Spike deceleration and bubble acceleration in the ...

PHYSICS OF PLASMAS 17, 122704 共2010兲

Spike deceleration and bubble acceleration in the ablative Rayleigh–Taylor instability W. H. Ye (叶文华兲,1,2,a兲 L. F. Wang (王立锋兲,2,3,b兲 and X. T. He (贺贤土兲1,2,c兲 1

Department of Physics, Zhejiang University, Hangzhou 310027, People’s Republic of China CAPT, Peking University, Beijing 100871, People’s Republic of China and LCP, Institute of Applied Physics Computational Mathematics, Beijing 100088, People’s Republic of China 3 SMCE, China University of Mining and Technology, Beijing 100083, People’s Republic of China 2

共Received 6 July 2010; accepted 14 September 2010; published online 7 December 2010兲 The nonlinear evolutions of the Rayleigh–Taylor instability 共RTI兲 with preheat is investigated by numerical simulation 共NS兲. A new phase of the spike deceleration evolution in the nonlinear ablative RTI 共ARTI兲 is discovered. It is found that nonlinear evolution of the RTI can be divided into the weakly nonlinear regime 共WNR兲 and the highly nonlinear regime 共HNR兲 according to the difference of acceleration velocities for the spike and the bubble. With respect to the classical RTI 共i.e., without heat conduction兲, the bubble first accelerates in the WNR and then decelerates in the HNR while the spike holds acceleration in the whole nonlinear regime 共NR兲. With regard to the ARTI, on the contrary, the spike first accelerates in the WNR and then decelerates in the HNR while the bubble keeps acceleration in the whole NR. The NS results indicate that it is the nonlinear overpressure effect at the spike tip and the vorticity accumulation inside the bubble that lead to, respectively, the spike deceleration and bubble acceleration, in the nonlinear ARTI. In addition, it is found that in the ARTI the spike saturation velocity increases with the perturbation wavelength. © 2010 American Institute of Physics. 关doi:10.1063/1.3497006兴 I. INTRODUCTION

In the inertial confinement fusion 共ICF兲, the energy absorbed by the target material drives an ablation that leads to the mass off the shell outer surface and generates a lowdensity blow-off plasma, resulting in the expected shell acceleration.1 Unfortunately the Rayleigh–Taylor instability 共RTI兲 occurs during this regime when a less dense fluid 共the ablation plasma兲 accelerates another fluid of higher density 共the unabated material兲. On account of this instability, any perturbation either from the shell surfaces or from the laser beams would grow exponentially in time during the linear regime. After a short linear growth, the perturbation enters the nonlinear growth regime, where the “bubbles” of the lighter fluid rise through the denser fluid, and the “spikes” of the heavier fluid penetrates down through the lighter fluid. On the one hand, in the acceleration stage, the integrity of the dense shell is severely compromised by the RTI bubbles approaching the inner shell surface. On the other hand, in the deceleration stage, the growth of the hot spot is dramatically dependent on the cold spikes of the shell at the hot spot boundary. Therefore, the ICF targets must be designed to keep the ablative RTI 共ARTI兲 growth at an acceptable level since it can break up the implosion shell during the acceleration stage and destroy the central ignition hot spot during the deceleration stage. The estimate for when these will happen naturally depends on the bubble 共spike兲 velocity through the dense 共light兲 fluid. In the frame of the classical theory1 关i.e., without heat conduction 共HC兲兴, when the heavier fluid of density ␳h is a兲

Electronic mail: [email protected]. Electronic mail: [email protected]. c兲 Electronic mail: [email protected]. b兲

