WEIBEL, TWO-STREAM, FILAMENTATION, OBLIQUE, BELL, BUNEMAN...WHICH ONE GROWS FASTER?

The T_zz terms reads,

$$\begin{equation} T_{zz}=1-\frac{R(1+\alpha)}{x^2}-\frac{\alpha }{(x-Z_z)^2 \gamma _b^3}-\frac{1}{(x+\alpha Z_z)^2\gamma _p^3}. \end{equation} \tag{ 9 }$$

The proton contribution stems from the term ∝R/x², the beam one from the second term with numerator α, and the return-current with γ_p = (1 − v²_p/c²)^−1/2 accounts for the last factor. The dispersion equation for these modes is independent of the magnetic field since the resulting electrostatic modes have k ∥ E and only generate velocity perturbations along the magnetic field. This equation has been analyzed elsewhere (Bret & Dieckmann 2008b). For a diluted beam, the interaction of the two electron beams produces two-stream unstable modes with maximum growth rate (Mikhailovskii 1974)

$$\begin{equation} \delta =\frac{\sqrt{3}}{2^{4/3}}\frac{\alpha ^{1/3}}{\gamma _b}\quad \mathrm{for}\quad Z_z=1, \end{equation} \tag{ 10 }$$

and the interaction of the return current with the proton yields unstable Buneman modes at much smaller wavelength with,

$$\begin{equation} \delta =\frac{\sqrt{3}}{2^{4/3}}R^{1/3}\quad \mathrm{for}\quad Z_z=1/\alpha. \end{equation} \tag{ 11 }$$

These expressions change when the density ratio approaches unity because the beam and the return current become symmetric for α = 1 (see Section 3.3).

The unstable electromagnetic modes arise here from the T_xx ± ıT_xy terms of Equation (8) which read,

$$\begin{eqnarray} T_{xx}\pm \imath T_{xy}&=& x^2-\frac{Z_z^2}{\beta ^2}+ \frac{\alpha (Z_z-x)}{ \gamma _b(x-Z_z)\mp \Omega _B}\nonumber\\ && -\frac{x+\alpha Z_z}{\gamma _p(x+\alpha Z_z)\mp \Omega _B}-\frac{R x (1+\alpha)}{x\pm R \Omega _B}. \end{eqnarray} \tag{ 12 }$$

A straightforward calculation from the general dispersion relation for circularly polarized wave with such wave vectors (Ichimaru 1973; Achterberg 1983; Amato & Blasi 2009) yields the very same result. These modes are the present configuration analogs to the ones explained by Bell (2004, 2005). The main difference with the scenarios in which they are usually involved is that they are presently driven by an electronic current rather than by a baryonic one.

Both dispersion equations yield unstable modes here. The most relevant feature with respect to the mode hierarchy is that growth rates are bounded by RΩ_B, regardless of the parameters involved. This does not imply the growth rate can go to infinity because unstable solutions exist only in a limited region of the phase space (α, γ_b, Ω_B, R).

The T_xx − ıT_xy term can be partially assessed analytically neglecting the x² and the return-current factor. The most unstable modes are found at Z_z ∼ Ω_B/γ_b, with x ∼ RΩ_B. The maximum growth rate varies like $\sqrt{\alpha }$, and is always bounded by

$$\begin{equation} \delta < R\Omega _B-\Omega _B^2R^{3/2}. \end{equation} \tag{ 13 }$$

The T_xx + ıT_xy factor is more involved and has been partially treated numerically. Unstable modes are found here at $Z_z\sim \frac{1}{2}\alpha /\Omega _B$, with maximum growth rates varying like α, and still bounded by Equation (13). Note that these modes are stable for γ_b ≲ αΩ²_B so that instability in the diluted beam regime demands a low Lorentz factor.

Figure 1 displays the growth rate of these four unstable modes in terms of Z_z for some parameters triggering them all. One can observe how the relevant spectrum extends over 3 orders of magnitude. This point, and its consequences, are discussed in the conclusion.

