Same-sided successive-shock HED instability experiments

Due to the ubiquitous nature of shocks interacting with multiple features and interfaces, inertial confinement fusion (ICF) and high-energy density physics (HEDP) systems experience complicated seeds for instability growth and, thus, mix.^1–4 Understanding instability growth and mix from complex seeds is especially important in ICF implosions, as mixing of the cold ablator material into the deuterium–tritium (DT) fuel is a primary conjectured degradation mechanism of the fuel burn,^5–7 as is parasitic loss of energy to hydrodynamic instabilities reducing the effective compression.^8,9 To design a robust ICF platform for studying mix and burn, we must have confidence our codes correctly model all relevant sources of mix.

Successive shocks, or shocks from the same direction, are a common phenomenon that contribute to Richtmyer–Meshkov (RM) instability^10,11 growth in ICF systems. Single-shell ICF implosions are often driven deliberately with a multiple-shock drive,^2,12 to keep the fuel entropy as low as possible during compression. In these systems, the shocks are often designed to coalesce right after breaking out of the DT ice layer, at the edge of the DT gas hot spot. Thus, imperfections at both the ablator/DT ice interfaces and the DT ice/DT gas interface seed RM instability growth for each shock. In contrast, Double Shell⁴ ICF implosions are driven with a single shock. However, the multi-layer nature of the target results in a complex shock profile, due to the many reflected and re-reflected waves created each time a shock interacts with an interface. In the current National Ignition Facility (NIF) Double Shell designs,¹³ successive shocks occur in multiple places, most importantly at the interface between the high-Z inner shell and the DT fuel, potentially mixing high-Z material into the burning fuel and degrading the capsule yield.

However, despite the presence of successive shocks in these systems, there are very few studies of successive shock dynamics. The lack of validation data are due to the difficulty in creating successive shocks of similar strengths with the drivers used in traditional low-energy-density experiments, e.g., pistons, membranes, or high-explosives. Successive shock hydrodynamics accesses a rich, complex space of RM instability growth,¹⁴ but models and hydrodynamic simulations for successive shocks are mostly unvalidated. The ability to create shocks of similar strengths and actively control the time delay between shocks is crucial to ensuring that experiments can access multiple instability growth scenarios. Controlling the magnitude of vorticity deposited by the first and second shocks is crucial to observing the range of vorticity competition behavior theorized for successive shocks, specifically the transition between the “continued growth,” “freeze out,” and “re-inversion” growth scenarios, which will be discussed in Sec. II.

In this paper, we present the minimum criteria for assessing all RM growth scenarios from successive shocks, as predicted by the vorticity competition model from ideal fluid theory (Sec. II). We then cover the details of a laser-driven shock-tube platform (Sec. III) at the National Ignition Facility (NIF)¹⁵ capable of creating controllable successive shocks (Sec. IV) that can access the three vorticity competition cases for successive shocks. Next, we present the steps of a data analysis routine to consistently find and fit the interface contour across data sets with non-trivial noise and interface contrast variation (Sec. V). Finally, we present experimental results (Sec. VI) and comparison to both a modified analytic fluid model (Sec. VI A) and hydrodynamics simulations (Sec. VI B) demonstrating not only that we can access and control all successive shock RM growth cases for the first time but also that both the model and simulations are in good agreement with the experiments. All together this confirms that the superposition treatment of RM for shocks from the same direction is theoretically sound for linear growth systems where there is no significant mode coupling.

Using ideal, incompressible fluid theory for the linear growth of a single-mode perturbation, Mikaelian¹⁴ predicts 15 different theoretical combinations of RM growth under two successive shocks or a rarefaction then a shock. Looking at just successive shocks, this simplifies to five cases. Since successive shocks effectively deposit two RM instabilities on the same interface, we can think of treating the perturbation evolution after the second shock as due to a superposition (time-dependent vector sum) of the vorticity deposited by each shock. Then, the five growth cases are the accessible combinations of the vorticity evolution after the first shock and the net vorticity evolution after the second shock. There are two vorticity deposition cases possible from a single shock:

1. positive vorticity (in-phase perturbation growth) and

2. or negative vorticity (perturbation inversion and out-of-phase growth).

Similarly, the net vorticity after the second shock falls into three cases:

1. net vorticity keeps the same sign,

2. net vorticity zeroes out, and

3. net vorticity changes sign,

We can take advantage of the heavy-to-light RM configuration to access all three of the net vorticity deposition cases, since in this configuration the second shock sees a similar density gradient as the first shock. If we allow the perturbation to invert before the second shock arrival, then the sign of the net vorticity after the second shock can vary. In the strong shock systems typical of HED experiments, the perturbation inversion occurs very quickly after first shock arrival, making it relatively easy to design a system where the second shock interacts with a post-inversion interface. Thus, we can, respectively, describe the three net vorticity cases in a heavy-to-light system in terms of the perturbation growth as follows (Fig. 1):

1. continued growth (first shock vorticity dominates),

2. freeze out (vorticity cancels out), and

3. shrink and re-invert (second shock vorticity dominates).

We note that the heavy-to-light, second shock arrival post-inversion configuration makes up three of the five growth cases with successive shocks. The two remaining cases, heavy-to-light with second shock arrival pre-inversion and light-to-heavy, are both cases where the sign of the vorticity deposition is the same for both shocks. When an interface is reshocked (i.e., the second shock is counter-propagating relative to the first), the vorticity from the first and second shocks can likewise compete. The vorticity competition cases can be accessed when the first shock is light-to-heavy, and when the first shock is heavy-to-light and the second shock arrives pre-inversion. However, these cases are both difficult to access due to the limited duration of support for light-to-heavy shocks in HED,^16,17 and the strong shocks used in HED systems, which leads to a short or non-existent pre-inversion portion of heavy-to-light RM growth. The most accessible configuration for reshock in HED is a heavy-to-light first shock with the second shock arriving post-inversion, in which the two shocks deposit vorticity of the same sign. Thus, the heavy-to-light successive shock case is the most straightforward way to access vorticity competition in the HED regime.

While the vorticity superposition model is illuminating for linear RM growth, the assumption breaks down for non-linear growth. During the linear growth stage, all the vorticity remains in the bubble/spike features of the perturbation, where it is easily influenced by the vorticity from the next shock. However, in non-linear growth, the vorticity begins to migrate along the interface, so that vorticity deposited by the subsequent shock will no longer match in profile. This will lead to incomplete cancelations of the two vorticities. As the nonlinear evolution of vorticity tends to collect the vorticity in rings near the spike and bubble tips (leading to the so-called mushroom capping), the vorticity deposited by a second shock will have a stronger relative effect on the remaining stems. This can result in the stem thinning after the successive shock, leading to potential vortex detachment instead of full perturbation inversion. Thus, it is apparent that our model cannot handle complex mix due to non-linear RM growth under successive shocks and more sophisticated tools, like hydrodynamic simulations and turbulence models will be necessary for understanding mix from successive shocks in HED and ICF systems. In addition, these types of systems are often compressible and have multi-mode perturbation spectra, where different modes can access different vorticity competition cases simultaneously, which adds further complexities to the models needed to treat the systems and the level of experimental data required for validation.

We should be able to determine how well we understand linear RM growth under successive shocks just by investigating the three vorticity competition cases, since all the different successive shock growth cases are just different realizations of the three cases. Our experiments use a heavy-to-light interface configuration and have strong shocks, which results in perturbation inversion under the first shock. Since the perturbation inverts under the first shock, the second shock's vorticity will always have the opposite sign making it possible to access all three vorticity competition cases with the same configuration as long as both shocks are of similar magnitude.

