HydroandMorphoDynamicModelingofBreakingSolitaryWavesOveraFineSandBeach

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Hydro- and morpho-dynamic modeling of breaking solitary waves over a ne sandbeach. Part II: Numerical simulation

Heng Xiao a,1, Yin Lu Young b,, Jean H. Prvost a

a Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544, USAb Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI 48109, USA

a b s t r a c ta r t i c l e i n f o

Article history:

Received 12 December 2008

Received in revised form 29 November 2009Accepted 13 December 2009

Available online 21 December 2009

Communicated by J.T. Wells

Keywords:

tsunami

solitary wave

sediment transport

mobile bed

morpho-dynamic modeling

wavesoil interaction

A comprehensive numerical model is developed to predict the transient wave propagation, sediment

transport, morphological change, and the elastodynamic responses of seabed due to breaking solitary waves

runup and drawdown over a sloping beach. The individual components of the numerical model are rst

validated against previous analytical, numerical, and experimental results. The validated numerical model is

then used to simulate breaking solitary wave runup and drawdown over a ne sand beach, where the

experimental results are presented in (Young et al. 2010b. Hydro- and morpho-dynamic modeling of

breaking solitary waves over a ne sand beach. Part I: Experimental study). The strengths and weaknesses of

the model are assessed through comparisons with the experimental data. Based on the results, sediment

transport mechanisms and waveseabed interactions in the nearshore region are discussed.

2009 Elsevier B.V. All rights reserved.

1. Introduction and literature review

Tsunamis could lead to great lossesamong the population near the

coast within the reach of wave runup and subsequent ooding. In

addition to hydrodynamic and debris impact, tsunamis could also

erode the coastlines and the soil supporting structural foundations

and roadways. In severe cases, structure collapse or soil failure

(liquefaction and slope instability) can occur. To fully assess theimpact of tsunamis, a comprehensive numerical model describing the

nearshore transformation of the waves, the wave-induced sedimenttransport, and the transient responses of the sediment bed is needed.

In this section, previous research in each of the three aspects listed

above is rst reviewed, and the research needs are identied.

1.1. Wave propagation

The propagation of waves has been a subject of research for over a

century. Classical works include the analytical model by Carrier and

Greenspan (1958)and the extensions along this line (e.g. Tuck and

Hwang, 1972). More recently,Carrier et al. (2003)developed a more

exible analytical solution for wave runup on a plane beach.However,

analytical solutions often suffer from severe limitations such as simple

geometries, restricted initial conditions, and small steepness ratios

due to the complexity of the problem (Carrier and Greenspan, 1958;

Carrier et al., 2003). Therefore, analytical solutions usually only serve

as benchmark cases for numerical models.

Most numerical models for water wave transformations fall into

one of the following three categories according to the assumptions

(Lin and Liu, 2000): depth-averaged models, potential ow models,

and Navier Stokes (NS) equation based models, arranged in the order

of increasing exibility, complexity, and computational cost. Potential

ow models are not suitable for simulating breaking waves since theirrotationalow assumption is not valid during wave breaking. NS-

based models (including Reynolds averaged NS simulations, largeeddy simulations, and direction numerical simulations), on the other

hand,are computationally expensive, and surface tracking or interface

capturing methods are needed to locate the moving free surface. The

objective of this work is to improve the understanding and themodeling capability of the global interactions between waves,

sediments, and soil beds, and thus a depth-averaged model is

sufcient to simulate the wave propagation and transformation.

1.2. Cross-shore sediment transport

Sediment transport models are important in studying the

evolution of beach proles. Many models of different complexities

have been proposed, validated, and used. Schoonees and Theron

(1995)gave a comprehensive review and evaluation of ten cross-

Marine Geology 269 (2010) 119131

Corresponding author.

E-mail address:[email protected](Y.L. Young).1 Current address: Institute of Fluid Dynamics, ETH, Zurich, Switzerland.

0025-3227/$ see front matter 2009 Elsevier B.V. All rights reserved.

doi:10.1016/j.margeo.2009.12.008

Contents lists available at ScienceDirect

Marine Geology

j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / m a r g e o


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shore sediment transport and morphological change models used in

coastal engineering research and practices. Sediment transport

models are also used in the studies of river dynamics.Cao and Carling(2002)reviewed various empirical relations used for the modeling of

alluvial rivers. Despite signicant research efforts from both commu-nities, the mechanisms of the sediment transport processes are still

not fully understood due to the wide variety of hydrodynamic,

morphological, and geological conditions. The assumptions based onthe data and conditions for one study may not be applicable for

