Interplay between the Weibel instability and the Biermann battery in realistic laser-solid interactions

A novel setup allows the Weibel instability and its interplay with the Biermann battery to be probed in laser-driven collisionless plasmas. Ab initio particle-in-cell (PIC) simulations of the interaction of short (≤ ps) intense (a0 ≥ 1) laser-pulses with overdense plasma targets show observable Weibel generated magnetic fields. This field strength surpasses that of the Biermann battery, usually dominant in experiments, as long as the gradient scale length is much larger than the local electron inertial length; this is achievable by carefully setting the appropriate gradients in the front of the target e.g. by tuning the delay between the main laser pulse and the pre-pulse. PACS numbers: 52.38.-r, 52.35.Qz, 52.65.Rr, 52.72.+v 1 ar X iv :1 90 7. 10 43 3v 1 [ ph ys ic s. pl as m -p h] 2 4 Ju l 2 01 9 The origin and evolution of magnetic fields starting from initially unmagnetized plasmas is a long-standing question, which has implications not only in astrophysics (e.g. Gammaray-bursts, TeV-Blazar, etc) [1–4] but also in laboratory plasmas (e.g. fast ignition) [5–7]. Magnetic field growth in astrophysical conditions is often attributed to the turbulent dynamo mechanism, which requires an initial seed field. The dominant processes responsible for magnetogenesis, i.e. the generation of these initial fields, are still under strong debate. Among the known mechanisms, the Biermann battery and the Weibel or current filamentation instability are two major candidates [8–14]. The Biermann battery acts in the presence of temperature and density gradients perpendicular to each other [15, 16]. In contrast, the Weibel instability is driven by temperature anisotropies [17, 18]. These key mechanisms have been reproduced using scaled experiments governed by similar physical laws [19, 20]. The interplay between the Biermann battery effect and the Weibel instability in the laboratory is both of fundamental interest and relevant to understand magnetogenesis. Recent developments in laser technology (intensities in excess of 10W/cm with laser pulse durations shorter than 1 ps and high-resolution diagnostics) open the possibility to probe such processes through laser-solid interactions [19, 21–24]. In these experiments, the magnetic field generation is often attributed to the Biermann battery [20, 25, 26]. The Biermann field grows linearly as B(t) ≈ −(tc/nee)∇ne ×∇Te ≈ (tc/e)(kBTe/LTLn), where Ln ≡ ne/∇ne and LT ≡ Te/∇Te are the density and temperature gradient scale lengths, respectively, kB is the Boltzmann constant, ne and Te are the electron density and temperature, e is the elementary charge, and c is the speed of light in vacuum. Theoretical and computational studies have demonstrated magnetic field generation via the Biermann battery [27, 28] in the context of hydrodynamical systems. Recently, Schoeffler et.al [29, 30] investigated the kinetic effects of the Biermann battery in a collisionless expanding plasma, finding that for sufficiently large gradient scale length L ∼ Ln ∼ LT the Weibel instability competes with the Biermann battery. The relative importance of the Biermann battery can be adjusted by changing the scale length of the density and temperature gradients. The saturated Biermann battery generated field obeys the scaling: B √ 8πPplasma = β−1/2 e ∼ de L , (1) where Pplasma is the plasma pressure, de ≡ c/ωp and ωp = (4πene/me) are the respective

Magnetic field growth in astrophysical conditions is often attributed to the turbulent dynamo mechanism, which requires an initial seed field. The dominant processes responsible for magnetogenesis, i.e. the generation of these initial fields, are still under strong debate.
Among the known mechanisms, the Biermann battery and the Weibel or current filamentation instability are two major candidates [8][9][10][11][12][13][14]. The Biermann battery acts in the presence of temperature and density gradients perpendicular to each other [15,16]. In contrast, the Weibel instability is driven by temperature anisotropies [17,18]. These key mechanisms have been reproduced using scaled experiments governed by similar physical laws [19,20]. The interplay between the Biermann battery effect and the Weibel instability in the laboratory is both of fundamental interest and relevant to understand magnetogenesis.
