Magnetic field amplification by the Weibel instability at planetary and astrophysical high-Mach-number shocks
Artem Bohdan, Martin Pohl, Jacek Niemiec, Paul J. Morris, Yosuke Matsumoto, Takanobu Amano, Masahiro Hoshino, Ali Sulaiman
aa r X i v : . [ a s t r o - ph . H E ] F e b Magnetic field amplification by the Weibel instability atplanetary and astrophysical high-Mach-number shocks
Artem Bohdan, ∗ Martin Pohl,
1, 2
Jacek Niemiec, Paul J. Morris, YosukeMatsumoto, Takanobu Amano, Masahiro Hoshino, and Ali Sulaiman DESY, DE-15738 Zeuthen, Germany Institute of Physics and Astronomy, University of Potsdam, DE-14476 Potsdam, Germany Institute of Nuclear Physics Polish Academy of Sciences, PL-31342 Krakow, Poland Department of Physics, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522, Japan Department of Earth and Planetary Science, the University of Tokyo,7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Department of Physics and Astronomy, University of Iowa, IA, USA (Dated: February 9, 2021)Collisionless shocks are ubiquitous in the Universe and often associated with strong magneticfield. Here we use large-scale particle-in-cell simulations of non-relativistic perpendicular shocksin the high-Mach-number regime to study the amplification of magnetic field within shocks. Themagnetic field is amplified at the shock transition due to the ion-ion two-stream Weibel instability.The normalized magnetic-field strength strongly correlates with the Alfv´enic Mach number. Mockspacecraft measurements derived from PIC simulations are fully consistent with those taken in-situat Saturn’s bow shock by the Cassini spacecraft.
Collisionless shocks are ubiquitous in the Universe, andthey are observed in planetary systems, supernova rem-nants (SNRs), jets of active galactic nuclei, galaxy clus-ters, etc. In contrast to fluid shock waves, where dissipa-tion at the shock front is mediated by binary collisions,collisionless shocks are shaped by collective particle in-teractions with interaction length much shorter than thecollisional mean free path [1, 2]. Collisionless shocks areusually magnetized, and magnetic fields play a key rolein their physics. The jump condition for the magneticfield [3] and the internal shock structure [4] strongly de-pends on the shock obliquity. Magnetic field turbulencenear the shock is a key ingredient of diffusive shock ac-celeration (DSA, [5–10]) and also shapes non-thermal X-ray emission. Amplified magnetic fields (at scales muchlarger than the upstream ion gyroradius) have been in-ferred from observations of SNRs through the detectionof non-thermal X-ray rims [11–14], fast temporal variabil-ity of X-ray hot spots [15], and the γ -ray/X-ray flux ratio[16]. We know various possible mechanisms for magneticfield amplification at these scales: cosmic-ray driven non-resonant modes [17, 18], fluid vorticity downstream ofthe shock seeded by upstream density inhomogeneities[19, 20], cosmic ray pressure-driven magnetic field ampli-fication [21, 22], and also inverse cascading of relativelyshort-scale Alfv´en waves [23].Here we study magnetic field amplification on scalessmaller than the upstream ion gyroradius at high-Mach-number quasi-perpendicular shocks. In-situ measure-ments by the Cassini spacecraft [24, 25] reveal the de-tailed magnetic field structure of Saturn’s bow shock withresolution below the ion gyroradius. The Alfv´enic Machnumber of this shock can reach values around 200 which issimilar to that of SNR shocks. [25] demonstrated that thenormalized overshoot magnetic-field strength displays a strong positive correlation with M A across the entirerange of measured M A . Particularly strong amplificationis observed at shocks at which shock