3.2.1. Damped mushroom-shape RTI spikes

The damped spherical mushroom spikes of diameter ~15–30 µm and two vortex rings surrounding the spike – one scarcely seen and the other larger deformed-ring – are shown in Figure 4a–4c. The size of deformed ring increases from (a) to (b) and to (c), while of the surroundings decreases; the deformed ring becomes larger and stretches the smaller one (Fig. 4c). The horseshoe type base-plane structures are self-organized (SO) into the rosette-like configuration, which is slightly perturbed by the weak reshocks. The horseshoe structures are the parabolic-like solitary waves, similar to that in the CR (Lugomer, Reference Lugomer2016b ). The straight-line structures with the bell-shaped profile are – the line solitons – while the others are cnoidal waves (Fig. 4d). Some of the line solitons in Figure 4 (a – upper left; b and c – down right) interact between themselves giving rise to the resonant “Y-junction” configuration. The roll-up of the waves into vortex filaments occurs in the segments where the Re number reaches the critical value, Re ≥ Re _crit (~2.3 × 10³), while those with Re < 10³ show the formation of nonlinear waves.

Fig. 4. SEM micrograph of the small partially-coherent-domains inside random flow field showing the SO of the horseshoe structures into the rosette-like configuration (a–c). The symmetry of the rosettes is disturbed due to partial coherence of the flow field in the domains. Note that every rosette comprises a damped mushroom-shape spike and two vortex rings. The size of damped mushroom-shape spike increases from (a) to (b) and to (c). Simultaneously, the stronger ring increases in size and becomes more deformed at the expense of the weaker one. (d) SEM profiles of radial structures of the rosettes: The bell shape profile indicates solitary waves (i, ii); the hump-like indicates cnoidal waves (iii).

The amplitude of the spikes h _s ~ 20–25 µm, and of the bubbles h _B ~ 18–20 µm gives h _S/h _B ~ 1, indicating approximately the same growth rate of damped mushroom spikes and bubbles.

The comparison of damped spherical mushroom spikes in Figure 4a–4c with those obtained by 3D numerical simulation of RMI turbulent mixing for the lower Atwood and the Mach numbers, shows a great similarity. In particular, the single-mode initial perturbation with A = 0.65 and Ma = 1.2 (Long et al., Reference Long, Krivets, Greenough and Jacobs2009), generates the spike with a damped mushroom cap and two vortex rings, one of them just above the base-plane. The deformed vortex ring around the spike stretches the weaker one into deformed configuration, similar to that in Figure 4a–4c. Generally, the characteristics of the rings depend on the A number. At the very low A (A = 0.15) flow instability possess symmetric features in the formation of vortex rings (Chapman & Jacobs, Reference Chapman and Jacobs2006), whereas with a higher A the flow losses this symmetry and at A = 0.65 the vortex ring becomes deformed (Long et al., Reference Long, Krivets, Greenough and Jacobs2009).

3.2.2. Spherical RTI mushroom-shape spikes

Many domains of the NCR show spherical mushroom spike, which is an indicator that the dynamics is isotropic in the plane normal to acceleration. Below the mushroom the vortex ring (or even two rings) of radius R ~ 15 µm and the core diameter of σ ~ 3–4 µm, surrounding the spike can be seen (Fig. 5a). Based on the above discussion the rings surrounding the spike generated at A ≥ 0.65 should be deformed and seen at the high resolution only (Long et al., Reference Long, Krivets, Greenough and Jacobs2009). However, the rings in Figure 5a formed at A ~ 0.85–0.65 are rather regular indicating dependence on some other parameters, as described latter on.

Fig. 5. Small partially-coherent-domain inside random flow field. (a) SEM micrograph showing the spherical mushroom-shape spike with two vortex rings below the mushroom cap. The structures in the base-plane (interface: saddle point plane), are the radial straight-line structures identified as the nonlinear waves and cylindrical vortex filaments organized into polygonal (square-like) web. The borders of the polygonal web divide the base-plane into four angular segments (i–iv). For the characteristics of the border-lines between the segments see text. Illustration of the cross-section through the vortex filaments by digital laser-induced florescence (LIF) streak images: (b) Digital LIF streak image showing low mixing for almost circular cross-section through the single cylindrical vortex filament rolled up at the density interface for Re = 1750. (c) Almost circular cross-section through two pairing cylindrical vortex filaments for Re = 1750. (Courtesy of Professor Koochesfahani M.M. and Dimotakis, P.E. Reproduced with permission of Cambridge University Press from, Koochesfahani & Dimotakis (Reference Koochesfahani and Dimotakis1986). Copyright Cambridge University Press, 1986).

