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## Abstract

Two aspects of the effects of eddies on ocean circulation have proven difficult to parameterize: eddy effects in regions of neutrally stable (or convecting) fluid and the mixing of passive tracers. The effects of linearized eddies, although a restrictive parameter regime, can be straightforwardly computed in these cases. The eddy effects in areas of neutral stability—for example, mixed layers—blend naturally into those in the stably stratified water below, although losing the concept of bolus velocity. Instead, the mixed layer density is advected by an extra overturning velocity and is diffused laterally by a diffusion that is the same as the eddy diffusion at the top of the stably stratified fluid. Passive tracers are advected by the bolus velocity and mixed by the same diffusivity as is used to compute the bolus velocity at that location, so that two different diffusivities are not needed.

## Abstract

Two aspects of the effects of eddies on ocean circulation have proven difficult to parameterize: eddy effects in regions of neutrally stable (or convecting) fluid and the mixing of passive tracers. The effects of linearized eddies, although a restrictive parameter regime, can be straightforwardly computed in these cases. The eddy effects in areas of neutral stability—for example, mixed layers—blend naturally into those in the stably stratified water below, although losing the concept of bolus velocity. Instead, the mixed layer density is advected by an extra overturning velocity and is diffused laterally by a diffusion that is the same as the eddy diffusion at the top of the stably stratified fluid. Passive tracers are advected by the bolus velocity and mixed by the same diffusivity as is used to compute the bolus velocity at that location, so that two different diffusivities are not needed.

## Abstract

This paper considers the interaction between a bottom-trapped low-frequency, reduced-gravity maid Kelvin wave propagating along a coastal wall, and a smooth ridge extending away from the coastline. Although the full problem appears intractable, it is shown that simple bounds may be placed on the amplitude of the Kelvin wave after it has passed the region of topography. The upper bound is found to be a good estimate for cases examined here. For a ridge of width one or two deformation radii, the reduction in amplitude of the Kelvin wave, induced by scattering along the ridge, is roughly equal to the fractional depth remaining in the undisturbed fluid layer at the highest point of the ridge; the reduction in energy is of course given by the square of this quantity. The bounds are found by considering the (approximate) conservation of mass between the incoming and transmitted Kelvin waves and the range of topographic waves on the rider, and also the (exact) conservation of energy. The full, and very complicated, interaction problem near the intersection of the ridge and the coastal wall does not need to be solved.

The effects of changing width of the topography are examined. Narrow ridges (with widths much ten than a deformation radius) permit Kelvin waves to pass them without loss of amplitude in the limit of vanishing width. Broad ridges (with widths much larger than a deformation radius) can have two effects depending on size of the frequency. When the frequency is small enough so that a small but finite number of topographic modes are possible, there is no loss in amplitude of the Kelvin wave. For smaller frequencies, where there are many topographic modes possible, a finite amount of amplitude is lost to topographic waves. Thus ridges of width of order a deformation radius or wider are the most efficient scatterers of coastal wave energy; four successive ridges of height one-half that of the resting depth of the layer would reduce the transmitted wave energy to less than 2% of its initial value.

Poor numerical resolution can strongly overestimate the transmitted wave amplitude. Since present general circulation models cannot resolve all the various modes discussed here, this overestimation must occur, and may be quite drastic. Additionally, the effects of mild numerical damping are discussed, and compared with the ideal fluid case. When the damping is Laplacian, the short topographic waves are damped, with two results: the flow field resembles that for a step topography, and the transmitted wave amplitude is very strongly over-estimated, despite the diffusion.

## Abstract

This paper considers the interaction between a bottom-trapped low-frequency, reduced-gravity maid Kelvin wave propagating along a coastal wall, and a smooth ridge extending away from the coastline. Although the full problem appears intractable, it is shown that simple bounds may be placed on the amplitude of the Kelvin wave after it has passed the region of topography. The upper bound is found to be a good estimate for cases examined here. For a ridge of width one or two deformation radii, the reduction in amplitude of the Kelvin wave, induced by scattering along the ridge, is roughly equal to the fractional depth remaining in the undisturbed fluid layer at the highest point of the ridge; the reduction in energy is of course given by the square of this quantity. The bounds are found by considering the (approximate) conservation of mass between the incoming and transmitted Kelvin waves and the range of topographic waves on the rider, and also the (exact) conservation of energy. The full, and very complicated, interaction problem near the intersection of the ridge and the coastal wall does not need to be solved.

