Amplitude of Rossby Wavetrains on a Sphere

Journal of the Meteorological Society of Japan. Ser. II

The Rossby wave propagation in various basic flows is studied, based on the numerical time-integration of a barotropic model on the sphere. First, the propagation in idealized basic flows is examined. When the zonally uniform basic flow has a strong jet, the jet stream acts like a waveguide for the Rossby wave. When the basic flow has a component of zonal wavenumber 1 (or 2), the Rossby wave emanating from the entrance region of the jet rapidly propagates eastward to the jet exit region. Then, the wave becomes stagnant and its amplitude is increased in the exit region. When the wave component superposed in the basic flow is strong, the basic flow has barotropically unstable modes. Nevertheless, this instability is weak so that the propagation patterns are not influenced by it. The barotropic conversion of kinetic energy from the basic field is evaluated to be important in this wave amplification in the jet exit region. Then, it can be understood that this wave amplification is produced both by the stagnation of wave energy from the energy source and by the subsequent barotropic conversion in the jet exit region. Second, the wave propagation is examined for monthly-mean basic flows derived from wintertime observational data. It is found that the propagation property of the Rossby wave is highly dependent on the basic wind distribution. In one case (December 1986) the Rossby wave, which has propagated eastward through the Asian jet, is blocked and becomes stagnant near the jet exit region. In contrast, in the other case (January 1984) near the jet exit the propagation path splits into eastward and equatorward paths.

Meteorology and Atmospheric Physics, 1984

Summary The non-linear interaction between large-scale Rossby waves is considered making use of a spectral model of a barotropic atmosphere in spherical geometry. It is found that, if an arbitrary number of zonal components is taken into account it is possible to predict the non-linear energy and enstrophy flux between Rossby waves whose interaction cannot be predicted by single triadic resonance.

Physics Letters A, 1996

The transformation to normal canonical variables is found in the rigid top approximation for barotropic Rossby waves of arbitrary amplitude in a thin fluid layer on a sphere. This transformation is used to derive an expression for the matrix of a three-wave interaction. Canonical variables describing barotropic Rossby waves in a fluid with a free surface on a sphere are also found. In addition, canonical variables describing Rossby waves in a thin fluid layer inside a rotating paraboloid are obtained both in the rigid lid approximation and in the presence of a free surface. The problem with such a geometry is of interest in connection with the use of rotating parabolic setups in laboratory experiments on modeling nonlinear waves and vortices in geophysics (Rossby waves) and in a plasma (drift waves).

Physics Letters A, 2008

A numerical simulation of the Charney-Obukhov equation modified by the presence of a sheared zonal flow is carried out. The zonal flow is assumed to propagate longitudinally and is sheared along the meridians. It is shown that owing to the nonlinear interaction of the sheared zonal flow with the initially given disturbances the energy of the zonal flow is accumulating into the formations which are broken into several pieces. As a result new solitary vortex structures arise to produce the structural turbulence

Quarterly Journal of the Royal Meteorological Society

This work revisits the theory of the mixed Rossby-gravity (MRG) wave on a sphere. Three analytic methods are employed in this study: (a) derivation of a simple ad hoc solution corresponding to the MRG wave that reproduces the solutions of Longuet-Higgins and Matsuno in the limits of zero and infinite Lamb's parameter, respectively, while remaining accurate for moderate values of Lamb's parameter, (b) demonstration that westward-propagating waves with phase speed equalling the negative of the gravity-wave speed exist, unlike the equatorial-plane, where the zonal velocity associated with such waves is infinite, and (c) approximation of the governing second-order system by Schrödinger eigenvalue equations, which show that the MRG wave corresponds to the branch of the ground-state solutions that connects Rossby waves with zonally symmetric waves. The analytic conclusions are confirmed by comparing them with numerical solutions of the associated second-order equation for zonally propagating waves of the shallow-water equations. We find that the asymptotic solutions obtained by Longuet-Higgins in the limit of infinite Lamb's parameter are not suitable for describing the MRG wave even when Lamb's parameter equals 10 4. On the other hand, the dispersion relation obtained by Matsuno for the MRG wave on the equatorial-plane is accurate for values of Lamb's parameter as small as 16, even though the equatorial-plane formally provides an asymptotic limit of the equations on the sphere only in the limit of infinite Lamb's parameter.

