Inefficient magnetic-field amplification in supersonic laser-plasma turbulence

We report a laser-plasma experiment that was carried out at the LMJ-PETAL facility and realized the first magnetized, turbulent, supersonic plasma with a large magnetic Reynolds number ($\mathrm{Rm} \approx 45$) in the laboratory. Initial seed magnetic fields were amplified, but only moderately so, and did not become dynamically significant. A notable absence of magnetic energy at scales smaller than the outer scale of the turbulent cascade was also observed. Our results support the notion that moderately supersonic, low-magnetic-Prandtl-number plasma turbulence is inefficient at amplifying magnetic fields.

Understanding the kinematics and dynamics of magnetic fields in supersonic plasma turbulence is a challenge that has both its own intrinsic merit and important astrophysical applications. Compared to the thorough characterizations of supersonic turbulent boundary layers arising from aerofoils [1,2], which have been tested by numerous computer simulations [e.g., 3,4] and wind-tunnel experiments [5,6], current theories of magnetized, supersonic, boundary-free plasma turbulence have a much weaker empirical foundation. At the same time, the wide range of physical processes that can arise in such a system promises an exceptionally rich collection of complex phenomena for study. In the astrophysical context, magnetic fields are believed to play a significant role in the turbulent, supersonic dynamics of the interstellar medium (ISM); understanding the complex interactions between the fields, shocks and vortices present in such an environment is a necessary component of a comprehensive picture of the ISM, encompassing important topics such as star formation [7][8][9][10]. Magnetized, moderately supersonic plasma turbulence is also thought to emerge in solar and stellar convection zones [11,12].
One key question concerning the relationship between magnetic fields and supersonic plasma turbulence in the ISM is how the fields attain their observed dynamical strengths. The equivalent question in subsonic plasma turbulence has been studied in greater depth, mostly within the framework of magnetohydrodynamics (MHD). Analytical theory [13][14][15][16][17], simulations [18][19][20][21][22], and recent experiments [23][24][25][26] give a consistent picture, showing that chaotic bulk motions of plasma (with characteristic scale L and velocity u rms ) can amplify any small seed magnetic field initally present in the plasma provided the magnetic Reynolds number Rm ≡ u rms L/η is greater than a certain critical value Rm c (here, η is the plasma resistivity). This critical value is usually significantly larger than unity [27].
For 1 Rm < ∼ Rm c , the magnitude δB of the magnetic field post amplification is related to the magnitude B 0 of the initial seed field via δB ∼ Rm 1/2 B 0 [21]. However, if Rm > Rm c , magnetic-field amplification of seed fields proceeds unabated until the magnetic-energy density of the amplified field reaches equipartition with the kinetic-energy density of the stochastic motions responsible for the amplification; this field-amplification mechanism is known as the fluctuation dynamo. Another important parameter for magnetic-field amplification is the magnetic Prandtl number Pm ≡ Rm/Re (where Re is the fluid Reynolds number): dynamo is less efficient for Pm 1 than for Pm > ∼ 1 [16,21].
Compared to the subsonic case, there exist far fewer studies of magnetic-field amplification in supersonic plasma turbulence. Numerical studies of supersonic MHD turbulence [28][29][30][31] indicate that the fluctuation dynamo is still capable of operating. The efficacy of the mechanism, both in terms of the characteristic growth rates of magnetic fields and saturated magnetic/kinetic energy ratios, is a function of the turbulent Mach number Ma turb ≡ u rms /c s (where c s is the plasma's sound speed): it is less effective for Ma turb > ∼ 1 than for Ma turb 1. Physically, this has been attributed to a number of factors: reduced energy available to the solenoidal stretching motions necessary for dynamo action on account of some of the driving kinetic energy flux being directed towards compressive motions, irrespective of the driving mechanism [28,29,[32][33][34]; a steepened turbulent velocity spectrum [35]; and enhanced dissipation of magnetic fields in shocks [36]. In the laboratory, there has only been one previous experiment that successfully realized boundary-free, supersonic plasma turbulence [37]; however, Rm achieved in that experiment was much smaller than unity, prohibiting significant magnetic-field amplification.
In this paper, we report a new experiment that managed to create supersonic, high-Rm plasma turbulence for the first time in the laboratory. The experiment was performed on the Laser Megajoule (LMJ) facility in Bordeaux [38]. The platform employed for the experiment is illustrated in Figure 1. Similarly to previous laser-plasma experiments investigating the fluctuation dynamo in subsonic plasma, which were carried out on the Omega Laser Facility [25,26], a turbulent plasma was created by colliding inhomogeneous, asymmetric, counter-propagating rear-side-blow-off plasma jets. Spatial inhomogeneity is introduced by placing grids in the paths of each jet prior to their collision; the jet asymmetry follows directly from using asymmetric grids. In order to reach the supersonic regime, three major design modifications to the previous Omega experiments were introduced. The thickness of the foils irradiated by the LMJ drive beams was reduced, and the beam energy per foil increased fourfold: both changes led to increased initial jet velocities. In addition, aluminium rather than plastic foils were used in the experiment; the resulting enhancement in radiative cooling reduced the plasma's temperature both before and after jet collision. Both modifications were anticipated to increase Ma turb , a claim supported by threedimensional, three-temperature radiation-MHD simulations performed concurrently to the experiment using the FLASH code [40,41].
