## 1. Introduction

The barotropic circulation induced by the tide in a semienclosed basin behaves as a damped gravity wave system, with reflection at the closed boundary. Many, if not all, oceanic basins are stratified, however, and this stratification adds a baroclinic component to the tidal circulation. Tidal circulation in fjords (Allen and Simpson 1998, 2002; Elliot et al. 2003; Valle-Levinson et al. 2007) provides one example of interfacial tides. The observations described by Allen and Simpson (1998) for Upper Loch Linnhe are particularly relevant to this work because the first mode linear internal tide is so evident, with axial currents near ½ m s^{−1} or over 10 times larger than the expected barotropic current. The loch is too narrow to support either barotropic or baroclinic lateral fluctuations. Axial sections suggest that the baroclinic wavelength is about three-fourths the loch length. On a considerably larger scale, internal tidal motions have been described in the Strait of Juan de Fuca–Strait of Georgia system (Herlinveaux and Tully 1961; Foreman et al. 1995; Ott and Garrett 1998; Ott et al. 2002; Thomson et al. 2007; Martin et al. 2005). The observations of Herlinveaux and Tully (1961) document isotherm displacements with amplitude reaching up to 50 m, or about 100 times larger than the barotropic tide at the entrance.

The subject of baroclinic tides in semienclosed estuaries has received considerable interest (see, e.g., Farmer and Smith 1980; Webb and Pond 1986; Arneborgh and Liljebladh 2001). Those analyses consider the normal modes of a baroclinic system with a rigid lid and thus cannot describe how the surface or barotropic tide couples to and forces the baroclinic tide. Instead, the baroclinic modes are fit, in some fashion, to observations. The approach described here overcomes this substantial limitation by considering a coupled barotropic and baroclinic model of elongated basins to a co-oscillating tide.

This model exhibits near-standing wave patterns for the interface as well as the surface, as long as friction is not too large. For simplicity, horizontal density gradients are excluded. The central objective of this work is to show how, in a semienclosed two-layer basin, tidal forcing of the surface at the entrance drives both a barotropic component that is similar to the tidal wave that exists without stratification, as well as a baroclinic wave system. That wave system has characteristic scales that are considerably shorter than the barotropic system and, consequently, a much richer spatial structure.

## 2. The model

This model extends the constant density three-dimensional fluctuating tidal model described by Winant (2007) to two immiscible, but frictional, layers.^{1} A semienclosed elongated basin (Fig. 1) is forced by a co-oscillating surface tide. The length *L** (dimensional variables are denoted by asterisks) and the width 2*B** are much larger than the maximum depth *H**. The *x* coordinate points away from the adjacent ocean, and *y* points to the left of *x*. The vertical coordinate *z* is measured positive up from the undisturbed surface. The constant width basin, considered symmetric about the *x*–*z* plane, extends laterally from *y** = −*B** to *y** = *B**. At the sides (*y** = ±*B**), the depth is assumed to be very small. Variables in the top layer all bear the subscript 1; for example, the top layer density is *ρ*_{1}. Away from the sides, the upper layer has constant depth, which is given by *h*_{1}* = *ħH**. Near the sides, the surface layer extends all the way to the bottom. The bottom layer (where it exists) extends from *z** = −*ħH** to *z** = −*h** = −(*h*_{1}* + *h*_{2}*). For simplicity, the depths *h*_{1} and *h*_{2} are assumed to depend on *y* only. The position of the free surface relative to *z** = 0 is denoted by *η**. The position of the interface relative to *z** = −*ħH** is denoted by *ζ**. The constant 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, the tide in the adjacent ocean forces the sea level at

_{h}*x*= 0 to fluctuate with amplitude

*C**

*N*

_{0}(

*y*)ℜ(

*e*

^{−iσ*t*}), where max(

*N*

_{0}) = 1.

