1.Graduate School, China Academy of Engineering Physics, Beijing 100088, China 2.Institute of Applied Physics and Computational Mathematics, Beijing 100094, China 3.Center for Applied Physics and Technology, Peking University, Beijing 100871, China 4.Science and Technology on Plasma Physics Laboratory, Laser Fusion Research Center, China Academy of Engineering Physics, Mianyang 621900, China
Fund Project:Project supported by the Science Challenge Project (Grant No. TZ2016005), the National Key Programme for S& T Research and Development in China (Grant No. 2016YFA0401100), NSAF (No. U1730449), the National Natural Science Foundation of China (Grant No. 11975055 and 11575030)
Received Date:18 September 2019
Accepted Date:25 November 2019
Published Online:05 February 2020
Abstract:When evaluating the plasma parameters in inertial confinement fusion, the flux-limited local Spitzer-H?rm (S-H) model in radiation hydrodynamics simulations may be invalid when electron temperature gradient is too large. In other publications, the electron distribution function (EDF) could be explained by comparing the energy equipartition rate $R_{\rm eq}=\dfrac{1}{2}m_{\rm e}v_{\rm te} ^2\nu_{\rm ee}$ with the heating rate $R_{\rm heat}=\dfrac{1}{2}m_{\rm e}v_{\rm os} ^2\nu_{\rm ei}$. When the condition $R_{\rm heat}\sim R_{\rm eq}$ is satisfied, the EDF deviates from Maxwell equilibrium distribution, and is well fitted to the super-Gaussian distribution $f({{ v}})=C_m{\rm e}^{-(v/v_m)^m}$ with the index m ($2<m<5$). The number of energetic electrons of the super-Gaussian distribution is less than that of the Maxwell distribution, which plays an important role in electron heat flux, especially for electrons of 3.7$v_{\rm te}$. So electron heat flux of the super-Gaussian distribution is smaller than that of the Maxwell distribution. In this paper, EDF and electron heat flux in laser-produced Au plasma are simulated by using 1D3V PIC code (Ascent). It is found that in the coronal region, the laser intensity is larger, and the electron temperature is lower than the high-density region. So $\alpha=Z(v_{\rm os}/v_{\rm te})^2>1$, $R_{\rm heat}>R_{\rm eq}$, the EDF is well fitted to super-Gaussian distribution, where the index m is evaluated to be 3.34. In this region, the large electron temperature gradient leads to a small temperature scale length ($L_{\rm e}=T_{\rm e}/(\partial T_{\rm e}/\partial x)$), but the low e-e and e-i collision frequencies lead to a large electron mean-free-path ($\lambda_{\rm e}$). So the Knudsen number $\lambda_{\rm e}/L_{\rm e}$ is evaluated to be 0.011, which is much larger than the critical value $2\times10^{-3}$ of the S-H model, flux-limited local S-H electron heat flux is invalid. As a result, the limited-flux S-H predicts too large an electron heat flux, which results in much higher electron temperature of radiation hydrodynamics simulation than that of SG experiments. This heat flux inhibition phenomenon in coronal region cannot be explained by the flux-limited local S-H model, and non-local electron heat flux should be considered. In the high density region, the laser intensity is weaker, and the electron temperature is higher, so $\alpha=Z(v_{\rm os}/v_{\rm te})^2<1$, $R_{\rm heat}<R_{\rm eq},$ but EDF is still well fitted to super-Gaussian distribution, where the index m is evaluated to be 2.93. In this region, $L_{\rm e}$ is larger, $\lambda_{\rm e}$ is smaller, so the Knudsen number is smaller, which is evaluated to be $7.58\times10^{-4}<2\times10^{-3}$. As a result, The flux-limited local S-H electron heat flux is valid. However, the electron heat flux depends on the flux limiting factor ($f_{\rm e}$) that varies with laser intensity and electron temperature. Keywords:electron distribution function/ Knudsen number/ heat flux inhibition/ flux limiting factor
