The study of Rayleigh–Taylor instability in the deceleration phase of inertial confinement fusion implosions is carried out using the three-dimensional (3-D) radiation-hydrodynamic Eulerian parallel code DEC3D. We show that the yield-over-clean is a strong function of the residual kinetic energy (RKE) for low modes. Our analytical models indicate that the behavior of larger hot-spot volumes observed in low modes and the consequential pressure degradation can be explained in terms of increasing the RKE. These results are derived using a simple adiabatic implosion model of the deceleration phase as well as through an extensive set of 3-D single-mode simulations using the code DEC3D. The effect of the bulk velocity broadening on ion temperature asymmetries is analyzed for different mode numbers –12. The jet observed in low mode is shown to cause the largest ion temperature variation in the mode spectrum. The vortices of high modes within the cold bubbles are shown to cause lower ion temperature variations than low modes.
I. INTRODUCTION
In inertial confinement fusion (ICF) implosions, a spherical shell of cryogenic deuterium and tritium (DT) ice is imploded at high velocities (300–400 km/s) to compress the DT gas to achieve high central temperatures (∼5 keV) and high areal densities (∼1–2 g/cm2) at stagnation. The DT fusion reactions1,2 at such high temperatures produce 3.5-MeV alpha particles and 14.1-MeV neutrons. The energies of alphas continuously transfer to the DT plasma within the hot spot, first dragged by electrons along straight lines3–5 and later scattered by ions through diffusion.6 Typical alpha slowing-down time scales (∼10−15 s) are short compared to hydrodynamic time scales (∼10−13 s), and the motion of alpha particles can be well approximated as instantaneous stopping inside the hot spot. When the rate of alpha heating balances all hot-spot power losses, including the thermal conduction and radiation losses, the hot spot ignites and drives a burn wave propagating outward through the surrounding dense shell.7 One-dimensional (1-D) ICF models8,9 provide the basic framework for understanding the implosion dynamics. However, the presence of nonuniformities due to the multiple-beam–overlapping pattern, beam-to-beam energy imbalance, beam mispointing, and target surface roughness, in general, leads to the growth of Rayleigh–Taylor (RT) instabilities in the acceleration and deceleration phases of ICF implosions. It is crucial to understand the degradation of implosion performance and neutron yield caused by three-dimensional (3-D) physics.
The issues of linear and nonlinear growth rates of deceleration-phase RT instabilities were addressed in early 3-D simulations.10–12 Due to the advancement in the shock-capturing hydrodynamic schemes and parallel computation, modern ICF computer codes13 provide well-resolved 3-D features and improved prediction capabilities.
In this work, we show that the degraded compression of the hot spot through PdV work is a strong function of residual kinetic energy (RKE) of the compressed shell. In spherical geometry, the ratio of the hot-spot radius R to the wavelength is used to classify low modes with and high modes with .7,14 The properties of low mode and intermediate mode using two-dimensional (2-D) simulations14 show that the hot-spot properties described by the burn-averaged and volume-averaged definitions can be different. Early work by Scott et al.15 using 2-D HYDRA simulations indicates that the effect of low-mode capsule shape asymmetries can result in a significant reduction in converting the implosion shell kinetic energy into the hot-spot internal energy. The mechanism of depleting the hot spot internal energy and Doppler velocity broadening of the neutron energy spectrum caused by mode 1 asymmetric flow was first reported by Spears et al.16 The effect of low mode areal density variation was studied by Gu et al.,17 in which the factor of areal density asymmetry was included to improve the metric for the ignition threshold factor (ITF). The effects of residual kinetic energy on reducing the capsule performance were studied by Kritcher et al.18 using 2-D HYDRA simulations to construct a correction to the ITF metric based on residual kinetic energies, areal densities, and asymmetries of the hot-spot shape. The degradation in hot-spot pressure, volume, temperature, and burn width was investigated in 2-D by Bose et al.14 The 3-D analytical deceleration-phase hot-spot model by Sanz et al.19,20 predicts the growth rates of low and intermediate modes through the linearized perturbation analysis.
The motivation of this work is to numerically and analytically evaluate the effects of residual kinetic energy on the yield degradation using both 2-D and 3-D simulations. The mode spectral range of –12 is systematically studied. We show that the yield-over-clean is a strong function of the residual kinetic energy in the compressed shell. The results are in agreement with the 2-D numerical simulations for NIF (National Ignition Facility) indirect-drive targets with low-mode asymmetries performed by Kritcher et al.,18 showing that residual kinetic energy (RKE) at the time of peak compression is well correlated to the yield degradation. We examine this result through an analytical model and extend the analysis of RKE to three dimensions using the radiation-hydrodynamic code DEC3D.21–23 A synthetic single-mode database is built using DEC3D simulations to study the yield degradation caused by RT instabilities in the deceleration phase of OMEGA cryogenic DT implosions. Low modes are shown to have larger total residual kinetic energies than intermediate and high modes at the same initial velocity perturbations. The RKE's are shown to be dominated mainly by RT spikes for high modes. The high-velocity jet in low mode is observed to contribute significant RKE. The yield degradation for high modes in terms of reduction in burn volumes in a way that is similar to the 2-D result of Ref. 14 is discussed. The effect of RKE on ion temperature asymmetry for different single modes is analyzed using the neutron transport code IRIS3D.24 The effects of vortices causing ion temperature asymmetries along different lines of sight (LOS) are discussed.
