 Good morning, my name is Alberto Canestrelli. I'm doing a postdoc at Boston University with Professor Sergio Fagrazi. This was supposed to be a tag team presentation with either Sergio Fagrazi or Andrea Dalpaos, but neither Sergio nor Andrea were able to be here today. So in the first part of the presentation, I'm gonna show some simplified models that Sergio and Andrea developed on the tidal channel morphodynamics. And in the second part of the presentation, I will show you some numerical modeling I did with the complete set of equations solving the hydrodynamics field and the morphodynamics. So what do we mean with the tidal channel? The tidal channel is a stream that is mainly affected by tide and in which freshwater and river sediment discharge can be considered negligible. And why Canestrelli can be considered like a transition zone between the river environment and the marine environment. And usually when we study with models, either tidal channels or estuaries, the horizontal scales are much larger than the vertical depth. So vertical acceleration component, the vertical acceleration component is relatively small so we can assume a hydrostatic pressure distribution on the vertical. If a vertical certification is important, we usually use a 3D model. If the water is vertically well mixed, we usually use a two dimensional model that means that the three dimensional equations are vertical averaged. And so the model provides vertical average quantities as a result. So this is the classical two dimensional shallow water equation equations. The first two are the momentum equation along x and along y. While the third equation is the continuity equation, these equations, eta is the water surface and d is the water depth. And zb is the bottom elevation. So I want to show you how these equations can be simplified in the case we consider a tidal channel flanked by a salt marsh if we assume that the tidal propagation across the intertidal areas, flanking the channel is dominated by friction and this is usually true because the salt marshes are widely vegetated and the vegetation largely increase the friction of the salt marshes with respect to the friction into the channel. If this first hypothesis is usually satisfied, then if we consider that the variation of water surface from the salt marsh to the average water elevation at time t is much smaller than the depth on the salt marsh and if we consider that the salt marsh bottom topography is nearly flat, we can simplify the system of equation, we can neglect the inertial terms on the left, inertial terms on the left, we can neglect the turbulent fluxes on the right and so we can basically assume a balance between the water surface gradient and the friction on the salt marshes. So if we put together these simplified momentum equations with the continuity equation, we find out we obtain this Poisson equation. Moreover, if you wanna fully describe the water surface and the flow feeling the entire salt marsh, entire system made of consisting of salt marshes and channels, we can assume that the pilot propagation within the channel network is instantaneous compared to the propagation across the shallow salt marshes or tidal flats. We are here considering small areas like salt marshes of the order of hundreds of meters. So in this for short tidal basin, this is usually a good approximation. And prescribing the no flux condition around the boundary. So having the instantaneous propagation of the tidal wave along the channels, we can resolve these equations with the following boundary conditions. What do we obtain? We obtain the water surface on the salt marshes. This is an exaggerated water surface. These difference of the water levels are actually really small of the order of millimeters. If you imagine like a basin of hundreds of meters, the differences between the marshes and the channel are very small. And once you have the water surface elevations, you can compute the, through the steepest descent method, you can compute the direction of drainage. These are the direction of drainage. And knowing the direction of drainage at each point of the salt marsh, you can find out the watershed of each tidal channel. So here on the right, we have the salt marshes on the north part of the Venice lagoon in Italy. And we compare the maximum discharges computed by the simplified model with the discharges computed by a finite element model. So this is the grid of the finite element model. And this is the comparison between the discharge on the simplified model and the discharges computed by the finite element model. And these are the maximum discharges. You can see there's a good agreement between the two values. And this is very important because we consider that in the tidal channels, the maximum discharges are the formative discharges. And these maximum discharges usually occur when the water surface is lowering on the salt marshes and the salt marshes are drained by the tidal channels. And you have the maximum exiting discharge when the water surface is right above the salt marshes surface. As it has been shown in a field by different researchers. So how do you describe statistically a tidal network? We can describe the tidal network through the unchanneled path length or overmarch pathways. So if you imagine that you parachute on