Effects of collisions on the generation and suppression of temperature anisotropies and the Weibel instability

The expansion of plasma with nonparallel temperature and density gradients and the generation of a magnetic ﬁeld via the Biermann battery is modeled using particle-in-cell simulations that include collisional effects via Monte Carlo methods. A scaling of the degree of collisionality shows that an anisotropy can be produced and drive the Weibel instability for gradient scales shorter than the mean free path. For larger collision rates the Biermann battery dominates as the cause of magnetic-ﬁeld generation. When the most energetic particles remain collisionless the Nernst effect causes the Biermann ﬁeld to be dragged with the heat ﬂux piled up and enhanced. DOI: 10.1103/PhysRevResearch.2.033233


I. INTRODUCTION
Identifying the mechanisms responsible for the generation of various magnetic fields present throughout the universe is a major topic of study in astrophysics [1].Two common candidates are the Biermann battery [2,3], which is driven by violent interactions with unmagnetized plasmas that leave the temperature and density gradients misaligned, and the Weibel instability [4], driven by temperature anisotropies.The Weibel instability in particular is only possible in collisionless systems, often found in astrophysics due to high temperatures and low densities of plasmas in space.In laser-plasma interaction experiments on earth, both the Biermann battery [3,[5][6][7] and the Weibel instability [7][8][9][10] play an important role.The sudden heating of the plasma by a laser leads to the gradients required for the Biermann battery, and although the densities can be large, the temperatures can be sufficiently high that the plasma is collisionless, can become anisotropic, and thus become unstable to the Weibel instability.
A natural question is; what level of collisionality is required for the Weibel instability to be suppressed?It was shown in Refs.[11,12] that for a collisionless system, the Weibel instability is the dominant magnetic field for sufficiently large gradient scales, L/d e > 100, where L is the temperature or density gradient length scale, d e = c/ω pe is the electron skin depth, ω pe = 4πn e e 2 /m e is the plasma frequency, m e , e, n e are the respective electron mass, charge, and density, and c is the speed of light.However, this is no longer the case for a sufficiently collisional plasma (e.g.Ref. [13]), where the Biermann field becomes the dominant field.Here we show, using PIC simulations, what level of collisionality is required for the Biermann field to dominate over the Weibel instability.
Although the original Weibel formulation assumes a collisionless plasma, the instability has been formulated for a semi-collisional system showing a dependence on the electron collision rate ν e [14,15].
Here γ W 0 is the collisionless growth rate of the Weibel instability [4], which depends on the perpendicular electron temperature T e⊥ , ω pe , and the temperature anisotropy A = T e /T e⊥ − 1, where T e and T e⊥ are the two temperatures in a bi-Maxwellian distribution.Here, parallel is defined by the direction that has a different temperature, and we have assumed T e > T e⊥ .The collision rate is ν e = ν 0 (1 + Z) where T e ≡ (2T e⊥ + T e )/3 is the electron temperature, ln Λ C is the Coulomb logarithm, and Z is the degree of ionization.ν e can be divided into electron-electron collisions ν ee = ν 0 , and electron-ion collisions ν ei = ν 0 Z.It turns out, however, that the modifications to γ W due to collisions are not relevant in most regimes.The growth rate drops to zero even when these modifications are negligible, because the instability is driven by A and collisions cause A to decay.The ratio of the collisional term in Eq. (1) to γ W 0 is ∼ ν e /(ω p v T /c) = (d e /L T )ν e L T /v T , where L T ≡ T e /dT e /dx is the temperature gradient scale, and v T = T e /m e is the thermal velocity.As long as L T d e , the collision term can be neglected when ν e L T /v T ∼ 1, which we will show is when the Weibel instability is suppressed.
The growth rate drops to zero when the time scale of the anisotropy generation t Ag reaches the time scale of collisional relaxation time of the anisotropy t Ar .Ref. [16] showed that a gradient in an isotropic Maxwellian temperature leads to a temperature anisotropy A, saturating at a time scale t Ag ≈ L T /v T with a value A ∼ 1. Refs.[17,18] showed that the relaxation rate is The factors of 1 and √ 2Z are a result of the respective electron-electron and electron-ion collisions.One finds ν A ∼ ν e using Eqs.(3)(4), since F 2 3 2 7 2 (x) is a constant of order unity (approaching 1 for small A) as long as A 1. Remember the anisotropy is expected to reach a maximum of A ∼ 1.The Weibel instability is thus suppressed if i.e. if the gradient scale is bigger than the mean free path of an electron.Even if the Weibel instability is fully suppressed, it has been suggested that an instability known as the thermomagnetic instability [19] may also generate filamentary magnetic fields due to a parallel density and temperature gradient due to a combination of the Righi-Leduc and Biermann battery effects, for collisional systems.However, Ref. [20] showed that this instability is suppressed due to the Nernst effect, and that the growth of the Biermann battery is reduced.The Nernst effect, when only the most energetic electrons are frozen into the magnetic field, is expected to drag Biermann generated fields in the direction of heat flux allowing them to pile up [21][22][23][24][25][26][27].

