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J . Fluid Mech. (1987), vol. 185, p p . 523-545 Printed in &eat Britain 523 The runup of solitary waves By COSTAS EMMANUEL SYNOLAKIS School of Engineering, University of Southern California, Los Angeles, California 90089-0242, USA (Received 22 August 1986 and in revised form 9 May 1987) This is a study of the runup of solitary waves on plane beaches. An approximate theory is presented for non-breaking waves and an asymptotic result is derived for the maximum runup of solitary waves. A series of laboratory experiments is described to support the theory. It is shown that the linear theory predicts the maximum runup satisfactorily, and that the nonlinear theory describes the climb of solitary waves equally well. Different runup regimes are found to exist for the runup of breaking and non-breaking waves. A breaking criterion is derived for determining whether a solitary wave will break as it climbs up a sloping beach, and a different criterion is shown to apply for determining whether a wave will break during rundown. These results are used to explain some of the existing empirical runup relationships. 1. Introduction The problem of determining the runup of solitary waves on plane beaches usually arises in the study of the coastal effects of tsunamis. Tsunamis are long water waves of small steepness generated by impulsive geophysical events on the ocean floor or at the coastline. Solitary waves are believed to model some important aspects of the coastal effects of tsunamis well. The process of long-wave generation and propagation is now well understood. The process of long-wave runup and reflection is not. Although there is consensus that a suitable physical model for this process is the formalism of a long wave propagating over constant depth and encountering a sloping beach, there is little agreement on the appropriate analytical formulation. The analytical studies of this problem can be classified in two general groups depending on the approximation of the equations of motion that was used in each study to determine the runup. One group has studied variants of the Boussinesq equations. The state-of-the-art numerical solutions of these equations for solitary-wave runup are those of Pedersen & Gjevik (1983) and Zelt (1986) who solved the equations in Lagrangian coordinates, and of Kim, Liu & Ligett (1983) who used boundary-integral methods. The other group has used the shallow-water-wave equations. These two nonlinear equations result directly from the Boussinesq equations, if the effects of dispersion and vertical accelerations are neglected. A comprehensive review of various regular, weak and apparent solutions of these equations may be found in Meyer (1986). The classic solution of their linear form was given by Lewy (1946) for the problem of periodic waves climbing up a sloping beach. Carrier & Greenspan (1958) derived a nonlinear transformation to reduce the two equations to a single linear equation and they solved several initial-value problems. Keller & Keller (1964) solved the linear problem of a periodic wave propagating first over constant depth and then up a 524 C. E . Synolakis sloping beach. Carrier (1966) used the Carrier & Greenspan transformation to calculate in closed form the runup of a wave generated by a bottom displacement and propagating over relatively arbitrary bathymetry. The state-of-the-art numerical solution of the nonlinear equations was achieved with the work of Hibberd & Peregrine (1979) ; they were able to calculate the runup of a uniform bore on a plane beach. Despite the quality of the analytical work, fundamental unresolved questions about the runup of long waves still exist. The empirical relationship between the normalized runup and the normalized wave height that has been established in the series of laboratory investigations of Hall & Watts (1953), Camfield & Street (1969), and Kishi & Saeki (1966) remains unexplained analytically. The results of the available numerical solutions have not been compared with detailed amplitude- evolution data from the laboratory, and, as a consequence, there is little conclusive information about the relative importance of dispersion and nonlinearity during runup. There is no realization of the differences in the runup behaviour of breaking and non-breaking waves, and this has led to numerical results for non-breaking waves to be compared with laboratory data for breaking waves. Compared with recent advances in periodic-wave runup (Guza & Thornton 1982; Holman 1986), the understanding of solitary-wave runup has been fragmented and incomplete. In the present study an exact solution to an approximate theory will be presented for determining the runup and the amplitude evolution of long waves on plane beaches. A series of experiments will also be described and the resolution of some of these questions will be attempted. 