Magnetic fields are ubiquitous in universe, space, and laboratory plasmas. Especially, self-generated magnetic fields are important to know the mind of nature. The formation of Weibel-mediated collisionless shock is studied theoretically as a structure formation by the linear plasma wave growth, nonlinear saturation, and mode–mode coupling. Following a series of computer simulations and experimental studies of the physics, a simple model equation is proposed here to describe the time evolution of magnetic turbulence. Weibel instability is saturated by magnetic pressure, and thicker filaments continue to be generated by current coalescence (magnetic reconnection) mechanism. The model equation concludes the fact that the filament spacing increases linearly in time, and the magnetic energy power spectrum is given as $ B k 2\u2009\u221d\u2009 k \u2212 2$. The time evolution of the turbulence is characterized with the cascade toward smaller k. Such inverse cascade is well-known in 2D hydrodynamic turbulence such as a typhoon or hurricane formation and is known to have Kolmogorov spectrum $ k \u2212 5 / 3$. Although only a small difference in power, the physics of inverse cascades is very different as shown in the present paper. With use of Alfvén current limit condition, the criteria of collisionless shock formation are evaluated. The present theory is compared to corresponding experiments done with Omega and NIF lasers and a variety of PIC simulations. The theory is also applied to evaluate the strength of magnetic field near the shock front of the supernova remnant SN1006. The enhancement of magnetic field of about 25 *μ*G is concluded in the present theory. Finally, a universality of the model equation is shown by applying the theory to the turbulent mixing due to Rayleigh–Taylor instability at the contact surface of two fluids in a gravitational or inertial force, which is very important in compressing plasma such as inertial confinement fusion by implosion. It is shown that the well-known evolution physics, mixing layer of the two fluids grows in proportion to (time)^{2}, can be explained with the same model equation.

## I. INTRODUCTION

Since the plasmas consist of charged particles, the magnetic field is easily generated if the plasmas are in non-equilibrium state thermodynamically. For example, when an electron beam is injected into a plasma at rest for the purpose of heating the plasma, perturbed electric current generates magnetic field and the beam orbits are affected by Lorentz force due to the self-generated magnetic field. We are required to study the stability of such a system with charged beam or multi-beams in plasmas.

Such an instability analysis is the most popular subject in plasma physics, and we can find the subject of “two-stream instability (TSI)” or “beam instability.”^{1} However, most of the textbooks explain this instability as one-dimensional problem in (x, v) space. Since it is a one-dimensional problem, only electrostatic TSI is discussed. Then, the dispersion relation to the frequency ω(k), where k is the wavenumber of Fourier mode of electrostatic field E(x), is derived by standard linear perturbation method.

Nature, however, has the property that any unstable dynamical system changes to the stable state with motion in three dimensions than two or one dimension. This means one-dimensional assumption may exclude the reality of the easiest path due to the other instabilities. In the present paper, one will find that three-dimensional freedom is important to know the instability of such TSI system. Therefore, we have to take into account the magnetic field in the instability analysis.

The magnetic field is very important to study many topics in laboratory, space, and astrophysical plasmas. Especially, magnetic field turbulence is one of the most important subjects in plasma physics. It is possible to enumerate several different physics causing the magnetic turbulence, while in the present paper, we focus on the magnetic field turbulence induced by nonlinear physics of Weibel instability in collisionless plasmas.

Weibel instability is an instability spontaneously excited in the plasma with non-Maxwellian distribution, for example, a multi-beam system or anisotropic distribution.^{2,3} The linear and nonlinear stages of Weibel instability have been studied from the early time of plasma physics research theoretically and computationally.^{4,5} Weibel instability was a good challenging subject for developing the particle-in-cell (PIC) simulation methods. It is known that thin current filaments appear due to the most unstable k-mode in Weibel instability. Then, one sees that these filaments coalescence to form thicker filaments according to the well-known magnetic reconnection phenomena. Finally, the resultant magnetic fields become in a turbulent state and saturate.

Weibel-mediated magnetic turbulence has been studied, for example, by Davidson.^{6} As an application of the physics, Weibel instability has been studied, for example, in fast-ignition laser fusion, where imploded fuel core is heated by relativistic electrons generated by an ultra-intense laser.^{7} The linear to nonlinear evolution in the fast-ignition is well studied with PIC simulation as plotted in Fig. 1.^{8} The time evolution of the magnetic field is shown from Figs. 1(b)–1(d), and Fig. 1(a) shows the initial uniform electron beam size by plotting the radial electric field. The relativistic electron beam injected into high-density plasma become unstable to Weibel instability, and magnetic reconnection makes the structure bigger. This is a typical phenomenon of electron Weibel instability.

In astrophysics, very short gamma-ray signal far from the universe, called gamma-ray bursts (GRBs), has been observed and studied intensively with PIC simulations. The key physics was explained with a model that relativistic electron–positron plasma is generated in a collapsing compact object with Lorentz factor ∼100 to interact with each other to generate strong magnetic field via Weibel instability.^{9} With the strong magnetic field, the GRB signal stems from synchrotron radiation from highly relativistic plasma as gamma-ray.

The standard model of cosmic-ray acceleration is the so-called diffusive shock acceleration (DSA).^{10,11} DSA can explain the power law spectrum of particle energies in many observation data in the range 10^{9}–10^{21} eV. Therefore, DSA has been appreciated as the standard theory for the origin of cosmic rays in our galaxy and far from our galaxy.^{12} In the DSA model, magnetic turbulence and collisionless shock are essential.

Supernova remnants (SNRs) are the most plausible source of cosmic rays in our galaxy,^{13} and active galactic nuclei (AGNs) are a possible candidate for ultra-high energy cosmic rays.^{14} Many astrophysicists have been challenged to explain the origin of collisionless shock and accompanying magnetic turbulence.^{15} Since the e–p plasma has only the mass of electron and the growth rate is larger because of the relativistic velocity of flows, it is relatively easy to use PIC simulation to study from the linear to nonlinear stage to obtain the collisionless shock structure.^{11} However, it has been regarded almost impossible to study such physics in laboratory.

