Spatial-temporal evolution of the current filamentation instability

The spatial-temporal evolution of the purely transverse current filamentation instability is analyzed by deriving a single partial differential equation for the instability and obtaining the analytical solutions for the spatially and temporally growing current filament mode. When the beam front always encounters fresh plasma, our analysis shows that the instability grows spatially from the beam front to the back up to a certain critical beam length; then the instability acquires a purely temporal growth. This critical beam length increases linearly with time and in the non-relativistic regime it is proportional to the beam velocity. In the relativistic regime the critical length is inversely proportional to the cube of the beam Lorentz factor $\gamma_{0b}$. Thus, in the ultra-relativistic regime the instability immediately acquires a purely temporal growth all over the beam. The analytical results are in good agreement with multidimensional particle-in-cell simulations performed with OSIRIS. Relevance of current study to recent and future experiments on fireball beams is also addressed.

observed in counterstreaming plasmas in high power laser experiments [11][12][13]. Experiments on laser wakefield acceleration have also reported the filamentation of the accelerating particle beam as it interacts with the background plasma [14]. These instabilities provide an efficient way of restoring the isotropy in collisionless plasmas, since the energetic particles scatter off the self generated magnetic fields by which the longitudinal momentum is transferred to the transverse momentum.
The available theoretical models for CFI are restricted mainly to a purely temporal analysis [2,3,15] and do not capture any spatial characteristics of the instabilities, which can be very relevant for finite size systems [11][12][13][16][17][18].
In this paper we obtain the relativistic spatial-temporal solutions for the unstable transverse CFI modes in cold plasmas. Our work and approach are inspired by Refs. [19,20]. A single differential equation is derived to model the instability, considering only the electron response, ignoring the finite transverse dimension effects, considering a semi-infinite plasma slab and including the effects of a beam density ramp. For a step-like Heaviside beam profile analytical solutions are obtained for physically relevant and realistic initial conditions. We further obtain the quasi-static and asymptotic behavior of the solutions. The theoretical results are compared with multidimensional particle-in-cell (PIC) simulations using OSIRIS [21]. Such spatial-temporal analysis, shown in the later part of this paper, is relevant to the jets emitted by the x-ray binaries where the velocities of the jets are in relativistic range ∼ 0.6c [22] where spatial effects in the CFI modes are significant, or to the fireball-like beams [16][17][18] interacting with the plasma.

