Geostrophic Adjustment Beyond the Traditional Approximation

We examine geostrophic adjustment in a model rotating ocean when the angular speed of rotation \({\varvec{\Omega}}\) does not coincide in direction with the gravity; the traditional and hydrostatic approximations are not used. Two models are considered: the barotropic one and the stably-neutrally stratified (SNS) fluid consisting of a stratified upper layer and a homogeneous lower layer. Wave spectra in the models contain gyroscopic and (in the SNS fluid) internal waves. Linear adjustment results in a tendency of any localized initial state to a geostrophically balanced steady motion parallel to the layer boundaries. In the barotropic fluid and homogeneous layer in the SNS fluid the motion is columnar, the columns are parallel to \({\varvec{\Omega}}\); in the stratified layer the geostrophic mode is not columnar. Using the perturbation theory we study the non-linear adjustment for the small Rossby number and aspect ratio. During the adjustment an arbitrary perturbation is uniquely split into slow quasi-geostrophic (QG) and fast ageostrophic components. In the barotropic model the slow component is described by the 2D fluid dynamics equation for the geostrophic streamfunction. In the SNS fluid the component is governed by two coupled nonlinear equations of QG potential vorticity in the upper and lower layers. The ageostrophic part consists of the inertial oscillations modulated by amplitude depending on coordinates and the slow time, and (in the SNS fluid) internal waves. The internal waves decay due to dispersion and the residual flow is a sum of the QG slow component and inertial oscillations coupled to the QG flow.