Are inlets responsible for the morphological degradation of Venice Lagoon?

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 111,, doi: /2005jf000334, 2006 Are inlets responsible for the morphological degradation of Venice Lagoon? N. Tambroni 1 and G. Seminara 1 Received 10 May 2005; revised 14 March 2006; accepted 12 April 2006; published 13 September [1] Through the centuries, Venice Lagoon has undergone morphological changes that can be attributed to both natural events and human actions. The lagoon has progressively deepened, and it is claimed to lose roughly one million cubic meters of sediments each year. In the ongoing debate concerning the possible means to counteract this morphodynamic degradation, inlet geometry is considered a major factor controlling the exchange of sediments. Our aim is to explore the causes of this loss. We focus first on sand, as this is the type of sediment present on the bottom of the near-inlet regions. We employ a simple model of the inlet hydrodynamics to estimate the net exchange of sand associated with the sequence of tidal events recorded for several years. Results suggest that in the absence of an excess supply from the sea, the yearly loss of sand through Venice inlets is an order of magnitude smaller than the total sediment loss usually claimed. We then show that this estimate is only slightly affected by the sand supply from wave resuspension in the far field whose effect is simply to store sediments in the near-inlet region. We finally argue that most of the sediment loss is wash load carried by the ebb currents overloaded by very fine sediments resuspended by wind in the inner lagoon and unable to settle within the channel network. Citation: Tambroni, N., and G. Seminara (2006), Are inlets responsible for the morphological degradation of Venice Lagoon?, J. Geophys. Res., 111,, doi: /2005jf Introduction 1 Department of Environmental Engineering, University of Genova, Genoa, Italy. Copyright 2006 by the American Geophysical Union /06/2005JF [2] Venice Lagoon has undergone significant changes in the last century and it is feared nowadays that the lagoon is gradually assuming the characters of a bay. Various factors have contributed to this state: the diversion of rivers discharging into the lagoon, the construction of long jetties bounding the inlets, the processes of sea level rise and land subsidence. [3] The two major consequences of the above changes are an increased frequency of high waters threatening the survival of Venice and a progressive loss of salt marshes threatening the survival of the lagoon. In the most recent estimate of the overall sediment balance of Venice Lagoon [Magistrato alle Acque di Venezia and Consorzio Venezia Nuova (MAV-CVN), 2002] it has been claimed that, in the decade , about m 3 of sediments per year were eroded from salt marshes and shoals: roughly 10 6 m 3 /yr led to aggradation of the canals, m 3 /yr were removed from the canals by dredging and m 3 /yr were lost to the sea. This picture, rough as it may be, helps the reader to appreciate why, along with the major problem of defending Venice from flooding due to high water, the next issue in the agenda is to attempt counteracting the morphodynamic degradation of the lagoon. This is a goal fairly difficult to achieve and calls for a deeper understanding of the exact nature of the mechanisms controlling the long-term exchange of sediments between enclosed basins and adjacent seas. [4] In this paper we investigate these mechanisms, starting with an analysis of the exchange of sand through tidal inlets. We concentrate on sand, as this is the type of sediment present on the bottom of the near-inlet regions. Why do we focus on tidal inlets? In the ongoing debate concerning the possible means to counteract the morphodynamic degradation of Venice Lagoon, it has been claimed that the inlet geometry is a major factor controlling the exchange of sediments. This claim is essentially based on two arguments: (1) The construction of jetties has definitely enhanced the degree of ebb-flood asymmetry experienced by the flow field in the near-inlet region, suggesting that a net exchange of sediments through the inlet in each tidal cycle may be driven by purely hydrodynamic factors. (2) The inlets have been moved a few hundred meters away from the shore: hence a reduced amount of sediment resuspended by wave breaking in the surf zone would now be able to reach the inlets during the flood phase. [5] If the latter arguments were correct one would be tempted to propose modifying the inlet geometry accordingly. On the other hand, this is quite a delicate issue, as the tidal inlets of Venice Lagoon have various important but conflicting functions. They must be wide and deep as well as stable enough to allow for navigation; however, they also regulate the exchange of water between the lagoon and the sea, hence their widths and depths affect the amplitude of 1of19

