Nonlocal Heat Transfer in a Laser Inertial Confinement Fusion for the Direct Irradiation Scheme

High gradients of electron temperature appear in plasma corona under direct laser irradiation of inertial confinement fusion targets. This results in nonlocality of heat transport. Such effect influence the efficiency of laser absorption, redistribute heat fluxes and could preheat plasma ahead the front shock wave, therefore alter the compression adiabat. Ignition requires a specially tuned compression dynamics, so such an effect should be taken into account. Target simulations with nonlocal models show the decrease of compression efficiency and hot-spot parameters degradation compared to local models: Spitzer–Härm model with and without flux-limiter.

The scheme of direct laser irradiation of a target to reach inertial confinement fusion (ICF, laser-induced nuclear fusion is considered in this work) is widely discussed at present [1]. The ignition condition can be ensured only under a certain dynamics of target compression. One of the necessary conditions is to maintain a low shock adiabat, which can be significantly complicated by the undesired preheating by nonthermal electrons appearing in the hot laser corona. Such preheating is discussed primarily for hot electrons generated because of parametric instabilities initiated by laser radiation propagating in the plasma corona [2]. At the same time, undesired preheating of the central part of the target is also possible at high temperatures and high temperature gradients because of the nonlocal heat flux q_e responsible for the kinetic transport of fastest electrons from absorption region to the center of the target. These electrons present a small fraction of electrons’ bulk and that undergo weak collisions. The velocity of such electrons is several times higher than thermal velocities, i.e., much lower than the velocities of fast parametrically generated electrons having energies of ~30–100 keV. In megajoule facilities, where the corona is heated to temperatures of about 5 keV, the energy of penetrating hot electrons can also reach tens of keV. However, in contrast to parametric instabilities, which are initiated only when laser intensity threshold is fulfilled and are specially suppressed in ICF, the nonlocality of the heat flux is almost always manifested under the conditions characteristic of ICF. The effect is enhanced with an increase in the temperature of the corona in the laser target, which was discussed in several publications and was demonstrated in some experiments [3–5]. The nonlocality-induced redistribution of heat flows not only results in preheating but also can change the absorption efficiency in the laser corona and affect the formation of shock waves, directly influencing the dynamics of target compression.

The localized laser energy deposition near the critical density in the scheme of the direct irradiation of the target leads to high temperature gradients, particularly at the initial stage of the laser pulse. This results in the violation of the classical Fourier law (for the plasma, the Spitzer–Härm law [6]), according to which the heat flux is proportional to the temperature gradient, ${\mathbf{q}_{{{\text{SH}}}}} = - {{\kappa }_{{{\text{SH}}}}}\nabla {{T}_{e}}$, which is valid only for su-fficiently low temperature gradients, ${{L}_{T}} \approx$ ${{(d\ln {{T}_{e}}{\text{/}}dx)}^{{ - 1}}}$, when

$${{\lambda }_{{ei}}}(T) < 0.06{{Z}^{{ - 1/2}}}{{L}_{T}}$$

(1)

(see [7]). Here, λ_ei is the mean free path of electrons and Z is the degree of ionization of the plasma. Calculations within the classical Spitzer–Härm model beyond the region of its applicability usually give nonphysically high heat fluxes exceeding the maximum possible Knudsen heat flux corresponding to the collisionless transport ${{q}_{{\max }}} \sim {{n}_{e}}{{v}_{{Te}}}{{T}_{e}}$, where ${{n}_{e}}$ and ${{v}_{{Te}}}$ are the density and thermal velocity of electrons, respectively. The early models were proposed to limit the heat transfer in the form of the limit heat flux ${{q}_{{{\text{lim}}}}} \sim f{{n}_{e}}{{v}_{{Te}}}{{T}_{e}}$, preventing even the passage to the Knudsen limit, where f is the phenomenological -limiting factor within a simple local model: $1{\text{/}}{{q}_{e}} = 1{\text{/}}{{q}_{{{\text{SH}}}}} + 1{\text{/}}{{q}_{{{\text{lim}}}}} \equiv 1{\text{/}}q_{e}^{f}$. The limiting factor is usually taken in the range of 0.03–0.2 according to empirical data [8]. Such a simple model sometimes cannot describe real heat transfer because the limiting factor is not universal but is at least a functional of the space–time distribution of the temperature. Correspondingly, the correctness of describing heat fluxes at high energy depositions within such simplified model is doubtful.

