Ion effects on the scaling of magnetic field amplification in plasmas with the system size

Magnetic field amplification during the nonlinear stage of the current filamentation instability excited by ultra-relativistic electron beams is investigated with a two-dimensional electromagnetic particle-in-cell (PIC) simulation code, with special attention paid to the effects of plasma ions and the system size. The effect of plasma ions is shown to be significant and enhanced magnetic field amplification and beam energy deposition are found due to plasma cavity expansion and merger. When the system size in the transverse direction (perpendicular to the beam propagation direction) is enlarged by a factor of m, the transverse magnetic field energy is found to increase by a factor of m 2 in the case of the plasma with movable ions, in contrast to m with immovable ions. The results are also confirmed by three-dimensional PIC simulations.


Introduction
Instabilities are of paramount importance in plasma physics. Similar instabilities occur in vastly different plasmas, from astrophysical to laboratory and fusion plasmas. The current filamentation instability (CFI) [1,2] between electron beams and plasmas has attracted much attention in recent decades due to the effective energy deposition [3] of beam electrons and strong magnetic field generation [4,5], which show important applications in high energy density physics [6] and astrophysics [7][8][9]. In astrophysics, the phenomena responsible for gamma-ray bursts and their associated afterglow are largely unknown. One proposed theory for the afterglow is the fireball theory [10], in which as matter (consisting of electrons, positrons, and ions) is ejected from the gamma-ray bursts relativistic collisionless shocks occur between this matter resulting in the generation of large magnetic fields and radiation, which could possibly be explained through the occurrence of the CFI. In high energy density physics, it is shown that the CFI can be employed as a novel channel of gamma-ray radiation with extreme energy density electron beams and solid-density plasmas [6].
One of the typical characteristics of CFI is the strong magnetic field generation. Both the linear and nonlinear stages of the instability are detailed investigated using PIC simulations [11][12][13][14][15][16][17][18][19][20][21]. During the linear stage of CFI, the magnetic field energy grows exponentially and then becomes saturated when beam electrons inside the filament are trapped in the magnetic field. Polomarov et al [13] showed that in the highly nonlinear stage, the mergers of super-Alfvénic and hollow-current density filaments decrease the magnetic field energy significantly. It should be noted that plasma ions show significant effects on CFI [3,[14][15][16][22][23][24]. Honda et al [3,16] have proved that the collective beam stopping of a mildly relativistic electron beam (REB) (with an energy of MeV) is more effective with immovable plasma ions. An enhanced instability growth rate and further amplification of magnetic field on ion time scale were found due to ion response [22]. In addition, during the nonlinear stage of CFI, the directional drift of beam filaments in plasmas with immovable ions was also observed due to the formation of asymmetric transverse magnetic fields [23]. With the development of high-power lasers, high-energy electron beams with GeV energy become widely available [25,26]. Compared to the plasma with immovable ions, it is shown that such an REB propagating in a hydrogen plasma may arise a new secondary instability [15,27] after the saturation of the linear CFI, which further amplifies the magnetic field energy by two orders of magnitude. Moreover, the spatiotemporal dynamics of the oblique two-stream instability (i.e. the mixed mode between two-stream and CFI) were investigated with theory and PIC simulations, taking into account the effects of finite beam length and radius [20,28]. Experimental investigations of CFI were also performed in the laboratory with accelerator electron beam [29] and the beam from laser-plasma interactions [30][31][32][33].
In actual situations, the system size of beam-plasma interactions may be much larger than one can simulate. To take the situation into account, PIC simulations are usually performed for CFI investigations, in which the periodic boundary condition is frequently adopted in the direction perpendicular to the beam propagation direction. However, the effects of the system size on magnetic field amplification are still unclear, especially for the case of ultra-REBs. It should be noted that the magnitude and spatial scale of the amplified magnetic fields determine the power and maximum energy of the radiated photons by beam electrons.