Along with such intensive research in astrophysics and cosmic rays, we have promoted emerging field of laboratory astrophysics with intense laser facilities.^{16} The early proposal of model experiments has mainly focused on radiation hydrodynamics physics. This is because laboratory astrophysics is proposed to do a model experiment of the hydrodynamic instability of Supernova 1987a.^{16}

Then, we paid attention to the fact that Weibel instability in GRBs is a common topic with anomalous transports in laser plasmas in the 1980s.^{17} With the intention of extending laboratory astrophysics to collisionless plasma physics, we have proposed to find out if laser-produced high-velocity ablation plasma can be a test bed to study the nonlinear Weibel instability and the resultant collisionless shock formation in the universe.

The intense laser can generate the ablating plasma with expanding velocity of at highest ten thousand km/s, but of course, this is non-relativistic and ion-electron plasma flow. We challenged ourselves to study the possibility of collisionless shock formation for the case of counterstreaming laser ablation plasmas.^{18} After many simulation data with different conditions, a scaling law was found to clarify the condition with which the collisionless shock formation is possible with an already existing laser facility. It is concluded that only National Ignition Facility (NIF) at LLNL, USA,^{19} has the possibility to demonstrate Weibel-mediated collisionless shock formation.^{20}

Just after the completion of the NIF facility, LLNL announced that 15% of NIF operation will be used for the “discovery science”^{21} and opened to the international community. Our proposal of the collisionless shock formation with NIF has been approved, and a big international collaboration group has been established for the experiment, simulation, and theory.

Many papers have been published on experimental data and analysis as well as large computing. Finally, we also observed particle acceleration by the collisionless shock experimentally with NIF.^{22} The acceleration process is speculated due to the diffusive-shock acceleration (DSA) by multi-bouncing of electrons by turbulent magnetic field.

A lot of works demonstrated the physics of Weibel instability from linear to nonlinear phase experimentally and computationally. However, we have not come to find a simple model including most important physics elements to understand the core physical mechanism to govern the time evolution of Weibel instability and collisionless shock formation, yet.

In this paper, we try to understand the core physics of the nonlinear physics of Weibel instability, the property of magnetic turbulence, and the physical mechanism of the formation of collisionless shock. We have found a simple master equation to govern the linear growth of the Weibel instability, filament coalescence physics (magnetic reconnection), and criteria of the timing of collisionless shock formation. The results of the mathematical model are compared to the experimental data and PIC simulation results published already. It is shown that a simple “toy” model can explain most of important physics obtained in the experiment and PIC simulations.

Inverse cascade and structure formation are also a process of the fluid turbulence in generating a typhoon, as shown in Fig. 2. The typhoon or hurricane is formed by the coalescence of small vortices in two-dimensional space because the size is much bigger than the height of atmosphere and only anti-clock wise rotational vortices are generated by Coriolis force in the north hemisphere.^{23} The tornado is the case at a small scale, and the three-dimensionality is essential to form it via the cascade process. Note that the present magnetic turbulence is always formed by the inverse cascade because it is driven by the instability with larger growth rate at larger k.

The theoretical result is applied to check the observation data of supernova remnant (SNR) SN1006. The amplified magnetic field derived by the present theory is found to coincide with the speculated data. Finally, we apply the present model to another turbulence, a very different turbulence, turbulent mixing of materials due to Rayleigh–Taylor instability. It is concluded that the time evolution of mixing layer can be explained.

In Sec. II, linear analysis of Weibel instability is shown for electron and ion plasma case separately. The ions and electrons are assumed cold for simplicity. In Sec. III, effect of magnetic pressure is found to saturate the growth of Weibel instability. It is shown that the saturated amplitude of magnetic field energy density is proportional to $ k \u2212 2$. In Sec. IV, the results obtained with PIC simulation of non-relativistic Weibel instability are shown to confirm that the collisionless shock is formed and sustained by the magnetic turbulence generated by Weibel instability. In Sec. V, the dispersion relation of Weibel instability expected in laser experiments is obtained for finite temperature case with temperature measured in our laser experiment. Then, the comparison of the nonlinear evolution to the analysis based on the quasi-linear theory proposed in Ref. 24 is carried out.

In Sec. VI, the inverse cascade observed in the PIC simulation is explained with magnetic reconnection and modeling of the cascade is explained. In Sec. VII, experimental results with large-scale lasers are shown to see how the spacing of the filaments grow in time. In addition, other two laser experiments on magnetic turbulence are shown to confirm the power law of magnetic energy spectra. In Sec. VIII, a simple mathematical model is proposed where instability, saturation, and inverse-cascade processes are modeled with one equation. Using the theoretical result, we evaluate the condition of collisionless shock formation. The result is compared to a collisionless shock of a supernova remnant SN1006 in our galaxy.

## II. WEIBEL INSTABILITY

It is instructive to show a simple calculation to obtain the linear growth rate of Weibel instability. For simplicity, consider Weibel instability generated in a very early time only by the perturbing motion of electrons by the thermal noise of magnetic field. In addition, assume that the electron beam plasma is cold and uniform density is n_{0}. The cold ion plasma of the same density n_{0} is at rest in the background and the electron plasma flows in the ion plasma with the relative flow velocity **u**_{0} being constant before Weibel instability is induced. This flow kinetic energy is the energy source to induce the magnetic instability.

**B**and

**E**induced by Weibel instability have to satisfy the Maxwell equation

**j**, (1) and (2) govern the propagation of electromagnetic waves (light). In the plasma, the induced current

**j**in Eq. (2) modifies the dispersion relation. In the present case, the induced current is given by electron motions. This assumption is appropriate in very early time, because ion mass is much higher and the ion current contributes to Weibel instability a long time after the electron Weibel instability.

**E**and

**B**, and $n$ is the electron density. The equations giving the time evolution of the electron plasma can be modeled with the fluid description.

**B**= (0, 0, and B). Then, we know that

**E**= (E

_{x}, E

_{y}, and 0) is required. According to the linear wave theory, all physical values are expended with Fourier-Laplace (FL) modes with (k,ω),

_{x}and E

_{y}, are also expanded as Eq. (6). Note that, finally, we derive a dispersion relation, and ω is given as a function of k. So, for simplicity, we retain only the subscript “k” meaning FL component.

