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- Author or Editor: Parker MacCready x

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## Abstract

Subtidal adjustment of estuarine salinity and circulation to changing river flow or tidal mixing is explored using a simplified numerical model. The model employs tidally averaged, width-averaged physics, following Hansen and Rattray, extended to include 1) time dependence, 2) tidally averaged mixing parameterizations, and 3) arbitrary variation of channel depth and width. By linearizing the volume-integrated salt budget, the time-dependent system may be distilled to a first-order, forced, damped, ordinary differential equation. From this equation, analytical expressions for the adjustment time and sensitivity of the length of the salt intrusion are developed. For estuaries in which the up-estuary salt flux is dominated by vertically segregated gravitational circulation, this adjustment time is predicted to be *T*
_{ADJ} = (1/6)*L*/* u*, where

*L*is the length of the salt intrusion and

*is the section-averaged velocity (i.e., that due to the river flow). The importance of the adjustment time becomes apparent when considering forcing time scales. Seasonal river-flow variation is much slower than typical adjustment times in systems such as the Hudson River estuary, and thus the response may be quasi steady. Spring–neap mixing variation, in contrast, has a period comparable to typical adjustment times, and so unsteady effects are more important. In this case, the stratification may change greatly while the salt intrusion is relatively unperturbed.*u

## Abstract

Subtidal adjustment of estuarine salinity and circulation to changing river flow or tidal mixing is explored using a simplified numerical model. The model employs tidally averaged, width-averaged physics, following Hansen and Rattray, extended to include 1) time dependence, 2) tidally averaged mixing parameterizations, and 3) arbitrary variation of channel depth and width. By linearizing the volume-integrated salt budget, the time-dependent system may be distilled to a first-order, forced, damped, ordinary differential equation. From this equation, analytical expressions for the adjustment time and sensitivity of the length of the salt intrusion are developed. For estuaries in which the up-estuary salt flux is dominated by vertically segregated gravitational circulation, this adjustment time is predicted to be *T*
_{ADJ} = (1/6)*L*/* u*, where

*L*is the length of the salt intrusion and

*is the section-averaged velocity (i.e., that due to the river flow). The importance of the adjustment time becomes apparent when considering forcing time scales. Seasonal river-flow variation is much slower than typical adjustment times in systems such as the Hudson River estuary, and thus the response may be quasi steady. Spring–neap mixing variation, in contrast, has a period comparable to typical adjustment times, and so unsteady effects are more important. In this case, the stratification may change greatly while the salt intrusion is relatively unperturbed.*u

## Abstract

A method for calculating subtidal estuarine exchange flow using an isohaline framework is described, and the results are compared with those of the more commonly used Eulerian method of salt flux decomposition. Concepts are explored using a realistic numerical simulation of the Columbia River estuary. The isohaline method is found to be advantageous because it intrinsically highlights the salinity classes in which subtidal volume flux occurs. The resulting expressions give rise to an exact formulation of the time-dependent Knudsen relation and may be used in calculation of the saltwater residence time. The volume flux of the landward transport, which can be calculated precisely using the isohaline framework, is of particular importance for problems in which the saltwater residence time is critical.

## Abstract

A method for calculating subtidal estuarine exchange flow using an isohaline framework is described, and the results are compared with those of the more commonly used Eulerian method of salt flux decomposition. Concepts are explored using a realistic numerical simulation of the Columbia River estuary. The isohaline method is found to be advantageous because it intrinsically highlights the salinity classes in which subtidal volume flux occurs. The resulting expressions give rise to an exact formulation of the time-dependent Knudsen relation and may be used in calculation of the saltwater residence time. The volume flux of the landward transport, which can be calculated precisely using the isohaline framework, is of particular importance for problems in which the saltwater residence time is critical.

