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- High Power Laser Science and Engineering
- Vol. 12, Issue 3, 03000e24 (2024)

Abstract

Keywords

1. Introduction

The Rayleigh–Taylor instability (RTI) is ubiquitous in high energy density physics, such as laboratory astrophysics experiments and inertial confinement fusion (ICF)^{[}1^{–}3^{]}. The RTI occurs when a light fluid supports a heavy fluid in external gravity field, which features the growth of perturbation amplitude on the interface between two fluids^{[}4^{,}5^{]}. In laser fusion experiments, the perturbation seeded by target defect or drive asymmetry could be significantly amplified by the RTI^{[}6^{]}. Due to the energy transfer creating a continuous density profile on the interface, the linear growth is stabilized compared to the classical case and a cutoff wavelength appears when the perturbation wavelength is sufficiently short^{[}7^{]}. Therefore, the ablative RTI (ARTI) is commonly called to emphasize the importance of the mass ablation^{[}8^{,}9^{]}. The ARTI mainly occurs at the ablation front separating the compressed target from the blow-off corona plasma and the inner interface between the fuel shell and the hot spot^{[}10^{,}11^{]}. This instability could destroy the shell integrity and limit the implosion efficiency during the acceleration stage. What is worse, it would aggravate the material mixing, reduce the effective size of the hot spot and even result in ignition failure^{[}12^{]}. Control of the ARTI at an acceptable level is crucial to improve fusion performance.

As an inevitable process in ICF and the general laser–plasma interaction, the intense magnetic field could be spontaneously self-generated by a number of mechanisms, including but not limited to the thermoelectric effect and anisotropic velocity distribution of hot electrons, with relative importance depending on the interaction parameters^{[}13^{–}15^{]}. For the interaction with solid targets of nanosecond lasers, the Biermann battery effect is regarded as the primary source, caused by nonparallel gradients between temperature and density^{[}16^{,}17^{]}. The mechanism behind magnetic field generation is the loss of electron energy, resulting in the breakdown of local neutrality. In recent years, the self-generated magnetic field has been studied analytically, experimentally and numerically by many researchers. Li *et al.*^{[}18^{]} utilized the monoenergetic proton radiography method to measure the electromagnetic field generated during the interaction of a solid target and long-pulse laser beams. Due to the fact that the target surface cannot be perfectly smooth, the perturbation-induced magnetic field attracts great attention. As early as the 1970s, Mima *et al.*^{[}19^{]} demonstrated the presence of the magnetic field in a plasma subject to the RTI from the perspectives of theory and simulation. The first experimental demonstration was published in 2012, which reported the generation of a several-tesla magnetic field during the linear stage and motivated further investigation of RTI-induced magnetic fields^{[}20^{,}21^{]}. The magnetic field strength could reach up to the megagauss level during the nonlinear growth phase in an ablatively driven plasma^{[}22^{]}. The self-generated magnetic field could play a stabilizing or destabilizing role depending on the Froude number during the linear stage^{[}23^{]}. Zhang *et al.*^{[}24^{]} also found that the Nernst compressed magnetic field reduced the bubble width and boosted the bubble velocity during the nonlinear stage in ARTI relevant to ICF implosion. During the deceleration stage of ICF implosion, the self-generated magnetic field is as intense as thousands of teslas with a large Hall parameter and anisotropic heat flux would promote spike penetration^{[}25^{]}. During the stagnation phase, the magnetic field, initially appearing on the inner interface of the cold shell, could be pushed into the hot spot by the low-mode perturbation and degrade the fusion energy^{[}26^{]}.

When a long-pulse laser irradiates targets, the pressure perturbation is generated by the laser imprint and further enhanced by the self-generated magnetic field due to a combination of the Nernst advection and the Righi–Leduc (R-L) heat flux^{[}27^{]}. This mechanism occurs in the early stage and the enhanced perturbation could be regarded as a seed perturbation for the hydrodynamic instability. The self-generated magnetic field indirectly feeds back on the hydrodynamic process through electron magnetization rather than the magnetic pressure^{[}28^{]}. Walsh *et al.*^{[}25^{,}26^{]} demonstrated that the magnetized heat flux is highly significant both in the deceleration and stagnation stages of ICF. However, the importance in the acceleration stage still remains unknown. In this paper, we analyze the importance of the R-L heat flux to the ARTI in a laser irradiating thin targets based on the fully extended magnetohydrodynamic simulations. The rest of this paper is structured as follows. Section 2 briefly outlines the numerical code and the extended-magnetohydrodynamic model used for the results presented here. In Section 3, the ARTI evolution with a self-generated magnetic field included is studied. The simulations show that an increase in the linear growth rate and the amplitude is mainly attributed to the R-L effect deflecting the total heat flux. The less heat flux concentrated at the spike tip effectively lowers ablative stabilization and promotes instability growth. The importance of the nonlocal effect is discussed theoretically in Section 4. Finally, Section 5 gives the conclusion to the whole paper.

