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## Abstract

An analytical model has been constructed to study the propagation of free waves of subinertial frequency in an infinite wedge filled with a uniformly stratified fluid. The problem is found to transform into the corresponding surface gravity wave problem in a nonrotating homogeneous fluid with the roles of the surface and bottom boundaries interchanged. Analytical solutions are thus available for waves that are either progressive or trapped in the cross-wedge direction, forming respectively continuous and discrete spectra in frequency space. The separation occurs when the nondimensional wave frequency a (scaled by the inertial frequency *f*) equals the Burger number *S*, defined here as (*N*/*f*) tanθ^{*}, where *N* is the Brunt-Väisälä frequency and tanθ ^{*} is the bottom slope. Since an infinite wedge has no intrinsic length scale, the only relevant nondimensional parameters are the wave frequency σ and the Burger number *S*. Thus, stratification and bottom slope play the same dynamical role, and the analysis is greatly simplified.

For the progressive waves, asymptotic solutions are obtained for both the far field and small *S*. Since the surface boundary condition is neglected in the far field, the solution there is similar to the edge wave solution found by Rhines (1970) in an infinitely deep ocean. The asymptotic solution for small *S*, on the other hand, clearly shows the refraction phenomenon and the presence of amplitude minimum as the apex is approached. Since the asymptotic solutions cheek very well with the calculations of the general solution, the qualitative behavior of the progressive waves are fairly predictable over the parameter range *S* ≫ O(1). The various wave properties associated with the general solution can be understood to a great extent by assuming quasi-geostrophy. The rigid upper surface is found to account for the onshore heat flux generated by these incoming waves.

For the trapped waves, the eigenfrequencies decrease when *S* decreases and approach the value (2*n* + 1)^{−1} when *S* approaches zero where *n* is the mode number. The modal structure broadens as *S* increases to some critical value above which no such coastally trapped modes exist.

## Abstract

An analytical model has been constructed to study the propagation of free waves of subinertial frequency in an infinite wedge filled with a uniformly stratified fluid. The problem is found to transform into the corresponding surface gravity wave problem in a nonrotating homogeneous fluid with the roles of the surface and bottom boundaries interchanged. Analytical solutions are thus available for waves that are either progressive or trapped in the cross-wedge direction, forming respectively continuous and discrete spectra in frequency space. The separation occurs when the nondimensional wave frequency a (scaled by the inertial frequency *f*) equals the Burger number *S*, defined here as (*N*/*f*) tanθ^{*}, where *N* is the Brunt-Väisälä frequency and tanθ ^{*} is the bottom slope. Since an infinite wedge has no intrinsic length scale, the only relevant nondimensional parameters are the wave frequency σ and the Burger number *S*. Thus, stratification and bottom slope play the same dynamical role, and the analysis is greatly simplified.

For the progressive waves, asymptotic solutions are obtained for both the far field and small *S*. Since the surface boundary condition is neglected in the far field, the solution there is similar to the edge wave solution found by Rhines (1970) in an infinitely deep ocean. The asymptotic solution for small *S*, on the other hand, clearly shows the refraction phenomenon and the presence of amplitude minimum as the apex is approached. Since the asymptotic solutions cheek very well with the calculations of the general solution, the qualitative behavior of the progressive waves are fairly predictable over the parameter range *S* ≫ O(1). The various wave properties associated with the general solution can be understood to a great extent by assuming quasi-geostrophy. The rigid upper surface is found to account for the onshore heat flux generated by these incoming waves.

For the trapped waves, the eigenfrequencies decrease when *S* decreases and approach the value (2*n* + 1)^{−1} when *S* approaches zero where *n* is the mode number. The modal structure broadens as *S* increases to some critical value above which no such coastally trapped modes exist.

## Abstract

In previous studies of tidal generation of mean flow over varying topography, the rectification mechanism has generally invoked bottom friction as a source of tidal flux of momentum and vorticity (hence referred as“friction” mechanism). The author proposes a different mechanism based on horizontal mixing of potential vorticity. Drawing analogy from tidal dispersion of passive tracers, this mixing is parameterized through a diffusivity (hence called “diffusivity” mechanism) that is quadratic in the tidal amplitude. In this, Part 1, the mean along-isobath flow near a shelf break is determined for a homogeneous ocean and contrasted with that induced by friction mechanism. In Part 2, the effect of a front will be considered.

