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Rayleigh–Taylor Instability

Intuition

Try to float water on top of oil: it won't stay. A heavy fluid resting on a light one is an inverted pendulum — in perfect equilibrium, and doomed. Any ripple lets heavy fluid slump down and light fluid rise; gravitational potential energy is released, the ripple grows, and soon "fingers" of the dense fluid spike downward while bubbles of light fluid rise. That is the Rayleigh–Taylor (RT) instability.

Result

From the general interface dispersion relation with no shear (\(U_1 = U_2 = 0\)), fluid 1 (upper) density \(\rho_1\), fluid 2 (lower) \(\rho_2\):

\[s = \sqrt{g k\,\frac{\rho_1 - \rho_2}{\rho_1 + \rho_2}}\]
  • Heavy over light (\(\rho_1 > \rho_2\)): \(s\) real and positive → exponential growth. The combination \(\mathcal{A} = \frac{\rho_1 - \rho_2}{\rho_1 + \rho_2}\) is the Atwood number; growth rate \(s = \sqrt{gk\mathcal{A}}\).
  • Light over heavy (\(\rho_2 > \rho_1\)): \(s\) imaginary → the interface oscillates:
\[\omega = \sqrt{g k\,\frac{\rho_2 - \rho_1}{\rho_1 + \rho_2}}\]

these are interfacial gravity waves. With \(\rho_1 \to 0\) (air over water), \(\omega = \sqrt{gk}\), giving the deep-water phase speed

\[c = \frac{\omega}{k} = \sqrt{\frac{g}{k}}\]

— longer waves travel faster, which is why long swell arrives at the beach before the storm that made it.

Stabilization

Surface tension penalizes interface curvature and stabilizes short wavelengths: only modes with

\[k^2 < \frac{g(\rho_1 - \rho_2)}{\gamma}\]

grow. This is why small drops hang stably from a ceiling but a large painted surface "rains": there's a critical patch size beyond which some unstable wavelength fits.

Where you see it

An overturned glass of water (air pushing up into water) · Crab Nebula filaments (supernova ejecta decelerated by interstellar medium — effective gravity from deceleration) · inertial-confinement fusion capsules (the great engineering enemy) · salt fingers in the ocean · mushroom clouds.

Equivalence principle bonus: "gravity" can be any acceleration of the interface. Accelerating a light fluid into a heavy one is RT-unstable even sideways — this is why ICF implosions and decelerating supernova shells share the same physics.

Common mistakes

  • Thinking static equilibrium implies stability. The stratified state satisfies hydrostatics exactly; stability is a separate, perturbative question — see hydrodynamic stability.
  • Confusing RT and Kelvin–Helmholtz: RT needs a density inversion (or acceleration), no shear; KH needs shear, no inversion. (Real flows often have both — RT spikes develop KH curls on their flanks.)
  • Using the linear growth rate for late times. Exponential growth is only the beginning; nonlinear spikes/bubbles follow different (power-law) laws.

Knowledge graph position

Prerequisites: Hydrodynamic stability, Buoyancy. Leads to: gravity-wave theory, turbulent mixing.

Quiz

Q1 (computational). Water (\(\rho = 1000\)) over air (\(\rho = 1.2\)), disturbance wavelength 10 cm, ignore surface tension. Growth rate?

Answer

\(k = 2\pi/0.1 \approx 62.8\ \text{m}^{-1}\), Atwood \(\mathcal{A} \approx 0.9976\). \(s = \sqrt{9.8 \times 62.8 \times 0.9976} \approx 24.8\ \text{s}^{-1}\) — e-folding in 40 ms. Overturned glasses empty fast.

Q2 (conceptual). Why can you nevertheless carry an overturned glass sealed by a card (or keep water in a thin straw with a finger on top)?

Answer

Rigid boundaries/surface tension suppress the unstable modes: in a narrow tube, the longest wavelength that fits is shorter than the surface-tension cutoff \(k_c = \sqrt{g\Delta\rho/\gamma}\), so no growing mode exists (and air pressure supplies the mean force).

Q3 (multiple choice). The deep-water gravity-wave dispersion \(\omega = \sqrt{gk}\) is obtained from the RT analysis by taking:

  • (a) \(\rho_1 \gg \rho_2\) (b) \(\rho_1 \to 0\) (light fluid above) (c) \(g \to 0\) (d) \(k \to 0\)
Answer

(b). Stable stratification with a negligible upper fluid = free-surface water waves.