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  • View in gallery

    The axial and lateral component of the Reynolds stress at a section located midchannel (x = 0.5) for α = 0.025, κ = 1.5. The view is toward the closed end of the basin: f and δ are noted above each pair of frames. Contour intervals are evenly spaced; shaded areas correspond to negative values.

  • View in gallery

    Plan view of the forcing transport terms for α = 0.025, κ = 1.5, and δ = 0.5. The components of the Stokes transports and are labeled as [Sx] and [Sy]. The closed end of the basin is at the top. Contour intervals are evenly spaced. Shaded areas correspond to negative values.

  • View in gallery

    As in Fig. 2, but of the axial and lateral component of the sea level gradient.

  • View in gallery

    Plan view of the transport streamfunction for three different values of δ, with and without rotation, for α = 0.025, κ = 1.5. The closed end of the basin is at the top. The arrows illustrate the direction of the flow. Contour intervals are evenly spaced; shaded areas correspond to negative values.

  • View in gallery

    Sections illustrating the Eulerian mean velocities at different axial positions in the basin for α = 0.025, κ = 1.5, f = 0, and for three different values of δ. The viewer is looking toward the closed end of the basin. The axial velocity is negative in the shaded area, and the axial velocity contour interval is 0.25. Lateral and vertical velocities are represented by arrows.

  • View in gallery

    As in Fig. 5, but with f = 0.5.

  • View in gallery

    As in Fig. 5, but for the Lagrangian velocities.

  • View in gallery

    As in Fig. 7, but with f = 0.5.

  • View in gallery

    The trajectory of fluid parcels released at the entrance (x = 0), at positions marked by the asterisk. For each value of δ and f, the left frame represents an end view of the basin and the right frame is a plan view with the closed end at the top of the frame. For f = 0 parcels are only released on the right side of the basin because of the symmetry of the flow around the midplane.

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Three-Dimensional Residual Tidal Circulation in an Elongated, Rotating Basin

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  • 1 Integrative Oceanography Division, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California
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Abstract

The three-dimensional residual circulation driven by tides in an elongated basin of arbitrary depth is described with a small amplitude, constant density model on the f plane. The inclusion of rotation fundamentally alters the residual flow. With rotation, fluid is drawn into the basin on the right side of an observer looking toward the closed end (in the Northern Hemisphere) and the return flow is on the opposite side. A lateral circulation is superposed on the axial flow, with upwelling over the deeper part of each section and downwelling near the sides. The residual flow is driven by a combination of advective terms, including the lateral advection of axial momentum associated with the Coriolis acceleration and Stokes forcing. Tidally averaged fluid parcel trajectories are determined by integrating the Lagrangian mean velocities. With or without rotation these trajectories vary considerably depending on small differences in initial position as well as on basin shape and other parameters of the problem.

Corresponding author address: Clinton D. Winant, Integrative Oceanography Division, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 92093. cdw@coast.ucsd.edu

Abstract

The three-dimensional residual circulation driven by tides in an elongated basin of arbitrary depth is described with a small amplitude, constant density model on the f plane. The inclusion of rotation fundamentally alters the residual flow. With rotation, fluid is drawn into the basin on the right side of an observer looking toward the closed end (in the Northern Hemisphere) and the return flow is on the opposite side. A lateral circulation is superposed on the axial flow, with upwelling over the deeper part of each section and downwelling near the sides. The residual flow is driven by a combination of advective terms, including the lateral advection of axial momentum associated with the Coriolis acceleration and Stokes forcing. Tidally averaged fluid parcel trajectories are determined by integrating the Lagrangian mean velocities. With or without rotation these trajectories vary considerably depending on small differences in initial position as well as on basin shape and other parameters of the problem.

Corresponding author address: Clinton D. Winant, Integrative Oceanography Division, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 92093. cdw@coast.ucsd.edu

1. Introduction

In a semienclosed basin tides drive both periodic fluctuations and a mean or residual flow that can be comparable in amplitude to the fluctuating component and is responsible for time-averaged circulation patterns and dispersion. The residual flow results from nonlinear interactions between fluctuating variables. When the nonlinear products are weak, the problem can be treated as a regular expansion in which the nonlinear problem is replaced by an ordered set of linear problems. Early analyses of tidally driven residual circulation (Abbott 1960; Johns 1970) were based on boundary layer approximations that may not be valid in shallow estuaries where the region of viscous influence may extend all the way from the bottom to the surface. Ianniello (1977, 1979) first overcame this limitation and derived two-dimensional analytic solutions for the residual flow in a constant-depth frictional basin. Solutions depend on the amplitude of the tide relative to the basin depth, on the length of the basin relative to the tidal wavelength, and on friction. Following Longuet-Higgins (1969), Ianniello (1977) distinguishes between the Eulerian mean velocity, the time average of the local velocity at some point, and the Lagrangian velocity, the time-averaged velocity of a fluid parcel. The mass transported in an unsteady flow is more readily described in terms of the Lagrangian velocity. In all cases considered by Ianniello (1977, 1979), the Eulerian residual currents are seaward almost everywhere, while the weaker Lagrangian current change direction with depth.

The assumption of invariance in the lateral direction is a major restriction on the result of Ianniello (1977, 1979). Robinson (1981) shows how the nonlinear interaction between fluctuating flows and variable topography can generate a residual circulation. Li and O’Donnell (1997, 2005) overcome the constant depth limitation by developing a two-dimensional model of the vertically integrated flow that includes laterally variable bathymetry. Their analysis shows that the vertically integrated residual circulation varies laterally and depends on the basin shape as well as the parameters identified by Ianniello (1977). For a channel that is shorter than a quarter wavelength, the residual flow is away from the ocean in the deep channels and in the opposite direction over the shoals, as observed in short tidal basins (Winant and Gutiérrez de Velasco 2003). For longer channels, there is a separate system of gyres near the open end that circulates in the opposite direction: water flows toward the ocean in the channels.

