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## Abstract

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## 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

Relaxation toward observed values is frequently undertaken in ocean models for numerous reasons. In level models, relaxation of some quantity takes the form of a linear “nudging” term proportional to the difference between observed and computed value of that quantity. In isopycnic models, relaxation of tracers and/or layer depth toward observed values is often employed as an equivalent. This note shows that relaxation of temperature and salinity—and hence density—in a level model is *not* equivalent to relaxation either of those tracers or of layer thickness in an isopycnic model. Comparison of layer thickness tendencies in the two model types shows that these differ by the ratio of observed vertical density gradient to model vertical density gradient. Only in the special case where the model remains close to observations are the two methods the same to leading order. It is not obvious whether isopycnic or level relaxation is to be preferred.

## Abstract

Relaxation toward observed values is frequently undertaken in ocean models for numerous reasons. In level models, relaxation of some quantity takes the form of a linear “nudging” term proportional to the difference between observed and computed value of that quantity. In isopycnic models, relaxation of tracers and/or layer depth toward observed values is often employed as an equivalent. This note shows that relaxation of temperature and salinity—and hence density—in a level model is *not* equivalent to relaxation either of those tracers or of layer thickness in an isopycnic model. Comparison of layer thickness tendencies in the two model types shows that these differ by the ratio of observed vertical density gradient to model vertical density gradient. Only in the special case where the model remains close to observations are the two methods the same to leading order. It is not obvious whether isopycnic or level relaxation is to be preferred.

## Abstract

A simple, uniformly stratified, linear model is developed to examine the effects on upwelling and internal Kelvin wave propagation of small, slow, longshore varying topography and coastline. The condition of no normal flow at the bottom yields correction terms with responses that propagate as Kelvin waves. For the first problem considered, a uniform wind stress is turned on abruptly. The response is fully three-dimensional with a zone of upwelling (downwelling) to the south of a ridge (canyon) near the shore. As time passes, the zone moves poleward and becomes centered over the topography. A complicated cyclonic and anticyclonic circulation is associated with a shoreward (seaward) flow over the ridge (canyon). If the basic state (i.e., the flow in the absence of topography) had no poleward undercurrent, the sign of the response is altered.

The second problem considered the modification of an internal Kelvin wave by isolated topography. Energy is scattered into all vertical modes (i.e., the natural decomposition of the flat-bottom response with respect to the vertical). Most energy goes into neighboring modes. The response consists of a steady contribution over the topography and a traveling, free Kelvin wave. For high incoming modes (those with many zero crossings in the vertical), little energy is scattered., most of what is scattered goes into the steady contribution. For low incoming modes, much energy is lost, divided about equally between steady and traveling responses. Although this problem can only be thought of as a first attempt at understanding scattering of baroclinic coastal waves by topography, it may help to explain why only low-mode Kelvin waves are observed.

## Abstract

A simple, uniformly stratified, linear model is developed to examine the effects on upwelling and internal Kelvin wave propagation of small, slow, longshore varying topography and coastline. The condition of no normal flow at the bottom yields correction terms with responses that propagate as Kelvin waves. For the first problem considered, a uniform wind stress is turned on abruptly. The response is fully three-dimensional with a zone of upwelling (downwelling) to the south of a ridge (canyon) near the shore. As time passes, the zone moves poleward and becomes centered over the topography. A complicated cyclonic and anticyclonic circulation is associated with a shoreward (seaward) flow over the ridge (canyon). If the basic state (i.e., the flow in the absence of topography) had no poleward undercurrent, the sign of the response is altered.

The second problem considered the modification of an internal Kelvin wave by isolated topography. Energy is scattered into all vertical modes (i.e., the natural decomposition of the flat-bottom response with respect to the vertical). Most energy goes into neighboring modes. The response consists of a steady contribution over the topography and a traveling, free Kelvin wave. For high incoming modes (those with many zero crossings in the vertical), little energy is scattered., most of what is scattered goes into the steady contribution. For low incoming modes, much energy is lost, divided about equally between steady and traveling responses. Although this problem can only be thought of as a first attempt at understanding scattering of baroclinic coastal waves by topography, it may help to explain why only low-mode Kelvin waves are observed.