_{0}= 0. Let us see how the flow gives another solution.

Since the dispersion relation Eq. (17) is the fourth-order equation to ω, the plasma will have two different modes; one is the normal electromagnetic field whose plasma frequency or cutoff frequency increases effectively by the flow effect in u_{0} ≪ ω/k. This is just the Doppler effect.

_{e}is kept constant, the electrostatic field by the charge separation from the ion plasma reduces the growth of Weibel instability. After easy mathematics, a new dispersion relation is obtained by replacing the blanket term on RHS of Eq. (17) in the form,

After a relatively rapid growth of electron Weibel instability, heavier ions are also affected by the magnetic field, and the ion Weibel instability is induced. It is expected that the ion Weibel instability is more dominant in the growth of magnetic energy because the source of Weibel instability is the kinetic energy of flow, the mass ratio of which is larger in the ion plasma for the same flow velocity.

It is easy to obtain the dispersion relation for the case of counterstreaming plasma to be used in real experiment with big laser facilities. In Fig. 3, the modification of electron and ion flow in Weibel magnetic field is shown schematically for the case of counterstreaming plasma to explain the mechanism of positive feedback sustaining Weibel instability. It is clear that both anti-directional flows help to generate current channels to enhance the strength of magnetic field.

Let us consider what happens after the linear phase where the exponential growth of Weibel instability is observed.

## III. NONLINEAR SATURATION OF WEIBEL INSTABILITY

**u**

_{0}in the x-direction,

**B**in the z-direction, and the wavenumber

**k**in the y-direction, the force balance Eq. (28) is approximately written to be

_{e}is a typical distance of the magnetic filament spacing, and the gradient of $ B 2$ in Eq. (28) is replaced with 2π/L

_{e}. It is noted that the saturated magnetic energy density increases as the filament spacing increases. It is important to point out that the relation Eq. (29) looks independent of the k-dependence of the linear growth rate of the Weibel instability Eq. (19).

Now, we consider the ion Weibel instability. This has also been studied intensively related to the collisionless shock formation in the universe.^{25} Assume a collisionless plasma consists of two components like a plasma shock passing in plasmas at rest. The electron Weibel instability saturates soon, and after a long time, the ions gradually get perturbed due to the ion Weibel instability.

_{i}is a typical magnetic filament distance by the ion Weibel instability, and ρ

_{i}is the ion inertial length (ρ

_{i}= cω

_{pi}

^{−1}). Equation (32) is similar to Eq. (29), but on the RHS in Eq. (32), the quantities are all from the ions, and consequently, the saturated magnetic field is stronger than the case of electron Weibel instability in Eq. (29). Note that regardless of electron or ion Weibel instability, saturated spectrum of the magnetic energy is proportional to $ k \u2212 2$.

## IV. PARTICLE-IN-CELL (PIC) SIMULATIONS

To find the possible condition to observe the nonlinear evolution of Weibel instability in non-relativistic, counterstreaming plasma, 2D PIC simulation has been carried out.^{18,20} Because of less computational power then, Weibel instability was studied for the cold ion-electron plasma flow with u_{0}/c = 0.1, 0.2, and 0.45 and the mass ratio of 20–100. This is not the real condition of experiment to be proposed, while we can study the physics until collisionless shock formation.

In order to model the counterstreaming plasma and study Weibel instability grow, the simulation starts when the two flowing collisionless plasma meet at the right boundary. The simulation is carried out for the left half plasma assuming the reflective boundary condition. The reflecting particles are regarded as the flow from the right boundary.

Note that this simulation was unique then because we expected to see the formation of collisionless shock and no periodic condition was adopted. Regarding nonlinear Weibel instability, it has been studied computationally in laboratory and space plasmas, while many papers have been published by assuming the periodic boundary condition in the direction of flow (x in the present case). The PIC simulation with periodic boundary condition saturates before the shock wave is formed.

In Fig. 4, typical snap shots of the simulation result are shown in (x–y) plane for ion density (a), current density (b) in the x-direction, and the magnetic field (c) in the z-direction.

The plasma flow from the right boundary has already penetrated beyond the left boundary and the small filaments stemming from the Weibel instability are observed. The instability soon goes into nonlinear phase and saturation, and filament coalescence is observed from the left to the right. Note that the snap shots also represent the time evolution from the left to the right.

The thickness of the magnetic field and current filaments also increases from the left to the right. This filament coalescences are expected from the nonlinear saturation mechanism shown in Eq. (31). However, further mode–mode coupling becomes important during and after the saturation phase. The mode–mode coupling nonlinearity is driven by the magnetic reconnection mechanism, and is to be discussed later in modeling the physics of the simulation and experimental results.

In Fig. 5, the time evolution of power spectrum of magnetic field energy density is plotted in k-space. It is observed that in the early time a broad spectrum of the magnetic field fluctuation of numerical noise is seen, while they start to grow in time. The modes in larger k-region soon saturate as mentioned above. The peak wavenumber of the spectra moves to smaller k modes and the saturated amplitude in larger-k region remains almost constant. The power law is established in the saturated high-k region with $ k \u2212 2$ dependence as predicted in Eq. (32).

In Fig. 6, the density, flow velocity, and the field energy densities are plotted as functions of the x-direction after averaging them over the y-direction in Fig. 3. It is clear that a shock wave is formed with a relatively wide transition region of density. It is well-known in fluid shock that in the shock front, namely, the transition region, the kinetic energy of flow is converted to the thermal energy and entropy increases.

The seeds of magnetic field fluctuation with smaller k are generated continuously by the mode–mode coupling between larger k modes. The magnetic energy flows in the k-space to smaller k-region like the inverse cascade to produce the typhoon in Fig. 2. After structure formation of collisionless shock, the spectrum becomes almost stationary in the frame moving with the shock front. Then, shock jump relation is satisfied like the Ranking–Hugoniot relation in hydrodynamic shock wave.^{26} Note that the density jump by the collisionless shock in Fig. 6 is 3, not 4, because the simulation is in 2D and the number of freedom is 2, not 3. The density jump is given as $ \gamma + 1 / ( \gamma \u2212 1 )$, and the specific heat $\gamma = N + 2/N$, where N is the number of freedom.