The effects of changing width of the topography are examined. Narrow ridges (with widths much ten than a deformation radius) permit Kelvin waves to pass them without loss of amplitude in the limit of vanishing width. Broad ridges (with widths much larger than a deformation radius) can have two effects depending on size of the frequency. When the frequency is small enough so that a small but finite number of topographic modes are possible, there is no loss in amplitude of the Kelvin wave. For smaller frequencies, where there are many topographic modes possible, a finite amount of amplitude is lost to topographic waves. Thus ridges of width of order a deformation radius or wider are the most efficient scatterers of coastal wave energy; four successive ridges of height one-half that of the resting depth of the layer would reduce the transmitted wave energy to less than 2% of its initial value.

Poor numerical resolution can strongly overestimate the transmitted wave amplitude. Since present general circulation models cannot resolve all the various modes discussed here, this overestimation must occur, and may be quite drastic. Additionally, the effects of mild numerical damping are discussed, and compared with the ideal fluid case. When the damping is Laplacian, the short topographic waves are damped, with two results: the flow field resembles that for a step topography, and the transmitted wave amplitude is very strongly over-estimated, despite the diffusion.

## Abstract

This paper examines the implications for eddy parameterizations of expressing them in terms of the quasi-Stokes velocity. Another definition of low-passed time-averaged mean density (the modified mean) must be used, which is the inversion of the mean depth of a given isopycnal. This definition naturally yields lighter (denser) fluid at the surface (floor) than the Eulerian mean since fluid with these densities occasionally occurs at these locations. The difference between the two means is second order in perturbation amplitude, and so small, in the fluid interior (where formulas to connect the two exist). Near horizontal boundaries, the differences become first order, and so more severe. Existing formulas for quasi-Stokes velocities and streamfunction also break down here. It is shown that the low-passed time-mean potential energy in a closed box is incorrectly computed from modified mean density, the error term involving averaged quadratic variability.

The layer in which the largest differences occur between the two mean densities is the vertical excursion of a mean isopycnal across a deformation radius, at most about 20 m thick. Most climate models would have difficulty in resolving such a layer. It is shown here that extant parameterizations appear to reproduce the Eulerian, and not modified mean, density field and so do not yield a narrow layer at surface and floor either. Both these features make the quasi-Stokes streamfunction appear to be nonzero right up to rigid boundaries. It is thus unclear whether more accurate results would be obtained by leaving the streamfunction nonzero on the boundary—which is smooth and resolvable—or by permitting a delta function in the horizontal quasi-Stokes velocity by forcing the streamfunction to become zero exactly at the boundary (which it formally must be), but at the cost of small and unresolvable features in the solution.

This paper then uses linear stability theory and diagnosed values from eddy-resolving models, to ask the question:*if climate models cannot or do not resolve the difference between Eulerian and modified mean density, what are the relevant surface and floor quasi-Stokes streamfunction conditions and what are their effects on the density fields*?

The linear Eady problem is used as a special case to investigate this since terms can be explicitly computed. A variety of eddy parameterizations is employed for a channel problem, and the time-mean density is compared with that from an eddy-resolving calculation. Curiously, although most of the parameterizations employed are formally valid only in terms of the modified density, they all reproduce only the Eulerian mean density successfully. This is despite the existence of (numerical) delta functions near the surface. The parameterizations were only successful if the vertical component of the quasi-Stokes velocity was required to vanish at top and bottom. A simple parameterization of Eulerian density fluxes was, however, just as accurate and avoids delta-function behavior completely.

## Abstract

This paper examines the implications for eddy parameterizations of expressing them in terms of the quasi-Stokes velocity. Another definition of low-passed time-averaged mean density (the modified mean) must be used, which is the inversion of the mean depth of a given isopycnal. This definition naturally yields lighter (denser) fluid at the surface (floor) than the Eulerian mean since fluid with these densities occasionally occurs at these locations. The difference between the two means is second order in perturbation amplitude, and so small, in the fluid interior (where formulas to connect the two exist). Near horizontal boundaries, the differences become first order, and so more severe. Existing formulas for quasi-Stokes velocities and streamfunction also break down here. It is shown that the low-passed time-mean potential energy in a closed box is incorrectly computed from modified mean density, the error term involving averaged quadratic variability.

The layer in which the largest differences occur between the two mean densities is the vertical excursion of a mean isopycnal across a deformation radius, at most about 20 m thick. Most climate models would have difficulty in resolving such a layer. It is shown here that extant parameterizations appear to reproduce the Eulerian, and not modified mean, density field and so do not yield a narrow layer at surface and floor either. Both these features make the quasi-Stokes streamfunction appear to be nonzero right up to rigid boundaries. It is thus unclear whether more accurate results would be obtained by leaving the streamfunction nonzero on the boundary—which is smooth and resolvable—or by permitting a delta function in the horizontal quasi-Stokes velocity by forcing the streamfunction to become zero exactly at the boundary (which it formally must be), but at the cost of small and unresolvable features in the solution.