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2. The simulation model

In this paper, numerical simulations are performed by using the open-source code FLASH^{[}29^{]}. FLASH is a highly parallel and multi-dimensional finite-volume Eulerian code, which solves the single-fluid governing equations and advances hydrodynamic evolution based on a directionally unsplit staggered mesh solver^{[}30^{,}31^{]}. The code has been extended to three temperature treatments and is coupled with a variety of physical processes to improve the capability of simulating high energy density physics. Without considering external gravity, viscous force and radiation transport, the governing equations are described by the following equations:

3. Two-dimensional simulations of the ablative Rayleigh–Taylor instability with the magnetic field included

As the RTI occurs on the ablation front in the acceleration stage, we firstly carry out two-dimensional (2D) simulations of a laser irradiating a thin target without perturbation to obtain the hydrodynamic behavior of the ablation front. The simulation domain is based on the *x*–*y* Cartesian coordinate. Periodic and outflow boundary conditions are imposed on the *x*- and *y*-directions, respectively. A schematic diagram is depicted in Figure 1. The target material is CH with an initial density of *y*-axis. A laser beam, operated at a wavelength of ^{[}34^{]}.

Figure 1.Schematic diagram of a laser driven RTI with the self-generated magnetic field included.

When a laser beam is switched on, the energy deposition leads to the ablated material coming off the outer surface of the target and generates blow-off corona plasma. The shock driven by the ablation pressure propagates towards the target inside, and the material behind the shock front is compressed into higher density. Figure 2(a) shows the density distributions along the *y*-axis at different times. At ^{[}35^{]}. When the reflected rarefaction wave arrives at the ablation front, the ablation front begins to be accelerated at

Figure 2.(a) The density distribution along the

In the configuration of a laser irradiating a planar target, it was experimentally found that the magnetic field was concentrated on a hemispherical shell surrounding the ablative plasma bubble with the maximum amplitude appearing near the edge and falling to zero at the center^{[}18^{]}. The magnetic field has a toroidal configuration with scale length comparable to the spot size. If a perturbation occurs in the laser-irradiation region, the Biermann battery effect would be significantly enhanced and the RTI-induced magnetic field could reach higher strength. This mechanism can be seen in Figure 1.

In the follow-up paper, the 2D simulations upon introducing perturbation are further carried out and the self-generated magnetic field is simultaneously taken into account. Velocity perturbation taking the form of

The perturbation amplitude firstly grows exponentially in time. After a short linear stage, the RTI grows into the nonlinear regime and a bubble-spike structure is formed^{[}36^{]}. When the bubble-spike amplitude is comparable to its perturbation wavelength, the RTI evolves into the highly nonlinear regime. Figure 3 shows the spatial distributions of the density and magnetic field at different times. Since the interface between the cold-dense target and corona plasma is not sharp, we define the position with the minimum density scale length as the ablation front, which is shown by the black-solid line in Figure 3 and the follow-up figures, so that the instability amplitude could be easily tracked in numerical simulations. At the early time, the peak-to-valley (P-V) amplitude

Figure 3.The spatial distributions of the density (a)–(d) and the magnetic field (e)–(h) at different times during the evolution of the RTI.

The amplitude evolution is considered as an important measurement of the growth of the RTI, as shown in Figure 4. It is usually thought that linear saturation occurs when the classical amplitude reaches

Figure 4.The peak-to-valley amplitude evolves over time without a magnetic field (red-solid line). The blue-dashed line is the amplitude difference between the amplitude with and without the magnetic field included.

The Hall parameter ^{[}32^{]}. The magnetized electron thermal conduction can be expressed as follows:

Figure 5(a) shows the spatial distribution of Hall parameters at

Figure 5.Spatial distributions of (a) the Hall parameter , (b) the ratio of perpendicular to parallel thermal transport coefficients and (c) the ratio of cross to parallel thermal transport coefficients at . Only the right-hand side of the spike is shown.

Figure 6(a) displays the total heat flux with a magnetic field (pink-dashed-dot line), which is superimposed on the density distribution at *z*-axis and without *x*- and *y*-components in 2D simulations, there is no component of the heat flow along the magnetic field lines. In the region above

Figure 6.At , a comparison of heat flux streamlines with (without) a magnetic field (a), and with (without) the Nernst effect (b) overlaid on the density spatial distribution. Only the right-hand side of the spike is shown. The pink-dashed-dot lines in (a) and (b) are identical.

Although the magnetization is still at a low level with

Figure 7.At , (a) the velocity of the spike tip (red-circle line) and the bubble vertex (blue-delta line), and (b) the linear growth rate (red-circle line) and the amplitude (blue-delta line) for different factors.

