2. Physics models

The simulations on 3-D ARTI and self-generated magnetic fields are carried out in planar geometry using the hydrodynamic code ART. Specifically designed for modelling ARTI in ICF-relevant scenarios, ART has been used in a number of recent works (Betti & Sanz Reference Betti and Sanz2006; Yan et al. Reference Yan, Betti, Sanz, Aluie, Liu and Frank2016; Zhang et al. Reference Zhang, Betti, Yan, Zhao, Shvarts and Aluie2018, Reference Zhang, Betti, Yan and Aluie2020, Reference Zhang, Li, Xin, Yan, Wan, Zhang and Zheng2022; Xin et al. Reference Xin, Yan, Wan, Sun, Zheng, Zhang, Aluie and Betti2019, Reference Xin, Liu, Jiang, Yan, Li, Wan, Sun and Zheng2023; Li et al. Reference Li, Yan, Zhao, Zheng, Zhang and Lu2022, Reference Li, Yan, Zhao, Zheng, Zhang and Lu2023, Reference Li, Yan, Zhao, Wu, Wang and Zou2024; Fu et al. Reference Fu, Zhang, Cai and Zhu2023b; Liu et al. Reference Liu, Zhang, Xin, Pu, Li, Tao, Sun, Yan and Zheng2023). Code ART solves the single-fluid equations in 2-D/3-D Cartesian coordinates, and this paper focus on the 3-D geometry. The hydrodynamic equations are

(2.1)$$\begin{gather} \frac{\partial \rho}{\partial t} +\boldsymbol{\nabla} \boldsymbol{\cdot} (\rho \boldsymbol{v})=0, \end{gather}$$

(2.2)$$\begin{gather}\frac{\partial\rho \boldsymbol{v}}{\partial t}+\boldsymbol{\nabla} \boldsymbol{\cdot} (\rho \boldsymbol{vv} )={-}\boldsymbol{\nabla} p +\rho\boldsymbol{g}, \end{gather}$$

(2.3)$$\begin{gather}\frac{\partial \epsilon}{\partial t}+\boldsymbol{\nabla}\boldsymbol{\cdot} [(\epsilon+p)\boldsymbol{v}] =\rho\boldsymbol{v}\boldsymbol{\cdot}\boldsymbol{g}-\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{q}, \end{gather}$$

where $\rho$ is the mass density, $\boldsymbol {v}$ is the macroscopic single-fluid velocity of the plasma, $p$ is the plasma thermal pressure, and $\boldsymbol {g}$ is the acceleration. The equation of state of an ideal gas is used, the plasma energy is $\epsilon =[{p}/{(\gamma -1)}]+[{\rho v^2}/{2}]$, and $\gamma ={5}/{3}$ is the specific heat ratio. As discussed in § 1, the magnetic forces have been neglected in (2.2). The low viscosity in high-energy-density plasma flows commonly neglected in the simulations on hydrodynamic instabilities (Betti & Sanz Reference Betti and Sanz2006; Wang et al. Reference Wang, Ye, Sheng, Don, Li and He2010; Zhang et al. Reference Zhang, Betti, Yan, Zhao, Shvarts and Aluie2018; Li et al. Reference Li, Yan, Zhao, Zheng, Zhang and Lu2022) is not considered in this work either. Although ARTI is in a regime with high Reynolds numbers ($Re \sim 10^5$ evaluated using the viscosity coefficient in Huba Reference Huba1998), the ablative effect in ARTI flattens the fine structures typically generated in CRTI, and prevents the transition to a turbulent state (Zhang et al. Reference Zhang, Betti, Yan, Zhao, Shvarts and Aluie2018).

