Nonlinear bubble competition of the multimode ablative Rayleigh–Taylor instability and applications to inertial confinement fusion

The Rayleigh–Taylor instability (RTI)^1,2 arises in many natural and engineering systems, where a heavy fluid is accelerated against a light fluid. For example, the RTI phenomenon is observed in supernova explosions,³ the oceanic overturning circulation,⁴ and laboratory fusion systems confined by inertia⁵ or magnetic fields.^6,7 The RTI is one of the major obstacles to achieving ignition via inertial confinement fusion (ICF).^8,9 In ICF, a cold dense shell is accelerated by the pressure induced by ablation of the shell into a hot low-density plasma. Mass ablation originates from the heat deposited at the interface by either electron heat conduction (in direct-drive ICF) or x-ray irradiation (in indirect-drive ICF). During the acceleration stage, interface modulations arising from initial surface roughness or irradiation non-uniformity (such as laser imprint) seed the RTI. Since the growth of RTI is affected by mass ablation off the unstable interface, it is often referred to as Ablative RTI (ARTI) as opposed to the classical RTI (CRTI).¹ In ICF implosions, the growth of ARTI can compromise the integrity of the imploding shell, reduce the final compression, and mix ablator material into the fuel, resulting in severe degradation of the implosion performance. Performance degradation depends mostly on the penetration of the ARTI bubbles into the dense shell. Understanding and predicting the ARTI bubble-front growth is crucial to determine the hydrodynamic stability boundary for target design and viability of ICF ignition.

The ablation effect on the single mode RTI has been extensively investigated in the past, with some of the main results summarized in Table I. Mass ablation can stabilize the RTI in its linear phase and results in a linear cutoff wave number k_cf [satisfying γ_abl(k_cf) = 0] in the unstable spectrum.^10,11 All modes with wave number k larger than k_cf are linearly stable.^12–14 There also exists a wave number with the maximum growth rate in the ARTI, which is in contrast to the classical RTI where the growth rate is a monotonically increasing function of k.¹ Unlike the stabilization effect during the linear phase, recent studies have shown that mass ablation can generate vorticity inside the bubble and further destabilize the RTI in the nonlinear phase in both two- (2D) and three-dimensional (3D) ARTI (Table I). Vorticity provides a centrifugal force to the bubble vertex and accelerates the nonlinear terminal bubble velocity above the classical value, especially for the small scale 3D bubbles.^15,16 Recent results have also shown that all linearly stable RTI modes beyond the nonlinear cutoff k_cf can be destabilized by finite amplitude perturbations driving intense vorticity inside the bubble.¹⁷ Therefore, mass ablation has a destabilizing effect on the nonlinear single mode RTI despite its stabilizing role in the linear regime.

Realistic RTI perturbations consist of superposition of multiple modes. It has been shown that multimode CRTI interactions can generate progressively larger scale structures and reach a self-similar regime with a bubble penetration distance h_b proportional to α_bA_TS.^20–27 Here, $S=(∫gdt)2$ has the unit of length. The self-similar dimensionless coefficient α_b is expected to be constant in time. The value of α_b is important in many nonlinear systems.²⁸ For example, it is crucial to the formation of a molecular cloud structure in Eagle Nebula.²⁹ It also determines the maximum in-flight aspect ratio (IFAR) to preserve shell integrity in ICF implosions.^30–32 The self-similar CRTI bubble interface growth is achieved in two limiting ways: (i) the bubble merger regime, which is dominated by the nonlinear coupling of saturated shorter wavelength modes. In the bubble merger regime, α_b is insensitive to the initial perturbations and a universal α_b ∼ 0.02–0.03 is expected for 3D CRTI.^21,22 (ii) The bubble competition regime, which is dominated by exponential growth and saturation of long wavelength modes. In this case, α_b increases logarithmically with initial perturbation amplitude h₀ and can be quantified by^23,26,27

The coefficients C and k₀ are two free parameters determined by comparison with experiments or numerical simulations. According to the theory developed by Dimonte and collaborators,^23,24,26,27 the nonlinear multimode CRTI first enters the bubble competition regime before transitioning into the bubble merger regime.

