2.1. General dispersion relation

The generalised dispersion relation of the plasma instabilities in counterpropagating beams in plasma with a general perturbation wavevector is derived in Califano, Pegoraro & Bulanov (Reference Califano, Pegoraro and Bulanov1997). The transverse mode of the perturbation represents the Weibel instability. However, it is noted that in highly relativistic beams and asymmetric beams the fastest-growing mode may not be the Weibel instability but the scope of this work is determining the growth rates of transverse magnetic fields, potentially leading to Fermi acceleration. Consequently, the reduced transverse dispersion relation for the Weibel instability for an electron–ion plasma is examined following the work by Pegoraro et al. (Reference Pegoraro, Bulanov, Califano and Lontano1996) which is easier to deal with analytically.

An initially uniform plasma contains two counterpropagating electron beams with an immobile neutralising ion background. The electron beams propagate along the $x$-axis of the system and have initial unperturbed number densities defined as $n_{0\alpha }$ with $\alpha = 1, 2$ denoting the electron species. The ion background has density $n_i = \sum _\alpha n_{0\alpha }$ which neutralises the global electron charge of the system. The global net current density is zero. The initial unperturbed electron beam velocities are defined as $v_{0x\alpha }$ with their three-momenta given by $\boldsymbol {v_\alpha } = c \boldsymbol {p_\alpha }/(m^2c^2 + p^2_\alpha )^{1/2}$. All variables are assumed to depend on $y$ and time. From this, the relativistic dynamics of the electrons are described by the following equations with $\boldsymbol {A}$ the vector potential field defined by the magnetic field $\boldsymbol {B} = \boldsymbol {\nabla } \times \boldsymbol {A}$:

(2.1)\begin{gather} p_{x \alpha} - \frac{e}{c}A_x = p_{0x \alpha}, \end{gather}

(2.2)\begin{gather}\frac{\partial p_{y \alpha}}{\partial t} + v_{y \alpha}\frac{\partial p_{y \alpha}}{\partial y} ={-}e \left(E_y - \frac{v_{x \alpha}}{c} B_z \right), \end{gather}

(2.3)\begin{gather}\frac{\partial n_\alpha}{\partial t} + \frac{\partial (n_\alpha v_{y\alpha})}{\partial y} = 0. \end{gather}

The electron momenta and densities are coupled with the following Maxwell's equations (Griffiths Reference Griffiths2013):

(2.4)\begin{gather} \frac{\partial E_y}{\partial y} = 4{\rm \pi} e\left(n_i - \sum_\alpha n_\alpha\right), \end{gather}

(2.5)\begin{gather}\frac{\partial B_z}{\partial y} ={-}\frac{4{\rm \pi} e}{c}\sum_\alpha n_\alpha v_{x\alpha} + \frac{1}{c}\frac{\partial E_x}{\partial t}, \end{gather}

(2.6)\begin{gather}\frac{1}{c}\frac{\partial B_z}{\partial t} = \frac{E_x}{\partial y}, \end{gather}

(2.7)\begin{gather}-\frac{\partial^2}{\partial y^2}E_x ={-}\frac{1}{c}\frac{\partial}{\partial t} \left(\frac{\partial B_z}{\partial y} \right). \end{gather}

Maxwell's equations and relativistic dynamics equations are linearised using a plane-wave approximation of the form $\sim \exp (-{\rm i}\omega t + {\rm i}ky)$ for the velocities, densities and fields with $\omega$ being the perturbation angular frequency and $k$ the perturbation wavevector. The applied transverse electric perturbation field is given by $\boldsymbol {E_{1}} = E_0 \exp (-{\rm i}\omega t + {\rm i}ky) \hat {\boldsymbol {x}}$. In this linearisation process, the following operations become ${\partial }/{\partial t} \rightarrow -{\rm i}\omega$ and $\boldsymbol {\nabla } \rightarrow {\rm i}k \hat {\boldsymbol {y}}$ for ease. It is found that the first-order components for these variables give the following:

