2. Simulations

We simulate a relativistic cold electron–proton plasma of uniform density and a relativistic factor $\gamma _0=20$ flowing through a background plasma at rest with identical electron and proton densities. Both electrons and protons are initialized with a small isotropic thermal velocity $\beta _{th}\approx 10^{-3}$ (${=}\sqrt {T/m c^2}$) to initiate the WI/CFI. The flow velocity (along $z$) is perpendicular to the simulation plane (see figure 1). The computational domain is $512\times 512 (c/\omega _{pe})^2$ with periodic boundaries. The cell size is ${\rm \Delta} x={\rm \Delta} y=0.1 c/\omega _{pe}$, time step ${\rm \Delta} t=0.045 \omega _{pe}^{-1}$ and each cell contains $3\times 3$ particles per cell per species, where $\omega _{pe}=(4{\rm \pi} n_0\,\textrm {e}^2/m_e)^{1/2}$ is the plasma frequency, $n_0$ is the initial plasma density, $m_e$ is the electron mass, $e$ is the electron charge and $c$ is the velocity of light. We include a uniform out-of-plane (along $z$) initial magnetic field ($B_z$) measured in the frame of the background plasma corresponding to a magnetization of the flowing plasma $\sigma \equiv B_{z}^2/4{\rm \pi} \gamma _0m_en_0c^2$ which we varied from $\sigma = 0.0$ to $0.2$ for both signs of $B_z$. The WI/CFI in interpenetrating plasma flows gives rise to current filaments with an associated azimuthal magnetic field ($B_\phi \hat {\boldsymbol {e}}_\phi$) surrounding them. Because the electrons respond at time scales faster than the protons, a space charge radial electric field ($E_r\hat {\boldsymbol {e}}_r$) develops at the edge of the electron filaments. The maximum electron WI/CFI growth rate is given by $\varGamma _{WI/CFI}\approx |\beta _0|/\sqrt {\gamma _0}\omega _{pe}$ (Silva et al. Reference Silva, Fonseca, Tonge, Mori and Dawson2002; Shukla et al. Reference Shukla, Stockem, Fiúza and Silva2012), where $\beta _0$ is the flow velocity normalized to $c$. For our $\gamma _0=20$ this yields $\varGamma \approx 0.2\omega _{pe}$, and simulations show that the magnetic field saturates in a few 100$\omega _{pe}^{-1}$.

Figure 1. (a) The spatial distribution of radiated power from electrons in a magnetized current filament of an electron–proton plasma ($m_i/m_e=1836$) with initial magnetization $\sigma =0.05$ and at time $t=9000\omega _{pe}^{-1}$. (b) The transverse magnetic field vectors (represented by arrows) arising due to WI/CFI for the same simulation as in (a). Trajectories of 50 electrons trapped in the current filament are shown from time $t_i=8910\omega _{pe}^{-1}$ to $t_f=9000\omega _{pe}^{-1}$, with colour scales representing their longitudinal momentum ($\boldsymbol {p}_z$). (c) The fields and tracks of electrons for the same time period in an unmagnetized electron–proton plasma.

At the saturation of WI/CFI, there is a strong emission of radiation from the filament edges. This is illustrated in figure 1(a), which shows the radiated power ($P$) from the electrons in a single magnetized current filament. The radiated power is calculated using the relativistic generalization of Larmor's formula, $P=2/3(\textrm {e}^2/c)\gamma _e^6[\boldsymbol {\dot {\beta }}_e^2-(\boldsymbol {\beta }_e\times \boldsymbol {\dot {\beta }}_e)^2]$, where $\boldsymbol {\beta }_e$ is the electron velocity vector normalized to $c$, $\boldsymbol {\dot {\beta }}_e$ is the normalized electron acceleration and $\gamma _e$ is the electron Lorentz factor (Jackson Reference Jackson2012). The peak value of $P$ at the filament edges is nearly ten times larger than that within the filament. This is because additional mutually perpendicular strong electric and magnetic fields resulting from WI/CFI lead to a relatively stronger $\boldsymbol {\dot {\beta }}_e$ perpendicular to $\boldsymbol {\beta }_e$ and increased kinetic energies of the electrons at the filament edges, whereas the electrons at the centre gyrate only under the influence of the initial magnetic field and the longitudinal momentum ($p_z$) is much larger than the transverse momentum ($p_x,p_y$). Hence, the electrons at the filament edge primarily contribute to the radiated energy.The electric field of the radiation emitted by an electron is given by (Jackson Reference Jackson2012)