The four radial structures divide the base-plane into four angular segments. The (wall) border-lime between the segments (i) and (ii) is the pair of two vortex filaments under an angle of α ≤ 3⁰ with the intersection in the nodal point. The segments (i) and (ii) show only a small depression (cavity) without a hole. The radial structures at the border of the segments (ii)–(iii) and (iii)–(iv) are joined in the nodal point at the base-plane. Finally, the border of the segments (i)–(iv) is the terrace-like wave formed by the pulsating flow due to the reshocks and accumulation of material. The surface depression and the hole in the segments (iii) and (iv) are formed by the bubble collapse near the solid target surface, so that the hole is created by the reentry jet (Lugomer, Reference Lugomer2016b ). Various radial structures in the base-plane in Figure 5a, like the nonlinear waves and the vortex filaments result from the inhomogeneous fluid acceleration. In such field, the fluid flow which does not reach the critical Re number forms the nonlinear waves, while that reaching the Re _crit is rolled up into vortex filaments or the Kelvin-Helmholtz (KH) rolls. The near-circular cross-section through the single KH roll for Re > Re _crit (~2.3 × 10³) in Figure 5b, formed at the mid-span plane of interfacial surface between the high-speed free-stream fluid (blue) and the low-speed one (red), reveals the low mixing (Koochesfahani & Dimotakis, Reference Koochesfahani and Dimotakis1986; Dimotakis, Reference Dimotakis2000; Zhou et al., Reference Zhou, Remington, Robey, Cook, Glendinning, Dimits, Buckingham, Zimmerman, Burke, Peyser, Cabot and Eliason2003). The cross-section through two pairing vortex rolls (Fig. 5c) (Koochesfahani & Dimotakis, Reference Koochesfahani and Dimotakis1986), is a good representation of the cross-section through the pair of vortex filaments at the border between the segments (i) and (ii) in Figure 5a. Thus, even in the late phase the transformation of the large-scale low-mixing structures into the small-scale turbulent mixing does not take place.

The absence of the small-scale turbulence at the late times is a complex problem and requires few more words. For the experiments performed in SCC, it expected to show the same behavior of RMI/RTI structures as those in the open configuration. It means that the large-scale low-turbulent structures formed at the early times of interaction should vanish and the small-scale structures should appear (characteristic for the late times) after reshocks. Such behavior by a number of respected authors (Cohen et al., Reference Cohen, Dennevik, Dimits, Eliason, Mirin, Zhou, Porter and Woodward2002; Anuchina et al., Reference Anuchina, Volkov, Gordeychuk, Es'kov, Ilyutina and Kozyrev2004; Miles et al., Reference Miles, Blue, Edwards, Greenough, Hansen, Robey, Drake, Kuranz and Leibrandt2005; Long et al., Reference Long, Krivets, Greenough and Jacobs2009; Youngs, Reference Youngs2013), etc. Even more, the transition to the small-scale highly-turbulent structures in their pictures is observed not only on the base-plane, but also on the mushroom-shape spikes. The examples are Figure 5 of Long et al. (Reference Long, Krivets, Greenough and Jacobs 2009 ) especially between 5.96 and 11.43 ms, and also Figure 6 (Anuchina et al., Reference Anuchina, Volkov, Gordeychuk, Es'kov, Ilyutina and Kozyrev2004) between 0.064 and 0.108 ms, as well as Figures 7 and 8 of Schiling and Latini between 8 and 11 ms, etc. Even more, they reveal the formation of the cascade from the large to the small-scale structures indicating self-similarity. The mushroom spikes in their above-motioned pictures become highly disturbed and show break down into small-scale high-mixing turbulence as the stochastic process.

Fig. 6. 3D numerical simulation of the perturbation development into the spherical mushroom-shape spike and diagonal structures in the time sequences t = 0.012 ms, t ₂ = 0.02 ms, t ₃ = 0.064 ms; t ₄ = 0.108 ms (Bottom). The enlarged mushroom-shape spike developed at t = 0.012 (the first calculation box) can be favorably juxtapositioned with the experimental one in Figure 4. (Reproduced with permission of Elsevier from, Anuchina et al. (Reference Anuchina, Volkov, Gordeychuk, Es'kov, Ilyutina and Kozyrev2004); Copyright Elsevier, 2004).

Fig. 7. SEM micrograph of the small partially-coherent domain inside random flow field showing the prolate ellipsoidal mushroom-shape spike without vortex rings below the mushroom. The straight-line structures in the base-plane (interface: saddle point plane) are the cnoidal waves (upper left), while the others (lower left and middle right) are the line solitary waves and vortex filaments SO into the irregular polygonal web. The upper structures are vortex ribbons, while the bottom structures are the line solitons, which interact and form the resonant “Y-junction” configuration.

Fig. 8. The steady state of a rising mushroom bubble for the ratio of densities, ρ_l/ρ_b = 40, as function of the Eõtvõs, Eo, and Morton, Mo, numbers. The values of the viscosity ratio, μ_l/μ_b, in the rows from left to right. Bottom row: 85; 151; 269; 479. Middle row: 88; 156; 277; 493. Top row: 88; 156; 277; 493 [the origin of the data is the paper: Unverdi & Trygvason (Reference Unverdi and Trygvason1992)].

However, Figures 4, 5a, 7 (taken at the end of experiment, i.e. at the late times), do not show the expected onset of small-scale structures. Instead, they show large spherical and ellipsoidal mushroom spikes with the smooth surface without any indication of break down into smaller structures. In the base-plane, they show nonlinear waves and vortex filaments of the core size of ~5–7 µm, and ~50 µm long as the large-scale coherent structures with quasi-regular or chaotic organization.