This paper then uses linear stability theory and diagnosed values from eddy-resolving models, to ask the question:*if climate models cannot or do not resolve the difference between Eulerian and modified mean density, what are the relevant surface and floor quasi-Stokes streamfunction conditions and what are their effects on the density fields*?

The linear Eady problem is used as a special case to investigate this since terms can be explicitly computed. A variety of eddy parameterizations is employed for a channel problem, and the time-mean density is compared with that from an eddy-resolving calculation. Curiously, although most of the parameterizations employed are formally valid only in terms of the modified density, they all reproduce only the Eulerian mean density successfully. This is despite the existence of (numerical) delta functions near the surface. The parameterizations were only successful if the vertical component of the quasi-Stokes velocity was required to vanish at top and bottom. A simple parameterization of Eulerian density fluxes was, however, just as accurate and avoids delta-function behavior completely.

## Abstract

No abstract available.

## Abstract

No abstract available.

## Abstract

Integra expressions are derived for the east-west velocity of propagation of isolated eddies on a beta plane. It is assumed that the eddies have no surface or floor expression, i.e., that both surface and floor are isopycnals. The results of Nof and Mory are generalized and demonstrate the crucial necessity for all such results that, on the hounding density surfaces, the linearized Bernoulli function depends only on the depth of that surface. Thus there are examples of isolated eddies satisfying the assumptions but which are not directly amenable to the analyses presented hitherto. Results for multiple layers (including a simple rule for the direction of propagation) and for continuously stratified eddies, subject to some assumptions, are given. A simple model fit to salt lenses observed by Armi and Zenk gives westward motion or order 1 cm s^{−1}, which is not unreasonable.

## Abstract

Integra expressions are derived for the east-west velocity of propagation of isolated eddies on a beta plane. It is assumed that the eddies have no surface or floor expression, i.e., that both surface and floor are isopycnals. The results of Nof and Mory are generalized and demonstrate the crucial necessity for all such results that, on the hounding density surfaces, the linearized Bernoulli function depends only on the depth of that surface. Thus there are examples of isolated eddies satisfying the assumptions but which are not directly amenable to the analyses presented hitherto. Results for multiple layers (including a simple rule for the direction of propagation) and for continuously stratified eddies, subject to some assumptions, are given. A simple model fit to salt lenses observed by Armi and Zenk gives westward motion or order 1 cm s^{−1}, which is not unreasonable.

## Abstract

Three exact, closed-form analytical solutions for the subtropical gyre are presented for the ideal fluid thermocline equations. Specifically, the flow is exactly geostrophic, hydrostatic, and mass and buoyancy conserving. Ekman pumping and density can be chosen as fairly arbitrary functions at the surface. No flow is permitted through the ocean's eastern boundary, or through its bottom. The solutions are continuous extensions of existing layered models. The first solution, discovered simultaneously with Janowitz's solution, uses a deep resting isopycnal layer; the surface density may only be a function of latitude for this solution. A second nonunique solution requires velocities to tend to zero at great depth, giving an additional degree of freedom which permits surface density to be specified almost arbitrarily. This second solution is unphysical in the sense that depth-integrated mass fluxes and energies are infinite. However, a small change in the solution (which returns surface density to a function of latitude only) permits solutions with finite fluxes once more. A third solution requires partial homogenization of the potential vorticity of fluid layers which, while overlying a deep resting iopycnal layer, are not directly ventilated from the surface. Again, fairly arbitrary surface density and Ekman pumping are permitted. All the problems reduce to linear homogeneous second-order differential equations when density replaces depth as the vertical coordinate. The importance of the bottom boundary for closing the problem is stressed.

## Abstract

Three exact, closed-form analytical solutions for the subtropical gyre are presented for the ideal fluid thermocline equations. Specifically, the flow is exactly geostrophic, hydrostatic, and mass and buoyancy conserving. Ekman pumping and density can be chosen as fairly arbitrary functions at the surface. No flow is permitted through the ocean's eastern boundary, or through its bottom. The solutions are continuous extensions of existing layered models. The first solution, discovered simultaneously with Janowitz's solution, uses a deep resting isopycnal layer; the surface density may only be a function of latitude for this solution. A second nonunique solution requires velocities to tend to zero at great depth, giving an additional degree of freedom which permits surface density to be specified almost arbitrarily. This second solution is unphysical in the sense that depth-integrated mass fluxes and energies are infinite. However, a small change in the solution (which returns surface density to a function of latitude only) permits solutions with finite fluxes once more. A third solution requires partial homogenization of the potential vorticity of fluid layers which, while overlying a deep resting iopycnal layer, are not directly ventilated from the surface. Again, fairly arbitrary surface density and Ekman pumping are permitted. All the problems reduce to linear homogeneous second-order differential equations when density replaces depth as the vertical coordinate. The importance of the bottom boundary for closing the problem is stressed.

