All Figures

$\begin{figure} \par {\includegraphics[width=8cm,clip]{7988fg1.eps} } \end{figure}$ Figure 1: Dynamical equilibrium solutions for $\kappa _{0}=4\times 10^{11}$ , $\kappa _{1}=10^{-2}$ and $\kappa _{2}=4\times 10^{9}$ . Plots show behaviour as function of position for the following quantities: top left, radial streaming speed $\sigma _0$ ; top right, total number density $\Sigma _0$ ; bottom left, differential azimuthal flow $\zeta _0$ ; bottom right, magnetic field. Solutions are calculated using a bespoke Runge-Kutta routine with the parameters in Table 1. The equilibrium calculated here was used to produce the solutions in Figs. 2-7.
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In the text

$\begin{figure} \par {\includegraphics[width=8cm,clip]{7988fg2.eps} } \end{figure}$ Figure 2: Spatial and temporal evolution of $\Delta$ , showing the plasma density evolution as a function of time $\tau$ and space $\xi$ associated with an electrostatic oscillation. This is the response after a $1\%$ initial charge density perturbation.
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In the text

$\begin{figure} \par {\includegraphics[width=8cm,clip]{7988fg3.eps} } \end{figure}$ Figure 3: Spatial and temporal evolution of the radial electric field, $\rho$ , of the plasma oscillation, consistent with Fig. 2. The nonlinear evolution of the field is visible at the edges of the oscillation site.
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In the text

$\begin{figure} \par {\includegraphics[width=8cm,clip]{7988fg4.eps} } \end{figure}$ Figure 4: Spatial and temporal evolution of the axial magnetic field, $\beta$ , showing clearly the propagation of an electromagnetic signal outwards from the electrostatic oscillation site. Note that there is no magnetic field fluctuation associated with a purely electrostatic phenomenon.
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In the text

$\begin{figure} \par {\includegraphics[width=8cm,clip]{7988fg5.eps} } \end{figure}$ Figure 5: Spatial and temporal evolution of the azimuthal electric field, $\theta$ , of the electromagnetic wave shown in Fig. 4.
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In the text

$\begin{figure} \par {\includegraphics[width=8.2cm,clip]{7988fg6.eps} } \end{figure}$ Figure 6: Spatial structure of the axial magnetic field, $\beta$ , for time step 800 (dotted line) and 1800 (solid line), corresponding to slices along the $\xi$ -axis in Fig. 4, showing that the wave is clearly propagating.
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In the text

$\begin{figure} \par {\includegraphics[width=8.2cm,clip]{7988fg7.eps} } \end{figure}$ Figure 7: Spatial structure of the azimuthal electric field, $\theta$ , for time step 800 (dotted line) and 1800 (solid line),corresponding to slices along the $\xi$ -axis in Fig. 5. Note that the phase of this component, taken with Fig. 6, is consistent with an electromagnetic wave.
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In the text

$\begin{figure} \par {\includegraphics[width=8.3cm,clip]{7988fg8.eps} } \end{figure}$ Figure 8: Dynamical equilibrium solutions for $\kappa _{0}=4\times 10^{7}$ , $\kappa _{1}=10^{-2}$ and $\kappa _{2}=4\times 10^{11}$ . Plots show behaviour as function of position for the following quantities: top left, radial streaming speed $\sigma _0$ ; top right, total number density $\Sigma _0$ ; bottom left, differential azimuthal flow $\zeta _0$ ; bottom right, magnetic field. Solutions calculated using a bespoke Runge-Kutta routine with the parameters in Table 2. The equilibrium calculated here used to produce solutions in Figs. 9-16.
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In the text

$\begin{figure} \par {\includegraphics[width=7.7cm,clip]{7988fg9.eps} } \end{figure}$ Figure 9: Spatial and temporal evolution of $\Delta$ for one period of an electrostatic oscillation after an initial $70\%$ charge density perturbation. Note the nonlinear evolution of the density and the formation of density spikes in contrast to that exhibited in Fig. 2. Note that this plot is of the region of interest between spatial grid points 1150 and 1850.
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In the text

$\begin{figure} \par {\includegraphics[width=7.7cm,clip]{7988fg10.eps} } \end{figure}$ Figure 10: Spatial and temporal evolution of the radial electric field, $\rho$ , of the plasma oscillation consistent with Fig. 9. Note that this plot is of the region of interest between spatial grid points 1150 and 1850.
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In the text