The tentative interpretation may be that interfacial RMI/RTI mixing in the SCC is somewhat different from the usual canonical turbulence, which is an equivalent of a stochastic process. In such process, the flow fluctuations are independent of the initial conditions, boundary conditions, and external forcing. For canonical turbulence to occur the conditions of isotropy, locality, homogeneity, and statistical steadiness should be fulfilled (Abarzhi, 2016, Private communication). The micrographs in Figures 4, 5a, and 7 indicate that such conditions in SCC are not fulfilled or they are disturbed, so that the RT and RM mixing flows do not form the small-scale structures. It seems that their sensitivity to the initial conditions suggests that mixing is more “chaotic” rather than stochastic processes (Abarzhi, 2016, Private communication).

The comparison of spherical mushroom spike in Figure 5a shows a great similarity with 3D numerical simulation of RMI/RTI for the multicomponent flow at the high Atwood and Mach numbers. A single-mode sinusoidal perturbation for A = 0.82 and Ma = 2.5, generates the spherical mushroom spike with vortex rings and four diagonal structures in the base-plane (Anuchina et al., Reference Anuchina, Volkov, Gordeychuk, Es'kov, Ilyutina and Kozyrev2004) (Fig. 6). The spherical mushroom spike appears at the short simulation times between t = 0.012 and 0.02 ms. At magnification, the mushroom spike with vortex rings in Figure 6 (down), shows the similarity with that in Figure 5a in more detail.

Diagonal structures in the base-plane (Fig. 6) are vortex structures which form four identical segments similar to that in Figure 5a. The shock-wave propagation through the sinusoidally perturbed interface causes the formation of vortex structures. In 3D simulation, the perturbation has different wavelengths in the diagonal directions from the cross-sections what leads to the earlier termination of the shock interaction with the interface in the diagonal sections (Anuchina et al., Reference Anuchina, Volkov, Gordeychuk, Es'kov, Ilyutina and Kozyrev2004). The vortex platens (the raised flat “walls” at the base-plane) are formed by the termination of the shock-wave refraction in the diagonal section of the simulation cell at t ₁ = 0.012 ms. The subsequent formation of reflected shock waves causes impact at the cross-platens at t ₂ = 0.020 ms, causing the roll-up and formation of vortex filament crosspieces (Fig. 6). At the latter stages (at t ₃ = 0.064 ms), the roll-up does not increase. Instead, the interaction of the next-generation waves among themselves and with the vortex formations takes place and initiates the final small-scale turbulent mixing as seen in Figure 6 at t ₄ = 0.108 ms (Anuchina et al., Reference Anuchina, Volkov, Gordeychuk, Es'kov, Ilyutina and Kozyrev2004).

Consider the growth rate of the experimental spherical mushroom spikes and bubbles. The spike amplitude h _S ~ 30 µm, and of the bubbles h _B ~ 18 µm (h _S/h _B ~ 1.66, h _B/h _S ~ 0.6), indicating that spikes grow somewhat faster than bubbles but slower than for the nonlinear multimodal perturbation since h _S/h _B < 3 (h _B/h _S ~ 0.3–0.4) (Alon et al., Reference Alon, Ofer, Shvarts, Young, Glimm and Boston1996). The fact that 3D-simulated RMI spherical mushroom spike in Figure 6 generated by the single-mode perturbation of Anuchina et al., is very similar to the experimental one in Figure 5a generated by the multimode perturbation, leads to the tentative conclusion. In the multimodal perturbation, only one – or few modes of close wavelengths (band of modes) – causes the formation of a spherical mushroom spike what makes it similar to that formed by the single perturbation mode. However, the another possible interpretation (which is more appropriate) indicates that interference of perturbation modes, which generates the resulting one, causes the formation of a spherical mushroom spike as shown by Abarzhi and co-workers (Pandian et al., Reference Pandian, Stellingwerf and Abarzhi2016; Stellingwerf et al., Reference Stellingwerf, Pandian and Abarzhi2016a , Reference Stellingwerf, Pandian and Abarzhi b ). By using the group theory analysis and the Smooth Particle Hydrodynamics numerical simulations they have shown that symmetry of the coherent RMI pattern depends besides the amplitude of the waves also on their relative phase. As a result, the wave interference influences symmetry, morphology, and growth rate of the spikes and bubbles at the interface.

3.2.3. Prolate ellipsoidal RTI mushroom-shape spikes

Some of the domains of the NCR show the RTI prolate ellipsoidal mushroom spike (with the small axis of ~20 µm and the long one of ~35–40 µm), without the KH vortex ring around the spike (Fig. 7). The comparison of elongated prolate mushroom spike with 3D simulation of such spikes is not very favorable. Namely, the simulation performed for the high A number (A = 0.9) gives the elongated 3D mushroom spike for the late times (t = 2.5) (He et al., Reference He, Zhang, Chen and Doolen1999). However, it is not the prolate ellipsoid, but the elongated spike spherical on top and flat on the bottom. Similarly, the 2D simulation generates also the elongated mushroom spike for the same late times (t = 2.5), but not the prolate ellipsoid (Zhang et al., Reference Zhang, Shu and Zhou2006). According to He et al., the elongated mushroom formation results from the symmetric variation of the horizontal growth rate, which increases with increasing height until the equatorial maximum is reached; then the horizontal growth rate starts to decrease thus forming the symmetric elongated mushroom (He et al., Reference He, Zhang, Chen and Doolen1999). The mushroom is symmetric with respect to the vertical and the horizontal cross-sections. In the case of simulation, the horizontal growth (velocity) is large and stays almost constant with growing vertical in the z-direction, until the top rounded surface starts to form. Thus, the mushroom is symmetric with respect to the vertical cross-section and asymmetric with respect to the horizontal one (He et al., Reference He, Zhang, Chen and Doolen1999). This rather general description is valid for all types of the elongated spikes, but does not elucidate their origin, especially not of the prolate one, and at present, its formation is not quite clear.

