Numerical Weather Prediction 2D: Grid Resolution & Forecast Skill

Forecast Model

Forecast grid resolution 24 × 24

CFL safety factor 0.80

Initial-condition error ε 0.05

Diffusion κ 0.006

Live Readouts

CFL number

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Forecast time

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Forecast RMSE vs truth

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How it works

A forecast field q(x,y,t) — a proxy for pressure/vorticity anomaly — is advected and diffused by a fixed jet-stream wind (u,v) on a doubly-periodic grid, solving the same 2D advection–diffusion equation real NWP models solve for each atmospheric variable:

∂q/∂t + u ∂q/∂x + v ∂q/∂y = κ∇²q

It is integrated explicitly with 1st-order upwind advection + central diffusion on a grid of spacing Δx = 1/N. Numerical stability requires the Courant–Friedrichs–Lewy condition:

CFL = max(|u|,|v|) · Δt / Δx  ≤  1

Grid resolution — sets Δx. A coarser grid has larger truncation error, so the same physical flow is represented less accurately — exactly the "Model Resolution" tradeoff real NWP centers face.
CFL safety factor — scales Δt below (or above) the stable limit, accounting for both the advection term and the diffusion term's own stability limit. Push it past 1.0 and the scheme goes numerically unstable — you'll see the right-hand panel erupt into checkerboard noise, a direct demonstration of the combined stability bound.
Initial-condition error ε — a small random perturbation added to the forecast's starting state, standing in for imperfect observations. Because the flow is chaotic, even a tiny ε makes the forecast diverge from the truth run over time — the "butterfly effect" behind ensemble forecasting.
Diffusion κ — physical/numerical smoothing; too little lets small-scale noise survive, too much erases real structure.

The left panel is a fixed, fine-resolution "truth" run of the same equation from the same initial state. Forecast RMSE is the root-mean-square difference between the forecast grid (right panel) and the truth field sampled (bilinearly) at the same points — a simplified version of how real forecast skill is scored against verifying analyses. Drag the canvas to pan and scroll/pinch to zoom into either grid.

Truth (fixed fine grid)Forecast (your settings)

⚠ CFL > 1 — numerically unstable

High pressure anomaly Low pressure anomaly

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