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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{4074F1.eps} \end{figure}$ Figure 1: As the SNR shock expands it reflects a fraction of the ISM protons. These protons move back into the upstream region and form beam 1. The upstream $\vec{B}$ rotates beam 1 which returns as beam 2. The simulation box covers a one dimensional region of x in front of the shock sufficiently small that we can assume it to be spatially homogeneous and one dimensional.
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In the text

$\begin{figure} \par\includegraphics[width=8.3cm,clip]{4074F2.eps} \end{figure}$ Figure 2: The initial velocity distribution showing the ISM electrons and protons, proton beam 1 and proton beam 2. The expanding SNR shock reflects a fraction of the ISM protons which form beam 1. The upstream $\vec{B}$ rotates beam 1 to create beam 2. Since the beam has been scattered by ESWs on its path through the foreshock, its thermal spread has increased. Both beams can grow ESWs with a speed similar to $v_{\rm b}$ which saturate by their interaction with the ISM electrons. By assuming that the plasma parameters do not change over the small box size we consider, we can represent this plasma by the 4 spatially homogeneous species shown.
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In the text

$\begin{figure} \par\includegraphics[width=17.3cm,clip]{4074F3.eps} \end{figure}$ Figure 3: The logarithmic amplitude of the most unstable mode $E(k_{\rm u},t)$ is plotted against time, in normalised units, for a variety of beam speeds. The dashed lines on each plot represent a linear fit over the region of exponential growth. Using this linear fit we obtain growth rates for the relativistic Buneman instability, in normalised units, of $\Omega \approx 0.0256, 0.0264, 0.0270, 0.0246, 0.0201$ and 0.0137 for beam speeds of $v_{\rm b} = 0.15c, 0.2c, 0.4c, 0.6c, 0.8c$ and 0.9c, respectively.
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{4074F4.eps} \end{figure}$ Figure 4: Contour plot of $\log (f_{\rm e}>10^{-6})$ at time t=150 for a system with an initial beam velocity of $v_{\rm b} = 0.2c$ . The exponential growth of the ESW is in the process of saturating via the trapping of electrons, forming a BGK mode at $p \approx + 0.2.$ The thermal spread of beam 2 inhibits its growth slightly, allowing the system to become dominated by one ESW. Hence, we do not observe electron trapping around $p \approx - 0.2$ . Here x is given in units of $2\pi / k_{\rm u}$ .
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{4074F5.eps} % \end{figure}$ Figure 5: Contour plot of $\log (f_{\rm e}>10^{-6})$ at time t=200 for a system with a relativistic initial beam velocity of $v_{\rm b} = 0.9c$ . The exponential growth of the ESW is in the process of saturating via the trapping of electrons, forming a pair of BGK modes at $p \approx \pm 3$ . The reduced significance of the higher temperature of species 4, with respect to species 3, at high beam speeds allows the growth of forwards and backwards propagating ESWs simultaneously. Here x is given in units of $2\pi / k_{\rm u}$ .
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In the text

$\begin{figure} \par\includegraphics[width=11.5cm,clip]{4074F6.eps} \end{figure}$ Figure 6: The time evolution of the ESW amplitudes with $k=k_{\rm u}$ that are driven by beam 1 for beam speeds of 0.15c, 0.2c, 0.6c and 0.9c. ESWs generated at non-relativistic beam speeds saturate smoothly and stabilise briefly. At $v_{\rm b}=0.6$ the ESW collapses abruptly and at relativistic beam speeds, $v_{\rm b} = 0.9c$ we observe a stabilisation of the ESW for a period of almost $1000\omega _{\rm pe}^{-1}$ , equivalent to almost 160 plasma periods. Approximate regions of stability are highlighted on each plot.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{4074F7.eps} \end{figure}$ Figure 7: The ESW spectrum for a beam speed of ${v_{\rm b} = 0.15c}$ : a) shows the frequency power spectrum of $k_{\rm c}=k_{\rm u}$ and b) shows the frequency power spectrum of $k_{\rm c}=k_{\rm u}/2$ as a function of time. The colour scale shows $\log_{10} W_2(\omega,t,k_{\rm c})$ . The ESW driven by beam 1 with $v_{\rm b} > 0$ corresponds to the wave with $\omega = \omega _{{\rm p},1}$ in a).
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In the text