It is important to insist that the saturated power spectrum of magnetic energy has a power law of $ B k 2\u2009\u221d\u2009 k \u2212 2$. Most of PIC simulations of magnetic turbulence have reported that they found the Kolmogorov-type spectrum $ k \u2212 5 / 3$ by using the analogy of Navier–Stokes turbulence. However, the spectrum $ k \u2212 5 / 3$ is mathematically derived in the inertial range where turbulent energy is conserved and transferred to larger k (cascade) or smaller k (inverse cascade) depending on where the source of vortices is given in the k-space. We will discuss the inverse cascade and $ k \u2212 2$ spectrum later in modeling the physics.

In the present collisionless shock, the magnetic turbulence induced by Weibel instability sufficiently bends the flow motion of ions randomly to terminate the directional flow. The bending of the flow plays the same role as viscosity in neutral fluid, stemming from the scattering of atom an molecule particles. As the result, a shock wave is formed without particle collision effect. This is the collisionless shock wave sustained by magnetic turbulence.

There are two different collisionless shocks; one is the electrostatic shock, and the other is turbulence-induced shock. Note that it is very different from the collisionless shock sustained by electrostatic field usually described by Sagdeev potential.^{1} Of course, the electrostatic shock has been experimentally studied.^{27} It is appropriate to say that the electrostatic shock wave is generated in one-dimensional system, while the latter is possible only in multi-dimensional system. It is widely accepted that most of the shock waves in the universe are induced by magnetic turbulence. The physics of the turbulence-induced collisionless shock formation is still an open question and has been studied intensively because the collisionless shocks are regarded to be an engine for accelerating charged particles, which are observed as cosmic rays on the earth.^{28}

In Fig. 6(b), the plasma flow is stagnated by the reflection of particles by magnetic force, and as a result, the density increases to form a shock structure. It is surprising to know in Fig. 6(c) that although it seems a strong magnetic field may grow to reflect the plasma flow, the magnetic field energy density is only about 1% of the kinetic energy density of the plasma flow. It is not the case where idealistic magnetic turbulence may follow the equipartition of energy and be of an order same as the plasma thermal energy.

The PIC simulation data have been used to obtain a scaling law to study a feasibility of a model experiment with high-power laser facilities in operation throughout the world. Inserting realistic flow velocity u_{0} =1000 km/s and plasma density n = 10^{20} cm^{−3}, it is found that the shock formation time is about 4 ns and the shock thickness is about 2 mm. This indicates that the plasma kinetic energy of 2 mm cube is about 100 kJ; consequently, only the NIF at LLNL has the potential to demonstrate the Weibel-mediated collisionless shock. Note that the scaling also says the generated magnetic field is about 1 MG. This value is much larger than *μ*G or mG speculated strength of magnetic field in the universe, because the plasma density is about 20-orders of magnitude higher in the experiment than in the universe.

In parallel with the experiment, more detailed calculation for finite temperature plasma has been carried out for the design and analysis of the real experiments, where laser-ablation plasma has relatively high temperature automatically.

## V. WEIBEL GROWTH RATE IN EXPERIMENTAL CONDITIONS

^{6}In the present case, the electrons and ions are assumed to be given with Maxwell distribution with their temperatures and the flow velocities. Vlasov equation for electrons and ions (α = e, i) is given as

^{24,29}to be

In the laser-produced ablation plasma experiment, high-Mach number flow of M ∼ 10 is produced to the ion sound velocity $ u 0 > v i$, but the electron thermal speed is much higher $ u 0 < v e$ and the electron Weibel instability is stable. As already mentioned, however, most of the magnetic energy stems from the ion flow kinetic energy; consequently, we can expect the formation of magnetic turbulence and shock formation only if the ion Weibel instability is unstable.

In the experiments using laser ablating plasmas with OMEGA laser,^{30} the plasma temperature is measured to be around T_{e} $\u2248$ T_{i} $\u2248$ 1 keV, and electron density is about 10^{19} cm^{−3} in a plastic (CH_{2}) plasma.^{31} The flow velocity of both counterstreaming plasmas is u_{0} = 1900 km/s. The experimental parameters are inserted in Eq. (38) and the growth rate was calculated. In Fig. 7, the dispersion relations are plotted for different parameters.

The temperature dependence is shown for T = 0.1 keV (green), 0.5 keV (magenta), 1 keV (blue), and 2 keV (red).^{31} The blue dashed curve is for pure carbon plasma. It is clear that the ion Weibel instability is excited in the experiment, although the electron Weibel is stabilized by the pressure effect. The dispersion relation of Eq. (24) is also plotted with the black curve. It is noted that the Weibel growth rate is large enough to observe the nonlinear stage of the instability, and the filament coalescence phenomenon has been clearly observed as shown later.^{31,32}

Another PIC simulation with finite temperature effect in the initial condition has been carried out by Ruyer *et al.*,^{24} and the so-called quasi-linear theory has been applied to explain the physical reason for the filament coalescence by assuming periodic boundary condition in the direction of flow. The periodic boundary condition prevents the formation of collisionless shock, but it is good to study nonlinear evolution of filaments. The detail of the simulation is given in Ref. 24, and we would like to briefly describe their study to show the difference in the physics for the filament coalescence phenomena.

In Fig. 8, a time evolution of the field energies is plotted for $ E x,\u2009 E y,\u2009 and \u2009 B z$. It is seen that in a very early time, $ E x$ (black line) grows exponentially, while it soon saturates and decreases its amplitude. This is confirmed in the inset figure. This instability is the so-called electrostatic two-stream instability (TSI), and it is essentially a one-dimensional phenomenon. Since the growth rate of TSI is larger than that of Weibel instability, TSI has been studied intensively and has been applied to space and astrophysical plasmas.

