On some stability properties of the discretization of damped propagation of shallow-water inertia–gravity waves on the Arakawa B-grid

We investigate a problem mentioned by Deleersnijder and Campin (Deleersnijder, E., & Campin, J. -M. (1993). Ocean Modelling 97, 2), who looked at the problem of the discretization of inertia–gravity waves in the staggered B-grid. We generalize this study by adding diffusion. General stability conditions are found with the help of Miller's theorem, and the paradox found in Deleersnijder and Campin (Deleersnijder, E., & Campin, J. -M. (1993). Ocean Modelling 97, 2) is discussed. It is argued that it stems from inappropriate application of boundary conditions in conjunction with a Coriolis force treatment which could produce mechanical work.