The Rayleigh–Taylor (RT) instability has been a great challenge for robust fusion ignition. In this paper, the evolution of the RT instability at the fuel inner interface during the coasting phase is investigated for the central ignition scheme [Hurricane *et al.*, Rev Mod Phys. **95**, 025005 (2023)] and the double-cone ignition (DCI) scheme [Zhang *et al.*, Philos. Trans. R. Soc. A. **378**, 20200015 (2020)]. It is found that the spherical convergent effect can be helpful for smoothing the disturbance by merging the spikes in the azimuthal direction. For the DCI scheme, the pressure gradient in the same direction with the density gradient at the fuel inner interface can further prevent the disturbance from growing. For the example case with an initial disturbance amplitude as large as 20 *μ*m, the DCI scheme can still reach a high-density isochoric plasma with an areal density of 2.18 g/cm^{2} at the stagnation moment, providing favorable conditions for fast ignition by the relativistic electron beam.

## I. INTRODUCTION

Inertial confinement fusion has been regarded as a possible way to provide a sustainable energy since the 1970s.^{1} The milestone of fusion ignition through the central ignition scheme has been reached for the first time on the National Ignition Facility (NIF) in 2022.^{2} However, the Rayleigh–Taylor (RT) instability at the inner interface of the fuel during the coasting phase is still one of the major challenges for a robust fusion ignition and high-gain burn. During the laser-driven acceleration phase, the RT instability will arise on the ablation front,^{3–5} which amplifies small disturbances caused by target fabrication defects or uneven laser irradiation.^{6} These disturbances propagate to the fuel inner interface^{7} and cause the interface to deviate from the perfect spherical shape.^{8} During the stagnation phase in the central ignition scheme, the instability on the inner interface of the deuterium and tritium (DT) fuel may grow rapidly because of the same direction of acceleration and the density gradient at the inner interface of the fuel. This may lead to a reduction in the fuel areal density^{9,10} hotspot temperature,^{11,12} and even cause material mixing,^{13–15} thus degrading the fusion ignition and energy gain. To achieve a high-gain fusion, the capsule is usually expected to have a high-quality DT ice layer with a roughness of $ \sigma rms\u2264$ 1 *μ*m at the interface.^{16}

As an alternative, the double-cone ignition (DCI) scheme^{17–22} has a potential to achieve high-gain fusion with relatively low drive energy. The whole process of the DCI scheme can be divided into four processes: compression, acceleration, collision, and fast heating. The feasibility of the above four processes has been validated experimentally in the last 3 yr in eight experimental campaigns. For example, the plasma jets at the cone tips were accelerated to over 200 km/s and compressed to 28 g/cm^{3} by a laser pulse optimized by the machine learning method.^{18} The temperature of the colliding plasma was increased from 50 eV to more than 300 eV during the collision of the supersonic imploding plasmas ejected from the cone tips.^{19,20} The heating effect by the relativistic electron beam was also observed using a PW laser beam of 500 J in 10 ps.^{20}

The eight experimental campaigns focus on different physical issues of the DCI scheme on the Shenguang-II upgrade laser facility. Some shots were initially aimed at the evolution of the RT instability, while the development of the RT instability during the coasting phase seemed to be negligible.^{19,20} These measurements triggered the study of this manuscript to understand the smoothing effect of spherical convergence on the development of the RT instability for the DCI scheme, due to the unique configuration of the DCI scheme in the coasting phase. Comparing the spherical convergent effect on the development of the RT instability of the DCI scheme and the central ignition scheme would be important for better understanding the evolution of the RT instability.

In this paper, we investigate the evolution and effects of the RT instability at the fuel inner interface during the coasting phase for both the DCI and the central ignition scheme. This paper consists of five sections. Section II presents the physical model of the MULTI-2D program, which is employed to simulate the evolution of the RT instability. Section III illustrates the development and effects of the RT instability during the coasting phase. Section IV analyzes the smoothing or growing mechanisms in the evolution of the RT instability. Finally, conclusions and discussions are given in Sec. V.

## II. PHYSICAL MODEL

^{23,24}Z-pinch fusion,

^{25,26}and heavy ion fusion.

