MODELLING THE SWASH ZONE FLOWS. Maurizio Brocchini, MCFA Member

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1 MODELLING THE SWASH ZONE FLOWS Maurizio Brocchini, MCFA Member D.I.Am., University of Genova, Via Montallegro 1, 1145 Genova, Italy. ABSTRACT An overview is given on recent research developments and ongoing studies on the modelling of the water motions occurring in the swash zone. 1. INTRODUCTION The swash zone (SZ hereinafter) is that part of the beach where waves move the shoreline, i.e. the instantaneous boundary between wet and dry (see figure 1). taken up by scientists of different disciplines: from the coastal engineers to the applied mathematicians, from the geomorpholigists to the biologists. The main features of SZ dynamics and the most important results on their modelling are briefly described with the following order. The net section is dedicated to an overview of SZ modelling at the time scale of the shortwaves. In section 3 the SZ role in generating/interacting with low frequency waves (LFW hereinafter) is illustrated. Finally, some remarks on ongoing research and suggestions for future studies are collected in section SZ MODELLING FOR BREAKING AND NON-BREAKING WIND WAVES As already mentioned most of the momentum and energy forcing of the SZ occurs at the short-wave scale. These waves propagate from the offshore region towards the coast and undergo a number of transformations mainly caused by the interaction with a seabed of decreasing depth. The most spectacular transformation of all is wave-breaking which can be seen as the ultimate stage of wave-steepening. However, breaking qualitatively differs from most of the other wave phenomena in two respects. Breaking is responsible for: 1. most of wave energy dissipation as intense turbulence is generated at the front face of the breaker; Figure 1: The swash zone on a beach. Although wind-waves or short-waves (typical period of about 10 seconds) are the major forcing for the SZ dynamics it has been recognized the importance of the SZ for the generation/transformation of long-period motions [2, 8, 14, 19]. Hence, modelling of SZ flows is greatly complicated by both intermittency (i.e. alternance in time and space of wet and dry phases, see for eample Schüttrumpf, Bergmann and Dette [17] and Brocchini []) and by the occurrence of motions with different time scales. Notwithstanding these difficulties very recently important efforts are being made to achieve a better, general understanding of the SZ water flows (E.U. SASME Project [12]). In particular modelling is being developed not only to predict run-up levels but also to represent the complete dynamics of this boundary region (Brocchini and Peregrine [9]; BP9 hereinafter). The challenge of understanding and reproducing the SZ dynamics has been gradually 2. momentum and energy transfer to low frequency modes like (LFW, longshore currents, rip-currents, shear waves, etc.). The importance of breaking is also reflected in SZ motions which are markedly different for non-breaking and breaking waves. These differences are most easily illustrated by means of the very few analytical solutions of the Nonlinear Shallow Water Equations (NSWE hereinafter) which are the most used set of equations for analysing SZ dynamics. In dimensionless form and neglecting seabed friction these equations read for a beach of equation (see figure 2 for some flow and coordinates definitions)!" #! %$'&!" ( # (2.1a) (2.1b) (2.1c) where subscripts are used to represent partial differentiation, and are respectively the onshore and longshore

