  The shallow water equations provide a useful analogue of the fully
compressible Euler equations since they have similar characteristics:
conservation laws, inertia-gravity and Rossby waves and a (quasi-) balanced
state. In order to obtain realistic simulation results, it is desirable that
numerical models have discrete analogues of these properties. Two prototypical
examples of such schemes are the 1981 Arakawa and Lamb (AL81) C-grid total
energy and potential enstrophy conserving scheme, and the 2007 Salmon (S07)
Z-grid total energy and potential enstrophy conserving scheme. Unfortunately,
the AL81 scheme is restricted to logically square, orthogonal grids; and the
S07 scheme is restricted to uniform square grids. The current work extends the
AL81 scheme to arbitrary non-orthogonal polygonal grids and the S07 scheme to
arbitrary orthogonal spherical polygonal grids in a manner that allows both
total energy and potential enstrophy conservation, by combining Hamiltonian
methods (work done by Salmon, Gassmann, Dubos and others) and Discrete Exterior
Calculus (Thuburn, Cotter, Dubos, Ringler, Skamarock, Klemp and others).
Detailed results of the schemes applied to standard test cases are deferred to
Part 2 of this series of papers.
