Shallow-Water Weather Grid Solver

Grid & Physics

Grid resolution 32 × 32

Coriolis parameter f 0.80

Eddy diffusion ν 0.05

CFL safety factor 0.40

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 mesh's vertical displacement and color), (u, v) is the horizontal wind field (the arrows), f is the Coriolis parameter, ν is an eddy-diffusion term standing in for the sub-grid turbulence parameterizations real models need, and r is a small fixed damping constant for numerical stability.

Every term is discretized on the grid with centered differences and stepped forward explicitly:

h[i,j] += dt·( -H·((u[i,j+1]-u[i,j-1])/2dx + (v[i+1,j]-v[i-1,j])/2dx) )
∇²u ≈ (u[i,j+1]+u[i,j-1]+u[i+1,j]+u[i-1,j]-4u[i,j]) / dx²

Because it's an explicit scheme, the time step is bounded by the CFL condition dt ≤ CFL·dx/√(gH) — raise the CFL slider too far and the grid spacing can no longer keep up with the gravity-wave speed, and the field visibly blows up. 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)

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