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Because we study small perturbations of the system composed by both the beam and the plasma, the present theory is perfectly valid up to α = 1. However, the physical interpretation of the calculations can no longer be derived considering the beam is a perturbation to the isolated background plasma. The system, beam+plasma, has its own proper modes that are quite different from those of the isolated plasma. Taking R = 0 for example, the dispersion equation has two branches exactly given by

$$\begin{equation} x^2=Z_z^2+\frac{1-\sqrt{1\pm 4 Z_z^2 \gamma _b^3}}{\gamma _b^3}, \end{equation} \tag{ 14 }$$

which definitely differs from the cold isolated plasma dispersion relation x = 1, i.e, ω = ω_p. For such a symmetric system, the Buneman modes at Z_z ∼ 1/α merge with the two-stream modes at Z_z ∼ 1, and depending on γ_bR^1/3, the resulting mode may be governed by the electrons or the ions dynamic (see Table 2).

For parallel wave vectors, electrostatic modes have been studied previously (Bret & Dieckmann 2008b), and two different regimes need to be considered depending on the product γ_bR^1/3. For γ_bR^1/3 ≲ 1/2, the fastest growing mode has

$$\begin{equation} \delta =\frac{1}{2\gamma _b^{3/2}}\quad \mathrm{for}\quad Z_z=\frac{\sqrt{3}}{2\gamma _b^{3/2}}. \end{equation} \tag{ 15 }$$

Note that this result is exact for R = 0. In the opposite limit γ_bR^1/3 ≳ 1/2, one has

$$\begin{equation} \delta =\frac{\sqrt{3}R^{1/6}}{2^{7/6}\gamma _b}\quad \mathrm{for}\quad Z_z=\sqrt{2R}. \end{equation} \tag{ 16 }$$

Regarding the electromagnetic modes, the two terms T_xx ± ıT_xy in Equation (8) yield the same growth rate because for α = 1, these two functions are equal through the transformation x → −x. Here again, the growth rate of these modes is always bounded by RΩ_B, but instability occurs only for γ_b ≲ 1/R and $\Omega _B\lesssim \sqrt{\gamma _b}$. As a consequence, the most unstable mode for a given couple (γ_b, Ω_B) cannot grow faster than $R\Omega _B<R\sqrt{\gamma _b}<\sqrt{R}$.

Figure 2 displays the growth rate of the two modes just discussed for parameter sets yielding a Bell governed system. Note that such situation demands a highly magnetized system. Although this plot gives the sensation that Bell-like modes can dominate some region of the parameters phase space, they were never found to govern the full two-dimensional spectrum. This example illustrates the importance of accounting for the full unstable k spectrum instead of focusing on a given wave vector orientation. While Bell-like modes grow slower for non-flow-aligned wave vectors, other modes grow faster in oblique directions. As a result, the unstable spectrum ends up governed by the latters, even if the formers can "reign" over the beam axis. This situation basically stems from the necessary smallness of the mass ratio R. Dealing with a proton beam would allow to reach R = 1, where Bell modes should definitely be found governing some portion of the phase space. A similar theory accounting for a proton beam would be needed to describe such situations.

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For the un-magnetized system with infinite ion mass, the resulting spectrum is now well understood, including within the framework of a full kinetic relativistic theory (Bret et al. 2008). There is a continuum of unstable modes bridging the two-stream instability (k_⊥ = 0) with the filamentation instability (k_∥ = 0). The magnetized version of the same system brings two additional oblique branches. For Ω_B > 1, the first one is found at Z_z ∼ Ω_B/γ_b, with a maximum growth rate varying like $\frac{1}{2}\sqrt{\alpha /\Omega _B}$. Still for Ω_B > 1, the second branch is located at $Z_z \sim \Omega _B/\gamma _b+\sqrt{1+\Omega _B^2}$ with a maximum growth rate varying like $\frac{1}{2}\sqrt{\alpha }/\Omega _B$ (Godfrey et al. 1975). Within the same region of the spectrum, the oblique modes already present at Ω_B = 0 near Z_z = 1 are now found at $Z_z\sim \sqrt{1+\Omega _B^2}$ with a maximum increment varying like α^1/3/γ_b. These later modes are commented in Section 4.3. At this stage, the unstable spectrum may look like the one pictured on Figure 3(a) for the parameters specified in caption.

