Synchrotron radiography of Richtmyer–Meshkov instability driven by exploding wire arrays

Hydrodynamic instabilities are of fundamental importance to geophysical and astrophysical flows,^1,2 planetary sciences,³ inertial confinement fusion,⁴ and other applications.^5–8 The Richtmyer–Meshkov instability (RMI) describes the baroclinic vorticity deposition that occurs during an impulsive acceleration of an interface separating fluids of different densities—usually due to the passage of a shockwave—and the resulting dynamics lead to the amplification of any perturbation on the interface. This process, unlike the Rayleigh–Taylor instability, can be realized in both light–heavy and heavy–light configurations. In its nonlinear stage, the RMI tends to form a turbulent mixing layer through its interaction with the Kelvin–Helmholtz instability. Due to the ubiquity of this phenomenon, it is interesting to understand the dependence of the fluid mixing initiated by the RMI on the initial conditions (ICs) of the interface.

Recent theoretical and numerical studies^9–15 have broadly focused on understanding the effect of two different types of ICs. In the first case, the RMI induced by a shock passing an arbitrary shaped interface was considered and shown to significantly affect the initial vorticity deposition and growth speed.^9,10 A corollary of this research was the optimization of the initial shape of the interface to suppress the RMI.¹¹ The second type of ICs—consisting of small broadband perturbations deposited on the interface by stochastic or manufacturing processes—was considered in connection to the persistence memory of the ICs in the nonlinear turbulent stage.⁶ Numerical simulations in both two (2D) and three (3D) dimensions show that the growth rate exponent θ of the turbulent mixing layer, as well as the mixing fraction, is strongly dependent on the ICs described by broadband and narrowband power spectra.¹³ 

Experimental investigations have focused on the first type of ICs, understanding the effect of the interface shape—likely due to its easier experimental diagnostics. The experiments mainly used shock tubes with initially separated gases (usually N₂ and SF₆). Example investigations include the shock interaction with a rectangular block,¹⁶ cylinder,¹⁷ two cylinders,¹⁸ inclined interfaces,^19–22 thin curtains,²³ and many other experimental configurations. The separation of the gas species in these experiments was achieved with a thin shaped nitrocellulose membrane, or with a gravity-assisted flow for membrane-less experiments.^24,25 The disadvantage of the former method is the potential disruption of the flow by the fragments of the membrane, while the latter suffers from the production of a diffusion layer that inhibits the RMI²⁶—both effects changing the ICs. Experiments with broadband ICs were largely conducted with statistically repeatable shear-flow experiments.^27–29 

More recently, RMI experiments utilized platforms traditionally used to study matter at extreme conditions (temperature T > 1 eV, pressure P > 1 Mbar), such as high-powered lasers³⁰ or magnetically driven pulsed power.³¹ The extreme pressures (pressure $≫$ yield strength) and shock speeds (Mach number M > 10) generated by these drivers allow manufacture of hydrodynamic targets from solid materials with a well-characterized initial topology. The experiments investigated the single-mode RMI growth at high values of M in planar,³² cylindrical,^30,31 and thin-layer configurations.³³ Despite the well-known ICs in these experiments, the fast shock speeds make it a challenge to obtain more than 1–2 radiographs per experiment and require repeating experiments with the same ICs to measure the growth over time. At the same time, the solid targets and convergent geometries often employed in these systems lead to optical opacity and a requirement for x-ray diagnostics.

In this article, we present a novel platform for RMI experiments in arbitrary geometries using the underwater electrical wire explosion (UEWE). UEWE occurs when a current pulse with duration in the range of $10 − 7 – 10 − 6$ s and the current density of $> 10 7$ A/cm² dynamically heats and explodes an array of thin metallic wires submerged in water. As the wires explode, they drive a strong shock traveling at multi-km s⁻¹ speeds and multi-kbar pressures into the surrounding medium.^34,35 The shock geometry can be adjusted to the requirement of the experiment by changing the wire array geometry. Previous research has explored planar,^36,37 cylindrical,^38,39 and spherical shockwave production.⁴⁰ Unlike the name suggests, the UEWE does not necessitate the use of water and can be successfully performed in any dielectric medium, including plastics and epoxy resins.⁴¹ Similar approaches were also used to explore shock interaction with water columns⁴² or reservoir stimulation.⁴³ 