The primary diagnostic on the experiment, CRACC (Cassette de Radiographie au Centre Chambre) [42], provides time-resolved proton imaging [43], which was used to measure magnetic fields and the electron number density in the plasma, as well as to determine the characteristic velocities of the initial jets. The proton imaging beam was generated by irradiating a gold foil with the highintensity PETAL beam (see Figure 1) [44]; via the target FIG. 1. Experimental set-up. Upper panel: annotated photograph of one of the targets used in our experiment. The parameters of both the LMJ and PETAL beams are indicated on the photograph. Aluminium foils (separated by 8 mm) are irradiated by the LMJ drive-beam lasers; they have a 3 mm diameter and 25 µm thickness. An annular CH washer (230 µm thick, 3 mm diameter, 400 µm hole) is placed over the foil to aid jet collimation. The grids (located 2 mm away from each foil, on the opposite side to the LMJ beams) are made of polyamide, have a thickness of 250 µm, and square holes (side length 300 µm) separated by 100 µm rods. The main target is rendered partially transparent, in order to show the location of the turbulent plasma (the yellow-purple region). The proton source is a 50 µm gold foil, and is protected from pre-plasma and return currents by an aluminium polycarbonate shield. It is located 3 cm from the turbulent plasma's centre. The RCF stack used to detect the protons after they pass through the plasma is placed 10 cm away on the opposite side, leading to a ×4.3 magnification. All length scales are shown with this magnification factor removed, i.e., on the plasma's scale. Lower panels: 8.5 MeV proton images (obtained from different experimental shots) at 15.7 ns (left), 19.7 ns (middle) and 22.7 ns (right) after the initiation of the LMJ drive beams. The proton flux normalisation is defined relative to the mean of the regions enclosed by red-dashed lines in each image. There was a 30% drop in delivered LMJ beam energy on both foils for the 15.7 ns experimental shot, and on one foil for the 22.7 ns shot; however, due to inefficient beam-energy absorption in the foil, we do not believe that our results are significantly affected by this [39].
normal sheath acceleration (TNSA) mechanism [45], this irradiation results in a highly directed proton beam with a thermal (∼ 3 MeV temperature) spectrum. The beam passed through the plasma generated by the LMJ drive beams, and subsequently was detected using a calibrated radiochromic film (RCF) stack [42]. The RCF stack was designed in such a way that protons with distinct energies were detected in separate layers of RCF (∼ 0.5 MeV energy resolution); this allowed for time-resolved measurements on each experimental shot, because slower beam protons passed through the plasma at later times than faster ones. The ∆t p ≈ 300 ps time delay between the fastest and slowest detected protons (8.5 MeV vs. 4.7 MeV) was too small to capture the full dynamical evolution of the plasma turbulence; to capture this evolution, we repeated our experiment, but with three different relative offsets between the LMJ and PETAL beams. The resulting proton images (for the 8.5 MeV protons) are shown in Figure 1 (see [39] for further information about the analysis of the RCF stack).
Detailed quantitative information about the magnetic fields present in the turbulent plasma can be obtained by analyzing the proton images. The theoretical basis for such analysis comes from the proton beam's high velocity and low density compared to that of the plasma with which it interacts prior to reaching the RCF stack; inhomogeneites in the detected proton flux can therefore be attributed to the action on the beam protons of the Lorentz forces arising from spatially varying magnetic fields present in the plasma [46]. Collisionless beam instabilities and deflections due to electric fields in this experiment have a negligible effect on the proton beam. This being the case, recent work [47] has shown that the two components of the path-integrated magnetic field that are perpendicular to the proton beam's direction can be reconstructed directly from these inhomogeneities, provided the proton beam, on account of its non-uniform distortion, does not self-intersect before reaching the detector. Further information on how this analysis was performed is given in [39].
The path-integrated field reconstructed from the 8.5 MeV proton image of the supersonic plasma jets prior to their collision is given in Figure 2 (top left); its time evolution can be used to determine the velocity u jet of the jets. The dominant component of the path-integrated field (characteristic magnitude ∼ 0.4 kG cm) inside the main bulk of each jet is aligned with the jets' predominant direction of motion, and oscillates strongly in the direction normal to it. To extract the velocity of the lefthand jet, we compare the path-integrated field recovered from 4.7 and 8.5 MeV proton images generated on the same experimental shot (see Figure 2, top right), corresponding to imaging times of 15.7 and 16.0 ns, respectively. While the morphology of both images is very similar, the characteristic extent in the parallel direction of the oscillatory path-integrated field structure is slightly greater at 16.0 ns; we illustrate this qualitatively using contour plots of the path-integrated field (Figure 2, bottom left). We attribute this finding to the motion of the magnetic fields inside the jet: these fields are frozen into the bulk flow provided the jet's magnetic Reynolds number Rm jet ≡ u jet L/η (where L = 0.04 cm is the grid periodicity) just after its interaction with the grid satisfies Rm jet 1 (an assumption supported by theoretical expectations concerning the initial jet properties -see [39]). The mean jet velocity u jet is obtained as follows: calculate average lineouts for five different regions (which are depicted in Figure 2, top right) for the path-integrated fields measured at each time (two sample lineouts are shown in Figure 2, bottom right); determine the mean spatial offset ∆x p between each temporal pair of lineouts; then estimate u jet via u jet ≈ ∆x p /∆t p . We find u jet = 290 ± 40 km/s; this value is consistent with a heuristic estimate determined from the known temporal delay between the LMJ drive-beam pulse's midpoint and the jet collision time, and the 4 mm distance from each foil to the target's centre.
