The self-similar nonlinear evolution of the multimode ablative Rayleigh–Taylor instability (RTI) and the ablation-generated vorticity effect are studied for a range of initial conditions. We show that, unlike classical RTI, the nonlinear multimode bubble-front evolution remains in the bubble competition regime due to ablation-generated vorticity, which accelerates the bubbles, thereby preventing a transition into the bubble-merger regime. We develop an analytical bubble competition model to describe the linear and nonlinear stages of ablative RTI. We show that vorticity inside the multimode bubbles is most significant at small scales with large initial perturbation. Since these small scales persist in the bubble competition regime, the self-similar growth coefficient α_{b} can be enhanced by up to 30% relative to ablative bubble competition without vorticity effects. We use the ablative bubble competition model to explain the hydrodynamic stability boundary observed in OMEGA low-adiabat implosion experiments.

## I. INTRODUCTION

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 inertia^{5} 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.

. | CRTI . | ARTI . |
---|---|---|

Linear dispersion relation | $\gamma cl=ATkg$^{1} | $\gamma abl=ATkg\u2212bATkVa$^{10} |

Nonlinear bubble velocity | $Ubcl=g(1\u2212rd)/Cgk$^{18,19} | $Ubrot=g(1\u2212rd)/Cgk+rd\omega 02/4k2$^{15} |

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_{b}A_{T}S.^{20–27} Here, $S=(\u222bgdt)2$ 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 h_{0} and can be quantified by^{23,26,27}

The coefficients C and k_{0} 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 shown^{37} 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 h_{0} and ablation velocity V_{a} can be evaluated through the ablative bubble competition theory,

using the coefficients (C = 0.6, b = 4.2, and k_{0} = 0.06 *μ*m^{−1}) for 2D and (C = 0.95, b = 4.2, and k_{0} = 0.06 *μ*m^{−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 h_{0}, 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.

## II. THE ARTI BUBBLE COMPETITION FOR DIFFERENT INITIAL PERTURBATION SPECTRA

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 *μ*m, which is defined as the distance between the inner and outer boundaries where the shell density equals 1/e of *ρ _{a}* (

*ρ*= 5.3 g/cm

_{a}^{3}is the maximum density near the ablation front Z

_{0}≈ 60

*μ*m.). The target is uniformly ablated from the left-hand-side with ablation velocity V

_{a}= 3.5

*μ*m/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−Z

_{0}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 g

_{0}= 100

*μ*m/ns

^{2}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

*μ*m

^{−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(m

_{1}-m

_{2}) is used to represent the power spectrum of the initial perturbation in mode number space. Here, m

_{1}-m

_{2}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

*μ*m in the Z direction. The grid size is 0.1

*μ*m 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}= 100

*μ*m. The normalized root mean square (RMS) velocity perturbation $Vrms=(\u2211i=1,NVp2(x)/N)1/2/Va$ 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 $hb=\alpha bS$ with $S=(\u222bgtdt)2$.^{24} For each power spectrum Ps(m_{1}-m_{2}), 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(m_{1}-m_{2}), 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_{b}A_{T}gt^{2} scaling law in the nonlinear phase and α_{b} increases with the initial perturbation amplitude. The dependence of α_{b} on initial perturbation amplitude h_{0} 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 $hb=h0\u2009exp\u2009(\gamma maxablt)+Vat$. The linear growth rate $\gamma maxabl$ 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. $\gamma maxabl$ 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_{b}gt^{2}) with α_{b} determined by h_{0}, g, and V_{a} through Eq. (2). Therefore, the time evolution of h_{b} can be determined by h_{0}, 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 h_{0} 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 h_{0} 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 h_{0} 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 (k_{0}h_{0} < 0.01), while α_{b} in some of simulations slightly deviates from Eq. (2) (within $\u2248\xb115%$) when the initial perturbation is large (k_{0}h_{0} > 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.