In this article, we describe a proof-of-concept RMI experiment conducted on the pulsed power driver (PPD) located at the European Synchrotron Research Facility Microtomography beamline (ID19). The combination of the PPD and the multi-frame synchrotron radiography enables measurement of up to 256 radiographs (60 ps exposure) per experiment in arbitrary geometries, including convergent geometries usually inaccessible to visible light. It opens the possibility of observing the development of the RMI in a single experiment, arbitrary geometry, and with well-characterized ICs.

The remainder of the article is organized as follows: in Sec. II, we describe the experimental setup, including details of the pulsed power generator, wire arrays, and x-ray diagnostics. In Sec. III, we detail the reasoning behind the choice of our hydrodynamic target and its pre-shot characterization. In Sec. IV, we show the results of an example RMI experiment and describe the observed dynamics. Finally, in Sec. V, we discuss the implications of this work and future research avenues.

The electrical setup consisted of a compact pulsed power generator with four 0.22 μf Maxwell capacitors connected in parallel and discharged through an externally triggered pancake spark gap switch. The spark gap switch was connected to the electrical load and triggered by a pulse provided by a PT-55 spiral generator. The high voltage trigger pulse was essential to lower the timing jitter of the system to < 10 ns, a timing accuracy required for reliable co-timing with experimental diagnostics, such as multi-MHz synchrotron radiography or streak cameras. On each shot, the capacitors were charged to + 30 kV (stored energy $∼ 400$ J). The discharge current was measured by a self-integrating Rogowski coil placed at the output of the generator and a current-viewing resistor (CVR) placed at the output of the load as shown in Fig. 1(a). The voltage was measured by a P6015A Tektronix voltage divider connected across the load. Additional details of the pulsed power setup were discussed in an earlier publication.⁴⁴ 

The experimental cell consisted of a wire array and the hydrodynamic target enclosed within a 50 mm diameter cylindrical plastic capsule filled with water. During the experiments, the chamber was filled with de-ionized water for two purposes. First, the de-ionized water prevented formation of a plasma channel along the wire surface and tamped the wire expansion, thus maintaining high energy density deposition into the wire material by localizing the current flow and maintaining high resistivity for a longer time. Second, the de-ionized water served as the working fluid into which the exploding wires launched shocks, which were used to drive the hydrodynamic experiments.

In Fig. 1(b), we show the load geometry used in the presented research: a planar wire array consisting of 19 copper wires of 64 μm diameter spaced 0.8 mm apart. The wires were connected to the high voltage electrode and grounded through the CVR. The number of wires and their spacing was chosen to produce a planar shockwave. At a propagation distance > 2 mm from the initial wire array position, numerical simulations of the shockwave performed in the magnetohydrodynamic code GORGON suggest a water density variation $< 0.5 %$ in the leading mm of the shockwave. The width of the array was chosen such that the resulting planar shockwave spanned the entire experimental field-of-view shown in Fig. 1(b) (red dotted rectangle). The total cross section of the wires and the wire length were optimized to maximize the energy density deposition rate into the wire material, which is realized for close to critically damped (slightly overdamped) discharge and results in stronger shockwave generation due to faster radial expansion of wires.⁴⁵ 

The cross section of wires, which determines the time of the wire explosion (i.e., when the vapor-low-ionized plasma phase transition occurs), was estimated using tabulated values of the current action integral for copper.⁴⁶ Here, we consider the time of explosion $τ e x$ to be $τ e x = 1.41 ( L C ) 1 / 2$⁠, where L and C are the total inductance and capacitance of the discharge circuit. The condition for the coincidence of peak current with the explosion time in terms of the current action integral $h ¯$ is $τ e x ≈ 2 h ¯ / j 2$⁠, where j is the current density per unit area and $h ¯$ is a tabulated value that primarily depends on the wire material for sufficiently fast discharges. In our experiments, the specific current action integral at peak voltage (corresponding to wire explosion) was $2 × 10 9$ A²s/cm⁴ compared to the tabulated value of $2 – 4 × 10 9 A 2 s / cm 4$⁠,⁴⁶ suggesting optimal wire explosion parameters for high energy density deposition. The length of the wires was 35 mm, adjusted empirically until the current pulse became almost critically damped, as shown in Fig. 2.

