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The numerical simulation of the hydrodynamics of Barbamarco Lagoon, Italy
Applied Numerical Mathematics 40 (2002) 273–289 www.elsevier.com/locate/apnum
The numerical simulation of the hydrodynamics of Barbamarco Lagoon, Italy Isabel Ramirez a,∗ , Jorg Imberger b a Centro de Investigacion Cientifica y de Educacion Superior en Ensenada, Km 107 carretera Tijuana,
Ensenada B. Cfa., Mexico b Department of Environmental Engineering, Centre for Water Research, The University of Western Australia,
Nedlands 6907, Australia
Abstract A three-dimensional numerical model in finite differences was used to simulate the transport and mixing in Barbamarco Lagoon, Italy. The field data for the boundary conditions and for the conditions within the lagoon are described in an earlier paper by Ramirez and Imberger [Hydrodynamics of a shallow lagoon: Barbamarco Lagoon, Italy, submitted, 2000]. In general, the model reproduced the main features of the circulation and stratification regimes. 2002 IMACS. Published by Elsevier Science B.V. All rights reserved.
1. Introduction The hydrodynamics of Barbamarco Lagoon are mainly influenced by the tides in the Adriatic Sea and fresh water discharged from the Po River, which enters the lagoon through a control gate. The role wind plays is mainly in redistributing the fresh water (Ramirez and Imberger [11] from now on called RI). Fresh water from the Po River influences the salinity of the lagoon in two ways. First, fresh water flows directly into the lagoon through a gate situated in the south-west boundary. Second, fresh water discharged into the Adriatic Sea is swept northwards along the coast and enters the lagoon through the entrance during the flood tide. The discharge of fresh water from the Po River into the ocean and into the lagoon, is determined by the river–ocean water level difference and by the river–lagoon water level differences. As described in RI, during times when the gate was opened, fresh water from the river discharged into the lagoon as a surface buoyant plume. Such plumes are common in fresh water river mouths and estuarine tidal channels [5,7–9]. Depending on the salinity of the Adriatic Sea, water entered the lagoon as a saline underflow during the flood tide. The brackish surface estuarine water was observed to exit the lagoon during ebb tide. * Corresponding author.
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On occasions the Po River water entering the Adriatic Sea would be swept northwards in front of the lagoon entrance and on such occasions Po River water was observed to reverse at the entrance (RI). The combined effect of the fresh water entering through the gate and the lagoon entrance led to a very variable stratification of the water column even in in the shallowest parts of the lagoon. The field data set thus offered an opportunity to severely test numerical models. The barotropic tidal circulation in Barbamarco Lagoon has been simulated before by Matticchio [10] using a depth averaged finite element model. His simulations describe the lagoonal circulation in response to the incoming tidal flow through Bocca Sud, the southern inlet of the lagoon. Three-dimensional models are now routinely used to simulate the salinity distribution and the circulation in coastal lagoons and estuaries. For instance, Umgiesser and Bergamasco [12] used a hydrostatic model to simulate various flow regimes in the coastal regime adjacent to the Po River outflow. Their model included forcing due to the Po River outflow, the wind and surface heat transport. The conclusion reached was that the bathymetry plays the most important role in determining the coastal circulation. The results obtained by D’Alpaos et al. [3] with a finite element model, applied in Barbamarco Lagoon, describe the fresh water displacements in conjunction with strong density stratification. The results also suggest that further fine tuning of model coefficients and local mesh refinements were still required in order to obtain a good comparison between model results and observations. The same finite element multi-layer model was applied by D’Alpaos et al. [3] to the Po River delta. The application of this model was focused on the complex irregular bathymetry and a range of boundary conditions. Their results showed the flexibility of a finite element model for use in practical situations where complex bottom bathymetry is present. In this more weakly stratified region, they found generally good agreement between the observed salinity distribution and the model predictions. A three-dimensional model for shallow water flow using a semi-implicit finite difference method for water flow has been developed by Casulli and Cheng [2]. This model (Tidal Residual Intertidal Mudflat, TRIM) can be applied to simulate the flooding and drying of tidal mud-flats. The objective of the present study was to apply TRIM with a new vertical mixing closure scheme to the observed data set in Barbamarco Lagoon, and assess the ability of this model, which is typical of a large class of models, to reproduce the complex variations in the salinity distribution. This model was applied here with the smallest grid size ever used in other simulations. The dynamics of Barbamarco Lagoon are typical of estuarine circulation. Relatively heavy saline water enters the lagoon through the Bocca Sud and slides over the bottom, under lighter fresh water coming from the Po River through the gate. The volume of fresh water in the lagoon is related to the river–lagoon head. During the field experiment, when the wind was in phase with the flood tide, it restrained the buoyant fresh water plume, which moved along the coastline instead of entering and spreading radially out into the lagoon. An extra volume of fresh water entered the lagoon, entering via Bocca Sud inlet as a result of the discharge from the Po River over the Adriatic Sea. 1.1. The site Barbamarco Lagoon is an inland water body on the east coast of the North Adriatic Sea, approximately 7 kilometers long and 2 kilometers wide at its widest section (Fig. 1). The lagoon is separated from the sea by a sand bar 3 kilometers long. Semidiurnal tides, with a range of one meter, propagate from the Adriatic Sea into the lagoon via two inlets; one in the north of the lagoon called Bocca Nord and one other in the south called Bocca Sud. The bathymetry of the lagoon is characterized by channels with a
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Fig. 1. Barbamarco Lagoon bathymetry. The channels are approximately four meters deep during the high tide.
maximum depth of 5 meters running across and along the lagoon, and by some shoals in the central part of the lagoon separating the channels. Barbamarco Lagoon can be dynamically separated into two parts. The southern part of the lagoon includes three main channels: channel one, connecting Bocca Sud to the Po River gate (Fig. 1); channel two, running perpendicular to the long axis of the lagoon starting at Bocca Sud; and channel three running parallel to the sand bar. The northern part of the lagoon includes the area from Bocca Nord to the main entrance, with one channel running parallel to the sand bar. A manually controlled gate of 8.0 m width and 3.3 m depth, allows fresh water from a Po River branch to flow into the southern part of the lagoon. The gate is normally used to regulate the freshwater flow and salinity in the lagoon.