The effect of ablation on the multimode ARTI bubble-front growth has been studied both theoretically and experimentally. By considering the ablation effect on reducing the RTI linear growth rate, it was argued that α_b in ARTI should be lower than in CRTI.^30,32 This result is consistent with some observation from laser direct-drive multimode ARTI experiments which exhibited α_b ≈ 0.04, at the lower limit of α_b = 0.04–0.08 range observed in CRTI experiments.³³ However, ARTI experiments by laser indirect-drive showed that multimode perturbations grow faster than predictions from Haan's multimode saturation model,^34,35 suggesting that some essential nonlinear physics is missing in the theoretical models. In both Refs. 30 and 32, ARTI models only consider the ablative stabilization effect in the linear phase but ignore the destabilization effect of ablation-generated vorticity in the nonlinear phase, which was shown to play a prominent role in more recent studies.^15–17,36 Here, we show that accounting for the vorticity effect may resolve the discrepancy between theory and experiments.

It was recently shown³⁷ that the self-similar multimode bubble-front penetration is dominated by bubble competition and α_b increases with the initial perturbation amplitude. The dependence of α_b on initial amplitude h₀ and ablation velocity V_a can be evaluated through the ablative bubble competition theory,

using the coefficients (C = 0.6, b = 4.2, and k₀ = 0.06 μm⁻¹) for 2D and (C = 0.95, b = 4.2, and k₀ = 0.06 μm⁻¹) for 3D.³⁷ The ablative bubble competition theory recovers the classical limit [Eq. (1)] when V_a = 0 and indicates that mass ablation can reduce α_b with respect to the classical value for the same initial perturbation amplitude. A similar dependence of α_b on V_a was also derived in the bubble merger regime without the dependence of α_b on initial perturbation.³² It is also shown that unlike in CRTI where small scale initial perturbations tend to reduce α_b by initiating an earlier transition into the bubble merger regime, small-scales in ARTI can keep the growth in the bubble competition regime because of ablation-driven vorticity. Due to the vorticity effect, α_b in ARTI can be higher than in the classical case for sufficiently large initial perturbations. The ablative bubble competition theory [Eq. (2)] is also supported by the recent multimode ARTI experiments showing that α_b depends on initial conditions and ablative corrections to α_b qualitatively agree with the theory.³⁸ 

In this work, the bubble competition of the multimode ARTI and the ablation-generated vorticity effects are investigated for different initial conditions. Building on the findings in Ref. 37, it is shown that the multimode bubble-front penetration is dominated by bubble competition for all investigated initial conditions. An analytical bubble competition model is developed to describe the linear and nonlinear growth of the multimode ARTI, which depends on the initial perturbation h₀, acceleration g, and ablation velocity V_a. It is also shown that the small-scale multimode perturbations can generate significant vorticity when the initial perturbation is large, preventing the transition of the ARTI from the bubble competition to the bubble merger regime. Due to vorticity acceleration of the bubble in the nonlinear stage, our results suggest that α_b can be enhanced by up to 30% relative to our ablative bubble competition model without the vorticity effect. The ablative bubble competition model is then applied to explain the hydrodynamic stability boundary observed in OMEGA low-adiabat implosion experiments.

This paper is organized as follows: in Sec. II, the ARTI bubble competition regime is investigated for different initial perturbation spectra. In Sec. III, the vorticity generation and its effect on enhancing the multimode bubble-front growth are studied. In Sec. IV, the bubble competition model is applied to assess the hydrodynamics stability boundary in laser direct-drive ICF. Section V is the summary.