In the Ianniello (1977, 1979) analyses, the basin geometry results in lateral velocity being everywhere zero. The vertically integrated lateral velocity is included in the Li and O’Donnell (1997, 2005) analyses: however, the lateral advection of momentum is estimated as the product of the vertically averaged lateral velocity by the lateral gradient of the vertically averaged axial velocity. There is no obvious way to relate that product to the local lateral advection, the product of the local lateral velocity by the lateral gradient of the local axial velocity. Chant (2002) points out that, even if lateral flows are an order of magnitude less than axial flow, lateral gradients can be much larger than axial gradients when the basin is narrow, with the result that lateral advection can dominate advection in momentum or salt budgets. For example, Kalkwijk and Booij (1986) develop a theory, confirmed by laboratory experiments, that shows how channel curvature modifies the residual circulation. Geyer (1993) describes how the residual circulation in Vineyard Sound, Massachusetts, is induced by curvature. Chant and Wilson (1997) describe the effect of stratification on secondary flows induced by curvature. Lacy and Monismith (2001) describe the secondary circulation in a curved channel in San Francisco Bay. These studies and others demonstrate how local lateral flows can affect and even dominate advection. Chant (2002) notes that, in a straight channel, the earth’s rotation acts in much the same way as channel curvature. While in a depth-averaged sense the Coriolis acceleration can be balanced by a lateral pressure gradient (in a constant density channel), any shear in the axial flow will result in depth-dependent forcing of the lateral circulation. If axial and lateral velocities are of comparable magnitude, lateral momentum advection is expected to dominate the advective terms if the channel aspect ratio α (the ratio of the half-width to the length) is small.

At higher frequencies than the tides, Xu and Bowen (1994) describe a model of steady circulation driven by surface gravity waves and winds, in which both rotation and friction are included, as a way to unify the effect of rotation on surface waves (Hasselmann 1970) with the various effects of friction (Longuet-Higgins 1953, 1969). Their model reveals the existence of a mean residual transport driven by waves in both down-wave and along-crest directions, comparable in magnitude to the steady wind-forced flow. Since both rotation and friction have to be represented to describe steady flows driven by surface gravity waves, it is to be expected that the same is true for tidal currents.

Lerczak and Geyer (2004) use a numerical model to determine how lateral circulation affects the dynamics of the circulation in a stratified, rotating, semienclosed basin. For weak stratification, they show that advection due to lateral currents is a dominant term in both the along-channel and cross-channel momentum equations over a tidal cycle and for the tidally averaged equations. They argue that, for weak to moderate friction, rotation is expected to drive a lateral flow. The relative amplitude of the fluctuating lateral to axial flow is estimated as one-eighth times the ratio of the Coriolis frequency to the frequency of the tide. This lateral circulation results in a pronounced lateral asymmetry in the residual currents with stronger inflow on the right bank (when looking away from the ocean in the Northern Hemisphere). In the tidally averaged, axial momentum equation, lateral advection acts as a driving term for the exchange flow and can be larger than the axial pressure gradient.

This paper describes an analytical model of the three-dimensional Eulerian and Lagrangian mean velocities for weakly nonlinear viscous flows forced by the tides on the f plane. The central conclusion is that rotation fundamentally alters the residual circulation, compared to the f = 0 case, for weak to moderate friction. With rotation, the residual circulation flows into the basin on the right side of an observer looking away from the ocean in the Northern Hemisphere and returns to the ocean along the other bank. When friction is large, the Coriolis acceleration is negligible and the residual circulation is consistent with the vertically averaged flows described by Li and O’Donnell (1997, 2005).

2. The model

The model presented here is based on the three-dimensional fluctuating flow model described by Winant (2007). This analytical model represents an estuary as an elongated basin, open at one end, on the f plane. The length L* (dimensional variables are denoted by asterisks) is much greater than the width 2B*, and the maximum depth H* is much smaller than either B* or L*. The origin of the coordinate system is at the surface, at the middle of the entrance section. The x coordinate extends away from the open end, so the closed end is located at x* = L*. Here y is to the left of x and the constant width basin, considered symmetric about the x axis, extends laterally from y* = −B* to y* = B*; z is measured positive up from the undisturbed surface: the bottom is located at z* = −h*, where h*(x, y) is the local depth; u, υ, and w are the three components of the velocity vector. The density ρ* is taken to be constant. The vertical eddy diffusivity K* is taken to be constant as well. This rules out any forcing due to covariation between K and the vertical shear, mechanisms that have been shown to drive residual circulations (Stacey et al. 2001; Jay and Musiak 1994). At the entrance (x = 0) the tide in the adjacent ocean forces the sea level at x, y = 0 to fluctuate with amplitude C* at frequency ω*; η* is the position of the free surface relative to z* = 0. Results from Winant (2007) relevant to this analysis are included in section b of appendix A.

The governing equations and boundary conditions are derived in appendix A. A perturbation expansion is used to transform the nonlinear equations of motion into an ordered set of linear inhomogeneous equations. The free surface boundary conditions are transformed into boundary conditions at z = 0 with a Taylor series expansion. In this way, the domain of the solution extends between the bottom and the mean surface, rather than the bottom and the actual surface. We shall see that this domain transformation results in two distinct forcing mechanisms that involve correlations between η and velocities for the residual flow in the fixed domain. The residual flow equations are obtained by averaging the equations and boundary conditions governing the first-order solutions [Eqs. (A31) through (A35)] over a tidal cycle.