(2.8)\begin{gather} v_{1y\alpha} = \frac{e}{{\rm i}\omega m\varGamma_\alpha}\left(E_y + \frac{k}{\omega}v_{0x\alpha}E_x \right), \end{gather}

(2.9)\begin{gather}v_{1x\alpha} = \frac{e}{{\rm i}\omega m\varGamma^3_\alpha}E_x , \end{gather}

(2.10)\begin{gather}n_{1\alpha} = \frac{e n_{0\alpha}k}{{\rm i}\omega^2 m\varGamma_\alpha} \left( E_y + \frac{k}{\omega}v_{0x\alpha}E_x\right), \end{gather}

(2.11)\begin{gather}E_y = 4 {\rm \pi}e^2 \frac{\sum_\alpha \dfrac{n_{0\alpha}v_{0x\alpha}}{m\varGamma_\alpha}\dfrac{k}{\omega}}{\left(\omega^2 - 4{\rm \pi} e^2\sum \dfrac{n_{0\alpha}}{m\varGamma_\alpha}\right)}E_x . \end{gather}

When the above definitions of the first-order quantities are introduced into (2.7) with the substitution of (2.5), a sixth-order dispersion relation is found for mode frequency $\omega$ and perturbation wavevector $k$:

(2.12)\begin{equation} (\omega^2 - \varOmega^{2}_a)[\omega^4 - \omega^2(k^2c^2 + \varOmega^2_b) - k^2c^2\varOmega^2_c] - k^2c^2\varOmega^4_d = 0, \end{equation}

where

(2.13)\begin{equation} \left. \begin{aligned} \varOmega^{2}_a & = \omega^2_{{\rm pe}} \sum_\alpha \frac{n_{0\alpha}}{n_i \varGamma_\alpha}, \quad \varOmega^{2}_b = \omega^2_{{\rm pe}} \sum_\alpha \frac{n_{0\alpha}}{n_i \varGamma^3_\alpha},\\ \varOmega^{2}_c & = \omega^2_{{\rm pe}} \sum_\alpha \frac{n_{0\alpha}v^2_{0x\alpha}}{n_i \varGamma_\alpha c^2}, \quad \varOmega^{2}_d = \omega^2_{{\rm pe}} \sum_\alpha \frac{n_{0\alpha}v_{0\alpha}}{n_i \varGamma_\alpha c}, \end{aligned} \right\} \end{equation}

with the non-relativistic plasma frequency $\omega _{\textrm {pe}} \equiv (4{\rm \pi} e^2 n_i/m)^{1/2}$ and the Lorentz factor $\varGamma _\alpha = (1 - v_{0x\alpha }^2/c^2)^{-1/2}$. The dispersion relation in (2.12) can be solved by a substitution of variables, $u = \omega ^2$, reducing the sextic dispersion relation to a cubic equation. This cubic equation can be solved numerically with numpy.roots function in Python. This function involves computing the eigenvalues of the companion matrix (Johnson & Horn Reference Johnson and Horn1985).

Three branches can be observed from the solutions to the dispersion relation, two oscillatory modes $(\mathrm {Im} (\omega ) = 0)$ and one exponentially growing mode with a non-vanishing imaginary component $(\mathrm {Im} (\omega ) \neq 0)$. The exponentially growing mode is responsible for the Weibel instability. By defining $\omega = \textrm {i}\gamma$, the growth rate $\gamma$ can be calculated from the exponentially growing mode as a function of the perturbation wavevector $k$. Curves for both non-relativistic and relativistic cases of two counterstreaming electron beams with a neutralising ion background are shown in figure 1, showing the growth rate of the Weibel instability as a function of $k^2$.

Figure 1. Instability growth rate for both a non-relativistic and relativistic case. The two beams have equal number density, i.e. $n_{0,1} = n_{0,2} = n_i/2$. In general, all dispersion relations follow the same pattern where the growth rate reaches a maximum at short perturbation wavelengths.

