Filamentation Instability of Counterstreaming Laser-Driven Plasmas

Filamentation due to the growth of a Weibel-type instability was observed in the interaction of a pair of counterstreaming, ablatively driven plasma flows, in a supersonic, collisionless regime relevant to astrophysical collisionless shocks. The flows were created by irradiating a pair of opposing plastic (CH) foils with 1.8 kJ, 2-ns laser pulses on the OMEGA EP Laser System. Ultrafast laser-driven proton radiography was used to image the Weibel-generated electromagnetic fields. The experimental observations are in good agreement with the analytical theory of the Weibel instability and with particle-in-cell simulations.

Figure 1

Experimental setup: plasma plumes are ablated from a pair of CH targets, separated by $2L=4.5\mathrm{mm}$, which collide and interact in the midplane. The electromagnetic fields formed due to instabilities in the interaction were radiographed with a laser-driven proton beam and imaged onto radiochromic film (RCF).

Figure 2

Radiochromic film images of the development of a striated instability at the midplane at 3.8, 4.8, and 5.8 ns relative to the start of the main driver laser pulse. The film records fluence of protons with energy of order 5–10 MeV, with darker features reflecting greater proton fluence. The striated features are well reproduced on neighboring film in the stack, reflecting the strong focusing of the diagnostic proton beam [20]. The insets show 1D traces of proton intensity along the long axis of the regions denoted in the film, averaged over the short direction, from which a quadratic background proton variation was subtracted to focus on the fluctuations. Typical filament wavelengths of $100–150\mu \mathrm{m}$ at 3.8 and 4.8 ns expand to wavelengths near $250\mu \mathrm{m}$ at 5.8 ns. The distances on the axes are those in the nominal object plane, which is perpendicular to the axis of the proton beam and the targets, and which goes through the center of the UV drive laser foci. The image has been sharpened in postprocessing to emphasize the striations. (Raw images are available in the Supplemental Material 28).

Figure 3

(a) Growth rate of Weibel instability and (b) wavelength of fastest growing mode versus ion temperature. Specifically, we solve the ion-pinch dispersion relation of Davidson et al. [6], using zero electron anisotropy due to the electron collisionality regime of the experiments. The growth rate is normalized to the counterstreaming speed $V$ and wavelength of maximum growth for each ion temperature. The blue curves are for a 60:40 H-C mixture close to the composition of the targets, and the red curves are for a 90:10 mixture reflecting fractionation. Solid curves are for $T_e=T_i$, and the dashed curves are for heated electrons with $T_e=0.25m_p{V}^2$. To compare wavelengths on an even footing, for both compositions the wavelengths are normalized to a nominal $d_{i\mathrm{CH}}$ calculated with the 60:40 H-C mixture: $d_{i\mathrm{CH}}\approx 1.29(m_p/{\mu }_0{n}_e{e}^2{)}^{1/2}$.

Figure 4

Particle-in-cell (PIC) simulation of growth of Weibel filaments between counterstreaming ablation flows. Top: evolution of the plasma density. Bottom: development of transverse magnetic filaments from the Weibel instability. To generate the counterstreaming ablation-flow geometry, plasma is added dynamically to small volumes at the left and right boundaries for time $t=[0,t_{\mathrm{laser}}]$. This sets up a pair of flows with ablationlike profiles for density [$n\approx n_{ab}\exp (-\Delta x/C_{s}t)$] and velocity ($V\approx C_s+\Delta x/t$), where $\Delta x$ is the distance from the boundaries, $C_s$ is the sound speed evaluated using the source temperature, and $n_{ab}$ is the peak density reached in the source region. The simulation uses two species, carbon ($Z=6$) and electrons, with heavy electrons ($Z{m}_e/m_C=1/100$), compared to the physical mass ratio, for computational reasons. The domain is $[-L,L]$ along $x$ and $[0,2L/3]$ along the transverse direction, which is included to allow multiple wavelengths of the Weibel instability to grow. We approximately match the ion-scale dimensionless parameters $L/d_{i,ab}\sim 180$ (experiment) versus 130 (simulation) and $t_{\mathrm{laser}}{C}_{s0}/L\sim 0.17$ (experiment) versus 0.21 (simulation). ($d_{i,ab}$ is the ion-skin depth calculated using the ablation density.) Interparticle collisions are modeled using a Monte Carlo binary collision operator, with the collisionality chosen so that ${\nu }_{\mathrm{ei}}/{\gamma }_{\mathrm{weib}}\sim 10$ during instability growth, as estimated in the experiment. The simulations were conducted with the massively-parallel, explicit particle-in-cell code PSC [29], using approximately $5.7\times {10}^9$ computational particles.