当$ \alpha\rightarrow0 $(无激光)时, $ m = 2 $, 电子分布函数回归麦克斯韦分布; 当$ \alpha\rightarrow\infty $(等离子体热速度趋于0) 时, $ m = 5 $, 电子分布函数表现为极端超高斯分布. 如图1所示, PIC模拟给出的电子分布函数与超高斯分布非常符合, 在低密度区域($ x = 100\lambda_0 $), Te= 1.53 keV, α = 1.49, m = 3.34, 在高密度区域($ x = 300\lambda_0 $), Te = 2.52 keV, α = 0.66, m = 2.93. 这与理论上低密度区域光强更大, 电子分布函数更偏离麦克斯韦分布的结果一致. 从图1可以发现, PIC统计的高能段电子数目高于超高斯分布, 而低于麦克斯韦分布. 实际EDF尾端处于$ 3 v_{\rm{te}}—4 v_{\rm{te}} $的电子数目低于麦克斯韦分布近似下的电子数目, 而这部分电子对电子热流贡献最大[16,17], 因此实际热流小于局域S-H热流$ {{Q}}_{\rm {SH}} $. 辐射流体通过简单的并联限流来减小$ {{Q}}_{\rm {SH}} $, 从而与实际电子热流更为符合. 图 1 电子的能量分布函数(PIC统计的电子分布函数严重偏离了同一个温度对应的麦克斯韦平衡分布, 而在低能段与相应的超高斯分布非常符合, 其中$0.02\ n_{\rm c}\;(100\lambda_0)$处Te = 1.53 keV, α = 1.49, m = 3.34; $0.23\;n_{\rm c}\;(300\lambda_0)$处Te = 2.52 keV, α = 0.66, m = 2.93; 在高能段, 电子分布函数介于麦克斯韦分布和超高斯分布之间, 这部分电子对热流贡献最大) Figure1. Electron distribution function (dotted lines) from PIC simulation in comparison with a Maxwell equilibrium distribution (dashed and dotted lines) and a super-Gaussian distribution (solid lines). The low-energy electron distribution from PIC simulation is well fitted to the super-Gaussian distribution. PIC simulation shows that the number of energetic electrons ($3 v_{\rm {te}} - 4 v_{\rm {te}} $) is more than that from Maxwell equilibrium distribution, but less than that from super-Gaussian distribution. These energetic electrons play an important role in electron heat flux.
为了研究辐射流体模拟中限流的局域S-H电子热流与实际电子热流是否一致, 本文统计了PIC模拟各个区域的热流, 一种按热流的定义式$ {{F}}_{\rm{e}} = n_{\rm{e}}\left\langle \epsilon_{\rm{e}}{{v}}_{\rm{e}}\right\rangle $进行统计, 另一种按限流的局域S-H电子热流((2)式)进行计算. 如图2所示, 在高密度区域, 限流的局域S-H电子热流具有一定的适用性, 但热流$ {{F}}_{\rm e} $严重依赖于限流因子$ f_{\rm{e}} $. 0.23nc—0.43nc (300λ0—350λ0)区域的限流因子取0.05较为合适, 而0.43nc—0.80nc (350λ0—400λ0)区域的限流因子则取0.09较为合适. 这与理论上的限流因子$ f_{\rm e} $随着驱动激光的增强而减小, 随着热阻增强而减小相一致. 辐射流体对空间尺度较大的等离子体中的限流因子取一个固定值会引入明显的误差. 在低密度区域, PIC模拟出现了热流受限现象: 按热流定义式$ {{F}}_{\rm{e}} = n_{\rm{e}}\left\langle \epsilon_{\rm{e}}{{v}}_{\rm{e}}\right\rangle $统计的热流远小于限流的局域S-H电子热流. 这与赵斌和郑坚[21]的Fokker-Planck模拟结果一致. 在0.12nc—0.23nc (250λ0—300λ0)之间甚至出现了热传导系数为负的现象, 需要用非局域热传导模型做出解释[18,19]. 图 2 等离子体中的电子热流${{F}}_{\rm e}$ (热流以$(1\;n_{\rm c}, 1\;{\rm{keV}})$对应的电子自由流$Q_{\rm{fs0}}=1.936\times10^{19}\;{\rm J}/({\rm{cm}}^2\cdot {\rm{s}})$为单位; 限流的S-H电子热流无法解释等离子体冕区存在的热流受限现象, 而与高密度区域的电子热流比较符合; 但电子热流严重依赖于限流因子$f_{\rm{e}}$, 需要根据不同位置的光强和电子温度调整$f_{\rm{e}}$的大小) Figure2. Electron heat flux (black line) from PIC simulation in comparison with that from flux-limited S-H model for $f_{\rm{e}}=0.05, 0.07, 0.09$. The unit of the electron heat flux ${{F}}_{\rm{e}}$ is the electron free stream Qfs0 (1nc, 1 keV) = 1.936 × 1019 J/(cm2·s). In the high density region, the electron heat flux from the limited S-H agrees well with PIC simulation result. But in the coronal region, the electron heat flux from the flux-limited S-H is much larger than that from PIC simulation.
