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

An interpretation of TRiSK-type schemes from a discrete exterior calculus perspective