An example of the current and resistive voltage waveforms measured in a typical experiment is shown in Fig. 2. The current waveform was measured by the CVR, while the resistive voltage waveform was calculated by subtracting the inductive voltage $L l d I / d t$ from the measured voltage, where L_l is the load inductance. The discharge was characterized by a peak current density of $j ∼ 5 × 10 11$ A/m² through the wire cross section. The skin depth of the discharge at standard conditions was $δ ∼ 130$ μm, which is $∼ 4$ times larger than the radius of the wires, implying uniform current distribution and energy deposition throughout the wire volume. The resistive voltage reached a maximum of $∼ 40$ kV approximately 300 ns after peak current. At the time of explosion, the corresponding deposited energy $∫ I ( t ) V R ( t ) d t$ in the wire material was $∼ 300$ J, showing a total efficiency in energy deposition of 75% in the first 1.5 μs of the discharge.

The main diagnostic in these experiments was multi-MHz x-ray radiography. Using x-ray imaging enables the use of optically opaque samples, such as foams, plastics, or low Z metals, as well as experiments in convergent geometries where the compression geometry does not allow a direct line of sight and obscures the dynamics—e.g., spherical geometry or side-on imaging of cylindrical geometries. In addition, it is possible to estimate approximate mass densities from the obtained x-ray radiographs through attenuation measurements. The x-ray imaging was performed by a polychromatic synchrotron beam produced by two axially aligned long-period undulators (λ = 32 mm), with its harmonics spanning energies between 20 and 50 keV and a mean energy of $E ¯ ∼ 30$ keV.

During the experiments, the synchrotron storage ring operated in a 16-bunch filling mode (60 mA current) and delivered a photon flux of approximately $2 × 10 7$ photons/mm² in a single 60ps long pulse every 176 ns. After propagating through the experimental volume, the x rays were converted to visible light by a fast single-crystal scintillator (LYSO:Ce, 40 ns decay time, Hilger Crystals, UK) as shown in Fig. 1(a). The resulting light was lens-coupled via pellicle beamsplitters to two high-speed cameras (HPV-X2, Shimadzu Corp., Japan). The cameras operated with an inter-frame time of 200 ns and an exposure time of 100 ns captured 128 frames each and were triggered in parallel to capture every synchrotron pulse (delay of 100 ns between camera triggers). If the cameras were set to trigger in series (one after another), the time window of the measurement would be doubled to 50 μs at the cost of missing a single synchrotron x-ray pulse every $∼ 1.3$ μs due to the mismatch between the camera acquisition frequency and the radio frequency of the storage ring. It is important to remark that the temporal resolution of this single-bunch imaging was set by the 60 ps long x-ray pulse, not the 100 ns camera exposure.

The spatial resolution of the measurement was set to 32 μm/pixel by the pixel size of the cameras and magnification of the camera optics, giving a field-of-view of $8 × 12.8$ mm² (V × H).

To visualize complex hydrodynamic flows, the x rays propagated 11 m past the target, reaching the “edge-enhancement mode” of phase-contrast imaging at the detector, i.e., propagation-based inline phase-contrast imaging⁴⁷ (source coherence length $d t ∼ 100$ μm). In this mode, interfaces between materials with different refraction indices are highlighted, while the bulk material density can still be analyzed assuming absorption-dominated radiography. More details about the imaging system can be found in previous publications.^48–51 In addition to the visualization, the x-ray radiographs were used to estimate mass densities of the samples during their hydrodynamic evolution via their x-ray absorption. Assuming linear response of the x-ray scintillator and the high-speed camera, the intensity of the x-ray beam after propagating through the water-filled capsule/sample was approximated by the Beer–Lambert law, $I = I 0 exp ( − κ ρ x )$⁠, where I₀ is the initial intensity of the x-ray beam, κ is the attenuation coefficient of the sample for a given photon energy, ρ is the sample density, and x is the thickness of the sample along the line-of-sight. In earlier research,³⁶ we estimated the effective attenuation coefficient for a water target of the same thickness to be approximately equal to the mass attenuation of water at 40 keV. The representative energy of 40 keV used to approximate the polychromatic beam is higher than the mean beam energy. This is due to the more effective attenuation of lower energy photons by the water in the sample, hence contributing less to the transmitted beam: an effect which is well-known as “beam hardening” in microtomography with polychromatic illumination.