Once collision between the jets has occurred, X-ray imaging from related experiments on other laser facilities [25,26] indicate that a turbulent plasma with higher characteristic temperatures and densities quickly coalesces; this coincides with a burst of self-emitted X-rays. The spectrum of these X-rays was measured in our experiment using the DMX diagnostic [48,49]. DMX is an absolutely calibrated, time-resolved broadband spectrometer with high temporal resolution ( 100 ps). The brightness temperature of the 10 lower energy channels (taking into account an X-ray emissive area corresponding to the collision zone) allow for the turbulent plasma's temperature to be extracted: T ≈ 100 eV.
Given our previous measurement of u jet , the characteristic turbulent velocity u turb in the interaction-region plasma can be estimated as follows. X-ray measurements from previous experiments [26] and FLASH simulations [41] indicate that, while the jet velocities are close to being uniform transversely, the density of either of the plasma flows is much larger at transverse spatial positions coincident with the locations of the grid holes through which that flow has passed than the density at the analogous position in the opposing flow. When the two plasma flows collide, conservation of momentum therefore dictates that the flow velocity in these transverse spatial locations will be close to the higher-density plasma flow's incident velocity. Taking into consideration the two-dimensional periodic reversals in the flow direction, and assuming that this flow profile is efficiently randomized by nonlinear interactions and/or Kelvin-Helmholtz instabilities, we conclude that u turb ≈ u jet / √ 2 ≈ 200 km/s. The sound speed in the plasma is c s = γ(Z + 1)T /m i ≈ 80 km/s, where γ is the adiabatic index, Z the plasma's ionization state, and m i the ion mass. Therefore, the turbulent Mach number is Ma turb ≈ 2.5, so the turbulence is supersonic.
The characteristic electron number density n e of the interaction-region plasma was determined by quantifying the effect of collisional scattering on the resolution of the sharp, large-amplitude proton-flux inhomogeneities ('caustics') present in the 4.7 MeV proton images. In the absence of collisions of the proton beam (and other finite-resolution effects), the Fourier spectrum of caustics is known to follow a characteristic power law ∝ k −1 [47]. However, Figure 3 shows that, for k > ∼ 250, the measured spectrum of the 4.7-MeV-proton flux inhomogeneities (at both t = 20.0 ns and t = 23.0 ns) is much steeper. Assuming that collisional scattering is the dominant process that limits the resolution of the proton images, the electron density can be estimated using known relations between the characteristic collisional-scattering angle and the image resolution [39]. We find n e ≈ 4-7 × 10 19 cm −3 , a value which is consistent with measurements from related experiments [25,26].
Using all this information, the viscosity and resistivity of the plasma -and thus the fluid and magnetic Reynolds numbers -are determined via known expressions for transport coefficients in an collisional, aluminium plasma [39]. We find that Re ≈ 10 6 , a Reynolds number which is certainly large enough to allow for the formation of a developed turbulent cascade. The magnetic Reynolds number is also significantly larger than unity, but is much smaller than Re: Rm ≈ 45, so Pm ≈ 4 × 10 −5 . The turnover time of the turbulence is τ L ≈ 2 ns, which is short compared to the lifetime of the interaction-region plasma.
The path-integrated magnetic-field maps extracted from 8.5 MeV proton images after the jet collision allow us to characterize both the seed fields initially present in the interaction-region plasma, and the stochastic field structures arising from the interaction of those seed fields with the supersonic plasma turbulence. The seed fields, which are generated at the laser spots by the Biermann battery [50] and subsequently advected into the interaction region, have a characteristic transverse scale comparable to that of the interaction region ( n⊥ ≈ 0.25 cm), while the correlation length of the stochastic fields is significantly smaller ( B ≈ 150 µm). We take advantage of this scale separation to extract distinct path-integrated field maps for the seed and stochastic magnetic fields in the experiment (see Figure 4). The extraction procedure for the large-and small-scale path-integrated magnetic fields is explained in [39].
We estimate the characteristic magnitude B 0 of the seed magnetic fields via a simple relation [46]: B 0 ≈ 10[B path,0 (kG cm)/2 kG cm][ n⊥ (cm)/0.25 cm] −1 kG, where B path,0 is the characteristic magnitude of the path-integrated seed magnetic field. The field magnitude obtained just after the collision has occurred . The (two-dimensional) streamlines of the perpendicular field are also depicted. Middle: magnitude of perpendicular pathintegrated stochastic (small-scale) magnetic fields. Top right: evolution of seed (blue) and stochastic (red) magnetic field over time. Bottom right: magnetic-energy spectra calculated in demarcated regions from maps of path-integrated stochastic field components. The anticipated resolution limit on our spectra imposed by collisional scattering of the 8.5-MeV proton beam at both times is also shown.
