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## Abstract

A model for a turbulent bottom boundary â€śslabâ€ť layer is described. The model is designed to fit under a standard, depth coordinate ocean general circulation model, with a view to improving its response both for local and climate problems. The depth of the layer varies temporally and spatially. Both analytical and numerical versions of the model conserve energy. The model is tested using a source of dense water on a slope, and performs satisfactorily, with the plume spreading far more than the equivalent case without a bottom layer.

## Abstract

A model for a turbulent bottom boundary â€śslabâ€ť layer is described. The model is designed to fit under a standard, depth coordinate ocean general circulation model, with a view to improving its response both for local and climate problems. The depth of the layer varies temporally and spatially. Both analytical and numerical versions of the model conserve energy. The model is tested using a source of dense water on a slope, and performs satisfactorily, with the plume spreading far more than the equivalent case without a bottom layer.

## Abstract

This paper examines how slowly varying topography induces changes in all aspects of long planetary wave propagation, including phase speed and surface signature, through steering effects. The approach introduces a method for the exact solution of the vertical topographic eigenvalue problem for arbitrary realistic stratification and ray theory in the horizontal. It is shown that, for observed stratifications, first internal mode topographic waves have phase speeds between about 0.4 and twice the local flat-bottom phase speed. Increases occur on the western and equatorward sides of hills. Focusing of ray trajectories and caustics are common features of the solutions. Despite a bias between slowdown and speedup, on average there is little speedup except in high latitudes (where long-wave theory is less applicable). Calculations are performed for five main ocean basins, assuming waves are generated at the eastern coastline, using smoothed topography. These calculations confirm the above findings: there are significant local effects on wave speed, but these largely cancel over the basin scale. Thus, topographic effects cannot explain recent observations, which demonstrate long planetary waves propagating about twice as fast as linear theory. The presence of mean flow, which induces changes to the planetary vorticity gradient, remains the prime candidate for the observed speedup.

## Abstract

This paper examines how slowly varying topography induces changes in all aspects of long planetary wave propagation, including phase speed and surface signature, through steering effects. The approach introduces a method for the exact solution of the vertical topographic eigenvalue problem for arbitrary realistic stratification and ray theory in the horizontal. It is shown that, for observed stratifications, first internal mode topographic waves have phase speeds between about 0.4 and twice the local flat-bottom phase speed. Increases occur on the western and equatorward sides of hills. Focusing of ray trajectories and caustics are common features of the solutions. Despite a bias between slowdown and speedup, on average there is little speedup except in high latitudes (where long-wave theory is less applicable). Calculations are performed for five main ocean basins, assuming waves are generated at the eastern coastline, using smoothed topography. These calculations confirm the above findings: there are significant local effects on wave speed, but these largely cancel over the basin scale. Thus, topographic effects cannot explain recent observations, which demonstrate long planetary waves propagating about twice as fast as linear theory. The presence of mean flow, which induces changes to the planetary vorticity gradient, remains the prime candidate for the observed speedup.

## Abstract

Propagating features and waves occur everywhere in the ocean. This paper derives a concise description of how such small-amplitude, large-scale oceanic internal disturbances propagate dynamically against a slowly varying background mean flow and stratification, computed using oceanic data. For a flat-bottomed ocean, assumed here, the linear internal modes, computed using the local stratification, form a useful basis for expanding the oceanic shear modes of propagation. Remarkably, the shear modal structure is largely independent of orientation of the flow. The resulting advective velocities, which are termed *pseudovelocities,* comprise background flow decomposed onto normal modes, and westward planetary wave propagation velocities. The diagonal entries of the matrix of pseudovelocities prove to be reasonably accurate descriptors of the speed and direction of propagation of the shear modes, which thus respond as if simply advected by this diagonal-entry velocity field. The complicated three-dimensional propagation problem has thus been systematically reduced to this simple rule.