The multimode ARTI simulations are carried out in planar geometry using the hydrodynamic code ART. The ART code solves the single-fluid equations in Cartesian coordinates with Spitzer-Harm thermal conduction.³⁹ Detailed information of the ART code can be found in Refs. 15, 16, and 36. The equilibrium density profile used in the multimode ARTI study is shown in Fig. 1(a). This equilibrium profile is the same as in the ARTI studies in Refs. 17 and 37, which corresponds to the acceleration phase of a typical direct-drive target. The thickness of the target is about Δ = 27 μm, which is defined as the distance between the inner and outer boundaries where the shell density equals 1/e of ρ_a (ρ_a = 5.3 g/cm³ is the maximum density near the ablation front Z₀ ≈ 60 μm.). The target is uniformly ablated from the left-hand-side with ablation velocity V_a = 3.5 μm/ns. The equilibrium density profile varies in the Z-direction due to the pressure from the rocket effect and can be approximated by ρ(r) = ρ_a[(2Δ−r)/2Δ]^3/2 (r = Z−Z₀ is the distance to the interface) during the acceleration phase. This is in contrast to CRTI studies, which often use a uniform density for the heavy and light fluids. The maximum density ρ_a is approximately constant in time during the acceleration phase for a fixed ablation pressure. Since the density exhibits a sharp variation with the heavy fluid density much larger than the light fluid, the Atwood number approaches unity (A_T ≈ 1). An initial effective gravity g₀ = 100 μm/ns² is used and adjusted in time during the simulation to balance the ablative pressure pushing on a steadily decreasing shell mass. Since the dependence of α_b on initial perturbations and ablation is quite similar between 2D and 3D simulations,³⁷ 2D simulations will be primarily used in this work. The linear dispersion relations of the classical and ablative RTI are compared in Fig. 1(b). Based on the simulation parameters, the linear cutoff wave number for the ARTI is k_cf ≈ 1 μm⁻¹. In the multimode simulations, the ARTI is seeded by a velocity perturbation initialized around the ablation front, similar to Ref. 37. The notation Ps(m₁-m₂) is used to represent the power spectrum of the initial perturbation in mode number space. Here, m₁-m₂ denotes the range of the perturbed mode numbers and s denotes the spectrum index. For example, P2(4–16) denotes the initial perturbation with modes 4 k_L ≤ k ≤ 16k_L and the mode amplitude decays as k⁻². Here, k_L = 2π/L_x and k = mk_L. L_x is the length of the simulation domain in the X direction and m is the mode number in X directions. The length of the simulation domain is L_z = 110 μm in the Z direction. The grid size is 0.1 μm in each direction and is sufficient for numerical convergence. The mode numbers corresponding to wave numbers k are also shown above Fig. 1(b) for L_x = 100μm. The normalized root mean square (RMS) velocity perturbation $Vrms=(∑i=1,NVp2(x)/N)1/2/Va$ is used to represent the overall initial perturbation amplitude.

In the multimode simulations, the bubble-front penetration distance h_b is defined as the distance from the vertex of the fastest-growing bubble to the initial ablation front in the frame of reference of the imploding shell. Since the acceleration of the target increases with time due to mass ablation, α_b is calculated by fitting the linear relation $hb=αbS$ with $S=(∫gtdt)2$⁠.²⁴ For each power spectrum Ps(m₁-m₂), we initialize the perturbations with random phases, as shown in the ensemble of 2D multimode ARTI simulations in Fig. 2(a). In each individual simulation, the ablation front is perturbed by P0(4–16) with the same initial amplitude V_rms = 0.77 but different random phases. These perturbed modes include wave numbers from the maximum linear growth rate to the linear cutoff. It is shown that h_b in each individual multimode simulation scales linearly with S in the nonlinear phase, indicating a self-similar nonlinear stage. The value of α_b varies slightly due to the random phases in each simulation. To eliminate the random phase effect and without loss of generality, h_b is calculated by averaging over the ensemble of simulations as in Fig. 2(a). For this P0(4–16) perturbation power spectrum, α_b is found to be 0.045 by linearly fitting the relation between the averaged h_b and S. We have determined that for any power spectrum Ps(m₁-m₂), a 4-member ensemble average is sufficient for a converged value of α_b.