To perform a successful hydrodynamic experiment, the target material needs to satisfy a set of requirements. First, the material must be machinable into a sample with a characteristic size of $∼ 10$ mm and geometric perturbations on its interface with size $∼ 200$ μm. Second, the target should not chemically interact with water and should maintain its density while submerged. Finally, the yield compressive strength of the target needs to be orders of magnitude lower than the shock pressure for the non-hydrodynamic part of the pressure tensor to be negligible.

Polyurea aerogels are mechanically robust with a compressive yield strength between 0.6 and 13 MPa (much lower than the $∼ 300$ MPa of the shock pressure generated by our approach), do not soak water, have a discernible foam microstructure only on $∼ 100$ nm length scales,⁵² and can be manufactured in the density range of 0.016–0.55 g/cm³. These properties make polyurea aerogels perfect hydrodynamic targets as they can be fabricated into a variety of shapes prior to experiments. Recently, research at MIT, LANL, and AWE ascertained the equation-of-state and Hugoniot for the polyurea aerogel in multi-shock experiments.^52–54 

In our experiments, a commercially available polyurea aerogel was machined on a 3-axis CNC machine using a set of solid carbide end mills with 0.5 mm corner radii to obtain a sinusoidal perturbation on its interface. The machining of the sample was enabled by the compressive yield strength $Y ∼ 0.6$ MPa at a low mass density of only 0.1 ± 0.02 g/cm³ [Ref. 62]. Each target was machined from a $30 × 20 × 15$ mm³ block with the transverse size (20 mm) larger than the field-of-view and the longitudinal size (30 mm) comparable to the wire length to minimize edge effects and increase the integrated x-ray attenuation along the line-of-sight.

To ascertain the ICs on the surface and within the aerogel volume, the samples were scanned using synchrotron-based microtomography prior to the experiments. A filtered pink beam generated by a wiggler insertion device (filters: 1.8 mm Diamond, 2.8 mm Aluminum, 0.14 mm Copper) with $∼ 30$ keV peak energy was used to illuminate the sample. The x-ray transmission and contrast were comparable to that used during the ultra-fast in situ measurements. For these measurements, an indirect x-ray detector was placed 1 m downstream from the sample to form propagation-based phase contrast on the air–aerogel interface. The indirect detector assembly consisted of a 500 μm LuAG:Ce scintillator optically coupled to a PCO Edge 5.5 detector. The resulting radiographs were $16.64 × 7.46$ mm² (V × H) in size with a pixel size of 6.5 μm. The tomography scans were performed over 360°, capturing 2200 projections in total. The direct phase retrieval algorithm⁵⁵ was used to treat the projections prior to tomographic reconstruction that was carried out with the filtered back-projection algorithm using the Nabu software.⁵⁶ The reconstructed volumes were post-processed for ring removal using a set of in-house developed codes.

An example of a single 2D slice of such scan is shown in Fig. 3. The scan was performed over a volume of $16.64 × 16.64 × 16.64$ mm³ with a voxel size of 6.5 μm³. The sample was volumetrically homogeneous on the spatial scale of the scan. The Fourier transform of the aerogel surface exhibited undesired multimode perturbations of $< a 0 / 60$ amplitude in the $λ < 400$ μm space, where a₀ is the initial amplitude of the pre-imposed sinusoidal perturbation and λ is the wavelength. In principle, the standardization of the sample could be improved to sub-μm resolutions over longer scanning times to inform more highly resolved numerical simulations.