(B 0 ≈ 10 kG at t = 19.7 ns after the LMJ drive beams are initiated) is consistent with related experiments [26]. Over one turnover time later than the collision (t = 22.7 ns), the seed fields decay considerably (B 0 ≈ 4 kG), which can be attributed to their dilution due to the interaction-region plasma's expansion, and turbulent diffusion.
The stochastic component of the magnetic field is characterised by its magnetic-energy spectrum E B (k), which describes the distribution of the magnetic energy amongst different length scales. We determine E B (k) from the path-integrated map of the stochastic magnetic field by assuming statistical homogeneity and isotropy; under these assumptions, it can be shown that is the one-dimensional spectrum of the path-integrated field [47]. The root-mean-square of the stochastic field, δB rms , can then be calculated directly from E B (k) as We find that at t = 19.7 ns, δB rms ≈ 6 kG, before subsequently attaining magnitudes comparable in strength to the initial seed fields (δB rms ≈ 10 kG at t = 22.7 ns -see Figure 4). The magnetic-energy spectra at both times have steep powerlaw tails E B (k) ∝ k −4 , with the spectral peaks at wavenumber k peak ≈ 2π/L, where L is the grid periodicity.
Our measurements suggest that amplification of the magnetic fields by the supersonic turbulence is quite limited, in spite of Rm being significantly greater than unity. The peak amplification factor of the seed field (at t = 22 ns) is ∼ 2.5 (significantly below the Rm 1/2 ≈ 6.3-times growth expected by simulations for incompressible flows if below critical [21]); this is similar to the amplification seen in previous subsonic plasma turbulence experiments at much lower Rm [24,39]. The magnetic-kinetic energy ratio is E mag /E kin ≈ 10 −4 , a value well below the saturation values found in simulations of MHD turbulent supersonic dynamos or in subsonic dynamo experiments [25,30,39]. This suggests that we did not reach the dynamo regime in our experiment, in turn providing a lower bound on Rm c for Pm 1. In summary, our results are broadly consistent with the expectation that magnetic-field amplification is less efficient in supersonic, low-Pm turbulence, as compared to moderate-Pm subsonic turbulence. In spite of this inefficiency, we believe that creating a laser-plasma turbulent dynamo in the supersonic regime in future experiments is feasible. FLASH simulations of the LMJ experiment, which (by assuming more efficient laser-target energy coupling than was attained in the experiment) realized characteristic kinetic and thermal energies ∼3-4 times greater than we report here, achieve Rm ≈ 750, and also show the key signatures of dynamo action [41]. This simulation finding suggests that exploring the transition to the dynamo regime in the laboratory is possible: a tantalizing prospect.
The research leading to these results received funding from the Engineering and Physical Sciences Research

EFFECT OF SUPPRESSED LASER PERFORMANCE ON RESULTS
As stated in the main text, in the shot during which the 15.7 ns proton image was obtained, a 30% drop in the laser energy (14 kJ per foil vs. 20 kJ per foil) delivered to both foils by LMJ was recorded, due to a technical failure; an identical drop on one foil was recorded in the shot during which the 22.7 ns proton image was obtained. Here, we explain why we believe that this change in delivered drive-beam energy does not significantly affect our results: specifically, that for our experimental parameters, the initial velocity of the rear-side-blow-off plasma jet is only weakly sensitive to the drive beam energy.
This claim can be justified as follows. Given a laser focal-spot diameter of 300 µm, a pulse length of 5 ns, and a total beam energy of 20 kJ per foil, the laser intensity I L is given by I L ≈ 5.6×10 15 W cm −2 . For a laser wavelength λ L = 351 nm and laser intensities I L ∼ 10 13 -10 15 W cm −2 , a simple physical argument [1] suggests that the ablation pressure P ab approximately scales with I L via P ab ∝ I 2/3 L -a scaling which has been (approximately) verified experimentally (see, for example, [2]). In turn, simple scaling arguments show that the characteristic velocity u jet (s) of the rearside-blow-off plasma jet at displacement s from its initial position is related to P ab by u jet (s) ∼ P ab s/δ t ρ t , where δ t is the thickness of the target foil onto which the laser is incident, and ρ t is the foil density. This implies that u jet (s) ∝ I 1/3 L . In short, a ∼ 30% reduction in the laser intensity only results in a modest (∼ 10%) reduction in the initial jet velocity. In fact, for the characteristic intensities employed in this experiment -for our laser parameters, [I L (W cm −2 )/10 15 W cm −2 ][λ L (µm)/1 µm] 2 ≈ 0.7 -the dependence of the ablation pressure on the laser intensity is likely even weaker, on account of various physical phenomena (nonlinear inverse Bremsstrahlung and resonant plasma instabilities) that reduce absorption efficiency as the intensity is increased further [3]. In consequence, the error induced in our results by the drop in delivered laser energy is less than 10%; this is smaller than the reported error of our measurements.