2. Basic equations and solutions Consider a topography consisting of a plane sloping beach of angle p, as shown in figure 1. The origin of the coordinate system is at the initial position of the shoreline and 2 increases seaward. Dimensionless variables are introduced as follows : 2 = xd, i0 = h, d , r" = qd, 92 = u(gd)i, t" = t (d /g)$ . q is the amplitude, u is the depth-averaged velocity, and h, is the undisturbed water depth. The topography is described by h,(x) = x t anp when x < cot@ ( 2 . 1 ~ ) and h,(x) = 1 when x > cotp. (2.1 b) Consider a propagation problem described by the shallow-water-wave equations h, + (hu), = 0 ( 2 . 2 ~ ) and ut+uu,+q, = 0, (2.2b) where h(x , t ) = h,(x) +?I(%, t ) . 2.1. Linear theory The system of equations (2.2) can be linearized by retaining the first-order terms only. The following equation results : Ttt - (Tzho), = 0. (2.3) The solution for constant depth is r when h,(x) = 1 , ( 2 . 4 ~ ) r ( x , t ) = e-ik(x+ct) + A eik(z-ct) Runup of solitary waves Maximum runup 525 FIQURE 1. A definition sketch for a sblitary wave climbing up a sloping beach. All variables except B are dimensional. and the solution for variable depth and finite at the shoreline is ~ ( x , t ) = B(k, /3) Jo(2k(x cot 8)’) e-ikct when h,(z) = x tan /3. (2.4b) A, is the amplitude of the incident wave, A, that of the reflected wave, B is the amplification factor, k is the wavenumber and c = 1. Keller & Keller (1964) presented another steady-state solution for the combined topography defined by (2.1). They matched the outer solution (2.4a) and its 2-slope to the inner solution (2.4b) and its x-slope, at x = X, = cot p, the toe of the beach, and they derived the following values for the reflected-wave amplitude and the amplification factor for an incident wave of the form ~(z, t ) = A, e-i(z+et) : and . J0(2k cot/3) A,( k, /3, A,) = A, exp [ - 2ik cot /3+ 2 arctan 2 exp [ - ik cot t!?l A, = J0(2k cotl)-iJ1(2k cotp)’ B(k’B’ ( 2 . 5 ~ ) (2.5b) 2.2. Exact solutions of the linear theory This formalism will now be used to study the behaviour of more general waveforms approaching the sloping beach. Since the governing equation (2.3) is linear and homogeneous, then, as Stoker (1947) pointed out, the standing-wave solution (2.4) can be used to obtain travelling-wave solutions by linear superposition. When a boundary condition is specified, the solution follows directly from the Fourier transform of the equation. For example, when the incident wave is of the form q(X,, t) = @(k) e-Ikct dk , then the transmitted wave to the beach is given by This solution is only valid when x 3 0; when x < 0, (2.3) does not reduce to Bessel’s equation. To obtain details of the motion in this case, one must solve the nonlinear set (2.2). When initial values are available, it is necessary to use a different method. The standard practice is to use Hankel transform techniques ; the substitution [ = xi transforms (2.3) into the equation : va+ ( 1/C)qs = 4 cot Prtt, and the solution follows 526 C. E. Synolakis directly as a Hankel transform integral (Carrier 1966; Tuck & Hwang 1972). Alternatively, one may use the expansion m ~ ( 6 , t ) = Z Cn(t)Jo(jn 5) n-1 where jn is the nth zero of J,. The coefficients of the Fourier-Bessel series can be found directly from the series expansion by using the orthogonality properties of the Bessel functions : n f l s is the integration variable. c,(t) can then be determined explicitly from and the appropriate initial conditions. A similar solution is presented in Carrier, Krook & Pearson (1966) for an equation that describes the motion of a hanging chain. 2.3. Nonlinear theory To solve the nonlinear set (2.2) for the sloping-beach case, h,(x) = x tan@, Carrier & Greenspan (1958) proceeded to consider the independent variables x, t as functions of the Riemann invariants of the hyperbolic system and, after some effort, they were able to deduce the following hodograph transformation, (referred to as the C&G transformation for short) : $a u=- , U ( 2 . 7 ~ ) t = c o t @ -+A , (2 1 ( 2 . 7 ~ ) and 7 = i $ A - $ 2 (2.7d) which reduces the set (2.2) to a single linear equation, (a@.,), = @AA * (2.8) The transformation is such that in the hodograph plane, i.e. the (u,A)-space, the shoreline is always at Q = 0; this can be deduced easily by setting a = 0 in (2.7b) and by observing that then x = -7 cot/3, which is an equality only valid at the shoreline tip. 2.4. Exact solutions of the nonlinear theory Equation (2.8) can be solved with standard methods. When a boundary condition is specified, then the method of choice is the Fourier transform technique. Defining the Fourier transform of $(a, A) as Y(a, f ) = $(u, A ) e-iAE dA, s_9, and, if Y(uo, &) = P ( f ) , then the bounded solution at u = 0 and u = 00 takes the form Runup of solitary waves 527 If