As seen in Fig. 8, however, the Weibel instability shown with the growth of $ B z$ (blue line) grows dominantly even after the saturation of TSI. The first maximum at $t \omega p e\u223c70$ is due to electron Weibel instability, and further growth to the saturation level is due to the ion Weibel instability. The electric field $ E y$ is induced by the charge separation of ion and electron as shown in Fig. 3 schematically.

^{6}the time evolution of $ B z 2$ is given to be

^{24}Solving Eq. (38) coupled with Eq. (43) and related quasi-liner equations to $ u 0 t$ and $ v \alpha (t)$, the modes with larger k is stabilized by the increases of the temperatures and the wavelength of most unstable mode decrease in k-space.

The quasi-linear theory gives the saturation of the instability due to the reduction of $\gamma k , \u2009 t$ by the modification of the plasma flow and temperatures in time. Note, however, that it does not include a highly nonlinear physics such as mode–mode coupling to be discussed in our model. This theory, quasi-linear theory, is well-known as week nonlinear theory. It cannot describe the whole nonlinear physics in the present problem.

Since the quasi-linear theory has led the wavelength with the maximum growth rate to longer k (smaller k), it is concluded in Ref. 24 that the quasi-linear theory can explain the phenomena of thickening of the filaments seen in PIC result. They obtain a simple model equation for the filament size, $ \lambda sat t,$ and it gives later time evolution as $ \lambda sat t\u2009\u221d\u2009 t 2.$^{33}

In their theory, the phenomenon of thickening of filaments over time is due to the wavenumber of the maximum growth rate becoming smaller over time. However, we will insist that both nonlinear effects of the saturation by magnetic pressure derived already and the magnetic reconnection stemming from the mode–mode coupling are essential. Then, our nonlinear theory to be explained soon predicts that the filament size evolves in time as $ \lambda sat t\u2009\u221dt$. We compare this with experimental data for a proof of our theory.

## VI. MAGNETIC RECONNECTION AND INVERSE CASCADE

From the linear to nonlinear phase of Weibel instability, there are at least two nonlinear effects important for the time evolution of the power spectrum of the magnetic energy density shown in Fig. 5. The quasi-linear theory is a week nonlinear theory, and the saturation is due to the quasi-linear diffusion in velocity space of the distribution function of the background plasmas. In the case of Weibel instability, the diffusion of the distribution function to the perpendicular direction in time stabilizes Weibel instability for larger k-mode.

The image of the three physical quantities in Fig. 4, corresponding to the time evolution of Weibel instability from left to right, suggests that the filamentary structure change and the filament sizes continuously become thicker, fatten in time. This evolution terminates when the collisionless shock wave is formed. In phenomenology, the quasi-linear theory may explain the time evolution; however, it is not true from the physical reason mentioned herein.

The same evolution is also observed in the case of three-dimensional simulation shown in Fig. 9.^{34} In Fig. 9, two bird-eye snap shots are shown for the electron density at Fig. 9(a), $ t \omega p e$= 824 and Fig. 9(b), $ t \omega p e=$ 1331. A collisionless shock is just formed at the right end layer in Fig. 9(b). It is clear that small filaments coalescence to form bigger ones from left to right as we saw in the two-dimensional simulation in Fig. 4.

The coalescence, inverse cascade, and collisionless shock formation in the nonlinear Weibel instability are common phenomena, regardless of two- or three-dimensional cases. A huge vortex of typhoon seen in Fig. 2 is also due to the inverse-cascade process from small vortexes, while it is formed by the two-dimensionality of atmosphere with several hundred km span but the height of several km. The structure formation of vortexes in the atmosphere of three dimensions is a tornado due to the cascade process of turbulence.

Such coalescence of filaments has been reported in many papers from early time,^{4} relating to relativistic electron beam ICF (inertial confinement fusion). In Fig. 10, the filament dynamics is well captured for electron Weibel instability in 2D PIC simulation, where the equi-contour is the amplitude of magnetic field. This is a study of the fast-ignition scheme, and the physics is studied to know how one-directional relativistic electrons penetrate in the dense plasma.^{35} It is clearly seen that three filaments at t = 120 coalescence to finally become one fat filament at t = 148. This can be also called magnetic reconnection from view of the change of magnetic field topology.

In the case of the present counterstreaming plasma, filaments are not one directional current, but flows are in the opposite directions to each other. In Fig. 11, a group of circle-like filaments in opposite directions are schematically drawn with colors showing bottom-to-top and top-to-bottom electric currents. When such current filaments are filled randomly in the plasma, the interaction by Lorentz force works among the current filaments. The purple and green colors are the cut view of the current filaments in 3D space, and the colors mean the direction of the currents “J” from top to bottom (purple) or bottom to top (green). The filaments are surrounded by magnetic field as shown with the water blue circle. The Lorentz force of $j\xd7B$ is attractive (red arrow) between the same directional filaments, while it is repulsive (purple arrow) between the anti-directional filaments.

Such random system with many current filaments in both directions is dynamically unstable, and the current filaments flowing to the same direction merge with each other with time, where the magnetic reconnection continuously takes place as schematically shown from left to right in Fig. 11. A mathematical modeling of such physics gives us the time evolution of the magnetic energy power spectrum in k-space as shown previously.

With use of the magnetic reconnection physics, the time evolution of magnetic power spectra plotted in Fig. 5 is consistent with the time evolution of inverse cascade. In addition, it is reasonable to speculate that the wavenumber dependence of the saturation condition Eq. (31) keeps the power spectrum with negative slope.

## VII. EXPERIMENTAL RESULTS

In order to confirm the physics and its time evolution scenario regarding the nonlinear ion Weibel instability and resultant collisionless shock formation,^{20} large-scale lasers have been used. The counterstreaming ablation plasmas are produced by irradiation of lasers to two target plates facing each other. This is a fundamental experiment to verify the fact that the collisionless shocks whose concept is widely used in astrophysics and space physics can be really produced by magnetic turbulence induced by a plasma instability.

The time evolution of the structure of the filaments has been measured in the collisionless shock experiments performed with OMEGA and NIF laser facilities.^{31,32,36} The configuration of the experiment and diagnostics is simple as schematically shown in Fig. 12. Two bunched beams are focused on two foils with a separation of 8 mm. The laser with 1 ns pulse duration and 4 kJ each irradiates the foil targets to generate collisionless counterstreaming two plasma flows.

