In the high-β cases (β ≈20−100) withMs=3, all the AIC, whistler, i-mirror and e-mirror waves can be triggered, as shown in the lower panels of Figure 32, leading to the generation of multi-scale waves from electron to ion scales. On the other hand, in the LM3.0β5 model (red solid lines), the e- mirror mode is stable, but other modes are unstable. In the LM3.0β1 model (gray solid lines), all the instabilities are stable with negative growth rates.
The sonic Mach number, Ms, is the key parameter that determines the temperature anisotropies in the transition of high-β ICM shocks (β ≈20−100), since the ion reflection fraction and the magnetic field compression are closely related toMs. Figure 33 shows the growth rates of the instabilities for Ms=2.0 (black), 2.3 (red), and 3.0 (blue), in the cases ofβ =20 (top), 50 (middle) and 100 (bottom).
AsMsincreases, bothAe andAi increase, so all the modes grow faster andkmshifts towards largerk, regardless ofβ.
Note that the AIC and whistler modes have γm at θm =0 independent of Ms, whereas θm de- creases with increasingMsfor the i-mirror and e-mirror modes (see also Table 7). In LM2.0β50 and LM2.0β100, the AIC instability is stable or quasi-stable, while the whistler and mirror modes can grow.
In the case of LM2.0β20, all the instabilities are stable (see black lines in top panels). In the models withMs=2.3−3 (red and blue lines), on the other hand, the four instabilities are unstable, and hence multi-scale plasma waves can be generated.
6.2 Nonlinear Evolution of Induced Waves in Periodic-Box Simulations
0 . 0 0 . 2 0 . 4 0 . 6 0 . 0
0 . 2 0 . 4 0 . 6
A I C t ~ τA I C
t ~ 3 τW I
c k l l/ωp i
c k l l/ωp i
ck⊥/ωpick⊥/ωpick⊥/ωpi
c k l l/ωp e
c k l l/ωp e
c k l l/ωp e c k l l/ωp e
ck⊥/ωpeck⊥/ωpe
ck⊥/ωpeck⊥/ωpe
c k l l/ωp e
ck⊥/ωpe
c k l l/ωp e
ck⊥/ωpe
t ~ τW I
i - m i r r o r
0 . 0 0 . 2 0 . 4 0 . 6
0 . 0 0 . 2 0 . 4 0 . 6
0 . 0 0 . 1 0 . 2 0 . 3
0 . 0 0 . 1 0 . 2 0 . 3
W I W I
i - m i r r o r
A I C
( i ) M 2 . 0 ( h ) M 2 . 0
( g ) M 2 . 0
( f ) M 2 . 3 ( e ) M 2 . 3
( d ) M 2 . 3
( c ) M 3 . 0 ( b ) M 3 . 0
c k l l/ωp i
1 0 - 5 1 0 - 4 1 0 - 3 1 0 - 2
δB(k)2 /B02
( a ) M 3 . 0
0 . 0 0 . 2 0 . 4 0 . 6
0 . 0 0 . 2 0 . 4 0 . 6
0 . 0 0 . 2 0 . 4 0 . 6
0 . 0 0 . 2 0 . 4 0 . 6
0 . 0 0 . 1 0 . 2 0 . 3
0 . 0 0 . 1 0 . 2 0 . 3
0 . 0 0 . 2 0 . 4 0 . 6
0 . 0 0 . 2 0 . 4 0 . 6
0 . 0 0 . 2 0 . 4 0 . 6
0 . 0 0 . 2 0 . 4 0 . 6
0 . 0 0 . 1 0 . 2 0 . 3
0 . 0 0 . 1 0 . 2 0 . 3
Figure 35: Power spectra of the magnetic field fluctuations, δB2y(k), in the period-box simulations for LM3.0β50 (top), LM2.3β50 (middle), and LM2.0β50 (bottom), plotted in the k∥-k⊥ (that is,kz−kx) plane. The results are shown att∼τWI(left),t∼3τWI(middle), andt∼τAIC(right). See the text for the remarks onτAICfor LM2.0β50. The gray star symbol marks the location of the maximum linear growth rate,γm, estimated from the linear analysis. In the models with Ms≥2.3, AIC, whistler and i-mirror waves appear, while those waves do not grow substantially in the model withMs=2.