1070-664X/2010/17共12兲/122704/6/$30.00

superposed over the lighter fluid of density ␳l in a gravitational field, −gyˆ , where g is the acceleration, a single-mode perturbation of wavelength ␭ with the perturbation amplitude ␩0 关i.e., the perturbation is in the form ␩共x兲 = ␩0 cos共kx兲兴 on the interface between two fluids yields its linear growth rate1 ␥cla = 冑ATkg, where k = 2␲ / ␭ is the perturbation wave number, and AT = 共␳h − ␳l兲 / 共␳h + ␳l兲 is the Atwood number. It is well known that the density gradient effects 共i.e., the interface width effects兲 reduce the RTI growth rate,2 which can be approximated by the formula,3–7 ␥cL = 冑ATkg / 共1 + ATkLm兲, where Lm = min关兩␳ / 共⳵␳ / ⳵x兲兩兴 is the minimum density gradient scale length. When the ablation is included, the linear growth rate of the ARTI, ␥abl, is remarkably reduced, especially for the short perturbation wavelength and a cut-off wavelength appears.1,8–18 Due to the removal of modulated material by ablation, the reduction of ␥abl in comparison with ␥cla, is simply expressed by the Takabe–Bodner formula,8,9 ␥T = ␣冑kg − ␤kva, where va is the ablation velocity 共i.e., the rate at which the material is removed兲, and ␣ and ␤ are the numerical adjustable coefficients depending on the flow parameters. A modified version of this given by Lindl10 including another stabilization factor, the width effect of the interface region, is in the form ␥mL = ␣冑kg / 共1 + kLm兲 − ␤kva, where Lm is the minimum scale length of density gradient at the ablation surface. The direct-driven 共DD兲 experiments11,12 have shown that the growth rate of the areal density perturbation is significantly reduced as compared with the calculated results of the Spitzer–Härm 共SH兲 electron thermal conductivity. Glendinning et al.12 recognized that the difference between the experiment and the LASNEX simulation is due to a change in the longitudinal density profile resulting from preheat by energetic electrons originating in the plasma corona that penetrate beyond the ablation front. Furthermore,

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Phys. Plasmas 17, 122704 共2010兲

Ye, Wang, and He

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Without considering the radiation, an ideal gas equation of state is used in our NS. The governing equations in a constantly accelerating reference frame for single-fluid and one-temperature are21–24

⳵␳ + ⵜ · 共␳V兲 = 0, ⳵t

共1a兲

⳵ 共␳V兲 + ⵜ · 共␳VV兲 + ⵜp = ␳g, ⳵t

共1b兲

⳵E + ⵜ · 关共E + P兲V兴 = ⵜ · 共␬ ⵜ T兲 + ␳V · g, ⳵t

共1c兲

where ␳, V, T, and g are, respectively, the density, velocity, temperature, and acceleration. E = CV␳T + ␳V · V / 2 is the total energy, p = ⌫␳T is the pressure, CV = ⌫ / 共␥h − 1兲 is the specific heat at constant volume, and ␥h equals to 5/3. The term of ⵜ · 共␬ ⵜ T兲 represents the electron heat conduction, which is turned off when we consider the classical RTI 共CRTI兲共i.e., without HC兲. The preheat model ␬共T兲 = ␬SH关1 + f共T兲兴 is introduced in our NS,14 where ␬SH is the SH electron heat conduction and f共T兲 interprets the preheating tongue effect in the cold plasma ahead of the ablation front. For temperatures greater than 0.10 MK, it has been shown that the ARTI growth rates, the fluid density, and temperature profiles at the ablation front obtained from the choice f共T兲 = ␣T−1 + ␤T−3/2, where T is in the unit of megakelvin, ␣ and ␤ are adjustable

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ρ V T

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ρ V T

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II. NUMERICAL METHODS AND BASIC FLOWS

(a)