These experiments are the first demonstration of two successive shocks, from the same direction, and of similar shock strength driving instability growth at a perturbed interface. The experiment setup uses a similar drive to our earlier demonstrations of a hybrid direct drive (DD) and indirect drive (ID) scheme,¹⁹ and a similar shock-tube setup to earlier Shear^20,21 and Mshock²² campaigns but driven only from one side like the Reshock experiments.²³ The target consists of a Au halfraum, a multi-layer ablator and tracer disk, a Beryllium shock-tube filled with a low-density 100 mg/cm³ CH or SiO₂ foam, an attached area backlighter (BL), and fiducials placed between the backlighter and the shock-tube [Fig. 2(a)]. The halfraum has a completely open, 4 mm diameter laser entrance hole (LEH) to prevent clipping of the direct drive beams pointed to the ablator. The halfraum is coated with 5–10 μm of plastic along the inner wall to tamp wall expansion due to any heating from the direct drive pulse.

We control the relative vorticity deposition between the two shocks by controlling the initial conditions that dictate the RM growth rates. For the first shock, we vary the growth rate by varying either the first shock strength (Sec. III A) or the machined perturbed surface wavelength(s) λ and amplitude(s) a₀ (Sec. III B). For the second shock, the initial conditions are set by the second shock strength and evolved perturbation profile, which is set by the growth rate after the first shock and the delay between the shock arrivals (Sec. III A). The studies presented here all use profiles with either a single- or two-mode sinusoidal perturbation, since we are focused on understanding the physics of basic vorticity competition under successive shocks as discussed in Sec. II. More complex perturbations will be the focus of future studies.

We use a hybrid direct/indirect laser drive scheme because it allows us flexibility in controlling the first and second shocks approximately independently. It is only the flexibility of a laser drive and the large amount of energy available at NIF that allows the Mshock campaign to be the first experiment able to probe a wide portion of the successive shock parameter space. As stated in Sec. II, to measure all three different vorticity competition cases, we need our drive to fulfill the following three criteria:

1. We must generate two shocks from the same direction.

2. The shocks must have similar shock strength for similar magnitude vorticity deposition. We want to be able to control the relative shock strength as one knob to vary the relative vorticity deposition.

3. The second shock must have a long (multiple ns) delay so the perturbation has enough time to grow under the first shock that we can measure the perturbation amplitude at the time of second shock arrival. The perturbation characteristics at second shock arrival determine the amount of vorticity deposited by the second shock.

Our direct/indirect drive scheme works by launching the first shock with a direct drive pulse and then launching the second shock with an indirect drive pulse. We control the relative shock strength by varying the direct drive power. The direct drive shock is driven by 16 laser beams irradiating the ablator [Fig. 2(b)]. We are limited to 16 beams since the four NIF near-polar quads (156.5°) are the only beams with a high enough angle of incidence to clear the LEH for the halfraum without clipping. The beams are pointed to create as uniform a laser intensity as possible, without clipping at the LEH or depositing significant energy on the halfraum wall. The direct drive pulse is a 1 ns square [Fig. 2(c)], with peak power of either 1.9 TW/beam or 1.14 TW/beam, where we will refer to these as the 100% power, or 60% power cases for the remainder of the manuscript.

We control the relative shock timings, or second shock delay, by varying the laser timing of the indirect drive (ID). The indirect drive consists of 60 outer cone beams (15 quads) pointed at the wall of the halfraum, with a 4 ns square pulse shape [Fig. 2(c)]. We adapted the pulse shape from the Shear experiments, where it is the first segment of the tiled Shear drive, but run at a lower power and energy of 1.194 TW/beam and 2.450 kJ/beam, respectively. We can theoretically vary the timing of the indirect drive widely, bound only by the pulse delay limits on NIF and the blow-in behavior of the gold halfraum wall. For this study, we use an indirect drive laser delay of either 6 or 7 ns, and we find that varying the ID delay by only 1 ns, in tandem with the change in direct drive energy, is sufficient to generate changes in vorticity competition behavior.

Over the course of the campaign, we have made several variations to the shock-tube and physics package of the target [Figs. 3(a) and 3(b)] to increase our radiography data-rate and develop an in situ drive measurement using the velocity interferometer system for any reflector (VISAR)²⁴ diagnostic instead of instead of relying on a separate radiography and VISAR-specific target designs¹⁹ to measure the shock propagation. Specifically, we demonstrated a VISAR+ imaging capability with a 4-strip x-ray framing camera, allowing us to quantify shock strengths and perturbation growth in a single shot, or at least across a single shot day without significant experiment setup variation (Fig. 4). The main changes are to the shock tube diameter, the foam material, and the addition of a 200-μm-thick silicon VISAR mirror and gold cone shielding assembly. The halfraum does not change over the course of all experiments in order to keep the drive similar across data sets.

All target types have a multi-layer ablator and tracer disk stack. The multi-layer nature of the ablator serves to mitigate both direct and indirect drive pre-heat, while preventing high-Z contaminants from ablating into the halfraum and interacting with the indirect drive beams. The ablator is composed of 50 μm of CH (polystyrene) facing the halfraum, which is the only material ablated into the halfraum while the drive lasers are on, followed by a 5 μm layer of gold and a layer of 3% iodine-doped CH (CHI) [Fig. 2(b)] to block both hot electrons and M-band radiation.²⁵ Attached to the top of the ablator is a polyamide-imide (PAI) disk with central, 200 μm-wide tracer strip of 3% iodine-doped CH, and perturbation machined onto the foam-facing interface in order to seed instability growth. The PAI/CHI tracer disk is the same as previous thin-layer Mshock experiments,²² where the CHI and PAI are mass matched at $≈ 1.4$ g/cm³ to minimize disturbance to the hydrodynamics at the PAI/CHI seams. The CHI has a significantly higher opacity than the PAI at 7.8 keV, the emission energy of the Ni backlighter (BL) used for all experiments presented in this manuscript. The higher CHI opacity allows preferential imaging of the small-scale instability growth and mix in the CHI and results in better resolution of the mix features, as has been previously demonstrated in multiple HED instability experiments. The axial thickness of the CHI layer and the PAI/CHI disk [Fig. 3(c)] varies slightly across the suite of experiments, from a total thickness of 320–370 μm, with the PAI/CHI tracer being 150, 200, or 250 μm thick. We moved to the larger tracer layer thicknesses to allow for more extensive mixing zones for future broadband roughness experiments, but for the single- and two-mode experiments, here the CHI/foam interface is always far enough from the low-transmission region that the perturbations are resolvable. Thus, for the purposes of this study, all the target variations of material thickness are interchangeable (documented in Table I).

The variation across individual shots between 100 mg/cc CH and SiO₂ foams has both advantages and disadvantages. The CH foam has a much higher contrast against the CHI tracer, both when the foam is shocked once (Fig. 5) and shocked twice. The contrast between the shocked and un-shocked foam is low for the first shock, but reasonable after the second shock catches up and merges with the first shock. The CH foam is better for measuring fine scale feature growth at the perturbed tracer/foam interface, but not as good for shock position tracking within the foam. The SiO₂ foam is primarily advantageous for use with the VISAR, since it is transparent to the VISAR wavelength²⁶ (659.5 nm) until shocked, allowing us to measure the shock parameters in the foam. The disadvantage of the SiO₂ foam is the lower contrast at the tracer/foam interface, with a reasonable contrast during the first shock phase, but the ability to only measure significant RM evolution under the second shock phase when the SiO₂ density has increased from both shocks. However, the contrast of the shocks in the foam is significant, allowing us to measure the first and second shock positions. The imaged shock positions are a complementary data set to the VISAR shock velocity data. Here, the VISAR measures the velocity history of the leading shock front: the first shock breakout and velocity evolution and then the first/second shock merger time and merged shock velocity evolution (Sec. IV). The imaged first and merged shock positions are good confirmation data points for checking the unfold of the shock velocity into shock trajectory. The second shock position pre-merger can only be captured in our images.