## Abstract

The problem of matching the nonlinear, frictional flow in a simple western boundary layer to a specified interior flow is considered. Two problems are discussed, using streamfunction as a coordinate across the boundary layer. First, a unidirectional flow is considered. The dissipation is considered to be some positive quantity, and it is shown that for a simple form of this, many different amounts permit a smooth match to the interior. The magnitude of the dissipation can be determined absolutely at the dividing point between in- and outflow. The dissipation south of this point must be smaller and north of this point must be larger; a simple equation describes the relationship between dissipations north and south of the dividing point. Second, a bidirectional boundary layer is permitted. A specific form of dissipation (a linear drag) is applied, with a constant coefficient. It is shown that in this case it still remains possible to match to a specified interior flow, although inertial overshoot occurs both into the next gyre polewards as well as equatorwards into the inflow region, if the drag is small enough. Thus, taken together with published results on Laplacian dissipation, these simple models suggest that western boundary layers are passive and can match to a specified interior flow without modifying that flow in any way (although this may not be the case for very low friction).

## Abstract

The problem of matching the nonlinear, frictional flow in a simple western boundary layer to a specified interior flow is considered. Two problems are discussed, using streamfunction as a coordinate across the boundary layer. First, a unidirectional flow is considered. The dissipation is considered to be some positive quantity, and it is shown that for a simple form of this, many different amounts permit a smooth match to the interior. The magnitude of the dissipation can be determined absolutely at the dividing point between in- and outflow. The dissipation south of this point must be smaller and north of this point must be larger; a simple equation describes the relationship between dissipations north and south of the dividing point. Second, a bidirectional boundary layer is permitted. A specific form of dissipation (a linear drag) is applied, with a constant coefficient. It is shown that in this case it still remains possible to match to a specified interior flow, although inertial overshoot occurs both into the next gyre polewards as well as equatorwards into the inflow region, if the drag is small enough. Thus, taken together with published results on Laplacian dissipation, these simple models suggest that western boundary layers are passive and can match to a specified interior flow without modifying that flow in any way (although this may not be the case for very low friction).

## Abstract

This paper discusses three distinct features of rotating, stratified hydraulics, using a reduced-gravity configuration. First, a new upstream condition is derived corresponding to a wide, almost motionless basin, and this is applied to flow across a rectangular sill and compared with the case of a zero potential vorticity upstream condition. For this geometry, it is shown that unidirectional flow permits more water to pass through the sill than bidirectional flow. Second, the general problem is considered of flow from any upstream configuration that passes through sills that vary slowly in depth cross sill (and so are effectively many deformation radii wide). Only two flow configurations permit any realistic amount of flux across the sill: either the fluid occupies a narrow region within the sill, with a small flux, or the fluid occupies a wide region, with sluggish geostrophic flow except for at boundary layers at each side. In the latter case, hydraulic control is not likely to occur. The zero potential vorticity limit, suitably modified, gives an upper bound to the net flux across the sill. Both configurations require bidirectional flow for all upstream conditions, so that unidirectional flow can be expected to occur only in relatively narrow sills. The relevance of providing upstream conditions for hydraulic flow is thus called into question. Third, the flux through four oceanic sills is recomputed, modeling the sills as parabolic or V-shaped. It is noted that general circulation models will not give a good representation of the flux in such cases.

## Abstract

This paper discusses three distinct features of rotating, stratified hydraulics, using a reduced-gravity configuration. First, a new upstream condition is derived corresponding to a wide, almost motionless basin, and this is applied to flow across a rectangular sill and compared with the case of a zero potential vorticity upstream condition. For this geometry, it is shown that unidirectional flow permits more water to pass through the sill than bidirectional flow. Second, the general problem is considered of flow from any upstream configuration that passes through sills that vary slowly in depth cross sill (and so are effectively many deformation radii wide). Only two flow configurations permit any realistic amount of flux across the sill: either the fluid occupies a narrow region within the sill, with a small flux, or the fluid occupies a wide region, with sluggish geostrophic flow except for at boundary layers at each side. In the latter case, hydraulic control is not likely to occur. The zero potential vorticity limit, suitably modified, gives an upper bound to the net flux across the sill. Both configurations require bidirectional flow for all upstream conditions, so that unidirectional flow can be expected to occur only in relatively narrow sills. The relevance of providing upstream conditions for hydraulic flow is thus called into question. Third, the flux through four oceanic sills is recomputed, modeling the sills as parabolic or V-shaped. It is noted that general circulation models will not give a good representation of the flux in such cases.