There is also qualitative experimental evidence that the suppressed laser performance does not affect our results: specifically, the central location of proton-flux inhomogeneities in the ∼22.7 ns proton image (see Figure 4). As stated earlier, the drive on one of the foils was 30% lower than on the other foil in the experimental shot during which this data was collected; if this drop in energy affected the initial velocity of one of the plasma jets, it would be anticipated that the collision of the two jets would happen off-center. However, the mean positions of the proton-flux inhomogeneities in the 19.7-ns and 22.7-ns proton images were, in fact, very similar.

Extracting proton flux measurements from RCF film pack
We convert the raw RCF images obtained in our experiment to (normalised) proton flux images using the following procedure. The RCF (HD-V2) films are electronically scanned, and then converted into 300-dots-per-inch RGB TIFF files. The red colour channel of these files is transformed to an optical density image, and then to a deposited energy (MeV/mm 2 ) image using an appropriate calibration for our chosen RCF film type. A GEANT-4 simulation [4] was then carried out, in order to determine the relationship between the energy deposited in each piece of film by a given proton, and that proton's energy. Curves showing this relationship ('the RCF response function') for the six distinct pieces of RCF film used in our experiment are shown in Supplementary Figure 1. Finally, an estimate of the proton flux is made by assuming that the majority of the energy deposited in each film comes from the protons that deposit the maximum energy associated with that particular piece of film. Table I shows this energy for each film layer. Such an estimate is justified (even though protons with a range of speeds deposit their energy in a given layer of the RCF stack) by the following argument. In any given film layer, the energy p,dep deposited by protons with incident energy p drops to < 50% of the peak deposited-energy value given in the final column of Table I for | p − p,max | > ∼ 0.25 MeV. As a result, the full-half-width-maximum (FHWM) ∆ p,max is much smaller than p,max . Protons with incident energy p < p,max − 0.5 MeV do not deposit any energy at all, because they do not reach the film layer. On the other hand, protons with incident energy p > ∼ p,max +0.25 MeV could, in principle, deposit 20-50% of the maximum deposited energy. However, in the experiment, the energy distribution of the proton-imaging beam is exponential, with a characteristic temperature of ∼3 MeV; as a result, the contribution of protons with higher energies is suppressed further, due to their lower number.
The proton images resulting from this (with flux values normalised to the mean value in the region between the grids) for each of our experimental shots are shown in Figures 2, 3 Figure 1 of the main text. Right: assumed initial 8.5 MeV proton-beam flux distribution employed for recovering path-integrated magnetic field (which is calculated using the approach described in the accompanying text).

Recovering path-integrated magnetic fields from proton flux measurements
In this section, we provide additional details about our analysis of the proton images. The reconstruction of the path-integrated magnetic field from a given proton image is carried out using the algorithm described in [5] (see also the Supplementary Information of [6]); however, here we clarify a few details of the analysis specific to this experiment.
First, we explain why we chose to present only the analysis carried out on the proton images arising from the sixth RCF film layer (8.5 MeV protons) for all of our experimental shots, and for the second RCF film layer (4.7 MeV) for the experimental shot characterizing our experiment prior to collision. As stated in the main text, quantitative analysis of a given proton image using the approach described in [5] is only possible if the proton beam does not self-intersect (due to deflections acquired in the plasma) prior to reaching the detector. Such self-intersection can be identified by certain features present in proton images (see [5] for an extended discussion). One such feature is the presence of localised proton-flux 'structures' with two key properties: first, typical values much higher than the mean proton flux in an image; second, the broadening of those structures in lower-energy proton images of the same fields. Such a feature can indeed be identified in our proton images for the (two) experimental shots subsequent to the collision of the jets in our experiment. This suggests that so-called 'caustics' are present in all but the highest-energy proton image, and thus the quantitative analysis technique proposed in [5] cannot be carried out reliably on the proton images derived from the first five film layers. By contrast, the variations in proton flux for the experimental shot prior to the jet collision do not have this feature, and so all film layers can, in principle, be analyzed. However, the first layer (see Supplementary Figure 2, top left) has other features suggestive of the film layer being damaged: so a quantitative analysis of such an image would likely lead to erroneous results. As the protons detected in second film layer arrive with the greatest time difference from the sixth layer, we chose to focus our analysis on these two layers for this shot.
Next, we discuss our approach to characterizing the initial proton beam's flux distribution -necessary input into any magnetic-field-reconstruction algorithm. It is known that proton beams arising from high-intensity laser sources can have significant spatial variation prior to any interaction with electromagnetic fields [7]. However, such variation is typically on much larger scales than strong variations in proton flux -so we assume that all variations on scales larger than the known physical scales of interest (i.e., the interaction region's size, n⊥ ≈ 0.25 cm) are variations in the initial proton flux. We determine these initial variations in a two-step procedure: we first calculate a (two dimensional) linear best fit; we then apply a low-pass filter to the difference between the flux distribution and the linear fit (with characteristic filtering scale = 1.5 n⊥ ). The result is illustrated in Supplementary Figure 5. The path-integrated field is then reconstructed using the usual technique described in [5]; in order to be consistent with Supplementary Figure 6: Magnitude of total path-integrated magnetic fields recovered from post-collision proton images. Left: total path-integrated magnetic field recovered from 8.5 MeV proton image taken at ∼19.7 ns after the initiation of the LMJ drive beams. The scale parameters and ⊥ of the anisotropic Gaussian low-and high-pass filters used to recover 'large-scale' and 'small-scale' components of the path-integrated field (shown in Figure 4 of the main text) are given by = 0.018 cm, ⊥ = 0.15 cm. Right: total path-integrated magnetic field recovered from 8.5 MeV proton image taken at ∼22.7 ns. The anisotropic Gaussian filter parameters used in this case are = 0.088 cm, ⊥ = 0.2 cm.
