Geostrophic adjustment in a closed basin with islands (original) (raw)
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Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow-water model
Journal of Fluid Mechanics, 2001
We develop a theory of nonlinear geostrophic adjustment of arbitrary localized (i.e. finite-energy) disturbances in the framework of the non-dissipative rotating shallowwater dynamics. The only assumptions made are the well-defined scale of disturbance and the smallness of the Rossby number Ro. By systematically using the multi-timescale perturbation expansions in Rossby number it is shown that the resulting field is split in a unique way into slow and fast components evolving with characteristic time scales f −1 0 and (f 0 Ro) −1 respectively, where f 0 is the Coriolis parameter. The slow component is not influenced by the fast one and remains close to the geostrophic balance. The algorithm of its initialization readily follows by construction. The scenario of adjustment depends on the characteristic scale and/or initial relative elevation of the free surface ∆H/H 0 , where ∆H and H 0 are typical values of the initial elevation and the mean depth, respectively. For small relative elevations (∆H/H 0 = O(Ro)) the evolution of the slow motion is governed by the well-known quasigeostrophic potential vorticity equation for times t 6 (f 0 Ro) −1. We find modifications to this equation for longer times t 6 (f 0 Ro 2) −1. The fast component consists mainly of linear inertia-gravity waves rapidly propagating outward from the initial disturbance. For large relative elevations (∆H/H 0 Ro) the slow field is governed by the frontal geostrophic dynamics equation. The fast component in this case is a spatially localized packet of inertial oscillations coupled to the slow component of the flow. Its envelope experiences slow modulation and obeys a Schrödinger-type modulation equation describing advection and dispersion of the packet. A case of intermediate elevation is also considered.
Lagrangian approach to geostrophic adjustment of frontal anomalies in a stratified fluid
Geophysical and Astrophysical Fluid Dynamics, 2005
Geostrophic adjustment of frontal anomalies in a rotating continuously stratified fluid is studied in the standard framework of strictly rectilinear fronts and jets. Lagrangian approach to this problem is developed allowing to analyze, in a conceptually and technically simple way, both major problems of the nonlinear adjustment: the existence of a smooth adjusted state for a given set of initial conditions and the attainability of the adjusted state during the adjustment process. Dynamical splitting into balanced (adjusted state) and unbalanced (inertia-gravity waves) motions becomes transparent in the Lagrangian approach. Conditions of existence of the balanced state in the unbounded domain are established. It is shown that nonexistence of a smooth adjusted state in the vertically bounded domains is generic and a parallel with the classical scenario of deformation frontogenesis is developed. Small perturbations around smooth adjusted states are then studied with special emphasis on the wave-trapping inside the jet/front. Trapped modes with horizontal scales comparable to the width of the jet are explicitly constructed for a barotropic jet and their evolution is studied with the help of the WKB-approximation for weakly baroclinic jets. Modifications of the standard scenario of adjustment due to subinertial (quasi-) trapped modes and implications for data analysis are discussed.
Journal of Fluid Mechanics, 2003
The problem of nonlinear adjustment of localized front-like perturbations to a state of geostrophic equilibrium (balanced state) is studied in the framework of rotating shallow-water equations with no dependence on the along-front coordinate. We work in Lagrangian coordinates, which turns out to be conceptually and technically advantageous. First, a perturbation approach in the cross-front Rossby number is developed and splitting of the motion into slow and fast components is demonstrated for non-negative potential vorticities. We then give a non-perturbative proof of existence and uniqueness of the adjusted state, again for configurations with nonnegative initial potential vorticities. We prove that wave trapping is impossible within this adjusted state and, hence, adjustment is always complete for small enough departures from balance. However, we show that retarded adjustment occurs if the adjusted state admits quasi-stationary states decaying via tunnelling across a potential barrier. A description of finite-amplitude periodic nonlinear waves known to exist in configurations with constant potential vorticity in this model is given in terms of Lagrangian variables. Finally, shock formation is analysed and semi-quantitative criteria based on the values of initial gradients and the relative vorticity of initial states are established for wave breaking showing, again, essential differences between the regions of positive and negative vorticity. † As usual in geophysical applications, the centrifugal force will be neglected and molecular dissipation will be absent in what follows.
Geostrophic Adjustment Problems in a Polar Basin
Atmosphere-Ocean, 2012
The geostrophic adjustment of a homogeneous fluid in a circular basin with idealized 22 topography is addressed using a numerical ocean circulation model and analytical process models. When the basin is rotating uniformly, the adjustment takes place via excitation of boundary propagating waves and when topography is present, via topographic Rossby waves. In the numerically derived solution, the waves are damped because of bottom friction, and a quasi-steady geostrophically balanced state emerges that subsequently spins-down on a long time scale. On the f-plane, numerical quasi-steady state solutions are attained well before the system's mechanical energy is entirely dissipated by friction. It is demonstrated that the 29 adjusted states emerging in a circular basin with a step escarpment or a top hat ridge, centred 30 on a line of symmetry, are equivalent to that in a uniform depth semicircular basin, for a given initial condition. These quasi-steady solutions agree well with linear analytical solutions for the latter case in the inviscid limit. On the polar plane, the high latitude equivalent to the β-plane, no quasi-steady adjusted state emerges from the adjustment process. At intermediate time scales, after the fast Poincaré andKelvin waves are damped by friction, the solutions take the form of steady-state adjusted solutions on the f-plane. At longer time scales, planetary waves control the flow evolution. An interesting property of planetary waves on a polar plane is a nearly zero eastward group velocity for the waves with a radial mode higher than two and the resulting formation of eddy-like small-scale barotropic structures that remain trapped near the western side of topographic features. Keywords geostrophic adjustment, polar circulation, Kelvin waves, vorticity waves.
