Introduction
Electromagnetic instabilities like Weibel instability (Weibel, Reference Weibel1959) and filamentation instability (Fried, Reference Fried1959) owing to their self-generated fields play a very important role in high-energy astrophysical phenomena such as particle acceleration and synchrotron emission. These instabilities are widely investigated (Kalman et al., Reference Kalman, Montes and Quemada1968; Davidson et al., Reference Davidson, Hammer, Haber and Wagner1972; Califano et al., Reference Califano, Pegoraro, Bulanov and Mangeney1998; Silva et al., Reference Silva, Fonseca, Tonge, Mori and Dawson2002; Bret, Reference Bret2006; Rowlands et al., Reference Rowlands, Dieckmann and Shukla2007) and being applied in several astrophysical scenarios, such as cosmic rays (Lucek and Bell, Reference Lucek and Bell2000), supernova remnants (SNRs), and jets from active galactic nuclei (AGN) (Medvedev and Loeb, Reference Medvedev and Loeb1999; Nishikawa et al., Reference Nishikawa, Hardee, Richardson, Preece, Sol and Fishman2003; Medvedev et al., Reference Medvedev, Fiore, Fonseca, Silva and Mori2005; Spitkovsky, Reference Spitkovsky2008; Ardaneh et al., Reference Ardaneh, Cai, Nishikawa and Lembege2015), and in laser–plasma experiments (Fiuza et al., Reference Fiuza, Fonseca, Tonge, Mori and Silva2012; Fox et al., Reference Fox, Fiksel, Bhattacharjee, Chang, Germaschewski, Hu and Nilson2013).
These instabilities release the excess of free energy stored in the plasma particle anisotropy, residing, for example, in a temperature anisotropy or in a counterstreaming motion of the plasma. A substantial fraction of the kinetic energy of plasma particles is converted into the magnetic energy generation and its amplification by these instabilities (Califano et al., Reference Califano, Pegoraro, Bulanov and Mangeney1998). The processes leading to the magnetic field generation in the intergalactic medium has been treated as an unsolved problem and remained a matter of debate (Kronberg, Reference Kronberg2002). Therefore, understanding the evolution of these instabilities in both its linear and nonlinear phases and the prediction of the amplitude of the Weibel-generated magnetic fields are of primary importance for a deeper insight into various astrophysical events, as well as laser–plasma related studies. Among several astrophysical scenarios, SNRs are ubiquitous and routinely observed in radio to gamma X-ray bands. X-ray observations have shown that electrons are accelerated to highly relativistic energies in SNR shocks (Nishikawa et al., Reference Nishikawa, Hardee, Richardson, Preece, Sol and Fishman2003). The X-ray observations of SNRs suggest the magnetic field amplification up to 10−2–10−3 μG in the vicinity of the collisionless shock front (Berezhko et al., Reference Berezhko, Puhlhofer and Volk2003a, Reference Berezhko, Ksenofontov and Volk2003b; Vink and Laming, Reference Vink and Laming2003; Bamba et al., Reference Bamba, Yamazaki, Yoshida, Terasawa and Koyama2005; Uchiyama et al., Reference Uchiyama, Aharonian, Tanaka, Takahashi and Maeda2007).
The presence of amplified magnetic fields in young SNR is well acknowledged now. However, Gamma Ray Bursts (GRBs) and AGN require a stronger magnetic field that cannot be achieved by shock compression solely. Various mechanisms have been proposed for magnetic field amplification in different astrophysical environments, some of them are cosmic ray-driven instability, whereby cosmic rays diffuse to the upstream region where the resonant and the nonresonant instabilities become unstable and cause fluctuations in the magnetic field (Bell, Reference Bell2004; Reville et al., Reference Reville, Kirk and Duffy2006, Reference Reville, Kirk, Duffy and O'Sullivan2007). The nonresonant instability has been investigated with particle-in-cell (PIC) simulations (Niemiec et al., Reference Niemiec, Pohl, Stroman and Nishikawa2008; Riquelme and Spitkovsky, Reference Riquelme and Spitkovsky2008; Ohira et al., Reference Ohira, Reville, Kirk and Takahara2009).