Journal of Computational Physics, 2006

We consider the flow of a thin layer of incompressible fluid on a rotating sphere, bounded internally by the surface of the sphere and externally by a free surface. Progressive-wave solutions are sought for this problem, without tangent plane simplifications. A linearized theory is derived for small amplitude perturbations about a base westerly flow field, allowing calculation of the linearized progressive wavespeed. This result is then extended to the numerical solution of the full model, to obtain highly non-linear large-amplitude progressive-wave solutions in the form of Fourier series. A detailed picture is developed of how the progressive wavespeed depends on wave amplitude. This approach reveals the presence of non-linear resonance behaviour, with different disjointed solution branches existing at different values of the amplitude. Additionally, we show that the formation of localized low pressure systems cut off from the main flow field is an inherent feature of the non-linear dynamics, once the amplitude forcing reaches a certain critical level.

The present study provides a consistent and unified theory for the three types of linear waves of the shallow-water equations (SWE) in a zonal channel on the ␤ plane: Kelvin, inertia-gravity (Poincaré), and planetary (Rossby). The new theory is formulated from the linearized SWE as an eigenvalue problem that is a variant of the classical Schrödinger equation. The results of the new theory show that Kelvin waves exist on the ␤ plane with vanishing meridional velocity, as is the case on the f plane, without any change in the dispersion relation, while the meridional structure of their height amplitude is trivially modified from exponential on the f plane to a one-sided Gaussian on the ␤ plane. Similarly, inertia-gravity waves are only slightly modified in the new theory in comparison with their characteristics on the f plane. For planetary waves (which exist only on the ␤ plane) the new theory yields a similar dispersion relation to the classical theory only for large gravity wave phase speed, such as those encountered in a barotropic ocean or an equivalent barotropic atmosphere. In contrast, for low gravity wave phase speed, for example, those in an equivalent barotropic ocean where the relative density jump at the interface is 10 Ϫ3 , the phase speed of planetary waves in the new theory is 2 times those of the classical theory. The ratio between the phase speeds in the two theories increases with channel width. This faster phase propagation is consistent with recent observation of the westward propagation of crests and troughs of sea surface height made by the altimeter aboard the Ocean Topography Experiment (TOPEX)/Poseidon satellite. The new theory also admits inertial waves, that is, waves that oscillate at the local inertial frequency, as a genuine solution of the eigenvalue problem.

Dynamics of Atmospheres and Oceans, 2006

An analytical theory is presented for the motion of a localized vortex in the presence of a zonal Rossby wave on the β-plane. In the framework of the equivalent-barotropic quasi-geostrophic model, the analytical method developed by Sutyrin and Flierl [Sutyrin G.G., Flierl G.R., 1994. Intense vortex motion on the β-plane: development of the beta gyres. J. Atmos. Sci. 51, 773-790] for intense vortices with piecewiseconstant potential vorticity is generalized to take into account a slowly propagating Rossby wave which modifies the background potential vorticity. The predictions of these asymptotic expansions are compared with the results of numerical simulations. The theory describes the vortex advection by the wave and the vortex drift due to the background potential vorticity gradient. The net vortex drift speed due to the wave is found to be smaller than the maximum wave velocity; this is due to the baroclinic ␤-effect and to the periodic structure of the background potential vorticity gradient. Besides known elliptical core deformations, triangular deformations are generated by the wave on the core boundary. Additionally, the planetary ␤-effect provides a predominantly westward vortex drift with nearly the same speed as the wave propagation speed. The asymptotic theory is shown to agree well the results of a numerical pseudo-spectral, high-resolution biperiodic model when the vortex velocity is much larger than the wave velocity. Both meridional and zonal vortex drifts are slightly overestimated when the wave velocity is comparable with the vortex velocity. Vortex size is shown to be more influential on vortex trajectory than the Rossby wave length. In particular, smaller vortices drift westward farther and faster than large ones. Vortex core deformations typically contain modes 2 and 3 with a stronger mode 3 component for more intense Rossby waves as predicted by theory.

2004

The effect of a baroclinic mean flow on long oceanic Rossby waves is studied using a combination of analytical and numerical solutions of the eigenvalue problem. The effect is summarized by the value of the nondimensional number

Tellus A, 1988

Numerical experiments are performed to examine the resonant growth of a topographically forced wave in a nonlinear model. Time-dependent simulations are made including damping mechanisms, wave-mean flow interactions, a zonal wind driving force and wave-wave interactions. The initial mean zonal wind permits the resonance of the barotropic mode, rather than the first internal mode as in a previous study. It is found that even when damping and nonlinear interactions are included, the growth rate and the amplitude achieved by the resonant barotropic wave are similar to those occurring in observed planetary wave amplification episodes, such as those seen in some developing blocking conditions. Eliassen-Palm flux cross-sections of one of the integrations are presented to illustrate the interaction between the forced wave and the mean zonal flow.