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At this junction, some comment is needed to clarify the meaning of "oblique" mode (with small "o"). Figures 3 and 4 make it clear that unstable modes with both k_x ≠ 0 and k_z ≠ 0 are numerous. Such kind of modes have so far been referred to in the literature as "oblique" (Watson et al. 1960; Niemiec et al. 2008; Ohira & Takahara 2008), "electromagnetic beam-plasma instability" (Califano et al. 1998b), "coupled two-stream Weibel" (Jaroschek et al. 2005), or "Mixed mode" (Frederiksen & Dieckmann 2008). The problem with such labeling is that "oblique modes" are here just too numerous for only one tag. In Tables 1 and 2, as in the rest of the paper, "oblique modes" bridging between two-stream and filamentation instabilities for Ω_B = 0 are labeled "oblique_B0." For Ω_B > 1, they evolve into what we presently call "oblique" modes (capital "O"), with both k_x ≠ 0 and k_z ≠ 0, and a growth rate scaling like γ⁻¹_b. Finally, modes found reaching their maximum growth rate at finite Z_z and Z_x = ∞ are labeled "upper-hybrid-like" modes after Godfrey et al. (1975).

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The spectrum is even richer when ions are "allowed" to move, as Buneman and Bell-like modes are triggered. The resulting spectrum is pictured on Figures 3(b) and (c) for R = 1/100 and 1/1836, respectively. An electron-to-proton mass ratio R = 1/100 may be relevant for PIC simulation where the mass ratio is usually incremented from its 1/1836 realistic value in order to speed up the ions dynamic, saving thus computer time. Regarding this last point, it is interesting to note that such a trick is possible because the growth rates associated with the moving ions increase with R, while the related scale length does not (see Equations (11)). This is reflected on Figures 3(b) and (c) where the dominant modes do not migrate with varying R. If such was not the case, some higher mass ratio could demand higher spatial resolution so that the computing time lost by a finer spatial resolution would not necessarily be compensated by the accelerated dynamic. Also on these Figures, the occurrence of the Buneman instability around Z_z ∼ 1/α = 10 is clear. The two-stream instability governs here the system even for R = 1/100, but the electron and ion spectrums are almost disconnected form each other only for R = 1/1836. Because of the smallness of the real mass ratio, both the time and length scales of each spectrum are disconnected from each other. In such a case, the overall spectrum is nearly the superposition of each sub-spectrum.

Without moving ions (R = 0), the most relevant features of the unstable spectrum in this case are oblique resonances at Z_z ∼ Ω_B/γ_b and Z_z ∼ Ω_B/2γ_b. Only the first one extends up to Z_x = ∞ and yields a maximum growth rate δ_m ∼ 1/Ω_B (Bret et al. 2006). The next paragraph is devoted to an overview of the second kind of modes. This large Z_x regime can be analytically investigated in the following way. We start deriving the dispersion equation in this limit setting α = 1, γ_b = γ_p and Z_x → ∞ in Equation (6). We then use the fact that unstable modes are found with ℜ(ω) ∼ 0 and develop the resulting dispersion equation to the second order in x. In this large Z_x limit, the system is found unstable only between,

$$\begin{equation} Z_{z1}=\frac{\Omega _B}{\gamma _b}\sqrt{1-\frac{2 \beta ^2 \gamma _b}{\Omega _B^2}}, \end{equation} \tag{ 17 }$$

and

$$\begin{equation} Z_{z2}=\frac{\Omega _B}{\gamma _b}. \end{equation} \tag{ 18 }$$

These results are exact and imply that filamentation instability is canceled as soon as Z_z1 > 0, i.e., $\Omega _B>\beta \sqrt{2\gamma _b}$. For Ω_B ≫ βγ^1/2_b, the unstable range decreases like Z_z2 − Z_z1 ∼ β²/Ω_B. The range of unstable wave vectors becomes therefore increasingly narrow, yielding a quasi-monochromatic unstable spectrum in this limit.