another.Most early sediment transport models employed energy ux

approaches such as the CERC equation (USACE, 1984). A more recent

development is the force balance approach, where sediment transport

uxes are modeled as functions of local hydrodynamic and bed

conditions, which is more suitable for wavesediment coupling. For

example, Cao et al. (2004) used empirical erosion and deposition

models based on local ow and bed conditions, and coupled the

sediment transport model with uid motions to predict erosioncaused by dam breakows.Simpson and Castelltort (2006)extended

the work ofCao et al. (2004)to 2D and predicted the morphologicalchanges caused by a train of long period waves running over a plane

beach. The simulation of Simpson and Castelltort (2006) showed

erosion immediately seaward of shoreline and deposition further

seaward of the erosion region. This pattern does not agree with the

experimental results ofKobayashi and Lawrence (2004) and Young

et al. (2010b-this issue)for solitary waves, where erosion was con-

sistently observed above thestill water level,and deposition zone was

formed slightly seaward of shoreline. Although a direct assessment

ofSimpson and Castelltort (2006)results is not possible since these

studies used different particle diameters, the difference between

predicted and observed morphological change patterns warrants fur-

ther investigations.Although force balance based models often formulated the

sediment transport rate as a function of the Shields parameter,which can be considered as a ratio of horizontal forces (viscous and

turbulent shear) and vertical forces (gravity and buoyancy) imposed

on a particle,several modications have been proposed.The studies of

Drake and Calantoni (2001), Hoefel and Elgar (2003) and Puleo et al.

(2003), among others, suggested that uid acceleration played animportant role in sediment transport. Madsen and Durham (2007)

studied the effect of subsurface horizontal pressure gradient induced

by breaking waves. Nielsen (1997) showed that there were two

competing effects caused by exltration, i.e. the reduction of effective

weight due to upward seepage and the reduction of horizontal shear

stresses on the particles due to the increase of the boundary layer

thickness (and opposite effects by inltration). He suggested a modi-

ed Shields parameter to accommodate the two effects. Later,Nielsenet al. (2001)conductedume experiments with nonbreaking waves,

which found that inltration impeded sediment motion for 0.2-mmquartz. Based on the modied Shields parameter formulation

introduced in Nielsen (1997), they concluded that seepage couldimpede or enhance sediment mobility depending on the particle

diameter and the hydraulic gradient. Similarly, Turner and Masselink

(1998)conducted eld measurements to quantify the two competingeffects. Simulations were conducted based on the formulation ofmodied Shields parameter proposed according to their eld

observations. The results showed that the inltration increasedtransport rate during runup and the exltration reduced transport

rate during drawdown, but the former dominated and thus the net

effect ofltration was enhancement of the net upslope transport of

sediment. In both cases, the effect of particle sizes was not studied,

and thus the experiments were not able lend direct support to the

theoretical derivations.

Pritchard and Hogg (2005) developed an analytical solution for the

sediment transport of suspended sediment by a swash event caused

by a bore collapse on a plane beach. With their exact solutions, the

sediment entrainment in the swash zone and the sediment contribu-

tion during the bore collapse are identied. The asymmetry between

uprush and backwash events and the settling lag effects are

investigated for their roles in determining the onshore and offshore

net sediment transport. Their solution revealed important physicalinsights in the swash zone sediment transportprocesses and provided

baseline benchmarks for future studies.

It can be seen that the sediment transport is a very complicatedprocess with many contributing factors. Although numerous modied

formulations have been proposed, many researchers still use theclassical models based solely on Shields parameters, and occasionally

on particle Reynolds number (e.g. Cao et al., 2004; Simpson and

Castelltort, 2006). This conservative approach is probably due to the

lack of systematic validation studies of the new models for different

hydrodynamic, morphological, and geological conditions. In addition,

most previous research efforts in sediment transport have been

directed toward wind waves, currents, or river ows. Much less has

been done for tsunami induced sediment transport and associated

waveseabed interactions in the nearshore region.

In this study, the erosion model rst proposed byMeyer-Peter andMller (1948)is adopted because of its simplicity and because it is

one of the most referenced models. Deposition and erosionuxes aremodeled separately since the two processes are governed by different

physical mechanisms (Cao et al., 2004; Simpson and Castelltort,

2006). The results obtained using Meyer-Peter and Mller (1948)

model is compared to those from the modied Shields parameter

formulation ofNielsen et al. (2001). Discussion of the comparison of

the two models is presented inSection 5.

1.3. Wave-induced seabed response

Waveseabed interactions have been heavily researched and the

literature is abundant. An excellent recent review was given byJeng

(2003), summarizing previous work on seabed dynamics induced by

waves, including analytical, numerical and experimental studies. This

problem is typically formulated in the framework of porous media

(Biot, 1941; Coussy, 2004), where the constituents (sand grains,

water, and for unsaturated cases, air) are assumed to be individual

continua, all interpenetrating each other and occupying the wholedomain, each being regarded as a phase.