If the subscript E denotes the average over one period of first-order variables (e.g., uE = is the Eulerian average axial velocity):
i1520-0485-38-6-1278-e1
i1520-0485-38-6-1278-e2
i1520-0485-38-6-1278-e3
The parameters α, δ, and κ, defined in Eq. (A7), represent the aspect ratio of the basin, the importance of friction, and the relative measure of the basin length to the tidal wavelength. The surface boundary conditions, applied at z = 0, are
i1520-0485-38-6-1278-e4
i1520-0485-38-6-1278-e5
where Sx = and Sy = are the two components of the Stokes transport.1 At the bottom z = −h
i1520-0485-38-6-1278-e6
The Eulerian flow is forced in part by the divergence of the tidal Reynolds stresses (Rx and Ry).2 This component is called uR. As noted above, the residual flow in this fixed domain is also forced by the divergence in the Stokes transport and by the boundary condition at z = 0 (Tx and Ty). In the Li and O’Donnell (1997, 2005) models the last two contributions add up to twice the Stokes transport. This problem differs from the problem solved by Ianniello (1977) because of the inclusion of the Coriolis terms and because of the nonzero boundary condition on the vertical shear of the horizontal velocities at z = 0.
The Reynolds stress divergence terms are determined by the zeroth-order flow, as given in section b of appendix A. Solutions are presented for a basin where the depth varies as a parabolic function of y, with a small constant depth rim chosen so that the depth is everywhere greater than the sea level fluctuation:
i1520-0485-38-6-1278-e7
The influence of basin geometry on the circulation presented here has been investigated by calculating results for three different depth profiles: the parabolic shape described above, a trough in which the depth increases linearly with distance from the side, and a Gaussian depth profile. The basin geometry changes details of the results, but not qualitatively. The axial and lateral components of the Reynolds stress for a parabolic depth profile are illustrated in Fig. 1 at the midpoint (x = 1/2) of the basin. Without rotation the axial Reynolds stress divergence (Rx) is dominated by . The vertical shear in the fluctuating axial velocity for this intermediate value of friction is responsible for the vertical and horizontal structure of Rx. When friction is small, Rx is relatively uniform over the entire section, the amplitude decreasing to zero in a thin boundary layer. Here Ry has the opposite sign on either side of the basin because, when f = 0, υ0 changes sign on either side of the basin. In this case Rx and Ry have comparable amplitudes.

The Reynolds forcing terms are qualitatively and quantitatively different for f = 1/2. Even though the distribution of the fluctuating axial velocity is relatively unaffected by including rotation (Winant 2007), the distribution of Rx is different because the and terms are comparable in magnitude or greater than . With rotation, if the aspect ratio is very small (α < 10−2), Rxα−1 and Ryα−2. Experience shows that for less extreme aspect ratios (10−2 < α < 1), Rx is of order one and Ryα−1, as shown in the middle frames of Fig. 1. In the remaining calculations Ry is taken to be O(1) when f = 0 and O(α−1) with rotation; Rx is taken to be O(1) with or without rotation. With rotation, the structure of both Reynolds stress terms depends on friction, as can be seen by comparing the middle and right frames of Fig. 1. For δ = 1/2 the lateral gradient in the fluctuating axial velocity, u0y, is spread over the width of the basin with the result that Rx is uniformly distributed as well. As friction decreases, the fluctuating axial velocity is uniform over most of the basin, and u0y is concentrated in narrow cores near the sides of the basin. As a result both Reynolds stress terms are concentrated near the basin sides as well. The differences in residual circulation patterns with and without rotation described below are due to the different magnitude and patterns illustrated in Fig. 1, as well as similar differences in the surface boundary conditions (Tx and Ty) and the Stokes transport.

Equations (2) and (3) can be combined into a single equation for the complex velocity ϒE = uE + iαυE:
i1520-0485-38-6-1278-e8
with boundary conditions
i1520-0485-38-6-1278-e9
Solutions for these velocities are presented in appendix B.

3. Tidally averaged sea level

The time average of Eq. (A12) is
i1520-0485-38-6-1278-e10
where the terms in parentheses represent the total time-averaged transports in the axial and lateral directions, respectively. It is convenient to think of these transports as consisting of a part driven by the three components of forcing, [uF], and a second part, [uη], that represents the transport due to the sea elevation gradients, set up to satisfy mass conservation for the residual flow. With the convention that an integral from the bottom to z = 0 can be represented by brackets, Eq. (10) becomes
i1520-0485-38-6-1278-e11
where
i1520-0485-38-6-1278-e12
is the axial forced transport and
i1520-0485-38-6-1278-e13
is the forced lateral transport. Both [uF] and [υF] are determined by the zeroth-order solution. The individual contributions of each term on the right-hand side of Eqs. (12) and (13) to [uF] and [υF] are illustrated in Fig. 2 for f = 0 and f = 1/2 and δ = 0.5. Without rotation, the axial transport forced by the Reynolds stress, [uR], changes sign near the basin entrance where (u0)x changes sign. Here [uR] is positive where Rx is negative; it is maximum along the x axis and decreases toward the sides because both the axial velocity and its axial derivative decrease away from y = 0: [uT] and Sx have very similar patterns. Both terms are positive with comparable magnitude everywhere except near the basin sides away from the entrance. For f = 0, the lateral transports have different sign on either side of the basin because Ry changes sign at y = 0 (cf. Fig. 1). The Reynolds forcing [υR] dominates the other terms and is comparable in magnitude to the axial transports.

Including rotation alters the pattern of the forced transports qualitatively. In the axial direction [uR] is no longer symmetric across the width of the basin, reflecting the structure of Rx (top middle frame of Fig. 1). As for the Reynolds stresses, the most important difference brought about by the inclusion of rotation is the magnitude of the terms in the lateral forcing. As long as α > 10−2, [υF], the dimensionless lateral forcing is O(α−1). The central point is that with rotation [υF]y is O(α−1) relative to [uF]x, as illustrated in the lower right three frames of Fig. 2. When f = 0 both [uF]x and [υF]y are of the same order. With rotation and for lower values of friction, the forced transports are concentrated near the basin sides, following the Reynolds stress pattern (right frames of Fig. 1).