It is found that although the mean flow is pointing in the same direction as that induced by friction mechanism (i.e., to the right when facing deep water in the Northern Hemisphere), it varies more slowly with the tidal amplitude. In the typical situation when the mean shear is small compared with the Coriolis parameter, this dependence is linear rather than quadratic, as is the case for the friction mechanism. This linear dependence compares more favorably with observations over Georges Bank.

## Abstract

In previous studies of tidal generation of mean flow over varying topography, the rectification mechanism has generally invoked bottom friction as a source of tidal flux of momentum and vorticity (hence referred as“friction” mechanism). The author proposes a different mechanism based on horizontal mixing of potential vorticity. Drawing analogy from tidal dispersion of passive tracers, this mixing is parameterized through a diffusivity (hence called “diffusivity” mechanism) that is quadratic in the tidal amplitude. In this, Part 1, the mean along-isobath flow near a shelf break is determined for a homogeneous ocean and contrasted with that induced by friction mechanism. In Part 2, the effect of a front will be considered.

It is found that although the mean flow is pointing in the same direction as that induced by friction mechanism (i.e., to the right when facing deep water in the Northern Hemisphere), it varies more slowly with the tidal amplitude. In the typical situation when the mean shear is small compared with the Coriolis parameter, this dependence is linear rather than quadratic, as is the case for the friction mechanism. This linear dependence compares more favorably with observations over Georges Bank.

## Abstract

A two-dimensional, two-layered frontal system is used to examine the wind-driven motion near a shelf-slope front. In the linear regime, the along-frontal current is characterized by barotropic perturbations. The front is dynamically passive and displaced according to purely kinematic constraints. The nonlinear solution shows that, even for a relatively small Rossby number, the frontal response to the oppositely directed along-frontal winds is highly asymmetric. When the wind is such that it forces surface water offshore, the model predicts ridging of the frontal interface, resembling some hydrographic observations. The model results suggest that the topographic shoaling of the deep onshore flow causes the generation of a cyclonic shear which, in a nonlinear regime, produces the observed ridging through geostrophic balance. It is reasoned that the increased entrainment above the pycnocline ridge could cut off the offshore shelf water and result in its export to the slope water regime. On the other hand, the apparent rigidity of the front as the surface water moves shoreward suggests a relative ineffectiveness for the surface slope water to penetrate through the frontal zone and contribute to mass or property balances on the shelf.

## Abstract

A two-dimensional, two-layered frontal system is used to examine the wind-driven motion near a shelf-slope front. In the linear regime, the along-frontal current is characterized by barotropic perturbations. The front is dynamically passive and displaced according to purely kinematic constraints. The nonlinear solution shows that, even for a relatively small Rossby number, the frontal response to the oppositely directed along-frontal winds is highly asymmetric. When the wind is such that it forces surface water offshore, the model predicts ridging of the frontal interface, resembling some hydrographic observations. The model results suggest that the topographic shoaling of the deep onshore flow causes the generation of a cyclonic shear which, in a nonlinear regime, produces the observed ridging through geostrophic balance. It is reasoned that the increased entrainment above the pycnocline ridge could cut off the offshore shelf water and result in its export to the slope water regime. On the other hand, the apparent rigidity of the front as the surface water moves shoreward suggests a relative ineffectiveness for the surface slope water to penetrate through the frontal zone and contribute to mass or property balances on the shelf.

## Abstract

A global-mean model is used here to elucidate possible bounds on the surface temperature of a simplified ocean–atmosphere system. Extending previous one-dimensional models, it has included as internal variables the low-level and high-level cloud covers and the turbulent wind at the surface. The main hypothesis for the model closure is that the conversion rate from the solar to the kinetic energy—or, equivalently, the rate of internal entropy production—is maximized, which has been applied with considerable success in past latitudinal models. From the model derivation, it is found that the surface temperature is narrowly bounded below by the onset of the greenhouse effect and above by the rapid increase of the saturation vapor pressure. Because both are largely intrinsic properties of water, the resulting surface temperature is mostly insensitive to detailed balances or changing external conditions. Even with a 50% change of the solar constant from its present-day value, the model temperature has varied by only about 10 K. The reason that the heat balances can be maintained is an internal adjustment of the low cloud cover, which offsets the solar effect. The model offers a plausible explanation of an equable climate in the geological past so long as there is a substantial ocean.