The first shear mode is dominated by westward propagation, and possesses a midlatitude speed-up over the undisturbed linear first-mode planetary wave. The pseudovelocity for the second shear mode, in contrast, while still dominated by westward propagation at lower latitudes, shows a gyrelike structure at latitudes above 30Â°. This resembles in both shape and direction the geostrophic baroclinic flow between about 500- and 1000-m depth, but are much slower than the flow at these depths. Features may thus be able to propagate some distance around a subtropical or subpolar gyre, but not, in general, at the speed of the circulation.

## Abstract

Propagating features and waves occur everywhere in the ocean. This paper derives a concise description of how such small-amplitude, large-scale oceanic internal disturbances propagate dynamically against a slowly varying background mean flow and stratification, computed using oceanic data. For a flat-bottomed ocean, assumed here, the linear internal modes, computed using the local stratification, form a useful basis for expanding the oceanic shear modes of propagation. Remarkably, the shear modal structure is largely independent of orientation of the flow. The resulting advective velocities, which are termed *pseudovelocities,* comprise background flow decomposed onto normal modes, and westward planetary wave propagation velocities. The diagonal entries of the matrix of pseudovelocities prove to be reasonably accurate descriptors of the speed and direction of propagation of the shear modes, which thus respond as if simply advected by this diagonal-entry velocity field. The complicated three-dimensional propagation problem has thus been systematically reduced to this simple rule.

The first shear mode is dominated by westward propagation, and possesses a midlatitude speed-up over the undisturbed linear first-mode planetary wave. The pseudovelocity for the second shear mode, in contrast, while still dominated by westward propagation at lower latitudes, shows a gyrelike structure at latitudes above 30Â°. This resembles in both shape and direction the geostrophic baroclinic flow between about 500- and 1000-m depth, but are much slower than the flow at these depths. Features may thus be able to propagate some distance around a subtropical or subpolar gyre, but not, in general, at the speed of the circulation.

## Abstract

One of the most successful theories to date to explain why observed planetary waves propagate westward faster than linear flat-bottom theory predicts has been to include the effect of background baroclinic mean flow, which modifies the potential vorticity waveguide in which the waves propagate. (Barotropic flows are almost everywhere too small to explain the observed differences.) That theory accounted for most, but not all, of the observed wave speeds. A later attempt to examine the effect of the sloping bottom on these waves (without the mean flow effect) did not find any overall speedup. This paper combines these two effects, assuming long (geostrophic) waves and slowly varying mean flow and topography, and computes group velocities at each point in the global ocean. These velocities turn out to be largely independent of the orientation of the wave vector. A second speedup of the waves is found (over that for mean flow only). Almost no eastward-oriented group velocities are found, and so features that appear to propagate in the same sense as a subtropical gyre would have to be coupled with the atmosphere or be density compensated in some manner.

## Abstract

One of the most successful theories to date to explain why observed planetary waves propagate westward faster than linear flat-bottom theory predicts has been to include the effect of background baroclinic mean flow, which modifies the potential vorticity waveguide in which the waves propagate. (Barotropic flows are almost everywhere too small to explain the observed differences.) That theory accounted for most, but not all, of the observed wave speeds. A later attempt to examine the effect of the sloping bottom on these waves (without the mean flow effect) did not find any overall speedup. This paper combines these two effects, assuming long (geostrophic) waves and slowly varying mean flow and topography, and computes group velocities at each point in the global ocean. These velocities turn out to be largely independent of the orientation of the wave vector. A second speedup of the waves is found (over that for mean flow only). Almost no eastward-oriented group velocities are found, and so features that appear to propagate in the same sense as a subtropical gyre would have to be coupled with the atmosphere or be density compensated in some manner.

## Abstract

Ray theory is used to predict phase and group velocities for long planetary waves under realistic, albeit slowly varying, oceanic conditions. The results are compared with local theory using fields smoothed to the same amount (9Â° latitude/longitude) as well as those with much less smoothing (1Â°). The agreement is excellent, showing that local theory forms a good proxy for ray theory results. The predicted speeds agree well with observations of planetary waves deduced from sea surface height data. The theory uses purely baroclinic mean flow; the inclusion of barotropic flow has little effect except at high latitudes.