The maximum growth rate is achieved in the short-wavelength limit where $k^2c^2 \gg \varOmega ^2_a, \varOmega ^2_b, \varOmega ^2_c, \varOmega ^2_d$. When this limit is applied to (2.12), equation (2.14) is returned, which shows no dependence on the wavevector $k$. In the long-wavelength limit where $k^2c^2 \sim 0$, the growth rate follows a linear dependence on the perturbation wavevector $k$ given by (2.15). These limits follow from Pegoraro et al. (Reference Pegoraro, Bulanov, Califano and Lontano1996). The limits are dependent on the $\varOmega ^2_i$ terms found in the dispersion relation (2.12), which are determined by the density and beam velocity parameters of the system

(2.14)\begin{gather} \gamma \approx \{[(\varOmega^2_a + \varOmega^2_c)^2 - 4\varOmega^4_d]^{1/2} - (\varOmega^2_a - \varOmega^2_c)\}^{1/2}/\sqrt{2}, \end{gather}

(2.15)\begin{gather} \gamma \approx kc[(\varOmega^2_a \varOmega^2_c - \varOmega^4_d)/(\varOmega^2_a \varOmega^2_b)]^{1/2}. \end{gather}

The collisional effects between plasma species can be ignored if the electron collision rate is much smaller than the Weibel instability growth rate. The instability growth rate, $\gamma$, can be determined in units of the plasma frequency, giving $\gamma = \gamma _{g} [\omega _{\textrm {pe}}]$, where $\gamma _g$ is the magnitude of the instability growth rate. Using the formulae for the electron collision rate and the electron plasma frequency from the NRL formulary (Huba Reference Huba2013), a condition for the collisional effects to be ignored can be determined,

(2.16)\begin{equation} \frac{n_e}{T^3_e} \ll \left( \frac{\gamma_g}{\ln{\varLambda}} \right)^2 9.52 \times 10^{18}, \end{equation}

where $\ln {\varLambda }$ is the Coulomb logarithm, $n_e$ is the electron number density in units of cm$^{-3}$ and $T_e$ is the electron temperature in units of eV. For example, if $\ln {\varLambda } = 10$ and electron density of $n_e = 10^{18} \, \textrm {cm}^{-3}$ are assumed whereas the instability growth rate is found to be $\gamma = 10^{-1} \ \omega _{\textrm {pe}}$, then the electron temperature must be $T_e \gg 10.2 \, \textrm {eV}$ for collisional effects to be ignored. Consequently, collisions are neglected in both the derived theoretical models and PIC simulations. If a low-density plasma is considered, then this model assumption is valid.

2.2. Three-electron-populations case

Extending from the work in § 2.1, the case of three electron populations with an immobile proton background is examined and the dispersion relation determined. This case involves two counterstreaming beams of electrons as previously and an additional third cold electron population with zero drift velocity with respect to the immobile proton background. Similarly, the number densities of the three populations must equate the number density of the proton background and net current density in the system is zero. What is found is a dispersion relation with the same functional form as the two-beam case (2.12) with an adjustment to the $\varOmega ^2_a$ and $\varOmega ^2_b$ terms

(2.17)\begin{equation} \left. \begin{aligned} \varOmega^{2}_a & = \omega^2_{{\rm pe}} \left[ \sum_{\alpha = 1,2} \left( \frac{n_{0\alpha}}{n_i \varGamma_\alpha} \right) + \frac{n_{0, 3}}{n_i} \right],\\ \varOmega^{2}_b & = \omega^2_{{\rm pe}} \left[ \sum_{\alpha = 1,2} \left(\frac{n_{0\alpha}}{n_i \varGamma^3_\alpha} \right) + \frac{n_{0, 3}}{n_i} \right]. \end{aligned} \right\} \end{equation}

The population with $\alpha = 3$ is the cold background electron population and has been shown explicitly for clarity. These terms therefore depend on the number density of the cold electron background. Here $\varOmega ^2_c$ and $\varOmega ^2_d$ remain unaffected by the background electron population as the drift velocity is $v_{03x} = 0$ by definition.

