# Sufficient conditions for Rayleigh-Taylor instability

I was reading the paper entitled "The Rayleigh—Taylor instability in astrophysical fluids" by Allen & Hughes (1984), and they discuss relativistically hot plasmas in the context of weak magnetic fields which are presumed to have no dynamic influence, so they take a fluid approach. In this paper, they analyze the RTI growth rate through a linear stability analysis. They state a few conditions for instability onset but I seem to not be grasping the physical meaning behind the following conditions/aspects of the RTI:

1. A growing instability only occurs for $\frac {\omega^2} {k^2 c^2} \ll 1$ as indicated between equations (27) and (28). Why must the phase speed of the instability be sufficiently slow?

2. After equation (40), it indicates that the instability can occur for $\rho_{01} \ll \rho_{02}$ which would indicate a negative Atwood number. But how is this possible? Does not the density gradient have to be opposite the direction of the effective gravity? Must not the Atwood number be necessarily positive for a Rayleigh-Taylor instability?

3. In (34) the authors state there is a theoretical maximum acceleration. What physically limits this? Is there a maximum rate at which energy can be transferred to the instability? Is the linear analysis the authors apply even valid in such a regime?

4. In their analysis of RTI outside black holes, why does approaching the Schwarzschild radius result in an infinite hydrostatic pressure (in this relativistic consideration)? Is this incorrect? The gravity here is still finite which would indicate there is no Rayleigh-Taylor instability as one approaches the Schwarzschild radius by simply inserting (55) into (28), but this seems unintuitive. What I'm further confused by is that the authors state at the top of the final page (page 621) that hydrostatic support cannot be obtained above a black hole which I presume is approaching the Schwarzschild radius from outside the black hole. But isn't this in contradiction with the infinite pressure indicated by (55)?

5. There are no equilibrium bulk flows and perturbations are subsonic in this analysis. But for a plasma, is it not essential to consider the developing magnetic fields due to even subsonic flows? Especially when these subsonic flows are still possibly relativistic, the authors dismiss the dynamic impact as unimportant.

• This basically won't make sense to anyone who hasn't read the paper – Kyle Kanos May 6 '18 at 18:22
• I add the references to equations in case one is interested in checking the claims. But it essentially boils down to: 1) Why intuitively must phase speed be much smaller than the speed of light for the RTI in general? 2) Can a negative Atwood number support the RTI? 3) What physically causes a maximum acceleration (or saturation) in linear RTI? 4) and 5) are a bit specific to the analysis in the paper, but are generally just regarding validity of conditions when applying linear stability analysis to RTI. – Mathews24 May 6 '18 at 18:37
• generally we prefer one question per post, not 5+, and links to articles as reference, rather than requirements. – Kyle Kanos May 6 '18 at 18:40
• Sure, I can split it. – Mathews24 May 6 '18 at 18:59