## Abstract

The three-dimensional tidal circulation in an elongated basin of arbitrary depth is described with a linear, constant-density model on the *f* plane. Rotation fundamentally alters the lateral flow, introducing a lateral recirculation comparable in magnitude to the axial flow, as long as friction is not too large. This circulation is due to the imbalance between the cross-channel sea level gradient, which is in near-geostrophic balance with the Coriolis acceleration associated with the vertically averaged axial flow, and the Coriolis acceleration associated with the vertically sheared axial flow. During flood condition, for example, the lateral Coriolis acceleration near the surface exceeds the pressure gradient, tending to accelerate the lateral flow, while the converse is true near the bottom. As a result, with rotation, fluid parcels tend to corkscrew into and out of the basin in a tidal period. The axial flow is only weakly modified by rotation. When friction is small, the axial velocity is uniform in each section, except in a narrow bottom boundary layer where it decreases to zero. The boundary layer thickness increases with friction, so that with moderate or large friction, axial velocities are sheared from bottom to surface. When friction is large, the local and Coriolis accelerations are both small and the dynamics are governed by a balance between friction and the pressure gradient.

## 1. Introduction

The circulation driven by tides in coastal embayments (estuaries and lagoons) and marginal seas is important not only for navigation, but because it contributes to and, in some cases, controls the net circulation within the bay as well as the exchange with the adjacent ocean. Tidal currents in shallow seas can be very large. The realization that the resulting dissipation of energy could be significant motivated early models (Street 1917; Taylor 1919, 1921) of the circulation in the Irish Sea. Those models include both friction and the rotation of the earth. Hendershott and Speranza (1971) generalize Taylor’s work by allowing the head of the bay to absorb a variable portion of the incident energy flux as a way of accounting for dissipation in shallow areas. That model reproduces well the tidal characteristics of the Irish Sea, the Adriatic Sea, and the Gulf of California. The classic treatment of co-oscillating tides found in subsequent texts (Defant 1961; Lamb 1932; Officer 1976; Proudman 1953) includes friction but usually neglects rotation and variable bathymetry. When the amplitude of the tide is smaller than the basin depth, sea level is governed by a damped second-order wave equation. For a constant-depth inviscid basin, the phase difference between the axial transport and the sea level is ±*π*/2. When friction is low, the solution consists of a nearly standing wave. The wave amplitude varies in a pattern of nodes and antinodes (here node is understood to designate a location at which the amplitude is a minimum rather than zero). Increasing friction generally decreases the amplitude and smears the node antinode pattern as the tidal wave decays with distance from the origin. In this case, the phase difference between sea level and transport is *π*/4. Ianniello (1977) was the first to develop a theory for the vertical structure of the amplitude and phase of the tide in a constant-depth estuary. That model also included a careful analysis of the residual circulation driven by the fluctuating flow. Li and Valle-Levinson (1999) have extended these models to include the effect of variable topography in the lateral direction (here the axial direction points away from the ocean and the lateral direction is perpendicular). Their analysis shows how the axial tidal flow varies laterally, and it reveals the existence of a small lateral flow, the magnitude of which is on the order of the axial velocity times the ratio of the width to the length of the basin. Their analysis ignores rotation, arguing, as does Gill (1982), that the condition for rotation effects to be small is that the width of the channel be small relative to the external Rossby radius. In this case, Gill (1982) further shows that rotation induces a small lateral tilt in the sea surface, in geostrophic balance with the axial transport.

Recent observations of the vertical profile of tidal currents (Geyer et al. 2000; Lerczak and Geyer 2004) have demonstrated that in the Hudson estuary there exists a vertically sheared lateral circulation with maximum amplitude of approximately 10%–15% of the axial velocities. Although the observed lateral currents are smaller than the axial flow, they are considerably larger than what is expected from models that ignore rotation. Lerczak and Geyer (2004) estimate the ratio of the lateral to along-channel tidal current to be ⅛ times the ratio of the Coriolis to the tidal frequency. These lateral currents are significant in that lateral fluid excursions are comparable to the width of the river and can thus significantly affect transport and longitudinal transport, as discussed by Chant (2002). A numerical model of the tidal circulation in straight, stratified estuaries (Lerczak and Geyer 2004) shows how stratification modifies the tidal and residual flow in an idealized estuary and how rotation further modifies the lateral flow. These observations and numerical simulations raise the possibility that, even if vertically integrated transports are only weakly affected by rotation in narrow basins, the local velocities may be significantly changed and can produce correlations that are significant forcing terms for the transport, even when stratification and channel curvature are small. Observations of tidal flow in the James River estuary (Valle-Levinson et al. 2000) reveal the existence of lateral flows with amplitude of 20% of the axial flow. The associated surface convergences have been attributed to varying phases of the axial flow as a function of depth, but models that ignore rotation predict convergence rates that are smaller than observations by a factor of 5–10. Chant and Wilson (1997) and Chant (2002) have described how channel curvature, an effect which is not included here, leads to secondary circulations that determine the residual flows as well as the longitudinal transport.