The meaning of Eq. (11) is that, in a steady state, the sea level must be distributed in such a way that the divergence of the total transport is zero. Since the transport driven by sea level gradients is given by Eq. (B6), Eq. (11) can be manipulated into a second-order ordinary differential equation for the tidally averaged sea level, ηE:
i1520-0485-38-6-1278-e14
where Pr and Pi are the real and imaginary parts of P1 (B7). Multiplying Eq. (14) through by α2 and reorganizing yields
i1520-0485-38-6-1278-e15
At the basin entrance (x = 0), the residual sea level ηE has to be specified. On the closed boundaries, the transport vanishes. Since the forced transport [uF] = 0 at the closed end (x = 1), or [uη] = 0,
i1520-0485-38-6-1278-e16
Similarly, [υF] = 0 at the lateral sides:
i1520-0485-38-6-1278-e17
A solution for the residual sea level is sought by expanding ηE as
i1520-0485-38-6-1278-e18
Introducing this expansion into Eq. (15) gives an ordered set of problems. To lowest order
i1520-0485-38-6-1278-e19
the boundary condition on lateral transport at the same order is
i1520-0485-38-6-1278-e20
This is satisfied if η(0)y = 0 or, equivalently, if η(0) is a function of x only.
If f = 0, since Pi = 0, the order α problem is trivial: η(1) = 0. When f ≠ 0, the [υF] forcing is O(α−1), and the order α problem is
i1520-0485-38-6-1278-e21
To the same order the boundary condition on lateral transport is
i1520-0485-38-6-1278-e22
because the forcing terms vanish at the sides. Equation (21) can be integrated across the width of the basin, and using the boundary condition at y = ±1:
i1520-0485-38-6-1278-e23
η(1) can then be evaluated by integrating Eq. (23) from one side of the channel.
When f = 0 the order α2 problem is
i1520-0485-38-6-1278-e24
with boundary conditions
i1520-0485-38-6-1278-e25
Integrating this expression across the width of the basin and using the condition (25) gives an ordinary differential equation for η(0):
i1520-0485-38-6-1278-e26
Using the boundary condition of no transport at x = 1, this equation can be integrated along x with the result, in view of expression (B8):
i1520-0485-38-6-1278-e27
which is equivalent to Eq. (20) in Li and O’Donnell (2005); η(2)y is then given by Eq. (24):
i1520-0485-38-6-1278-e28
When f ≠ 0, the order α2 problem is
i1520-0485-38-6-1278-e29
with boundary conditions
i1520-0485-38-6-1278-e30
In view of Eq. (23), Eq. (29) can be rewritten as
i1520-0485-38-6-1278-e31
Integrating this expression across the width of the basin and using the condition (30) gives an ordinary differential equation for η(0):
i1520-0485-38-6-1278-e32
where the angle brackets denote the lateral average of any quantity. If the depth is only a function of lateral position, Eq. (32) can be integrated once in x to give an expression for the axial sea level gradient:
i1520-0485-38-6-1278-e33
Equation (31) can then be solved for η(2)y:
i1520-0485-38-6-1278-e34
The two components of the sea elevation gradients computed for α = 0.025, κ = 1.5, δ = 0.5 are illustrated in Fig. 3 for f = 0 and f = 1/2. The axial sea elevation gradient, (ηE)x, is of O(1), both with and without rotation. Without rotation, (ηE)x is independent of y; with rotation it depends weakly on the lateral position. The lateral sea elevation gradient, (ηE)y, is O(α2) without rotation and in the sense to balance the bottom friction associated with the lateral mean current in the inner part of the basin and in balance with the lateral Stokes transport near the open end. With rotation, (ηE)y is O(α) and drives a transport that mostly opposes [υR]. It should be noted that, since (ηE)y is determined even at x = 0 in this solution, the sea level cannot be prescribed arbitrarily as it ought to be. This is because the regular expansion used to obtain these solutions has eliminated large wavenumber (in x) solutions that would be needed to adjust an arbitrary entrance condition to the solutions illustrated in Fig. 3. Thus, in reality the solution would adjust from that illustrated in Fig. 3 to the open boundary condition over a distance comparable to the basin width.

4. Transport and Eulerian local velocities

Equation (10) guarantees the existence of a transport streamfunction defined as
i1520-0485-38-6-1278-e35
and
i1520-0485-38-6-1278-e36
The transport streamfunction computed for different values of δ, with and without rotation, is illustrated in Fig. 4. Without rotation the streamfunction patterns for moderate and large friction are similar, although the amplitude decreases with increasing friction. This pattern is very similar to the two-cell pattern described by Li and O’Donnell (2005). For moderate friction uT and the Stokes transport dominate over [uR] near the entrance (Fig. 2). The lateral distribution of the forcing results in overall outflow near the center of the basin.

The integrated residual transport across the width of the basin is zero since the residual flow is in steady state. The direction of the axial component −ψy depends on the relative magnitudes of the different components as a function of lateral position. For instance, it is shown in appendix B that [uη] depends on the cube of the depth when f = 0. In general, the transport due to Stokes forcing depends on a lower power of depth, while [uR] depends on a higher power of depth. Near the entrance, where the combined Stokes forcing exceeds the Reynolds forcing, the (negative) residual transport driven by the sea elevation gradient overcomes the forced transport near the center of the channel, so the flow is directed toward the ocean. Farther inside the basin [uR], the major forced transport, is larger than [uη] in the central channel, with the result that net transport is toward the closed end. The direction of the residual transport is thus determined by small differences in the lateral distributions of the different transport components, subject to the condition that the residual axial transport, integrated across the basin width, is zero.

Rotation qualitatively alters the vertically integrated circulation except when friction is large. With rotation the Reynolds forcing is asymmetric and larger in the lateral direction (Fig. 1), resulting in asymmetric distribution of [uR] (Fig. 2). Since ηx remains nearly symmetric (Fig. 3), [uη] is also nearly symmetric, with the result that the combined transport forms a single large gyre. The inflow is to the right of an observer looking toward the closed end in the Northern Hemisphere, and the return flow is on the other side. With rotation the amplitude of the currents is larger (more than 10 times when friction is small) than when f = 0. This is not only due to the asymmetric nature of Rx, but also due to the fact that, when friction is relatively small, the lateral gradients in fluctuating axial velocity (u0y) are very large, as noted above. As a consequence, the transport is also concentrated near the sides of the basin.