Each target includes a set of spatial fiducials [Figs. 3(a) and 3(b)] that allow us to verify image resolution, measure interface and shock position, and check for perturbation inversion. The fiducial design is adapted from previous experiments,^22,23 where we use a comb of 150 μm wide teeth with different point profiles so each tooth can be identified whether the others are obscured by the moving instability layer. We also include a set of transmission fiducials consisting of three regions of 25, 50, and 75 μm thick Al, as seen outlined on the right-hand side of the images in Fig. 5. The transmission fiducials are designed to help to convert the radiographs to density maps in future multi-mode studies with significant mix. For the single- and two-mode studies presented here, these fiducials serve as secondary spatial fiducials. If we line up the fiducials in the as-shot images with our pre-shot radiographs (Fig. 5), we can measure the location of any of the evolved features relative to their initial positions. We can also line up the pre-shot perturbation peaks with the evolved perturbation peaks or valleys to check for inversion under the first shock and a potential re-inversion under the second shock.

Our hybrid drive scheme can generate successive shocks, and we can change the relative shock strengths and relative shock timings to vary vorticity deposition at the perturbed interface. As discussed in Sec. III A, we vary the relative shock strength by changing the direct drive power. We vary the relative shock timings by changing the indirect drive laser turn-on delay. We use the VISAR²⁶ platform from Sec. III B to measure the shock parameters to constrain the drive in our simulations and, thus, the shock conditions in our theoretical model and RM growth simulations. Because the initially transparent SiO₂ foam becomes optically reflecting upon shock compression, the VISAR laser probe reflects off (and reports the velocity history of) the leading shock front. We can, therefore, measure the first shock breakout and decay and, then, the first/second shock merger and merged shock decay (Fig. 6). We cannot directly measure the second shock, since that shock is behind the leading shock front of the first shock. Instead, we measure the transmitted shock after self-impedance match, referred to as the merged shock.

We demonstrate our ability to control the relative shock strengths by decreasing the direct drive from a power of 30.4 TW (100% power) to 18.2 TW (60% power), as seen in Fig. 2(c). The 100% power case gives us a larger velocity jump from the first shock when it breaks out of the tracer into the foam, $Δ v ≈ 42$ km/s, and a significantly smaller velocity jump due to the second shock catching up and merging with the first shock, $Δ v ≈ 14$ km/s [Fig. 6(a)]. The factor of 3 difference between the two velocity jumps indicates that the first shock is notably stronger than the second shock. In contrast, the 60% power case yields similar velocity jumps from the first shock breakout and first /second shock merger, $Δ v ≈ 31$ km/s and $Δ v ≈ 25$ km/s, respectively [Fig. 6(b)]. The factor of 1.24 difference indicates that these shocks are of very similar strengths, close enough to equal strength for our vorticity study purposes.

We demonstrate our ability to control the relative shock timing by varying the indirect drive turn-on delay from 6 to 7 ns and observing the change in shock merger time (Fig. 6). It is possible to vary the delay by several nanoseconds, as demonstrated in earlier platform development experiments,¹⁹ but for these experiments a 1 ns change in delay was sufficient to create a measurable difference in merger timing without allowing for significant evolution of the halfraum plasma that might change the laser coupling into the ablator or the backscatter out of the halfraum. We note that for each drive case, we keep the interface perturbation the same while varying only the indirect drive delay so that there is a negligible contribution to the change in merger time from any change in how long it takes the second shock to transition the perturbed interface. For the 100% power case, we use a completely flat interface, and for the 60% power case, we use a single mode with $A / λ = 9 / 180$⁠. We did thin the ablator/tracer stack by 50 μm in the 60% power case (Table I) to keep the direct drive shock breakout time consistent across both shock strength cases. For the 100% power, weaker second shock case, we find that the $Δ t = 1$ ns in the indirect drive laser delay generates a $Δ t = 6$ ns delay in the shock merger. This is consistent with the difference in shock strengths, since a weaker second shock will take longer to catch up to the stronger first shock. We can think of this as a 1 ns delay from the laser turn-on, and a 5 ns lag from the difference in shock strengths. For the 60% power case, the corresponding change in merger delay is only $Δ t = 1.5$ ns since the shocks are close enough in strength that we only get a 0.5 ns lag due to the shock velocity difference.

We note that the change in direct drive power also changes the merger delay for the indirect drive laser delays. As expected, the move from a stronger first shock to equal strength shocks significantly shortens the delay to the shock merger, as the second shock can catch up to the first shock faster. From an experiment perspective, the earlier shock merger time has the advantage of giving us a longer observable growth interval after the second shock transits the mixing layer, but before other transients in the system start appearing at $t ≈ 50$ ns.

Because of the two distinct drive pulses and long subsequent phase of hydrodynamic shock propagation, the system experiences multiple phases (i.e., direct drive, indirect drive, and a late-time cooling Hohlraum) in which the support of the shock by the drive is dominated by different physics processes under evolving conditions. This makes it difficult to predict how the shock will behave through the duration of the experiment without detailed modeling of the full physics involved, including the laser drive. Correctly accounting for these physics, however, allows for a self-consistent simulation that performs well at reproducing the experimentally observed shock behavior.

The xRAGE^27,28 code, an Eulerian radiation-hydrodynamics code maintained by Los Alamos National Laboratory, has the full physics capability needed to do this modeling, and we, therefore, use it to simulate the full, integrated system. xRAGE contains models for hydrodynamics, heat conduction, three-temperature plasma physics, radiation transport, and tabular equations of state (EOS). Finally, a laser deposition physics model adapted from the Mazinisin²⁹ package in the DRACO³⁰ code has been implemented in xRAGE.

We perform the simulation in 2D, with both the direct and indirect drives fully captured by the laser model. The initial setup is identical to the experimental schematics shown in Figs. 2(a) and 3, where we use SESAME³¹ tabular equations of state (EOS) to model the materials. All plastic components use a polystyrene EOS, while other materials use their corresponding EOS. The hydrodynamic mesh is resolved to 1 μm in regions where the shocks are propagating and dezoned as appropriate elsewhere. We use Lee-More³² conductivities to model heat conduction, a multigroup gray diffusion model for radiation transport, and the Mazinisin laser model to drive the simulation.

Data from the VISAR diagnostic provides the detailed shock information needed in order to benchmark the performance of the simulation, with the shock merger timing and the merged shock decay providing strong constraints on the drive physics. The details of the modeling will be discussed in depth in a separate manuscript, but xRAGE simulations of the drive are able to match the experimental shock profiles for both relative shock strength cases (Fig. 6), reproducing the shock velocities to within a couple of percent while simultaneously reproducing the breakout and merger events to within about a nanosecond. This was accomplished with a multiplier of 0.75 on the direct drive laser power in both cases and a multiplier of 0.6 and 0.72 on the indirect drive for the stronger first shock and equal-strength shock cases, respectively. In practice, we used the stronger first shock/weaker second shock VISAR measurements to refine our xRAGE drive modeling capability and then successfully used xRAGE to design the laser drive required to generate equal strength shocks. Due to the computational expense of simulating the full drive, which takes several weeks, we used drive information from the full laser-drive simulations to inform faster reduced-geometry simulations discussed in Sec. VI B.

We measure the instability evolution by fitting a single-mode or two-mode sinusoidal profile to the interfaces in our experimental images and then extracting the mode amplitude a. Each image yields the amplitude of the mode(s) for a single time point, which we will compare to both our analytic model and simulation outputs. Since some of the data sets span multiple foam materials, which alter the interface contrast and background noise profile, we had to develop an analysis procedure for selecting the interface contour consistently across those data sets. Similarly, we required that our interface fitting procedure be able to handle both single- and two-mode profiles so that the analysis was comparable across all data sets. The final data analysis is a multiple step process as shown in Fig. 7. Due to the image quality, we begin by filtering the images to remove noise. After the image is filtered, we take a contour of the perturbed interface in the image between the CHI/PAI and the CH foam. We then fit the mode profile to the interface contour.

