a. Summary of previous observations

The first observation of internal solitary waves generated in the Strait of Messina was made by the SAR aboard the American Seasat satellite on 15 September 1978. The three rings visible on the Seasat SAR image in the Tyrrhenian Sea north of the strait were interpreted as sea surface manifestations of a train of internal solitary waves propagating northward (Alpers and Salusti 1983). In the following years, oceanographic campaigns were carried out to measure internal solitary waves north and south of the strait. In November 1980 two northward propagating internal wave trains were detected in temperature data records at a position 25 km north of the strait. The time lag between the time of arrival of the two wave trains at this position was 11 h 30 min, which agrees well to the time interval of 11 h 22 min between the two successive tidal flow reversals at the strait’s sill, identified as the release times of the observed internal bores from the sill (Alpers and Salusti 1983; Griffa et al. 1986). During an oceanographic cruise from May to June 1982, Sapia and Salusti (1987) detected in temperature data records two trains of internal solitary waves north of the strait. During this cruise oceanographic measurements were also carried out south of the strait, which revealed the existence of large amplitude isolated signals in the temperature data records south of the sill and as far as to the coast of Syracuse. By using temperature sensors and a KODEN fish finder, Di Sarra et al. (1987) observed a 100-m deep depression of the interface between the water masses in the zone immediately south of the sill. The first observation of a well developed internal wave train south of the sill was made by the thematic mapper (TM) aboard the Landsat satellite (Artale et al. 1990). In October 1987 Nicolò and Salusti (1991) also observed in temperature data records three large amplitude internal wave trains south of the sill.

b. Synthetic aperture radar data from the ERS 1/2

Internal waves become visible on radar images because they are associated with variable surface currents that modify the surface roughness patterns via current–wave interaction. The radar is a surface roughness sensor: the rougher the sea surface, the higher the normalized radar cross section (NRCS) and thus the brighter the image intensity. According to a first-order radar imaging theory, the relative variation of NRCS associated with internal waves Δσ/σ₀ is related to the gradient of the surface velocity ∂u/∂x by

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where A denotes a positive function that depends, among others, on radar wavelength, radar incidence angle, angle between the radar look direction and the propagation direction of the internal wave, azimuth angle, and wind velocity at the sea surface (Alpers 1985). For a linear SAR system, Δσ/σ₀ is equal to the relative variation of the SAR image intensity (modulation depth). Thus, to first order, the variation of the image intensity is proportional to the gradient of the surface velocity or, as we will call it in the following, to the surface convergence, −∂u/∂x.

From 1991 to 1995, several SAR images have been acquired by the European Remote Sensing satellites ERS 1 and ERS 2 over the Strait of Messina and the adjacent sea areas. In this investigation, we have analyzed ERS 1/2 SAR images acquired during 160 satellite overflights over the sea areas north and south of the sill. On SAR images acquired during 77 satellite overflights, sea surface manifestations of internal waves could be delineated. Figure 1 shows an ERS 1 SAR image, which was acquired at 2113 UTC 17 August 1995 (orbit: 21 388, frames: 747/765). On this image, sea surface manifestations of three internal wave trains propagating southward and one propagating northward can be identified. Regarding the available ERS 1/2 SAR dataset of the Strait of Messina, one has to note that, due to the different orbits flown by the ERS 1/2 satellites, more SAR images were acquired over the Strait of Messina in the period from April to November than in the period from December to March (see Fig. 2).

The analysis of the available ERS 1/2 SAR images shows that sea surface manifestations of internal waves are observed more frequently during periods where a strong seasonal thermocline is known to be present, that is, during summer. Furthermore, sea surface manifestations of southward propagating internal waves can be delineated on ERS 1/2 SAR images more frequently than those of northward propagating ones (Fig. 3). In general, sea surface manifestations of southward propagating internal waves are stronger than those of northward propagating ones. The SAR image depicted in Fig. 1 shows evidence of such a north–south asymmetry in the sea surface manifestations of the internal wave field.

