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## Abstract

A simple theory is presented for steady geostrophic circulation of a stratified fluid in a rectangular basin with a circumpolar connection. The interior flow obeys the *β*-plane Sverdrup vorticity balance, and the circulation is closed by geostrophic boundary currents. The circulation is forced by surface thermal gradients and wind-driven Ekman transport near the latitudes of the circumpolar connection. A thermal circumpolar current arises in response to imposed surface thermal gradients and northward Ekman transport across the gap latitudes. The transport of this model circumpolar current depends on the imposed surface thermal gradients and the gap geometry, but not on the strength of the wind forcing. In contrast, the circulation induced in a related reduced-gravity model by Sverdrup transport into the gap latitudes has zero zonally integrated zonal transport. The thermal current arises as a consequence of the geostrophic constraint, which requires that the northern region fill with warm fluid until it reaches the sill depth, where return geostrophic flow can be supported. Thus, the structure of the middepth, midlatitude thermocline is directly influenced by the geometry of the gap. A similar constraint evidently operates in the Southern Ocean.

## Abstract

A simple theory is presented for steady geostrophic circulation of a stratified fluid in a rectangular basin with a circumpolar connection. The interior flow obeys the *β*-plane Sverdrup vorticity balance, and the circulation is closed by geostrophic boundary currents. The circulation is forced by surface thermal gradients and wind-driven Ekman transport near the latitudes of the circumpolar connection. A thermal circumpolar current arises in response to imposed surface thermal gradients and northward Ekman transport across the gap latitudes. The transport of this model circumpolar current depends on the imposed surface thermal gradients and the gap geometry, but not on the strength of the wind forcing. In contrast, the circulation induced in a related reduced-gravity model by Sverdrup transport into the gap latitudes has zero zonally integrated zonal transport. The thermal current arises as a consequence of the geostrophic constraint, which requires that the northern region fill with warm fluid until it reaches the sill depth, where return geostrophic flow can be supported. Thus, the structure of the middepth, midlatitude thermocline is directly influenced by the geometry of the gap. A similar constraint evidently operates in the Southern Ocean.

## Abstract

The growth of linear disturbances to stable and unstable time-periodic basic states is analyzed in an asymptotic model of weakly nonlinear, baroclinic wave–mean interaction. In this model, an ordinary differential equation for the wave amplitude is coupled to a partial differential equation for the zonal-flow correction. Floquet vectors, the eigenmodes for linear disturbances to the oscillatory basic states, split into wave-dynamical and decaying zonal-flow modes. Singular vectors reflect the structure of the Floquet vectors: the most rapid amplification and decay are associated with the wave-dynamical Floquet vectors, while the intermediate singular vectors closely follow the decaying zonal-flow Floquet vectors. Singular values depend strongly on initial and optimization times. For initial times near wave amplitude maxima, the Floquet decomposition of the leading singular vector depends relatively weakly on optimization time. For the unstable oscillatory basic state in the chaotic regime, the leading Floquet vector is tangent to the large-scale structure of the attractor, while the leading singular vector is not. However, corresponding inferences about the accessibility of disturbed states rely on the simple attractor geometry, and may not easily generalize. The primary mechanism of disturbance growth on the wave timescale in this model involves a time-dependent phase shift along the basic wave cycle. The Floquet vectors illustrate that modal disturbances to time-dependent basic states can have time-dependent spatial structure, and that the latter need not indicate nonmodal dynamics. The dynamical splitting reduces the “butterfly effect,” the ability of small-scale disturbances to influence the evolution of an unstable large-scale flow.