Nonetheless, the detailed physics of such wide magnetic field generations is still uncertain. It is assumed that the leading hypothesis for field amplification in GRB afterglows of SNRs is the relativistic Weibel instability (Medvedev and Loeb, Reference Medvedev and Loeb1999) where the instability extracts free energy from the anisotropy of the particles, and produces filamentary currents aligned with the normal direction; and these currents are responsible for the creation of transverse magnetic fields.
Furthermore, the extensive distributions of the magnetic field energy in GRBs put forward that including the microphysics, macroscopic properties of GRB environments are equally important for the generation of magnetic fields. Therefore, GRBs and their surrounding environment are poorly known. Among macroscopic structures present, there should be clump wind regions and shocked wind regions in the surrounding environments of GRBs. Each GRB is different in their structures and length scales. Moreover, the observational evidence of radio waves shows the electron density irregularities over a wide range of scale size of the interstellar medium (ISM), and the Kolmogorov power spectrum justifies the existence of large-scale density turbulence (Lee and Jokipii Reference Lee and Jokipii1976; Armstrong et al., Reference Armstrong, Cordes and Rickett1981; Armstrong et al., Reference Armstrong, Rickett and Spangler1995).
Sironi et al. have proposed the magnetic field amplification through macroscopic turbulence excited by the interaction of the shock with a clumpy pre-shock medium (Sironi and Goodman, Reference Sironi and Goodman2007). Theoretical estimations made for intergalactic plasmas also suggest micro-Gauss levels of the magnetic field of Weibel type which are consistent with the magnetic field values 10−7–10−5 G derived from Faraday rotation measure of the linearly polarized emission of galactic or extragalactic sources (Lazar et al., Reference Lazar, Schlickeiser, Wielebinski and Poedts2009a). Recently (Tomita and Ohira, Reference Tomita and Ohira2016), the second-stage magnetic field amplification is observed on the basis of Weibel-type plasma instability which is driven by density fluctuations.
Counterstreaming plasma systems with intrinsic temperature anisotropies are susceptible to the excitation of Weibel- and filamentation-type instabilities and are extensively studied in the homogeneous plasma medium (Bret and Deutsch Reference Bret and Deutsch2006; Lazar et al., Reference Lazar, Schlickeiser and Shukla2006, Reference Lazar, Schlickeiser and Shukla2008, Reference Lazar, Smolyakov, Schlickeiser and Shukla2009b; Stockem and Lazar, Reference Stockem and Lazar2008; Stockem et al., Reference Stockem, Lazar, Shukla and Smolyakov2009; Bret et al., Reference Bret, Stockem and Narayan2016; Dieckmann and Bret Reference Dieckmann and Bret2017, Reference Dieckmann and Bret2018). Here, we study the electron Weibel instability induced in weakly relativistic counterstreaming plasmas with the inhomogeneous distribution and with intrinsic anisotropies and compare with widely studied counterstreaming plasmas with homogeneous distribution. As mentioned earlier, in SNRs and in GRBs, such inhomogeneous outflows are always present and we can expect some role of the Weibel instability/filamentation instability. Therefore, in the context of SNRs and GRBs, it is shown using PIC simulation that the pre-shock density distribution plays an important role in the magnetic field generation. Moreover, such small-scale inhomogeneity also exits in the intracluster medium (ICM) and the circumstellar medium (CSM) which may further trigger the magnetic field after saturation in the downstream region of the relativistic shock. Numerical PIC simulations have been used to describe the dynamics of a region corresponding to the downstream of the shock, in which the upstream inhomogeneous plasma has been compressed along the shock propagation direction. The computational plasma domain is considered relatively smaller compared to the long-scale domain of the density fluctuations present in astrophysical environments.
