D., Richardson, J. D., et al. 2015, ApJ,809, 28, doi: 10.1088/0004-637X/809/1/28 Yang, Z., Lu, Q., Liu, Y. D., & Wang, R. 2018, ApJ, 857,36, doi: 10.3847/1538-4357/aab714Yang, Z. W., Lu, Q. M., Lemb`ege, B., & Wang, S. 2009,J. Geophys. Res., 114, A03111,doi: 10.1029/2008JA013785Zank, G. P., Li, G., Florinski, V., et al. 2006, Journal ofGeophysical Research (Space Physics), 111, A06108,doi: 10.1029/2005JA011524Zank, G. P., Pauls, H. L., Cairns, I. H., & Webb, G. M.1996, J. Geophys. Res., 101, 457, doi: 10.1029/95JA02860Zhao, X., Liu, Y. D., Hu, H., & Wang, R. 2019, ApJ, 882,122, doi: 10.3847/1538-4357/ab379b
Yang et al.
Table 1.
Setups of species parameters for the linear analysisRun
B/B N/N Species n % β // β ⊥ V dx /V A V dy /V A V dz /V A V ring /V A M + + − − + + + − − Note —(1) The species can be treated either magnetized ( M = 1) or unmagnetized ( M = 0). (2) J = 8 is usedfor the J − pole Pad´e expansion. (3) The electron VDF is contributed by a hot drifting Maxwellian core anda relatively cool Maxwellian ring. The ring velocity distribution and its drift across field are modeled as thatused by Umeda et al. (2012b) and Umeda & Nakamura (2018), respectively. The ion VDF is a superposition oftwo or three drifting Maxwellian subpopulations. As shown in Figure 2, “Inc.” and “Ref.” are abbreviations of“incident” and “reflected”, respectively. “New ref.” represents ions which are newly reflected at the ramp andmoving toward upstream. “Old ref.” refers to the ions that are reflected at earlier time, and they are gyratingback toward the downstream at this time. (4) V dx,y,z and V ring indicate the drift velocity components and theradius of modeled ring distributions. (5) We consider the nearly perpendicular wave modes. Wave normal angles(WNA) θ kB = 89 . ◦ and 89 ◦ are employed for regions A and B, respectively. (6) All parameters are issued fromthe simulation and averaged over the sampling period in the shock ramp rest frame. hock waves near the Sun Figure 1. (a) An overview of the time-evolving Y − averaged shock magnetic field B z . The sampling period for the waveanalysis is denoted by white lines. (b-d) Snapshots of electromagnetic fluctuations δE x , δE y and δB z at t = 3 . − ci within thesampling period. The ramp location X = 56 . d i is marked by black vertical lines. The Y − averaged B z / δE x ( x, y, t ) at Y = L y / X ramp = 0. (g-h) Hanning windows are used to compensate for the nonperiodicity of thedata, in both X and t directions. The same post-processing is performed on other fluctuation components before doing the 3-DFFT. Yang et al.
Figure 2. (a-b) Phase space plots X − V ex,y of electrons at t = 3 . − ci . The shock ramp location is marked by a vertical blackline. Sampling regions A and B are denoted on the top. The core and ring components are marked in panel (a). In order tosee these two components clearly, the particles located at Y = L y / ± . d i are sampled for these plots. (c-d) Similar plots forions. Incident ions (“Inc.”), freshly reflected ions (“New ref.”) and old gyro-reflected ions (“Old ref.”) are marked in panel (c).(e) Y − averaged number density profiles of reflected ions (red), incident ions (blue), and electrons (green). The Y − averagedmagnetic field B (black) is also shown for reference. Here, we only focus on the wave excitation region and separated thereflected ions ahead of the ramp as in Otsuka et al. (2019). (f-h) Corresponding 2-D number density profiles. hock waves near the Sun Figure 3. (a-c) A 3-D view of the dispersion relation diagram of fluctuating fields δE x , δE y and δB z in region A of the shockfront. The color indicates the amplitude of Fourier transformed fields in the κ x − κ y − ω space. The excited ES and EM wavesare marked. (d-f) Similar plots as in (a-c) for fluctuating fields in region B. On the upper edge of each panel, the black arrowindicates the shock normal direction n . Yang et al.