*f** is the Coriolis frequency so that

*f*represents the ratio of the basin length to the external Rossby radius of deformation, and

*σ*is a measure of the ratio of the basin length to the tidal wavelength. In addition to the barotropic parameters, solutions depend on the nondimensional density anomaly

*ρ*= 2(

*ρ*

_{2}* −

*ρ*

_{1}*)/(

*ρ*

_{2}* +

*ρ*

_{1}*), as well as the nondimensional baroclinic frequency,Although

*σ*is fixed for a semidiurnal tide, if the basin length and depth are fixed, then

*σ*′ can vary either if the thermocline depth

*ħ*or the density anomaly

*ρ*varies, as they do in natural systems. This implies that the system response has to be determined for a range of the parameter

*σ*′. Typically,

*σ*is on the order of one for most basins (because larger basins are also deeper), whereas

*σ*′ is two orders of magnitude larger.

*u*

_{1}and

*u*

_{2}represent the axial velocity in the top and bottom layers. The parameter

*ϵ*=

*C**/

*H** is the ratio of the amplitude of the tidal wave at the open end to the maximum depth. The same velocity has been chosen to nondimensionalize both horizontal velocities, based on the analysis of the equivalent rotating, constant density system (Winant 2007). Finally, surface and interface elevations are nondimensionalized as

*ϵ*becomes small, the horizontal momentum equations in the top layer becomeand in the bottom layer they becomewhere

*δ*=

*K**/

*σ**

*H**

^{2}

*z*= −

*ħ*), both the velocities and stresses have to be continuous,whereas at the bottom the velocities are zero,For periodic solutions, complex amplitudes are introduced asSolutions for the complex amplitudes

*U*

_{1,2}and

*V*

_{1,2}in terms of gradients of the complex amplitudes of the sea level and the interface are given in appendix A.

*U*

_{1,2}] and [

*V*

_{1,2}] can be eliminated by introducing expressions (A16)–(A19) to obtain a coupled set of elliptic partial differential equations (PDEs) for the surface and the interface:where

*Q*

_{0}

^{N}=

*Q*

_{1}

^{N}+

*Q*

_{2}

^{N}, etc., are defined in Eqs. (A20)–(A27):To quantify how barotropic fluctuations at the mouth can force a baroclinic circulation in the basin, the surface displacement at

*x*= 0 is prescribed, while the interface displacement is set to zero:In the case of a fjord, when the sill extends above the interface depth, the second condition is changed to force no transport out of the lower layer. At

*x*= 1, the upper- and lower-level transports are set to zero:On the lateral sides, we require zero transport in both layers:where

*y*

*B**/

*L** represents the horizontal aspect ratio of the basin:where

*ỹ*is the lateral extent of the lower layer, the location where the interface intersects the bottom (

*h*(

*ỹ*) =

*ħ*).

## 3. The sea level and the interface

Consider first the simple case of a two-dimensional, constant depth, frictionless basin, ignoring rotation, described in appendix B. The solution for *N* is given by Eq. (B11). Away from the immediate vicinity of the boundaries, the structure of the response is governed by *σ* and *σ*′. For most basins, *σ* is on the order of one or smaller (this is because longer basins are usually also deeper basins) so that wavelike barotropic modes are normally uniform in the lateral direction. Barotropic modes that have lateral variability (*l* > 0) are usually evanescent in *x*, confined near *x* = 0 and *x* = 1. In contrast, *σ*′ can reach very large values, so the baroclinic axial wavenumber *k*′* _{n}* can be real when

*l*> 0 and baroclinic modes with nontrivial lateral structure can even be resonant. This is the case for the second basin solution described in the following section.

Based on Eq. (B11), several points can be made. The first is that the solutions for the sea level and the interface consist of barotropic and baroclinic parts. The second point is that the barotropic component of the sea level usually dominates over the baroclinic component, whereas both components contribute comparable amounts to the interface amplitude. Third, the solutions are singular wherever *k _{n}* or

*k*′

*equals (2*

_{n}*n*+ 1)

*π*/2. The last point is that, when

*σ*′

*y*

*π*/2, nonevanescent modes can be excited that are laterally variable (as in the Strait of Juan de Fuca–Strait of Georgia system).

