The Use of Topographic Wave Modes to Solve for the Barotropic Mode of a Rigid-Lid Ocean Model

Abstract

Topographic wave modes are defined for the barotropic mode of a rigid-lid ocean, and the question is asked whether these might form an efficient basis for a description of the barotropic mode of a general ocean flow. The modes are shown to be incomplete, most particularly in their representation of barotropic potential vorticity over certain areas, so extra functions must be added to “patch up” the wave modes description. With the aid of two simple, flat bottom beta-plane models, it is shown that the, form of the extra function required depends on the position of the boundaries relative to contours of planetary barotropic potential vorticity, f/H. Where all the boundaries are along contours of f/H, the extra function is simply a function of f/H and time. Where a finite stretch of boundary runs parallel to f/H contours, a further additional function can (at least sometimes) produce a complete set, but when the boundary runs parallel to f/H contours for no finite distance, there is no simple way to augment the wave modes to produce a complete set. It is shown that this incompleteness is only in the representation of barotropic potential vorticity at the boundary and causes no finite error in the streamfunction, but it seems likely that the presence of this incompleteness spoils the efficiency of the sum of wave modes as a description of a general flow. It appears that topographic wave modes are the natural modes only for systems in which the boundary (at least partially) follows contours of planetary barotropic potential vorticity, f/H. The above deficiencies vanish when enough modes are considered to resolve frictional boundary layers if the no-slip boundary condition is applied. When boundary layers are too thin to resolve, however, use of the modes to represent the difference between rotating and nonrotating responses suggests the possibility of a novel way of modeling the approach to the inviscid limit. In both these cases, however, current technology limits the practicality of the method to cases where the spatial structure of the wave modes can he calculated analytically.