(2.1)\begin{equation} \boldsymbol{E}_\mathrm{rad}(\boldsymbol{r},t) =\frac{e}{c} \left[\frac{\boldsymbol{n}\times[(\boldsymbol{n}-\boldsymbol{\beta}_e)\times \boldsymbol{\dot{\beta}}_e]}{(1-\boldsymbol{\beta}_e\cdot\boldsymbol{n})^3R}\right]_{\textrm{ret}},\end{equation}

where $\boldsymbol {n}$ is the unit vector from the position of the charge to the observer at a distance $R$ from the particle. The quantities are evaluated at the retarded time $t^\prime =t-R/c$. The spectrum of the radiated electric field is determined from the Fourier transform of (2.1) (Jackson Reference Jackson2012):

(2.2)\begin{align} \boldsymbol{E}_\mathrm{rad}(\omega)&=\frac{e}{c}\left(\frac{1}{2{\rm \pi}}\right)^{1/2} \int_{-\infty}^{\infty}\frac{\boldsymbol{n}\times[(\boldsymbol{n}-\boldsymbol{\beta}_e)\times \boldsymbol{\dot{\beta}}_e]}{(1-\boldsymbol{\beta}_e\cdot\boldsymbol{n})^2R(t^\prime)}\nonumber\\ &\quad \times \exp({\textrm{i}\omega(t^\prime-\boldsymbol{n}\cdot\boldsymbol{r}(t^\prime)/c)})\,\textrm{d} t^\prime. \end{align}

The information about the instantaneous position, velocity and acceleration of the electrons can be obtained from the electron trajectories. The electron trajectories, superimposed on the magnetic fields due to WI/CFI for magnetized and unmagnetized plasmas, are shown in figures 1(b) and 1(c), respectively. The electrons have enhanced longitudinal momentum ($p_z$) at the filament edge, which indicates that most of the radiation emitted from them will be collimated in the direction perpendicular to the simulation plane. An initial magnetization causes the electrons to drift along the azimuthal direction, due to a combination of $\boldsymbol {E}\times \boldsymbol {B}$ and $\boldsymbol {\nabla }\boldsymbol {B}$ drifts, executing a correlated motion (figure 1b). On the other hand, in an unmagnetized plasma, the electrons scatter at all possible angles from the WI/CFI fields (figure 1c).

3. Model and comparison with simulations

To formulate a model describing the motion of charged particles in WI/CFI fields, we consider a cylindrical current filament with electrons flowing along the positive $z$ direction ($\hat {\boldsymbol {e}}_z$). The electric and magnetic fields can be described as $\boldsymbol {E}=E_rS(r)\hat {{e}}_r$ and $\boldsymbol {B}=B_\phi S(r)\hat {\boldsymbol {e}}_\phi +B_{z}\hat {\boldsymbol {e}}_z$, respectively, where $E_r$ and $B_\phi$ are the respective amplitudes of the $\boldsymbol {E}$ and $\boldsymbol {B}$ fields in the radial and azimuthal directions, $B_z$ is the initial magnetic field and $S(r)$ is the spatial profile. At saturation, the spatial profile of the fields from the simulation closely resembles a Gaussian with $\textrm {FWHM}\sim 2.5c/\omega _{pe}$. Hence, we substitute $S(r)=\exp [-(r-r_0)^2/2\eta ^2]$, where $r_0$ is the filament radius and $\eta =1/\sqrt {2}$. The equation for the electron trajectory obtained by combining the momentum and energy equation is

(3.1)\begin{equation} \frac{\textrm{d}\boldsymbol{\beta}_e}{\textrm{d} t}={-}\frac{1}{\gamma_e}[\boldsymbol{E}+ \boldsymbol{\beta}_e\times \boldsymbol{B}-\boldsymbol{\beta}_e(\boldsymbol{E} \cdot\boldsymbol{\beta}_e)],\end{equation}

where the normalizations $\boldsymbol {E}\equiv e\boldsymbol {E}/(m_e\omega _{pe}c)$; $\boldsymbol {B}\equiv e\boldsymbol {B}/(m_e\omega _{pe}c)$; and $t\equiv \omega _{pe}t$ have been used. The radiated electric field vectors can be calculated using (2.1) and (3.1) for a given field configuration and initial position and velocity of the particles.

