The Rayleigh–Taylor instability (RTI) - the growth of interfacial perturbations when a heavy fluid is accelerated into a light one - is the principal hydrodynamic obstacle in inertial-confinement-fusion (ICF) implosions and core-collapse supernova explosions, where the unstable interface is not planar but imploding. Despite mature theory for planar RTI, no general predictive theory exists for convergent RTI that covers multimode initial perturbations, the nonlinear regime beyond weakly nonlinear truncations, compressible flow, and realistic material response (viscosity, elasticity, plasticity, ablative drive), through the transition to turbulent mixing. Existing analytical treatments solve the convergent-geometry amplitude equations only for special background motions and stop at second or third order in amplitude; nonlinear models are single-mode; compressibility and material effects are treated linearly; turbulent mixing in converging geometry is handled only by large-scale simulation and calibrated engineering models.

Background

The Rayleigh–Taylor instability (RTI) develops at the interface between two fluids of different density whenever the lighter fluid pushes the heavier one - equivalently, whenever the effective gravity points from the heavy fluid into the light one (Taylor, 1950). Small perturbations grow exponentially, the interface deforms into rising bubbles of light fluid and falling spikes of heavy fluid, secondary shear (Kelvin–Helmholtz) instabilities roughen the fronts, and with broadband initial perturbations the flow becomes a turbulent mixing layer. In planar geometry this progression is comparatively well understood: linear growth rates are classical, weakly nonlinear (WN) amplitude equations describe harmonic generation and mode coupling, single-mode nonlinear evolution is captured by potential-flow bubble and buoyancy–drag models (in which bubble motion is governed by a balance of buoyancy and drag forces), and semi-empirical turbulence closures calibrated against experiments predict mixing-zone growth.

The situation is qualitatively different when the interface is converging - imploding inward in cylindrical or spherical geometry - which is precisely the configuration of the two flagship applications. In inertial confinement fusion (ICF), a millimetre-scale capsule of cryogenic deuterium–tritium fuel is imploded by laser-driven ablation or X rays; Rayleigh–Taylor growth first at the ablation front during acceleration and later at the fuel–shell interface during deceleration degrades compression and injects high-atomic-number contaminant into the hot spot, and in extreme cases breaks the implosion apart. In core-collapse supernovae, the rebounding shock propagates through the stellar core while the ejecta are decelerated by infalling material, and the resulting RT mixing redistributes nucleosynthesis products through the envelope, with observable consequences.

Convergent geometry changes the instability physics through the Bell–Plesset (BP) effects - the modifications of perturbation growth caused by the time dependence of the interface radius, geometrical convergence (perturbation amplitudes scale geometrically as the interface shrinks), and uniform compression of the fluids (Plesset, 1954; Epstein, 2004). BP effects are not a separate instability, but they alter growth rates and stability criteria, and - decisively for theory - they make every coefficient in the perturbation equations time dependent. Analytical work therefore proceeds by WN expansion: the interface shape and the velocity potentials (the perturbation flow is usually taken incompressible, irrotational and inviscid) are expanded in mode amplitudes, and coupled amplitude equations are collected order by order. In planar geometry such equations are solvable; in convergent geometry they have been solved only for special background motions - for example a uniformly converging cylinder (Wang et al., 2015), or two-mode perturbations on a uniformly radial background (Guo et al., 2018; Guo et al., 2020) - and the newest 3-D spherical multimode theory, which includes mode coupling and BP effects on a dynamic imploding background, is still truncated at second order and states in its own text that the evolution equations preclude closed-form analytic solutions (Li et al., 2026). The density contrast is conventionally measured by the Atwood number A=(ρh−ρl)/(ρh+ρl)A=(\rho_h-\rho_l)/(\rho_h+\rho_l), which enters every growth formula.