The local Eulerian velocities are illustrated in Fig. 5 for f = 0 and for three different values of δ. For high friction (δ = 1) the shifting pattern of axial velocity is explained by the relative magnitude of the different forcings. Near the entrance the forcing due to correlations with fluctuating sea level ([uT] and ) are larger than [uR]. As already discussed above in the context of the streamfunction, this forcing overcomes the pressure gradient forcing in shallower depth, resulting in positive Eulerian mean velocities at the sides of the basin. Near the closed end the Reynolds forcing overwhelms the sea level forcing, with the result that positive Eulerian mean velocities are found near the middle of the basin. For δ = 1 the transition between the two patterns occurs near x = 1/2, where the Eulerian mean is everywhere negative, illustrating the point that the Eulerian velocity does not integrate to zero over any section. In the plane of the section the velocities (represented by vectors) are mostly vertical, in a sense to connect regions of inflow to regions of outflow. Patterns of mean Eulerian velocities for moderate friction (δ = 1/2) are qualitatively similar to the case of high friction, the main difference being the lateral velocities that have a larger horizontal component. When δ = 0.1 friction is confined to narrow boundary layers near the bottom, and the Reynolds forcing is uniform everywhere else. The pattern of the flow driven by the Reynolds forcing is thus almost exactly the same as the flow driven by the sea level gradient (which also acts uniformly with depth), with the result that the pattern of the Eulerian mean is mostly determined by the Stokes forcing. In this case, as with the cases described above, the lateral flow is mostly in the sense to connect inflow and outflow.

With rotation ( f = 1/2) the pattern of Eulerian velocities is illustrated in Fig. 6. Just as for the streamfunctions (Fig. 4) the Eulerian velocities are significantly modified by rotation. When friction is high (δ = 1), the axial velocities are nearly the same as for the f = 0 case; however, the lateral velocities differ, with relatively large sweeping motion down the right side of the basin, and significant cross-basin flow at depth. For moderate friction (δ = 1/2) the structure of the axial flow is asymmetric across the width of the basin: flow enters on the right of an observer looking into the basin and returns to the ocean on the left side. The amplitude of all three Eulerian velocity components is significantly larger with rotation. In the lateral plane, for δ = 1/2, there is strong upwelling near the center of the basin, corresponding to two counterrotation gyres in the portion of the basin closest to the opening. This pattern can be explained in terms of the lateral momentum balance as follows: the lateral sea level gradient (lower right panel, Fig. 3) balances the vertically averaged lateral Reynolds stress Ry (lower right panel, Fig. 1). Near the surface the local Ry exceeds ηy, driving flow toward the side of the basin, whereas the reverse is true at depth. The resulting convergence at the bottom near y = 0 forces upwelling near the middle. Farther into the basin, the right-side gyre strengthens, and near the closed end there is a single gyre in the lateral plane. For low friction (δ = 0.1) the axial velocity is concentrated in two jets that extend from the side of the basin (y = ±1) to a distance δ from each side. The sense of the axial flow is as in the δ = 1/2 case: into the basin on the right. The location of these jets is explained by the lateral structure of the Reynolds forcing and, in particular, lateral terms such as . When friction is low, the fluctuating axial velocity u0 is nearly uniform across the basin width, except in a boundary layer of width δ where it decreases to zero, as the depth decreases, as noted earlier when the Reynolds stress patterns were described.

5. Lagrangian flow and the average trajectory of fluid parcels

The average velocity of a fluid parcel is given by the Lagrangian mean velocity; using indicial notation,
i1520-0485-38-6-1278-e37
where the Stokes velocity (uS)i is defined in appendix C.

The local Lagrangian velocities are illustrated in Fig. 7 for f = 0 and for three different values of δ. In contrast to the Eulerian velocities (Fig. 5), the axial Lagrangian velocity integrated over any section is zero. When friction is large, δ = 1, water enters the basin in a central core that extends over the whole basin length. The return flow is in a sheath near the sides and bottom of the basin. The axial Stokes velocity, the difference between the Eulerian and the Lagrangian velocities, is strong and positive near the center of the channel (where the tidal wave is near progressive) and weakly negative on the shallow sides, where it overcomes the positive axial Eulerian mean. The lateral flow is everywhere radial and outward, connecting inflow to outflow throughout the length of the basin. This picture of the Lagrangian flow contrasts with the sense of the vertically integrated transport calculated for the same parameters (lower left frame of Fig. 4), which shows transport occurring in two distinct cells. For moderate friction, δ = 1/2, the patterns of Eulerian and Stokes velocity are nearly opposite. Near the entrance their sum results in water flowing into the basin at the surface on the sides and returning to the ocean in the middle and near the bottom of the basin. This structure changes farther into the basin, near where the structure of the transport streamfunctions change (middle left frame of Fig. 4), where the inflow is near the surface in the center while the outflow is near the bottom. This pattern is similar to what occurs for δ = 1. At the lowest value of friction, δ = 0.1, the in–outflow pattern is complex. Near the entrance inflow occurs both in a central near-surface core and near the bottom, while outflow is sandwiched between these layers. Farther in there are still three distinct regions of axial flow, but the direction is reversed from the entrance, with inflow between two distinct outflow regions.

The local Lagrangian velocities are illustrated in Fig. 8 for f = 1/2. At the highest value of friction the structure of Lagrangian flow is similar to the case without rotation (Fig. 7), although rotation skews the pattern slightly. The lateral circulation is characterized by the same clockwise pattern as in the Eulerian velocity field (Fig. 6). For moderate friction, δ = 1/2, the asymmetry is much more developed, resulting in inflow on the right side of the basin and outflow on the other side. The amplitude of axial and lateral flow components is five times larger than without rotation. The lateral circulation consists of two counterrotating gyres aligned with the axis of the basin, with upwelling near the center and downwelling on either side, due to the same force balance described previously for the corresponding Eulerian mean flow. This pattern of axial and lateral velocities, fundamentally different in amplitude and structure from the no-rotation case, is the central result of this work. When friction is small, δ = 0.1, the largest Lagrangian velocities are found in narrow layers extending a distance δ from either side of the basin. In this case the flow structures that span each half basin (−1 ≤ y < 0 and 0 ≤ y < 1) for the δ = 0.5 case are squeezed into the shallow corners of the basin (−1 ≤ y < −(1 − δ) and (1 − δ) ≤ y ≤ 1).