2 tidal oscillations in the enclosed basin, posing a constraint somehow in conflict with the previous requirement. As mentioned before, control of sediment exchange might suggest inlet shapes able to reduce the ebb-flood asymmetry of the flow field encouraging littoral sediments to merge into the flood current at the inlet; however, such a feature would also encourage siltation of the inlets. Finally, pollution control would call for large volumes of exchanged water and inlet shapes enhancing the ebb-flood asymmetry of the flow field, in contrast with the just mentioned requirements posed by the need to enhance sediment supply from the sea. [6] Below, we argue that the exchange of sand through the inlets is not the main mechanism responsible for sediment export from the lagoon. In order to support this statement, we need to employ a sufficient simple model of the inlet hydrodynamics. In fact, evaluating the long-term exchange of sediments through the inlets is a major task, which can hardly be tackled using too detailed (e.g., twodimensional) models: these may be useful to estimate the net exchange of sediments in a few tidal cycles, but they are computationally too demanding to allow for long-term predictions. In this paper we extend a simple model of inlet hydrodynamics to the evaluation of sediment exchange through Venice inlets and obtain, by a non prohibitive numerical effort, an upper bound for the loss of sand experienced by the lagoon for periods of years or decades. [7] We start restricting our attention on the transport of sand (of grain size in the range mm) as this is the sediment present on the bottom of the near-inlet regions, available for entrainment in suspension and transport as bed load. Furthermore, we assume that sediment transport equals the transport capacity of the stream during both the flood and the ebb phases. This is a priori an oversimplified assumption, as it ignores the role of boundary conditions. In fact, (1) on one hand, the ebb currents may carry sediments much finer than sand, resuspended in the inner lagoon and unable to settle in the channel network (wash load in fluvial terms ); (2) on the other hand, sand resuspended in the surf zone during storm events can overload the flood currents beyond transport capacity. [8] Hence employing the above assumption, we will possibly achieve a correct estimate of the amount of sand, (rather than the total amount of sediments) lost to the sea during the ebb phase, while the amount of sand entering the lagoon carried by the flood currents may be underestimated. [9] In the second part of our analysis (section 6), we estimate to what extent sand resuspension in the surf zone modifies the above picture. We show that the excess supply of sand, overloading the flood currents, is unable to enter the lagoon: it rather settles in the region close to the inlet, encouraging inlet siltation. This is shown modeling the flood flow in the near-inlet region as a two-dimensional (2-D) irrotational plane flow, a simple scheme substantiated by our recent and less recent experimental observations [Blondeaux et al., 1982; Tambroni et al., 2005a]. The latter assumption simplifies the treatment considerably and allows us to couple the hydrodynamics with the transport of suspended sediments, evaluated through the solution of an advection-diffusion equation for sand concentration. We find that the effect of an excess supply in the far field is simply to store sediments in the near-inlet region: in other words, a very small fraction of the sand overload reaches the inlet. [10] Next, we attempt to obtain a rough estimate of the amount of very fine sediments lost to the sea as wash load. The role of the latter mechanism will be discussed in section 7 on the basis of a few available field measurements of concentration of suspended sediments in the shoals as functions of the wind speed. This estimate will be shown to be affected by a high degree of uncertainty due to the limited amount of available data, hence we will only be able to suggest (rather than conclusively prove) that the total sediment loss experienced by the lagoon is dominantly wash load. Finally, we will show that the construction of gates for protection of Venice from high waters will have a small beneficial effect reducing by a small amount the net export of sediments from the lagoon. 2. Physical Setting [11] Venice Lagoon is the largest Italian lagoon and one of the largest in Europe. It is located in the northern part of the Adriatic Sea ( N, E) and extends over an area of about 550 km 2. [12] Three inlets, Lido, Malamocco, and Chioggia (Figure 1c from right to left), connect the lagoon to the Adriatic Sea, entailing a subdivision into three subbasins. Over 90 per cent of the total variance in the average water flux through the inlets is due to tidal forcing. The dominant astronomical constituent is semidiurnal with typical tidal range averaging approximately 1 m. As far as the sea climate it is concerned, a wave statistics was obtained analyzing data recorded from October 1987 to December 2003 by Magistrato alle Acque di Venezia and Consorzio Venezia Nuova in a gauge station located in the Adriatic Sea adjacent to Venice Lagoon. Results reported by MAV-CVN [2004] show that there are two dominant wave sectors: NE and SE, corresponding to the prevailing wind directions in the area (Bora and Scirocco). The bulk of the wave heights is significantly lower than 2 m and only in rare occasions they exceed 4 m: the significant wave height with return period of five years is equal to 4.4 m. [13] Venice Lagoon formed as a consequence of the regression of the Adriatic coast line, induced by the 130 m sea level rise following the Last Glacial Maximum (LGM, 18 ka). Rivers flowing into the Adriatic Sea at that time formed a sequence of lagoons: all of them, except for Grado and Venice Lagoons, have since disappeared due to siltation. Venice Lagoon has survived also thanks to a number of actions undertaken by Venice Republic since the Renaissance. The first major project was the diversion of the three main rivers debouching into the lagoon (Brenta, Sile, and Piave): this stopped the ongoing siltation driven by fluvial sediments but also set the premises for a reversed evolution of Venice Lagoon into a bay. Figure 1a shows the lagoon configuration at the beginning of the 19th century when the first accurate survey was performed by Denaix [1811]. [14] At the middle of the 19th century the development of steam navigation prompted the need for deepening the natural inlets. This goal was achieved through the construction of long jetties bounding the inlets, which were moved a few hundred meters seaward. Figure 1b shows the lagoon 2of19