A schematic representation of the geometry of the magnetic field due to the current filaments and the motion of plasma electrons in the frame of the background plasma is shown in figure 2(a). The radiated electric field vectors obtained using (2.1) and (3.1) for electrons making pitch angles $\psi _1$ ($<{\rm \pi} /2$) and $\psi _2$ ($>{\rm \pi} /2$) with respect to the direction of the WI/CFI magnetic field at the instant of emission are shown in figure 2(b). The direction of rotation of the radiated electric field vectors shows a left-handed circular polarization for $\psi _1$ and right-handed circular polarization for $\psi _2$. The handedness of circular polarization depends on the pitch angle. The emission from a single electron whose velocity vector makes a pitch angle $\psi$ with the direction of the WI/CFI magnetic field vector at the instant of emission is equivalent to that emitted by a particle moving at a constant speed in a circular path. For the observer lying on the $z$ axis in the far field, it is reasonable to assume that the angle between the observer and the point of emission $\alpha \approx 0$. Hence, the electric field vectors of the emitted radiation will be oriented in the $xy$ plane. For a relativistic electron, the radiation lasts for a very short time and is limited within a small angle $\theta \sim 1/\gamma _e\ll 1$ for a fixed observer. Under this assumption, (2.2) reduces to $\boldsymbol {E}_\mathrm {rad}= (e/c)(2/3 {\rm \pi} )^{1/2} (\omega /\omega _0 R) [-\textrm {i}(1/\gamma _e^2+({\rm \pi} /2-\psi )^2) K_{2/3}(\xi )\widehat {\boldsymbol {e}_{x}}]$ $+ ({\rm \pi} /2-\psi ) \sqrt {(1/\gamma _e^2+({\rm \pi} /2-\psi )^2)} K_{1/3}(\xi )\widehat {\boldsymbol {e}_{y}} ]$, where $K_{2/3}$ and $K_{1/3}$ are the modified Bessel functions, $\xi =(\omega /\omega _0)(1+\gamma _e^2({\rm \pi} /2-\psi )^2)^{3/2}/3\gamma _e^3$, $\omega _0$ is the gyrofrequency and $\omega$ is the frequency of radiation (Jackson Reference Jackson2012). It is clear that the polarization of radiation is right-handed or left-handed according to $\psi \gtrless {\rm \pi} /2$ and is consistent with the results from the semi-analytical model shown in figure 2(b1,b2). For a plasma with no initial magnetic field component along $\hat {\boldsymbol {e}}_z$, the electrons have a symmetric pitch angle distribution for $\psi \gtrless {\rm \pi} /2$. Hence, the contributions to the right-handed and left-handed polarizations are equal, resulting in a net zero circular polarization. In a magnetized plasma, the initial magnetic field ($B_z$) along $\hat {\boldsymbol {e}}_z$ bends the electron trajectory in the $xy$ plane creating an anisotropy in pitch angle distribution. When $B_z$ is along positive $\hat {\boldsymbol {e}}_z$, the number of electrons with negative pitch angles exceeds the number with positive pitch angles resulting in a net right-handed circular polarization and vice versa when the direction of $B_z$ is reversed. To validate our model, we extracted trajectories of 1000 electrons from a current filament directly from PIC simulations and computed the radiation spectrum and the degree of circular polarization ($P_c$) of the radiation emitted from them using the post-processing radiation code jRad (Martins et al. Reference Martins, Martins, Fonseca and Silva2009a). Although the OSIRIS simulation is a two-dimensional one, we reconstruct the three-dimensional trajectories as $z=\int p_z(t)/\gamma _e(t)\,\textrm {d} t$. The spectrum was calculated on a two-dimensional virtual detector in the $xy$ plane at a distance $z=1.5\times 10^4c/\omega _{pe}$ and size $60\,000\times 60\,000(c/\omega _{pe})^2$ divided in $100\times 100$ cells to capture the entire emitting region. The detector captured a spectrum of frequencies ($\omega$) in the range $\omega =(10^0\text {--}10^3)\omega _{pe}^{-1}$ with a resolution of 256 cells per decade in the frequency axis. Radiation was calculated following the trajectories from time $t_i=6300\omega _{pe}^{-1}$ to $t_f=9000\omega _{pe}^{-1}$ during which the filament was in a steady state. The degree of circular polarization ($P_c$) was estimated using the relevant Stokes parameters, for which $P_c=V/I$, $V=2\langle Im[(\boldsymbol {\epsilon }_1\cdot \boldsymbol {E}_{\textrm {rad}})^*(\boldsymbol {\epsilon }_2\cdot \boldsymbol {E}_{\textrm {rad}})]\rangle$, $I=|\boldsymbol {\epsilon }_1\cdot \boldsymbol {E}_{\textrm {rad}}|^2+|\boldsymbol {\epsilon }_2\cdot \boldsymbol {E}_{\textrm {rad}}|^2$ and $\boldsymbol {\epsilon }_1$ and $\boldsymbol {\epsilon }_2$ are the unit vectors perpendicular to the direction of observation. The angular brackets $\langle \cdot \rangle$ represent the time average. Figures 3(a) and 3(c) show the spatial distribution of frequency-averaged radiated energy, $\langle I_{\textrm {rad}}\rangle _\omega =\int I_{\textrm {rad}}\,\textrm {d}\omega /\int \textrm {d}\omega$, from the electrons confined in the current filament of figures 1(b) and 1(c), respectively. Figures 3(b) and 3(d) show the corresponding $\langle P_c\rangle _\omega =\int P_cI_{\textrm {rad}}\,\textrm {d}\omega /\int I_{\textrm {rad}}\,\textrm {d}\omega$ for the filament. A strong right circular polarization with a peak value of $\langle P_c\rangle _\omega \approx 0.25$ (25 %) is observed for the magnetized filament. For the unmagnetized filament, a nearly equal distribution of left and right circular polarization is observed resulting in a net zero circularly polarized radiation flux. The total circularly polarized radiation flux $\langle P_c\rangle$ can be obtained using the formula $\langle P_c\rangle =\iint P_cI_{\textrm {rad}}\,\textrm {d} A\,\textrm {d}\omega /\iint I_{\textrm {rad}}\,\textrm {d} A\,\textrm {d}\omega$, where $A$ is the detector area. For the radiation in figure 1(b), $\langle P_c\rangle =0.117$ (11.7 %). To confirm that $\langle P_c\rangle$ is not affected by the time period on which it is averaged, the time of averaging ($6300\omega _{pe}^{-1}\text {--}9000\omega _{pe}^{-1}$) was divided into three equal segments of interval $900\omega _{pe}^{-1}$ and $\langle P_c\rangle$ was separately calculated for each of these intervals. We found that $\langle P_c\rangle$ was equal in all cases with a variation of ${\pm }0.1\,\%$, which indicates that our observations for a single filament are physically meaningful.