This analysis can be extended to the case when *f* ≠ 0, when the fundamental mode consists of Kelvin waves propagating in the axial direction, whereas the higher-order modes are Poincaré modes. As shown below, friction modifies these solutions because the no-slip boundary condition forces the axial velocity to be sheared, so the lateral Coriolis acceleration is also sheared. Winant (2007) shows that this forces a periodic lateral circulation of amplitude comparable to the axial flow. When friction and rotation are included, the coefficients *Q* and *R* in Eqs. (19) and (20) are complex. Even then the lateral problem is expected to have real eigenvalues that satisfy dispersion relations equivalent to expression (B9).

The richness of internal tides reported in basins such as Upper Loch Linnhe and the Strait of Juan de Fuca–Strait of Georgia system, where neither the rotation of the earth, the variation of the bathymetry, nor friction can be ignored, motivates exploring solutions to Eqs. (19)–(24). In this more complex case, it is expedient to solve this system of equations [Eqs. (19)–(24)] numerically. Solutions are described for the semidiurnal (*σ* = 2*f* ) tide in two basins. The first is an idealized representation of Upper Loch Linnhe (Allen and Simpson 1998) and the second is an idealized version of the Strait of Juan de Fuca–Strait of Georgia system. Characteristics of the two basins are summarized in Table 1. In both cases the, depth is a parabolic function of lateral position only: *h* = 0.01 + 0.99(1 − *y*^{2}). Both basins are forced at the entrance by a unit amplitude surface elevation (*N _{o}*(

*y*) = 1).

The transport out of the lower layer of basin 1 is required to be zero ([*U*_{2}](*x* = 0, *y*) = 0), corresponding to the sill, at the entrance of Upper Loch Linnhe, that extends above the interface. Because this basin is both short and narrow, the surface displacement is nearly horizontally invariant: the free surface tracks conditions at the entrance. The maximum amplitude of the interface response as a function of the baroclinic wavenumber *σ*′ is illustrated in the left frame of Fig. 2. As expected from the simple analysis presented above, the interface response is characterized by multiple peaks, corresponding to near resonances of the baroclinic mode. The baroclinic wavenumber *σ*′ is 6, which is close to the second peak in the response; nearly a full cycle of the interfacial mode is expected, in agreement with the observations (Allen and Simpson 1998). Cotidal charts for the interface are illustrated in Fig. 3 for three different values of the density anomaly *ρ*. In each case, the complex amplitude of the interface has two maxima along the central axis and the lateral variations are slight. The values of the different maxima and the phase at the closed end are sensitive functions of *ρ*. It is useful to emphasize that the open boundary condition applied on the interface is that the lower layer transport out of the basin is zero, so the interface displacements illustrated in Fig. 3 are only forced by the tide at the free surface.

For basin 2, the cotidal chart for the surface displacement *N* is illustrated in Fig. 4. The surface exhibits an amphidrome near the middle of the basin, in qualitative agreement with the *M*_{2} surface charts for the Strait of Juan de Fuca–Strait of Georgia system published by Foreman et al. (1995). Farther away from the ocean, the amplitude increases, reaching a maximum slightly in excess of the entrance value at the closed end.

Because there is no sill at the western entrance to the strait and because very little is known about the internal tide there, the interface amplitude is forced to zero at *x* = 0 to focus attention on interface displacements in the basin forced by the surface fluctuation at the entrance. For relatively small values of *σ*′ (less than 50), the interface amplitude illustrated in Fig. 2 exhibits resonances corresponding to laterally uniform modes, when *σ*′ is near multiples of *π*. The amplitude of the response increases by over an order of magnitude when *σ*′ > 60, or *y**σ*′ > 2, as lateral interfacial modes become excited. The largest peaks in the response correspond to near resonance of the first, third, and other odd lateral modes. This response is similar to the “seiching” mode described by Martin et al. (2005).