The trajectory of several parcels released at the basin entrance for each of the six flows considered above is illustrated in Fig. 9. Parcels are released at the basin entrance from positions where the axial Lagrangian velocity is positive (unshaded areas in Figs. 7 and 8). Without rotation the parcels trajectories are symmetric about the central plane of the basin. Trajectories on the left side are mirror images of the trajectories illustrated. For large friction, δ = 1, the trajectories are relatively simple. The parcel is advected into the basin along a central core. At some axial position it is carried laterally toward the boundary where it joins the return current. Parcels released near the surface at the center of the basin travel farthest toward the closed end. When friction is smaller, δ = 1/2, fluid enters the basin on either side of the entrance. Trajectories fall into one of two patterns. Fluid that has been drawn in near middepth (heavy solid line) soon drifts toward the center and changes direction, returning to the entrance after only a short stay in the basin. Parcels that enter closer to the surface first drift toward the center where they are carried toward the closed end. Then they move both toward the sides and the bottom and join outbound flow. When friction is low, δ = 0.1, parcels that enter the basin in the near-surface core soon drift toward the sides where they join the outflow: those parcels do not get far into the basin. Fluid drawn into the basin near the bottom at first drifts up and to the sides where it is advected toward the closed end. After one, or in some cases several, counterclockwise laps (in plan view) in the basin interior, these parcels return to the ocean at middepth. The contrast between particle paths for different values of friction and the contrast with the transport streamfunctions (Fig. 4) is striking. The transport does not necessarily (indeed rarely) give the particle trajectory because the flow is three-dimensional.

For f = 1/2, trajectories are no longer symmetric across the width of the basin. For δ = 1, two parcels (solid lines) that are released beneath the surface close to each other are advected to the closed end, but return on different sides of the basin. The parcel whose initial position was farther right returns on the left side of the basin, and vice-versa. For smaller values of friction, fluid is generally drawn into the basin on the right and returns to the ocean on the left. For δ = 1/2 there is a tendency for parcels to downwell on the right side, move back up to the surface near the middle of the basin, and eventually cross over to the left side and drift back toward the ocean with an opposite sense of lateral circulation. At the lowest value of friction considered, the circulation cells are concentrated near the basin sides, as expected from the Lagrangian velocity patterns. Even though the axial velocities inside these cores are quite large, fluid parcels do not reach the far end of the basin significantly faster than others because they spend so much time spinning around between areas of large and small axial velocity. Taken as a group, the trajectory of parcels illustrated in Fig. 9 are remarkably different and make the point that actual trajectories, and therefore dispersion and transport, are sensitive functions of the basin and parameters of the problem. This is a significant result of this analysis.

6. Discussion

In the Li and O’Donnell (1997) model, the fluctuating tidal flow is independent of depth, and as a result the Reynolds forcing is also depth independent. Since the axial pressure gradient only depends on axial position, the flow driven by the Reynolds forcing can be exactly canceled out by the pressure gradient–driven flow. As a result the residual flow is twice the Stokes transport. When f = 0, a similar balance occurs in this model for the lowest value of friction because the viscous boundary layers occupy a small fraction of the water column, so the fluctuating velocities and the Reynolds forcing are nearly depth independent and the residual velocities that they induce are almost exactly canceled by the flow driven by the pressure gradient. This explains why the transport streamlines illustrated in the top left frame of Fig. 4 are similar to the residual transport velocity in Li and O’Donnell (1997), with outflow near the middle of the channel and inflow on the sides. The fluctuating tidal flow in the Li and O’Donnell (2005) model varies as a function of both lateral and axial position, with the result that near the open end of the basin the flow is as in their earlier model, whereas farther inside the basin the residual circulation is in the opposite sense, away from the ocean near the center of the basin. In this inner section, the residual flow is mainly driven by the Reynolds forcing, as the sea level and axial velocity are nearly in quadrature. That circulation pattern is the same as found in this model for intermediate and high friction. The presence of closed transport streamlines raises the possibility that inner portions of the basin are effectively isolated from the ocean, but the fluid parcel trajectories calculated with this model demonstrate that such is not the case. For moderate and large friction (middle and lower left frames of Fig. 9) fluid that enters the basin near the surface penetrates all the way to the closed end.

The central result of the work presented here is that rotation profoundly alters the residual circulation driven by the tides. With rotation the tidal component of the estuarine transport is larger than predicted by vertically integrated models, or models that exclude rotation from the analysis. In the perspective presented here, there is a steady transport of ocean water into the basin on the right side of an observer looking away from the ocean (in the Northern Hemisphere) while estuarine water flows toward the ocean on the other side. This has practical implications for the transport of anthropogenic or other pollutants introduced into the estuary because it identifies locations where the residence time of the product in the basin is minimal. The reason for the asymmetry of the residual circulation is that the Coriolis-induced, tidally fluctuating flow produces a lateral advection of both components of momentum (which is neglected in vertically integrated models), even when the vertically integrated lateral transport is zero. This result is in accord with Lerczak and Geyer (2004) who note that, while lateral velocities are usually only 10% of the axial flow, they play a critical role in terms of transport and dynamics of estuaries. Smith (1976) demonstrated how lateral transport determines the longitudinal dispersion of a buoyant pollutant in a slowly varying current. More recently, Chant (2002) has shown that in the western end of Kill van Kull, near the entrance of Newark Bay, a centrifugally driven helical recirculation is an important component of the streamwise dynamics and dispersion. Taken together, these results all point to the fact that in any real elongated basin the residual circulation and dispersion is often controlled by the lateral secondary flows because lateral gradients, either in momentum or in transported properties, are so much larger than streamwise gradients. This underlines the importance of three-dimensional modeling when transport and dispersion need to be resolved. The residual circulation patterns described here for the case with rotation are consistent with commonly observed distinct ebb and flood channels near the entrance of estuaries.

The adoption of the simplest possible form for the vertical eddy diffusivity places a further restriction on the results presented here. It is simple to extend this model to the case when K* is a constant function of total water depth h*, and even to consider the eddy viscosity to be a function of z*, since the momentum Eqs. (A5) and (A6) remain linear ordinary differential equations. Calculations using a variety of such models lead to some quantitative but no qualitative difference in the results presented here. In reality, the vertical eddy diffusivity depends not only on location but on the strength of the flow, which results in a nonlinear problem. This effect has been explored to a limited extent by numerically simulating the flows described here, using the Regional Ocean Model System (ROMS) (Haidvogel et al. 2000), including a kω closure scheme to evaluate the eddy diffusivity. The model is fully nonlinear, including a quadratic formulation for the bottom friction. In this case the viscous effects are measured by the ratio CdU*max/(ω*H*), where U*max is a measure of the fluctuating axial velocity. Only small quantitative differences are found between the nonlinear simulations and results described above. In the simulations, the sense of the residual circulation, into the estuary near the right bank and back toward the ocean on the other side, is the same as in the work described here with comparable amplitudes. Jay and Musiak (1994) and Stacey et al. (2001) have shown that covariance between the eddy diffusivity and the vertical shear can lead to residual circulation as well. This process is not accounted for in the model presented here.

