Abstract
In earlier works the Krook model for nonlocal electron energy transport in laser produced plasmas was examined. This paper extends the earlier work by treating spherical configurations, specifically laser driven implosions. Additions to the theory due to spherical geometry are worked out. As in the planar case, the nonlocal effects manifest themselves both as flux limitation and preheat. Nonlocal transport does have an effect on the fusion gain of laser pellet implosions.
This work was supported by DoE/NNSA. We would like to thank Valeri Goncharov for discussions involving how frequently to calculate the thermal flux.
I. INTRODUCTION
II. MODIFICATIONS TO THE THEORY: ANALYTICAL
A. The vacuum boundary
B. The center
III. MODIFICATIONS TO THE THEORY: NUMERICAL
A. Limits on and
B. The local conductivity
C. Definition of
D. The energy groups
E. The diffusion approximation
F. Speeding up the simulation
IV. THE UNIVERSITY OF ROCHESTER EXPERIMENT
V. THE FUSION TEST FACILITY PELLET
VI. CONCLUSIONS
Key Topics
 Green's function methods
 25.0
 Boundary value problems
 8.0
 Plasma temperature
 8.0
 Boltzmann equations
 7.0
 Diffusion
 7.0
Figures
A schematic of the temperature profile in space used to calculate the nonlocal flux near the center.
A schematic of the temperature profile in space used to calculate the nonlocal flux near the center.
A schematic of the region of (and also ) space where the hyperbolic sine must be used instead of the exponential in Green’s function.
A schematic of the region of (and also ) space where the hyperbolic sine must be used instead of the exponential in Green’s function.
(a) A plot of as a function of (solid) and the analytic approximation to it (dashed). (b) A plot of as a function of (solid) and the analytic approximation to it (dashed).
(a) A plot of as a function of (solid) and the analytic approximation to it (dashed). (b) A plot of as a function of (solid) and the analytic approximation to it (dashed).
(a) A plot of temperature as a function of for a test problem. (b) A plot of the Spitzer thermal flux as a function of and plots of the Krook model for the cases of 5, 10, 15, 30, and 60 energy groups. The plot of five energy groups is the faint line and the plots of ten and up all lie on top of one another.
(a) A plot of temperature as a function of for a test problem. (b) A plot of the Spitzer thermal flux as a function of and plots of the Krook model for the cases of 5, 10, 15, 30, and 60 energy groups. The plot of five energy groups is the faint line and the plots of ten and up all lie on top of one another.
Plot of (a) and (b) for the UR/LLE experiment.
Plot of (a) and (b) for the UR/LLE experiment.
For the UR/LLE experiment, a plot of maximum pressure as a function of time for the case of Spitzer conductivity and for the Krook model for calculating the thermal flux every time step, every three time steps, and every ten time steps. For the Krook model, the maximum pressure is maximum for the calculation every time step and minimum for every ten time steps.
For the UR/LLE experiment, a plot of maximum pressure as a function of time for the case of Spitzer conductivity and for the Krook model for calculating the thermal flux every time step, every three time steps, and every ten time steps. For the Krook model, the maximum pressure is maximum for the calculation every time step and minimum for every ten time steps.
For the FTF, the temporal profile of laser power. Dotted, optimized for . Solid, optimized for Krook model.
For the FTF, the temporal profile of laser power. Dotted, optimized for . Solid, optimized for Krook model.
For the FTF, laser absorption as a function of time for case (solid line) and nonlocal transport Krook model (dashed line).
For the FTF, laser absorption as a function of time for case (solid line) and nonlocal transport Krook model (dashed line).
For the FTF, temperature profile at 8 ns for the case (solid) and for the Krook model (dashed).
For the FTF, temperature profile at 8 ns for the case (solid) and for the Krook model (dashed).
For the FTF, mass averaged fuel as a function of time for (solid) and Krook (dashed).
For the FTF, mass averaged fuel as a function of time for (solid) and Krook (dashed).
For the FTF, (a) maximum pressure as a function of time for the case (solid), optimized Krook (dashed), and nonoptimized Krook (, dash dot). (b) Maximum as a function of time for the case (solid) and Krook model (dashed). (c) Total inward kinetic energy as a function of time for the case (solid) and Krook model (dashed). Notice that one significant effect of the nonlocal transport, apparent in all three graphs, is that the implosion time is delayed.
For the FTF, (a) maximum pressure as a function of time for the case (solid), optimized Krook (dashed), and nonoptimized Krook (, dash dot). (b) Maximum as a function of time for the case (solid) and Krook model (dashed). (c) Total inward kinetic energy as a function of time for the case (solid) and Krook model (dashed). Notice that one significant effect of the nonlocal transport, apparent in all three graphs, is that the implosion time is delayed.
Plot of (a) and (b) for the FTF.
Plot of (a) and (b) for the FTF.
A plot of temperature at the vacuum interface for the FTF for for Krook with the exponential Green’s function and Krook with the sinh Green’s function.
A plot of temperature at the vacuum interface for the FTF for for Krook with the exponential Green’s function and Krook with the sinh Green’s function.
Tables
Bang time and neutron production for various thermal transport models for the UR/LLE experiment. SH stands for Spitzer–Harm, FP for Fokker–Planck, ES for Epperlein and Short, and BD for beam deposition.
Bang time and neutron production for various thermal transport models for the UR/LLE experiment. SH stands for Spitzer–Harm, FP for Fokker–Planck, ES for Epperlein and Short, and BD for beam deposition.
Result of calculating heat flux every one, three, and ten time steps for untuned Krook (zero current) model.
Result of calculating heat flux every one, three, and ten time steps for untuned Krook (zero current) model.
Summary of characteristics for various transport models. Same symbols are used in Table I. MV refers to the FP solution of a thermostatic problem as described by Matte and Virmont in Ref. 5.
Summary of characteristics for various transport models. Same symbols are used in Table I. MV refers to the FP solution of a thermostatic problem as described by Matte and Virmont in Ref. 5.
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