Rossby Normal Modes in Basins with Barriers* (original) (raw)

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.

Baroclinic Rossby Waves in Irregular Basins*

Journal of Physical Oceanography, 2002

The properties of baroclinic, quasigeostrophic Rossby basin waves are examined. Full analytical solutions are derived to elucidate the response in irregular basins, specifically in a (horizontally) tilted rectangular basin and in a circular one. When the basin is much larger than the (internal) deformation radius, the basin mode properties depend profoundly on whether one allows the streamfunction to oscillate at the boundary or not, as has been shown previously. With boundary oscillations, modes occur that have low frequencies and, with scale-selective dissipation, decay at a rate less than or equal to that of the imposed dissipation. These modes approximately satisfy the long-wave equation in the interior. Using both unforced and forced solutions, the variation of the response with basin geometry and dissipation is documented. The long-wave modes obtain with scale-selective dissipation, but also with damping that acts equally at all scales. One finds evidence of them as well in the forced response, even when the dissipation is weak and the corresponding free modes are apparently absent.

The Instability of Rossby Basin Modes and the Oceanic Eddy Field*

Journal of Physical Oceanography, 2004

have isolated a new class of basin modes particularly resistant to dissipation mechanisms that preferentially damp small scales, such as the type employed by Qiu (1997). While basin modes typically require the synthesis of long Rossby waves with westward propagating group velocity and short Rossby waves with eastward group velocity, these new modes closely resemble free, long Rossby waves with zonal wavelengths that are integral multiples of the basin width. Such

On turbulence and normal modes in a basin

Journal of Marine Research, 2002

The problem of forced, geostrophicturbulence in a basin is revisited. The primary focus is the time dependent eld, which is shown to be approximately isotropic (in contrast to the strongly zonally anisotropic elds seen in periodic domains). It is also approximately homogeneous, away from the boundaries. Phenomenologicalarguments suggest the isotropy occurs because the inverse cascade of energy is arrested by basin normal modes rather than by free Rossby waves. Peaks in the velocity spectra at modal frequencies are consistent with basin modes, as has been noted previously. We discuss which modes would be excited and whether dissipation or the mean ow would be expected to alter the modes and their frequencies. A relatively novel feature is the use of Eulerian velocity statistics to quantify the wave and turbulence characteristics. These measures are more suitable to this environment than measures like wavenumber spectra, given the inhomogeneities associated with the boundaries. With regards to the mean, we observe a linear^q& 2^c& relation in the region of the mean gyres (at the northern and southern boundaries), consistent with previous theories. This is of interest because our numerical advection scheme has implicit rather than explicit small scale dissipation, and requires no boundary conditions on the vorticity. The gyre structure is however somewhat different than in an (inviscid) Fofonoff-type solution, suggesting dissipation cannot be neglected.

Excitation of basin modes by ocean-atmosphere coupling

A conceptual model of the coupling between the upper-ocean wind-driven circulation and the mid-latitude atmospheric wind-stress illustrates that large-scale basin-wide oscillations with decadal period can be excited. These oceanic modes are also found in the absence of ocean-atmosphere feedback, but they are damped. The period of the oscillation and the spatial structure of the modes are essentially the same with and without coupling. These oscillations are distinct from the coupled modes of variability arising from a delayed negative feedback between the wind-driven flow and the wind-stress. They are ocean-only linear basin modes which become sustained by ocean-atmosphere coupling.

Multiple Oscillatory Modes of the Argentine Basin. Part II: The Spectral Origin of Basin Modes

Journal of Physical Oceanography, 2007

In this paper the spectrum of barotropic basin modes of the Argentine Basin is shown to be connected to the classical Rossby basin modes of a flat-bottom (constant depth), rectangular basin. First, the spectrum of basin modes is calculated for the Argentine Basin, by performing a normal-mode analysis of the barotropic shallow-water equations. Then a homotopy transformation is performed that gradually morphs the full-bathymetry geometry through a flat-bottom configuration into a rectangular basin. Following the eigenmodes through this transition establishes a connection between most of the basin modes and the classical Rossby basin modes of a rectangular geometry. In particular, the 20-day mode of the Argentine Basin is identified with the lowest-order mode of classical theory. Sensitivity studies show that the decay rate of each mode is controlled by bottom friction, but that it is insensitive to lateral friction; lateral friction strongly impacts the oscillation frequency. In addition, the modes are found to be only slightly sensitive to the presence of a background flow.

