fluid dynamics - Rayleigh-Taylor instability with negative Atwood number?

I was reading a paper entitled "The Rayleigh—Taylor instability in astrophysical fluids" by Allen & Hughes (1984) that indicates the instability can occur for $ \rho_{01} < \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?

Answer

Your intuition is correct; there's no such thing as a Rayleigh-Taylor instability with a negative Atwood number. That would imply that the density of the upper fluid, $\rho_{01}$, is less than the density of the lower fluid, $\rho_{02}$, which is clearly a stable situation with respect to the R-T instability.

So how did $\rho_{01} < \rho_{02}$ appear in the Allen and Hughes paper? I'm pretty sure it was just a typo. I read through the paper and the only place I saw anything that looked like a negative Atwood number was in section 4.2.2, where there's a sentence: In conclusion, it may be seen that the growth of R—T instabilities saturates for large accelerations, except in the limit $\rho_{01} \ll \rho_{02}$ where the growth remains of the usual form

$$ \omega = (gk)^{1/2} $$

But this sentence refers to an earlier paragraph in the same section that says, "Again, for $\eta$ ~ 1, compressibility has little effect and $\omega^2 \approx gk$." Since in the authors' notation, the Atwood number $\eta$ is defined as

$$ \eta = \frac{\rho_{01} - \rho_{02}}{\rho_{01} + \rho_{02}} $$

it is obvious that $\eta$ ~ 1 implies $\rho_{01} \gg \rho_{02}$ rather than $\rho_{01} \ll \rho_{02}$.