When ion motion is accounted for, part of the stability domain can still be calculated exactly following the same guidelines. With R ≠ 0, one finds

$$\begin{equation} Z_{z1,R}=Z_{z1}, \end{equation} \tag{ 19 }$$

and

$$\begin{equation} Z_{z2,R}=\frac{\Omega _B}{\gamma _b} \sqrt{\frac{1+R \big(\gamma _b+\Omega _B^2/2\big)}{1+R \Omega _B^2 \big/ 2}}. \end{equation} \tag{ 20 }$$

The maximum growth rate within this range remains close to 1/Ω_B, and another range of unstable wave vectors and similar growth rate appears at slightly larger Z_zs. These new modes, arising purely from the ion motion, will not be investigated here although they are accounted for in the forthcoming numerical evaluation of modes hierarchy.

Figures 4(a)–(c) shows the two-dimensional spectrum for the parameters chosen for Figure 2 with Ω_B = 60. The purpose of this choice is to observe how some oblique modes govern the full spectrum while Bell-like modes dominate the Z_z-axis. On Figure 2, the spectrum for Ω_B = 60 is clearly governed by these modes at Z_z ∼ 0.08. The same modes, together with the way they evolve when leaving the Z_z-axis, are perfectly visible on Figure 4(c) and one can check how they quickly stabilize for oblique wave vectors.

Sections 4.1 and 4.2 discussed unstable modes located at Z_x = ∞. The cold fluid approximation typically send some fast growing modes to Z_x = ∞, whereas temperature effects tend to stabilize these small wavelengths instabilities by preventing the pinching of too small filaments (Silva et al. 2002). But the occurrence of a magnetic field also triggers some truly locally most unstable oblique modes. The term "locally" means here that the very same mode, as defined by one root of the dispersion equation, reaches its maximum for a finite Z_x. For more clarity, Figure 4(d) displays a two-dimensional spectrum with α = 1, while the other parameters have been chosen to single out this mode.

For the diluted beam case, Table 1 mentions a maximum growth rate varying like α^1/3/γ_b. For the symmetric case α = 1, Table 2 indicates a maximum oblique growth rate ∼0.5/γ^1.15_b (numerical fit). Such scaling of this finite Z_x mode bears important consequences regarding modes hierarchy: a look at Tables 1 and 2 shows that the ultra-magnetized regime tends to stabilize every modes, except this one, the Buneman and the Two Stream. While more modes compete for moderate Ω_B, the large Ω_B limit is decided among these three. More details on the ultra-magnetized regime are given in Section 5.2.

Gathering the aforementioned data, the frontiers between the domains governed by different modes are sketched in terms of (γ_b, α) on Figure 5. Although α is limited to the range [0.1, 1], the lower part of the graph can easily be deduced from the diluted beam regime expressions reported in Table 1. This process is repeated for various Ω_Bs and R = 1/1836, 1/100.

Starting with Ω_B =0, the findings of Bret & Dieckmann (2008b) are confirmed numerically here. Filamentation dominates for large beam-to-plasma density ratios. The frontier visible in the lower-right corner pertains to the oblique/Buneman transition. The growth rate of the un-magnetized oblique competing mode in this case³ reads $\sqrt{3}/2^{4/3}(\alpha /\gamma _b)^{1/3}$ (Faĭnberg et al. 1970). The equation of the border is, therefore, simply

$$\begin{equation} \alpha =R\gamma _b. \end{equation} \tag{ 21 }$$

For α = R and γ_b = 1, Equations (10) and (11) are identical so that this frontier extends all the way down to α = R. Below this line, Buneman modes govern the linear phase.

