Evolution of solitary marginal disturbances in baroclinic frontal geostrophic dynamics with dissipation and time-varying background flow

Abstract

A two-layer frontal geostrophic flow corresponds to a dynamical regime that describes the low-frequency evolution of baroclinic ocean currents with large amplitude deflections of the interface between the layers on length-scales longer than the internal deformation radius within the context of a thin upper layer overlying a dynamically active lower layer. The finite-amplitude evolution of solitary disturbances in baroclinic frontal geostrophic dynamics in the presence of time-varying background flow and dissipation is shown to be governed by a two-equation extension of the unstable nonlinear Schrödinger (UNS) equation with variable coefficients and forcing. The soliton solution of the unperturbed UNS equation corresponds to a saturated isolated coherent anomaly in the baroclinic instability of surface-intensified oceanographic fronts and currents. The adiabatic evolution of the propagating soliton and the uniformly valid first-order perturbation fields are determined using a direct perturbation approach together with phase-averaged conservation relations when both dissipation and time variability are present. It is shown that the soliton amplitude parameter decays exponentially due to the presence of the dissipation but is unaffected by the time variability in the background flow. On the other hand, the soliton translation velocity is unaffected by the dissipation and evolves only in response to the time variability in the background flow. The adiabatic solution for the induced mean flow exhibits a dissipation-generated ‘shelf region’ in the far field behind the soliton, which is removed by solving the initial-value problem.