numerical weather prediction
When the forecaster says there is a 70 percent chance of rain on Thursday, that number did not come from looking at the sky. It came from a supercomputer that solved the equations of the atmosphere overnight. Numerical weather prediction is the daily, planet-scale practice of taking today's measured state of the atmosphere and integrating the physical equations forward in time to predict tomorrow's.
Under the hood it is a giant CFD problem on a rotating sphere. The atmosphere is covered by a 3D grid — today's global models use cells of roughly ten kilometers or finer — and at each grid point the model carries wind, pressure, temperature, and humidity. The governing equations (the Navier-Stokes equations plus thermodynamics and the laws for water vapor and radiation) are stepped forward by minutes at a time. Processes too small for the grid, like individual clouds and turbulence, are handled by parameterizations, which are essentially physically-motivated models of the unresolved scales. Two pieces are decisive: data assimilation, which blends millions of fresh observations from satellites, balloons, and stations into the best current starting state, and ensembles, where the model is run many times from slightly perturbed starts to estimate the spread, and hence the probability, of outcomes.
NWP is one of the quiet triumphs of computational science: a modern five-day forecast is about as accurate as a one-day forecast was forty years ago. Its hard limit is not the computer but the physics. The atmosphere is chaotic — Edward Lorenz's famous discovery that tiny differences in the start blow up exponentially — so even a perfect model run from imperfect data loses skill after roughly two weeks. That is why forecasts are probabilistic and why a single deterministic run far into the future should never be trusted as fact.
To forecast a hurricane's path, a center runs an ensemble of, say, 50 model integrations from slightly different starting states. If most members curve the storm out to sea but a few drive it ashore, the spread of tracks IS the forecast: the famous 'cone of uncertainty' is literally the cloud of ensemble paths, not a single predicted line.
A hurricane forecast cone is an ensemble of model runs — uncertainty made visible, not a single prediction.
Because the atmosphere is chaotic, forecast skill decays no matter how good the model: beyond about two weeks even a perfect model run from imperfect data is no better than climatology. A precise long-range single-run forecast is overconfidence, not insight.