In Section “Numerical method and code resolution”, we present the numerical method and code resolution. In Section “Simulation results”, we present the simulation results on Weibel instability, associated growth of the magnetic field components and resulting in the magnetic energy development. We also discuss the thermal anisotropy development for the e −/e + plasmas with homogeneous and inhomogeneous density distributions. Finally, the conclusion is presented in the section “Conclusion”.
Numerical method and code resolution
For our simulation, we consider the pre-existing isotropic plasma density fluctuations and study the probable effects on the downstream motion and its characteristics. First, we consider two symmetric counterstreaming electron–positron e −/e + plasma flows with initial drift velocities, 
$\vec{v}_0 = {\pm} v_0\hat{y}$ with v 0 = 0.5c(γ0 ≅ 1), that is, a weakly relativistic case. Each e −/e + plasma has the same density and the thermal velocity is equivalent to v th = 0.1c along the 
$\hat{y}$ direction. The initial particle distribution is Maxwellian and the initial conditions of the flows are unstable for the Weibel instability. Two different plasma distributions, n(x, y) are considered: the first is the spatially uniform homogeneous plasma; n(x, y) = n 0, and the second is the inhomogeneous plasma distribution which is closer to the reality and is of the form of n(x, y) = n 0[1 + δn/n 0 sin(2πx/L x)], where δn/n 0 is the amplitude of the density fluctuation, n 0 is the constant electron number density, and L x is the simulation box size along the 
$\hat{x}$ direction. The system has initially zero net current. The simulation box length is normalized by c/ωpe and the wavenumber by ωpe/c, where c is the speed of light in vacuum and 
${\rm \omega }_{{\rm pe}} = \sqrt {n_0e^2/m_{\rm e}{\rm \epsilon }_0}$ is the electron plasma frequency. Here, m e and e are the electron rest mass and charge, respectively.
To understand the magnetic field amplifications, PIC simulations are carried out using EPOCH (Arber et al., Reference Arber, Bennett, Brady, Lawrence-Douglas, Ramsay, Sircombe, Gillies, Evans, Schmitz, Bell and Ridgers2015) code and cplPIC (Hur and Suk, Reference Hur and Suk2011; Cho et al., Reference Cho, Kim, Suk, Ersfeld, Jaroszynski and Hur2015) code. In PIC simulations, the interplay of the electric and magnetic fields with the particles of a collisionless kinetic plasma is modeled self-consistently. The main set of equations associated with PIC codes are the Lorentz force as the equation of motion, and the Maxwell's equations determining the electric and magnetic fields. In our simulations, the cell size Δx = Δy = 0.1c/ωpe and the simulation time step 
$\Delta t = 0.05{\rm \omega }_{{\rm pe}}^{{-}1} \;$ are considered, respectively. The simulation domain L x × L y = 128c/ωpe × 128c/ωpe is chosen for this study and periodic boundary conditions are used. The initial density of the e −/e + particles per cell is 40. Each cell contains 40 electrons and 40 positrons on average.
Simulation results
Evolution of magnetic field energy and its saturation
For the electric and the magnetic field energy calculation, the box-averaged energies of the electric E i(x, y, t) and B i(x, y, t) field components are given by 
${\rm \epsilon }_{E_i}\lpar t\rpar = \lpar N_xN_y\rpar ^{{-}1}\sum\nolimits_{j\comma k} {{\rm \epsilon }_0\lsqb E_i\lpar j\Delta _x\comma \;k\Delta _x\comma \;t\rpar \rsqb } ^2/2$ and 
${\rm \epsilon }_{B_i}\lpar t\rpar = \lpar N_xN_y\rpar ^{{-}1}\sum\nolimits_{j\comma k} {\lsqb B_i\lpar j\Delta _x\comma \;k\Delta _x\comma \;t\rpar \rsqb } ^2/2{\rm \mu }_0$, respectively. The box-averaged kinetic energy density is 
${\rm \epsilon }_K\lpar t\rpar = \lpar N_xN_y\rpar ^{{-}1}\sum\nolimits_j {m_{{\rm cp}}c^2\lpar {\rm \gamma }_j-1\rpar }$ in our 2D PIC simulation.