The most accurate method to calculate the nonlocal heat flux based on the solution of the Fokker–Planck equation for electrons is too numerically expensive and cannot allow a complete simulation at hydrodynamic times. For this reason, a number of nonlocal transfer models were developed [9–16] and actively incorporated in hydrodynamic codes in order to describe both the flux in the region of maximum gradient and preheating of the plasma within a single approach. Some of them are based on the nonlocal convolution of the classical heat flux q_SH with the kernel G chosen ambiguously according to some kinetic reasons. For example, the convolution [7, 17, 18]

$${{q}_{{{\text{nl}}}}}(x) = \int G(x,x{\kern 1pt} '){{q}_{{{\text{SH}}}}}(x{\kern 1pt} ')dx{\kern 1pt} '$$

(2)

is usually used for the one-dimensional geometry. Here, the nonlocal kernel can be represented in the dimensionless form in terms of the function Ψ:

$$G(x,x{\kern 1pt} ') = \frac{{\Psi (\eta (x,x{\kern 1pt} '))}}{{2a\lambda _{{ei}}^{'}}},\quad \eta = \frac{{{\text{|}}x - x{\kern 1pt} '{\text{|}}}}{{a\lambda _{{ei}}^{'}}},$$

(3)

where η is the dimensionless difference coordinate, a is the model parameter specifying the characteristic spatial scale of nonlocality, and the prime in $\lambda _{{ei}}^{'}$ means that the free path is calculated at the point x'. Since the ICF plasma is inhomogeneous, the expression for η in Eqs. (3) is changed to

$$\eta (x,x{\kern 1pt} ') = \frac{1}{{{{n}_{e}}(x{\kern 1pt} ')a{{\lambda }_{{ei}}}(x{\kern 1pt} ')}}\left| {\int\limits_x^{x'} {{n}_{e}}(x{\kern 1pt} '')dx{\kern 1pt} ''} \right|,$$

(4)

where the dependence of the electron free path on the density profile along the trajectory is taken into account.

Within a linear nonlocal theory based on the exact solution of the linearized Fokker–Planck for small perturbations, the nonlocal kernel for the analytical model (marked as BB) was obtained in [7] in the form

$${{\Psi }_{{{\text{BB}}}}}(\eta ) = \frac{2}{\pi }\int \frac{{dp{\kern 1pt} \cos (\eta p)}}{{1 + {{p}^{{0.9}}}}}$$

(5)

with the expression ${{a}_{{{\text{BB}}}}} = 10\sqrt Z (Z + 5){\text{/}}(Z + 12)$ for the constant a in Eqs. (3) [7]. The exponent of p in the denominator is the approximation of the numerical result obtained for the kernel in the Fourier space for gradient scales down to the mean free path of electrons. This exponent determines the asymptotic behavior of the kernel (5) in the limit $p \to \infty$, which corresponds to the extreme heat flux almost coinciding with the kinetic Knudsen flow ${{q}_{{\max }}}$. Strictly speaking, the latter corresponds to ${{p}^{{ - 1}}}$, close to ${{p}^{{ - 0.9}}}$ (the difference between the results of calculations with these two exponents presented below is negligibly small). In the limiting case p → 0, this model corresponds to the classical Spitzer–Härm heat flux ${{q}_{{{\text{SH}}}}}$.

We compare the model with the kernel (5) with the simplest heuristic model first proposed in [17] with the kernel

$${{\Psi }_{{{\text{exp}}}}} = \exp ( - \eta )$$

(6)

and the combined nonlocal model with the kernel Ψ_BB at finite η values and the kernel Ψ_exp in the limit $\eta \to \infty$, i.e., with the kernel

$${{\Psi }_{2}}(\eta ) = \frac{{\ln (1 + 1{\text{/}}\eta + \eta )\exp ( - \eta )}}{{1.553}},$$

(7)

where the constant corresponds to the unit integral of Ψ with respect to η over $[0;\infty )$. It is noteworthy that the interpolation Ψ₂ model in the region of parameters does not lead to the preheating of the target because it has an exponential asymptotic behavior rather than the physically justified power-law asymptotic behavior as in the BB model: $\mathop {\lim }\limits_{\eta \to \infty } {{\Psi }_{{{\text{BB}}}}}(\eta ) \propto {{\eta }^{{ - 1.9}}}$, which is responsible for pronounced preheating tails in the heat flux corresponding to the power-law asymptotic behavior of the velocity distribution function of electrons, in particular, demonstrated in [19]; these tails are due to electrons arriving from more heated regions.

To directly compare the listed kernels, we used the same coefficient a in Eq. (3), although it is slightly different in [17]. The dependence of these kernels on the parameter η is shown in Fig. 1. The kernel Ψ_BB is larger than the other kernels in the limit $\eta \to \infty$. Because of the normalization condition $\int_{ - \infty }^\infty \Psi d\eta = 1$, the kernel Ψ₂ is larger than Ψ_BB in the region η ~ 1, which affects the results at a large temperature gradient (see below).

