Rayleigh–Taylor Instability
A heavy fluid sits above a lighter one in a gravitational field — the classic Rayleigh–Taylor configuration. Linear stability theory says a small sinusoidal disturbance of wavenumber k grows in time as exp(σt) with σ = √(A g k), where A = (ρ_h − ρ_l)/(ρ_h + ρ_l) is the Atwood number. The interface is built as a superposition of many such Fourier modes with random phases; each mode grows at its own σ until a tanh envelope saturates the amplitude, producing the characteristic mushroom-shaped fingers of heavy fluid penetrating into the light one.
Who it's for: Intro fluid instability, astrophysics (supernova remnants, ICF), and applied mathematics (linear stability, Fourier methods).
Key terms
- Rayleigh–Taylor instability
- Atwood number
- linear stability
- Fourier modes
- mushroom finger
- inertial confinement
How it works
When a heavier fluid sits **above** a lighter one in gravity, any tiny disturbance of wavelength λ grows exponentially. Short waves grow fastest (σ = √(A g k)) — that is why classic **mushroom fingers** appear and pinch off downward.
Key equations
Frequently asked questions
- Which wavelengths grow fastest?
- In the pure inviscid, surface-tension-free model σ = √(A g k) grows without bound with k, so very short wavelengths dominate. Real systems are regularised by viscosity and surface tension, which cut off small scales and select a finite "most unstable" wavelength.
- Why do the fingers look like mushrooms?
- Once the linear stage saturates, secondary Kelvin–Helmholtz roll-up at the sides of each finger curls the tip outward. Our simulator only fakes the saturation with a tanh envelope, but the spacing and growth of the fingers come from real linear theory.
- Where does this instability matter in real life?
- Anywhere a denser fluid pushes on a lighter one — supernova remnants, inertial-confinement fusion capsules, atmospheric inversions, and even cream poured into coffee.
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