Tsunamis -- the propagation of long waves onto a shelf
01 Nov 1978-
TL;DR: In this paper, a numerical method of solving the Boussinesq equations for constant depth using finite element techniques is presented, which is extended to the case of an arbitrary variation in depth (i.e., gradually to abruptly varying depth).
Abstract: The various aspects of the propagation of long waves onto a shelf (i.e., reflection, transmission and propagation on the shelf) are examined experimentally and theoretically. The results are applied to tsunamis propagating onto the continental shelf. A numerical method of solving the one-dimensional Boussinesq equations for constant depth using finite element techniques is presented. The method is extended to the case of an arbitrary variation in depth (i.e., gradually to abruptly varying depth) in the direction of wave propagation. The scheme is applied to the propagation of solitary waves over a slope onto a shelf and is confirmed by experiments. A theory is developed for the generation in the laboratory of long waves of permanent form, i.e., solitary and cnoidal waves. The theory, which incorporates the nonlinear aspects of the problem, applies to wave generators which consist of a vertical plate which moves horizontally. Experiments have been conducted and the results agree well with the generation theory. In addition, these results are used to compare the shape, celerity and damping characteristics of the generated waves with the long wave theories. The solution of the linear nondispersive theory for harmonic waves of a single frequency propagating over a slope onto a shelf is extended to the case of solitary waves. Comparisons of this analysis with the nonlinear dispersive theory and experiments are presented. Comparisons of experiments with solitary and cnoidal waves with the predictions of the various theories indicate that, apart from propagation, the reflection of waves from a change in depth is a linear process except in extreme cases. However, the transmission and the propagation of both the transmitted and the reflected waves in general are nonlinear processes. Exceptions are waves with heights which are very small compared to the depth. For these waves, the entire process of propagation onto a shelf in the vicinity of the shelf is linear . Tsunamis propagating from the deep ocean onto the continental shelf probably fall in this class.
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TL;DR: In this paper, a high-order numerical model based on the Boussinesq model was developed and applied to the study of two canonical problems: solitary wave shoaling on slopes and undular bore propagation over a horizontal bed.
Abstract: Fully nonlinear extensions of Boussinesq equations are derived to simulate surface wave propagation in coastal regions. By using the velocity at a certain depth as a dependent variable (Nwogu 1993), the resulting equations have significantly improved linear dispersion properties in intermediate water depths when compared to standard Boussinesq approximations. Since no assumption of small nonlinearity is made, the equations can be applied to simulate strong wave interactions prior to wave breaking. A high-order numerical model based on the equations is developed and applied to the study of two canonical problems: solitary wave shoaling on slopes and undular bore propagation over a horizontal bed. Results of the Boussinesq model with and without strong nonlinearity are compared in detail to those of a boundary element solution of the fully nonlinear potential flow problem developed by Grilli et al. (1989). The fully nonlinear variant of the Boussinesq model is found to predict wave heights, phase speeds and particle kinematics more accurately than the standard approximation.
902 citations
Cites background from "Tsunamis -- the propagation of long..."
...Details can be found in Grilli et al. (1989), Grilli (1993), and Grilli & Subramanya (1994). The velocity potential 4(x, t) is used to represent inviscid irrotational twodimensional flows in the vertical plane (x, z ) and the velocity is defined by u = V4 =...
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...Details can be found in Grilli et al. (1989), Grilli (1993), and Grilli & Subramanya (1994)....
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TL;DR: In this article, an approximate theory is presented for non-breaking waves and an asymptotic result is derived for the maximum runup of solitary waves on plane beaches, and a series of laboratory experiments is described to support the theory.
Abstract: 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.
866 citations
TL;DR: In this paper, a numerical code based on Nwogu's equations is developed, which uses a fourth-order predictor-corrector method to advance in time, and discretizes first-order spatial derivatives to fourthorder accuracy, thus reducing all truncation errors to a level smaller than the dispersive terms.
