Some generic properties of the nonlinear equations of fluid flow are demonstrated with simple illustrative problems. Properties of the shallow water model are then described, and the solutions shown to be close to that of a ‘balanced’ approximation to them.

In three dimensions, the generalisation of the concept of ‘balance’ leads to models from which sound waves have been filtered, in particular the incompressible and anelastic models. Properties of their solutions are described. In particular, analytic solution of the equations requires solution of an elliptic problem, which thus also has to be solved in numerical models using the equations. The hydrostatic equations can be solved without solving an elliptic problem, but it is shown that this means that the solutions break down for weak stratification. Use of the hydrostatic approximation in numerical models requires use of a numerical equivalent of a non-hydrostatic pressure to ensure stability.

Operational models are more correctly viewed as solving space-time averages of the equations. Both Eulerian and Lagrangian averaging procedures are illustrated. In particular, both suggest that the averaged variable representing the fluid trajectory is best treated as different from that representing the momentum.

Averaged equations can be related to filtered models in which all inertia-gravity waves are removed. While such models do not give a complete description of the atmosphere, since they exclude real waves, they can describe the motions that are wellresolved and predictable by operational models. Their properties are thus useful in designing models, particularly the way that the computation of the resolved flow is related to the sub-grid models which parametrise the unresolved motions.

**Keywords: **Nonlinear Equations Averaging Balance

- Introduction
- Observed behaviour
- Toy problems
- Shallow water equations
- Three dimensional equations
- Averaged equations
- Summary
- References

}, year = {2002}, journal = {Meteorological Training Course Lecture Series}, month = {2002}, publisher = {ECMWF}, url = { }, language = {eng}, }