## Abstract

Ray theory is used to predict phase and group velocities for long planetary waves under realistic, albeit slowly varying, oceanic conditions. The results are compared with local theory using fields smoothed to the same amount (9Â° latitude/longitude) as well as those with much less smoothing (1Â°). The agreement is excellent, showing that local theory forms a good proxy for ray theory results. The predicted speeds agree well with observations of planetary waves deduced from sea surface height data. The theory uses purely baroclinic mean flow; the inclusion of barotropic flow has little effect except at high latitudes.

## Abstract

The characteristics of an unforced, stratified *f*-plane geostrophic flow over topography are described, and scaling arguments are made to justify the use of such a flow as a first-order approximation to a real, large-scale circulation. Consideration of integral constraints then provides an insight into the ways in which second-order processes must balance the wind forcing. The importance of bottom pressure in this model is used to test the scalings and theory on a dataset taken from the Fine Resolution Antarctic Model. Two plots of bottom pressure, each with depth dependence filtered out in a different way, confirm the scalings producing the following conclusions: The effect of topography on the bottom boundary condition (no flow through the boundary) is important to the first-order (*f*-plane geostrophic) circulation; the turning of horizontal velocities with depth is limited, especially in regions of strong flow; and a picture of bottom pressure, appropriately filtered for depth dependence, contains a wealth of valuable information about the importance of second-order processes, demonstrating that they are most important in particular localized regions associated with topographic features.

## Abstract

The characteristics of an unforced, stratified *f*-plane geostrophic flow over topography are described, and scaling arguments are made to justify the use of such a flow as a first-order approximation to a real, large-scale circulation. Consideration of integral constraints then provides an insight into the ways in which second-order processes must balance the wind forcing. The importance of bottom pressure in this model is used to test the scalings and theory on a dataset taken from the Fine Resolution Antarctic Model. Two plots of bottom pressure, each with depth dependence filtered out in a different way, confirm the scalings producing the following conclusions: The effect of topography on the bottom boundary condition (no flow through the boundary) is important to the first-order (*f*-plane geostrophic) circulation; the turning of horizontal velocities with depth is limited, especially in regions of strong flow; and a picture of bottom pressure, appropriately filtered for depth dependence, contains a wealth of valuable information about the importance of second-order processes, demonstrating that they are most important in particular localized regions associated with topographic features.

## Abstract

Oceanic rings tend to have length scales larger than the deformation radius and also to he long-lived. This latter characteristic, in view of the former, is particularly curious as many quasigeostrophic and primitive equation simulations suggest such eddies are quite unstable. Large eddies eventually break into smaller deformation scale vortices, with the attendant production of considerable variability.

Here it is argued that the stability characteristics of oceanic eddies and rings are sensitive to the presence of deep flows. In particular, eddies in which the deep flow is counter to the sense of the shallow flows are often more unstable than eddies with no deep flow, while eddies with circulations in the same sense as the shallow circulation can experience an enhanced stability. For a given vertical shear, oceanic eddy stability can vary dramatically. (This is in contrast to quasigeostrophic theory, where stability properties are largely determined by vertical shear.) The onset of these mechanics is quite pronounced for Gaussian oceanic eddies. Linear â€ś*f*â€ť- plane stability calculations reveal a marked suppression of unstable growth rates for warm corotating eddies with relatively weak deep flows. Cold eddies also experience a suppression of instability in the corotating state, although relatively weak unstable modes have been found. Comparisons of *f*- and Î˛-plane numerical primitive equation experiments support these results, as well as demonstrate some relevant limitations. Finally, studies of dipolar eddies and non-Gaussian circular eddies are used to examine the generality of the results. We suggest such stability considerations may be partially responsible for the observed long lives of oceanic rings.

An examination of the unstable normal modes from the *f*-plane model demonstrates an intimate coupling between the suppression of instability and the appearance of multiple critical layers. The normal-mode energetics are used to demonstrate the role of upgradient momentum fluxes at the points of stabilization, and a heuristic argument involving critical layers is given.

