3 Tidal systems modelling: ASMITA model

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1 3 Tidal systems modelling: ASMITA model 3.1 Introdution For many pratial appliations, simulation and predition of oastal behaviour (morphologial development of shorefae, beahes and dunes) at a ertain level of auray require the use of mathematial models representing the basi hydrodynami and sediment transport proesses (Van Rijn, 1993). An important objetive of morphologial modelling is the long-term behaviour of morphologial systems in relation to human interferene and autonomous proesses. Stive et al. (1997) desribed a model representing the morphologial interation between a tidal basin and its adjaent oastal environment, inluding the ebb-tidal delta. From now on, we will fous on this model, the ASMITA model. First, variables desribing the system need to be defined by a shematisation of tidal basins. Then, we will explain the basi onepts of ASMITA model and its formulation. One of the main ideas in this model is the hypotheses of the existene of an equilibrium state: under relatively onstant hydrodynami foring onditions eah element an undergo a self-organisation leading to a (quasi-) equilibrium morphologial situation. Finally, we will onsider a tidal system and introdue the effet of a river disharge. This new ondition will modify the model equations, as it hange the boundary onditions of the system. We might keep in mind that rivers an supply large quantities of sediment into the oastal systems. 3.2 Preditability and sales Tidal inlet and basin morphology is the result of a stohastially fored, non-linear interation between the water and sediment motion and the bed topography. Understanding and prediting their funtioning implies dealing with a wide range of spae and time sales, with omplex multisale interations of the onstituent proesses, and with strong, partly stohasti variations of the foring. Besides omplexity, also the possibility of limited preditability has to be taken into aount, beause these systems seem to satisfy all onditions for inherently unpreditable behaviour (De Vriend, 1998). This would imply that large-sale behaviour annot be derived in a deterministi way from the small-sale proesses. 13

2 In order to takle the problem of limited preditability we introdue the onept of dealing with our interest on a asade of sales. Aording to Van Rijn (1993) these are: - mega-sale oastline hanges related to strutural erosion and deposition due to interation with neighbouring sediment-importing tidal inlet systems (10 to 100 km, 10 to 100 years) - maro-sale, yli oastline hanges related to the yli behaviour of morphologial features suh as sand banks, sand waves, rip and tidal hannels - meso-sale deaying oastline hange related to transition to a new equilibrium, often indued by man-made strutures (1 to 10 km, 1 to 10 years) - miro-sale flutuating oastline hanges related to stohasti and deterministi variations of morphologial features (0 to 1 km, 0 to 1 year) In our ase, we will to work with meso and maro sales, whih are relevant for medium-term oastal zone management and poliy formulation. 3.3 ASMITA model ASMITA (Aggregated Sale Morphologial Interation between a Tidal inlet and the Adjaent oast, Stive et al., 1998) model an be onsidered as an aggregation and an extension of ESTMORF model formulation for tidal basins (Wang et al, 1998). The aggregation onerns the fat that we haraterise the system elements by only one state variable. The extension onerns the inorporation of formulations for the ebb tidal delta and diretly adjaent oast as well, without modifying the basi onepts Shematisation of a tidal system: morphologial elements The ASMITA model is built on the idea that a tidal inlet system an be shematised into a number of morphologial elements. The most important ones are the tidal flats and the tidal hannels (together forming the tidal basin) and the ebb-tidal delta. Next figure shows this simplifiation: 14

3 Figure 3-1 Elements used in ASMITA onept After Van Goor (2001) Next, a short desription of eah element is presented. Ebb-tidal delta: Ebb-tidal deltas are sediment aumulations on the seaward side of tidal inlets, and they are also morphologially and dynamially onneted to the tidal inlets. In the tidal system, deltas are important beause they at as sediment reservoirs, so they an supply sediment when it is needed in the system. In ase of no sediment enough in the delta, it will be delivered by the adjaent oast. We may define the volume of the ebb-tidal delta via the no inletbathymetry (Dean and Walton, 1975) as if there was no inlet in the oast. In this ase, the oastal slope is assumed undisturbed by the oastal inlet and therefore assumed equal to the bathymetry of the adjaent barrier oast. The volume of the ebb-tidal delta is equal to the volume above this fititious no-inlet oast. This definition is also used in ASMITA. Channel One an say that the hannels are the veins of the system: they transport onsiderable amounts of water and sediment. During flood, water enters into the inlet, and brings a large amount of sediment. Part of this sediment will be deposited in the hannels and/or in the flats. Also, part of this sediment will be transported out of the system during the ebb. Referring to the shape of the hannels, one an say they follow a branhing 15