## Abstract

The adjustment of estuarine circulation and density to changes in river flow and tidal mixing is investigated using analytical and numerical models. Tidally averaged momentum and salinity equations in a rectangular estuary are vertically averaged over two levels, resulting in equations that are analytically tractable while retaining a broad range of time-dependent behavior.

It is found that both strongly stratified and well-mixed estuaries respond rapidly to either type of forcing change, while those of intermediate stratification respond more slowly. Intermediate estuaries also have the greatest sensitivity to change.

Exchange flow dominates the up-estuary salt flux in strongly stratified cases. Changing the river flow in such cases leads to an internal wave propagating the length of the estuary, which accomplishes much of the adjustment. The internal wave speed thus controls the adjustment time. Increased tidal mixing in strongly stratified cases initially decreases the exchange flow contribution to up-estuary salt flux by decreasing both the stratification and the vertical current shear. However, the decreased up-estuary salt flux leads to a loss of total salt in the estuary, and hence a greater longitudinal salinity gradient. The increasing gradient eventually restores the exchange-flow salt flux to near its original value. Well-mixed solutions have an advective–diffusive balance between river flow and longitudinal tidal mixing. In these cases the adjustment time corresponds to the time it takes the depth-averaged flow to travel the length scale of the salt intrusion, a result that applies to both types of changes considered.

In all cases the adjustment depends upon the dynamical feedback between the longitudinal salt flux and the longitudinal salinity gradient, which varies as the estuary gains or loses total salt.

## Abstract

The adjustment of estuarine circulation and density to changes in river flow and tidal mixing is investigated using analytical and numerical models. Tidally averaged momentum and salinity equations in a rectangular estuary are vertically averaged over two levels, resulting in equations that are analytically tractable while retaining a broad range of time-dependent behavior.

It is found that both strongly stratified and well-mixed estuaries respond rapidly to either type of forcing change, while those of intermediate stratification respond more slowly. Intermediate estuaries also have the greatest sensitivity to change.

Exchange flow dominates the up-estuary salt flux in strongly stratified cases. Changing the river flow in such cases leads to an internal wave propagating the length of the estuary, which accomplishes much of the adjustment. The internal wave speed thus controls the adjustment time. Increased tidal mixing in strongly stratified cases initially decreases the exchange flow contribution to up-estuary salt flux by decreasing both the stratification and the vertical current shear. However, the decreased up-estuary salt flux leads to a loss of total salt in the estuary, and hence a greater longitudinal salinity gradient. The increasing gradient eventually restores the exchange-flow salt flux to near its original value. Well-mixed solutions have an advective–diffusive balance between river flow and longitudinal tidal mixing. In these cases the adjustment time corresponds to the time it takes the depth-averaged flow to travel the length scale of the salt intrusion, a result that applies to both types of changes considered.

In all cases the adjustment depends upon the dynamical feedback between the longitudinal salt flux and the longitudinal salinity gradient, which varies as the estuary gains or loses total salt.

## Abstract

Lee wave generation and horizontal flow separation in stratified flow along a slope, with corrugations or a ridge running directly downslope, are explored using analytical and numerical methods. Both of these processes are important to the drag on alongslope currents. The analytical solution for steady wave generation by stratified flow along a corrugated slope is extended to the evanescent flow regimes. There are two evanescent regimes, having intrinsic frequencies either above the buoyancy frequency *N* (fast flow), or below *N* sin(*a*) (slow flow), for nonrotating fluid and slope angle, *a.* Streamlines of the low speed evanescent solution tend to follow isobaths, while those of wave solutions tend to flow up over ridges and down in canyons. An analytical expression is developed for the wave drag felt by an isolated ridge on a slope. For a Gaussian ridge of alongslope length *L,* the drag becomes small when *U*/*LN* > 1 (the fast flow regime), or when *U*/(*LN* sin *a*) < 1/2 (the slow flow regime).