The approach adopted in this work utilizes tools and knowledge

developed in both coastal engineering and soil mechanics com-

munities. Therefore, in this paper, sediment and soil are used

interchangeably to refer to the bed material. In both the physical

and numerical simulations, the beds consist of cohesionless sand

particles with negligible amounts of organic material, and the inter-particle pores are lled with varying amounts of water and air.

The structure formed by solid grains is referred to as matrix or

skeleton.

The responses of the soil matrix and the pore uid are intrinsically

coupled, that is, the change of deformation and stress state of one

phase would inuence those of other phases. Only under special

circumstances (e.g. rigid skeleton, incompressible uid, or one-

dimensional strain state) could they be uncoupled.Recently, research efforts have been directed toward advancing

the understanding of the constitutive behaviors of the sand bed under

repeated or extreme loads where nonlinear soil behavior, liquefaction,

and slope instability are important. For example, liquefaction of sand

beds due to progressive waves was studied experimentally with a

centrifuge (Sassa and Sekiguchi, 1999) and latter numerically with

nite element methods (Sassa and Sekiguchi, 2001). In addition, the

propagation of liqueed zones in sand bed under progressive wave

loading was studied using a simplied theoretical model and

validated experimentally by Sassa et al. (2001). However, these

studies were concernedwith progressive wave loading over a at bed.

Hence, the responses and failure of coastal slopes due to long wave

120 H. Xiao et al. / Marine Geology 269 (2010) 119131


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runup and drawdown were not considered. Young et al. (2009a)

investigated the inuence of solitary wave runup and drawdown on

the soil responses using an uncoupled wavesoil model. The wavemodel was the same as the one presented here, but the sediment

transport processes were not considered.In most tsunamis, there are only two or three waves that arrive

onshore. Although the wave periods are long, the event durations of

tsunamis are usually much shorter than those of storms. Therefore,the nonlinear behaviors of soil are not as signicant. The focus of this

work is on the waveseabed interactions. Hence, the soil is assumedto be linear elastic, and the us-p formulation is used, where the

displacements of soil skeleton (us) and the pore pressure (p) are fully

coupled (Coussy, 2004). The systemof equations for us andp is solved

with nite element method in a fully coupled approach (Prvost,

1997). More details are presented inSection 2.2.

1.4. Objectives

The objectives of this study are to develop and validate a

comprehensive numerical model to predict the transient hydrody-namics, sediment transport and bed prole changes, as well as bed

responses in the nearshore region under the impact of breaking

solitary waves. By complementing the experimental studies pre-

sented in (Young et al., 2010b-this issue), the goal is to improve the

understanding of sediment transport mechanisms and bed responses,

particularly dueto breaking solitary waves runup and drawdown over

a ne sand beach.

2. Mathematical formulation and numerical methods

2.1. Wave propagation coupled with sediment transport

Nonlinear shallow water equations (NSWE) have been used by

many researchers to study the long wave propagation near the coast

where the water is relatively shallow compared to the wave length

(e.g. Zoppou and Roberts, 2000; Kim et al.,2004; Wei et al.,2006). The

scalar transport equation is used to describe thesediment transportin

the water, and thus the coupled NSWE and sediment transport model

for a 2D problem can be written as (Cao et al., 2004):

U

t +

F

x =S 1

where the conservative variable U, the uxF, and the source term S

are, respectively,

U=h;hu;hcT 2

F=uh;u2h+ gh2 =2;hucT 3

and

S=

qeqd1;ghS0 + Sf

qeqdu

1 ; qeqd T 4

where t is the time; h is the water depth; u is the depth-averaged

velocity; cis the depth-averaged sediment concentration; is the bed

porosity;S0is the source term representing bottom slope, i.e. S0=dz/

dx, wherezis the bed elevation.Sfis the source term due to bottom

friction, which is modeled as follows:

Sf= n2u ju j

h4=3 5

wheren is Manning's roughness coefcient. In the dry portion of the

bed where the water depthhis almost zero,Sfis set to zero.qeandqd

are, respectively, the sediment erosion and deposition uxes, which

also needs to be modeled and will be discussed inSection 2.1.2.