## Abstract

The growth of linear disturbances to stable and unstable time-periodic basic states is analyzed in an asymptotic model of weakly nonlinear, baroclinic wave–mean interaction. In this model, an ordinary differential equation for the wave amplitude is coupled to a partial differential equation for the zonal-flow correction. Floquet vectors, the eigenmodes for linear disturbances to the oscillatory basic states, split into wave-dynamical and decaying zonal-flow modes. Singular vectors reflect the structure of the Floquet vectors: the most rapid amplification and decay are associated with the wave-dynamical Floquet vectors, while the intermediate singular vectors closely follow the decaying zonal-flow Floquet vectors. Singular values depend strongly on initial and optimization times. For initial times near wave amplitude maxima, the Floquet decomposition of the leading singular vector depends relatively weakly on optimization time. For the unstable oscillatory basic state in the chaotic regime, the leading Floquet vector is tangent to the large-scale structure of the attractor, while the leading singular vector is not. However, corresponding inferences about the accessibility of disturbed states rely on the simple attractor geometry, and may not easily generalize. The primary mechanism of disturbance growth on the wave timescale in this model involves a time-dependent phase shift along the basic wave cycle. The Floquet vectors illustrate that modal disturbances to time-dependent basic states can have time-dependent spatial structure, and that the latter need not indicate nonmodal dynamics. The dynamical splitting reduces the “butterfly effect,” the ability of small-scale disturbances to influence the evolution of an unstable large-scale flow.

## Abstract

The influence of localized regions of intensified vertical mixing on the stratification and circulation in a large-scale ocean model is investigated with idealized numerical experiments. Numerical solutions are obtained of a closed-basin, single-hemisphere ocean model based on the planetary geostrophic equations. Mesoscale eddy effects are minimized, and vertical mixing at the turbulent microscale is represented by a vertical diffusivity *κ*
_{υ}. Solutions with uniform *κ*
_{υ} are contrasted with a “localized mixing” solution, in which *κ*
_{υ} increases by two orders of magnitude from its interior value (0.2 × 10^{−4} m^{2} s^{−1}) in a region 500 km wide adjacent to the vertical eastern boundary. When *κ*
_{υ} is uniform, the stratification beneath the ventilated thermocline is characterized by a single vertical scale. In contrast, the localized vertical mixing supports a deep diffusive thermocline with two distinct vertical scales: an internal boundary layer centered at the base of the ventilated thermocline (roughly 1000-m depth) and an abyssal thermocline whose vertical scale is set in the region of large *κ*
_{υ}. This stratification is qualitatively similar to observed deep ocean stratification. In contrast to the Stommel–Arons meridional abyssal flow that arises in the model when *κ*
_{υ} is uniform and small, the localized mixing solution has primarily zonal flow in the abyssal interior, with meridional motion confined to boundary layers. An advective–diffusive balance is established in the region of enhanced mixing. The near-surface circulation is dominated by westward zonal flow in the southern half of the interior, northward flow along the western boundary, and eastward flow in the northern half of the interior, while the pattern of flow in the abyssal interior is essentially the reverse. The circulation is closed by upwelling in the mixing region and downwelling along the northern boundary. Meridional motion in the mixing region is consistent with the Sverdrup vorticity balance, with northward flow at depth and southward flow near the surface. The source water for the deep circulation is confined to a narrow range of the coldest temperature classes in the basin, while the middepth subtropical thermocline is filled with warmer deep water that enters the gyre as cold deep water and then is modified in the eastern mixing region.

## Abstract

The influence of localized regions of intensified vertical mixing on the stratification and circulation in a large-scale ocean model is investigated with idealized numerical experiments. Numerical solutions are obtained of a closed-basin, single-hemisphere ocean model based on the planetary geostrophic equations. Mesoscale eddy effects are minimized, and vertical mixing at the turbulent microscale is represented by a vertical diffusivity *κ*
_{υ}. Solutions with uniform *κ*
_{υ} are contrasted with a “localized mixing” solution, in which *κ*
_{υ} increases by two orders of magnitude from its interior value (0.2 × 10^{−4} m^{2} s^{−1}) in a region 500 km wide adjacent to the vertical eastern boundary. When *κ*
_{υ} is uniform, the stratification beneath the ventilated thermocline is characterized by a single vertical scale. In contrast, the localized vertical mixing supports a deep diffusive thermocline with two distinct vertical scales: an internal boundary layer centered at the base of the ventilated thermocline (roughly 1000-m depth) and an abyssal thermocline whose vertical scale is set in the region of large *κ*
_{υ}. This stratification is qualitatively similar to observed deep ocean stratification. In contrast to the Stommel–Arons meridional abyssal flow that arises in the model when *κ*
_{υ} is uniform and small, the localized mixing solution has primarily zonal flow in the abyssal interior, with meridional motion confined to boundary layers. An advective–diffusive balance is established in the region of enhanced mixing. The near-surface circulation is dominated by westward zonal flow in the southern half of the interior, northward flow along the western boundary, and eastward flow in the northern half of the interior, while the pattern of flow in the abyssal interior is essentially the reverse. The circulation is closed by upwelling in the mixing region and downwelling along the northern boundary. Meridional motion in the mixing region is consistent with the Sverdrup vorticity balance, with northward flow at depth and southward flow near the surface. The source water for the deep circulation is confined to a narrow range of the coldest temperature classes in the basin, while the middepth subtropical thermocline is filled with warmer deep water that enters the gyre as cold deep water and then is modified in the eastern mixing region.