Figure 4 shows a time versus distance diagram delineating the propagation of internal bores north and south of the sill as inferred from the analyzed ERS 1/2 SAR images. In this figure, the distance between the fronts of internal wave trains and the sill, referenced to the time of maximum northward tidal flow at Punta Pezzo, is depicted. This distance is measured along a mean path of the center of the wave fronts, which is inferred from the available ERS 1/2 SAR images. The lines are least square fits that yield for the northward propagating waves an average propagation speed of 1.00 m s⁻¹ and for the southward propagating waves an average propagation speed of 0.91 m s⁻¹. From this figure, the time of release of internal bores from the sill can be estimated: Southward propagating internal bores are released from the sill between 1 and 5 hours after maximum northward tidal flow at Punta Pezzo, northward propagating internal bores between 2 and 6 hours after maximum southward tidal flow at Punta Pezzo, that is, between 8 and 12 hours after maximum northward tidal flow at Punta Pezzo. Figure 5 shows the spatial separation between the first two internal solitary waves of northward and southward propagating wave trains as a function of the distance from sill as inferred from ERS 1/2 SAR images. This spatial separation ranges from 500 m to 1900 m for southward propagating wave trains, and from 350 m to 1000 m for northward propagating wave trains in the region considered here. Figure 5 shows that, in general, the spatial separation between the first two internal solitary waves of southward propagating wave trains is smaller in the period from July to September than in the period from October to June. A detailed comparison of the spatial separation between the first two internal solitary waves as inferred from ERS 1/2 SAR images and as calculated with our numerical model will be presented in section 7.

Figures 6 and 7 show two ERS 1 SAR images acquired at 0941 UTC 11 July 1993 (orbit: 10 387, frame: 2835) and at 2114 UTC 13 July 1995 (orbit: 20 887, frame: 765). On both images sea surface manifestations of northward propagating internal solitary waves are visible. Note the pronounced difference in shape, wavelength, and modulation depth of the sea surface manifestations of these two wave trains. The wave train visible on the image depicted in Fig. 6 has a strong surface signature and a weak wave front curvature. It propagates along the Calabrian coast in the direction of the strait axis. On the contrary, the wave train visible on the image depicted in Fig. 7 has a weak surface signature, a strong wave front curvature, and it propagates almost spherically from the northern mouth of the Strait of Messina into the Tyrrhenian Sea. The large variability in the sea surface manifestations of northward propagating internal waves is a general feature that we have noticed in the analysis of the ERS 1/2 SAR images.

a. Theory

The proposed model for the description of the water mass dynamics in the Strait of Messina is a two-layer model as schematically depicted in Fig. 8. In this figure, D is the water depth, h₁ and h₂ are the thicknesses of the upper and lower layers respectively, ζ₁ is the sea surface displacement, and ζ₂ is the distance of the interface from the mean sea surface. The model is capable of describing the flow of a two-layer water mass through a channel with a variable water depth and a variable width. The governing equations used in this investigation are the nonlinear primitive Boussinesq equations for two water layers. They are obtained from a perturbation expansion of the Laplace equation for the velocity potential for both layers in the case of shallow water and small wave amplitude. In terms of the expansion parameters μ² and ϵ, defined by μ² = k²H² and ϵ = aH⁻¹, where a and k denote the amplitude and the wavenumber of the internal wave respectively, and H denotes a characteristic depth scale; the model is valid for ϵ ≪ 1 and μ² ≪ 1. In the Boussinesq equations only terms O(μ², ϵ) are retained and terms O(μ⁴, μ²ϵ, . . .) are neglected. The hydrostatic shallow-water equations, which are used in the generation model, are obtained by neglecting also terms O(μ²). Furthermore, the model requires that the water depth D = D(x) and the channel width W = W(x) are weakly varying functions of the space variable x. In this model we assume that the channel has a rectangular cross section. The equations used in our model include horizontal momentum diffusion and vertical shear stress terms at the bottom and at the interface between the two water layers. In the following we denote the vertically and horizontally (perpendicular to the strait axis) averaged velocities and transports by u_i and Q_i = u_ih_iW respectively. The subscripts i = 1 and i = 2 refer to the upper and lower layer respectively. For the upper layer, the momentum and the continuity equations read

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while for the lower layer the corresponding equations read