our previous assumptions, we apply a Gaussian high-pass filter to the path-integrated fields with the same parameters at the low-pass filter applied to the original proton image. Thirdly, we briefly discuss the implications of assuming that the measured proton images are created using a monoenergetic beam (with energy p,max ) when we apply the field-reconstruction algorithm, despite our statement in the previous section that protons with a distribution of energies (the characteristic dispersion of that energy around the mean, ∆ p,max , being ∆ p,max ≈ 0.5-1 MeV) contribute significantly to the deposited energy in each film layer of the RCF film stack. The uncertainty in the imaging protons' energy leads to an uncertainty in the deflection angles of the protons composing a particular proton image; for deflections induced by magnetic fields (where the deflection angle, α p , satisfies α p ∝ −1/2 p ), this uncertainty is of order ∼∆ p,max /2 p,max . Therefore, the path-integrated magnetic field measurements derived from the second film layer have an ∼15% intrinsic uncertainty due to this effect, while those from the sixth film layer have an ∼5% uncertainty. The latter error is a small fraction of the error (∼20-30%) of magnetic-field measurements of the interaction-region plasma (which are all derived from the sixth film layer) arising from other sources of uncertainty in the experiment, suggesting that the monoenergetic assumption is a reasonable one for us to make.
The 'total' path-integrated magnetic fields recovered from the proton images obtained after collision of the plasma jets are shown in Supplementary Figure 6. In the main text, we report 'large-scale' -that is, at the scale of the interaction region -and 'small-scale' -at the characteristic scale of the turbulence, and at smaller scales -pathintegrated fields. In order to recover the large-scale (small-scale) path-integrated magnetic fields from the total path-integrated fields, we apply an additional low-pass (high-pass) filter. The choice of filter is complicated by the fact that the morphology of the interaction region is anisotropic: its parallel size n ≈ 400 µm is much smaller than its perpendicular size ( n⊥ ≈ 2.5 mm). We therefore apply an anisotropic Gaussian low-pass filter (scale parameters and ⊥ at particular times given in the caption of Supplementary Figure 6) to obtain the large-scale field, and an anisotropic Gaussian high-pass filter (same parameters) to extract the small-scale field. We note that since the parallel size of the interaction region is rather similar to the scale associated with the peak wavenumber of the magnetic-energy spectra, it is possible that some magnetic structures associated with the supersonic turbulence whose associated wavevectors are oriented in the parallel direction will be suppressed by the high-pass filter. However, for an isotropic stochastic magnetic field, the resulting underestimate of the RMS value of the field will only be a small correction (of order ∼ / ⊥ < ∼ 15%). To illustrate how the large-scale and small-scale path-integrated magnetic fields relate back to the proton images from which they were extracted, we show the proton images resulting from the large-and small-scale fields individually  Supplementary Figure 6, right). Bottom mid-right: proton-flux distribution arising from the large-scale path-integrated field at 22.7 ns (presented in Figure 4 of the main text, bottom left). Bottom far-right: proton-flux distribution arising from the small-scale path-integrated field at 22.7 ns (presented in Figure 4 of the main text, bottom mid-left).
in Figure 7. For comparison, we also show the original proton images, and the predicted images associated with the total path-integrated magnetic field. It is qualitatively clear that the stochastic, smale-scale flux structures in the original proton images are indeed reproduced by imaging the small-scale path integrated field; however, the presence of the large-scale field distorts both their shape and position.

Electron number density measurements using collisional scattering of imaging protons
Here, we elaborate on how we obtained the measurements of electron number density of the interaction-region plasma using collisional scattering of the proton imaging beam; we also provide a bound on the electron number density of the initial supersonic plasma jets on account of the absence of such scattering.
