♨️ Rayleigh-Bénard Convection — Thermal Cells

The stream-function/vorticity method requires solving a Poisson equation with Gauss-Seidel relaxation every single frame (20 iterations here), and that cost grows quickly with grid size. A 96×48 lattice is coarse enough to update in real time in the browser while still being fine enough to resolve the roll pattern, whose natural wavelength is roughly twice the layer depth.

The stirAt() function converts your pointer position into the nearest interior grid cell and adds +0.6 to its vorticity and +0.3 to its temperature (capped at 1). This lets you manually kick-start convection cells below Ra_c, or disrupt an existing roll pattern to watch it reorganise.

The bottom row is pinned to T = 1 (hot) and the top row to T = 0 (cold) every step, which is what drives the whole system. The side walls are insulated (zero temperature gradient) so no heat escapes sideways. All four walls are rigid and no-slip, enforced on vorticity with Thom's formula ω_wall = -2·ψ_adjacent, which reproduces zero velocity at the boundary without needing a ghost layer.

Instead of tracking pressure, the code uses the vorticity-stream function formulation: vorticity ω is transported and diffused directly, then ∇²ψ = -ω is solved (via 20 Gauss-Seidel iterations per frame) to recover a stream function ψ whose derivatives, u = ∂ψ/∂y and v = -∂ψ/∂x, are automatically divergence-free by construction — incompressibility is guaranteed by the math, not enforced with an extra pressure-correction step.

It checks two things: whether Ra/Ra_c is above 1 and whether the average kinetic energy of the velocity field has grown past a small threshold (1e-4). Near the threshold the ratio alone can be misleading — the perturbation added at seed time takes a moment to grow into visible rolls — so the badge only switches to "Convecting" once real motion is detected.