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Here g′ = gΔρ/ρ̄ denotes the reduced gravity, g is the acceleration of gravity, Δρ = ρ₂ − ρ₁ is the density difference between the two layers, ρ̄ = (ρ₁ + ρ₂)/2 is the mean water density, A_H is the horizontal eddy viscosity coefficient, τ_int is the vertical shear stress at the interface, and τ_bot is the bottom stress. The stresses τ_int and τ_bot are assumed to depend on the current velocities in the upper and lower layers, u₁ and u₂, respectively, in the following way:

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Here r_int and r_bot denote the nondimensional friction parameters at the interface and the bottom respectively. The last terms in the two momentum equations (2) and (4) describe the weakly nonhydrostatic effects in the Boussinesq equations. Note that in our model we have neglected the Coriolis force. In order to minimize reflections at the open boundaries, we have implemented in our model the Orlanski radiation condition (Orlanski 1976).

We have introduced in (2) and (4) horizontal diffusion terms in order to avoid the development of flow discontinuities at the transitions between subcritical and supercritical flow (e.g., see Woodward and Colella 1984; Oğuz et al. 1990). In our model we assume that the horizontal eddy viscosity coefficient A_H is constant and have chosen the value of A_H as the minimum value ensuring numerical stability.

b. The generation and propagation models

The numerical model presented in this paper is a composite of two models consisting of 1) a hydrostatic “generation model” equipped with movable lateral boundaries and 2) a weakly nonhydrostatic “propagation model.” The first model describes the dynamics of the water masses in the region close to the strait’s sill where the internal bores are generated (generation region), and the second model describes the dynamics of the water masses outside of this region where the internal bores may disintegrate into internal solitary waves (propagation region).

In the generation model, the weakly nonhydrostatic terms that appear in the momentum equations (2) and (4) are neglected. In this model, we allow the lower layer to intersect the bottom topography, which is achieved by introducing movable lateral boundaries for the lower layer. For those grid points where no lower layer exists, we treat the integration domain as composed of only one active upper layer. For further details about the treatment of movable lateral boundaries, the reader is referred to the papers of Backhaus (1976) and Jungclaus and Backhaus (1994). In order to maintain numerical stability, the interface depth has been low-pass filtered every 100 time steps in the sill region. The equations are discretized on a staggered Arakawa C-grid and the time-stepping procedure is performed by an explicit two-time-level scheme.

In the propagation model the complete set of equations (2)–(5), which include also the weakly nonhydrostatic terms, are solved. As in the generation model the equations are discretized on a staggered Arakawa C-grid. However, here the time-stepping procedure is performed by an implicit two-time-level scheme.

Due to the explicit scheme used in the generation model, the calculation is staggered in time. This means that, for every model iteration, the calculated displacement values ζ_i and the calculated transport values Q_i are determined for the times t − Δt/2 and t respectively, where Δt is the time step used in the calculations. On the other hand, in the implicit scheme used in the propagation model both the displacement values ζ_i and the transport values Q_i are determined for the same time t. The two-way coupling between the generation and propagation model for each iteration is implemented as follows: If ζ_i(t − Δt/2) and Q_i(t) are known for the whole integration domain, the explicit generation model yields ζ_i(t + Δt/2) for the whole integration domain and Q_i(t + Δt) for the generation region. The values Q_i(t + Δt) for the generation region are inserted as boundary conditions into the propagation model. Averaging ζ_i(t − Δt/2) and ζ_i(t + Δt/2) yields ζ_i(t). Using ζ_i(t), Q_i(t), and the boundary conditions Q_i(t + Δt), the implicit propagation model yields the transport values Q_i(t + Δt) in the propagation regions. This procedure is repeated step by step in time. The exact location of the two boundaries between the generation region and the propagation regions will be specified in section 6a.

a. Test of the generation model

In order to test the generation model, we have carried out numerical simulations for the same flow types as investigated by Baines (1984). He performed laboratory experiments for studying the stationary flow of a two-layer fluid over topography for different values of the internal Froude numbers F. The Froude number is defined as F = U/(g′H₂)^1/2, where U denotes the velocity of the undisturbed flow, which here is assumed to be the same in both layers, and H₁ and H₂ denote the undisturbed thicknesses of the upper and lower layer respectively. The channel width is assumed to be constant and the sill to have the following depth profile

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where D_max the maximum water depth, D_min the minimum water depth (at the sill), and L the horizontal length scale of the sill. The center of the sill is located at x = 0. In our simulations we have chosen: H₁ = 150 m, H₂ = 150 m, D_max = 300 m, D_min = 135 m, L = 1667 m, and g′ = 9.5 × 10⁻³ m s⁻².