Figure 2. (a) A schematic representation of the azimuthal magnetic field due to a cylindrical current filament with a flow along $z$ with 1 and 2 representing electron trajectories that, at a given time, make pitch angles $\psi _1<{\rm \pi} /2$ and $\psi _2>{\rm \pi} /2$, respectively, with the direction of the WI/CFI magnetic field vector at the time of emission. (b) The trajectories of two electrons from the semi-analytical model for an observer along $\hat {\boldsymbol {e}}_z$ (trajectory 1 ($\psi _1$) in blue and trajectory 2 ($\psi _2$) in red). The electric field vectors (arrows) are attached to each point on the trajectory, and shown more clearly in the zoom (b1,b2). It is clear that, at the time of dominant radiation where the radius of curvature is at a minimum, the radiation from trajectory 1 is left circularly polarized and from trajectory 2 is right circularly polarized because the electric field vectors are rotating clockwise and counterclockwise, respectively.

Figure 3. Simulation results from jRad illustrating the key properties of radiation and degree of circular polarization from 1000 electrons trapped in a current filament. (a,c) The frequency-averaged spatial distribution of radiated energy on the detector lying in the $xy$ plane (parallel to the simulation plane) from the current filament shown in figures 1(c) and 1(b), respectively. (b,d) The frequency-averaged degree of circular polarization corresponding to (a,c). For the unmagnetized filament, there is an almost equal contribution to left and right circular polarization, whereas a magnetized filament produces a strong right circularly polarized radiation with peak value ${\sim }25\,\%$. (e) The frequency-averaged spatial distribution of radiated energy on a detector similar to (c) from four magnetized current filaments (with initial magnetization $\sigma =0.05$) of electron–proton plasma shown in the inset (a zoom of this inset can be found in figure 5). (f) The frequency-averaged degree of circular polarization corresponding to (e) with the inset showing $\langle P_c\rangle$ for each of the corresponding filaments. A line-out indicated by a dashed line is also plotted as a solid curve in each panel.