Consider the growth rate of experimental spikes and bubbles. The amplitude of the spikes h _s ~ 30–35 µm and of the bubles h _B ~ 10 µm with h _S/h _B ~ 3–3.5, indicating three times faster growth rate of the prolate ellipsoidal spikes than of bubbles. This result is in agreement with the growth rates of RMI spikes and bubbles for the nonlinear multimodal perturbation, assuming that all incommensurate modes take place in the formation of structure (Alon et al., Reference Alon, Ofer, Shvarts, Young, Glimm and Boston1996). However, regarding the spike dynamics in the light of empirical, semi-empirical and engineering models (such as Layzer-type, drag model Alon et al., Reference Alon, Ofer, Shvarts, Young, Glimm and Boston1996; Zhang's model, Reference Zhang1998, etc.), more caution is required. As a matter of fact, at present there is no a rigorous theoretical description of the dynamics of nonlinear spikes. On the theory side, the empirical models often have many limitations (such as unphysical velocity fields, non-existing mass fluxes, etc.) that are not confirmed in experiments. Extensive use of adjustable parameters enables the models agreement with experiments (Abarzhi, 2016, Private communication).

The base-plane in Figure 7 (upper left) comprises the straight “walls” with the profile of cnoidal waves, while the other “walls” (lower left) as well as those on the right side of the micrograph have the bell-shape profile of line solitons. The line solitons observed at the lower side interact and form the resonant “Y-junction” configuration (Kodama, Reference Kodama2004; Oikawa & Tsuji, Reference Oikawa and Tsuji2006; Lugomer et al., Reference Lugomer, Maksimovic, Geretovszky and Szorenyi2013; Lugomer, Reference Lugomer2016a ). (For details see the paper, Lugomer, Reference Lugomer2016b , and for the mathematical aspects, Kodama, Oikawa, Biondini). A variety of wave-vortex structures indicates that the fluid dynamics in the base-plane is inhomogeneous and causes complex formations. An example is the anchored vortex filament with the spiral-like structure exposed to torsion (Fig. 7, lower right side). Another example is the flattened vortex filament (upper left side) transformed by twisting into twisted-ribbon (Lugomer & Fukumoto, Reference Lugomer and Fukumoto2010).

3.2.4. Damped, spherical, and elongated RTI mushroom spikes

The shape variation of damped, spherical, and the prolate ellipsoidal mushroom-shape spikes in Figures 4, 5a, and 7, cannot be attributed to the variation of the A number only. Description of the mushroom-shape variation requires characteristics of the system, like the density ratio (ρ_l/ρ_b) and viscosity ratio (μ_l/μ_b), where index, l, relates to liquid and, b, to the bubble (gas), to be taken into account. Variation of the RTI mushroom shapes has been studied by numerical simulation for bubbles in a viscous liquid (Hua & Lou, Reference Hua and Lou2007), in the RTI (Unverdi & Trygvason, Reference Unverdi and Trygvason1992), and under different flow conditions with incorporation of the force term (Shu & Yang, Reference Shu and Yang2013). These models describe the gas–liquid system in which the density (ρ_l) and the viscosity (μ_l), are used for definition of the additional parameters like the Eõtvõs number, Eo, and the Morton number, Mo. The Morton number involves the fluid properties only, and the Eõtvõs number is the nondimensional size of the bubble. The bubble shape can be characterized by the Morton number, Mo = gμ_l ⁴/ρ_lσ³, by the Eõtvõs number, Eo = ρ_l gd _l ²/σ (also called the Bond number), and by d _l, the effective radius of the bubble (Unverdi & Trygvason, Reference Unverdi and Trygvason1992). The rate of rise (growth) is expected as the Reynolds number in terms of the rise velocity, Re = ρUd _l/μ, where U is the velocity. Often the Weber number, We = ρUd _l/σ, is also used to characterize the bubble.

In the analysis of the mushroom-shape variation, we rely on the simulation of Unverdi and Trygvason for the mushroom bubble shapes in RTI (Unverdi & Trygvason, Reference Unverdi and Trygvason1992). The analysis is based on the Boussinesq approximation for the buoyant bubbles where two fluids have dependence on the surface tension and difference of densities of the fluid inside the bubble and the surrounding fluid (Unverdi & Trygvason, Reference Unverdi and Trygvason1992). The resulting diagram of the mushroom shapes as function of the Eo and Mo for the constant density ratio (ρ_l/ρ_b = 40), is shown in Figure 8. The diagram shows mushroom shapes for the Eo ranging from 1–10², and Mo number ranging from 10⁻⁷ to 10² (Unverdi & Trygvason, Reference Unverdi and Trygvason1992). The viscosity ratio, μ_l/μ_b, for every particular case is given in the caption to Figure 8.