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2. The model Earlier versions of models similar to TRIM [1] have been applied successfully in estuaries like San Francisco Bay and Venice [2]. Here we used the smallest grid size possible to obtain the optimum vertical resolution. The main features of the model include: the boundary conditions at an extreme edge may be specified as either no-slip, full-slip or open, and all other boundaries (i.e., the bottom) are no-slip. Inflows and outflows are modeled by raising or lowering the surface elevation at the relevant location. A full description of the model can be found in [1]. This model was selected because it represents a class of similar hydrostatic pressure models which have a fast execution time and thus tends generally to the conclusions reached in this study. The governing three-dimensional equations describing free surface flows of variable density in estuaries were derived from the Navier–Stokes equations after turbulent averaging and under the simplifying assumption that the pressure is hydrostatic.
∂u ∂u ∂u ∂η ∂ ∂u ∂ 2u ∂ 2u ∂u +u +v +w = −g +µ + 2 + ν + f v, 2 ∂t ∂x ∂y ∂z ∂x ∂x ∂y ∂z ∂z 2 ∂v ∂v ∂v ∂η ∂ ∂v ∂ v ∂ 2v ∂v +u +v +w = −g +µ + + ν − f u, ∂t ∂x ∂y ∂z ∂y ∂x 2 ∂y 2 ∂z ∂z where u, v and w are the velocity components in the x, y and z directions, respectively; t is the time; η is the water surface elevation; g is gravity acceleration; f is the Coriolis parameter and µ and ν are the coefficients of horizontal and vertical eddy viscosity. The salinity conservation equation is thus defined by
∂S ∂S ∂S ∂ 2S ∂ 2S ∂S ∂S +u +v +w = Kh + 2 + Kv , (1) 2 ∂t ∂x ∂y ∂z ∂x ∂y ∂z where Kh and Kv are the coefficients of horizontal and vertical eddy diffusivity, S(x, y, z, t) is the salinity. The hydrostatic equation is defined by ∂p = −ρg (2) ∂z with ρ = ρo + ρ (S),
(3)
where ρ is the density and ρo is the reference water density. The standard form of the boundary condition at the water surface (z = η) is defined by ∂η ∂η ∂η = wz=η − u −v . (4) ∂t ∂x ∂y The same formulation is used at the bottom, for z = −h, to define the kinematic bottom boundary conditions by ∂h ∂h +v = 0. (5) w+u ∂x ∂y The boundary conditions at the surface are specified by the wind stress by √ √ ρa cD U U 2 + V 2 ρa cD V U 2 + V 2 , τy = , (6) τx = ρo ρo
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where cD is the drag coefficient, ρa is the air density and U and V are the wind velocity components in the x and y direction. Integrating the continuity equation over depth and using a kinematic condition at the free surface leads to the free surface elevation equation, defined by ∂ ∂η + ∂t ∂x
η −h
∂ u dz + ∂y
η
v dz = 0.
(7)
−h
In this study, the horizontal grid spacing was constant, while the vertical spacing was allowed to vary. The vertical diffusion coefficient was constant and the density was calculated using the UNESCO equation of state. Compressibility effects in the equation of state for density are neglected since the area is shallow. These equations are solved over vertical layers to produce results for the three-dimensional velocity field, the surface elevation and salinity. The gradient of surface elevation in the momentum equations and the velocity in the free surface equation are discretized implicitly. The convection, Coriolis and horizontal viscosity terms in the momentum equations are discretized explicitly. The presence of islands and other permanently dry areas, as well as tidal flats, is accounted for the numerical scheme. The boundary shoreline, which varies with time, is defined as the no mass flux conditions. A very simple closure scheme was implemented where the Richardson number (Ri ) was computed and if Ri < 14 , the two adjacent layers were mixed. The bathymetry used has a grid with dx = 10 m and dy = 10 m. The vertical resolution was 22 layers of 0.25 m depth. A time step of 10 sec was selected as being the minimum for the required stability. The initial conditions of the simulation were: a lagoon with an homogeneous salinity of 24.3 psu, which represents the volume-averaged salinity in the lagoon before the gate was opened; and 32 psu at the open boundaries of Bocca Nord and Bocca Sud.
3. The forcing in the lagoon Tides, meteorological surface forcing, buoyant inflows induced mainly by river discharges, and estuary geometry all play an important role in the hydrodynamics of coastal lagoons [6]. Barbamarco Lagoon, at the Adriatic Sea, presents an almost laboratory-like environment, allowing study of the interaction and contribution of each of these variables to the general mean circulation. The results presented in RI from a field study, are used here to establish the initial conditions of the modeling and to provide the necessary information at the open boundaries. 3.1. Salinity at Bocca Sud In RI it was reported that the salinity at the Adriatic Sea in front of Barbamarco Lagoon played an important role in the flushing time of the lagoon. The river–ocean head determined the discharge of fresh water over the Adriatic Sea, causing low salinity water to be moved along the coast by the coastal currents, and enter the lagoon through Bocca Sud during the flood tide. To obtain the proper balance of fresh water in the lagoon, different salinities at Bocca Sud were tested. They were: first, a constant salinity of 35 psu; second, the time series of the salinity measured at Bocca Sud and linearly interpolated (Fig. 2(a)); and third, a time series of salinity adjusted to the measurements with tidal periodicity.
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Fig. 2. (a) The salinity at Bocca Sud inlet. (b) The water elevations of the Adriatic Sea at the inlets. (c) The discharge of the river calculated. (d) The wind measured at the power plant.
3.2. The tides Tides in the Adriatic Sea were semidiurnal, with a maximum amplitude of 0.5 m and produced a maximum tidal current along the channels in the lagoon of 0.4 ms−1 . During the period of the study, two high tides occurred during the day, with one almost twice as high as the other (Fig. 2(b)). Beside the important role played by the tide in the general circulation of the lagoon, it also played an important role on the averaged salinity in the lagoon. Because of the salinity at the Adriatic Sea inlets, brackish or oceanic water will enter the lagoon. Time series of the predicted tides for the Adriatic Sea were applied at the open boundaries Bocca Nord and Bocca Sud (Fig. 2(b)). 3.3. The discharge The inflow of fresh water into the lagoon through the Po River gate was measured during the field study and it was found to be correlated with the differential river–lagoon head by: Qg = 1.11 + 194.16(hr − hl ),
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where (hr − hl ) is in meters and Qg is in m3 s−1 . The discharge of the Po River was calculated with the time series of water level for the lagoon and the river is shown in Fig. 2(c). The discharge was introduced into the simulation by changing the flow at the open boundary at the Po River gate. 3.4. The wind The speed and direction of the wind measured during the field study presented an average wind speed of 5 ms−1 , with one strong event of 15 ms−1 that occurred on the 8th of July (Fig. 2(d)). The direction of the wind was mainly south and south-west. The wind acted to distribute the surface buoyant water into the northern boundary of the lagoon.