The central objective of this work is to present a linear analytical model of the three-dimensional tidal flow in an elongated basin as a way to unify the seemingly disparate models and observations described above. The basins considered range from small rivers to marginal seas, the only limitation being that the basin width be smaller than the external Rossby radius. This model uses a constant vertical eddy diffusivity but can be extended to cases in which the eddy diffusivity varies with total depth or with position in the water column. The bathymetry is allowed to change gradually as a function of position. One major conclusion is that rotation drives a lateral flow that is smaller than but of comparable magnitude to the axial flow when friction is weak to moderate. This lateral flow scales as the ratio of the Coriolis to tidal frequencies, a number of order 1 except in equatorial regions. When friction is large (or the basin is shallow), neither acceleration nor rotation are significant in the dynamics that control the circulation.

## 2. The model

Consider 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 maximum width 2*B**, 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, and so the closed end is located at *x** = *L**. The *y* coordinate is to the left of *x*, and the constant-width basin, considered to be symmetric about the *x* axis, extends laterally from *y** = −*B** to *y** = *B**. The *z* coordinate is measured positive up from the undisturbed surface: the bottom is located at *z** = −*h**, where *h**(*x*, *y*) is the local depth. The three components of the velocity vector are *u*, *υ*, and *w*. The density *ρ** is taken to be constant. The vertical eddy diffusivity is *K**. Lateral mixing is ignored because the ratio *K _{h}H**

^{2}/

*K**

*B**

^{2}(

*K*represents the horizontal eddy viscosity) is assumed to be small. At the entrance (

_{h}*x*= 0), the tide in the adjacent ocean forces the sea level at

*x*,

*y*= 0 to fluctuate with amplitude

*C** at frequency

*ω**. The position of the free surface relative to

*z** = 0 is

*η**.

Nondimensional (no asterisk) variables are introduced as follows:

and

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**^{2}*H** ≫ 1), the vertical component of the momentum equation is hydrostatic, and the continuity and horizontal components of the momentum equation are

where

*α* is the horizontal aspect ratio of the basin (always less than 1 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. In the limit when *α* is very small, the *y*-momentum equation [Eq. (5)] predicts zero lateral pressure gradient and, to next order, a geostrophic balance between the lateral pressure gradient and the Coriolis acceleration.

At the surface (*z* = *εη*), the kinematic boundary condition is

and the dynamic boundary conditions state that the pressure is constant and the shear stress is zero, *u _{z}* =

*υ*= 0. At the bottom (

_{z}*z*= −

*h*), the no-slip boundary condition,

*u*=

*υ*=

*w*= 0, is applied. This implies that the horizontal flow will always be sheared. As a result, when rotation is taken into account, the Coriolis acceleration is also sheared. On the closed sides (

*x*= 1 and

*y*= ±1) the velocity normal to the boundary must be zero.

This set of equations describes a system that depends on the five nondimensional parameters *α*, *δ*, *ε*, *f*, and *κ*, as well as the geometry of the channel. The problem is nonlinear because of the advective terms in the momentum equations and because the surface boundary condition is applied at a location that is specified by the solution. In the limit of small *ε*, these difficulties are removed because the relatively small advective terms can be ignored, and the surface boundary conditions are applied at *z* = 0. The resulting linear problem for the fluctuating variables (*u*_{0}, *υ*_{0}, *w*_{0}, and *η*_{0}) is

with boundary conditions

For periodic solutions, complex amplitudes are introduced as

the momentum equations become

with boundary conditions

This problem is solved in three steps. First, the momentum equations are solved, as shown in the appendix, to give expressions relating local velocities and transports to the sea elevation gradients:

Second, as shown in the following section, these expressions are introduced into the continuity equation, and a solution is found for the fluctuating sea level. In the third step, the transports and horizontal velocities are determined from the sea level, and the vertical velocity can then be evaluated, as shown in section 4.