For small Eo (large surface tension or small bubble), the RTI bubbles are almost spherical (bottom row) (Fig. 8). However, for small Eo and small Mo the bubble front is slightly flatter than the back (bottom row, left), while for the small Eo and higher Mo the bubbles become more spherical (bottom row, right).

For higher Eo (transition zone from small to large bubbles), the bubbles attain somewhat flat shape (middle row). If the Mo is low (middle row, left), then the bubbles become ellipsoidal (Fig. 8). With increasing Mo the bubble front becomes semi-circular while bottom becomes more flat (middle row, right). For the high Mo the bubbles become ellipsoidal.

For the highest Eo (small surface tension or large bubbles), the solution is not truly steady state; the skirts of the bubbles are pulled off with a separated flow increase continuously in length (top row, left) (Fig. 8). The vertical rise of the bubble is independent of the Mo number; for largest bubbles the pressure drag and the velocity are independent of the viscosity (Unverdi & Trygvason, Reference Unverdi and Trygvason1992).

The Eo and Mo numbers for the mushroom shapes in Figures 4, 5a, and 7 may be estimated from the comparison with the calculated shapes in Figure 8. The damped mushroom-shapes in Figure 4a–4c (A ~ 0.65), are similar to the shapes in Figure 8, for Eo ≳ 10 and 10⁻⁴ ≤ Mo ≤ 10⁻¹. The prolate ellipsoidal mushroom shape in Figure 7 is similar to the bubbles-with-skirts for Eo ≳ 10² and, 10⁻¹ ≲ Mo ≲ 1, while the spherical one in Figure 4a (A = 0.85) is similar to the spherical mushroom for Eo ≳ 10² and Mo ~ 10².

3.2.5. Low-mixing structures in the base-plane

The base-plane structures in Figures 4, 5a, and 7 – mostly the nonlinear waves and vortex filaments – are the large-scale formations with diameter (core size σ ~ 3–6 µm), and the length L ~ 30–40 µm. These coherent formations are organized into various configurations: the nonlinear horseshoe waves – into the rosette-like structures, and the line solitary waves – into the polygonal web. The horseshoe waves are similar to those in the CR (Lugomer, Reference Lugomer2016b ), and identified as the parabolic-like solitary waves, which were studied by various mathematical groups on the basis of the cyllindric Kadomtsev–Petviashvilli equation (Klein et al., Reference Klein, Matveev and Smirnov2007; Khusnutdinova et al., Reference Khusnutdinova, Klein, Matveev and Smirnov2013; Lugomer, Reference Lugomer2016b ). The straight-line structures with the bell shape profile are – the line solitons – while the others are cnoidal waves [Fig. 4(iv)]. The line solitons interact between themselves giving rise to the “Y-junction” or the “X-junction”, etc. as the resonant and nonresonant configurations, respectively (Kodama, Reference Kodama2004; Oikawa & Tsuji, Reference Oikawa and Tsuji2006; Biondini, Reference Biondini2007; Lugomer, Reference Lugomer2016b ). The other radial structures are vortex filaments formed by the roll-up of the nonlinear waves. The roll up into KH filament-rolls occurs in the segments where the Re number reaches the critical value Re ≥ 10³. These base-plane structures show only the low mixing, which stays to the late times and does no show transition to the small-scale turbulent mixing.

The 3D RMI simulations reveal transition from the large-scale low-mixing to the small-scale well-developed turbulent mixing with randomly oriented structures at the late times (Anuchina et al., Reference Anuchina, Volkov, Gordeychuk, Es'kov, Ilyutina and Kozyrev2004; Long et al., Reference Long, Krivets, Greenough and Jacobs2009). Such structures are characteristic for the statistically random distribution of velocity, density, or temperature (Zhou et al., Reference Zhou, Remington, Robey, Cook, Glendinning, Dimits, Buckingham, Zimmerman, Burke, Peyser, Cabot and Eliason2003). Thus, the late time, low-mixing, large-scale structures of nonlinear waves, and vortex filaments in Figures 4, 5a, and 7, can be compared with the similar simulated ones existing only for the short times. In the experiment, the large-scale structures persist to the late times when the solidification of target surface takes place (~200 µs after pulse termination). The transition to the small-scale turbulent mixing is frustrated (prevented) most probably because of the interaction with the nonstationary flow in the base-plane (Zhou et al., Reference Zhou, Remington, Robey, Cook, Glendinning, Dimits, Buckingham, Zimmerman, Burke, Peyser, Cabot and Eliason2003); and because of the collapse of bubbles, which generates the shock waves into the surrounding fluid that wipe out the small-scale structures. The existence of the large-scale structures even at the late times indicates the lack of turbulent mixing with statistically random distribution of velocity or density. Thus, the paradigm of the nonlinear low-mixing wave-vortex structures characterizes the RMI/RTI in the NCR.