This work is devoted to the magnetic field amplification during the nonlinear stage of CFI, taking into account the effects of plasma ions and the system size. Significant differences in the magnetic field amplification and beam energy deposition between the plasmas with movable and immovable ions are found. A parameter scan of the electron beam energy E be and the system size was performed to study how the beam-plasma interaction system parameters affect the magnetic field amplification. It is found that when the system size in the transverse direction is enlarged by a factor of m, the transverse magnetic field energy is found to increase by a factor of m 2 in the case of the plasma with movable ions, in contrast to m with immovable ions. The paper is organized as follows. The simulation model is described in section 2. The magnetic field amplification during the nonlinear stage of CFI and the effects of plasma ions and system size are discussed in sections 3 and 4. Finally, we summarize the work in section 5.

Simulation model
In this paper, a two-dimensional (2D) electromagnetic PIC code IBMP [23,34,35] is adopted and simulations are performed using the transverse simulation plane, which is perpendicular to the beam propagation direction. The initial plasma density n pe0 and temperature are set to be 10 23 m −3 and 2 eV, respectively. For the adopted parameters, the plasma can be considered to be collisionless. The electron beam with an initial momentum u = γ 0 m e v be0 moves along the z-direction (where γ 0 = 1000 is the initial Lorentz factor of the electron beam, m e is the electron rest mass, and v be0 is the initial velocity of beam electrons). The initial beam-to-background plasma density ratio is n be0 /n pe0 = 0.1. In addition, the background plasma electron flow with velocity v pe0 ∼ ( n be0 /n pe0 ) c (where c is the speed of light) is set to neutralize the beam current at the beginning. Periodic boundary conditions are adopted in the simulations. Electrons and ions are treated initially uniformly in the simulation boxes to conserve neutrality. For the reference simulation in section 3, we consider a simulation area of s = 51.2 × 51.2(c/ω pe ) 2 with 512 × 512 cells, and 50 particles per cell for each species, which are varied to check the numerical convergence of the code and stability of the results.

Magnetic field amplification
We begin by comparing the distributions of the electron beam density n be , plasma electron density n pe , and transverse magnetic field B ⊥ = √ B 2 x + B 2 y for the plasmas with movable and immovable ions, as indicated in figure 1. For the movable ions case, a hydrogen plasma with m i = 1836m e (where m i is the mass of ions) is adopted in the simulations. As the CFI develops, the initially uniform electron beam in the simulation region breaks up into filaments pinched by self-generated magnetic fields (figures 1(a) and (e)). At the same time, plasma electrons at the beam filament regions are expelled outwards due to the space charge of the beam electrons and the magnetic force between beam and plasma return currents, forming an electron plasma-void (i.e. plasma cavity) region with n pe = 0 (figures 1(b) and (f)).
Due to the mass difference between plasma ions and electrons, charge separation is formed when plasma electrons are expelled outwards and the uniform stationary ions background limits the outward movement of plasma electrons in the case of the plasma with immovable ions. However, in the movable ions situation, plasma ions can respond effectively to the movement of plasma electrons and satisfy: n pi ≈ n be for r < R and n pi ≈ n pe for r > R (where r is the radial distance to the filament center, R is the filament radius and n pi is the density of plasma ions), as can be clearly seen in figure 1(d). In this way, plasma electrons can be expelled outward gradually and hence the cavity expands gradually, forming a gap between the beam filaments and plasma electrons. As the cavities expand and approach each other, the cavities merge and the size of the cavity increases further, showing a significant difference from the case of immovable ions. Comparing figures 1 (b) and (f), a significantly larger cavity size and higher plasma electron density contrast can be clearly observed in the case of movable ions.  Comparisons of the transverse magnetic energy EB ⊥ (a) and the average beam electron energy E be (b) for the movable (blue solid lines) and immovable ions (red dashed lines) cases. EB ⊥ and E be are normalized to the initial energy of the electron beam E be0 . The growth rate of the instability ΓCFI is also shown in (a) for comparison.
