Where Shear Starts to Curl

A nearly flat interface starts to buckle, then rolls into billows once velocity shear has something to amplify.

A web search for Kelvin-Helmholtz instability shear layer clouds NASA Met Office pulls together the three contexts this sketch depends on: the historical note that Helmholtz described the instability in 1868 and Kelvin extended the analysis in 1871, the Met Office explanation of the cloud form that makes the phenomenon easy to recognize, and NASA material on the same rolling structure in atmospheric and plasma flows. The shared mechanism is velocity shear. Two layers move past each other at different speeds, a small ripple appears at the boundary, and the flow feeds that ripple until it becomes a visible curl.

The Kelvin-Helmholtz artifact keeps the setup spare enough to make that escalation readable. It begins with 300 vortices distributed along a nearly flat midline and perturbs their vertical positions by only a few hundredths of the unit domain. Once the update rule starts accumulating induced velocities from neighboring vortices, that small perturbation turns the line from a boundary into a mixing layer. The screen shifts from a thin seam into repeated billows that look like cloud crests, surf, or the bands on Jupiter because those shapes all come from the same instability class.

for (let k = -1; k <= 1; k++) {
  const dx = xi - (xj + k);
  const r2 = dx*dx + dy*dy + DELTA2;
  u += GAM_2PI * dy / r2;
  v -= GAM_2PI * dx / r2;
}

The loop sums the circulation each vortex receives from every other vortex, including periodic image copies at x±1, so the domain reads like a continuous strip instead of a box with dead edges. The DELTA2 term applies Rosenhead regularization to prevent the Biot-Savart kernel from spiking at short range and turning the simulation numerically brittle. The artifact can keep the motion crisp while local interactions pile up into larger rolls.

The sketch stages the condition that produces billows and lets the form arrive on its own. Open the full artifact page or jump straight to the gallery sketch.