The heat flux $\boldsymbol {q}$ contributes to the energy equation (2.3) via a thermal conduction term $-\boldsymbol {\nabla } \boldsymbol {\cdot } \boldsymbol {q}$, which is treated separately in a Strang-splitting way (Strang Reference Strang1968) by solving the heat conduction equation

(2.4)\begin{equation} \rho c_v\,\frac{\partial T}{\partial t}={-}\boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{q}, \end{equation}

after solving the pure hydrodynamic equations (2.1)–(2.3) without the $-\boldsymbol {\nabla } \boldsymbol {\cdot } \boldsymbol {q}$ term. Here, $T$ is the kinetic temperature including the Boltzmann constant, and $c_v$ is the constant-volume specific heat capacity. An MUSCL-Hancock scheme (van Leer Reference van Leer1984) with an HLLC (Harten, Lax & van Leer Reference Harten, Lax and van Leer1983) approximate Riemann solver is used as the hydrodynamic solver with second-order accuracy in both space and time to solve the hydrodynamics equations. For each computational time step, the pure hydrodynamic equations (2.1)–(2.3) without the $-\boldsymbol {\nabla } \boldsymbol {\cdot } \boldsymbol {q}$ term are advanced first, followed by an intermediate update of $T$ and $p$ with the help of the equation of state. Equation (2.4) is then solved to update $T$ once again, using a temporally implicit scheme to avoid the strict time step ($\Delta t$) requirement of explicit diffusion equation solvers. Then $p$ and $\epsilon$ are updated with the newly calculated $T$ using the equation of state before entering the next time step. The single-temperature approximation of the plasma is applied so that the electron temperature $T_e$ and ion temperature $T_i$ are equal (i.e. $T_e=T_i=T$), since the relaxation time (${\sim }10^{-4}$ ns evaluated using the formula in Huba Reference Huba1998) for electron–ion temperature equilibration on the ablation front is much shorter than the time scale of the ARTI evolution. For the magnetic-free cases, the classical Spitzer–Harm model (Spitzer & Härm Reference Spitzer and Härm1953) is used to calculate the heat flux $\boldsymbol {q}$ in (2.4) as

(2.5)\begin{equation} \boldsymbol{q}_{sh} ={-}\kappa_{sh}\,\boldsymbol{\nabla} T, \end{equation}

where $\kappa _{sh}$ is the Spitzer–Harm conductivity without flux limiter. The heat flux limiters that are often used in high-energy-density fluid simulations in various regimes to cap the unphysically large heat fluxes given by (2.5) and to match experiments are not applied in the simulations in this paper, since the maximum ratio of $q$ to the free-stream heat flux in the simulations always reads below 0.06, a commonly used flux-limiting threshold (Bell Reference Bell1985; Craxton et al. Reference Craxton2015). Equation (2.5) also does not include the non-local electron heat transport that is attracting research interest. Our previous simulations (Li et al. Reference Li, Yan, Zhao, Zheng, Zhang and Lu2022, Reference Li, Yan, Zhao, Zheng, Zhang and Lu2023) showed that the non-local electron heat transport can mitigate ARTI growths. Recent work by Campbell et al. (Reference Campbell2022) demonstrated the need to account for suppression of Biermann battery generation due to non-local effects by comparing experimental data and simulations. However, modelling the non-local electron heat transport is computationally costly in 3-D simulations, and beyond the scope of this paper. In the simulations, we set the single fluid to be a DT plasma with the number ratio 1 : 1. The advantage of using a DT plasma in the simulations is that it can avoid complex physics such as radiation transport coming from higher-Z materials.

The equation of magnetic field ($\boldsymbol {B}$) evolution can be derived readily from Ampère's law, Faraday's law, and the momentum equation of electrons (Nishiguchi et al. Reference Nishiguchi, Yabe, Haines, Psimopoulos and Takewaki1984), and formulated in Gaussian units as

(2.6)\begin{equation} \frac{\partial\boldsymbol{B}}{\partial t}= \underbrace{\boldsymbol{\nabla}\times(\boldsymbol{v}\times\boldsymbol{B})}_{{I}} +\underbrace{\frac{c}{e}\,\boldsymbol{\nabla}\times\left(\frac{\boldsymbol{\nabla} p_e}{n_e}\right)}_{{II}} -\underbrace{\frac{c}{4{\rm \pi} e}\,\boldsymbol{\nabla}\times\left[\frac{(\boldsymbol{\nabla}\times\boldsymbol{B})\times\boldsymbol{B}}{n_e}\right]}_{{III}} -\underbrace{\frac{c}{e}\,\boldsymbol{\nabla}\times\frac{\boldsymbol{R}}{n_e}}_{{IV}}, \end{equation}