4. Results 4.1. The distribution of the salinity The variation in the salinity distribution simulated with TRIM agreed well with the main features measured with the fine scale profiler throughout the lagoon during the field experiment conducted in July 1994. The results from the model are presented in a similar format as the results from the field study presented in RI: a plan view of the salinity averaged over the top half meter of the lagoon and the salinity distribution along the main channel. To compare the field data with the model results, it is necessary to bear in mind that, due to logistical constraints, the plots resulting from the experiment are the result of 3 hours sampling, and the results from the model are an output taken in the middle of that sampling time. The simulation started on the 5th of July, two days before the gate connecting the lagoon with the Po River was opened. During this time, tides and wind were the only forces causing the lagoon water to mix with the Adriatic Sea water. The initial conditions of salinity in the lagoon were 24.3 psu, vertically and horizontally homogeneous. Meanwhile the boundary conditions in the Adriatic Sea at the inlets Boca Nord and Boca Sud were a uniform salinity of 32 psu. During the time the gate was closed, the salinity simulated with TRIM changed slowly, and heavier Adriatic Sea water intruded over the bottom of the lagoon creating a partially stratified lagoon (Fig. 3(a)). The salinity observed during the measurements presented a more stratified lagoon and the inflow of fresh water through Bocca Sud was more noticeable (Fig. 3(d)). In the model results the isohalines were more vertically oriented indicating that more mixing was occurring in the simulation compared with that on the field. Also, from the model results it was observed that, between the flood and the ebb tide the lagoon became well mixed with an average salinity of 26 psu. After two days, on the 7th of July, the Po River gate was opened to produce the same conditions at the open boundaries as were produced in the field. Within one hour a surface buoyant plume began to intrude into the lagoon. The combined effects of the flood tide coming into the lagoon, and the south-west wind at the time, restrained the intrusion of the fresh water plume, moving the brackish water northward along the western perimeter of the lagoon (Fig. 4(a)). In the field experiment the intrusion of the fresh water plume was moved about one kilometer northwest along the coastline, forced by the wind (Fig. 4(c)). However, in the simulation results the fresh water intrusion was more dispersed and the buoyant surface plume was considerably thinner than the measurements.
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Fig. 3. Salinity contours simulated for the 5th of July at 11.8 am. (a) Horizontal plan view of the top 0.25 cm layer of the lagoon. The discharge, in m3 /s, from one hour before the simulation is presented in the bottom left corner. (b) A vertical section of salinity along the channel one. The tide is presented with the amplitude in meters. (c) Salinity contours measured on the lagoon at 12:00 hrs on 5th of July. The contours represent the salinity averaged over the top 50 cm. The wind velocity averaged over the four hours prior to the sampling time is presented by an arrow at the right top corner. (d) The salinity in transect along channel one. The tide for the day of the measurements is presented with a bar indicating the length of the sampling period.
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Fig. 4. Salinity contours simulated for the 7th of July at 8 hrs. (a) and (b) are presented as in Figs. 3(a) and (b). The salinity contours measured in the lagoon at 10:00 hrs on 7th of July. (c) and (d) have the same description as in Figs. 3(c) and (d).
The ebb tide and the low intensity of the wind in the next 6 hours created favorable conditions for the advancement of the buoyant plume during the afternoon of the same day. This feature was reproduced in the simulation (Fig. 5(a)) and compares well with the field observations as shown in Fig. 5(c).
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Fig. 5. Salinity simulated for the 7th of July at 15 hrs. (a) and (b) are presented as in Figs. 3(a) and (b). The salinity contours of the lagoon at 16:00 hrs on the 7th of July. (c) and (d) have the same description as in Figs. 3(c) and (d).
On the other hand, in the simulation, the buoyant overflow was less influenced by the wind, being of a more radially spreading nature and moving the fresh water considerably further away on the surface layer. The fresh water volume discharged into the lagoon on the 8th of July was forced by the wind to border the lagoon coastline. Meanwhile the ebb tide, in the opposite direction to the wind, caused the mixing to increase, deepening the top layer to 2 meters (Fig. 6(a)). This result from the simulation agreed with
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Fig. 6. Salinity simulated for the 8th of July at 15 hrs. (a) and (b) are presented as in Figs. 3(a) and (b). The salinity contours of the lagoon at 15:00 hrs on the 8th of July. (c) and (d) are presented as in Figs. 3(c) and (d).
that observed in the field data, showing a good resolution of horizontal distribution of salinity. Along the channel the salinity was lower indicating more mixing occurring in the simulation, but the shape of the distribution was similar. The two layer stratification with 15 psu at the top and 25 psu salinity at the bottom was also well reproduced by the model (Figs. 6(b) and (d)). The maximum discharge observed on the 9th of July, combined with the flood tide and the wind, restricted the advancement of the buoyant plume overflow creating a sharp front between the Po River
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Fig. 7. Salinity contours simulated for the 9th of July at 10 hrs. (a) and (b) are presented as in Figs. 3(a) and (b). The salinity measured of the lagoon at 10:00 hrs on the 9th of July. (c) and (d) have the same description as in Figs. 3(c) and (d).
and the lagoon water. The low salinity water in the Adriatic Sea influenced the stratification presenting a more distributed buoyant flow. This pattern agreed well with the field measurements both with respect to the shape of the intrusion and the plume size (Figs. 7(a) and (c)). However, the averaged salinity produced by the simulation (Fig. 7(a)) in the south of the lagoon was higher than that in the measurements (Fig. 7(c)).
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Fig. 8. Salinity contours of the lagoon at 10:00 hrs on the 10th of July. (a) and (b) are presented as in Figs. 3(a) and (b). The salinity measured for the 10th of July at 9 hrs. The same description as in Figs. 3(c) and (d).
On the 10th of July, during the morning, a maximum discharge was observed with similar characteristics as the day before (Fig. 8(a)). The prevailing south-east wind pushed the fresh water northwest, bordering the coastline. The reduction of volume created by the wind allowed Adriatic Sea water to replace that volume, resulting in an oceanic environment with 30 psu at the bottom half of the lagoon. The fresh water was observed to be advected further away in the surface compared with the field data, and in general the fresh water was mixed faster in the simulation (Fig. 8(c)).
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Fig. 9. Salinity contours simulated for the 11th of July at 12 hrs. (a) and (b) are presented as in Figs. 3(a) and (b). Wind, tide and discharge are presented as in Fig. 3(a).