^{27}The latest version can solve multi-layer, multi-material, and multi-physics problems on two-dimensional cylindrical geometry unstructured grids and has a second-order arbitrary Lagrange–Eulerian (ALE) module. The role of the ALE module is to remap physical quantities from distorted grids to better grids so that simulations will not be blocked by grid distortions. The ALE module is essential for calculating implosions that involve the development of hydrodynamic instabilities. The tabulated equation of state and opacity are generated by the program MPQEOS

^{28}and SNOP,

^{29}respectively. The radiation magnetohydrodynamic equations used by the MULTI-2D program are as follows:

Equations (1)–(5) are the mass conservation equation, the momentum conservation equation, the energy conservation equation, the magnetic field evolution equation, and the radiation transport equation, respectively. *ρ* is density, *m* is mass, *V* is volume, ** u** is velocity,

*P*is thermal pressure,

*P*is artificial viscosity pressure,

_{ν}**is current density,**

*J***is magnetic field,**

*B**e*is specific internal energy,

**is electron heat flow,**

*q*_{e}**is artificial heat flow,**

*q*_{v}*S*is the external source term (such as laser, x ray, and α particle deposition),

_{ext}*η*is electrical resistivity, and $ \mu 0$ is vacuum magnetic permeability.

**is the radiation intensity, which impacts the hydrodynamics by the radiation emission and absorption of the matter. The net radiation energy deposition enters Eq. (2) through the external source term $ S ext$.**

*I***is the unit vector in the direction of radiation propagation, $ I P$ is Planck intensity, and**

*n**λ*is radiation mean free path. More information about the physical model and mathematic algorithms of the MULTI-2D program can be found in Refs. 30 and 31.

## III. EVOLUTION OF THE RT INSTABILITY AT FUEL INNER INTERFACE

Figures 1(a) and 1(b) show the initial simulation conditions for the DCI and the central ignition scheme. According to axial-symmetric geometry (cylindrical R-Z coordinates), only 1/4 of the target is simulated. The simulation region has a size of 500 × 500 *μ*m^{2} and a grid of 200 × 200 meshes. The gold cone has an opening angle of 100 deg, a density of 19.2 g/cc, a temperature of 0.001 eV, and a thickness of 30 *μ*m. The simulations including laser–plasma interaction are conducted by the one-dimensional hydrodynamic code MULTI-IFE. The plasma parameters at the turn-off moment of the laser pulse in the one-dimensional simulation are taken as the initial condition of two-dimensional simulations. The initial peak density of DT fuel is 18 g/cc, the initial temperature is 10 eV, and the initial velocity is −340 km/s (point to the spherical center). The disturbance on the inner interface of fuel is set as *R*(*μ*m) = 20 cos(18*θ*) + 200. The initial perturbation amplitude of $\Delta r$ =20 *μ*m and the perturbation number of *n* =18 are selected to reflect the feed-in effect of the external interface perturbation^{7,8} and to illustrate clearly the evolution of RT instability within the simulation time. The region inside the disturbed interface is set as vacuum in the DCI case since the coronal plasma will be peeled off by the gold cone and will not reach this region. In the central ignition case, this region is filled with stationary gas with a density of 0.5 g/cc and a temperature of 2000 eV. The fusion reaction is turned off during the simulations to compare the evolution of the RT instability in similar conditions.

Figures 1(c) and 1(d) show the density distributions at *t* = 0.3 ns. At this time, the low-density plasma front in the DCI configuration has begun to collide and the disturbance on the inner interface of the fuel has almost vanished. The RT instability in the central ignition configuration is entering the nonlinear stage. The bubble structure generated by the low-density hotspot begins to invade the fuel shell, and the spike structure generated by high-density fuel begins to invade the low-density hotspot. Figures 1(e) and 1(f) show the density distributions at *t* = 0.65 ns. At this time, the high-density plasmas in the DCI scheme are colliding and forming an isochoric distribution near the spherical center. The DT fuel shell in the central ignition configuration reaches the stagnation moment when the main DT fuel stops radial movement at about *t* = 0.65 ns, which can also be seen in Fig. 2(b).

