 Right. So I'm going to talk about improving our predictions for Delta forms and processes. And you might think after Becca's talk that what's to be improved. But I don't think any of you would want me or be in the position of telling the Army Corps of Engineers here in the U.S. how to engineer the Mississippi Delta to maximize wetlands. Do we blow a hole in the levee or what do we do? I don't think we're there yet. So I'm going to tell you also about some of the limitations that Delta has and what we're trying to do to improve that. I'm a spokesperson for a whole host of scientists and I apologize to them in advance. If I muddle their ideas, I'll direct them to you. So part of the problem here is that there's a whole range of Delta types as Rebecca showed. Here from the Lina Delta with literally hundreds if not thousands of active distributaries to this Delta, the Blue Nile in Tana Lake in Ethiopia, one distributary like this. Now Rebecca would argue that this is largely a function of grain size and I don't contest that. But the question is, is that a necessary and sufficient condition for this effect? So of all the questions you could ask about this, this group of scientists asked these three. The first one was, how do they do it? How do they self-organize out of this turbulent expanding plane jet, the quintessential statistical process? This beautifully evolved set of forms with characteristic length scales. What are the feedback loops that set these in response times of the system? And then ultimately when we develop confidence there, and of course this all has to happen in five years because that's the length of the grant, how will they respond to perturbations in climate change in particular but sea level rise and so on? So our approach was to pose the problem like this, that we have a five-year effort to develop high-resolution quantitative models here. We're trying to span a number of timescales and you all know what problems arise there. It's all funded through NSF's Frontiers and Earth System Dynamics. We argue that it's a beautiful example of an earth system, a dynamical earth system. And specifically we're after in the end not only the dynamics but the sustainability of these in the face of sea level rise and declining sediment feed. Okay, so it's a collaboratory which is a buzzword of course but it does in fact consist of two laboratories. There is the field laboratory, Wax Lake Delta. This is Wax Lake Delta into a Chappell Eye Bay down on the LA coast. It's instrumented to within an inch of its life now. There's acoustic Doppler curve profilers doing real-time data collection, nitrate samplers, water stage gauges, temperature, salinity, bird counts, whatever you want. And if you want any of those data they're being put on the web and you should talk with Dave Morick who's the contact here at the University of Texas for that. The second part that I'm going to talk about, the second laboratory is this virtual modeling laboratory which notionally at least is housed here at the systems facility and the group of us who are interested in advancing these models for development hypothesis testing are trying to use the Wax Lake data and interact with our collaborators there although telling them what kind of data you want and what you get is of course a very different thing. I'll tell you about three kinds of models that are under development here. There are reduced complexity models of which maybe two or three I'll talk about. A multi-dimensional ecomorphodynamic model, basically a Dell 3D successor. And then if I have time ecologic models of which there are two at the moment, a vegetation model that has vegetation growth succession and so on interacting with sedimentation. I won't talk about that. I'll talk about one that's a fish population model if I have time. Okay, so let's talk about the first of these reduced complexity models. It's called Delta RCM. It's a cellular model. This is the research team. Paola Pasalapa is the lead on it. So you want to talk with her at the University of Texas if you have any questions. And the focus as it should be for a bottle of this sort is on the large scale system dynamics. Like Andrew said, you've got different scales here that require possibly different kinds of models. There's a domain of square cells here in which the discharge vector, bed elevation, water surface elevation, and sediment transport rates are calculated. Pretty standard kind of formulation. But what I think is a little different about this model is the way they are calculating the water and sediment fluxes. They're thinking of water and sediment as parcels in a Lagrangian point of view and that the parcels are routed stochastically according to some simplified physical laws. At each time step, there's the calculation of the routing probabilities for the water and then the update of the water flow field and surface elevation, then the routing probabilities for the sediment, update of the bed, and then that feeds back into the routing of water. All right. To give you a flavor for what these probabilities are, the probability of water flowing from cell I into any of the adjacent cells depends upon both the water depths, of course, in each of those cells, but also upon a combination of where the cell is. Is it down water surface gradient from the cell? And flow inertia is also important in this case, as you can imagine. And so the actual vector strength of the velocity is taken into account to calculate the probability for water passing from cell A to cell B and then the water surface profile is updated. The probabilities for the sediment, there's sand parcels that can pass through a cell and that's limited by the local flow field, just the cube of the velocity, which is a fairly standard transport, you know, bulk sediment transport formulation. And any number of these parcels that can't be transported given that flow strength in the cell are deposited in the cell. The mud is treated a little differently. The amount deposited is a difference between the local flow velocity and threshold cubed. And