## Abstract

The narrow regions of intense vertical mixing to great depths in the ocean are discussed, with emphasis on such a region or “chimney” observed recently in the Weddell gyre. It is deduced that such chimneys are the result of surface wintertime cooling. Application of two models of the resulting vertical convection shows that the entire of the area of the Weddell gyre is prone to overturning; yet only one narrow region apparently did so.

This shows that a preconditioning process is responsible, which preselects a narrow area and reduces its vertical stability. Baroclinic instability of the mean flow is capable of producing cyclonic and anticyclonic eddies with horizontal length scales of the same width as the observed chimneys. At the center of the cyclonic eddies, the vertical stratification is greatly reduced in the top 300 m, thus acting as an efficient preselection mechanism, at the onset of winter cooling.

Such a theory can also explain the appearance of bottom water in the Greenland Sea, where no examples of vertical homogeneity have ever been observed. Estimates of the number of chimneys necessary to form the bottom water make it likely (82%) that no chimney would have been observed to date.

## Abstract

The narrow regions of intense vertical mixing to great depths in the ocean are discussed, with emphasis on such a region or “chimney” observed recently in the Weddell gyre. It is deduced that such chimneys are the result of surface wintertime cooling. Application of two models of the resulting vertical convection shows that the entire of the area of the Weddell gyre is prone to overturning; yet only one narrow region apparently did so.

This shows that a preconditioning process is responsible, which preselects a narrow area and reduces its vertical stability. Baroclinic instability of the mean flow is capable of producing cyclonic and anticyclonic eddies with horizontal length scales of the same width as the observed chimneys. At the center of the cyclonic eddies, the vertical stratification is greatly reduced in the top 300 m, thus acting as an efficient preselection mechanism, at the onset of winter cooling.

Such a theory can also explain the appearance of bottom water in the Greenland Sea, where no examples of vertical homogeneity have ever been observed. Estimates of the number of chimneys necessary to form the bottom water make it likely (82%) that no chimney would have been observed to date.

## Abstract

A simple two-level model is designed to simulate the “thermocline equations,” applicable for large-scale steady oceanic flow. The model serves two functions. First, it replaces problems with the interpretation of slablike dynamics (e.g., Luyten *et al*., 1983) by using continuously horizontally varying buoyancy, but at the cost of reducing the vertical resolution drastically. The equations used are geostrophy (plus a small linear drag to close a Stommel-like western boundary layer), mass conservation, and buoyancy conservation with a small but necessary horizontal diffusion. (Inclusion of vertical diffusion has little effect). The ocean is driven by an Ekman layer, whose functions are to provide a given surface input of mass (through Ekman pumping) and buoyancy (through a specified buoyancy in the Ekman layer), i.e., to maintain the same boundary conditions as in classical thermocline studies. Sidewall conditions are not well understood and are almost certainly over-specified in this formulation. Second, the model works towards the development of a simple numerical model which can permit rapid, cheap evaluation of the ocean circulation on climatic timescales.

The depth integrated flow is known from the Ekman pumping, so that the only unknown flow is the (single) baroclinic mode, which may be derived from the thermal wind equations as the density field is advected and diffused. The time taken to a steady solution is a few hundred years for a two-gyre basin of side 4000 km.

Despite the apparent simplicity of the model, the solution is fairly realistic and quite complicated. The solution involves convective adjustment in the northern (cool) part of the basin. The area occupied by convection increases with the amplitudes of both buoyancy forcing and Ekman pumping. There is a strong western boundary current, that separates farther south of its equivalent North Atlantic latitude, and flows toward the northeast corner of the basin where there is strong downwelling as the flow is returned in the lower level. The average of the level densities serves as an approximate streamfunction for the baroclinic flow that spins up initially like a long Rossby wave response of a linear ocean to wind forcing. Transfer from the southern to the northern gyre is produced by diffusion and ageostrophic effects in midocean, and not at the western boundary.

