The time-dependent Rayleigh-Taylor instability in interstellar shells and supershells, including the eROSITA bubbles

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.