Numerical Simulation of Long-Shore Currents Induced by Regular Breaking Wave

Journal of Coastal Research Special Issue 5 15- Florida, USA ISSN 79- Numerical Simulation of Long-Shore Currents Induced by Regular Breaking Wave Yong-Ming Shen, Jun Tang, and Wenrui Huang State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 113, P. R. China, ymshen@dlut.edu.cn Department of Civil & Environmental Engineering, College of Engineering, Florida A&M University and Florida State University, 55 Pottsdamer St., Tallahassee, FL 331-, USA, whuang@eng.fsu.edu ABSTRACT The hydrodynamics of coastal zones are extremely complicated, being influenced greatly by shallow water waves and currents induced by wave breaking. This paper presents numerical simulations of long-shore currents induced by the breaking of oblique incident waves in shallow coastal zones. The wave numerical model is based on parabolic mild slope equation, and so the wave radiation stress required for the generation of wave-induced currents are calculated based on the variables in the parabolic mild slope equation, and the long-shore currents have been numerically simulated based on these. The numerical models are validated against experimental data, and the results suggest that the long-shore current velocity and wave set-up increase with the increasing incident wave amplitude and offshore slope steepness, as well as the wave set-up increase with the increasing incident wave period. ADDITIONAL INDEX WORDS: regular wave, parabolic mild slope equation, radiation stress, long-shore currents INTRODUCTION The hydrodynamics of shallow coastal zones are often extremely complicated, as evident by the distortion and breaking of water waves, and the presence of near-shore currents induced by wave breaking. Many authors have studied and developed theories for the wave-induced near-shore currents (including long-shore and rip currents). LONGUET-HIGGINS and STEWART (19, 19) were the first to explain in terms of the wave radiation stresses the generation mechanisms for the wave-induced currents; later, LONGUET-HIGGINS (197a, 197b) analyzed longshore currents using the concept of wave radiation stresses. MEI (197), BETTESS and BETTESS (19), COPELAND (195), and DING et al. (199) studied various aspects of the radiation stresses for linear water waves and proposed formulas for them. LIU and DALRYMPLE (197), DODD (199), and HALLER and KIRBY (1999) considered the bottom frictional stresses of longshore currents. More recently, CHEN and SVENDSEN (3), MILLER and DEAN (), and PRASAD and SVENDSEN (3) examined the effect of boundary conditions on near-shore currents. This paper present a numerical study on the application of the parabolic mild slope equation for waves to long-shore currents induced by the breaking of waves propagating to the coastal zone. The numerical model is validated by comparing computed results with experimental data. Since it is difficult to ascertain a wave angle for a wave which may undergo apparent transformation as it propagates to shore, the wave radiation stresses imposed on a near-shore current by the waves are difficult to determine as well. In this regard, the wave radiation stresses can be calculated based on the parabolic mild slope equation. It is demonstrated here that this coupled numerical model works efficiently. GOVERNING EQUATIONS Parabolic Mild Slope Equation The classical mild slope equation put forward by BERKHOFF (197) has now been extensively employed in the study of propagation of water waves in mild slope areas. Several authors have studied and made progress on numerically solving the elliptical mild slope equation (e.g. LI, 199; ZHAO and

1 ANASTASIOU, 199; TANG et al., ). If the waves propagate in a principal direction, and only the forward propagation of waves is considered, the parabolic mild slope equation can be used to simulate a relatively large wave field in an economical and efficient manner. The parabolic mild slope equation that was derived by KIRBY (19) reads as follows: Wave-Induced Current Model In mild slope coastal regions where the water column is shallow, the movement of water particles is assumed to be uniform across the water layer, and therefore can be described by the two-dimensional depth-integrated equations of motion: A icg 1 D 1 C b g Cg + i( k ak) Cg + D A + + A x E x b1 A i k CCg + a1 b1 + ωk x y ω k ( Cg ) b 1 k x x A + CC g = ω k kc g (1) η + + + + = t x 1 ρ ( U( h η) ) ( V( h η) ) U U U η 1 S S xx xy + U + V + g + + t x x ρ( h+ η) x ( ) ( τ τ ) h + η A = ηx bx mx (3) () where = C ( cosh ( kh) + tanh ( kh) ) sinh ( ) 3 D k C g kh ω H H Db = ρ g α π b A(x,y) denotes the complex wave amplitude, i is the imaginary unit, k(x,y) is the wave number, ω is the wave angular frequency, C=ω/k is the wave velocity, C g = ω/ k is the wave group velocity, kx ( ) is the average of k(x,y) over the y-direction (it is assumed that the wave propagates principally in the x-direction), h is the still-water depth, a, a 1, and b 1 are the coefficients of the rational approximation determined by the varying aperture width θ a, D is the wave nonlinear effect term, D b is the wave energy loss term due to wave breaking effect and D b = if none wave breaking occurs, and a 1.5 (VAN RIJN and WIJNBERG, 199). It is further assumed that the wave begins to break if the wave height H exceeds the breaking wave height H b, which is usually defined as follows(van RIJN and WIJNBERG, 199): Hb.π = min( γ h, tanh kh ) () k where γ.. is the wave breaking index. For the parabolic mild slope equations, only the lateral boundary conditions are needed besides the incident boundary. The lateral boundary condition for Eq. () is prescribed as follows: V V V η 1 Syx Syy + U + V + g + + t x y ρ( h+ η) x 1 ηy by Am y = ρ ( ) ( τ τ ) h + η where U and V are the wave-induced current velocities in the x- and y-directions, respectively; η is the wave set-up or set-down; S xx, S xy, S yx and S yy are wave radiation stresses; τ ηx and τ ηy are the friction stresses in the water surface in the x- and y-directions, respectively; τ bx and τ by are the friction stresses in the water bottom in the x- and y-directions, respectively; A mx and A my are the lateral mixing stresses in the x- and y-directions, respectively. LONGUET-HIGGINS and STEWART (19, 19) were the first to propose and to use the concepts of wave radiation stresses to explain surf beats. Later, MEI (197), BETTESS and BETTESS (19), COPELAND (195), and DING et al. (199) derived various formulae for the wave radiation stresses based on different wave theories. For a simple wave field in which the propagating direction of waves is well defined, the radiation stress formulas proposed by LONGUET-HIGGINS and STEWART (19, 19) can be readily used to determine the wave-induced currents. In contrast, for a complicated wave field in which the waves undergo transformation, the wave direction is not distinct, and the radiation stress formulas proposed by LONGUET-HIGGINS and STEWART (19, 19), which work for a pure progressive wave, become unfit to use. In the present study, several alternative formulas proposed by ZHENG et al. () on the basis of the parabolic mild slope equation, and are more applicable to complicated wave fields, are applied to calculate the wave radiation stresses. The wave radiation stress formulas as proposed by ZHENG et al. () are as follows: (5) A = icak r sina where c r 1. is wave reflection coefficient on the lateral boundaries, a is wave angle along the boundaries. ρg A 1 kh kh Sxx = + ika 1 A + + x k sinh kh s inh + ika + k A k x kh tanhkh 1 A A + kh ()

17 S yy S g A 1 kh kh 1 A sinh sinh ρ = + + + k kh kh + ika + k A k x y kh tanhkh 1 A A xy ρg A A 1 kh = Re ika 1 + + x k sinhkh (7) () where A(x,y) is the complex wave amplitude corresponding to that appearing in the parabolic mild slope equation (1), A is the conjugate of A(x,y), and the other variables are the same as those appearing in Eq. (1). conditions are assumed to be at the state of rest, which is: η = (1) U = (15) V = (1) The boundary conditions are prescribed as follows: of the offshore open sea boundary, no-flow condition is imposed by setting the velocity components and the wave set-up component equal to zero, assuming that the offshore open sea boundary is far enough from the wave surf zone, and no current other than the wave-induced current flows into the computational regions: The following expressions for the bottom friction stresses under waves and weak currents (where the waves are nearly perpendicular to the currents) are used: τb x = ρcuu f (9) π τb y = ρcuv f (1) π where μ is the bottom velocity amplitude, which can be given by the linear wave theory as u =πa wb /T, and a wb =H/(sinhkh) is the bottom horizontal trajectory amplitude, c f is the empirical wave friction coefficient in the