Laboratory observations of the morphodynamic evolution of tidal channels and tidal inlets

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 110,, doi: /2004jf000243, 2005 Laboratory observations of the morphodynamic evolution of tidal channels and tidal inlets N. Tambroni, M. Bolla Pittaluga, and G. Seminara Department of Environmental Engineering, University of Genova, Genova, Italy Received 28 September 2004; revised 30 May 2005; accepted 21 June 2005; published 12 November [1] This paper tackles the problem of morphodynamic equilibrium of tidal channels and tidal inlets. We report a laboratory investigation of the process whereby an equilibrium morphology is established in a tidal system consisting of an erodible channel connected through an inlet to a tidal sea. Observations suggest that a morphodynamic equilibrium is eventually established both in the inlet region and in the channel. The latter exhibits a weakly concave bed profile seaward, a weakly convex profile landward, and the formation of a beach close to the landward end of the channel. A second set of observations concerns the formation and development of both small- and large-scale bed forms. In particular, small-scale forms are found to develop in the channel and in the basin, while larger-scale forms, i.e., tidal bars, develop in the channel. A last observation concerns the formation of an outer delta in the sea basin. Results concerning the long-term equilibrium of the bed profile in the channel compare fairly satisfactorily with recent theoretical results. The nature and characteristics of the observed small-scale forms appear to be consistent with theoretical predictions and field observations concerning fluvial ripples and tidal dunes; bars show features in general accordance with recent results of a stability theory developed for tidal bars. The hydrodynamics of the inlet region exhibits a strongly asymmetric character, as observed in the field and predicted in early theoretical works, while the overall characteristics of the outer delta conform to available empirical relationships. Citation: Tambroni, N., M. Bolla Pittaluga, and G. Seminara (2005), Laboratory observations of the morphodynamic evolution of tidal channels and tidal inlets, J. Geophys. Res., 110,, doi: /2004jf Introduction [2] Morphological processes occurring in tidal channels and tidal inlets are of obvious practical relevance for the management of lagoons and estuaries. In particular, the issue of whether such complex systems may reach an equilibrium state has lately attracted considerable attention in the scientific community [Schuttelaars and de Swart, 1996, 2000; Lanzoni and Seminara, 2002]. (Some aspects of the theoretical investigation of Lanzoni and Seminara [2002] are briefly reviewed in Appendix A.) In contrast, attempts to pursue the same goal on the basis of controlled laboratory observations are not known to the present authors. This is not surprising as the timescale of morphodynamic evolution is typically much larger than the hydrodynamic timescale. Moreover, the evolution becomes slower and slower as equilibrium is approached: hence as it will appear from the scaling arguments discussed in Appendix A and from the experimental observations reported below, the time required to approach conditions sufficiently close to equilibrium in the laboratory is of the order of days or weeks. [3] In spite of these complexities, controlled laboratory observations of the morphodynamic evolution of tidal Copyright 2005 by the American Geophysical Union /05/2004JF systems are worth pursuing. In fact, controlled experiments with simple boundary conditions provide a check of some of the main mechanisms which drive the evolution process, a goal quite difficult to achieve on the basis of field observations whose interpretation is generally complicated by the large scale of the processes, the more irregular natural geometries and the simultaneous presence of a variety of features whose role cannot be readily isolated. [4] It is the aim of the present work to perform qualitative and quantitative observations of the temporal development of bed morphology in a model of a tidal channel connected to an adjacent sea through a tidal inlet and to attempt clarifying the observed phenomena through a comparison with recent theoretical findings on various aspects of the process. [5] The plan of the paper is as follows. The next section is devoted to a brief description of the experimental apparatus and the experimental conditions. Sections 3, 4, 5 and 6 report experimental results compared with theoretical predictions on the morphodynamic evolution of channel, inlet, small-scale and large-scale forms (bars), respectively. Section 7 concludes the paper with some final remarks. 2. Experimental Apparatus [6] Experiments must reproduce various ingredients of a real tidal system. The first ingredient is sediment transport. 1of19

2 Figure 1. Grain size distribution of crushed hazelnuts employed as sediments in the physical model. Below, we assume cohesionless sediments, an assumption suitable, e.g., for the larger channels of Venice Lagoon (one of the environments we are particularly interested in). Moreover, we assume the size of sediments in the prototype to be in the sand-silt range, such that transport occurs both as bed load and as suspended load. The second ingredient is the geometry of tidal channels: they are typically landward convergent and meandering. We will reproduce channel convergence but ignore channel curvature. The latter feature affects the lateral equilibrium of the bed [Solari et al., 2002] but presumably it does not crucially affect its longitudinal equilibrium. The third ingredient to be reproduced is the forcing effect of the sea at the channel inlet which determines the hydrodynamic boundary condition for the tidal wave propagating through the channel. The ability of the sea to exchange sediments with the channel through the inlet is determined by the geometry of the latter. The sediment concentration in the far field, which is in turn affected by the wave climate, may also play some role which cannot be reproduced and will be disregarded in the present experiments. A further ingredient is important in nature, namely the presence of tidal flats adjacent to the channel: we will leave aside the analysis of this feature, which affects both the hydrodynamics of tide propagation and the sediment balance in the main channel. While it will not be too hard to model the former effect in future extensions of the present work, the exchange of sediments between the channel and the adjacent flats is crucially dependent on particle resuspension typically driven in nature by the action of wind waves, a delicate mechanism which is not easy to reproduce in the laboratory. Finally, the presence of small- and large-scale bed forms must also be included in the experiments as they may affect the net transport of sediments and consequently the establishment of an equilibrium morphology. [7] Taking into account the constraints posed by the cited requirements, experiments were carried out on a large indoor platform above which a straight channel was built using a plastic material. The channel was closed at one end and connected at the other end to a rectangular basin representing the sea. Finally, an oscillating discharge was supplied to the basin from a tank where the apparatus for tide generation was installed. The latter consisted of a cylinder set in motion by a piston held by a steel frame and controlled by an oleodynamic mechanism driven by a control system, which generated the desired law of motion. Such a law was first predicted theoretically by solving the differential equation A* dh* dt* ¼ W* c ðz* Þ dz* dt* þ Q* channel; where A* is the area of the free surface of the basin, W* c is the area of the intersection between the floating cylinder and the free surface of the basin, z* is the elevation of the cylinder axis, t* is time and Q* channel is the flow discharge entering the basin through the channel inlet. Equation (1) was solved numerically for z*(t*) assuming the desired law h*(t*) for the free surface oscillation in the basin, namely, ð1þ h* ðt* Þ ¼ a* 0 cosð2pt*=tþ; ð2þ with T tidal period and a* 0 amplitude of the tidal wave. [8] Note that small variations of a* 0 may be experienced throughout the experiments due to the morphodynamic evolution of the system. During the experiments, we have continuously monitored, by means of a sound probe, the amplitude of the oscillations at the inlet and carefully checked that such variations kept small enough (less than 10%). Figure 2. Sketch of the experimental apparatus. 2of19