In order to understand comprehensively the influence of the magnetic field on RTI growth, the thermal driven terms in Equation (2) are re-arranged into a form that is similar to the advection velocity. The transport of the magnetic field can be written in a physically motivated form as follows:

Magnetic field advection is a balance between the frozen-in-flow with ions and the thermally driven effects, and the latter provides an additionally convective velocity along the heat flux^{[}39^{,}40^{]}. The Nernst effect convects the magnetic field down the temperature gradient and the cross-gradient Nernst tends to advect the magnetic field towards the spike base. The Nernst flux limiter is equal to the thermal flux limiter and is chosen to be *y*-component of the plasma velocity as well as thermally driven velocity at *x*-axis. The total advection velocity is closer to zero, meaning that the self-generated magnetic field rapidly accumulates locally. This phenomenon is beneficial for an increase in the field strength. Figure 8(b) shows a comparison of the evolution of the peak magnitude of the magnetic field over time. When the Nernst effect is included, the magnetic field is compressed and amplified near the ablation front. At

Figure 8.(a) Distributions of the

Figure 9(a) shows a comparison between the linear growth rates without and with the self-generated magnetic field for different wavelengths. In order to theoretically predict the linear growth rate, the laser irradiating the planar target without perturbation is simulated to get the distribution of density and pressure near the ablation front. Then the fitting method in Ref. [41] is employed to obtain variables including the density scale length

Figure 9.(a) Comparison of the linear growth rate. The black-solid line is the theoretical prediction. The red circles are from the simulations without a magnetic field, while the blue squares correspond to cases with a magnetic field included. The generation rate of the self-generated magnetic field (b), the percentage increase (c) and the derivative of the amplitude difference (d) evolve over time for different wavelengths.

Fr | |||
---|---|---|---|

0.25 | 0.31 | 162.60 | 3.54 |

Table 1. The averaged values used to theoretically predict the linear growth rate for different wavelengths.

Figure 9(c) shows the percentage increase in perturbation amplitude during the whole process for three wavelengths. The percentage increase is defined as the amplitude difference divided by the amplitude without a magnetic field included. The percentage gradually increases from

In order to simplify the simulation model, the radiation is neglected in the simulations. As for the perturbation wavelength of

4. The importance of the nonlocal effect

The nonlocal thermal transport has the potential to reduce the nonlinear growth of high-mode perturbation with a short wavelength^{[}43^{]}. Currently, the FLASH code does not support a self-consistent coupling between the nonlocal effect and the magnetic field. However, it is still necessary to evaluate the importance of the nonlocal effect. The Knudsen number is an indicator used to quantify the nonlocal effect. It is defined as the ratio of the electron mean free path and the temperature scale length, *y*-axis at *y*-axis is plotted in Figure 10(b), along with the mass density. The Knudsen number around the critical density surface is about

Figure 10.At , (a) the distributions of the temperature gradient scale length (red-solid line) and the electron mean free path (blue-dashed line), (b) the density (red-solid line) and the Knudsen number (blue-dashed line) along the

In addition to the thermal transport, the Biermann battery effect and the Nernst effect are both dependent on the temperature gradient and would be influenced by the nonlocal effect, called nonlocal suppression. The suppression factor for the Biermann battery effect is

Our previous work points out that the nonlocal effect is dependent on both the laser intensity and the laser frequency, and ^{[}46^{]}. The laser intensity threshold for considering the nonlocal effect is

5. Conclusion

To conclude, the self-generated magnetic field and R-L heat flux in the ARTI in a laser irradiating thin target are studied through 2D extended-magnetohydrodynamic simulations. Although the strength of the self-generated magnetic field could reach up to hundreds of teslas during the evolution of the RTI, the plasma is still in a low magnetization state due to cold temperature and high density near the ablation front. The simulations show that the R-L heat flux, additionally generated by the self-generated magnetic field, has a non-ignorable impact during the whole RTI evolution in the acceleration stage. The R-L effect deflects the total heat flux along the spike and towards the spike base. This deflection reduces the heat deposition near the spike tip, lowers the ablative stabilization, allows the spike to penetrate further into the conduction region and results in an increase in the spike-bubble amplitude. The simulations show that the magnetic field increases the linear growth rate by a factor of about 10% compared to the case without a self-generated magnetic field considered. Our results reveal the importance of R-L heat flux and promote deep understanding of the feedback of the self-generated magnetic field on instability evolution, especially during the acceleration stage in ICF.

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Ye Cui, Xiao-Hu Yang, Yan-Yun Ma, Guo-Bo Zhang, Bi-Hao Xu, Ze-Hao Chen, Ze Li, Fu-Qiu Shao, Jie Zhang. The importance of Righi–Leduc heat flux to the ablative Rayleigh–Taylor instability during a laser irradiating targets[J]. High Power Laser Science and Engineering, 2024, 12(3): 03000e24

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