In the CFI, a filament merger event leads to a significant increase in the magnetic field energy and the reduction of the beam energy. Time evolutions of the transverse magnetic field energy E B ⊥ and beam electron energy E be are shown in figure 2 for the two plasma cases. The growth rate of the instability is [12,28,36,37], where β = v be0 /c, α is the beam-to-plasma density ratio, as indicated in figure 2(a). Good agreement is found between the theory and simulation results. The saturation occurs when the beam electron bouncing frequency in the magnetic field is on the same order of the instability growth rate [38]. The transverse magnetic field energy in the plasma with movable ions is about two orders of magnitude higher than that with immovable ions (figure 2(a)), indicating an enhanced magnetic field amplification by cavity expansion and merger. The enhanced magnetic field amplification can be expected from the number of filament merger events that occurred in the plasmas. The merger of plasma cavities leads to the merger of beam filaments. Time evolutions of the number of beam filaments in the simulation regions are shown in figure 3(a) for the two cases. For the beam energy adopted here, the change of beam longitudinal velocity (i.e. the velocity along the beam propagation direction) due to beam energy loss is negligible, and hence the total beam current in the simulation region. At the given time, the number of current filaments left in the simulation region for the movable ion case is less and the average filament current is higher, leading to an enhanced magnetic field generation. The number difference due to cavity expansion and merger can be found after a beam travel time of 3000ω −1 pe in the figure. From figure 1(a), one can see that there is only one beam filament in the simulation region for the case of movable ions, in contrast to five for the immovable case in figure 1(e). Thus the maximum transverse magnetic field is shown to be 4 MG for the movable ions case (figures 1(c) and 3(b)), which is about 2 times higher than that of the plasma with immovable ions (figures 1(g) and 3(b)).
The enhanced magnetic field amplification can also be expected from the spatial scale of the transverse magnetic field. The transverse magnetic fields are distributed in the plasma cavity regions. Figures 1(d) and (h) show the spatial profile of the transverse magnetic field B y for the two plasma cases, in which the boundary effect is removed for illustration. From the figures, the magnetic field reaches the maximum at the edge of the beam filament and then decreases gradually up to the boundary of the plasma cavity. Due to the cavity expansion, the spatial scale of the transverse magnetic field is much larger than the plasma skin depth and reaches 46c/ω pe , which is about 2.5 times larger than that in the plasma with immovable ions. Together with a higher magnitude and larger spatial scale, one thus can expect an enhanced magnetic field amplification in the plasma with movable ions.

Scaling with the system size
To consider the influence of the system size on magnetic field amplification, we further performed PIC simulations with the simulation areas m × s (where m = 1 and s = 51.2 × 51.2(c/ω pe ) 2 are the reference simulation in section 3, and m = 2, 4 and 5 are the simulation cases of expanding the reference size by a factor of 2, 4 and 5, respectively). An overview of the results is shown in figure 4: it is found that after the simulation area is expanded by a factor of m, the transverse magnetic field energy in the plasma with movable ions increases by a factor of m 2 ( figure 4(a)), while in the immovable ion case increases by a factor of m ( figure 4(b)). With immovable ions, due to the low merger rate of the current filaments, the number of beam filaments during the nonlinear stage of the CFI increases linearly with the system size. Judging the filament number from the magnetic field distributions in figures 5(e)-(h), it can be found that when the simulation region is expanded by a factor of 2, 4 and 5, the number of current filaments increases by a factor of 2.8, 4.4 and 5.8, respectively. When the system size is enlarged, the initial beam current also increases by a corresponding factor while keeping the initial beam density n be0 and energy unchanged. In addition, the spatial scale and amplitude of the transverse magnetic field excited by each current filament are comparable with different system sizes, as can be seen from figures 5(e)-(h). Therefore, the transverse magnetic field energy in the plasma with immovable ions increases linearly with the system size and the magnetic field energy density (i.e. the field energy divided by simulation area) keeps unchanged, as indicated in figure 4(b).