## 3. Sea level fluctuations

The continuity Eq. (8) can be integrated from the bottom to the surface (*z* = 0). Along with the linearized boundary condition at the surface, this gives

or, in terms of the complex amplitudes defined in Eq. (12),

[*U*] and [*V*] can be eliminated by introducing Eqs. (17) and (18) and multiplying all terms by −*iα*^{2}*κ*^{2}:

At the basin entrance, the sea level is specified, and on the closed boundaries, the transport vanishes:

On the scale of the basin, it is reasonable to expect that *N*, *N _{x}*, and

*N*will be of comparable magnitude, and a solution for the sea level is sought by expanding

_{xx}*N*as

To determine velocities and transports to lowest order, *N*^{(0)}, *N*^{(1)}, and N^{(2)} are required. Introducing the expansion in Eq. (24) into Eq. (21) gives an ordered set of problems.

To lowest order,

the boundary condition on lateral transport at the same order is

This is equivalent to requiring [*V*] = 0 to lowest order and is satisfied if *N*^{(0)}_{y} = 0 or equivalently if *N*^{(0)} is a function of *x* only. In that case, the order-*α* problem becomes

to the same order, the boundary condition on lateral transport is

When friction is small, this expression represents the near-geostrophic balance between lateral tilt of the sea level and axial transport. It can be integrated once across the width of the basin, and, using the boundary condition at *y* = ±1,

*N*^{(1)} can then be evaluated as

where the constant of integration is evaluated so that *N*^{(1)} (*y* = 0) = 0.

The order-*α*^{2} problem is

with boundary conditions

where *M*_{0} = *f* ^{2}*Q*^{2}_{0}/*P*_{0} − *P*_{0}. Integrating this expression across the width of the basin and using the condition in Eq. (32) gives an ordinary differential equation for *N*^{(0)}:

where the angle brackets denote the lateral average of any quantity; in this case

Equation (34) is an ordinary linear differential equation for *N*^{(0)}. In the case in which neither the depth nor the width is a function of *x*, that is, when 〈*M*_{0}〉 is a (complex) constant, it has the closed-form solution

where *μ* = 〈*M*_{0}〉^{−1/2}. The following discussion is confined to this case (constant width, with depth being a function of *y* alone). More complicated basin shapes are discussed in section 5. When friction is small (*δ* → 0), *M*_{0} tends to *h*, with a small imaginary part proportional to *δ*. If 〈*h*〉 represents the lateral average of the nondimensional depth, the real part of *μ* = 〈*h*〉^{−1/2}, independent of *f*. When friction is large (*δ* → ∞), *M*_{0} tends to −*P*_{0}, and *μ* → (3)^{1/2}(1 + *i*)*δ*〈*h*^{3}〉^{−1/2}/2, also independent of *f*. In between these limits, it is found that *μ* depends only weakly on *f*, with the result that the lowest-order solution for the sea level, *N*_{0}, is practically not affected by rotation. Equation (36) is qualitatively similar to solutions for sea level given previously (Ianniello 1977, 1979; Li and O’Donnell 1997, 2005; Li and Valle-Levinson 1999). If *μ* were real, Eq. (36) would describe a wave standing in the basin, with amplitude fixed at the open end and zero sea level gradient normal to the closed end, to ensure no transport through that boundary.

Solutions are presented for a basin in which 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:

The phase and amplitude of *N*^{(0)} are illustrated in Fig. 1 for *f* = 0.5. Results for different values of *f* are nearly identical; *κ* is taken as 1.5, chosen so that the basin is slightly longer than one-quarter wavelength of the frictionless tidal wave. For a maximum basin depth of 10 m, the corresponding length for a semidiurnal tide would be just over 100 km. Three values of *δ* corresponding to low, moderate, and high friction are explored. If *δ* is small (0.1), the real part of *μ* is much greater than the imaginary part, and the tidal wave is nearly a standing gravity wave. There is a node (where the amplitude is minimum) in elevation at one-quarter of the gravity wavelength from the closed end. The phase of the sea level changes by *π* in this vicinity. For small friction, the amplitude is always maximum at the closed end. Friction also determines the minimum amplitude of the wave and the distance over which the phase changes from 0 to *π*.