The largest difference between the two-beam case and the three-populations case occurs in the relativistic regime. In the two-beam case, $\varOmega ^2_a$ and $\varOmega ^2_b$ tend to zero as the Lorentz factors $\varGamma _\alpha$ increase due to electron inertia. For the three-population case, $\varOmega ^2_a$ and $\varOmega ^2_b$ follow $\varOmega ^{2}_a \approx \varOmega ^{2}_b \rightarrow \omega ^2_{\textrm {pe}} ({n_{0, 3}}/{n_i})$ in the relativistic limit, which is significant if the stationary density is a large fraction of the total electron density.

The perturbation wavelength limits formulae (2.14)–(2.15) were applied to the three-electron-population case by substituting the new $\varOmega ^2_a$ and $\varOmega ^2_b$ terms, this is shown in figure 2(a). A large density ratio of $n_{\textrm {back}}/2n_{\textrm {beam}} = 10$ is chosen because there is significant change to the original two-electron-beam model used to derive the original limits in Pegoraro et al. (Reference Pegoraro, Bulanov, Califano and Lontano1996). Both the short- and long-wavelength limits still agree with the new dispersion relation for the three-electron-populations case.

Figure 2. Growth rate as a function of the perturbation wavenumber for (a) the three-electron-populations case, and (b) the lepton-beam case and the plasma-flow case, where the short- and long-wavelength limits are shown by the red and green dashed lines, respectively. The three-electron-populations case has a density ratio of $n_{\textrm {back}}/2n_{\textrm {beam}} = 10$ and beam Lorentz factors $\varGamma _{\alpha } = 3.57$. The lepton-beam and plasma-flow cases have the parameters $n_j/n_p = 0.5$ and $\varGamma _j = 100$. The short-wavelength limit of the lepton-beam case is a factor of $\sqrt {2}$ larger than the same limit in the plasma-flow case.

There are competing modes for the Weibel instability in the three-electron-population case due to beam–beam and beam–background interactions. This is determined by comparing the dispersion relations of the two-beam, one-beam-on-background and the three-population cases. At high background densities of $2n_{\textrm {back}}/n_{\textrm {beam}} \geq 1$, the beam–background interaction dominates leading to the three-electron-population case to follow the one-beam-on-background case for all $k$ values. Meanwhile, at low background densities of $2n_{\textrm {back}}/n_{\textrm {beam}} < 0.01$, the beam–beam interaction dominates leading to the three-electron-population case following the two-beam case for all $k$ values. In the density regime between those endpoints, the beam–background interaction dominates for $k < 1$ whereas the beam–beam interaction dominates for $k > 1$.

In the cold-fluid formalism, the short-wavelength limit, where $k \to \infty$, gives a saturated maximum instability growth rate (Bret et al. Reference Bret, Gremillet and Dieckmann2010). The short-wavelength limit can be analysed analytically to retrieve the scaling with beam velocity and density parameters when working with equal density and speed beams for the three-electron-populations case. For this case, $\varOmega ^2_d = 0$ and (2.14) reduces to

(2.18)\begin{equation} \frac{\gamma}{\omega_{{\rm pe}}} \approx \beta \sqrt{\frac{1}{\varGamma_b (R + 1)} } , \end{equation}

where $\gamma$ is the maximum growth rate, $\varGamma _b$ is the beam Lorentz factor, $\beta = v/c$ and $R = n_{\textrm {back}}/2n_{\textrm {beam}}$. Equation (2.18) can be applied to the low- and high-beam-density regimes to give

(2.19)\begin{equation} \frac{\gamma}{\omega_{{\rm pe}}} \approx \left\{ \begin{array}{@{}ll} \beta \sqrt{\dfrac{1}{R \varGamma_b}}, & \text{for }R \gg 1, \\ \beta \sqrt{\dfrac{1}{\varGamma_b}}, & \text{for }R \ll 1, \end{array} \right. \end{equation}

which are comparable to the scalings found in table I in Bret et al. (Reference Bret, Gremillet and Dieckmann2010) for electron beams incident on electron–ion plasmas. Consequently, in the three-electron-populations case the maximum growth rate is dependent on the beam-to-background-density ratio and the beam Lorentz factor in the low-beam-density regime. Meanwhile, in the high-beam-density regime the maximum growth rate decouples from the density ratio and is wholly dependent on the beam Lorentz factor.