Figure 2(b) shows the simulations of the multimode ARTI bubble-front penetration h_b for perturbation P0(4–16) and different initial perturbation amplitudes, which were a main focus in Ref. 37. It was shown in Ref. 37 that h_b follows the α_bA_Tgt² scaling law in the nonlinear phase and α_b increases with the initial perturbation amplitude. The dependence of α_b on initial perturbation amplitude h₀ and ablation velocity V_a can be quantified by the bubble competition theory [Eq. (2)]. Building upon the findings in Ref. 37, it is further shown here that the linear and nonlinear bubble-front penetration h_b can be re-constructed by the analytical bubble competition model described here: in the linear phase (t < t_NL), the bubble penetration is determined by $hb=h0 exp (γmaxablt)+Vat$⁠. The linear growth rate $γmaxabl$ and the corresponding dominant wavelength λ_max are determined by the ARTI mode with the maximum linear growth rate. This is consistent with the fact that the mode with the maximum linear growth rate grows faster than the other modes in the linear phase. $γmaxabl$ and λ_max are further determined by V_a and g through the ARTI linear dispersion relation (Table I). In the nonlinear phase (t > t_NL), the multimode ARTI is dominated by the bubble competition (h_b = α_bgt²) with α_b determined by h₀, g, and V_a through Eq. (2). Therefore, the time evolution of h_b can be determined by h₀, g, and V_a for both the linear and nonlinear stages. In the simulations, the maximum linear growth rate of the multimode ARTI is found to be γ ≈ 2.5 ns⁻¹, close to the theoretical maximum linear growth rate. Meanwhile, the initial surface perturbation amplitude h₀ can be estimated by extrapolating the linear phase of the averaged multimode amplitude to t = 0. As shown in Fig. 2(b), although the linear phase and nonlinear growth (α_b) are different for simulations with different initial perturbation amplitudes, analytical h_b evolution closely agrees with simulations if the nonlinear transition time t_NL is defined as the point when h_b reaches 0.2λ_max. This result is consistent with current understanding of bubble competition during which the multimode bubble-front grows exponentially in the linear phase before reaching the self-similar nonlinear state.²³ In Sec. IV, we use our analytical bubble competition model to explain the hydrodynamic stability boundary observed in OMEGA implosion experiments.^40,41

The bubble competition regime in the multimode ARTI is further investigated using simulations for varying initial conditions. Figure 3 compares the dependence of α_b on h₀ from simulations with the ablative bubble competition model of Eq. (2). Figure 3 extends the results of Ref. 37 by including additional ARTI simulations with different perturbation spectra and also different simulation domains (L_x). As discussed in Ref. 37, the nonlinear CRTI transitions into the bubble merger regime with α_b saturating for sufficiently large initial perturbations. Note that the multimode CRTI with P0(20–40) transitions to the bubble merger at lower h₀ than in the P2(5–20) case and the saturated α_b is at its lower bound. This result indicates that large-amplitude small scale perturbations can reduce the CRTI nonlinear bubble-front penetration by facilitating the transition from the bubble competition to bubble merger regime.²⁷ Unlike the classical case, the nonlinear multimode ARTI remains in the bubble competition regime for all of the investigated initial conditions. This is because the ablation-generated vorticity can accelerate the bubble and keep the nonlinear ARTI in the bubble competition regime even for small scale perturbations. Therefore, the ARTI can reach higher α_b values than in the classical case when the initial perturbation is sufficiently large. Figure 3 also shows that α_b in all the multimode ARTI simulations is in very good agreement with the ablative bubble competition model [Eq. (2)] when the initial perturbation is small (k₀h₀ < 0.01), while α_b in some of simulations slightly deviates from Eq. (2) (within $≈±15%$⁠) when the initial perturbation is large (k₀h₀ > 0.01). This is because vorticity can affect α_b by enhancing the nonlinear bubble velocity, which becomes significant for large initial perturbation. Vorticity generation and its effect on enhancing the multimode ARTI bubble-front growth are investigated in detail in Sec. III.

