Shallow-Water Weather Grid Solver (2D)

Grid & Physics

Grid resolution 32 × 32

Coriolis parameter f 0.80

Eddy diffusion ν 0.05

CFL safety factor 0.40

Drag the grid to pan, scroll/pinch to zoom. Click a cell to inject a low there directly; shift-click for a high.

Live Readouts

Sim time

0.0 s

Step Δt

0.0 ms

Max wind |v|

0.00

RMS pressure h'

0.000

How it works

This is a real finite-difference core for the linearized shallow-water equations — the same conservation-of-mass / conservation-of-momentum system a numerical weather model is built from, just at toy resolution:

∂h/∂t = -H(∂u/∂x + ∂v/∂y)                (mass continuity)
∂u/∂t = -g∂h/∂x + f v - r u + ν∇²u        (momentum, x)
∂v/∂t = -g∂h/∂y - f u - r v + ν∇²v        (momentum, y)

h is the pressure-height perturbation (the heatmap color), (u, v) is the horizontal wind field (the white arrows), f is the Coriolis parameter, ν is an eddy-diffusion term standing in for the sub-grid turbulence real models parameterize, and r is a small fixed damping constant for numerical stability.

Stability note: a plain forward-time / centered-space stepping of this system (as a naive port would do) is a textbook-known unconditionally unstable scheme — a standalone numerical check confirmed it blows up to infinity within a few hundred to few thousand steps at every tested CFL setting, not just above some threshold, contrary to what "stay under the CFL bound and it's stable" would suggest. This build instead uses a Lax–Friedrichs correction (each cell's own previous value is replaced by the average of its four neighbors before the update) plus a tightened, diffusion-aware step-size cap — the combination genuinely respects the safety factor across the exposed parameter range.

Inject a low-pressure cell and watch the pressure dip radiate outward as gravity waves while the Coriolis term deflects the inflowing wind sideways until it settles into a rotating, roughly circular flow around the low — geostrophic adjustment, the same mechanism that spins up real cyclones.

High pressure h' > 0 Low pressure h' < 0 Wind vector (u, v)

drag: pan · wheel: zoom
click: inject low · shift+click: high