Stability of stationary barotropic modons by Lyapunov's direct method (original) (raw)
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The theory of the ?-plane baroclinic topographic modons
2003
Form-preserving, uniformly translating, horizontally localized solutions (modons) are considered within the framework of nondissipative quasi-geostrophic dynamics for a two-layer model with meridionally sloping bottom. A general classification of the beta-plane baroclinic topographic modons (-BTMs) is given, and three distinct domains are shown to exist in the plane of the parameters. The first domain corresponds to the regular modons with the translation speed outside the range of the phase speeds of linear waves. In the second domain, modons cannot exist: only non-localized solutions are permissible here. The third domain contains both linear periodic waves and the so-called anomalous modons traveling without resonant radiation.
Coherent magnetic modon solutions in quasi-geostrophic shallow water magnetohydrodynamics
Journal of Fluid Mechanics
A class of exact solutions of the magnetohydrodynamic quasi-geostrophic equations (MQG), which result from rotating shallow water magnetohydrodynamics in the limit of small Rossby and magnetic Rossby numbers is constructed analytically. These solutions are magnetic modons, steady-moving dipolar vortices, which are generalizations of the well-known quasi-geostrophic modons. It is shown that various configurations of magnetic modons are possible: with or without external magnetic field, and with or without internal magnetic field trapped inside the dipole. By using the modon solutions as initial conditions for direct numerical simulations of the MQG equations, it is shown that they remain coherent for a long time, running over about a hundred deformation radii without change of form, provided the external and internal magnetic fields are not too strong, and even if a small-amplitude noise is added to initial conditions.
Il Nuovo Cimento C, 1994
It is well known that Serrin's universal stability criterion, by means of the energy method, gives an upper bound for the Reynolds number in order for a generic perturbation to exponentially decay in time. The original framework of the criterion is referred to the Navier-Stokes equations for a three-dimensional fluid in a periodic or bounded domain. In this paper the problem of the asymptotic non-linear stability of a quasi-geostrophic flow in a zonal channel is investigated. Two kinds of dissipation mechanisms are considered: the lateral diffusion of vorticity and the Ekman bottom dissipation. By using a procedure similar to Serrin's, a non-linear asymptotic stability criterion is obtained. The stability is governed by the maximum of the shear of the basic zonal current and the intensity of the dissipation parameters.
Propagation of barotropic modons over topography
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This is a broad survey of the interaction of modons with topography in a one-layer, quasigeostrophic model. Numerical simulations of modons interacting with ridges, hills, random topography and other obstacles are presented. The behavior of the modon is compared to numerical simulations of a two-point-vortex model, which proves a useful guide to the basic trajectory deflection mechanism. Under sufficiently strong but quasigeostrophically valid topographic perturbations, the modon is shown to fission into two essentially independent, oppositely-signed vortices. In the breakup of a modon near a hill it is found that the positive vortex migrates to the top of the hill. The resulting correlation between the positive vorticity trapped over the hill and the topography is in sharp contrast with the theories of turbulent flow over topography and generation of flow over topography by large scale forcing, both of which describe the development of vorticity anticorrelated with topography. A heuristic explanation of this new behavior is provided in terms of the dynamics of /3-45 Downloaded By: [Carnevale, George F.] At: 16:54 19 January 2008 46 G. F. CARNEVALE et al.
Eastward-moving convection-enhanced modons in shallow water in the equatorial tangent plane
Physics of Fluids
We report a discovery of steady long-living slowly eastward moving large-scale coherent twin cyclones, the equatorial modons, in the shallow water model in the equatorial beta-plane, the archetype model of the ocean and atmosphere dynamics in tropics. We start by constructing analytical asymptotic modon solutions in the non-divergent velocity approximation, and then show by simulations with a high-resolution numerical scheme that such configurations evolve into steady dipolar solutions of the full model. In the atmospheric context, the modons persist in the presence of moist convection, being accompanied and enhanced by specific patterns of water-vapour condensation.