Two hours before the gate was closed on the 11th of July, the lagoon presented a marine water environment caused by the level difference between the lagoon and the ocean, and the flooding tide pushing saline water through the south inlet. A surge of fresh water created by a large discharge before the gate was closed, and the oceanic water created an horizontally stratified lagoon with salinities ranging from 1 psu at the gate to 35 psu at the Bocca Sud inlet (Fig. 9). The results from the simulation show an intermediately mixed lagoon with salinities from 33 psu at the bottom and 18 psu at the top, instead of
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the mainly marine environment (35 psu of salinity) observed in the field. Thus, the mixing coefficients in the model were providing more mixing in the lagoon than was observed. After the gate was closed, remnants of fresh water were mixing in the surface of the lagoon during the next two days. In summary, TRIM reproduced the main features of the circulation in Barbamarco Lagoon, as well as the main mixing characteristics of interactions occurring in the lagoon between two water bodies. These were: the plume shape extent of the buoyant plume created by Po River discharge; the effect of the wind on the distribution of the fresh water; and the estuarine circulation at the Bocca Sud entrance. 4.2. Fresh water volume balance The total volume of fresh water in the estuary is calculated from the freshness [4], V=
η
f dz dA,
(8)
A −h
where η is the surface elevation, and A is the area of the lagoon. The freshness is defined as f=
So − S , So
(9)
where S is the salinity of the water and So is a salinity reference. The averaged salinity found in the Adriatic Sea at the mouth of the lagoon during the study was used as a reference, So = 32 psu, to calculate the freshness of each cell of the lagoon. For the field study, the salinity was calculated using the conductivity and temperature measured with a fine scale profiler (RI). The salinity profiles were depth averaged every 25 centimeters and these values extrapolated to the cells around the sampling sites to obtain the freshness of each cell in the lagoon. Thus, the total volume of fresh water in the lagoon was calculated using Eq. (8), integrating the fresh water in each cell using a grid resolution of dx = 10 m, dy = 10 m, and h = z the depth of the bathymetry. The field data show that fresh water had accumulated from the 5th to the 9th of July, and started flushing into the ocean from that time until the end of the experiment. The cumulative volume of fresh water discharged into the lagoon through the gate between the 5th and the 9th of July clearly matched the fresh water volume calculated by Eq. (9) (Fig. 10(a)). It should be borne in mind that the volume of fresh water in the lagoon is an approximation over 3 hours sampling time. After the 9th of July, the volume of fresh water in the lagoon decreased until at the end of the experiment the average salinity in the lagoon was lower than that found at the beginning of the experiment. The volume of fresh water in the lagoon during the simulation was calculated in the same way as it was calculated with the field data. The freshness of each cell was calculated based on the salinity obtained through the mixing in TRIM, then all the grid cells in the lagoon were integrated. The calculations of the fresh water volume in the simulation show a fresh water variation with a periodicity similar to the tide period (Fig. 10(b)). The pattern of the freshness in the lagoon calculated from the field measurements was hidden in the signal in tidal periodicity obtained by the simulations. The alias so introduced became evident when the signal was subsampled at the times corresponding to the experiment sampling time. The loss of fresh water following the 9th of July was due to the flushing of the lagoon through Bocca Sud.
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Fig. 10. (a) Fresh water volume calculated with the salinity profiles in broken line, and accumulated discharge from the Po River in continuous line. (b) The fresh water volume from the simulation (continuous line), and subsampled at the experiment (+). The volume calculated with the salinity profiles in broken line.
5. Conclusions The distribution of the salinity simulated using TRIM showed good agreement with the field observations presented in RI. The results provided the details of the hydrodynamics in the shallow lagoon. Tides were the main variable driving the mean circulation and the flushing of freshwater out of the lagoon. Meanwhile the effect of the wind was to redistribute the fresh water in the surface layer throughout the lagoon. The effect of the wind was more evident when its directions was acting in the same direction as the tidal flow, favoring the stratification. When the direction of the wind was in the
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opposite direction to the tidal flow it favored mixing and deepening the surface plume. The estuarine circulation produced by the differences in density between the lagoon and the ocean was well reproduced by TRIM. There were times when the stratification produced in the simulation was weaker than that in the field observations indicating the necessity of some improvement in the closure scheme. The freshness of the lagoon was well reproduced and provided a better understanding of the flushing time of a shallow coastal lagoon. The results from the fresh water balance in the lagoon revealed a higher frequency variability of the processes occurring in the lagoon hidden by the lower frequency sampling of the field measurements.
Acknowledgements The authors wish to thank Vincenzo Casulli for allowing the use of TRIM, and Abi Javam, for help in learning about the model. Isabel Ramirez was supported by The National Council of Science and Technology of Mexico and The Centro de Investigacion Cientifica y Educacion Superior en Ensenada. This paper forms the Centre for Water Research reference ED-1280-PD.
References [1] V. Casulli, E. Cattani, Stability, accuracy and efficiency of a semi-implicit method for three-dimensional shallow water flow, Comput. Math. Appl. 27 (1994) 99–112. [2] V. Casulli, R.T. Cheng, Semi-implicit finite difference methods for three-dimensional shallow water flow, Internat. J. Numer. Methods Fluids 15 (1992) 629–648. [3] L. D’Alpaos, A. Defina, B. Matticchio, Multilayer model for shallow water flows and density currents applied to a lagoon in the Po River delta, in: International Conference on Barrages, Cardiff, UK, 1996. [4] H.B. Fischer, E.J. List, R.C.Y. Koh, J. Imberger, N. Brooks, Mixing in Inland and Coastal Waters, Academic Press, New York, 1979. [5] R.W. Garvine, J. Monk, Frontal structure of a river plume, J. Geophys. Res. 79 (1974) 2251–2259. [6] L. Huzzey, J. Brubaker, The formation of longitudinal fronts in a coastal plain estuary, J. Geophys. Res. 93 (1988) 1329–1334. [7] J. Imberger, Tidal jet frontogenesis, Mech. Engrg. Trans. 12 (1983) 51–55. [8] D.A. Luketina, J. Imberger, Characteristics of a surface buoyant jet, J. Geophys. Res. 92 (1987) 5435–5447. [9] D. Luketina, J. Imberger, Turbulence and entrainment in a buoyant surface plume, J. Geophys. Res. 94 (12) (1989) 619–636. [10] B. Matticchio, Semi-implicit finite difference methods for three-dimensional shallow water flow, Master’s Thesis, University of Padova, Italy, 1992. [11] I. Ramirez, J. Imberger, Hydrodynamics of a shallow lagoon: Barbamarco Lagoon, Italy, submitted, 2000. [12] G. Umgiesser, A. Bergamasco, The spreading of the Po plume and the Italian coastal current, in: ISDGMCNR, Venice, Italy, 1994.