First, we outline the relationship between the effective proton image resolution and the plasma's electron number density. A beam of protons (initial speed V 0 ) travelling through a plasma with electron number density n e diffuses in the direction(s) transverse to its initial motion due to small-angle Coulomb collisions. The characteristic spread in velocities ∆v ⊥ in the proton beam associated with these collisions evolves according to where v is the speed of the proton beam after time t of its interaction with the plasma, ν p|Al ⊥ is the characteristic perpendicular diffusion rate due to Coulomb collisions with aluminium ions, and ν p|e ⊥ is the characteristic perpendicular diffusion rate due to Coulomb collisions with electrons [8]. For the protons used in our experiment, whose velocity is V 0 > ∼ 3 × 10 9 cm/s, the beam protons' velocities greatly exceed both the plasma's thermal ion and electron velocities (assuming a jet-plasma temperature T ≈ 100 eV); as a consequence, ν p|Al ⊥ and ν p|e ⊥ are given by the fast test particle rates: where p is the proton energy (in eV), n i the aluminium ion number density, n e the electron number density, Z the mean aluminium charge, log Λ p|Al the proton-ion Coulomb logarithm, and log Λ p|e the proton-electron Coulomb logarithm. If Coulomb collisions are sufficiently weak for the protons' velocities to be only slightly perturbed before they traverse the whole plasma (assumed path length n⊥ ), it follows from (1) that where t cross ≈ n⊥ /V 0 is the time taken by the proton beam to traverse the plasma. The characteristic scattering angle ∆θ coll associated with collisional interactions is then given by For any proton scattering process with characteristic scattering angle ∆θ 1, it can be shown using kinetic theory [5] that the proton-flux distribution measured on the detector in a proton-imaging diagnostic set-up is the convolution of the proton-flux distribution in the absence of that scattering process with a Gaussian point-spread function, whose full-half-width maximum (FHWM) a res is given by a res ≈ 2.2r s ∆θ, where r s is the distance from the plasma to the detector. Assuming the scattering process is collisional scattering, (5) gives Now substituting (3), we obtain a res ≈ 0.045 r s 10 cm where Z and the Coulomb logarithms are calculated using the characteristic parameters of the beam and plasma. Taking account of the proton-imaging magnification factor M = 13/3, this means that wavenumber cutoff k res due to collisional broadening is given by With the relationship between the proton-image resolution and electron number density determined, we now explain how we construct a predictive model of the high-wavenumber tail of the spectrum of the sharp, large-amplitude proton-flux inhomogeneities (known as caustics) in the 4.7 MeV proton images at a given electron number density. We first calculate the one-dimensional spectrum E δΨ (k) of the relative flux distribution -that is, the spectrum of δΨ ≡ (Ψ − Ψ 0 )/Ψ 0 , where Ψ is the actual proton-flux distribution, and Ψ 0 is the mean proton flux -for a particular region of the 4.7 MeV proton images (the regions used are shown in Figure 3 of the main text). We then assume that the measured spectrum can be expressed in the following manner: where where Eδ Ψ,0 is a normalisation constant chosen so that the the low-wavenumber part of the measured and true spectra coincide [5]. We can then use (9) to give an expression for E 4.7 MeV δΨ (k) in terms of the two constants Eδ Ψ,0 and E noise ( p = 4.7 MeV), which can be determined directly from the data, and k res ( p = 4.7 MeV), which is a function of n e : This is the model used in Figure 3 of the main text to predict the 4.7 MeV relative flux spectrum for a particular electron number density. In addition to our measurements of the electron number density n e of the interaction-region plasma, we can also place an upper bound on n e in each plasma jet prior to collision, based on the absence of collisional scattering of the proton imaging beam. The effective resolution of the 4.7 MeV pre-collision proton images (Figure 8, middle) was the same as for the 8.5 MeV images (Figure 8, left). The characteristic magnitude of the small-scale flux inhomogeneities evident in both images -likely associated with electromagnetic collisionless microinstabilities arising at early times in the experiment [9], before plasma densities rise sufficiently for collisional damping to suppress them -is small, and thus, in the absence of collisions of the proton beam, the Fourier spectrum Eδ Ψ (k) of the flux inhomogeneities in both images is the same, up to a scaling factor equal to the ratio of the beam-proton energies (viz., Eδ Ψ (k) ∝ −1 p ) [5]. This is indeed the case for our data (see Figure 8, right). Writing down an expression analogous to (9) for the 8.5 MeV protons, we can then construct a model for the measured spectrum of small-scale flux inhomogeneities in the 4.7 MeV proton images, given a particular electron number density and the equivalent spectrum in the 8.5 MeV proton images: For electron number densities n e ≥ 10 19 cm −3 , Figure 8, right, shows that the high-wavenumber tail of the 4.7 MeV proton-flux spectrum would be suppressed; we therefore conclude from the absence of any suppression that n e 10 19 cm −3 . This is consistent with the estimates of the electron number density obtained in the FLASH simulation, in which n e ≈ 10 18 cm −3 at the front of each jet [10].

Calculation of interaction-region plasma parameters
Table II presents a summary (both formulae and values) of key plasma parameters for our experiment, including all those referenced in the main text. The formulae used are derived from [8,11,12]; the opacities are calculated using data tables given by [13]. For our reported parameters, the plasma may appear to be well described as being optically thin (λ P is much larger than the largest spatial dimension of the interaction-region plasma, n⊥ ≈ 0.25 cm), and the radiative cooling time is much longer than the turnover time of the plasma turbulence. However, we caution that the Planck opacity for aluminium increases by around three orders of magnitude over the temperature interval T = 20 eV to T = 100 eV. As a consequence of this, the radiative cooling time for a plasma with the same mass density as given in Table II, but with T = 50 eV, is considerably smaller than that at T = 100 eV: τ rad (T = 50 eV) ≈ 1.7 ns (see Supplementary Figure 9, left). As well as this, for T > ∼ 100 eV, the Planck opacity increases as the temperature does (until T ≈ 175 eV); the Supplementary Figure 9: Radiative properties of aluminium plasma relevant to our experiment. Left: radiative cooling time (calculated using the formula given in Supplementary Table II), plotted as a function of temperature.