The following sections of this paper: (1) describe the simulation method and the DEC3D hydrocode; (2) discuss the effect of residual kinetic energy on ion temperature asymmetries; (3) derive a simple 3-D analytical hot-spot model that shows YOC is a strong function of RKE; (4) discuss the yield degradation for high modes in terms of reduction in burn volumes; and (5) present our conclusions.
II. THE DEC3D HYDROCODE AND THE SIMULATION DATABASE
Implosion 77068 is considered because it is one of the best-performing implosions to date on OMEGA, with a hot-spot pressure >50 Gbar was inferred.25 The target has an 8-μm thickness Carbon-Deuterium (CD) ablator, an outside radius of 430 μm, and a DT-ice thickness of 50 μm. The implosion has adiabat α = 3.2, defined by the mass-averaged ratio of the fuel pressure to the Fermi-degenerate pressure P/PFermi, and a convergence ratio of about 20. It was shown that the core conditions for a hydrodynamic-equivalent 77068 implosion at a NIF energy scale could lead to significant alpha heating.26 The DEC3D single-mode database is generated by introducing initial radial velocity perturbations on the inner shell surface located at m for OMEGA shot 77068 at the beginning of the deceleration phase. The time ns is when the shell has the maximum implosion velocity. As shown in Fig. 1, initial one-dimensional (1-D) profiles for at t0 are obtained from the 1-D LILAC simulation of 77068; LILAC is a 1-D Lagrangian radiation-hydrodynamic code27 routinely used for target designs at the Laboratory of Laser Energetics. Its main capabilities include refractive ray tracing, cross-beam energy transfer, nonlocal electron thermal transport,28 and first-principle equation of state.29 Initial transverse velocities are zero. The initial perturbation is applied on the radial velocity component in forms of spherical-harmonic modes
Legendre modes –12 are studied, and the initial radial velocity perturbation is , where is the fluid velocity. The spherical-harmonic functions are normalized and orthogonal. In this work, 3-D even Legendre modes with for even -modes are studied. The remaining Legendre modes are –12 with m = 0. The perturbation levels in our simulation database ranging from 1% to 14%. Figure 2 shows the 3-D electron temperature contour surface at 1 keV for all single modes in the DEC3D single-mode simulation database. The burn surface defined at 1 keV has the property of vanishing small enthalpy flux exchanged that leads to the conservation of the hot spot adiabatic parameter between 3-D and 1-D hot spot. f(r) is a shape function that is unity at r0 and decays radially away from r0 to avoid loading too much initial energy perturbation on the sharp interface at r0
The initial mass-density profile for OMEGA shot 77068 at the beginning of deceleration phase ns. The initial radial velocity perturbation is applied on the inner shell surface m. The blue line shows the shape function f(r) that is unity at r0.
The initial mass-density profile for OMEGA shot 77068 at the beginning of deceleration phase ns. The initial radial velocity perturbation is applied on the inner shell surface m. The blue line shows the shape function f(r) that is unity at r0.
The summary of DEC3D single-mode simulation database for the shot 77068: (1) HLLC approximate Riemann solver with PPM high-resolution method; (2) HYPRE implicit thermal diffusion; and (3) high angular resolution 128 × 256 for θ and zones. A 3-D contour surface at keV is shown for each single mode. The first, second, and third rows are 3-D odd Legendre modes, 2-D even Legendre modes with m = 0, and 3-D even Legendre modes with , respectively.
The summary of DEC3D single-mode simulation database for the shot 77068: (1) HLLC approximate Riemann solver with PPM high-resolution method; (2) HYPRE implicit thermal diffusion; and (3) high angular resolution 128 × 256 for θ and zones. A 3-D contour surface at keV is shown for each single mode. The first, second, and third rows are 3-D odd Legendre modes, 2-D even Legendre modes with m = 0, and 3-D even Legendre modes with , respectively.
DEC3D is a single-fluid two-temperature hydrocode. The electron pressure equation is solved as a scalar advection equation30 in the single-fluid HLLC (Harten-Lax-van Leer-contact) approximate Riemann solver. An ideal-gas equation-of-state is used to express the total fluid pressure as a function of electron and ion number densities and temperatures . Ideal gas is a good approximation for deceleration-phase simulations since the high-density shell at the time of peak compression is weakly degenerate –5.
DEC3D is a full spherical, 3-D parallel Eulerian code. The spherical mesh provides a low level of numerical noise for simulating RT instabilities in spherical geometry. The implementation of 3-D spherical mesh is challenging. The mesh near the origin and along the pole is needed be finer because of the decreasing size of arc lengths of finite-volume cells in the θ and directions defined by , where and are the discretized angles and the discretized radius of the finite-volume cells. For example, at the high angular resolution of 128 × 256 in θ and directions, the sizes of arc lengths for the first finite-volume cell are near the pole and near the origin. The macro-zoning technique is applied to map the fine mesh onto a coarser mesh, defined by the size of arc lengths of finite-volume cells comparable to the size of discretized radius, to avoid the small-time-step issue due to the limitation of the Courant condition. During the macro-zoning calculation in every time step, two cells are recombined to form a coarser cell in the θ direction followed by another recombination in the direction. The grid spacing in the radial direction is uniform, but it shrinks during the deceleration phase and expands in the disassembly phase.
