Abstract
The problem of nonlinear saturation of baroclinic waves in two-layer models is studied and it is shown that Shepherd's rigorous bound on the wavy disturbance growth due to instabilities of parallel shear flow can be improved significantly, in some cases, by exact calculation of the averaged Arnol'd's invariant. Shepherd's bound for the Phillips' β-plane two-layer model with constant potential vorticity gradient is achievable at the minimum critical shear as the supercriticality parameter ε → 0. The underlying reason for such an achievable bound for the wavy disturbance is that the condition leading to the Arnol'd's stability theorem is both necessary and sufficient. Based on such an achievable bound, (2β/3F)1/2 is deduced as the maximum wave amplitude at the minimum critical shear as the supercriticality parameter ε → 0. When Arnol'd's invariant is applied to an f- plane two-layer model, the bound derived from Arnol'd's invariant is not as powerful a constraint on the amplitude of the evolving wavy disturbance. The reason is that the opposite signs of potential vorticity gradients in upper and lower layers are not a sufficient condition for instability.