$\begin{figure} \par {\includegraphics[width=7.7cm,clip]{7988fg11.eps} } \end{figure}$ Figure 11: Spatial and temporal evolution of the axial magnetic field, $\beta$ , showing clearly the propagation of an electromagnetic signal outwards from the electrostatic oscillation site. Note the hallmark features indicative of the nonlinear response of the plasma. This is more clearly exhibited in Fig. 16.
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In the text

$\begin{figure} \par {\includegraphics[width=7.7cm,clip]{7988fg12.eps} } \end{figure}$ Figure 12: Spatial and temporal evolution of the azimuthal electric field, $\theta$ , of the electromagnetic wave shown in Fig. 11. Note the hallmark features indicative of the nonlinear response of the plasma. This is more clearly exhibited in Fig. 15.
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In the text

$\begin{figure} \par {\includegraphics[width=8.1cm,clip]{7988fg13.eps} } \end{figure}$ Figure 13: Spatial structure of $\Delta$ for the time steps 20(solid line) and 1000 (dotted line). The nonlinear evolution of the plasma oscillation and the onset of the density instability is clearly shown.
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In the text

$\begin{figure} \par {\includegraphics[width=8.1cm,clip]{7988fg14.eps} } \end{figure}$ Figure 14: Spatial structure of $\Delta$ for the time steps 1620(solid line) and 2000 (dotted line). The nonlinear evolution of the plasma oscillation and the onset of the density instability is clearly shown. Note the gradual flow of the plasma oscillation downstream by the dynamical background plasma.
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In the text

$\begin{figure} \par {\includegraphics[width=8.2cm,clip]{7988fg15.eps} } \end{figure}$ Figure 15: Spatial structure of the azimuthal electric field, $\theta$ , for time step 1620 (solid line) and 2000 (dotted line). Note the nonlinear behaviour consistent with Fig. 14.
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In the text

$\begin{figure} \par {\includegraphics[width=8.2cm,clip]{7988fg16.eps} } \end{figure}$ Figure 16: Spatial structure of the axial magnetic field, $\beta$ , for time step 1620 (solid line) and 2000 (dotted line). Note the nonlinear behaviour consistent with Figs. 14 and 15.
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In the text

$\begin{figure} \par {\includegraphics[width=8.3cm,clip]{7988fg17.eps} } \end{figure}$ Figure 17: Variation of the azimuthal electric field $\theta$ of the radiated electromagnetic wave with electrostatic mode amplitude. $\vert\beta _{0}'\vert=3.5\times 10^{-7}$ (arbitrary units) at the centre of the oscillation site.
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In the text

$\begin{figure} \par {\includegraphics[width=8.1cm,clip]{7988fg18.eps} } \end{figure}$ Figure 18: Variation of the phase and amplitude of the electric field component, $\theta$ , of the EM wave with initial perturbation amplitude in the range 40% to 70% of background density.
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In the text

$\begin{figure} \par {\includegraphics[width=8.3cm,clip]{7988fg19.eps} } \end{figure}$ Figure 19: Variation of the phase and amplitude of the electric field component, $\theta$ , of the EM wave with initial perturbation amplitude in the range 5% to 30% of background density.
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In the text

$\begin{figure} \par {\includegraphics[width=8.4cm,clip]{7988fg20.eps} } \end{figure}$ Figure 20: Variation of the azimuthal electric field $\theta$ of the radiated electromagnetic wave with background magnetic field gradient, $\beta _{0}'$ , measured at the centre of the oscillation site, after an initial $10\%$ density perturbation.
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In the text

$\begin{figure} \par {\includegraphics[width=8.4cm,clip]{7988fg21.eps} } \end{figure}$ Figure 21: Plot of axial magnetic field $\beta$ showing injected EM wave (left-hand structure) propagating towards EM wave generated by electrostatic coupling.
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In the text

$\begin{figure} \par {\includegraphics[width=8.4cm,clip]{7988fg22.eps} } \end{figure}$ Figure 22: Nonlinear response of plasma to EM wave interaction, when the outward directed injected EM wave has passed beyond the edge of the ES oscillation site. Plot shows the difference between the cases where the two EM waves are independently propagating, and when they interact nonlinearly. The solid line is the case where the ratio of the EM wave wavelength to the width of the density perturbation is unity. The dotted and dashed lines correspond to a ratio of 2 and 3 respectively. Note the EM wave amplitude is constant for all ratios. The residual nonlinear response consists of three main features: the reflected feature (left-hand side); the oscillation feature (centre); and a transmitted feature, corresponding to the injected EM wave (right-hand side).
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In the text