self-reformation isevident [24]. Reasons for such behaviour are unknownand they are the objective of our study.Leroy’s calculations [26] for perpendicular shocks com-bined with hybrid simulations suggest that the overshootmagnetic-field strength ( B over ) can be estimated as B over ≈ . B M / , (1)where B is the upstream field strength. The prefactor0 . M A shocks demonstrate strong amplification of the upstreammagnetic field due to the ion-ion filamentation/Weibel in-stability [32, 33], which results from the interaction of up-stream and shock-reflected ions. The mediation of high- M A shocks by the Weibel instability is also confirmed bylaboratory experiments [34] and in-situ measurements ofthe Earth’s bow shock at M A ≃
39 [35]. In this letterwe discuss a mechanism of magnetic-field amplificationthat is based on a realistic description of perpendicularnonrelativistic high- M A shocks and can explain the cor-relation between field strength and M A observed withCassini at Saturn’s bow shock.To tackle this issue we use 2D PIC simulations withan in-plane magnetic-field configuration which permitsa good approximation of realistic 3D shocks [29, 36].We perform shock simulations using an optimized fully-relativistic electromagnetic 2D code with MPI paral-lelization developed from TRISTAN [37–39]. Shocks areinitialized with a modified flow-flow method [40]. The TABLE I. Parameters of simulation runs. Listed are: the ion-to-electron mass ratio, m i /m e , the Alfv´enic and sonic Machnumber, M A and M s , the electron plasma beta, β e . Somevalues are shown separately for the left (runs *1) and the right (runs *2) shock. Results for runs marked by a ’ † ’ are notdiscussed in this letter because of the strong numerical noiseat the shock upstream. All runs use the in-plane magneticfield configuration, ϕ = 0 o .Runs m i /m e M A M s β e ∗ ∗ ∗ ∗ · − · − · − · − · − · − · − † , H2 50 100 4870 154 5 · − † , I2 50 150 7336 232 5 · − collision of two counterstreaming electron-ion plasmaflows, each described with 20 particles per cell perspecies, spawns two independent shocks propagating inopposite directions. The inflow speed of two beams is v L = v R = v = 0 . c . The plasma temperature fortwo flows differs by a factor of 1000, therefore electron plasma beta (the ratio of the electron plasma pressureto the magnetic pressure) is 5 · − and 0.5 for the left (runs *1) and the right (runs *2) shocks, respectively.The large-scale magnetic field, B , is perpendicular tothe shock normal ( θ Bn = 90 o ) and lies in the simula-tion plane (the in-plane configuration, ϕ = 0 o ). Theadiabatic index is Γ ad = 5 /
3, the shock compressionratio is about 4, and the shock speed in the upstream frame is v sh = 0 . c . The Alfv´en velocity is v A = B / p µ ( N e m e + N i m i ), where µ is the vacuum per-meability; N i and N e are the ion and electron numberdensity. The sound speed reads c s = (Γ ad k B T i /m i ) / ,where k B is the Boltzmann constant and T i is the iontemperature. The Alfv´enic, M A = v sh /v A , and sonic, M s = v sh /c s , Mach numbers of the shocks are defined inthe conventional upstream frame (Table 1).The ratio of the electron plasma frequency, ω pe = p e N e /ǫ m e , to the electron gyrofrequency, Ω e = eB /m e , is in the range ω pe / Ω e = 8 . −
80. Here, e is the electron charge, and ǫ is the vacuum permittiv-ity. The temporal and spatial resolutions are δt = ω − and ∆ = λ se , where λ se is the electron skin depth.The transverse box size is L y = (8 − λ si , where λ si = p m i /m e λ se is the ion skin depth. The simulationtime is about T ≈ − , where Ω i = eB /m i .Our simulations cover a wide range of physical pa-rameters: M A = 22 . − m i /m e = 50 −
400 and β e , R = 5 · − − .
