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## 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.

## 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

Both the 6-year time-mean flow and the eddy kinetic energy in the Fine Resolution Antarctic Model are found to be approximately self-similar in the vertical, with the velocity showing a rapid decay with depth. A simple argument is given as to why this would be expected to occur in this and other numerical models, and an analysis is given as to the degree of steering of the mean flow by topography.

## Abstract

Both the 6-year time-mean flow and the eddy kinetic energy in the Fine Resolution Antarctic Model are found to be approximately self-similar in the vertical, with the velocity showing a rapid decay with depth. A simple argument is given as to why this would be expected to occur in this and other numerical models, and an analysis is given as to the degree of steering of the mean flow by topography.

## 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.

## Abstract

The three-dimensional circulation of a steady, frictional, homogeneous ocean forced by a wind stress varying sinusoidally in the northward direction is examined, with reference to the effects of continental slopes. It is found that the local circulation pattern is considerably altered by the presence of slopes, although the global pattern is not. Moderately strong eastern boundary currents, of order 1–3 cm sec^{−1}, are predicted, flowing in the same direction as those observed. Areas of secondary up- or downwelling are found near the bottom of the continental slope, even in the absence of a local long-shore wind stress, again agreeing qualitatively with observations.

The northward, intense (linear) return flow on the western boundary is again modified. It consists of a wide weak flow over the slope, and a narrower strong flow over the flat bottom; both flows have a northward flux of the same order, however.

## Abstract

The three-dimensional circulation of a steady, frictional, homogeneous ocean forced by a wind stress varying sinusoidally in the northward direction is examined, with reference to the effects of continental slopes. It is found that the local circulation pattern is considerably altered by the presence of slopes, although the global pattern is not. Moderately strong eastern boundary currents, of order 1–3 cm sec^{−1}, are predicted, flowing in the same direction as those observed. Areas of secondary up- or downwelling are found near the bottom of the continental slope, even in the absence of a local long-shore wind stress, again agreeing qualitatively with observations.

The northward, intense (linear) return flow on the western boundary is again modified. It consists of a wide weak flow over the slope, and a narrower strong flow over the flat bottom; both flows have a northward flux of the same order, however.

## 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

It is shown that the usual practice of forcing ocean models by linear interpolations of monthly mean data values does not produce a forcing whose mean over a month is the data value required. For wind stress data this can yield monthly mean errors, coherent over a basin, of as much as 0.4 × 10^{−1} N m^{−2}, with 30%–40% of values being in error by more than 10%. A simple method is given to avoid the difficulty, which involves no change to model computer code and no increase in the amount of data stored internally.

## Abstract

It is shown that the usual practice of forcing ocean models by linear interpolations of monthly mean data values does not produce a forcing whose mean over a month is the data value required. For wind stress data this can yield monthly mean errors, coherent over a basin, of as much as 0.4 × 10^{−1} N m^{−2}, with 30%–40% of values being in error by more than 10%. A simple method is given to avoid the difficulty, which involves no change to model computer code and no increase in the amount of data stored internally.

## 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

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, 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.