3 Figure 1 3of19

4 Figure 2. Loss of salt marsh areas. Red lines bound the regions occupied by salt marshes in The green regions inside the red lines represent the actual salt marsh areas surviving in 1972 (courtesy of L. D Alpaos). configuration at the beginning of the 20th century when the works at Lido and Malamocco had been completed, while those at Chioggia had not yet been undertaken. [15] Various further man-induced actions were undertaken to enhance the industrial development of the city. In particular, deep channels were excavated through the central part of the lagoon while parts of the lagoon were filled up in order to reclaim land for industrial uses. Several new fishing ponds (valli da pesca) were obtained by constructing fences around portions of the lagoon which were no longer subject to tidal motion (the total surface area of fishing ponds increased from about m2 at the end of the 19th century to about 108 m2 in 1930). Furthermore, natural phenomena, namely sea level rise and subsidence, have led to a pronounced process of sinking of the city (roughly 23 cm in the last century). [16] Along with the increased frequency of high waters, the main consequence of the above processes has been a general erosion of the lagoon which still persists: the salt marsh area has decreased (Figure 2) from about 110 km2 around 1790 to 30 km2 at the end of the 20th century; the average depth of shoals has increased in the last century by 60 cm, 40 cm and 30 cm in the Malamocco, Lido and Chioggia basins, respectively. Part of the eroded sediments has deposited in the canals, which have recently undergone siltation: surveys performed in 1989 and 1999 suggest an average yearly reduction of the volume of canals ranging about 106 m3/yr. [17] On the contrary, since the construction of the jetties, the three inlets have undergone progressive deepening: only in the last two decades sediment deposition at Lido and Chioggia has inverted the above tendency. This is shown in Figure 3, plotting the cross-sectional area of each inlet as a function of time. The (average) cross-sectional area of the inlet is defined as inlet volume (the volume of water bounded by the bed, the mean water level and the side banks) divided by inlet length. Data are based on surveys of bed topography performed by MAV-CVN. The average Figure 1. Pictures showing Venice Lagoon at various stages of its development: (a) the pattern obtained from the first complete survey performed by Denaix [1811], (b) the lagoon configuration in 1910 when jetties bounding two of the three inlets (Lido and Malamocco from right to left) had been constructed, and (c) Venice Lagoon at the present time. 4 of 19

5 Figure 3. Temporal evolution of the cross-sectional area undergone by each of Venice inlets in response to the construction of the inlet jetties (data from MAV-CVN [2002]). flow depth of each inlet is defined as average crosssectional area divided by channel width. 3. Estimating the Sand Loss: Formulation of the Mathematical Model [18] The hydrodynamics of tidal inlets is a major subject of research widely explored in the literature. An estimate of the average inlet velocity was obtained, using simple concepts of steady flow hydraulics, as early as 1928 by Brown [1928]. Keulegan [1967] was the first to give an analytical solution of the 1-D equations, although disregarding the inertial term in the momentum equation. Others, since then, have obtained a variety of analytical solutions for the latter equations using different approximations: Dean [1971] linearized the dissipation term and neglected inertia, Kondo [1975] removed the second hypothesis and found a more complete solution although the dissipation term was still linearized. More recently, numerical models have provided more detailed pictures using 1-D, 2-D, or 3-D approaches [Tambroni et al., 2005b; D Alpaos and Martini, 2005]. Though feasible, these numerical models are computationally demanding to allow for long-term predictions. We will then employ a simple, yet physically sound, model which revisits the recent analytical formulation of Marchi [1990], essentially based on the 1-D model proposed by Bruun [1978]. Having determined the inlet hydrodynamics, we will then proceed to evaluate the exchange of sand experienced through the inlets in a period of five years: this exercise will prove quite instructive. [19] We model each of the three Venice inlets as a straight rectangular channel which connects the portion of the lagoon drained by the inlet with the open sea. Since the length scale characteristic of each basin as well as the lengths of inlet channels are small compared with the tidal wavelength, we assume that the free surface in the basin is effectively horizontal. With this assumption, following the 1-D approach reported in Appendix A, we obtain a known equation governing the temporal development of the free surface oscillation h ~ 2 in the basin, subject to the tidal forcing at the seaward end cross section, where the temporal development of the free surface oscillation ~ h 1 is imposed. This equation, in dimensionless form, reads ~h 1 ~ h 2 ¼ d 1 d 2 ~ h 2 d~t 2 þ d 2 d ~ h 2 d~t d h ~ 2 d~t ; where ~ h 1 and ~ h 2 are scaled by the amplitude a 0 of the forcing oscillation, while ~t is timescaled by the inverse angular frequency w 1 of the tide. [20] Moreover, d 1 and d 2 are dimensionless parameters measuring the relative role played in the momentum equation by local inertia and dissipation compared with gravity. The relationship between the latter parameters and those introduced by Bruun [1978] (a, b) and Keulegan [1967] (K), are as follows: ð1þ d 1 ¼ a 2 ; d 2 ¼ a 2 b ¼ K 2 : ð2þ [21] Below, we perform two exercises. (1) In the former exercise, we assume a simple harmonic forcing and investigate the general characteristics of the inlet hydrodynamics (section 4). (2) In the latter, we force the system with the actual tidal oscillations recorded in the period in the Adriatic Sea close to Venice inlets: this allows us to determine the sequence of tidal currents which have determined the exchange of sediments through the inlets (section 5). [22] With the above boundary conditions, we then solve numerically equation (1) for the free surface oscillation in the basin and evaluate the instantaneous cross-sectionally averaged velocity ~U at the inlet (see equation (A10)). [23] The maximum value attained by the (dimensionless) speed ~U max in the inlet for a prescribed tidal forcing is then found as a function of the dimensionless parameters d 1 and d 2 : ~U max ¼ fd ð 1 ; d 2 Þ: ð3þ Once the inlet hydrodynamics is known, we can proceed to estimate the exchange of sand through the inlets. The total 5of19