In this section, we present the results of an experiment measuring the single-mode planar RMI in the small amplitude regime. This well-established test problem serves to benchmark and highlight the capabilities offered by the presented experimental method.

We remind the reader of Fig. 1(b), where the conceptual design of the experiment including the field-of-view (highlighted in a red dashed rectangle) is shown. In Fig. 4, we show a small selection of radiographs obtained in a single experiment. The radiographs show the development of a sinusoidal interface (⁠ $λ 0 = 4.4$⁠, $a 0 = 0.2$ mm) separating the water (1 g/cm³, dark region) from the aerogel (0.1 g/cm³, bright region). The RMI is characterized by a pre-shock Atwood number of $A 0 = ( ρ 0 − ρ 1 ) ( ρ 0 + ρ 1 ) = − 0.8$⁠, where ρ₀ is the initial aerogel density and ρ₁ is the initial water density.

Upon the wire array explosion, each individual wire launched a cylindrical shock into its surroundings as visible in Fig. 4 at t = 0.18 μs. By t = 0.88 μs, the cylindrical shocks had merged to form a stable planar interface propagating at 2.2 ± 0.3 km s⁻¹ toward the water–aerogel interface. At impact, numerical simulations suggest that the leading mm of the shock can be modeled as an unsupported planar shock with a variation in the density of $< 0.5 %$⁠. A detailed investigation of the formation and planarity of the shock behind the interface produced by this method will be discussed in a separate publication. Due to the planarity of the shock, the compressed water between the wire array and the shock could be characterized by an average post-shock water density of 1.06 ± 0.01 g/cm³—calculated by considering the difference in the x-ray intensity transmitted through the compressed and ambient water. Using the measured density jump and shock speed, the pressure behind the planar shockwave in water was estimated via the Rankine–Hugoniot conditions for mass and momentum conservation as $P s w = 300 ± 100$ MPa, which is significantly larger than the compressive yield strength of the aerogel $Y aerogel ∼ 0.6$ MPa. As such, it is expected that strength effects will not play a significant role and the shock/target interaction will be considered hydrodynamically.

At t = 1.06 μs, the shockwave arrived at the minima of the interfacial perturbation, isentropically released to zero pressure, and drove a rarefaction wave back into the water. Shortly after, the shockwave arrived at the maxima, released to zero pressure, and also drove a reflected rarefaction wave. The combined rarefaction waves can be seen in subsequent radiographs (⁠ $1.41 < t < 1.58$ μs) as a reflected (bright) wave propagating toward the wires at a speed of $2 ± 0.3$ km s⁻¹, where the error on the speed measurement is predominantly due to the uncertainty in the interfacial position measurement. In the regions where the reflected rarefaction waves converged (i.e., focused), we observed a sudden appearance of low-density structures at times t > 3.87 μs. The high x-ray transmission through these features is consistent with the formation of cavities within the water volume. These features can be explained by a nucleation of cavitation bubbles in the water, which, unlike gases, can support negative pressures (i.e., tension).⁵⁷ 

In Fig. 5(a), we show the perturbation amplitude on the interface obtained from the radiographs as a function of laboratory (physical) time t and dimensionless time $t ′ = [ λ 0 ( λ 0 / 2 π a 0 A Δ u ) ] −1 t$⁠. The variable $t ′$ is present in the scale-free Euler equations, where $( λ 0 / 2 π a 0 A Δ u )$ represents the characteristic growth rate of the perturbation predicted by the Richtmyer's impulsive model. In Fig. 5(b), we show the mean position of the interface as a function of laboratory time. The evolution of the interface after shock impact is described by a combination of compressibility effects, linear hydrodynamics, and non-linear hydrodynamic effects. Between t = 2.9 and t = 8.5 μs, the evolution of the interface perturbation is not shown as its amplitude is close to zero and is hard to distinguish from the compaction wave launched into the aerogel. The time origin t = 0 is chosen such that the first radiograph with t > 0 captures the wire explosion. Making this choice, the shockwave arrives at the interface perturbation at t = 1.06 μs when the interface is suddenly shock-compressed.