Geostrophic adjustment on the mid-latitude β-plane
2023
Analytical and numerical solutions of the linearized rotating shallow water equations are combined to study the geostrophic adjustment on the midlatitude β plane. The adjustment is examined in zonal periodic channels of width L y = 4R d (narrow channel, where R d is the radius of deformation) and L y = 60R d (wide channel) for the particular initial conditions of a resting fluid with a step-like height distribution, η 0. In the one-dimensional case, where η 0 = η 0 (y), we find that (i) β affects the geostrophic state (determined from the conservation of the meridional vorticity gradient) only when b = cot(φ 0) R d R ≥ 0.5 (where φ 0 is the channel's central latitude, and R is Earth's radius); (ii) the energy conversion ratio varies by less than 10 % when b increases from 0 to 1; (iii) in wide channels, β affects the waves significantly, even for small b (e.g., b = 0.005); and (iv) for b = 0.005, harmonic waves approximate the waves in narrow channels, and trapped waves approximate the waves in wide channels. In the two-dimensional case, where η 0 = η 0 (x), we find that (i) at short times the spatial structure of the steady solution is similar to that on the f plane, while at long times the steady state drifts westward at the speed of Rossby waves (harmonic Rossby waves in narrow channels and trapped Rossby waves in wide channels); (ii) in wide channels, trapped-wave dispersion causes the equatorward segment of the wavefront to move faster than the northern segment; (iii) the energy of Rossby waves on the β plane approaches that of the steady state on the f plane; and (iv) the results outlined in (iii) and (iv) of the one-dimensional case also hold in the two-dimensional case.
Journal of Fluid Mechanics, 2004
High-resolution shock-capturing finite-volume numerical methods are applied to investigate nonlinear geostrophic adjustment of rectilinear fronts and jets in the rotating shallow-water model. Numerical experiments for various jet/front configurations show that for localized initial conditions in the open domain an adjusted state is always attained. This is the case even when the initial potential vorticity (PV) is not positive-definite, the situation where no proof of existence of the adjusted state is available. Adjustment of the vortex, PV-bearing, part of the flow is rapid and is achieved within a couple of inertial periods. However, the PV-less lowenergy quasi-inertial oscillations remain for a long time in the vicinity of the jet core. It is demonstrated that they represent a long-wave part of the initial perturbation and decay according to the standard dispersion law ∼t −1/2. For geostrophic adjustment in a periodic domain, an exact periodic nonlinear wave solution is found to emerge spontaneously during the evolution of wave perturbations allowing us to conjecture that this solution is an attractor. In both cases of adjustment in open and periodic domains, it is shown that shock-formation is ubiquitous. It takes place immediately in the jet core and, thus, plays an important role in fully nonlinear adjustment. Although shocks dissipate energy effectively, the PV distribution is not changed owing to the passage of shocks in the case of strictly rectilinear flows.
Unstably stratified geophysical fluid dynamics
Dynamics of Atmospheres and Oceans, 1997
The linear dynamics of the unstably stratified geophysical flows is investigated with a two-layer formulation. A 'convective' deformation radius classifies the dynamics into three regimes: 1. the scales smaller than the deformation radius: the dynamics characterized by unstable inertial-gravity modes; 2. the scales larger than the deformation radius: a quasi-geostrophic regime; 3. the scales close to the deformation radius, where the dynamics transits from the inertial-gravity regime to the quasi-geostrophic regime. The Rossby wave can propagate eastward in the unstably stratified quasi-geostrophic regime. The baroclinic instabilities are basically realized as a larger-scale extent of the inertial-gravity instabilities, but the former can be isolated from the latter in a limit of small /3-effect, with a very deep lower layer. The results suggest that the convectively unstable Jovian atmospheric dynamics can be well described as a quasi-geostrophic system.
A Geostrophic Adjustment Model of Two Buoyant Fluids
Journal of Physical Oceanography, 2012
A combination of analytical calculations and laboratory experiments has been used to investigate the geostrophic adjustment of two buoyant fluids having different densities in a third denser ambient fluid. The frontal position, the depth profile, and the horizontal and vertical alignments of the two buoyant fluids at the final equilibrium state are determined by the ratio of the baroclinic Rossby radii of deformation Γ1 = λ31/λ21 and Γ2 = λ32/λ21 and the Burger numbers B1 = λ31/L1 and B2 = λ32/L2 of the two buoyant fluids, where [Formula: see text] is the baroclinic Rossby radius of deformation between fluids i and j. The buoyant fluids 1 and 2 have densities ρ1 and ρ2 (ρ1 < ρ2), respectively; the ambient denser fluid has density ρ3; g′ is the reduced gravity; H and L are the buoyant fluids’ initial depth and width, respectively; and f is the Coriolis parameter. Laboratory rotating experiments confirmed the analytical prediction of the location of the two fronts. After reaching g...
Finite volume simulation of the geostrophic adjustment in a rotating shallow-water system
SIAM J. Sci. Comput, 2008
The goal of this article is to simulate rotating flows of shallow layers of fluid by means of finite volume numerical schemes. More precisely, we focus on the simulation of the geostrophic adjustment phenomenon. As spatial discretization, a first order Roe-type method and some higher order extensions are developed. The time discretization is designed in order to provide suitable approximations of inertial oscillations, taking into account the hamiltonian structure of the system for these solutions. The numerical dispersion laws and the wave amplifications of the schemes are studied and their well-balanced properties are analyzed. Finally, some numerical experiments for 1d and 2d problems are shown.