However, in the case of movable ions, the current filaments can finally merge into one filament with a radius much larger than plasma skin depth, as shown in figures 5(a)-(d). First, we consider the spatial scale of the transverse magnetic field with different system size. Considering the merger of two identical ultra-relativistic beam filaments with current I 0 and radius R 1 into a single filament with current I and radius R ′ 1 as shown in figure 6(a). The radii of the plasma cavities (which are surrounded by plasma return currents indicated by the red circle in the figure) before and after the merger are R 2 and R ′ 2 , respectively. The filament   currents are assumed to be uniformly distributed and the current density keeps unchanged before and after the merger. From the conservation of beam particles and filament current in the system before and after the merger, one can obtain I = 2I 0 and R ′ 1 = √ 2R 1 . The situation holds for the size of plasma cavity R ′ 2 = √ 2R 2 and therefore R ′ 2 /R ′ 1 ≈ R 2 /R 1 (i.e. the ratio of plasma cavity size to filament size keeps unchanged before and after the merger). Assume that the radius of the final beam filament in the reference simulation is R 1 . Then, enlarging the size of the reference simulation by a factor of 2 can be considered as a further merger of two identical filaments with radius R 1 into one filament with radius R ′ 1 = √ 2R 1 . By analogy, after expanding the system size by a factor of m, the radii of the final current filament and plasma cavity increase by a factor of √ m. Since the transverse magnetic fields distribute in the plasma cavity regions, the spatial scale of the transverse magnetic field increases by a corresponding factor √ m. From figure 7, the spatial scale of the transverse magnetic field in the four cases are: 51c/ω pe for m = 1; 72c/ω pe for m = 2; 94c/ω pe for m = 4; 104c/ω pe for m = 5. It fits well with the √ m scaling: 72c/ω pe ≈ √ 2 × 51c/ω pe ; 94c/ω pe ≈ √ 4 × 51c/ω pe ; 104c/ω pe ≈ √ 5 × 51c/ω pe . The magnetic field energy can be calculated as follows. From Ampere's law, the transverse magnetic field around the current filament shown in figure 6 can be expressed as: where µ 0 is the permeability of vacuum. The magnetic field increases linearly to the maximum at the edge of the beam filament and then decreases gradually up to the boundary of the plasma cavity. Within the thin layer of plasma return current, the magnetic field decreases rapidly to zero, as indicated in figure 7. Then the magnetic field energy distributed in the plasma cavity of radius R 2 due to the filament current of I 0 and radius R 1 can be calculated as: As analyzed before, the ratio of plasma cavity to filament radius (R 2 /R 1 ) keeps unchanged during the filament merger and hence the magnetic field energy E B ⊥ ∼ I 2 0 . For the ultra-REBs adopted in this work, the beam energy loss due to filament mergers is small compared to its initial energy and the current of the final merged filament I 0 increases linearly with the system size. As indicated in the simulations: I m=1 = 88.743 kA, I m=2 = 176.83 kA ≈ 2I m=1 , I m=4 = 351.78 kA ≈ 4I m=1 and I m=5 = 439.76 kA ≈ 5I m=1 . Here I m=1 represents the current of the final merged filament in the reference simulation with the size s = 51.2 × 51.2(c/ω pe ) 2 , and I m=2 , I m=4 and I m=5 represent the cases of enlarging the reference size by a factor of 2, 4 and 5, respectively. Therefore, enlarging the system size by a factor of m leads to amplification of the magnetic field energy E B ⊥ by a factor of m 2 , showing agreement with the simulation results in figure 4(a).   figure 4(a). It should be noted here that the filament currents in the simulations performed in this work are much smaller than Alfvén current due to the high Lorentz factor of the electron beams. For the beam and plasma parameters adopted here, enlarging the reference system size in section 3 by a factor of ∼200 would cause the filament current to exceed the Alfvén current limit. As indicated in the work by Polomarov et al [13], the beam filaments can carry super-Alfvénic currents and develop hollow-current density profiles. The merger of two super-Alfvénic hollow-current filaments would lead to a significant magnetic energy decay. The saturation limit of the filament number left in the simulation region during the nonlinear stage is 1 provided that the initial beam current is smaller than the Alfvén current.
The scaling of the magnetic field amplification with the system size for the case of movable ions is also confirmed with three-dimensional (3D) PIC simulations, using the code EPOCH [39]. The initial beam and plasma parameters are consistent with the 2D simulations. The electron beam propagates along the z-axis and the simulation sizes are 10 × 10 × 20 directions. Different from the transverse 2D simulations, the system is initially dominated by the oblique two-stream instability [28,36] (i.e. the coupled two-stream and CFI) in the 3D simulations, as the chevron-shaped pattern imprinted on the beam density profile shown in figure 9(a). Although differences are found during the initial stage, we mainly focus in this work on the magnetic field amplification during the nonlinear stage of CFI, and the key process such as the cavity expansion and merger show consistence between 3D and 2D simulations. As shown in figures 9(b) and (c), the merger of the chevron-shaped electron beams into a single filament and the plasma cavity around the filament can be clearly observed, showing consistence with 2D simulation results. The variations of the (final) filament current density, transverse magnetic field, and plasma cavity along the beam propagation direction are not significant. Most importantly, the scaling of magnetic field amplification with the system size is still valid in 3D simulations, as indicated in figure 10.