2.1.1. Boussinesq equations: corrections for dispersion effects

When the wave is propagating from a distance far away from theshoreline, the dispersion effect is important and the NSWE is not

suitable. Therefore, a Boussinesq model accounting for the dispersion

is necessary. Among others, Madsen et al. (1991) and Madsen andSrensen (1992) derived a Boussinesq-type equation with the

advantage of enhanced linear characteristic and relatively fewadditional terms (Borthwick et al., 2006). To account for dispersion

effects, the conservative variable U and the source termS need to be

modied while the uxF remains the same. The modied conserva-

tive variable and source term are denoted as Uand S, respectively,

and have the following form:

U*=

h

hu+ B+ 1

3

d2huxx+

1

3ddxhux

|{z}

Boussinesq terms

hc

266666664

377777775

6

and

S*=

qeqd1

ghS0 + Sfqeqdu

1 + Bgd

3xxx + 2Bgd

2xx

|{z}

Boussinesq terms

qeqd

266666664

377777775

7

where B is the dispersion coefcient, which is taken as 1/15

(Borthwick et al., 2006); is the wave elevation; and d =h is

the still water level. The subscripts x , xx , and xxx indicate the rst-,

second-, and third-order derivatives, respectively. The additional

terms does not need special treatments and the numerical schemesfor NSWE can be easily extended to Boussinesq equations.

In this study,the Boussinesq equationsabove are solvedbefore the

wave breaks. During and after the wave breaking, the Boussinesqterms are turned off to avoid numerical instability caused by the high

order derivatives in the dispersion terms, and the NSWE is thus

recovered. The breaking criterion is dened as the water surface slope

being greater than 20, or equivalently d/dx >0.36 (Borthwick et al.,

2006).

2.1.2. Empirical modeling of erosion and depositionuxes

The deposition ux is modeled as a function of the near-bed

sediment concentration, Ca, and the particle settling velocity, 0, inquiescent water, followingCao et al. (2004):

qd = 01CamCa 8

where the exponentm =2.0 is adopted. The local near-bed sediment

concentration is modeled as Ca=c, where =min (2.0, (1)/c).The upper limit is set to ensure that the volume fraction of sand in the

water does not exceed that in the bed.The erosion ux is formulated as:

qe = An0c

ffiffiffi

p = Re1:2p if> c

0 if c

( 9

where is the Shields parameter; Repis the particle Reynolds number

dened as Rep = d50ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis1gd50

p =, that is, the Reynolds number

based on meanparticlediameter (d50) andthe relative velocity between

a freely settling particle and its surrounding calm water. The particle

Reynolds number characterizes the ow type around the particle

(laminar for smallRepand turbulent for largeRep). The fall velocity, 0,

121H. Xiao et al. / Marine Geology 269 (2010) 119131


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is the terminal velocity of a single particlewhen released into quiescent

water (that is, when the gravityof the particle and the water resistance

balance).An is a dimensionless coefcient; s =s/w;s is the density ofthe particle; w is the density of the water; d50 is the mean particle

diameter.c=0.045 is the critical Shields parameter, below which noerosion occurs.is Shields parameter dened as:

=u2*

s1

gd50

10

where u* is the friction velocity and is dened as u*=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0= w

p =ffiffiffiffiffiffiffiffiffi

ghSfp

with0=wghSfbeing the bed shear stress. The erosion ux in

Eq. (9) follows that of Meyer-Peter and Mller (1948) except for a

constant dimensionless factor An, which is used to account for the

inuence of bed compactness (porosity).The erosion ux model presented above was originally proposed

for steady state river-type ows. The hydraulic gradient in the bed(and thus the inltration and exltration) was not considered. To

account for this effect, Nielsen et al. (2001) introduced a modied

Shields parameter,m:

m =

u2* 1w

u*

gd50 s1wkc :

11

The two dimensionless constants are =16 and =0.4, which

were determined by previous experimental studies (Conley and

Douglas, 1994);wis the exltration velocity, which can be calculated

according to the hydraulic gradient at the bed surface according to

Darcy's law, which in turn can be obtained from the soil solver

presented inSection 2.2;kcis the hydraulic conductivity of the soil.

2.1.3. Morphological evolution

All the morphological changes are assumed to be caused by wave-

induced sediment transport, and thus the bed evolves solely due toerosion and deposition uxes. Therefore, the following equation is

used to describe the elevation change of the bed:

dz

dx =

qdqe1

12

wherezis the bed surface elevation.

2.1.4. Treatment of the wave breaking process

Wave breaking plays an important role in the wave propagation andsediment transport processes because of its energy dissipation and

sediment entrainment capability. However, detailed modeling of thebreaking process poses great difculties for numerical models. In this

study, the wave-breaking process is not modeled explicitly, but the

energy dissipation caused by wave breaking is accounted for implicitly.

Specically,the breaking wave is represented as a discontinuityof water

depth and velocity in the solution. Whenever discontinuities occur in

the solution, the numerical methods as detailed inSection 2.1.5woulddissipate more energy than for smooth solutions. Good correlations

between wave breaking and energy dissipation have been observed in

our simulations, but further investigations are needed to compare the

numerical energy dissipation and the physical energy dissipation.The sediment entrainment due to wave breaking is not explicitly

accounted for. The sediment entrainment in the surf zone (includingthewave-breaking zone and bore runup region) is treated in thesame

wayas in theswashzone using the erosion model in Eq.(9). However,

since the wave-breaking zone is generally associated with larger

Shields parameter, the erosion ux in this region is thus larger than

elsewhere. This can be interpreted as an implicit treatment of wave

breaking induced sediment suspension, although the detailed physics

is not modeled. This is illustrated in Section 4.2with a plot of Shields

parameters (Eqs. (10) and (11)) during wave breaking inFig. 9.

