CR acceleration at supercritical shocks has been explored through plasma simulations mainly forβ≲1 cases (i.e.,MA∼Ms). In the typical hot ICM plasma, however,β∼100, so shocks have relatively high Alfvén Mach number,MA≈10Ms≈20−30, but low sonic Mach number,Ms≈2−3. Note that for these ICM shocks,Mf≈Ms. To our knowledge, the supercriticality of weak (Ms≈2−4) quasi-parallel shocks in such highβ environment has not yet been studied by PIC or hybrid simulations.
PIC simulations ofMs=3 shocks in highβ ICM plasmas were considered by [23, 24], focusing mainly on electron acceleration at quasi-perpendicular shocks. They showed that for the shock model withβ =20,θBn=63◦, andMs=3, about 20 % of incoming ions are reflected at the shock and gain a small amount of energy via a few cycles of SDA. Those energized ions overcome the potential barrier and advect downstream along with the magnetic fields. Note that their simulations were not intended to study ion acceleration in the DSA regime in sufficiently long ion gyro-time scales.
Excitation of magnetic turbulence by protons streaming upstream of quasi-parallel shocks is an in- tegral part of injection and acceleration of CR particles. There are two dominant modes: (1) resonant streaming instability which excites left-handed circularly polarized waves [20], and (2) nonresonant current-driven instability which excites right-handed circularly polarized wave [90]. Using hybrid sim- ulations of quasi-parallel shocks inβ ∼1 plasma, [27] showed that the resonant streaming instability is dominant in the precursor of shocks withMA≲30, while the nonresonant current-driven instability operates faster at stronger shocks withMA≳30. The magnetic field amplification factor increases with increasing Alfvén Mach number as⟨B/B0⟩2∝MA. Both instabilities amplify primarily transverse com- ponents of the magnetic field, so they generate locally perpendicular fields in the shock foreshock and downstream region, which in turn facilitate SDA of the reflected ions and reflect subsequently arriving ions. Eventually, excited turbulent waves act as scattering centers both upstream and downstream of the shock, which are required for the Fermi I acceleration.
In this Section, we examine the physics of ‘shock criticality’ in weak ICM shocks by using PIC simulations. To identify ion injection and early stage acceleration, we study shock structures and ion energy spectra. In order to understand the nature of CR ion-driven instabilities and turbulent magnetic field amplification, we perform Fourier analysis of upstream self-excited magnetic field components.
We then discuss dependence of ion injection and CR ion-driven instabilities on the pre-shock conditions such asMs,β, andθBn.
Table 2: Model Parameters for the Simulations Ms≈Mf MA v0/c θBn β Te=Ti[K(keV)] mmi
e Lx[c/wpe] Ly[c/wpe] tend[Ω−1ci ]b
M3.2c 3.2 29.2 0.052 13◦ 100 108(8.6) 100 2×104 2 90.2
M2.0 2.0 18.2 0.027 13◦ 100 108(8.6) 100 2×104 2 90.2
M2.15 2.15 19.6 0.0297 13◦ 100 108(8.6) 100 2×104 2 90.2
M2.25 2.25 20.5 0.0315 13◦ 100 108(8.6) 100 2×104 2 90.2
M2.5 2.5 22.9 0.035 13◦ 100 108(8.6) 100 2×104 2 90.2
M2.85 2.85 26.0 0.0395 13◦ 100 108(8.6) 100 2×104 2 90.2
M3.5 3.5 31.9 0.057 13◦ 100 108(8.6) 100 2×104 2 90.2
M4 4.0 36.5 0.066 13◦ 100 108(8.6) 100 2×104 2 90.2
M3.2-θ23 3.2 29.2 0.052 23◦ 100 108(8.6) 100 2×104 2 90.2