From Ω_B = 1, the evolution is three-fold. To start with, the two-stream instability governs an increasing weakly relativistic region (lower-left corner) as its magnetized growth rate remains the same, while oblique modes are made less unstable. By virtue of the same kind of effects, Buneman modes gain weight on the lower-right corner because their growth rate too does not vary with the magnetic field. Finally, filamentation domain shrinks as the magnetic field progressively shuts it down (Cary et al. 1981; Stockem et al. 2007). Regarding the border of the two-stream region, a look at the Z_x component shows it switches from 0 to ∞ when crossing the limit. A comparison between the growth rate values in this region and the expressions gathered in Table 1, shows that the two-stream mode competes with the upper-hybrid-like one located at $Z_z\sim \Omega _B/\gamma _b+\sqrt{1+\Omega _B^2}$. The two-stream/oblique frontier is thus defined for α ≪ 1 by

$$\begin{equation} \frac{\sqrt{3}}{2^{4/3}}\frac{\alpha ^{1/3}}{\gamma _b}=\frac{\sqrt{\alpha }}{2 \Omega _B} \Leftrightarrow \alpha =\frac{27 \Omega _B^6}{4 \gamma _b^6}. \end{equation} \tag{ 22 }$$

Regarding the Buneman/upper-hybrid-like frontier, the growth rate $\sim \frac{1}{2}\sqrt{\alpha }/\Omega _b$ simply decreases with α until the Buneman mode overcomes it. This transition thus occurs for

$$\begin{equation} \frac{\sqrt{3}}{2^{4/3}}R^{1/3}=\frac{\sqrt{\alpha }}{2 \Omega _B} \Leftrightarrow \alpha =\frac{3R^{2/3}}{2^{2/3}}\Omega _B^2. \end{equation} \tag{ 23 }$$

Below this intersection, two-stream modes are directly competing with the Buneman ones. This later frontier is thus defined by

$$\begin{equation} \alpha =R\gamma _b^3. \end{equation} \tag{ 24 }$$

The partition defined by these three equations in the diluted regime is thus pictured on Figure 6 for Ω_B larger than 1 but smaller than ∼R^−1/3 (see Section 5.2 for this limitation).

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For larger values of Ω_B, the previous trends amplify. The oblique domain (central region) is bounded toward the low αs because oblique growth rates are scaled like α^1/2 (or α^1/3), while the Buneman growth rate does not vary with α. The evolution is of course faster with R = 1/100 than 1/1836 because the Buneman grows faster in the second case.

From Ω_B ∼ 5.8, the Buneman modes gain more "territory" according to the aforementioned mechanisms. Noteworthily, new modes appear to govern some portion of the phase space at rather high density ratio and moderate Lorentz factor. For R = 1/1836, the inserted spectrum maps on Figure 5 show that a small part of the phase space (blue one) pertains to the oblique mode described in Section 4.3. This "middle region" is analytically involved to explore and such results could only be derived numerically so far. The same mode governs a similar region for R = 1/100, but in that case, another kind of oblique mode (green regions) also intervenes. Why can such modes lead the linear phase for R = 1/100 and not R = 1/1836? The solution lies in the mode coupling already observed in Section 4. Figures 3(a) and (c) show how the Buneman modes interfere with the rest of the spectrum according to the value of R. For R = 1/100, interferences are stronger than for R = 1/1836 because the spectrum is wider. While the beam is diluted (as is the case for these plots), coupling effects are limited because the various modes are well separated in the k space. As the beam-to-plasma density ratio α increases, the Buneman spectrum progressively merges with the rest of the unstable modes. Interferences thus become potentially more intense, and all the more that R is large. It turns out that the green region in Figure 5 arises from a mode coupling which is much less excited for R = 1/1836 than for R = 1/100. We find here that tuning the electron-to-proton mass ratio in order to speed up the system dynamic may bear qualitative consequences by changing the nature of the dominant mode.

The outcome of the ultra-magnetized regime is relevant for pulsars physics, for example. For diluted beams, an increasing magnetic field progressively stabilizes every mode except the two-stream, the Buneman, and the oblique one (see Table 1). Both two-stream and oblique are scaled like α^1/3/γ_b, but the pre-factors favor two-stream modes. Competition is eventually between Buneman and two-stream with a frontier defined by Equation (24). The magnitude of Ω_B needed to trigger this picture can be derive by setting α = 1 in Equation (23) as if the "triple-point" pictured in Figure 6 was rejected to α = 1. Such criterion places the ultra-magnetized regime beyond

$$\begin{equation} \Omega _B\sim R^{-1/3}, \end{equation} \tag{ 25 }$$

which fits well what is observed in Figure 5. Let us now detail the symmetric case and detail the mode hierarchy in terms of (γ_b, Ω_B).