DEC3D is integrated with HYPRE,31 a library of high-performance preconditioners, to solve large, sparse nonsymmetric systems for implicit thermal, radiation, and alpha diffusions. The nature of the strong spatial variation of opacity results in highly nonsymmetric systems in the radiation diffusion; the hybrid generalized minimal residual solver32 preconditioned with algebraic multigrid33 is used to solve 3-D multigroup radiation diffusion and thermal and one-group alpha diffusion without directional splitting. The multigrid method34 provides a fast convergence in the diffusion of high-energy photons with long mean free paths. The Spitzer thermal conductivity formula35,36 is used in the electron and ion thermal diffusions.
DEC3D implements the finite-volume moving-mesh algorithm37 and integrates a HLLC approximation Riemann solver38 on a uniformly shrinking or expanding spherical mesh in the radial direction. DEC3D applies a HLLC approximate Riemann solver to obtain the first-order hydrodynamic solution. The third-order solution is achieved by the piecewise-parabolic method (PPM)39 for strong shock-capturing capability and robust nonlinear hydrodynamic simulations.
The 3-D computational domain in directions is decomposed along the radial direction into subdomains . Each subdomain has radial zones . The radial domain decomposition is convenient for generating the coarser mesh and integrating with HYPRE in a parallel fashion. For the perturbed simulations, the yield converges when the angular zone is about 128 × 256 in directions. In this work, the resolution of the DEC3D single-mode database is in directions, about 25 zones per wavelength for mode 10, which is sufficient for Legendre modes –12 studies. The perturbed simulation of single-mode is shown in Fig. 3.
The 3-D electron temperature contour at 1 keV at stagnation for the single-mode . The resolution is for zones, respectively.
The 3-D electron temperature contour at 1 keV at stagnation for the single-mode . The resolution is for zones, respectively.
The benchmark test for a full 3-D, deceleration-phase, clean 1-D unperturbed simulation for shot 77068 is shown in Figs. 4 and 5 to summarize the integrated numerical performance of DEC3D. Figure 4 shows the good agreement between DEC3D and LILAC in the solutions for mass density, radial velocity, electron temperature and total pressure at the time of stagnation. In the velocity profile, the steep spatial gradient across the return shock at m is well resolved by the third-order PPM, while the LILAC solution is more diffusive due to the use of numerical viscosity. The performance of HYPRE for the thermal diffusion is validated in the electron temperature profile. A slight larger central total pressure in DEC3D than LILAC is observed, which results in a slight increase in the ion temperature and neutron production.
The benchmark test for a full 3-D, deceleration-phase, clean simulation for shot 77068. Comparison of solutions between LILAC (black circles) and DEC3D (solid red curves) for (a) density, (b) total pressure, (c) radial velocity, and (d) electron temperature profiles at stagnation using resolution in directions. The 1-D burn radius at stagnation for shot 77068 is m, and the return shock is located at m.
The benchmark test for a full 3-D, deceleration-phase, clean simulation for shot 77068. Comparison of solutions between LILAC (black circles) and DEC3D (solid red curves) for (a) density, (b) total pressure, (c) radial velocity, and (d) electron temperature profiles at stagnation using resolution in directions. The 1-D burn radius at stagnation for shot 77068 is m, and the return shock is located at m.
The benchmark test for neutron productions without alpha heating. Comparison of solutions between LILAC (black circles) and DEC3D (solid red curves) for the temporal history of neutron rate. The fusion reactivity scales with ion temperatures Ti in a power law for the temperature range keV in the BUCKY2 fusion reactivity model.
The benchmark test for neutron productions without alpha heating. Comparison of solutions between LILAC (black circles) and DEC3D (solid red curves) for the temporal history of neutron rate. The fusion reactivity scales with ion temperatures Ti in a power law for the temperature range keV in the BUCKY2 fusion reactivity model.
III. RESIDUAL KINETIC ENERGY AND ION TEMPERATURE ASYMMETRY
Sketches of burn volumes for low and high modes on a 2-D plane are shown in Fig. 6. For low modes, bubbles are hot enough to sustain DT fusion reactions.14,46 The volume of hot bubbles is a part of the burn volume that contributes to neutron productions. As the shell implodes and RT spikes grow, the rate of change in the burn volume is determined by the boundary velocity of the burn surface. For high modes, the boundary velocity is not equal to the fluid velocity of the perturbed inner shell surface but depends on the spatial profile of the neutron rate at each moment. Inside the burn volume, physical properties are approximately smooth in space. For high modes, bubbles are too cold for fusion reactions, the burn volume is reduced to the clean volume surrounded by RT spikes, and the boundary velocity is not the same as the fluid velocity on the perturbed inner shell surface as shown in Fig. 6.