Fig. 1.

(a) (Color online) Behavior of the kernel of various models.

The described nonlocal heat transfer models are implemented in our radiation hydrodynamic codes ERA [20] and FRONT [21] developed to calculate ICF targets. To verify these numerical codes based on the above nonlocal heat transfer models, we use the Fokker–Plank kinetic simulation results for heat wave propagation:

$$\begin{gathered} \frac{{\partial {{f}_{e}}}}{{\partial t}} + v\mu \frac{{\partial {{f}_{e}}}}{{\partial x}} - \frac{{e{{E}_{x}}}}{{{{m}_{e}}}}\left\{ {\frac{1}{{{{v}^{2}}}}\frac{\partial }{{\partial v}}\left( {{{v}^{2}}\mu {{f}_{e}}} \right)} \right. \\ \left. { + \;\frac{1}{v}\frac{\partial }{{\partial \mu }}\left( {\left( {1 - {{\mu }^{2}}} \right){{f}_{e}}} \right)} \right\} = I[{{f}_{e}}], \\ \end{gathered} $$

(8)

where ${{f}_{e}}(t,x,v,\mu )$ is the distribution function of electrons over the coordinate x and the velocity v ($v$ is the magnitude of the velocity and μ is the cosine of the angle between the velocity and the x axis), E_x is the self-consistent electric field, and I[ f_e] is the collision integral (we use the electron–electron and electron–ion collision integrals in the Landau form). It is convenient to separate the algorithm of the numerical solution of the kinetic equation (8) for electrons based on the split method for physical processes (separate steps for transfer and collisions). The first step includes the solution of the kinetic equation with the nonlinear collision integral I[ f_e] using a fully conservative explicit difference scheme [22]. The second step is the solution of the difference equation for f_e with a simple explicit upwind scheme. The self-consistent electric field E_x was calculated with an asymptotically correct scheme [23] adapted for the two-dimensional geometry in the velocity space. The electric field at the initial time satisfies the Poisson equation, and the grid values of the plasma density and the electric current at each time satisfies the continuity equation. This scheme allows one to calculate with a high accuracy the self-consistent electric field even at spatial scales significantly exceeding the Debye radius.

As an illustrative example for comparison of different nonlocal transfer models, we consider the evolution of the initial stepwise temperature distribution with a finite gradient

$$T(x) = \left\{ \begin{gathered} {{T}_{1}},\quad x < {{x}_{1}} \hfill \\ {{T}_{1}} + \frac{{{{T}_{0}} - {{T}_{1}}}}{{{{x}_{2}} - {{x}_{1}}}}(x - {{x}_{1}}),\quad {{x}_{1}} \leqslant x < {{x}_{2}} \hfill \\ {{T}_{0}},\quad {{x}_{2}} \leqslant x. \hfill \\ \end{gathered} \right.$$

(9)

Here, the parameters of the uniform plasma with the electron density n₀ = 10²¹ cm^–3 and $Z = 1$ in the calculation region [0, L], where $L = 2400{{\lambda }_{0}}$, were chosen as follows: T₁ ≫ T₀ = 0.6 keV, x₁ = 492λ₀, x₂ = 500λ₀, where λ₀ = 5.2 μm is the mean free path of electrons at the chosen T₀ and n₀ values. The calculations were performed with two initial temperature ratios T₁/T₀ = 5 and 20. In both cases, the expansion of the plasma at the initial stage is kinetic because the mean free path of electrons is much longer than the threshold value of the gradient length for nonlocality of heat transfer (1).

The kinetic calculation was performed on a uniform grid with the division of the calculation region into 12 000 × 600 × 20 cells in the x coordinate, the magnitude of the velocity, and the cosine of the angle between the velocity and the x axis, respectively. The characteristic time of electron–ion collisions was ${{\tau }_{{ei}}} \approx 0.5$ ps, a constant time step of $5 \times {{10}^{{ - 3}}}{{\tau }_{{ei}}}$ was chosen, and the total calculation time was 400τ_ei = 200 ps. The initial distribution function in each spatial cell was taken Maxwellian corresponding to the initial profile of the electron temperature and a uniform distribution of the electron and ion densities. Stationary Maxwellian distributions with the temperatures T₁ and T₀ were set at the left and right boundaries, respectively. The same boundary temperatures were used in the hydrodynamic calculations. In all kinetic and hydrodynamic model simulations, ions were considered immobile, forming a neutralizing background for the electron subsystem, and a constant Coulomb logarithm of 10 was taken.

