Continuous equation sets and approximations

Desire for computational efficiency motivates approximation of the governing PDEs. Accuracy and self-consistency are major issues, and casual approximation can have entirely spurious consequences. The only notable approximation made in the Newtonian momentum equations on which the UM is based is the representation of the mean geopotentials of apparent gravity as spheres. In contrast, most dynamical cores worldwide additionally incorporate the hydrostatic approximation and a 'shallow atmosphere' approximation that implies a counter-intuitive, non-Euclidean geometry and involves an incomplete representation of the Coriolis force.

Various modes of wave-like oscillation occur when small disturbances are applied in an otherwise resting atmosphere. Normal mode analysis of their evolution allows discrimination between different numerical discretization. For example, the particular 'staggering' of velocity components and thermodynamic variables within a computational grid determines the quality of treatment of Rossby waves (especially near the limit of resolution).

Stable flows provide valuable test-beds for the formulation and coding of numerical models. The continuous PDEs have exact analytical solutions that correspond to global, axisymmetric, meridionally sheared flows in geostrophic and cyclostrophic balance under gravity, extended recently to include a family of unsteady flows. However, classical proof of the stability of certain flows of this type only applies to the shallow water equations. How far the proof of stability can be extended to exact solutions of the full equations is not at present clear.

Even with available supercomputers, precise solution of the governing PDEs is impracticable in realistic situations. However, accurate solutions of simplified versions of the governing equations can be computed. The validity of the solutions produced by the dynamical core can be checked by requiring them to reproduce the solutions of simplified equations to the expected degree of accuracy. It is, therefore, important to identify appropriate simplified equations and to estimate their accuracy.

Key aims

• To maintain the Met Office's leading position in the development and numerical application of accurate and conceptually straightforward continuous equation sets in geophysical fluid dynamics.

• To continue to use normal mode analysis to inform the choice of grid staggering in numerical models.

• To seek 3D flows that are necessarily stable and may be used in numerical model testing.

• To validate the accuracy of the dynamical core by showing that it preserves the key properties of relevant simplifications of the governing equations.