The 3-D hot-spot models for low and high modes. The red part is the burn volume, while the blue part is the shell. As the shell implodes, the burn volume is shrinking with the boundary velocity . For high modes, the cold bubbles shown by the white part do not contribute DT fusion reactions, and the actual burn volume is smaller than the total hot-spot volume bounded by the perturbed inner shell surface.
The 3-D hot-spot models for low and high modes. The red part is the burn volume, while the blue part is the shell. As the shell implodes, the burn volume is shrinking with the boundary velocity . For high modes, the cold bubbles shown by the white part do not contribute DT fusion reactions, and the actual burn volume is smaller than the total hot-spot volume bounded by the perturbed inner shell surface.
Figure 7 compares the configuration of the hot spot between low and high modes. The core parts for the high mode and low mode at high temperatures keV are shown approximately spherical in shape, so the nonradial components of the shell velocities do not participate in the core compression. In 1-D, the shell decelerates as the hot-spot pressure force pushes against the shell
where D/Dt is the material derivative. The term is the driving factor that decelerates the shell and converts the shell's kinetic energy into the hot-spot internal energy. θ is the inclination angle between the pressure gradient vector and the shell's velocity vector in the presence of perturbations. Components of the velocity field perpendicular to do not participate in the core compression. When the shell's velocity field is less convergent with respect to 1-D, the hot spot gains less internal energy through the PdV work relation , where ε is the hot spot's internal energy density and is the heat source or sink term. The convergence of a perturbed velocity field directly reflects the difference in core compression between different single modes. The velocity fields at stagnation for low mode and high mode are compared in Fig. 7.
The mass density profile for (a) low mode and (b) high mode at stagnation with electron temperature contours at 1, 2, 3, and 4 keV. Black arrows indicate the fluid velocity field. Since the cores at temperatures keV are approximately spherical in shape, the core compression is contributed mainly by the radial velocity component. For high mode , the bubbles are cold, characterized by low neutron production rates.
The mass density profile for (a) low mode and (b) high mode at stagnation with electron temperature contours at 1, 2, 3, and 4 keV. Black arrows indicate the fluid velocity field. Since the cores at temperatures keV are approximately spherical in shape, the core compression is contributed mainly by the radial velocity component. For high mode , the bubbles are cold, characterized by low neutron production rates.
For high modes, nonstagnating RT spikes contribute to a large fraction of the residual kinetic energy. Figure 8 shows the spatial distribution of kinetic energy density on RT spikes for the high mode . Long-wavelength perturbations induce a large spatial scale of high-velocity flows inside the hot spot and more nonradial velocity components in the compressed shell than high modes. Figure 9 shows the dominant jet of the low mode at speed km/s toward the negative z direction. As shown in Fig. 9, the core region at keV is advected by the jet with the maximum flow velocity. For low modes, both the nonstagnating RT spike and the high-velocity jet account for the residual kinetic energy at stagnation, as shown by Fig. 8.
The kinetic energy density profiles for (a) low mode and (b) high mode at stagnation. For low mode , both the RT spike and the jet contribute to residual kinetic energy. For high mode , the residual kinetic energy is dominated mainly by the nonstagnating RT spikes. The blue spherical outline is the residual kinetic energy of the unshocked part of the shell.
The kinetic energy density profiles for (a) low mode and (b) high mode at stagnation. For low mode , both the RT spike and the jet contribute to residual kinetic energy. For high mode , the residual kinetic energy is dominated mainly by the nonstagnating RT spikes. The blue spherical outline is the residual kinetic energy of the unshocked part of the shell.
Velocity magnitude for low mode at stagnation. A jet at 500 km/s is flowing along the negative z direction. The yield-over-clean is 0.74 and the burn-averaged hot-spot velocity magnitude is 166 km/s in this simulation. The red curve indicates the core region at a 2.3-keV electron temperature, showing that the core region is advected by the jet with the maximum flow velocity.
Velocity magnitude for low mode at stagnation. A jet at 500 km/s is flowing along the negative z direction. The yield-over-clean is 0.74 and the burn-averaged hot-spot velocity magnitude is 166 km/s in this simulation. The red curve indicates the core region at a 2.3-keV electron temperature, showing that the core region is advected by the jet with the maximum flow velocity.
In ICF implosions, target offset, stalk, laser beam power unbalance, and DT ice roughness are the main sources of mode non-uniformity.16,40,41 The large hot-spot residual kinetic energies associated with low modes lead to significant variations in measuring ion temperatures along different lines of sight (LOS). The signature of implosions with large hot-spot RKE's are characterized by the shift of mean neutron energy , the broadening of the width of the neutron energy spectrum , and the deviation of shape from the Maxwellian distribution of fusion ions. Neutrons born from each fluid element are characterized by a Gaussian energy spectrum centered at the neutron birth energy MeV. For a stationary fusion plasma, the width of the spectrum is a function of thermal ion temperature Ti given by , where and are the neutron and alpha particle masses, respectively. The effect of collective bulk motion of nonstagnating fusing ions within the hot spot causes the shift in the neutron mean energy estimated in Refs. 42–44
where θ is the angle between the spectrometer LOS and the direction of the bulk velocity . For neutrons born from non-stationary fluid elements, the mean neutron energy is . The sign of depends on the angle θ between the direction of velocity field vectors and the LOS. From Eq. (4), the bulk velocity 500 km/s in the mode simulation as shown in Fig. 9 causes a maximum mean energy shift of about 271 keV, when the detector is aligned in the same direction as the bulk velocity, i.e., θ = 0% or 2% mean energy shift with respect to E0.