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the experiments of Shigemori and Sakaiya et al.12,13 suggests that the nonlocal electron heat transport plays an important role in the suppression of RTI at the ablation front. Subsequently, Ye et al.14 found that the ARTI growth rate formula, ␥mY = 冑Akg / 共1 + AkLm兲 − 2kva, agrees well with the experiments and simulations, and is appropriate for the preheating case in DD experiments in ICF. The observation of density profile in DD experiment19 and simulation12,13 indicates that Lm is markedly improved by the preheat ablation effect. In addition, the recent self-consistent linear theory15–18 identified another physical mechanism of stabilization, which is expressed as ␥SC = 冑ATkg − AT2 k2v2a / rD − 共1 + AT兲kva for the case of large Froude number. Here, rD is the effective density ratio of the blow-off plasma to the ablation front. The main difference between ␥SC and all the versions of the Takabe– Bodner formula 共e.g., ␥T, ␥mL, and ␥mY 兲 is the negative term under the square root. Recently, a new phase of the nonlinear bubble evolution is discovered by Betti et al.,20 which is attributed to the vorticity accumulation inside the bubble resulting from the mass ablation and vorticity convection off the ablation front. The focus of the present work investigates the single-mode velocities for the bubble and the spike including the interface thickness effects and the preheat ablation effects, which are both relevant to the typical ICF experiments by numerical simulation 共NS兲. In order to separate the influence of the interface width and the ablation, we, therefore, consider situations with and without HC.

Density ρ (g/cm )

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FIG. 1. 共Color online兲 Basic flows for the WP 共a兲 and SP 共b兲 cases. The minimum scale length of density gradient at the ablation surface for the WP and SP cases are 0.23 and 1.83 ␮m, respectively.

parameters, agree well with the experiments.14 In this paper, we choose two preheat cases, i.e., the weak preheat 共WP兲 case with ␣ = 0.86 and ␤ = 0.24, and the strong preheat 共SP兲 case with ␣ = 8.6 and ␤ = 1.6. With the ARTI, we obtain the steady state flow fields, as shown in Figs. 1共a兲 and 1共b兲 by integrating Eq. 共1兲 in the accelerating reference frame of the ablation front from the peak density to the isothermal sonic points on both sides of the ablation front.25 The averaged ablation parameters for the WP 共SP兲 case are as follows: the peak density ␳a = 5.25 g / cm3 共4.25 g / cm3兲, the ablation velocity va = 1.19 ␮m / ns 共1.58 ␮m / ns兲, the temperature at the peak density Ta = 8.23⫻ 10−2 MK 共8.37⫻ 10−2 MK兲, and the acceleration g = 17.85 ␮m / ns2 共16.57 ␮m / ns2兲, which are obtained from the one-dimensional calculation,25 which are adopted for the above integration. With the CRTI, the density profile is from the SP case, while the pressure is computed via the hydrostatic equation in the form p = p0 + 兰rl ␳gdx, where p0 = 20 Mbar is the static pressure, g = 16.57 ␮m / ns2 and the integral domain is from the left boundary to the right boundary within the calculation zone.


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FIG. 2. 共Color online兲 Linear growth rate curves ␥共k兲 for WP, SP, and classical cases. The cut-off wavelengths are 3.2 and 6.0 ␮m for the WP and SP cases, respectively.

III. SPIKE DECELERATION AND BUBBLE ACCELERATION

Since the transition between the fluids is no longer sharp, we shall generate a reasonably defined interface in the fluid so that the spatial evolution of the perturbed interface can be tracked and analyzed. For this purpose, we shall define the interface as the location where the density gradient is the largest in the one-dimensional steady density field. The linear growth rate can be obtained by fitting the amplitude of the fundamental mode within the linear growth regime. The linear grow rate curves ␥共k兲 are shown in Fig. 2. Considering the CRTI, the linear growth rate decreases with the perturbation wavelength ␭. Regarding the ARTI, the linear growth rate is remarkably reduced in comparison with the CRTI, especially for the short perturbation wavelength and there is a cut-off perturbation wavelength that is about 3.2 ␮m 共6.0 ␮m兲 in our WP 共SP兲 preheat model. The temporal evolutions of the normalized amplitudes ␩ / ␭, normalized velocities v / 冑g␭, and normalized accelerations a / g for the spike and the bubble with ␭ = 20 ␮m and ␩0 = 0.0069 ␮m are plotted in Fig. 3. In the CRTI, the spike length is always greater than the bubble’s while the bubble length can surpass the spike’s in the ARTI, as shown in Fig. 3共a兲. It is seen in Fig. 3共b兲 that in the nonlinear regime 共NR兲 the bubble velocity in the ARTI can exceed the spike’s, while this does not happen in the CRTI. As shown in Fig. 3共c兲, we can see that in the NR there is a spike 共bubble兲 acceleration 共deceleration兲 stage for the CRTI, while conversely there is a bubble 共spike兲 acceleration 共deceleration兲 phase in the ARTI. Because of the opposite behaviors of the acceleration velocities for bubble and spike in the ARTI and the CRTI, it is suggested that the NR of the RTI can be divided into the weakly nonlinear regime 共WNR兲 and the highly nonlinear regime 共HNR兲 according to the difference of acceleration velocities for the spike and the bubble. For the CRTI, the bubble first accelerates in the WNR and then decelerates in the HNR resulting in a bubble saturation velocity,26,27 while