In order to further investigate the role of vorticity in the nonlinear multimode ARTI, Fig. 4 compares the wavelength λ_b and vorticity ω_o inside the tip of the dominant fastest-growing bubbles between cases with initial spectra P2(5–20) and P0(20–40) with varying perturbation amplitudes. Both P2(5–20) cases in Fig. 4 are dominated by large scale modes near m = 5 (λ = 20 μm), which are linearly unstable [see Fig. 1(b)]. We have found in Fig. 3 that the P2(5–20) cases have α_b = 0.043 when k₀h₀ = 0.024 and α_b = 0.049 when k₀h₀ = 0.054. On the other hand, cases P0(20–40) are dominated by small scale modes (λ < 5 μm), which are linearly stable [Fig. 1(b)]. Figure 3 shows α_b = 0.028 when k₀h₀ = 0.002 and α_b = 0.058 when k₀h₀ = 0.035 for the P0(20–40) cases. Figure 4(a) shows how λ_b scales linearly with h_b in the deep nonlinear phase (h_b > 10 μm) for all the simulations, consistent with the self-similar evolution of the multimode bubbles. In the P2(5–20) cases, the fastest-growing bubble wavelength is initially λ_b ≈ 12 μm and grows to λ_b ≈ 20 μm. In contrast, for P0(20–40), λ_b grows from ≈3 μm to ≈14 μm, indicating that the bubbles remain generally smaller than those in P2(5–20). The simulation results indicate that the relation between λ_b and h_b is more sensitive to the perturbation spectrum than the initial amplitude.

Vorticity generation of the multimode simulation is shown in Fig. 4(b). The vorticity in P2(5–20) is ≈5 ns⁻¹ at the beginning and exhibits a negligible increase in the deep nonlinear stage even when the initial perturbation amplitude is large. The vorticity evolution seems insensitive to the initial perturbation amplitude in the P2(5–20) cases. On the other hand, the overall vorticity in P0(20–40) is higher than that in P2(5–20) and increases with h_b in the deep nonlinear phase. The nonlinear vorticity in P0(20–40) also increases with initial perturbation amplitude. For k₀h₀ = 0.035, ω_o can reach about 3 times the value in P2(5–20) despite P2(5–20) having a larger initial perturbation. In Fig. 4(b), we compare the vorticity evolution of multimode ARTI with that of single mode. We find that the vorticity in P2(5–20) is close to that of the λ = 20 μm single mode ARTI, while vorticity in the P0(20–40) and k₀h₀ = 0.035 case reaches similar levels to that of λ = 5 μm single mode in the deep nonlinear phase. It was found^15,16 that in single mode ARTI, vorticity acceleration of the nonlinear bubble is quite significant for the small scale modes [see also Fig. 6(a)]. The nonlinear bubble velocity can be enhanced by more than 50% for the λ < 5 μm modes. These linearly stable small scale modes can be even nonlinearly destabilized by the vorticity effect induced by finite amplitude perturbations.³⁷ However, vorticity seems to have a negligible effect on large scale bubbles with wavelength around λ = 20 μm. This comparison can explain the observation that α_b is larger in P0(20–40) than in P2(5–20) for large initial perturbation amplitudes, while at small initial perturbation amplitudes, both spectra agree closely with the bubble competition model. We expect that the vorticity effect on enhancing small scale multimode bubble-front growth should be more significant in 3D than in 2D due to higher vorticity levels in 3D.^16,42

The vorticity mode structure is further compared between P2(5–20) and P0(20–40) for the large initial perturbation cases (Fig. 5). In the early nonlinear phase, the size of the dominant bubbles is around 15 μm in P2(5–20). The bubble region has a larger size compared to the spike region similar to the spike structure in single mode simulations.^15,17 The size of the dominant bubbles is much smaller in P0(20–40) compared to that in P2(5–20). These small scale bubbles in P0(20–40) are destabilized by large initial perturbation amplitude. There also exist some nearly unperturbed regions in P0(20–40), where the initial perturbation amplitude is too small to destabilize the small scale ARTI modes. In the deep nonlinear phase, the size of the fast-growing bubbles increases as the bubbles penetrate into the shell. The spikes in P2(5–20) become narrower during the bubble competition phase and the vorticity inside the bubbles increases slightly [Fig. 5(b)]. In P0(20–40), the nearly unperturbed regions further develop and turn into spikes as the bubbles grow. The resulting spikes are wider than those in P2(5–20). It is also shown that the bubble vorticity in P0(20–40) increases significantly in the deep nonlinear phase and is more intense than in P2(5–20). As the bubbles penetrate into the shell, vorticity generation can be further enhanced when the tips of spikes approach each other, resulting in the enhancement of convection into the bubbles.

