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.

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

A = (ρ_h − ρ_l) / (ρ_h + ρ_l)

σ(k) = √(A g k) (linear theory, no surface tension or viscosity)

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.