As friction increases to *δ* = 0.5, the apparent wavelength shortens (as illustrated by the movement of the node toward *x* = 1), because the real part of *μ* increases. The increasing imaginary part of *μ* results in lower amplitude at the closed end and a more progressive change in the phase over the length of the basin, relative to the low-friction case. For larger friction, *δ* = 1, the real and imaginary parts of *μ* are close to equal in magnitude, and the node is no longer apparent. For very large friction, the wave is completely damped before reaching the closed end. In this case, the boundary condition at the closed end must be replaced by a condition that *N*^{(0)} vanishes for large *x*, and the solution is

where *γ* is a parameter of order 1 that depends weakly on the geometry of the basin.

where *R** = (*g***H**)^{1/2}/*f** is the Rossby radius of deformation. This correction to sea level for geostrophy is proportional to the ratio of the width of the channel to the Rossby radius. The fact that successive terms in Eq. (24) are expected to become smaller implies that the solution described here is not expected to be valid when the channel width is greater than *R**.

The phase and amplitude of *N* are illustrated in Fig. 2 for *f* = 0.5, *α* = 0.25, *κ* = 1.5, and three values of *δ*. When friction is small, the sea level exhibits an amphidrome, where the amplitude is a minimum and around which the phase progresses, at the same axial position where the corresponding minimum in *N*^{(0)} was located (Fig. 1) and slightly off center. The phase pattern is explained as follows. The amphidrome is located where *N*^{(0)} is a minimum and where *N*^{(1)} is a maximum. If *α* is large enough, the lateral tilt in sea level from geostrophy and represented by *N*^{(1)} overwhelms the laterally uniform change represented by *N*^{(0)}, and so the sea level phase changes by *π* somewhere between *y* = ±1. The exact location depends on friction. If friction is low, the minimum in *N*^{(0)} is small close to the node, so that the main contributor to sea level is due to *N*^{(1)}. In that case, the amphidrome is located near *y* = 0. As friction increases, the minimum *N*^{(0)} amplitude increases, with the result that the amphidrome shifts toward positive *y*, in the Northern Hemisphere. This sea level pattern is qualitatively similar to the sea level described by Taylor (1921) for the Irish Sea and to the results of Hendershott and Speranza (1971) for other locations. In those earlier results, the sea level is idealized as a combination of incident and reflected Kelvin waves that propagate around the basin in the sense shown by the phase pattern. Hendershott and Speranza (1971) show that, in the Northern Hemisphere, dissipation moves the amphidrome to the left of the axis, or toward positive *y*, consistent with the results illustrated in Fig. 2.

As friction increases, the sea level patterns change in a manner consistent with the reasoning above. For *δ* = 0.5, the minimum amplitude is located close to the *y* = 1 boundary of the basin. For *δ* = 1, the pattern is consistent with a virtual amphidrome, located some distance to the left of the basin. These patterns show considerable spatial structure, but their underlying cause is straightforward: the axial variation is determined by the dynamics of the tidal wave, modified by friction, while the lateral structure is governed by the requirement for a lateral sea level gradient to balance the Coriolis acceleration associated with the axial flux of mass.

Equation (40) satisfies boundary conditions at *y* = ±1 and makes the transport [*U*] zero to order 1 at the closed end. It does not, however, satisfy the condition that sea level be uniform at the open end (*x* = 0), because of the lateral sea level tilt required by geostrophy. The scaling [Eqs. (1) and (2)] used to derive the ordered set of equations for *N* assumes motions of axial extent comparable to the basin length. It is also possible to find solutions for which the lateral and axial scales are comparable, by introducing a new dimensionless axial coordinate *x*′ = *x**/*B**. In this case, in the limit of large *α*, the equation for *N* becomes

In the simplest case, in which the depth is constant and *f* = 0, this reduces to Laplace’s equation, in which case it is clear that the solution consists of normal modes, trigonometric in one direction and exponential in the other. Because of the symmetry of the basin around the *x* axis, solutions are trigonometric in *y* and damped or evanescent in *x*. The details are beyond the scope of this work, but it is clear that evanescent modes of this kind can be used to reconcile the solution for *N* given by Eq. (40) to any prescribed lateral variation at the open end.

Last, to determine the lateral velocity *V*, *N*^{(2)}_{y} is required. Subtracting Eq. (34) from Eq. (33) and integrating from the axis of symmetry (*y* = 0) gives an expression for *N*^{(2)}_{y} in terms of *N*^{(0)} and *N*^{(1)}:

where

Equation (30) gives a relation between *N*^{(2)}_{y} and *N*^{(0)}_{xx}:

## 4. Transport and local velocities

To leading order, the horizontal transports are

The phase and amplitude of [*U*] are illustrated in Fig. 3 for *f* = 0.5, *α* = 0.25, *κ* = 1.5, and three values of *δ*. Patterns of axial transport change little with varying *f* in the range 0 ≤ *f* ≤ 1. For low and intermediate friction, the maximum axial transport occurs in the middle of the channel, near the surface elevation node, and decays to zero at the closed end, as required by the boundary condition. For the highest value of friction considered, the axial velocity decays monotonically from the entrance to the closed end at a rate determined by friction. Because the sea level amplitude decreases with increasing friction (Fig. 1), the axial transport amplitude diminishes as well. In the lateral direction, the axial transport decreases from the center to the sides of the basin. This pattern is consistent with the results of Li and Valle-Levinson (1999). For *δ* = 0.1, the phase of the axial transport is uniform over most of the basin, lagging surface elevation at the entrance by close to *π*/2. Because the surface elevation changes phase by almost *π* near the node (Fig. 1), the axial flow leads the elevation by *π*/2 in the inner half of the basin, as expected for a standing wave. The phase of the transport decreases toward the sides of the basin, where friction dominates over local acceleration, in such a way that transport is nearly in phase with the axial elevation gradient. As friction increases, this lateral change in phase spreads out farther away from the sides.

The dimensionless lateral transport [*V*] is zero at the center and at the sides of the basin. The direction of [*V*] is opposite on either side of the basin because *G* is an antisymmetric function of *y*. The scaling [Eq. (2)] implies that the dimensional lateral transport scales as the dimensional axial transport times the horizontal aspect ratio (*α*), consistent with the analysis of Li and Valle-Levinson (1999). This means that in elongated basins the lateral transports are small. To leading order in *α*, the sea level [*N*^{(0)}] does not vary laterally, and the axial transport does. The role of the lateral transport is thus to provide the mass required to raise and lower sea level off the central axis of the basin, consistent with Eq. (20).

Excluding terms of order *α* or higher, the complex amplitudes of the local velocities are given by

The vertical velocity is obtained by integrating the continuity equation either from the bottom up or from the surface down. In practice, the latter is computationally preferable:

To facilitate comparison with earlier work (e.g., Lerczak and Geyer 2004), local velocities (*u*_{0}, *υ*_{0}, and *w*_{0}) are described, rather than their complex amplitudes. Further, the velocities are scaled as

where |*η*_{0}| represents the local amplitude of the fluctuating sea level. Because of the kinematic surface boundary condition on the vertical velocity [Eq. (11)], *w̃*_{0} is always −1 at the surface one-quarter period after local high water, independent of the axial location or the other parameters in the problem. Although the magnitude of the velocities varies considerably with friction, the scaled velocities have comparable magnitude. They are illustrated in Fig. 4 for *f* = 0 and three different values of *δ* at six different times over one-half cycle, beginning at high water. Velocities in the following half period are minus the velocities illustrated. The section is located at midbasin (*x* = 0.5).

When friction is relatively small (*δ* = 0.1), and the tidal wave is close to standing, *ũ*_{0} is nearly constant over the section except in the bottom boundary layer where it decreases to satisfy the no-slip condition at *z* = −*h*. At high water, the axial velocities in the section are weak and change sign, with little net flow—a slack-flow condition. As time progresses, the axial flow becomes negative throughout the section, corresponding to an ebb condition. The largest ebb flow leads low water by *π*/2. Increasing friction changes the axial flow pattern in two ways. First, the timing of the axial flow changes relative to local sea level; at the highest value of friction considered here, the maximum ebb current occurs at a phase of *π*/4 before low water, as expected for a frictional system. Second, the shear in the axial velocity is distributed throughout the section rather than being confined near the bottom. Because the amplitude of the sea level variation |*η*_{0}| decreases with increasing friction (Fig. 1), the local velocities (as opposed to the scaled velocities illustrated in Fig. 4) decrease with increasing friction as well.

When *f* = 0, Eqs. (47) and (48) show that *V* is of the same order as *U* and the dimensional lateral velocities are of order *α* times the axial velocities—small for an elongated basin. This explains why the lateral circulation in Fig. 4 is dominated by the vertical velocities that are required to move the surface up and down. The scaled vertical velocities are equal to −1 one-quarter period after high water. Off the central axis of the basin there is a weak lateral flow that provides some of the mass required to change the sea level on the shallow sides of the basin, where the axial velocities are relatively small. Just as in the case of the axial velocities, the local vertical and lateral flows decrease with increasing friction.

