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Adrian Hines and Andrew J. Willmott

Abstract

Analytical and numerical models are presented for linear quasigeostrophic buoyancy-driven flow forced bya time periodic pulsating point mass source in a continuously stratified, incompressible β-plane ocean withconstant Brunt–Väisälä frequency. The source represents the seasonal introduction of dense water into the abyssalocean and is located on a linear sloping bottom of arbitrary orientation. The ocean domain is horizontallyunbounded and of infinite depth. Rayleigh friction is incorporated into the horizontal momentum equations andappears at order Rossby number in the quasigeostrophic expansions. In the density equation the influence ofRayleigh friction and Laplacian friction are each considered in turn.

Analytical solutions are obtained in the case of 1) a midlatitude β plane with no bottom slope and 2) an fplane with a bottom slope. In both of these problems the fluid is initially at rest and the mass source is switchedon and maintained. A three-dimensional radiating field of baroclinic Rossby waves is generated, which arebottom trapped in the second problem. If the time between successive mass pulses is sufficiently long to enablethe free waves to dominate the solution, it is found that the azimuthal wavelength of the bottom-trapped vorticitywave decreases thereby producing a series of elongated vortices. The present generation of ocean generalcirculation models would be unable to resolve this bottom-trapped flow.

Numerical solutions are presented for the case of a sloping bottom of arbitrary orientation on a β plane whenthe time periodic source exists for all time. During each cycle of the forcing a bottom-trapped anticyclonicvortex is generated at the source and propagates in a direction dictated by the relative role of the planetary andtopographic beta effects. The horizontal distance that the vortex propagates before decaying is larger whenLaplacian mixing is incorporated in the density equation rather than Rayleigh damping. A study of how slopemagnitude and orientation influences the solution is presented.

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Andrew J. Willmott and Estanislao Gavilan Pascual-Ahuir

Abstract

The eigenfrequencies of freely propagating barotropic, divergent, planetary waves and gravity waves in a spherical polar cap are presented using an approximation in which full spherical geometry is retained in the derivation of the wave amplitude equation. Subsequently, the colatitude angle in the coefficients of the wave amplitude equation is fixed, thereby allowing the eigenvalue problem to be solved using analytical methods. The planetary wave frequencies are compared with published results that adopt the polar-plane approximation to solve the equivalent free-wave problem. Low-order planetary wave frequencies calculated in this study agree well with the polar-plane approximation results. The sensitivity of the wave frequencies to the choice of the fixed colatitude in the coefficients of the wave amplitude equation is discussed.

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Lawrence A. Mysak and Andrew J. Willmott

Abstract

A general theory for forced barotropic long trench waves in the presence of linear bottom friction is presented. Two specific forcing mechanisms are considered: (i) transverse fluctuations in a western boundary current as it flows across a trench, and (ii) a traveling wind system that moves parallel to the trench. The mechanisms (i) and (ii) are applied to the Japan-Kuril trench and Aleutian trench, respectively. In the case of the Japan-Kuril trench it is found that 3-month period fluctuations in the Kuroshio are able to generate currents along the trench of 0 (10 cm s−1) and coastal sea level variations of O (7 cm). In the case of the Aleutian trench, traveling wind systems in the northeast Pacific may produce a near resonant response. Such a response consists of velocity fluctuations of 0 (10 cm s−1) along the trench and of 0 (4 cm s−1) across the trench, the coastal sea level fluctuations can be up to 12 cm. While these estimates should be regarded as tentative because of the uncertainty in the value of the bottom friction coefficient, they nevertheless suggest that trench wave motions could produce significant long-time scale velocity and sea level fluctuations in the North Pacific trenches.

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Andrew J. Willmott and Richard E. Thomson

Abstract

The authors examine barotropic nondivergent shelf waves generated on an exponential continental shelf that has an abrupt change in width. Three types of forcing are considered: 1) a tidal period volume flux through a gap in the coastline located along the discontinuity, 2) an alongshore propagating wind stress over the continental shelf, and 3) an alongshore propagating perturbation in the streamfunction at the edge of the continental slope. Dimensional results for the linearized models are derived for the northwest coast of Vancouver Island, British Columbia, where the shelf abruptly widens into Queen Charlotte Sound. Because of the change in wave scales and numbers of shelf modes possible on either side of the coastline discontinuity, the response for the discontinuities width shelf differs markedly from that for a uniform width shelf. Results show that shelf wave energy generated by fortnightly tidal flow through the gap (or coastal strait) is radiated in a narrow “beam” across the broader portion of the shelf. Diurnal period motions are trapped near the mouth of the strait and do not contribute significantly to the shelf current. The use of realistic periods and wavelengths (5–10 days and 500–1000 km) for the alongshore forcing terms yields propagating eddylike circulation patterns that closely resemble the flow patterns commonly seen in satellite thermal imagery over the narrow portion of the Vancouver Island shelf. At low forcing frequencies a distinct “shadow zone” with relatively weak barotropic response is found over the wide portion of the shelf in the vicinity of the coastal discontinuity.

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Darryl G. Murphy and Andrew J. Willmott

Abstract

The interaction of a prescribed channel mode Rossby wave with a meridional line barrier situated in an infinitely long, zonally aligned channel is considered. An analytical solution for the scattered field is presented for a barrier that spans half the channel width. The solution is obtained using the Wiener-Hopf technique. For other barrier widths the scattered field is determined by numerically solving the Rossby wave equation. In all cases the scattered field is described in terms of propagating cross-channel eigenmodes, together with an infinite number of evanescent modes. The solutions are interpreted in terms of the energy density and energy flux associated with each scattered field eigenmode. Mode one dominates the energy balance in the scattered field for most barrier widths in response to a mode-one incident wave. However, for a particular range of barrier widths the second cross-channel mode is found to be the most energetic. The energy density of each scattered field mode is larger on the eastern side of the barrier than on the west. The implication of the model results for Rossby wave propagation in the Southern Ocean are examined.