To reveal features of heat flux formation in the entire domain of the kinetic solution, we estimate the contributions of different velocity groups of electrons (with velocities in the range of ${{v}_{1}} < v < {{v}_{2}}$) to the heat flux

$${{q}_{v}}({{v}_{1}},{{v}_{2}}) = \frac{1}{2}\int\limits_{{{v}_{1}}}^{{{v}_{2}}} dv{{v}^{5}}\int\limits_{ - 1}^1 d\mu \mu {{f}_{e}}(v,\mu ).$$

(10)

The quantity q_tot = ${{q}_{v}}$(0, ∞) is the total heat flux at a given point of the space. Figure 2 shows the calculated ${{q}_{v}}({{v}_{1}},{{v}_{2}})$/q_tot values at a time of 50 ps. It is clearly seen that the heat flux at different points of the space is determined by electrons with different velocities. In particular, in the region of the highest temperature gradient (400–600λ_ei), the heat flux is determined by electrons with velocities $< 4{{v}_{T}}$. The contribution of high-energy particles to the heat flux increases with the distance from the main front of the heat wave. Electrons with $v > 5{{v}_{T}}$ dominate in the region of 1500–2000λ_ei. This differentiation of contributions of particles with different energies to the heat flux along the profile of the heat wave (because of a strong energy dependence of the mean free path of electrons $\lambda \propto {{E}^{2}}$) indicates that the electron distribution function is time dependent, which leads to a difference between the results of the kinetic and hydrodynamic simulations. This difference decreases with a decrease in the temperature gradient and the formation of the quasistationary velocity distribution of particles, which is almost established for a lower temperature gradient at times about the end of kinetic calculations (200 ps). At this time, the maximum ratio λ_ei/L_T in the region of the largest temperature gradient is 0.03 and 0.2 for T₁/T₀ = 5 and 20, respectively (see Figs. 3, 4).

Fig. 2.

(Color online) Contribution of various electron groups to the heat flux divided by the total flow q_tot at the time t = 50 ps versus the velocity of electrons.

Fig. 3.

(Color online) Profiles of the relative temperature T/T₀, relative excess of the temperature over the background T/T₀ – 1, and dimensionless heat flux $q{\text{/}}{{n}_{0}}{{v}_{{{{T}_{0}}}}}{{T}_{0}}$ at the time t = 200 ps for the problem with T₁/T₀ = 5 according to the calculations within (FP) the Fokker–Planck equation, (SH) Spitzer–Härm model, (Ψ_BB) model with the kernel Ψ_BB, (Ψ₂) model with the kernel Ψ₂, and (f = 0.15) model with the limiting factor f = 0.15.

Fig. 4.

(Color online) Same as in Fig. 3 but for the problem with T₁/T₀ = 20.

In the former case (Fig. 3), the implemented hydrodynamic regime corresponds to the local heat flux, which is evidenced by good agreement of the temperature and heat flux profiles (in the region x ≤ 700λ_ei of the highest temperature gradient) for almost all considered models. A noticeable difference in the temperature and heat flux profiles is observed only near the front of the heat wave. The local model predicts a sharp heat front, which is strongly smoothed for all nonlocal models and kinetic calculation. In this case, the nonlocal BB model predicts the largest preheating, which decreases in the presence of the exponential tail in the kernel (which is closer to the kinetic calculation). The limitation of the heat flux changes the velocity of the front of the heat wave.

An increase in the initial temperature gradient increases the difference between the used models and kinetic simulation, particularly in the region of preheating ahead of the front of the heat wave (see Fig. 4), although the results obtained within all nonlocal models in the region of the highest temperature gradient (x ≤ 1200λ_ei) do not noticeably differ from the kinetic calculation. In this case, the model with an exponential phenomenological asymptotic behavior gives even a higher heat flux (and, correspondingly, a larger preheating) than the nonlocal BB model. The pure exponential model provides an unstable solution in this regime (see, e.g., [18]). Thus, the test calculations for the propagation of the heat wave with a high temperature gradient show some differences between the existing models even in a simplified problem of nonlocal heat transfer. These differences are apparently due to the quasistationary transfer approximation used in these models. To determine whether these differences are significant for the dynamics of targets, we performed simulations reported below.

Very high temperature gradients can be expected in the case of a high intensity of laser irradiation, e.g., in the currently popular scenario of shock ignition [24]. At the same time, the parameters of the plasma co-nsidered in the first case are closer to the direct irradiation scheme planned in the Russian megajoule facility [25].

