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## Abstract

It is argued that the ideal fluid thermocline equations have “weak” (i.e., nondifferentiable) solutions that satisfy no mass-flux boundary conditions at the East. This conclusion is based on a local analysis of the eastern “corner” of a subtropical gyre. Specifically we suppose that the surface density is uniform while the density on the eastern boundary is either uniform (but different from that of the surface) or else is linearly stratified. The surface density is injected into the interior by specified Ekman pumping. In the absence f dissipation the resulting solution would have a discontinuity is “smoothed” and becomes an internal boundary layer which separates the light fluid originating at the surface from the denser fluid which abuts the eastern boundary.

This solution, which is of the similarity type, illustrates the applicability of solutions of the ideal fluid thermocline problem with discontinuities. It is these discontinuities which enable ideal fluid solutions to satisfy eastern boundary conditions. Thus, contrary to statements in the literature, there is no a priori need for an eastern boundary layer which exchanges mass with an ideal interior.

This concept of a weak solution is implicit in recent theories of the large-scale oceanic circulation. For example, in the continuously stratified, quasigeostrophic model developed by Rhines and Young, the solution is singular at the boundary between the moving pool of homogenized potential vorticity and the motionless shadow region. Analogous surfaces of discontinuity enable the models discussed in previous studies to satisfy eastern boundary conditions. The present study makes this assumption more explicit and shows how one particular dissipative mechanism (vertical density diffusion) heals the singularity.

## Abstract

It is argued that the ideal fluid thermocline equations have “weak” (i.e., nondifferentiable) solutions that satisfy no mass-flux boundary conditions at the East. This conclusion is based on a local analysis of the eastern “corner” of a subtropical gyre. Specifically we suppose that the surface density is uniform while the density on the eastern boundary is either uniform (but different from that of the surface) or else is linearly stratified. The surface density is injected into the interior by specified Ekman pumping. In the absence f dissipation the resulting solution would have a discontinuity is “smoothed” and becomes an internal boundary layer which separates the light fluid originating at the surface from the denser fluid which abuts the eastern boundary.

This solution, which is of the similarity type, illustrates the applicability of solutions of the ideal fluid thermocline problem with discontinuities. It is these discontinuities which enable ideal fluid solutions to satisfy eastern boundary conditions. Thus, contrary to statements in the literature, there is no a priori need for an eastern boundary layer which exchanges mass with an ideal interior.

This concept of a weak solution is implicit in recent theories of the large-scale oceanic circulation. For example, in the continuously stratified, quasigeostrophic model developed by Rhines and Young, the solution is singular at the boundary between the moving pool of homogenized potential vorticity and the motionless shadow region. Analogous surfaces of discontinuity enable the models discussed in previous studies to satisfy eastern boundary conditions. The present study makes this assumption more explicit and shows how one particular dissipative mechanism (vertical density diffusion) heals the singularity.

## Abstract

The stability of the western boundary layer is studied by idealizing it as a parallel flow and solving the Orr–Sommerfeld equation, generalized to include the gradient of planetary vorticity. The critical Reynolds number, at which the idealized flow first becomes unstable, is found to be between 20 and 100 depending on the details of the profile. The modes themselves are trapped within the boundary jet because their phase speeds exceed that of the fastest free Rossby wave with the same meridional wavenumber. However, in the important case of a jet with a broad exponential decay, corresponding to a highly inertial flow, we find that the phase speed of the critical mode exceeds that of a free Rossby wave by a very small amount. Consequently, the trapped mode has a very slowly decaying oscillatory tail and so is much wider than the basic state that supports it. The Reynolds stresses in the tail region induce a mean Eulerian flow opposite in direction to the basic state jet. However, Stokes drift is substantial so that the mean Lagrangian flow is an order of magnitude smaller than the mean Eulerian.

## Abstract

The stability of the western boundary layer is studied by idealizing it as a parallel flow and solving the Orr–Sommerfeld equation, generalized to include the gradient of planetary vorticity. The critical Reynolds number, at which the idealized flow first becomes unstable, is found to be between 20 and 100 depending on the details of the profile. The modes themselves are trapped within the boundary jet because their phase speeds exceed that of the fastest free Rossby wave with the same meridional wavenumber. However, in the important case of a jet with a broad exponential decay, corresponding to a highly inertial flow, we find that the phase speed of the critical mode exceeds that of a free Rossby wave by a very small amount. Consequently, the trapped mode has a very slowly decaying oscillatory tail and so is much wider than the basic state that supports it. The Reynolds stresses in the tail region induce a mean Eulerian flow opposite in direction to the basic state jet. However, Stokes drift is substantial so that the mean Lagrangian flow is an order of magnitude smaller than the mean Eulerian.

## Abstract

A sub-basin scale recirculation can be driven by imposing low values of potential vorticity in the northwest corner of a β-Plane box. Mesoscale eddies parametrized by lateral potential vorticity diffusion, carry this anomaly into the interior and establish the mean flow. While the structure of the flow is not sensitive to details of the boundary forcing or to the size of the diffusion coefficient, κ, the amplitude and length scale are. For instance, as, κ is reduced, the maximum transport scales as κ^{½} and the Reynolds number as κ^{−½}.

## Abstract

A sub-basin scale recirculation can be driven by imposing low values of potential vorticity in the northwest corner of a β-Plane box. Mesoscale eddies parametrized by lateral potential vorticity diffusion, carry this anomaly into the interior and establish the mean flow. While the structure of the flow is not sensitive to details of the boundary forcing or to the size of the diffusion coefficient, κ, the amplitude and length scale are. For instance, as, κ is reduced, the maximum transport scales as κ^{½} and the Reynolds number as κ^{−½}.

