1. Beach modelling III: Morphodynamic models – Description Adonis F. Velegrakis Dept Marine Sciences University of the Aegean

2. Synopsis 1 Beach morphodynamic models 1.1 Purpose 1.2 ‘Static’ and dynamic ‘bottom-up’ models 2 Static models 2.1 The Bruun (1962, 1988) model (rule) 2.2 The Edelman (1972) model 2.3 The Dean (1991) model 3 Dynamic models 3.1 Basic structure 3.2 Hydrodynamic models 3.3 Sediment transport modules 3.4 Morphological module 4 Beach retreat 1-D models 4.1 The SBEACH model 4.2 The Leont’ yev model 4.3 The Boussinesq Model

3. 1.1 Beach morphodynamic models: Purpose Diagnosis/prediction of beach response to (wave forcing) and sea level changes Basic principle: when the sea level and/or the wave forcing changes the beach profile is forced to change to a new profile Beach erosion/retreat models can be differentiated into ‘static’ models ‘dynamic’ / ‘bottom-up’ models

4. 1.2 Beach morphodynamic models: ‘Static’ models In static models, beach erosion/retreat is assessed through the solving of one or of a system of equations In these models, hydrodynamic and sediment dynamic processes are not (fully) considered Therefore, (most of) static models are used to predict effects of long-term sea level rise (ASLR) on the cross-shore beach profile; thus, these models are 1-D models Here, we will considered 3 of such models, i.e. the models Bruun (1962, 1988), Edelman (1972) and Dean (1991)

5. 1.2 Beach morphodynamic models: Dynamic ‘bottom- up’ models The basic ‘ingredient’ of the dynamic beach morphodynamic models is the coupling of hydrodynamic and sediment transport models The results of the coupled models are then used to determine morphological changes using e.g. some form of the sediment continuity equation (see below)

6. 2.1 Static models: The Bruun (1962, 1988) model (rule) Bruun model assumptions: The active cross-shore beach profile attains an ‘equilibrium’ profile Under ASLR, the equilibrium profile migrates onshore, causing erosion in the sub- aerial and deposition in the sub-marine beach Deposition occurs between the (new) shoreline and the closure depth and Seabed elevation increase due to deposition equals sea level rise Governing expression: where s, coastal retreat; l, distance to the closure depth; hc, closure depth; α, sea level rise; and Bh, berm elevation Note: There is no control by sediment size or wave characteristics, except by the most energetic waves of the year that define the closure depth (see Presentation 2) Much has been written for and against the validity of assumptions of the Bruun model (e.g. Pilkey et al., 1993; Cooper και Pilkey, 2004; Zhang et al., 2004)

7. Fig.1 Schema showing the parameters of the Bruun model. Key: S, coastal retreat; l, distance to the closure depth; hc the closure depth; α, sea level rise; Bh, berm elevation; ht, hc + B (after Slott, 2003).

8. 2.2 Static models: The Edelman (1972) model This model can be used for more realistic profiles and higher and shorter-term sea level rises (e.g. storm surges) According to the model, beach profiles maintain their basic morphology. The governing expression is: Where s, beach retreat; α, sea level rise; Β(t), instantaneous height of the profile above the current level; and hb and wb depth at wave breaking and width of the active beach inshore of wave breaking, respectively. Thus, beach retreat is controlled by the wave parameters Replacing and integrating leads to: Where Βo, initial berm elevation.

9. 2.3 Static models: The Dean (1991) model The Dean (1991) model uses the beach equilibrium profile defined by h = A x m, where Α a parameter controlled by the beach sediment grain size. Beach retreat is given by: Where hb the depth at wave breaking; Hb, wave height at breaking; Wb, width of the surf zone defined as wb = (hb/A) 3/2, where Α is a scale parameter (Α = 2.25 (ws 2 /g) 1/3 ) controlled by sediment grain size (ws, sediment settling velocity). Thus, beach retreat is controlled by both the sediment size and the wave parameters