The 20 different lines plotted correspond to this curve at different mass densities, with ρ min = 10 −5 g cm −5 , and ρ max = 10 −3 g cm −3 , and equal logarithmic spacing. Right: same as left, but showing photon mean free path.
cooling function increases by nearly two orders of magnitude, giving τ rad (T = 175 eV) ≈ 1.4 ns. Because the plasma turbulence is both supersonic and has a large Peclet number, it is to be expected that there be both significant density and temperature variations in the interaction-region plasma; as a consequence, we cannot rule out the possibility that radiative cooling plays a significant role in the turbulent plasma's dynamics. We do, however, observe that the characterisation of the plasma as being optically thin remains appropriate, even in the presence of significant temperature variations. At T = 50 eV, the photon mean free path λ P is given by λ P ≈ 0.5 cm (see Figure 9, right), which remains much larger than the scale of temperature variations (L ≈ 400 µm).

Plasma parameters of supersonic plasma jets prior to collision
As discussed in the main text, we do not have a measurement of the temperature T jet of either jet prior to collision. However, we can estimate T jet in the following manner, using a combination of physical arguments and bespoke FLASH simulations of the LMJ experiment. The latter, which assumed a more efficient laser-foil coupling efficiency than was likely realised in practice, and thus obtained greater jet velocities by a factor of ∼2, suggest that the initial jet Mach number Ma jet of the jet in the experiment satisfies the following bound: Ma jet < ∼ 10. This being the case, we use the known jet velocity (u jet ≈ 290 km/s) to estimate where we have used a Saha ionization model to estimate Z jet at T = 20 eV. If we combine this bound with the physical requirement that T jet T = 100 eV (in other words, assume that the plasma jets experience significant heating when they collide), we conclude that 20 eV < ∼ T jet < ∼ 50 eV. Once these bounds are established, we observe that for the characteristic jet densities (ρ jet ∼ 10 −5 − 10 −4 g/cm 3 ), the radiative cooling time τ rad,jet is approximately τ rad,jet ∼ 3 − 10 ns. This is comparable to the time taken for the jet to travel the distance grid ≈ 0.2 cm from the grids in the centre of the target (t grid ≈ grid /u jet ≈ 7 ns). Thus, we conclude that each jet experiences significant cooling as it travels towards its counterpart, and so a reasonable estimate for T jet is the lower bound (13): T jet ∼ 20 eV.
Using the formula for η given in Table II, we can now estimate the magnetic Reynolds number of either jet: Rm jet ≈ u jet L/η jet ≈ 20. This justifies the assumption made in the main text that the magnetic field is effectively frozen into the flow of each jet over the timescale ∆t p ≈ 300 ps separating the two proton species used for imaging the jets.

The role of the Biermann battery in our experiment
The Biermann battery is known to generate significant magnetic fields in colliding laser-plasma jet experiments; here, we elaborate on their importance in our experiment. In the main text, we claim that the subsequent amplification of a stochastic component of the magnetic field observed in our experiment can be attributed to the action of supersonic motions, rather than to the action of the Biermann battery alone. To justify this claim, we consider the induction equation governing the evolution of the magnetic field, where k B is Boltzmann's constant, e is the elementary charge, and c is the speed of light, and compare the respective sizes of the inductive term and the Biermann term just after collision: |u × B| |ck B ∇n e × ∇T e /en e | ∼ δn e n e We conclude that the terms are comparable, provided that both the variations in density and in temperature are comparable to their mean values.

RELATIONSHIP OF RESULTS TO PREVIOUS EXPERIMENTS
In the main text, we reference several previous laser-plasma experiments that reported turbulent amplification of magnetic fields [6,14,16,17], and one which reported creating boundary-free supersonic turbulence without observing amplification [15]; here, we outline in more detail the relationship between these experiments and the experiment reported in this paper. Here, L is the driving scale of turbulent motions in the plasma; we note that in [14], the reported magnetic Reynolds number is instead defined with the scale length of the entire turbulent plasma.
have all been calculated using consistent definitions of relevant quantities (e.g., the length scales used in the fluid and magnetic Reynolds numbers); we caution readers that the conventions (and notation) employed in the original papers are not all consistent with each other. For convenience, we refer to each experiment via an identifier: the name of the laser facility at which the experiment was carried out. While amplification of magnetic fields is seen in all of the experiments, it is much more significant in the NIF, OMEGA A and OMEGA B experiments than in our LMJ experiment, or in the Vulcan experiment. The obvious difference between these two sets of experiments is the values of Rm and Pm. More specifically, for the LMJ and Vulcan experiments, Rm is likely to be below the critical value Rm c required for the turbulent dynamo to operate in the Pm 1 regime. In the OMEGA and NIF experiments, that critical threshold seems to have been successfully surpassed (indeed, in OMEGA B and NIF we anticipate that the greater than order-unity value of Pm reduces Rm c significantly).
The similar degree of amplification seen in the LMJ experiment and in the Vulcan A experiment, in spite of the significantly smaller values of Rm obtained in the latter, suggests that another factor must be affecting the efficacy of the amplification mechanism in our experiment. As discussed in the main text, a plausible candidate is the turbulent Mach number. The Vulcan B experiment suggests that the turbulent Mach number of the plasma increases