The phase and amplitude of the interface *I* are illustrated in Fig. 5 and three values of the baroclinic frequency *σ*′. The central cotidal chart for *ρ* = 10^{−3} is near the first maximum amplitude illustrated in Fig. 2. The maximum interface amplitude is 90 times the sea level amplitude at the entrance of the basin. The variability is lateral: when the interface rises on one side, it falls on the other. This suggests that the lateral eigenvalue is *n* = 1, *l*_{1} = 120, and *y**l*_{1} = 2.4. Cotidal charts on the right and left of Fig. 5, for different values of *ρ*, demonstrate a combination of lateral and axial variability. The left frame (*ρ* = 4 × 10^{−3}, *σ*′ = 60, and *y**σ*′ = 1.2) exhibits a pattern of oscillations in the axial direction, with near 10 cycles. This is consistent with *n* = 0, *l*_{0} = 0, *k*_{0} = *σ*′. In the right-hand chart of Fig. 5, *ρ* = 8 × 10^{−4}, *σ*′ = 134. Because *σ*′ > *l*_{1}, *l*_{1} = 120 and *k*_{1} = ^{2} − 120^{2}

To illustrate how sensitive the solutions are to the stratification parameters, the interface *ζ* when the surface is high at the entrance (the end of ebb tide) is mapped for three different values of the density anomaly in Fig. 6. The amplitude and phase of the interface change markedly, even for small changes in *ρ*. This suggests that, without any change in a surface tide, the amplitude and phase of the interface can change by large amounts in response to changing *ρ*. This could explain the observation that the interface fluctuations as well as the baroclinic currents are not locked in phase with the barotropic tide.

## 4. Velocities

Given the amplitudes of the sea level and the interface, the horizontal velocities can be computed from Eqs. (A9) and (A10). The vertical velocity is then obtained by integrating the mass conservation equation up from the bottom.

The complex amplitudes of the axial and vertical velocity components in basin 1 are illustrated in Fig. 7 in such a way as to facilitate comparison with Allen and Simpson (1998). The axial velocity would be of order one for a constant density basin. Instead, for this value of *σ*′, the axial velocities are for the most part one order of magnitude greater, with a phase change of *π* at the depth of the interface. These large internal velocities are explained by the large interface displacement that arises because of convergences and divergences in the horizontal velocity field. The upper layer axial velocities are relatively larger than in the bottom layer because the top layer is nearly 10 times smaller than the bottom layer. Given the boundary conditions, the vertical velocity has to be one at the surface. In the constant density case, it would increase monotonically from zero at the bottom to that value at the surface. In this two-layer system, the vertical velocity is also an order of magnitude greater than without the interface, with a maximum at the interface depth: baroclinicity dramatically changes the amplitude and structure of the velocity field. In real flows, these features would lead to enhanced mixing and dispersion. For this basin, the lateral *y* velocities are small. The agreement between velocities predicted by this model and observations in Upper Loch Linnhe is remarkable.

The complex amplitudes of the axial *U*, lateral *V*, and vertical *W* velocity components at five different sections for basin 2 are illustrated in Fig. 8. For the case illustrated, *y**σ*′ = 2. This value has been chosen to emphasize the lateral variability in the flow, or more accurately, the case where *k*′ is small. The amplitude of the axial velocity (top row) is uniform in the lower layer (beneath *z* = −*ħ* = −0.35), adjusting to the no-slip condition in a boundary of thickness comparable to *δ*. The phase is close to *π*/2, corresponding to peak flood lagging high water at the entrance by a half cycle (because the basin is between one-quarter and one-half wavelength: *σ* = 1.9 > *π*/2). In the upper layer, the axial velocity is about 10% less than in the bottom layer, and the phase is about 10° less than in the lower layer. The decrease of axial velocity with increasing depth is consistent with the “distinctive vertical structure” reported by Martin et al. (2005).

The lateral velocities are comparable to *U*. They change phase across the interface: when the upper layer sloshes in one direction, the lower layer sloshes in the opposite direction. The vertical velocities are largest near the sides of the basin. The nondimensional *W* are about 10 times larger than the nondimensional axial velocities. In the lower layer, *V* is *π*/4, or lagging the axial velocity. The vertical velocities are in phase with *U* on the positive *y* side of the basin (left side in each section of Fig. 8).

