A forecast is a calculation. A weather model takes the physical laws for moving air, chops the atmosphere into boxes, and steps every box forward a few minutes at a time. Three things limit the answer: the equations, the grid, and chaos.
At every point the model tracks wind (u, v, w), pressure p, density ρ and temperature T. Six unknowns need six equations, called the primitive equations: two for horizontal momentum, one hydrostatic balance, one for mass, one for heat, and the gas law that ties them together. Real models add equations for water vapor, cloud and rain. Pick an equation to see what its key term does.
How to read this: Seen from above: with nothing else pushing, the Coriolis push (gold) keeps bending the air's path (amber) into a circle.
Try this: pick "mass" and set Wind in and Wind out equal: the level stops moving.
A computer can't hold a smooth field, so the model keeps one number per grid box and replaces derivatives with differences between neighbors. Here the simplest piece of a weather model carries a storm along in a steady wind, ∂q/∂t + u ∂q/∂x = 0, on a loop. The gold curve is the exact answer: the storm should slide along unchanged. The green curve is what the grid computes.
How to read this: the bump is a storm being blown to the right around a loop; its height is how much storm there is. Gold is where it truly is; green is what the computer gets from its boxes. Green turning red means the calculation has broken.
The upwind scheme qi ← qi − C(qi − qi−1) copies part of the upstream box into each box. That mixing acts like extra diffusion with strength K = uΔx(1 − C)/2, now —. Lax–Wendroff adds a second-order correction that keeps the peak but makes ripples and negative rain behind it. Real models use higher-order schemes, but they all obey the same speed limit.
How to read this: four top-down rain maps of the same storms. The first is the real thing. The others show what a model sees when each square box can hold only one average value.
Try this: shrink Storm cell size and watch the coarsest panel lose the storms entirely.
Each box holds the average of everything inside it, so a thunderstorm wide nearly vanishes into a box. A common rule of thumb says a model needs about 4 to 7 grid spacings to represent a feature well. Halving Δx gives 4 times the columns, and the CFL limit then forces a time step half as long, so the work grows about 8 times. Going from to costs about — times more, before adding vertical levels.
Even a perfect model starts from imperfect observations. W00 · Hurricanes shows two runs of the Lorenz 1963 system splitting apart. Here a whole ensemble runs on Lorenz-96, a toy "weather around a latitude circle" with 40 values that drift, interact and decay. Every member starts from the same best guess plus a small random error. The truth is one more run that the forecast never sees.
How to read this: each line is a forecast of "the weather" at 40 spots around the globe; the right edge wraps around to the left. Watch the green forecasts peel away from the gold truth as days pass.
How to read this: how well the forecast matches reality (1 = perfect, 0 = no better than guessing) as it looks further ahead. Below the red line (ACC 0.6, a standard score) the forecast isn't worth much.
Try this: drag Starting error to the far left and see how many days of skill it buys.
The first term moves values around the ring like wind, −xi is friction, and F is steady heating. Integrated with 4th-order Runge–Kutta. Using the usual scaling for this model, 0.05 time units count as 6 hours. "Useful skill" means the anomaly correlation (ACC) stays above 0.6, the threshold forecast centers commonly use.
Errors grow exponentially, so a 10 times smaller starting error buys only a fixed number of doublings. Drag the starting error and watch the skill line shift by roughly the same number of days each decade. In the real atmosphere small-scale errors grow within hours and spread to larger scales, so useful skill for day-to-day weather typically runs out somewhere around a week to ten days, and the practical limit is thought to be about two weeks. Averaging the ensemble cancels part of the random error, which is why the gold line outlasts a single run.
Lewis Fry Richardson worked out the first numerical forecast by hand and published it in 1922. In 1950 Jule Charney, Ragnar Fjørtoft and John von Neumann ran the first computer forecast on ENIAC.