Modutional instability in the presence of forcing and dissipation
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On the Baroclinic Instability of Nonplanar Flows
Nuovo Cimento Della Societa Italiana Di Fisica C-Physics and Astronomy, 1985
Different ocean models with one or two layers having constant static stability and supporting constant-shear flows, whose directions are allowed to change with depth, are examined in the framework of the linear nonzonal baroclinic stability theory and in the absence of the fl-effect. The analysis is reduced to solving a simple Sturm-Liouville boundary value problem in one dimension. A fairly general dispersion relation is found which correctly reproduces several special cases analysed by other authors. This relation shows a fair variety of possible behaviours for stability curves of two-layer models. The results show that the presence of a nonplanar shear-flow may have profound consequences on the stability character of the stationary geostrophic flow. In fact, it appears that the stability properties are strongly dependent on the propagation angle of the disturbance so that wave numbers which appear stable in the usual zonal theory may result unstable on such a nonzonal flow, and vice versa.
A numerical study of non‐geostrophic baroclinic instability
Atmosphere-Ocean, 1987
The stability properties of a zonal current with a constant vertical shear are investigated with a view to extending earlier calculations obtained for only weakly nongeostrophic conditions. The numerical approach that is adopted allows consideration of a range of parameters. This makes it possible to obtain a stability diagram in which the growth rate of the perturbations is presented as a function of their zonal wavelengths and of the Rossby number Ro, the ratio of the vertical shear to the product of the width of the domain, and the Coriolis parameter. An equation is derived for the neutral curve at small Rossby numbers. The numerical results show that when the flow is only weakly non-geostrophic both the neutral curve and the wavelength of maximum instability shift to longer scales as Ro is increased. A short discussion of the (nearly) symmetric instabilities appearing in the model when the Richardson number is sufficiently small illustrates the influence of the side boundaries on these modes. RÉSUMÉ La stabilité d'un courant zonal ayant un cisaillement vertical constant est étudiée afin de parfaire certains calculs précédents valables seulement pour des écoulements faiblement agéostrophiques. L'approche numérique adoptée permet de considérer un évantail de paramètres. Cela rend possible le calcul d'un diagramme de stabilité, où le taux de croissance des perturbations est présenté en fonction de leur longueur d'onde et du nombre de Rossby, défini comme étant le rapport entre le cisaillement vertical et le produit de la largeur du domaine et du paramètre de Coriolis. L'équation de la courbe de stabilité marginale, valable pour de petites valeurs de Ro, est développée. Les résultats numériques indiquent que lorsque l'écoulement n 'est que faiblement agéostrophique la courbe de stabilité marginale et la longueur de l'onde de croissance maximum se déplacent vers les ondes longues lorsque Ro augmente. Une courte présentation des instabilités (presque) symétriques du modèle illustre l'influence des limites latérales du modèle sur ces modes d'instabilité.
STABILITY IN THE WEAK VARIATIONAL PRINCIPLE OF BAROTROPIC FLOWS
Arxiv preprint astro-ph/9501080, 1995
I find conditions under which the "Weak Energy Principle" of Katz, gives necessary and sufficient conditions. My conclusion is that, necessary and sufficient conditions of stability are obtained when we have only two mode coupling in the gyroscopic terms of the perturbed Lagrangian. To illustrate the power of this new energy principle, I have calculated the stability limits of two dimensional configurations such as ordinary Maclaurin disk, an infinite self gravitating rotating sheet, and a two dimensional Rayleigh flow which has well known sufficient conditions of stability. All perturbations considered are in the same plane as the configurations. The limits of stability are identical with those given by a dynamical analysis when available, and with the results of the strong energy principle analysis when given. Thus although the "Weak Energy" method is mathematically more simple than the "Strong Energy" method of Katz, Inagaki and Yahalom )1993) since it does not involve solving second order partial differential equations, it is by no means less effective.
Shear instabilities in shallow-water magnetohydrodynamics
2013
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