The relatively unvarying phase of the axial velocity combined with the reversing direction of the lateral velocity *V* with depth means that this baroclinic component is very different from a mode 1 internal wave, where both components would have to change direction in such a way as to maintain clockwise rotation at all depths. In this case, the sense of rotation changes at the interface, which is consistent with combined surface and interfacial Kelvin waves.

## 5. Discussion

The central result of this work is to demonstrate that the interface, when forced by the surface tide, responds in a broadly similar way to the surface, supporting a near-standing wave pattern characterized by smaller wavelengths than the surface. The fact that either the surface or the interfacial mode can be resonant means that estuaries and straits can have very large tidal velocities, even when the surface tide is far from resonance, as observed both in Upper Loch Linnhe and the Strait of Juan de Fuca–Strait of Georgia system.

### a. Fjords: Basin 1 and Upper Loch Linnhe

Upper Loch Linnhe is a strongly stratified Scottish fjord forced at the entrance by a barotropic tide that ranges in amplitude between 1 and 3 m (Allen and Simpson 1998, 2002). The sill depth is less than the usual depth of the thermocline, so the lower layer is effectively isolated from the open ocean. Observations demonstrate that the internal tide behaves as a standing wave, and the corresponding amplitude of the currents in the surface layer is reported to be 0.57 m s^{−1}, whereas the barotropic component is an order of magnitude smaller.

Observed maximum vertical velocities (Allen and Simpson 2002; Fig. 5, middle) are reported to be 10^{−2} m s^{−1}, corresponding to a nondimensional value [Eq. (4)] of 20, which is somewhat larger than the results illustrated in Fig. 7. This seems reasonable agreement given the sensitivity of the solutions to the exact value of the density anomaly and interface depth. The structure of the observed vertical velocity pattern is similar to the pattern of interface displacement illustrated in Fig. 3: maximum values at the closed end and a third of the way in from entrance. The solutions are reasonably insensitive to lateral position.

Allen and Simpson (1998, 2002) interpret their observations in terms of normal modes associated with linear internal nonrotating wave theory, and the more complete theory presented here confirms the validity of that interpretation. The benefit of this added complexity is that, in this model, the coupling between the barotropic and interfacial tides is explicitly treated.

### b. Straits: Basin 2 and the Strait of Juan de Fuca–Strait of Georgia system

The Strait of Juan de Fuca–Strait of Georgia system is strongly stratified system, characterized by surface tidal amplitudes on the order of ½ m and large internal tides of amplitude up to a maximum of 50 m (Herlinveaux and Tully 1961). The barotropic tide has been modeled in detail by Foreman et al. (1995). The most striking feature is an amphidrome located near Victoria on Vancouver Island. That amphidrome is reproduced in the solution for *N* illustrated in Fig. 4, as is the ratio of the amplitude of the surface tide near the closed end to the amplitude at the entrance (1.45).

Martin et al. (2005) offer several models to explain the large internal tides and conclude that “much of the baroclinic structure can be explained as a cross-channel, internal seiche that is locally driven by reversing Ekman forcing in the bottom boundary layer.” The “reversing Ekman forcing” is the corkscrew motion described by Winant (2007) that is driven by the vertical shear in the lateral Coriolis acceleration. As Martin et al. (2005) note, this lateral circulation can excite cross-channel internal waves, or seiches. However, the analysis presented here demonstrates that this motion is almost never strictly two dimensional. Instead, it is a two-dimensional view of a wave motion that is better described as a frictional, near-standing internal Kelvin wave. The authors report they “did not find any significant along-channel shift in the semidiurnal signal.” This is consistent with a standing Kelvin mode, as long as the axial wavenumber is small enough that the observations are confined within a half wavelength.