To examine the ventilation of the lower subtropical level of the ocean, trajectories were examined for water particles emitted from the downwelling Ekman layer. Those released in the southern half of the subtropics have quite complex tracks, with a tendency for anticyclone circulation for several years followed by a cross-gyre movement to the subpolar gyre and circuitous routes back to the subtropics. The net result seems to be little direct ventilation. Particles released nearer the gyre boundary also show little tendency to direct lower ventilation. Adding random walks to the particle tracks to simulate the horizontal diffusivity shows that diffusion made little qualitative difference apart from an expected smearing out of the tracks.

## Abstract

A simple two-level model is designed to simulate the “thermocline equations,” applicable for large-scale steady oceanic flow. The model serves two functions. First, it replaces problems with the interpretation of slablike dynamics (e.g., Luyten *et al*., 1983) by using continuously horizontally varying buoyancy, but at the cost of reducing the vertical resolution drastically. The equations used are geostrophy (plus a small linear drag to close a Stommel-like western boundary layer), mass conservation, and buoyancy conservation with a small but necessary horizontal diffusion. (Inclusion of vertical diffusion has little effect). The ocean is driven by an Ekman layer, whose functions are to provide a given surface input of mass (through Ekman pumping) and buoyancy (through a specified buoyancy in the Ekman layer), i.e., to maintain the same boundary conditions as in classical thermocline studies. Sidewall conditions are not well understood and are almost certainly over-specified in this formulation. Second, the model works towards the development of a simple numerical model which can permit rapid, cheap evaluation of the ocean circulation on climatic timescales.

The depth integrated flow is known from the Ekman pumping, so that the only unknown flow is the (single) baroclinic mode, which may be derived from the thermal wind equations as the density field is advected and diffused. The time taken to a steady solution is a few hundred years for a two-gyre basin of side 4000 km.

Despite the apparent simplicity of the model, the solution is fairly realistic and quite complicated. The solution involves convective adjustment in the northern (cool) part of the basin. The area occupied by convection increases with the amplitudes of both buoyancy forcing and Ekman pumping. There is a strong western boundary current, that separates farther south of its equivalent North Atlantic latitude, and flows toward the northeast corner of the basin where there is strong downwelling as the flow is returned in the lower level. The average of the level densities serves as an approximate streamfunction for the baroclinic flow that spins up initially like a long Rossby wave response of a linear ocean to wind forcing. Transfer from the southern to the northern gyre is produced by diffusion and ageostrophic effects in midocean, and not at the western boundary.

To examine the ventilation of the lower subtropical level of the ocean, trajectories were examined for water particles emitted from the downwelling Ekman layer. Those released in the southern half of the subtropics have quite complex tracks, with a tendency for anticyclone circulation for several years followed by a cross-gyre movement to the subpolar gyre and circuitous routes back to the subtropics. The net result seems to be little direct ventilation. Particles released nearer the gyre boundary also show little tendency to direct lower ventilation. Adding random walks to the particle tracks to simulate the horizontal diffusivity shows that diffusion made little qualitative difference apart from an expected smearing out of the tracks.

## Abstract

We consider the problem of a low-frequency, two-layer, coastal Kelvin wave which impinges on a topographic ridge or valley at some angle to the coastline, with the aim of bounding the transmission of the Kelvin wave beyond the topography (or, put alternatively, of bounding the scattering of energy into topographic waves along the ridge). The width of the topographic feature is assumed to be of order the internal deformation radius. It is not necessary to solve the very complicated interaction problem near the junction of the ridge and the coastline. Instead, a simple series of eigenvalue o.d.e.'s must be solved.