The mass density, velocity, and ion thermal temperature profiles for mode in Fig. 7 are post-processed by IRIS3D, a Monte Carlo-based neutron transport code. The hot-spot ion temperature is inferred from the width of Doppler-broadened primary neutron energy spectrum43
The effect of Doppler broadening of the width of neutron energy spectrum is24
where is the variance in the component of the fluid velocity along the direction of the detector, denoted by the unit vector , and is the variance of energy for primary neutrons. The nonvanishing contribution to results from velocity components parallel to . The presence of hot-spot RKE leads to larger inferred ion temperatures than thermal ion temperatures. The effect of Doppler broadening on the width of the neutron energy spectrum is the same for two detectors located on the opposite directions.
The flow effect on inferred ion temperature variations depends not only on the unique flow pattern within the hot spot of each single mode but also on the spatial distribution of neutron productions. Because the variance is weighted over the burn distribution, the relative importance of vortex structure depends on the difference in neutron production rates between the core part and the bubble regions. As shown in Fig. 8, high modes do not have explicit jet flowing inside the hot spot and have a small spatial scale of vortices located inside the cold bubbles, where the neutron production rates are small. Therefore, high modes, in general, have smaller inferred ion temperature variations than low modes.
DT ion temperatures are inferred by IRIS3D for mode using six detectors along different LOS's: , as shown in Fig. 10. Since the vortex structure for mode has a rotational symmetry along the z axis, the inferred ion temperatures at the LOS's at are about the same. These four detectors are located perpendicular to the jet, resulting in negligible inferred ion temperature variations or zero variance. The LOS's at have the largest parallel velocity components and cause about a 1.25-keV inferred ion temperature variation. The blue curve in Fig. 10 shows a small inferred ion temperature variation for the high mode . The averaged inferred ion temperature over six LOS's on the blue curve is 3.67-keV, which is close to the neutron-averaged ion temperature (3.64 keV) measured at that time for high mode .
The inferred DT ion temperatures at stagnation by IRIS3D using six detectors at different LOS's at for low mode (red curve) and high mode (blue curve).
The inferred DT ion temperatures at stagnation by IRIS3D using six detectors at different LOS's at for low mode (red curve) and high mode (blue curve).
The vortices of high modes within the cold bubbles do not contribute significant inferred ion temperature variations because of the low neutron production weight in the variance. The high-velocity jet in mode corresponds to the largest inferred ion temperature variation in the mode spectrum. The ion temperature asymmetries in the mode spectrum –12 are summarized in Fig. 11, showing a decreasing inferred ion temperature variation with Legendre mode number. It implies that large ion temperature variations observed in experiments indicate the presence of low modes.45 Large ion temperature variations along different LOS's are caused by large-scale high-velocity flows induced by long-wavelength perturbations.
The inferred DT ion temperatures at stagnation by IRIS3D using 16 detectors at different LOS's including and other 10 typical nTOF (neutron time of flight) diagnostics on OMEGA.
The inferred DT ion temperatures at stagnation by IRIS3D using 16 detectors at different LOS's including and other 10 typical nTOF (neutron time of flight) diagnostics on OMEGA.
IV. A 3-D HOT-SPOT MODEL
A simple 3-D hot-spot model is shown in Fig. 6. For high modes, the cold bubbles do not contribute DT fusion reactions and the actual burn volume is smaller than the perturbed hot-spot volume.14,46 The volume-averaged quantities with hot-spot electron temperatures greater than 1 keV are denoted by . Here, we assume that the 1-keV contour encloses the burn region and other colder parts of the distorted hot spot do not significantly contribute to the fusion yield. The neutron-averaged or burn-averaged quantities are denoted by . In this work, the analytical yield degradation model is first derived for low modes using the volume-averaged definition. For low modes, because neutrons produced within the warm bubbles in low modes are of the same order of magnitude of neutrons produced within the core region.14 The neutron-averaged quantities are used to characterize the yield degradation model for high modes because the burn volume for high modes is much smaller than the total perturbed hot-spot volume due to the formation of cold bubbles14 as shown in Fig. 12.
The 3-D hot-spot shapes for the electron-temperature contour surface (1, 1.5, 2, and 2.5 keV) for high mode at stagnation. The length scale is the same in all figures. The left-most hot-spot shape includes the cold bubbles as shown in Fig. 7. It corresponds to the total perturbed hot-spot volume while the right-most figure is the core region for DT fusion reactions.
