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- Author or Editor: Joseph Pedlosky x

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## Abstract

An adiabatic, inertial, and quasigeostrophic model is used to discuss the interaction of surface Ekman transport with an island. The theory extends the recent work of Spall and Pedlosky to include an analytical and nonlinear model for the interaction. The presence of an island that interrupts a uniform Ekman layer transport raises interesting questions about the resulting circulation. The consequential upwelling around the island can lead to a local intake of fluid from the geostrophic region beneath the Ekman layer or to a more complex flow around the island in which the fluid entering the Ekman layer on one portion of the island's perimeter is replaced by a flow along the island's boundary from a downwelling region located elsewhere on the island. This becomes especially pertinent when the flow is quasigeostrophic and adiabatic. The oncoming geostrophic flow that balances the offshore Ekman flux is largely diverted around the island, and the Ekman flux is fed by a transfer of fluid from the western to the eastern side of the island. As opposed to the linear, dissipative model described earlier, this transfer takes place even in the absence of a topographic skirt around the island. The principal effect of topography in the inertial model is to introduce an asymmetry between the circulation on the northern and southern sides of the island. The quasigeostrophic model allows a simple solution to the model problem with topography and yet the resulting three-dimensional circulation is surprisingly complex with streamlines connecting each side of the island.

## Abstract

An adiabatic, inertial, and quasigeostrophic model is used to discuss the interaction of surface Ekman transport with an island. The theory extends the recent work of Spall and Pedlosky to include an analytical and nonlinear model for the interaction. The presence of an island that interrupts a uniform Ekman layer transport raises interesting questions about the resulting circulation. The consequential upwelling around the island can lead to a local intake of fluid from the geostrophic region beneath the Ekman layer or to a more complex flow around the island in which the fluid entering the Ekman layer on one portion of the island's perimeter is replaced by a flow along the island's boundary from a downwelling region located elsewhere on the island. This becomes especially pertinent when the flow is quasigeostrophic and adiabatic. The oncoming geostrophic flow that balances the offshore Ekman flux is largely diverted around the island, and the Ekman flux is fed by a transfer of fluid from the western to the eastern side of the island. As opposed to the linear, dissipative model described earlier, this transfer takes place even in the absence of a topographic skirt around the island. The principal effect of topography in the inertial model is to introduce an asymmetry between the circulation on the northern and southern sides of the island. The quasigeostrophic model allows a simple solution to the model problem with topography and yet the resulting three-dimensional circulation is surprisingly complex with streamlines connecting each side of the island.

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## Abstract

A quasigeostrophic, two-layer model is used to study the baroclinic circulation around a thin, meridionally elongated island. The flow is driven by either buoyancy forcing or wind stress, each of whose structure would produce an antisymmetric double-gyre flow. The ocean bottom is flat. When the island partially straddles the intergyre boundary, fluid from one gyre is forced to flow into the other. The amount of the intergyre flow depends on the island constant, that is, the value of the geostrophic streamfunction on the island in each layer. That constant is calculated in a manner similar to earlier studies and is determined by the average, along the meridional length of the island, of the interior Sverdrup solution just to the east of the island.

Explicit solutions are given for both buoyancy and wind-driven flows. The presence of an island of nonzero width requires the determination of the baroclinic streamfunction on the basinâ€™s eastern boundary. The value of the boundary term is proportional to the islandâ€™s area. This adds a generally small additional baroclinic intergyre flow. In all cases, the intergyre flow produced by the island is not related to topographic steering of the flow but rather the pressure anomaly on the island as manifested by the barotropic and baroclinic island constants. The vertical structure of the flow around the island is a function of the parameterization of the vertical mixing in the problem and, in particular, the degree to which long baroclinic Rossby waves can traverse the basin before becoming thermally damped.

## Abstract

A quasigeostrophic, two-layer model is used to study the baroclinic circulation around a thin, meridionally elongated island. The flow is driven by either buoyancy forcing or wind stress, each of whose structure would produce an antisymmetric double-gyre flow. The ocean bottom is flat. When the island partially straddles the intergyre boundary, fluid from one gyre is forced to flow into the other. The amount of the intergyre flow depends on the island constant, that is, the value of the geostrophic streamfunction on the island in each layer. That constant is calculated in a manner similar to earlier studies and is determined by the average, along the meridional length of the island, of the interior Sverdrup solution just to the east of the island.

