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- Author or Editor: Peter D. Killworth x

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## 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 “*f*”- 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 β-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 “*f*”- 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 β-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 Wentzel–Kramers–Brillouin–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 Wentzel–Kramers–Brillouin–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

An eigenvalue problem for the dispersion relation for planetary waves in the presence of mean flow and bottom topographic gradients is derived, under the Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) assumption, for frequencies that are low when compared with the inertial frequency. Examples are given for the World Ocean that show a rich variety of behavior, including no frequency (or latitudinal) cutoff, solutions trapped at certain depths, coalescence of waves, and a lack of dispersion for most short waves.

## Abstract

An eigenvalue problem for the dispersion relation for planetary waves in the presence of mean flow and bottom topographic gradients is derived, under the Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) assumption, for frequencies that are low when compared with the inertial frequency. Examples are given for the World Ocean that show a rich variety of behavior, including no frequency (or latitudinal) cutoff, solutions trapped at certain depths, coalescence of waves, and a lack of dispersion for most short waves.

## Abstract

Mechanisms are considered which may induct the large (over 10^{5} km^{2}) area of open water, or polynya, which frequently occurs within the Weddell Sea winter sea-ice. We propose that when surface cooling and ice formation decrease the temperature and increase the salinity of the surface water (the latter by salt rejection during ice formation) in a preconditioned area, static instability with intense vertical mixing can occur. The upwelled warm, salty deep water can then supply enough heat to melt the ice, or prohibit its formation, even in the middle of winter.

A simple two-level model is derived to test this theory and is found to agree well with observations. The process is found to be irregular due to different times of ice onset from one year to the next, and to a lesser extent from variations in surface heating and cooling. Further, it is shown that unless the freshwater input exactly balances the increased salinity from the overturn each year, the system will either gain or lose salt yearly and eventually stabilize permanently (i.e., attain a steady-state condition).

The model is insensitive to short term stochastic variations in surface heat flux or freshwater input rates, but is somewhat sensitive to longer scale variations in the net freshwater input and also to the depth of the pycnocline (i.e., preconditioning). It is suggested that upwelling may raise the pycnocline until convection can occur and the polynya form. The preconditioned area then advects westward with the mean flow. Permanent stability finally is attained but the preconditioned area is advected into the western boundary current of the Weddell subpolar gyre and destroyed. Prior to destruction, topographical features may quantitatively affect both the movement and occurrence of the polynya.

Regardless of the preconditioning mechanism, if overturning is responsible for the polynya, this would contribute a minimum of 10^{6} m^{3} s^{−1} to the total deep water formation, and constitute the largest area of deep open-ocean convection yet discovered.

## Abstract

Mechanisms are considered which may induct the large (over 10^{5} km^{2}) area of open water, or polynya, which frequently occurs within the Weddell Sea winter sea-ice. We propose that when surface cooling and ice formation decrease the temperature and increase the salinity of the surface water (the latter by salt rejection during ice formation) in a preconditioned area, static instability with intense vertical mixing can occur. The upwelled warm, salty deep water can then supply enough heat to melt the ice, or prohibit its formation, even in the middle of winter.

A simple two-level model is derived to test this theory and is found to agree well with observations. The process is found to be irregular due to different times of ice onset from one year to the next, and to a lesser extent from variations in surface heating and cooling. Further, it is shown that unless the freshwater input exactly balances the increased salinity from the overturn each year, the system will either gain or lose salt yearly and eventually stabilize permanently (i.e., attain a steady-state condition).

The model is insensitive to short term stochastic variations in surface heat flux or freshwater input rates, but is somewhat sensitive to longer scale variations in the net freshwater input and also to the depth of the pycnocline (i.e., preconditioning). It is suggested that upwelling may raise the pycnocline until convection can occur and the polynya form. The preconditioned area then advects westward with the mean flow. Permanent stability finally is attained but the preconditioned area is advected into the western boundary current of the Weddell subpolar gyre and destroyed. Prior to destruction, topographical features may quantitatively affect both the movement and occurrence of the polynya.