We filter all images by first removing any aberrant signal from dead or hot pixels on the camera and then applying a non-local mean (NLM) filter to remove diagnostic noise. The hot pixels removal is a find and replace algorithm, which replaces any pixels outside of a set number of standard deviations, 3.5σ in this case, from the peak of the image histogram. The hot/dead pixels are replaced with an average of the values of the four surrounding pixels. This procedure is repeated until there are no more pixels with values outside the set number of standard deviations. The noise removal uses the Matlab³³ “imnlmfilt” algorithm on the natural log of the image. The exact parameters of the NLM vary between whether the image is on film or a CCD to account for the difference in resolution per pixel across the two diagnostic variations. We also add a three pixel Gaussian blur using “imgaussfilt” before the NLM filter in the case of targets with a SiO₂ foam to account for the lower overall transmission and, thus, higher relative camera noise. The images are cropped to an area near the interface such that regional artifacts, like the edge of the tube or the fiducial teeth, or extra interfaces, like the shock or ablator/tracer interface, are excluded as much as possible from the area used to calculate the interface contour.

The contour finding is the most sensitive step in the analysis due to variability in interface contrast and image noise levels originating from shot-to-shot variations, such as foam material and backlighter performance. These shot-to-shot variations mean that we cannot just use a consistent contour level value across every image in a data set but instead have developed a method for selecting a similar relative contour level value from the interface gradient in each image. First we enhance the interface contrast using a variation of the unsharp mask method, where we subtract a blurred version of the image from the original. In this case, we blur the image using a Gaussian blur of 50 pixels and scale the blurred image intensity by 0.5 before subtraction. We then normalize the resulting image to make comparison more consistent across all images in the data set. Second, we calculate a sample interface profile from an average line-out (20 pixels wide) across the interface in the image, where the sample region is as shown in Figs. 8(a) and 8(e) and the corresponding line-outs are shown in Figs. 8(b) and 8(f). The line-outs are taken near the center of the image, allowing for a ±50 pixel variation if the interface is too close to the image edge or there is a significant image artifact that would impede calculating the interface gradient.

Once we have a gradient profile, we choose a consistent point along the gradient as the contour level. To do this, we take the derivative of the line-out, transforming the gradient into a peak in the derivative profile. In a high-contrast and low-noise image, the derivative profile is strongly peaked with little other structure in the profile tails or the peak itself, Fig. 8(b). However, when the image is lower contrast with higher noise, we see more structure in the derivative profile, Fig. 8(f). Fitting a two Gaussian function to the derivative profile allows us to fit both the gradient peak and any significant background structure. We then take the peak corresponding to the gradient to be the one that is (1) within the image, as large backgrounds can be fit to a Gaussian significantly wider and center outside the image, and (2) the peak closest to the center of the interface. Once we have calculated the derivative peak position, we use that to calculate the contour level from a corresponding value in the initial line-out. We take the contour level to be the line-out value $f ( x ) = f ( x peak + 0.25 σ )$⁠, where σ is the corresponding Gaussian full-width at half-maximum (FWHM). We find that setting the contour value at this position yields the best consistency in calculating an interface contour that is continuous across the entire width of the image for all images.

After we have calculated the contour level, we find the actual interface contour C using the Matlab “contour” routine on the interface image. On a clean image, this tends to generate a single contour [Fig. 8(c)], while a noisy image can generate several contours [Fig. 8(g)]. In the case that there exist several contours at the calculated contour level, we must select the one that corresponds best to the interface. Our first selection criterion is to choose the contour with a width closest to the image vertical width. This is not necessarily the longest contour, since contours can wrap into a closed loop with diameters close to but smaller than the overall width than the image. If there are multiple contours that span the width, then we choose the contour with the least amount of folding back on itself, calculated as the contour with the fewest instances of multiple values along the horizontal for any vertical position. After selecting the best interface contour, we then remove any interface curvature by fitting and subtracting a third order polynomial.

We fit the interface profile with either a single- or two-mode sinusoidal function corresponding to the seed wavelengths of the perturbation [Figs. 8(d) and 8(h)], since this remains a valid approximation when the RM instability is in the linear regime. Since the perturbation wavelength can also evolve slightly over time, due to things such as interface curvature, we use a Lomb–Scargle periodogram Matlab function “plomb” to extract the dominant perturbation wavelengths. If the calculated wavelength deviated by more than 25% from the seeded wavelength, then we fit using the seeded wavelength instead. However, since significant wavelength deviation was mainly observed after the arrival of other transients in the system, enforcing limits on the wavelength variation mainly serves as a test of data applicability at late-times in the experiment. We also enforce a ±10% bound on the wavelength in the Matlab “fit” algorithm. The amplitude of the fitted sine wave is the instability amplitude or mixing half width.

We calculate the uncertainty in the perturbation amplitude as a combination of the fit confidence interval values and the peak amplitude variance. The fit function calculates the 95% confidence interval values, U_CI for the mode amplitude. We calculate the peak amplitude variance by dividing the contour into discrete wavelengths and then calculating the difference between the fitted peak amplitude and the data maximum and minimum values in that interval: $Δ a peak = | a fit − C max |$ and $Δ a valley = | a fit − C min |$⁠. We track the variance of the peaks and valleys separately to help to account for any bubble/spike asymmetry. We take the variance as the average of the differences, $U var = Δ a / N$⁠, where N is the number of peaks or valleys. However, if the fitted amplitude is below the image noise floor of 10 μm, then we just calculate the variance as the signal standard deviation. We still treat C > 0 and C < 0 separately to give separate peak and valley values. The final uncertainty is the confidence interval uncertainty added in quadrature with the peak variance, $U = U C I 2 + U var 2$⁠. However, if the total uncertainty is below the noise floor, then we set the uncertainty equal to the noise limit of ±5 μm.

The only difference between fitting a single- and two-mode function to the contour is the addition of a masking factor in the fitting weights for the two-mode function. In the two-mode perturbation presented in Sec. VI D, the longer wavelength mode sees significant enough growth to begin to manifest the canonical “bunny ear” morphology seen in these mode harmonic profiles. That morphology has enough non-linearity that simply fitting both modes at once does not give an accurate fit to the small mode. Thus, to fit the small mode, we exclude any data in approximately the bottom eighth of the contour to bias the fit toward the small mode in the bunny ears. We do this using the weighting function in the “fit” algorithm by setting the weights to zero for any excluded data, $W ( C < C min / 4 ) = 0$⁠. We find that this masking does not significantly change the fit to the large mode, such that we can still fit both modes at once.

The fundamental philosophy of our instability and turbulence experiments is to begin with a simple system and build toward the more complex systems we want to study and to build our understanding and confidence that our models are handling each mixing mechanism correctly:

• Single-mode: Check that the analytic model and simplest hydro simulations are working

• Two-mode/Multi-mode: Models need to capture that different modes can have different growth cases post-successive shock and that mode-coupling may alter the growth

• Broadband roughness: Models need to capture growth across a wide spectrum of scales, which results in a highly mixed system for use with a turbulent mix model

In this paper, we cover the results from our single- and two-mode experiments, establishing that we can vary the vorticity competition between first and second shock RM growth. We demonstrate all three vorticity competition cases and compare them to both an analytic model based on Mikaelian's general theory (Sec. VI A) and xRAGE hydrodynamic simulations (Sec. VI B). In an ideal version of our experiment, we could demonstrate all vorticity competition cases by only varying the initial interface profile for a single set of shock conditions. In practice, we do demonstrate post-second shock continued growth, freeze out, and perturbation shrinking across our single- and two-mode experiments with our initial stronger first shock/weaker second shock drive (Secs. VI C and VI D). Unfortunately, the time to post-second shock re-inversion is too long in these experiments, and transients end the experiment before we can confirm that the perturbation both inverts and then grows up again at late-times. This re-inversion and recovery is a key theoretical outcome that is unique to successive shocks. The possibility of re-inversion and recovery also has implications for the evolution of mixing in an ICF system. Re-inversion and recovery potentially allow for significant intervals of both decreased and increased mixing after a successive shock, making the time-dependent mixing profile more complex. While our initial drive could not measure re-inversion and recovery, moving to our equal shock strength drive allows us to shorten the growth interval under the first shock, increase the growth rate under the second shock, and make the re-inversion to occur sooner. Hence, with the equal shock strength drive, we do measure the re-inversion and the perturbation recovery (Sec. VI E), providing a complete demonstration of the predicted successive shock growth cases.