2.1.5. Numerical methods for the wavesediment system

Finite volume method is used to solve the system of equationsdescribing the wavesediment interactions. A second order Godunov-

type scheme with shock-capturing weight averaged ux (WAF) is

used (Fraccarollo and Toro, 1995; Kim et al., 2004). At the interface oftwo wet cells, an approximate Riemann solver HLLC is used

(Fraccarollo and Toro, 1995), which is based on the HartenLaxvanLeer (HLL) Riemann solver developed by Hartenet al. (1983) but with

improved performance for contact discontinuity problems (such as

the sediment concentration in this system). At the drywet interface,

in place of the HLLC Riemann solver, an exact Riemann solver with

front speed from the analytical solution is used (Toro, 2000). A

threshold water depth is chosen to be 0.1% of themaximum still water

depth, below which the cell is considered as dry.

To ensure numerical stability, the CourantFriedrichsLewy (CFL)

number is limited to a value smaller than 1. The sediment terms

tend to impair the convergence performance of the system, and thussmaller step size is necessary.

To simulate the far eld boundary at sea, a transmissive boundarycondition is implemented. To achieve this, the two Riemann in-

variants r1

and r2

are extrapolated along characteristic lines at the

transmissive boundary. Specically, for shallow water ows:

r1 u2ffiffiffiffiffiffigh

p =const along

dx

dt =u

ffiffiffiffiffiffigh

p

r2 u+ 2ffiffiffiffiffiffigh

p =const along

dx

dt =u +

ffiffiffiffiffiffigh

p :

8>>>: 13

2.2. Soil deformation and pore pressure

The soil skeleton and the pore water in the sediment bed are

modeled in the framework of poromechanics theory (Coussy, 2004).

The soil deposit is assumed to be fully saturated (with either water orair). The following equations are solved:

+1s + ff= 0 14

dfdt

+ fqf= 0 15

with

d

dt =bv

s+

1

N

dpfdt

: 16

The following relation is assumed for pore uid ux according to

Darcy's law:

q

f

=

k

f

pfff: 17

Biot's coefcientb and modulusNare dened as (Coussy, 2004):

b= 1KmKs

; and 1

N=

b0Ks

18

whereis the total stress tensor of the mixture; is the bed porosity

(same as that dened in wavesediment simulator inSection 2.1);0is the initial porosity;sandfare the solid and uid density;fis the

body force (gravity in this study);pfis the poreuid pressure; qfis the

mass ux of the pore uid; vs is the solid velocity; k is the intrinsic

permeability tensor of the soil skeleton;fis the dynamics viscosity of



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the pore uid;KmandKsare the bulk moduli of the uid and the solid

constituents, respectively.

For saturated porous media, the change ofuid density is relatedto the change of pore pressure by

dff

=Cfdpf 19

whereCfis the pore uid compressibility.

The density of the pore uid fis taken to be the water density(w) in the wet region below the water table and to be the air density(a) in the region above the water table. The pore water compress-

ibility is obtained by accounting forthe airdissolved in thepore water,which results in the increase of effective uid compressibility

compared with pure water. The compressibility of pore water is

estimated according toVerruijt (1969)as follows:

Cf=SrCw+ 1Sr

pa20

whereSr is the degree of saturation of water in the pores, Cwis the

compressibility of pure water (4.61010 m2/N), andpais the abso-

lute pressure of the air (the atmospheric pressure, 105 Pa). The time

step size is determined according to t=0.003Tc, where Tc is the

characteristic consolidation time scale dened as:

Tc= L2d

K+4

3G

fk

21

where Kand G are the bulk and shear moduli of the soil skeleton,

which can be expressed in terms of Young's modulus Eand Poisson

ratioof the soil as K=E/[3(12)] andG = E/[2(1+)].Ldis the

shortest drainage path (represented by the maximum soil depth).

The solid stress and velocity are expressed in terms of solid

displacementeldus as follows:

= sbpf 22

s

=C : s

23

s = us

vs = us

t

24

and the uid pressure is written as

pf =p0 + pe 25

where C is the constitutive tensor (fourth-order); The symbol :denotes the contraction product of two tensors; s strain of the

skeleton;()us= (us+ us)/2is thesymmetric part of tensorus;

is the second order unit tensor;p0 is the initial uid pressure;peis the

excess pore pressure; Eq. (22) shows that the total stress of the

mixture () is decomposed into effective stress acting on the soilskeleton (s) and pressure carried by the pore uid (pf). In Eq. (25),

the pore pressure, pf, is decomposed into an initial hydrostatic

component,p0, and an excessive component,pe.

