ALBERT

Description: The propagation of weakly nonlinear, long internal wave fronts in a contraction is considered in the transcritical limit as a model for the establishment of virtual controls. It is argued that the appropriate equation to describe this process is a variable coefficient Korteweg-de Vries equation. The solutions of this equation are then considered for compressive and rarefaction fronts. Rarefaction fronts exhibit both normal and virtual control solutions. However, the interaction of compressive fronts with contractions is intrinsically unsteady. Here the dynamics take two forms, interactions with the bulk of the front and interactions with individual solitary waves separating off from a front trapped downstream of the contraction.

Description: The evolution of an intense barotropic vortex on the β-plane is analysed for the case of finite Rossby deformation radius. The analysis takes into account conservation of vortex energy and enstrophy, as well as some other quantities, and therefore makes it possible to gain insight into the vortex evolution for longer times than was done in previous studies on this subject. Three characteristic scales play an important role in the evolution: the advective time scale Ta (a typical time required for a fluid particle to move a distance of the order of the vortex size), the wave time scale Tw (the typical time it takes for the vortex to move through its own radius), and the distortion time scale Td (a typical time required for the change in relative vorticity of the vortex to become of the order of the relative vorticity itself). For an intense vortex these scales are well separated, Td ≪ Tw ≪ Td, and therefore one can consider the vortex evolution as consisting of three different stages. The first one, t ≤ Tw, is dominated by the development of a near-field dipolar circulation (primary β-gyres) accelerating the vortex. During the second stage, Tw ≤ t ≤ Td, the quadrupole and secondary axisymmetric components are intensified; the vortex decelerates. During the last, third, stage the vortex decays and is destroyed. Our main attention is focused on exploration of the second stage, which has been studied much less than the first stage. To describe the second stage we develop an asymptotic theory for an intense vortex with initially piecewise-constant relative vorticity. The theory allows the calculation of the quadrupole and axisymmetric corrections, and the correction to the vortex translation speed. Using the conservation laws we estimate that the vortex lifetime is directly proportional to the vortex streamfunction amplitude and inversely proportional to the squared group velocity of Rossby waves. For open-ocean eddies a typical lifetime is about 130 days, and for oceanic rings up to 650 days. Analysis of the residual produced by the asymptotic solution explains why this solution is a good approximation for times much longer than the expected formal range of applicability. All our analytical results are in a good qualitative agreement with several numerical experiments carried out for various vortices.

Description: The process of nonlinear geostrophic adjustment in the presence of a boundary (i.e. in a half-plane bounded by a rigid wall) is examined in the framework of a rotating shallow water model, using an asymptotic multiple-time-scale theory based on the assumed smallness of the Rossby number ε. The spatial scale is of the order of the Rossby scale. Different initial states are considered: periodic, 'step'-like, and localized. In all cases the initial perturbation is split in a unique way into slow and fast components evolving with characteristic time scales f-1 and (εf)-1, respectively. The slow component is not influenced by the fast one, at least for times t ≤ (fε)-1, and remains close to geostrophic balance. The fast component consists mainly of linear inertia-gravity waves rapidly propagating outward from the initial disturbance and Kelvin waves confined near the boundary. The theory provides simple formulae allowing us to construct the initial profile of the Kelvin wave, given arbitrary initial conditions. With increasing time, the Kelvin wave profile gradually distorts due to nonlinear-wave self-interaction, the distortion being described by the equation of a simple wave. The presence of Kelvin waves does not prevent the fast-slow splitting, in spite of the fact that the frequency gap between the Kelvin waves and slow motion is absent. The possibility of such splitting is explained by the special structure of the Kelvin waves in each case considered. The slow motion on time scales t ≤ (εf)-1 is governed by the well-known quasigeostrophic potential vorticity equation for the elevation. The theory provides an algorithm to determine initial slow and fast fields, and the boundary conditions to any order in ε. For the periodic and step-like initial conditions, the slow component behaves in the usual way, conserving mass, energy and enstrophy. In the case of a localized initial disturbance the total mass of the lowest-order slow component is not conserved, and conservation of the total mass is provided by the first-order slow correction and the Kelvin wave. On longer time scales t ≤ (ε2f)-1 the slow motion obeys the so-called modified quasi-geostrophic potential vorticity (QGPV) equation. The theory provides initial and boundary conditions for this equation. This modified equation coincides exactly with the 'improved' QGPV equation, derived by Reznik, Zeitlin & Ben Jelloul (2001), in the step-like and localized cases. In the periodic case this equation contains an additional term due to the Kelvin-wave self-interaction, this term depending on the initial Kelvin wave profile.

Description: We consider the non-stationary dynamics of an intense localized vortex on a β-plane using a shallow-water model. An asymptotic theory for a vortex with piecewise- continuous potential vorticity is developed assuming the Rossby number to be small and the free surface elevation to be small but finite. Analogously to the well-known quasi-geostrophic model, the vortex translation is produced by a secondary dipole circulation (β-gyres) developed in the vortex vicinity and consisting of two parts. The first part (geostrophic β-gyres) coincides with the β-gyres in the geostrophic model, and the second (ageostrophic β-gyres) is due to ageostrophic terms in the governing equations. The time evolution of the ageostrophic β-gyres consists of fast and slow stages. During the fast stage the radiation of inertia-gravity waves results in the rapid development of the β-gyres from zero to a dipole field independent of the fast time variable. Correspondingly, the vortex accelerates practically instantaneously (compared to the typical swirling time) to some finite value of the translation speed. At the next slow stage the inertia-gravity wave radiation is insignificant and the β-gyres evolve with the typical swirling time. The total zonal translation speed induced by the geostrophic and ageostrophic β-gyres tends with increasing time to the speed of a steadily translating monopole exceeding (not exceeding) the drift velocity of Rossby waves for anticyclones (cyclones). This cyclone/anticyclone asymmetry generalizes the well-known finding about the greater longevity of anticyclones compared to cyclones to the case of non-stationary evolving monopoles. The influence of inertia-gravity waves upon the vortex evolution is analysed. The main role of these waves is to provide a 'fast' adjustment to the 'slow' vortex evolution. The energy of inertia-gravity waves is negligible compare to the energy of the geostrophic β-gyres. Yet another feature of the ageostrophic vortex evolution is that the area of the potential vorticity patch changes in the course of time, the cyclonic patch contracting and the anticyclonic one expanding.