Higher-order corrections for Rossby waves in a zonal channel on the [beta]-plane (original) (raw)
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Higher-order corrections for Rossby waves in a zonal channel on the β-plane
Quarterly Journal of the Royal Meteorological Society, 2007
A formal analytic perturbation expansion in the β term is carried out for the Rossby wave solution of the shallow-water equations in a zonal channel on the β-plane. Apart from a quantization of the meridional wave number, the presence of zonal boundaries alters, to first order, both the velocity and the geopotential structures of the wave but does not alter the phase speed of the wave. The ageostrophic component of the velocity field is identical in first order with that of the unbounded β-plane and is therefore not related to the presence of boundaries. In contrast, the first-order correction to the geostrophic velocity component is inherently related to the presence of walls as it ensures the vanishing of the total meridional velocity on the boundaries. This first-order correction to the geostrophic field yields only a third-order correction in the Rossby phase speed, as can be expected from symmetry considerations.
Annales Geophysicae, 2012
Using the shallow water equations for a rotating layer of fluid, the wave and dispersion equations for Rossby waves are developed for the cases of both the standard βplane approximation for the latitudinal variation of the Coriolis parameter f and a zonal variation of the shallow water speed. It is well known that the wave normal diagram for the standard (mid-latitude) Rossby wave on a β-plane is a circle in wave number (k y , k x) space, whose centre is displaced −β/2ω units along the negative k x axis, and whose radius is less than this displacement, which means that phase propagation is entirely westward. This form of anisotropy (arising from the latitudinal y variation of f), combined with the highly dispersive nature of the wave, gives rise to a group velocity diagram which permits eastward as well as westward propagation. It is shown that the group velocity diagram is an ellipse, whose centre is displaced westward, and whose major and minor axes give the maximum westward, eastward and northward (southward) group speeds as functions of the frequency and a parameter m which measures the ratio of the low frequency-long wavelength Rossby wave speed to the shallow water speed. We believe these properties of group velocity diagram have not been elucidated in this way before. We present a similar derivation of the wave normal diagram and its associated group velocity curve for the case of a zonal (x) variation of the shallow water speed, which may arise when the depth of an ocean varies zonally from a continental shelf.
On the meridional structure of extra-tropical Rossby waves
A B S T R A C T The common derivation of Rossby waves is based on the quasi-geostrophic approximation. A simple non-harmonic approximation for extratropical Rossby waves on the sphere is proposed, in which the meridional coordinate is a parameter instead of a continuous variable. It is shown that, in contrast to the quasi-geostrophic solution, to first order the meridional structure of these non-harmonic Rossby waves becomes irrelevant for determining the dispersion relation in this theory. The proposed approximation accurately reproduces numerical results obtained from runs of an ocean general circulation model initiated from several initial meridional structures and captures the latitudinal dependence of the phase speed of these waves. The proposed theory yields explicit expressions for the dispersion relation and for the meridional structure of the waves.
Dynamics of Atmospheres and Oceans, 1992
The exchange of fluid mass between quasi-geostrophie and ageostrophic motions during the reflection of Rossby waves from a coast. I. The case of an infinite rectilinear coast. Dyn. Atmos. Oceans, 16: 305-329. When the problem of the reflection of spatially localized Rossby waves from a coast is treated using the quasigeostrophic (QG) approximation, the total fluid mass and the along-shore circulation calculated from the geostrophic height field are not conserved. To understand the correct mass balance and the degree to which the QG equations and boundary conditions may be in error, we analyze an initial-value problem for the Laplace tidal equations on a t-plane in the asymptotic fimit c ~ 1, where ~ is the ratio of the spatial scale of the motion to the Earth's radius. It is shown that there is a coupling between QG and O(c) fields. Physically, the coupfing occurs by a peculiar adjustment process in the O(c) approximation in which' fast gravity waves are permanently generated to build up a quasi-stationary edge Kelvin wave. Different temporal scales (large for O(1) Rossby waves and small for the 0(c) gravity waves make comparable the contributions of the waves to the mass and circulation balance equations. However, QG analysis itself describes the reflection of Rossby waves correctly, but is incomplete, and for satisfactory balances one has to take into account the fields of both orders of the approximation. Applications of the results to closed basins, baroclinicity, and variable bottom topography are discussed. It is conjectured that an interaction of strong oceanic eddies with a coast (continental slope) may give rise to noticeable along-shore jet currents.