where $e$ is the elementary charge carried by an electron, $c$ is the speed of light in vacuum, $n_e$ is the number density of the electrons, and $p_e$ is the isotropic pressure of the electrons. Isotropic electron pressure has been used in (2.6), since the pressure anisotropy that is crucial in collisionless plasmas is not important in collisional plasmas (Egedal, Le & Daughton Reference Egedal, Le and Daughton2013; Tubman et al. Reference Tubman2021) as in this regime. Here, $\boldsymbol {R}=\boldsymbol {R}_{{u}}+\boldsymbol {R}_{{T}}$ as the transfer of momentum from ions to electrons caused by collisions consists of two parts: (i) the thermal force $\boldsymbol {R}_{{T}}$ due to the gradient of the electron temperatures; and (ii) the friction force $\boldsymbol {R}_{{u}}$ due to the relative velocity of electrons and ions. These $\boldsymbol {R}_{{T}}$ and $\boldsymbol {R}_{{u}}$ are usually formulated in a Braginskii-like form (Braginskii Reference Braginskii1965) as

(2.7)$$\begin{gather} \boldsymbol{R}_{{T}}={-}\beta_{{\parallel}}^{uT}\,\nabla_{{\parallel}}T_e -\beta_{{\perp}}^{uT}\,\nabla_{{\perp}}T_e -\beta_{\land}^{uT}\boldsymbol{b}\times\boldsymbol{\nabla} T_e, \end{gather}$$

(2.8)$$\begin{gather}\boldsymbol{R}_{{u}}={-}\alpha_{{\parallel}}\boldsymbol{u}_{{\parallel}} -\alpha_{{\perp}}\boldsymbol{u}_{{\perp}} +\alpha_{\land}\boldsymbol{b}\times \boldsymbol{u}, \end{gather}$$

where $\boldsymbol {b} \equiv \boldsymbol {B}/|\boldsymbol {B}|$ is the unit direction vector parallel to the magnetic field, and $\boldsymbol {u} =\boldsymbol {u}_{{e}}-\boldsymbol {u}_{{i}}$ is the relative velocity of electrons and ions that can be associated with the magnetic field via Ampère's law $\boldsymbol {u}=-c\,\boldsymbol {\nabla }\times \boldsymbol {B}/(4{\rm \pi} n_ee)$. Also, $\nabla _{\parallel }T_e$ and $\nabla _{\perp }T_e$ are the components of the temperature gradient parallel and perpendicular to the direction of $\boldsymbol {B}$, respectively, and $\boldsymbol {u}_{\parallel }$ and $\boldsymbol {u}_{\perp }$ are the components of $\boldsymbol {u}$ parallel and perpendicular to the direction of $\boldsymbol {B}$, respectively. Here, $\beta _{\parallel }^{uT}$, $\beta _{\perp }^{uT}$, $\beta _{\land }^{uT}$, $\alpha _{\parallel }$, $\alpha _{\perp }$ and $\alpha _{\land }$ are the plasma transport coefficients that are detailed in Appendix A. The Braginskii formulation on the transport coefficients is a classical model that has been used widely in collisional astrophysical and high-energy-density plasmas (Velikovich, Giuliani & Zalesak Reference Velikovich, Giuliani and Zalesak2019; Sadler, Walsh & Li Reference Sadler, Walsh and Li2021; Tapinou et al. Reference Tapinou, Wheatley, Bond and Jahn2023; Kopp & Yanovsky Reference Kopp and Yanovsky2024), including ARTI-relevant problems (García-Rubio et al. Reference García-Rubio, Betti, Sanz and Aluie2021; Zhang et al. Reference Zhang, Li, Xin, Yan, Wan, Zhang and Zheng2022). Improved transport coefficients with less physical approximation than Braginskii's were obtained later (Epperlein & Haines Reference Epperlein and Haines1986; Ji & Held Reference Ji and Held2013; Davies et al. Reference Davies, Wen, Ji and Held2021) through fitting numerical solutions of the Fokker–Planck equation with different fitting strategies. It was reported in Davies et al. (Reference Davies, Wen, Ji and Held2021) that the fitted transport coefficients can give physically incorrect results under certain conditions. Davies et al. (Reference Davies, Wen, Ji and Held2021) clarified that the only serious error in Braginskii's coefficients is an overestimation of perpendicular resistivity ($\alpha _{\perp }$) at intermediate Hall parameters ($\chi >2$ up to $\sim$100), while the errors in the Epperlein and Haines coefficients are particularly severe, giving increasing $\beta _{\parallel }^{uT}-\beta _{\perp }^{uT}$ and $\alpha _{\parallel }-\alpha _{\perp }$ for $\chi < 1$ when they should decrease linearly to zero. In this work, we still utilize the classical Braginskii coefficients in most of the simulations, also to be consistent with our previous work on 2-D ARTI (Zhang et al. Reference Zhang, Li, Xin, Yan, Wan, Zhang and Zheng2022) for a fair comparison. Moreover, a few simulation cases (cases xiv–xvi) are performed with the Davies et al. (Reference Davies, Wen, Ji and Held2021) transport coefficients to show the difference introduced by different transport models.