In Ref. 37, the dependence of α_b on initial perturbation h₀ and ablation velocity V_a [Eq. (2)] was derived using the ablative bubble competition model and the classical terminal bubble velocity. However, vorticity acceleration of the nonlinear bubble was not taken into account. Here, we generalize the model to assess the vorticity effect on enhancing the multimode bubble-front growth using the ARTI nonlinear bubble velocity $Ubrot$(see Table I). We follow the derivation in Ref. 37, starting with Takabe's formula: $γabl≈gk−bkVa=γcl(1−bV̂a)$¹ for the ARTI linear dispersion relation. Here, $V̂a=k/gVa$ is the non-dimensional ablation velocity and the coefficient b ≈ 4 in the regime of interest. $γcl=gk$ is the linear growth rate of CRTI in the A_T ≈ 1 limit. Since the linear stage develops over a relatively short time, we assume a temporally constant acceleration g for simplicity. The bubble penetration distance is $hb=h0 exp (γablt)+Vat$ in the linear stage. We further assume that the ARTI reaches the nonlinear terminal bubble velocity $Ubrot$ instead of the classical terminal bubble velocity $Ubcl$ when it develops into the nonlinear phase at t_NL. The time t_NL of this linear-to-nonlinear transition occurs when $∂hb/∂t=Ubrot$⁠. Therefore, the nonlinear occurrence time is

The associated bubble penetration distance is

Here, $M=ln (Ubrot−Vah0γabl)−1$⁠. At t > t_NL, the nonlinear bubble penetration of the fastest growing bubble satisfies

Since the evolution of the nonlinear ARTI bubble-front is self-similar, it satisfies the condition $∂hb/∂k=0$⁠. Therefore, from Eq. (5) we find that

Here, we assumed $∂M/∂k=0$ and $∂rd/∂k=0$ since M and r_d depend weakly on k. We also assumed $Ubrot≫Va$ since it is usually satisfied in the regime of interest. Substituting Eq. (6) into Eq. (5) with $γabl=gk(1−bV̂A)$⁠, we find

with

Here $αbabl$ is given by Eq. (2). This result indicates that vorticity modifies the multimode ARTI bubble-front growth through the coefficient β^rot mainly determined by $Ubrot/Ubcl$⁠. In the $Ubrot=Ubcl$ limit, $αbrot$ returns to the ablative bubble competition model without the vorticity effect [Eq. (2)].

The terminal bubble velocity of the single mode is further compared between the ablative case and the classical case for both 2D and 3D [Fig. 6(a)]. It is shown that $Ubrot$ approaches $Ubcl$ for large scale (small k) modes, as shown in previous studies.^16,17 The vorticity effect on enhancing the nonlinear bubble velocity increases as the mode scale decreases (k increases). $Ubrot$ can reach about $2.5Ubcl$ for sufficiently small scale modes. Considering that $1<Ubrot/Ubcl<2.5$ is usually true for both 2D and 3D ARTI, it is expected that the vorticity effect can enhance the multimode bubble-front penetration (α_b) by up to 30% compared to the case without vorticity [Fig. 6(b)]. It should be noted that Eqs. (2) and (8) express the same dependence of α_b on h₀ and V_a. Therefore, both expressions account for the role of mass ablation in suppressing bubble penetration speed. The main difference is that unlike Eq. (2), Eq. (8) accounts for an additional effect of ablation-driven vorticity on enhancing the bubble penetration via the coefficient $βrot ≳1$⁠. However, using Eq. (8) to fit the simulation results with different perturbation spectra is not straightforward. The vorticity effect is sensitive to perturbation spectra at large initial perturbation (Fig. 4) and the dominant wavenumber is time-dependent, rendering the quantification of β^rot complicated. Considering that the variation of β^rot ≈ 1 in regimes of interest [Fig. 6(b)], Eq. (2) is much easier to use for quantifying our ARTI simulation results. Indeed, it is shown in Fig. 3 that although our ARTI simulations include the additional vorticity effect, our results can be well fitted by Eq. (2) within ±15%. As the nonlinear bubbles penetrate into the shell, the small scale bubbles will eventually develop into large scale bubbles. In this regime, the vorticity effect on enhancing the multimode bubble-front growth may become less effective.