10. 3.1 Dynamic models: Basic structure They perform calculations at different locations (nodes) of the beach (profile) and simulate its evolution in the desired time step. They consist of the following sub-models (modules): The hydrodynamic sub-model (module) which estimates beach hydrodynamic conditions (waves and wave-induced currents) with input parameters the seabed morphology (bathymetry), the offshore wave conditions, and the sediment characteristics (as bed friction control) The sediment dynamic sub-model (module) currents (and their interaction) on the basis of the hydrodynamci conditions estimated by the hydrodynamic module The sediment dynamic sub-model (module), which estimates sediment transport due to waves, wave-induced currents (and their interaction) on the basis of the hydrodynamci conditions estimated by the hydrodynamic module The morphological sub-model (module) that estiamates the new morphology on the basis of the sediment transport patterns estimated by the sediment dynamic model

11. 3.2 Hydrodynamic models I. Objective: to provide a synoptic picture, in various temporal scales, of the coastal hydrodynamics of the study area They use numerical analysis, which solves through approximations complex mathematical problems The solving method is called algorithm, and its suitability depends on 2 criteria (i) speed and (ii) accuracy Two types of potential errors: Those originating from inaccuracies in the input information (e.g. coastal bathymetry inaccuracies) and Those inherent in the algorithm

12. A hydrodynamic (circulation or wave) model requires: Bathymetric information of the best possible resolution Information on the model forcing (e.g. wind, tides, water density etc) Information on the bed type (sediment texture and forms) On the basis of the above, a model is constructed using (a)spatial and temporal discretisation techniques and (b)Resolution techniques Spatial discretisation refers to the division of the area into boxes or meshes; the hydrodynamic equations are numerically solved in 3, 2 or 1 coordinates with the models being 3D, 2DH (depth-averaged), 2DV (longitudinal) και 1DH. Temporal discretisation refers to the time-step of the solution and depends on the process to be modelled (e.g. waves, tides etc) The resolution techniques refer to the type of the mesh (finite differences- finite elements). 3. 2 Hydrodynamic models II

13. 3. 2 Hydrodynamic models III In order to estimate flows, hydrodynamic models solve a system of equations, i.e.: the momentum equations (Navier Stokes) and the mass conservation (continuity) equation Their requirements are good bathymetric data, and good information on the forcing (winds, density, tides etc). Problems with open domain boundaries. Boundary conditions must be established, which can be acquired by wider domain models Using numerical analysis, they can define flow vectors within the domain Several accomplished hydrodynamic models ((3-D, 2-D etc) are available (e.g. POM)

14. 3. 2 Hydrodynamic models IV Coastal wave models have a different construction (see below) The ultimate forcing is the wind, which generates offshore waves that they are driven inshore changing by the seabed friction Although waves in the open sea transfer only energy (not mass), inshore waves can generate currents (i.e. mass transport), generating wave- induced coastal circulation (flow) Waves and wave-induced currents can interact with natural and/or artificial structures inducing secondary circulation

15. 3.3 Sediment dynamic modules Τhe modules use as inputs the hydrodynamic model results and estimate sediment transport for each spatial step. Sediment transport can take place as bedload, suspended load and, under particular conditions, as sheet flow. Estimations can consider total sediment transport in a wave period (time-averaged), or sediment transport in shorter (intra-wave) temporal scales. They can describe sediment mobility and/or sediment transport patterns i.e. coastal sediment circulation (flow) There are issues with the non-linearity of sediment transport and particularly with the complex transfer function linking hydrodynamics to sediment mobility and sediment transport rate

16. Sediment transport due to wave - current interaction Wave current interaction may change significantly sediment transport in a non-linear way. Wave current interaction also changes sediment transport direction. Generally: (i)For bedload transport, the transport direction is controlled by the (non-linear) combination of the magnitudes/directions of the shear stresses (force per unit area) due to currents (τ c ) and waves (τ w ) (ii)For suspended sediment transport, transport direction is controlled by the current direction

17. 3.4 Morphological module Dynamic models of beach retreat can be generally differentiated into (Roelvink and Brøker, 1993): (i) profile development models and (ii) process-based models In most models, beach morphological development is estimated using the sediment continuity (conservation) equation (analogous to water continuity equation)