The generated magnetic field has been observed as the perturbation of straight path of many protons generated by capsule implosion. For the energetic proton generation, D^{3}He fusion fuel is imploded by 1 ns laser with 9 kJ. The image of the protons after passing through the plasma region is affected by Lorentz force by generated magnetic field. These data can be used to observe the magnetic field. Since the proton beam is generated during a short time of about 1 ps, it is possible to obtain a snap shot in two-dimensional space. For this purpose, the Omega laser facility was used, and a direct imaging of the magnetic field filaments has been performed successfully by use of proton backlight technique.

As clearly seen in two data on the right in Fig. 12, the image of proton on CR37 detector has no perturbation in the single foil case, while filament image is obtained for the two-foil case. This is clear evidence of Weibel instability with filamentary structure of magnetic field in the counterstreaming ablation plasmas.

^{32,36}It is clear from both data that the filament spacing increases linearly in time. The gradient of the lines in both figures, indicating an effective velocity of the spacing increase, is reported to be from Figs. 13(a) and 13(b), respectively, as follows:

The statistical distribution of the filaments as a function of their spacing is also measured and shown at a given time in Fig. 14.^{32} It is too early to say that the shorter part of the filament spacing distribution is like a function of k^{−2}, namely, λ^{2} as given in Eq. (31). Note that some relation to explain the full spectrum of Fig. 14 is required.

The second example of experimental evidence is about the power spectrum $ k \u2212 2$ in Eq. (31) in electron Weibel instability. In Refs. 37 and 38, on the other hand, a direct measurement of turbulence spectrum in giant magnetic fields has been reported from a laser-plasma experiment. In the experiment, magnetic turbulence is generated in an over-dense, hot plasma by relativistic intensity (10^{18}W∕cm^{2}) femtosecond laser pulses. A magneto-optic measurement at femtosecond time intervals is used to observe snap shots of magnetic field by use of polarization of an observation laser beam. The spatial profiles of the magnetic field show randomness. Detailed two-dimensional particle-in-cell simulations delineate the underlying interaction between forward currents of relativistic energy “hot” electrons created by the laser pulse and “cold” return currents of thermal electrons. Electron Weibel instability is observed, while heavier ions are almost at rest.

The time evolution of power spectra from 0.2 to 7.0 ps are plotted in Fig. 4 of Ref. 37, where the authors concluded that inverse cascade makes the spectrum in power law. It is found that the power law is $ B k 2\u2009\u221d\u2009 k \u2212 2$ as suggested in Eq. (31) due to the balance of linear force and nonlinear magnetic pressure force to electrons.

The third example is also an experimental observation of the $ k \u2212 2$ power law in laser generated magnetic turbulence. The power spectrum of magnetic turbulence is also reported in a different experiment to study Biermann battery effect generating magnetic field.^{39} This resembles the case of Weibel instability, and the saturation condition Eq. (28) can also be expected even in MHD (magneto-hydrodynamic) plasma. It is useful to see the turbulent magnetic spectrum. The present theory has been developed for the case of Weibel instability, while the formulation of the power spectrum Eqs. (32) and (31) is also applicable to the magnetic turbulence by Biermann battery effect. Once a wide spectrum of the magnetic field is produced by this effect as shown in Ref. 39, the over-saturation of the magnetic field components soon reduces to the level satisfying Eq. (32) in each k. Then, the power spectrum of $ k \u2212 2$ is observed in the experiment after a relatively long time. The spectrum in Fig. 4 in Ref. 39 is transferred to that of k as indicated in Supplement.^{39} It is concluded that the transformation gives $ k \u2212 2$ magnetic power spectrum.^{40}

## VIII. THEORETICAL MODELING

^{31}The resultant growth rate for the parameters in the experiment are approximated by a linear function of k for $k \rho i<10\u2009 \rho i = c / \omega p i$,

We already know that the dominant wavenumber of the Fourier component of magnetic energy moves from large k to small k with time, namely, the inverse cascade progression. Since the initial perturbations of all modes are small enough, it is realistic that most of the small k-modes are seeded by the nonlinearity of the saturated amplitudes with larger k-values. This ansatz is reasonable from the dynamics of magnetic reconnection explained in Fig. 11. This is also because the linear growth rate in Eq. (45) is larger for larger k.

Consider a simple case where k_{1} and k_{2} modes saturate earlier than k mode, because both k_{1} and k_{2} > k. During and/or after the saturation given in Eq. (31) for k_{1} and k_{2} modes, assume that the magnetic reconnection occurs to form a larger-scale filament with k. Since the amplitude of k-mode is smaller than the saturation amplitude of Eq. (31), the amplitude grows exponentially in time. The series of the processes is schematically shown in more simplified form in Fig. 9. After the linear growth of the mode 2πk (=λ), the saturation by magnetic pressure terminates the liner growth and the formation of the seed of the linear growth of k/2 (=2 λ) mode via magnetic reconnection. These three steps are repeated as shown in Fig. 9 to evolve the magnetic energy spectrum as shown in Fig. 3.

^{23}or drift wave turbulence in Tokamak.

^{41}It is known that such standard inverse cascade is formed as “inertial range” of turbulent spectrum.

^{23}That is, only the nonlinear term transfers the energy of turbulence from larger k region to smaller k region, and there is no energy generation and loss in the inertial range. The k-dependence of the nonlinear energy transfer velocity in k-space results in a power law spectrum. It is informative to point out the difference to the well-known Kolmogorov spectrum. The Kolmogorov spectrum is a power law in fluid kinetic energy spectrum in the form

Different from the standard inverse cascade with Kolmogorov spectrum, the present inverse cascade, like dynamics of magnetic energy, is driven by the instability growth in all k-space. The magnetic field energy flows to the lower k direction due to the magnetic reconnection, but it should be noted that the magnetic energy not only flows in k-space but also it is generated at all k-points. Let us consider a theoretical model including all phenomena of the ion Weibel instability.