Recent developments in laser technology (intensities in excess of 10 19 W/cm 2 with laser pulse durations shorter than 1 ps and high-resolution diagnostics) open the possibility to probe such processes through laser-solid interactions [19,[21][22][23][24]. In these experiments, the magnetic field generation is often attributed to the Biermann battery [20,25,26]. The Biermann field grows linearly as B(t) ≈ −(tc/n e e)∇n e × ∇T e ≈ (tc/e)(k B T e /L T L n ), where L n ≡ n e /∇n e and L T ≡ T e /∇T e are the density and temperature gradient scale lengths, respectively, k B is the Boltzmann constant, n e and T e are the electron density and temperature, e is the elementary charge, and c is the speed of light in vacuum. Theoretical and computational studies have demonstrated magnetic field generation via the Biermann battery [27,28] in the context of hydrodynamical systems. Recently, Schoeffler et.al [29,30] investigated the kinetic effects of the Biermann battery in a collisionless expanding plasma, finding that for sufficiently large gradient scale length L ∼ L n ∼ L T the Weibel instability competes with the Biermann battery. The relative importance of the Biermann battery can be adjusted by changing the scale length of the density and temperature gradients. The saturated Biermann battery generated field obeys the scaling: where P plasma is the plasma pressure, d e ≡ c/ω p and ω p = (4πe 2 n e /m e ) 1/2 are the respective electron skin depth and plasma frequency, and m e is the electron rest mass. For systems where L/d e < 100 the dominant magnetic field is generated via the Biermann mechanism. In contrast when L/d e ≥ 100, the Weibel instability generates magnetic fields that are stronger and grow faster than that of the Biermann battery.
In this Letter, we carry out a numerical and theoretical study using particle-in-cell (PIC) simulations to investigate magnetic fields generated by the Weibel instability in the interaction of a short (ps) high intensity (a 0 ≥ 1) laser pulse and a plasma with sufficiently large L. Until now, the large simulation domains and long simulation times required to capture these mechanisms have impeded detailed exploration of this regime. Our simulation results reveal that by tuning the delay between an ionizing pre-pulse and the main pulse, and defining the spot size of the laser such that L/d e ≥ 100, the Weibel generated magnetic field magnitude surpasses the usually observed Biermann field, and can be directly observed in current laser-plasma interaction experiments.
We simulate the interaction of an ultraintense laser pulse with a fully ionized unmagnetized electron-proton plasma with realistic mass ratio (proton mass m i = 1836 m e ) using the OSIRIS framework [31][32][33]. The laser is s-polarized (i.e. the electric field is perpendicular to the simulation plane) and has a peak intensity I L = 10 19 W/cm 2 (normalized vector potential a 0 = 2) with a wavelength λ 0 = 1.0 µm. We choose s-polarization to isolate the out-of-plane Biermann and Weibel magnetic fields from the laser field. Furthermore, s-polarization in 2D better approximates 3D conditions, as both conditions have been shown to produce less heating than with p-polarization in 2D [28,34]. We have performed 2D simulations with similar laser parameters using p-polarized laser confirming the conclusion predicted by Ref. [28]. We define ω p and d e using a reference plasma density  We focus our observations on the magnetic field at the front surface of the target, choosing the length of the target long enough that the back side does not influence the front (we have checked that the particles reflecting from the back do not reach the region x 1 < 1150 d e where significant heating occurs until after t = 2812.60 ω −1 p ), and n e = 0 at the right wall to avoid significant particle loss at the boundary. We choose a step function at x 1 = 1750 d e to minimize the length and save computational time (see Fig. 1(a)) . Figure 1 shows, in the simulation where L n = 400 d e , that the laser produces temperature gradients that are not aligned with the density gradient associated with L n . The laser enters the simulation domain from the left and at time t 1200.50 ω −1 p penetrates the plasma up to 1000 d e (Fig. 1(a)). The interaction of the laser with the plasma resonantly heats the electrons, consistent with the scaling of ref. [36] (Fig. 1(b)). The temperature is defined as is the temperature tensor, u i is the normalized proper velocity, γ = √ 1 + u 2 , and f (u) is the velocity distribution function. By time t 2812.60 ω −1 p , the laser has created a conical shaped channel (see Fig. 1(c)) and induced a large thermal gradient with L T = 1000 d e pointing radially towards the axis of the laser beam (see Fig. 1(d)). The temperature gradient is not aligned with the density gradient along x 1 allowing the Biermann battery to generate a toroidal B-field.
The average temperature along the line at x 1 = 700 d e is T e x2 = 0.34 m e c 2 (see Fig. 1).