Most estuaries, lagoons, and embayments are stratified, whereas the model presented here is for a constant density fluid. Lerczak and Geyer (2004) describe the results of numerical simulations of the tidal and residual flow in an idealized stratified estuary including the effect of rotation. That work shows that both the fluctuating flow as well as the residual flow is determined to a large extent by the lateral circulation, driven in turn by the Coriolis acceleration and the density difference along the axis of the basin. The latter mechanism, sometimes described as tidal straining (Simpson et al. 1990), consists of two counterrotating gyres on either side of the basin that drive downwelling in the central, deeper basin and upwelling on the sides. Rotation drives a single circulation cell, as in Winant (2007). The combined result for the fluctuating component is two asymmetric gyres. The density-driven residual flow brings heavier water into the basin at depth with a compensating return at the surface. With rotation the residual outflow and inflow are no longer separated in the horizontal but along a skewed axis, a combination of the density-driven circulation and the Coriolis-driven residual flow described here.

Acknowledgments

The manuscript benefitted from critical comments by Aurelien Ponte and Arnoldo Valle-Levinson. Several conversations with Stephen Henderson shed considerable light on the Stokes velocity field. Comments by two anonymous reviewers considerably clarified the manuscript. This work was sponsored by the National Science Foundation Grant OCE-0425029.

REFERENCES

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APPENDIX A Model Equations

Nondimensional (without asterisks) variables are introduced as follows:
i1520-0485-38-6-1278-ea1
and
i1520-0485-38-6-1278-ea2
where ω* is the frequency of the tide, f* is the Coriolis frequency, and ϵ = C*/H* is the ratio of the amplitude of the tidal wave at the open end to the maximum depth. In this nondimensional system, the surface is located at z = ϵη. For shallow water waves (g*/ω*2H* ≫ 1), the vertical component of the momentum equation is hydrostatic, the continuity equation is
i1520-0485-38-6-1278-ea3
and the vertically integrated continuity equation, combined with the surface boundary condition, is
i1520-0485-38-6-1278-ea4
The horizontal components of the momentum equation are
i1520-0485-38-6-1278-ea5
i1520-0485-38-6-1278-ea6
where
i1520-0485-38-6-1278-ea7
Lateral mixing is ignored because the ratio KhH*2/K*B*2 (Kh represents the horizontal eddy viscosity) is assumed to be small. Here α is the horizontal aspect ratio of the basin (always less than one for an elongated geometry); δ, the relative amplitude of the periodic boundary layer to the maximum depth, measures the relative importance of friction to local acceleration; and κ is a relative measure of the length of the basin to the wavelength of the tidal wave. At the surface (z = ϵη) the kinematic boundary condition is
i1520-0485-38-6-1278-ea8
and the dynamic boundary conditions state that the pressure is constant and the shear stress is zero, uz = υz = 0. At the bottom (z = −h), u = υ = w = 0. On the closed sides (x = 1 and y = ±1) the velocity normal to the boundary must be zero.

Perturbation and Taylor series expansion

Introducing the perturbation expansion
i1520-0485-38-6-1278-ea9
into the equations and neglecting terms of order ϵ2 or greater gives
i1520-0485-38-6-1278-ea10
Introducing the perturbation expansion (A9) into the vertically integrated continuity Eq. (A4) and taking the limit as ϵ → 0 gives two equations. To lowest order:
i1520-0485-38-6-1278-ea11
and to order ϵ
i1520-0485-38-6-1278-ea12
The horizontal components of the local momentum equations become
i1520-0485-38-6-1278-ea13
i1520-0485-38-6-1278-ea14
With the perturbation expansion, the surface boundary conditions become
i1520-0485-38-6-1278-ea15
i1520-0485-38-6-1278-ea16
at z = ϵη where terms of order ϵ2 or greater have been neglected.
A Taylor series expansion is used to transform the boundary condition at z = ϵη into a condition at z = 0:
i1520-0485-38-6-1278-ea17
or, with the perturbation expansion,
i1520-0485-38-6-1278-ea18
i1520-0485-38-6-1278-ea19
Using mass conservation, the last expression can we rewritten as
i1520-0485-38-6-1278-ea20

Because lateral friction is ignored, only the velocity normal to the sides of the basin is required to go to zero. At the open end the transport or, equivalently, the sea level is specified.

Lowest-order problem

In the limit as ϵ → 0, the equations and boundary conditions that govern the zeroth-order problem are identical to the problem solved in Winant (2007). The solution for the zeroth-order axial velocity is
i1520-0485-38-6-1278-ea21
with equivalent forms for the sea level and the other velocity components. For a basin in which the depth is a function of lateral position only, the complex amplitudes given in Winant (2007) are
i1520-0485-38-6-1278-ea22
where μ = 〈M0−1/2, M0 = f2Q20/P0P0, and the angled brackets signify the average value over the basin width. The velocities are given by
i1520-0485-38-6-1278-ea23
i1520-0485-38-6-1278-ea24
where
i1520-0485-38-6-1278-ea25
i1520-0485-38-6-1278-ea26
i1520-0485-38-6-1278-ea27
i1520-0485-38-6-1278-ea28
and
i1520-0485-38-6-1278-ea29
The vertical velocity is obtained by integrating the continuity equation from the bottom up:
i1520-0485-38-6-1278-ea30