Regardless of the preconditioning mechanism, if overturning is responsible for the polynya, this would contribute a minimum of 10^{6} m^{3} s^{−1} to the total deep water formation, and constitute the largest area of deep open-ocean convection yet discovered.

## Abstract

The classic Stommel–Arons problem is revisited in the context of a basin with a general bottom topography containing an equator. Topography is taken to be smoothly varying and, as such, there are no vertical side walls in the problem. The perimeter of the abyssal basin is thus defined as the curve along which the layer depth vanishes. Because of this, it is not required that the component of horizontal velocity perpendicular to the boundary curve vanish on the boundary. Planetary geostrophic dynamics leads to a characteristic equation for the interface height field in which characteristics typically originate from a single point located on the eastern edge of the basin at the equator. For a simple choice of topography it is possible to solve the problem analytically. In the linear limit of weak forcing, the solution exhibits an intensified flow on the western edge of the basin. This flow is pan of the interior solution and is thus not a traditional, dissipative western boundary current.

When the fully nonlinear continuity equation is considered, the characteristics form a caustic on the western side of the basin. Characteristics cannot be integrated through the caustic so that a boundary layer involving higher-order dynamics is required. Approaching the caustic from the cast, velocities predicted by the interior solution become infinite. Because of this and because of global man budget considerations, the boundary layer is shown to lie east of the caustic. The position of the boundary layer is, however, not unique. Away from the equator, it is straightforward to append a Stommel boundary layer to the interior solution.

Next, localized mass sources are considered. Their dynamics (planetary geostrophy is assumed, except where boundary currents are required) include caustics and boundary layers in a manner similar to the sink-driven problem described above. For topography such that characteristics have a positive westward component, a caustic lies on the poleward side of the source and extends to the equator. Again, to balance the mass budget, a dissipative boundary layer is required inside the caustic. This, together with all the characteristics leaving the source region, intersects the equator at a finite angle. There, velocities become infinite, while the layer depth goes to zero. Although the details of the implied boundary layer are left unresolved, it is argued that the boundary current crosses the equator at a small angle to the eastward direction.

## Abstract

The classic Stommel–Arons problem is revisited in the context of a basin with a general bottom topography containing an equator. Topography is taken to be smoothly varying and, as such, there are no vertical side walls in the problem. The perimeter of the abyssal basin is thus defined as the curve along which the layer depth vanishes. Because of this, it is not required that the component of horizontal velocity perpendicular to the boundary curve vanish on the boundary. Planetary geostrophic dynamics leads to a characteristic equation for the interface height field in which characteristics typically originate from a single point located on the eastern edge of the basin at the equator. For a simple choice of topography it is possible to solve the problem analytically. In the linear limit of weak forcing, the solution exhibits an intensified flow on the western edge of the basin. This flow is pan of the interior solution and is thus not a traditional, dissipative western boundary current.

When the fully nonlinear continuity equation is considered, the characteristics form a caustic on the western side of the basin. Characteristics cannot be integrated through the caustic so that a boundary layer involving higher-order dynamics is required. Approaching the caustic from the cast, velocities predicted by the interior solution become infinite. Because of this and because of global man budget considerations, the boundary layer is shown to lie east of the caustic. The position of the boundary layer is, however, not unique. Away from the equator, it is straightforward to append a Stommel boundary layer to the interior solution.

Next, localized mass sources are considered. Their dynamics (planetary geostrophy is assumed, except where boundary currents are required) include caustics and boundary layers in a manner similar to the sink-driven problem described above. For topography such that characteristics have a positive westward component, a caustic lies on the poleward side of the source and extends to the equator. Again, to balance the mass budget, a dissipative boundary layer is required inside the caustic. This, together with all the characteristics leaving the source region, intersects the equator at a finite angle. There, velocities become infinite, while the layer depth goes to zero. Although the details of the implied boundary layer are left unresolved, it is argued that the boundary current crosses the equator at a small angle to the eastward direction.

## Abstract

No Abstract Available.

## Abstract

No Abstract Available.