The ignition of the DCI scheme relies on injecting fast electrons along the heating direction (the horizontal direction in Fig. 1), which is perpendicular to the colliding direction of the plasma jets. Figure 2(a) shows the evolution of the density distribution in the heating direction and the density profile at the stagnation moment (*t* = 0.8 ns) for the DCI scheme. The density profile at the stagnation moment forms an isochoric distribution composed of a high-density platform and rapidly declining edge (*ρ _{max}* = 420 g/cc and

*ρR*= 2.18 g/cm

^{2}). Figure 2(b) presents the evolution of the density distribution in the horizontal direction and the density profile near the stagnation moment (

*t*= 0.65 ns) for the central ignition scheme. The density distribution is an isobaric distribution composed of a low-density hotspot (

*ρ*= 53 g/cc) and a high-density fuel shell (

*ρ*= 123 g/cc and

*ρR*= 0.99 g/cm

_{shell}^{2}). Due to the lack of central gas, the DT fuel in the DCI scheme can be compressed closer to the spherical center to obtain a higher fuel areal density. The fusion energy gain can be expressed as $ G fusion= q D T M D T E laser \rho R \rho R + 7,$

^{6}where $ q D T$ is the energy released by unit mass DT fuel, $ M D T$ is the DT fuel mass, $ E laser$ is the drive laser energy, and $\rho R$ is the fuel areal density. From the expression of fusion energy gain, a higher fuel areal density obtained by the DCI scheme is favorable to obtain a higher energy gain.

Figure 3(a) shows the effects of the RT instability evolution on the plasma temperature and areal density in the DCI scheme. Since the DCI scheme relies on fast electron heating of high-density plasma to achieve ignition, the temperature corresponding to the highest density plasma is chosen as the plasma temperature and the areal density in the fast ignition direction is chosen as the areal density. As can be seen from Fig. 3(a), since the interface perturbations are smoothed during the coasting phase, the RT instability has little effect on the evolution of plasma temperature and areal density for the DCI scheme.

Figure 3(b) shows the effects of the RT instability development on the plasma temperature and areal density in the central ignition scheme. Since the central ignition scheme relies on a hotspot to ignite, the temperature of the hotspot is chosen as the plasma temperature. Because the central ignition scheme needs to ensure the integrity of the main fuel shell, we pay more attention to areal density in the minimum areal density direction in this section. The first temperature peak in Fig. 3(b) represents the moment when the shock converges at the spherical center. This shock is generated by the impact of the imploding fuel onto the central gas. The second temperature peak corresponds to the moment when the hotspot starts to cool down. The hotspot temperature decreases when the sum of heat conduction power and radiation cooling power is larger than the power of mechanism work applied by the main fuel. The hotspot temperature with initial disturbance (*T* = 4720 eV) is lower than the case without RT instability seeds (*T* = 5160 eV). This is because the spike structure generated by the cold DT shell will invade into the hotspot and cause the hotspot temperature to decrease. Since the hotspot is cooler, the main fuel can be compressed closer to the spherical center and a higher peak areal density can be obtained. However, the RT instability still causes the areal density to be lower than the ideal case when the hotspot temperature is highest around *t* = 0.55 ns. Luckily, one can adopt different kinds of methods^{32–34} to mitigate the development of the RT instability in the central ignition scheme and still achieve successful fusion ignition.

## IV. ANALYSIS OF THE SMOOTHING MECHANISMS FOR THE RT INSTABILITY

In this section, we demonstrate that there are two mechanisms that make the disturbance at the fuel inner interface smooth during the coasting phase in the DCI scheme. First, the fuel inner interface in the DCI configuration is a fluid–vacuum interface. The pressure gradient direction aligns with the density gradient on the fluid–vacuum interface, so that it does not satisfy the RT instability growth condition and the initial disturbance will not grow. Second, the effect of spherical convergence causes the initial disturbance to be smoothed due to the merging of spikes.

In Fig. 4, the evolution of the RT instability in planar geometry is compared with the situation under the equivalent conditions corresponding to the DCI scheme and the central ignition scheme. The initial perturbation wavenumber and plasma velocity in the planar cases are equivalent to the initial wavenumber ( $ k planar\u223c k spherical= l R$) and imploding velocity in the spherical cases as shown in Fig. 1.