From Mikaelian's general theory discussed in Sec. II, for this experiment, we can use the particular case of a single, heavy-to-light, interface transited by two shocks.¹⁴ At the hydrodynamically linear level^34, $( a / λ < 0.4 )$⁠, the first and second RM processes are treated on an equal footing. The first RM process initiates linear growth (⁠ $a ̇$⁠) of the perturbation, scaling with Richtmyer's linear estimate of amplitude growth based on the initial amplitude a₀, $a ̇ = k A Δ V a 0$⁠, where A is the Atwood number and $Δ V$ is the interface jump velocity. When the second shock comes through, a second RM process occurs, using the dynamic value of a at the second shock time. This new process generates a new growth rate impulse to the perturbation $Δ a$⁠, which is added to the previous growth rate to determine a new linear growth rate that the system subsequently follows. If it seems odd to superimpose growth rates, we can recall that the RM instability can usefully be thought of as the self-advecting motion of a vortex sheet.^34,35 The effect of each shock is to induce new vorticity along the density discontinuity, irrespective of whether or not it is in motion. If a phase inversion of the perturbation occurs between the two shocks, the second shock's deposition will locally be opposite in sign to the first shock's, and the resultant vorticity can reduce or reverse the interface velocities.

Richtmyer's linear estimate of amplitude growth, $a ̇ = k A Δ V a 0$⁠, must be improved by several correction factors to extend Richtmyer's derivation for shocks with negligible compression to account for some of those effects. Taking into account for the change in initial conditions affected by the shock itself, it is appropriate to use an average of the pre- and post-shock amplitudes and Atwood numbers.^36,37 Furthermore, between the two shocks, the shock is outpacing the interface at the speed of only $v s ≈$ 12 km/s. As the amplitude growth $a ̇ 1$ approaches vs, there will be additional distortion in the growth from the proximity to the shock (as a limiting factor, $a ̇ 1$ cannot exceed vs). This reduces the effective growth of the amplitude from $a ̇ 1$ to $a ̇ 1 ′$ (below, we apply Holmes et al.'s prescription, which has been experimentally demonstrated to be a good model for the marginally linear regime we are targeting^38,39) but not the underlying impulse $a ̇ 1$ (again, it may be helpful to think of this as a measure of vorticity at the interface). After the second shock passes, both vs is larger while $a ̇$ is decreased, and this effect reduces in importance.

While these corrections seem adequate to describe the system coarsely, there are still omissions and ambiguities left to account for, due to the simplifying limitations of the model. For instance, actual RM does not instantaneously reach its linear growth rate, but has a period of acceleration, not accounted for here.⁴⁰ Relatedly, the interval τ is ambiguous to the finite transit time of the shock across the perturbation, which can be quite substantial for large a₁, and the results can be sensitive to those details for large enough $a ̇ 1$⁠. It is an open question in this framework whether τ should be taken when the shock first reaches the perturbation bubble, when it exits the spike, or at some point in the interior. The simple model also assumes that the interface remains sharp for both shock events, while the true interface could be diffusing before (due to upstream preheating from the laser) or between the two events (due to plasma diffusivity), which would likewise lead to reductions in the perturbation growth rates.⁴¹ Many of these corrections can be absorbed into the simple model by a rescaling of, for example, the local Atwood numbers at the times of the shocks, but this shows the limitations of making such a simple model fully predictive of the expected behavior. The present model is, however, adequate for sketching the process and, once tuned to reproducing one case, accounting quantitatively for changes in perturbation parameters (i.e., a₀, k) in otherwise hydrodynamically equivalent experiments. We would expect fully compressible models for the RM of each shock could properly account for some of these issues, but they are more complex and would have to be fundamentally reformulated to account for additional waves in the system.⁴² Below, we will instead construct fully hydrodynamic numerical simulations to account for some of these possibilities and serve as a more predictive design capability.

In the HED regime, simulation work plays an important role in guiding and analyzing experiments. As discussed in Sec. IV A, simulations can provide a great wealth of information, but this comes at a cost. Highly detailed simulations such as those done to examine the drive and the full target geometry require a large computer and large amounts of computational time. Incorporating the laser dynamics into the xRAGE²⁷ simulations takes a large commitment of resources. It is, therefore, desirable to have another way of running simulations that is simplified. This allows for a quicker way to run parameter scans, give a quick first pass analysis to compare to experimental results, and guide experimental choices, such as timing windows for diagnostics. The physics package described in Figs. 3(a) and 3(b) is complex and, as shown in Sec. IV A, can be complicated to simulate. By narrowing focus and using the internal stack geometry as shown by Fig. 3(c), a reduced geometry simulation can be derived. We use just the internal geometry, as shown in Fig. 9, as the starting point for a second set of simulations, which will be described presently. This geometry is formulated by using only the shock tube stack and cutting out the Hohlraum, Be flange, and outer Be tube. A volume of Hohlraum gas is kept to allow for the blow off of the ablator material during both the direct and indirect drives. This cuts out much of the complexity but still allows for investigation of the physics we wish to interrogate. To drive the first shock, a direct energy deposition routine within xRAGE is used to deposit the requisite energy in the outer most section of the ablator. This is shown in Fig. 9 by the small slice denoted as “Ablator deposition.” For the second shock, a frequency dependent spectral (FDS) source is used to drive the stack from the lower boundary where the Hohlraum gas is. This generates two shocks that propagate through the system, driving the RM instability with the selected perturbation in the PAI tracer and shock tube material interface. The results of these reduced geometry simulations can be found in the subsequent Secs. VI C and VI E.

This blackbody spectrum is then modified with a functional form to account for the high energy M-band photons when the spectrum is above a cutoff frequency ν₀, which is around 2 keV. Furthermore, after this addition to the blackbody spectra is made, the entire distribution is re-normalized, with the multiplier N, to ensure that an equal amount of energy is still in the source. While this is a physically true description of the energy generated from a Hohlraum or halfraum, it ignores many physics effects. Due to the missing physics, it is an iterative process, where power multipliers are applied to the FDS to align the simulated shocks carefully to the experimental data. Specifically, we tune the power multipliers until the simulated shocks match the experimental first shock arrival, first shock velocity, as long as we can match enough details of the shock profiles to reproduce the gross vorticity competition (see Sec. VI C), the reduced geometry simulation is a reasonable reproduction of the experiment that does allow for faster computational throughput. The reduced simulations also allow us to test sensitivity to various physics effects in isolation, where they may be coupled in an integrated system, or their effects buried beneath obscuring features, such as shock or interface curvature.

Our first study verifies that not only do different post-second-shock growth cases exist but also that they can be accessed just by changing the perturbation seed at second shock arrival. In this study, we choose two perturbations of similar expected linearity at second shock arrival, by requiring that $2 a 0 / λ ≤ 10$% for both perturbations. The two perturbations are $a 0 / λ = 2.5$μm/150 μm and $a 0 / λ = 14 μ$m/275 μm.

In these experiments, we use the stronger first shock/weaker second shock described in Sec. III A. This drive puts the first shock arrival at the interface at $t ≈ 11$ ns, and the second shock traverses the interface at $t ≈ 28$–32 ns. The second shock arrival time at the interface depends on the amplitude of the perturbation, where the shock will reach the edge of a larger perturbation sooner, and then there will be a period where the shock crosses the extended interface and compresses the perturbation. Since both experiments use the same drive, even with minor differences in shock arrival timing, we can assume that they see the same nominal shock parameters for both shocks.

Figure 10 shows the results of the experiments with both perturbations. The two perturbations behave similarly under the first shock, showing linear growth. When the second shock arrives, both perturbations also re-compress as the shock transits the layer. However, after the second shock, the behavior of the two perturbations diverges. The $a 0 / λ = 14 μ$m/275 μm perturbation continues to grow, consistent with the first shock vorticity dominating. The $a 0 / λ = 2.5 μ$m/150 μm perturbation decreases, which is consistent with the case where second shock vorticity dominates. Hence, just changing the perturbation, and, thus, the perturbation conditions at second shock transit, is enough to alter the vorticity dominance and post-second shock perturbation behavior.