Multiple oscillatory modes of the Argentine Basin. Part II. The spectral origin of the basin modes

Journal of Physical Oceanography, 2007

Low-Frequency Basin Modes in a Two-Layer Quasigeostrophic Model in the Presence of a Mean Gyre Flow*

Journal of Physical Oceanography, 2005

The spectrum of baroclinic basin modes is investigated in a two-layer wind-driven quasigeostrophic model through weakly nonlinear multiple time-scale expansion in the Burger number. The baroclinic basin modes are mainly advected by a barotropic steady Sverdrup flow. Emphasis is given to the regularizing influence of dispersion rather than to dissipation. In the inviscid large-scale limit, that is, for basin scale considerably larger than the Rossby radius of deformation, all of the basin modes are neutral. Their typology is then numerically examined (with some necessary dissipation), and their frequency and spatial properties are discussed. Three types of modes arise for some wind forcing strong enough to produce a recirculating gyre with closed geostrophic contours: the classical Rossby basin modes deformed by the mean flow (shadow modes), stationary modes, and recirculating pool modes, the two latter being trapped in the closed-contours pool. Focus is made here on the recirculating modes that could have very low frequencies for moderate recirculating gyres. Strong gyres lead to higher frequencies, and recirculating modes resonate with deformed Rossby basin modes. * Woods Hole Oceanographic Institution Contribution Number 10999.

Topographic Waves in Basins with Complex Shapes and Complex Bathymetries

Advances in Geophysical and Environmental Mechanics and Mathematics, 2011

In the last two chapters, construction of analytical solutions to the topographic wave (TW)-equation in enclosed basins subject to the no-flux boundary condition was possible only for basins of simple geometries and simple bathymetries. The situations were generally such that the linear boundary value problems could be constructed and solved by the method of separation of variables leading to two-point-eigenvalue problems of ordinary differential equations with homogeneous boundary conditions, which could be expressed in terms of simple functions. However, unless the bathymetry was approximately expressible by very simple exponential or power law functions, the differential equations soon took forms, which were no longer expressible by common functions of mathematical physics, or the mathematical expressions for the solution would be so tedious to handle, that they are very likely better solved numerically. As an example, we presented the solutions of the few lowest order TW-modes in a circular basin with parabolic radial profile in terms of hypergeometric polynomials (see Chap. 20, formulae (20.23)). It is also known that the interior of a circle can be transformed by a conformal mapping onto the interior of a rectangle. This transformation involves, among others, elliptic integrals of the first kind. So, the solution in the rectangle will be a composition of hypergeometric polynomials and elliptic integrals of the first kind. Such solution techniques served their purposes in times prior to electronic computation. Today, more flexibility is demanded, such that mathematical expressions can be optimally matched to the realistic bathymetries. Indeed, we have so far not been able to describe qualitatively how TW-modes look like for a rectangular long basin even when its bathymetry is very simple. In fact it is claimed that an elongated basin of more general than rectangular or elliptical shape possesses also TW-modes which are characteristically different from those hitherto constructed. The identification of these other solutions is physically important; and it will solve the Lake of Lugano controversy explained in Chap. 19.

Rossby Wave–Coastal Kelvin Wave Interaction in the Extratropics. Part I: Low-Frequency Adjustment in a Closed Basin

Journal of Physical Oceanography, 1999

The formation of an island circulation is investigated both theoretically and numerically in light of the dynamics of coastal Kelvin waves and Rossby waves. An island circulation is formed in three stages. First, the direction of the circulation is initiated by the coastal Kelvin wave; second, the transport of the circulation is established by the short Rossby wave dissipated against the eastern coast of the island; and finally, the basinwide circulation pattern is completed by the long Rossby wave radiated from the western coast of the island. An island circulation can be forced by either a local alongshore wind or a remote vorticity forcing to the east of the island; the initial Kelvin wave is directly forced by the alongshore wind in the former case, but indirectly forced by a planetary wave incident on the eastern coast of the island in the latter case. A comparison is also made between the spinup of an island circulation and a basin circulation. In addition, this spinup study also provides an alternative derivation of the island rule in light of the dynamics of Kelvin and Rossby waves. The implication for understanding the temporal response of an island circulation to a variable forcing is also discussed.

Rossby Normal Modes in Basins with Barriers* (original) (raw)

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