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The evolution of an unstable baroclinic jet, subject to a small perturbation, is examined numerically in a quasi-geostrophic two-layer β-channel model. After a period of initial wave growth, wave breaking leads to turbulence within each layer, and to the eventual equilibration of the flow. The equilibrated flow must satisfy certain dynamical constraints: total momentum is conserved, the total energy is bounded and the flow must be realizable via some area-preserving (diffusive) rearrangement of the initial potential vorticity field in each layer. A theory is introduced that predicts the equilibrated flow in terms of the initial flow parameters. The idea is that the equilibrated state minimizes available potential energy, subject to the constraints on total momentum and total energy, and the further ‘kinematic’ constraint that the potential vorticity changes through a process of complete homogenization within well-delineated regions in each layer. Within a large region of parameter space, the theory accurately predicts the cross-channel structure and strength of the equilibrated jet, the regions where potential vorticity mixing takes place, and total eddy mass (temperature) fluxes. Results are compared with predictions from a maximum-entropy theory that allows for more general rearrangements of the initial potential vorticity field, subject to the known dynamical constraints. The maximum-entropy theory predicts that significantly more available potential energy is released than is observed in the simulations, and that an unphysical ‘exchange’ of bands of fluid will occur across the channel in the lower layer. The kinematic constraint of piecewise potential vorticity homogenization is therefore important in limiting the ‘efficiency’ of release of available potential energy in unstable baroclinic flows. For a typical initial flow, it is demonstrated that if the dynamical constraints alone are considered, then over twice as much potential energy is available for release compared to that actually released in the simulations.

The turbulent equilibration of an unstable baroclinic jet