6 Figure 4. Frequency of tidal events in the Adriatic Sea near Venice inlets plotted versus the height of the tidal oscillation (defined as the difference between the maximum and minimum free surface elevations in a tidal cycle). sediment flux transported as both bed load and suspended load is evaluated at each instant of time throughout the tidal cycle using Engelund and Hansen s [1967] predictor, in dimensional form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5=2 ðs 1Þgds 3 ; Qs ¼ B 0:05C 2 t * ð4þ where B is the inlet width, g is gravity, ds is the mean grain diameter of the sand mixture and s is its relative density (rs/r). Moreover, C is the flow conductance and t* is the instantaneous value of the average Shields stress in the cross section, which reads U2 t* ¼ 2 : C ðs 1Þgds ð5þ Time integration of the total load finally provides the sought estimate of the temporal evolution of the volume of sand exchanged through each inlet. [26] 1. Each curve displays a peak. For the value of the tidal amplitude chosen to construct Figure 6 the peak value of Umax is associated with a value of the inlet depth Dc smaller than the value presently experienced by that inlet. This is an important observation. In fact, as pointed out by Marchi [1990], while the branch of the curve corresponding to values of D > Dc describe stable conditions, the branch of the curve corresponding to values of D < Dc describe unstable conditions (reduction of the flow depth is there associated with a reduction of the flow speed, hence with a decreased sediment transport capacity which encourages deposition and a further reduction of the inlet depth). [27] 2. If we examine the static equilibrium limit, defined by the inability of tidal currents to transport (either in suspension or as bed load) the sediments available in the bed, the threshold conditions are obtained by stipulating that the maximum current speed must not exceed the critical speed for sediment motion defined by Shields criterion. The latter conditions are plotted in Figure 6 for a realistic range of sediment sizes, suggesting that the present configuration of each of Venice inlets is quite far from static equilibrium, which would require flow depths much larger than the present ones. This is not surprising as inlet equilibrium must be viewed as a dynamic state: we have recently investigated numerically as well as through laboratory experiments the establishment of inlet equilibrium in a system channel-inlet-sea [Lanzoni and Seminara, 2002; Tambroni et al., 2005a]. Both theory and experiments show that equilibrium is reached when the net flux of sediments in a tidal cycle vanishes everywhere: a quite slow and delicate process, which is not simply controlled by the local conditions at the inlet. [28] To complete our understanding of the inlet hydrodynamics, we must address the issue of flood and ebb dominance of the flow field. Figure 5 shows that flow dominance depends on the values of d1 and d2: in particular, it appears that, for the typical values that the latter parameters attain at each of Venice inlets under an M2 forcing, flow is weakly flood dominated only at Lido. The character of the flow plays an important role in the process of sand 4. Inlet Hydrodynamics and Inlet Equilibrium [24] Let us then perform our first exercise. We assume a simple M2 forcing tide characterized by an amplitude a0 equal to 0.5 m, a value fairly typical of tidal events in the Adriatic Sea close to Venice inlets (see Figure 4). With the above boundary condition, we have evaluated the function f (see the relationship (3) for the maximum inlet speed) as a function of the dimensionless parameters d1 and d2. Results are plotted in Figure 5. Using the scale for U (see equation (A10)), we can transform (3) into a dimensional relationship between the maximum value of the inlet velocity in a tidal cycle and the inlet depth: such dependence is plotted in Figure 6 for values of the relevant parameters appropriate to each of Venice inlets. Note that the present results agree qualitatively with those obtained by Marchi [1990] whose approach differed from the present one for some approximations introduced in order to allow for a fully analytical treatment of the problem. Those approximations led to a slight underestimation of the velocity at each inlet. [25] A glance at Figure 6 allows some general observations of relevance for the issue of inlet equilibrium. Figure 5. Function f, i.e., the maximum dimensionless inlet velocity, plotted as a function of the dimensionless parameters d1 and d2 for a pure M2 forcing tide with amplitude a0 = 0.5 m. Also plotted are the conditions corresponding to each of Venice inlets. 6 of 19

7 Figure 6. Maximum inlet velocity plotted versus the inlet depth for a pure M 2 forcing tide with amplitude a 0 = 0.5 m. Also plotted is the critical speed for sediment motion for sizes in the range mm. transport through the inlet. As discussed in the next section, flow dominance depends strongly on the harmonic content of the tidal forcing, a feature which typically varies throughout the year and modifies significantly the picture suggested by Figure 5. [29] One last point, which still deserves some considerations, is that in the presence of multiple inlets, interactions may occur whereby hydrodynamics and sediment transport in each inlet may also depend on variations occurring at the other inlets. We have investigated the above idea following van de Kreeke s [1990] approach: in other words, we now model the lagoon as a single basin with surface S tot, connected to the sea through three straight rectangular channels. [30] The formulation of inlet hydrodynamics does not differ substantially from the one already discussed (Appendix A). The main difference is that in this case all the inlets discharge into the same basin hence the free surface elevation h 2 at the inner boundary of each inlet takes the same value. Following an approach similar to that reported in Appendix A we end up with three differential equations governing the flow velocity U i at each inlet: du i þ g h 2 h 1 þ U iju i j dt L i Ci 2D ¼ 0; i where h 1 is the tidal forcing, g is gravity and D i, C i and L i are depth, flow conductance and length of the ith inlet, respectively. In order to find a solution for U i, equations (6) must be solved along with a further equation imposing continuity, which may be written in the form with B i inlet width. X 3 i¼1 U i D i B i ¼ S tot dh 2 dt ð6þ ð7þ [31] We then assume a simple harmonic forcing tide h 1 and solve the system of equations (6) and (7) numerically. [32] The maximum flow velocity at Malamocco is plotted in Figure 7 as a function of the local depth, for given values of depths at Chioggia and Lido (8.5 m and 8.9 m, respectively). The curve obtained for an isolated inlet (Figure 6) is also reported in Figure 7a for comparison. Figures 7b and 7c show the same plots for Lido and Chioggia inlets, respectively. Results may be summarized as follows. (1) At Malamocco the peak value of U max is roughly the same whether or not we take into account the presence of the other inlets; in the case of multiple inlets the peak value of U max decreases at Lido while it increases at Chioggia. (2) If the effect of multiple inlets is taken into account the peak value of U max at each inlet is associated with a value of the inlet depth D c larger than in the single inlet approach. (3) Moreover, the branch of the curve with D > D c decays much faster as the flow depth increases. Also, note that equilibrium can be empirically associated with the validity of O Brien s [1931] relationship between the tidal prism P i and the cross-sectional area: D i B i ¼ constp n i : with n = 0.85 and, if D i, B i in m and P i in m 3, const = m 0.55, as suggested by Marchi [1990] for Venice Lagoon. Support to the validity of (8) has recently been provided by the theoretical investigation of Lanzoni and Seminara [2002] and the laboratory observations of Tambroni et al. [2005a]. The equilibrium values of the flow velocity as a function of the inlet depth obtained from (8) are plotted in Figure 7. van de Kreeke s [1990] model suggests that Lido and Chioggia inlets are both unstable and tend to fill up, while Malamocco is in a near critical condition. However, the assumption of a single uniform free surface elevation for all ð8þ 7of19