the salt marsh and you follow the steepest descent direction of the water surface, you can compute for each point the unchanneled path length. So this is the unchanneled path. These are the unchanneled path length computed for different salt marshes in the Venice lagoon. This is a semi log plot. You can see that this is the probability, distribution probability of the unchanneled path length. You can see that there's a linear, kind of a linear behavior of the probability. That means that the unchanneled length are typically characterized by the tendency to develop an exponential decays. So there's no power law. And so you can say that there is an absence of scale free network features. Also between adjacent tidal marshes, you have that the mean of the distribution is different among adjacent sites. So this is the more photodynamic model. So how do we arrive at the description of the planimetric evolution of the tidal channels from the water surface gradient? So if we suppose that the time scale of initial network incision is much larger than the time scale of channel meandering of the marsh platform growth and changes in external forcing, like relative sea level rise, we can decouple the initial network incision from slower processes. And if we consider that the headward channel growth is driven by exceedances of a critical stress. And if you consider instantaneous adaptation of the channel cross-section to the local tidal prism, we can find the evolution of the tidal channels in the salt marsh. So this is just an example of how the tidal channels carve the water, the bottom surface of the salt marshes. We consider that the landscape forming events are due to the spring peak discharges. And to assess, to compare the result with the configuration that we find in the real world, we use the PDF of un-channel flow length. So what about the hypothesis I just talked to you about? Like this is, we said that we consider that the channels carve the salt marshes through the headward process. And you can see that this is the computer by the sheer stress on two salt marshes of the Venice lagoon. You can see that the higher values of the sheer stress are on the tip, on the landward end of the tidal channels and on the meandering band. And the headward growth character of network development has also been shown by you can set out on some marshes of the muddy bay in the South Carolina. You can see that in 1968, the channel was in this location and in like 30, about 34 years, the channel carved the salt marsh through a headward process. And we also consider that the section of the tidal channel instantaneous adapt to an equilibrium section that is given by these power law that relates the watershed area with, sorry, the area of the section with the tidal prism. So this is a comparison with a spontaneous tidal creek network that generated that carved salt marshes in the Venice lagoon. So this is in this area, oh, sorry, in this area, a restored salt marsh was built and it has been seen, so this is the study site. And these are the photographs of the study site. You can see that very fast from 2000 to 2002. This is an initial channel that was carved on purpose and these are the natural channels that formed after two years. So you can see that the term scale of the formation of these tidal channels is really short in this case, 2002 year. And also this is 20 meters. So this length is about 100 meters. So we can consider it a short basin. This is the, these are the modeling results. And on the upper line, you can see the results obtained with the shear stress, critical shear stress of 0.15 Pascal. On the lower line, the results obtained with the shear stress, critical shear stress of 0.25 Pascal. With the lower shear stress, you clearly have an increase of tidal channels on the salt marsh. And we also have different behavior using a different value of these. This is a statistical parameter. Without entering on the detail, we can say that if you use a small parameter T, you have a kind of the deterministic carving of the salt marsh's surface. While if you increase this parameter, you have more a random character. This is due to the fact that you random choose the value of the maximum shear stress of the pixel having the maximum shear stress in the area. In this case, you always choose the pixel having the maximum shear stress. Here you can choose also another pixel. So for example, in this case, this pixel was selected and it was not the pixel having the maximum shear stress. And this is the real network. So from a statistical point of view, this is a comparison between the channel length, the PDF of the channel length on the generated network and on the real network. There's a good agreement between the distribution as this kind of the same behavior. The decrease at the end is due to the fact that you have just a few points with a large channel length. So you can reproduce point-wise data channels but from a statistical point of view, the generated tidal networks resemble the real network. Now I'm gonna show you some results about the numerical modeling of the long-term profile of the tidal channel compared with the experimental results. So this is the first experiment, physical laboratory experiment that was done to study the evolution, long-term evolution of the bottom elevation