## Abstract

The diffusivity dependence of internal boundary layers in solutions of the continuously stratified, diffusive thermocline equations is revisited. If a solution exists that approaches a two-layer solution of the ideal thermocline equations in the limit of small vertical diffusivity *κ*
_{υ}, it must contain an internal boundary layer that collapses to a discontinuity as *κ*
_{υ} → 0. An asymptotic internal boundary layer equation is derived for this case, and the associated boundary layer thickness is proportional to *κ*^{1/2}_{υ}*κ*
_{υ} → 0.

## Abstract

The diffusivity dependence of internal boundary layers in solutions of the continuously stratified, diffusive thermocline equations is revisited. If a solution exists that approaches a two-layer solution of the ideal thermocline equations in the limit of small vertical diffusivity *κ*
_{υ}, it must contain an internal boundary layer that collapses to a discontinuity as *κ*
_{υ} → 0. An asymptotic internal boundary layer equation is derived for this case, and the associated boundary layer thickness is proportional to *κ*^{1/2}_{υ}*κ*
_{υ} → 0.

## Abstract

Two idealized, three-dimensional, analytical models of middepth meridional overturning in a basin with a Southern Hemisphere circumpolar connection are described. In the first, the overturning circulation can be understood as a “pump and valve” system, in which the wind forcing at the latitudes of the circumpolar connection is the pump and surface thermodynamic exchange at high northern latitudes is the valve. When the valve is on, the overturning circulation extends to the extreme northern latitudes of the basin, and the middepth thermocline is cold. When the valve is off, the overturning circulation is short-circuited and confined near the circumpolar connection, and the middepth thermocline is warm. The meridional overturning cell in this first model is not driven by turbulent mixing, and the subsurface circulation is adiabatic. In contrast, the pump that primarily drives the overturning cell in the second model is turbulent mixing, at low and midlatitudes, in the ocean interior. In both models, however, the depth of the midlatitude deep layer is controlled by the sill depth of the circumpolar gap. The thermocline structures in these two models are nearly indistinguishable. These models suggest that Northern Hemisphere wind and surface buoyancy forcing may influence the strength and structure of the circumpolar current in the Southern Hemisphere.

## Abstract

Two idealized, three-dimensional, analytical models of middepth meridional overturning in a basin with a Southern Hemisphere circumpolar connection are described. In the first, the overturning circulation can be understood as a “pump and valve” system, in which the wind forcing at the latitudes of the circumpolar connection is the pump and surface thermodynamic exchange at high northern latitudes is the valve. When the valve is on, the overturning circulation extends to the extreme northern latitudes of the basin, and the middepth thermocline is cold. When the valve is off, the overturning circulation is short-circuited and confined near the circumpolar connection, and the middepth thermocline is warm. The meridional overturning cell in this first model is not driven by turbulent mixing, and the subsurface circulation is adiabatic. In contrast, the pump that primarily drives the overturning cell in the second model is turbulent mixing, at low and midlatitudes, in the ocean interior. In both models, however, the depth of the midlatitude deep layer is controlled by the sill depth of the circumpolar gap. The thermocline structures in these two models are nearly indistinguishable. These models suggest that Northern Hemisphere wind and surface buoyancy forcing may influence the strength and structure of the circumpolar current in the Southern Hemisphere.