First-order problem

The equations for the next order are obtained by subtracting the lowest-order equations from the full expansion [(A10)(A14)]. If the equations and boundary conditions governing the fluctuating flow are subtracted from the full equations, the resulting problem, in the limit, is
i1520-0485-38-6-1278-ea31
i1520-0485-38-6-1278-ea32
i1520-0485-38-6-1278-ea33
The surface boundary conditions, applied at z = 0, are
i1520-0485-38-6-1278-ea34
while at the bottom, z = −h,
i1520-0485-38-6-1278-ea35

APPENDIX B The Residual Velocity

It is convenient to think of the residual velocity as consisting of two parts, one, ϒη, due to the pressure gradient that arises so that the total flow conserves mass, while the other is driven by the Reynolds stress and the surface boundary condition. Because the system is linear, the Eulerian circulation can thus be expressed as ϒE = ϒη + ϒT + ϒR. The residual velocity forced by the pressure gradient satisfies
i1520-0485-38-6-1278-eb1
i1520-0485-38-6-1278-eb2
The solution for ϒη is
i1520-0485-38-6-1278-eb3
where
i1520-0485-38-6-1278-eb4
and δE = δ/f is the Ekman depth. For f = 0, p1 is real and is given by
i1520-0485-38-6-1278-eb5
The Eulerian transport driven by the residual pressure gradients is then
i1520-0485-38-6-1278-eb6
where
i1520-0485-38-6-1278-eb7
For f = 0 this is evaluated as
i1520-0485-38-6-1278-eb8
A negative P1 corresponds to downgradient transport [uη].
The second component is the residual velocity forced by the boundary condition at z = 0:
i1520-0485-38-6-1278-eb9
i1520-0485-38-6-1278-eb10
The solution for ϒT is
i1520-0485-38-6-1278-eb11
where
i1520-0485-38-6-1278-eb12
For f = 0, q1 is real and given by
i1520-0485-38-6-1278-eb13
The integral from the bottom to the mean surface of the velocity driven by P is then
i1520-0485-38-6-1278-eb14
where
i1520-0485-38-6-1278-eb15
For f = 0 this is evaluated as
i1520-0485-38-6-1278-eb16
A positive Tx, corresponding to a negative correlation between η0 and u0zz|z=0, drives a positive transport [uT].
The third component of the residual velocity is forced by the Reynolds stress and solves
i1520-0485-38-6-1278-eb17
i1520-0485-38-6-1278-eb18
If Rx and Ry were independent of z, ϒR would have the same z dependence as ϒη. For arbitrary Rx and Ry, the solution for ϒR can, in principle, be found analytically, as was done by Ianniello (1977), since the forcing terms are known from the solution to the lowest order in ϵ problem. In practice it is much simpler to solve numerically by evaluating Rx and Ry for each x, y, and z and then using an implicit algorithm to solve for ϒR(z) at each x and y. The corresponding Eulerian transport is
i1520-0485-38-6-1278-eb19
A negative Rx drives a positive [uR], just as a negative sea elevation gradient drives a positive [uη].

APPENDIX C The Stokes Velocity

For small displacements, and using indicial notation, the Stokes velocity vector (uS)i = (uS, υS, wS) is defined as (Longuet-Higgins 1969)
i1520-0485-38-6-1278-ec1
where (u0)i = (u0, υ0, w0) is the vector representing the fluctuating velocities and the vector
i1520-0485-38-6-1278-ec2
represents the instantaneous position of a fluid parcel, that is, the time integral of the fluctuating velocity. In terms of the complex phase and amplitude of the zeroth-order velocities [Eq. (A21)], Xi = Re(iUeit). Each component of the Stokes velocity can then be evaluated knowing U, V, W, as, for instance,
i1520-0485-38-6-1278-ec3
with equivalent expressions for the two other components.
The transport associated with the x component of the Stokes velocity is
i1520-0485-38-6-1278-ec4
In three dimensions, the vertical integral of the axial component of the Stokes velocity is thus in general not equal to the Stokes transport.

Fig. 1.
Fig. 1.

The axial and lateral component of the Reynolds stress at a section located midchannel (x = 0.5) for α = 0.025, κ = 1.5. The view is toward the closed end of the basin: f and δ are noted above each pair of frames. Contour intervals are evenly spaced; shaded areas correspond to negative values.

Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3819.1

Fig. 2.
Fig. 2.

Plan view of the forcing transport terms for α = 0.025, κ = 1.5, and δ = 0.5. The components of the Stokes transports and are labeled as [Sx] and [Sy]. The closed end of the basin is at the top. Contour intervals are evenly spaced. Shaded areas correspond to negative values.

Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3819.1

Fig. 3.
Fig. 3.

As in Fig. 2, but of the axial and lateral component of the sea level gradient.

Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3819.1

Fig. 4.
Fig. 4.

Plan view of the transport streamfunction for three different values of δ, with and without rotation, for α = 0.025, κ = 1.5. The closed end of the basin is at the top. The arrows illustrate the direction of the flow. Contour intervals are evenly spaced; shaded areas correspond to negative values.

Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3819.1

Fig. 5.
Fig. 5.

Sections illustrating the Eulerian mean velocities at different axial positions in the basin for α = 0.025, κ = 1.5, f = 0, and for three different values of δ. The viewer is looking toward the closed end of the basin. The axial velocity is negative in the shaded area, and the axial velocity contour interval is 0.25. Lateral and vertical velocities are represented by arrows.

Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3819.1

Fig. 6.
Fig. 6.

As in Fig. 5, but with f = 0.5.

Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3819.1

Fig. 7.
Fig. 7.

As in Fig. 5, but for the Lagrangian velocities.

Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3819.1

Fig. 8.
Fig. 8.

As in Fig. 7, but with f = 0.5.

Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3819.1

Fig. 9.
Fig. 9.

The trajectory of fluid parcels released at the entrance (x = 0), at positions marked by the asterisk. For each value of δ and f, the left frame represents an end view of the basin and the right frame is a plan view with the closed end at the top of the frame. For f = 0 parcels are only released on the right side of the basin because of the symmetry of the flow around the midplane.

Citation: Journal of Physical Oceanography 38, 6; 10.1175/2007JPO3819.1

This is not to be confused with the Stokes velocity (appendix C).

This extends the meaning of Reynolds stress to include correlations between fluctuating tidal velocities.

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