- M. A. Alkhadra

# Rayleigh–Taylor Instability

Updated: Oct 6, 2018

The next time you go out on a Friday night, try performing the following experiment (and analysis), which is certain to earn you new friends. Partially fill a glass with some distilled alcoholic beverage—such as vodka—and wait for the fluid to settle. (As a disclaimer, I am not promoting the consumption of alcoholic beverages; I simply use this example in the spirit of a professor of fluid mechanics I had in graduate school.) Then, slowly add a second layer of a denser liquid—such as tomato juice—while observing the consequences at the interface between the two liquids.

If you are able to visualize the situation carefully (and quickly), you will probably notice that the denser liquid forms droplets that resemble mushrooms drowning in the less dense liquid. If not, then this __video__ may be useful. The phenomenon in the attached video is known as the Rayleigh–Taylor instability after Lord Rayleigh and Sir G. I. Taylor. This hydrodynamic instability occurs at the interface between two fluids of different densities as the lighter fluid pushes upward against the heavier fluid. The basis of this upward push involves a disturbance of the liquid–liquid interface, followed by baroclinic torque due to the misalignment of gradients in pressure and in density at the perturbed interface (**Figure 1**). The rest of this post aims to explain the previous sentence.

The Rayleigh–Taylor instability occurs at the interface between two fluids of different densities as the lighter fluid pushes upward against the heavier fluid.

The Rayleigh–Taylor instability is most commonly described using the concept of __vorticity__, which is the tendency of a fluid to rotate or spin. Now, imagine the effect of a disturbance at the (ideally quiescent) interface separating two immiscible liquids, the top liquid being heavier than the bottom. The interface will locally adopt a sinusoidal configuration, as shown in **Figure 1**, which throws the gradients in pressure and in density out of alignment. The dominant pressure gradient in our model system exists because of gravity and is thus purely hydrostatic. (This is the same pressure gradient that you experience as you get deeper in the swimming pool.) Moreover, our model system naturally exhibits a density gradient by virtue of the liquids we chose to mix. At this point, the misalignment of gradients in pressure and in density are more prominent at the perturbed interface. This misalignment then results in a baroclinic torque that produces vorticity, which in turn will increase the misalignment of the gradients in pressure and in density. Further misalignment leads to additional vorticity, and so on, until the system becomes unstable. If you are a mathematically oriented person, then you will appreciate an explanation that involves the vorticity equation for a baroclinic fluid (this is also where the real winning of friends takes place):

where *D*()/*Dt* is the material derivative, *ω* is vorticity, *u* is velocity, *ρ* is density, *P* is pressure, *τ* is the stress tensor, and *B* is the sum of external body forces. The cross product ∇*ρ* × ∇*P* is the baroclinic term that accounts for changes in vorticity due to intersecting gradients of density and of pressure. The Rayleigh–Taylor instability terminates with the development of regions of turbulence and mixing, which have far-reaching implications in many natural and man-made flows. A particularly beautiful example that arises naturally is found in young supernovae as well as in the Crab Nebula shown in **Figure 2** and in this __video__.