FIG. 3. 共Color online兲 Normalized amplitudes 共a兲, normalized velocities 共b兲, and normalized accelerations 共c兲 for the spike and the bubble vs time for a 20 ␮m perturbation wavelength with an initial amplitude 0.0069 ␮m. Solid, dotted, and dash dot dotted lines represent the bubbles for WP, SP, and classical cases, respectively. Dashed, dash dotted, and short dotted lines correspond to the spikes for WP, SP, and classical cases, respectively.

the spike keeps acceleration in the whole NR. Instead, for the ARTI, the spike first accelerates in the WNR and then decelerates in the HNR leading to a spike saturation velocity, while the bubble holds acceleration in the whole NR. The


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(a)

(c)

(b)

(d)

FIG. 4. 共Color online兲 Density contours 共a兲 and, respectively, the corresponding vorticity 共b兲, absolute value of heat flux 共c兲, and pressure 共d兲 contours for the SP case of a 20 ␮m perturbation wavelength with an initial amplitude 0.0069 ␮m at ␥t = 7.7, 8.8, 9.8, and 11.0. The units of density, vorticity, heat flux, and pressure are g / cm3, 10 ns−1, 1013 W / cm2, and megabar, respectively.

nonlinear behavior in the CRTI has been well understood26,27 while the new highly nonlinear behaviors in the ARTI has not been fully understood. Here, we try to understand it by analyzing the nonlinear evolutions of the density, vorticity, absolute value of heat flux, and pressure. Figure 4 shows the density contours and the corresponding contours of vorticity, absolute value of heat flux, and pressure for the SP case with ␭ = 20 ␮m and ␩0

= 0.0069 ␮m at ␥t = 7.7, 8.8, 9.8, and 11.0, where the vorticity is defined as 共⳵Vy / ⳵x − ⳵Vx / ⳵y兲 and the absolute

value of heat flux is defined as 兩q兩 = ␬冑共⳵T / ⳵x兲2 + 共⳵T / ⳵y兲2. The density contours and the corresponding vorticity contours in the CTRI for a 20 ␮m perturbation wavelength with ␩0 = 0.0069 ␮m at ␥t = 7.7, 8.4, 9.0, 9.7, 10.4, and 11.1, are illustrated in Fig. 5.


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(a)

(b)

FIG. 5. 共Color online兲 Density contours 共a兲 and the corresponding vorticity contours 共b兲 for the classical case of a 20 ␮m perturbation wavelength with an initial amplitude 0.0069 ␮m at ␥t = 7.7, 8.4, 9.0, 9.7, 10.4, and 11.1. The units of density and vorticity are g / cm3 and 10 ns−1, respectively.