Scaled velocities computed for *f* = 0.5 are illustrated in Fig. 5. The local axial velocities with rotation are nearly the same as when *f* = 0 for all values of friction considered. However, the pattern of lateral circulation is profoundly different from the nearly vertical motion illustrated in Fig. 4, when friction is either low (*δ* = 0.1) or moderate (*δ* = 0.5). At middepth, the vertical velocities computed with rotation can exceed those computed for *f* = 0 by a factor of 10 for the parameters considered here. In the low-friction case (*δ* = 0.1) the largest lateral velocities are confined within the bottom boundary layer, where the shear in the axial velocity is large. The pattern of recirculation is most evident for intermediate friction (*δ* = 0.5), because it is evenly distributed throughout the section. In that case, the lateral recirculation pattern is nearly in phase with the axial-flow, maximum-counterclockwise rotation near maximum ebb (when the phase is 2*π*/3).

This large recirculation owes its existence to the first term within the curly brackets in Eq. (48), which is inversely proportional to the aspect ratio *α* of the basin. The scaling [Eq. (2)] implies that with rotation the dimensional lateral velocities are of comparable magnitude to the axial velocities, whereas without rotation they are of order *α* times the axial velocities. Large lateral flows require large vertical flows to satisfy continuity and the boundary conditions, so that both vertical and lateral flows are larger when rotation is included. This recirculation can be understood on the basis of the horizontal momentum equations [Eqs. (9)–(10)] when friction is small or moderate. In that case, taking the vertical derivative of either equation eliminates the pressure gradient. Because the tidal frequency is similar in amplitude to the Coriolis frequency, vertical gradients of both dimensional velocity components have to have similar amplitude. A similar argument explains why including rotation has little effect when friction is large, because in that case the dominant balance in the horizontal momentum equation is between pressure gradients and vertical stress divergence. In that case, neither the local acceleration nor the Coriolis acceleration plays an important role in the dynamics.

## 5. Discussion

Vertical mixing is parameterized in this model by a constant vertical eddy diffusivity. Ianniello (1977) gives careful consideration to the consequences of different distributions of eddy viscosity and concludes that his model is “not overly sensitive to the details of the eddy viscosity.” Field observations (Friedrichs and Hamrick 1996) suggest that the eddy diffusivity depends on the total depth, and laboratory observations of channel flow (Nezu and Rodi 1986) suggest further that it should vary as a quadratic function of distance over the bottom. It is straightforward to extend the theory presented above to the case in which the vertical eddy diffusivity is a function of depth. The solution to the momentum equation described in the appendix is only slightly modified in the case in which *K** is a constant function of depth, with *p*_{0}, *q*_{0}, and their integrals having slightly different dependence on the local depth *h*. The examples presented above have all been evaluated for the simple case in which *K** is linearly dependent on depth. This results in only small quantitative changes in sea level and velocities. Solutions can also be found when the mixing varies as a function of position in the water column, by solving Eq. (A3) numerically, as long as *K** > 0 everywhere. Sea level and velocities are then computed just as in the case presented here. Implementing this model again only makes small changes to the results discussed above. In a real estuary, it is likely, however, that mixing not only depends on position in the water column and depth but varies in time and as a function of the velocity, itself a function of position. In that case the momentum equations are nonlinear, and solutions are best found using a three-dimensional time-dependent numerical model, which is beyond the scope of the work reported here.

The solution for the lowest-order sea level *N*^{(0)} is fundamentally the same as the solution given by Ianniello (1977) and Li and Valle-Levinson (1999). When friction is low, the wave is nearly standing, and the phase of the sea level leads transport (near the central axis) by ±*π*/2, depending on the basin length. As friction increases, the wave amplitude is damped with distance from the origin and the phase difference between transport and sea level becomes *π*/4. To see whether these solutions produce useful results for real estuaries, Eq. (36) has been applied to a basin that is 300 km long and 20 m deep, as a crude idealization of Chesapeake Bay. The predicted amplitude and phase are compared in Fig. 6 with harmonic constants published by the National Oceanic and Atmospheric Administration (obtained online at http://tidesandcurrents.noaa.gov). The vertical eddy diffusivity taken as 4 × 10^{−3} m^{2} s^{−1} was chosen to give the best possible fit to the *M*_{2} amplitude. For the *M*_{2} harmonic, (with *δ* = 0.378) the model replicates the observed variation of phase as well as amplitude, including a node located near the middle of the bay. For the same eddy diffusivity, the model underpredicts the *K*_{1} amplitude (with *δ* = 1/2) near the closed end and represents the phase of that constituent well, except very near the closed end, where the modeled phase difference is about 20° less than observed. Considering how simple the model geometry is, the comparison suggests that Eq. (36) does represent the varying influence of friction and geometry to a first approximation.

