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Numerical weather prediction (NWP) forecasts the weather by solving the physical equations of the atmosphere on a computer, starting from the best estimate of its present state. The work has three main steps: combine observations into a starting picture (data assimilation), step the equations forward in time on a grid, and turn the raw output into forecasts people can use. Because the atmosphere is chaotic, small errors in the starting state grow, which limits useful day-to-day forecasts to roughly two weeks. Ensemble forecasts run the model many times from slightly different starting states to show how certain a forecast is.
Treat the forecast as an initial-value problem: if you know the state now and the laws of change, you can step forward. Have students step a simple equation forward by hand for a few time steps to feel how time steps and grid spacing work. Then show an ensemble "spaghetti plot" of one contour line from many runs, and discuss why the lines spread apart with forecast lead time.
At the start of the twentieth century, the Norwegian scientist Vilhelm Bjerknes argued that forecasting could be treated as a physics problem. If you know the state of the atmosphere now and the laws that govern it, you can calculate its future. Those laws include Newton's second law for moving air, conservation of mass, the first law of thermodynamics, and the gas law, along with equations for water vapor.
Lewis Fry Richardson tried this by hand during the First World War and published the result in 1922. It took him weeks to compute a six-hour forecast, and the answer was badly wrong. The idea worked once computers arrived. In 1950, Jule Charney, Ragnar Fjørtoft, and John von Neumann ran the first computer forecast on the ENIAC, using a simplified equation for the flow at mid-levels.
A modern forecast cycle has three parts. First comes data assimilation. Millions of observations from weather balloons, satellites, aircraft, ships, and surface stations are blended with a short-range forecast to estimate the state of the atmosphere at every grid point. Second comes the model run. The globe is divided into a three-dimensional grid, and the equations are stepped forward in short time steps. Processes smaller than a grid box, such as most clouds, turbulence, and radiation in detail, are handled by parameterizations. Third comes post-processing, which corrects known model biases and turns the output into temperature, wind, and precipitation forecasts.
There is a limit to how far ahead this can work. In 1963, Edward Lorenz showed that a simple model of convection was extremely sensitive to its starting values. Tiny differences grew until two runs looked nothing alike. The real atmosphere behaves the same way. Even with an excellent model, day-to-day forecasts lose their skill after about two weeks.
Forecasters respond with ensembles. Instead of one run, a center runs dozens, each starting from a slightly different but realistic state. When the members agree, confidence is high. When they spread apart, the forecast is uncertain, and the ensemble can give probabilities, such as a 70 percent chance of more than 10 mm of rain. Better observations, better assimilation, and faster computers have steadily lengthened the range of useful forecasts.