Comparing the data to our analytic theory from Sec. VI A, we see that the model is in good agreement with both the experimental data sets [Fig. 10(a)], and it displays the same change in post-second shock perturbation behavior. The input conditions for the model are given in Table II and use the same Atwood numbers (A_n), shock parameters (⁠ $Δ V n$⁠, τ), and compression values (γ_n). Only the perturbation inputs differ. The model predicts a post-second shock re-inversion for the $a 0 / λ = 2.5 μ$m/150 μm case, but we do not see the re-inversion in the data. However, we note that the perturbation does not grow above the noise limit before the end of the experiment.

xRAGE simulations with both the full laser package (Sec. IV A) and the reduced geometry (Sec. VI B) for the $a 0 / λ = 14 μ$m/275 μm case show continued growth post-second shock, similar to both the data and analytic theory, but with a slightly smaller second-shock compression and post-second shock growth. Again, for either simulation type, laser package or reduced geometry, both simulations of that configuration use the same shock conditions and timing [Fig. 6(a)] and only the perturbations differ. The under-prediction of compression is likely due to equation-of-state (EOS) uncertainties for the CHI, and the reduced growth is likely due to the differences between the experimental and simulated shock profiles.

In contrast to the analytic model, the xRAGE simulations do a better job of matching the $a 0 / λ = 2.5 μ$m/150 μm case [Fig. 10(b)]. The simulations for the $a 0 / λ = 2.5 μ$m/150 μm case show a freeze out of the interface more consistent with the data than the analytic prediction of re-inversion. We theorize that this is due to the simulations naturally incorporating complex mix dynamics, such as the growth of fine-scale surface mix features (Fig. 11) or material diffusion, which could easily disrupt or obscure the growth of the seeded mode while it remains at small amplitude.xRAGE simulations inherently incorporate high-frequency modes, in addition to the seeded perturbation, due to imprint by the initial simulation mesh and by the adaptive mesh refinement (AMR). The former results in small-scale structure of the interface profile itself, while the latter can produce artificial gradients in the hydrodynamic state as the system evolves. For example, if there is a pressure gradient in a given region, the gradient will be sampled differently by cell resolutions of different sizes resulting in slightly different values of the pressure in each cell. If two such differently resolved cells are adjacent, as can happen at a location where the AMR is dezoning the mesh, this can result in artificial pressure imbalances across the cells on the scale of the mesh resolution that can affect the interface.

In the laser package simulations shown here, the coarsest mesh resolution in the materials adjacent to the perturbed interface is 2 μm and allows mesh refinement down to 0.5 μm at the interface. Reduced geometry simulations use an initial grid resolution of 1 μm and allowed adaptive mesh refinement down to 0.25 μm. We note that experimental surface roughness from either machining the perturbation or the foam pore size is on the order of $∼ 0.1 μ$m so the mesh imprint is a reasonable scale to approximate surface roughness effects in the simulations.

Right before second shock arrival at the interface (Fig. 11), both the laser package and reduced model simulations show some growth of the high-frequency mode but not enough to significantly alter or obscure the primary $a 0 / λ = 2.5 μ$m/150 μm mode. We see this in both the vorticity and density plots for both simulation types at 25 ns. The vorticity is still mostly symmetric, but with an opposite sign, across the perturbation peak, indicating that vorticity corresponds to the large primary mode. Little vorticity exists in small-scale vortices corresponding to the roughness, in either simulation type.

The vorticity deposition and evolution changes significantly post-second shock, where the vorticity images show significant small-scale vortex development. The vortices correspond to the surface density fluctuations seen in the density images. At 31 ns, right after second shock passage through the interface, the high-frequency perturbations are still spatially small compared to the seed mode (including their compression by amplitude of the second shock), but we can see that the vorticity field is already fragmented to a much shorter period. Then, these short modes grow significantly faster under the second shock than the first shock. At late-times post-second shock (37 and 43 ns), the laser package simulations show that the high-frequency mode has grown to a similar or greater amplitude as the primary mode. Examining the laser package vorticity plot at 43 ns, we see that the high-frequency surface roughness structures have spread to cover most of the extent of the mixing region, even as some small curvature remains from the primary mode along the bottom of the interface. The density plot at 43 ns shows that the small-scale density fluctuations are prominent, but there is still a distinct interface curvature from the primary mode.

The reduced geometry simulations also show increased, high-frequency mode growth post-second shock, but less than the laser package simulations. The difference in high-frequency growth may be due to the shock curvature in the laser package simulations, which presents as shear-like roll-over of the perturbations.

The 1D analytic theory by design only describes the RM growth of the single seeded mode. Thus, it is consistent with expectations that the simulations better reproduce the experiments when secondary mixing processes are prominent enough to compete with the seeded RM growth. Further refinement of the simulations will likely increase the agreement between the simulation and the experimental data. However, the simulations still reproduce gross difference in post-second shock growth behavior for the two single-mode perturbations, giving us confidence that xRAGE is reproducing the physics of successive shock vorticity competition.

Since the model and simulations show good agreement with our single-mode experiments, the next test is whether the model performs well for a two-mode system. Specifically, we want to test a simple system where the modes can both display different post-second shock growth cases and have the potential for mode-coupling effects at weak non-linearity, potentially deviating from the RM superposition assumption of the model during the linear phase. We designed our experimental two-mode perturbation using our linear model such that one of the modes freezes and the other shrinks post-second shock. We use the same stronger first shock/weaker second shock drive cases as Sec. VI C for direct comparison between the data sets. The two-mode perturbation experiments still show good agreement with the model, by behaving as designed with no significant deviation from the predicted model behavior. Since the modes do not show significant mode coupling or non-linearity, we can assert that one of the modes from these experiments demonstrates the final post-second shock growth cases where the vorticity approximately cancels.

The two-mode perturbation uses modes modified from the single-mode experiments in Sec. VI C, as shown in Fig. 12(a). We decrease the wavelength and $a 0 / λ$ ratio of the $a 0 / λ = 14 μ$m/275 μm perturbation to a mode with $a 0 , 1 / λ 1 = 9 μ$m/270 μm. These changes reduce the perturbation growth under the first shock and leave the perturbation weakly non-linear at second shock arrival such that the model should still apply. We then choose the second mode to be an in-phase harmonic of the first mode, similar to previous work done on single-shock RM experiments.⁴⁵ The second mode is then the first harmonic of the larger mode with $λ 2 = 135 μ$m, which is close enough to $λ=150μ$ m that we can use the previous experiments as a reasonable guide. We want the linearity of the two modes under the first shock to be equal to each other so they display similar growth. However, since the total perturbation amplitude is a sum of the amplitudes of both modes because they are in-phase, we must consider this sum when defining the mode amplitudes. We require $a 0 , tot ≈ 13.5 μ$m to meet the linearity condition of $2 a 0 , tot / λ 1 ≈ 10$%, similar to the single-mode experiments. Individual mode amplitudes of $a 0 , 1 = 9 μ$m and $a 0 , 2 = 4 μ$m then have similar growth parameters of $2 a 0 , i / λ i ≈ 6$%. The larger amplitude of the $a 0 , 2 / λ 2 = 4 μ$m/135 μm perturbation compared to the $a 0 / λ = 2.5 μ$m/150 μm single-mode perturbation results in the same post-second shock growth case, but the perturbation amplitude is larger giving a cleaner measurement above the diagnostic noise limit. Finally, we oriented the perturbation so the higher harmonic occurs on the spikes and gives us the bunny ears split spike effect because that is easier to resolve and verify in the images.