presence of currents (typically in the order of 1-). The lateral mixing stresses are defined as follows: A A mx my U U μ μ = + (11) x x V V μ μ = + (1) x x where μ is the lateral mixing factor which is defined in this paper as: η 1,j = (17) U 1,j = (1) V 1,j = (19) where the subscripts for the variables denote the positions of the boundary grids at the offshore open sea boundary. The onshore boundary conditions require that there is no flow into the beach as the model approaches a steady state, and the slip boundary condition is used: η x ni,j = () U ni,j = (1) V x ni,j = () where the subscripts of the variables denote the positions of the boundary grids on the onshore boundary. Along the lateral open sea boundaries, continuous boundary conditions are imposed, on assuming a gentle bottom slope on these boundaries: η = (3) μ = Nx gh (13) l where x l is the distance between the breaking point and the shore line, N.1 is a non-dimensional coefficient, and μ can be taken as a constant equal to the value at the wave breaking point. As is discussed below, the value of N can affect the distribution of wave-induced currents, and hence the stability of the wave-induced current model. Wave characteristics are determined following the procedure of solving the parabolic mild slope equation (1). The initial and boundary conditions for the wave-induced current model are specified as follows. Here, U is set to be the normal velocity to the beach boundary, and V is the velocity parallel to the beach boundary. In all applications of the model, the initial U = () V = (5) A finite difference scheme has been used to solve the problem formulated above. A regular staggered uniform grid system has been adopted, and is depicted in Figure1, where U is evaluated on the position of, and V is evaluated on the position of, while η and other variables are evaluated on the position of. The ADI (alternating direction implicit) scheme is applied to discretize the time-space domain for the initial-boundary value problem, in which a one-time step is divided into two half-steps. Essentially, in the first half-time

1 step, the explicit method is applied in the x-direction, while the implicit method is applied in the y-direction. This is followed by application of the explicit method in the y-direction, and the implicit method in the x-direction in the second half-time step. A tri-diagonal matrix for η is hence formed, which can be solved readily in each half-time step. For computational stability the bottom friction and lateral mixing terms are lagged on half-step in time in each advancement through the half-time step. and plane slope, H is the incident wave height, T is the incident wave period, γ is the wave breaking parameter, c f is the empirical friction coefficient for a bottom under both waves and currents, and N=.1 is the non-dimensional coefficient required in computing the lateral mixing factor in Eq. (13). J+1 J J-1 I-1 I I+1 Figure. Top view of experimental set up (units: m). Figure 1. Variable nodes in a staggered grid system. Table 1. Calculation parameters in the numerical model The nonlinearity of the problem makes it impossible to deduce an analytical stability criterion in terms of the time step t that is required in numerically solving the wave-induced current model. Here, the Courant stability criterion is employed to guide the choice of a time step t, which according to EBERSOLE and DALRYMPLE (19) is given by: ( Δ x) + ( Δy) Δt () gh max where x and y are the spatial discretizations in the x- and y-directions, respectively. EXPERIMENTAL VALIDATION OF THE NUMERICAL MODEL An experimental study on wave-induced currents has been carried out in the State Key Laboratory of Coastal and Offshore Engineering of the Dalian University of Technology, and the experimental results, reported by WANG (1), are used here to validate the numerical model described above. The experimental bottom topography is set up as shown in Figure. In the experiment, longshore currents are generated as the incident waves break on climbing up the inclined plane. The parameters used in the numerical model in each experimental case are listed in Table 1, where h is the water depth in flat regions, θ is the angle between the incident wave case plane slope h / m θ / ( o ) H / m T/ s c f γ case 1 1 :.5 3.5 1..9.7 case 1 :.5 3.9 1..9.7 case 3 1 :.5 3.5..9.7 case 1 : 1.1 3.3 1.5.5. case 5 1 : 1.1 