an initial condition is available instead, then one may use Hankel transform methods (Carrier 1966), or proceed as suggested in $2.2. To complete the solution of (2.8) an appropriate initial or boundary condition has to be specified explicitly. Carrier & Greenspan (1958) presented a general solution to an initial-value problem where the initial velocity u(x, to) is zero. Spielvogel (1974) used this solution to derive the evolution of a wave during rundown, assuming an initially exponentially shaped runup profile. Carrier & Greenspan also presented certain solutions with u(x, to) =I= 0, but for very specific initial conditions.. Tuck & Hwang (1972) also solved the initial-value problem. In general, it is difficult to specify initial or boundary data on the sloping beach without making restrictive assumptions about the solution ; a boundary condition requires specification of the solution at (zo, Vt), and an initial-condition specification at (to,Vx), but in practice the solution is only known at (zo 2 cotp, t < to), where to is the time when the wave reaches the z-location xo. Even when boundary or initial conditions are available in the (x,t)-space, the process of deriving the equivalent conditions in the (u, A)-space is not trivial. These difficulties have restricted the use of the Carrier & Greenspan formalism to problems that can be reduced directly to those solved by Carrier & Greenspan. This is unfortunate because some of the problems described can be circumvented. Carrier (1966) demonstrated how to specify a boundary cohdition when reflection from the beach is negligible. Another method will be presented here to specify a boundary condition including reflection. 2.5. Approximate solution of the nonlinear theory Carrier (1966) pointed out that far from the shoreline nonlinear effects are small. The transformation equations can then be simplified by neglecting O(uz) terms. To the same order, $,, + kz and $Ju 4 +A. Using these approximations, the set of equations (2.7) becomes U ' u=- $u 'I = a$,,, x = &uz cot/?, t = -$A cotp. (2.10) These equations are uncoupled and allow direct transition from the (a,A) to the (z, t)-space. One method for specifying a boundary condition in the physical space is to use the solution of the equivalent linear problem, as given by (2.8). This is formally correct to the same order of approximation as (2.10). The obvious choice for the specification is the seaward boundary ; it is desirable to use the linearized form of the equation of motion at the furthest possible location from the initial position of the shoreline where the Carrier & Greenspan formalism is valid. This is the point z = X, = cot /? and it corresponds to the point cr = uo = 4 in the (c,h)-space. Then (2.10) implies that v(Xo, t ) = *$,,(4, A). The boundary condition F(&) in the (a, /\)-space is then determined from (2.29) by repeated application of the Fourier integral theorem. Assuming that $(uo, A) + 0 as A+ f co, then the solution of (2.8) follows: - . * . the substitution K = (2/X0)& was used to simplify the expression. 528 C. E. Synolakis 2.6. Runup invariance in the linear and nonlinear theory It is interesting to compare the predictions of the linear and nonlinear theory for the maximum runup and minimum rundown. It will be proved that they are identical. The maximum runup according to the linear theory is the maximum value attained by the wave amplitude at the initial position of the shoreline x = 0, or @(k) exp [ - ik(X, + ct)] dk J0(2kX0) - iJ1(2kX0) 'I(0,t) = 2 (2.12) In the nonlinear theory the maximum runup is given by the maximum value of the amplitude at the shoreline 7(xs,A). (xs is the x-coordinate of the shoreline and it corresponds to t~ = 0.) To determine q(xs,A) one can use (2.7d) to obtain (2.13) @(K) exp [ - iKX,( 1 -+A)] dK-$:; J,(2KX0) - iJ,(2KX0) us = dxs/dt is the velocity of the shoreline tip. Setting us = 0 and t~ = 0 in the transformation equations (2.7) reduces them to At the point of maximum runup the velocity of the shoreline tip becomes zero. u = 0, 'I = : $ A , x = -'I cotp, t = -y cotp. (2.14) Substitution of these values in (2.13) reduces it to (2.12), proving that the maximum runup predicted by the linear theory is identical with the maximum runup predicted by the nonlinear theory. The same argument applies at the point of minimum rundown where the shoreline tip also attains zero velocity. This invariance was first noted by Carrier (1971) who observed that the maximum runup is given correctly by a linear theory (presumably (2.8)) as the maximum value of g = a$*. However, Carrier did not present an actual comparison between linear and nonlinear theory for polychromatic waveforms, and this result has been largely unrecognized. It is unexpected, because, as will be shown later, the amplitude-evolution data derived using the linear and the nonlinear theory differ most at the initial shoreline (see, for example, figure 5 ) . 