In Fig. 4, between t = 1.06 and t = 1.41 μs, the shockwave compresses the interface perturbation from $a 0 / λ = 0.045$ to $a 0 / λ = 0.034$⁠; an effect visible in Fig. 5(a) as a sudden drop in $a ( t ) / λ$⁠. It is remarkable that the imaging system can resolve the compression phase and does not require any of the assumptions in estimating the post-shock amplitude common for other approaches.³¹ After compression, the perturbation continues decreasing in amplitude, passes through zero at $t ∼ 5$μs, and grows in an inverted direction until the end of the measurement.

In an idealized experiment, the dynamics following compression would be asymptotically described by a linear RMI theory if $a ( t ) / λ < 1$ and $t ≫ τ ch$⁠, where $τ ch$ is a characteristic time required to attain the asymptotic linear solution. In the experiment, the interface develops a non-uniform structure just after impact that appears to be formed by many small vertical jets launched from the water interface into the aerogel (see inset in Fig. 4 at t = 6.51 μs). These structures are likely explained by jetting, a strongly non-linear process that occurs due to high-frequency multi-mode perturbations deposited on the aerogel interface during the machining process. Despite being under-resolved, the approximate wavelength of the jets appears to agree with the small perturbations (⁠ $λ < 400$ μm) seen in the tomograph of the aerogel sample during characterization. Microscopically, these jets are expected to be different even in experiments with the same nominal ICs; however, they should be correctly captured by numerical simulations using the 3D tomograph of the aerogel sample as its ICs. In the following analysis, we will assume that the non-linear processes are small and do not strongly affect the linear growth of the perturbation.

At the beginning of the RMI growth, between t = 1.1 and t = 3.1 μs, the measured growth speed of the perturbation was not constant and did not appear to follow the asymptotic linear theory. This effect can be explained by the proximity of the compaction wave to the perturbation effectively decreasing the RMI growth rate.⁵ In the small amplitude theory, the RMI instability initially accelerates from $d a ( t ) / d t = 0$ to reach the asymptotic limit at $t ≫ τ ch$⁠, where $τ ch$ represents a characteristic time over which the presence of reflected and transmitted shocks affects the instability growth. For the heavy–light configuration, the characteristic time $τ ch$ is given by $τ ch = ( λ 0 / 4 π ) [ ( 1 − A + ) / U 1 + ( 1 + A + ) / U 2 ]$⁠, where U₁ is the speed of the reflected shock and U₂ is the speed of the transmitted shock.⁵⁹ Substituting the shock speeds for the transmitted compaction wave and reflected rarefaction wave speeds, the characteristic time for the measured experiment is $τ ch ∼ 2$ μs. This estimate is consistent with the experiment reaching the asymptotic linear growth speed by $t ∼ 8$ μs, despite being at an early dimensionless time $t ′$⁠.

In this article, we presented a novel, shape-agnostic, membrane-less approach to shock-driven hydrodynamic experiments involving a combination of a pulsed power driven electrical wire explosion to drive high pressure shockwaves in water (P > 100 MPa) and synchrotron multi-frame x-ray imaging diagnostic. The technique offers a number of advantages, including the ability to shape the experimental geometry, produce high pressures that enable the use of liquid and solid materials as hydrodynamic targets, and co-time the experiment with multi-MHz radiography to acquire all data within a single experiment, eliminating uncertainties regarding repeatability of the ICs.

Currently, the synchrotron x-ray diagnostic at the European Synchrotron Radiation Facility ID19 can acquire up to 256 radiographs, spaced 176 ns apart, and with an exposure time of 60ps on each experiment. The spatial resolution each radiograph is 32 μm/pixel with the field-of-view spanning $8 × 12.8$ mm² (V × H). This resolution can be increased up to 3.2 μm/pixel at the cost of proportional decrease in the field-of-view. These characteristics make this approach a valuable diagnostic for single-experiment, multi-frame measurements—especially in the late nonlinear development of hydrodynamic instabilities where the knowledge and the repeatability of the ICs may play a significant role.