The numerical simulation of the hydrodynamics of Barbamarco Lagoon, Italy Isabel Ramirez a,∗ , Jorg Imberger b a Centro de Investigacion Cientifica y de Educacion Superior en Ensenada, Km 107 carretera Tijuana,
Ensenada B. Cfa., Mexico b Department of Environmental Engineering, Centre for Water Research, The University of Western Australia,
Nedlands 6907, Australia
Abstract A three-dimensional numerical model in finite differences was used to simulate the transport and mixing in Barbamarco Lagoon, Italy. The field data for the boundary conditions and for the conditions within the lagoon are described in an earlier paper by Ramirez and Imberger [Hydrodynamics of a shallow lagoon: Barbamarco Lagoon, Italy, submitted, 2000]. In general, the model reproduced the main features of the circulation and stratification regimes. 2002 IMACS. Published by Elsevier Science B.V. All rights reserved.
1. Introduction The hydrodynamics of Barbamarco Lagoon are mainly influenced by the tides in the Adriatic Sea and fresh water discharged from the Po River, which enters the lagoon through a control gate. The role wind plays is mainly in redistributing the fresh water (Ramirez and Imberger [11] from now on called RI). Fresh water from the Po River influences the salinity of the lagoon in two ways. First, fresh water flows directly into the lagoon through a gate situated in the south-west boundary. Second, fresh water discharged into the Adriatic Sea is swept northwards along the coast and enters the lagoon through the entrance during the flood tide. The discharge of fresh water from the Po River into the ocean and into the lagoon, is determined by the river–ocean water level difference and by the river–lagoon water level differences. As described in RI, during times when the gate was opened, fresh water from the river discharged into the lagoon as a surface buoyant plume. Such plumes are common in fresh water river mouths and estuarine tidal channels [5,7–9]. Depending on the salinity of the Adriatic Sea, water entered the lagoon as a saline underflow during the flood tide. The brackish surface estuarine water was observed to exit the lagoon during ebb tide. * Corresponding author.
E-mail address: [email protected] (I. Ramirez). 0168-9274/02/$22.00 2002 IMACS. Published by Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 2 7 4 ( 0 1 ) 0 0 0 7 9 - 4
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On occasions the Po River water entering the Adriatic Sea would be swept northwards in front of the lagoon entrance and on such occasions Po River water was observed to reverse at the entrance (RI). The combined effect of the fresh water entering through the gate and the lagoon entrance led to a very variable stratification of the water column even in in the shallowest parts of the lagoon. The field data set thus offered an opportunity to severely test numerical models. The barotropic tidal circulation in Barbamarco Lagoon has been simulated before by Matticchio [10] using a depth averaged finite element model. His simulations describe the lagoonal circulation in response to the incoming tidal flow through Bocca Sud, the southern inlet of the lagoon. Three-dimensional models are now routinely used to simulate the salinity distribution and the circulation in coastal lagoons and estuaries. For instance, Umgiesser and Bergamasco [12] used a hydrostatic model to simulate various flow regimes in the coastal regime adjacent to the Po River outflow. Their model included forcing due to the Po River outflow, the wind and surface heat transport. The conclusion reached was that the bathymetry plays the most important role in determining the coastal circulation. The results obtained by D’Alpaos et al. [3] with a finite element model, applied in Barbamarco Lagoon, describe the fresh water displacements in conjunction with strong density stratification. The results also suggest that further fine tuning of model coefficients and local mesh refinements were still required in order to obtain a good comparison between model results and observations. The same finite element multi-layer model was applied by D’Alpaos et al. [3] to the Po River delta. The application of this model was focused on the complex irregular bathymetry and a range of boundary conditions. Their results showed the flexibility of a finite element model for use in practical situations where complex bottom bathymetry is present. In this more weakly stratified region, they found generally good agreement between the observed salinity distribution and the model predictions. A three-dimensional model for shallow water flow using a semi-implicit finite difference method for water flow has been developed by Casulli and Cheng [2]. This model (Tidal Residual Intertidal Mudflat, TRIM) can be applied to simulate the flooding and drying of tidal mud-flats. The objective of the present study was to apply TRIM with a new vertical mixing closure scheme to the observed data set in Barbamarco Lagoon, and assess the ability of this model, which is typical of a large class of models, to reproduce the complex variations in the salinity distribution. This model was applied here with the smallest grid size ever used in other simulations. The dynamics of Barbamarco Lagoon are typical of estuarine circulation. Relatively heavy saline water enters the lagoon through the Bocca Sud and slides over the bottom, under lighter fresh water coming from the Po River through the gate. The volume of fresh water in the lagoon is related to the river–lagoon head. During the field experiment, when the wind was in phase with the flood tide, it restrained the buoyant fresh water plume, which moved along the coastline instead of entering and spreading radially out into the lagoon. An extra volume of fresh water entered the lagoon, entering via Bocca Sud inlet as a result of the discharge from the Po River over the Adriatic Sea. 1.1. The site Barbamarco Lagoon is an inland water body on the east coast of the North Adriatic Sea, approximately 7 kilometers long and 2 kilometers wide at its widest section (Fig. 1). The lagoon is separated from the sea by a sand bar 3 kilometers long. Semidiurnal tides, with a range of one meter, propagate from the Adriatic Sea into the lagoon via two inlets; one in the north of the lagoon called Bocca Nord and one other in the south called Bocca Sud. The bathymetry of the lagoon is characterized by channels with a
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Fig. 1. Barbamarco Lagoon bathymetry. The channels are approximately four meters deep during the high tide.
maximum depth of 5 meters running across and along the lagoon, and by some shoals in the central part of the lagoon separating the channels. Barbamarco Lagoon can be dynamically separated into two parts. The southern part of the lagoon includes three main channels: channel one, connecting Bocca Sud to the Po River gate (Fig. 1); channel two, running perpendicular to the long axis of the lagoon starting at Bocca Sud; and channel three running parallel to the sand bar. The northern part of the lagoon includes the area from Bocca Nord to the main entrance, with one channel running parallel to the sand bar. A manually controlled gate of 8.0 m width and 3.3 m depth, allows fresh water from a Po River branch to flow into the southern part of the lagoon. The gate is normally used to regulate the freshwater flow and salinity in the lagoon.