2 ; 5 " * +, -. ) Figure 2: Basic flow and coordinates definitions. coordinates,, -. /0 * is the total water depth and and are respectively the onshore and longshore components of the depth-averaged velocity (horizontally 2D flow or 2DH). a linear equation, the nonlinearity being transferred to the hodograph relations. Linearity of the governing equation allows for mode superposition as described in BP9. In the same paper the solution of Carried and Greenspan is also etended to predict the longshore velocity component in the case of weakly-2dh flow. Contour plots of figure 3 illustrates the structure of this solution for a specific pair of modes which do not give wave breaking. The solution of BP9 also allows for direct computation of the mean longshore velocity (over the period of short waves) associated with the run-up/run-down cycle (see figure 4). It is clear that such velocity is non-negligible even for non-breaking waves and is essentially caused by the intermittency of the SZ motions. E F (a) (b) Figure 7& 4: Mean G18:9 longshore H velocity,8i& J& in the swash zone for, 5 and 5 (; for all the cases). The seaward limit of the swash zone is given by the intersection of the curves and the -ais. (c) (d) Figure 23 4" 3: Superposition of two waves. Contour + -. plots of: (a) the hodograph coordinates in the plane, (b) the free surface elevation *, (c) the onshore velocity and (d) the longshore velocity. Dimensionless amplitudes 7&8:9 =18 9 =18:9 and7& 8:A frequencies are 5', ;< and 5?, 4 respectively. Lines of constant run from left to right whilst lines of constant 2 run from top to bottom. This canonical beach problem only allows for a few analytical solutions. The most famous is that of Carrier and Greenspan [11] valid for a unidirectional BC&D non-breaking, standing wave of amplitude 5 and frequency &. This can be obtained by using a clever hodograph transformation which reduced the 1DH version of (2.1) to As anticipated SZ water flows due to breaking waves are essentially different from those just described as it can be seen by analysing the run-up solution of Shen and Meyer [15]. In this case very thin sheets of water (the water surface becoming tangent to the seabed) are pushed up the beach by the breaking waves in contrast to the wedge-type water masses which characterize the SZ of non-breaking waves. Although different in character SZ flows of both nonbreaking and breaking waves induce a mean longshore mass flu which is proportional to the square of the dimensionless amplitude of the incoming waves [8]. 3. THE SZ: A LOCUS FOR MODIFICATION/GENERATION OF LOW FREQUENCY MOTIONS It is well known that both nonlinearity and groupiness of short-waves are the major responsible for the generation of long waves. These waves can be either bound to a group of short waves (i.e. propagating at the group velocity [13]) or free. Essentially these can either be bound long waves freed from the group structure by wave breaking which dissipates the short-waves (see figure 5) or the result of short-waves interacting with each other in the SZ [19].

3 Figure 5: A wave group (thin dashed) on a beach, with the Fourier (heavy dashed) and wavelet (solid) filtered signals magnified and superimposed. [Courtesy of T. Barnes (see [3])]. M 3N PO Q Figure : Bores interaction in the swash zone. Input signal: Carrier and Greenspan wave of 5 and ;. Top: perspective view of the space-time plot of surface elevation. Middle: characteristics (thin), bore paths (heavy) and shoreline (heavy). Bottom: incident (thin) and outgoing (heavy and magnified of about 9 times) Riemann invariants.

4 s c r z w - The first mechanism is particularly significant in the case of a saturated surf-zone, in that case the low frequency motion of the shoreline is dominated by the run-up of low frequency cross-shore standing waves. Since in shallow waters the wave velocity is proportional to the square root of the total water depth, large waves travel faster than small waves and catch them up hence producing a single wave. This is the mechanism by which frequency downshift occurs as described by Mase [14]. This type of LFW generation is more important in the case of an unsaturated surf-zone [2]. Figure gives an eample of how the generation of free LFW by shortwaves coalescence in the SZ is typically modelled in the framework of the NSWE. The phenomenon of LFW generation in the SZ is strongly governed by the ratio of the two most important time scales which are the natural swash period R3S (time for a run-up/run-down cycle of a wave of given amplitude) and the group period R3T (sum of the periods of the waves in the group). For RTUVR S the amplitude of the LFW propagating from the SZ is maimum while it rapidly decreases for both R3T#WXR S and R3T#YZR S [19] (see figure 7). LFW Amp. (a) R S R3T LFW Amp. Figure 7: The dependence of the LFW amplitude on R"[ : \18I& (a) the amplitude of the input short waves is 5, \18:] (b) the amplitude of the input short waves is 5. Parametric effect of varying either R (thin) or ^ (thick) is shown. Adapted from Watson et al. [19]. (b) 4. DISCUSSION OF ONGOING AND FUTURE RESEARCH ON MODELLING SZ FLOWS It is becoming more and more clear that modelling SZ flows is essential both for the physical significance of the phenomena occurring in this narrow region and for the importance of the SZ as a boundary region for nearshore numerical models, be they flow solvers at the scale of the short-waves [4, 10, 1] or solvers which model the flow with a wave-averaged description [18]. Notice that when the full SZ is included in a numerical computation it not only involves a larger domain of integration with a special boundary condition at the shoreline but also frequently determines the maimum allowable time-step. Hence, an integral representation of the SZ could be very useful. This representation has already R S R3T been made available in BP9 and is now the object of considerable research efforts. _ 1` "a envelope of the run-downs ` Figure 8: Sketch of the domain of integration used to define the integral model equations. The model is based on the integration across the SZ of a conservation form of the NSWE. _ The SZ boundaries are position of the highest run-up and the lower boundary ` which can be taken either as the lowest run-down position in a group of waves or the envelope of the run-down locations (see figure 8). ` If integral properties of the water shoreward of are considered the following set of equations can be obtained (see BP9 for more details): b3c b"i b"i b"i b"r d e-. Df hg $ b j I-. kf hg $ c bm e-. l$ b n $#o bm j I-. kf hg $ n b. $qo e-. p b h tsu I-. nd I-. kf vg $ i I-. k$ b s'd g e-. p y${z I-. }8 (4.2a) (4.2b) (4.2c) (4.2d) These equations introduce a number of new flow properties (both local and integral) which are listed in table 1. Name Epression Flow property d'~ j3~k i ~ m ~k o?~ ~ ~, ' k~k ƒ ƒh! hg? hg d ~' v g j3~k? vg ƒ n vg ˆ ~' v g ~ ˆ ~' Local water depth Local mass flow Local momentum flu tensor Local energy density Water volume in SZ Water momentum in SZ Integrated momentum flu tensor Water energy in SZ Friction force in SZ Frictional work in SZ Table 1: Definition of the flow properties adopted in equations (4.2).

