Stability of ablation flows in inertial confinement fusion: Nonmodal effects

Fast transient growth of hydrodynamic perturbations due to nonmodal effects is shown to be possible in an ablation flow relevant to inertial confinement fusion (ICF). Likely to arise in capsule ablators with material inhomogeneities, such growths appear to be too fast to be detected by existing measurement techniques, cannot be predicted by any of the methods previously used for studying hydrodynamic instabilities in ICF, yet could cause early transitions to nonlinear regimes. These findings call for reconsidering the stability of ICF flows within the framework of nonmodal stability theory.

Figure 1

Self-similar ablation-wave solution to Eq. (1) for $\gamma =5/3$, $(\mu ,\nu )=(2,13/2)$, and boundary condition parameters $({\mathcal{B}}_{\varphi },{\mathcal{B}}_p)=(0.8,0.31)$. Dimensionless spatial profiles in the coordinate $x$ at time $t_0=1$ of the fluid density $\rho$, longitudinal velocity $v_x$, and heat flux ${\phi }_x$. Correspondence with the actual physical extent of the wave, relative to the shock front, at the chosen reference time of a simulated capsule implosion is also indicated (top axis) (see the Appendix).

Figure 2

Schematic layouts in the complex plane of eigenvalues and numerical ranges corresponding to (a) modal stability with slower nonmodal transient decay, (b) modal instability with faster nonmodal transient growth, and (c) modal stability with nonmodal transient growth.

Figure 3

Intensity map, in the plane $(x,{\varkappa }_x)$, of $log{\sigma }_0$, ${\sigma }_0>0$, obtained from Eq. (5) for $k_{\perp }=1.2$: (a) regions of nonmodal growth but modal stability (color) and regions of modal instability (black) and (b) ${\sigma }_0-max\text{Re}\left(\mathrm{\Lambda }\right)$ everywhere. Same horizontal axis conventions as in Fig. 1.

Figure 4

Amplification of the perturbation norm, $\parallel \widehat{\mathbf{U}}{\parallel /\parallel \widehat{\mathbf{U}}\parallel |}_{t_0}$, (thick lines, left axis) and of the optical depth perturbation, $\widehat{\text{OD}}/\widehat{\text{OD}}{|}_{t_0}$ (thin lines, right axis), for initial conditions Eq. (7) at (a) a midpoint location within the compression region, $x_{\circ }=-0.06$, and (b) a location immediately upstream to the ablation layer, $x_{\circ }=-0.11$, with ${\varkappa }_x=1758$, ${\varkappa }_{\perp }=1.40$ (solid) or ${\varkappa }_{\perp }=131.5$ (dash). Remarkable events in the evolution of $\parallel \widehat{\mathbf{U}}\parallel$ are identified by letters A to D.

Figure 5

Evolution of $\widehat{\mathbf{U}}$ in the compression region for initial conditions Eq. (7) at $x_{\circ }=-0.06$, in the case $({\varkappa }_x,{\varkappa }_{\perp })=(1758,1.40)$. Same conventions as in Fig. 1 with shock (sf) and ablation (af) front trajectories. (a) Local Euclidean norm. Components (absolute value) in longitudinal pseudo-characteristic variables [23]: (b) forward and (d) backward acoustic waves, (c) entropy waves. (e) Transverse potential vorticity. Labeled circles in (a) relate to the events identified in Fig. 4.

Figure 6

Time evolutions of the ablation front deformation, normalized by $\parallel \widehat{\mathbf{U}}\parallel {|}_{t_0}$, for initial conditions Eq. 4 at (a) a midpoint location within the compression region, $x_{\circ }=-0.06$, and (b) a location immediately upstream to the ablation layer, $x_{\circ }=-0.11$, with ${\varkappa }_x=1758$, ${\varkappa }_{\perp }=1.40$ (solid) or ${\varkappa }_{\perp }=131.5$ (dash).