Term I of (2.6) is usually known as the convection term that freezes the magnetic field along with the plasma. Term II is the baroclinic term (also known as the Biermann battery) generating the self-magnetic field through the misaligned density and pressure gradients, since $c\,\boldsymbol {\nabla }\times (\boldsymbol {\nabla } p_e /n_e)/e=c\,\boldsymbol {\nabla } p_e \times \boldsymbol {\nabla } n_e /{(en_e^2)}$. Term III is often referred to as the Hall term, which is neglected since the ratio of III to II approximately equals $1/\beta \ll 1$ in the regimes covered by this work. Term IV brings the effects of collisions, including the magnetic dissipation related to $\boldsymbol {R}_{{u}}$ and the Nernst effect related to $\boldsymbol {R}_{{T}}$. Bringing in the expressions for $\boldsymbol {R}_{{T}}$ and $\boldsymbol {R}_{{u}}$, term IV of (2.6) can be written as

(2.9)\begin{align} -\frac{c}{e}\,\boldsymbol{\nabla}\times\frac{\boldsymbol{R}}{n_e} &=\frac{c}{e}\,\boldsymbol{\nabla}\times\left(\frac{\beta_{{\parallel}}^{uT}\,\boldsymbol{\nabla} T_e }{n_e}\right) -\frac{c}{e}\,\boldsymbol{\nabla}\times\left[\frac{(\beta_{{\parallel}}^{uT} -\beta_{{\perp}}^{uT})(\boldsymbol{b} \times\boldsymbol{\nabla} T_e)\times\boldsymbol{B}}{B n_e}\right]\nonumber\\ &\quad -\frac{c}{e}\,\boldsymbol{\nabla}\times\left(\frac{\beta_{\land}^{uT}\,\boldsymbol{\nabla} T_e\times \mathbf{B} }{ B n_e}\right)-\frac{c^2}{4{\rm \pi} e^2}\,\boldsymbol{\nabla}\nonumber\\ &\quad \times\left\{\frac{\alpha_{{\parallel}}\,\boldsymbol{\nabla}\times\boldsymbol{B}-\alpha_{\land}\boldsymbol{b} \times(\boldsymbol{\nabla}\times\boldsymbol{B})-(\alpha_{{\perp}}-\alpha_{{\parallel}})\boldsymbol{b} \times[\boldsymbol{b}\times(\boldsymbol{\nabla}\times\boldsymbol{B})]}{n_e^2}\right\}. \end{align}

The first term on the right-hand side of (2.9) has no contribution in a fully ionized plasma (Sadler et al. Reference Sadler, Walsh and Li2021), while the combination of the second and third terms can be rewritten as $\boldsymbol {\nabla }\times [(\boldsymbol {V}_{{N}}+\boldsymbol {V}_{{CN}})\times \boldsymbol {B}]$, where $\boldsymbol {V}_{{N}}$ and $\boldsymbol {V}_{{CN}}$ are often referred as the Nernst velocity and the cross-gradient Nernst velocity in the forms

(2.10)$$\begin{gather} \boldsymbol{V}_{{N}}={-}\frac{c \beta_{\land}^{uT}}{e B n_{e}}\,\boldsymbol{\nabla} T_e, \end{gather}$$

