Sufficient conditions for Rayleigh-Taylor instability

[TheCanadian]

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.

 

[nbryan5]

Is this assumption valid?1. The reason why the phase speed of the instability must be sufficiently slow is because the growth rate of the instability is proportional to the phase speed - the slower the phase speed, the greater the growth rate. This is due to the fact that when the phase speed of the instability is greater than the characteristic speed of the medium, the instability is unable to transfer energy efficiently to the medium and thus the instability will not grow. 2. For the RTI, the Atwood number is defined as the ratio of the density of the two fluids (A and B). The Atwood number must be positive in order for an RTI to occur, which means that the density of fluid A must be greater than the density of fluid B. In this paper, the authors are considering the case where the density of fluid B is greater than the density of fluid A, so the Atwood number is negative. However, this does not mean that the instability does not occur; it just means that the instability has a different character than the traditional RTI.3. The theoretical maximum acceleration is limited by the sound speed of the medium. This is because when the acceleration of the instability exceeds the sound speed, the instability is no longer able to transfer energy to the medium and thus the growth rate of the instability is decreased.4. The hydrostatic pressure approaching the Schwarzschild radius is not infinite, but rather the pressure gradient becomes infinite. This is because the gravity near a black hole is so strong that the pressure gradient of the medium becomes very large near the Schwarzschild radius. This means that the linear analysis that the authors apply is not valid near the Schwarzschild radius.5. The authors are assuming that the magnetic field has no dynamic influence, which is a valid assumption for weak magnetic fields. However, for stronger magnetic fields, the magnetic field should be taken into account as it can have a significant effect on the dynamics of the plasma.