$\begin{figure} \par {\includegraphics[width=8.3cm,clip]{7988fg23.eps} } \end{figure}$ Figure 23: Plot of the magnetic energy of the resultant reflected feature as a function of the wavelength of the injected wave normalised to the width of the ES oscillation site. Note that the resultant is calculated by subtracting the full nonlinear calculation from the linear superposition of the two non-interacting cases (that is, the electrostatic oscillation alone, and the injected electromagnetic wave on its own). Notice that for a given density perturbation, the residual reflected feature is a strong function of wavelength of the injected EM wave peaking close to where the wavelength matches the ES site width. However, this peak response drifts with ES amplitude towards longer wavelengths, reflecting the essential nonlinearity of the coupling.
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In the text

$\begin{figure} \par {\includegraphics[width=8.3cm,clip]{7988fg24.eps} } \end{figure}$ Figure 24: Same as for Fig. 23.
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In the text

$\begin{figure} \par {\includegraphics[width=8.3cm,clip]{7988fg25.eps} } \end{figure}$ Figure 25: Plot of the amplitude of the central oscillating feature as a function of the wavelength of the injected EM wave normalised to the width of the ES oscillation site.
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In the text

$\begin{figure} \par {\includegraphics[width=8.4cm,clip]{7988fg26.eps} } \end{figure}$ Figure 26: Plot of the magnetic energy of the residual reflected feature as a function of the injected EM wave amplitude normalised to the to the value of the background magnetic gradient at the centre of the ES oscillation site.
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In the text

$\begin{figure} \par {\includegraphics[width=8.4cm,clip]{7988fg27.eps} } \end{figure}$ Figure 27: Plot of the amplitude of the central oscillating feature as a function of the amplitude of the injected EM wave normalised to the to the value of the background magnetic gradient at the centre of the ES oscillation site.
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In the text