The main contribution to loss of energy by the Kelvin wave comes from long waves along the ridge. Whether this loss is significant depends crucially on whether the topography is high enough to intersect a density surface (in this case, the interface between the two layers). If the topography remains solely in the lower layer, then the Kelvin wave continues with negligible loss of energy in the limit of very small frequency.

In a continuously stratified fluid, topography of any height would cut through an infinite number of density strata, so that a more realistic model would permit the topography to intersect the interface. This case is also considered, and results in a finite loss of energy from the Kelvin wave to topographic waves along the ridge (as in the one-layer reduced gravity case considered in an earlier paper). As a rough guide, the amplitude of the transmitted wave is reduced by an amount approximately equal to the fractional depth of the fluid blocked by the topography. Thus, models that do not permit topography to break through a density interface give qualitatively different answers from those which do—which should be considered when second-generation ocean models are being consructed.

It is found that, even using a supercomputer, available numerical resolution cannot adequately represent the topographically trapped waves, so topographical scattering processes will inevitably be badly misrepresented in numerical models. The case of a continuously stratified fluid is also briefly considered, although solutions would be considerably more complicated to produce.

## Abstract

We consider the problem of a low-frequency, two-layer, coastal Kelvin wave which impinges on a topographic ridge or valley at some angle to the coastline, with the aim of bounding the transmission of the Kelvin wave beyond the topography (or, put alternatively, of bounding the scattering of energy into topographic waves along the ridge). The width of the topographic feature is assumed to be of order the internal deformation radius. It is not necessary to solve the very complicated interaction problem near the junction of the ridge and the coastline. Instead, a simple series of eigenvalue o.d.e.'s must be solved.

The main contribution to loss of energy by the Kelvin wave comes from long waves along the ridge. Whether this loss is significant depends crucially on whether the topography is high enough to intersect a density surface (in this case, the interface between the two layers). If the topography remains solely in the lower layer, then the Kelvin wave continues with negligible loss of energy in the limit of very small frequency.

In a continuously stratified fluid, topography of any height would cut through an infinite number of density strata, so that a more realistic model would permit the topography to intersect the interface. This case is also considered, and results in a finite loss of energy from the Kelvin wave to topographic waves along the ridge (as in the one-layer reduced gravity case considered in an earlier paper). As a rough guide, the amplitude of the transmitted wave is reduced by an amount approximately equal to the fractional depth of the fluid blocked by the topography. Thus, models that do not permit topography to break through a density interface give qualitatively different answers from those which do—which should be considered when second-generation ocean models are being consructed.

It is found that, even using a supercomputer, available numerical resolution cannot adequately represent the topographically trapped waves, so topographical scattering processes will inevitably be badly misrepresented in numerical models. The case of a continuously stratified fluid is also briefly considered, although solutions would be considerably more complicated to produce.

## 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

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 behavior of a reduced-gravity cylinder of fluid, released from rest in a rotating system, is considered. The eventual steady state, found by normal principles of conservation of angular momentum, mass, and potential vorticity, is shown to have less energy than the initial state. This energy deficit can be accounted for by time-dependent motions, instabilities, and dissipative effects (waves cannot propagate energy to infinity in this system since the active fluid is of finite extent). We show here that an extra feature, hitherto unconsidered, comes into play. The time-dependent motion allows occasional wave-breaking events, which can act as a mechanism to remove the energy deficit on short (i.e., inertial) time scales. Such a process has not been parameterized in ocean circulation models.

## Abstract

The behavior of a reduced-gravity cylinder of fluid, released from rest in a rotating system, is considered. The eventual steady state, found by normal principles of conservation of angular momentum, mass, and potential vorticity, is shown to have less energy than the initial state. This energy deficit can be accounted for by time-dependent motions, instabilities, and dissipative effects (waves cannot propagate energy to infinity in this system since the active fluid is of finite extent). We show here that an extra feature, hitherto unconsidered, comes into play. The time-dependent motion allows occasional wave-breaking events, which can act as a mechanism to remove the energy deficit on short (i.e., inertial) time scales. Such a process has not been parameterized in ocean circulation models.