## III. VORTICITY EFFECT ON THE MULTIMODE BUBBLE-FRONT GROWTH

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 *μ*m), 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 k_{0}h_{0} = 0.024 and α_{b} = 0.049 when k_{0}h_{0} = 0.054. On the other hand, cases P0(20–40) are dominated by small scale modes (λ < 5 *μ*m), which are linearly stable [Fig. 1(b)]. Figure 3 shows α_{b} = 0.028 when k_{0}h_{0} = 0.002 and α_{b} = 0.058 when k_{0}h_{0} = 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 *μ*m) 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 *μ*m and grows to λ_{b} ≈ 20 *μ*m. In contrast, for P0(20–40), λ_{b} grows from ≈3 *μ*m to ≈14 *μ*m, 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 k_{0}h_{0} = 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 *μ*m single mode ARTI, while vorticity in the P0(20–40) and k_{0}h_{0} = 0.035 case reaches similar levels to that of λ = 5 *μ*m 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 *μ*m 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 *μ*m. 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 *μ*m 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 h_{0} 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: $\gamma abl\u2248gk\u2212bkVa=\gamma cl(1\u2212bV\u0302a)$^{1} for the ARTI linear dispersion relation. Here, $V\u0302a=k/gVa$ is the non-dimensional ablation velocity and the coefficient b ≈ 4 in the regime of interest. $\gamma cl=gk$ 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 $hb=h0\u2009exp\u2009(\gamma ablt)+Vat$ 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 $\u2202hb/\u2202t=Ubrot$. Therefore, the nonlinear occurrence time is

The associated bubble penetration distance is

Here, $M=ln\u2009(Ubrot\u2212Vah0\gamma abl)\u22121$. 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 $\u2202hb/\u2202k=0$. Therefore, from Eq. (5) we find that

Here, we assumed $\u2202M/\u2202k=0$ and $\u2202rd/\u2202k=0$ since M and r_{d} depend weakly on k. We also assumed $Ubrot\u226bVa$ since it is usually satisfied in the regime of interest. Substituting Eq. (6) into Eq. (5) with $\gamma abl=gk(1\u2212bV\u0302A)$, we find

with

Here $\alpha babl$ 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, $\alpha brot$ 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 h_{0} 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 $\beta rot\u2009\u22731$. 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.

## IV. APPLICATION OF THE BUBBLE COMPETITION MODEL TO ICF HYDRODYNAMIC STABILITY BOUNDARIES

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 R_{0}. 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 *μ*m/ns)^{5/3} for OMEGA's cryogenic implosions. Therefore, we have IFAR_{max} ≈ 23(V_{a}/3.5 *μ*m/ns)^{1.83}. Since the initial perturbation h_{0} 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 h_{0} 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 h_{0}, 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 R_{0}, h_{0}, 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 $dM/dt=\u22124\pi R2\rho aVa$ and Eq. (11) for the given R_{0} 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 R_{0} = 400 *μ*m to 1/3R_{0} with g = 200 *μ*m/ns^{2}. The dependence of IFAR_{max} on h_{0} 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 h_{0} increases since α_{b} increases logarithmically with h_{0}. IFAR_{max} is also estimated for the CRTI by h_{b} = α_{b}^{cl}gt^{2} 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 h_{0} ≈ 0.007 *μ*m. The inferred α_{b} ≈ [0.039, 0.043] for 2.5 < α_{F} < 3.5 also agrees with the nonlinear ARTI experiments on OMEGA^{33} 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 $hb=\alpha bgt2$ with α_{b} determined by Eq. (2). Then, the evolution of the effective shell mass is

Since gt = v = −dR/dt, we find $0.8d(R2\Delta )/dt=2\alpha bR2dR/dt$. Therefore, mass conservation implies that $0.8R2\Delta \u22122\alpha bR3/3$ = *constant*. Considering that the shell accelerates from R_{0} to R_{ca} during the acceleration phase and the initial shell thickness Δ_{0} 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 $Crca\u2009\u223c\u20093\u22124$ for typical ICF implosion, we find $R0/\Delta 0\u223c1.2/\alpha b$ if the ARTI is initially in the nonlinear phase.

## V. SUMMARY

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 h_{0} is around 0.007 *μ*m, 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.

## ACKNOWLEDGMENTS

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.

## DATA AVAILABILITY

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