The equations above are solved with a nite element program,DYNAFLOW (Prvost, 1983a). The details of DYNAFLOW can be found

in (Prvost, 1983b, 1997), where various validation cases were alsopresented.

As explained inSection 1, the nonlinear constitutive behavior of

the soil skeleton and the potential of bed slope failure are neglectedin

the present calculations. Interested reader should refer to (Young

et al., 2009a) on this topic. In the present study, the wave loads on the

bed are assumed to be small enough and the duration of the loading is

short so that the soil behaves linear elastically and no failure model is

necessary. The focus is on the nearly saturated sand bed in the

nearshore region. Since the rate of wave loading and unloading ismuch faster than the movement of the pore uid, the subsurface

water table is assumed to be at and stationary throughout the waveloading cycle. The dynamics of the vadose zone (dened as the

unsaturated portion above the initial subsurface water table) where

capillary effects and the interferences between water and air ows doplay important roles is a subject of future work.

2.3. Coupling between the components

Based on the experimental observations, the amount of the

morphological change is negligibly small compared to the extent of the

soil domain and thus the bed surface prole in the soil simulation is not

updated. However, the bed prole changes may not be negligible

compared to the depth of the water column, particularly near the

maximum runup region. Hence, the wavesediment transport and bed

evolution models arefully coupled. The amount of mass andmomentum

exchange between the wave and the subsurface ow is negligiblebecause the inltration and exltration velocities are very small

comparedto the depth-averaged wavevelocities.Therefore, the coupling

between the wave andthe subsurface hydrodynamics is neglectedin the

present calculation.

3. Validation and calibration of numerical models

3.1. Validation of wave simulator

The NSWE wave simulator has been validated against numerous

cases. A benchmark case composed from the analytical solution of

Carrier et al. (2003)is shown here. In this case, a nonbreaking solitary

wave runs up over a smooth bed with a constant slope. The wave

proles from three different time instances and the horizontal

excursion history of the shoreline are shown in Fig. 1. The simulationresults agree very well with the analytical solutions.

The wave solver has been also validated against the experimentalstudies by Synolakis (1987), where a solitary wave with relative wave

height ofH/d =0.3 runs up over a rigid,impermeableslope of 1:19.85.The initial water depth is d =1 m. The toe of the slope is located at

x =19.85 m and the initial wave is centered at x = 24.42 m. The

comparisonis shown in Fig. 2. Alsoshownin thegureis thenumerical

solution of NSWE by Wei et al. (2006). When the dispersion terms are

turned off, the current solutions are almost identical to that ofWei et

al. (2006), whichis expectedsincebothmodels used similar numerical

methods.The current Boussinesq-NSWE model predicted that wave-

breaking occurs at around the normalized timetffiffiffiffiffiffiffiffiffiffig= d

p 19, which is

very close to the experimental results reported in ( Synolakis, 1987),

while a pure NSWE solver tends to predict earlier breaking (Wei etal.,

2006). In the pre-breaking snapshot at tffiffiffiffiffiffiffiffiffiffig= d

p = 15 (top left plot in

Fig. 2), the improvement is signicant when the dispersion effects are

included. However, both numerical solutions agreed well with

experimental measurements post-breaking.

3.2. Calibration of the sediment transport model

Various sediment transport models were investigated by compar-

ing the numerical predictions with the experimental results of

Kobayashi and Lawrence (2004) and Young et al. (2010b-this issue).

The combination of deposition and erosion uxes in Eqs. (8) and (9)was found to give good results and was adopted. To accommodate

different bed conditions, a coefcientAn is introduced into the Meyer-

Peter and Mller (1948)model (see Eq. (9)), which depends only on

the bed compactness (porosity). Note thatAn is not a fudge factor. The

wavesediment interactions as given in Eqs. (1), (8), and (9)

represent a highly nonlinear and coupled process. Hence, it is not



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possible tot thenumerical predictions to theexperimental results byadjusting a single parameter if the essential dynamics of the coupled

processes were not modeled correctly.

The numerical model with erosion and deposition described in

Section 2.1 was used to simulate the case of three 30-cm solitary

waves runup and drawdown over a wave-modied 1:15 slope bed.

The bed proles before and after the waves, as well as the amount of

morphological change are shown inFig. 3.An=0.35 was used in this

calculation, and is also used for the 60-cm solitary wave validation

study shown later inSection 4.Asshown in Fig. 3, theerosion and deposition patterns compare well

with the experimental measurements, especially considering the

complexity of the process. The deposition region, the erosion region, aswell as the transition point of the two regions are all captured correctly.