To estimate the effect of heat flux nonlocality on the efficiency of compression of the direct irradiation target, we consider the geometry, target composition, and laser pulse profile as in [25]. The spherical target consists of three layers: the inner region is filled with DT gas, the next layer is DT ice, and the outer layer consists of plastic CH ablator. The description of the compression of the target in the one-dimensional approximation makes it possible to exclude the additional multidimensional factors affecting its dynamics such as the inhomogeneity of irradiation and hydrodynamic instabilities. The target dynamics is described by a hydrodynamic model of two-temperature plasma with the electron and ion heat conductions, radiative transfer, and laser energy deposition taking into account the inverse bremsstrahlung absorption mechanism. The equation of states and mean free paths are evaluated within the RESEOS model [26] (in the ERA code) or within the ideal plasma model with complete ionization (in the FRONT code). The latter is a quite strong approximation but is allowable because the target consists of materials with a relatively small charge and, consequently, the ionization state at high temperatures of the plasma formed after irradiation is close to maximal.

The calculations with different models with different nonlocal kernels give different results for the dynamics and structure of the dense shell (see Fig. 5). The stronger the nonlocality (more pronounced tails in the function G: SH → Ψ₂ → Ψ_BB), the higher the temperature in the dense shell and the lower the density peak. The same is valid for a low-dense plasma surrounded by this shell. In this case, its preheating is determined not only by nonlocality but also by a significant effect of radiative transfer.

Fig. 5.

(Color online) Profiles of the density and electron temperature at the time t = 8 ns calculated with the ERA code for (SH) local heat transfer model with the limiting factor f = 0.15 and two nonlocal models with the kernels Ψ_BB and Ψ₂.

The results of calculations of the target are summarized in Table 1 for all three considered heat transfer models (the results in the table are obtained with the ERA code; the results within the FRONT code are qualitatively the same). The effect on compression is the strongest in the heat transfer model with the power-law kernel Ψ_BB: the calculations with this model give a significant decrease in the maximum density and the parameter ρR at the maximum compression time. This is due to a noticeable spreading of the dense shell because of its increased temperature, which is a result of a more efficient heat transfer from the laser absorption region inside the target. The latter is also confirmed by a higher temperature in the gas cavity. The Ψ₂ model predicts a smaller decrease in the parameters of the hot region. All models give close values for the fraction of the absorbed laser energy. The summarized parameter indicating the efficiency of the target is the “margin for ignition” (for details, see [27, 28]). The calculation with the local model with the limited heat flux according to [25] indicates the efficiency of the target (its ignition), which is also predicted by the nonlocal Ψ₂ model with the exponential tail (the ignition threshold is insignificantly lower). The calculation with the nonlocal BB model gives a much lower margin for ignition W_Q < 1 (not enough high for confident ignition), which is due to the preheating of the shell and central part of the target. The calculations with the considered nonlocality models demonstrate different magnitudes of influence on the parameters and indicate the necessity of the quantitative kinetic inclusion of the nonlocal effect in simulations of the ignition laser ICF.

Table 1. Parameters of the target calculated with the ERA code: the fraction of the absorbed laser energy K_a, electron temperature T_ec at the critical density at the time t = 8 ns, the maximum density ρ_max and maximum ion temperature T_{i, max} in the hot region of the target, the surface density at the maximum compression time (ρR)_max, and the margin for ignition W_Q

To summarize, it has been demonstrated that the effect of nonlocality is really dangerous for direct irradiation targets, where the corona is hotter than that in the indirect scheme since the mean free path of electrons in the former case is longer at almost the same characteristic spatial scales in the target. The effect should be stronger at high laser intensities in the shock ignition scheme. It occurs even when the development of parametric instabilities, which are currently considered as the main sources of hot electrons, can be impossible. The effect of nonlocal transfer in the reported simulations is sufficient to change the parameters in the hot region of the target at the time of its maximum compression and to affect the ignition process. This is due both to a change in the heat flux near the dense shell, which modifies its structure and dynamics, and to the preheating of the inner part of the gas cavity. It is also remarkable that nonlocal mo-dels with kernel involve only moments of the distribution function of particles, which incompletely describes kinetics transport. Furthermore, it is assumed that heat transport is quasistationary; i.e., the dynamics of electron propagation is disregarded. The effect of nonlocality can be accurately taken into account only in the full-scale kinetic simulation, which is currently impossible for the studied targets. At the same time, it is necessary to develop a hybrid approach based on Fokker–Planck calculations for short times in individual spatial regions that are co-nsistently incorporated in the global hydrodynamic simulation. This work explicitly shows how simplified nonlocal models can be further developed and ver-ified.