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## 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

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

Using linear wave theory, the rate at which energy is converted into internal gravity waves by the interaction of the barotropic tide with topography in an ocean is calculated. Bell's formula for the conversion rate is extended to the case of an ocean of finite depth *H* with weak two-dimensional topography *h*(*x,*
*y*) and arbitrary buoyancy frequency *N*(*z*). Approximate solutions are computed using the WKB method, which reduce to the previous result for an ocean of infinite depth with constant stratification. The conversion rate for a finite-depth ocean can be substantially smaller than the infinite-ocean prediction when the length scale of the topography is of the same order as the horizontal wavelength of the internal tide. The conversion rate for two-dimensional Gaussian seamounts is calculated. Using observed statistics for the distribution of seamounts, the authors estimate 1/4 GW of conversion for a square of ocean floor of side 1000 km.

## Abstract

Using linear wave theory, the rate at which energy is converted into internal gravity waves by the interaction of the barotropic tide with topography in an ocean is calculated. Bell's formula for the conversion rate is extended to the case of an ocean of finite depth *H* with weak two-dimensional topography *h*(*x,*
*y*) and arbitrary buoyancy frequency *N*(*z*). Approximate solutions are computed using the WKB method, which reduce to the previous result for an ocean of infinite depth with constant stratification. The conversion rate for a finite-depth ocean can be substantially smaller than the infinite-ocean prediction when the length scale of the topography is of the same order as the horizontal wavelength of the internal tide. The conversion rate for two-dimensional Gaussian seamounts is calculated. Using observed statistics for the distribution of seamounts, the authors estimate 1/4 GW of conversion for a square of ocean floor of side 1000 km.

## Abstract

No abstract available.

## Abstract

No abstract available.

## Abstract

The authors study the stability of a barotropic sinusoidal meridional flow on a *β* plane. Because of bottom drag and lateral viscosity, the system is dissipative and forcing maintains a basic-state velocity that carries fluid across the planetary vorticity contours; this is a simple model of forced potential vorticity mixing. When the Reynolds number is slightly above the stability threshold, a perturbation expansion can be used to obtain an amplitude equation for the most unstable disturbances. These instabilities are zonal flows with a much larger length scale than that of the basic state.

Numerical and analytic considerations show that random initial perturbations rapidly reorganize into a set of fast and narrow eastward jets separated by slower and broader regions of westward flow. There then follows a much slower adjustment of the jets, involving gradual meridional migration and merger. Because of the existence of a Lyapunov functional for the dynamics, this one-dimensional inverse cascade ultimately settles into a steady solution.

For a fixed *β,* the meridional separation of the eastward jets depends on the bottom drag. When the bottom drag is zero, the process of jet merger proceeds very slowly to completion until only one jet is left in the domain. For small bottom drag, the steady-state meridional separation of the jets varies as (bottom drag)^{−1/3}. Varying the nondimensional *β* parameter can change the instability from supercritical (when *β* is small) to subcritical (when *β* is larger). Thus, the system has a rich phenomenology involving multiple stable solutions, hysteritic transitions, and so on.

## Abstract

The authors study the stability of a barotropic sinusoidal meridional flow on a *β* plane. Because of bottom drag and lateral viscosity, the system is dissipative and forcing maintains a basic-state velocity that carries fluid across the planetary vorticity contours; this is a simple model of forced potential vorticity mixing. When the Reynolds number is slightly above the stability threshold, a perturbation expansion can be used to obtain an amplitude equation for the most unstable disturbances. These instabilities are zonal flows with a much larger length scale than that of the basic state.

Numerical and analytic considerations show that random initial perturbations rapidly reorganize into a set of fast and narrow eastward jets separated by slower and broader regions of westward flow. There then follows a much slower adjustment of the jets, involving gradual meridional migration and merger. Because of the existence of a Lyapunov functional for the dynamics, this one-dimensional inverse cascade ultimately settles into a steady solution.

For a fixed *β,* the meridional separation of the eastward jets depends on the bottom drag. When the bottom drag is zero, the process of jet merger proceeds very slowly to completion until only one jet is left in the domain. For small bottom drag, the steady-state meridional separation of the jets varies as (bottom drag)^{−1/3}. Varying the nondimensional *β* parameter can change the instability from supercritical (when *β* is small) to subcritical (when *β* is larger). Thus, the system has a rich phenomenology involving multiple stable solutions, hysteritic transitions, and so on.

## Abstract

A vertically sheared eastward jet in the equatorial Pacific in late 1991 and early 1992 carried relatively fresh water from the western Pacific overriding the saltier surface layer of the central region. Salinity anomalies of about −1.0 psu were observed over a period of several months in a surface layer 50 m thick near the equator. Below this fresh layer them was a steep halocline having very little temperature stratification, so that the density changes were dominated by salinity. In December 1991, eastward surface velocities in the fresh jet at 170°W were 100 cm s^{−1} with a shear of about 40 cm s^{−1} in the top 100 m; the core of the jet was about 200 km in width, centered at 1.5°S. The jet decayed and vanished over the next few months, though the surface halocline remained.