4. Comparison of numerical predictions and

experimental measurements

4.1. Model setup and parameters

The numerical model presented above was used to simulate the

case of a 60-cm breaking solitary wave runup and drawdown over a

wave-modied 1:15 plane slope beach. The wave conditions and

bathymetry in the simulation followed exactly with the experiments,

which has been described in detail in ( Young et al., 2010b-this issue).The experimental setup is briey summarized inFig. 4for complete-

ness. The left panel shows the general setup. The right panel showsthe nominal and actual bed proles as well as the locations of the

sensors where numerical results are compared with experimental

measurements.In the numerical simulation, a solitary wave with initial height of

H=60 cm was centered atx0=10 m att=0 s over an initial offshorewater depthd = 1 m. The initial condition was estimated according to

Tivoli and Synolakis (1995):

x; 0= Hsech2ffiffiffiffiffiffiffiffi

3H

4d3

s xx0

24

35 26

ux; 0= x; 0ffiffiffiffiffiffiffiffiffiffig= d

q : 27

The computational domain ranged from x =0 m to x =42 m. A

uniform mesh was used with 1000 cells with size x =0.042 m. The

CFL number Cn=0.5 was used. The time step size was dynamically

adjusted according to the maximum velocity (vmax) in the system and

the CFL number, i.e. dt=Cnx/vmax. Manning's coefcient (n) wastaken to be 0.008. Since the wave did not reach the right boundary at

x =42 m during the simulation and during the experiment, the water

depth andow velocity were set to zero at this boundary. On the left

boundary at x =0 m, a blended reective and transmissiveboundary was used, assuming 75% refection.

For the soil simulation, as with most soil mechanics problems, it isassumed here that the bulk modulus of the soil grain was much larger

than that of the soil matrix, and thus 1/N0 and b1. Other material

parameters used in the computation are presented inTable 1.

The unsaturated zone was modeled by using the uid properties

(density and viscosity) of air. This treatment essentially assumed no

inltration or exltration occurred at the bed surface and thus might

not give accurate results in the unsaturated zone, but it would not

signicantlyaffect theaccuracy of the nearly saturated zone below the

water table.At the top boundary of the bed, on which the wave force acts, the

pressure was set to the hydrostatic pressure corresponding to the

instant wave height of the water column, and the effective stress wasset to be zero. Therefore, the boundary conditions on the top were

pfx= fghx; t 28

sn= pfx; t 29

where n is the outward normal of the bed surface. At other

boundaries, where the bed was in contact with the concrete wall

and ume bottom, respectively, the displacements normal to the wall

is set to be zero. The pressure gradients (and thus uxes) normal to

the wall was also set to be zero. Initial pressure was set to be

hydrostatic. The displacement was initialized as follows: The domainwas rstallowed to consolidate under gravity with thepresenceof the

hydrostatic pressure from the initial water column until the system

reached steady state, and then the displacements of the whole soil

domain were set to be zero.

4.2. Comparison to experiments and evaluation of the numerical model

Fig. 5shows the numerical and experimental time series of water

surface elevations at four representative locations in the nearshore

region. The water surface elevations were measured by wave gauges

(WG) or distance sonics (DS). WG8 was located at x =23 m, where

wave breaking occurred. Wave gauge 10 was located x =25 m. The

Fig. 1.Comparison of the NSWE predictions to the analytical solution of (Carrier et al.,

2003). (a) Wave proles at three time instances. (b) Runup history.



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wave plunged between WG8 and WG10. After the wave breaking,

large amounts of bubble rich white water (referred to as bore) was

formed and rushed up onshore. WG12 and DS2, located at x =27 m

(nominal shoreline) and x = 29 m, respectively, measured the

height of the bore. WG10 and PPS14 shared the same cross-

shore location at x =25 m; WG12 and PPS58 shared the same

cross-shore location of x =27 m. The time series of the pore

pressure will be presented later. The readers are referred to

(Young et al., 2010b-this issue) for details of the experiment and

the sensor deployment.

Fig. 2. The wave proles predicted by the current hybrid NSWE-Boussinesq model compared to the predictions of a pure NSWE model ( Wei et al., 2006) and experimental

measurements bySynolakis (1987).



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Fig. 5shows that the general trend of the water surface elevationvariations are captured satisfactorily. All three wave gauges (WG 8, 10,

and 12) show two distinct peaks resulted from wave runup anddrawdown, occurred at approximately t=3 s and t=13 s, respectively.