Explicit solutions are given for both buoyancy and wind-driven flows. The presence of an island of nonzero width requires the determination of the baroclinic streamfunction on the basinâ€™s eastern boundary. The value of the boundary term is proportional to the islandâ€™s area. This adds a generally small additional baroclinic intergyre flow. In all cases, the intergyre flow produced by the island is not related to topographic steering of the flow but rather the pressure anomaly on the island as manifested by the barotropic and baroclinic island constants. The vertical structure of the flow around the island is a function of the parameterization of the vertical mixing in the problem and, in particular, the degree to which long baroclinic Rossby waves can traverse the basin before becoming thermally damped.

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## Abstract

The response of a weakly stratified layer of fluid to a surface cooling distribution is investigated with linear theory in an attempt to clarify recent numerical results concerning the sinking of cooled water in polar ocean boundary currents.

A channel of fluid is forced at the surface by a cooling distribution that varies in the down-channel as well as the cross-channel directions. The resulting geostrophic flow in the central region of the channel impinges on its boundaries, and regions of strong downwelling are observed. For the parameters of the problem investigated, the downwelling occurs in a classical Stewartson layer but the forcing of the layer leads to an unusual relation with the interior flow, which is forced to satisfy the thermal condition on the boundary while the geostrophic normal flow in the interior is brought to rest in the boundary layer.

As a consequence of the layerâ€™s dynamics, the resulting long-channel flow exhibits a nonmonotonic approach to the interior flow, and the strongest vertical velocities are limited to the boundary layer whose scale is so small that numerical models resolve the region only with great difficulty. The analytical model presented here is able to reproduce key features of the previous nonlinear numerical calculations.

## Abstract

The response of a weakly stratified layer of fluid to a surface cooling distribution is investigated with linear theory in an attempt to clarify recent numerical results concerning the sinking of cooled water in polar ocean boundary currents.

A channel of fluid is forced at the surface by a cooling distribution that varies in the down-channel as well as the cross-channel directions. The resulting geostrophic flow in the central region of the channel impinges on its boundaries, and regions of strong downwelling are observed. For the parameters of the problem investigated, the downwelling occurs in a classical Stewartson layer but the forcing of the layer leads to an unusual relation with the interior flow, which is forced to satisfy the thermal condition on the boundary while the geostrophic normal flow in the interior is brought to rest in the boundary layer.

As a consequence of the layerâ€™s dynamics, the resulting long-channel flow exhibits a nonmonotonic approach to the interior flow, and the strongest vertical velocities are limited to the boundary layer whose scale is so small that numerical models resolve the region only with great difficulty. The analytical model presented here is able to reproduce key features of the previous nonlinear numerical calculations.

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## Abstract

The Rossby normal modes of a two-layer fluid in a meridional channel of width *L*âˆ— are altered by the presence of a meridional flow with a small vertical shear. The stability of the modes in the presence of the weak shear is considered. It is found that the joint presence of the Rossby modes and the vertical shear leads to baroclinic instability even for arbitrarily small values of the shear.

The results are used to explain previous numerical calculations of the persistent instability of meridional flows when the ratio *Î²L*
^{2}
_{
D
}
*V* > 1, where *V* is the magnitude of the shear, *Î²* is the planetary vorticity gradient, and *L*
_{
D
} is the deformation radius. If the flow were zonal it would be stable for such weak shears.

The growth rates are weak when *Î²L*
^{2}
_{
D
}
*V* â‰« 1 and each unstable mode exists in a narrow range of meridional wavenumber. The asymptotic results qualitatively agree with the earlier numerical results at moderate values of the same parameter.

## Abstract

The Rossby normal modes of a two-layer fluid in a meridional channel of width *L*âˆ— are altered by the presence of a meridional flow with a small vertical shear. The stability of the modes in the presence of the weak shear is considered. It is found that the joint presence of the Rossby modes and the vertical shear leads to baroclinic instability even for arbitrarily small values of the shear.

The results are used to explain previous numerical calculations of the persistent instability of meridional flows when the ratio *Î²L*
^{2}
_{
D
}
*V* > 1, where *V* is the magnitude of the shear, *Î²* is the planetary vorticity gradient, and *L*
_{
D
} is the deformation radius. If the flow were zonal it would be stable for such weak shears.

The growth rates are weak when *Î²L*
^{2}
_{
D
}
*V* â‰« 1 and each unstable mode exists in a narrow range of meridional wavenumber. The asymptotic results qualitatively agree with the earlier numerical results at moderate values of the same parameter.

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## Abstract

The response of a basin with a topographic barrier to spatially localized and time periodic forcing is considered. The barrier, which almost completely divides the full basin into two adjacent subbasins, is offered as a model of either a planetary island in the wind-driven circulation or a portion of the midocean ridge in the abyssal circulation.