$\begin{figure} \par {\includegraphics[width=8cm,clip]{7988fg2.eps} } \end{figure}$	Figure 2: Spatial and temporal evolution of $\Delta$ , showing the plasma density evolution as a function of time $\tau$ and space $\xi$ associated with an electrostatic oscillation. This is the response after a $1\%$ initial charge density perturbation.
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$\begin{figure} \par {\includegraphics[width=8cm,clip]{7988fg3.eps} } \end{figure}$	Figure 3: Spatial and temporal evolution of the radial electric field, $\rho$ , of the plasma oscillation, consistent with Fig. 2. The nonlinear evolution of the field is visible at the edges of the oscillation site.
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$\begin{figure} \par {\includegraphics[width=8cm,clip]{7988fg4.eps} } \end{figure}$	Figure 4: Spatial and temporal evolution of the axial magnetic field, $\beta$ , showing clearly the propagation of an electromagnetic signal outwards from the electrostatic oscillation site. Note that there is no magnetic field fluctuation associated with a purely electrostatic phenomenon.
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$\begin{figure} \par {\includegraphics[width=8cm,clip]{7988fg5.eps} } \end{figure}$	Figure 5: Spatial and temporal evolution of the azimuthal electric field, $\theta$ , of the electromagnetic wave shown in Fig. 4.
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$\begin{figure} \par {\includegraphics[width=8.2cm,clip]{7988fg6.eps} } \end{figure}$	Figure 6: Spatial structure of the axial magnetic field, $\beta$ , for time step 800 (dotted line) and 1800 (solid line), corresponding to slices along the $\xi$ -axis in Fig. 4, showing that the wave is clearly propagating.
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$\begin{figure} \par {\includegraphics[width=8.2cm,clip]{7988fg7.eps} } \end{figure}$	Figure 7: Spatial structure of the azimuthal electric field, $\theta$ , for time step 800 (dotted line) and 1800 (solid line),corresponding to slices along the $\xi$ -axis in Fig. 5. Note that the phase of this component, taken with Fig. 6, is consistent with an electromagnetic wave.
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$\begin{figure} \par {\includegraphics[width=7.7cm,clip]{7988fg9.eps} } \end{figure}$	Figure 9: Spatial and temporal evolution of $\Delta$ for one period of an electrostatic oscillation after an initial $70\%$ charge density perturbation. Note the nonlinear evolution of the density and the formation of density spikes in contrast to that exhibited in Fig. 2. Note that this plot is of the region of interest between spatial grid points 1150 and 1850.
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$\begin{figure} \par {\includegraphics[width=7.7cm,clip]{7988fg10.eps} } \end{figure}$	Figure 10: Spatial and temporal evolution of the radial electric field, $\rho$ , of the plasma oscillation consistent with Fig. 9. Note that this plot is of the region of interest between spatial grid points 1150 and 1850.
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$\begin{figure} \par {\includegraphics[width=7.7cm,clip]{7988fg11.eps} } \end{figure}$	Figure 11: Spatial and temporal evolution of the axial magnetic field, $\beta$ , showing clearly the propagation of an electromagnetic signal outwards from the electrostatic oscillation site. Note the hallmark features indicative of the nonlinear response of the plasma. This is more clearly exhibited in Fig. 16.
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$\begin{figure} \par {\includegraphics[width=7.7cm,clip]{7988fg12.eps} } \end{figure}$	Figure 12: Spatial and temporal evolution of the azimuthal electric field, $\theta$ , of the electromagnetic wave shown in Fig. 11. Note the hallmark features indicative of the nonlinear response of the plasma. This is more clearly exhibited in Fig. 15.
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$\begin{figure} \par {\includegraphics[width=8.1cm,clip]{7988fg13.eps} } \end{figure}$	Figure 13: Spatial structure of $\Delta$ for the time steps 20(solid line) and 1000 (dotted line). The nonlinear evolution of the plasma oscillation and the onset of the density instability is clearly shown.
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$\begin{figure} \par {\includegraphics[width=8.1cm,clip]{7988fg14.eps} } \end{figure}$	Figure 14: Spatial structure of $\Delta$ for the time steps 1620(solid line) and 2000 (dotted line). The nonlinear evolution of the plasma oscillation and the onset of the density instability is clearly shown. Note the gradual flow of the plasma oscillation downstream by the dynamical background plasma.
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$\begin{figure} \par {\includegraphics[width=8.2cm,clip]{7988fg15.eps} } \end{figure}$	Figure 15: Spatial structure of the azimuthal electric field, $\theta$ , for time step 1620 (solid line) and 2000 (dotted line). Note the nonlinear behaviour consistent with Fig. 14.
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$\begin{figure} \par {\includegraphics[width=8.2cm,clip]{7988fg16.eps} } \end{figure}$	Figure 16: Spatial structure of the axial magnetic field, $\beta$ , for time step 1620 (solid line) and 2000 (dotted line). Note the nonlinear behaviour consistent with Figs. 14 and 15.
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$\begin{figure} \par {\includegraphics[width=8.3cm,clip]{7988fg17.eps} } \end{figure}$	Figure 17: Variation of the azimuthal electric field $\theta$ of the radiated electromagnetic wave with electrostatic mode amplitude. $\vert\beta _{0}'\vert=3.5\times 10^{-7}$ (arbitrary units) at the centre of the oscillation site.
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$\begin{figure} \par {\includegraphics[width=8.1cm,clip]{7988fg18.eps} } \end{figure}$	Figure 18: Variation of the phase and amplitude of the electric field component, $\theta$ , of the EM wave with initial perturbation amplitude in the range 40% to 70% of background density.
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$\begin{figure} \par {\includegraphics[width=8.3cm,clip]{7988fg19.eps} } \end{figure}$	Figure 19: Variation of the phase and amplitude of the electric field component, $\theta$ , of the EM wave with initial perturbation amplitude in the range 5% to 30% of background density.
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$\begin{figure} \par {\includegraphics[width=8.4cm,clip]{7988fg20.eps} } \end{figure}$	Figure 20: Variation of the azimuthal electric field $\theta$ of the radiated electromagnetic wave with background magnetic field gradient, $\beta _{0}'$ , measured at the centre of the oscillation site, after an initial $10\%$ density perturbation.
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$\begin{figure} \par {\includegraphics[width=8.4cm,clip]{7988fg21.eps} } \end{figure}$	Figure 21: Plot of axial magnetic field $\beta$ showing injected EM wave (left-hand structure) propagating towards EM wave generated by electrostatic coupling.
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$\begin{figure} \par {\includegraphics[width=8.3cm,clip]{7988fg24.eps} } \end{figure}$	Figure 24: Same as for Fig. 23.
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$\begin{figure} \par {\includegraphics[width=8.3cm,clip]{7988fg25.eps} } \end{figure}$	Figure 25: Plot of the amplitude of the central oscillating feature as a function of the wavelength of the injected EM wave normalised to the width of the ES oscillation site.
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$\begin{figure} \par {\includegraphics[width=8.4cm,clip]{7988fg26.eps} } \end{figure}$	Figure 26: Plot of the magnetic energy of the residual reflected feature as a function of the injected EM wave amplitude normalised to the to the value of the background magnetic gradient at the centre of the ES oscillation site.
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$\begin{figure} \par {\includegraphics[width=8.4cm,clip]{7988fg27.eps} } \end{figure}$	Figure 27: Plot of the amplitude of the central oscillating feature as a function of the amplitude of the injected EM wave normalised to the to the value of the background magnetic gradient at the centre of the ES oscillation site.
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