Note that the density contours in the SP case are smooth showing no characteristic Kelvin–Helmholtz 共KH兲 mushroom structures at the spike heads, as shown in Fig. 4共a兲, while the KH mushroom structures are fully developed in the classical case, as shown in Fig. 5共a兲. Accordingly, with and without the noticeable KH mushroom structures are the significant difference between the ARTI and the CRTI. The vortex climbs toward the bubble vertex in the SP case, as shown in Fig. 4共b兲, while it does not occur in the CRTI, as shown in Fig. 5共b兲. The phenomenon of the vorticity accumulation inside the bubble is first reported by Betti et al.,20 which is used to explain the bubble acceleration in the HNR of the ARTI. Our results is in general agreement with their. Comparing Figs. 4共b兲 and 5共b兲, it is seen that the vorticity is strongly suppressed by the HC. The distribution of the heat flux round the spike tip forms a cone that encloses the spike tip, as shown in Fig. 4共c兲. One can expect that, in the surroundings of the high temperature heat flow cone, it is hard for the spike tip to become decentralized to form the classical KH mushroom structures. Detailed research in these areas will be pursued and reported in our future work. We shows the density contours and the corresponding vorticity contours for the WP case with ␭ = 20 ␮m and ␩0 = 0.0069 ␮m at ␥t = 8.4, 9.0, 9.7, and 10.0, in Fig. 6. For the WP case, the characteristic KH mushroom structure appears in the NR regime, although these KH mushroom structures are not very typical, as shown in Fig. 6. Furthermore, it is found that the vorticity accumulation inside the bubble is reduced in comparison with the SP case, as shown in Fig. 4共b兲, but increased in comparison with the classical case, as

FIG. 6. 共Color online兲 Density contours 共a兲 and the corresponding vorticity contours 共b兲 for the WP case of a 20 ␮m perturbation wavelength with an initial amplitude 0.0069 ␮m at ␥t = 8.4, 9.0, 9.7, and 10.0. The units of density and vorticity are g / cm3 and 10 ns−1, respectively.


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celerates in the HNR while the spike holds acceleration in the whole NR. On the other hand, for the ARTI, instead, the spike first accelerates in the WNR and then decelerates in the HNR while the bubble keeps acceleration in the whole NR. Our NS results indicate that it is the nonlinear overpressure effect15–18 at the spike tip and the vorticity accumulation inside the bubble20 that lead to, respectively, the spike deceleration and bubble acceleration in the nonlinear ARTI. It is noteworthy that the effects of the interface width and the preheating ablation are not sufficiently included in the previous works, especially for the analytical investigations15–18,20,26,27, which are fully considered in our numerical simulations presented here. ACKNOWLEDGMENTS FIG. 7. 共Color online兲 Normalized spike saturation velocities vs perturbation wavelengths for the SW case.

shown in Fig. 5共b兲. It is therefore concluded that the vorticity convection in the WP case is less pronounced in comparison with the SP case but more pronounced in comparison with the classical case. As noted above, in the linear growth regime, the overpressure term 共i.e., AT2 k2v2a / rD兲 can increase the ablative pressure at the spike tip providing an “restoring force.” One can expect this effect can be strengthened in the NR. Because in the NR, the spike tip becomes more close to the hot laser absorption zone, where not only ⵜT but also the heat conduction coefficient ␬共T兲 are improved. As can be seen in Fig. 4共d兲, the maximum pressure around the spike tip is about 24.09, 24.72, 25.01, and 25.42 Mbar for ␥t = 7.7, 8.8, 9.8, and 11.0, respectively, while the initial pressure at the interface is only 22.05 Mbar, as shown in Fig. 1共b兲. It is seen that the restoring force ⬀k2 as noted above. Therefore, one can expect that the spike saturation velocity ⬀␭2. We show the normalized spike saturation velocities in Fig. 7. The saturation velocity increases with the perturbation wavelength, as shown in Fig. 7, which can be preliminarily understood as mentioned above. Detailed research in these areas will be pursued and reported in our future work. IV. CONCLUSION

The nonlinear evolutions of the ARTI with preheat has been studied by NS in the present work. In order to better understand the ablation effect and ablation surface thickness effect, we also investigate the CRTI for comparison. We find a new regime of spike deceleration in the nonlinear evolution of the ARTI. Subsequently, it is therefore suggested that the nonlinear evolution of the RTI can be divided into the WNR and the HNR according to the difference of acceleration velocities for the spike and the bubble. On the one hand, for the CRTI, the bubble first accelerates in the WNR and then de-

This work was supported by the National Basic Research Program of China 共Grant No. 2007CB815103兲 and the National Natural Science Foundation of China 共Grant Nos. 10935003, 10775020, and 11075024兲. 1

