Geostrophic adjustment with gyroscopic waves: barotropic fluid without the traditional approximation

Abstract

AbstractWe study geostrophic adjustment in rotating barotropic fluid when the angular speed of rotation $\boldsymbol{\Omega}$ does not coincide in direction with the acceleration due to gravity; the traditional and hydrostatic approximations are not used. Linear adjustment results in a tendency of any localized initial state towards a geostrophically balanced steady columnar motion with columns parallel to $\boldsymbol{\Omega}$. Nonlinear adjustment is examined for small Rossby numbers Ro and aspect ratio $H/L$ ($H$ and $L$ are the layer depth and the horizontal scale of motion), using multiple-time-scale perturbation theory. It is shown that an arbitrary perturbation is split in a unique way into slow and fast components evolving with characteristic time scales $(\mathit{Ro}f)^{-1}$ and $f^{-1}$, respectively, where $f $ is the Coriolis parameter. The slow component does not depend on depth and is close to geostrophic balance. On times $O(1/f\, \mathit{Ro})$ the slow component is not influenced by the fast one and is described by the two-dimensional fluid dynamics equation for the geostrophic streamfunction. The fast component consists of long gyroscopic waves and is a packet of inertial oscillations modulated by an amplitude depending on coordinates and slow time. On times $O(1/f\, \mathit{Ro})$ the fast component conserves its energy, but it is coupled to the slow component: its amplitude obeys an equation with coefficients depending on the geostrophic streamfunction. Under the traditional approximation, the inertial oscillations are trapped by the quasi-geostrophic component; ‘non-traditional’ terms in the amplitude equation provide a meridional dispersion of the packet on times $O(1/f\, \mathit{Ro})$, and, therefore, an effective radiation of energy from the initial perturbation domain. Another important effect of the non-traditional terms is that on longer times $O(1/f\, \mathit{Ro}^{2})$ a transfer of energy between the fast and the slow components becomes possible.