Three cases with different Froude numbers are studied: (a) F = 0.033, (b) F = 0.51, and (c) F = 1.25, which are shown in Figs. 9, 10, and 11 respectively. Case a (F = 0.033, subcritical) describes a flow type where the lower layer flow is totally blocked by the sill (Fig. 9). Case b (F = 0.51, near critical) describes a flow type where an internal hydraulic jump is formed downstream of the sill (Fig. 10). For both cases, a new, larger upstream lower layer thickness will be achieved, which represents the adjustment of the system to the condition of critical flow immediately upstream of the sill (Baines 1984). Case c (F = 1.25, supercritical) describes a flow type that is everywhere supercritical. After a transient phase, a stationary symmetrical bump is formed over the sill and no upstream disturbances are generated (see Fig. 11). Figure 12 shows the first half-hour of the evolution of the interface for case b. This figure illustrate the “run-up” of the lower layer on the upstream side of the sill. Note that, while bottom and interface friction have been neglected in these simulations, only in the last two simulations has an eddy viscosity been introduced in order to maintain numerical stability. The simulated flow types agree at least qualitatively with the flow types found experimentally by Baines (1984). Due to the periodic tidal flow in the Strait of Messina, these different flow types can occur in the sill region. We feel confident, therefore, that our model is capable of describing correctly the interaction of a tidal flow with a sill where the lower layer may intersect the bottom topography.

b. Test of the propagation model

It is well known that, from internal disturbances propagating in a weakly nonhydrostatic, nonlinear medium as described by the Boussinesq equations (2)–(5), in the case of constant depth and width, waves of constant shape may evolve, which are called internal solitary waves (Keulegan 1953; Long 1956; Benjamin 1966; Benny 1966; Miles 1979). Different equations, with different degrees of accuracy, have been derived for describing the propagation of internal solitary waves. In this section we compare our numerical solutions of the Boussinesq equations (2)–(5) for solitary waves with the analytical solutions of the Korteweg–de Vries (KdV) equation (Korteweg and de Vries 1895; Djordjevic and Redekopp 1978), the extended KdV equation (Koop and Butler 1981), the Benjamin–Ono equation (Benjamin 1967; Ono 1975), and the Miyata equations (Miyata 1988). In this comparison we assume, for simplicity, that the water depth and the channel width are constant and that there is no diffusion and no friction. In the simulations carried out for this comparison we have chosen the following values for the parameters: g′ = 20 × 10⁻³ m s⁻², H₁ = 150 m, and H₂ = 850 m. Solitary wave solutions of the KdV equation with different amplitudes have been used as initial conditions for our numerical simulations. As the numerical model has produced waves of constant shape, such shape, together with the corresponding phase speed, has been used for the comparison of our numerical solutions with the different analytical solutions.

Among the aforementioned equations, only the Miyata equations can describe internal wave forms whose amplitudes are of the same order as the characteristic water depth H = H₁H₂/(H₁ + H₂) (Miyata 1988). For the case of constant water depth, Miyata (1988) found an analytical solitary wave solution of these equations.

Figure 13 depicts the wavelength–amplitude relationship in nondimensional form calculated with the various models. Here the wavelength L of a solitary wave (equivalent-square-wavelength) is defined by

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where A denotes the amplitude (peak to trough) of the solitary wave. Figure 13 shows that the solitary wave solutions of our propagation model agree quite well with the solutions of the Miyata equations. The wavelength of solitary waves calculated with the Miyata equations as well as from our model increases with increasing amplitude for large amplitude waves (A/H ≳ 1). This is a consequence of the fact that, as the wave amplitude increases, the nonlinear effects due to the motion in the lower layer increase and tend to compensate the nonlinear effects due to the motion in the upper layer. Since nonlinear effects are responsible for the wave steepening, the attenuation of the coupled nonlinear effect due to the motion in both layers must lead to an increase of the wavelength. Figure 14 depicts the phase speed–amplitude relationship in nondimensional form calculated with the various models. Although the Boussinesq model is derived by assuming that the amplitude of the wave is small, we see that the wavelength–amplitude and the phase speed–amplitude relationships calculated numerically with our propagation model agree quite well with the relationships calculated analytically with the Miyata equations (see Figs. 13, 14). Therefore, we feel confident that our propagation model is also capable of describing correctly the propagation of internal solitary waves in the Strait of Messina, where the internal solitary waves have amplitudes that often are not small compared to the characteristic water depth.