The 3-D hot-spot shapes for the electron-temperature contour surface (1, 1.5, 2, and 2.5 keV) for high mode at stagnation. The length scale is the same in all figures. The left-most hot-spot shape includes the cold bubbles as shown in Fig. 7. It corresponds to the total perturbed hot-spot volume while the right-most figure is the core region for DT fusion reactions.
The survey of deceleration-phase single-mode energetics at stagnation for shot 77068 is shown in Fig. 13 using the burn-averaged and volume-averaged definitions. The following behaviors are observed at the same 14% large initial velocity perturbation in the mode spectrum. The 3-D single modes with are shown to have more total residual kinetic energy than 2-D single modes. The burn-averaged shell internal energy is shown to increase with Legendre mode number because of the inclusion of cold bubbles for high modes. The burn-averaged hot-spot internal energy is shown to decrease with mode numbers because of the reduction in burn volumes. For the hot spot defined by keV, the internal energies both the hot spot and the shell do not show significant variation among different modes. A trend indicates that low modes have more total residual kinetic energies at stagnation than high modes. The low mode is found to have the largest hot-spot kinetic energy within the burn volume at stagnation.
Summary of single-mode energetics at stagnation for the same 14% initial velocity perturbations. Plot (a) is the burn-averaged hot spot and (b) is the -keV hot spot. In (a) and (b), the blue dots are , the black dots are , and the red dots are . The 3-D Legendre modes with are denoted by on the x axis. (c) Measurement of with respect to 1-D at stagnation using the hot-spot definition with keV shows the conservation of adiabatic parameter for all modes. In (c), the red dots are Legendre modes with m = 0, while blue dots are 3-D Legendre modes with . The 1-D energies at stagnation are J, J, J, J, and J.
Summary of single-mode energetics at stagnation for the same 14% initial velocity perturbations. Plot (a) is the burn-averaged hot spot and (b) is the -keV hot spot. In (a) and (b), the blue dots are , the black dots are , and the red dots are . The 3-D Legendre modes with are denoted by on the x axis. (c) Measurement of with respect to 1-D at stagnation using the hot-spot definition with keV shows the conservation of adiabatic parameter for all modes. In (c), the red dots are Legendre modes with m = 0, while blue dots are 3-D Legendre modes with . The 1-D energies at stagnation are J, J, J, J, and J.
Another remarkable feature is observed in the simulation database. It suggests that the subsonic flow approximation for the energy equation of hot spot is an appropriate assumption in both 1-D and 3-D implosions, leading to the following simplified form of the energy equation in the hot spot:
The sum of rates of alpha heating, heat-conduction loss, and radiation loss is given by . A direct volume integration over the burn volume D(t) with a boundary moving at velocity is given by Leibniz's rule
where and is the rate of heat loss through the burn surface. The surface integral measures the enthalpy flux across the burn surface because the boundary of the burn surface is not moving at the same velocity as the fluid. The boundary is defined as the interior domain enclosed by the distorted hot-spot surface. Equation (8) is the subsonic flow approximation of the hot-spot energy equation derived for burn-averaged quantities , which is valid for both low and high modes. In Fig. 6, the burn volume for low and high modes is shown in red. includes the region of warm bubbles for low modes because of but excludes the cold bubbles for high modes. In this work, we use the volume-averaged quantities to describe low modes and the burn-averaged quantities for high modes .
The hot-spot energy equation can be rewritten as
is the hot-spot adiabatic parameter, which depends on the rate of heat gain or loss term and the relative motion between the fluid velocity and the velocity of the burn surface through the term . At the same time, the hot-spot entropy increases as a result of heat-transferring processes within the hot spot such as radiation, electron and ion heat conduction, and equilibration.
When the hot-spot surface is defined by keV, the velocity of the burn surface is about the same as the fluid velocity on the perturbed inner shell surface, i.e., . It leads to the first property of vanishing enthalpy flux in Eqs. (8) and (9) for both low and high modes. The second property is vanishing on the perturbed inner shell surface for both low and high modes because the heat leaving the hot spot is recycled in the form of internal and kinetic energies of the plasma ablated off the hot-spot's inner shell surface.47 Therefore, in the absence of alpha heating and radiation loss , the hot-spot adiabatic parameter is conserved robustly for both low and high modes14,48 for volume-averaged quantities with keV. This property is validated in Fig. 13
where is the bang time. In the burn-average definition, however, the hot-spot adiabatic parameter decreases with Legendre mode numbers because of the reduction in burn volumes. The conservation of adiabatic parameter for low modes leads to the first general 3-D relation that connects the hot-spot pressure and the hot-spot volume in terms of the hot-spot internal energetics48
and
Equations (11) and (12) are validated in Fig. 14. When and are important in the presence of strong alpha heating or dominant radiation losses, the hot spot adiabatic parameter is not constant in time. A time-dependent integrating factor modifies the adiabatic parameter
The exponent factor appears as a multiplier modifying the expression of the hot spot pressure and the hot spot volume in Eqs. (11) and (12) only. The following analysis of the total energy conservation and the mass ablation rate remain unchanged. The conservation of total energy provides the second general 3-D relation that connects the hot-spot's internal energies to the shell and the hot-spot's kinetic energies at stagnation. The shell's initial kinetic energy is converted into the hot-spot's internal energy through PdV work
and are the total kinetic energy measured in the simulation domain at the beginning of the deceleration phase at time t0 and at stagnation , respectively. For shot 77068, the initial total kinetic energy is large enough to neglect the initial shell and hot-spot internal energies in Eq. (14). We define the normalized residual kinetic energy as
Besides is the initial value, other variables are measured at stagnation. For simplicity, we drop the subscript for and t0 and divide both sides of Eq. (14) with respect to 1-D values
where the simplified labels are defined by and . The right-hand side of Eq. (16) can be expanded into
where the normalized residual shell internal energy is defined as . Equation (17) can be simplified by retaining only the leading term and neglecting the quadratic terms leading to
From the survey of single-mode energetics shown in Fig. 13, the change of volume-averaged shell internal energies measured at stagnation for various single modes is significantly less than the change in the total residual kinetic energies that justifies the approximation . The changes in hot-spot pressure and hot-spot volume can be expressed as a unique function of the total residual kinetic energy by rewriting Eqs. (11) and (12)
and
Equations (19) and (20) provide the fundamental explanation for 3-D hydrodynamic behavior for low modes. The larger hot-spot volume and lower hot-spot pressure observed in low modes are caused by the increasing total residual kinetic energy. Equations (19) and (20) are validated by DEC3D simulations in Fig. 14 using the hot-spot definition with keV. Observe that the scalings are not affected significantly by neglecting the quadratic terms and in Eq. (17).