18. 4 Beach retreat 1-D dynamic models For the purpose of the present (RiVAMP) training, 1-D dynamic models have been selected, as they are more manageable It must, however, be understood that such models diagnose/predict beach morphological changes without taking into account lateral sediment transport i..e. longshore and/or oblique sediment transport Beach response to sea level changes is a non-linear process, depending mainly on: the rate of ASLR and the magnitude/duration of storm surges the coastal slope/morphology the impinging (and generated-infragravity) wave energy and the nature (texture/composition) of beach sediments Note: Our knowledge on costal erosion processes is still incomplete and, thus, predictions are associated with a large degree of uncertainty

19. Wave module Sediment transport module k w k w :empirical coefficient of wave dissipation Ε F Ε F : wave energy flow Ε Fs Ε Fs : constant wave energy flow K s K s : empirical coefficient of sediment transport rate, D e D e : Energy dissipation, D eq D eq : energy dissipation in equilibrium qiη ενέργεια διάχυσης σε ισορροπία ε ε: coefficiet related to the sediment transport rate for the bed slope term 4.1 The SBEACH model (Larson and Κraus, 1989) Governing expressions

20. module 4.2 The Leont’ yev model: Hydrodynamic module It is based on the energetics approach of Battjes and Janssen (1978), i,e. on the assumption that cross-shore changes in wave energy flow in each profile location equal wave energy losses due to bottom fiction φ: wave angle E w : wave energy c g : wave group celerity and D e : wave energy dissipation

21. (Leont’yev 1996): The beach profile is divided into zones, with sediment transport varying along the profile as (Leont’yev 1996): Wave refraction zone: q R = 0 και q = q W Surf zone: Surf zone: q = q W + q R Swash zone: q W = 0 και q = q R 4.2 The Leont’ yev model: Sediment transport module

22. Sediment transport rate due to wave-current interaction q W, in the refraction zone is: Bedload/suspended sediment load : friction factor w s : sediment settling velocity φ: angle of approach ε s : effectiveness coefficient F e, B e : energy losses due to bed friction and turbulence, respectively 4.2 The Leont’ yev model: Sediment transport module

23. Sediment transport in the swash zone: : maximum sediment transport c 3 =0.2-0.3 Η 0 : offshore wave height 4.2 The Leont’ yev model: Sediment transport module

24. Beach morphological change is defined by the sediment continuity equation Sediment porosity is also considered 4.2 The Leont’ yev model: Morphological module

25. This state-of-the-art model is not included in the training tool as (a)It is very heavy (a day to run) (b)It requires extensive expertise and (c)It is still under development Nevertheless, as it is used for model intercalibration (see Presentation 4), a brief description is given The model, that has been initially developed by Karambas and Koutitas (2002), includes: A wave model that is based on Boussinesq equations for dispersive, non- linear waves A sediment transport module, that calculates: 1.bedload and sheet flow using the expressions of Dibajnia et al. (2001), 2.suspended sediment load through the energetics approach 3.sediment transport at the swash zone using an expression based on Meyer- Peter & Muller 4.3 Boussinesq model

26. Mom. equations U: horizontal velocity ζ: sea level rise, h: normal sea level d = h + ζ τ b : bed shera stress Β i = 1/15 Ε v :eddy coefficient Μ u :effect of the non-normal velocity distribution Continuity equation 4. 3 Boussinesq hydrodynamic module

27. The expression for the estimation of the sediment transport sheet flow q b due to non-monochrmatic waves is: bedload, : velocities under crest and trough, : the associated durations, : percentages of sediment that are directly transported, : percentages of sediment that remain after the half period : settling velocity 4. 3 Boussinesq sediment dynamic module

28. Suspended load : mean wave energy diffusion due to breaking : current velocity : parameter that relates nera bed value to mean value : effectiveness coefficient 4. 3 Boussinesq sediment dynamic module

29. Sediment transport rate ate the swash zone is estimated by a modified Meyer-Peter & Muller model:bedload : non dimensional transport rate : Shields parameter : porosity : angle of repose : bed slope : parameter that takes different value whne the water moves onshore 4. 3 Boussinesq sediment dynamic module

30. It estimates profile changes using: Numerical solution of the continuity equation An additional gravity term to take into account the bed slope 4. 3 Boussinesq morphological module

31. Thank you!! See you later

32. Fig. 2 Edelman (1972) model. Key:, Key: S, beach retreat; α, sea level rise; Β(t) the instantaneous height of the beach profile above the current level; and hb και wb the breaking wave zone depth and the width of the active beach inshore of the wave breaking, respectively (after CEM, 2008).