3.3.1. Deformation of mushroom-shape spikes

Deformation of the mushroom-shape spikes and breakup of their symmetry is shown in Figures 9a(i, ii), which can be compared with the simulated ones for the initial RT mixing layer (Zhang et al., Reference Zhang, Shu and Zhou2006). Their 2D simulation of the shock effect on RT structures at Ma = 6, shows the evolution of deformation and symmetry breakup in t = 1.6, 1.85, 2.3, 2.6, and 3 (Fig. 9b). The great similarity with mushroom structures in Figure 9a is observed for 1.6 < t ≤ 2.6. The deformation occurs when the interface with the shock wave moves downward (from above) (Zhang et al., Reference Zhang, Shu and Zhou2006). If the Ma number of the shock is low the symmetric organization of two mushroom structures is not perturbed. Increasing the Ma number the symmetry is broken and the structures enter turbulent mixing and merging. The insight into this process is obtained from the temperature, entropy, and vorticity measurement (Zhang et al., Reference Zhang, Shu and Zhou2006). In the moment when the shock hits the interface (t = 1.85), the temperature field is divided along the shock location. The mushroom structures have not yet changed from their original entropy values, and the vorticity is small. After the passage of the reshock (t = 1.9), the temperature and the entropy of the mushroom structure increase rapidly with enhanced vorticity (Zhang et al., Reference Zhang, Shu and Zhou2006). However, the temperature and the entropy are different at different locations. Below the mushroom they are lower and the flow structures are stable. At the top of the mushroom the entropy and temperature are higher indicating the higher energy level and instability that leads to the breakdown of flow structures into smaller ones (Zhang et al., Reference Zhang, Shu and Zhou2006).

Fig. 9. The effect of a strong reshock on the RT-mushrooms and horizontal base-plane structures. (a) SEM micrograph of RTI mushroom-shape spikes showing deformation, breaking of the symmetry between them and disordered fluid flow in the base-plane. (b) Numerical simulation of strong shock effects on the RTI mushrooms. The shock hits the RT interface at t = 1.85 and causes strong deformation, which evolves in time: (a) t = 1.9; (b) t = 2.3; (c) t = 2.6; (d) t = 3.9. Note the similarity between deformation of mushrooms in Figure 7a and simulated ones for 1.6 < t ≤ 2.6. [Courtesy of Dr. Y-T. Zhang. Reproduced with the permission of AIP Publishing from Zhang et al. (Reference Zhang, Shu and Zhou2006); Copyright AIP, 2006.]

The simulation gives the evidence that the motion of the interface continues until the mushroom structures are broken into small turbulent pieces and merged with the base-plane vorticity making a small-scale fully turbulent mixing layer with stochastic distribution of temperature, density, and entropy (Zhang et al., ). In contrast, Figure 9a shows that the downward motion of the interface is stopped at some small distance above the base-plane without of the onset of small-scale turbulent mixing. Thus, the mushroom-shape structures – although deformed – stay above the base-plane without transition to fully developed turbulence.

3.3.2. Formation of stripes and ribbons in the base-plane

The base-plane structures, namely the “egg carton” walls, under reshoks also suffer deformation and symmetry breakup [Fig. 10(i)]. The most dramatic effect is vanishing of the “walls”, the increase of their size and spreading, as well as their splitting into three or more stripes. Under spanwise perturbation the stripes are transformed into vortex ribbons [Fig. 10(ii, iii)]. The longer ribbons tend to tangling due to transversal perturbation [Figure 10(iv)].

Fig. 10. SEM micrographs of the domains experiencing strong reshocks show breakdown of the “egg carton” web in the base-plane. Transformation of the former “walls” of the web between the RTI mushroom spikes into various structures by spreading and deformation (i), by splitting and the roll up into vortex ribbons due to transversal perturbation (ii, iii), and tendency to tangling of ribbons due to spanwise perturbation (iv).

3.3.2.2. Interaction of flow stripes and ribbons with bubble-cavities

Figure 11a(i) shows two vortex filament ribbons (incoming from above), one directed to the left and other the right side, which enter the cyclic motion around the cavity of a collapsed bubble; one forming a loop and other spiraling down to the hole in the center of the cavity. The ribbon structures in the vicinity merge with the ribbon loop. More complex configuration arising from the ribbon trapping with formation of closed loops and knotted structures around cavities (point defects) can be seen in Figure 11a(ii).

Fig. 11. Trapping of the filament-ribbon at the hole (assumed the “point defect”) with the coiling SO. (a) SEM micrographs of the filament-ribbon trapped at the hole (cavity) which form the open-loop structure (i), and the multiple coils of closed-loop ribbons around two holes at few layers giving rise to the 3D coil SO (ii). (b) The elongated cavity with a hole (most similar to the real situation), representing a point defect surrounded by a liquid shell (trapping region) of radius R _t. Trapping of the filaments and ribbons is caused by the potential V _t. The trapping force F _t = −∂/∂x (V _t), where V _t = Δp and Δp is the pressure difference between the point defect and the surrounding area. The corresponding trapping potential V _t and the effect of the trapping force on the ribbon configuration as a kind of landscape is schematically shown in the lower picture diagram, where the trapping potential is shown circular (symmetric) for simplicity.

