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## 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 new inverse method for finding the large-scale ocean circulation is described. Unlike most previous methods it uses no horizontal gradient information, and is designed for widely spaced data. The method assumes that density, (linear) potential vorticity and Bernoulli function are all approximately conserved on a streamline, so that the Bernoulli function depends solely on density and potential vorticity, both of which are known from data. The requirement that Bernoulli functions should match at points where density and potential vorticity match then leads to a heavily overdetermined problem for the surface pressure field, solved by a singular-value decomposition.

The method is tested on an analytical solution due to Welander, and on the results of a numerical circulation model of Cox and Bryan, before use on climatological data, both over the main North Atlantic and over the beta triangle area. Analysis shows that for closely spaced data, the Bernoulli method reduces to beta-spiral dynamics, but with additional constraints from nonneighboring stations. The method, which is essentially nonlocal in character, as it depends on following flow streamlines, seems to be fairly robust both to noise in the data and to station spacing and selection.

## Abstract

A new inverse method for finding the large-scale ocean circulation is described. Unlike most previous methods it uses no horizontal gradient information, and is designed for widely spaced data. The method assumes that density, (linear) potential vorticity and Bernoulli function are all approximately conserved on a streamline, so that the Bernoulli function depends solely on density and potential vorticity, both of which are known from data. The requirement that Bernoulli functions should match at points where density and potential vorticity match then leads to a heavily overdetermined problem for the surface pressure field, solved by a singular-value decomposition.

The method is tested on an analytical solution due to Welander, and on the results of a numerical circulation model of Cox and Bryan, before use on climatological data, both over the main North Atlantic and over the beta triangle area. Analysis shows that for closely spaced data, the Bernoulli method reduces to beta-spiral dynamics, but with additional constraints from nonneighboring stations. The method, which is essentially nonlocal in character, as it depends on following flow streamlines, seems to be fairly robust both to noise in the data and to station spacing and selection.

## Abstract

This paper examines the motion and propagation of an isolated of anomalous water on a beta-plane, considered previously by Nof (1981). His perturbation analysis is extended to show the following:

1) Only westward propagation can occur, induced by the beta-effect; the eddy's speed must be ten less than two-thirds of the long Rossby-wave speed (unless the potential vorticity of the eddy is somewhere negative, which would be unlikely).

2) The eddy must be at least 2√2 deformation radii in radius.

3) The shape and velocity structure of the eddy has a simple structure, which is calculated for one range of cases.

4) The unperturbed eddy (on an *f*-plane) is stable to small disturbances making it likely that the eddy can propagate great distances before decaying.

## Abstract

This paper examines the motion and propagation of an isolated of anomalous water on a beta-plane, considered previously by Nof (1981). His perturbation analysis is extended to show the following:

1) Only westward propagation can occur, induced by the beta-effect; the eddy's speed must be ten less than two-thirds of the long Rossby-wave speed (unless the potential vorticity of the eddy is somewhere negative, which would be unlikely).

2) The eddy must be at least 2√2 deformation radii in radius.

3) The shape and velocity structure of the eddy has a simple structure, which is calculated for one range of cases.

4) The unperturbed eddy (on an *f*-plane) is stable to small disturbances making it likely that the eddy can propagate great distances before decaying.

## 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

No abstract available.

## Abstract

No abstract available.

## 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

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

## Abstract

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

## Abstract

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

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

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

## Abstract

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