Finite-amplitude baroclinic instability of a mesoscale gravity current in a channel

A finite amplitude theory is developed for the evolution of marginally unstable modes for a mesoscale gravity current on a sloping bottom. The theory is based on a nonquasigeostrophic, baroclinic model of the convective destabilization of gravity currents which allows for large amplitude isopycnal deflections while filtering out barotropic instabilities. Two calculations are presented. First, a purely temporal amplitude equation is derived for marginally unstable modes not located at the minimum of the marginal stability curve. These modes eventually equilibrate with a new finite amplitude periodic solution formed. Second, the evolution of a packet of marginally unstable modes located at the minimum of the marginal stability curve is presented. These two models are dramatically different due to Fundamental physical differences. For marginally unstable modes not located at the minimum of the marginal stability curve, it is possible to determine the evolution of a single normal mode amplitude. For the marginally unstable mode located at the minimum of the marginal stability curve the entire gravity current forms a nonlinear critical layer leading to an infinity of coupled amplitude equations. IF this system is truncated, on an ad hoc basis, to include only the fundamental harmonic and its accompanying mean flow, there exists a steadily-travelling solitary cold-core eddy solution.