Meridional component of oceanic Rossby wave propagation
Dynamics of Atmospheres and Oceans, 2005
A three-dimensional spectral analysis of Topex altimeter data reveals a large meridional component k y of the wavevector k for baroclinic Rossby waves of all timescales. Its existence necessitates some refinements in our estimates of certain basic properties of the Rossby wave field. In particular, by taking into account an actual off-zonal direction of k (often exceeding 70 •), one finds that the wavelength, phase speed, and group velocity of mid-latitude Rossby waves (with periods less than 2 years) are much smaller than they appear to be on the assumption of a purely zonal wavenumber vector. Because of a shorter wavelength (yielding kL as high as 0.6, where L is the Rossby radius of deformation), these waves are essentially dispersive. Their group velocity vector may depart from zonal by more than 30 •. An important intrinsic feature of the wave spectrum confirmed by our analysis is a broadband distribution with respect to k y. Some of the dynamical implications of the large k y /k x ratio are discussed.
Linear and nonlinear Rossby waves in basins both with and without a thin meridional barrier
2002
The linear and nonlinear Rossby wave solutions are examined in homogeneous square basins on the /-plane both with and without a thin meridional barrier. In the presence of the meridional barrier the basin is almost partitioned into two; only two small gaps of equal width, d, to the north and south of the barrier allow communication between the eastern and western sub-basins. Solutions are forced by a steady periodic wind forcing applied over a meridional strip near the eastern side. Bottom friction is present to allow the solutions to reach equilibrium. The linear solution for the basin containing the barrier is determined analytically and the nonlinear solutions for both basins are found numerically. In the linear solution with the barrier present, particular attention was paid to the resonant solutions. We examined the effects of varying the symmetry of the forcing about the mid-latitude, the frequency of the periodic forcing and the strength of the bottom friction. For each solution we focus on how the no net circulation condition, which is central to any solution in a barrier basin, is satisfied. The nonlinear solutions were studied for both basin configurations. In each case the transition from the weakly nonlinear solution to the turbulent solution was examined, as the forcing frequency and forcing strength were varied. Only integer multiples of the forcing frequency are present in the weakly nonlinear solutions. The turbulent solutions were accompanied by the appearance of many other frequencies whose exact origins are unknown, but are probably the result of instabilities. A hysteresis was found for the turbulent solutions of both the barrier-free and barrier basins. In the weakly nonlinear solutions of the barrier basin it was predicted and confirmed that there is never a steady net flow from sub-basin to sub-basin. It was also 4shown that with a symmetric forcing all modes oscillating with an odd multiple of the forcing frequency are symmetric and all modes oscillating with even multiples of the forcing frequency are antisymmetric. I would like to thank my advisor, Joe Pedlosky, for introducing me to the the problem of studying Rossby modes for various basin configurations and for helping me publish my pregenerals research (presented in chapter 3 of this thesis) in II Nuovo Cimento. I am especially grateful for the way Joe has made himself readily available for discussions and for his promptness in returning my work and providing feedback. I would also like to thank my committee members Mike Spall, Joe LaCasce, Nelson Hogg, and Glenn Flierl. In particular, I thank Mike Spall for giving me his modified version of MICOM when I began my research. Special thanks are due to thank my classmate Pablo Zurita who has been a good friend and has helped me immensely in the last few years. Through numerous discussions we have had, I have learnt a great deal me much about computers and numerical computation. This not only helped me to get MICOM running, but has provided me with skills that will be very helpful in the future. I am also grateful to Pablo for allowing me to run MICOM and store my results on his computer. I am indebted to Brian Arbic who directed the computer Huron my way when he went to Princeton. Huron has been a great benefit to me this last year. I would alsolike to thank Richard Wardle, Francis Poulin, and Samar Khatiwala for interesting discussions regarding this thesis. Last, but not least, I would like to thank my parents, sister, and Adrian for all their encouragement.