M3.2-θ33 3.2 29.2 0.052 33◦ 100 108(8.6) 100 2×104 2 90.2
M3.2-θ63 3.2 29.2 0.052 63◦ 100 108(8.6) 100 2×104 2 90.2
M2.0-β30 2.0 10.0 0.027 13◦ 30 108(8.6) 100 2×104 2 165
M2.0-β50 2.0 12.9 0.027 13◦ 50 108(8.6) 100 2×104 2 128
M3.2-β30 3.2 16.0 0.052 13◦ 30 108(8.6) 100 2×104 2 165
M3.2-β50 3.2 20.6 0.052 13◦ 50 108(8.6) 100 2×104 2 128
M2.0-m400 2.0 18.2 0.013 13◦ 100 108(8.6) 400 2×104 2 22.6
M2.0-m800 2.0 18.2 0.009 13◦ 100 108(8.6) 800 2×104 2 22.3
M3.2-m400 3.2 29.2 0.026 13◦ 100 108(8.6) 400 2×104 2 22.6
M3.2-m800 3.2 29.2 0.018 13◦ 100 108(8.6) 800 2×104 2 22.3
M2.0-r2 2.0 18.2 0.027 13◦ 100 108(8.6) 100 2×104 2 22.3
M2.0-r0.5 2.0 18.2 0.027 13◦ 100 108(8.6) 100 2×104 2 22.3
M3.2-r2 3.2 29.2 0.052 13◦ 100 108(8.6) 100 2×104 2 22.3
M3.2-r0.5 3.2 29.2 0.052 13◦ 100 108(8.6) 100 2×104 2 22.3
M2.0-2D 2.0 18.2 0.027 13◦ 100 108(8.6) 100 2×104 60 34.6
M3.2-2D 3.2 29.2 0.052 13◦ 100 108(8.6) 100 2×104 60 34.6
a: See the Section "Numerics" for model-naming convention b:Ω−1ci =mic/(eB0)is the ion gyration period.
c: The fiducial model.
246 246
2 0 0 0 2 5 0 0 3 0 0 0 3 5 0 0 4 0 0 0
01234
1 5 0 0 2 0 0 0 2 5 0 0 3 0 0 0
01234
M 2 . 0 - 2 D
[ / p e ]
x c ω
M 3 . 2 - 2 D
[ / p e ]
x c ω
202 e m v
φ
0BB
Figure 7: Stack plots of total magnetic field strength,B(x), and electric potential for eφ(x), averaged over the transverse direction in M3.2-2D (left) and M2.0-2D (right) models for five different time epochs (fromwpet=0.8×105(purple) to 1.2×105(red)).
In typical PIC simulations, due to severe requirements for computational resources, ‘ions’ with a reduced mass ratiomi/me≪1836 are adopted to represent the real proton population. Thus, hereafter we will refer ‘ions’ as positively charges particles with a reduced mass ratiomi/me=100−800.
The flow Mach numberM0of the upstream bulk flow is specified as M0≡ v0
cs = v0 p2ΓkBTi/mi
, (4)
where cs is the sound speed in the upstream medium, Γ=5/3 is the adiabatic index, and kB is the Boltzmann constant. Here, thermal equilibrium is assumed for the incoming flow, so the ion temperature Ti is the same as the electron temperatureTe. In the weakly magnetized limit, the sonic Mach number, Ms, of the induced shock is related withM0as follows:
Ms≡vsh
cs ≈M0 r
r−1. (5)
Herevsh=v0·r/(r−1)is the upstream flow speed in the shock rest frame and r= Γ+1
Γ−1+2/Ms2 (6)
is the Rankine–Hugoniot compression ratio across the shock.
The strength of the uniform background magnetic field B0 in the x-y plane is parameterized by plasma betaβ as follow:
β =8πnkB(Ti+Te) B20 = 2
Γ MA2
Ms2. (7)
−3.78
ix i
p m c
iy i
p m c
0.0
−0.63
−1.26
−1.89
−2.52
−3.15
0.2 0
−0.2 0.2
0
−0.2
0.2 0
−0.2 0.2
0
−0.2
(a)
(b)
(e)
(f)
ix i
p m c
iy i
p m c
iz i
p m c
0
ni
n
sh[c/ pe] x−x ω
−500 −250 0 250 500 −500 −250 0 250 500
3 2 1 0 0.2
0
−0.2
3 2 1 0 0.2
0
−0.2
(c) (g)
(h) (d)
iz i
p m c
0
ni
n
sh[c/ pe] x−x ω
M3.2-2D M2.0-2D
Figure 8: Shock structure of the M3.2-2D (left panels) and M2-2D (right panels) atwpet≈4.5×104 (Ωcit≈12). Here,xshdenotes the shock position and the length scale is expressed in the unit ofc/ωpe. From top to bottom, the ion momentum phase space plots x−pix, x−piy & x−piz (normalized by micand colorbar indicates logf(pi,α)), and the one-dimensional profile of ion density (normalized by number density of incoming plasma) are shown.