## Abstract

A global-mean model is used here to elucidate possible bounds on the surface temperature of a simplified ocean–atmosphere system. Extending previous one-dimensional models, it has included as internal variables the low-level and high-level cloud covers and the turbulent wind at the surface. The main hypothesis for the model closure is that the conversion rate from the solar to the kinetic energy—or, equivalently, the rate of internal entropy production—is maximized, which has been applied with considerable success in past latitudinal models. From the model derivation, it is found that the surface temperature is narrowly bounded below by the onset of the greenhouse effect and above by the rapid increase of the saturation vapor pressure. Because both are largely intrinsic properties of water, the resulting surface temperature is mostly insensitive to detailed balances or changing external conditions. Even with a 50% change of the solar constant from its present-day value, the model temperature has varied by only about 10 K. The reason that the heat balances can be maintained is an internal adjustment of the low cloud cover, which offsets the solar effect. The model offers a plausible explanation of an equable climate in the geological past so long as there is a substantial ocean.

## Abstract

Through a simple model, it is demonstrated that earth's sphericity (the beta effect) imposes a severe constraint on the discharge pattern near the equator. Using either bottom or lateral friction to counter the beta effect in the vorticity balance, the flow in the far field is confined to boundary layers either along the solid boundaries that are open on the anticyclonic side or along the equator to the west of the point source. Thus, for all possible orientation of the radial boundaries flanking the point source, there are either one or two branches receptive of the discharge in the far field depending on whether the open angle spanned by the two boundaries excludes the east direction. Even when the branching is permissible, it is further argued, based on symmetry of the governing equation, that the outflow is strongly favored toward the branch that deviates more from the east direction. Numerical solutions show that the bulk of the diversion of the discharge to this branch occurs within one frictional scale of the point source. Since this distance, as crudely estimated, can be of order 100 km, the model can explain the sharp deflection of the Amazon outflow to the northern coast over the shelf proper without requiring asymmetric external forcings.

## Abstract

Through a simple model, it is demonstrated that earth's sphericity (the beta effect) imposes a severe constraint on the discharge pattern near the equator. Using either bottom or lateral friction to counter the beta effect in the vorticity balance, the flow in the far field is confined to boundary layers either along the solid boundaries that are open on the anticyclonic side or along the equator to the west of the point source. Thus, for all possible orientation of the radial boundaries flanking the point source, there are either one or two branches receptive of the discharge in the far field depending on whether the open angle spanned by the two boundaries excludes the east direction. Even when the branching is permissible, it is further argued, based on symmetry of the governing equation, that the outflow is strongly favored toward the branch that deviates more from the east direction. Numerical solutions show that the bulk of the diversion of the discharge to this branch occurs within one frictional scale of the point source. Since this distance, as crudely estimated, can be of order 100 km, the model can explain the sharp deflection of the Amazon outflow to the northern coast over the shelf proper without requiring asymmetric external forcings.

## Abstract

A reduced-gravity model is used to examine the dynamics of dense water descending a continental slope. The model solves for the geostrophically adjusted state before it is subjected to significant frictional decay. For such bottom-mounted flow, it is argued that frictional torque would dominate the net vorticity balance to equalize the edge flows, resulting in double velocity cores. Constrained by the geostrophic balance, the dense water thus may settle only over a concave bottom and is sheetlike, covering typically the whole slope rise. As such, the adjustment is characterized by a spreading rather than sinking of the layer—with little descent of the upper edge but a swift downslope current propelling the lower edge. Through the mechanical energy balance, it is found in addition that a greater density anomaly would increase the total entrainment flux to more strongly dilute the original anomaly, yielding a product water that is less varied in the water-mass properties. Model predictions compare favorably with some observed dense outflows, in support of the entrainment and friction control of the geostrophic adjustment.