Numerical experiments are performed for stratified flow along a slope with an isolated ridge. The ridge height is varied and the pressure drag on the ridge evaluated for flow in the wave-generating and slow evanescent regimes. The slow-flow case gives rise to a large region of horizontal (isopycnal) flow separation with proportionally stronger relative vorticity in the wake than the wave-generating case. Pressure drag coefficients based on projected frontal area are similar for both cases, increase linearly with the corrugation amplitude, and level off around a value of 1–1.2.

## Abstract

Lee wave generation and horizontal flow separation in stratified flow along a slope, with corrugations or a ridge running directly downslope, are explored using analytical and numerical methods. Both of these processes are important to the drag on alongslope currents. The analytical solution for steady wave generation by stratified flow along a corrugated slope is extended to the evanescent flow regimes. There are two evanescent regimes, having intrinsic frequencies either above the buoyancy frequency *N* (fast flow), or below *N* sin(*a*) (slow flow), for nonrotating fluid and slope angle, *a.* Streamlines of the low speed evanescent solution tend to follow isobaths, while those of wave solutions tend to flow up over ridges and down in canyons. An analytical expression is developed for the wave drag felt by an isolated ridge on a slope. For a Gaussian ridge of alongslope length *L,* the drag becomes small when *U*/*LN* > 1 (the fast flow regime), or when *U*/(*LN* sin *a*) < 1/2 (the slow flow regime).

Numerical experiments are performed for stratified flow along a slope with an isolated ridge. The ridge height is varied and the pressure drag on the ridge evaluated for flow in the wave-generating and slow evanescent regimes. The slow-flow case gives rise to a large region of horizontal (isopycnal) flow separation with proportionally stronger relative vorticity in the wake than the wave-generating case. Pressure drag coefficients based on projected frontal area are similar for both cases, increase linearly with the corrugation amplitude, and level off around a value of 1–1.2.

## Abstract

Experiments are performed using a two-layer isopycnic numerical model in a zonal channel with a large meridional topographic ridge in the lower layer. The model is forced only by a steady meridional volume transport in the upper layer, and develops a current structure similar to the Antarctic Circumpolar Current. Meridional volume flux across time-mean geostrophic streamlines is found to be due to a combination of the geostrophic eddy bolus flux and the lateral Reynolds stress. The proportion of each depends on the strength of the forcing. The Reynolds stress increases with the forcing, while the bolus flux is relatively constant. Topography localizes the eddy fluxes at and downstream of the topography, where eddy energies are greatest. The strength of the zonal transport is governed by the onset of baroclinic instability and so is relatively insensitive to the strength of the meridional transport.

## Abstract

Experiments are performed using a two-layer isopycnic numerical model in a zonal channel with a large meridional topographic ridge in the lower layer. The model is forced only by a steady meridional volume transport in the upper layer, and develops a current structure similar to the Antarctic Circumpolar Current. Meridional volume flux across time-mean geostrophic streamlines is found to be due to a combination of the geostrophic eddy bolus flux and the lateral Reynolds stress. The proportion of each depends on the strength of the forcing. The Reynolds stress increases with the forcing, while the bolus flux is relatively constant. Topography localizes the eddy fluxes at and downstream of the topography, where eddy energies are greatest. The strength of the zonal transport is governed by the onset of baroclinic instability and so is relatively insensitive to the strength of the meridional transport.

## Abstract

The turbulent bottom boundary layer for rotating, stratified flow along a slope is explored through theory and numerical simulation. The model flow begins with a uniform current along constant-depth contours and with flat isopycnals intersecting the slope. The boundary layer is then allowed to evolve in time and in distance from the boundary. Ekman transport up or down the slope advects the initial density gradient, eventually giving rise to substantial buoyancy forces. The rearranged density structure opposes the cross-slope flow, causing the transport to decay exponentially from its initial value (given by Ekman theory) to near zero, over a time scale proportional to *f*/(*N*α)^{2}, where *f* is the Coriolis frequency, *N* is the buoyancy frequency, and α is the slope angle. The boundary stress slowing the along-slope flow decreases simultaneously, leading to a very “slippery” bottom boundary compared with that predicted by Ekman theory.

