Relate residual kinetic energy expressions from the thin-shell Hamiltonian model and Hurricane et al. (2022)

Determine the precise relationship between the residual kinetic energy at stagnation predicted by the thin-shell Hamiltonian quasilinear model for spherical inertial-confinement-fusion implosions—expressed in terms of Legendre-mode amplitudes of the areal-density asymmetry—and the alternative residual-kinetic-energy expression formulated in terms of the harmonic-weighted averaged areal density by Hurricane et al. (Phys. Plasmas 29, 012703, 2022), so as to reconcile and unify these two formulations.

Background

The paper develops a thin-shell Hamiltonian model and a quasilinear analysis that yields an explicit expression for the residual kinetic energy at stagnation in terms of the Legendre components of areal-density asymmetries. This expression generalizes prior piston-model results by accounting for contributions from all Legendre modes.

Hurricane et al. (2022) provided a different formula for residual kinetic energy based on a harmonic-weighted averaged areal density. The authors state that clarifying the relationship between these two approaches is deferred, indicating an unresolved connection between formulations that are both used to interpret performance degradation in ICF implosions.

Establishing this relationship would improve theoretical consistency and facilitate comparison of models with experimental observables that are often decomposed into Legendre modes or harmonic-weighted measures.