3 Table 1. Hydrodynamic and Geometrical Parameters of the Laboratory Model Experiment 1 Experiment 2 L* c, m L* s,m B* 0,m B* s, m L* b,m inf 31 D* 0, m U* 0, m/s C T, s d* s, mm r s, kg/m V* od,m P*, m R/S J R p T 0 */T [9] A layer of cohesionless granular material of sufficient thickness was laid on the bottom of both the flume and the basin, in order to have a flat bottom as initial condition for the experiments. The granular material was chosen light enough to be entrained into suspension throughout most of the tidal cycle with the values of friction velocity typically generated in the present experiments. The final choice was to use crushed hazelnut shells characterized by a density of 1480 kg/m 3 and median grain size d* s = 0.31 mm. Tests were also made to ascertain the absence of cohesive effects. The grain size distribution of this material was measured by sieving and is reported in Figure 1. [10] Two experiments were performed using different geometrical settings for the channel and for the basin as well as different hydrodynamic conditions. The first experiment was characterized by a straight channel with constant width and sharp inlet, with the walls of the channel joining the basin walls at 90. The main changes introduced in the second experiment concerned the shape of the channel, now convergent, and the shape of the inlet, modified by smoothing out the junctions between the walls of the channel and the basin (Figure 2). The latter feature was introduced in order to reduce the scour holes at the inlet which were so deep as to reach the underlying platform in the first experiment. The new shape of the inlet in the second experiment decreased the strong flow acceleration induced by streamline convergence during the flood phase. [11] The initial hydrodynamic characteristics were chosen such to generate a flow field able to mobilize sediments both as bed load and as suspended load throughout a large part of the tidal cycle. The parameters chosen for both the experiments are reported in Table 1 and are related to the values of the two selected prototypes (Rotterdam Estuary and Dei Bari channel of the Venice Lagoon) following the procedure discussed in Appendix A (Table 2). The initial temporal distribution of the average flow speed measured at the inlet gave rise to corresponding distributions for the Shields stress J and the Bagnold parameter N(=u * /W s ) such that the stream was indeed observed to mobilize sediments throughout most of the tidal cycle. [12] During both the experiments the water surface elevation was continuously monitored by means of ultrasound probes in five equidistant cross sections along the channel. The probes have a resolution of 0.2 mm and are able to perform five measurements per second. A whirl probe was used to measure the local velocity in the absence of sediments during the preliminary tests; for the rest of the experiments the surface velocity field in the basin was measured using a particle tracking technique. In particular, the free surface was seeded with polyethylene pearls 3 mm in size. The position of these floating tracers were then recorded using a digital camera, at a frame rate of 18 Hz and with an image size equal to Table 2. Hydrodynamic and Geometrical Parameters Characterizing the Laboratory Model in Experiment 1 Related to a Prototype Satisfying the Scaling Rules (A19) (A22) Exactly and to a Prototype Reproducing the Hydrodynamics (i.e., Equations (A20) and (A21)) Exactly and Sediment Transport (i.e., Equations (A19) and (A22)) Only Approximately a Model Prototype a (Rotterdam) Scale Factor a Prototype b (Venice Lagoon) Scale Factor b L* c, m ,210 (37,000) l c = (9826) l c = 400 L* b,m inf inf (56,000) l b = 1500 inf (6850) l b = 400 D* 0, m (11.5) l = (5.0) l =60 U* 0, m/s (0.48) j = (0.20) j =1.0 C (21) c = (28.5) c =2 T, s ,200 (43,200) t = ,200 t = 240 d* s, mm d = d = 0.17 r s, kg/m (2650) s = (2650) s = 3.44 V* od,m 3 l 2 c l/d 3 = l 2 c l/d 3 = P*, m l c ljt = l c ljt = (0.09) e = (0.14) e = 0.6 K 0 0 (0.66) l c /l b = 1 0 (1.43) l c /l b =1 R/S (0.65) el c /(lc 2 ) = (0.34) el c /(lc 2 )=1 J j 2 /(c 2 sd) = j 2 /(c 2 sd) = 0.44 R p s 1/2 d 3/2 = s 1/2 d 3/2 = 0.13 T 0 */T l c l/s 1/2 d 3/2 = l c l/s 1/2 d 3/2 = 792 a The values of the Rotterdam Estuary [Lanzoni and Seminara, 1998] and of the Dei Bari channel of the Venice Lagoon [Lanzoni and Seminara, 2002] are also reported. 3of19

4 Figure 3. Comparison between the cross-sectionally averaged bed profiles observed in experiment 1 at different times and those calculated by Lanzoni and Seminara [2002]: (a) t* = 80T; (b) t* = 200 T; (c) t* = 250 T; (d)t* = 370 T; (e)t* = 1000 T; and (f) t* = 2000 T pixel, covering a 1.2-m-long reach of the basin. Postprocessing the recorded images by means of a software performing image analysis, enabled us to identify the tracers, reconstruct their trajectories and estimate their velocity. In order to detect the migration of small-scale bed forms, the bottom evolution at a fixed location of the channel axis of a cross section 12 m far from the inlet was monitored in time for about 4 hours using a profile indicator. The latter consists of a probe which is continuously maintained at a fixed distance of 0.5 to 2 mm from the bed level and is able to follow and record temporal changes of the bottom elevation. Moreover, the bed topography in the channel and the basin, was scanned at different times (after cycles in experiment 1, and after cycles in experiment 2) by means of a laser system (60 mm resolution). Scanning was performed along the channel in a sequence of cross sections 5 cm distant from one another. At each cross section measurements of bottom elevation were performed every 0.5 cm. The mean longitudinal bed profile was then obtained by performing a lateral averaging of bottom elevation at each cross section. In order to allow a sufficient accurate analysis of small-scale bed forms, at each stage of the experiment a more detailed survey of bottom elevation was performed, reducing the distance between adjacent cross sections to 1 cm. Flow depths were readily calculated once the free surface elevation and the bottom elevation were known. 3. Evolution of the Bed Profile [13] We now first outline some significant observations on the morphodynamic evolution of the bed profile and then discuss its features in the light of recent theoretical results of Lanzoni and Seminara [2002]. 4of19