so, like Andrew's models, there's sand and there's mud and they're treated differently here. So the sediment parcel either is passed or not, it accumulates, and the bed is updated. So here's just one example to show you the way the flow field works. I'm not going to show you the complete simulation of the delta, but here is a flat sloping plane with a bump on it. Now you could think of that bump as a river mouth bar that's growing. And so here is a plot of the normalized bump height, normalized by the water depth there. So a bump height of one, it's right at the water surface. Zero, there's no bump there at all. So this is normalized bump height. And here's the normalized velocity at the apex of the bump. Now I think you all appreciate that the Delft 3D solution, which is the triangles here, looks like this, that normalized velocity is one here when there's no bump. And then as the bump grows, the flow velocity over the bump accelerates because you have a constriction of the flow and the flow speeds up. There comes a critical bump height around, well, what is it here, about 0.5 or so, where the back pressure causes deviation of the flow velocity vectors upstream and the water begins to flow around the bump and the flow velocity over the bump precipitously declines. This model produces these results, which is at least in spirit quite similar. Extended to the Wax Lake domain here, this is the Wax Lake net. Here's Delft's solution for a particular set of boundary and initial conditions. It's a steady state flow with the Corps of Engineers hydrographic data and I'll show you another example of this in a minute. And here's Delta RCMs. It's over predicting here in these shallower areas the flow, but otherwise the general pattern is the same. And I will refer you to Paola for more on that model. The second model takes a very different approach to this whole idea of how to better predict consequences in deltas. And it's by the team of Effie Fuffalogeorgiou and her students up in Minnesota. Now, if you know Effie, this is a pretty much an abstract mathematical model and it's a network-based framework for understanding delta vulnerability. And the basic idea is this, that the delta net is a series of a distributary nodes, bifurcations, and channels, links is abstracted into a matrix, an adjacency matrix, and then you can perform operations on that matrix to answer questions about distant connectivity, sub-networks within the system, and downstream regions that are influenced by upstream parts of the net. I'll try to give you a flavor for this. Here's, again, Wax Lake Delta, and you can abstract that net here and label these links, one, two, three, four. Now the actual labeling does matter to some extent, but it becomes internally consistent given a particular labeling. And then you compute your adjacency matrix in the following manner. If there is a connectivity between link i and link j, then the adjacency matrix has a one in that ij location in the matrix, otherwise it has a zero. All right, and there's a second matrix and that is this degree matrix D, which is a diagonal matrix in which, along the diagonal dii, reflects the number of links directly downstream from link i. And so this operation here, where V represents a particular sub-network, will extract out for you all the sub-networks in the delta. And this operation, where this is the union of all V vectors, will extract out downstream regions of influence. Not only F, you could think about deltas this way, I certainly don't. And then you can find hotspots of change where there are links where a flux reduction would cause the most drastic reduction in the shoreline. So I thought it's a pretty novel way to think about this and obtainable from aerial imagery. All right, the third model is Gary Parker's group at Illinois with Matt, his student, and Enrica, who is now, where's Enrica? South Carolina maybe. Here's Wax Lake Delta again, and they're taking one Sukkim's model, which is this delta restoration model, which you've seen published, in which they're arguing that, hey, you blow a hole in the levee and you have the lip height just right and you can create wetlands to save New Orleans. You've seen that published in AGUEOs and so on. Well, they're improving that because one of the realizations about these low-slope sandbed rivers, of which certainly this qualifies but also the Mississippi itself, is that these are bedrock alluvial rivers. Bedrock in quotes meaning that they are entrenched within fairly, fairly stiff Pleistocene and early Holocene muds that act like bedrock and that the bed of these rivers is not covered by alluvium completely. Well, this creates a couple of problems. Number one, you've got to account for that in channel formation and number two, all of our transport laws are capacity laws that assume that there's an infinite amount of material on the bed to interact with. Well, that's not true in these rivers. So you've got to start calculating under capacity flows. And Gary is showing here where in Wax Lake Delta these parts of the channels are underlain by just stiff Bay mud and then the alluvium is out here and part of his approach is going to be with an aerial fraction cover of alluvium. All right, and then he's going to apply this to a model in which they're not going to try to simulate the whole Delta form like Delft does but abstract the results from it and use rules for this 1 over 2 cascade in reduction and discharge and bifurcation length that Rebecca talked about. And if any of you know Gary Parker, you will know better than I why he has a duck down here in the corner. All right, now I want to talk about the creation of a successor maybe to Delft 3D. I think you all would agree that Delft 3D flow version 6.0 which is available here in open source on the system's website and from Deltares is a state-of-the-art in morphodynamic modeling of this form. And the group that's working on this is Alberto Canastrelli who sits over here so I will defer to him. Burt Yoggers and Bert is here someplace from Deltares and so this is a joint Deltares