Abstract
TRiSK-type numerical schemes are widely used in both atmospheric and oceanic
dynamical cores, due to their discrete analogues of important properties such
as energy conservation and steady geostrophic modes. In this work, we show that
these numerical methods are best understood as a discrete exterior calculus
(DEC) scheme applied to a Hamiltonian formulation of the rotating shallow water
equations based on split exterior calculus. This comprehensive description of
the differential geometric underpinnings of TRiSK-type schemes completes the
one started in \cite{Thuburn2012,Eldred2017}, and provides a new understanding
of certain operators in TRiSK-type schemes as discrete wedge products and
topological pairings from split exterior calculus. All known TRiSK-type schemes
in the literature are shown to fit inside this general framework, by
identifying the (implicit) choices made for various DEC operators by the
different schemes. In doing so, unexplored choices and combinations are
identified that might offer the possibility of fixing known issues with
TRiSK-type schemes such as operator accuracy and Hollingsworth instability.