Relation between hot-spot pressure, hot-spot volume, hot-spot internal energy, and residual kinetic energies for an adiabatic implosion using hot-spot definition with keV for Legendre modes –12. The blue curves indicate the analytic model relations for (a) , (b) , (c) , and (d) . The shorthand notations are .
Relation between hot-spot pressure, hot-spot volume, hot-spot internal energy, and residual kinetic energies for an adiabatic implosion using hot-spot definition with keV for Legendre modes –12. The blue curves indicate the analytic model relations for (a) , (b) , (c) , and (d) . The shorthand notations are .
The fusion reactivity scales with ion temperatures as a power law for ion temperatures Ti = 1–5 keV, as shown in Fig. 5. This leads to a simple estimation of neutron yield or , where n is the hot-spot ion number density and τ is the burn width. The yield can be expressed in terms of the hot-spot pressure and hot-spot volume by substituting the ideal gas relation and the hot-spot mass . Because the hot-spot pressure P and the hot-spot volume V are related to the hot-spot internal energies and residual kinetic energies through adiabatic conditions and energy conservation, the yield scaling describes the general 3-D hot-spot conditions, regardless of linear or nonlinear RT instabilities. The yield's dependence on comes from the fusion reactivity scaling , which changes to for fusion plasma49 ranging from 6 to 20 keV. The yield-over-clean is approximated as
The yield degradation can be shown to be a strong function of the residual kinetic energy of the compressed shell by substituting Eqs. (11), (12), and (18) into P and V terms. The yield-over-clean is further simplified into
The rate of change in the hot-spot mass is given by , where va is the mass ablation velocity on the inner shell surface. We assume that for low modes, the ablation rate scaling is similar to the predictions of the 1-D theory. One-dimensional approximations47 for the mass ablation velocity and the perturbed hot-spot surface area are used. The total gain of hot-spot mass caused by thermal mass ablation on the inner shell surface over a characteristic time of the fusion is
where τ is the burn width. A simple scaling for the perturbed hot-spot mass is obtained by substituting the ion temperature in terms of P and V, where is the DT ion mass
As shown in Fig. 15, 1-D approximations for the mass ablation rate and hot-spot surface area provide a reasonable estimation for the perturbed hot-spot mass for low modes. Equation (24) can be validated using the hot-spot mass scaling relation in Ref. 50
where t = 0 is the beginning time of the deceleration-phase, for the ideal gas, for Spitzer thermal conductivity, and for the self-similar flow solution for the 1-D deceleration phase model in the absence of alpha and radiation transport. The subscript h denotes the volume-averaged hot-spot quantities. By approximating the time integration over a characteristic burn time τ, the hot-spot mass at stagnation resulting from the mass ablation off the inner shell surface is
By eliminating the hot-spot mass term in Eq. (22)
The effect of ablation off the inner shell surface relaxes the dependence of yield degradation on residual kinetic energies from to , where . The burn truncation, about , is typically a small effect and can be neglected. Therefore, the YOC is reduced to a strong function of the residual kinetic energy of the compressed shell, and hydrodynamic instabilities play an important role in causing the yield degradation to scale as
(a) The 1-D scaling relation for the hot-spot mass at stagnation for low modes –5 using the hot-spot definition keV. The shorthand notations are . (b) Comparison of simulated YOC against RKE and the analytic models for low modes –6 and (c) for high modes –12. The solid blue curve is the yield degradation model , while the dashed blue curve is yield degradation model .
