= Solution
The <Rayleigh-Taylor instability> occurs when a denser fluid is supported against an effective <acceleration> by a lighter fluid. Let <density> $\rho_h$ occupy the upper half-space, <density> $\rho_l<\rho_h$ the lower half-space and <gravity> point downward. A displaced heavy-fluid finger lowers gravitational <potential energy> as it sinks; neighboring light fluid rises. For incompressible inviscid half-spaces without <surface tension>, a disturbance $\eta\propto e^{\sigma t+ikx}$ has
$$
\boxed{\sigma^2=gk\frac{\rho_h-\rho_l}{\rho_h+\rho_l}.}
$$
The <density> ratio in this expression is the <Atwood number>. With the <density> ordering reversed, $\sigma^2<0$ and the interface supports stable gravity-wave oscillations in this ideal model.
\Image[/past-exam-of-the-astrophysical-fluid-dynamics-course-of-the-university-of-cambridge/2012/paper-1-rayleigh-taylor-interface.png]
{title=Rayleigh-Taylor growth: a heavy layer above a light layer develops growing fingers}
<Surface tension> suppresses sufficiently short <wavelengths>, and <viscosity> changes the growth dynamics. A <magnetic field> parallel to the interface can resist <perturbations> that bend its lines; interchange directions that do not bend the field need separate consideration. In astrophysics, effective deceleration can make dense ejecta or a swept-up shell supported by a lower-density hot interior unstable. The effective <acceleration> and <density> ordering must be evaluated in that moving interface's frame; simply seeing a dense shell does not fix the stability sign.
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