5 Figure 4. Net volume of sediments exchanged between the channel and the adjacent basin V e scaled by the volume of water initially stored in the channel V 0 (dotted line) and ratio between V e and the volume of sediments deposited in the landward portion of the channel V d (solid line) plotted at different times for experiment 1. The net volume of sediments exchanged between the channel and the adjacent basin V e scaled by the volume of water initially stored in the channel V 0 calculated by Lanzoni and Seminara [2002] (dash-dotted line) is also reported Experimental Results [14] Let us first refer to the experiment 1. The major observation was that sediments were scoured in the seaward portion of the channel, driven landward and deposited in the inner part of the channel. This mechanism is associated with the flood dominant character of the flow field in the initial stage of the morphodynamic evolution process. In fact, the temporal distribution of the cross-sectionally averaged flow speed is increasingly asymmetric proceeding from the inlet landward, with the ebb phase longer than the flood phase and the flood maximum invariably larger than the ebb maximum. As a result, a fairly sharp front develops in the bed profile and migrates landward. Once the front reaches the inner end of the channel, a dry and wet region forms and leads quite rapidly (say after 400 cycles) to the development of a shore. The temporal evolution of the cross-sectionally averaged bed profile is plotted in Figure 3, where D* 0 is the uniform flow depth at the beginning of the experiment, h* is the local and instantaneous value of the average elevation of the mobile bed and x* is the landward oriented longitudinal axis with the origin located at the inlet of the channel. The apparent noise displayed by such profiles is associated with the presence of small-scale bed forms, while larger oscillations are the laterally averaged expression of bars. [15] Comparison between the bed profiles at 1000T and 2000T shows that the pattern reached at the end of the experiment was likely not far from equilibrium. However, approaching equilibrium, the net sediment flux in a tidal cycle progressively decreases, the evolution process increasingly slows down so that further improving the degree of closeness of the bed profile to equilibrium would require prohibitively large times even in the laboratory. In fact, though weak, a net sediment flux was still present at the final stage of the experiment, as shown by the weak migration of small-scale bed forms. The final bottom profile was slightly concave seaward and slightly convex landward, consistently with field observations concerning tidal channels of Venice Lagoon reported by Lanzoni and Seminara [2002]. [16] The temporal evolution of the net volume of sediments exchanged between the channel and the adjacent basin V e scaled by the volume of water initially stored in the channel V 0 is shown in Figure 4. Note that the net exchange was positive (from the basin into the channel) in the first half of the experiment, then it changes sign to keep negative until the end of the experiment. Also shown is the temporal evolution of V e scaled by the volume of sediments deposited in the landward portion of the channel V d. The latter quantity describes to what extent deposition has been driven by redistribution of sediment within the channel. In fact, a vanishing value of V e /V d implies that deposition is wholly driven by sediment redistribution, while the opposite limit V e /V d! 1 implies that deposition is wholly due to sediment supply from the sea. The observed bed evolution is partly the result of a rearrangement of the sediments stored in the channel bed, which degraded seaward and aggraded landward; while part of the evolution was driven by the net exchange of sediments between the channel and the basin. However, in the final equilibrium configuration only 10% of the total aggradation originates from the net exchange of sediments with the adjacent basin. [17] The evolution of the bed profile in the second experiment (Figure 5) was somewhat similar in many respects to that observed in the first experiment, even though it displayed a more intense deposition in the landward portion of the channel and the fluctuations of the longitudinal mean bed profile had longer amplitude and wavelength. The temporal evolution of the net volume of sediments exchanged between the channel and the adjacent basin V e scaled by the volume of water initially stored in the channel V 0 is shown in Figure 6. In this case the net exchange was negative (from the channel to the basin) until the end of the experiment, except in the very initial stage where it was positive. In analogy with Figure 4, it is also reported the temporal evolution of V e now scaled by the volume of sediments eroded in the seaward portion of the channel V r. The latter quantity describes to what extent erosion has been driven by redistribution of sediment within the channel. In fact, a vanishing value of V e /V r implies that erosion is wholly driven by sediment redistribution, while the opposite limit V e /V r! 1 implies that erosion is wholly due to sediment exchange with the sea. Unlike the first experiment, the final equilibrium is now significantly affected by such an exchange, as it drives 55% of the total bed degradation Discussion [18] Attempts to determine the long-term equilibrium of the cohesionless bed of tidal channels on a theoretical basis have recently been proposed [Schuttelaars and de Swart, 1996, 2000; Lanzoni and Seminara, 2002]. Somewhat related work, concerning the equilibrium of cohesive intertidal mudflats, is due to Pritchard et al. [2002]. As discussed 5of19

6 Figure 5. Comparison between the cross-sectionally averaged bed profiles observed in experiment 2 at different times and those calculated by Lanzoni and Seminara [2002]: (a) t* =20T;(b)t*=60T; (c)t*= 100 T; (d)t* = 500 T; (e)t* = 1000 T; and (f) t* = 5000 T. in Appendix A, Lanzoni and Seminara [2002] investigated the hydrodynamics and morphodynamics of a straight, convergent channel closed at one end and connected at the other end with a tidal sea. They solved numerically the basic one-dimensional (1-D) mass and momentum conservation equations coupled with the evolution equation of the bed interface [Exner, 1925]. [19] The observed amplitudes and phases of the tidal wave propagating in the channel compare fairly satisfactorily with theoretical predictions as shown in Figure 7. Similarly, Figures 3 and 5 show a detailed comparison between the bed profiles observed in experiment 1 and experiment 2 and the corresponding profiles calculated by the approach of Lanzoni and Seminara [2002] at different times. Various observations arise from the latter comparison: (1) though the general trend of the evolution process seems adequately reproduced in the physical model, however the observed bed profile tends to equilibrium faster than the calculated one: such discrepancy arises from the presence of a net exchange of sediments between the channel and the basin, driven by the inlet hydrodynamics, which is more intense than that predicted by the 1-D model, obviously unable to reproduce the effect of streamline convergence occurring in the flood phase; (2) the above discrepancy tends to reduce in the course of the experiment, and this is reflected in the fact that the equilibrium configuration predicted theoretically is fairly close to the experimental observations, except in a neighborhood of the channel inlet where a pronounced localized scour is driven by streamline convergence and by 3-D effects acting at the inlet edges. [20] This suggests that a sound model of the morphodynamic evolution of tidal channels must include an at least 2-D (possibly 3-D) description of the flow field in the inlet region. A simpler 1-D model of the kind proposed by Lanzoni and Seminara [2002] can be employed to predict 6of19

7 Figure 6. Net volume of sediments exchanged between the channel and the adjacent basin V e scaled by the volume of water initially stored in the channel V 0 (dotted line) and ratio between V e and the volume of sediments eroded in the seaward portion of the channel V r (solid line) plotted at different times for experiment 2. the final equilibrium bed profile throughout most of the channel, being aware that the scour depth at the inlet is underpredicted. 4. Morphodynamic Evolution of the Inlet Region 4.1. Experimental Results [21] In Figure 8 the surface velocity measured in the basin during the initial phase of the first experiment is shown. The flow field in the inlet region turns out to be highly asymmetric throughout the tidal cycle: during the flood phase, the channel effectively acts as an unsteady sink which gives rise to a nearly irrotational 2-D flow pattern; during the ebb phase, vorticity is continuously shed from the separation points at the sharp edges of the inlet and gives rise to an unsteady turbulent jet characterized by the formation of a large-scale recirculating cell consisting of a pair of counterrotating vortices which leaves the generation area under the effect of the induced velocity that each vortex determines on the other. [22] In the present context our interest is concentrated on the exchange of sediments and on the mobile character of the bed. The latter has a strong effect on both the inlet hydrodynamics and on its morphodynamics. In fact, the strong seaward current that the counterrotating vortices generate close to the channel axis leads to an intense scour of the bed and to the generation of a submerged channel which progresses seaward until it reaches a quasi-equilibrium condition. Simultaneously, the sediments entrained by the scouring action of the vortices deposit laterally, giving rise to the formation of sort of submerged levees confining the submerged channel. This is shown in the Figures 9a and 9b where the bed elevation measured in experiment 1 at two different cross sections of the basin, located at increasing distances from the inlet, is reported. [23] Note that the bed evolution displays a somewhat oscillating and unstable character: this feature arises from the complex interaction between the large-scale coherent structures generated by the shear layer issued from the sharp edges of the inlet and the mobile bed. This is clarified by the sequence of topography fields measured at different times (Figure 10). An important feedback of the morphodynamic evolution of the inlet region is a significant reduction of the ebb-flood flow asymmetry: in fact, the formation of the submerged channel keeps the flood flow field more aligned with the channel axis. [24] The smooth, diverging shape of the inlet in experiment 2 and the associated diverging character of the streamlines gives rise to sediment deposition in the form of an elongated central bar located seaward and levees bounding the scoured region adjacent to the inlet. This is shown in Figure 11. The observed pattern corresponds to the so-called outer delta which forms in the sea region opposite to tidal inlets. The volume of sediments stored in the outer delta is reported in Table 1. Moreover, the shape of the inlet as well as the morphodynamic evolution of the bed described above caused a further reduction of the ebb-flood flow asymmetry Discussion [25] The problem of estuarine circulation has been the subject of several investigations starting from the work of Stommel and Farmer [1952]. Moreover, the fundamental issue concerning the conditions for inlet stability has been widely explored [Bruun, 1978; Eysink, 1990]. On the basis Figure 7. Comparison between experimental and theoretical [Lanzoni and Seminara, 2002] relative amplitudes and phases of the first overtide of the surface elevation at different cross sections along the channel after 570 cycles in experiment 1. 7of19