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2. The model Earlier versions of models similar to TRIM [1] have been applied successfully in estuaries like San Francisco Bay and Venice [2]. Here we used the smallest grid size possible to obtain the optimum vertical resolution. The main features of the model include: the boundary conditions at an extreme edge may be specified as either no-slip, full-slip or open, and all other boundaries (i.e., the bottom) are no-slip. Inflows and outflows are modeled by raising or lowering the surface elevation at the relevant location. A full description of the model can be found in [1]. This model was selected because it represents a class of similar hydrostatic pressure models which have a fast execution time and thus tends generally to the conclusions reached in this study. The governing three-dimensional equations describing free surface flows of variable density in estuaries were derived from the Navier–Stokes equations after turbulent averaging and under the simplifying assumption that the pressure is hydrostatic.
∂u ∂u ∂u ∂η ∂ ∂u ∂ 2u ∂ 2u ∂u +u +v +w = −g +µ + 2 + ν + f v, 2 ∂t ∂x ∂y ∂z ∂x ∂x ∂y ∂z ∂z 2 ∂v ∂v ∂v ∂η ∂ ∂v ∂ v ∂ 2v ∂v +u +v +w = −g +µ + + ν − f u, ∂t ∂x ∂y ∂z ∂y ∂x 2 ∂y 2 ∂z ∂z where u, v and w are the velocity components in the x, y and z directions, respectively; t is the time; η is the water surface elevation; g is gravity acceleration; f is the Coriolis parameter and µ and ν are the coefficients of horizontal and vertical eddy viscosity. The salinity conservation equation is thus defined by
∂S ∂S ∂S ∂ 2S ∂ 2S ∂S ∂S +u +v +w = Kh + 2 + Kv , (1) 2 ∂t ∂x ∂y ∂z ∂x ∂y ∂z where Kh and Kv are the coefficients of horizontal and vertical eddy diffusivity, S(x, y, z, t) is the salinity. The hydrostatic equation is defined by ∂p = −ρg (2) ∂z with ρ = ρo + ρ (S),
(3)
where ρ is the density and ρo is the reference water density. The standard form of the boundary condition at the water surface (z = η) is defined by ∂η ∂η ∂η = wz=η − u −v . (4) ∂t ∂x ∂y The same formulation is used at the bottom, for z = −h, to define the kinematic bottom boundary conditions by ∂h ∂h +v = 0. (5) w+u ∂x ∂y The boundary conditions at the surface are specified by the wind stress by √ √ ρa cD U U 2 + V 2 ρa cD V U 2 + V 2 , τy = , (6) τx = ρo ρo
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where cD is the drag coefficient, ρa is the air density and U and V are the wind velocity components in the x and y direction. Integrating the continuity equation over depth and using a kinematic condition at the free surface leads to the free surface elevation equation, defined by ∂ ∂η + ∂t ∂x
η −h
∂ u dz + ∂y
η
v dz = 0.
(7)
−h
In this study, the horizontal grid spacing was constant, while the vertical spacing was allowed to vary. The vertical diffusion coefficient was constant and the density was calculated using the UNESCO equation of state. Compressibility effects in the equation of state for density are neglected since the area is shallow. These equations are solved over vertical layers to produce results for the three-dimensional velocity field, the surface elevation and salinity. The gradient of surface elevation in the momentum equations and the velocity in the free surface equation are discretized implicitly. The convection, Coriolis and horizontal viscosity terms in the momentum equations are discretized explicitly. The presence of islands and other permanently dry areas, as well as tidal flats, is accounted for the numerical scheme. The boundary shoreline, which varies with time, is defined as the no mass flux conditions. A very simple closure scheme was implemented where the Richardson number (Ri ) was computed and if Ri < 14 , the two adjacent layers were mixed. The bathymetry used has a grid with dx = 10 m and dy = 10 m. The vertical resolution was 22 layers of 0.25 m depth. A time step of 10 sec was selected as being the minimum for the required stability. The initial conditions of the simulation were: a lagoon with an homogeneous salinity of 24.3 psu, which represents the volume-averaged salinity in the lagoon before the gate was opened; and 32 psu at the open boundaries of Bocca Nord and Bocca Sud.
3. The forcing in the lagoon Tides, meteorological surface forcing, buoyant inflows induced mainly by river discharges, and estuary geometry all play an important role in the hydrodynamics of coastal lagoons [6]. Barbamarco Lagoon, at the Adriatic Sea, presents an almost laboratory-like environment, allowing study of the interaction and contribution of each of these variables to the general mean circulation. The results presented in RI from a field study, are used here to establish the initial conditions of the modeling and to provide the necessary information at the open boundaries. 3.1. Salinity at Bocca Sud In RI it was reported that the salinity at the Adriatic Sea in front of Barbamarco Lagoon played an important role in the flushing time of the lagoon. The river–ocean head determined the discharge of fresh water over the Adriatic Sea, causing low salinity water to be moved along the coast by the coastal currents, and enter the lagoon through Bocca Sud during the flood tide. To obtain the proper balance of fresh water in the lagoon, different salinities at Bocca Sud were tested. They were: first, a constant salinity of 35 psu; second, the time series of the salinity measured at Bocca Sud and linearly interpolated (Fig. 2(a)); and third, a time series of salinity adjusted to the measurements with tidal periodicity.
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Fig. 2. (a) The salinity at Bocca Sud inlet. (b) The water elevations of the Adriatic Sea at the inlets. (c) The discharge of the river calculated. (d) The wind measured at the power plant.
3.2. The tides Tides in the Adriatic Sea were semidiurnal, with a maximum amplitude of 0.5 m and produced a maximum tidal current along the channels in the lagoon of 0.4 ms−1 . During the period of the study, two high tides occurred during the day, with one almost twice as high as the other (Fig. 2(b)). Beside the important role played by the tide in the general circulation of the lagoon, it also played an important role on the averaged salinity in the lagoon. Because of the salinity at the Adriatic Sea inlets, brackish or oceanic water will enter the lagoon. Time series of the predicted tides for the Adriatic Sea were applied at the open boundaries Bocca Nord and Bocca Sud (Fig. 2(b)). 3.3. The discharge The inflow of fresh water into the lagoon through the Po River gate was measured during the field study and it was found to be correlated with the differential river–lagoon head by: Qg = 1.11 + 194.16(hr − hl ),
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where (hr − hl ) is in meters and Qg is in m3 s−1 . The discharge of the Po River was calculated with the time series of water level for the lagoon and the river is shown in Fig. 2(c). The discharge was introduced into the simulation by changing the flow at the open boundary at the Po River gate. 3.4. The wind The speed and direction of the wind measured during the field study presented an average wind speed of 5 ms−1 , with one strong event of 15 ms−1 that occurred on the 8th of July (Fig. 2(d)). The direction of the wind was mainly south and south-west. The wind acted to distribute the surface buoyant water into the northern boundary of the lagoon.