Then the Alfvén Mach number of the shock,MA, is defined as:
MA≡vsh
vA = v0 B0/√
4πnmi
r
r−1 =MA,0 r
r−1, (8)
wheren=ni=neis the number density of the incoming plasma,vA=B0/√
4πnmiis the Alfvén speed along the background magnetic field, andMA,0=v0/vAis the flow Alfvén Mach number. The fast mode Mach number is defined as
Mf≡ vsh
vf = vsh q
c2s+v2A
, (9)
where vf is the fast mode speed for the wave propagating perpendicular to the magnetic fields (i.e.
magnetosonic speed). Forβ ≫1,Mf≈Ms, sincevA≪cs.
The orientation of magnetic field is given by the obliquity angleθBn, which is the angle between the shock normal direction (ˆn=ˆx) andB0. Thus the background magnetic field is given by
B0=B0(cosθBnˆx+sinθBnˆy). (10) Initial electric field is zero, but the incoming magnetized plasma carries a uniform magnetic fieldB0and the motional electric field,E0=−v0/c×B0, is induced, wherecis the speed of light. Since ion injection and acceleration depends only weakly on the obliquity angle as shown by [26], we chooseθBn=13◦for our fiducial model.
PIC simulations follow kinetic plasma processes on different length scales for different species:
the electron skin depth, c/wpe, and the ion skin depth, c/wpi, where wpe=p
4πe2n/me and wpi = p4πe2n/mi are the electron plasma frequency and the ion plasma frequency, respectively. Here the simulation results are presented in term ofx/[c/wpe]andt/[w−1pe]. On the other hand, the shock structure varies and evolves at length scales of the Larmor radius for ions with the particle speedv0,
rL,i≡miv0c
eB0 =MA,0 rmi
me c
wpe, (11)
and in timescales of the ion gyration periodΩ−1ci =mic/(eB0) =rL,i/v0.
The simulations are performed in two-dimensional computational domains. The longitudinal dimen- sion,Lx, corresponds to 2×104c/wpe, which is represented byNx=2×105cells with a grid resolution of∆x=0.1c/wpe. The transverse dimension,Ly, comes in two different modes: Ny=20 cells for “al- most 1D" simulations andNy=600 cells for 2D simulations (∆y=∆x). We place 32 particles per cell (16 per species). The time step iswpe∆t=0.045.
Based on the previous studies on one-dimensional PIC simulations for strong shocks [25], we assume
“almost 1D" simulations would be good enough to investigate ion injection at weak ICM shocks (see Section "shock structures and ion injection" for the dependence of the simulation results on transverse box size). We also find that simulations with different spatial resolutions give essentially the same results.
The model parameters of our simulations are summarized in Table 2. We considerβ =30−100 andkBT =kBTe=kBTi =0.0168mec2=8.6 keV, relevant for typical ICM plasmas [10, 11]. For given
1 0 - 4 1 0 - 3 1 0 - 2 1 0 - 1 1 00
1 0 - 4 1 0 - 3 1 0 - 2 1 0 - 1 1 0 0
1 0- 5 1 0- 4 1 0- 3 1 0- 2 1 0- 1
1 0 - 4 1 0 - 3 1 0 - 2 1 0 - 1 1 00
1 0- 5 1 0- 4 1 0 - 3 1 0 - 2 1 0 - 1
1 0 - 4 1 0 - 3 1 0 - 2 1 0 - 1 1 0 0
( b )
( a ) M 3 . 2
M 3 . 2 - 2 D
M 2 . 0 M 2 . 0 - 2 D
( c )
~ 1 . 7 2
p
ωp et ~ 4 5 0 0 0 ωp et ~ 8 3 6 0 0 ωp et ~ 1 3 0 0 0 0 ωp et ~ 2 3 0 0 0 0 ωp et ~ 3 3 7 5 0 0
( d )
1γ −
1γ −
d
)1 d N
( −
~ 1 . 7 2
p
M 3 . 2 M 3 . 2 -θ2 3
M 3 . 2 -θ3 3
M 3 . 2 -θ6 3
d
)1 d N
( −
Figure 9: Panels (a) & (b) show downstream ion energy spectrum at wpet ≈1.3×105 (Ωcit ≈35) in M3.2 and M2.0 models. Downstream ion spectra of 1D M3.2 model for 5 different time epochs fromwpet≈4.5×104 (Ωcit≈12) to 3.4×105(Ωcit≈90) are displayed in panel (c). Downstream ion spectra atwpet≈3.4×105ofMs≈3.2 shocks for 4 different obliquity angles are plotted in panel (d).