8 Figure 8. Surface velocity measured in the basin in the initial stage of experiment 1, when the bed was still flat, showing the asymmetry between the (left) flood and (right) ebb flow fields. of empirical observations, O Brien [1969] proposed a well known relationship between the minimum cross-sectional area at tidal inlets and the so-called tidal prism, later substantiated by further field observations [Jarrett, 1976] and theoretical work [e.g., Lanzoni and Seminara, 2002]. However, detailed theoretical analysis of the hydrodynamics of tidal inlets are not known to the present authors except for the early work of Blondeaux et al. [1982], who, assuming the bed to be fixed and flat, modeled this complex flow field as 2-D irrotational in the flood phase and as inviscid rotational during the ebb phase. Moreover, using vortex shedding techniques, the latter authors were able to reproduce successfully the formation of the recirculating cell, which was also observed in their experiments. The latter picture has implications for the inlet morphodynamics. In fact, flow asymmetry drives a net exchange of sediments between the inlet and the adjacent sea, depending on wave climate and littoral currents. [26] An empirical relationship correlating the volume of sediments deposited in the outer delta V* od to the tidal prism P* has been proposed by Bruun [1978] and reads V* od ¼ P* 1:23 ; ð3þ with V* od and P* both expressed in m 3. [27] As mentioned above, the present experiments indeed reproduce the formation of an outer delta: detailed mapping of the bottom topography then allows a comparison with the above relationship. Following a procedure similar to the one reported in Appendix A referring to experiment 1, the observed value of V* od, conveniently rescaled in the prototype ( m 3 ), has the correct order of magnitude when compared with the value ( m 3 ) predicted by equation (3). 5. Small-Scale Bed Forms 5.1. Experimental Results [28] Immediately after the start of the first experiment, small-scale bed forms were observed to develop throughout the whole length of the channel (Figure 12), while in experiment 2 they formed everywhere except for the regions close to the inlet and to the landward end of the channel, extending to most of the channel as the intensity of sediment transport decreased. [29] The wavelength L* r of small-scale bed forms in both experiments varied over the range of (15 20 cm) while their amplitude H* r attained values ranging about 1 cm. Both amplitudes and wavelengths exhibited some fluctuations along the channel, but showed very weak correlation with flow depth (Figure 13). Indeed the best fit relationships between wavelength and flow depth for experiment 2 in the initial stage can be written in the form L* r ¼ 0:62D* þ 0:01; ð4þ Figure 9. Evolution of the bed elevation measured in experiment 1 at two different cross sections of the sea basin, located at increasing distances from the inlet: (a) 2x*/B* = 1; (b) 2x*/B* = 3. 8of19

9 Figure 10. Topography fields measured in experiment 1 at different times in the inlet region using the laser scanner device. They show localized patches of scour and deposition driven by the interaction between the large-scale coherent structures generated by the instability of the shear layers shed by the inlet edges and the mobile bed. Bottom elevation is scaled by the initial mean flow depth. (a) t* =20T. (b) t* = 370 T. (c)t* = 2000 T. See color version of this figure at back of this issue. with a correlation coefficient r equal to Similarly, H* r ¼ 0:04D* þ 0:009 r ¼ 0:17: ð5þ [30] Small-scale bed forms displayed initially a quasi 2-D pattern replaced by complex 3-D patterns in the landward portion of the channel after the first hours of experiment. Moreover, they responded promptly to variations of the hydrodynamic conditions and of sediment transport. In particular, the damping effect of suspended load on bed form development was clearly detected, since during the flood phase they were washed out to form again during the ebb phase which was characterized by lower values of suspended load. Such a behavior suggests that, under the experimental conditions, the timescale of bed form growth and decay is much smaller than the tidal period. In other words, bed forms adjust, with some delay, to the instantaneous hydrodynamic conditions. In these terms they may be described as having a fluvial, rather than coastal, nature. [31] As the experiment proceeded and equilibrium was approached, the intensity of sediment transport decreased: 9of19

10 Figure 11. Evolution of the pattern of bed elevation measured in the sea basin at different times in experiment 2. (a) t* =20T. (b)t* =60T. (c)t* = 100 T. See color version of this figure at back of this issue. as a result, bed forms persisted throughout the tidal cycle. This allowed us to follow their development in detail and ascertain that they migrated landward during the flood phase and seaward during the ebb phase. However, the former prevailed over the latter, hence a net landward migration was detected, a feature readily explained as a consequence of the flood dominant character of tide propagation. Figure 14, which shows the temporal development of bed elevation in the middle of a cross section located at a distance of 12 m from the channel inlet, displays the passage of four bed forms characterized by a net migration speed ranging about 0.4/0.5 m/hr. [32] An interesting feature displayed by the shape of these bed forms was the presence of a distinct correlation between the orientation of their asymmetry and the flood, or ebb, dominance of the tidal wave. This is clarified in Figure 15 which shows the bed elevation measured after 60 tidal cycles in experiment 2 along the channel axis in three different reaches of the channel, the first (Figure 15a) located in the ebb dominated portion of the channel, the third (Figure 15c) located in the flood dominated reach and the second (Figure 15b) in between the latter two reaches. Note that dominance was ascertained by examining the numerical solution for the velocity field Discussion [33] The reader will have noted that we have so far avoided to classify these small-scale bed forms as ripples 10 of 19