The examples described so far have been limited to a basin in which the depth varies as a function of lateral position alone. If the depth varies also as a linear function of *x*, as, for example, *h* = (1 − *x*)*ℏ︀*(*y*), where *ℏ︀*(*y*) is a function of *y* alone, the solution to Eq. (34) when *δ* = 0 is

where *J*_{0} is the Bessel function of the first kind, of order zero. Equation (34) can be solved numerically for nonzero values of *δ*. Results are illustrated in Fig. 7. The sloping bottom modifies the tidal wave in two opposing ways. The local phase speed of the wave is smaller in shallow depth, tending to increase the amplitude, but frictional effects are greater there also, tending to decrease the amplitude. The frictionless solution, given by Eq. (51), reaches maximum amplitude at the closed end, and the phase of the sea elevation changes near *x* = 0.6. In contrast, when *δ* > 0 the solutions are all damped near the closed end, because as the bottom shoals, friction becomes dominant. For *δ* = 0.1, the amplitude initially increases between the node and the closed end. For larger friction, the amplitude decays from the entrance. In the case of *δ* = 0.5, the tidal oscillation never reaches the closed end.

The amphidrome pattern illustrated in the left panel of Fig. 2 is reminiscent of maps of tidal properties of semienclosed seas, such as the Adriatic (Polli 1960; Gill 1982; Hendershott and Speranza 1971) and the Gulf of California (Morales and Gutiérrez 1989; Hendershott and Speranz 1971). How well does the model presented here [Eq. (40)] reproduce the tidal characteristics in those basins? The Adriatic is represented as a basin of uniform width equal to 170 km and length 700 km. The depth, illustrated in the left frame of Fig. 8, includes the south Adriatic pit and gradually slopes to near zero at Venice, Italy, the northern end of the sea. The parameters *κ* and *α* are determined by the geometry, and *K** was taken as 2 × 10^{−3} m^{2} s^{−1}, chosen to give the best fit to the phase of the *M*_{2} constituent. Cotidal and corange lines for both *M*_{2} and *K*_{1} constituents are illustrated in Fig. 8. The agreement with the maps of Polli (1960) is excellent. In the case of the Gulf of California, the depth is illustrated in the left frame of Fig. 9. A large value of the vertical eddy diffusivity, *K** = 0.2 m^{2} s^{−1}, is required to move the amphidrome over to the left of the basin and to produce a good fit to the observations. Cotidal and corange lines are illustrated in Fig. 9. Here also the agreement with observations (Morales and Gutiérrez 1989) is excellent for both constituents. The very different values of vertical eddy diffusivity required to produce agreement in both basins may be explained by the different amplitudes of the tides: the amplitude of the *M*_{2} constituent at San Felipe, at the northern end of the Gulf of California, is 2 m or 10 times that at Venice, and depths in the Gulf of California are several times as large as in the Adriatic. The increase in *K** is consistent with the Hendershott and Speranza (1971) analysis that shows that the power dissipated in the Gulf of California is about 100 times that in the Adriatic.

The central new result of this analysis is the identification of the lateral circulation induced by the rotation of the earth, illustrated in Fig. 5. Equation (48) shows that, to leading order, the complex amplitude of the lateral velocity is proportional to *f*/*α*. The lateral momentum balance [Eq. (10)] shows that the lateral velocity results from the combined effects of the Coriolis acceleration associated with the axial flow and the lateral sea level gradient. Because the axial flow is sheared in the vertical direction and the sea level gradient is depth independent, the Coriolis acceleration exceeds the sea level gradient near the surface and vice versa. This imbalance drives the lateral circulation. If friction is weak (*δ* is small), the shear in the axial flow is confined in a very narrow bottom boundary layer and the lateral flow is expected to be weak. At the opposite end of the spectrum, when friction is large, the momentum balance is between friction and sea level gradients, the Coriolis terms and the local accelerations are small, and lateral flows are also expected to be weak. The ratio *αV*/*fU*, a measure of the amplitude and phase of the dimensional lateral velocity relative to *f* times the dimensional axial velocity, midway along the channel and at the midpoint on the surface (*x* = 0.5, *y* = *z* = 0), is illustrated in Fig. 10. For values of *f* ≠ 1, the amplitude of this ratio initially increases with friction, reaching a maximum value near 0.2, then decreases back to zero, as expected from the argument above. In the special case in which *f* = 1, the ratio tends to 0.2 as friction goes to zero. In this limit (*δ* = 0), corresponding to inviscid inertial waves, the velocities are infinite, because Eqs. (A9) and (A10) are singular. Figure 10 suggests that in the range 0.1 < *δ* < 1 the dimensional lateral velocity near the surface can be estimated as 0.2*f* times the axial velocity. For semidiurnal tides at midlatitudes this corresponds to a lateral flow about 10% as large as the axial flow, consistent with observations in the Hudson River (Geyer et al. 2000; Lerczak and Geyer 2004).

