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 ?, while keeping it as well as the velocity field u as depth-independent. In spite of the widespread use of this `slab' model, there has been neither a discussion on the range of validity of the system nor an explanation of points such as the existence of peculiar zero-frequency normal modes, the nature of the instability of a uniform u flow, and the lack of explicit vertical shear associated with horizontal density gradients. These questions are addressed here through the development of a subinertial model with more vertical resolution, i.e., one where the buoyancy ? varies linearly with depth. This model describes satisfactorily the problem of baroclinic instability with a free boundary, even for short perturbations and large interface slopes. An enhancement of the instability is found when the planetary β effect is compensated with the topographic one, due to the slope of the free boundary, allowing for a `resonance' of the equivalent barotropic and first baroclinic modes. Other low-frequency models, for which buoyancy stratification does not play a dynamical role, are invalid for short perturbations and have spurious terms in their energy-like integral of motion.