5. Hence, we can compare our simulation results with data for Saturn’s bow shock for M A > x sh ,is defined as position of the shock overshoot. Bune-man waves are visible as small-scale density ripples at x − x sh ≈ (8 − λ si . The Weibel instability is repre-sented by density filaments at x − x sh ≈ (2 − λ si . Thedownstream region is at x − x sh < − λ si . This structureis representative for all runs and for the high- M A regimein general [30, 31, 36, 42–45]. Earlier linear analysis [30]and its adaptation to our study [44] both indicate thathigh- M A shocks are Weibel-instability mediated.Figure 1(b) displays the density and magnetic-fieldprofiles at the shock transition of run B2, averaged intime over two cycles of shock reformation. The plasmacompression reaches N over /N ≈ N over depends on M A . The field strength increasestwice as much, indicating substantial noncompressionalmagnetic-field amplification.The B y profile almost coincides with that expected forsimple compression of B y according to the density profile.The modest increase of B y around ( x − x sh ) /λ si ≈ (0 − B x into B y when magnetic loops elongated in x -direction break upinto chains of magnetic vortices [44]. As expected, B x and B z grow due to folding of magnetic field by theWeibel modes whose wave vector is perpendicular tothe relative velocity of shock-reflected and incoming up-stream ions [30, 31]. Further straightening of magnetic-field lines leads to convergence of the density and mag-netic field profiles at the shock downstream.We define the shock region as a sector of width L sh = r gi , up / x sh , where r gi , up = M A λ si . The nu-merical coefficient is chosen to match the shock width andthe average ion gyroradius at the shock transition layer;its exact value has little, if any, impact on the resultsdiscussed here. The shock region for run B2 is markedwith dashed lines in Figure 1.In Figure 2 we present the amplitude (panel a) and en-ergy density (panel b) of the magnetic field in the shockregion, averaged over the shock self-reformation cycle andwith error bars reflecting the level of temporal variation.The normalized field strength, | B sh | /B , grows with in-creasing M A . The Weibel growth rate is about Γ ≈ . ω pi regardless of the shock parameters [44]. Shock-self refor-mation limits the time available for the Weibel instabil-ity to develop to about Ω − , implying that the numberof exponential growth cycles is proportional to M A fora given shock speed. Exponential growth of the ampli-tude of Weibel filaments is not observed though even atlow M A . In fact, the Weibel instability quickly becomesnonlinear, and the magnetic-field strength defies an ana-lytical derivation. Here we can only estimate it as (green FIG. 1. Density and magnetic field in run B2. Panel (a): iondensity in logarithmic units. Panel (b): red line - the profileof normalized ion density, green line - the profile of normal-ized magnetic field, magenta line - B x /B , dark blue line - B y /B , light blue line - B z /B . Profiles are calculated in theshock reference frame and averaged over the shock reforma-tion cycle. The shock region is marked by dashed lines. x sh is the shock position. line in Fig 2(a)) | B sh | ≈ p M A B . (2)The normalized energy density of the magnetic field canbe expressed as U sh , B U sh , i = B µ N i m i v ≈ M A , (3)which is a descending trend (green line in Fig 2(b)), thatwell reproduces the energy density observed in the simu-lations. A lower limit for the normalized magnetic energydensity should be provided at very high M A or unmag-netized shocks. For the latter the fraction of magneticenergy in the shock region is about U B = 0 . U sh , i [46], which with Eq. 3 is expected at M A ≈ | B sh | ≈ B .The magnetic field remains amplified for only a fewion gyroradii behind the shock, and far downstream thefield strength is 4 B . Our simulation time is too shortto fully capture the entire relaxation especially for high M A . The data we have suggest that the length scale ofrelaxation is roughly proportional to | B sh | / | B | .We use the analytical description presented in [44] toclarify the relation between the Weibel growth rate andthe choice of plasma parameters, namely, the upstreamplasma beta, the mass ratio, and the shock speed. Runs*1 and *2 differ by the upstream plasma temperature. Atthe shock foot, however, the temperature of the plasmaconstituents is similar on account of partial thermaliza-tion, which leads to similar Weibel growth rates. Runs FIG. 2. The normalized magnetic-field strength (panel (a))and the magnetic energy density normalized by the upstreamion energy density (panel (b)), both evaluated in the shockregion defined in Fig. 1. The blue and red color correspondsto left ( β = 5 · − ) and right ( β = 0 .
