Numerical Weather Prediction: 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. Push it past 1.0 and the scheme goes numerically unstable — you'll see the field erupt into checkerboard noise, a direct demonstration of the CFL 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 translucent floor plane 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 and the truth field sampled (bilinearly) at the same points — a simplified version of how real forecast skill is scored against verifying analyses.

⚠ CFL > 1 — numerically unstable

High pressure anomaly Low pressure anomaly Truth field (floor)

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