(a) The 1-D scaling relation for the hot-spot mass at stagnation for low modes –5 using the hot-spot definition keV. The shorthand notations are . (b) Comparison of simulated YOC against RKE and the analytic models for low modes –6 and (c) for high modes –12. The solid blue curve is the yield degradation model , while the dashed blue curve is yield degradation model .
from pure hydrodynamic instabilities without thermal transport and that includes ablation driven by thermal losses are compared in Fig. 15 for low modes –6 and high modes . The yield is shown to decrease monotonically with residual kinetic energies for both low and high modes when the hot spot is defined by keV. Long-wavelength perturbations are shown to be well approximated by the adiabatic implosion model. The energetic behaviors of pressure degradation, increasing hot-spot volumes, and yield degradation between low modes and are indistinguishable because the conservations of energy and the hot-spot adiabatic parameter are general properties for arbitrary distorted hot spot in 3-D.
Here, we describe the hot spot for high modes using the burn-averaged hot-spot pressure and volume . For high modes , the hot-spot adiabatic parameter is not conserved because of the reduction in burn volumes. Figure 12 shows the decreasing hot-spot volume with electron temperatures. The cold bubbles shown by the 1-keV contour surface do not contribute to a significant fraction of neutron productions compared with the core part shown by the 2.5-keV contour surface. We use our single-mode simulation database to validate the yield degradation model for high modes in terms of reduction in burn volume derived in Refs. 46 and 51
The hot-spot pressure and ion temperature between 3-D and 1-D are assumed equivalent in Ref. 51. The 3-D effect in Ref. 51 is introduced in terms of reduction in burn volumes as a result of the nonlinear growth in RT spikes. On average about 10% variation in hot-spot pressures for Legendre modes, is observed in our simulation database. The slight increase in shown in Fig. 16 is due to the perfect symmetry of the RT spikes converging to the center. This property is lost for low modes since the pressure degradation is a strong function of RKE. Figure 16 validates Eq. (29) where the yield degradation for high modes is dominated by the reduction in burn volume in agreement with the 2-D results of Refs. 14 and 46.
(a) For low mode , the degradation of hot-spot pressure and the increasing hot-spot volume are strong functions of residual kinetic energies. (b) For the high mode , the burn volume is reduced and the core pressure is increased due to the growth of converging RT spikes. The burn-averaged quantities are used in (a) and (b). (c) The simulated YOC is compared against the hot-spot volume for high modes –12. The blue curve is the yield degradation model , where .
(a) For low mode , the degradation of hot-spot pressure and the increasing hot-spot volume are strong functions of residual kinetic energies. (b) For the high mode , the burn volume is reduced and the core pressure is increased due to the growth of converging RT spikes. The burn-averaged quantities are used in (a) and (b). (c) The simulated YOC is compared against the hot-spot volume for high modes –12. The blue curve is the yield degradation model , where .
It is difficult to directly measure the residual kinetic energy in experiments. However, flows in hot spot affect the ion temperature measurement. Through multiple line of sights temperature measurements, the anisotropic part of the velocity variance in Brysk ion temperature43,45,52 can be extrapolated and used to infer the magnitude of the hot spot residual kinetic energy. Together with the correlation between the shell areal density variation and the shell residual kinetic energy, a statistical model of the total residual kinetic energy in terms of ion temperature measurement and areal density measurement variations obtained by post-shot modeling can be derived to access the actual yield degradation in experiments through ion temperature and areal density measurement asymmetries.
V. CONCLUSION
We have developed a 3-D radiation-hydrodynamic code DEC3D to study the yield degradation caused by Rayleigh-Taylor instabilities in the deceleration phase of inertial confinement fusion. A systematic investigation using DEC3D's synthetic single-mode database indicates that the yield degradation caused by low- and mid-mode nonuniformities is a strong function of the residual kinetic energy. This result agrees with a simple YOC model assuming the hot spot satisfying adiabatic implosion model and subsonic flow approximation. The dependence of the YOC on residual kinetic energy (RKE) is also in agreement with 2-D HYDRA simulation results from Kritcher et al. The simulated YOC is well approximated by the pure hydrodynamic instability curve and the thermal-driven curve . The degradation of hot-spot pressure and increasing hot-spot volume for low modes can be explained in terms of increasing residual kinetic energies. For high modes, the yield degradation is dominated by the reduction in burn volumes .14,46 The burn volumes for high modes are significantly reduced due to the growth of cold bubbles. At the same initial perturbation amplitudes, low modes are shown to have more total residual kinetic energies than high modes and 3-D single modes are shown to have more total residual kinetic energies than 2-D single modes. The jet of low mode is shown to cause the largest ion temperature asymmetries in the mode spectrum. Low modes are shown to cause large ion temperature asymmetry, while modes exhibit negligible Ti variations. The vortices of high modes in cold bubbles do not cause large ion temperature asymmetry because of the low neutron production rates within the cold bubbles.
ACKNOWLEDGMENTS
The author would like to acknowledge Dr. Otto L. Landen for reviewing and offering advices on the manuscript of the paper.
This material was based upon the work supported by the Department of Energy National Nuclear Security Administration under Award No. DE-NA0001944. Partial supported was provided by DOE Office of Fusion Energy Sciences Grant No. DE-SC0014318. This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees makes any warranty, express, or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the U.S. Government or any agency thereof.