Nonlinear theory of geostrophic adjustment. Part 2. Two-layer and continuously stratified primitive equations

Abstract

This paper continues the work started in Part 1 (Reznik, Zeitlin & Ben Jelloul 2001) and generalizes it to the case of a stratified environment. Geostrophic adjustment of localized disturbances is considered in the context of the two-layer shallow-water and continuously stratified primitive equations in the vertically bounded and horizontally infinite domain on the $f$-plane. Using multiple-time-scale perturbation expansions in Rossby number $\hbox{\it Ro}$ we show that stratification does not substantially change the adjustment scenario established in Part 1 and any disturbance of well-defined scale is split in a unique way into slow and fast components with characteristic time scales $f_0^{-1}$ and $(f_0 \hbox{\it Ro})^{-1}$ respectively, where $f_0$ is the Coriolis parameter. As in Part 1 we distinguish two basic dynamical regimes: quasi-geostrophic (QG) and frontal geostrophic (FG) with small and large deviations of the isopycnal surfaces, respectively. We show that the dynamics of the FG regime in the two-layer model depends strongly on the ratio of the layer depths. The difference between QG and FG scenarios of adjustment is demonstrated. In the QG case the fast component of the flow essentially does not ‘feel’ the slow one and is rapidly dispersed leaving the slow component to evolve according to the standard QG equation (corrections to this equation are found for times $t\,{\gg}\, (f_0 \hbox{\it Ro})^{-1}$). In the FG case the fast component is a packet of inertial oscillations produced by the initial perturbation. The space-time evolution of the envelope of inertial oscillations obeys a Schrödinger-type modulation equation with coefficients depending on the slow component. In both QG and FG cases we show by direct computations that the fast component does not produce any drag terms in the equations for the slow component; the slow component remains close to the geostrophic balance. However, in the continuously stratified FG regime, as well as in the two-layer regime with the layers of comparable thickness, the splitting is incomplete in the sense that the slow vortical component and the inertial oscillations envelope evolve on the same time scale.