11 or dunes. This is because the unsteady character of the flow field and the distortion of the ratio morphodynamic over hydrodynamic timescales in the laboratory model makes such a classification neither easy nor generally agreeable. In fact, sound theoretical analyses of the formation of small-scale bed forms in tidal environments, which would help us interpreting the experimental observations, are unfortunately not available to our knowledge. There is, however, an extensive literature on field observations of tidal bed forms (in particular, Allen [1980], Aliotta and Perillo [1987], and Dalrymple and Rhodes [1995]), which is worth considering in the light of the present experimental results. [34] Let us first examine how our small-scale bed forms would be classified in a fluvial context. Of course, one would first need decide which stage of the tidal cycle should be considered as the formative stage. This is a difficult and possibly badly posed question, though attempts to answer it have been made [Boothroyd and Hubbard, 1975; Dalrymple et al., 1978; Rubin and McCulloch, 1980]. Before we engage ourselves in such a debate, let us note that both the empirically based bed form phase diagrams [e.g., Dalrymple and Rhodes, 1995, Figures 13.1A and 13.1B] as well as the most reliable theoretical results of Sumer and Bakioglu [1984] would indicate that the observed bed forms are ripples even referring to the maximum speed experienced in the tidal cycle. In particular, the formation criterion for fluvial ripples of Sumer and Bakioglu [1984] can be written in the form k s u * n < 57:5; ð6þ Figure 12. Picture of the small-scale bed forms pattern observed along the channel in the initial phase of experiment 1. where k s is the absolute bed roughness, which can be estimated as (2.5 d* s ) and u * is the friction velocity. [35] At incipient sediment transport, when ripples form, the friction velocity of experiment 1 can be estimated to be about m/s, hence the parameter (k s u * /n) is about 6.5, well within the range of the formation criterion (6). The most unstable wavelength L* r predicted by Sumer and Bakioglu [1984] is known to underestimate sharply the wavelength of ripples experimentally observed. In fact, this is found in our case: with the present values of the relevant parameters the theoretical value of L* r is equal to 165n/u *, and is about 2 cm to be compared with the experimental value of cm. Note that in the latter comparison the fact that the sediment employed in the experiments was lighter than quartz can be approximately accounted for assuming a somewhat smaller equivalent sediment size (about 0.1/0.2 mm (see Table 2)). [36] On the other hand, the observed bed forms are definitely not coastal ripples, namely those features which arise as a result of an instability of a sandy bottom, driven in Figure 13. Spatial distribution of the dimensionless wavelength L* r of (left) small-scale bed forms and (right) height H* r along the channel in the initial phase of experiment of 19

12 Figure 14. Temporal development of bed elevation at a cross section located at a distance of 12 m from the channel inlet in experiment 1 showing the passage of four smallscale bed forms migrating landward. coastal regions by the action of a gravity wave. In fact, the work of Blondeaux [2001] has clarified that the latter forms have wavelengths scaling with the amplitude of the fluid displacement at the edge of the wave boundary layer. In the present experiments the boundary layer fills the whole flow depth, hence the relevant fluid displacement would be of the order of tens of meters! [37] If one had to follow the suggestion of various authors [Boothroyd and Hubbard, 1975; Dalrymple et al., 1978; Rubin and McCulloch, 1980; Middleton and Southard, 1984] that the criteria developed for current bed forms also apply to tidal bed forms using appropriate effective values for the hydrodynamic properties, then our conclusion, based on the previous comparison should be that the observed bed forms are indeed of fluvial ripples type. However, we must recall that the morphodynamic process is much faster in the model than in the prototype. In particular, detailed observations suggest that, in the initial stage of the morphodynamic evolution process, bed forms form at incipient transport, grow and develop an asymmetry as the speed increases, but they disappear at the flow peak when transport in suspension is also at its peak. This behavior resembles strongly the dune behavior in unsteady flow [Fredsoe, 1979]. Such an interpretation would find support in the classical empirical diagram of Simons and Richardson [1961] which suggests that the bottom response to a variation of friction velocity would let the bed form regime undergo various stages: flat bed at very low speeds, incipient transport, then ripple formation, their development into dunes and finally their disappearance as flow and transport in suspension reach their peaks. [38] As regards the size of the observed bed forms, a comparison between Figure 15 and Dalrymple and Rhodes [1995, Figure 13.5] would suggest that they fall in the lower range of the dune behavior. The ripple index is roughly 15 20, a value appropriate to steep small dunes: their shape (see Figure 15) is weakly asymmetric, a characteristic shared by tidal dunes [Dalrymple and Rhodes, 1995, pg. 381]. As regards the very low correlation coefficient of the relationship between wavelength and flow depth (as well as between height and flow depth) this is also a feature of dunes observed in the field [Dalrymple and Rhodes, Figure 15. Profile of three sequences of small-scale bed forms measured in experiment 2 along the channel axis in (a) ebb-dominated, (b) symmetrical, and (c) flood-dominated reaches, showing the distinct correlation between the latter feature and bed form asymmetry. 12 of 19

13 Figure 16. Pattern of bed topography in the seaward half of the channel in experiment 1 at different times, showing the presence of alternate bars. Note that the cross-sectionally averaged bed profile has been filtered out. Bottom elevation is expressed in millimeters. (a) t* = 570 T. (b)t* = 1000 T. (c)t* = 2000 T. See color version of this figure at back of this issue. 1995, pg. 394]. The average wavelength of our observed bed forms does not experience any detectable change throughout the tidal cycle as well as during the morphological evolution of the bed profile. On the contrary, as already pointed out, bed form height does change significantly during the tidal cycle, at least in the initial phase of the experiment, when sediment transport is sufficiently intense. The reader then appreciates that most of the features of our observed bed forms are shared by tidal dunes. [39] The above discussion clarifies the difficulty one meets in classify the observed bed forms as ripples or dunes. However, in tidal environment such a distinction has not been thoroughly analyzed, which might suggest its moderate importance. 6. Bars 6.1. Experimental Results [40] In both the experiments bars formed after about 50 tidal cycles. In the first experiment bars spread initially throughout most of the channel (Figure 16), while in the second they initially concentrated in the middle reach (Figure 17). Indeed, an alternate riffle and pool pattern can clearly be detected in Figure 16a. Note that in the latter plots the average bed profile has been subtracted, while the contribution of small-scale bed forms has been retained. As the experiment proceeded the amplitude of bars H* BM decreased in the landward reach and increased seaward (Figures 16b and 16c). [41] Note that, as shown in Figure 18, the relative bar height (scaled by local mean flow depth) increases landward, an observation which is consistent with theoretical arguments (see below). The average bar wavelength was estimated to fall over the range of three to six channel widths. No significant bar migration was detected throughout the experiments Discussion [42] In a few recent papers [Seminara and Tubino, 1998, 2001] the mechanism which controls the formation of free bars in tidal channels was investigated. In particular, it turned out that the formation of tidal bars arises from an instability of the cohesionless bed, conceptually similar to the well known mechanism underlying the formation of fluvial bars. The unsteady, oscillatory character of tidal flow prevents bar migration, except for a weak propagation possibly driven by the presence of residual currents and/or asymmetry of the basic tidal wave [Garotta and Bolla Pittaluga, 2004]. Essentially, bars form whenever the aspect ratio of the channel exceeds a threshold value depending mainly on the intensity of sediment transport measured by the Shields parameter. In the tidal case the threshold value of the aspect ratio varies over the range of few units. The theory is also able to predict the most unstable bar wavelength, which is typically about few channel widths. 13 of 19