_{k,0}is the initial value of the amplitude of the mode k. Taking the time derivative of Eq. (48), the following equation is obtained:

As a continuous process, we assume that Eq. (49) is an equation giving the linear growth and filament coalescence dynamics after the saturation limit given in Eq. (32). That means Eq. (49) is used as a master equation governing the time evolution of the magnetic field near the saturation point, magnetic reconnection process, and the time variation of the wavenumber k [ = k(t)], where these nonlinear physics are dominant. We assume the solution of time evolution of the magnetic energy density like the solid black lines schematically shown in Fig. 15.

Compare the present model to the other studies insisting the time evolution of the spacing of filaments in PIC simulations. Another mathematical model has been proposed based on hierarchical and self-similar processes of magnetic coalescence. For example, it is reported in Ref. 42 that the spacing of the magnetic fields grows in time as a power law of time for relativistic Weibel case, but exponential form for non-relativistic case. In the relativistic limit, the measurement of 2D PIC simulation, the power law on time, is concluded 0.8 as shown in Fig. 16.^{42} This model may need further study based on the first principles of the physics. In Fig. 16, fitting curves with a power of 0.8 and 1.0 of Eq. (51) are plotted by black and red lines, respectively. It can be also said that the 2D PIC simulation also roughly shows the present relation Eq. (51).

## IX. COLLISIONLESS SHOCK FORMATION

The studies on the nonlinear saturation physics of the magnetic turbulence have been published from early time,^{6} where it is concluded that the magnetic field saturation of Weibel instability occurs when the linear growth rate is equal to the bounce frequency of particles in the magnetic channel. It can be said that the so-called Alfvén current limit^{43} represents the saturation of the growth of the magnetic field and current filament.^{44} This is evaluated with the condition that ion Larmor orbit is confined in the magnetic field, the magnetic channel in the present case.

In Fig. 17, charged particle orbits in a uniform column current flow in z-direction are plotted. The magnetic field is produced around the z-axis self-consistently. The orbits a and b start at z = 0 point near the z-axis, and c and d are a bit far from the axis. The orbits e and f plotted with the dotted lines are bent by the self-magnetic field and return backward. Roughly speaking, the charged particle with a certain distance from the center cannot carry the current, and there is the maximum current carried by a big filament current. This is Alfvén current limit.

_{B}and ε

_{K}are the energy densities of the magnetic field and ion flow, respectively. We assume that the termination condition of the growth of magnetic field amplitude is roughly given by

This fraction of the energy conversion to magnetic energy is reported as 1.5%,^{44} and in many other papers^{24} as roughly $\sigma $ = 1%–10% even for the case of highly relativistic gamma-ray burst (GRB) simulations.^{25} In the present simulation in Fig. 6, this fraction is very small, $\sigma \u2009\u223c1%$ compared to Eq. (61). Probably, the present simple model for magnetic field saturation and the termination condition of cascade Eq. (60) is too rough to predict the ion motion in the magnetic field to explain the computational result. It is, however, seen in Fig. 5 that the inverse cascade is terminated when the peak of the magnetic energy arrives at $ k \rho i\u2009\u223c1$, as suggested by Eq. (63), and no more growth of the inverse cascade is observed.

Here, it is useful to see the electron and ion distribution near the collisionless shock front to see the force balance to keep a stationary shock wave. In Fig. 18, the distributions of the x-components of electron and ion velocities are plotted as a function of x-space. It is clear that electrons are heated near the shock front and a substantial fraction of ions are reflected back by the random magnetic fields. The electron thermalization is due to induced electric field by random magnetic field in Eq. (1). Since the electrons are randomly scattered inelastically, they are heated and gain more energy from the ions via coupling with magnetic field.

On the other hand, the larger inertia and Larmor orbit of ion cause the direct reflection of substantial fraction of ions toward the upstream region as seen in Fig. 18. The momentum transfer by such reflection of ions balances the pressure by heated electrons to keep the shock wave structure stationary. Most of the collisionless shock interactions with charged particles may be kept as the physics shown above, and the turbulent magnetic field always accompanies the shock waves driven by the magnetic turbulence. It is not an exaggeration to say that the structure of a shock wave is maintained by the magnetic turbulence, and such a collisionless shock wave is due to a structure formation in a complex system.

### A. Application to SNR shock

Let us try to apply the present theory to the collisionless shocks in the universe. We choose the supernova remnants (SNRs) because x-ray astronomy observation has provided precise parameters of some of them. Let us apply this to the case of SNR SN1006, whose explosion was observed about 1000 years ago. Its present x-ray image expanding over 40 light years is shown in Fig. 19.^{45} It is reported that the density is about 1 cm^{−3} and the shock velocity is 3000 km/s by observations.^{45} After the clear observation of x-ray image by Chandra x-ray satellite, it became a big debate why as to how the cosmic-ray electrons with TeV can be confined to a thin precursor region of the shock wave.^{46} It is now widely accepted that the magnetic field near the shock front is amplified by some mechanisms.^{47}

With observation data, Coulomb mean-free-path was evaluated to be about 100 light years, and the system is collisionless. However, we can see clearly the edge of the spherical shock. The x-ray intensity profile is observed only near the shock front. The x-ray is speculated to be emitted by cosmic rays accelerated and confined by strong magnetic field. Our speculation is shown as the source of strong magnetic field and the formation of collisionless shock wave in a very narrow region near the shock front.^{45}

The present ion Weibel instability is a possible candidate and the amplitude of the magnetic field at the shock front. Inserting the shock formation condition Eq. (61) with the observation parameters, we can evaluate the amplified magnetic field.

_{s}= 3000 km/s).

^{45}Inserting Eq. (64) into Eq. (59) with Eq. (61) and assuming $ n i=0.06\u2009 cm \u2212 3,$

^{48}we obtain the amplified magnetic field strength as

^{48}

Of course, the ion Weibel instability is one of candidates to explain the formation of collisionless shocks in the universe, and it may provide magnetic turbulence indispensable for the generation of cosmic rays in the diffusive shock acceleration model.^{28} It is better to point out another candidate for magnetic turbulence, the so-called Bell instability.^{15} It should be noted that the former does not require any background magnetic field, and magnetic turbulence is generated in any space where some explosions or jet generate counterstreaming plasma in collisionless condition. However, the latter demands background magnetic field for magnetic instability. The Weibel mediated magnetic turbulence is more ubiquitous in the universe.

