There is a current explosion of interest in new numerical methods for atmosphere modeling. A driving force behind this is the need to be able to efficiently simulate complex systems of partial differential equations on massively parallel computer systems. The previous generation of numerics depends on structured, orthogonal meshes which suffer from the ``pole problem": as the model resolution is increased the resolution explodes at one or more special points in the grid. Recently developed finite element discretizations avoid this issue while maintaining the exact force balances which are essential for accurate atmosphere simulations. However, the efficient solution of the resulting saddle-point systems arising from semi-implicit methods is a major source of difficulty.

In this talk, we present solution techniques for the saddle-point systems arising in geophysical flows. The procedure known as ``hybridization'' mirrors standard finite difference techniques, where the resulting equations are algebraically manipulated to produce a sparse elliptic operator. We demonstrate this within the context of different atmospheric models and their implementation in open-source software available as part of the Firedrake Project. We also discuss the application of hybridization within the next-generation dynamical core project at the UK Meteorological Office (Gung-Ho!).