## Abstract

The motion of fluid parcels in a two-dimensional kinematic model of a meandering jet is investigated using Melnikov's method. The study is motivated by a recent analysis of float trajectories in the Gulf Stream. The results indicate that the efficiency of cross-jet exchange induced by fluctuating meander amplitudes depends strongly on the frequency of the fluctuations. For high frequencies (≳0.04 cpd), exchange between the core of the jet and regions of “trapped” fluid moving with the meander is preferred, while for low frequencies (≲0.04 cpd), exchange between the “trapped” fluid and the slow-moving fluid surrounding the jet is preferred. Propagating waves superimposed on the meandering jet can efficiently cause exchange between regimes when their phase speeds roughly match the basic flow velocities along the regime boundaries. Numerical results suggest that exchange across the center of the jet is less efficient than exchange between adjacent regimes so that the meandering jet will tend to stir fluid along each of its sides but preserve gradients across the jet core.

## Abstract

The motion of fluid parcels in a two-dimensional kinematic model of a meandering jet is investigated using Melnikov's method. The study is motivated by a recent analysis of float trajectories in the Gulf Stream. The results indicate that the efficiency of cross-jet exchange induced by fluctuating meander amplitudes depends strongly on the frequency of the fluctuations. For high frequencies (≳0.04 cpd), exchange between the core of the jet and regions of “trapped” fluid moving with the meander is preferred, while for low frequencies (≲0.04 cpd), exchange between the “trapped” fluid and the slow-moving fluid surrounding the jet is preferred. Propagating waves superimposed on the meandering jet can efficiently cause exchange between regimes when their phase speeds roughly match the basic flow velocities along the regime boundaries. Numerical results suggest that exchange across the center of the jet is less efficient than exchange between adjacent regimes so that the meandering jet will tend to stir fluid along each of its sides but preserve gradients across the jet core.

## Abstract

A recently proposed reduced-gravity model of the warm-water branch of the middepth meridional overturning circulation in a rectangular basin with a circumpolar connection is extended to include time dependence. The model describes the balance between gain of warm water through northward Ekman advection across the circumpolar current, loss of warm water through eddy fluxes southward across the current, net gain or loss of warm water through diabatic processes north of the current, and changes in the thickness of the warm-water layer. The steady solutions are the same as those found previously, when the previous parameterization of diabatic fluxes is used. Time-dependent solutions are considered for the approach of the solution to a new equilibrium when the forcing or parameters are abruptly changed and then held fixed. An initial adjustment occurs through a combination of boundary and equatorial adjustment, followed by planetary wave propagation. The longer-term adjustment to equilibrium consists primarily of the slow change in eastern boundary thickness of the warm layer, which controls the mean depth of the entire layer. An approximate analytical solution of the time-dependent equations yields an explicit expression for the intrinsic time scale of the long-term adjustment, which depends on the eddy and diabatic flux parameters and on the equilibrium solution toward which the time-dependent solution adjusts. Numerical solutions are also considered with a second, advective–diffusive diabatic flux parameterization. These solutions differ quantitatively but not qualitatively from those with the original parameterization. For the range of parameter values considered, the adjustment time scale has dimensional values of several decades to more than a century, but the meridional flux of warm water may respond to changes in external parameters or forcing much more rapidly than this time scale for equilibration of the eastern boundary thickness and thermocline structure.

