fluid dynamics - Why is inverting a cup with water in it an example of the Rayleigh-Taylor instability?

Sorry if this is considered a duplicate, but I have further questions that are based on an old post. I was told that air pressure is strong enough to hold up water. Is this true? The vapour pressure of water at room temperature is 0.0313 atm which is far smaller than air pressure. But air pressure is 101325 N/m$^2$ and a tall column of liquid water with a surface area of 0.001m$^2$ and height of 1000m is 997 kg. This translates into a net gravitational force of 9771 N or 9771000 N/m$^2$ which is greater than that of air pressure in this contrived example.

Also, in the experiment of inverting a cup with water where there is a flat sheet cover on the opening of the cup, does the flat layer simply suppress amplitude oscillations of the water at the interface between the water and layer and thus stop Rayleigh-Taylor instabilities from growing? Does the sheet have to be perfectly flat? What is stopping the sheet from falling down? Are there important water-surface interactions that keep the surface from falling? Is this valid for any liquid (e.g. oil) and any surface (e.g. denser than air) that is not porous? Is it still consider the Rayleigh-Taylor instability once the evolution become nonlinear?

Answer

Yes, it is true, depending on the size of the vessel.

If you fill a vessel with water and place a massless, impermeable membrane across its mouth, then whether or not the water will fall out when you invert the vessel depends on the balance of fluid pressure pushing out and atmospheric pressure pushing in. The water will stay in the cup if $$P_{atm} \geq \rho_{water}gD$$

where $D$ is the depth of the cup. Roughly speaking, the trick works if $ D\leq 10 \text{ m}$, which is a pretty reasonable requirement.

Note that the pressure balance does not depend on the presence of the membrane. However, the presence of the membrane prevents the onset of the Rayleigh-Taylor instability by not allowing the air to "bubble up" through the water.

Because the membrane resists deformation, there is no instability; in the absence of such protection, the arrangement of water-above-air becomes unstable to perturbations in the air/water interface and the experimenter gets all soggy.