A central part of weather forecasting is simulating the underlying PDEs
numerically. The Met Office currently use a "C-grid" approach, in which
variables are staggered at different horizontal locations. This gives good
representation of horizontal wave propagation, but requires a
latitude-longitude grid to obtain other desirable properties. Unfortunately,
the excessive concentration of gridpoints close to the poles ultimately causes
inefficient execution on massively-parallel supercomputers. Shortly before I
started my PhD, Colin Cotter (Imperial) proposed mixed finite element methods
as a way of obtaining these desirable properties without being restricted to
the latitude-longitude grid. Such methods were introduced by Brezzi,
Nedelec and others, in the 70s and 80s, and have recently been unified by
Doug Arnold and collaborators under the label "finite element exterior
calculus". We have used Arnold's ideas to extend our method in the vertical
direction using tensor-product finite elements, producing a finite-element
analogue of the Charney-Phillips vertical staggering.