## X. UNIVERSALITY OF THE THEORY

We can enumerate many cases where instability grows exponentially to saturate due to a nonlinear force; then, mode–mode coupling becomes dominant to cascade the unstable fields like B or E and unstable particle fields like density and velocity to finally come to the turbulent state. If the essential physics mechanism is the same, we may apply the present model to another turbulence. Let us briefly apply the present model to the problem of turbulent mixing of materials induced by Rayleigh–Taylor (RT) instability. This is one of the most important physics in implosion of the inertial confinement fusion (ICF).

It is better to see an example of RT turbulent mixing generated in a tank with a rocket loaded on the top of the tank.^{49,50} The left image in Fig. 20 is a tank at rest flipped upside down so that the heavier fluid comes to the upside and the lighter fluid is downside so that the readers can imagine the physics easily. The “g” is the gravity on the earth and the light fluid is on heavy fluid initially as the stationary state. The rocket on the top (bottom) of the fluid tank is fired abruptly to generate ten times the gravity (G = 10g) to accelerate the tank downward (upward).

As seen in the picture on the left in Fig. 20, the contact surface of two fluids is smooth with no long wavelength perturbation. On the right figure, on the other hand, large-scale mixing layer (black region) is seen. The black area is the turbulent mixing zone. In the transmission diagnostics, the black area is where the light for measurement is diffusely reflected and not transmitted. The background physics is simply understood by the fact that the “nature” takes the shortest way to stabilize the unstable system. The excess potential energy by G is converted to the kinetic energy of fluids as fast as possible.

^{51}

^{52}It is interesting to compare their model to such model derived based on an intuitive insight. Note that Eq. (68) is a saturation condition of a single mode, and Haan has proposed the so-called Haan's saturation model

^{53}as used in Ref. 52.

## XI. CONCLUSION

Magnetic field is ubiquitous in the universe, space, and laboratory plasma. It is usually observed as magnetic turbulence. Many physical mechanisms have been proposed for inducing magnetic turbulence. In the present paper, the generation of magnetic field and its nonlinear development due to the ion Weibel instability are considered. It has been confirmed experimentally and computationally with the use of counterstreaming ablation plasmas generated by intense lasers.

The power spectrum moves in time from large k to small k. Such evolution in power spectrum is common in a 2D hydrodynamic system, and a typhoon or a hurricane is a well-known example of the inverse cascade in turbulence. Although both behaviors in power spectra look similar, the physics is very different. It is well-known that 2D fluid turbulence shows the steady state spectrum as $ k \u2212 5 / 3$.

This Kolmogorov spectrum is the same as in 3D fluid turbulence for the inertia rage. In the case of the Weibel turbulence, the power law $ k \u2212 2$ is formed via balance of the linear growth and saturation. Different from Kolmogorov spectrum, the power law −2 is determined not only conserving energy from in the k-space like the inertial range, but also k-dependent instability supplies turbulent energy to all unstable k-modes. This effect causes the difference from Kolmogorov spectrum.

The model equation also predicts how the growth of the filament spacing increases linearly in time $(\u27e8L\u27e9\u221d\u2009t)$. In modeling the physics, the growth of current filaments due to the magnetic reconnection is explicitly used to obtain a simple master equation to the time evolution of the spacing. This relation is well confirmed by Omega laser experiments. In addition, PIC simulations also support this time dependence.

It is clear in PIC simulation that at a certain time such time evolution terminates because of the formation of collisionless shock. The criterion of a collisionless shock formation is considered theoretically, and it is concluded as follows. When the filament spacing becomes almost the ion inertia length, the ions flows are trapped in a filament by Larmor rotation and further current cannot continue to increase the intensity of current filament. This is the same physics as Alfvén current limit. Then, the ion flows start to stagnate or reflect to form a shock-like structure. It is surprising that such phenomenon is induced by the magnetic energy density about 1%–5% of the flow kinetic energy of ions as seen in many of PIC simulations.

The present theoretical result is applied to estimate the strength of magnetic field near the shock front of supernova remnant (SNR) SN1006. It is concluded that it is about 20–80 *μ*G for plasma density 0.06 cm^{−3}. This value is about 100 times enhanced compared to the ambient magnetic field. This suggests that the Weibel instability is a leading candidate to mediate a collisionless shock in SNR and confine electrons and ions on both sides of the shock front. Since the magnetic field is turbulent, it may also contribute the diffusive shock acceleration to generate cosmic rays as suggested in NIF experiment. It is interesting to study that the magnetic turbulence with power law $ k \u2212 2$ provides the accelerated particle energy spectrum proportional to (energy)^{−2}.

Finally, we found a universality of the master equation in applying the model to the turbulent mixing of two fluids in the Rayleigh–Taylor unstable system. It is found that with the same dimensionless factor in the model, the well-known experimental and computational results are predicted with this universal master equation, although the physics is very different and dispersion relation is also different in both cases. It is expected to extend the present master equation to the other turbulences.

## ACKNOWLEDGMENTS

I would like to thank T. E. Cowan and R. Sauerbrey for supporting this research in HZDR. I also thank Y. Sakawa, Y. Kuramitsu, B. Remington, and H.-S. Park for a long collaboration on the Weibel-mediated collisionless shocks. I also thank Pisin Chen for his hospitality to modify the present paper, and Min Chen who provided me the opportunity for publication of the present paper. This research has been done for the HiBEF project of HZDR (Dresden) and HED in Europe XFEL (Hamburg) as a proposal of future experiment in HiBEF.

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to disclose.

### Author Contributions

**Hideaki Takabe:** Conceptualization (lead); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Validation (equal); Writing – original draft (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*Introduction to Plasma Physics and Controlled Fusion*

*Method in Nonlinear Plasma Theory*

*Fluid Mechanics*