FESD project. And I will give an example here of a Delft 3D flow on Wax Lake Delta although you don't need it now after Rebecca's talk but you have Messela who is here sitting over on this side and his students are investigating the interactions in the hydraulics and morphology in the end with our ecologists on the project the nutrient loading on this Delta. And the grid is here, the boundary condition is here where we somewhat know the discharge and sediment inflow. The tides and water surface stage are known down here in Achafulae Bay and this is the grid and you really do have to incorporate all of this area because it's a big swampy wetland that stores water significantly. The grid is 25 by 25 expanding out here to 100 by 100 10 sigma layers in the vertical. And I just show one example because you get all kinds of results out of these. If you look at a line of section right here with that red line and look at two stations along that line at which we have data you'll see that the Delft prediction of the flow velocity here with normalized depth matches pretty well the ADCP data here although might underpredict a bit. Same for this station and the suspended sediment concentrations are measured down there by the field folks is predicting pretty well. Now I think you already know that this is a really good vetted engineering model but it has problems particularly for building morphodynamic Delta. So let's talk about those problems. And number one the simulations sometimes can be a very strong artifact of the underlying grid structure. In this case you get these right angle bends in the distributaries of the final form Delta and we don't want that of course. This is an artifact of the grid. Number two you would expect that these channels should be hydraulically similar to real channels. Okay so here's a Gary Parker kind of non-dimensional plot of bankfull channel width divided by well scaled by grain size and bankfull channel discharge scaled and here are sand bed rivers up here here are gravel bed rivers down here there's this plot you all know that the width scales is roughly the square root of the discharge. Delta has larger widths for larger discharges that's true but the slope isn't right. And part of the problem is this which leads to problem number three the algorithm for eroding the channel banks is ad hoc and in Delta at the moment. Here's the problem here's a wet cell here's a dry cell and let's say that this is a cohesive bank cell we're going to call this a pretty steep bay and here's the channel. Let's say discharge wants to increase in that channel well the channel should widen right but there's no way for Delta to widen this channel because there's no water up here to erode and there's no lateral erosion mechanism in Delta. And so the work around at the moment is to say right if a meter is going to be eroded here if that's what's predicted in the flow field let's take a proportion of that and erode it from here and ultimately we'll be able to widen the channel and that proportionality constant is a user specified number well we can't have that right. The echomorphodynamic interactions are primitive in Delta now and so we need some improvements in that. How much does vegetation growth trap sediment create turbulence maybe in erode sediment on the part tops. Alright and so we're going to call this for one of a better word at the moment d3d plus and I'll tell you how far we've gotten along this line. So the first is a mass conservative staggered three-dimensional shallow water model this is the team Alberto Aucke and Bert I'm the cheerleader and then I'll talk about the sub grid vegetation flow interaction module that's Phasing, James, Doug and William Nardin. Okay. Now here's the MRSL boundary technique. Right so you have a rectangular grid you preserve that because the ADI method is very efficient so we preserve that but you have a boundary that really should go through here and you don't want the jaggies on here to represent that boundary you want a smooth boundary and so you have a solid region here and a fluid region here and the first thing you do is you superimpose a spline on top of that grid and you keep track of that spline that's a little computation overhead but it turns out we think to be well worth it and then there's this ghost cell technique where you actually try to specify a value out here for the momentum equations that leads to the right solution for the flow field along that wall and so it's an artificial construction this is used a lot actually in mechanical engineering when you want to get turbine blade dynamic turbine blade solutions for compression in jet engines for example and that's all I'm going to say about that you talk with Burt and Alberto about it if you want to know more so the momentum equations are solved this way and the cut cells are used to console the continuity of mass and sediment by the way alright so is this worth it? here's a typical example this is contemporary Dell 3D Cartesian coordinate system and we have a channel that's 45 degrees to the actual orientation of the grid you can't see it in the back for sure but there's jaggies all along the edge here because the edge is simulated by either a dry cell or a wet cell this is water level here and this is flow velocity this is the immersed boundary solution this is the Cartesian present Dell 3D solution you look at the water level in the back you can't see the numbers here but you think well that's not so bad you know red red that's high water flows that way it's flowing down it's steady non-uniform flow the velocity field doesn't look so good but here's the real test if you plot longitudinal velocity here and you plot the water surface elevation here there's an analytic solution of course for this steady non-uniform flow and it's this line right here which you can't even see because it falls right on the immersed boundary solution but the present Dell 3D over predicts the water height because it needs a higher slope to take care of I don't know what you'd call a numerical friction along the edge of the