The simulations presented in this work consider initially uniform beam-plasma systems with periodic boundary conditions, which is a reasonable approximation for kinetic scales associated with astrophysical plasmas. To investigate the evolution of CFI in the laboratory, the beam transverse size should be much larger than the plasma skin depth. As indicated in the work by Allen et al, the CFI was investigated in the laboratory with a 60 MeV accelerator electron beam and a capillary discharge plasma, in which the transition from plasma lens effect to CFI excitation is observed when the transverse beam size σ r > 2.2 (c/ω pe ) [29]. In the longitudinal direction, the beam length should be smaller than plasma wavelength λ p = 2π(c/ω pe ) to suppress the excitation of the oblique two-stream instability. We also performed simulations with finite-size beams (not shown here). The results show that after the excitation of the oblique two-stream instability, tilted beam filaments are formed and beam electrons are strongly deflected in the transverse direction, probably due to the hosing instability excitation in the later stage. Parts of electrons in the beam tail regions are strongly deflected in the transverse direction and absorbed by the chamber wall, which is not desirable for experimental investigations. A more detailed study with finite-size beams is deferred to future work.

Conclusion
In conclusion, magnetic field amplification in plasmas during the nonlinear stage of the CFI excited by ultra-REBs is validated with 2D and 3D PIC simulations, with special attention paid to the effects of plasma ions and the system size. In the plasma with immovable ions, the number of beam filaments and hence the magnetic field energy are shown to increase linearly with the system size. While for the plasma with movable ions, the regions of plasma cavity expand gradually under the action of magnetic pressure, and enhanced magnetic field amplification and beam energy deposition are found due to plasma cavity expansion and merger. The beam filaments in the system are shown to finally merge into one filament with a radius much larger than plasma skin depth. For the ultra-REBs adopted in this work, the beam energy loss during the filament merger is small compared to its initial energy and the current of the final filament increases linearly with the system size. As a result, when the system size in the transverse direction (perpendicular to the beam propagation direction) is enlarged by a factor of m, the transverse magnetic field energy increases by a factor of m 2 in the case of the plasma with movable ions, in contrast to m with immovable ions.
In actual beam-plasma interactions, the real system size may be much larger than one can simulate and only parts of the system are considered in the simulations with periodic boundary conditions. In high energy density physics and astrophysics, the magnetic field generation during the CFI is an important issue. The magnitude and spatial scale of the saturated magnetic fields in plasmas determine the power and maximum energy of the radiated photons by beam electrons. In the cases of the plasmas with ion motion, due to plasma cavity expansion and merge, the spatial scale of the magnetic field increases significantly, and the magnetic field energy is shown to increase quadratically with the transverse system size. Understanding the long-term nonlinear evolution and saturation behavior of CFI, especially in terms of field energy and length scale, is critical for astrophysical plasmas. PIC simulations have shown that CFI is important in the formation of relativistic collisionless shocks [40] and non-thermal particle acceleration [41][42][43]. However, plasma streaming instabilities typically saturate at small scales when compared to the gyroradii of the beam particles. These kinetic scales are much smaller than the magnetic coherence length required to explain polarized gamma-ray burst emission [44,45]. The results in this work can have important implications for the magnetization of the precursor of collisionless shocks. It will enable a dilute beam of shock-accelerated particles to drive strong magnetic fields far ahead of the shock, where plasma micro-instabilities are very inefficient. These large-scale magnetic fields are then expected to be advected toward the shock, modifying its structure and affecting non-thermal particle acceleration, radiation emission [46,47], and the magnetic field decay in the downstream region [27,48].

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).