14 Figure 17. Pattern of bed topography in the seaward portion of the channel in experiment 2 at different times, showing the presence of alternate bars. The cross-sectionally averaged bed profile has been filtered out. Bottom elevation is expressed in millimeters. (a) t* = 200 T. (b)t* = 1000 T. (c)t* = 2000 T. See color version of this figure at back of this issue. [43] Note that, in the theory of Seminara and Tubino [2001] the basic state was a purely oscillatory flow field, i.e., spatial variations associated with the tidal wave were ignored. This is consistent with the spatial scale of bars being much smaller than the tidal wavelength as well as channel length. In other words the basic state is slowly varying in space at the scale of bars. Fully nonlinear numerical solutions of bar development in channels with finite length have been proposed by Hibma et al. [2003, 2004]. However, it is difficult to perform any comparison of our experimental observations with the latter works, as the results are given there in dimensional form referring to specific tidal configurations. [44] The presence of bars in our experiments is consistent with the theoretical predictions of Seminara and Tubino [2001]. The observed wavelengths (about three channel widths) are somewhat smaller than theory predicts. In fact, Figure 19 shows the wave number selected as a function of the width to depth ratio for the values of physical parameters appropriate to experiment 2. The latter curve suggests that the critical value of bar wavelength should be about six channel widths (l ffi 0.5). Since the theory of Seminara and Tubino [2001] was linear, it was obviously unable to predict the bar amplitude. However, the observed landward increase of the relative bar height (Figure 18) is consistent with theoretical knowledge on fluvial bars. Indeed the relative amplitude of the latter bed forms has been shown [Colombini et al., 1987] to be an increasing function of the aspect ratio of the channel, a quantity which increases landward in our experiments. 7. Concluding Remarks [45] Results of the present experiment provide some understanding of the general picture whereby an equilib- Figure 18. Relative bar height scaled by the local mean flow depth in experiment of 19

15 Figure 19. Values of the dimensionless wave number l corresponding to the maximum bar growth rate for a given aspect ratio of the channel b calculated on the basis of the theory of Seminara and Tubino [2001], using the values of physical parameters appropriate to experiment 2 (d* s = 0.31 mm, r s /r = 1.48, D* 0 = 0.09 m, U* 0 = 0.3 m). The dimensionless wave number l is scaled by the inverse of half channel width. rium configuration may be established in a tidal channel communicating through an inlet with a tidal sea: the basic observation emerging from the experiment is the fact that the bed elevation established close to the inlet at equilibrium results from the readjustment of the whole profile. In other words, except for an immediate neighborhood of the inlet where the flow acceleration induced by streamline convergence during the ebb phase gives rise to a locally enhanced scour, the equilibrium cross section develops near the inlet as a result of nonlocal effects. It appears as though a relationship might exist between the length of a tidal channel at equilibrium and the flow depth established at the entrance, for given values of the hydrodynamic and sedimentologic parameters. This is a conjecture which will deserve to be systematically investigated in the near future. A strong interaction between flow field and bottom topography also occurs in the seaward portion of the inlet region as discussed in section 4. [46] Finally, bed forms have been observed to develop and evolve in the channel throughout the experiments. We have shown that small-scale bed forms exhibit features shared by fluvial ripples and tidal dunes while larger bed forms are alternate bars. [47] Though the above set of observations appears to provide a fairly complete picture which proves to be generally consistent, however the comparison between experimental observations and theoretical predictions has pointed out that further theoretical developments are needed to remove the inability of 1-D models to describe the effect of strong streamline convergence occurring during the flood phase in the near inlet region: a feature which leads to scour and net exchange of sediments through the inlet, largely in excess of those predicted by the 1-D model. The recent work of Tambroni et al. [2004] suggests that significant improvement can be achieved in this respect by coupling the 1-D model for the channel with a 2-D model for the inlet. [48] Various features of the natural process disregarded in the present investigation will also require attention in the near future. In particular one may reasonably wonder whether channel equilibrium may still exist in the presence of intertidal storage areas, which are known to generate a residual sediment flux through the mechanism described by Schijf and Schonfeld [1953]. Further investigations should analyze the possible influence of channel meandering [Marani et al., 2002; Solari et al., 2002] and the role of sediment sorting, which is known to lead to progressive fining in the landward direction, with the inner portions of lagoons often composed of cohesive sediments. The morphological equilibrium of salt marshes is a more delicate problem, which still awaits to be fully explored: here a complex interplay among various mechanisms occurs, namely the biological production of fine matter, the wind driven resuspension of sediments and the tidally induced exchange of sediments between the marshes and the adjacent shoals. [49] Finally, as regards the mechanisms controlling the morphological equilibrium of tidal inlets in the field, the recent preliminary results of Tambroni and Seminara [2004] concerning the inlets of Venice Lagoon, suggest that an important role is also played by the harmonic content of the tidal forcing and by the relative phase of overtides. The latter effect was demonstrated by means of a simple 1-D model. On the contrary, ascertaining the role of flow asymmetry and of the possible presence of significant littoral currents carrying sediments resuspended in the surf zone during storm events, requires the use of more refined 2-D models of the type developed by Tambroni et al. [2004]. Appendix A: Scaling Arguments and Similarity Rules [50] We now report briefly the derivation of the relevant dimensionless parameters governing the phenomenon. This will help us clarify which scaling rules may be adopted to insure similarity with the real world. [51] Let us consider a straight tidal channel of length L* c (hereafter a star apex will denote dimensional quantities) and mean flow depth at the mouth of the estuary D* 0 (Figure A1). The channel is assumed to consist of cohesionless, nearly uniform sediments. We model the cross section as rectangular and assume the local channel width B* to vary along the landward oriented longitudinal axis x* according to the following classical exponential law [Langbein, 1964]: B* ¼ B* 0 exp x* ; ða1þ L* b where L* b is an e-folding length (hereafter called convergence length ) and B* 0 is the channel width at the inlet. In the absence of tidal flats adjacent to the main channel, the basic one-dimensional equations which govern mass and momentum conservation for the fluid phase and the 15 of 19