A simple extension of the familiar 1½-layer model can account for the initial development of the sheared eastward jet. The surface pressure gradient in this initial value problem, tending to accelerate the fluid eastward, diminishes with depth because there is a zonal salinity gradient in the initially mixed layer. The depth dependence of the pressure gradient causes the accelerating flow to be vertically sheared, resulting in a tilting over of the isohalines. The shear progressively unmixes the mixed layer. The vertically integrated part of this solution is the Yoshida jet. The depth-dependent part of the solution results from a local conversion of potential to kinetic energy as the tilting isohalines lower the center of gravity of the surface layer. For added realism, generalizations of the model include wind forcing and a meridional salinity gradient.

While not discounting the conventional explanation of westerly wind stress in driving the eastward jet, it is shown that the tilting/shearing mechanism can be comparable to wind stress and is important in the production of salinity barrier layers. Fresh equatorial jets may provide a key to a better understanding of the physics of tropical ocean circulation and air-sea interaction during El Niño.

## Abstract

A vertically sheared eastward jet in the equatorial Pacific in late 1991 and early 1992 carried relatively fresh water from the western Pacific overriding the saltier surface layer of the central region. Salinity anomalies of about −1.0 psu were observed over a period of several months in a surface layer 50 m thick near the equator. Below this fresh layer them was a steep halocline having very little temperature stratification, so that the density changes were dominated by salinity. In December 1991, eastward surface velocities in the fresh jet at 170°W were 100 cm s^{−1} with a shear of about 40 cm s^{−1} in the top 100 m; the core of the jet was about 200 km in width, centered at 1.5°S. The jet decayed and vanished over the next few months, though the surface halocline remained.

A simple extension of the familiar 1½-layer model can account for the initial development of the sheared eastward jet. The surface pressure gradient in this initial value problem, tending to accelerate the fluid eastward, diminishes with depth because there is a zonal salinity gradient in the initially mixed layer. The depth dependence of the pressure gradient causes the accelerating flow to be vertically sheared, resulting in a tilting over of the isohalines. The shear progressively unmixes the mixed layer. The vertically integrated part of this solution is the Yoshida jet. The depth-dependent part of the solution results from a local conversion of potential to kinetic energy as the tilting isohalines lower the center of gravity of the surface layer. For added realism, generalizations of the model include wind forcing and a meridional salinity gradient.

While not discounting the conventional explanation of westerly wind stress in driving the eastward jet, it is shown that the tilting/shearing mechanism can be comparable to wind stress and is important in the production of salinity barrier layers. Fresh equatorial jets may provide a key to a better understanding of the physics of tropical ocean circulation and air-sea interaction during El Niño.

## 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.

## Abstract

Two models of advection-diffusion in the oscillatory, sheared-velocity field of an internal wave are discussed. Our goal is to develop intuition about the role of such currents in horizontal ocean mixing through the mechanism of shear dispersion. The analysis suggests simple parameterizations of this process, i.e., those in Eqs. (7), (36) and (42). The enhanced horizontal diffusion due to the interaction of the vertical diffusion and vertical shear of the wave field can be described by an “effective horizontal diffusivity” which is equal to the actual horizontal diffusivity plus a term equal to the mean-square vertical shear of horizontal displacement times the vertical diffusivity, provided the vertical length scale of the horizontal velocity field is not too small. In the limit of small vertical length scale the expression reduces to Taylor's (1953) result in which the effective horizontal diffusivity is inversely proportional to the actual vertical diffusivity.

The solutions also incidentally illuminate a variety of other advection-diffusion problems, such as unsteady shear dispersion in a pipe and enhanced diffusion through wavenumber cascade induced by steady shearing and straining velocity fields.

These solutions also serve as models of horizontal stirring by mesoscale eddies. Simple estimates of mesoscale shears and strains, together with estimates of the horizontal diffusivity due to shear dispersion by the internal wave field, suggest that horizontal mesoscale stirring begins to dominate internal-wave-shear dispersion at horizontal scales larger than 100 m.

## Abstract

Two models of advection-diffusion in the oscillatory, sheared-velocity field of an internal wave are discussed. Our goal is to develop intuition about the role of such currents in horizontal ocean mixing through the mechanism of shear dispersion. The analysis suggests simple parameterizations of this process, i.e., those in Eqs. (7), (36) and (42). The enhanced horizontal diffusion due to the interaction of the vertical diffusion and vertical shear of the wave field can be described by an “effective horizontal diffusivity” which is equal to the actual horizontal diffusivity plus a term equal to the mean-square vertical shear of horizontal displacement times the vertical diffusivity, provided the vertical length scale of the horizontal velocity field is not too small. In the limit of small vertical length scale the expression reduces to Taylor's (1953) result in which the effective horizontal diffusivity is inversely proportional to the actual vertical diffusivity.

The solutions also incidentally illuminate a variety of other advection-diffusion problems, such as unsteady shear dispersion in a pipe and enhanced diffusion through wavenumber cascade induced by steady shearing and straining velocity fields.

These solutions also serve as models of horizontal stirring by mesoscale eddies. Simple estimates of mesoscale shears and strains, together with estimates of the horizontal diffusivity due to shear dispersion by the internal wave field, suggest that horizontal mesoscale stirring begins to dominate internal-wave-shear dispersion at horizontal scales larger than 100 m.