II. SIMULATION SETUP
In order to verify and quantify the scalings of Eq. ( 6), we performed several simulations of an expanding bubble of plasma using the OSIRIS framework [28,29], while including collisional effects [30,31], and varying the collisionality ν 0 L T /v T (via the density).The bubble has a peak density n 0 and expands into a background n b = 0.1n 0 , with a peak initial temperature at the center of the box T e0 /m e c 2 = 0.04 varying only along the x direction to a background temperature T eb = 0.0025T e0 , and a realistic mass ratio m i /m e = 1836, the same as the simulations in Ref. [12] with where and L n = L T /2.n 0 is the reference density used to define ω pe and d e and is used along with the reference temperature T e0 to calculate v T and ν 0 .The distributions of density and temperature are shown in Fig. 1 with the gradients highlighted.Unless otherwise specified, each simulation uses 198 particles per cell on a 12000 × 12000 grid (1500.0× 1500.0 d 2 e ).The simulations are run for 1800.0 ω −1 pe , with a timestep dt = 0.07 ω −1 pe .The Coulomb logarithm ln Λ C is calculated automatically depending on the local parameters.Only the collisions between electrons and ions are included.There is an 8 point average over the magnetic fields generated.
In realistic experimental setups, where collisions become important, the temperatures are lower and the system sizes are larger than we simulate here.As these parameters are more computationally expensive, we instead vary the density, allowing for a scaling to realistic parameters.Note that for the parameters that we simulate, ln Λ C varies significantly due to the small values if Λ C .Once v T /c exceeds 2α (T e > 108 eV), where α is the fine structure constant, Λ C grows more slowly with respect to temperature.Therefore, for a given collision rate, Λ C is smaller for higher temperatures.Furthermore, for larger system sizes, equal collisionalities occur at smaller collision rates, and thus at larger Λ C .
In our simulations the velocity distribution does not necessarily remain bi-Maxwellian, and we measure the anisotropy using the temperature tensor T ij ≡ (u i u j /γ)f (u)/ f (u) calculated in the species rest frame.u i is the proper velocity, γ = √ 1 + u 2 , and f (u) is the velocity distribution function.T e and T e⊥ < T e are eigenvalues of the temperature tensor.We only consider the in-plane temperatures and assume the out-of-plane temperature is also T e⊥ , which has been verified for our simulations to be a reasonable assumption.

III. SIMULATION RESULTS
In Fig. 2, the out-of-plane magnetic field B z for three representative collisionalities is presented.The first case (Fig. 2a) is like the previous collisionless studies, where the Weibel magnetic field dominates compared to the fields due to the Biermann battery.Only the growth rate and the strength of the saturated field are modified by the collisions.The second case (Fig. 2b) is the transition scale where the Weibel fields are suppressed, but still visible, and the Biermann field is the dominant field.In the third case (Fig. 2c), only the most energetic electrons remain collisionless, leading to pile-up of Biermann generated fields via the Nernst effect [21][22][23][24][25][26][27].To the best of our knowledge, this is the first time the Nernst effect has been demonstrated using PIC simulations.
We can make a prediction where to expect the electron Weibel instability, depending on the gradient length scale, temperature, density, and charge state of the ions.The transition occurs when ν e L T /v T ≈ 1, as predicted from Eq. ( 6).The three cases in Fig. 2 occur at ν 0 L T /v T = 0.837, 5.32, and 26.5 respectively.We calculate the local ν A at the location where the instability occurs in the collisionless case (x/d e , y/d e = 150, 100) by measuring the parameters averaged within a box of 20 × 10d 2 e at time t c ω pe = 907 when the measured growth rate reaches its maximum.Here, T e⊥,loc /m e c 2 = 0.0244, A = 0.56, n loc /n 0 = 0.385, and the gradient length scale L T,loc /L T = 0.2563.We thus find the respective cases occur at ν A,loc L T,loc /v T,loc = 2.09, 13.5, and 63.7.We therefore confirm that the Weibel instability is suppressed when the anisotropy relaxation time is smaller than the generation time (t Ar > t Ag ), but it is not completely suppressed until t Ar t Ag .The transition to a regime where no Weibel exists occurs in the simulation with ν 0 L T /v T = 5.32 (Fig. 2b).This simulation has no electron-electron collisions, Z = 1, and the local L T,loc = 55nm, lnΛ C = 4.0, and v T,loc /c = 0.170 (T e,loc = 14.8 keV ).Using this nu- merical value of ν 0 L T /v T and the scaling from Eq. ( 6), our simulation result can be scaled to more experimentally relevant densities and temperatures, and the general transition density can be expressed in an engineering formula.
n tr,loc = 5.42 × 10 21 cm −3 T e,loc 1.0 keV The other two cases correspond to a density n loc = 0.1n tr,loc (Fig. 2a) and 10n tr,loc (Fig. 2c).We show the effects of collisions on the Weibel growth in Fig. 3, plotting the evolution of the maximum magnetic field and exponential fits of the growth rate.There is a significant change in growth rate between the essentially collisionless case at ν 0 L T /v T = 0.00175, and ν 0 L T /v T = 0.837 (0.0098 to 0.0065ω pe ).The growth rate effectively goes to zero in the case with ν 0 L T /v T = 5.32.The measured growth rates of the magnetic field for each simulation are reported in Table I, along with the local parameters used to calculate the theoretical Weibel growth rate.Table II shows the measured wavenumber of the instability, the predicted fastest growing wavenumbers, and the growth rates calculated using these wavenumbers, providing evidence that the observed filaments are due to Weibel instability.
The growth rate of the Weibel instability depends on Circles are calculated from magnetic field energy, and stars from the maximum magnetic field.A theoretical estimate of the growth rate is plotted in black using the anisotropy from Eq. ( 9).A, which depends on the collisionality as collisions inhibit the anisotropy growth.Due to the exponential decay of A predicted by Eq. ( 3) for a constant ν A , a good approximation of the A dependence on the local collisionality is: In addition to the simulations presented so far with L T /d e = 400, we have simulated several more simulations with L T /d e = 200 (half the system size, with constant resolution), where we have also measured the growth rate.In Fig. 4 the measured growth rates normalized to the collisionless growth rates are presented as a function of the local collisionality ν A,loc L T,loc /v T,loc .ν A,loc is calculated as previously assuming a constant anisotropy A 0 = 0.56 and perpendicular temperature T e⊥,loc /m e c 2 = 0.0244.A theoretical prediction for the growth rate is given by the Weibel growth rate using the anisotropy from Eq. ( 9) (black curve in Fig. 4), which agrees with the measured results.Fig. 4 also gives evidence that this scaling with collisionality is independent of L T for constant ν 0 L T /v T .