16 Figure A1. Sketch of the tidal channel and notations. evolution equation of the bed interface in dimensionless form then read as @t ðudþ KUD þ þ þ R UU j j S c 2 D Kq s ¼ 0: ða2þ ða3þ ða4þ In (A2), (A3), and (A4), t denotes time, D is the local flow depth, H is the water surface elevation relative to the still water level, U is the local velocity averaged over the cross section, c is a normalized Chezy coefficient, q s is the sediment flux per unit width and h is the local and instantaneous value of the average elevation of the mobile bed. Furthermore, we have set D* ¼ D* ðx*; t* ÞþH* ðx*; t* Þ; ða5þ and the variables have been scaled as follows: t* ¼ w* 1 t; D* ¼ D* 0 D; H* ¼ a* 0 H; U* ¼ U* 0 U; C ¼ C 0 c; ða6þ t* ¼T* 0 t; h* ¼ D 0 *h; p q* s ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi Dgd s * 3 q s ; x* ¼ L* c x : ða7þ Here w*, a* 0, D* 0, U* 0 and C 0 denote the angular frequency of the tidal wave, the wave amplitude of the forcing tide, the average flow depth at the initial state, a typical value of the cross-sectionally averaged flow speed and a characteristic flow conductance respectively, while L* c is the longitudinal scale relevant for the hydrodynamics. Note that, in the case of infinitely long estuaries treated by Lanzoni and Seminara [1998], this scale depends on the dynamic balance prevailing in the momentum equation for the particular tidal flow considered. In the present case, continuity suggests that the relevant scale is, indeed, the channel length L* c. Moreover, g is gravity, d* s is grain size taken to be uniform and D is the excess relative density (s 1) with s = r s /r and r, r s water and sediment densities respectively. Finally, T * 0 is a morphological timescale associated with the evolution of the bed profile, which reads T * 0 ¼ ð1 pþ D 0 *L c* p ffiffiffiffiffiffiffiffiffiffiffiffiffi ; ða8þ Dgd s * 3 with p being sediment porosity. [52] The dimensionless parameters appearing in (A2), (A3), and (A4) are defined as follows: ¼ a* 0 ; K¼ L c* D* 0 L b * ; S¼ 2pL 2 c* ; L 0 * R¼ L c* C0 2D 0 * S; t ¼ 1 T : ða9þ 2p T * 0 16 of 19

17 p Here L* 0 (=2p ffiffiffiffiffiffiffiffi gd 0 * /w*) denotes the length of an inviscid small amplitude tidal wave propagating on constant flow depth D* 0. The parameter K is a relative measure of the effect of channel convergence on mass conservation; the parameters S and R weigh the relative importance of local inertia and friction with respect to gravity. Note that the above results have been obtained determining the velocity scale U* 0 by balancing the second and third term of the continuity equation (A2), to find U* 0 = w*l* c. Also note that the parameter t involving the ratio between the tidal period T and the morphodynamic timescale T * 0 is typically very small (about 10 3 in the experiment, much smaller in nature): hence the first term of equation (A2) can be safely ignored. Estuaries can be classified as strongly or weakly dissipative depending on whether the ratio R/S is much larger or much smaller than one. [53] In order to complete the formulation, closures are needed to evaluate the total load q s. As regards the bed load component of q s, fairly established semiempirical relationships are available in the fluvial literature which may be extended to tidal channels simply relating the load to the instantaneous and local values of the Shields stress J [Shields, 1936] and of the particle Reynolds number R p [Yalin, 1972], respectively defined as follows: J ¼ t 0 * ðr s rþgd s * ða10þ pffiffiffiffiffiffiffiffiffiffiffiffiffi R p ¼ Dgd* s 3 ; n ða11þ t* 0 being the local and instantaneous value of the average bottom stress and n the kinematic viscosity of the fluid phase. [54] The evaluation of the suspended load, strictly requires the solution of a classical advection-diffusion equation for the sediment concentration. However, the dominant balance in the latter equation in a slowly varying context (where advection is small) involves gravitational settling and vertical turbulent diffusion [Bolla Pittaluga and Seminara, 2003]: the dimensionless parameter governing such balance is the so-called Rouse number Z, which reads Z ¼ W s ku * ; ða12þ having denoted by k, W s and u * the von Karman constant, the settling speed of sediment particles and the friction velocity respectively. [55] Scaling factors for length, flow depth, velocity, time, convergence length, grain size, relative density, flow conductance and relative amplitude of the tidal oscillation can be introduced as follows (stars for dimensional quantities are omitted below): l c ¼ L c L 0 c ; l ¼ D 0 D 0 0 ; j ¼ U 0 U 0 0 d ¼ d s ð ds 0 ; s ¼ r s=rþ 1 r 0 s =r ; c ¼ C 0 1 C0 0 ; t ¼ t t 0 ; l b ¼ L b L 0 ; ða13þ b ; e ¼ 0 ; ða14þ 17 of 19 where the apex refers to the model (throughout Appendix A the word model is referred to the laboratory model). [56] Note that the model considered herein is necessarily distorted as it proves impossible to find cohesionless natural sediments as small as it would be required if the same scaling factor characterizing the depth scale were used for sediment size. The size distortion will be partially compensated by a simultaneous distortion of the relative density of sediments, achieved by choosing a granular material much lighter than quartz. In the present experiments we have employed crushed hazelnuts (s = 1.48, hence s = 3.44). [57] Let us now discuss the scaling rules appropriate to model the hydrodynamics and morphodynamics of tidal channels. We note that gravity, driving the tidal wave, may be balanced by local inertia, friction or both depending on the ratio R/S being much smaller than one, much larger than one or of order one. Moreover, convective inertia is O() smaller than local inertia. Recalling that the velocity scale U* 0 reads w*l* c, the continuity equation (A2) suggests that kinematic similarity is achieved provided j ¼ el c t l b ¼ l c : ða15þ In order to achieve dynamic similarity we must impose that the ratio between friction and local inertia attains the same value in the model and in the prototype, hence R=S R 0 =S 0 ¼ el c lc 2 ¼ 1: ða16þ The latter condition is sufficient provided convective inertia (hence ) is small. [58] Equations (A15) and (A16) set the scaling rules for tidal hydrodynamics under fixed bed conditions. Note that the slope of the free surface i (=(@H*)/(@x*)) is not conserved in general, as it must satisfy the scaling rule i i ¼ el S 0 l c S ¼ el c 0 t : 2 ða17þ [59] Strict sediment transport similarity requires conserving the Shields parameter J and the particle Reynolds number R p, hence J J 0 ¼ j2 c 2 sd ¼ 1 R p R p 0 ¼ s 1=2 d 3=2 ¼ 1 : ða18þ It is easy to show that (A18) automatically insures that the Rouse number Z is also conserved while some distortion of advective effects is unavoidable in the model, a not too severe approximation in slowly varying flows. The relationships (A15), (A16), and (A18) can be rearranged in the form l c ¼ t e cs1=3 ; l ¼ el c c 2 ; j ¼ el c t ; d ¼ s 1=3 ; ða19þ ða20þ ða21þ ða22þ