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Andrew J. Willmott and Lawrence A. Mysak

Abstract

A thermodynamic reduced-gravity ocean model forced by the steady-state surface wind stress and a Haney-type heat flux was used to determine the climatological ice-edge position, ice thickness, ocean circulation, and upper-ocean temperature in a high-latitude meridional channel. The ice model used is purely thermodynamic; however, a parameterization is used to allow the surface wind stress to be transmitted to the water below the ice. The temperature distribution of the upper ocean is specified along the southern zonal boundary of the model domain, and the heat equation is integrated from this boundary poleward along streamlines for the mass transport. As a column of warm water moves poleward, its temperature tends to decrease since the air temperature monotonically decreases to the north. At high latitudes the steady-state heat balance between horizontal advection and cooling to the atmosphere can no longer hold in an ice-free ocean. Thus, an ice layer forms to insulate the ocean from the very cold air temperatures at these latitudes.

The model is applied to the Greenland and Norwegian Seas, between 60° and 80°N, and between the east coast of Greenland and 15°E. Exterior to a narrow western boundary layer, the predicted ice-edge position compares favorably with the climatological 90% ice concentration isoline obtained from an analysis of 32 years of Arctic sea ice data.

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Daniel G. Wright and Andrew J. Willmott

Abstract

Simple models are developed to describe the abyssal circulation in a circumpolar ocean driven by localized annual sources of water representing convection events. Models are based on a geostrophic reduced-gravity formulation and are located on a zonally periodic beta plane.

Nonlinear analytical solutions are first obtained for the evolution of an initially prescribed water mass distribution in the presence of a zonal mean flow and topography that is a function of the meridional coordinate only. For large time, our model suggests that diffusion results in a zonally uniform interface height with associated geostrophic currents if the initial water mass has nontrivial zonally averaged meridional structure. Possible influence of neglected effects on this large time behavior are discussed. Solutions are also presented for the evolution of an abyssal water mass introduced into an area where no equally dense water existed previously, and for a water mass added to a preexisting abyssal layer of equal density. The examples help clarify the varying roles of planetary and topographic beta as well as the role of horizontal diffusion. The initial value problem is then extended to allow for repeated additions of abyssal water. The resulting complicated flow field is discussed in terms of simple principles. For large times the active layer depth and the zonally averaged zonal current grow without bound in this model, pointing to the need for a loss of abyssal water if a statistical equilibrium is to be achieved.

Finally a quasi-linear model is developed that includes water loss parameterized by a Newtonian damping term and allows for more general bottom topography and background flow. Horizontal diffusion is not included in this model so wave breaking can occur, after which time the model is invalid. The model is used to investigate the influence of a meridional ridge on the abyssal circulation.

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Andrew J. Willmott and Arlene A. Bird

Abstract

The dispersion relation is derived for trapped freely propagating barotropic long trench waves on a midlatitude β-plane. It is found that a critical wavenumber kc, which depends on trench orientation and wave frequency, partitions the behavior of each mode. Leaky modes occur when the wavenumber k of a particular mode satisfies k<kc, in which case the mode takes the form of a linear barotropic Rossby wave in the ocean interior which radiates energy offshore. Coastally trapped solutions occur when k>kc. For this latter case the solutions are spatially damped as they propagate along the trench. Dispersion curves are presented for the coastally trapped solutions along the Japan, Kuril and Peru trenches. Surfaces of the mass transport streamfunction are also displayed for both evanescent and propagating solutions along the Japan, Kuril and Peru trenches. The theory suggests that leaky trench waves might be a generating mechanism for barotropic Rossby waves in the Pacific Ocean basin.

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Andrew J. Willmott, Neil R. Edwards, and Peter D. Killworth

Abstract

The response of the deep ocean to periodic and steady forcing by mass sources is considered, in the presence of fluid loss, diffusion, and topography, which may or may not have regions of closed planetary vorticity. There can be a preexisting deep layer, or the forcing may produce one. The long-time response to forcing, which varies annually, is shown to have only a small periodic component, essentially because the mass loss by diapycnic transfer is weak. This is in qualitative agreement with observations of overflows, which show little seasonal signal. Ridges do not form an effective block to the flow, which can bypass the ridges, approximately following lines of constant planetary vorticity. A brief discussion of how fluid leaks out of closed planetary vorticity regions by diffusion is included.

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Neil R. Edwards, Andrew J. Willmott, and Peter D. Killworth

Abstract

A frictional geostrophic model is used to examine how the stability of the thermohaline circulation is affected by idealized topographic variations and the presence or absence of wind stress. If the flow exhibits collapses, the authors consider how topography and wind stress affect the ensuing oscillations. Large-scale slope up toward the north or the west can significantly destabilize the circulation by modifying the barotropic flow and reducing the depth of convection. Wind stress stabilizes the circulation by deepening the thermocline in the subtropical gyre. Wind driving can also radically reduce the period of oscillations by destabilizing the northern halocline in the collapsed phase. The overall period of the oscillation is usually governed by the time taken for diffusive warming to destabilize the deep ocean.

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