## Abstract

Oceanic rings tend to have length scales larger than the deformation radius and also to he long-lived. This latter characteristic, in view of the former, is particularly curious as many quasigeostrophic and primitive equation simulations suggest such eddies are quite unstable. Large eddies eventually break into smaller deformation scale vortices, with the attendant production of considerable variability.

Here it is argued that the stability characteristics of oceanic eddies and rings are sensitive to the presence of deep flows. In particular, eddies in which the deep flow is counter to the sense of the shallow flows are often more unstable than eddies with no deep flow, while eddies with circulations in the same sense as the shallow circulation can experience an enhanced stability. For a given vertical shear, oceanic eddy stability can vary dramatically. (This is in contrast to quasigeostrophic theory, where stability properties are largely determined by vertical shear.) The onset of these mechanics is quite pronounced for Gaussian oceanic eddies. Linear â€ś*f*â€ť- plane stability calculations reveal a marked suppression of unstable growth rates for warm corotating eddies with relatively weak deep flows. Cold eddies also experience a suppression of instability in the corotating state, although relatively weak unstable modes have been found. Comparisons of *f*- and Î˛-plane numerical primitive equation experiments support these results, as well as demonstrate some relevant limitations. Finally, studies of dipolar eddies and non-Gaussian circular eddies are used to examine the generality of the results. We suggest such stability considerations may be partially responsible for the observed long lives of oceanic rings.

An examination of the unstable normal modes from the *f*-plane model demonstrates an intimate coupling between the suppression of instability and the appearance of multiple critical layers. The normal-mode energetics are used to demonstrate the role of upgradient momentum fluxes at the points of stabilization, and a heuristic argument involving critical layers is given.

## Abstract

The momentum budget of the Antarctic Circumpolar Current (ACC) is analyzed using data from the Fine Resolution Antarctic Model (FRAM), using density as a vertical coordinate, since density is approximately conserved on streamlines. This steady budget is balanced. Volume, heat, and salt budgets are also computed within density layers, although these remain time dependent. The leading-order momentum balance is of approximate equality of the form drag on the top and bottom surfaces of each undulating density layer, but because density layers outcrop at the surface, this does not imply that the surface wind stress is simply transferred downward by form drag without change of amplitude. Restricting attention to the net form drag on a layer, this is found to be balanced by Coriolis force and, if the layer outcrops, the amount of wind stress put into the layer where it outcrops. Between 40% and 60% of the density layers at any latitude outcrop somewhere at the surface, so that wind stress can be moved directly into these layers, totally unlike the quasigeostrophic situation. Reynolds stress divergence and cross-isopycnal momentum transport are negligible. In layers dense enough to ground at the ocean floor, the form drag changes sign several times, following the sign changes in the northward volume flux through the Coriolis term. These north-south fluxes are produced by time-dependent filling or emptying of fluid layers south of the ACC. This shows that although FRAM and other marginally eddy resolving models reach apparent statistical steady dynamical states in about a decade, this is illusory: the long-time thermodynamic behavior affects the dynamics. It is shown that balances from a time-averaged dataset are not accurate guides to the time-averaged balances.

## Abstract

The momentum budget of the Antarctic Circumpolar Current (ACC) is analyzed using data from the Fine Resolution Antarctic Model (FRAM), using density as a vertical coordinate, since density is approximately conserved on streamlines. This steady budget is balanced. Volume, heat, and salt budgets are also computed within density layers, although these remain time dependent. The leading-order momentum balance is of approximate equality of the form drag on the top and bottom surfaces of each undulating density layer, but because density layers outcrop at the surface, this does not imply that the surface wind stress is simply transferred downward by form drag without change of amplitude. Restricting attention to the net form drag on a layer, this is found to be balanced by Coriolis force and, if the layer outcrops, the amount of wind stress put into the layer where it outcrops. Between 40% and 60% of the density layers at any latitude outcrop somewhere at the surface, so that wind stress can be moved directly into these layers, totally unlike the quasigeostrophic situation. Reynolds stress divergence and cross-isopycnal momentum transport are negligible. In layers dense enough to ground at the ocean floor, the form drag changes sign several times, following the sign changes in the northward volume flux through the Coriolis term. These north-south fluxes are produced by time-dependent filling or emptying of fluid layers south of the ACC. This shows that although FRAM and other marginally eddy resolving models reach apparent statistical steady dynamical states in about a decade, this is illusory: the long-time thermodynamic behavior affects the dynamics. It is shown that balances from a time-averaged dataset are not accurate guides to the time-averaged balances.