The Transmission and Transformation of Baroclinic Rossby Waves by Topography*
Journal of Physical Oceanography, 2000
The transmission of westward propagating baroclinic Rossby waves incident on a gappy meridional barrier is studied in the context of the two-layer, quasigeostrophic model. The meridional barrier models the presence of very steep topography such as the midocean ridge system or extensive island arcs. The nature of the transmission depends strongly on the nature of the gaps in the meridional barrier. If the gaps extend throughout the depth of the fluid, the Rossby waves propagate through the barrier, as a consequence of Kelvin's theorem, with no change in vertical structure. On the other hand, if the gaps in the barrier are partial and extend only over a single layer, there is a significant transformation of the vertical structure of the wave field as it traverses the barrier. In particular, waves of baroclinic vertical structure in the model are transformed on the western side of the barrier into barotropic waves that radiate from the segment of the barrier between two such gaps. Such segments act as antennae radiating barotropic energy into the western subbasin. It is suggested that recent observations of signal enhancement of Rossby waves at the midocean ridge system in the Pacific may be related to such transformation of wave structure. The problems of free waves and forced waves in open regions and normal modes in closed basins are described.
Propagation of Meridional Circulation Anomalies along Western and Eastern Boundaries
Journal of Physical Oceanography, 2013
Motivated by the adjustment of the meridional overturning circulation to localized forcing, solutions are presented from a reduced-gravity model for the propagation of waves along western and eastern boundaries. For wave periods exceeding a few months, Kelvin waves play no role. Instead, propagation occurs through short and long Rossby waves at the western and eastern boundaries, respectively: these Rossby waves propagate zonally, as predicted by classical theory, and cyclonically along the basin boundaries to satisfy the no-normal flow boundary condition. The along-boundary propagation speed is cLd/δ, where c is the internal gravity/Kelvin wave speed, Ld is the Rossby deformation radius, and δ is the appropriate frictional boundary layer width. This result holds across a wide range of parameter regimes, with either linear friction or lateral viscosity and a no-slip boundary condition. For parameters typical of contemporary ocean climate models, the propagation speed is coincidental...
Nonlinear Rossby adjustment in a channel: beyond Kelvin waves
Journal of Fluid Mechanics, 1989
Nonlinear advective adjustment of a discontinuity in free-surface height under gravity and rotation is considered, using the method of contour dynamics. After linear wave-adjustment has set up an interior jet and boundary currents in a wide (9 one Rossby radius) channel, fluid surges down-channel on both walls, rather than only that wall supporting a down-channel Kelvin wave. A wedgelike intrusion of low potential vorticity fluid on this wall, and a noselike intrusion of such fluid on the opposite wall, serve to reverse the sign of relative vorticity in the pre-existing currents. For narrower channels, a coherent boundary-trapped structure of low potential vorticity fluid is ejected at one wall, and shoots ahead of its parent fluid. The initial tendency for the current to concentrate on the 'right-hand' wall (the one supporting a down-channel Kelvin wave in the northern hemisphere) is defeated as vorticity advection shifts the maximum to the left-hand side. Ultimately fluid washes downstream everywhere across even wide channels, leaving the linearly adjusted upstream condition as the final state. The time necessary for this to occur grows exponentially with channel width. The width of small-amplitude boundary currents in linear theory is equal to Rossby's deformation radius, yet here we find that the width of the variation in velocity and potential vorticity fields deviates from this scale across a large region of space and time. Comparisons of the contour dynamics solutions, valid for small amplitude, and integration of the shallow-water equations a t large amplitude, show great similarity. Boundary friction strongly modifies these results, producing fields more closely resembling the linear waveadjusted state. Observed features include those suggestive of coastally trapped gravity currents. Analytical results for the evolution of vorticity fronts near boundaries are given in support of the numerical experiments.
Rossby Wave Instability and Apparent Phase Speeds in Large Ocean Basins
Journal of Physical Oceanography, 2007
The stability of baroclinic Rossby waves in large ocean basins is examined, and the quasigeostrophic (QG) results of LaCasce and Pedlosky are generalized. First, stability equations are derived for perturbations on large-scale waves, using the two-layer shallow-water system. These equations resemble the QG stability equations, except that they retain the variation of the internal deformation radius with latitude. The equations are solved numerically for different initial conditions through eigenmode calculations and time stepping. The fastest-growing eigenmodes are intensified at high latitudes, and the slower-growing modes are intensified at lower latitudes. All of the modes have meridional scales and growth times that are comparable to the deformation radius in the latitude range where the eigenmode is intensified. This is what one would expect if one had applied QG theory in latitude bands. The evolution of large-scale waves was then simulated using the Regional Ocean Modeling Sy...