(2.11)$$\begin{gather}\boldsymbol{V}_{{CN}}={-}\frac{c(\beta_{{\parallel}}^{uT}-\beta_{{\perp}}^{uT})( \boldsymbol{b}\times\boldsymbol{\nabla} T_e)}{e B n_e}, \end{gather}$$

respectively. Here, $\boldsymbol {V}_{{N}}$ is along the direction opposite to $\boldsymbol {\nabla } T_e$ so that it convects the magnetic field in the direction of the heat flow; $\boldsymbol {V}_{{CN}}$ is in the direction of $-\boldsymbol {b}\times \boldsymbol {\nabla } T_e$, which causes the magnetic field to convect along the isothermal line. Also, $V_{N}$ is much larger than $V_{CN}$ where $\chi \sim {T_e^{3/2}B}/{n_e} \ll 1$, thus the contribution of $V_{N}$ is dominating with moderate magnetic fields. The last term on the right-hand side of (2.9) reflects the diffusion of the magnetic field in different directions. Both the Nernst effect and the magnetic field diffusion are implemented in our simulations.

The intense magnetic field changes the process of electron heat conduction by magnetizing the plasma. While the magnetic field is not expected to be strong enough to significantly affect the implosion hydrodynamics via the momentum equation in ICF-relevant plasmas, it may be strong enough to magnetize the plasma and make the heat flux $\boldsymbol {q}$ anisotropic via the Lorentz forces applied on the electrons moving in different directions with respect to the local magnetic field. The magnetic field leads to a flux-limiting effect perpendicular to $\boldsymbol {B}$ compared to the classical Spitzer–Harm heat flux $\boldsymbol {q}_{sh}$, which was derived in a magnetic-free plasma. In addition, the magnetic field generates a heat flux component known as the Righi–Leduc heat flux perpendicular to both $\boldsymbol {B}$ and $\boldsymbol {\nabla } T$. The heat flux in a magnetized plasma to feed (2.4) reads as

(2.12)\begin{equation} \boldsymbol{q}_{mag} ={-}\kappa_{{\parallel}}\,\nabla_{{\parallel}}T_e -\kappa_{{\perp}}\,\nabla_{{\perp}}T_e -\kappa_{\land}\boldsymbol{b}\times\boldsymbol{\nabla} T_e, \end{equation}

where $\kappa _{\parallel }$, $\kappa _{\perp }$ and $\kappa _{\land }$ are the conduction coefficients in magnetized plasma. The detailed forms of $\kappa _{\parallel }$, $\kappa _{\perp }$ and $\kappa _{\land }$ are given in Appendix A. Along the direction parallel to $\boldsymbol {B}$, the magnetic field has no modification on the electron conduction, and $\kappa _{\parallel }$ is identical to $\kappa _{sh}$. Also, $\kappa _{\perp }$ is smaller than $\kappa _{\parallel }$ where $B\neq 0$, $\kappa _{\perp }$ retreats to $\kappa _{sh}$ where $B=0$, and $\kappa _{\perp }$ decreases as $\chi$ increases, leading to a flux-limiting effect perpendicular to the magnetic fields. The term $-\kappa _{\land } \boldsymbol {b}\times \boldsymbol {\nabla } T_e$, known as the Righi–Leduc heat flux, makes a special contribution to the heat conduction in a magnetized plasma. Both (2.6) and (2.12) have been implemented in ART, with the option to turn on and off different terms to be able to investigate a specific physical process. In this paper, $\boldsymbol {q}_{mag}$ and $\boldsymbol {q}_{sh}$ are switched in the calculation of $\boldsymbol {\nabla } \boldsymbol {\cdot } \boldsymbol {q}$ in (2.4) in the simulations to compare the 3-D ARTI evolution with and without the magnetized heat fluxes and investigate the magnetic feedback on ARTI.

3. Simulation settings

In the ART simulations, ARTI is initialized to grow from small perturbations on top of a quasi-equilibrium state abstracted from a typical profile of direct-drive NIF targets as shown in figure 2. In figure 2(a), the red box outlines the region on the target surface where ARTI occurs, which has been selected as the computational domain for the simulations in this paper. A planar approximation is valid as long as the target thickness, mode wavelength and conduction zone region are much smaller than the target radius. This condition is satisfied during most of the acceleration stage. The initial hydrodynamic profiles of a DT ablator used in case i in table 1, shown in figure 2(c), are similar to those of a direct-drive ignition target (McKenty et al. Reference McKenty, Goncharov, Town, Skupsky, Betti and McCrory2001) during the acceleration phase of an implosion driven by the 1.5 MJ, 351 nm wavelength lasers.