5) shocks, respectively.The green dotted line in panel (a) reflects | B sh | /B = 2 √ M A ,and that in panel (b) shows U sh , B /U sh , i = 4 M − . that differ only in the mass ratio also show the samemagnetic-field amplification level. We use the plasmaparameters observed in the shock foot of run F2 to calcu-late the Weibel instability growth rate for different massratios, keeping all kinetic and thermal parameters con-stant. We find that the growth rate of the most un-stable mode remains the same within ∼
10% margin(Fig. 3(a)). Therefore we conclude that the upstreamplasma beta and the mass ratio do not play a significantrole in magnetic-field amplification.We also explore how the behavior of the Weibel in-stability depends on the shock speed, which in the sim-ulations is two orders of magnitude higher than at Sat-urn’s bow shock. Figure 3(b) shows the Weibel instabilitygrowth rates for three values of the shock velocity: 0 . c ,0 . c and 0 . c . The last case with v sh = 780 km/sis very close to the speed of Saturn’s bow shock, whichis about 400 km/s [47]. For v sh = 0 . c , we use plasmaparameters from run F2. For the two other cases we ac-cordingly rescale the velocity and the temperature of theplasma flow. To be noted from Figure 3(b) is that thenormalized peak growth rate is proportional to the shockspeed: Γ max ∝ v sh ω pi or Γ max ∝ M A Ω i (4)This finding matches the result of earlier, simplified cal-culations [4]. Eq. 4 shows that the number of exponen-tial growth cycles available for Weibel modes scales in-versely with the Mach number, whatever the shock speed. FIG. 3. Growth rate of Weibel modes for five mass ratios (a)and three shock speeds (b).
Therefore M A is the only upstream parameter that de-fines magnetic-field amplification at the shock transition.The intrinsic shock dynamics also affects the magnetic-field amplification level. [24] showed that 16 shocks outof 54 shock crossings undergo shock reformation, and themeasured B max /B ( B max is the maximal magnetic fieldmeasured during a shock crossing by the spacecraft) atthese shocks is 1.42 times that at the other 38 shocks.This behaviour is likely explained by the differences inion reflection at the shock ramp between reforming andnon-reforming shocks. With shock self-reformation, theion reflection rate is time dependent and swings periodi-cally [40], reaching larger values than for a non-reformingshock where the ion reflection rate is steady. This resultsin a stronger magnetic-field amplification in reformingshocks, on account of the higher growth rate of Weibelmodes and stronger plasma compression at the shockramp. Therefore, B max /B is higher for shocks at whichshock reformation is observed. In all of our simulationsshock reformation is clearly visible. To properly comparewith the full set of in-situ measurements, which includesboth reforming and non-reforming shocks, we thereforereduce the peak field strength measured in the simula-tions by a factor of 1 . / (1 . n r +(1 − n r )) = 1 .
26, where n r = 16 /
54 is the fraction of reforming shocks in the in-situ data of [24].The largest set of magnetic-field measurements at Sat-urn’s bow shock [25] contains 422 shock crossings duringwhich the shock was quasi-perpendicular, θ B n > o , andfor which B max /B is indicated by gray crosses in Fig-ure 4. We derive B max /B from PIC simulation dataassuming that a virtual spacecraft crosses a simulatedshock with a straight trajectory. On the spacecrafts tra- FIG. 4. Cassini measurements [25] indicated by gray crossesand PIC simulation data displayed with blue and red dotsfor left ( β = 5 · − ) and right ( β = 0 .
5) shocks, respec-tively. The yellow dash-dotted line is an earlier prediction, B over /B ≈ . M / / .