## Abstract

A reduced-gravity model is used to examine the dynamics of dense water descending a continental slope. The model solves for the geostrophically adjusted state before it is subjected to significant frictional decay. For such bottom-mounted flow, it is argued that frictional torque would dominate the net vorticity balance to equalize the edge flows, resulting in double velocity cores. Constrained by the geostrophic balance, the dense water thus may settle only over a concave bottom and is sheetlike, covering typically the whole slope rise. As such, the adjustment is characterized by a spreading rather than sinking of the layer—with little descent of the upper edge but a swift downslope current propelling the lower edge. Through the mechanical energy balance, it is found in addition that a greater density anomaly would increase the total entrainment flux to more strongly dilute the original anomaly, yielding a product water that is less varied in the water-mass properties. Model predictions compare favorably with some observed dense outflows, in support of the entrainment and friction control of the geostrophic adjustment.

## Abstract

To demonstrate some effects of a seamount an oceanic flows, we have considered a uniform, two-layer flow passing a right circular cylinder of arbitrary height in a rotating fluid. In the case of vanishing stratification, we first generalize previous results of low obstacles to an obstacle of finite height, and then show how the frictional regime provides a transition from partial to total blocking as the obstacle top approaches the surface.

In the case of general stratification, we have discerned various dynamical regimes according to blockage of the flows, characterized by distinctive interface signatures. For example, as the obstacle top rises through the water column, the axisymmetric doming of the interface first gives way to a reduced fore-and-aft symmetry when the lower layer is partially blocked, then becomes a net depression when the lower layer is totally blocked, and finally returns to its unperturbed level as both layers become totally blocked. We have derived the critical stratification below which there may be overlapping Taylor columns, and hence possible ventilation of the lower layer if surface cooling occurs. For typical oceanic conditions, this critical stratification corresponds to a baroclinic deformation radius measuring about one-half of the obstacle radius.

By generalizing the model results to a multiple-layer fluid, we have deduced mesoscale features similar to that observed over the Emperor Seamounts. When the model is applied to the Maude Rise in the Weddell Sea, it can explain the extensive ventilation of the water above the rise, with possible implications on the initiation and maintenance of the Weddell polynya.

## Abstract

To demonstrate some effects of a seamount an oceanic flows, we have considered a uniform, two-layer flow passing a right circular cylinder of arbitrary height in a rotating fluid. In the case of vanishing stratification, we first generalize previous results of low obstacles to an obstacle of finite height, and then show how the frictional regime provides a transition from partial to total blocking as the obstacle top approaches the surface.

In the case of general stratification, we have discerned various dynamical regimes according to blockage of the flows, characterized by distinctive interface signatures. For example, as the obstacle top rises through the water column, the axisymmetric doming of the interface first gives way to a reduced fore-and-aft symmetry when the lower layer is partially blocked, then becomes a net depression when the lower layer is totally blocked, and finally returns to its unperturbed level as both layers become totally blocked. We have derived the critical stratification below which there may be overlapping Taylor columns, and hence possible ventilation of the lower layer if surface cooling occurs. For typical oceanic conditions, this critical stratification corresponds to a baroclinic deformation radius measuring about one-half of the obstacle radius.

By generalizing the model results to a multiple-layer fluid, we have deduced mesoscale features similar to that observed over the Emperor Seamounts. When the model is applied to the Maude Rise in the Weddell Sea, it can explain the extensive ventilation of the water above the rise, with possible implications on the initiation and maintenance of the Weddell polynya.

## Abstract

In the Antarctic, dense shelf water is formed in coastal polynyas and is differentiated from the fresher surface water by the wind-induced ice motion that displaces offshore the ice melt from production zones. Where the shelf water discharges into the deep ocean, the Antarctic Slope Front (ASF) is V shaped, separating the shelf and surface waters (referred to as “frontal” waters) from the Circumpolar Deep Water (CDW). To elucidate basic constraints on frontal properties, a minimal model of homogeneous water masses forced by offshore wind and freshwater input in a perpetual winter is considered. With the surface water stirred by—and hence aligned with—the ice cover, there is little leakage of ice or meltwater from the frontal system, so ice production and melt merely redistribute heat and salt between frontal waters. As such, the heat loss to the atmosphere needs to be supplied by entraining CDW, which then necessitates the shelf water discharge on account of the mass balance. Because of the freshwater input, the discharged shelf water may not be saltier than the CDW, which thus may descend the slope to form bottom water only because of its coldness. With both frontal waters cooled to the freezing point, dominant balances are formulated to determine their salinity and exchange rate with the ambient CDW. Although extremely crude, the model derivations are favorably compared with observations, which thus may provide physically based parameterizations for the bottom-water formation that can be incorporated into global models.

