Baroclinic instability in a reduced gravity, three-dimensional, quasi-geostrophic model

The classical quasi-geostrophic model in an active layer with an arbitrary vertical structure is modified by adding a boundary condition at the interf ace with a passive (motionless) lower layer: the difference between isopycn al and interface elevations is a Lagrangian constant, so that a particle in this boundary remains there and conserves its density. The new model has t he appropriate integrals of motion. in particular, a free energy quadratic and positive definite in the deviation from a state with a uniform flow, ma de up of the internal and 'external' potential energies (due to the displac ement of the isopycnals and the interface) and the kinetic energy. Eady's model of baroclinic instability is extended with the present system, i.e. including the effect of the free lower boundary. The integrals of mot ion give instability conditions that are both necessary and sufficient. If the geostrophic slope of the interface is such that density increases in op posite directions at the top and bottom boundaries, then the basic flow is nonlinearly stable. For very weak internal stratification (as compared with the density jump at the interface) normal modes instability is similar to that of a simpler model, with a rigid but sloping bottom. For stronger stra tification, though, the deformation of the lower boundary by the perturbati on field also plays an important role, as shown in the dispersion relation, the structure of growing perturbations, and the energetics of the instabil ity. The energy of lo;ng growing perturbations is mostly internal potential , whereas short ones have an important fraction of kinetic energy and, for strong enough stratification, external potential.