Dedication: John Patrick Tully 1906–1987

Abstract

A shallow‐water control‐volume model is used to demonstrate features affecting the accuracy of numerical ocean‐atmosphere calculations. This model provides a context for illustrating the importance of consistent use of fields defined at points and fields averaged over cell volumes, cell faces and other surfaces. The accurate transformation between various averaged fields is demonstrated. Application of semi‐Lagrangian and COSMIC advection within N‐cycle time stepping is explored and compared with a control volume scheme that is time step limited. A fourth‐order accurate implicit solution for the potential update is demonstrated using two applications of a second‐order full multigrid solver with a defect correction. The defect correction does not make a significant correction to the potential update for our control‐volume formulation. It follows, therefore, that a single application of the second‐order solver effectively gives fourth‐order accuracy. On the other hand, potential gradients do need to be calculated using fourth‐order differencing or accuracy will suffer. Furthermore, calculating potential gradients at second order within an otherwise fourth‐order accurate spatial differencing scheme can result in unphysical eddies developing.