Figure 1a shows energies corresponding to E x, E y along with B z components for the inhomogeneous distribution case for δn/n 0 = 0.75. The B z is the only component that grows in the considered geometry in response to the Weibel instability and the magnitude of E x, E y energies is lower than B z energy. Figure 1b demonstrates the time evolution of the box-averaged energies of B z component for the counterstreaming e −/e + plasmas. For a homogeneous plasma case (black line), the amplified energy of B z shows exponential growth in the linear regime up to 
$t\sim 10{\rm \omega }_{{\rm pe}}^{{-}1}$. However, in the nonlinear regime, the magnetic energy simply decays which is similar to the previous findings (Tomita and Ohira, Reference Tomita and Ohira2016; Grassi et al, Reference Grassi, Grech, Amiranoff, Pegoraro, Macchi and Riconda2017). While in an inhomogeneous plasma case (blue line, red line, and green line), the magnetic energy of B z initially shows similar behavior in the linear stage as in the case of homogeneous plasma. However, after the saturation at 
$t\sim 10{\rm \omega }_{{\rm pe}}^{{-}1}$, magnetic energy of B z again gets amplified around 
$t\sim 270{\rm \omega }_{{\rm pe}}^{{-}1}$ (blue-dotted line) for δn/n 0 = 0.5. Hence, the magnetic field is amplified for longer time in the inhomogeneous e −/e + plasmas even though the total kinetic energy is the same as in the case of the homogeneous e −/e + plasmas. To confirm this result, we examine simulations with different density inhomogeneities, δn/n 0 = 0.75 (red line) and 1.0 (green-dotted line), respectively. We find similar growth of the magnetic energy with enhanced magnitude at the second peak as shown in Figure 1b.
Fig. 1. (a) The logarithmic plot of the temporal evolution of the field energies of E x,E y, and B z for δn/n 0 = 0.75. (b) The logarithmic plot of the temporal evolution of the transverse magnetic field energy (B z component) for different plasma density inhomogeneities: δn/n 0 = 0 (black line), 0.5 (blue-dotted line), 0.75 (red line), and 1.0 (green-dotted line), respectively.
Such an inhomogeneous plasma distribution is commonly found in GRBs where isotropic density fluctuations post-shock becomes anisotropic in the shock transition region. Finally, these distributions cause anisotropy in velocity distributions as well in the downstream region due to the temperature anisotropy (Tomita and Ohira, Reference Tomita and Ohira2016). However, the scale of the plasma environment for GRBs are on much larger scales than the plasma scale considered here in the simulation. Therefore, large-scale simulations are required to confirm the above scaling which is impossible in PIC simulation.
Figures 2a and 2b show the 2D profile of the magnetic field B z, component at two different simulation times 
$t\sim 10{\rm \omega }_{{\rm pe}}^{{-}1}$ and 
$t\sim 270{\rm \omega }_{{\rm pe}}^{{-}1}$, respectively, in the inhomogeneous e −/e + plasma case. From Figure 2a, one can notice that in the beginning, small magnetic filaments are developed only in a high-density region, 
$0 \lt x/c{\rm \omega }_{{\rm pe}}^{{-}1} \lt 60$, that is, left-hand side of the simulation domain. Whereas, in the right-hand side of the simulation domain 
$60 \lt x/c{\rm \omega }_{{\rm pe}}^{{-}1} \lt 126$, magnetic filaments are completely absent. However, at 
$t\sim 270{\rm \omega }_{{\rm pe}}^{{-}1}$as shown in Figure 2b, the previous smaller structures on the left-hand side of the simulation domain, grow with time and form large-scale magnetic structures along the counterstreaming e −/e + plasmas flow direction 
$\hat{y}$. Interestingly, on the right-hand side of the simulation box, 
$60 \lt x/c{\rm \omega }_{{\rm pe}}^{{-}1} \lt 126$, which is a low-plasma density region, the magnetic field structures with finite magnitude and finite scale appear along the 
$\hat{x}$ direction. Therefore, we find that the magnetic field fluctuations start to grow after the saturation of the first Weibel instability. As mentioned earlier, there is no net magnetic field in the right-hand side of the simulation domain at t = 0. Therefore, the second growth of the magnetic field is also expected to be due to Weibel instability. The Fourier spectrums corresponding to Figures 2a and 2b are shown in Figures 2c and 2d, respectively. One can see that the larger values of k x and k y of B z shown in Figure 2c confirm that the shortest magnetic field structures are produced in the initial simulation stage at 
$t\sim 10{\rm \omega }_{{\rm pe}}^{{-}1}$. Similarly, Figure 2d represents the Fourier spectrum corresponding to Figure 2b, which shows the smaller values of k x and k y of B z. This means larger magnetic structures are developed nearby 
$t\sim 270{\rm \omega }_{{\rm pe}}^{{-}1}$.