8 Figure 7. Maximum velocity at (a) Lido, (b) Malamocco, and (c) Chioggia plotted versus depth. The dotted line describes the case of independent inlets. The dash-dotted line in Figure 7a accounts for the effects of Chioggia and Malamocco on Lido inlet assuming the flow depth at Chioggia and Malamocco to be fixed and equal to the one presently observed, and it is similar in Figures 7b and 7c. The solid line provides the equilibrium values of the flow velocity as a function of the inlet depth according to O Brien s [1931] law. the basins is not necessarily the most appropriate one, as each of the inlets has a drainage basin of its own; while the local free surface elevation is definitely continuous through the basins, however the average elevation of different basins may reasonably be slightly different. [33] Finally, it is of some interest to compare the results obtained by the two approaches presented above with some field data. Figure 8 shows the temporal dependence of the average cross-sectional velocity predicted at each Venice inlet using the interactive model of van de Kreeke [1990] compared with that obtained by the present approach. It turns out that both models predict roughly the same results at Chioggia, while slightly larger differences emerge at Malamocco and Lido. Both predictions are in fairly good agreement with field observations at each inlet during an M 2 dominated event although, at Malamocco, the measured speeds appear to be better reproduced by the present model. These results have motivated our choice to use the present model for the hydrodynamics to evaluate the net exchange of sand at the inlets. This notwithstanding, we will show in the next section that the estimate of the yearly volume of sand lost from Venice Lagoon would not change significantly if the interactive approach were used. 5. An Estimate of an Upper Bound for the Yearly Volume of Sand Lost From Venice Lagoon [34] The second exercise we performed was to assume for the forcing function the sequence of tidal oscillations recorded in the last few years by the CNR gauge station located in the sea region adjacent to Venice Lagoon and evaluate the sediment flux following the approach discussed in section 3. The temporal evolution of the net volume of sand exchanged through each inlet in the period is plotted in Figure 9. [35] In order to estimate the importance of inlet interactions, the same calculations were also performed using the van de Kreeke s [1990] model (Figure 10). A comparison between Figures 9 and 10 suggests that the presence of the other inlets, accounted for by van de Kreeke s approach, implies that the net volume of sand exchanged at Malamocco increases while it decreases at Lido. However, the general features of the temporal dependence of the net sand exchange and the total volume of the yearly loss of sand from Venice Lagoon are not significantly altered by the effect of inlet interaction. For this reason, in the following we will refer to results obtained by the present model, being aware that similar conclusions would be reached if inlet interactions were accounted for. [36] A first observation clearly emerges from Figure 9: the net volume of sand exchanged at the inlets displays a repetitive seasonal dependence, being invariably negative in spring and autumn and positive (entering the lagoon) roughly in winter and summer. In order to clarify the mechanism underlying the above behavior we have attempted to correlate the net exchange of sediments with the characteristics of the tide. [37] Figure 11 shows the clear correlation between the intensity of the sand exchange at each of Venice inlets and the amplitude of the forcing tidal oscillation for a sequence of tidal events that occurred in In particular, significant exchange of sand is associated with forcing tides 8of19

9 Figure 8. Comparison between the temporal evolution of the average cross-sectional velocity observed at (b) Lido, (c) Malamocco, and (d) Chioggia inlets during an event that occurred in September 1970 [Di Paola et al., 1979] (asterisks) and the velocity distribution calculated under a pure (a) M 2 forcing tidal oscillation with amplitude a 0 = 0.5 m using the present approach (dotted line) and the van de Kreeke s [1990] approach (dash-dotted line), which accounts for the inlets interaction. characterized by differences between maximum and minimum free surface elevations larger than a threshold value averaging approximately 1 m. However, from Figure 11 one is unable to predict the direction of the sand exchange. In Figure 12 the net volume of sand exchanged through Venice inlets is plotted versus the maximum tidal amplitude experienced during tidal cycles grouped according to the phase difference j between the diurnal and semidiurnal components of the tide. Figure 9. Temporal evolution of the net volume of sand exchanged through (a) Lido, (b) Malamocco, and (c) Chioggia inlets in the years neglecting the effects of inlet interactions. 9of19