The barrier completely blocks the flow between the two adjacent subbasins except for the possibility of flow through two small gaps at the termini of the barrier. The barrier has nonzero thickness, and scale-dependent lateral friction acts in the gap channels to impede the flow from one subbasin to the next. Bottom friction also acts uniformly on the flow in the basin. The degree to which localized forcing is able to excite large-scale motions in the adjacent subbasin is shown to be connected to the structure of the forcing and its frequency.

In the absence of forcing and friction a set of full basin normal modes exist. The degree to which the forcing is able to resonate with such modes determines the degree to which energy can be transmitted from one subbasin to the other. Friction in the gaps reduces both the amplitude of that transmission and smooths the peaks of the response curve of the motion as a function of frequency in both subbasins. However, even for substantial friction, a considerable amount of large-scale variability can be excited in the adjacent basin. The quantitative dependence of the response on the degree of friction, the length of the channels representing the gaps, and the meridional structure of the forcing are discussed.

In cases where the western boundary of the basin is nonreflecting, so that no full basin normal modes are possible, substantial energy transmission is still demonstrated. Whether resonance occurs or not, the necessity for energy transmission is closely related to the existence of the integral circulation constraint around the island barrier and the possibility of resonance acts mainly to set the level of the response.

## Abstract

The response of a basin with a topographic barrier to spatially localized and time periodic forcing is considered. The barrier, which almost completely divides the full basin into two adjacent subbasins, is offered as a model of either a planetary island in the wind-driven circulation or a portion of the midocean ridge in the abyssal circulation.

The barrier completely blocks the flow between the two adjacent subbasins except for the possibility of flow through two small gaps at the termini of the barrier. The barrier has nonzero thickness, and scale-dependent lateral friction acts in the gap channels to impede the flow from one subbasin to the next. Bottom friction also acts uniformly on the flow in the basin. The degree to which localized forcing is able to excite large-scale motions in the adjacent subbasin is shown to be connected to the structure of the forcing and its frequency.

In the absence of forcing and friction a set of full basin normal modes exist. The degree to which the forcing is able to resonate with such modes determines the degree to which energy can be transmitted from one subbasin to the other. Friction in the gaps reduces both the amplitude of that transmission and smooths the peaks of the response curve of the motion as a function of frequency in both subbasins. However, even for substantial friction, a considerable amount of large-scale variability can be excited in the adjacent basin. The quantitative dependence of the response on the degree of friction, the length of the channels representing the gaps, and the meridional structure of the forcing are discussed.

In cases where the western boundary of the basin is nonreflecting, so that no full basin normal modes are possible, substantial energy transmission is still demonstrated. Whether resonance occurs or not, the necessity for energy transmission is closely related to the existence of the integral circulation constraint around the island barrier and the possibility of resonance acts mainly to set the level of the response.

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## Abstract

The first-order effects of nonlinearity on the thickness and frictionally driven flux in the Ekman layer are described for the case of an Ekman layer on a solid, flat plate driven by an overlying geostrophic flow as well as the Ekman layer on a free surface driven by a wind stress in the presence of a deep geostrophic current. In both examples, the fluid is homogeneous. Particular attention is paid to the effect of nonlinearity in determining the thickness of the Ekman layer in both cases. An analytical expression for the Ekman layer thickness as a function of Rossby number is given when the Rossby number is small. The result is obtained by insisting that the perturbation expansion of the Ekman problem in powers of the Rossby number remains uniformly valid. There are two competing physical effects. The relative vorticity of the geostrophic currents tends to reduce the width of the layer, but the vertical velocity induced in the layer can fatten or thin the layer depending on the sign of the vertical velocity. The regularized expansion is shown to give, to lowest order, expressions for the flux in agreement with earlier calculations.

## Abstract

The first-order effects of nonlinearity on the thickness and frictionally driven flux in the Ekman layer are described for the case of an Ekman layer on a solid, flat plate driven by an overlying geostrophic flow as well as the Ekman layer on a free surface driven by a wind stress in the presence of a deep geostrophic current. In both examples, the fluid is homogeneous. Particular attention is paid to the effect of nonlinearity in determining the thickness of the Ekman layer in both cases. An analytical expression for the Ekman layer thickness as a function of Rossby number is given when the Rossby number is small. The result is obtained by insisting that the perturbation expansion of the Ekman problem in powers of the Rossby number remains uniformly valid. There are two competing physical effects. The relative vorticity of the geostrophic currents tends to reduce the width of the layer, but the vertical velocity induced in the layer can fatten or thin the layer depending on the sign of the vertical velocity. The regularized expansion is shown to give, to lowest order, expressions for the flux in agreement with earlier calculations.