本文进一步统计了等离子体各个位置的电子温度和激光到达各个位置的相对光强, 如图3和图4所示. 一方面, 在低密度区域, 光强衰减速度一致, 无热流的简单光路追踪理论模型的电子温度与PIC较为符合, 而采用限流的局域S-H电子热流的RDMG程序给出的电子温度则明显高于PIC模拟结果. 这是因为RDMG程序的限流的局域S-H电子热流加热了低密度区域等离子体, 而PIC模拟发生了热流受限现象. 另一方面, 低密度区域的温度梯度比高密度区域更大, 温度梯度标长更小; 低密度区域的密度远低于高密度区域, $ \nu_{\rm{ee}}, \nu_{\rm{ei}} $更小, 电子平均自由程$ \lambda_{\rm{e}} = v_{\rm{te}}/(\nu_{\rm{ee}}+\nu_{\rm{ei}}) $更大, 因此低密度区域克努森数$ \epsilon = \lambda_{\rm{e}}/L_{\rm{e}} $更大. 当克努森数$ \epsilon $大于S-H理论的临界值$ 2\times10^{-3} $时, 局域S-H电子热流完全不适用[16,20]. $ \epsilon\,(x \!=\! 100 \lambda_0) \!=\! 0.011$$ > 2 \times 10^{-3}, \;\epsilon\,(x = 300 \lambda_0) = 7.58 \times 10^{-4} < 2\times10^{-3}, $ 因此局域S-H电子热流在低密度区域完全不适用, 即使采用并联限流的处理方式也不能使理论热流与实际热流相符合; 而高密度区域限流的局域S-H电子热流具有一定的适用性. 热流受限导致辐射流体高估了等离子体冕区的热流, 从而高估了冕区的电子温度, 造成SRS散射光光谱出现红移[7]. 而高密度区域RDMG和无热流光路追踪理论模型的电子温度高于PIC模拟结果是因为理论IBA系数$ \kappa_{{\rm s}} = \dfrac{\nu_{{\rm {ei}}}}{\rm c}\dfrac{n_{{\rm e}}/n_{{\rm c}}}{\sqrt{1-n_{{\rm e}}/n_{{\rm c}}}} $随密度急剧增大, 而实际IBA系数随电子-离子碰撞频率$ \nu_{{\rm {ei}}} $的变化相对平缓[23]. 图 3 电子温度的空间分布(上方三条粗线对应$5000T_0$, 下方三条细线对应$1300T_0$, 蓝色点线Theory表示无热流的光路追踪理论模型; 等离子体冕区温度梯度大, 克努森数超过临界值, 限流的局域S-H电子热流无法准确描述电子热流, 造成了RDMG对冕区电子热流的高估, 从而高估了电子温度; 高密度部分区域由于实际IBA系数低于理论值, 导致RDMG和无热流光路追踪模型估计的电子温度高于PIC结果) Figure3. Electron temperature from PIC simulation (solid lines) in comparison with that from radiation hydrodynamic simulation RDMG (dashed lines) and that from the optical path tracking model without heat flux (dashed and dotted lines). In the coronal region, the limited-flux S-H predicts too large electron heat flux and results in too high electron temperature from RDMG.
图 4 等离子体中的光强衰减图(上方三条粗线对应$5000T_0$, 下方三条细线对应$1300T_0$, 蓝色点线Theory表示无热流的光路追踪理论模型; 由于高密度区域实际IBA系数低于理论值, 导致RDMG和无热流光路追踪模型高估了能量沉积, 激光衰减更快而无法到达临界面) Figure4. Decay curves of laser intensity from PIC simulation (solid lines) in comparison with that from radiation hydrodynamics simulation RDMG (dashed lines) and that from the optical path tracking model without heat flux (dashed and dotted lines). In the high density region, the inverse Bremsstraws absorption coefficient is smaller than the classical expression $\kappa_{\rm{s}}=\dfrac{\nu_{\rm{ei}}}{c}\dfrac{n_{\rm{e}}/n_{\rm{c}}}{\sqrt{1-n_{\rm{e}}/n_{\rm{c}}}}$, so the laser intensity from PIC simulation is higher than that from RDMG and the optical path tracking model.
图 1 电子的能量分布函数(PIC统计的电子分布函数严重偏离了同一个温度对应的麦克斯韦平衡分布, 而在低能段与相应的超高斯分布非常符合, 其中
图 2 等离子体中的电子热流
图 3 电子温度的空间分布(上方三条粗线对应
图 4 等离子体中的光强衰减图(上方三条粗线对应