II. THEORY
We consider a two dimensional (2D) slab geometry, where a relativistic beam with velocity v 0bẑ and density n 0b F (z, t), where F (z, t) is the initial density profile of the beam, is propagating in a stationary plasma comprised of cold electrons and immobile ions with homogeneous plasma density n 0p . We analyze the stability of a transverse CFI mode with wavenumber k, and vector potential A = A(ψ, τ )ẑ exp[ikx], where ψ = v 0b t − z and τ = t, which satisfies the Coulomb gauge condition · A = 0 by solving the wave equation ( 2 − ∂ 2 t /c 2 ) A = −4π J/c. Under the slow envelope approximation |∂ ψ A| |kA|, the governing equation for the vector potential of the electromagnetic wave driven by a current density J z can be written as where J = −e[n 0b F (ψ) v 1b + n 0p v 1p + n 1b v 0b ] is the current density driving the vector potential /c 2 is the beam Lorentz factor, and c is the velocity of light in vacuum. The suffixes 0 and 1 are the 0 th and 1 st order perturbed values of the plasma (p) and beam (b) parameters defined as plasma electron velocity and density v p = v 1p , n p = n 0p + n 1p , and beam electron velocity and density v b = v 0b + v 1b , n b = n 0b F (ψ) + n 1b . The chosen vector potential perturbation will generate a magnetic field Resorting to the fluid equations of motion of a two-species-plasma (plasma electrons and beam electrons), using the continuity equation and the equation of momentum conservation for the relativistic beam and the stationary background plasma electrons, and restricting to the first order values in the weakly coupling limits by ignoring the (∂ τ + v 0b ∂ ψ ) term with respect to kv 0b , the perturbed quantities can be written as, Incorporating Eq. (2) in Eq. (1) by taking the second order τ derivative of Eq. (1) and further neglecting the higher order derivatives ∂ 2 τ , ∂ ψ ∂ τ , ∂ 2 ψ when compared with k 2 c 2 , we obtain, , ω pb = n 0b e 2 /(m 0 ) and ω pp = n 0p e 2 /(m 0 ). Considering an infinite beam [F (ψ) = 1] and ignoring the second term in Eq. (3) we retrieve the well known purely temporal evolution of the system with growth rate Γ 0 . Interestingly, the equation obtained by Mori et al. [19] (Eq. 10 in Ref. [19]) to analyze the spatial-temporal evolution of Raman forward scattering has the same form as Eq. for ψ > 0 is the Heaviside function. Respecting causality, we can impose A = 0 for τ < 0, and define the double Laplace transform of A(τ, ψ) as Thus, by doing the double Laplace transformation of Eq. (3), according to Eq. (4), we obtain the field expression in Laplace space as where A(0, 0), A(0, β), A(α, 0) and ∂ τ A(0, β) are the Laplace transforms of A(τ = 0, ψ = 0), A(τ = 0, ψ), A(τ, ψ = 0) and ∂ τ A(τ = 0, ψ) respectively, which are the required initial conditions.
The field A(τ, ψ) can be obtained by performing inverse Laplace transformation of A(α, β), defined as where σ (α,β) are chosen such that the contour from ∞ − iσ (α,β) to −∞ − iσ (α,β) lies below all the singularities. For the sake of simplicity we consider the following realistic initial conditions, which considers that at τ = 0, there is an initial constant noise source throughout the beam and for τ > 0 the beam front (ψ = 0) always encounters fresh plasma, and hence a constant noise source. The noise source for most instabilities are considered to be associated with the thermal fluctuations, and if thermal fluctuations have no time or space dependence, the constant noise source assumption holds correct. Longitudinally modulated or time dependent noise amplitude can be some of the forms of noise source that should be considered and the detailed analysis of the effect of different noise sources on the CFI spatial-temporal evolution will be addressed elsewhere.
The above conditions yield A(0, β) = A n /(iβ), A(α, 0) = A n /(iα), A(0, 0) = A n and ∂ τ A(0, β) = 0, which leads to the solution of Eq. (3), by inverting Eq. (5), as where I j is the j th order modified Bessel function of the first kind [23]. Neglecting the term ∂ 2 τ in Eq. (3) leads to the quasi-static solutions, which are valid at the beam front for ψ Qτ , as  plasma. The wavenumber of the CFI mode is k = 0.628ω pp /c taken to get maximum Q = 0.32c at γ 0b = 1.25.
The partial differential equation governing the CFI [Eq. is also evident in the simulation results to be discussed later in this paper. It is worth mentioning here that the asymptotic approach, used extensively for spatial-temporal analysis of the longitudinal beam-plasma instabilities [24], overestimates the growth and does not seem to give correct spatial characteristics for the transverse instabilities, specifically for the CFI discussed in here.
At the beam front, for ψ Qτ the quasi-static solutions given by Eq. (9) figure 1 (b)]. Beyond this length the instability grows with spatially constant temporal growth rate Γ 0 .
As observed from the previous discussions, and from Eqs. (3), (8), (9) and (10), the spatial-temporal behavior depends on the cross coupling coefficient Q. To address this, in figure 2 we analyze the dependence of Q on the beam velocity v 0b for different CFI wavenumbers k at n 0b = n 0p . For kc ω pp , ω pb , the maximum value of Q is achieved for v 0b = 0.5c (γ 0b = 1.15), and varies as Q max 0.65n 0b /(n 0p k 2 c 2 ). For kc ω pp = ω pb , Q max 0.4 at v 0b = 0.6c (γ 0b = 1.25). In the non-relativistic scenario γ 0b 1 and kc ω pp = ω pb , Q = v 0b , which is shown as a dashed line in figure 2. At higher γ 0b 1, k >> ω pp /c or n 0b n 0p , Q tends to 0, and the instability acquires a purely temporal behavior.
We have also considered a beam profile with F