26 (cf. Eq. 1), corrected for shock ref-ormation. The green dashed line is the behavior found in ourPIC simulations, B max /B = 5 . (cid:0) √ M A − (cid:1) . jectory we calculate B max /B and then we average it overall possible shock crossing points and the speed and flightdirection of the virtual spacecraft. Hereby we account forboth the temporal and the spatial variations of B max /B .The results are shown in Figure 4 as blue and red dotswith error bars. Note, that we already applied to bothour results and Leroy’s model the downward correctionby the factor 1 .
26 that we discussed in the precedingparagraph as compensation for shock reformation.Figure 4 demonstrates a good match between in-situmeasurement and simulation data. A good fit of the sim-ulation data is shown as green dashed line in Fig. 4, B max B = 5 . (cid:16)p M A − (cid:17) , (5)which also well describes the in-situ measurements for M A &
10. This is not proof that magnetic fields are de-fined by Weibel instability at 10 < M A <
20, but at leastEq. 5 can be used to estimate the field strength. For com-parison, the yellow dash-dotted line in Fig. 4 shows thescaling of Eq. 1, which also was confirmed with recent 2Dsimulations [28]. However, 2D simulations cannot alwayscapture realistic shock physics, the out-of-plane magneticfield configuration utilized in [28] misses the Weibel in-stability, which changes the magnetic field amplificationphysics compared to our in-plane
2D simulations and the3D simulations of [29]. Although Eq. 1 matches the datareasonably well for M A <
60, even that may be a coin-cidence because this model relies on simplified 1D shockphysics. In our view, Eq. 5 is a better and physicallymotivated approximation for B max /B at shocks with M A & B max /B , the shock refor-mation period, T reform ≈ . − , is the same in our sim-ulations and in the Cassini data [24]. Also the magnetic-field relaxation distance is similar with about one shockwidth, further suggesting similar physical processes atplay in PIC simulations and real bow shocks.We find no evidence for magnetic-field amplificationby ion beam cyclotron instabilities. They would requiremore time to develop, T ≫ Ω − , and usually these in-stabilities are observed at quasi-parallel shocks where theshock-reflected ions can move far upstream.We have established a strong connection between theWeibel instability and magnetic-field amplification athigh- M A shocks. The results of our PIC simulationsare fully consistent with in-situ measurements of Sat-urn’s bow shock. As M A is the only relevant parameter,our findings on field amplification inside the shock tran-sition layer should also apply to SNR shocks. Weibelmodes can increase the local synchrotron emissivity by afactor ( B sh /B ) , which may reach a thousand. Largerenhancements arise in the X-ray band beyond the syn-chrotron peak frequency, but overall the effect is likelyunobservable with current facilities due to low resolution.However, the interaction of Weibel modes with other am-plification processes may introduce significant changes inthe shock structure and it should be taken into accountin further studies.Electron pre-acceleration [36, 44] and heating [45]strongly depend on the structure and strength of themagnetic field. At quasi-perpendicular shocks, wherestochastic shock drift acceleration (SSDA) is expectedto operate [29, 48], strong magnetic field generated bythe Weibel instability limits the mean free path and in-creases the cyclotron frequency of electrons, and so thecut-off energy of SSDA may depend on M A .Also due to the strong magnetic field at the shock tran-sition particles require larger momenta for injection intoclassical DSA, they repeatedly cross the shock withoutsignificant deflection in the shock internal structure. TheLarmor radius in the amplified field (Eq. 2) should thenbe much larger than the shock width, r sh ∝ r gi , up ∝ B − ,which implies for the injection momentum p inj ∝ r sh B sh ∝ p M A , (6)at Weibel-mediated shocks.The work of J.N. has been supported by Nar-odowe Centrum Nauki through research project2019/33/B/ST9/02569. M.P. acknowledges supportby DFG through grant PO 1508/10-1. This researchwas supported by PLGrid Infrastructure. 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