Table 1.

Parameters and physical options of the simulation cases in this paper. All simulation cases are conducted with grid resolution 10 grid points per micron. The Mach numbers $Ma$ are calculated on the bottom boundaries. If the $\boldsymbol {q}_{mag}$ option is off, then $\boldsymbol {q}_{sh}$ is used in (2.4). If the $\boldsymbol {q}_{mag}$ option is on, then $\boldsymbol {q}_{mag}$ is used in (2.4) instead.

As shown in figure 2(b), the cold and dense unablated DT shell is placed on top of the ablated DT plasma with a higher temperature but a lower density. The initial ablation front (the interface between the dense and the ablated plasma) is located at $z_0=70\ \mathrm {\mu } {\rm m}$, with the peak density $\rho _a=5.3$ g cm$^{-3}$ reached on the ablation front. The quasi-equilibrium state is obtained by integrating the 1-D hydrodynamic equilibrium equations in the frame of reference of the shell from the ablation front towards both sides. Above the ablation front, the shell is performing like a solid piston, so $v_z = V_a$ is initially set in the shell. In the ablation area below the ablation front, the 1-D profiles can be found by neglecting the derivatives on $t$, $x$ and $y$ in (2.1)–(2.3). The ablation front is kept approximately fixed in the middle of the computational domain by balancing the ablative pressure with a dynamically adjusted effective gravity $\boldsymbol {g}$. This is equivalent to solving the fluid equations in the frame of reference of the accelerated shell. The initial gravity is ${g}(0)=100$ $\mathrm {\mu }$m ns$^{-2}$. Since the shell mass decreases due to ablation, the effective acceleration $g(t)$ is slowly and automatically adjusted in time during the simulation to keep the ablation front approximately fixed in space, i.e. $g(t) = [(p + \rho u^2)_{bot}-(p + \rho u^2)_{top}]/M_{tot}$, where the subscripts ‘bot’ and ‘top’ indicate the integral values at the bottom and top boundaries, respectively, and $M_{tot}$ is the total mass of the remaining plasma in the computational domain. The quasi-equilibrium hydrodynamic profiles together with a time-dependent but spatially uniform gravity $g(t)$ are used to mimic an already well-established quasi-equilibrium ICF plasma slowly evolving under the isobaric assumption also used in the analytical ARTI theories (Goncharov et al. Reference Goncharov, Betti, McCrory, Sorotokin and Verdon1996a,Reference Goncharov, Betti, McCrory and Verdonb). The ablation velocity is $V_a = 3.5$ $\mathrm {\mu }$m ns$^{-1}$, and the ablation pressure (i.e. the pressure at the ablation surface) is $p_a = 140$ Mbar. In this quasi-equilibrium state, the unablated plasma flows in from the top boundary approximately at the ablation velocity, and the ablated plasma flows out through the bottom boundary at a very high velocity, as shown in figure 2(c).

Our simulations do not include the underdense region where the lasers are interacting with the plasma, therefore we do not handle laser absorption directly. Instead, the laser energy transported towards the ablation front is simulated by a constant bottom boundary heat flux $Q_{bottom} = 6.0\ \mathrm {MW}\ \mathrm {\mu } \mathrm {m}^{-2}$, which is calculated self-consistently with the Spitzer–Harm model on the basis of the 1-D hydrodynamic profiles. A characteristic Mach number that is also the maximum in the simulation zone is evaluated on the bottom boundary as ${Ma}=0.43$, indicating a subsonic outflow. The Mach number decreases rapidly from the bottom boundary towards the ablation front neighbourhood where ARTI grows (typically $Ma < 0.3$). The Atwood number is $A_T= 0.66$, and the Froude number is ${Fr}=4.61$. In addition to the above parameters associated with the initial hydrodynamic profiles in figure 2(c), we also use different $V_a$, $P_a$ and $g_0$ to explore the ARTI and magnetic evolution in a broader parameter space. The detailed simulation parameters are listed in table 1.