III. SIMULATIONS
In order to confirm and to explore the theoretical results presented above, we have performed 2D PIC simulations using OSIRIS [21]. We consider a scenario where a relativistic beam is propagating through a cold plasma, where the beam and the plasma are both comprised of an electron-proton neutral plasma (the temperature of the background plasma is set to zero).
The simulation box, with dimensions 250 × 100(c/ω pp ) 2 , is divided into 12500 × 5000 cells with 3 × 3 particles per cell per species. The beam has a gaussian density ramp at the front, where L is the length of the gaussian ramp at the beam front. When L → 0 this profile mimics a sharp rising beam front with a Heaviside function profile which was considered to obtain the analytical solutions [Eq. (8)]. We seed the instability, in order to analyze a single CFI mode with wavenumber k seed , with a small magnetic field perturbation of the form here to predict the spatial-temporal growth of the CFI along the beam, as expected. In the simulations the instability starts to grow after a relaxation time (τ relax ≈ 1.63ω 1 pp for this particular simulation) necessary for the self-consistent electromagnetic fields and the electromagnetic noise to adjust to the initial flow condition. Thus, for the comparison with the theory, the time is re-normalized to τ = τ sim − τ relax , where τ sim is the simulation time. One can observe in figure 3 that the theoretical estimate for the CFI magnetic field, given by Eq. (8), matches well with the magnetic field profile observed in the simulations. We analyze the variation of saturation length L sat with time τ in figure 3 (e) obtained from the simulations. The rate at which L sat increases with time τ is Q = 0.32c, which is equal to the theoretical value of Q = ∂ τ L sat , as predicted by our model. Beyond the beam length L sat , the magnetic field amplitude is spatially constant and grows temporally with growth rate Γ 0 , as predicted by the theory. longitudinal electrostatic instability. We attribute this to the fact that, in a similar way as for the CFI, the longitudinal instabilities also have a spatial-temporal nature [24]. This also demonstrates that a full understanding of this scenario requires the combined analysis of CFI and longitudinal electrostatic instabilities.
The magnetic field snapshots in the x − z plane, shown in figure 4 indicate a sharper rise in the magnetic field at the beam front for higher γ 0b [ figure 4 (b)] as compared to lower γ 0b [ figure 4 (a)], which further validates the theory, since at high γ 0b the cross coupling term Q, and thus the saturation length L sat , decreases as ∼ 1/γ 3 0b for a given time τ . Since Γ 0 ∝ 1/ √ γ 0b , the field amplitude at the back of the beam in figure 4 (b) (high γ 0b ) is weaker as compared to the field in figure 4 (a) ( low γ 0b ), also as predicted by the theory.
As observed in figure 5, where the transversely averaged B y is compared for various density ramps, on increasing the ramp size L, and thus reducing the initial seed for the longitudinal modes, the simulation results show that these longitudinal modulations on the purely transverse CFI modes (at the wavelength ≈ λ p ) can be suppressed. As the time progresses (not shown in the paper) the longitudinal modes as well as other faster growing CFI modes start to play an important role and their interplay in the nonlinear stage becomes significant.

IV. DISCUSSION AND CONCLUSIONS
To summarize, in this paper we have derived a single differential equation modeling the spatial-temporal evolution of the purely transverse current filamentation instability. For relativistic fireball electron-positron beams [17,18] undergoing current filamentation in an electron-ion plasma, the cross coupling coefficient Q is enhanced by a factor of 2 (Q → 2Q) and the purely temporal growth rate is enhanced by a factor of √ 2 (Γ 0 → √ 2Γ 0 ) due to the contribution from the current driven by the velocity and density perturbations of the positrons in the beam. However, in the relativistic regime this enhancement is not sufficient to balance the 1/γ 3 0b dependence of Q on the beam Lorentz factor γ 0b . As a result for an ultra-relativistic 29 GeV fireball beam [17] with n 0b = n 0p , Q 2 × 10 −14 c, and thus the spatial evolution to the CFI can be attributed only to the density gradient scale length. For the recent experiments with a 60MeV electron beam [16], Q 1.2 × 10 −6 c, thus again suggests only purely temporal growth of the CFI is present along the beam. However, in the case of moderately relativistic fireball beams Q can be significantly enhanced. For instance, in the case of Sarri et al. [18] with γ 0b = 15, n 0b = 10n 0p and considering a density ramp of L = 0.22c/ω pp , the cross coupling coefficient can be Q ≈ 0.01c, which suggests that in the linear regime of the CFI, for τ 220ω −1 pp the CFI spatially grows beyond the density ramp size L and spatially saturates with L sat Qτ L.
Based on our analysis we further observe that the spatial-temporal nature of the instability also has an effect for finite beam-plasma interaction time τ int and beam size L beam . In fact, depending on the relation between these parameters, i.e. either L beam < Qτ int or > Qτ int , a weaker or stronger filamentation of the beam can be expected which will further affect the nonlinear growth of the instability. Moreover, the study of spatial-temporal evolution of the beam-plasma instabilities can also lead to a better understanding and characterization of the Weibel mediated collisionless shock formation process in laboratory and astrophysical plasmas.