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## Abstract

The bottom boundary layer of a stratified flow on a coastal continental shelf is examined using the model of Chapman and Lentz. The flow is driven by a surface stress, uniform in the alongshore coordinate, in a downwelling-favorable direction. The stress diminishes in the offshore direction and produces an Ekman pumping, as well as an onshore Ekman flux. The model yields an interior flow, sandwiched between an upper Ekman layer and a bottom boundary layer. The interior has a horizontal density gradient produced by a balance between horizontal diffusion of density and vertical advection of a background vertical density gradient. The interior flow is vertically sheared and in thermal wind balance. Whereas the original model of Chapman and Lentz considered an alongshore flow that is freely evolving, the present note focuses on the equilibrium structure of a flow driven by stress and discusses the vertical and lateral structure of the flow and, in particular, the boundary layer thickness. The vertical diffusivity of density in the bottom boundary layer is considered so strong, locally, as to render the bottom boundary layerâ€™s density a function of only offshore position. Boundary layer budgets of mass, momentum, and buoyancy determine the barotropic component of the interior flow as well as the boundary layer thickness, which is a function of the offshore coordinate. The alongshore flow has enhanced vertical shear in the boundary layer that reduces the alongshore flow in the boundary layer; however, the velocity at the bottom is generally not zero but produces a stress that locally balances the applied surface stress. The offshore transport in the bottom boundary layer therefore balances the onshore surface Ekman flux. The model predicts the thickness of the bottom boundary layer, which is a complicated function of several parameters, including the strength of the forcing stress, the vertical and horizontal diffusion coefficients in the interior, and the horizontal diffusion in the boundary layer. The model yields a boundary layer over only a finite portion of the bottom slope if the interior diffusion coefficients are too large; otherwise, the layer extends over the full lateral extent of the domain.

## Abstract

The bottom boundary layer of a stratified flow on a coastal continental shelf is examined using the model of Chapman and Lentz. The flow is driven by a surface stress, uniform in the alongshore coordinate, in a downwelling-favorable direction. The stress diminishes in the offshore direction and produces an Ekman pumping, as well as an onshore Ekman flux. The model yields an interior flow, sandwiched between an upper Ekman layer and a bottom boundary layer. The interior has a horizontal density gradient produced by a balance between horizontal diffusion of density and vertical advection of a background vertical density gradient. The interior flow is vertically sheared and in thermal wind balance. Whereas the original model of Chapman and Lentz considered an alongshore flow that is freely evolving, the present note focuses on the equilibrium structure of a flow driven by stress and discusses the vertical and lateral structure of the flow and, in particular, the boundary layer thickness. The vertical diffusivity of density in the bottom boundary layer is considered so strong, locally, as to render the bottom boundary layerâ€™s density a function of only offshore position. Boundary layer budgets of mass, momentum, and buoyancy determine the barotropic component of the interior flow as well as the boundary layer thickness, which is a function of the offshore coordinate. The alongshore flow has enhanced vertical shear in the boundary layer that reduces the alongshore flow in the boundary layer; however, the velocity at the bottom is generally not zero but produces a stress that locally balances the applied surface stress. The offshore transport in the bottom boundary layer therefore balances the onshore surface Ekman flux. The model predicts the thickness of the bottom boundary layer, which is a complicated function of several parameters, including the strength of the forcing stress, the vertical and horizontal diffusion coefficients in the interior, and the horizontal diffusion in the boundary layer. The model yields a boundary layer over only a finite portion of the bottom slope if the interior diffusion coefficients are too large; otherwise, the layer extends over the full lateral extent of the domain.

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## Abstract

The time-dependent response of an ocean basin to the imposition of cooling (or heating) is examined in the context of a quasigeostrophic, two-layer model on the beta plane. The focus is on the structure and magnitude of the vertical motion and its response to both a switch-on forcing and a periodic forcing. The model employed is a time-dependent version of an earlier model used to discuss the intensification of sinking in the region of the western boundary current. The height of the interface of the two-layer model serves as an analog of temperature, and the vertical velocity at the interface consists of a cross-isopycnal velocity modeled in terms of a relaxation to a prescribed interface height, an adiabatic representation of eddy thickness fluxes parameterized as lateral diffusion of thickness, and the local vertical motion of the interface itself. The presence of time dependence adds additional dynamical features to the problem, in particular the emergence of low-frequency, weakly damped Rossby basin modes. If the buoyancy forcing is zonally uniform the basin responds to a switch-on of the forcing by coming into steady-state equilibrium after the passage of a single baroclinic Rossby wave. If the forcing is nonuniform in the zonal direction, a sequence of Rossby basin modes is excited and their decay is required before the basin achieves a steady state. For reasonable parameter values the boundary layers, in which both horizontal and vertical circulations are closed, are quasi-steady and respond to the instantaneous state of the interior. As in the steady problem the flow is sensitive to small nonquasigeostrophic mass fluxes across the perimeter of the basin. These fluxes generally excite basin modes as well. The basin modes will also be weakly excited if the beta-plane approximation is relaxed. The response to periodic forcing is also examined, and the sensitivity of the response to the structure of the forcing is similar to the switch-on problem.