The energy spectra shown in all panels are taken from the downstream region[1.5−2.5]rL,i behind the shock position. The black and purple dashed lines indicate the test-particle power-law spectrum and the thermal Maxwellian distribution in the postshock region, respectively. In (c) and (d), the injection energy,Einj≈5×10−3mic2for the M3.2 model is marked as the orange dashed line.
1 0- 4 1 0- 3 1 0- 2 1 0- 1 1 00
1 0- 4 1 0- 3 1 0- 2 1 0- 1 1 00
1 0- 5 1 0- 4 1 0- 3 1 0- 2 1 0- 1
1 0- 4 1 0- 3 1 0- 2 1 0- 1 1 00
1 0- 5 1 0- 4 1 0 - 3 1 0 - 2 1 0 - 1
1 0- 4 1 0- 3 1 0- 2 1 0- 1 1 00
( a ) M 2 . 0 P o s t - s h o c k
N e a r d o w n s t r e a m F a r d o w n s t r e a m
~ 2
p
( b ) M 2 . 2 5
~ 1 . 8 8
p
~ 1 . 7 2
p
( c ) M 2 . 5
1γ −
d
)1 d N
( −
d
)1 d N
( −
1γ −
( d ) M 3 . 2
Figure 10: Ion energy spectra measured in the shock downstream atwpet≈3.4×105 (Ωcit≈90) for M2.0, M2.25, M2.5, and M3.2 models. Here, energy spectra are taken from three different positions:
Post-shock (black;(0−1)rL,ibehind the shock), near downstream (red;(1−2)rL,ibehind the shock), far downstream (blue;(5−6)rL,ibehind the shock). The black and purple dashed lines show the test-particle spectrum expected for a shock with given Ms and thermal Maxwellian distribution in the postshock region, respectively. In (b), (c) and (d), the injection energy,Einjfor each model is marked as the orange dashed line.
β andcs, the incident flow velocity, v0, is specified to induce the shock with the sonic Mach number, Ms∼2−4, which is characteristic for cluster merger shocks [33]. The model M3.2 in the first row of Table 2 represents the fiducial model in 1D with the following parameters: Ms=3.2, θBn=13◦, β =100, and mi/me=100. Models with different Ms are named with the combination of the letter
‘M’ and the sonic Mach numbers (for example, M2.25 model hasMs=2.25). Models with parameters different from the fiducial model have names that are appended by a character for the specific parameter and its value. For example, M3.2-θ33 model hasθBn=33◦, while M3.2-m400 model hasmi/me=400.
We refer M3.2-2D and M2-2D models as 2D runs with the larger transverse dimension. M3.2-r2 and M3.2-r0.5 models are considered to explore the effects of different spatial resolution.
The last two columns of Table 2 show the end of simulation time for each model in units w−1pe and Ω−1ci . For the fiducial model M3.2,tendwpe≈3.4×105, which corresponds totendΩci≈90. The ratio of the ion gyration period to the electron oscillation period scales aswpe/Ωci∝(mi/me)p
β. So with a smaller mass ratiomi/me a shorter simulation time is required to see ion acceleration at early stage of Fermi I acceleration. On the other hand, withβ=100, it would take 10 times longer simulation time to reach the similar stage of ion acceleration, compared to simulations withβ =1.