Fig. 2. 2D snapshots of the B z component in an inhomogeneous e −/e + plasma system at two different simulation times: (a) at 0 < x < 128c/ωpe and (b) at 0 < x < 128c/ωpe and the corresponding k-spectra of B z: (c) at 
$t\sim 270{\rm \omega }_{{\rm pe}}^{{-}1}$ and (d) at 
$t\sim 270{\rm \omega }_{{\rm pe}}^{{-}1}$, respectively.
Weibel instability due to thermal anisotropy generation
In this section, we study the evolution of temperature anisotropy A = (T y/T x − 1) in homogeneous and inhomogeneous plasma distributions both which is shown in Figure 3. The A is an important parameter to understand the magnetic field amplification via anisotropic density fluctuations present in the downstream region of the shock in GRBs. The A is calculated with the simulation time by taking averages of electron momentum components p x and p y over the whole simulation domain along the simulation time. At t = 0, the counterstreaming e −/e + plasmas have finite thermal temperature T y > T x along the 
$\hat{y}$ direction only. This initial anisotropic velocity distribution, v thy = 0.1c and v thx = 0, makes the system Weibel unstable.
Fig. 3. The evolution of temperature anisotropy A = (T y/T x − 1) along the simulation time for different plasma density inhomogeneities: δn/n 0 = 0 (black line), 0.5 (blue line), 0.75 (cyan line), and 1.0 (red line), respectively.
For these initial conditions, for the homogeneous plasma distribution in the simulation box 0 < x < 128c/ω pe, thermal anisotropy A (black line) starts decreasing from its maximum value, and finally approaches toward zero as the simulation progresses with time. The system is isotropized slowly with the simulation time. However, in the inhomogeneous plasma distribution case, shown for δn/n 0 = 0.5, A (blue line) shows a similar variation initially, however, different in magnitude. As the simulation time grows, it shows later oscillatory pattern post-saturation and becomes negative in the downstream region. The size of these oscillation becomes larger. The similar pattern was observed for 0.75 (cyan line) and 1.0 (red line), respectively. The inset of Figure 3 shows the temperature anisotropy temperature anisotropy (TA) along the simulation time for the δn/n 0 = 0.75 case in the high-plasma density region, 20c/ωpe < x < 42c/ωpe (purple line), and in the low-plasma density region, 80c/ωpe < x < 110c/ωpe (cyan line). Both TA curves show similar oscillator patterns as the plasma is isotropized by the magnetic field; however, the scale length is relatively smaller.
As mentioned earlier, the simulation domain length used here is relatively shorter than the scale length of the density clumps present in SNRs, ICM, and CSMs where high-density clumps with uneven scale length are supposed to be always present. These clumps are strongly compressed by the collisionless shocks by the factor Γd/r (Tomita and Ohira, Reference Tomita and Ohira2016) in the normal direction (in our case, it is the 
$\hat{x}$ direction) at the downstream rest frame, where r and Γd stand for the shock compression ratio and the Lorentz factor of the downstream flow at the shock rest frame, respectively.
