All numerical atmospheric models are built around calculations derived from primitive equations that describe atmospheric flow. Vilhelm Bjerknes discovered the relationships and thereby became the father of numerical weather prediction. Conceptually, the equations can be thought of as describing how a parcel of air would move in relationship to its surroundings. For instance, we learn at a young age that hot air rises. The hydrostatic vertical momentum equation explains why and quantifies under what condictions hot air would stop rising. (As the air rises it expands and cools until it reaches hydrostatic equilibrium.) The other equations consider other types of motion and heat transfer.
Unfortunately, the equations are nonlinear, which means that you can't simply plug in a few numbers and get useful results. Instead, weather models are simulations which divide the atmosphere into three-dimensional grids and calculate how matter and energy will flow from one cube of space into another during discrete time increments. Actual atmospheric flow is continuous, not discrete, so by necessity the models are approximations. Different models make different approximations appropriate to their specific purpose.
Numerical models have been improving over time for several reasons:
- More and better input data,
- Tighter grids, and
- Better approximations.
Increasing computational power has allowed models to use smaller grid boxes. However, the number of computations increases exponentially with the number of boxes and the process suffers diminishing returns. On the input end of things, more and better sensors improve the accuracy of the initial conditions of the model. Synoptic scale and mesoscale models take input from General Circulation Models, which helps set reasonable intial conditions. On the output end, Model Output Statistics do a remarkable job of estimating local weather by comparing the current model state with historical data of times when the model showed similar results. Finally, ensemble models take the output of several models as input and produce a range of possibly outcomes.