Even with such a relatively small amplitude, the lateral circulation can strongly affect and even dominate transport. The advective terms in either the time-averaged momentum or transport equations include correlations such as and . For elongated estuaries, axial gradients are expected to be smaller than lateral gradients (by a factor of order *α*), so that if the length of the basin is more than 4 times the full width, the lateral advective term in the transport, frequently neglected in models of residual circulation, will dominate over the advective term. The phase difference between the two components modifies the amplitude of the correlation. If the phase between *U* and *V* were exactly *π*/2, as in an inviscid inertial wave, the correlation would be zero independent of amplitude, but the results illustrated in Fig. 10 show that this is almost never the case. The residual circulation and transport due to the fluctuating flow described here are explored in a separate report.

## 6. Summary and conclusions

The tidal circulation in an elongated basin of width less than the external Rossby radius, but otherwise of general shape, has been described with a linear, three-dimensional barotropic *f*-plane model. The solution depends on the geometry of the basin and on four nondimensional parameters: *ε* is the amplitude of the tidal elevation at the open end relative to the maximum depth, *κ* measures the length of the basin relative to the tidal wavelength, *δ* is the relative height of the time-dependant boundary layer to the basin depth, and *f* is the ratio of the Coriolis frequency to the tidal frequency. When *f* = 0, sea level behaves as predicted by several earlier models. When the bathymetry is independent of axial position and friction is low, the solution consists of a nearly standing wave. The wave amplitude varies in a pattern of nodes and antinodes. The phase difference between the axial transport and the sea level is ±*π*/2. Axial velocities are nearly constant except in a narrow near-bottom boundary layer. Increasing friction generally decreases the amplitude and smears the node–antinode pattern, as the tidal wave decays with distance from the origin. In this case, the phase difference between sea level and transport is *π*/4 and local axial velocities are sheared throughout the water column. Lateral velocities are small and vertical velocities increase in amplitude from the bottom to the surface.

Rotation affects the fluctuating flow in two ways. The sea level tilts across the width of the basin, in geostrophic balance with the axial transport, in a manner consistent with the model of the Adriatic presented by Gill (1982). The major conclusion presented here is that rotation also drives a lateral flow that is smaller than but of comparable magnitude to the axial flow, when friction is weak to moderate. This lateral flow scales as the ratio of the Coriolis to tidal frequencies, a number of order 1 except in equatorial regions. This flow is expected to play an important role in residual circulation and transport.

## Acknowledgments

I am grateful to Bill Young who showed me how to obtain Eq. (34) from Eq. (33). I also thank Myrl Hendershott for innumerable insightful discussions. Arnoldo Valle-Levinson and Aurelien Ponte provided helpful suggestions on the original draft. This work was sponsored by the National Science Foundation Grant OCE-0425029. Very helpful comments from anonymous reviewers significantly clarified this paper.

## REFERENCES

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### APPENDIX

#### Solving the Horizontal Momentum Equations

The momentum equations [Eqs. (4) and (5)] can be solved to yield expressions that relate the complex amplitudes of the local velocities and transports to the gradient of the sea level as follows. Equation (13) can be manipulated to express *V* in terms of *U:*

In a similar way, Eq. (14) can be used to express *U* in terms of *V*:

with boundary conditions

The solution is

where

and

When *f* = 1, these expressions remain finite (as long as *δ* ≠ 0) and are evaluated as

Equation (A1) can then be used to give the other component:

The expressions for *U* and *V* remain finite when *f* = 1, so long as *K** ≠ 0. The limits of *U* and *V* for *f* = 0 are

as given by Lamb (1932).

The vertically integrated transports are

where

and

where *P*_{0} and *Q*_{0} are complex functions of *f*, *δ*, and *h*. In the limit as *δ* tends to zero, both *P*_{0} and *Q*_{0} tend toward *h*/( *f* ^{2} − 1), and their ratio remains close to 1 for *δ* < 0.5. As *δ* increases, *Q*_{0} decreases as *δ*^{−4} and *P*_{0} tends to the limit

## Footnotes

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