## Abstract

The time-dependent response of an ocean basin to the imposition of cooling (or heating) is examined in the context of a quasigeostrophic, two-layer model on the beta plane. The focus is on the structure and magnitude of the vertical motion and its response to both a switch-on forcing and a periodic forcing. The model employed is a time-dependent version of an earlier model used to discuss the intensification of sinking in the region of the western boundary current. The height of the interface of the two-layer model serves as an analog of temperature, and the vertical velocity at the interface consists of a cross-isopycnal velocity modeled in terms of a relaxation to a prescribed interface height, an adiabatic representation of eddy thickness fluxes parameterized as lateral diffusion of thickness, and the local vertical motion of the interface itself. The presence of time dependence adds additional dynamical features to the problem, in particular the emergence of low-frequency, weakly damped Rossby basin modes. If the buoyancy forcing is zonally uniform the basin responds to a switch-on of the forcing by coming into steady-state equilibrium after the passage of a single baroclinic Rossby wave. If the forcing is nonuniform in the zonal direction, a sequence of Rossby basin modes is excited and their decay is required before the basin achieves a steady state. For reasonable parameter values the boundary layers, in which both horizontal and vertical circulations are closed, are quasi-steady and respond to the instantaneous state of the interior. As in the steady problem the flow is sensitive to small nonquasigeostrophic mass fluxes across the perimeter of the basin. These fluxes generally excite basin modes as well. The basin modes will also be weakly excited if the beta-plane approximation is relaxed. The response to periodic forcing is also examined, and the sensitivity of the response to the structure of the forcing is similar to the switch-on problem.

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## Abstract

The stability of the marginally stable baroclinic wave in a uniform zonal shear flow is studied. Although thesingle, marginally stable wave on a horizontally uniform baroclinic current has little or no cross-streamvariation, the present paper shows that such a baroclinic wave is itself unstable to waves whose meridionalscale is of the order of the Rossby deformation radius.

The implications of this result for oceanic mesoscale dynamics is discussed since the instability, whichoccurs in the form of a triad resonance, may provide an amplitude limiting mechanism ignored in single-wave stability analyses.

## Abstract

The stability of the marginally stable baroclinic wave in a uniform zonal shear flow is studied. Although thesingle, marginally stable wave on a horizontally uniform baroclinic current has little or no cross-streamvariation, the present paper shows that such a baroclinic wave is itself unstable to waves whose meridionalscale is of the order of the Rossby deformation radius.

The implications of this result for oceanic mesoscale dynamics is discussed since the instability, whichoccurs in the form of a triad resonance, may provide an amplitude limiting mechanism ignored in single-wave stability analyses.

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## Abstract

The finite-amplitude dynamics of a resonant triad of baroclinic waves in a slightly unstable barocliniccurrent is examined. The wave in the triad which is linearly unstable on the current is assumed to havenegligible cross-stream variation and therefore lacks a self-equilibrating mechanism. It is shown, however,that the triad cooperatively equilibrates although for certain initial conditions the amplitude of the wavesmay become quite large. The triad possesses the deformation radius as its characteristic scale in bothhorizontal directions. It is suggested, therefore, that this triad instability may be a mechanism for theproduction of intense mesoscale oceanic eddies. However, the dependence of the result on the choice ofinitial conditions does not make it possible to definitely scale the amplitudes with fixed and external oceanicparameters.

## Abstract

The finite-amplitude dynamics of a resonant triad of baroclinic waves in a slightly unstable barocliniccurrent is examined. The wave in the triad which is linearly unstable on the current is assumed to havenegligible cross-stream variation and therefore lacks a self-equilibrating mechanism. It is shown, however,that the triad cooperatively equilibrates although for certain initial conditions the amplitude of the wavesmay become quite large. The triad possesses the deformation radius as its characteristic scale in bothhorizontal directions. It is suggested, therefore, that this triad instability may be a mechanism for theproduction of intense mesoscale oceanic eddies. However, the dependence of the result on the choice ofinitial conditions does not make it possible to definitely scale the amplitudes with fixed and external oceanicparameters.