$\begin{figure} \par\includegraphics[width=7.6cm,clip]{4074F8.eps} \end{figure}$ Figure 8: Contour plot of $\log (f_{\rm e}>10^{-6})$ for ${v_{\rm b} = 0.15c}$ at the time t=3500: the initial electron BGK modes have collapsed and they can be shown to form a plateau distribution at $p / m_{\rm e} c < 0.3$ . Centred at $p / m_{\rm e} c \approx 0.52$ we find BGK modes driven by the beat wave.
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In the text

$\begin{figure} \par\includegraphics[width=8.2cm,clip]{4074F9.eps} \end{figure}$ Figure 9: The ESW spectrum for a beam speed of $v_{\rm b} = 0.2c$ : a) shows the frequency power spectrum of $k_{\rm c}=k_{\rm u}$ and b) shows the frequency power spectrum of $k_{\rm c}=k_{\rm u}/2$ as a function of time. The colour scale shows $\log_{10} W_2(\omega,t,k_{\rm c})$ . The ESW driven by beam 1 with $v_{\rm b} > 0$ corresponds to the wave with $\omega = \omega _{{\rm p},1}$ in a).
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In the text

$\begin{figure} \par\includegraphics[width=7.6cm,clip]{4074F10.eps} \end{figure}$ Figure 10: Contour plot of $\log (f_{\rm e}>10^{-6})$ for $v_{\rm b} = 0.2c$ at the time t=2000: the initial electron BGK modes have collapsed and form a plateau distribution. Centred at $p / m_{\rm e} c \approx 0.75$ we find BGK modes driven by the beat wave.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{4074F11.eps} \end{figure}$ Figure 11: The ESW spectrum for a beam speed of ${v_{\rm b} = 0.4c}$ : a) shows the frequency power spectrum of $k_{\rm c}=k_{\rm u}$ and b) shows the frequency power spectrum of $k_{\rm c}=k_{\rm u}/2$ as a function of time. The colour scale shows $\log_{10} W_2(\omega,t,k_{\rm c})$ . The ESW driven by beam 1 with $v_{\rm b} > 0$ corresponds to the wave with $\omega = \omega _{{\rm p},1}$ in a).
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{4074F12.eps} \end{figure}$ Figure 12: The ESW spectrum for a beam speed of ${v_{\rm b} = 0.6c}$ : a) shows the frequency power spectrum of $k_{\rm c}=k_{\rm u}$ and b) shows the frequency power spectrum of $k_{\rm c}=k_{\rm u}/2$ as a function of time. The colour scale shows $\log_{10} W_2(\omega,t,k_{\rm c})$ . The ESW driven by beam 1 with $v_{\rm b} > 0$ corresponds to the wave with $\omega = \omega _{{\rm p},1}$ in a).
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In the text

$\begin{figure} \par\includegraphics[width=7.6cm,clip]{4074F13.eps} \end{figure}$ Figure 13: Contour plot of $\log (f_{\rm e}>10^{-6})$ for ${v_{\rm b} = 0.6c}$ at the (early) time t=200: the initial electron BGK modes have just been formed and the trapped electrons reach a momentum up to $p / m_{\rm e} c \approx 3$ .
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In the text

$\begin{figure} \par\includegraphics[width=7.6cm,clip]{4074F14.eps} \end{figure}$ Figure 14: The electron momentum distribution for ${v_{\rm b} = 0.6c}$ at the (late) time t=3500: the initial electron BGK modes have collapsed and the electrons that are trapped by the sideband mode reach a momentum up to $p / m_{\rm e} c \approx 7$ .
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{4074F15.eps} \end{figure}$ Figure 15: The ESW spectrum for a beam speed of $v_{\rm b} = 0.8c$ : a) shows the frequency power spectrum of $k_{\rm c}=k_{\rm u}$ and b) shows the frequency power spectrum of $k_{\rm c}=k_{\rm u}/2$ as a function of time. The colour scale shows $\log_{10} W_2(\omega,t,k_{\rm c})$ . The ESW driven by beam 1 with $v_{\rm b} > 0$ corresponds to the wave with $\omega = \omega _{{\rm p},1}$ in a).
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In the text