The observed amplitude of the *M*_{2} tide near the Pacific Ocean is about 0.6 m (Foreman et al. 1995), reaching up to a maximum near 1 m near the northern end of the Georgia Strait. In contrast, observed semidiurnal interface displacements are quite variable in time, ranging from the maximum of 50 m, or 100 times the surface amplitude at the entrance (Herlinveaux and Tully 1961), to an average amplitude of 5 m (Martin et al. 2005, their Fig. 12a). These amplitudes are entirely consistent with the model presented here: the maximum amplitude of *I* in Fig. 2 is 180. Variations in the amplitude of the response are consistent with the reported spring–neap variations in the stratification due to tidal mixing.

Vertical profiles of ellipse parameters (as defined by Martin et al. 2005) are illustrated in Fig. 9 for a site located at *x* = 0.2, *y* = 0.7 (roughly corresponding to site 99SA) in the observations. The profiles in Fig. 9 are in the same format as in Fig. 9 of Martin et al. (2005). The agreement between this model and the observations appears to be quite good: the major axis is larger in the lower layer, and the phase increases up through the bottom boundary layer and then decreases slightly. The minor axis is positive in the bottom boundary layer, then it becomes negative in the lower layer, and then positive again in the upper layer. The agreement is quantitatively good as well: velocities are nondimensionalized by the product *ϵσ***L** = (5 × 10^{−3})(1.4 × 10^{−4})(5 × 10^{5}) ≈ 0.4 m s^{−1}. The quantitative values of the phases agree as well.

Although the agreement described above is quite good, it is only fair to point out (as two anonymous reviewers have) that the model simplifies features of the Strait of Juan de Fuca–Strait of Georgia system so much that some of the agreement may be coincidental. The Victoria Sill region, which might be expected to modify the internal wave structure, is not represented in the model basin. It may be that because most of the circulation associated with the baroclinic mode is predominantly lateral, rather that perpendicular to the sill, the real topographic effect is not of great importance, but further study would have to be made to verify this.

### c. Internal tides on the open coast

The main difference between the two idealized basins described above is their length, as measured by the parameter *σ* (Table 1). Because of this difference, the parameter *y**σ*′ is above unity for basin 2, resulting in vigorous lateral oscillation of the interface. This raises the question of what this model would predict for the behavior of a truly large basin, of oceanic proportions, in terms of motion of the interface, ignoring variations in the Coriolis parameter. Because both the depth and the horizontal dimension of ocean basins are larger than a strait, the parameter *σ* is of comparable amplitude; however, typical ocean basins have comparable widths and lengths, so the parameter *y**σ*′ is expected to be considerably larger. This suggests that the baroclinic wavelength will be much smaller than the basin width. Because these are Kelvin waves, they are expected to be trapped to the coast. This opens the question of whether internal tides on the open coast could be associated with baroclinic wave systems, just as the surface tide is associated with a barotropic Kelvin wave system.

Lerczak et al. (2003) describe internal wave observations on the Southern California shelf and point out that the observed structure is “not at all what the mode 1 picture would predict,” because the sense of current rotation (always clockwise polarized in a mode 1 internal wave) changes with depth in the observations (similar to the behavior illustrated in Fig. 9 above). Dale and Sherwin (1996) and Dale et al. (2001) extend baroclinic coastal-trapped wave theory to superinertial and near-inertial frequencies and point out that “in the ocean, tides provide forcing at discrete superinertial frequencies and could produce resonant response leading to an alongshelf dependence of the internal tide.” These results provide motivation to apply the method described here to the open continental shelf.

## 6. Summary and conclusions

The tidal circulation in an elongated basin of width less than the external Rossby radius, but otherwise general shape, has been described with a linear, coupled barotropic and baroclinic model on the *f* plane. The solution shows that, for moderate friction, both the surface and interface support standing wave patterns, even when forcing is confined to the surface alone. The barotropic mode is governed by the nondimensional parameter *σ* = *σ***L**/*g***H***σ*′ = *σ*/*ρħ*(1 − *ħ*

The discussion suggests that, in broad terms, the baroclinic circulation falls into one of three patterns. If the basin width is less than the baroclinic wavelength (basin 1), the variability is confined to the axial direction. If the basin width is comparable or greater than the baroclinic wavelength, the circulation exhibits lateral variability and, as in the case of basin 2, that response can be near resonant, resulting in very large internal tides. Finally, it is surmised that, for basins of oceanic proportions, the internal tidal patterns will be trapped within a coastal region of width on the order of a few baroclinic wavelengths.