4. Results 4.1. The distribution of the salinity The variation in the salinity distribution simulated with TRIM agreed well with the main features measured with the fine scale profiler throughout the lagoon during the field experiment conducted in July 1994. The results from the model are presented in a similar format as the results from the field study presented in RI: a plan view of the salinity averaged over the top half meter of the lagoon and the salinity distribution along the main channel. To compare the field data with the model results, it is necessary to bear in mind that, due to logistical constraints, the plots resulting from the experiment are the result of 3 hours sampling, and the results from the model are an output taken in the middle of that sampling time. The simulation started on the 5th of July, two days before the gate connecting the lagoon with the Po River was opened. During this time, tides and wind were the only forces causing the lagoon water to mix with the Adriatic Sea water. The initial conditions of salinity in the lagoon were 24.3 psu, vertically and horizontally homogeneous. Meanwhile the boundary conditions in the Adriatic Sea at the inlets Boca Nord and Boca Sud were a uniform salinity of 32 psu. During the time the gate was closed, the salinity simulated with TRIM changed slowly, and heavier Adriatic Sea water intruded over the bottom of the lagoon creating a partially stratified lagoon (Fig. 3(a)). The salinity observed during the measurements presented a more stratified lagoon and the inflow of fresh water through Bocca Sud was more noticeable (Fig. 3(d)). In the model results the isohalines were more vertically oriented indicating that more mixing was occurring in the simulation compared with that on the field. Also, from the model results it was observed that, between the flood and the ebb tide the lagoon became well mixed with an average salinity of 26 psu. After two days, on the 7th of July, the Po River gate was opened to produce the same conditions at the open boundaries as were produced in the field. Within one hour a surface buoyant plume began to intrude into the lagoon. The combined effects of the flood tide coming into the lagoon, and the south-west wind at the time, restrained the intrusion of the fresh water plume, moving the brackish water northward along the western perimeter of the lagoon (Fig. 4(a)). In the field experiment the intrusion of the fresh water plume was moved about one kilometer northwest along the coastline, forced by the wind (Fig. 4(c)). However, in the simulation results the fresh water intrusion was more dispersed and the buoyant surface plume was considerably thinner than the measurements.
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Fig. 3. Salinity contours simulated for the 5th of July at 11.8 am. (a) Horizontal plan view of the top 0.25 cm layer of the lagoon. The discharge, in m3 /s, from one hour before the simulation is presented in the bottom left corner. (b) A vertical section of salinity along the channel one. The tide is presented with the amplitude in meters. (c) Salinity contours measured on the lagoon at 12:00 hrs on 5th of July. The contours represent the salinity averaged over the top 50 cm. The wind velocity averaged over the four hours prior to the sampling time is presented by an arrow at the right top corner. (d) The salinity in transect along channel one. The tide for the day of the measurements is presented with a bar indicating the length of the sampling period.
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Fig. 4. Salinity contours simulated for the 7th of July at 8 hrs. (a) and (b) are presented as in Figs. 3(a) and (b). The salinity contours measured in the lagoon at 10:00 hrs on 7th of July. (c) and (d) have the same description as in Figs. 3(c) and (d).
The ebb tide and the low intensity of the wind in the next 6 hours created favorable conditions for the advancement of the buoyant plume during the afternoon of the same day. This feature was reproduced in the simulation (Fig. 5(a)) and compares well with the field observations as shown in Fig. 5(c).
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Fig. 5. Salinity simulated for the 7th of July at 15 hrs. (a) and (b) are presented as in Figs. 3(a) and (b). The salinity contours of the lagoon at 16:00 hrs on the 7th of July. (c) and (d) have the same description as in Figs. 3(c) and (d).
On the other hand, in the simulation, the buoyant overflow was less influenced by the wind, being of a more radially spreading nature and moving the fresh water considerably further away on the surface layer. The fresh water volume discharged into the lagoon on the 8th of July was forced by the wind to border the lagoon coastline. Meanwhile the ebb tide, in the opposite direction to the wind, caused the mixing to increase, deepening the top layer to 2 meters (Fig. 6(a)). This result from the simulation agreed with
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Fig. 6. Salinity simulated for the 8th of July at 15 hrs. (a) and (b) are presented as in Figs. 3(a) and (b). The salinity contours of the lagoon at 15:00 hrs on the 8th of July. (c) and (d) are presented as in Figs. 3(c) and (d).
that observed in the field data, showing a good resolution of horizontal distribution of salinity. Along the channel the salinity was lower indicating more mixing occurring in the simulation, but the shape of the distribution was similar. The two layer stratification with 15 psu at the top and 25 psu salinity at the bottom was also well reproduced by the model (Figs. 6(b) and (d)). The maximum discharge observed on the 9th of July, combined with the flood tide and the wind, restricted the advancement of the buoyant plume overflow creating a sharp front between the Po River
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Fig. 7. Salinity contours simulated for the 9th of July at 10 hrs. (a) and (b) are presented as in Figs. 3(a) and (b). The salinity measured of the lagoon at 10:00 hrs on the 9th of July. (c) and (d) have the same description as in Figs. 3(c) and (d).
and the lagoon water. The low salinity water in the Adriatic Sea influenced the stratification presenting a more distributed buoyant flow. This pattern agreed well with the field measurements both with respect to the shape of the intrusion and the plume size (Figs. 7(a) and (c)). However, the averaged salinity produced by the simulation (Fig. 7(a)) in the south of the lagoon was higher than that in the measurements (Fig. 7(c)).
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Fig. 8. Salinity contours of the lagoon at 10:00 hrs on the 10th of July. (a) and (b) are presented as in Figs. 3(a) and (b). The salinity measured for the 10th of July at 9 hrs. The same description as in Figs. 3(c) and (d).
On the 10th of July, during the morning, a maximum discharge was observed with similar characteristics as the day before (Fig. 8(a)). The prevailing south-east wind pushed the fresh water northwest, bordering the coastline. The reduction of volume created by the wind allowed Adriatic Sea water to replace that volume, resulting in an oceanic environment with 30 psu at the bottom half of the lagoon. The fresh water was observed to be advected further away in the surface compared with the field data, and in general the fresh water was mixed faster in the simulation (Fig. 8(c)).
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Fig. 9. Salinity contours simulated for the 11th of July at 12 hrs. (a) and (b) are presented as in Figs. 3(a) and (b). Wind, tide and discharge are presented as in Fig. 3(a).