Beyond the WN regime, nonlinear analyses in convergent geometry exist only for single-mode perturbations - nonlinear evolution models for cylindrical interfaces, finite-thickness shells and spherical bubbles (Zhao et al., 2019; Zhao et al., 2020; Lu et al., 2025; Zhang et al., 2026) - while compressibility (Yu and Livescu, 2008) and viscous/elastic/plastic material response (Sun et al., 2021; Piriz et al., 2022; Zeng et al., 2024; Wang et al., 2025) have been treated only linearly. The transition to turbulent mixing at a converging interface is today computed only by large-scale multidimensional simulation of individual designs (Haines et al., 2014; Joggerst et al., 2010), or by engineering mixing models whose closure content is transplanted from planar experiments (Liu et al., 2024); experiments demonstrating controlled RTI growth in cylindrical implosions (Sauppe et al., 2020) are recent and few. What does not exist is a predictive theory that carries a multimode perturbation spectrum on a compressible, converging interface with realistic material response from the linear stage through nonlinear saturation into turbulent mixing - the shared obstacle identified, from different angles, by early 2-D multi-wavelength simulation studies (Atzeni and Guerrieri, 1993), by the convergent-geometry WN analyses whose amplitude equations lack general solutions, and by linear material-effect analyses that defer nonlinear saturation, turbulent mixing, plasticity, three-dimensionality and compressibility to future work. The field's own review literature states the same objective: predictive control of instabilities and mixing in imploding systems is the pivotal open fluid-dynamics problem for ICF (Zhou, 2017; Zhou, Sadler and Hurricane, 2025).

Problem Statement

Construct a general predictive theory of the Rayleigh–Taylor instability at a converging (imploding) interface - cylindrical or spherical - that carries the evolution from the linear growth stage, through nonlinear saturation and mode coupling, to the transition to turbulent mixing, and that remains valid for multimode initial perturbations, compressible flow, and realistic material response (viscosity, elasticity and plasticity; ablative drive where relevant). These components jointly serve one objective - predictive description of convergent-interface RTI and mixing - rather than four independent deliverables, and a solution must realize all four: the amplitude-equation solution feeds the nonlinear multimode description, which in turn supplies the physical content any turbulent-mixing closure for converging geometry must encode. Concretely, the theory should: (i) solve, or systematically reduce, the convergent-geometry perturbation (amplitude) equations for an arbitrary time-dependent radial background motion - not only the special uniformly converging or self-similar backgrounds treated so far - in both cylindrical and spherical geometry, retaining the Bell–Plesset coupling of perturbations to the moving interface and going beyond the second-order truncations of prior work, either as a closed system truncated at a stated order N ≥ 3 in the perturbation amplitude or as an all-order systematic scheme, with the neglected higher-order mode couplings and the resulting truncation error stated explicitly in either case; (ii) describe nonlinear multimode evolution - harmonic generation, mode competition, and the geometry-dependent spectral energy transfer - rather than single-mode interfaces; (iii) predict the growth of the mixing zone and its transition to turbulence in convergent geometry, with closure content derived for converging flow and incorporating the compressible and material-response effects named in the objective rather than transplanted unmodified from planar experiments; and (iv) reduce, in the appropriate limits, to the established planar theory, the linear Bell–Plesset results, the existing weakly nonlinear convergent-geometry solutions, and the single-mode nonlinear models. For (i), a systematic reduction must take the form of a closed, evaluable system of the reduced amplitude equations - an explicit closed set of ODEs for the mode amplitudes, with no unresolved time-dependent coefficients - valid for an arbitrary time-dependent radial background, with the truncation or closure and its error stated as required in (i) and with the finite closed set related to the multimode coupling of (ii) through that stated truncation, and returning the known special-background solutions (e.g. the uniformly converging-cylinder results) when such a background is imposed; component (iv) is a consistency requirement on the construction, not an independent deliverable. The construction must remain valid for the conditions named in the opening objective - multimode initial perturbations, compressible flow, and viscous, elastic and plastic (and, where relevant, ablative) material response. Component (ii) must retain explicit mode-coupling terms that reproduce the established weakly nonlinear single-mode and multimode limits when the coupling is switched off. Component (iii) must meet three requirements: its closure content must depend explicitly on the time-dependent interface radius, its derivatives and the convergence ratio (the Bell–Plesset terms, which have no planar analogue), rather than being transplanted unmodified from planar experiments; it must predict the onset of the turbulent mixing regime - the transition time (or perturbation amplitude) at which the mixing layer attains self-similar growth, defined as a mixing-layer width following a power law in time with a constant dimensionless growth coefficient, and its evolution ceases to depend on the initial perturbation spectrum; and its mixing-zone growth and closure must reduce to the planar mixing-zone growth law (the buoyancy–drag and calibrated mixing-zone closures described in the Background) as the convergence (Bell–Plesset) terms are switched off. Where published convergent-geometry benchmarks exist - for example the laser-driven cylindrical implosion experiments of Sauppe et al. (2020) and high-resolution multidimensional simulations - the theory must be expressed in a form whose observables permit quantitative comparison with their reported results, without being tied to any particular dataset.

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