When α reaches unity, Table 2 indicates that the competition is between the merged two-stream/Buneman, the upper-hybrid-like and the oblique modes. However, a comparison of the growth rates is possible as long as these modes are well separated in the k space. When overlapping, the growth rate interfere, and analytical predictions become very involved. A closer look shows that the region

$$\begin{equation} \Omega _B \lesssim \sqrt{2R}\gamma _b, \end{equation} \tag{ 26 }$$

is thus defined. This limit lies below the cancellation threshold for the filamentation instability, $\Omega _B=\beta \sqrt{2\gamma _b}$, at least up to γ_b = 1/R. Indeed, the numerical exploration of this region shows that filamentation governs the system almost as long as it is not canceled. This allows for the tracing of the filamentation domain in Figure 8, together with the other limits explained in this section. As filamentation vanishes, it evolves smoothly into the upper-hybrid-like modes with k_∥ ∝ Z_z ≠ 0.

When condition (26) is fulfilled, it is possible to directly compare the growth rate of the three competing modes. The resulting situation is rendered in Figure 7 and varies strongly with R. The origin of such R-sensitivity is the fact that only the two-stream/Buneman growth rate depends on the mass ratio in the large γ_b limit. While its γ_b variation is monotonous, the variation of the oblique modes is not, rendering the determination of the hierarchy highly non-trivial for moderately relativistic beam.

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Figure 7 shows that for R = 1/100, oblique modes are only allowed to overcome the others as long as Ω_B is not too large. Numerically, it is found that for Ω_B ≳ 300, they are overcome by two-stream/Buneman regardless of the beam energy. Beyond this threshold, upper-hybrid-like modes govern if

$$\begin{equation} \Omega _B < \frac{2^{7/6}}{\sqrt{3}} \frac{\gamma _b}{R^{1/6}}. \end{equation} \tag{ 27 }$$

The resulting hierarchy map is display on Figure 8 and uncover an intriguing bubble-like oblique domain.

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When decreasing R down to its realistic value 1/1836, Figure 7 shows how the large γ_b part of the two-stream/Buneman curve is shifted down by virtue of its R^1/6/γ_b scaling (see Table 2). Even for Ω_B = 500, the oblique growth rate (plain blue curve) surpasses the two-stream/Buneman from γ_b ∼ 5 to ∼200. Whether such a situation holds for any Ω_B is an open question, but it has been checked that it does at least up to Ω_B = 10³. In order to draw the hierarchy map, let us first determine the Lorentz factor for which two-stream/Buneman modes overcome the oblique one. The corresponding value of γ_b fulfills

$$\begin{equation} \frac{1}{2}\frac{1}{\gamma _b^{1.15}} =\frac{\sqrt{3}}{2^{7/6}} \frac{R^{1/6}}{\gamma _b}\Rightarrow \gamma _b\sim \frac{0.055}{R^{10/9}}, \end{equation} \tag{ 28 }$$

yielding γ_b ∼ 234 for R = 1/1836. The hierarchy map here is the one pictured in Figure 8 for R = 1/1836. For Ω_B > 2 × 234^1.15 = 1060, the system is governed with γ_b increasing by two-stream/Buneman modes, then oblique, then two-stream/Buneman again, then upper-hybrid-like and finally, filamentation. For smaller Ω_Bs (yet, larger than ∼4), the last two-stream/Buneman step is just skipped, as the system goes directly from the oblique to the upper-hybrid-like regime.

It is therefore not easy to define some ultra-magnetized regimes here. Unlike the diluted beam case where only two modes are left to compete for a reasonably high magnetic field defined by Equation (25), we find here that every mode involved is likely to play a role, regardless of the magnetization. For R = 1/100, one could define a so-called ultra-magnetized regime from the top of the oblique "Bubble" located at Ω_B ∼ 300. For larger magnetization, only three modes are left to compete. But such a definition is no longer possible for R = 1/1836.