2.3. Lepton beam and plasma beam cases

The lepton beam into a background plasma case is also examined with a similar linearisation process. The electron–positron beam has both net zero charge density and zero current density, i.e. $n_{ e^-,0j} = n_{e^+,0j} = {n_{0j}}/{2}$ and $v_{e^-,0j} = v_{e^+,0j} = v_{0xj}$ and is travelling in the positive $x$ direction. The background electron–proton plasma has zero net charge $(n_e = n_p)$ and zero electron drift velocity with the protons being frozen. There is no restriction between the density of the lepton beam and the density of the background electron–proton plasma as individually they satisfy the neutrality requirements. When the linearisation process is applied to Maxwell's equations and the relativistic dynamics equations, a comparable dispersion relation to (2.12) is found albeit with a change in the $\varOmega ^2_i$ terms which are as follows

(2.20)\begin{equation} \left. \begin{aligned} \varOmega^{2}_a & = \omega^2_{{\rm pe}} \left( \frac{n_{0j}}{n_p \varGamma_j} + 1 \right), \quad \varOmega^{2}_b = \omega^2_{{\rm pe}} \left( \frac{n_{0j}}{n_p \varGamma^3_j} + 1 \right),\\ \varOmega^{2}_c & = \omega^2_{{\rm pe}} \frac{n_{0j}v^2_{0xj}}{n_p \varGamma_j c^2}, \quad \varOmega^{2}_d = \omega^2_{{\rm pe}} \frac{n_{0j}v_{0xj}}{n_p \varGamma_j c}. \end{aligned} \right\} \end{equation}

The terms are normalised to the background electron–proton plasma frequency, in this case $\omega _{\textrm {pe}}$. Similarly to the three-electron-populations case, the $\varOmega ^2_a$ and $\varOmega ^2_b$ terms have a component that is unchanged for differing beam velocities due to the electron density in the background plasma. In the ultrarelativistic limit, $\varOmega ^2_a \approx \varOmega ^2_b \rightarrow \omega ^2_{\textrm {pe}}$, which means the electron inertia becomes wholly dependent on the electron–proton plasma density.

The perturbation wavelength limits formulae (2.14)–(2.15) were applied to the lepton-beam case by substituting the new $\varOmega ^2_i$ terms, which is shown in figure 2(b). A density ratio of $n_j/n_p = 0.5$ is chosen so that the beam density is a significant fraction of the background plasma density, hence the Weibel instability growth rate is appreciable. A beam Lorentz factor of $\varGamma _j = 100$ is chosen such that model is in the relativistic beam regime. There is great agreement between the short- and long-wavelength limits with the dispersion relation for the lepton-beam case.

For a plasma beam incident on a background electron–proton plasma, the derivation of the dispersion relation follows from the lepton-beam case with the change of positrons to protons in the beam. This means the positive charges within the beam are now approximately 1836 times heavier than the electrons although their initial velocities are equal to maintain current density neutrality. The protons in the flow are not frozen. The $\varOmega ^2_i$ terms derived from the dispersion relation are as follows:

(2.21)\begin{equation} \left. \begin{aligned} \varOmega^{2}_a & = \omega^2_{e} \left( \frac{n_{0j}}{2 n_p \varGamma_j} + 1 \right) + \omega^2_p \frac{n_{0j}}{2 n_p \varGamma_j}, \quad \varOmega^{2}_b = \omega^2_{e} \left( \frac{n_{0j}}{2 n_p \varGamma^3_j} + 1 \right) + \omega^2_p \frac{n_{0j}}{2 n_p \varGamma^3_j},\\ \varOmega^{2}_c & = \frac{n_{0j}v^2_{0xj}}{2 n_p \varGamma_j c^2}(\omega^2_e + \omega^2_p), \quad \varOmega^{2}_d = \frac{n_{0j}v_{0xj}}{2 n_p \varGamma_j c}(\omega^2_e + \omega^2_p), \end{aligned} \right\} \end{equation}

with non-relativistic electron and proton plasma frequencies given by $\omega ^2_e = ({4 {\rm \pi}e^2 n_p})/{m_e}$ and $\omega ^2_p = ({4 {\rm \pi}e^2 n_p})/{m_p}$, respectively. With respect to the lepton beam $\varOmega ^2_i$ terms, the plasma-flow terms show a factor of 1/2 wherever the density ratio $n_{0j}/n_p$ is found and there is a small addition to all terms due to the perturbation of the protons in the beam $(\omega ^2_e/\omega ^2_p \approx 1836)$. The overall behaviour of the dispersion is comparable to the lepton-beam case with the difference in the short-wavelength limit growth rate shown in figure 2(b). The lepton-beam case shows a factor $\sqrt {2}$ increase in the short-wavelength limit growth rate compared with the plasma beam case in the relativistic or low $n_{0j}/n_p$ density ratio regimes with the same beam Lorentz factors. This is due to the maximum instability growth rate being a function of the density ratio $n_{0j}/n_p$ as seen in table I of Bret et al. (Reference Bret, Gremillet and Dieckmann2010) in the low-beam-density regime. Therefore, the inherent doubling of the density ratio when comparing the lepton-beam case to the plasma-flow case leads to the $\sqrt {2}$ factor in maximum growth rate.

3.1. PIC simulations and growth rates

The PIC simulations are performed using the open-source code Smilei (Derouillat et al. Reference Derouillat, Beck, Pérez, Vinci, Chiaramello, Grassi, Flé, Bouchard, Plotnikov and Aunai2018). The simulation box is set with $20 \lambda _k$ in both the $x$ and $y$ directions where $\lambda _k$ is the wavelength of the applied electric perturbation field given by $\lambda _k = 2{\rm \pi} /k$. The perturbation wavevector $k$ is normalised to the reference frequency $\omega _r/c$ where in the counterstreaming-beam cases, $\omega _r$ is given by the plasma frequency of all electron populations $\omega ^2_{\textrm {pe}} = 4{\rm \pi} n_{e,total} e^2/m_e$. In the beam cases, $\omega _r$ is given by the background electron plasma frequency $\omega ^2_{\textrm {pe}} = 4{\rm \pi} n_{p} e^2/m_e$. The grid spacing $d$ for both directions is $\lambda _{k}/8$ for most of the simulations and the timestep is half the grid spacing, $\Delta t = 0.5*d/c$. The grid spacing $d = \lambda _{k}/16$ is sometimes used to determine the growth rate at low values of $k$ where the timestep becomes too large to resolve the linear regime. The number of particles per cell for both electrons and protons is 256. The boundary conditions for both fields and particles are periodic. Perturbations are applied via a transverse electric field $E_y$ of sinusoidal form with wavelength $\lambda _k$ and an amplitude of $0.001 \, m_e c \omega _{\textrm {pe}} /e$ in plasma units. With the parameters defined as such, the timestep and grid spacing are different for each perturbation wavevector $k$ and this is taken into account when the diagnostics are analysed.