3.3..5. The computations are run until the results approach a steady state. Comparisons between the experimental and numerical results of the waves, wave set-up, and wave-induced longshore currents for cases 1 through 5 are shown in Figures 3 7, respectively, and the numerical results of the wave-induced longshore current fields are shown in Figure. It is clear from Figures 3 7 that the computed results generally agree with the measured ones in all the eight cases. The results also suggest that the long-shore current velocity and wave set-up increase with the increasing incident wave amplitude and offshore slope steepness; as well as the wave set-up increase with the increasing incident wave period. It is found that the empirical friction coefficient c f may have an appreciable effect on the velocity of the wave-induced current and the associated wave set-up. However, this coefficient is in general difficult to evaluate directly and is usually estimated by empiricism according to the specific bottom topography. Also, the input value of N for the lateral mixing factor can materially affect the stability of the model and the distribution of the wave-induced current velocity near the wave breaking line. The results reveal that a greater value of N can lead to a broader distribution of the wave-induced current near the wave breaking

H (m) H (m) H (m) 19 line, and therefore a smaller longshore current velocity and set-up at the breaking point...5.1.5.15.1.5..3..1 1 1 -.5 1 1 1 1 numerical results experimental results Figure 3. Comparison between the numerical and experimental results for case 1...5.1.5.15.1.5..3..1 1 1 -.5 1 1 1 1 numerical results experimental results Figure. Comparison between the numerical and experimental results for case...5.1.5.15.1.5..3..1 1 1 -.5 1 1 1 1 numerical results experimental results Figure 5. Comparison between the numerical and experimental results for case 3.

H (m) H (m).... 1 1.1.5 -.5 1 1.5..15.1.5 1 1 numerical results experimental results Figure. Comparison between the numerical and experimental results for case...1... 1 1.5 -.5 1 1.5..15.1.5 1 1 numerical results experimental results Figure 7. Comparison between the numerical and experimental results for case 5. 1 1 1 1 1 1.5 (m/s).5 (m/s) 1 1 1 1 1 1 1 1 1 1 (a) case 1 (b) case

1 1 1 1 1 1 1.5 (m/s).5 (m/s) 1 1 1 1 1 1 1 1 1 1 (c) case 3 (d) case 1 1 1.5 (m/s) 1 1 1 1 1 (e) case 5 Figure. Numerical results of the wave-induced longshore current fields. CONCLUSIONS Waves and wave-induced currents dominate the hydrodynamics in a shallow coastal zone, as they play a central role in the transport of sediments and the morphological change of coast lines. The numerical modeling of wave-induced long-shore currents in a shallow mild slope zone has been performed in the present study. Experimental measurements of waves and wave-induced long-shore currents produced as waves propagate obliquely to the shore have been used to validate the numerical model,and the numerical results suggest that the long-shore current velocity and wave set-up increase with the increasing incident wave amplitude and offshore slope steepness, as well as the wave set-up increase with the increasing incident wave period. Since the wave propagation direction is difficult to ascertain for a wave which may undergo obvious transformation as it propagates in shallow coastal zones, the wave radiation stresses are evaluated in this paper based on the theory of the parabolic mild slope equation. The approach taken is feasible both in terms of economy and efficiency for application to a relatively extensive coastal region. ACKNOWLEDGMENTS This research was financially supported by the National Basic Research (973) Program of China under Grant No. 5CB7 and the National Natural Science Foundation of China under Grant Nos. 579 and 5779. LITERATURE CITED BERKHOFF, J.C.W., 197. Computation of combined refraction-diffraction. Proceeding of the 13th Conference on Coastal Engineering (Vancouver, Canada), pp.71-9. BETTESS, P., and BETTESS, J., 19. A generalization of the radiation stress tensor. Applied Mathematical Modelling, 1-15. CHEN, Q., and SVENDSEN, I.A., 3. Effects of cross-shore boundary condition errors in nearshore circulation modeling. Coastal Engineering, 3-5. COPELAND, G. J. M., 195. Practical radiation stress calculations connected with equations of wave propagation. Coastal Engineering 9, 195-19.