Figures 4(a) and 4(b) suggest that the disturbance on the plasma interface will neither grow nor decay in planar geometry, when the pressure gradient and density gradient at the interface are in the same direction. Figures 4(c) and 4(d) present the RT instability evolution in the planar geometry under the equivalent conditions of the central ignition case shown in Fig. 1. Since the directions of pressure gradient and density gradient at the interface are opposite, the initial disturbance amplitude of the interface increases continuously. The growth of disturbance amplitude during the linear phase satisfies the exponential form $A= A 0 e a k / ( 1 + k L ) t,$^{6} where $ A 0$ is the initial disturbance amplitude, $a$ is acceleration, *k* is the disturbance wavenumber, *L* is the characteristic length of density affected by diffusion effect,^{35} and *t* is evolution time. After the development of the RT instability enters the nonlinear phase [Fig. 4(d)], the shape of disturbance changes from the initial sinusoidal distribution to a bubble–spike structure. In addition, a faster propagating shock can also be observed in front of the disturbed interface, which increases the gas temperature from initial 2000 to 3200 eV.

Figure 5(a) shows the distribution of plasma density with the disturbed interface in the DCI scheme at *t* = 0.2 ns. Within the green dashed line, the density disturbance has been smoothed. Thus, it can be concluded that the spherical convergent effect makes the spikes merge each other and cause the disturbance to be smoothed. Figure 5(b) shows the plasma density distribution with the spherical convergent effect in the central ignition scheme at *t* = 0.2 ns. Since the pressure of the central gas is higher than that of the main fuel, the high-pressure central gas not only causes the interface disturbance to grow radially, but also prevents the spikes from merging with each other in the angular direction.

We also studied the development of the RT instability with different initial wavelengths. For short-wavelength disturbances, the disturbed interface is easily smoothed due to the combined effects of spherical convergence, heat conduction, and x-ray radiation transport. For medium-wavelength disturbance modes, the spikes are far away in the angular direction and need stronger spherical convergence to merge each other. In the DCI scheme, the impact of the RT instability on the stagnated plasma is still very limited since the disturbance is smoothed by spherical convergence. In the central ignition scheme, the spherical convergent effect still makes the spikes tend to merge, but the high-pressure central gas makes it difficult for the spikes to merge. This causes a large modulation in the areal density of the main fuel in the angular direction.

Figure 6(a) shows the impacts of the RT instabilities with different initial disturbance wavelengths on the angular distribution of fuel areal density at the stagnation moment for the DCI scheme and the central ignition scheme. As can be seen, when the initial disturbance wavelength increases from 70 *μ*m (*n* = 18) to 209 *μ*m (*n* = 6), the fuel areal density in the DCI scheme is almost unaffected, while the areal density modulation for the central ignition scheme increases from 14% to 20%. The areal density modulation is calculating by dividing the range of areal density by its average value. Figure 6(b) shows the fuel density distribution near the stagnation moment for the central ignition scheme when the initial disturbance wavelength is 209 *μ*m (*n* = 6), where the black line represents the contour line with density of 100 g/cc. It can be observed that the spikes generated by the RT instability have not yet merged because of the high-pressure central gas. The density at the spike front is relatively lower (∼70 g/cc) than the peak of the main fuel (>100 g/cc) due to the electron heat conduction and x-ray radiation transport.

^{36,37}to semi-quantitatively explain the evolution of interface disturbance under the DCI conditions.

The above BP equation is obtained by solving Laplace's equation of velocity potential for incompressible and irrotational fluid in spherical coordinates, where $ a l m$ is the amplitude of the spherical harmonic function $ Y l m$ with mode *l* and *m*, $ a \u0307 l m$ is the growth rate of corresponding disturbance amplitude, *R* is the average radius of the interface, $ R \u0307$ is the imploding velocity, $ R \xa8$ is the imploding acceleration, and $ \rho 1$ and $ \rho 2$ are densities of outer and inner fluids, respectively. The interface equation can be expanded in spherical harmonics as $F r , \theta , \phi , t=r\u2212R t\u2212 \u2211 a l m Y l m \theta , \phi =0$, where *R*(*t*) is average radial position of interface.

*l*and

*m*, $ a \u0307 l m , 0$ is the initial growth rate of corresponding disturbance amplitude. Equation (7) shows that the evolution of perturbation amplitude $ a l m$ is completely determined by the initial condition of $ a \u0307 l m , 0$ during the coasting phase. If the initial growth rate $ a \u0307 l m , 0<0$, the disturbance amplitude $ a l m$ will keep decaying. If the initial growth rate $ a \u0307 l m , 0>0$, the disturbance amplitude $ a l m$ will keep growing.