5 Some preliminary numerical eperiments have shown they aremore than adequate for reproducing SZ flows by just prescribing flow properties at the boundary,` i.e. without resolving any equation in the SZ domain (see BP9). The integral model is currently being used also for reproducing SZ dynamics on the basis of eperimental data [1]. An eample of application of the integral model to eperimental data is shown in figure 9 in which the measured shoreline is compared to that predicted by the integral SZ model on the basis of the measured local mass flow at `. (m) t(s) Figure 9: Eample of comparison of measured (solid line) and predicted shoreline (dashed line). It is clear that the integral model is potentially very useful for both practical and research applications. To this purpose research is underway to incorporate the effects of seabed friction, previously neglected, in the model. This can be accomplished by prescribing a suitable parametrization of the frictional forces in terms of the model variables. Archetti and Brocchini [1] have shown that a suitable parametrization for unidirectional flow propagation is: o f i f i ŠhŒ ŒVŽ c ` with Ž V < M (4.3) where < is a dimensionless bed friction coefficient of the order of,8:,& and M is the beach slope. In the same paper it is also shown that inclusion of frictional contributions does improve the perfrormances of the integral model at least for gently sloping beaches where frictional effects are more significant. Work is still in progress to etend the parametrization of the frictional forces to the case of 2DH flow conditions. However, the most important contribution of the integral model is to provide the basis for prescribing shoreline boundary conditions for wave-averaged models. Among them we can mention the quasi-3d flow solver SHORE- CIRC [18] which is one of the most complete and reliable wave-averaged models currently available. However, though well developed both theoretically and numerically wave-averaged flow models do not properly take into account swash zone motions. Indeed their shoreward boundary is incorrectly defined, this being the intersection of the mean water level with the beach face. It is easy to show that in the case of periodic wave this point coincides with the maimum run-up location. Moreover, wave-averaged models always rely on wave drivers which are used to prescribe the short-wave forcing for the long-wave motion. Unfortunately such wave drivers use wave theories, like linear wave theory, not compatible with the local dynamical conditions. This problem is most important in the limit of vanishing water depths i.e. in the SZ where linear theory is obviously wrong. It is therefore auspicable that new shoreline boundary conditions for the mean flow be derived to account for SZ dynamics. To this purpose Bellotti and Brocchini [5] are working on the implementation of the mean shoreline boundary conditions for wave-averaged models described in BP9 and based on an integral description of the SZ. Preliminary results of numerical computations reveal that the boundary conditions described in BP9 are well posed. It is also shown what the logical steps of the implemetation procedure of these conditions should be and what are the most delicate numerical issues involved in the implementation. The above discussed two lines of research will hopefully lead to significant improvement of the mathematical/numerical modelling of SZ dynamics. It is, however, clear that a number of issues are still open and must be solved before an adequate modelling of SZ flows can be achieved. Proper description (i.e. specific for the SZ) of phenomena like turbulence, percolation through the seabed, bed friction is still needed and represent interesting challenges. 5. ACKNOWLEDGEMENTS Part of this research work was undertaken during the author s stay at the University of Bristol. This was made possible by the E.U. economical support through a Human Capital and Mobility Programme Grant (contract number CHBI-CT-93078). Thanks are also etended to the Marie Curie Fellowship Association of which the author is Member (membership N.2938).. REFERENCES [1] R. Archetti and M. Brocchini An integral swash zone model with friction: an eperimental and numerical investigation, Submitted to Coastal Engng. (2001) [2] T.E. Baldock, P. Holmes and D.P. Horn, Low frequency swash motion induced by wave grouping, Coastal Engng (1997). [3] T.C.D. Barnes The generation of low frequency waves on a beach, Ph.D Dissertation. University of Bristol, Bristol, U.K. (199). [4] G. Bellotti and M. Brocchini, The shoreline boundary conditions for Boussinesq-type models, Accepted for publication in Int. J. for Num. Meth. in Fluids (2001). [5] G. Bellotti and M. Brocchini, Swash zone modelling for wave-averaged solvers: an eample of implementation, in preparation (2001). [] M. Brocchini, Flows with freely moving boundaries: the swash zone and turbulence at a free surface, Ph.D Dissertation. University of Bristol, Bristol, U.K. (199).

