  The Rayleigh-Taylor (RT) instability is omnipresent in the physics of
inversely density-stratified fluids subject to effective gravitational
acceleration. In astrophysics, a steep stratification of the ambient medium can
fragment a bubble shell faster due to a strongly time-dependent RT instability,
causing the classical constant gravity models to fail. We derive the
time-dependent instability criteria analytically for the cases of constant,
exponential, and power-law accelerations, verifying them through
high-resolution numerical simulations. Our results show that (1) even in the
linear phase there is a term opposing exponential growth, (2) non-linear growth
approaches asymptotically the solution found by Fermi and von Neumann, (3) the
interpenetrating spikes and bubbles promote a significant mixing, with the
fractal dimension of the interface approaching 1.6, only limited by numerical
diffusion, and (4) the probability density function (PDF) for the passive
scalar to study mixing becomes increasingly sharper peaked for power-law and
exponential acceleration. Applying our solutions to stellar wind bubbles, young
supernova remnants (SNRs), and superbubbles (SBs), we find that the growth rate
of the RT instability is generally higher in the shells of wind-blown bubbles
in a power-law stratified medium than in those with power-law rising stellar
mechanical luminosities, Tycho-like than Cas A-like SNRs, and one-sided than
symmetric SBs. The recently observed eROSITA bubbles indicate smooth rim
surfaces, implying that the outer shell has not been affected by RT
instabilities. Therefore the dynamical evolution of the bubbles suggests
maximum final ages that are significantly above their current age, which we
estimate to be about 20 Myr.