In order to seed 3-D ARTI, the velocity perturbations ($\boldsymbol {v}_{{p}}$) are initialized around the ablation front in a divergence-free form as

(3.1)$$\begin{gather} v_{px}=v_{p0} \sin(k x) \exp({-}k\vert z-z_0 \vert), \end{gather}$$

(3.2)$$\begin{gather}v_{py}=v_{p0} \sin(k y)\exp({-}k\vert z-z_0 \vert), \end{gather}$$

(3.3)$$\begin{gather}v_{pz}=v_{p0} [\cos(k x)+\cos(k y)]\exp({-}k\vert z-z_0 \vert), \end{gather}$$

where $v_{p0}$ is the magnitude of the initial velocity perturbation, set as $v_{p0} = 0.5$ $\mathrm {\mu }$m ns$^{-1}$. A typical simulation is carried out with a simulation box 140 $\mathrm {\mu }$m in the $z$ direction, while the widths in the $x$ and $y$ directions are chosen to be $\lambda$. A uniform Cartesian grid is used, with resolution 10 grid points per micron, and the grid independence is checked to ensure numerical convergence. Periodic boundary conditions are applied in the $x$ and $y$ directions for both the hydrodynamic quantities and the magnetic field. The inflow/outflow boundary conditions are used on the top/bottom boundaries in the $z$ direction for hydrodynamics to facilitate the ablated plasma flow to leave the simulation zone smoothly. The adiabatic boundary condition is applied for the top boundary, i.e. $\partial T/\partial z=0$. And the fixed-flux boundary condition is applied for the bottom boundary, i.e. $-\kappa \,\partial T/\partial z=Q_{bottom}$. The top and bottom boundary conditions for the magnetic field are set as $\partial (v_z \boldsymbol {B}) /\partial z = 0$.

5. Summary

The self-generated magnetic field in 3-D single-mode ARTI is investigated numerically with the parameters relevant to direct-drive ICF. This study finds that 3-D ARTI can produce much stronger magnetic fields, reaching magnitudes up to a few thousand teslas, compared to its 2-D counterpart. Similar to the 2-D cases, the inclusion of the Nernst effect significantly alters magnetic field convection and amplifies magnetic fields from the linear to the nonlinear stage of ARTI. The magnetic field is compressed towards the spike tip by the Nernst effect, reaching approximately twice the peak magnitude in the non-Nernst case. Moreover, it is found that the influence coming from $\boldsymbol {V}_{{CN}}$ on the magnetic field convection is less significant than that from $\boldsymbol {V}_{{N}}$. The Davies et al. (Reference Davies, Wen, Ji and Held2021) transport model causes minimal changes on the ARTI growths, but mostly alters the magnetic field distribution and reduces $B_{peak}$ compared to the Braginskii (Reference Braginskii1965) model.

In many 3-D cases, the Hall parameter can reach a characteristic value $\chi _c = 0.4$, beyond which the magnetized heat flux deviating significantly from the unmagnetized classical Spitzer–Harm heat flux is able to affect ARTI evolution. Unlike the magnetic field significantly accelerating the bubble growth in the short-wavelength 2-D modes, the magnetic field mostly accelerates the spike growth but has little influence on the bubble growth in 3-D ARTI. The spike acceleration due to magnetic feedback is attributed to the ablation reduction and the pressure reduction near the spike tip. The accelerated growth of spikes in ARTI due to the self-generated magnetic field is expected to enhance mixing of the materials near the ablation front, threatening the shell integrity. Moreover, the kinetic energy carried by the spikes and bubbles does not contribute well to the implosion and is harmful to the shell compression and implosion performance in ICF.

This work is focused on NIF direct-drive ICF-relevant parameters. The ICF fluids are influenced by sophisticated physics, and other laser configurations and/or target designs generally yield different hydrodynamic profiles. However, common key features exist in most direct-drive ICF fluids near the ablation front: high energy density, strong ablation, large $\beta$, etc. The findings of this work are expected to be representative in capturing the physical picture of the self-generated magnetic field in ARTI in the plasmas with these common features. By scanning $V_a$, $p_a$ and $g$, we have also explored a broader parameter space in which the trends of the magnetic field evolution obtained are expected to provide referential information to other ICF designs. The simulations also cover a range of key hydrodynamic parameters ($A_T$, $Fr$ and $Ma$) that are relevant to other hydrodynamic or astrophysical scenarios. The self-generated magnetic fields in hydrodynamic instabilities in various regimes are important topics to be explored in the future.