The classical quasi-geostrophic model in an active layer with an arbitrary vertical
structure is modified by adding a boundary condition at the interface with a passive
(motionless) lower layer: the difference between isopycnal and interface elevations is a
Lagrangian constant, so that a particle in this boundary remains there and conserves
its density. The new model has the appropriate integrals of motion: in particular,
a free energy quadratic and positive definite in the deviation from a state with a
uniform flow, made up of the internal and ‘external’ potential energies (due to the
displacement 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 motion give instability
conditions that are both necessary and sufficient. If the geostrophic slope of the
interface is such that density increases in opposite 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
stratification, though, the deformation of the lower boundary by the perturbation
field also plays an important role, as shown in the dispersion relation, the structure of
growing perturbations, and the energetics of the instability. The energy of long growing
perturbations is mostly internal potential, whereas short ones have an important
fraction of kinetic energy and, for strong enough stratification, external potential.
Wave excitation in three-dimensional discs by external potential