In this section, the ablative bubble competition model [Eq. (2)] is applied to explain the hydrodynamic stability boundary observed in laser direct-drive implosion experiments on the OMEGA laser. In ICF, the hydrodynamic stability with respect to short wavelength modes (λ ≪ R, where R is the capsule radius) is the inflight aspect ratio, IFAR, defined as IFAR = R_2/3/Δ_2/3 in 1D implosions. Here, R_2/3 and Δ_2/3 are the in-flight capsule inner radius and shell thickness at 2/3 of the initial inner radius R₀. Here, the word “initial” means at the beginning of the acceleration phase. To prevent performance degradation by hydrodynamic instabilities, an upper bound in the IFAR, i.e., IFAR_max, is set to ensure that the RTI bubbles do not penetrate through the shell. A maximum IFAR (IFAR_max) is often used to represent the hydrodynamic stability boundary in ICF implosions. In OMEGA experiments, the experimentally inferred hydrodynamic stability boundary for the low-adiabat (α_F < 3.5, α_F is the ratio of the pressure to the Fermi pressure) cryogenic implosions is approximated by IFAR_max ≈ 20(α_F/3)^1.1.⁴¹ The degradation of implosion performance is mainly attributed to high-mode perturbations caused by laser imprint and mass modulations in the target.⁴⁰ Since α_F scales as V_a^5/3,⁴³ it is found that α_F ≈ 3.4(V_a/3.5 μm/ns)^5/3 for OMEGA's cryogenic implosions. Therefore, we have IFAR_max ≈ 23(V_a/3.5 μm/ns)^1.83. Since the initial perturbation h₀ and ablation velocity V_a can affect the nonlinear bubble-front penetration speed α_b through the ablative bubble competition model [Eq. (2)], they also affect the magnitude of IFAR_max.

Assuming that IFAR_max is mainly determined by the ARTI growth during the acceleration phase with an approximately constant acceleration g, the dependence of IFAR_max on h₀ and V_a for the typical OMEGA low-adiabat cryogenic implosion can be estimated. Since the density profile in the shell can be approximated by ρ(r) = ρ_a[(2Δ−r)/2Δ]^3/2 [see Fig. 1(a)], then the effective shell mass is determined by R and Δ,

in the thin shell limit. The effective shell mass M can be evaluated by

where the right-hand-side represents the rate of mass degradation from the bubble-front penetration. The shell inner radius is evaluated by

During the acceleration phase, the linear and nonlinear multimode bubble-front penetration h_b through the shell is evaluated by the analytical bubble competition model (Sec. II), i.e.,

α_b is determined by h₀, g, and V_a through Eq. (2) in 3D with the coefficient determined by the 3D ARTI simulations. In this case, the time dependence of Δ and R can be numerically evaluated by Eqs. (10)–(12) for the given R₀, h₀, V_a, and g. The minimum initial inflight shell thickness Δ_0min to avoid hydrodynamic instabilities is found by enforcing that Δ = 0 at the end of the acceleration phase. Then, Δ_2/3 due to mass ablation in 1D implosions is calculated by $dM/dt=−4πR2ρaVa$ and Eq. (11) for the given R₀ and Δ_0min. The maximum limit of IFAR_2/3 = R_2/3/Δ_2/3, i.e., IFAR_max, can then be obtained.