a. Model parameters

In the model calculations presented in this section we approximate the bottom topography of the Strait of Messina by the depth profile and the channel width, which are plotted in Figs. 15a and 15b. The model domain covers a distance of 70 km in the alongchannel direction. However, in the plots only a range of 52 km around the sill is shown. All quantities are plotted as a function of the alongchannel coordinate x. At the sill (x = 0) the water depth is 98 m. The strait is narrowest at a location 1.5 km north of the sill where the channel width is 2.8 km (see Fig. 15). The boundaries between the generation region in the sill area and the propagation regions north and south of the sill area are chosen such that their locations has no significant influence on the evolution of northward and southward propagating internal waves. After several test runs we set the boundary north of the sill region at x_n = 5 km and the boundary south of the sill region at x_s = −4 km. For x_s ≤ x ≤ x_n, the generation model is used, outside of this region the propagation model is used. The two-layer system is forced by imposing barotropic, semidiurnal tidal displacements at the two open boundaries. The imposed tide has been chosen in accordance with Vercelli’s data (Vercelli 1925): The amplitudes at the northern and southern boundary are 17 and 10 cm respectively, the phase difference is 174° and the semidiurnal tidal period is 12.4 h. The other model parameters used in the simulations are given in Table 1.

In order to explain the peculiarities of the observed internal wave field in the Strait of Messina, two different numerical simulations with different parameters of the undisturbed interface depth H₁ and of the reduced gravity g′ are presented in this section. The first numerical simulation refers to a scenario in which a strong seasonal thermocline is present in the Strait of Messina. The interface depth is 30 m and the reduced gravity is 14.7 × 10⁻³ m s⁻² (Table 2, thermocline). The second numerical simulation refers to a scenario in which no seasonal thermocline is present in the Strait of Messina; here the interface between the two water masses, the TSW and LIW, which exist throughout the year in the area, is present. The interface depth is 150 m and the reduced gravity is 4.9 × 10⁻³ m s⁻² (Table 2, no thermocline). The simulations have been carried out by running the model for several tidal cycles until an almost periodic solution developed.

b. Simulation with a seasonal thermocline

In the simulation presented in this section we assume that the stratification of the water body in the Strait of Messina is due to a seasonal thermocline located at a depth of 30 m. The reduced gravity associated with this seasonal thermocline is assumed to be 14.7 × 10⁻³ m s⁻². Our simulation yields the result that in the sill region the amplitudes of the total tidal transport, the tidal transport in the upper layer, and the tidal transport in the lower layer have the values 0.55, 0.15, and 0.4 Sv (1 Sv ≡ 1 × 10⁶ m³ s⁻¹) respectively. The mean transports in the upper and lower layer are 0.02 Sv into the Ionian Sea and 0.06 Sv into the Tyrrhenian Sea, respectively.

Figure 16 shows the interface depth for six different times separated by 1/6 of the tidal cycle with period T. In the bottom panels of Fig. 16 the depth profile used in the simulations is depicted. The arrows indicate the direction and the strength of the tidal flow at the sill. The light gray shaded areas mark the regions where the composite Froude number G, defined by

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is larger than 1. In these regions the flow is supercritical and it is connected to the subcritical flow on the downstream side by an internal hydraulic jump.