## Abstract

The part of the meridional overturning circulation driven by time-varying winds is usually assumed to be an Ekman flux within a mixed layer, and a depth- and laterally independent return flow beneath. For a simple linear frictional ocean model, the return flow is studied for a range of frequencies from several days to decades. It is shown that while the east–west integral of the return flow is usually, but not always, almost independent of depth, the spatial distribution of the return flow varies strongly with both horizontal and vertical position. This can have important consequences for calculations of the northward heat flux, which traditionally assumes a spatially uniform return flow.

## Abstract

The part of the meridional overturning circulation driven by time-varying winds is usually assumed to be an Ekman flux within a mixed layer, and a depth- and laterally independent return flow beneath. For a simple linear frictional ocean model, the return flow is studied for a range of frequencies from several days to decades. It is shown that while the east–west integral of the return flow is usually, but not always, almost independent of depth, the spatial distribution of the return flow varies strongly with both horizontal and vertical position. This can have important consequences for calculations of the northward heat flux, which traditionally assumes a spatially uniform return flow.

## Abstract

A geostrophic adjustment model is used to find out how water can cross the equator, and how far it can reach, while conserving its potential vorticity, in the context of geostrophic adjustment. A series of problems is considered; all but the last permit variation north–south only. The first problem discusses the equatorial version of the classic midlatitude adjustment problem of a one-layer, reduced gravity fluid in the Southern Hemisphere which is suddenly permitted to slump away from its initially uniform height distribution. Fluid which crosses the equator reaches farther northward than it began south of the equator. The configuration in which fluid reaches the farthest north requires fluid starting as far south as is possible subject to water actually crossing the equator. Particles move north a distance of at most 2.32 deformation radii. This problem is then extended in turn to a one-layer fluid occupying all space, whose depth changes abruptly from one value to another, and to the linearized problem which is fully tractable analytically. A second layer, with a rigid lid, is also discussed. In common with many adjustment problems in which wave radiation to infinity is prohibited, although one may seek a steady final state, such a state is not achieved in these problems. However, wherever possible it is shown that the long-time average of the time-dependent problem is the steady state solution already found. An extension is then made to include east–west variation and the effect of side walls. It is found that the one-dimensional solutions describe the fluid behavior for much longer than would be anticipated. In these adjustment problems, cross-equatorial flow occurs in two ways. First, particles cross the equator a short distance as in the one-dimensional problem, and are then advected some way eastward. Second, particles cross the equator in the western boundary layer, where dissipation act to change the sign of the potential vorticity and so permits long northward migration.

## Abstract

A geostrophic adjustment model is used to find out how water can cross the equator, and how far it can reach, while conserving its potential vorticity, in the context of geostrophic adjustment. A series of problems is considered; all but the last permit variation north–south only. The first problem discusses the equatorial version of the classic midlatitude adjustment problem of a one-layer, reduced gravity fluid in the Southern Hemisphere which is suddenly permitted to slump away from its initially uniform height distribution. Fluid which crosses the equator reaches farther northward than it began south of the equator. The configuration in which fluid reaches the farthest north requires fluid starting as far south as is possible subject to water actually crossing the equator. Particles move north a distance of at most 2.32 deformation radii. This problem is then extended in turn to a one-layer fluid occupying all space, whose depth changes abruptly from one value to another, and to the linearized problem which is fully tractable analytically. A second layer, with a rigid lid, is also discussed. In common with many adjustment problems in which wave radiation to infinity is prohibited, although one may seek a steady final state, such a state is not achieved in these problems. However, wherever possible it is shown that the long-time average of the time-dependent problem is the steady state solution already found. An extension is then made to include east–west variation and the effect of side walls. It is found that the one-dimensional solutions describe the fluid behavior for much longer than would be anticipated. In these adjustment problems, cross-equatorial flow occurs in two ways. First, particles cross the equator a short distance as in the one-dimensional problem, and are then advected some way eastward. Second, particles cross the equator in the western boundary layer, where dissipation act to change the sign of the potential vorticity and so permits long northward migration.