Nonlinear effects in two-layer large-amplitude geostrophic dynamics. Part 1. The strong-beta case

Baroclinic large-amplitude geostrophic (LAG) models, which assume a leading -order geostrophic balance but allow for large-amplitude isopycnal deflecti ons, provide a suitable framework to model the large-amplitude motions exhi bited in frontal regions. The qualitative dynamical characterization of LAG models depends critically on the underlying length scale. If the length sc ale is sufficiently large, the effect of differential rotation, i.e. the be ta-effect, enters the dynamics at leading order. For smaller length scales, the beta-effect, while non-negligible, does not enter the dynamics at lead ing order. These two dynamical limits are referred to as strong-beta and we ak-beta models, respectively. A comprehensive description of the nonlinear dynamics associated with the s trong-beta models is given. In addition to establishing two new nonlinear s tability theorems, we extend previous linear stability analyses to account for the finite-amplitude development of perturbed fronts. We determine whet her the linear solutions are subject to nonlinear secondary instabilities a nd, in particular, a new long-wave-short-wave (LWSW) resonance, which is a possible source of rapid unstable growth at long length scales, is identifi ed. The theoretical analyses are tested against numerical simulations. The simulations confirm the importance of the LWSW resonance in the development of the how. Simulations show that instabilities associated with vanishing potential-vorticity gradients can develop into stable meanders, eddies or b reaking waves. By examining models with different layer depths, we reveal h ow the dynamics associated with strong-beta models qualitatively changes as the strength of the dynamic coupling between the barotropic and baroclinic motions varies.