IV. CONCLUSION
Using particle-in-cell simulations, we have placed a limit where collisions will inhibit the generation of the electron Weibel instability in the expansion of a hot plasma, when ν e L T /v T ∼ 1.While in [11] it was shown that magnetic fields from the Weibel instability will be larger than the Biermann field for L T /d e > 100, we now show this additional limit due to collisions, where the Biermann field again dominates.
Although the simulations presented here are all 2D, the results should not differ greatly in 3D.For the collisionless case a 3D simulation showed similar results for L T /d e = 50 [12].For larger system sizes we expect Weibel filaments with wavenumbers also out of the 2D simulation plane, but besides that, the results should remain similar to 2D.
We do not observe the thermomagnetic instability, confirming Ref. [20], but we also do not observe any of the predicted reduction of the Biermann battery growth.This is likely because we start from a Maxwellian distribution, where the Biermann battery should grow rather than evolve to a such a state by plasma heating and expansion.
This still remains a simplified model, and assumes that the interaction will generate these temperature gradients on a quick enough time scale that this model is valid.The effects of the laser magnetic fields and heating processes often occur at the same time as the Biermann and Weibel magnetic fields grow.It has been shown that for an intense short pulse laser, where the plasma becomes relativistically hot, the Weibel field can be observed [32].

FIG. 1 .
FIG. 1. Map of the initial (a) density n and (b) electron temperature Te.The gradients are highlighted in white.

FIG. 2 .
FIG. 2. Map of magnetic field Bz at tωpe = 1797.6.The top panel shows the simulation with a collisionality ν0LT /vT = 0.837, where the Weibel instability still exists, but grows slower and saturates at a lower intensity field.The middle panel shows a simulation with ν0LT /vT = 5.32, where the Weibel instability is significantly damped, and the Biermann field is visible.The bottom panel shows a simulation with ν0LT /vT = 26.5,where there are no traces of the Weibel instability and a pileup of magnetic flux dragged by the Nernst effect is present.

FIG. 4 .
FIG.4.Measured growth rate of the Weibel instability normalized to the growth rate measured in the collisionless case for a range of simulations with different collisionalities.Simulations with LT /de = 400 (200) are indicated in blue (red).Circles are calculated from magnetic field energy, and stars from the maximum magnetic field.A theoretical estimate of the growth rate is plotted in black using the anisotropy from Eq. (9).

TABLE I .
Measured growth rate γm and parameters determining the theoretical growth rate γt at the location where the instability occurs (x/de, y/de = 150, 100) averaged within a box of 20 × 10d 2 e at time tc where the measured growth rate reaches its maximum.The local density is 0.385 n0.

TABLE II .
The measured wavenumbers k, theoretical fastest growing mode kmax, and the theoretical growth rates γt given these wavenumbers.