On the validity of layered models of ocean dynamics and thermodynamics with reduced vertical resolution

With the purpose of studying the upper part of the ocean, the shallow water equations (in a 'reduced gravity' setting) have been extended in the last decades by allowing for horizontal and temporal variations of the buoyancy field theta, while keeping it as well as the velocity field u as depth-inde pendent. In spite of the widespread use of this 'slab' model, there has bee n neither a discussion on the range of validity of the system nor an explan ation of points such as the existence of peculiar zero-frequency normal mod es, the nature of the instability of a uniform u flow, and the lack of expl icit vertical shear associated with horizontal density gradients. These que stions are addressed here through the development of a subinertial model wi th more vertical resolution, i.e., one where the buoyancy theta varies line arly with depth. This model describes satisfactorily the problem of barocli nic instability with a free boundary, even for short perturbations and larg e interface slopes, An enhancement of the instability is found when the pla netary beta effect is compensated with the topographic one, due to the slop e of the free boundary, allowing for a 'resonance' of the equivalent barotr opic and first baroclinic modes. Other low-frequency models, for which buoy ancy stratification does not play a dynamical role, are invalid for short p erturbations and have spurious terms in their energy-like integral of motio n. (C) 1999 Elsevier Science B.V. All rights reserved.