Mathematical models represent the heart of the Numerical Weather Prediction process. The physical/mathematical weather prediction problem can be reduced to an “initial condition problem” and consists of the solution of a set of equations integrated starting from the initial state defined in the analysis. Atmospheric behavior is in fact described by specific physics equations, in essence, reducible to conservation principles of momentum, atmospheric species mass and energy.
The equations used in weather prediction are known as primitive equations; they summarize our knowledge about the dynamics and thermodynamics of the atmosphere. In their solving through numerical methods, since the “analytic” approach is unusable, primitive equations are not solved in their general theoretical form, but they are rewritten in an approximated form, suitable for the computational implementation; this passage is the first possible source of “error” for the expected values of meteorological variables. Resolution techniques for prediction equations of NWP models in general use a discrete representation of the atmospheric variables on a regular spatial grid (grid points models) or express the functions as a sum of harmonics, through Fourier analysis (spectral models). The atmosphere is represented at different levels through grid points that, seen in space, form cells. In this way continuous physical fields become discretized; it follows just an approximated representation of the atmospheric fields and an incorrect calculation by gradients finite difference techniques and of the superior order derivatives (truncation error).