Since the interface speed at given density includes the implosion speed of fuel, the growth speed of disturbance amplitude, and the diffusion speed of plasma into vacuum, the growth rate $ a \u0307 l m , 0$ of disturbance amplitude is not easy to be determined directly. During the data processing, we can treat it as an undetermined coefficient and obtain the theoretical curve by fitting the data with the least squares method. Since the disturbed interface is *R*(*μ*m) = 20 cos(18*θ*) + 200 and simulation is cylindrically symmetric, *m* is equal to 0. The perturbed interface can be expanded by the spherical harmonic functions of *l* $\u2264$ 18, and *m* = 0.

Figure 7 shows the amplitude evolution of *l* = 18 and *m* = 0 disturbance mode on different density interfaces and the average position of interfaces. It can be found that the implosion speed of fuel during the coasting phase under the DCI configuration is indeed almost constant ( $ R \xa8=0)$. When the fuel approaches the spherical center, the compression effect becomes significant and the theory described by Eq. (7) becomes invalid. In Fig. 7(a), the interface of *ρ* = 0.5 g/cc approaches the spherical center earlier and the simulation results fit well with the theoretical curve until *t* = 0.16 ns. While in Fig. 7(b), the interface of 1 g/cc approaches the collision center a little later and fits well with the theoretical curve within 0.2 ns. Therefore, when the compression effect is not obvious in the early stage of the coasting phase, the BP equation is valid for explaining the smoothing of the RT instability in the DCI scheme.

## V. SUMMARY AND DISCUSSIONS

In this work, we have simulated the development of the RT instability at the fuel inner interface during the coasting phase in the DCI scheme and the central ignition scheme. It is found that the spherical convergent effect helps to smooth the interface disturbance by merging the spikes in the DCI scheme, while the high-pressure central gas in the central ignition scheme makes it difficult for the spikes to merge. A high-density isochoric distribution plasma with a density of 420 g/cc and an areal density of 2.18 g/cm^{2} can be achieved in the DCI scheme. Compared to the central ignition scheme, the plasma parameters at the stagnation moment in the DCI scheme (such as areal density and fuel temperature) are much less sensitive to the initial disturbance during the coasting phase. Additionally, the pressure gradient in the same direction with the density gradient in the DCI scheme prevents the disturbance from growing and the disturbance can be smoothed more effectively by the spherical convergence. The analysis based on the Bell–Plesset equation also indicates that the disturbance at the fuel inner interface in the DCI scheme will keep decaying during the coasting phase.

It should be noted that the initial disturbance on the fuel inner interface during the coasting phase caused by the RT instability at the outer surface of the ablator during the acceleration phase has not been considered in this paper. This is appropriate for the case when the DT fuel shell is relatively thick. In the future, we will study the laser-driven implosions with the consideration of the RT instability on both the ablator outer surface and the fuel inner interface, to provide a more accurate reference for the designs of the future laser fusion experiments.

## ACKNOWLEDGMENTS

This work was supported by the Strategic Priority Research Program of Chinese Academy of Sciences (Nos. XDA25010100 and XDA25051200), the National Natural Science Foundation of China (Nos. 12205185 and 12105173), Shanghai Municipal Science and Technology Key Project (No. 22JC1401500), and the Shanghai Pujiang Program (No. 22PJ1407900).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

Y. Y. Lei and F. Y. Wu contributed equally to this work.

**Yang Yi Lei:** Conceptualization (equal); Investigation (equal); Visualization (equal); Writing – original draft (equal). **Fuyuan Wu:** Conceptualization (equal); Investigation (equal); Software (equal); Writing – review & editing (equal). **Rafael Ramis:** Investigation (equal); Software (equal); Writing – review & editing (equal). **Jie Zhang:** Conceptualization (equal); Investigation (equal); Project administration (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

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

## REFERENCES

*The Physics of Inertial Fusion: Beam Plasma Interaction, Hydrodynamics, Hot Dense Matter*

_{2}implosions on OMEGA laser system