Finally, let us add a remark regarding the accuracy of the borders calculations. While it is perfectly possible to numerically compute hierarchy maps 5 and 8 with higher definition, such progress is not imperative because two different systems located on each side of a given border are likely to evolve in quite a similar way during the linear phase. Let us assume the border between two mode domains A and B goes through α = α₀, in such a way that A dominates for α < α₀. For α = α₀, the growth rates δ_A and δ_B are strictly equal. If α = α₀ ± ε, then δ_A − δ_B = ∓ε'. If both A and B are excited with the same initial amplitude, the time required for the fastest to overcome the slowest only by a factor e is ∼ω⁻¹_p/ε'. Depending on the system considered, this time may well exceed the duration of the linear phase for ε' small enough. It would thus be inappropriate to claim that only A or B is relevant on its respective side of the border. Even if the dominant mode evolution can be discontinuous, the system dynamic stemming from the growth of the entire spectrum should evolve smoothly when crossing a frontier. One can focus on either A or B only far away from the borders so that the precise calculation of its location is not crucial.

Here, we perform the calculation for one magnetized and one non-magnetized case. The system location in the parameters phase space is designated by the red crosses on Figure 5, and the growth rate maps are displayed on Figures 9(a). For both scenarios, the Buneman modes are clearly visible at Z_z ∼ 8 = 1/α, but cannot compete with the two-stream ones at Z_z ∼ 1 because the beam is not relativistic enough. With such a beam-to-plasma density ratio, a much higher relativistic factor is required for the two-stream/Buneman transition.

For the non-magnetized case (upper Figure 9(a)), this weakly diluted beam system is governed by non-magnetized "oblique_B0" modes at Z_z ∼ 1. Although not dominant, the filamentation instability⁴ plays an important role as its growth rate is quite close to the largest one. The introduction of the magnetic field (lower Figure 9(a)) damps filamentation as well as the oblique mode, resulting in a two-stream driven system. Noteworthily, Figure 5 shows that a small variation of the Lorentz factor can trigger a dominant mode transition for Ω_B = 1. For R = 1/100, the system is close to a triple-point where two-stream, Buneman and upper-hybrid-like modes grow the same speed. The system behavior in this case is therefore very sensitive to the parameters choice.

A word of caution is needed here before concluding with respect to the solar flares environment. The simulation performed in Karlicky (2009) accounts for an electronic temperatures of 21.4 × 10⁶ K (21.4 MK). Indeed, analysis of soft and hard X-ray flare data indicate that plasma temperatures up to 40 MK are obtained in large flares (Aschwanden 2002). Furthermore, velocity distributions can be highly anisotropic due to stronger heating along the magnetic field lines (Fisk 1976; Miller 1991; Miller et al. 1997). For a mode with wave vector k growing at growth rate ω_i, a temperature spread Δv can be neglected providing |Δv|ω⁻¹_i ≪ k⁻¹ (Faĭnberg et al. 1970). Such a condition simply ensures that during one e-folding time, particles are traveling almost the same path when compared to the wavelength considered. Focusing on the fastest growing oblique_B0 mode for the non-magnetized case (upper Figure 9(a)), we find,

$$\begin{equation} \frac{|\Delta \mathbf {v}|}{c}\ll \frac{\sqrt{3}}{2^{4/3}}\left(\frac{n_b/n_p}{\gamma _b}\right)^{1/3}. \end{equation} \tag{ 29 }$$

Accounting for |Δv| = 0.08c (i.e., T = 20 MK), the condition above translates 0.08 ≪ 0.31. The only change when considering the two-stream dominated magnetized case (lower Figure 9(a)) is that the Lorentz factor on the right-hand side has the exponent −1 instead of −1/3. In such a case, the condition reads 0.08 ≪ 0.25. The cold approximation therefore seems reasonable here, although condition (29) is only weakly fulfilled. An interesting consequence of Equation (29) is that the condition is not homogenous over the unstable spectrum. A kinetic theory of the full unstable spectrum is currently under elaboration, together with a rigorous mode-dependent definition of the kinetic-fluid transition (A. Bret & L. Gremillet 2009, in preparation). For a clearer assessment of the accuracy of the cold approximation in the present case, the insert in the upper Figure 9(a) displays a kinetic calculation of the most relevant portion of the unstable spectrum, assuming electronic beam and plasma temperatures of 20 MK. Such a calculation has been performed neglecting ion motion, which is perfectly valid here. One can check how the portion of the unstable spectrum only slightly departs from its cold counterpart, while the largest growth rate switches from 0.3ω_p down to 0.22ω_p. Note that similar calculation for the magnetized case is yet to be done.