## Abstract

A recently proposed reduced-gravity model of the warm-water branch of the middepth meridional overturning circulation in a rectangular basin with a circumpolar connection is extended to include time dependence. The model describes the balance between gain of warm water through northward Ekman advection across the circumpolar current, loss of warm water through eddy fluxes southward across the current, net gain or loss of warm water through diabatic processes north of the current, and changes in the thickness of the warm-water layer. The steady solutions are the same as those found previously, when the previous parameterization of diabatic fluxes is used. Time-dependent solutions are considered for the approach of the solution to a new equilibrium when the forcing or parameters are abruptly changed and then held fixed. An initial adjustment occurs through a combination of boundary and equatorial adjustment, followed by planetary wave propagation. The longer-term adjustment to equilibrium consists primarily of the slow change in eastern boundary thickness of the warm layer, which controls the mean depth of the entire layer. An approximate analytical solution of the time-dependent equations yields an explicit expression for the intrinsic time scale of the long-term adjustment, which depends on the eddy and diabatic flux parameters and on the equilibrium solution toward which the time-dependent solution adjusts. Numerical solutions are also considered with a second, advective–diffusive diabatic flux parameterization. These solutions differ quantitatively but not qualitatively from those with the original parameterization. For the range of parameter values considered, the adjustment time scale has dimensional values of several decades to more than a century, but the meridional flux of warm water may respond to changes in external parameters or forcing much more rapidly than this time scale for equilibration of the eastern boundary thickness and thermocline structure.

## Abstract

The vertical structure of coastal-trapped disturbances in several idealized models of a stably stratified lower atmosphere is examined. The vertical structure and phase speeds of the trapped modes depend on the resting stratification and the height of the orographic step. The presence of a stable layer above the boundary layer inversion increases the gravest-mode phase speed and supports the existence of higher vertical modes. Trapped wave solutions for the step orography are obtained for a lower atmosphere with constant buoyancy frequency. The modes are primarily concentrated below the step but penetrate weakly into the stratified region above the step. The phase speed of the gravest trapped mode is greater than the gravest-mode Kelvin wave speed based on the height of the step. Results from a linear two-layer model suggest that the observed vertical structure of isotherms at the leading edge of a 10–11 June 1994 event may arise during a transition from a directly forced, barotropic, alongshore velocity response to a regime influenced by wave propagation, as the coastal-trapped vertical modes excited by the mesoscale pressure gradients begin to disperse at their respective phase speeds. The results suggest also that the observed vertical structure of alongshore velocity, with largest velocities in the stable layer above the boundary layer, may arise from drag at the sea surface.

## Abstract

The vertical structure of coastal-trapped disturbances in several idealized models of a stably stratified lower atmosphere is examined. The vertical structure and phase speeds of the trapped modes depend on the resting stratification and the height of the orographic step. The presence of a stable layer above the boundary layer inversion increases the gravest-mode phase speed and supports the existence of higher vertical modes. Trapped wave solutions for the step orography are obtained for a lower atmosphere with constant buoyancy frequency. The modes are primarily concentrated below the step but penetrate weakly into the stratified region above the step. The phase speed of the gravest trapped mode is greater than the gravest-mode Kelvin wave speed based on the height of the step. Results from a linear two-layer model suggest that the observed vertical structure of isotherms at the leading edge of a 10–11 June 1994 event may arise during a transition from a directly forced, barotropic, alongshore velocity response to a regime influenced by wave propagation, as the coastal-trapped vertical modes excited by the mesoscale pressure gradients begin to disperse at their respective phase speeds. The results suggest also that the observed vertical structure of alongshore velocity, with largest velocities in the stable layer above the boundary layer, may arise from drag at the sea surface.

## Abstract

A model for hydraulically supercritical atmospheric marine-layer flow along a smoothly varying coastline is formulated and solved numerically. The model is motivated by a recent comparison of CODE observations to a simple hydraulic theory, which suggested the presence of an expansion fan and a compression jump downstream of topographic features. The marine layer is modeled as a homogeneous rotating fluid layer decelerated by surface friction and forced by imposed upper-level pressure gradients. The equations are solved by a characteristic-based gridpoint scheme. The results indicate that the expansion fan is a robust feature that persists under most conditions in the present more realistic model, but is dramatically altered in structure by the presence of friction, while the jump may weaken rapidly offshore due mainly to offshore variations of the layer height upstream of the jump. The agreement between observations and model predictions is good enough to suggest that a first-order description of the dynamics has been attained in which friction dramatically alters the character of the supercritical flow features. The supercritical flow features cause variations in wind stress of 10%–50% over tens of kilometers.