For typical OMEGA cryogenic implosions, the shell is accelerated from R₀ = 400 μm to 1/3R₀ with g = 200 μm/ns². The dependence of IFAR_max on h₀ and V_a for the OMEGA implosion parameters is assessed in Fig. 7 using the analytical bubble competition model. It is shown that the implosions with lower adiabat require smaller IFAR_max for the same initial perturbation level because of the weaker ablation stabilization (α_F ∼ V_a^5/3). For the same adiabat α_F, IFAR_max decreases as h₀ increases since α_b increases logarithmically with h₀. IFAR_max is also estimated for the CRTI by h_b = α_b^clgt² with the lower bubble merger limit α_b^cl = 0.03 for reference (dashed line in Fig. 7). Due to the ablation-generated vorticity that prevents the transition of the ARTI from the bubble competition to bubble merger region, IFAR_max in the ICF implosions can be smaller than the estimation from the CRTI theory for sufficiently large initial perturbations. The comparison between the analytical results and the experimental stability boundary IFAR_max ≈ 20(α_F/3)^1.1 indicates that the OMEGA low-adiabat implosions have a similar level of initial perturbation amplitude h₀ ≈ 0.007 μm. The inferred α_b ≈ [0.039, 0.043] for 2.5 < α_F < 3.5 also agrees with the nonlinear ARTI experiments on OMEGA³³ and recent direct-drive ARTI experiments with all the laser smooth technics.³⁸ This result shows that the stability boundary observed in OMEGA low-adiabat implosions can be self-consistently explained by the multimode ARTI bubble competition during the acceleration phase of the implosion. This explanation is supported by recent experiments on OMEGA, which shows that significant material mixing of the ablator into the fuel is observed only when the implosion design is beyond the stability boundary. Multidimensional simulations indicate that the mix is caused by laser imprint induced hydrodynamic instability.⁴⁴ 

If the initial perturbation is significant so that the ARTI is initially in the nonlinear phase, the bubble-front penetration is $hb=αbgt2$ with α_b determined by Eq. (2). Then, the evolution of the effective shell mass is

Since gt = v = −dR/dt, we find $0.8d(R2Δ)/dt=2αbR2dR/dt$⁠. Therefore, mass conservation implies that $0.8R2Δ−2αbR3/3$ = constant. Considering that the shell accelerates from R₀ to R_ca during the acceleration phase and the initial shell thickness Δ₀ becomes Δ_ca = 0 after the acceleration (hydrodynamic stability boundary), we find

with $Crca=R0/Rca$ being the convergence ratio during the acceleration phase. Since $Crca ∼ 3−4$ for typical ICF implosion, we find $R0/Δ0∼1.2/αb$ if the ARTI is initially in the nonlinear phase.

In summary, the nonlinear evolution of the multimode ARTI and the ablation-generated vorticity effect are investigated numerically and analytically. It is found that the nonlinear ARTI is dominated by bubble competition with α_b dependent on initial perturbation and ablation velocity. The linear and nonlinear evolution of the ARTI bubble-front penetration can be quantified by an analytical bubble competition model. The significant vorticity generation is observed when the initial perturbation is large and on a small scale. Besides showing that the vorticity effect can keep the small scale ARTI in the bubble competition regime, it is also assessed that the vorticity effect can enhance α_b by up to 30% relative to the predictions of the bubble competition model without the vorticity effect, consistent with the α_b variation at large initial perturbation for different perturbation spectra. Due to the dependence of α_b on the initial perturbation amplitude and vorticity effect, ablative stabilization of the nonlinear ARTI in ICF may not be as effective as expected if the initial perturbation is significant. The ablative bubble competition theory is applied to explain the hydrodynamic stability boundary observed in OMEGA low-adiabat implosion experiments. The inferred initial perturbation level h₀ is around 0.007 μm, and the inferred α_b is consistent with the observation in direct-drive multimode ARTI experiments. We conclude that capsule surface roughness and laser imprint should be further reduced in order to improve the performance of low-adiabat implosions.

This research was funded by U.S. DOE FES Grant Nos. DE-SC0014318 and DE-SC0020229. H.A. was also supported by U.S. NASA Grant No. 80NSSC18K0772, U.S. NNSA Award Nos. DE-NA0003856 and DE-NA0003914, and U.S. DOE Grant No. DE-SC0019329. H. Zhang was also supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11975056 and Science Challenge Project (SCP) No. TZ2016005. R. Yan was also supported by the Strategic Priority Research Program of Chinese Academy of Sciences No. XDA25050400, NSFC Nos. 11772324, 11621202, and SCP No. TZ2016001. Computing time was provided by the National Energy Research Scientific Computing Center (NERSC) under Contract No. DE-AC02-05CH11231. This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.

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