Abstract: The extended Boussinesq equations derived by Nwogu (1993) significantly improve the linear dispersive properties of long-wave models in intermediate water depths, making it suitable to simulate wave propagation from relatively deep to shallow water. In this study, a numerical code based on Nwogu's equations is developed. The model uses a fourth-order predictor-corrector method to advance in time, and discretizes first-order spatial derivatives to fourth-order accuracy, thus reducing all truncation errors to a level smaller than the dispersive terms retained by the model. The basic numerical scheme and associated boundary conditions are described. The model is applied to several examples of wave propagation in variable depth, and computed solutions are compared with experimental data. These initial results indicate that the model is capable of simulating wave transformation from relatively deep water to shallow water, giving accurate predictions of the height and shape of shoaled waves in both regular and irregular wave experiments.
546 citations
TL;DR: In this paper, the mean flow and turbulence in a wave flume for a spilling breaker and a plunging breaker were studied, and the results indicated that there are fundamental differences in the dynamics of turbulence between spilling and plunging breakers, which can be related to the processes of wave breaking and turbulence production.
384 citations
Journal Article•
TL;DR: In this article, the mean flow and turbulence in a wave flume for a spilling breaker and a plunging breaker were studied, and the results indicated that there are fundamental differences in the dynamics of turbulence between spilling and plunging breakers, which can be related to the processes of wave breaking and turbulence production.
Abstract: Undertow and turbulence in the surf zone have been studied in a wave flume for a spilling breaker and a plunging breaker. Fluid velocities across a 1 on 35 sloped false bottom were measured using a fiber-optic laser-Doppler anemometer, and wave decay and set-up were measured using a capacitance wave gage. The characteristics of mean flow and turbulence in spilling versus plunging breakers were studied. The mean flow is the organized wave-induced flow defined as the phase average of the instantaneous velocity, while the turbulence is taken as the deviations from the phase average. It was found that under the plunging breaker turbulence levels are much higher and vertical variations of undertow and turbulence intensity are much smaller in comparison with the spilling breaker. It was also found that turbulent kinetic energy is transported seaward under the spilling breaker and landward under the plunging breaker by the mean flow. The study indicates that there are fundamental differences in the dynamics of turbulence between spilling and plunging breakers, which can be related to the processes of wave breaking and turbulence production. It is suggested that the types of beach profile produced by storm and swell waves may be the results of different relationships between mean flow and turbulence in these waves.
363 citations
References
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TL;DR: In this paper, a method for solving the initial value problem of the Korteweg-deVries equation is presented which is applicable to initial data that approach a constant sufficiently rapidly as
Abstract: A method for solving the initial-value problem of the Korteweg-deVries equation is presented which is applicable to initial data that approach a constant sufficiently rapidly as $|x|\ensuremath{\rightarrow}\ensuremath{\infty}$. The method can be used to predict exactly the "solitons," or solitary waves, which emerge from arbitrary initial conditions. Solutions that describe any finite number of solitons in interaction can be expressed in closed form.
3,896 citations
Book•
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01 Jan 1966
TL;DR: The importance of basic principles is recognized in this article in two ways : first, by devoting the opening chapters to a fairly leisurely discussion of introductory principles, including a recapitulation of the underlying arguments derived from the parent subject of fluid mechanics; and second, by takingnevery opportunity in the later chapters to refer back to this earlier material in order to clarify particular applications as they arise.