Furthermore, to check for the effect of multiple filaments, we computed the radiation spectra and $P_c$ from 1000 electrons distributed equally in four filaments of a magnetized electron–proton plasma of figure 1(b). Figures 3(e) and 3(f) show $\langle I_{\textrm {rad}}\rangle _\omega$ and $\langle P_c\rangle _\omega$, respectively, for the four filaments described in the inset of figure 3(e). The radiation spectra and circular polarization are similar to those of the single filament. In addition, $\langle P_c\rangle$ obtained from each of the filaments (inset of figure 3f) is computed separately and found to be nearly equal (${\sim }0.115$). This shows that the anisotropy in pitch angle distribution depends only on the fields and not the filament shape. An important observation is that WI/CFI in magnetized plasmas generates local islands of anisotropic pitch angle distribution. Hence, circular polarization can be observed even in the case of WI/CFI-driven shocks in magnetized spherical jets irrespective of the position of observation in contrast to the model described earlier (Nava, Nakar & Piran Reference Nava, Nakar and Piran2015), which is valid only for collimated jets. To investigate the role of initial magnetization of the plasma, $\langle P_c\rangle$ was calculated from 1000 electrons confined in a single filament of electron–proton plasma with initial magnetizations varying from 0.0 to 0.2 for both signs of $B_z$. It was found that $\langle P_c\rangle$ increased initially with magnetization and then started to saturate beyond $\sigma = 0.05$ converging to ${\sim }13\,\%$. The level of circular polarization depends on the electron pitch angle distribution, and we have confirmed that the external magnetic field changes this distribution such that finite levels of circular polarization become possible. Specifically, in unmagnetized scenarios, the pitch angle distribution is symmetric about ${\rm \pi} /2$. Thus, the total level of circular polarization vanishes in this case. In magnetized scenarios, instead, the pitch angle distribution is offset due to the particle drifts that appear in the presence of the external magnetic fields. As the pitch angle distribution becomes asymmetric, the level of circular polarization increases. Other processes can also lead to finite circular polarization levels, for instance (Sinha et al. Reference Sinha, Keitel and Kumar2019) where the generation of circular polarization in a different physical configuration is attributed to an asymmetric energy dissipation mechanism instead of the topological changes in pitch angle distribution discussed here.

For high magnetizations (${\gtrapprox }0.05$), the cyclotron motion completely dominates over the azimuthal velocity vector leading to a saturation of the pitch angle distribution which results in the saturation of $\langle P_c\rangle$. The handedness of circular polarization changes with the sign of $B_z$ because the velocity vector due to cyclotron motion reverses with the sign of $B_z$ resulting in an anisotropy in the opposite direction. Furthermore, to understand the effect of plasma composition, i.e. ion–electron mass ratios ($m_i/m_e$), we performed simulations for the same magnetizations for plasmas with $m_i/m_e=1$ and 918, and compared the degree of circular polarization shown in figure 4. We observed that the values of $\langle P_c\rangle$ for $m_i/m_e=918$ and 1836 were almost equal. This is because the ion response time scales are very large when compared to electron time scales, resulting in the radiation primarily being emitted only by the electrons and the circular polarization arising only due to the anisotropies in electron pitch angle distribution. For electron–positron plasmas, both species contribute to the emitted radiation. They rotate in opposite directions due to their opposite charge and create anisotropies in mutually opposite directions resulting in a net cancellation of the circular polarization. Although the level of circular polarization is strongly dependent on mass ratio for intermediate values of $m_i/m_e$, we did not include this range of mass ratios in figure 4. In this range, it is not possible to find a steady-state filament for comparable time intervals, because filaments evolve rapidly compared to the electron motion. Although a study showing the scaling of polarization as a function of mass ratio may be possible, and could allow for a three-dimensional study at reduced mass ratio, this is beyond the scope of this paper.

Figure 4. Simulation results from jRad illustrating the normalized flux of circularly polarized photons (averaged over frequency and spatial domain) from 1000 electrons trapped in a typical current filament arising from interpenetrating flows of electron–positron and electron–ion plasmas for varying magnetizations. Note that here negative $\sigma$ represents negative $B_z$.