33. initial morphology Offshore wave conditions sediment size hydrodynamicssediment transport new morphology If time < wished final morphology Fig. 3 Generalised flow diagram of dynamic models

34. Fig. 3 Sketch showing models with different spatial discretisation

35. Fig. 4 Sketch showing nested models of different spatial discretisation. The output of the larger scale-small discretisation area is used for the boundary conditions at the open boundaries of the coastal models.

36. Fig. 5 Modelled currents in the Gera Gulf, Lesbos, E. Med (Stagonas, 2004)

37. Fig. 6 Depth-integrated 2-D tiadal flow model for a region of the English Channel Bastos et al., 2003)

38. Fig. 7. 2-D numerical model results for wave heights (a) and wave-induced currents (b) at the Negril beach. Conditions: Offshore wave height (Hrms) = 2.8 m, Tp=8.7 s. Waves approach from the northwest. Note the diminishing wave heights and changed nearshore flow patterns at the lee of the shallow coral reefs (RiVAMP, 2010)

39. Fig. 8. Wave height contours Ηs (m) at beach with the breakwater (Karambas et al., 2007)

40. Fig. 9. 3-D schema of wave transmission over a submerged breakwater, breaking and wave run-up on the beach from a pseudo 3-D Boussinesq model (Koutsouvela, 2010),

41. Fig.10 East Lesbos beach, E. Mediterranean. Wave generated currents and seabed morphology development under weighed N, NE and E waves, in the presence of a breakwater. Solid line, initial bathymetry; strippled line, final bathymetry (Karambas et al., 2007).

42. Fig. 11 Velocity field and morphological change (red, initial bathymetry and grey, final bathymetry) for wave transmission coefficient Kt = Ht/Hi = 0.8. Step = 1 m (Koutsouvela, 2010) Initial distance of structure from coastline =150 m, LG, structure spacing = 120 m, LB, breakwater length = 120 m. Monochromatic waves, Ηο =1.5 m, T = 8s Note the development of salients (accretion) behind the structures, but alos the very strong (and dangerous currents) between the structures

43. Fig. 12 Sediment transport due to tidal currents in a region of the northern coast of the English Channel (Bastos et al., 2003).

44. Fig 13. Potential resuspension time under: (a, c, e) tidal currents alone (spring tides); and (b, d, f) tidal currents (spring tides) and waves approaching from the west (significant wave height Hs = 1 m, Period = 8 s). Potential resuspension time has been estimated for: particles sizes 0.040 mm (a, b); 0.100 mm (c, d); and 0.200 mm (e, f). Seabed mobility under: (g) spring tidal currents alone and (h) spring tidal currents and waves. Both potential resuspension and seabed mobility times are expressed as percentages of the total tidal cycle time, during which bed shear stress exceeds the critical shear stress for the initiation of resuspension/movement. Origin (0, 0) of grid is at 49.293 N 0, 2.363 W 0.(Velegrakis et al., 1999)

45. In all sedimentary environments, if there is a difference between the sediment input and output through a control volume, then this should represent either bed deposition or erosion. If i x is the sediment input, i x + δ x the sediment output and q the sediment that settles through the water column, then for a time δ t : (i x - i x +δ x ) δ x δ t + q δ x δ x δ t = δ z δ x δ x Dividing by δ x δ x δ t (i x - i x +δ x )/δ x + q = δ z /δ t And if δ x → 0 and δ t → 0 then -  I /  x + q =  z /  t Sediment continuity equation

46. The SBEACH model (Larson and Κraus, 1989). Fig. 14 Discretisation of the SBEACH model (after CEM, 2008). It approximates the sediment continuity equation with finite differences and a step-mesh of discretisation. Vertical changes in the water depth h are defined by the horizontal gradients of sediment transport rate q.

47. Fig. 15 Coastal wave zones. Longshore transport in the coastal zone occurs mainly in the surf and swash (wave run up) zones (After SEPM, 1996). Key: h, water depth; H, wave height; L, wave length. Surf zone Swash zone >