$\begin{figure} \par\includegraphics[width=8cm,clip]{4074F16.eps} \end{figure}$ Figure 16: The ESW spectrum for a beam speed of $v_{\rm b} = 0.9c$ : a) shows the frequency power spectrum of $k_{\rm c}=k_{\rm u}$ and b) shows the frequency power spectrum of $k_{\rm c}=k_{\rm u}/2$ as a function of time. The colour scale shows $\log_{10} W_2(\omega,t,k_{\rm c})$ . The ESW driven by beam 1 with $v_{\rm b} > 0$ corresponds to the wave with $\omega = \omega _{{\rm p},1}$ in b).
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{4074F17.eps} \end{figure}$ Figure 17: Contour plot of $\log (f_{\rm e}>10^{-6})$ for $v_{\rm b} = 0.9c$ at the time t=4000: no well-defined BGK mode is visible in the electron distribution since the sideband unstable modes have a superluminal phase speed.
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{4074F18.eps} \end{figure}$ Figure 18: The electron momentum distributions at the simulation's end times.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{4074F1.eps} \end{figure}$	Figure 1: As the SNR shock expands it reflects a fraction of the ISM protons. These protons move back into the upstream region and form beam 1. The upstream $\vec{B}$ rotates beam 1 which returns as beam 2. The simulation box covers a one dimensional region of x in front of the shock sufficiently small that we can assume it to be spatially homogeneous and one dimensional.
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$\begin{figure} \par\includegraphics[width=7.6cm,clip]{4074F8.eps} \end{figure}$	Figure 8: Contour plot of $\log (f_{\rm e}>10^{-6})$ for ${v_{\rm b} = 0.15c}$ at the time t=3500: the initial electron BGK modes have collapsed and they can be shown to form a plateau distribution at $p / m_{\rm e} c < 0.3$ . Centred at $p / m_{\rm e} c \approx 0.52$ we find BGK modes driven by the beat wave.
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$\begin{figure} \par\includegraphics[width=7.6cm,clip]{4074F10.eps} \end{figure}$	Figure 10: Contour plot of $\log (f_{\rm e}>10^{-6})$ for $v_{\rm b} = 0.2c$ at the time t=2000: the initial electron BGK modes have collapsed and form a plateau distribution. Centred at $p / m_{\rm e} c \approx 0.75$ we find BGK modes driven by the beat wave.
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$\begin{figure} \par\includegraphics[width=7.6cm,clip]{4074F13.eps} \end{figure}$	Figure 13: Contour plot of $\log (f_{\rm e}>10^{-6})$ for ${v_{\rm b} = 0.6c}$ at the (early) time t=200: the initial electron BGK modes have just been formed and the trapped electrons reach a momentum up to $p / m_{\rm e} c \approx 3$ .
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$\begin{figure} \par\includegraphics[width=7.6cm,clip]{4074F14.eps} \end{figure}$	Figure 14: The electron momentum distribution for ${v_{\rm b} = 0.6c}$ at the (late) time t=3500: the initial electron BGK modes have collapsed and the electrons that are trapped by the sideband mode reach a momentum up to $p / m_{\rm e} c \approx 7$ .
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$\begin{figure} \par\includegraphics[width=7.8cm,clip]{4074F17.eps} \end{figure}$	Figure 17: Contour plot of $\log (f_{\rm e}>10^{-6})$ for $v_{\rm b} = 0.9c$ at the time t=4000: no well-defined BGK mode is visible in the electron distribution since the sideband unstable modes have a superluminal phase speed.
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$\begin{figure} \par\includegraphics[width=8.4cm,clip]{4074F18.eps} \end{figure}$	Figure 18: The electron momentum distributions at the simulation's end times.
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