Figure 34 shows the magnetic field fluctuations, δBy(upper panels) andδBz(lower panels), in the x-z plane (simulation plane) at three different times in the LM3.0β50 model. Here, the growth time scales,τWIandτAIC, are estimated byγmof each mode in Table 7. Att∼τWI, the transverse component, δBy, appears on electron scales and the waves containing it propagate parallel toB0in panel (a), but the longitudinal component,δBz, does not grow significantly in panel (d). In this early stage, the dominant mode is the whistler mode, while the e-mirror mode is much weak to be clearly manifested. As Ae
decreases in time due to the electron scattering off the excited waves, the whistler waves decay as shown in panel (b). On the time scale ofτAIC, both the AIC and i-mirror instabilities grow and become dominant. It is clear that the AIC-driven waves, shown in panel (c), are parallel-propagating, while the i-mirror-driven waves, shown in panel (f), are oblique-propagating; the blue arrow in the bottom-left corner of panel (f) denotes the wavevector of the i-mirror-driven mode with the maximum growth rate.
Figure 35 shows the time evolution of the power spectrum for the magnetic field fluctuations, δB2y(k), for LM2.0β50, LM2.3β50, and LM3.0β50 att∼τWI,t ∼3τWI, andt∼τAIC . Again, the growth time scale of each mode is estimated withγmlisted in Table 7, except for the LM2.0β50 model, in which the AIC instability is stable, and so the output time of panel (i) is chosen at the evolutionary stage similar to that of LM2.3β50. In the cases ofMs=2.3 and 3, whistler waves are excited dominantly at quasi-parallel propagating angles att∼τWI. After the initial linear stage, the energy of the whistler waves is transferred to smaller wavenumbers and the waves gradually decay, as shown in panels (b) and (e). On the time scale of∼τAIC, AIC waves and i-mirror waves appear dominantly at quasi-parallel and highly oblique angles, respectively, as shown in panels (c) and (f). This is consistent with the evolution- ary behavior which we have described with Figure 34. For the AIC and whistler instabilities, the linear predictions forkm with the maximum growth rate (gray star symbols) agree reasonably well with the peak locations of the magnetic power spectrum realized in the PIC simulations. But the linear estimates for the i-mirror mode are slightly off, becauseγmis obtained without the electron anisotropy, as stated through the linear analysis results in the previous section. In summary, the results of the periodic-box simulations are quite consistent with the linear predictions described earlier. Also we note that the results of our PIC simulations are in good agreement with those of [172], in which PIC simulations were carried out to explore the evolution of the instabilities due to the temperature anisotropies in space plasmas with β ∼1. The bottom panels of Figure 35 confirm that waves do not grow noticeably in the LM2.0β50 model.
In these periodic-box simulations, the electron-scale waves develop first and then decay as Ae is relaxed in the early stage, followed by the growth of the ion-scale waves due toAi. In the shock transition region, by contrast, temperature anisotropies are to be supplied continuously by newly reflected-gyrating ions and magnetic field compression, hence multi-scale plasma waves from electron to ion scales are expected to be simultaneously present.
Table8:ShockCriticalityoftheSimulatedShockModelsandStabilityoftheLinearAnalysisModels SimulatedShockModelShockCriticalityLinearAnalysisModelAICWIion-mirrorelectron-mirror M2.0β20subLM2.0β20stablestablestablestable M2.0β50subLM2.0β50stableunstablequasi-stableunstable M2.0β100subLM2.0β100quasi-stableunstableunstableunstable M2.3β20superLM2.3β20unstableunstableunstablestable M2.3β50superLM2.3β50unstableunstableunstableunstable M2.3β100superLM2.3β100unstableunstableunstableunstable M3.0β1subLM3.0β1stablestablestablestable M3.0β5superLM3.0β5unstableunstableunstablestable M3.0β20superLM3.0β20unstableunstableunstableunstable M3.0β50superLM3.0β50unstableunstableunstableunstable M3.0β100superLM3.0β100unstableunstableunstableunstable
6.3 Implications for Shock Simulations