## Abstract

A model for hydraulically supercritical atmospheric marine-layer flow along a smoothly varying coastline is formulated and solved numerically. The model is motivated by a recent comparison of CODE observations to a simple hydraulic theory, which suggested the presence of an expansion fan and a compression jump downstream of topographic features. The marine layer is modeled as a homogeneous rotating fluid layer decelerated by surface friction and forced by imposed upper-level pressure gradients. The equations are solved by a characteristic-based gridpoint scheme. The results indicate that the expansion fan is a robust feature that persists under most conditions in the present more realistic model, but is dramatically altered in structure by the presence of friction, while the jump may weaken rapidly offshore due mainly to offshore variations of the layer height upstream of the jump. The agreement between observations and model predictions is good enough to suggest that a first-order description of the dynamics has been attained in which friction dramatically alters the character of the supercritical flow features. The supercritical flow features cause variations in wind stress of 10%–50% over tens of kilometers.

## Abstract

The generation of continental shelf currents by wind forcing is investigated by analytical and numerical methods. The investigation is motivated by observations from the Coastal Ocean Dynamics Experiment. A central assumption is that the vertical structure of the response over the inshore half of the shelf is controlled by the vertical distribution of the turbulent stress. This suggests a two-layer model of the wind-driven circulation, in which the upper layer represents a surface wind-mixed layer, and the lower layer represents the remainder of the fluid. The response of this idealized dynamical model to wind forcing is examined and compared with observations in the 2–7-day period band. For the alongshore velocity gain relative to local wind stress, an onshore surface maximum and an offshore interior maximum are robustly reproduced by the model. These features are evidently related to a dynamical transition over the inner half of the shelf, in which the alongshore wind stress is balanced more by acceleration of near-surface alongshore flow and less by time-dependent Ekman transport as the coast is approached. This differs from a previous hypothesis, based on a linear model in which the turbulent stress was confined to infinitesimally thin surface and bottom boundary layers, which related the alongshore flow structure to the cross-shore profile of the alongshore wind amplitude. In the present model, the cross-shore velocity variances are roughly comparable to those observed over the onshore half of the shelf. This also contrasts with the previous model results, which underpredicted cross-shore velocity variances by more than an order of magnitude. However, the present agreement is probably fortuitous, as the enhanced lower-layer cross-shore flow is frictionally driven, and should probably be confined to a bottom boundary layer as it was in the previous model. The results demonstrate that the response of these models over the inner half of the shelf depends strongly on a poorly understood coastal boundary condition.

## Abstract

The generation of continental shelf currents by wind forcing is investigated by analytical and numerical methods. The investigation is motivated by observations from the Coastal Ocean Dynamics Experiment. A central assumption is that the vertical structure of the response over the inshore half of the shelf is controlled by the vertical distribution of the turbulent stress. This suggests a two-layer model of the wind-driven circulation, in which the upper layer represents a surface wind-mixed layer, and the lower layer represents the remainder of the fluid. The response of this idealized dynamical model to wind forcing is examined and compared with observations in the 2–7-day period band. For the alongshore velocity gain relative to local wind stress, an onshore surface maximum and an offshore interior maximum are robustly reproduced by the model. These features are evidently related to a dynamical transition over the inner half of the shelf, in which the alongshore wind stress is balanced more by acceleration of near-surface alongshore flow and less by time-dependent Ekman transport as the coast is approached. This differs from a previous hypothesis, based on a linear model in which the turbulent stress was confined to infinitesimally thin surface and bottom boundary layers, which related the alongshore flow structure to the cross-shore profile of the alongshore wind amplitude. In the present model, the cross-shore velocity variances are roughly comparable to those observed over the onshore half of the shelf. This also contrasts with the previous model results, which underpredicted cross-shore velocity variances by more than an order of magnitude. However, the present agreement is probably fortuitous, as the enhanced lower-layer cross-shore flow is frictionally driven, and should probably be confined to a bottom boundary layer as it was in the previous model. The results demonstrate that the response of these models over the inner half of the shelf depends strongly on a poorly understood coastal boundary condition.