4 pattern. Near the inlet, the depth and width of the hannels is largest and lose to the basin, they get smaller. Flat The flats are defined as the area that dries during low water and inundates during high water in a tidal yle, so all areas between MHW and MLW (Eysink, 1990). Eysink found that, aording to general lassifiations of oastal features, the presene of tidal flats mainly depends on the tidal range, basin area, shape of the basin and the orientation of the basin aording to the dominating wind diretion. Large tidal basins, speially when they are orientated in the diretion of the dominant wind, allow signifiant wave ation around high water beause of the onsiderable feth lengths. As mentioned before, we assume that the morphologial state of eah element an be desribed by aggregated state variables: the total sand volume of the delta, the total water volume below low sea level and the area of the intertidal zone in the lagoon basin (Stive et al., 1998). This shematisation of the tidal basin bring us to the definition of the tidal prism (P). Moreover, the tidal range (h), that is to say, the differene between high and low mean sea levels; the area at the mean high water, the so alled basin area (Ab); the area of the hannels (A) and the flat areas (Af) play a role in obtaining the tidal prism, as well as the volume of the flats (Vf). Next, we present a figure where the previous indiations beome lear (the volume of the hannel (V) is been also inluded in the following sheme, although it is not neessary to alulate the tidal prism, but this will help to later explanations): A b MHW h V f V f MLW V Figure 3-2 Tidal prism After Van Goor (2001) 16

5 P = ha V (3.1) b f where Ab = A + Af (3.2) Impliitly, we assume that the basin is small ompared to the length of the tide waves, hene, the water level hange "simultaneously" everywhere in the basin. Thus, the volume of eah element is defined as it follows: Delta: sediment volume above a fititious no-inlet oast (Dean and Walton, 1975) Flats: sediment volume in the tidal flat between MLW and MHW (Vf) Channel: water volume below the MLW (Vh), see figure Morphologial development in tidal basins: equilibrium onept Morphologial development in tidal basins is mainly related to the tide. One of the main hypotheses assumed in the model is that eah element in the system tends to evolve towards a morphologial equilibrium (a steady state where there will be no morphologial hanges with time), whih is determined by the loal, long term averaged hydrodynami and morphometri onditions. Theoretial arguments for the existene of suh equilibrium were given by Dronkers (1998), but also supported by various field investigations. An empirial relation is required for eah element to define the morphologial equilibrium state, they an be derived from literature, Eysink (1990). In this state, the volume in eah element is alled the equilibrium volume ( V ie ), and it depends on: ie (, ) V = f P h (3.3) Morphologial hanges towards an equilibrium state our by tide-residual sediment transport. Then, the long-term (time sale muh larger than tidal period) mass-balane an be onsidered for every element: dv T n ± = ni dt (3.4) i 17

6 The left-hand side of this equation represents the erosion rate within an element. The positive sign orresponds to a wet volume (wet volume inreases when erosion is larger) and the negative sign orresponds to a dry volume (sand volume derease when erosion inreases). The righthand side represents the sum of transports from the element to its adjaent elements in the system, inluding the outside world (if that is the ase). The rate of volume hange is assumed to be proportional to the differene between the loal equilibrium onentration and the atual onentration (Galappatti and Vreugdenhil, 1985): dv dt n ± = wa s n ne n (3.5) where A w n s ne n 2 is the horizontal area of the element m is the vertial exhange rate [ m/s] is the loal equilibrium onentration - [] is the atual onentration - [] For eah element in the system, a loal equilibrium onentration ne is defined, so that if the element is in morphologial equilibrium, the atual onentration equals to the loal equilibrium onentration, hene, n = ne. When they differ from eah other, morphologial hanges take plae. Erosion within an element will our when the atual sediment onentration is smaller than the equilibrium onentration. Aretion will our when the atual onentration of an element is larger than the equilibrium onentration. If we think about the delta and the flat areas (both defined as a dry volume), when n is smaller than ne, the derivative of the volume beomes negative, hene, there is erosion as the amount of sediment in the elements is dereasing with time. In the hannel (defined by a wet volume), when the equilibrium onentration is smaller than the atual onentration, the derivative is positive; sedimentation will take plae. A key element in the modelling onept is the global equilibrium onentration E. The definition is based on the following arguments. When all elements in the system are in equilibrium, a onstant sediment onentration is present in the whole system. This overall equilibrium onentration is idential to the global equilibrium onentration. Therefore, i = ie = E. 18