10 Figure 10. Temporal evolution of the net volume of sand exchanged through (a) Lido, (b) Malamocco, and (c) Chioggia inlets in the years accounting for the effect of inlet interactions. [38] Results in Figure 12 confirm that the intensity of the net exchange of sand increases with the amplitude of the tidal oscillation. Moreover, in the absence of further components, a fairly clear correlation emerges between the sign of the net sand flux and the phase difference j: if0<j < p (p < j <2p) the net volume of sand exchanged through the inlets is positive (negative) and its modulus reaches a maximum when averages approximately p/2 (3p/2). The above findings are quite reasonable as the tide turns out to be flood- or ebb-dominant depending on whether the phase difference between the diurnal and semidiurnal components of the tide satisfies the condition 0 < j < p or the alternative condition p < j <2p. The presence of further harmonics in the forcing spectrum makes the picture less straightforward than just described and justifies the exceptions emerging in the correlation presented in Figure 12. [39] A second major observation arising from Figure 9 is that the yearly loss of sand from Venice Lagoon, averaging approximately 50,000 m 3, is about an order of magnitude smaller than the overall loss of sediments usually claimed Figure 11. Correlation between (b) the intensity of the sand exchange at each of Venice inlets and (a) the amplitude of the forcing tidal oscillation for a sequence of tidal events that occurred in the first three months of of 19

11 of tidal inlets with cohesionless bottom [Tambroni et al., 2005a] which confirmed earlier observations performed on fixed bed models [Blondeaux et al., 1982]. Let us represent the boundary of Venice inlets as shown in Figure 13 which specifically refers to Malamocco. We denote by z*(= x + iy) the complex variable describing the physical plane, with x and y the Cartesian dimensionless coordinates scaled by half the inlet width, and by z (= x + ih) the complex variable describing the transformed region. [41] Following Blondeaux et al. [1982], we then use the Scharwz-Christoffel conformal transformation to map the flow region of the physical plane z into the upper half plane z through the following relationship: Figure 12. Net volume of sand exchanged in a tidal cycle plotted versus the maximum amplitude experienced in that cycle. Data are grouped according to the phase difference j between the diurnal and semidiurnal components of the forcing tide. [MAV-CVN, 2002]: a result clearly suggesting that the major contribution to the yearly sediment loss from Venice Lagoon is not associated with an exchange of the sand available in the bed close to the inlets, amenable to transport either as bed load or as suspended load. However, before the latter conclusion can be firmly reached, we need to check whether an excess supply of sand resuspended in the surf zone may modify the picture. 6. An Analysis of the Effect of Sand Supply in the Far Field [40] In order to accomplish the latter goal, we model the flood field as plane irrotational: this assumption, which allows for an analytical treatment, has received some support from laboratory observations on physical models z* ¼ 2 p ffiffiffiffiffi ab pk i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð z aþðz þ bþþ 2 2ab þ arcosh ð a b Þz p zða þ bþ d i; where a, b and k are the parameters describing the transformation, which depend on the inlet geometry. With the notation of Figure 13, the values of the relevant parameters describing Malamocco inlet are e ¼ 1:83; d ¼ 5:44; a ¼ 3:65; b ¼ 9:4; k ¼ 3:87: ð10þ We assume that the flood flow in the physical plane is driven by a given temporal dependence of the flow discharge Q(t) atx! 1along with a littoral current of speed ( V l, V l > 0) uniformly distributed in the far field. In the transformed plane, this corresponds to the presence of a sink of intensity Q(t) (with Q(t) < 0) superimposed on a uniform flow parallel to the x axis, with velocity V l, whose complex potential W P is readily written in the form W P ¼ Q p ln z þ V lz: ð9þ ð11þ Figure 13. Sketch of the (a) physical and (b) transformed domains for Malamocco inlet. 11 of 19

12 Figure 14. Velocity distribution in the sea region close to Malamocco inlet plotted assuming an M 2 tidal forcing with a 0 = 0.5 m and a uniform littoral current in the far field with intensity V l = 0.1 m/s. In the physical plane, the complex velocity v* associated with the latter form of the complex potential in the transformed plane, is obtained as follows: dw P dz ¼ dw P dz* dz* ¼ v* dz dz dz! v* ¼ V l þ dz* dz Q pz ; ð12þ with the quantity (dz*/dz) readily calculated by differentiating the transformation (9). [42] Using the latter procedure we end up with an expression for the complex velocity of the form v*[z(z*)] = v x (z*) iv y (z*). Calculations were performed assuming constant depth equal to the average inlet depth, using for Q the values obtained from our 1-D model forced by a pure M 2 tide (section 4) and assuming for V l a typical value for the intensity of the littoral current along the Venice coast where it averages approximately 0.1 m/s. In Figure 14 we have plotted the velocity distributions at various times during the flood phase of the tidal cycle. They show that the spatial scale of the region where the presence of the inlet is felt is of the order of few inlet widths. Moreover, it is not surprising that, within the framework of the present irrotational scheme, the velocity peaks at the edges of the inlet jetties where geometrical discontinuities are present: the scheme obviously fails in an immediate neighborhood of the edges where flow separation occurs. [43] Having determined the structure of the tidally driven flow field in the near-inlet region, we can proceed to estimate the concentration distribution built up in response to the far field conditions through advection, settling and turbulent diffusion: our aim is to ascertain to what extent our previous assumption, namely that sediment transport through the inlet is determined by the transport capacity of the stream, may be violated in the flood phase due to a deficit or excess of sediment supply from the far field. [44] Let us then write the advection-diffusion equation for the mean volumetric concentration of suspended sediment c in the Cartesian coordinates x, y, z, where x and y coincide with the horizontal coordinates represented in Figure 13 and z is a vertical coordinate. Hence W @z ¼ @c D Tz : In (13) u, v, w are the components of the 3-D mean velocity vector. We evaluate it from the knowledge of the plane irrotational velocity field determined above by noting that the flow can be taken as a slowly varying sequence of locally and instantaneous uniform near-horizontal flows characterized by depth-averaged velocity equal to the plane irrotational velocity. The vertical component of flow velocity is readily estimated: jwj dh 1 dt a 0w 10 4 m=s: ð14þ 12 of 19