Given this temperature and the maximum density n 0 = 1.1 × 10 22 cm −3 , we conservatively estimate the collisionality. The ratio of L n to the electron collisional mean free path l e [37], L n /l e = 0.00047 1, therefore we neglect collisions. ing the typical experimental resolution (see e.g. [35]). The boundary between Biermann and Weibel regimes is estimated at the location where L T (x 1 )/d e (n e (x 1 )) ≈ 100 [29,30], where d e (n e (x 1 )) is the local electron inertial length. Remarkably, this transition occurs precisely at x 1 = 700 d e , indicated by the dotted vertical line in Fig. 2(a), as d e (n e (x 1 )) = 10 d e and L T (x 1 ) = 1000 d e (see Fig. 1(c-d)).  after the laser has passed this region (see Fig. 3(b)), the laser magnetic fields are no longer present. Here, we observe an exponential growth of the magnetic field (Γ sim = 0.0015 ω p with a corresponding wave-vector k 0.15 d −1 e , agreeing reasonably with theory from ref. [38]). The spatiotemporal evolution of the laser magnetic field energy shown in Fig. 3 (b) shows that the end of the laser pulse passes the region where we calculate the growth rate (x 1 < 900 d e ) at t = 1950 ω −1 p (322 fs). Meanwhile, the expansion of the hot energetic electron population generated via laser-heating contributes to the average anisotropy in the velocity distribution (see Fig. 3(c)) [39]. The anisotropy A ≡ T hot /T cold − 1, where T hot and T cold are the respective larger and smaller eigenvalues of the temperature tensor T ij , provides the free energy that drives the Weibel instability.
The time varying spectrum of B 2 3 in Fig. 3(d) shows the contribution of the Weibel instability and the Biermann battery to the magnetic field energy. The spectra are obtained by performing a Fourier transform over the entire system for the out-of-plane magnetic fields, and then averaging over all directions of k. With the log scale it is not obvious that the energy contained in the Weibel magnetic fields is comparable to that of the Biermann. The Biermann magnetic field energy (kd e < 0.025) remains about five times higher than the Weibel magnetic fields energy (kd e > 0.025) after t = 2370 ω −1 p . We performed a parameter scan for L n /d e = 0, 80, 160, 240, 320, and 400. Note that by the time the laser reaches the target at t ∼ 1250 ω −1 p , the length scale rises by ∼ k B T e0 /m i t ∼ 1.3 d e , given T e0 = 1 keV. Therefore, for L n /d e = 0, the effective density scale length is 1.3 d e . Fig. 4(a-d) shows B 3 at time t = 2023.70 ω −1 p (when the Weibel generated magnetic fields saturate in the L n /d e = 400 case, see Fig. 4(e)) for a selection of L n /d e .
With a target of sufficiently large L n /d e > 160, a region of Weibel generated magnetic fields is visible (see Fig. 4(a) where L n /d e = 320). However, for L n /d e ≤ 160, the Biermann magnetic field dominates, and no region exists where the Weibel instability is prominent (see Fig. 4(b-d)).
Thin filaments in B 3 explained by the current filamentation instability (CFI) [7,40,41] are observed in many experiments [42,43] where a laser hits a plasma target with a sharp density profile. Fig. 4(d) shows these filaments (without the low-pass filter). Unlike the Weibel generated field described in this work, a sharp relativistic electron beam provides the free energy rather than the thermal expansion of the plasma. In our simulations, the CFI field is much weaker than both the Weibel and Biermann fields for other L n /d e . Furthermore, in this Letter, we focus on the region with density and temperature gradients that lead to the Biermann battery and Weibel instability, rather than deep inside the target where these thin filamentary fields are found.
The magnetic energy-density produced from the laser-interaction depends on L n . Fig. 4(e) shows the temporal evolution of the average out-of-plane magnetic energy-density B 2 3 (with low-pass filter) in the region between x 1 = 1250 d e − 1.875 L n and x 1 = 1250 d e + 1.25 L n for each simulation (see highlighted regions in Fig. 4(a-d)). Weibel fields are observed when L n /d e > 160, saturating at t ∼ 2000 ω −1 p . For all cases, the Biermann field grows and saturates after t ≥ 2150 ω −1 p . The dashed line shows B 2 3 (without low-pass filter) in the range x 1 = [950 − 1250] d e associated with the zoomed region in Fig. 4(d), which peaks at p . This CFI magnetic field is much smaller than the dominant fields for bigger L n . In Fig. 4(f), the peak B 2 3 is shown as a function of L n /d e . The maximum B 2 3 occurs at L n /d e = 160, the transition between the Biermann and Weibel regimes. Magnetic fields can be measured using the synchrotron radiation in addition to the conventional method of proton radiography [45]. For the parameters of this study, radiation will have wavelength estimated between 190 − 1200 nm, while for higher power lasers, this signal would become stronger and approach x-ray frequencies. The detailed prediction of the radiation spectra, which can in principle be performed using radiation algorithms [46,47], will be left for future work.