6 [7] M. Brocchini, Eulerian and Lagrangian aspects of the longshore drift in the surf and swash zones, J. Geophys. Res (1997). [8] M. Brocchini, The run-up of weakly-two-dimensional solitary pulses, Nonlin. Proc. in Geophys (1998). [9] M. Brocchini and D.H. Peregrine, Integral flow properties of the swash zone and averaging, J. Fluid Mech (199). [10] M. Brocchini, R. Bernetti, A. Mancinelli and G. Albertini, An efficient solver for nearshore flows based on the WAF method, Accepted for publication in Coastal Engng. (2001). [11] G.F. Carrier and H.P. Greenspan, Water waves of finite amplitude on a sloping beach, J. Fluid Mech (1958). [12] Europen Union Surf And Swash Zone Mechanics Project, Contract number MAST3-CT [13] M.S. Longuet-Higgins and R.W. Stewart, Radiation stresses in water waves: a physical discussion, with applications, Deep-Sea Res. 11, (194). [14] H. Mase, Frequency down-shift of swash oscillations compared to incident waves, J. Hydraulic Res. - IAHR (1995). [15] M.C. Shen and R.E. Meyer, Climb of a bore on a beach 3: Run-up, J. Fluid Mech (193). [1] H.A. Schäffer, P.A. Madsen and R.A. Deigaard A Boussinesq model for waves breaking in shallow waters, Coastal Engng (1993). [17] H. Schüttrumps, H. Bergmann and H.H. Dette, The concept of residence time for the description of wave run-up, wave set-up and wave run-down, Proc. 24th Int. Conf. Coastal Engng. - ASCE (1994). [18] A.R. Van Dongeren, F.E. Sancho, I.A. Svendsen and U. Putrevu, SHORECIRC: a quasi 3-D nearshore model, Proc. 24th Int. Conf on Coastal Engng. - ASCE (1994). [19] G. Watson, T.C.D. Barnes and D.H. Peregrine, The generation of low frequency waves by a single wave group incident on a beach, Proc. 24th Int. Conf. Coast. Engng. - ASCE (1994). [20] G. Watson, D.H. Peregrine and E.F. Toro, Numerical solution of the shallow-water equations on a beach using the weighted average flu method, In Computational Fluid Dynamics 92 - Vol. 1 (ed. Ch. Hirsh et al.). Elsevier Science Publishers (1992).

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