Point defects and trapping potential. The point defects are assumed all structures on the target surface such as holes, droplets, or other structures of spherical or hemispherical shape with diameter ~15–20 µm (or more), which is much larger than the core size of vortex filament (3–7 µm, usually). In close interaction with rigid spheres and holes (cavities- collapsed soft spheres or bubbles) – vortex filaments become trapped giving rise to complex configurations (Lugomer et al., Reference Lugomer, Fukumoto, Farkas, Szorenyi and Toth2007). The trapping results in the pinning of filament at the rigid sphere, which can be interpreted on the basis of the Pedrizzetti model (Pedrizzetti, Reference Pedrizzetti1992). More complex behavior such as looping and spiraling of filament around the sphere as the result of trapping, can be interpreted on the basis of the Shwarz model (Schwarz, Reference Schwarz1985). The model describes vortex filament capture by a sphere or equivalently by a hemispherical protrusion, because around the point defect the boundary field exists, which tends to move the filament around the sphere (Schwarz, Reference Schwarz1985). When the filament gets close enough, a cusp is pulled out by the boundary field of a sphere, causing the vortex to approach the sphere infinitely being finally pinned (Schwarz, Reference Schwarz1985).

Pinning of filaments and ribbons (strings) on material defects is well known to occur in various physical, chemical, and biological systems. Generally speaking, the filaments or the stringlike structures are pinned to the defects in various systems, an example of which is pinning of the magnetic flux strings in superconductors (Tonomura et al., Reference Tonomura, Kasai, Kamimura, Matsuda, Harada, Nayakama, Shimoyama, Kishio, Hanaguri, Kitazawa, Sasase and Okayasu2001; Blatter et al., Reference Blatter, Geshkenbein and Koopmann2004), pinning of carbon nanotubes to defects on silicon substrate (Tsukruk et al., Reference Tsukruk, Ko and Peleshanko2004), pinning of spiral vortex filaments like various biological strings in excitable media (Pazo et al., Reference Pazo, Kramer, Pumir, Kanani, Efimov and Krinsky2004), etc. When completing the list with recently observed pinning of magnetic flux filaments in the Sun's penumbra (Thomas et al., Reference Thomas, Weissl, Tobias and Brummel2002), one can say that pinning is a common phenomenon taking place in the filamentary organized matter from the atomic to the astrophysical scales.

Trapping of vortex ribbon at the cavity (the point defect) can be described by the trapping potential, V _t, originating from the pressure difference when the filament ribbon is in the close vicinity of a defect, either because of self-induction, or because of shear flow. The pressure difference or the hydrodynamic force is responsible for trapping at the distances R ≤ R _t, where R _t is the trapping radius (Fig. 11b) (Lugomer et al., Reference Lugomer, Fukumoto, Farkas, Szorenyi and Toth2007). The topmost illustration in Figure 11b shows the point defect (elongated cavity with a hole) formed by the bubble collapse. Trapping of the filament ribbon by the trapping potential V _t is schematically shown in Figure 11b (middle). The trapping (drag) force is F _t = −∂/∂x (V _t), where V _t = Δp, and Δp is is the pressure difference between the point defect and the surrounding area. The trapping potential, V _t, and the effect of the trapping force on the ribbon configuration is schematically shown in the lower picture diagram; the trapping potential is shown with circular symmetry for simplicity. The ribbon looping around a hole occurs via continuous bending similar to the spiraling of an elastic rod. The process is associated with the vortex-ribbon splitting because of the strong strain field surrounding a hole (Lugomer et al., Reference Lugomer, Fukumoto, Farkas, Szorenyi and Toth2007).

The high trapping potential of a cavity (hole) (Fig. 11) causes the formation of ribbon coils, of closed-loop structures as well as spiraling down cavity. It also causes the formation of closed loops with more or less complex knots, but the dense winding and merging with the nearby structures causes their configuration difficult to be resolved. Various fluid mechanic phenomena, like vortex filament bending, looping, and spiraling on point defects (spheres and cavities) have been studied in laser–matter (indium) interactions (Lugomer et al., Reference Lugomer, Fukumoto, Farkas, Szorenyi and Toth2007). Also, the other phenomena such as vortex filament reconnection, merging, and undulation (Lugomer et al., Reference Lugomer, Fukumoto, Farkas, Szorenyi and Toth2007), as well as the formation of ribbons and ribbon helicoids, etc. (Lugomer & Fukumoto, Reference Lugomer and Fukumoto2010) have been studied in laser–matter interactions on various targets. In these interactions, it was naturally to assume viscous phenomena like other authors (Pedrizzetti, Reference Pedrizzetti1992), while some of them assume inviscid phenomena (Schwarz, Reference Schwarz1985).

Generally speaking, interaction of filaments with (small) material defects is well known to occur in various physical, chemical, and biological systems. An example is the spiraling of waves and vortex filaments around defects in active media obtained as the solution of Ginsburg–Landau equation (Pazo et al., Reference Pazo, Kramer, Pumir, Kanani, Efimov and Krinsky2004). Although the forces in these systems are different, the effects on the filaments with formation of complex structures as well as their topology are the same. Interestingly, similar behavior was also found in the steady configurations of a vortex filament embedded in a point-source or point-sink flow in three dimensions, which shows looping bending and spiraling quite similar to the interaction of filaments and point defects (Fukumoto, Reference Fukumoto1997).