In this study, where the initial magnetic field is zero, the Weibel instability becomes the main driver for the magnetic field fluctuations and dissipates the upstream bulk flow in the shock transition region, so that the upstream plasma is heated isotopically. At post-shock, electrons with a large v x escape from the high-density clump in the 
$\hat{x}$ direction, but it takes longer time to escape in the shock-tangential direction, that is, 
$\hat{y}$-axis. As a result, a temperature anisotropy appears at the high-density clump in the downstream region. Therefore, in the high-density clump, the temperature in the shock normal direction becomes lower than that in the shock-tangential direction. On the other hand, outside the high-density clump, the temperature in the shock normal direction becomes higher than that in the shock-tangential direction because there are escaping particles with a large v x. Hence, the upstream density fluctuations generate the temperature anisotropy in the shock downstream region.
To understand the Weibel instability due to temperature anisotropy, the dispersion relation for the linear phase is given by 
$k^2 +{\sigma }^2 = {-}\lsqb {1 + 1/2\lsqb {1 + A + 2{\lpar {v_0/v_{{\rm th}\bot }} \rpar }^2} \rsqb {Z}^{\prime}\lpar {i{\rm \sigma }/kv_{{\rm th}\bot }} \rpar } \rsqb$ (Lazar, Reference Lazar2008; Stockem et al., Reference Stockem, Lazar, Shukla and Smolyakov2009), where ω = iσ, and Z ′(ζ) = −2[1 + ζZ(ζ)] is the first derivative of the well-known plasma dispersion function 
$Z\lpar {\rm \zeta } \rpar = {\rm \pi }^{{-}1/2}\int_{-\infty }^\infty {d{\rm t}{\rm exp}\lpar {-t^2} \rpar /\lpar {t-{\rm \zeta }} \rpar }$ with ζ = ω/kv th⊥ (Fried and Conte, Reference Fried and Conte1961). For small temperature anisotropy A ≪ 1, the maximum growth of Weibel instability can be deduced as γmax = (4/27π)1/2[A 3/2/(1 + A)](v th/c)ωpe for k = (A 1/3)ωpe/c (Davidson et al., Reference Davidson, Hammer, Haber and Wagner1972), where k is the wave vector of the most unstable mode and parallel to the direction of the lower temperature. For A = 0.3 and v th = 0.5c, the mode with k ≈ 0.3ωpe/c shows the maximum growth rate of γmax ≈ 10−2ωpe, directly proportional to the plasma density ωpe, and closely match with our simulations.
In continuation to the discussion of the temperature anisotropy, Figure 4 shows the velocity distribution from the linear stage to the nonlinear one. The initial velocity distribution of counterstream electrons is in equilibrium state free-from external forces that is, the Maxwellian distribution centered around the velocity v 0 = 0.5c (blue line and blue-dotted line) computed at time 
$t\sim 0{\rm \omega }_{{\rm pe}}^{{-}1}$ [see Figs. 4(a) and 4(b)]. However, as the simulation time grows, the magnetic field generates and the plasma distribution becomes anisotropic. Hence, the velocity dispersion occurs and we get the shifted Maxwellian velocity distribution at 
$t\sim 100{\rm \omega }_{{\rm pe}}^{{-}1}$ (red line and red-dotted line). This analysis of the electron velocity distributions in the self-generated fields initially shows that more deviations from the forward-directed motion occur only after field saturation. One can notice as the simulation time grows, the width of the velocity distribution keeps increasing and it is isotropized (cyan line and cyan-dotted line) near zero velocity at 
$t\sim 400{\rm \omega }_{{\rm pe}}^{{-}1}$ in the inhomogeneous case [Fig. 4(b)]. However, in the case of the homogeneous plasma, the isotropized mechanism takes much longer time.