At maximum northward tidal flow, the interface at the sill is lifted upward while, north of the sill, a strong interfacial depression is present (Fig. 16a). At slack water the depression (internal bore) starts propagating southward and approaches the sill. Due to amplitude dispersion its steepness increases (Fig. 16c). As a consequence, frequency dispersion is strengthened and the coupled effect of these two dispersions causes the bore to disintegrate, thus giving birth to a train of internal solitary waves (Figs. 16d and 16e). At maximum southward tidal flow, the interface at the sill is again lifted upward, while south of the sill a strong interfacial depression is present (Fig. 16d). As discussed for the evolution of the southward propagating internal bore, this depression passes the sill and disintegrates into a train of northward propagating internal solitary waves (Fig. 16a). Note that, close to the sill, the shapes of the northward and southward propagating internal solitary waves are very similar (compare Fig. 16a with 16d), whereas farther away their shapes are quite different (compare Fig. 16b with 16e). This north–south asymmetry of the wave field is mainly caused by the north–south asymmetry of the channel width. Northward propagating internal waves are attenuated more strongly than southward propagating ones because in the northern section the channel is wider.

Figure 17 shows the simulated surface convergence field in a space–time diagram for two tidal cycles. As discussed in section 3b, the surface convergence associated with internal waves is proportional to the measured relative variation of the NRCS. Our simulation with a seasonal thermocline yields the result that internal solitary wave trains are generated that propagate northward as well as southward. The persistent surface convergence patterns in the sill region are due to the strong interfacial depressions caused by the internal tide dynamics.

c. Simulation with no seasonal thermocline

In the simulation presented in this section we assume that the stratification of the water body in the Strait of Messina is solely due to the presence of the TSW and LIW, which are separated at a depth of 150 m. The reduced gravity associated with this stratification is assumed to be 4.9 × 10⁻³ m s⁻². Our simulation yields the result that in the sill region the amplitudes of the total tidal transport, the tidal transport in the upper layer, and the tidal transport in the lower layer have the values 0.75, 0.4, and 0.25 Sv respectively. The mean transports in the upper and lower layer are 0.02 Sv into the Ionian Sea and 0.06 Sv into the Tyrrhenian Sea respectively.

Figure 18 shows the interface depth for six different times separated by 1/6 of the tidal cycle with period T. The arrows indicates the direction and the strength of the tidal flow at the sill. The light gray shaded areas mark the regions where the composite Froude number G is larger than 1. At maximum northward tidal flow, the interface at the sill is lifted upward, while north of the sill a strong interfacial depression is present (Fig. 18a). As long as the northward tidal flow persists, the depression north of the sill moves slightly northward (Fig. 18b). At slack water, the depression starts propagating southward. At maximum southward tidal flow (see Fig. 18d), the interfacial depression (internal bore) has already passed the sill. As discussed in section 6b, the internal bore disintegrates into a train of internal solitary waves. The number of solitary waves of the wave train increases with time (Figs. 18f, 18a, and 18b).

At maximum southward tidal flow (Fig. 18d), a strong interfacial depression develops south of the sill (onto which the southward propagating bore is now superimposed). At maximum northward tidal flow, this depression has passed the sill (Fig. 18a) and propagates northward. However, its shape is very smooth and cannot give birth to a train of internal solitary waves in the model domain (Figs. 18b and 18c).

In this case, the difference in the behavior of the southward and northward propagating internal waves cannot be ascribed only to the north–south asymmetry of the channel width. Here the north–south asymmetry of the depth profile plays an important role. As discussed for the amplitude–wavelength relationship of solitary waves (section 5b), for a very thick lower layer, as encountered south of the sill, nonlinear effects in the upper layer flow determine the steepening of the internal bore. For a thinner lower layer, as encountered north of the sill, nonlinear effects in the flow of such layer become important and tend to compensate nonlinear effects in the upper layer flow. As a consequence, the shape of the northward propagating internal bore is less steep.

In Fig. 19 the simulated surface convergence field is depicted in a space–time diagram for two tidal cycles. This space–time diagram clearly delineates the asymmetry of the convergence patterns north and south of the strait’s sill. Whereas southward propagating internal waves are associated with alternating surface convergences and divergences, northward propagating internal waves are associated with a single weak surface convergence band.

At the end of this section, we want to discuss the necessity of using a generation model equipped with movable lateral boundaries, as the one we implemented for the present study. Actually, Fig. 18 shows that, in this simulation, the interface never intersects the bottom topography. Thus, for this particular case, the implementation of movable boundaries in the generation model is uniquely needed to prescribe, as initial condition, an undisturbed interface between the two water layers. Moreover, the result that the periodic solution does not show intersection between the interface and the bottom topography was a priori unknown.