channel same with the longitudinal velocity so we think that this is a step forward it's particularly dramatic here where you do an infinite riverband you have a periodic boundary condition here the flow goes round and round and round for a 2D vertically integrated hydrodynamic problem you know that the flow velocity or that the water surface slope has to be radial to change the velocity vectors there and the immersed boundary method does thank you dear immersed boundary method here this is the exact solution the present Cartesian system doesn't look so good okay and the beauty of doing this implementation now is that we can have lateral bank erosion according to whatever rule you want just say excess bank shear stress cause lateral motion of your spline surface through that grid okay vegetation this is the vegetation formulation at the moment is that the Baptiste formulation with rigid cylinders stem densities and so on can be simulated the flow velocity through the vegetation is uniform and then the log profiles on top of that so there's a lot here that I'm short changing these researchers but I want to show you what difference this makes now this vegetation isn't growing yet the growing module I'm not going to talk to you about today in the interest of time here is a plot of relative vegetation height it's relative to the water level and let's talk about a bar like a river mouth bar on top of a bar and this is the ratio between the sand that's deposited but in a vegetated case over the non-vegetated case and so one right here with no vegetation that's the reference state now as the vegetation height increases to be roughly the water depth over the bar the amount of sand that's deposited increases relative to the standard state with no vegetation that kind of makes sense you get increased friction you slow the flow the sediment settles out that's all good but notice what happens here and what happens there is that as the vegetation continues to increase the friction on the top of the bar more and more the water is shunted into the channels it doesn't even go over the bar so you don't get any sediment up there and so it's a bipolar state of yes vegetation will enhance sedimentation up to a point now what about actually keeping the vegetation there well Fay and James are worried about turf erosion and so here you have a block and you do a calculation of the combined wave current shear stress on that block of peat or turf and if that wave current shear stress is greater than the sum of the root strength and the cohesion from the sediment itself you'll rip up the sod, rip up the turf and so that makes a difference because here's a map of wax lake delta and in the back all you need to know is that the reds are where in the presence of the vegetation without the turf being ripped up there's more sedimentation up here on the edges of the islands than not and so we can now quantify the amount that roots protect the vegetation marshes from erosion I'll finally end here with this fish model now an individual based fish model means we're going to track every fish and the objectives are to look at the interaction between say some structure of the delta like the hypsometry of the mouth bars and the population density because you can't engineer deltas without wondering what you're going to do to the ecology, to the fish population and so you can evaluate restoration scenarios in terms of fish productivity if this thing has any truth in it at all the model starts with a delt 3D bathymetry with vegetation, that's important you've got to have the temperature of the water at every location, the salinity of the water and of course the water stages as they flood the tops of the bars and then there's the fish model which I won't go into the rules but it's kind of a standard population model for fish, Paul says it's on a grid like this there's five species of fish the delta t is small because fish grow fast I guess they feed and grow, swim about, reproduce and die which just about sums up life and the end result, here's an example here's a distributary just a snapshot within the domain of the delft run and you can see that these dots here just representing the population density in the cell of how many fish are there just the number of fish it's a little denser over here just because this is a deeper water area there are some that have lost their way and gotten up in the shallow water of the bar who knows what they're doing and then the more species are over here so that's just a flavor of this I'll finish by saying these models are being tested against the Wax Lake data and as we speak they're going to be posted here on the systems website we really encourage all of you to help us out there's any part of this you want to do of course there's no money left and to use these models and improve them they're going to be in the systems open repository if you have any questions about this I can orchestrate whom you should speak with thank you if you really want to know why we're doing this let me just show that my favorite cartoon character, Calvin and Hobbes nice talk can you elaborate on the importance of stratification? stratification? yes or is it already in the simulation? so stratification in the water column in the basin so buoyant plumes? no you know none of us has explored in any systematic way how delta form changes there's a lot of work that's in the literature about the role of hyperpignaal jets and plumes but understand something that I didn't even say here is there is no mechanism in delft for foreset collapse and so delft you know a true morphodynamic model for a delta should have some kind of a foreset stability module in it and so that's low hanging fruit for any of you uh-oh, Alberto I didn't have the time yet to analyze all the results but basically what I found is that the stratification basically reduced the active depth of the jet and basically makes the jet more stable so it diffuses more and it announces better position this is the first thing I can tell you but I still need more time to analyze all the data thanks