## Abstract

The turbulent bottom boundary layer for rotating, stratified flow along a slope is explored through theory and numerical simulation. The model flow begins with a uniform current along constant-depth contours and with flat isopycnals intersecting the slope. The boundary layer is then allowed to evolve in time and in distance from the boundary. Ekman transport up or down the slope advects the initial density gradient, eventually giving rise to substantial buoyancy forces. The rearranged density structure opposes the cross-slope flow, causing the transport to decay exponentially from its initial value (given by Ekman theory) to near zero, over a time scale proportional to *f*/(*N*α)^{2}, where *f* is the Coriolis frequency, *N* is the buoyancy frequency, and α is the slope angle. The boundary stress slowing the along-slope flow decreases simultaneously, leading to a very “slippery” bottom boundary compared with that predicted by Ekman theory.

## Abstract

A method is presented for calculating a complete, numerically closed, mechanical energy budget in a realistic simulation of circulation in a coastal–estuarine domain. The budget is formulated in terms of the “local” available potential energy (APE; ). The APE may be split up into two parts based on whether a water parcel has been displaced up or down relative to its rest depth. This decomposition clearly shows the different APE signatures of coastal upwelling (particles displaced up by wind) and the estuary (particles displaced down by mixing). Because the definition of APE is local in almost the same sense that kinetic energy is, this study may form meaningful integrals of reservoir and budget terms even over regions that have open boundaries. However, the choice of volume to use for calculation of the rest state is not unique and may influence the results. Complete volume-integrated energy budgets over shelf and estuary volumes in a realistic model of the northeast Pacific and Salish Sea give a new way to quantify the state of these systems and the physical forces that influence that state. On the continental shelf, upwelling may be quantified using APE, which is found to have order-one seasonal variation with an increase due to winds and decrease due to mixing. In the Salish Sea estuarine system, the APE has much less seasonal variation, and the magnitude of the most important forcing terms would take over 7 months to fully drain this energy.

## Abstract

A method is presented for calculating a complete, numerically closed, mechanical energy budget in a realistic simulation of circulation in a coastal–estuarine domain. The budget is formulated in terms of the “local” available potential energy (APE; ). The APE may be split up into two parts based on whether a water parcel has been displaced up or down relative to its rest depth. This decomposition clearly shows the different APE signatures of coastal upwelling (particles displaced up by wind) and the estuary (particles displaced down by mixing). Because the definition of APE is local in almost the same sense that kinetic energy is, this study may form meaningful integrals of reservoir and budget terms even over regions that have open boundaries. However, the choice of volume to use for calculation of the rest state is not unique and may influence the results. Complete volume-integrated energy budgets over shelf and estuary volumes in a realistic model of the northeast Pacific and Salish Sea give a new way to quantify the state of these systems and the physical forces that influence that state. On the continental shelf, upwelling may be quantified using APE, which is found to have order-one seasonal variation with an increase due to winds and decrease due to mixing. In the Salish Sea estuarine system, the APE has much less seasonal variation, and the magnitude of the most important forcing terms would take over 7 months to fully drain this energy.

## Abstract

In the few previous measurements of topographic form drag in the ocean, drag that is much larger than a typical bluff body drag estimate has been consistently found. In this work, theory combined with a numerical model of tidal flow around a headland in a channel gives insight into the mechanisms that create form drag in oscillating flow situations. The total form drag is divided into two parts: the inertial drag, which is derived from a local potential flow solution, and the separation drag, which accounts for flow features such as eddies. The inertial drag can have a large magnitude, yet it cannot do work on the flow because its phase is in quadrature with the velocity. The separation drag has a magnitude that is nearly equal to the bluff body drag and accounts for all of the energy removed from the flow by the topography. In addition, the dependence of the form drag on the tidal excursion distance and the aspect ratio of the headlands were determined with a series of numerical experiments. This theory explains why form drag can be so large in the ocean, and it provides a method for separating the pressure field into the parts that can and cannot extract energy from the flow.