DING, P.X.; KONG, Y.Z., and SHI, F.Y.,199. Radiation stress of water waves and its calculation. Journal of East China Normal University 1, -7 (in Chinese with English abstract). DODD, N., 199. On the destabilization of a longshore current on a plane beach: bottom shear stress, Critical conditions, and onset of instability. Journal of Geophysical Research 99(C1), 11-. EBERSOLE, B.A., and DALRYMPLE, R.A., 19. Numerical modeling of nearshore circulation. Proceeding of the 17th Conference on Coastal Engineering (Sydney, Australia), pp. 71-75. WANG, S.P., 1. Study on the Longshore Currents. Dalian:Dalian University of Technology, Master s thesis (in Chinese). ZHAO, Y., and ANASTASIOU, K., 199. Modeling of wave propagation in the nearshore region using the mild slope equation with GMRES-BASED iterative solvers. International Journal for Numerical Methods in Fluids 3, 397-11. ZHENG, Y.H.; SHEN, Y.M., and QIU, D.H.,. Calculation of wave radiation stress in combination with parabolic mild slope equation. China Ocean Engineering 1(), 95-5. HALLER, H.T., and KIRBY, J.T., 1999. Nonlinear evolution of shear instabilities of the longshore current: A comparison of observations and computations. Journal of Geophysical Research 1(C11), 5953-59. KIRBY, J.T., 19. Rational approximations in the parabolic equation method for water waves. Coastal Engineering 1, 355-37. LI, B., 199. A generalized conjugate gradient model for the mild slope equation. Coastal Engineering 3, 15-5. LIU, P.L.F., and DALRYMPLE, R.A., 197. Bottom frictional stresses and longshore currents due to wave with large angles of incidence. Journal of Marine Research 3(), 357-375. LONGUET-HIGGINS, M.S., 197a. Longshore currents generated by obliquely incident sea waves, 1. Journal of Geophysical Research 75, 77-79. LONGUET-HIGGINS, M.S., 197b. Longshore currents generated by obliquely incident sea waves,. Journal of Geophysical Research 75, 79-1. LONGUET-HIGGINS, M.S., and STEWART, R.W., 19. Radiation stress mass transport in gravity waves, with application to surf beats. Journal of Fluid Mechanics 13, 1-5. LONGUET-HIGGINS, M.S., and STEWART, R.W., 19. Radiation stress in water waves: a physical discussion with applications. Deep-Sea Research 11, 59-5. MEI, C.C., 197. A note on the averaged momentum balance in two-dimensional water waves. Journal of Marine Research 31(), 97-1. MILLER, J.K., and DEAN, R.G.,. A simple new shoreline change model. Coastal Engineering 51, 531-55. PRASAD, R.S., and SVENDSEN, I.A., 3. Moving shoreline boundary condition for nearshore models. Coastal Engineering 9, 39 1. TANG, J.; SHEN, Y.M.; ZHENG, Y.H., and QIU, D.H.,. An efficient computational model for solving the mild slope equation. Coastal Engineering 51, 13-15. VAN RIJN, L.C., and WIJNBERG, K.M., 199. One dimensional modelling of individual waves and wave-induced longshore currents in the surf zone. Coastal Engineering, 11-15.