Abstract: PrefaceAlthough this book was originally conceived as a text for use by the civilnengineering student in advanced courses either in his senior year or at graduatenlevel, it is also designed to have some appeal to the practicing engineer.Open channel flow, like any topic of engineering interest, is defined andnclassified partly by its possession of certain characteristic applications andnpartly by the principles that are invoked to deal with them. This particularnsubject is so rich in the variety and interest of its practical problems that anyntextbook on the subject is in danger of becoming a mere catalogue of applicationsnand routine techniques devised for dealing with them. But it has to benremembered that mastery of this subject, as of any other, demands a grasp ofbasic principles no less than a facility in routine operations. The practicing nengineer is reminded of this fact whenever he turns from the familiar numericsnof backwater curves and flood-routing procedures to some unusual transitionnproblem whose solution requires a good grasp of fundamentals.The importance of basic principles is recognized in this text in two ways :nfirst, by devoting the opening chapters to a fairly leisurely discussion of introductorynprinciples, including a recapitulation of the underlying argumentsnderived from the parent subject of fluid mechanics; and second, by takingnevery opportunity in the later chapters to refer back to this earlier materialnin order to clarify particular applications as they arise. It is hoped that thenpracticing engineer, as well as the student, will find this kind of treatmentnhelpful, and a compensation for the fact that not every application is pursuednthrough every possible variant that occurs in practice. Further compensationnwill, it is also hoped, be found in the fairly complete system of references andnin the unusually large number of applied topics dealt with.This insistence on the importance of principles does not imply that theynshould be given a status and significance independent of the applications theynpossess. The engineer invokes principles in order to deal with problems thatnarise in practice, and when dealing with these general principles he stillnremains in touch with the physical events which have prompted the need to generalize. This notion has dictated the structure of many chapters in thisnbook, particularly Chapters 2 and 3. In each of these, a typical basic problemnis discussed first; the theory is then developed to solve this problem, and isnfinally generalized to cover other problems as well. n n n n
2,297 citations
TL;DR: In this paper, the growth of an undular bore from a long wave is described, which forms a gentle transition between a uniform flow and still water, and a physical account of its development is followed by the results of numerical calculations.
Abstract: If a long wave of elevation travels in shallow water it steepens and forms a bore. The bore is undular if the change in surface elevation of the wave is less than 0·28 of the original depth of water. This paper describes the growth of an undular bore from a long wave which forms a gentle transition between a uniform flow and still water. A physical account of its development is followed by the results of numerical calculations. These use finite-difference approximations to the partial differential equations of motion. The equations of motion are of the same order of approximation as is necessary to derive the solitary wave. The results are in general agreement with the available experimental measurements.
961 citations
"Tsunamis -- the propagation of long..." refers background or methods in this paper
...For a l l of t hese s t u d i e s t h e s o l u t i o n was obta ined f o r a n harmonic wave wi th a s i n g l e frequency i n t h e s teady state. Kaj iura (1961) proposed a method of s o l u t i o n f o r s lopes of...
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...equat ions a f t e r Boussinesq (1872) and a r e a s fol lows:...
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...vary ing depth, a v a r i a b l e depth form of t h e KdV equat ion can be der ived and, us ing t h e same techniques as were used by Gardner e t aZ. (1967) t o s o l v e t h e KdV i n cons t an t depth, asymptot ic s o l u t i o n s f o r t h e s o l i t a r y waves which emerge on t h e she l f can be obtained....
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...C lea r ly , i f f C is a l i n e a r func t ion , t h e Midpoint Rule and t h e Trapezoidal Rule C a r e i d e n t i c a l , however, i f f i s a nonl inear func t ion , Eqs. (3.86) and (3.89) a r e q u i t e d i f f e r e n t . Although both t h e Midpoint Rule and t h e Tr(apezoida1 Rule a r e uncondi t iona l ly s t a b l e and second order accu ra t e , t h e Midpoint Rule is p re fe r r ed f o r nonl inear problems because t h e s t a b i l i t y a n a l y s i s more c l o s e l y p a r a l l e l s t h e s t a b i l i t y a n a l y s i s f o r l i n e a r problems and thus r e s u l t s in a more d e f i n i t e s ta tement of uncondi t iona l s t a b i l i t y . The d e t a i l s of t h i s and o the r a s p e c t s of t h e s t a b i l i t y a n a l y s i s of t h e Midpoint Rule and t h e Trapez'oidal Rule f o r non l inea r problems a r e given by Hughes (1977)....
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...i i ) A continuous s l o p e determined such t h a t t h e b a s i c equat ion is transformed i n t o a n equat ion which g ives s imple expressions f o r t h e r e f l e c t i o n and t ransmiss ion c o e f f i c i e n t s . Worig e t aZ. (1963) and Dean (1964) obta ined t h e s o l u t i o n f o r a s l o p e 0x1 which t h e depth v a r i e s l i n e a r l y a s a func t ion of t h e d i s t a n c e...
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TL;DR: In this article, a review of recent work and new developments for the penalty function/finite element formulation of incompressible viscous flows is presented, in the context of the steady and unsteady Navier-Stokes equations.
548 citations