The experiment confirms both that the two modes are weakly non-linear with minimal mode coupling at second shock arrival and that they display distinctly different post-second shock growth. Examining the images in Fig. 12(b), we see that both modes grow under the first shock and begin to evolve the canonical bunny ear structure right before second shock arrival, as characterized by the development of roundness in the perturbation lobes. After the second shock, we see that the longer wavelength mode persists, but the smaller wavelength mode decreases and seems to disappear by the end of the data set. These observations from the images are supported by measurements of the individual mode amplitude evolution, as shown in Fig. 12(c). Both modes grow under the first shock, in good agreement with the model calculations, where the model initialization parameters are given again in Table II. We note that the masking process for the two-mode fit discussed in Sec. V was primarily necessary for images just pre-second shock arrival, where the bunny ear structure was the most pronounced. The added complexity of fitting the perturbation morphology at these times supports the conclusion that the modes show weak mode coupling at second shock arrival. Post-second shock, the longer wavelength mode shows minimal growth and the shorter wavelength mode decreases down to the diagnostic noise limit, again in good agreement with the model. The shorter mode also shows some minor indication of recovering at late-time as the model predicts, but again the experiment ends before the mode amplitude grows significantly above the diagnostic noise. Thus, the RM superposition approximation for successive shocks is still valid for multiple-mode systems as long as all modes remain linear or weakly non-linear with minimal mode-coupling.

Since the mode-coupling is minimal in the experiments, we can treat the evolution of each mode as a separate realization of vorticity competition and compare them directly to the single-mode experiments. Between the modes from the two-mode perturbation and the $a 0 / λ = 14 μ$m/275 μm single-mode, we have demonstrated the entire set of successive shock vorticity competition cases as shown in Fig. 13. The $a 0 , 1 / λ 1 = 9 μ$m/270 μm mode shows small enough post-second shock growth that it approximates the freeze out or velocity cancelation case. Further modifications of the perturbation could be made to decrease the growth even more, but with experimental uncertainties achieving exact vorticity cancelation is challenging. The model comparison for all three cases still uses the same Atwood numbers (A_n), shock parameters (⁠ $Δ V n$⁠, τ), and compression values (γ_n) with only the perturbation inputs differing.

Equal-strength shocks, by which we mean shocks with approximately equal pressure jumps, allow us to demonstrate the first proof of post-successive shock re-inversion and recovery. The theory predicts that for a long enough post-second shock growth interval, when the second shock vorticity dominates; then, the perturbation amplitude will decrease, invert, and then grow back up with a change in phase. Since the perturbation initially inverts under the first shock, we then refer to the behavior post-second shock as a re-inversion since it puts the perturbation back in-phase with the initial profile. Both the single-mode and two-mode experiments with a stronger first shock and weaker second shock demonstrated that perturbations could decrease post-second shock, but the perturbation never had enough time to recover above the diagnostic noise. Thus, those experiments do not confirm either the amplitude recovery or the change in phase.

We designed the equal strength shock drive (Sec. III A) to decrease the time interval between first and second shock arrival at the interface and leave more time for post-second shock perturbation growth before transients end the experiment around 50 ns. The first shock still arrives at the interface at $t ≈ 11$ ns. These experiments use two different indirect drive delays, so the second shock arrives at the interface at t = 21 ns or 22.5 ns for an indirect drive delay of 6 and 7 ns, respectively. Thus, $Δ t ≈ 10$ ns between the first and second arrival for the equal strength shock drive, compared to $Δ t ≈ 20$ ns for the stronger first shock/weaker second shock drive.

These experiments use a $a 0 / λ = 9 μ$m/180 μm single-mode perturbation, which still meets the $2 a 0 / λ ≈ 10$% linearity criterion. We designed the perturbation to grow significantly under the first shock to yield good resolution of the perturbation amplitude, while still showing post-second shock re-inversion behavior. The decrease in first shock strength and an increase in second shock strength increase how much larger the second shock vorticity ω₂ deposition is than the first shock vorticity ω₁, compared to the previous experiments. The increase in $Δ ω = ω 2 − ω 1$ then increases the post-second shock perturbation growth rate and decreases the time interval to re-inversion.

Figure 14(a) shows experimental validation of post-second shock re-inversion and recovery for the 6 ns indirect drive delay. The perturbation again decreases to the diagnostic noise level, similar to previous experiments, but in this case has enough time to grow again to resolvable amplitudes. The 7 ns indirect drive delay data shows the same trend [Fig. 14(b)], but with a slightly faster post-second shock growth. The greater growth rate is due to the greater amplitude at second shock arrival, from the 1.5 ns shock arrival delay. The greater growth rate results in a slightly faster inversion and greater perturbation amplitude at late-times. The model behavior is consistent with both data sets and shows the same sensitivity of post-second shock growth rate to second shock arrival time. Thus, the model captures not only the approximate growth case behavior but also the experiment sensitivities.

While the perturbation amplitude curves show a post-second shock amplitude decrease and then recovery, the 1D nature of the measurement does not include phase information. To confirm actual perturbation inversion, we examine the images of the interface evolution, as shown in Fig. 15. For every experiment, we take a pre-shot radiograph of the target, which allows us to compare the as-shot interface position and phase to the initial machined perturbation. Because the pre- and as-shot images are acquired on different systems, we re-scale the pre-shot images and use the spatial fiducials to line up the two images. As predicted, the data confirms that the interface inverts under the first shock, and is inverted right before the 2nd shock arrival. Right after the second shock arrives and while the perturbation is decreasing, the experimental perturbation phase is still inverted. We note that at the time before the predicted re-inversion, the interface is still very discrete with little development of a density gradient due to surface mix or material moving into the spaces between the peaks.

During the perturbation recovery interval (t > 30 ns), the data show re-inversion (back in phase with the initial profile) for the 7 ns indirect drive delay case as predicted by the theory. In contrast, we do not see another phase inversion for the 6 ns indirect drive delay case even though we observe continued growth of the perturbation. However, we observe significant low-density feature growth during the recovery interval. The low-density features are characteristic of jet formation in the perturbation valleys due to the shock passage.⁴⁶ In the 7 ns delay case, we see that the contour corresponds to a lower-contrast/lower-density interface between the CH foam and the CHI. We observe that there exists another higher-contrast interface that appears to be between higher- and lower-density CHI regions that is out-of-phase with the CH/CHI interface. However, the contour for the 6 ns delay case corresponds to the higher-contrast interface, with some indication that lower-density perturbations may exist beyond that interface in the foam region. Thus, the disagreement in phase between the 6 and 7 ns delay cases is likely an artifact of the differences in the low-density structure formation. Jet formation due to shock-passage is an inherently higher dimensional (>1D) phenomenon, which would not be captured by the 1D theory. While the agreement between theory and experiment of the post-second shock growth rates shows that the theory is a good tool for understanding the general principles of RM growth under successive shock, observation of these 2D density features indicates that we need more than the theory to understand the details of instability growth and mix in these type of systems.

Successive shocks, or shocks from the same direction, are a common phenomenon that contribute to Richtmyer–Meshkov instability growth in ICF systems. However, despite the presence of successive shocks in these systems, there are very few studies of successive shock dynamics due to the difficulty in creating successive shocks of similar strengths with the drivers used in traditional low-energy-density experiments, e.g., pistons, membranes, or high-explosives. Controlling the magnitude of vorticity deposited by the first and second shocks is crucial to observing the range of vorticity competition behavior theorized for successive shocks, specifically the transition between the continued growth, freeze out, and re-inversion growth scenarios. We should be able to determine how well we understand linear RM growth under successive shocks just by investigating the three vorticity competition cases, since all the different successive shock growth cases are just different realizations of those three cases.

These experiments are the first demonstration of two successive shocks of similar shock strength driving instability growth at a perturbed interface. We use a hybrid direct/indirect laser drive scheme to launch the first shock with a direct drive pulse and then launch the second shock with an indirect drive pulse. The hybrid drive allows us flexibility in controlling the first and second shocks' timing and strengths approximately independently. It is only the flexibility of a laser drive and the large amount of energy available at NIF that allows the Mshock campaign to be the first experiment able to probe a wide portion of the successive shock parameter space. The experiments use a similar shock-tube setup to earlier Shear and Mshock campaigns but driven only from one side like Reshock experiments. Since our experiments use a heavy-to-light interface configuration and have strong shocks, this results in perturbation inversion under the first shock and allows us to access all three vorticity competition cases needed to span successive-shock RM dynamics.