Two hours before the gate was closed on the 11th of July, the lagoon presented a marine water environment caused by the level difference between the lagoon and the ocean, and the flooding tide pushing saline water through the south inlet. A surge of fresh water created by a large discharge before the gate was closed, and the oceanic water created an horizontally stratified lagoon with salinities ranging from 1 psu at the gate to 35 psu at the Bocca Sud inlet (Fig. 9). The results from the simulation show an intermediately mixed lagoon with salinities from 33 psu at the bottom and 18 psu at the top, instead of
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the mainly marine environment (35 psu of salinity) observed in the field. Thus, the mixing coefficients in the model were providing more mixing in the lagoon than was observed. After the gate was closed, remnants of fresh water were mixing in the surface of the lagoon during the next two days. In summary, TRIM reproduced the main features of the circulation in Barbamarco Lagoon, as well as the main mixing characteristics of interactions occurring in the lagoon between two water bodies. These were: the plume shape extent of the buoyant plume created by Po River discharge; the effect of the wind on the distribution of the fresh water; and the estuarine circulation at the Bocca Sud entrance. 4.2. Fresh water volume balance The total volume of fresh water in the estuary is calculated from the freshness [4], V=
η
f dz dA,
(8)
A −h
where η is the surface elevation, and A is the area of the lagoon. The freshness is defined as f=
So − S , So
(9)
where S is the salinity of the water and So is a salinity reference. The averaged salinity found in the Adriatic Sea at the mouth of the lagoon during the study was used as a reference, So = 32 psu, to calculate the freshness of each cell of the lagoon. For the field study, the salinity was calculated using the conductivity and temperature measured with a fine scale profiler (RI). The salinity profiles were depth averaged every 25 centimeters and these values extrapolated to the cells around the sampling sites to obtain the freshness of each cell in the lagoon. Thus, the total volume of fresh water in the lagoon was calculated using Eq. (8), integrating the fresh water in each cell using a grid resolution of dx = 10 m, dy = 10 m, and h = z the depth of the bathymetry. The field data show that fresh water had accumulated from the 5th to the 9th of July, and started flushing into the ocean from that time until the end of the experiment. The cumulative volume of fresh water discharged into the lagoon through the gate between the 5th and the 9th of July clearly matched the fresh water volume calculated by Eq. (9) (Fig. 10(a)). It should be borne in mind that the volume of fresh water in the lagoon is an approximation over 3 hours sampling time. After the 9th of July, the volume of fresh water in the lagoon decreased until at the end of the experiment the average salinity in the lagoon was lower than that found at the beginning of the experiment. The volume of fresh water in the lagoon during the simulation was calculated in the same way as it was calculated with the field data. The freshness of each cell was calculated based on the salinity obtained through the mixing in TRIM, then all the grid cells in the lagoon were integrated. The calculations of the fresh water volume in the simulation show a fresh water variation with a periodicity similar to the tide period (Fig. 10(b)). The pattern of the freshness in the lagoon calculated from the field measurements was hidden in the signal in tidal periodicity obtained by the simulations. The alias so introduced became evident when the signal was subsampled at the times corresponding to the experiment sampling time. The loss of fresh water following the 9th of July was due to the flushing of the lagoon through Bocca Sud.
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Fig. 10. (a) Fresh water volume calculated with the salinity profiles in broken line, and accumulated discharge from the Po River in continuous line. (b) The fresh water volume from the simulation (continuous line), and subsampled at the experiment (+). The volume calculated with the salinity profiles in broken line.
5. Conclusions The distribution of the salinity simulated using TRIM showed good agreement with the field observations presented in RI. The results provided the details of the hydrodynamics in the shallow lagoon. Tides were the main variable driving the mean circulation and the flushing of freshwater out of the lagoon. Meanwhile the effect of the wind was to redistribute the fresh water in the surface layer throughout the lagoon. The effect of the wind was more evident when its directions was acting in the same direction as the tidal flow, favoring the stratification. When the direction of the wind was in the
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opposite direction to the tidal flow it favored mixing and deepening the surface plume. The estuarine circulation produced by the differences in density between the lagoon and the ocean was well reproduced by TRIM. There were times when the stratification produced in the simulation was weaker than that in the field observations indicating the necessity of some improvement in the closure scheme. The freshness of the lagoon was well reproduced and provided a better understanding of the flushing time of a shallow coastal lagoon. The results from the fresh water balance in the lagoon revealed a higher frequency variability of the processes occurring in the lagoon hidden by the lower frequency sampling of the field measurements.
Acknowledgements The authors wish to thank Vincenzo Casulli for allowing the use of TRIM, and Abi Javam, for help in learning about the model. Isabel Ramirez was supported by The National Council of Science and Technology of Mexico and The Centro de Investigacion Cientifica y Educacion Superior en Ensenada. This paper forms the Centre for Water Research reference ED-1280-PD.
References [1] V. Casulli, E. Cattani, Stability, accuracy and efficiency of a semi-implicit method for three-dimensional shallow water flow, Comput. Math. Appl. 27 (1994) 99–112. [2] V. Casulli, R.T. Cheng, Semi-implicit finite difference methods for three-dimensional shallow water flow, Internat. J. Numer. Methods Fluids 15 (1992) 629–648. [3] L. D’Alpaos, A. Defina, B. Matticchio, Multilayer model for shallow water flows and density currents applied to a lagoon in the Po River delta, in: International Conference on Barrages, Cardiff, UK, 1996. [4] H.B. Fischer, E.J. List, R.C.Y. Koh, J. Imberger, N. Brooks, Mixing in Inland and Coastal Waters, Academic Press, New York, 1979. [5] R.W. Garvine, J. Monk, Frontal structure of a river plume, J. Geophys. Res. 79 (1974) 2251–2259. [6] L. Huzzey, J. Brubaker, The formation of longitudinal fronts in a coastal plain estuary, J. Geophys. Res. 93 (1988) 1329–1334. [7] J. Imberger, Tidal jet frontogenesis, Mech. Engrg. Trans. 12 (1983) 51–55. [8] D.A. Luketina, J. Imberger, Characteristics of a surface buoyant jet, J. Geophys. Res. 92 (1987) 5435–5447. [9] D. Luketina, J. Imberger, Turbulence and entrainment in a buoyant surface plume, J. Geophys. Res. 94 (12) (1989) 619–636. [10] B. Matticchio, Semi-implicit finite difference methods for three-dimensional shallow water flow, Master’s Thesis, University of Padova, Italy, 1992. [11] I. Ramirez, J. Imberger, Hydrodynamics of a shallow lagoon: Barbamarco Lagoon, Italy, submitted, 2000. [12] G. Umgiesser, A. Bergamasco, The spreading of the Po plume and the Italian coastal current, in: ISDGMCNR, Venice, Italy, 1994.