Fig. 4. The velocity distributions of counterstream electrons computed by the simulations at three different simulation time settings: 
$t\sim 0{\rm \omega }_{{\rm pe}}^{{-}1}$, 
$t\sim 100{\rm \omega }_{{\rm pe}}^{{-}1}$, and 
$t\sim 400{\rm \omega }_{{\rm pe}}^{{-}1}$, respectively, in (a) homogeneous and (b) inhomogeneous e −/e + plasmas. Here, e u denotes electrons motion along the 
$+ \hat{y}$ and e d denotes electrons motion along the 
$-\hat{y}$ direction.
To check the authenticity of the previous simulation results, shown in Figure 5, we show the plots of 
$-\hat{y}$ averaged, normalized plasma density distribution along the 
$\hat{x}$ direction at four different simulation time settings. The black line represents the initial density distribution at t = 0 which is clearly a sinusoidal variation. The red line shows the plasma density distribution taken at time 
$t\sim 110{\rm \omega }_{{\rm pe}}^{{-}1}$, just after the second peak of the magnetic field energy plot (see Fig. 1). Therefore, the large-scale sinusoidal density distribution turns out on a small-scale density filamentation, approaching toward the equilibration state. The electrons from the high-density region move toward the low-density region making the system in the equilibration state. The blue line taken at 
$t\sim 200{\rm \omega }_{{\rm pe}}^{{-}1}$ represents the shock transition region. The electrons move toward the low-density region, rapidly increasing the number of electrons in the low-density region. The magenta line taken at 
$t\sim 400{\rm \omega }_{{\rm pe}}^{{-}1}$ shows the system reached to the equilibration stage, as supported by the temperature anisotropy calculations shown in Figure 4. It would be interesting to study the highly relativistic collisionless shock generation in an inhomogeneous e −/e + plasma medium in future.
Fig. 5. Plots of the −y averaged normalized plasma density of the inhomogeneous e −/e + plasma along the 
$\hat{x}$ direction at different simulation time settings. Plasma density is normalized by the mean upstream plasma density.
Conclusion
2D PIC simulation has been carried out for a counterstreaming electron–positron (e −/e +) plasma system having intrinsic temperature anisotropy with the homogeneous and inhomogeneous density distributions. Such types of distributions are relevant for the relative motions of filaments and expected to exist in astrophysical environments.
Our simulation results show that in a pre-existing, upstream homogeneous density fluctuations, the initial strong temperature anisotropy T y ≫ T x gradually drops with time and becomes zero (isotropized system) at the end of the simulation. Subsequently, the Weibel instability-driven magnetic field energy is amplified exponentially up to (~10−1) in a linear stage and then decays gradually to post-saturation. However, in the case of pre-existing, upstream inhomogeneous density fluctuations, we find that the finite temperature anisotropy exists at post-saturation and the magnetic field is re-amplified 
$\lpar {t\sim 270{\rm \omega }_{{\rm pe}}^{{-}1} } \rpar$ in the downstream region. The increase in the magnetic field in the downstream region can be larger depending on how strong is the inhomogeneity factor of the upstream density fluctuations. The hot particles are shifted to the low-density region from the high-density region, which causes temperature increment in the 
$\hat{x}$ direction resulting in the velocity dispersion. Therefore, the density fluctuations cause a finite temperature anisotropy and the magnetic field amplification in the downstream region. The upstream isotropic density structures are compressed by collisionless shock and made to be anisotropic density structures causing the temperature anisotropy in the downstream region. Weibel instability extracts free energy from the anisotropy of the particles, producing filamentary currents aligned with the normal, and these currents are responsible for the creation of transverse magnetic fields and its amplification in the magnetic energy. Hence, the generated magnetic fields cover large regions. This analysis can be applied to wide range scenarios of astrophysical plasmas such as supernova remnants and GRBs whereby pre-existing large-scale upstream turbulence causes the magnetic field amplification in the downstream region.
Acknowledgments
This work was supported by the National Research Foundation of Korea (NRF); the Ministry of Science, ICT and Future Planning (MSIP) (NRF-2016R1A5A1013277); and the CHEA project, Korea.
