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Kelvin–Helmholtz Billow Tank

Learning goal

Make the interface dispersion relation visible. It is a wall of algebra with four physical regimes hiding inside it; a slider that crosses the stability boundary turns it into a single idea you can hold.

Things to try

  1. Air over water, raise ΔU slowly. Nothing happens, nothing happens — and then at 6.60 m/s a band of wavenumbers goes unstable. That number is not fitted; it is $\(\Delta U_c = \sqrt{\frac{2(\rho_1+\rho_2)}{\rho_1\rho_2}\sqrt{g\gamma(\rho_2-\rho_1)}}\)$ and it is why a glassy sea stays glassy in a light breeze and roughens rather suddenly at around force 4. (The real onset is messier — wind generates waves by other mechanisms too — but the scale is right and the mechanism is real.)

  2. Look at the wavelength at onset. \(\lambda_c = 2\pi\sqrt{\gamma/g\Delta\rho} = 1.71\) cm. This is the same capillary–gravity crossover that sets the size of the smallest wind ripples on a pond — the scale at which surface tension hands over to gravity. Two independent phenomena, one length.

  3. Turn surface tension off. The stability threshold collapses to zero: any shear at all destabilises some wavenumber. Look at what the readout says about the fastest-growing mode — there isn't one. Growth increases without bound as \(k\to\infty\), which means the pure vortex sheet is ill-posed: arbitrarily small wavelengths grow arbitrarily fast. Surface tension (or viscosity, or a finite shear-layer thickness) is what makes the problem well posed. This is the single most important caveat in the whole chapter, and it is easy to miss on paper.

  4. Fresh water over salt. Density contrast is tiny (998 vs 1025) and there is no surface tension between miscible fluids, so the interface is destabilised by very gentle shear. This is why oceanic KH billows are routinely imaged by sonar in the thermocline, and why clear-air turbulence exists at atmospheric inversions — the same instability, with cloud occasionally kind enough to make it visible.

  5. Equal densities. Gravity drops out entirely. Pure shear, unstable at any ΔU once tension is off — the classic textbook vortex sheet.

  6. Watch the growth-rate curve, not the tank. The shaded band is where \(\mathrm{Re}(s) > 0\). Note that it is bounded on both sides when tension is on: long waves are stabilised by gravity (the \(kg\Delta\rho\) term) and short waves by tension (the \(\gamma k^3\) term). Instability lives in the middle, which is why billows have a characteristic size.

An honest caveat about the animation

The tank shows \(h(x,t) = h_0e^{\mathrm{Re}(s)t}\cos(kx + \mathrm{Im}(s)t)\) — linear theory. Real KH billows roll up into vortex cores and eventually break, and this model cannot produce that: it is only valid while the amplitude is small compared with the wavelength. Once the readout says the amplitude has saturated, the picture is a cartoon and the dispersion plot below is the part still telling the truth.

Kelvin–Helmholtz instability · Interface dispersion relation · Rayleigh–Taylor · Hydrodynamic stability · Vorticity — a shear layer is a vortex sheet · Interchange instability (PC368) — the magnetised cousin