13 Figure 15. (top) Depth-averaged sediment concentration field in the sea region close to (a) Lido, (b) Malamocco, and (c) Chioggia inlets at the peak of the flood phase driven by a pure M 2 forcing tide of amplitude a 0 = 0.5 m and a uniform littoral current with intensity V l = 0.1 m/s. (bottom) Depth-averaged sediment concentration in excess with respect to the local equilibrium value at (d) Lido, (e) Malamocco, and (f) Chioggia. Values are expressed in ppm. Hence jwj is much smaller than the horizontal speed of the fluid velocity and is also negligible compared with the settling speed W s of sand particles that averages approximately few centimeters per second. We then write where ½u; vš ¼ v x ; v y Fz=z0 ð Þ; ð15þ Fz=z ð 0 Þ ¼ D 0 lnðz=z 0 Þ Z D0 lnðz=z 0 Þdz z 0 ; ð16þ having denoted by z 0 the reference elevation for the no slip condition, a quantity which can be expressed in terms of the mean flow conductance C 0 as follows: z 0 ¼ D 0 expð kc 0 1Þ: ð17þ Further simplification of (13) can be achieved noting that horizontal turbulent diffusion is negligible compared with the vertical turbulent diffusion. In fact, (1) the horizontal components of the turbulent diffusivity vector [D Tx, D Ty ] have orders of magnitude similar to that of the vertical component D Tz ; (2) the length scale of the horizontal variations of concentration is of the order of the inlet width, as opposed to the vertical variations which occur on the scale of flow depth. In conclusion, the advection diffusion equation may be reduced to the simpler þ v xfz=z ð þ v yfz=z ð @c D ð18þ The latter equation can be solved once a closure relationship for the vertical diffusivity and appropriate boundary conditions are adopted. In the present work, given the slowly varying character of the concentration field we will assume that D Tz ¼ ku * z 1 z : ð19þ D 0 The boundary conditions associated with (18) impose (1) no flux at the free surface, ½q nš z¼d0 ¼ W s c þ D Tz ¼ z¼d 0 and (2) gradient boundary condition at the bed, ð20þ ½q nš z¼a ¼ W s ðc e cþ; ð21þ 13 of 19

14 Figure 16. (top) Temporal evolution of the average cross sectional velocity at the inner boundary of Lido inlet. Considered stages of the decaying phase: t/t = 11/12 (a), t/t = 12/12 (b), and t/t = 1/12 (c). (middle) Depth-averaged sediment concentration field driven by a pure M 2 forcing tide of amplitude a 0 = 0.5 m and a uniform littoral current with intensity V l = 0.1 m/s at t/t = 11/12 (a 0 ), t/t = 12/12 (b 0 ), and t/t = 1/12 (c 0 ). (bottom) Depth-averaged sediment concentration in excess with respect to the local equilibrium value at t/t = 11/12 (a 00 ), t/t = 12/12 (b 00 ), and t/t = 1/12 (c 00 ). Values are expressed in ppm. where c e is the equilibrium concentration at the reference elevation a. We employ van Rijn s [1984] formula to estimate the reference concentration at the bed, assuming the latter to be plane or dune covered depending on whether or not the local and instantaneous value of the Shields stress satisfies van Rijn s [1984] criterion. [45] Finally, we assume a significant wave height of 3 m, assign the concentration at the breaking line using Bijker s [1968] approach, which accounts for the contemporaneous presence of waves and currents, and solve the equation (18) numerically for each inlet of Venice Lagoon using an explicit finite difference scheme. [46] In Figures 15a, 15b, and 15c we have plotted the concentration distributions driven by a pure M 2 forcing tide of amplitude a 0 = 0.5 m at Lido, Malamocco and Chioggia inlets respectively, at the peak of the flood phase. Results suggest that, at Malamocco and Chioggia, the concentration distribution induced by wave breaking is able to overload the current above transport capacity only close to the breaker region, i.e., outside the inlets. In other words the sediment overload deposits mostly outside the inlets. At Lido, part of the sediments resuspended by waves in the near-inlet region is able to enter the inlet (see Figure 15d), a result which agrees with field observations suggesting the presence of large deposits of sand along the inner side of the northern jetty. As the flood decays (Figure 16) the sediment supply from the sea decreases and the picture remains roughly the same, with smaller values of concentration. In conclusion, the dominant effect of sand resuspension by the action of breaking waves during storms is simply to store sediments in the near-inlet region. 7. An Estimate of the Role of Wash Load [47] It is appropriate at this stage to go back to our main, as yet unresolved, question: how to reconcile the current estimate of the yearly loss of sediments from Venice Lagoon based on field observations (around 500,000 m 3 ) with the Table 1. Frequency Distribution of Wind Events Recorded in Venice Lagoon Speed, m/s Percent Events of 19