The computational Weibel instability growth rates are determined from the electromagnetic energy density diagnostics of the PIC simulations. In CGS units, the electromagnetic energy density is $u = ({1}/{8 {\rm \pi}})(E^2 + B^2)$, using the $B_z$ component for the Weibel instability and the introduction of the plane wave perturbation, the following equation can be found

(3.1)\begin{equation} \log(u) = 2\gamma t + \log \left(\frac{B_0}{8{\rm \pi}} \right), \end{equation}

where $u$ is the electromagnetic energy density given by $B_z$, $\gamma$ is the Weibel instability growth rate, $t$ is time and $B_0$ is the amplitude of the plane wave perturbation derived from Faraday's law (2.6) on the applied perturbative electric field. The gradient of a linear fit to a log plot of $u$ gives $m = 2\gamma$, which is used to determine the growth rates of the Weibel instability from the PIC simulations at different values of $k$. These PIC determined growth rates are compared with the growth rates determined from the derived dispersion relations to check the accuracy of the models.

3.2. PIC simulations for the three-electron-populations case

In these simulations, the two counterstreaming electron beams are initialised with equal density and magnitude in momentum for the sake of simplicity. In the first set of simulations, the ratio of the counterstreaming beam densities and the background electron population is fixed with $n_{\textrm {back}}/2n_{\textrm {beam}} = 0.5$. The Lorentz factor for the counterstreaming beams is varied to see its effect on the Weibel instability growth rate. The comparisons between the linear dispersion theory and PIC code determined growth rates are shown in figure 3 along with the maximum growth rate scaling from (2.18). It should be noted that in the non-relativistic regime, the electrostatic field also grows via an instability. The growth rates for both the transverse Weibel instability and the longitudinal electric field instability were found to be similar thereby the majority electromagnetic energy comes from both the transverse magnetic and longitudinal electrostatic fields.

Figure 3. Instability growth rates for the three-electron-populations case with differing Lorentz factors. The ratio between the added beam densities and the background density is $n_{\textrm {back}}/2n_{\textrm {beam}} = 0.5$. The beam Lorentz factors are determined from the rest frame of the immobile protons and the crosses are the growth rates in the PIC simulations. The dotted lines give the short-wavelength limit from (2.18).

Focussing on the Weibel instability growth rate, it shows a strong linear dependence with $k$ in the long-wavelength limit, which contrasts with the original two population case in the relativistic regime seen in figure 1. The introduction of the background electron plasma with a comparable density to the beams, which affects the $\varOmega ^2_a$ and $\varOmega ^2_b$ terms, causes the strong linear dependence to remain in this limit. This is due to competing instabilities for the beam–beam interaction and the beam–background interaction. It is found qualitatively that the beam–beam Weibel instability dominates at high $k$ values and low background plasma densities, whereas the beam–background Weibel instability dominates at low $k$ values and high background plasma densities. For $n_{\textrm {back}}/2n_{\textrm {beam}} = 0.5$ found in figure 3, the beam–background interaction dominates, which produces the strong linear dependence for lower values of $k$. Overall the dispersion relation for this case is slightly overestimating the growth rates determined from the PIC simulations.

The ratio between the counterstreaming beam density and the cold electron background density is changed to see how this would affect the growth rate of the Weibel instability while maintaining the beam Lorentz factor constant. The results from the PIC simulations and comparison with the linear dispersion theory are shown in figure 4 for a non-relativistic and relativistic case. When the background electron population density is small, the functional dependence between the growth rate and $k$ shows similar behaviour in the long-wavelength limit as it approximates the relativistic two-counterstreaming-beams case seen in figure 1. This behaviour vanishes when the density of the background electron population is comparable to the beam densities. Again, the dispersion relation is slightly overestimating the PIC determined growth rates.

Figure 4. Instability growth rates for (a) a non-relativistic case of $| v_{0x}|= 0.1c$ and (b) a relativistic case of $| v_{0x} | = 0.96c$ with density variations defined by the ratio $n_{\textrm {back}}/2n_{\textrm {beam}}$. For the relativistic case, as the density ratio is increased, the growth rate of the instability decreases. In the non-relativistic case, no difference is observed for the density ratios used in the PIC simulations.