## Abstract

In the few previous measurements of topographic form drag in the ocean, drag that is much larger than a typical bluff body drag estimate has been consistently found. In this work, theory combined with a numerical model of tidal flow around a headland in a channel gives insight into the mechanisms that create form drag in oscillating flow situations. The total form drag is divided into two parts: the inertial drag, which is derived from a local potential flow solution, and the separation drag, which accounts for flow features such as eddies. The inertial drag can have a large magnitude, yet it cannot do work on the flow because its phase is in quadrature with the velocity. The separation drag has a magnitude that is nearly equal to the bluff body drag and accounts for all of the energy removed from the flow by the topography. In addition, the dependence of the form drag on the tidal excursion distance and the aspect ratio of the headlands were determined with a series of numerical experiments. This theory explains why form drag can be so large in the ocean, and it provides a method for separating the pressure field into the parts that can and cannot extract energy from the flow.

## Abstract

Observational and model estimates of the form drag on Three Tree Point, a headland located in a tidal channel of Puget Sound, Washington, are presented. Subsurface, Three Tree Point is a sloping ridge. Tidal flow over this ridge gives rise to internal lee waves that lead to wave drag and enhanced mixing. At the same time, horizontal flow separation produces a headland eddy that distorts the surface height field in the lee of the point. Two observational methods for estimating the portion of the form drag associated with deformation of the surface height field, referred to here as the “external” form drag, are also introduced. Drogued drifters and ship-mounted acoustic current profiles from different days are used to indirectly map the flood-tide surface height field. Data are derived from a depth shallow enough that baroclinic pressure gradient forcing may be neglected, and yet deep enough that wind stress may also be ignored. This leaves an approximate balance between the acceleration and surface height pressure gradient, permitting, in this case, two independent estimates of the surface height (to within a constant). These fields are used to calculate the external form drag at the headland. Drag estimates from both observational datasets agree well. External form drag decreases offshore of the headland as expected, and is highly dependent on tidal phase, with maximum drag leading peak flood currents by 1–2 h at this location. Form drag is much larger than model estimates of the frictional drag, implying that it is the dominant mechanism extracting energy from the barotropic tide. A kinematic argument is also presented to show why the external form drag should increase in importance relative to the frictional drag as the topographic slope and tidal excursion increase.

## Abstract

Observational and model estimates of the form drag on Three Tree Point, a headland located in a tidal channel of Puget Sound, Washington, are presented. Subsurface, Three Tree Point is a sloping ridge. Tidal flow over this ridge gives rise to internal lee waves that lead to wave drag and enhanced mixing. At the same time, horizontal flow separation produces a headland eddy that distorts the surface height field in the lee of the point. Two observational methods for estimating the portion of the form drag associated with deformation of the surface height field, referred to here as the “external” form drag, are also introduced. Drogued drifters and ship-mounted acoustic current profiles from different days are used to indirectly map the flood-tide surface height field. Data are derived from a depth shallow enough that baroclinic pressure gradient forcing may be neglected, and yet deep enough that wind stress may also be ignored. This leaves an approximate balance between the acceleration and surface height pressure gradient, permitting, in this case, two independent estimates of the surface height (to within a constant). These fields are used to calculate the external form drag at the headland. Drag estimates from both observational datasets agree well. External form drag decreases offshore of the headland as expected, and is highly dependent on tidal phase, with maximum drag leading peak flood currents by 1–2 h at this location. Form drag is much larger than model estimates of the frictional drag, implying that it is the dominant mechanism extracting energy from the barotropic tide. A kinematic argument is also presented to show why the external form drag should increase in importance relative to the frictional drag as the topographic slope and tidal excursion increase.