## Abstract

The local response of an ocean with slowly varying mean flow, stratification, and topography to two sources of disturbance is examined, concentrating on whether the resulting surface elevations are observable. The first is the ocean response to surface forcing (Ekman pumping or buoyancy forcing). For typical amplitudes of random forcing, while much of the ocean response is small (surface elevations less than 1 mm), there are sufficient near resonances (or pseudoresonances involving a critical layer) to produce elevations of 1 cm or more in much of the ocean. The second source is baroclinic instability. The fastest linear growth rate, as well as those for specific wavelengths, is computed globally. Almost all of the ocean is baroclinically unstable, and the most unstable waves are found to possess a small wavelength (often less than 10 km) with a disturbance concentrated near the surface: *e*-folding times *O*(20 days) are frequently found. However, the phase speed for the disturbances is almost everywhere slower westward than free planetary waves with mean flow and topography. Since the free waves propagate at speeds similar to observations, instability may be a good source of variability but is probably not responsible directly for observed wave propagation.

## Abstract

The local response of an ocean with slowly varying mean flow, stratification, and topography to two sources of disturbance is examined, concentrating on whether the resulting surface elevations are observable. The first is the ocean response to surface forcing (Ekman pumping or buoyancy forcing). For typical amplitudes of random forcing, while much of the ocean response is small (surface elevations less than 1 mm), there are sufficient near resonances (or pseudoresonances involving a critical layer) to produce elevations of 1 cm or more in much of the ocean. The second source is baroclinic instability. The fastest linear growth rate, as well as those for specific wavelengths, is computed globally. Almost all of the ocean is baroclinically unstable, and the most unstable waves are found to possess a small wavelength (often less than 10 km) with a disturbance concentrated near the surface: *e*-folding times *O*(20 days) are frequently found. However, the phase speed for the disturbances is almost everywhere slower westward than free planetary waves with mean flow and topography. Since the free waves propagate at speeds similar to observations, instability may be a good source of variability but is probably not responsible directly for observed wave propagation.

## Abstract

The one-dimensional examples of the dispersion relation for planetary waves under the Wentzelâ€“Kramersâ€“Brillouinâ€“Jeffreys (WKBJ) assumption given in Part I are extended to two dimensions and analyzed globally. The dispersion relations are complicated, and there is a nontrivial lower bound to the frequency given by the column maximum of what would be the local Doppler shift to the frequency. This generates short waves of a much higher frequency than would be expected from traditional theory; these waves can have larger phase velocities than long waves but do not appear to have faster group velocities. The longer waves possess phase speeds in excellent agreement with recent remotely sensed data. Waves cannot propagate efficiently across ocean basins, suggesting that mechanisms other than eastern boundary generation may be playing a role in the ubiquitous nature of planetary waves.

## Abstract

The one-dimensional examples of the dispersion relation for planetary waves under the Wentzelâ€“Kramersâ€“Brillouinâ€“Jeffreys (WKBJ) assumption given in Part I are extended to two dimensions and analyzed globally. The dispersion relations are complicated, and there is a nontrivial lower bound to the frequency given by the column maximum of what would be the local Doppler shift to the frequency. This generates short waves of a much higher frequency than would be expected from traditional theory; these waves can have larger phase velocities than long waves but do not appear to have faster group velocities. The longer waves possess phase speeds in excellent agreement with recent remotely sensed data. Waves cannot propagate efficiently across ocean basins, suggesting that mechanisms other than eastern boundary generation may be playing a role in the ubiquitous nature of planetary waves.