7 Conentrations and volumes are related by the following equation: V n ie ie = E Vi (3.6) n > 0 for wet volumes: hannel n < 0 for dry volumes: delta and flats The power n is larger than one, usually the value of this power is 2, based on the usual assumption in sediment transport formulas that the sediment transport is proportional to the third power of the flow veloity (Stive, 1997). When the loal equilibrium onentration in an element is smaller than the global equilibrium onentration, a tendeny to arete exists; when it is higher than the global equilibrium onentration, there is a tendeny to erode. Let's onsider a ase where the loal equilibrium onentration in an element defined by a dry volume is larger than E. The atual volume of sediment will be larger than the equilibrium volume, so it means there is an exess of sediment in the element (ompared to the equilibrium state) so it tends to erode; and vie-versa if the loal onentration is smaller than E. Let's think now about the hannel, sine a wet volume defines it. If one onsider a situation where the loal onentration is larger than the equilibrium onentration, the atual volume in the hannel will be smaller than the equilibrium one. This orresponds to a situation where there is an exess of sediment in the element, so then a tendeny to erode exists. Morphologial hanges in the elements let the system evolve to an equilibrium state. Sediment transport make this hanges our. There is a tendeny to erode or to arete depending on the loal equilibrium onentration and the overall equilibrium onentration. However, and as mentioned before when the parameter Tni was introdued, there exists also an exhange of sediment between adjaent elements in the system, and also between the system and the outside world. It an be expressed as it follows: T = (3.7) ni ni n i 19

8 where ni is the horizontal exhange parameter between elements n and i As the equation shows, sediment exhange between two elements ours depending on the differene in sediment onentration between those elements. This sediment transport will always take plae from high onentration to lower onentration. Hene, if an element has a larger onentration ompared to its adjaent element, the sediment is transported towards the element with the lower onentration. This transported sediment an settle in the latter element or be transported again to the next element (it might have an even lower onentration). An important hypotheses is that the outside world is able to provide or aept sediment unonditionally as demanded by the tidal inlet system. And vieversa, the morphologial development of the tidal inlet does not influene the outside world. To omplete this explanation, we an summarise sediment exhange our depending on: the surplus or defiit of sediment in individual morphologial elements (determined by how the atual state deviates from the equilibrium state of this element) the apaity to exhange sediment between elements (determined by the differene in onentration between different elements) the availability of sediment in the system boundary If we establish a mass balane between the horizontal transport and the vertial exhange, see equation (3.5) and equation (3.7) we obtain: n Tni = ni ( n i ) = ws An ( ne n ) = ± (3.8) i i dv dt From here, we an relate horizontal and vertial sediment exhange. 3.4 Original ASMITA model equations Before writing the equations for a tidal system with a hannel, flat areas and a delta, we will present a sheme showing equilibrium relations and sediment exhange between elements: 20

9 ( ) f f ( ) d d ( ) od d E outside world wa f f f fe wa ( e) wa( ) d d d de flat wet volume dry volume hannel delta Figure 3-3 Sediment exhange lines After Kragtwijk (2001) Hene, onsidering these exhanges of sediment between elements, the mass-balane for eah element an be written as: ( ) ( ) w A ( ) + = (3.9) od d E d d sd d de d ( ) ( ) w A ( ) + = (3.10) f f d d s e ( ) w A ( ) = (3.11) f f sf f fe f Hereby we present the rest of the model equations when the system is formed by a hannel, flats and delta, that means rewriting equations (3.5) and (3.6), applied to eah element in the system: dv dt dv dt dv dt d f = w A (3.12) sd d d de = w A (3.13) s e = w A (3.14) sf f f fe 21