S. Atzeni and J. Meyer-ter-Vehn, The Physics of Inerial Fusion: Beam Plasma Interaction, Hydrodynamics, Hot Dense Mater 共Oxford University Press, Oxford, 2004兲. 2 K. O. Mikaelian, Phys. Rev. Lett. 48, 1365 共1982兲. 3 K. O. Mikaelian, Phys. Rev. A 33, 1216 共1986兲. 4 D. H. Munro, Phys. Rev. A 38, 1433 共1988兲. 5 K. O. Mikaelian, Phys. Rev. A 40, 4801 共1989兲. 6 A. B. Bud’ko and M. A. Liberman, Phys. Fluids B 4, 3499 共1992兲. 7 L. F. Wang, W. H. Ye, and Y. J. Li, Phys. Plasmas 17, 042103 共2010兲. 8 S. Bodner, Phys. Rev. Lett. 33, 761 共1974兲. 9 H. Takabe, K. Mima, L. Montierth, and R. L. Morse, Phys. Fluids 28, 3676 共1985兲. 10 J. D. Lindl, P. Amendt, R. L. Berger, S. G. Glendinning, S. H. Glenzer, S. W. Haan, R. L. Kauffman O. L. Landen, and L. J. Suter, Phys. Plasmas 11, 339 共2004兲. 11 K. Shigemori, H. Azechi, M. Nakai, M. Honda, K. Meguro, N. Miyanaga, H. Takabe, and K. Mima, Phys. Rev. Lett. 78, 250 共1997兲. 12 S. G. Glendinning, S. N. Dixit, B. A. Hammer, D. H. Kalantat, M. H. Key, J. D. Kilkenny, J. P. Knauer, D. M. Pennington, B. A. Remington, R. J. Wallace, and S. V. Weber, Phys. Rev. Lett. 78, 3318 共1997兲. 13 T. Sakaiya, H. Azechi, M. Matsuoka, N. Izumi, M. Nakai, K. Shigemori, H. Shiraga, A. Suna-hara, H. Takabe, and T. Yamanaka, Phys. Rev. Lett. 88, 145003 共2002兲. 14 W. H. Ye, W. Y. Zhang, and X. T. He, Phys. Rev. E 65, 057401 共2002兲. 15 J. Sanz, Phys. Rev. Lett. 73, 2700 共1994兲. 16 A. P. Piriz, J. Sanz, and L. F. Ibañez, Phys. Plasmas 4, 1117 共1997兲. 17 V. N. Goncharov, R. Betti, R. L. McCrory, P. Sorotokin, and C. P. Verdon, Phys. Plasmas 3, 1402 共1996兲. 18 R. Betti, V. N. Goncharov, R. L. McCrory, and C. P. Verdon, Phys. Plasmas 5, 1446 共1998兲. 19 S. Fujioka, H. Shiraga, M. Nishikino, K. Shigemori, A. Sunahara, M. Nakai, H. Azechi, K. Nishihara, and T. Yamanaka, Phys. Plasmas 10, 4784 共2003兲. 20 R. Betti and J. Sanz, Phys. Rev. Lett. 97, 205002 共2006兲. 21 L. F. Wang, X. Chuang, W. H. Ye, and Y. J. Li, Phys. Plasmas 16, 112104 共2009兲. 22 L. F. Wang, W. H. Ye, and Y. J. Li, Phys. Plasmas 17, 052305 共2010兲. 23 L. F. Wang, W. H. Ye, and Y. J. Li, Europhys. Lett. 87, 54005 共2009兲. 24 L. F. Wang, W. H. Ye, Z. F. Fan, and Y. J. Li, Europhys. Lett. 90, 15001 共2010兲. 25 Z. Fan, J. Luo, and W. Ye, Phys. Plasmas 16, 102104 共2009兲. 26 V. N. Goncharov, Phys. Rev. Lett. 88, 134502 共2002兲. 27 K. O. Mikaelian, Phys. Rev. E 67, 026319 共2003兲.