The model presented here is supported both qualitatively and quantitatively by field observations in a narrow basin (Upper Loch Linnhe in Scotland) and in an intermediate basin (the Strait of Juan de Fuca–Strait of Georgia system), even though the idealized basins used as models have very simplified bathymetry.

## Acknowledgments

This work was sponsored by the National Science Foundation Grant OCE-0726673. Suggestions by Alejandro Souza, Aurelien Ponte, and two anonymous reviewers are gratefully acknowledged.

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## APPENDIX A

### Solving the Horizontal Momentum Equations

*ħ*, the boundary condition at the bottom is the same as in the constant density problem,If the depth is greater than

*ħ*, the condition at the interface (

*z*= −

*ħ*) iswhile at the bottom (of the lower layer)The two parametersare defined such that

*q*,

^{N}*r*,

^{N}*q*, and

^{I}*r*specify the

^{I}*z*dependence of the velocities and the superscripts

*N*and

*I*identify whether the forcing is due to sea level or interface gradients. These functions are evaluated asThe expressions for

*U*and

*V*remain finite when

*f*= 1, as long as

*K** ≠ 0.

*Q*

_{1}

^{N},

*Q*

_{2}

^{N},

*R*

_{1}

^{N},

*R*

_{2}

^{N},

*Q*

_{1}

^{I},

*Q*

_{2}

^{I}and

*R*

_{1}

^{I},

*R*

_{2}

^{I}depend on the local layer thicknesses, as well as

*δ*and

*f*:where

*h*

_{1}= min(

*h*,

*ħ*) and

*h*

_{2}= max(

*h*−

*ħ*, 0).

## APPENDIX B

### A Simple Solution for the Sea Level and Interface

*h*= 1) and friction and rotation are ignored, Eqs. (19) and (20) simplify toAt the entrance, the boundary conditions areOn the three other boundaries, there is no transport,Subtracting Eq. (B1) from Eq. (B2) gives an expression for

*I*in terms of

*N*,When Eq. (B6) is substituted into Eq. (B1), a fourth-order partial differential equation for

*N*is obtained,This problem is the equivalent to the vibrating membrane problem. If solutions of the form

*N*(

*x*,

*y*) = ∑

*are sought, where*

_{n}X_{n}Y_{n}*X*=

_{n}*e*and

^{iknx}*Y*=

_{n}*e*, Eq. (B7) becomesThere are several possibilities,where terms of order

^{ilny}*ρ*

^{2}have been neglected. The left-hand root corresponds to the barotropic mode, whereas the right-hand root is the baroclinic mode.

*Y*, and eigenvalues are

_{n}*l*= (

_{n}*nπ*)/(2

*y*

*n*= 0, 1 …;

*n*even corresponds to symmetric eigenvectors (

*Y*= cos

_{n}*l*), whereas

_{n}y*n*odd corresponds to antisymmetric eigenvectors (

*Y*= sin

_{n}*l*). A solution for

_{n}y*N*that satisfies the boundary at the closed end [Eq. (B3)] can be written aswhere

*x̃*= 1 −

*x*and

*k*,

_{n}*k*′

*are given by Eq. (B9). Resonances are expected whenever cos*

_{n}*k*or cos

_{n}*k*′

*, with*

_{n}*k*,

_{n}*k*′

*real, are zero.*

_{n}*A*, etc., are determined by the boundary conditions at

_{n}*x*= 0 and the orthogonal property of eigenvectors,For arbitrary

*N*(

_{o}*y*), eigenvectors corresponding to both positive and negative

*k*

_{n}^{2}, (

*k*′

*)*

_{n}^{2}will be included in the solution. When the

*x*component of the wavenumber is complex, those modes are evanescent in

*x*.

Parameters used for two idealized basins.

^{1}

There are important differences in nomenclature between Winant (2007) and this paper.