10 fe e de V f = E V fe V e = E V V d = E Vde r r r (3.15) (3.16) (3.17) Notie that, ompared to equation (3.6), we have onsidered r as the power in previous expressions, and r is always positive. To be in agreement with equation (3.6), in ase n is negative, nominator and denominator are inverted. From equations above we an see there are different kind of parameters, whih are needed to know how elements evolve with time. We an lassify these input data in different groups, suh as: geometrial parameters (these are physial parameters), proess parameters (we annot measure them diretly) and numerial parameters (needed to make the model run). Geometrial parameters Areas and initial volumes Initial volumes form the boundary ondition for solving the model equations. Tidal range Although the tidal range is not expliitly used in the model, it determines among other things the equilibrium volumes, as they depend on the tidal prism (P). Proess parameters Equilibrium volumes They an be obtained from ertain empirial relations. In general, they depend on the tidal prism and the tidal range, see equation (3.3). 22

11 Horizontal exhange oeffiient This orresponds to the long-term horizontal transport between adjaent elements. The value of this parameter must be given, and it bears the dimension m 3 /s, like a disharge. Equilibrium onentration The equilibrium onentration is a neessary input to make the model run. It an be obtained as the integral of the vertial distribution of the suspended sediment onentration (Van Rijn et al., 2001). However, in ASMITA model, onentrations are dimensionless, and the equilibrium onentration beomes an empirial parameter. Vertial exhange oeffiient It represents the vertial exhange and it is measured by m/s. Vertial exhange represents both the long-term residual sedimentation and erosion. Numerial parameters r parameter The value of this parameter is 2. Time-step and number of time-steps They also must be given as an input parameter in the model. Its value depends on our interest. 3.5 New ondition in the system: river disharge. How we do onsider the influene of a river: The effet of the river will be taken into aount onsidering two new onditions: an input of sediment (S) and a water disharge (Qw). Then, the lines of sediment exhange will be modified. The idea is to onsider that the disharge flows to the hannel, and from there goes, throw the delta, to the outside world, in other words, there is no diret influene to the flat areas. Furthermore, we have onsidered that tides do not affet the river, hene in our approah, sediment annot be transported from the hannel into the river; the disharge is always positive and independent of the tide. The 23

12 river is not a new element in the system but it forms a new boundary ondition for the hannel, as it adds a water flow and a sediment input New equations and other onsiderations Next we present a sheme to understand how the river has been onsidered in the tidal system: S Q w Q w Q w outside world wa f f f fe ( ) f f ( ) d d wa Q w Q w d e wa( ) d d d de flat hannel delta dry volume wet volume ( ) od d E Figure 3-4 Exhange lines in ase of river disharge After this report With the above onsiderations, the mass balane equations beome: ( ) ( ) Q Q w A ( ) + + = (3.18) od d E d d d sd d de d ( ) ( ) Q S w A ( ) ( ) w A ( ) + + = (3.19) f f d d s e = (3.20) f f sf f fe f One might notie that we have substituted writing. Q w for Q, just to simplify the Indeed, these are the only equations that hange in the model, equations that relate equilibrium volumes with onentrations remain the same, as the hypotheses of the equilibrium state existene do not disappear. 24

13 Keeping with the idea that the river is like a new boundary ondition, we ould think about whih is the ondition that determines the behaviour of the system: is the global equilibrium onentration the onentration at the steady state in eah element or does the onentration in the river have an influene in this steady state? In other words, we wonder whih will be the onentration at the equilibrium state for eah element. Looking at the hannel, one ould think there are two equilibrium possibilities for it. One ould be reahed when the onentration in the hannel would equal the onentration of the river, as the river represents a boundary ondition for the hannel, and, on the other hand, the hannel ould also be in equilibrium when its onentration would equal the outside world onentration, as the equilibrium tendeny of eah element shows. But, of ourse, one element tends to evolve to only one situation, so it is lear that the hannel will evolve to a ertain onentration whih will put it in an equilibrium state. Whether this equilibrium situation is given with a onentration equal to the global equilibrium onentration or equal to the river onentration still has to be studied (and this is what we will do in next hapters). However, to take into aount the fat that there are two boundary onditions for the hannel, we ould onsider that the outside world equilibrium onentration hanges due to the river, so that a new global equilibrium onentration might be onsidered. There are different ways to obtain a new equilibrium onentration, but, as we do not know the real influene of the river in the boundary onditions, the easiest way to obtain this new onentration ould be just alulating an average of both onentrations: Enew, ( + ) Eriver Esea = (3.21) 2 Nevertheless, we will not onsider a new global equilibrium onentration in order to see more learly whih is the influene of a river in a tidal basin. But, of ourse, this ould also be onsidered in later works as a way to represent morphology hanges with the ASMITA model. 25