3. The solitary-wave solution The results of the previous section will now be applied to derive a result for the maximum runup of a solitary wave climbing up a sloping beach. A solitary wave centred at x = X , at t = 0 has the following surface profile: (3.1) H d ~ ( x , 0) = - sech2 y(z -X, ) , where y = (3H/4d)4. The function @(k) associated with this profile is derived in Synolakis (1986) and it is given by @(k) = Qk cosech (ak) e'"1, (3-2) where a = n/2y. Substituting this form into (2.12) and defining as &(t) the dimensional surface elevation at the initial position of the shoreline, then the following relationship results : dk exp [ik(X, -X,-ct)] J0(2kX0) - iJ1(2kX0) !!@ = "r k cosech (ak) d 3 -m (3.3) Runup of solitary waves 529 This integral can be calculated with standard methods of applied mathematics ; its convergence and evaluation is discussed in the Appendix. The integration result is The series can be simplified further by using the asymptotic form for large arguments of the modified Bessel functions. When 4X,y B 1, then (-l)n+1n~exp[-2y(X,+X,-ct)n]. (3.5) This form of the solution is particularly helpful for calculating the maximum runup. The series in (3.5) is of the form z (-l)n+lnixn; its maximum value occurs at x = 0.481 = e-0.732. This value defines the time t,, when the wave reaches its maximum runup as m n-1 t,, = - X1+X,-- C Y 0-366). Y The value of the series (3.5) at t,, is s, and s,, = 0.15173. Defining as W the maximum value of ' ( t ) and evaluating the term 8(n~3)bm, then the following expression results for the maximum runup : - = 2.831 (cot/?)t W d (3.7) This equation will be henceforth referred to as the runup law. It is formally correct when (H/d)f >> 0.288 tan/? - the assumption implied when using the asymptotic form of the Bessel functions - and when the series converges as discussed in the Appendix. The same result can also be derived by calculating $A from the nonlinear-theory solution (2.11) and then using the appropriate equations (2.14) for the shoreline motion. To derive surface profiles in the entire flow domain i t is necessary to use the nonlinear theory and solve the transformation equations (2.7). The solution is given by u = - $n g J,(&nvX,) exp [iK(XI -X, +- K cosech ( a ~ ) J0(2Kx,) - u1(2Kx,) 'no)1dK-~2, (3.8b) x = cot/?($-+ ( 3 . 8 ~ ) and t = cot/3($-:n). (3 .8d) 530 C . E . Synolakis - 13 ' 0 FIQURE 2. The function f l A ( c , A ) defined by (3.9) for a solitary wave with H/d = 0.019 up a beach with co tp = 19.85. The integrals in these equations can be evaluated directly for given (a, A) using the formalism of the Appendix. The function $A is shown in figure 2 as a function of (a,A) for the case when H / d = 0.019, X, = 19.85 and X, = 37.35. The amplitude evolution predicted by (3.8b) is discussed in 85.2, where it is compared with laboratory data obtained with the methods discussed next. 4. Experimental equipment A series of laboratory experiments was conducted to investigate the validity of the results derived in the previous sections. The experiments were performed in the 40 m wave-tank facility of the W. M. Keck Laboratories of the California Institute of Technology. The facility consists of a wave tank, a wave generation system and a wave measuring system. The experimental equipment is detailed in Hammack (1972), Goring (1978) and Synolakis (1986). The wave tank has glass sidewalls and dimensions 37.73 m x 0.61 m x 0.39 m. At one end of the tank and at a distance of 14.68 m from the wave generator a sloping beach was constructed; it consisted of a ramp made out of anodized aluminium panels with a hydrodynamically smooth surface. The ramp was supported with a wooden truss which was surveyed periodically to ensure that it conformed to the Runup of solitary wuvcs 53 1 initial 1 : 19.85 slope. For brevity, the ramp will be referred to as the laboratory beach. The wave generation system consists of a piston attached to vertical wave plate which moves horizontally and displaces the adjacent fluid thereby generating waves. The piston is driven by a doubly actuated hydraulic cylinder controlled by a servo- valve which can be programmed to produce any desired piston motion. The system was interfaced with a PDP11/23 microprocessor which provided the required piston trajectory. The generation algorithm developed by Goring ( 1978) and generalized by Synolakis (1986) was used to compute the trajectory. The laboratory equipment produced near-perfect solitary waves ; near-perfect refers to waves that conform to the Boussinesq profile (3.1) and have a tail that does not exceed 2% of the wave height H. To measure wave heights in the constant-depth region parallel-wire resistance- type wave gauges were used. The gauges are constructed of steel wire of diameter 0.025 cm and can be calibra