## Abstract

The question posed in the title of this paper is answered in the affirmative by investigating a two-layer, quasi-geostrophic model of the wind-driven circulation. The two layers model the thermocline rather than the whole depth of the ocean. The wind stress is balanced by interfacial and bottom drag. This is perhaps the simplest baroclinic extension of Stommel's (1948) barotropic circulation model. It differs from an earlier model of Welander (1966) in that the vortex stretching nonlinearity is of primary importance.

In this model the dynamics of the frictional western boundary layer determine the vertical structure of the wind-driven flow in the Sverdrup interior. Thus, in a sense, the boundary layer is “active” and cannot be appended to an arbitrary interior flow; rather it partially determines the interior circulation by setting the functional relationship between the streamfunction and the potential vorticity in the lower layer.

In previous studies (Rhines and Young 1982b) this functional relationship has been calculated using a generalized Prandtl-Batchelor theorem. This result does not apply to the present calculation because every lower layer streamline passes through a frictional boundary layer.

## Abstract

The question posed in the title of this paper is answered in the affirmative by investigating a two-layer, quasi-geostrophic model of the wind-driven circulation. The two layers model the thermocline rather than the whole depth of the ocean. The wind stress is balanced by interfacial and bottom drag. This is perhaps the simplest baroclinic extension of Stommel's (1948) barotropic circulation model. It differs from an earlier model of Welander (1966) in that the vortex stretching nonlinearity is of primary importance.

In this model the dynamics of the frictional western boundary layer determine the vertical structure of the wind-driven flow in the Sverdrup interior. Thus, in a sense, the boundary layer is “active” and cannot be appended to an arbitrary interior flow; rather it partially determines the interior circulation by setting the functional relationship between the streamfunction and the potential vorticity in the lower layer.

In previous studies (Rhines and Young 1982b) this functional relationship has been calculated using a generalized Prandtl-Batchelor theorem. This result does not apply to the present calculation because every lower layer streamline passes through a frictional boundary layer.

## Abstract

Analytical estimates of the rate at which energy is extracted from the barotropic tide at topography and converted into internal gravity waves are given. The ocean is idealized as an inviscid, vertically unbounded fluid on the *f* plane. The gravity waves are treated by linear theory and freely escape to *z* = ∞. Several topographies are investigated: a sinusoidal ripple, a set of Gaussian bumps, and an ensemble of “random topographies.” In the third case, topographic profiles are generated by randomly selecting the amplitudes of a Fourier superposition so that the power spectral density is similar to that of submarine topography. The authors' focus is the dependence of the conversion rate (watts per square meter of radiated power) on the amplitude of the topography, *h*
_{0}, and on a nondimensional parameter *ϵ*∗, defined as the ratio of the slope of an internal tidal ray to the maximum slope of the topography. If *ϵ*∗ ≪ 1, then Bell's theory indicates that the conversion is proportional to *h*^{2}_{0}*ϵ*∗ < 1 and show that the enhancement above Bell's prediction is a smoothly and modestly increasing function of *ϵ*∗: For *ϵ*∗ → 1, the conversion of sinusoidal topography is 56% greater than Bell's *ϵ*∗ ≪ 1 estimate, while the enhancement is only 14% greater for a Gaussian bump. With random topography, the enhancement at *ϵ*∗ = 0.95 is typically about 6% greater than Bell's formula. The *ϵ*∗ ≪ 1 approximation is therefore quantitatively accurate over the range 0 < *ϵ*∗ < 1, implying that the conversion is roughly proportional to *h*^{2}_{0}*ϵ*∗ is increased, the radiated waves develop very small spatial scales that are not present in the underlying topography and, when *ϵ*∗ approaches unity, the associated spatial gradients become so steep that overturns must occur even if the tidal amplitude is very weak. The solutions formally become singular at *ϵ*∗ = 1, in a breakdown of linear, inviscid theory.

## Abstract

Analytical estimates of the rate at which energy is extracted from the barotropic tide at topography and converted into internal gravity waves are given. The ocean is idealized as an inviscid, vertically unbounded fluid on the *f* plane. The gravity waves are treated by linear theory and freely escape to *z* = ∞. Several topographies are investigated: a sinusoidal ripple, a set of Gaussian bumps, and an ensemble of “random topographies.” In the third case, topographic profiles are generated by randomly selecting the amplitudes of a Fourier superposition so that the power spectral density is similar to that of submarine topography. The authors' focus is the dependence of the conversion rate (watts per square meter of radiated power) on the amplitude of the topography, *h*
_{0}, and on a nondimensional parameter *ϵ*∗, defined as the ratio of the slope of an internal tidal ray to the maximum slope of the topography. If *ϵ*∗ ≪ 1, then Bell's theory indicates that the conversion is proportional to *h*^{2}_{0}*ϵ*∗ < 1 and show that the enhancement above Bell's prediction is a smoothly and modestly increasing function of *ϵ*∗: For *ϵ*∗ → 1, the conversion of sinusoidal topography is 56% greater than Bell's *ϵ*∗ ≪ 1 estimate, while the enhancement is only 14% greater for a Gaussian bump. With random topography, the enhancement at *ϵ*∗ = 0.95 is typically about 6% greater than Bell's formula. The *ϵ*∗ ≪ 1 approximation is therefore quantitatively accurate over the range 0 < *ϵ*∗ < 1, implying that the conversion is roughly proportional to *h*^{2}_{0}*ϵ*∗ is increased, the radiated waves develop very small spatial scales that are not present in the underlying topography and, when *ϵ*∗ approaches unity, the associated spatial gradients become so steep that overturns must occur even if the tidal amplitude is very weak. The solutions formally become singular at *ϵ*∗ = 1, in a breakdown of linear, inviscid theory.