The corresponding spectrum is plotted on Figure 9(b). With a small β and a beam-to-plasma density ratio of 1, this case is also sensitive to the parameters (see the star on Figure 8). We find here that non-magnetized "oblique_B0" modes govern the system, though filamentation is far from being shut down. But such a weakly relativistic symmetric system can switch between oblique or filamentation regimes through very small parameter variations. When accounting for kinetic effects, two-stream/Buneman modes can also compete because they are least sensitive to temperature. Indeed, for R = 0, these three modes grow almost exactly the same way for a beam with γ_b = 1.1 and temperature 100 keV (Bret et al. 2008). Such a result was found for a plasma temperature of 5 keV, but similar conclusions can be drawn with different plasma thermal spread. The discussion at the end of Section 5 is relevant in this case as this system is bordering various frontiers.

These kind of structures are currently extensively studied by means of PIC simulations (Medvedev & Loeb 1999; Silva et al. 2003; Nishikawa et al. 2003; Milosavljevic et al. 2006; Chang et al. 2008; Spitkovsky 2008a, 2008b; Martins et al. 2009), due to their role in the Fireball model for gamma-ray bursts (Piran 2000). The typical scenario arising from these studies is two-fold: to start with, two relativistic plasmas shells collide and the resulting instability eventually generates a quasi-steady propagating shock. These plasma shells can be pair plasmas (Silva et al. 2003; Chang et al. 2008), or electron–ion plasmas (Spitkovsky 2008a; Martins et al. 2009). The un-magnetized pair plasma case can be analyzed here setting R = Ω_B = 0, and the system is found clearly governed by the filamentation instability (see Figure 9(c)). This conclusion supports the emphasis put on this instability by the aforementioned authors.⁵ Moreover, such domination is quite robust: as long as n_b/n_p = 1, γ_b = 10 and Ω_B = 0, the only alternative to the filamentation instability are the non-magnetized oblique modes. Accounting for kinetic effects, a transition to such a regime demands a beam temperature of the order 10⁶ keV (Bret et al. 2008). Figure 8 also suggests that some magnetized version of the system with $\Omega _B>\gamma _b\sqrt{2R}$ could trigger a transition to the magnetized oblique regime. But the bigger threat to the domination of filamentation seems to be the beam-to-plasma density ratio. As evidenced by Figure 5, this instability dominates the ultra-relativistic regime only for strictly symmetric systems. At γ_b = 10 and 100, filamentation governs only for n_b/n_p ≳ 0.6 and 0.85, respectively. The symmetric system hypothesis has been so far related to the simulation of equivalent density colliding shells, within the Fireball framework. Since a slight change in the density ratio may trigger a dominant mode transition, subsequent work should be needed to test the robustness of the shock formation in this respect.

The second stage of the scenario involves particles which are accelerated by the shock (Spitkovsky 2008b), escape it, and interact with the upstream medium generating more instabilities. Considering now Ω_B = 0, γ_b = 100, and α = 10⁻², Figure 5 shows that oblique_B0 modes dominate for R = 1/1836 while Buneman modes do so for R = 1/100. A more detailed evaluation of the spectrum should be done from this point, accounting for the non-thermal energy dispersion of the escaping particles, and for the upstream ion temperature. Sorting out this issue may be important because an interaction governed by the purely electrostatic Buneman instability is less likely to feed the magnetic turbulence needed in the Fireball scenario for synchrotron radiation emission. At any rate, the small beam-to-plasma density ratio implied in this second stage should prevent the filamentation instability from playing any prominent role in the linear phase.