Regarding the spiraling around sphere, it was assumed that the boundary field of point defect acts to distort the filament such as to generate a self-induced motion toward the sphere (defect). The frictional component of the motion also causes the filament to spiral in, and in fact becomes the main factor when the filament is far away. As the result, an initially straight filament will undergo an inward-spiraling motion (Schwarz, Reference Schwarz1985).

For the analysis of dynamics, such spiraling and knotted filament ribbons can be assumed as the “…. closed solution curves of the vortex filament equation and the periodic problem connected with the nonlinear Schrödinger (NLS) equation….” (Calini & Ivey, Reference Calini and Ivey2001; Ivy, Reference Ivy2005). The knot types of finite gap solution are related with the corresponding NLS potential, but this is out of scope of this paper. For this aspect of the problem the reader is directed to the mathematical literature (Calini & Ivey, Reference Calini and Ivey2001; Ivy, Reference Ivy2005).

Figure 12 shows a 3D flow separation of incoming fluid into “A”, “B”, and “C” ribbon branches of which “B” and “C” are bended under ~90° with respect to the ribbon flow “A”. The ribbon flow “A” makes an open loop around the cavity (collapsed bubble) in the backward direction schematically shown in the excerpt (upper right). The other excerpt (upper left), schematically shows the separation of the incoming flow into two branches (left and right) that form the open-loops around the cavities. The trapping potential V _t is not as strong as in the case in Figure 11 and only changes the direction of the ribbon flow. The ribbon “A” makes a loop around the cavity with the hole also formed by the bubble collapse (Lugomer, Reference Lugomer2016b ). The ribbon “B” is strongly bended around the bubble (hole) with tendency to form a loop. Generally, such trapping at the point defect (the spike or bubble) starts with approaching to the cavity (sphere); the vortex ribbon first aligns itself along the circumference in a straight form; then, closer to the cavity (sphere) it becomes arc bended. Finally, at very small distance the filament is turned backward and the arc-bended segment becomes a loop. (Lugomer et al., Reference Lugomer, Fukumoto, Farkas, Szorenyi and Toth2007). It may be also merged with the nearby ribbon as seen at the right side of the micrograph. This ribbon (bypassing the hole) is exposed to the local forces, which cause twisting and the formation of knots (Lugomer & Fukumoto, Reference Lugomer and Fukumoto2010).

Fig. 12. SEM micrograph of vortex ribbon trapping by the trapping potential of the cavities with the hole. The trapping potential is not very strong, so that ribbons make only the open loop around the holes. The incoming flow is separated into three ribbon flow branches: flow A, flow B, and flow C. Every one of them enters close interaction with the hole inside the trapping distance radius, R _t. The flow separation and looping around the holes (cavities) as the point defects is schematically shown in the excerpt (upper left). The looping around the bubbles with the change of direction of the ribbon flow is schematically shown in the excerpt (upper right).

3.3.3.3. Interaction of flow stripes and ribbons with mushroom-shape spikes

Interaction of vortex filament-ribbons with two deformed RTI mushroom-shape spikes as well as merging of two mushroom spikes are shown in Figure 13. This figure clearly shows two mushroom spikes at small distance connected by a bridge what indicates merging. This effect is not a common occurrence in laser experiments, but appears in the experiments performed in the SCC. It was not observed in the whole region, but only in some domains. In such domains, all other structures in the base-plane are deformed or tangled. Since the merging of spikes does not normally appear (in contrast to bubble merging) this phenomenon seems to be the effect of most probably of the strong reshocks in such domains. An argument in favor is the similarity of this picture to Figure 9 (especially to Fig. 9a), and to the numerically simulated sequences of mushroom spikes, which approach each other under reshock (Fig. 8b) (Zhang et al., Reference Zhang, Shu and Zhou2006).

Fig. 13. SEM micrograph of vortex ribbon trapping at deformed mushroom-shape spike. A 3D flow separation of the incoming flow into few ribbons (upper, center) trapped by the trapping potential of the spikes. One ribbon branch is pinned at the spike, while the other two at the right side make loops around the spike. The trapping potential causes a large loop around the spike. The ribbon flow (upper right) is splitted into one smaller ribbon branch, which makes a loop to the right and turns backward, while the other ribbon extends along the other structures been deformed sinusoidally.

This figure also shows the filament-ribbons incoming from above which are layered one above the other and form 3D configuration. One of the ribbons is pinned at the mushroom spike (upper left) what can be interpreted on the basis of the Schwartz model for the filament pinning at protrusion (Schwarz, Reference Schwarz1985; Lugomer et al., Reference Lugomer, Fukumoto, Farkas, Szorenyi and Toth2007). The pinned ribbon makes the static irregular structure in the base of the spike (Fig. 13). It seems that there is a deformed ring around the RTI with a damped mushroom cap. The other ribbon (center) makes a large loop around the spike because its trapping potential is not strong enough to capture the ribbon. The vortex filament in Figure 13 (upper right) is splitted into two branches; one, which makes a loop backward around the cavity (dark), and another filament branch which spreads up into a wide ribbon that passes along the other structures being wavy-like deformed due to transversal oscillation. Obviously, the mixing is relatively higher than in the domains of the small reshock effects, but it is still the large-scale and not the small-scale turbulent mixing.