15 Figure 17. (a) Wind speed V w plotted versus the significant wave height H s measured in some shoals of Venice Lagoon. (b) Depth-averaged sediment concentration C avg [ppm] measured in the same shoals plotted versus the significant wave height H s. Lines of best fit (solid lines) and 95% confidence interval lines (dashed lines) are also plotted. Data were kindly made available by MAV-CVN [1992]. much smaller estimate (an order of magnitude smaller) of the yearly loss of sand through the inlets obtained from our calculations. [48] The resolution of the latter apparent contradiction lies in the appreciation that wind storms are able to resuspend very fine material in the shallow regions of the lagoon. Moreover, fine material is made available through the progressive retreat of the banks of salt marshes, as a result of shoal deepening and bank collapse. A part of these sediments is fine enough to be transported by tidal currents as wash load : in other words, they do not settle in the lagoon and are carried by ebb currents to the inlets where they are lost due to flow asymmetry. In fact, in the ebb phase the presence of inlet jetties gives rise to the formation of an unsteady jet with a pair of counter rotating vortices, each driven by the induced velocity of the other. This mechanism, observed in the laboratory [Blondeaux et al., 1982; Tambroni et al., 2005a] and in the field, is such that the exported water reaches distances of the order of tens of inlet widths, where it merges into the littoral currents and abandons the inlet region. On the contrary, the sea region feeding the nearly irrotational flood flow has a scale of the order of the inlet width and extends to regions on the sides of the inlet, which are not affected by the ebb jets. Hence the amount of wash load reentering the inlets at each cycle can be estimated as rather small (of the order of few percents). [49] A reliable estimate of the amount of sediments lost to the sea due to the latter process would require a careful analysis of the mechanism of wind driven sediment resuspension in shallow flows along with extensive field observations. However, the feasibility of the above interpretation can be at least qualitatively substantiated. In fact, let us consider particles of sizes ranging about, say, few tens of microns: they are characterized by settling speeds averaging approximately a fraction of a mm/s, hence these particles need many hours to deposit and may well reach the inlet to be lost into the sea by the ebb currents. A simple calculation of the order of magnitude of the volume of sediments which can leave the Venice Lagoon owing to the above processes may be performed considering the data reported in Table 1 for the annual frequency-speed distribution of wind events recorded in Venice Lagoon [MAV-CVN, 1992]. [50] The field measurements reported in Figure 17a allow us to associate to each wind event an estimate of a corresponding significant wave height in the shoals as well as a value for the depth-averaged sediment concentration (Figure 17b). It is not surprising that the latter plots show considerable scatter as the shoals are characterized by variable depth and the effectiveness of wind action depends also on the available fetch which may vary depending on wind direction. Hence, bearing in mind that these data can only allow rough estimates, we evaluate the annual mean volumetric concentration of sediments in the shoals of Venice Lagoon which turns out to be typically of the order of 20 ppm. [51] The above field measurements have provided a further important piece of information, namely a typical grain size distribution of sediments resuspended in the shoals (Figure 18). This distribution is characterized by a d 20 ranging about 20 mm. If we then assume that all the particles characterized by settling speeds smaller than, say, a fraction of one mm/s are unable to settle inside the lagoon and are lost to the sea, we may somewhat arbitrarily set at 20 mm the maximum size of wash load particles. Then, the annual mean volumetric concentration of wash load turns to be of the order of 5 ppm. Let us finally consider the water volume exchanged annually with the sea. Figure 19 shows Figure 18. A typical grain size distribution of the sediments suspended in the shoals of Venice Lagoon (data kindly made available by Magistrato alle Acque di Venezia and Consorzio Venezia Nuova). Note that almost 20% of the total amount of the suspended sediments have a diameter smaller than about 20 mm. 15 of 19

16 Figure 19. Ebb water volume exchanged with the sea in the period plotted as a function of the tidal amplitude exceeded by the corresponding events. that this quantity, calculated through our model for the period as a function of the tidal amplitude exceeded by the events considered, is of the order of m 3 /yr. It then appears that sediment volumetric concentrations of the order of 5 ppm would be sufficient to drive a yearly loss of very fine sediments of the order of 10 6 m 3. This estimate must be taken with great caution as it is based on highly scattered field data (the confidence interval of our estimated average concentration is 1 30 ppm). Moreover the choice of the size of 20 mm can only be taken as indicative: larger particles resuspended in shoals close to the inlets may well behave as wash load, while particles coming from the innermost shoals may be able to deposit before reaching the inlets. [52] However, in spite of its highly qualitative nature, the above estimate suggests a likely conclusion: in order to reduce the amount of sediments exported by the lagoon, the major issue to be confronted is the resuspension of sediments by severe wind events acting on the shoals, which have deepened considerably in the last century. Reintroducing at least part of the fluvial sediments carried by rivers originally discharging into the lagoon (notably the Brenta river) might give some contribution and would be worth investigating. [53] One last point deserves some attention: one may wonder what role would be played by the construction of the gate barriers designed to protect Venice Lagoon in relation to the exchange of sediments through the inlets. Figure 20. (a) Percentage volume of sand lost to the sea and (b) percentage ebb water volume exchanged with the sea in the period plotted as a function of the maximum free surface elevation exceeded by the corresponding events. 16 of 19