18 which establishes the relationships between the physical characteristics of the model and of the prototype. Let us clarify this statement. [60] The periods employed in the model were constrained by the need to achieve sufficiently intense tidal currents able to mobilize sediments throughout most of the tidal cycle. We were then forced to use periods T lower than 200 s (t = 216). Assuming c = 2 (a reasonable assumption according to our experimental observations) and with the values of s (=3.44) and t (=240) appropriate to experiment 1, from the relations (A19) (A22) we find el c ¼ 724; l ¼ 181; j ¼ 3:02; d ¼ 0:67: ða23þ Once a scale factor l c has been chosen, equation (A23) sets the ratio between the relative amplitudes of the forcing tide in the model and in the prototype: for example choosing a value of l c about 1500 (as in Table 2), it follows that the dimensionless amplitude in the model is roughly twice as large as in the prototype (e = 0.48). In other words, nonlinearity of the tidal wave in the prototype is weaker than in the model. Note that with the values of the parameters chosen above, kinematic, dynamic and sediment transport similarities are fully satisfied and the features observed in the laboratory model are immediately related to those expected in a real tidal environment satisfying the scaling rules (A23). The Rotterdam estuary turns out to be an example of tidal channel well represented by our model. It is also worth to notice that the ratio between the morphological and the hydrodynamic timescale (T * 0 /T) is not conserved, even in the case of exact similarity, the morphodynamic evolution being (fortunately) much faster in the model than in the prototype. [61] The scaling rules can also be employed to reproduce the hydrodynamics (i.e., equations (A20) and (A21)) exactly and sediment transport (i.e., equations (A19) and (A22)) only approximately. An example may illustrate the above concept. If j = 1, t = 240 and e = 0.6 (assumptions appropriate to average channels of Venice Lagoon) then (A21) gives l c = 400. With c 2 = 4 then (A20) gives the value of l(=60) which ensures that scaling rules for the hydrodynamics are exactly reproduced. On the contrary, with d =1/6 (A18) gives J = 0.44J 0 and R p = 0.13R 0 p: hence in the prototype the role of transport in suspension is larger than in the model and the total transport is not exactly reproduced. However, the only effect of the latter approximations is to alter the ratio between the morphodynamic timescale in the model and in the prototype in a predictable fashion. [62] Acknowledgments. The construction of the experimental apparatus employed in the present experiment was funded by the Italian Ministry of University and of the Scientific and Technological Research under the National Project Idrodinamica e morfodinamica di ambienti a marea. Support has also come from the University of Genova and from CORILA (Consorzio per la Gestione del Centro di Coordinamento delle Attivitá di Ricerca inerenti il Sistema Lagunare di Venezia) (initiated under the research program , Linea 3.2 Idrodinamica e Morfodinamica and Linea 3.7 Modelli Previsionali and terminated under the research program , Linea 3.14 Modelli di erosione e sedimentazione nella laguna di Venezia and Linea 3.18 Tempi di residenza e dispersione idrodinamica nella laguna di Venezia). This work is also part of the Ph.D. thesis of Nicoletta Tambroni, to be submitted to the University of Genova in partial fulfillment of her degree. The authors are grateful to L. Solari, C. Zucca, V. Garotta, and P. Latona for providing assistance in the development of the experimental tests. Finally, thanks are due to M. Toffolon for providing numerical assistance. References Aliotta, S., and G. M. E. 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Southard (1984), Mechanics of Sediment Movement, 2nd ed., Short Course Notes, vol. 3, 401 pp., Soc. for Sediment. Geol., Tulsa, Okla. O Brien, M. P. (1969), Equilibrium flow areas of inlets on sandy coast, J. Waterw. Harbours Coastal Eng. Div. Am. Soc. Civ. Eng., 95, Pritchard, D., A. J. Hogg, and W. Roberts (2002), Morphological modelling of intertidal mudflats: The role of cross-shore currents, Cont. Shelf Res., 22, Rubin, D. M., and D. S. McCulloch (1980), Single and superimposed bedforms: A synthesis of San Francisco Bay and flume observations, Sediment. Geol., 26, Schijf, J. B., and J. C. Schonfeld (1953), Theoretical considerations on the motion of salt and fresh water, in Proceedings of the Minnesota International Convention, pp , Am. Soc. of Civ. Eng., Tulsa, Okla. Schuttelaars, H. M., and H. E. de Swart (1996), An idealized long-term morphodynamic model of a tidal embayment, Eur. J. Mech. B, 15, Schuttelaars, H. M., and H. E. de Swart (2000), Multiple morphodynamic equilibria in tidal embayment, J. Geophys. Res., 105, 24,105 24,118. Seminara, G., and M. Tubino (1998), On the formation of estuarine free bars, in Physics of Estuaries and Coastal Seas, edited by J. Dronkers and M. Scheffers, pp , A. A. Balkema, Brookfield, Vt. 18 of 19

19 Seminara, G., and M. Tubino (2001), Sand bars in tidal channels. Part 1: Free bars, J. Fluid. Mech., 440, Shields, A. (1936), Application of similarity principles and turbulence research to bed-load movement, Mitt. Preussischen Versuchsanstalt Wasserbau Schiffbau, 26, Simons, D. B., and E. V. Richardson (1961), Forms of bed roughness in alluvial channels, J. Hydraul. Div. Am. Soc. Civ. Eng., 87, Solari, L., G. Seminara, S. Lanzoni, M. Marani, and A. Rinaldo (2002), Sand bars in tidal channels. Part 2: Tidal meanders, J. Fluid Mech., 451, Stommel, H., and H. G. Farmer (1952), On the nature of estuarine circulation, report, Woods Hole Oceanogr. Inst., Woods Hole, Mass. Sumer, B. M., and M. Bakioglu (1984), On the formation of ripples on an erodible bed, J. Fluid Mech., 144, Tambroni, N., and G. Seminara (2004), A simple model for the evaluation of the long term net exchange of sand through the inlets of Venice Lagoon, paper presented at the Atti XXIX Convegno di Idraulica e Costruzioni Idrauliche, Trento, 7 10 Sept. Tambroni, N., P. Stansby, and G. Seminara (2004), On the formation of outer-deltas at tidal inlets, paper presented at the Atti XXIX Convegno di Idraulica e Costruzioni Idrauliche, Trento, 7 10 Sept. Yalin, M. S. (1972), Mechanics of Sediment Transport, Elsevier, New York. M. Bolla Pittaluga, G. Seminara, and N. Tambroni, Department of Environmental Engineering, University of Genova, Via Montallegro 1, Genova, I-16145, Italy. (miki@diam.unige.it; sem@diam.unige.it; nicotam@diam.unige.it) 19 of 19

20 Figure 10. Topography fields measured in experiment 1 at different times in the inlet region using the laser scanner device. They show localized patches of scour and deposition driven by the interaction between the large-scale coherent structures generated by the instability of the shear layers shed by the inlet edges and the mobile bed. Bottom elevation is scaled by the initial mean flow depth. (a) t* =20T. (b) t* = 370 T. (c)t* = 2000 T. 9of19

21 Figure 11. Evolution of the pattern of bed elevation measured in the sea basin at different times in experiment 2. (a) t* =20T. (b)t* =60T. (c)t* = 100 T. 10 of 19

22 Figure 16. Pattern of bed topography in the seaward half of the channel in experiment 1 at different times, showing the presence of alternate bars. Note that the cross-sectionally averaged bed profile has been filtered out. Bottom elevation is expressed in millimeters. (a) t* = 570 T. (b)t* = 1000 T. (c)t* = 2000 T. 13 of 19

23 Figure 17. Pattern of bed topography in the seaward portion of the channel in experiment 2 at different times, showing the presence of alternate bars. The cross-sectionally averaged bed profile has been filtered out. Bottom elevation is expressed in millimeters. (a) t* = 200 T. (b)t* = 1000 T. (c)t* = 2000 T. 14 of 19