on the tidal channel in the lab. The apparatus consists of an oscillating cylinder that generates a sinusoidal oscillation in the basin with an amplitude of 0.032 meter and a period of 120 seconds. The tidal wave propagates on the basin and it enters a channel having an exponential decreasing width. The bad material is a cohesion less and it consists of a hazelnut shell, broken hazelnut shells with an average diameter of 0.3 millimeters and this is the density. This is the mathematical numerical model. These are the equations you saw before. The only peculiarity is the treatment of the wetting and drying processes. This is important because on the physical experiment, we have that a beach forms at the end of the tidal channel so it's very important to deal with wetting and drying processes. So we are considered that the bottom elevation are distributed according to a Gaussian probability density function where the amplitude of ground irregularity are two times the variance of the distribution. For the morphodynamic use, the classical X and R equation for computing the variation of bottom elevations as function of the gradient of the bottom of bed load transport plus the source terms due to the position and erosion. And this is the classical diffusion advection equation for suspended load. We compute the entrainment minus the deposition in this way and this is the intensity of bed load rate expressed as function of the bed load and the sediment transport is computed by the formulation proposed by Van Rijn. So this is the hydrodynamics in the basin. So first of all, I wanna recall that this is a long tidal channel and it was scaled, like it scales well with the Rotterdam estuary that is of a size of 20 kilometers. So you cannot assume an instantaneous propagation of tidal wave in this channel. So the full numerical, the full set of equation has to be solved numerically. And this is the result. This is the tidal wave. There are some irregularities because the cylinder was not able to create a perfect sinusoidal wave but you can see that numerical model is able to see, to recreate the increase of tidal range inside the channel. And also the fact that during the flood, so the flood time is shorter than the ab time of the ab phase. So this implies that the velocities during the flood are larger than the velocity during the ab. This is in the inner part of the channel. So going landward, you pass, you go from a situation of almost equal value between flood and ab and then you go to a situation of flood dominated the channel. And this is the flow field of the numerical model in the basin in proximity of the inlet. So this is the maximum ab, this is the end of the ab, beginning of the ab, and this is the maximum flood. You can see that during the ab, this is basically a jet on standing stagnant water. And you have that the jet occupies basically the area of the inlet, the section of the inlet. This is the end of the ab and this is the beginning of the flood and the maximum flood. You can see that during the flood, you have a classical example of potential flow. So you can see the difference. During the maximum ab, you have like this is the section, the active section, during the maximum flood, the active section is larger. So you can see as the, during the ab, the velocity are larger, the velocities are larger than during the flood. And this symmetry on the velocity fields creates a scour in proximity of the inlet. And you can see this on the evolution, morphodynamic evolution of the channel. Close to the inlet, you have a scour and you have a accumulation of sediment at the end of the channel due to the flood dominated character that you have landward. So this is the, this is the means level. This is the bad profile after 20 cycles, after 500 cycles and after 5000 cycles. And the model was run both the physical and numerical model were run for more than 5000 cycles, but not noticeable differences were noticed. So the physical apparatus was stopped. And the equilibrium configuration was reached. And you can see how the numerical model is able to predict in a reasonable way the profile, the monotonic increase of bad profile in the channel. You can see as in the end of, at the landward part of the channel, you have information of a beach. And so the wetting and drying processes are very important in the model. So this is evolution. We, here we did some American experiments considering also the presence of tidal flats flying in the tidal channel. So we consider a hypothetical configuration like this. This is a six meter deep tidal channel flanked by 0.5 deep tidal flats. And we prescribe a water, a sinusoidal water level, seaworth and a concentration equal to the equilibrium concentration during the fall phase. During the ab phase, the concentration is calculated by the numerical model. No flux conditions are prescribed for both water and solid phase on the other three boundaries. These are the parameters we consider very fine sand with a diameter of 0.064 millimeters. And the sinus solution varies from 30 to 80 meter at the boundary of the tidal flats. We neglect the presence of a vegetation or a suspension of sediment by wind wave motion. This is the longitudinal profile computed by the model on the channel, on the main channel. And you can see that after 5,000 cycles, the