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We describe a new approach to modelling three-dimensional rotating stratified flows under the Boussinesq approximation. This approach is based on the explicit conservation of potential vorticity, and exploits the underlying leading-order geostrophic and hydrostratic balances inherent in these equations in the limit of small Froude and Rossby numbers. These balances are not imposed, but instead are used to motivate the use of a pair of new variables expressing the departure from geostrophic and hydrostratic balance. These new variables are the ageostrophic horizontal vorticity components, i.e. the vorticity not directly associated with the displacement of isopycnal surfaces. The use of potential vorticity and ageostrophic horizontal vorticity, rather than the usual primitive variables of velocity and density, reveals a deep mathematical structure and appears to have advantages numerically. This change of variables results in a diagnostic equation, of Monge-Ampère type, for one component of a vector potential ϕ, and two Poisson equations for the other two components. The curl of ϕ gives the velocity field while the divergence of ϕ is proportional to the displacement of isopycnal surfaces. This diagnostic equation makes transparent the conditions for both static and inertial stability, and may change form from (spatially) elliptic to (spatially) hyperbolic even when the flow is statically and inertially stable. A numerical method based on these new variables is developed and used to examine the instability of a horizontal elliptical shear zone (modelling a jet streak). The basic-state flow is in exact geostrophic and hydrostratic balance. Given a small perturbation however, the shear zone destabilizes by rolling up into a street of vortices and radiating inertia-gravity waves.

A method to decompose geophysical flows into a balanced flow (defined by its potential vorticity, PV) and an imbalanced component (inertia-gravity waves, IGWs) is introduced. The balanced flow, called the optimal potential vorticity (OPV) balance, is a solution of an IGW-permitting dynamics in which the amount of IGWs is minimal. The residual IGWs are those spontaneously generated by the vortical flow during the numerical integration in which the PV anomaly grows slowly over a sufficiently long ramp period toward a prescribed PV field. The OPV balanced flow is obtained, iteratively, in a cycle of backward and forward integrations where IGWs are removed and PV is restored in every loop. The method is applied to the flow of unsteady vortices in the three-dimensional baroclinic non-hydrostatic dynamics on the f-plane and to the single-layer shallow-water dynamics on the sphere. Both applications show that the iterative method converges strongly, after only a few iterations, to the balanced flow.

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