## Abstract

In the Antarctic, dense shelf water is formed in coastal polynyas and is differentiated from the fresher surface water by the wind-induced ice motion that displaces offshore the ice melt from production zones. Where the shelf water discharges into the deep ocean, the Antarctic Slope Front (ASF) is V shaped, separating the shelf and surface waters (referred to as “frontal” waters) from the Circumpolar Deep Water (CDW). To elucidate basic constraints on frontal properties, a minimal model of homogeneous water masses forced by offshore wind and freshwater input in a perpetual winter is considered. With the surface water stirred by—and hence aligned with—the ice cover, there is little leakage of ice or meltwater from the frontal system, so ice production and melt merely redistribute heat and salt between frontal waters. As such, the heat loss to the atmosphere needs to be supplied by entraining CDW, which then necessitates the shelf water discharge on account of the mass balance. Because of the freshwater input, the discharged shelf water may not be saltier than the CDW, which thus may descend the slope to form bottom water only because of its coldness. With both frontal waters cooled to the freezing point, dominant balances are formulated to determine their salinity and exchange rate with the ambient CDW. Although extremely crude, the model derivations are favorably compared with observations, which thus may provide physically based parameterizations for the bottom-water formation that can be incorporated into global models.

## Abstract

Motivated by the observed branching of the Equatorial Undercurrent in the Gulf of Guinea, an idealized model is developed here to examine the termination of an equatorial jet in a gulf. Similarity solutions are, found that can satisfy the boundary conditions along an idealized gulf coast and hence represent realistic flow fields. It is found that friction tends to retard the northern branch, resulting in a southward flow at the equator, and that as the incident jet intensifies, the cross-equatorial flow is increasingly prohibited, resulting in a greater return flow within the gulf.

## Abstract

Motivated by the observed branching of the Equatorial Undercurrent in the Gulf of Guinea, an idealized model is developed here to examine the termination of an equatorial jet in a gulf. Similarity solutions are, found that can satisfy the boundary conditions along an idealized gulf coast and hence represent realistic flow fields. It is found that friction tends to retard the northern branch, resulting in a southward flow at the equator, and that as the incident jet intensifies, the cross-equatorial flow is increasingly prohibited, resulting in a greater return flow within the gulf.

## Abstract

To provide possible dynamical interpretations of the GulfStream-induced circulation in the Middle Atlantic Bight (MAB), the inshore flow driven by a steady and straight jet in a homogeneous ocean is considered via similarity solutions.

To isolate the curvature effect of the coastal boundary, the author first considers the case of a constant-depth ocean, from which various nonlinear flow regimes are discerned. When applied to the MAB, the model can explain the observed intrusion of the Gulf Stream water just downstream of Cape Hatteras where the coastline curves convexly.

Over larger scales of the MAB, the scale analysis suggests the importance of the topography in the vorticity balance. When the topography is included, the similarity solution shows the strong flow to be confined offshore, flanked inshore by a weak counterflow, consistent with the observed slope sea gyre. There is in addition a flow convergence toward the inshore edge of the jet, consistent with the observed occurrence of the shelf water there and the inferred shoreward flux of nutrients across the jet axis.

## Abstract

To provide possible dynamical interpretations of the GulfStream-induced circulation in the Middle Atlantic Bight (MAB), the inshore flow driven by a steady and straight jet in a homogeneous ocean is considered via similarity solutions.

To isolate the curvature effect of the coastal boundary, the author first considers the case of a constant-depth ocean, from which various nonlinear flow regimes are discerned. When applied to the MAB, the model can explain the observed intrusion of the Gulf Stream water just downstream of Cape Hatteras where the coastline curves convexly.

Over larger scales of the MAB, the scale analysis suggests the importance of the topography in the vorticity balance. When the topography is included, the similarity solution shows the strong flow to be confined offshore, flanked inshore by a weak counterflow, consistent with the observed slope sea gyre. There is in addition a flow convergence toward the inshore edge of the jet, consistent with the observed occurrence of the shelf water there and the inferred shoreward flux of nutrients across the jet axis.