To establish this technique, we conducted a small amplitude planar RMI experiment at the ESRF. As this type of experiment has a well-established theory with abundant experimental evidence, it serves as a good point-of-comparison against other hydrodynamic platforms. In the experiment, a planar shockwave propagating in water at 2.2 ± 0.3 km s⁻¹ was driven by an underwater wire array explosion produced by a $∼ 30$ kA discharge in $∼ 1000$ ns. The shockwave impacted a sinusoidal perturbation on a water–aerogel interface characterized by a pre-shock Atwood number of $A 0 = − 0.8$⁠. Radiography of the subsequent dynamics obtained a very highly resolved RMI growth, including the period of perturbation compression, compaction wave detachment, and linear RMI growth. Shortly after the shock impact, the perturbation growth did not agree with the asymptotic linear RMI theory due to the proximity of the compaction wave (in the aerogel sample) to the perturbation. When the compaction wave propagated sufficiently far, but the interface perturbation still satisfied the linear RMI criterion, it was shown that the growth of the interface was consistent with the asymptotic linear theory. The agreement suggests that this approach can capture purely hydrodynamic phenomena using liquid and solid materials; however, the consistency between the shown experiments and the small amplitude RMI theory does not guarantee ideal hydrodynamic behavior in the nonlinear stage. Importantly, the temporal resolution of the measurement was sufficient to capture the compression of the perturbation during the shock impact and so did not require any assumptions to estimate the post-shock perturbation amplitude. The x-ray radiography also unveiled other related phenomena that could be investigated, such as cavitation in liquids caused by converging rarefaction waves.

In the future, we aim to use this technique to explore the nonlinear growth of the RMI with well-characterized initial conditions in both planar and cylindrical geometries. To extend the measurement into the nonlinear regime, the ICs can be adjusted by decreasing the perturbation wavelength λ₀ and increasing the time between radiographs. These changes should allow the experiment to capture a larger range of the dimensionless instability evolution. It is also possible to initiate the experiments in the regime, where $a 0 / λ 0 ∼ 1$ to obtain significant velocity shear, thus initiating the growth of the Kelvin–Helmholtz instability and amplified fluid-mixing early in the experiment. To diagnose the nonlinear experiments, it is necessary to improve the on-shot characterization of the x-ray source or use a monochromatic source, such as an x-ray free-electron laser to directly measure the mass mixing fraction during the experiment—a metric that is much more suitable for the characterization of the nonlinear mixing stage than the usually measured instability amplitude (mixing zone width).⁶¹ Due to the versatility of this approach, other experiments are also possible, including but not limited to, thin-layer RMI experiments or experiments involving multiply re-shocked interfaces.

This research was sponsored by First Light Fusion Ltd, EPSRC, U.S. Department of Energy under Collaborative Agreement No. DE-NA0003764. Beamtime was kindly granted for the experiments by the ESRF user program under Award Nos. HC-4679 and HC-4455.

The authors have no conflicts to disclose.

Jergus Strucka: Conceptualization (lead); Data curation (equal); Formal analysis (lead); Investigation (equal); Methodology (lead); Writing – original draft (equal). Alexander Rack: Supervision (equal); Writing – review & editing (equal). Yakov E. Krasik: Investigation (equal); Resources (equal); Supervision (equal); Writing – review & editing (equal). Jeremy P. Chittenden: Conceptualization (equal); Supervision (equal); Writing – review & editing (equal). Simon N. Bland: Conceptualization (equal); Data curation (equal); Investigation (equal); Project administration (equal); Resources (equal); Supervision (lead); Writing – review & editing (equal). Bratislav Lukic: Investigation (equal); Writing – review & editing (equal). Marlene Koerner: Formal analysis (equal). Jack W. D. Halliday: Supervision (equal); Writing – review & editing (equal). Yifan Yao: Investigation (supporting). Kassim Mughal: Investigation (supporting). Daniel Maler: Investigation (supporting). Sergey Efimov: Investigation (supporting). J. Skidmore: Resources (equal); Supervision (equal); Writing – review & editing (equal).

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