We developed two different simulation capabilities to model our successive shock system. Both use the LANL xRAGE Eulerian radiation-hydrodynamics code, but one models the entire laser physics and halfraum dynamics of the drive and the other is constrained to only model the shock propagation and instability growth inside the shock tube. Both are benchmarked to the experimental shock velocities to reproduce the shock profiles driving the instability growth within the uncertainties of the EOS that were used. The xRAGE simulation with the laser package is used to design changes to the drive necessary to reach different shock conditions since it reproduces the shock profiles to high fidelity. However, the laser package simulations are very computationally expensive, so the reduced geometry/shock-tube only simulations allow a quicker way to run parameter scans, give a quick first pass analysis to compare to experimental results, and guide experimental choices, such as timing windows for diagnostics. The reduced simulations allow us to confirm that the re-inversion physics occurs for the expected reasons of vorticity competition driven by the two shocks and not because of a different feature in the actual experiment.

The fundamental philosophy of our instability and turbulence experiments is to begin with a simple system and build toward the more complex systems we want to study and to build our understanding and confidence that our models are handling each mixing mechanism correctly. In this paper, we cover the results from our single- and two-mode experiments, establishing that we can vary the vorticity competition between first and second shock RM growth. We demonstrate all three vorticity competition cases and compare them to both an analytic model based on Mikaelian's general theory and xRAGE hydrodynamic simulations.

Our first study verifies that not only do different post-second-shock growth cases exist but also that they can be accessed just by changing the perturbation seed at the second shock arrival. Using two different perturbations, $a 0 / λ = 14$/275 and $a 0 / λ = 2.5 μ$m/150 μm, with the same drive, we are able to change the relative second shock vorticity competition to access both the continued growth and freeze out RM growth cases, respectively. By comparing with our analytic theory, we see that the model is in reasonable agreement with both the experimental data sets, and it displays the same change in post-second shock perturbation behavior. However, the analytic model predicts a re-inversion instead of freeze out growth scenario for the $a 0 / λ = 2.5 μ$m/150 μm perturbation, but the data is close enough to the diagnostic noise limit that the comparison is slightly ambiguous.

In contrast, the reduced geometry simulations reproduce gross difference in post-second shock growth behavior for the two single-mode perturbations and are a better match to the two growth scenarios. The $a 0 / λ = 2.5 μ$m/150 μm simulation agrees well with both the experimental data and analytical theory up to first shock and also predicts the freeze out seen in the experimental data. The $a 0 / λ = 14 μ$m/275 μm simulation matches the continued growth scenario. This gives us confidence that xRAGE is reproducing the physics of successive shock vorticity competition.

Since the model and simulations show good agreement with our single-mode experiments, the next test is whether the model performs well for a two-mode system. The two-mode perturbation experiments show good agreement with the model, by behaving as designed with no significant deviation from the predicted model behavior, where the long wavelength mode was designed to freeze out and the shorter wavelength mode was designed to shrink and re-invert. Since the modes do not show significant mode coupling or non-linearity, we can assert that the RM superposition approximation for successive shocks is still valid for multiple-mode systems as long as all modes remain linear or weakly non-linear with minimal mode-coupling. These experiments also demonstrate a clearer example of mode freeze out, with the mode freezing at an amplitude significantly larger than the camera noise limit.

Equal-strength shocks allow us to demonstrate the first proof of full evolution of the third vorticity competition, post-successive shock “re-inversion and recovery.” The theory predicts that for a long enough post-second shock growth interval, when the second shock vorticity dominates, the perturbation amplitude will decrease, invert, and then grow back up with a change in phase. These experiments use two different indirect drive delays of 6 and 7 ns, respectively, and the same $a 0 / λ = 9 μ$m/180 μm single-mode perturbation. In both cases, we observe that after the successive shock, the perturbation amplitude decreases to the diagnostic noise level, like previous experiments, but in this case, the perturbation has enough time to grow again to resolvable amplitudes. We also observe that the slight delay in second shock arrival at the interface leads to an increase in the post-second shock perturbation growth rate and a decrease in the time interval to re-inversion, which is consistent with greater second shock vorticity deposition expected for shock interaction with a large amplitude perturbation at shock arrival time. The analytic theory behavior is consistent with both data sets and shows the same sensitivity of the post-second shock growth rate to second shock arrival time. Thus, the model captures not only the approximate growth case behavior but also the experiment sensitivities.

While the perturbation amplitude curves show a post-second shock amplitude decrease and then recovery, the 1D nature of the measurement does not include phase information. To confirm actual perturbation inversion, we examine the images of the interface evolution and observe significant low-density feature growth during the recovery interval characteristic of jet formation in the perturbation valleys due to the shock passage. We observe that one case does show phase re-inversion, while the other does not; however, it shows perturbation recovery. This disagreement in phase between the cases is likely an artifact of the differences in the low-density structure formation, indicating that there are more subtleties present in RM growth under successive shocks than captured in basic theory. While the agreement between theory and experiment of the post-second shock growth rates shows that the theory is a good tool for understanding the general principles of RM growth under successive shock, observation of these 2D density features indicates that we need more than the theory to understand the details of instability growth and mix in these types of systems. Some of the remaining discrepancy between ignition metrics and predictions may turn out to be attributable to the sharp sensitivity in the multi-shock process observed here.

The authors would like to thank the LANL MST-7, GA, and LLNL target fabrication teams for their support. The authors would also like to thank the crew and support staff of the National Ignition Facility for operational and technical support. Additional thanks is due to Menolly Benedict of New Mexico Tech for her help in vetting the analytic equations. Los Alamos National Laboratory, an affirmative action/equal opportunity employer, is operated by Triad National Security, LLC for the National Nuclear Security Administration of U.S. Department of Energy under Contract No. 89233218CNA000001. Part of this work was conducted by Lawrence Livermore National Laboratory under the auspices of the U.S. Department of Energy under Contract Nos. DE-AC52-07NA27344.

The authors have no conflicts to disclose.

Elizabeth C. Merritt: Conceptualization (supporting); Data curation (lead); Formal analysis (lead); Funding acquisition (supporting); Investigation (lead); Methodology (equal); Project administration (equal); Resources (equal); Software (supporting); Supervision (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Nikolaus S. Christiansen: Resources (supporting). Marius Millot: Formal analysis (supporting). Lynn Kot: Investigation (supporting); Resources (supporting). Theodore Sonne Perry: Investigation (supporting). David Meyerhofer: Methodology (supporting); Writing – review & editing (supporting). Forrest W. Doss: Conceptualization (lead); Formal analysis (supporting); Funding acquisition (lead); Investigation (supporting); Methodology (equal); Project administration (equal); Resources (equal); Software (lead); Supervision (equal); Visualization (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Carlos Alex Di Stefano: Formal analysis (supporting); Investigation (supporting); Methodology (supporting); Software (supporting); Visualization (supporting); Writing – original draft (supporting). Ryan Sacks: Formal analysis (supporting); Methodology (supporting); Software (supporting); Visualization (supporting); Writing – original draft (supporting). Alexander Martin Rasmus: Data curation (supporting); Formal analysis (supporting); Investigation (supporting); Methodology (supporting); Software (supporting); Writing – review & editing (supporting). Joseph M. Levesque: Data curation (supporting); Formal analysis (supporting); Investigation (supporting); Software (supporting); Visualization (supporting); Writing – review & editing (supporting). Kirk A. Flippo: Data curation (supporting); Investigation (supporting); Methodology (supporting). Harry F. Robey: Investigation (supporting); Methodology (supporting); Writing – review & editing (supporting). Derek W. Schmidt: Investigation (supporting); Resources (supporting).

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