An example of the transverse magnetic field $B_z$ progression for the three-electron- populations case is shown in figure 5. At the earlier stages of the PIC simulation, filamentation occurs which causes the bands seen in the first figure. As the magnetic field saturates, the turbulent phase begins causing the filaments to break up. This is more evident in the lepton-beam case.

Figure 5. Snapshots of $B_z$ magnetic field from PIC simulations for the three-population-case at (a) $t= 235.6 \, \omega ^{-1}_{\textrm {pe}}$ and (b) $t = 4712.4 \, \omega ^{-1}_{\textrm {pe}}$. The beam Lorentz factor is $\varGamma = 2$, wavevector $k = 0.2$ and density ratio $n_{\textrm {back}}/2n_{\textrm {beam}} = 0.5$. All quantities are in plasma units.

3.3. PIC simulations for lepton and plasma beams into background electron–proton plasma

The effect of the Lorentz factor on the Weibel instability growth rate is explored by fixing a beam-to-background density ratio and varying the Lorentz factor. The comparison between the PIC growth rates and the dispersion relations are shown in figure 6(a). The functional dependence between the growth rate and $k$ in the long-wavelength limit follows from the non-relativistic two-counterstreaming-beams case even with high beam Lorentz factors. This shows that the behaviour in the relativistic two-counterstreaming-beams case in the long-wavelength limit is unique amongst all the models explored here. The short-wavelength limit shows inverse proportionality to the square root of the beam Lorentz factor as expected from a cold fluid approximation.

Figure 6. Instability growth rates compared with PIC simulations for the lepton-beam case and plasma-flow case. (a) Relation between the growth rates and the beam Lorentz factor. The ratio between the beam density and the background density is fixed at $n_{j}/n_{p} = 0.5$. The maximum instability growth rate is inversely proportional to $\sqrt {\varGamma _j}$. (b) Relation between the growth rates and the density ratio. The beam Lorentz factor is fixed at $\varGamma _j = 35.36$. The crosses show the PIC determined growth rates for the lepton-beam case and the circles represent the PIC determined growth rates for the plasma beam case.

Conversely, the beam Lorentz factor is fixed $(\varGamma _j = 35.36)$ and the density ratios are varied for both cases. The growth rates from the PIC simulations and the linear dispersion relation are compared in figure 6(b). Regardless of the density ratio between the beam and the background plasma, the growth rate shows the linear dependence in the long-wavelength limit and the maximum growth rate in the short-wavelength limit. Furthermore, the maximum growth rate is proportional to $\sqrt {n_j/n_p}$ The dispersion relation shows good agreement with the PIC determined growth rates.

From the above, the linear dispersion relation is accurate in determining growth rates for a lepton beam inbound on a background electron–proton plasma and a plasma beam on a background electron–proton plasma. Furthermore, the lepton-beam case shows greater Weibel instability growth rates compared with the plasma beam case by a factor of $\sqrt {2}$ in most cases as can be seen in figure 2(b). This is explained by the doubling of the beam-to-background-density ratio between the lepton-beam and electron–proton-beam cases seen in (2.20) and (2.21). The numerical methods used to determine the dispersion relation can be extended to predict Weibel instability growth rates.

An example of the transverse magnetic field $B_z$ progression in time for the lepton-beam case is shown in figure 7. At the earlier stages of the PIC simulation, the filamentation occurs and grows exponentially as seen in figure 7(a) as with the three-electron-populations case. As the magnetic field saturates, the turbulent phase is shown clearly where islands of magnetic field intensity are found in figure 7(b).

Figure 7. Snapshots of $B_z$ magnetic field from PIC simulations for the lepton-beam case at (a) $t=314.16 \, \omega ^{-1}_{\textrm {pe}}$ and (b) $1413.72 \, \omega ^{-1}_{\textrm {pe}}$. The beam Lorentz factor is $\varGamma = 2$, wavevector $k = 0.2$ and density ratio $n_{j}/n_{p} = 0.5$. All quantities are in plasma units.