sediment waiver reaches the end of the channel. You have a scour near the inlet, the position, land work, a sediment front propagating land work. And the value of oxygen is lower and slower as the equilibrium is approached. So this is the velocity along the channel with the initial topography and the final topography. You can see that with the initial topography, you go from a situation where the channel is ab-dominated at the inlet, then you go toward a situation where the channel is slow-dominated, land work. You have an almost linear decrease of the maximum and minimum velocity along the channel. While with the final topography, you have a stronger reduction of ever-flood asymmetry and the velocity tends to be spatially constant along the channel. So you see how the feedback between the bed evolution and the hydrodynamics create a situation which the erosional power of the channel tends to be constant along the channel itself. And so the same applies for the shear stress, great differences along the channel, small differences toward the end, the same for the suspended sediment transport. And this is the net tidally average sediment discharge. You can see that it's large at the beginning and it tends to be very small at the end of the simulation. This is the tidal range. At the beginning, the tidal range is like diminishing toward the end of the channel. This is the end of the channel. But after at the end of the simulation, the tidal range is about constant along the channel. This is the fact that the tidal flats are carved by secondary channels. And these channels allow the system to drain the water faster and less energy is needed to drain the tidal flats. And so the tidal wave can propagate faster in both the main tidal channels and in the secondary tidal channels. This is a section of a secondary tidal channel. You can see that you have a classical increase of bed elevation. This is the bank elevation. You have, you can see that you have like an upward concave profile on the channel and levy due to the position of sediment coming from the channel to the tidal flat, as it's usually found in the real world. This is the, you can see also you can have a funneling, a main channel, you can reproduce a main channel funneling as in the real tidal channel. This is the exponential decreasing width in the channel. And a almost linear relationship can be seen between the cross-sectional area of the creek in lead and the creek watershed area as it was shown in Rinaldo Tal. One minute, 30 seconds. Okay, I want to just show you some numerical results on the fly river estuary. The fly river is the second river in Papua New Guinea. This is the fly river is dividing in three reaches. This is the lower fly, this is an estuary and it's 400 kilometers long. There's almost constant water discharge and a very high sediment discharge of 85 million tons per year. And because the basin is characterized by rapid rate of erosion. This is the tidal regime. You have five meters of tide during the spring, during the spring tide and one meter during the nip tide. This is the tidal influence reach 400 kilometers. We assume that the well mixed conditions are present in the estuary. And also the wave energy in the disability channels is minimal. This is the computational mesh, water discharge and tidal wave. This is the bathymetry. These are the results for the discharges and the tidal range. You can see the decrease of tidal range in the channel. You can see the discharges are way higher at the mouth of the channel and tends to the constant value at the landward at the average junction, the junction with the upstream tributary. This is the digitalization of the width along the fly river. You can see a break on the width in correspondence of the point where the discharges, the water flux becomes one monodirectional. And this is very interesting. Now just last slide and I'm done. So this is what I'm doing now is just trying to recreate the long-term profile of the estuarine region with a simplified geometry. I prescribe the average annual discharge concentration with 50% of sun, 85% of mud and the tide salinity on the downstream area. I start with it from a constant slope this is the upper part of the river. This is the estuary. This is, you can see there's a break. This is the depth obtained by the mathematical charts. And this is the, you just have the pelvic elevation but you can see there's a break here. You have like a first slope on the estuary like downstream decreasing elevation. Then you have an increase of elevation and then you have the delta front. And this is what you obtain with a nip spring tide with a, I used the depth remodel and this is what you obtain with a M2 component with one meter. You can see how if you use just the nip tide you have a high deposition along the estuary. While if you use the nip spring tide you're not that far, there are some differences of course but you are not that far from the present equilibrium configuration. And this is very important because it means that it's a strong deposition during the nip tide and the sediments are flushed out the estuary during the spring tide. And I'll let you the conclusions for you to read. Sorry. There's anyone with a very eager question? I really would like to move on to the next slide. Any question? Just one. Okay.