The heights of the two peaks were predicted by the numerical model

with good accuracy.The exception wasthe second peakof WG10,which

wasslightlyunderpredicted.This wasprobablybecause it waslocated at

a region where interactions between the waves reected by the beachandthe waves running onshore were signicant.This effectcouldnot be

modeled by the depth-averaged models (NSWE or Boussinesq). The

wavewave interactions also explainthe small wavelength components

present in the experiment but missing in the numerical results.

However, in general, the agreement was very good considering

complexity in the physical processes involving wave-breaking, bore

generation and collapse, and wavewave interactions.

The cross-shore ow velocities and sediment concentrations arepresented inFig. 6. The ow velocities and sediment concentrations

were measured by Acoustic Doppler Velocimeters (ADV) and OpticalBack-Scattering sensors (OBS). The velocity measurements are shown

in three representative locations: x =29 m (ADV5; 2 m onshore),

x =28 m (ADV4; 1 m onshore), and x =23 m (ADV8; 4 m offshore).

OBS4 and ADV8 were located 9 cm above the bed surface.

The experimental results for ADV5 and ADV4 shown inFig. 6still

contain some noises at t5 s. As explained in (Young et al., 2010b-this issue), this was because the two ADVs were initially dry. The data

Fig. 3. Morphological evolution caused by three consecutive 30-cm solitary waves

running over a nominally 1:15 beach (with 15min between the waves to allow thewater to return to a calm state). (a) The initial bed pro le and that after three waves

(numerical and experimental). (b) The amount of morphological change by three

waves (numerical and experimental). The envelopes in thin dash lines are the total

deposition and erosion from the numerical simulation, the sum of which being net

change (solid line).

Fig. 4. (a) Schematic of the experimental setupand denition. Not to scale. (b) Nominal

1:15 bedprole(dashedline), actualbed prole (solid line), and the sensors (symbols).

Not to scale. Only the sensors relevant to the numerical study are shown. OBS4 is co-located with ADV8 (at x =23 m, 9 cm above bed surface).

Table 1

Physical and computational parameters used in the sediment/soil simulation.

Young's modulus of skeleton (E) 1.5 108 Pa

Poisson's ratio of skeleton () 0.2

Density of soil grains (s) 2650 kg/m3

Intrinsic permeabilityof skeleton (k) 1.51011 m2

Porosity of soil () 0.4

Compressibility of pore water (Cf) 2.5 107 m2/N

Compressibility of air (Ca) 1.0 105 m2/N

Density of air (w) 1.0 103 kg/m3

Density of water (a) 1.03 kg/m3

Dynamics viscosity of water (w) 1.0 103 kg m/s

Dynamics viscosity of air (a) 1.8 105 kg m/s

Soil domain (L D) 30 m (horizontal) 2 m (vertical)

Element shape Four-node quadrilateral

Element type Linear elasticity and scalar diffusion

(coupled)

Number of elements 300 (horizontal)40 (vertical)=12,000

Element width (y) 10 cm

Element height (x) Varying f rom 2 .3 cm ( top) t o 9 .1 cm ( bottom)

Time step size (t) 0.047 s



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obtained during thetime when they were suddenly submergedby thewater were of poor quality. Between t=5 s and 12 s when the

experimental data were of good quality, the agreement of between

the numerical and the experimental results were quite good except

for a slightshift. In particular, thedeceleration/acceleration rate of the

runup and the initial stage of the drawdown were well predicted. The

deceleration/acceleration rate could be valuable for evaluating the

forces acting on the coastal structures during tsunami impact. Another

quantity of interests to engineers (for the same reason) is the

maximum runup velocity at the two onshore locations, which

occurred at approximately t=5 s. After t=12 s, as the waterretreated seaward during the drawdown and the water level dropped

below the ADVs, no signal was collected by the ADVs until thereected wave from the wave maker arrived at this location.

Thebottomright panel ofFig. 6 shows that thesediment suspensiontime predicted by the numerical simulation correlates well with the

experiments,although theexact amount of entrainedsediment wasnot

correctly predicted by the numerical model. The second peak in theexperimental results is due to the sediment carried by the water during

the drawdown. The discrepancy between measured and predicted

sediment concentrations could be due to both the limitations of the

capability of the numerical model and the accuracy of the experimental

measurements (e.g., signicant variations among different runs due to

the turbulent mixing). In particular, the numerical results for water

velocities and sediment concentrations were obtained based on the

depth-averaged values at the corresponding cross-shore locations,while the experimental results were obtained from specic measure-

ment points. Due to the highly turbulent sediment mixing at this

location, the two values could be quite different for sediment

concentrations. It was visually observed during the experiments that

the sediment concentration distribution varies signicantly along the

depth of the water column. On the other hand, the overall good

agreement between the numerical simulation and the experimental

measurements on the velocity (ADV5 in the top left panel, ADV4 in thetop right panel, and ADV8 in the bottom left panel, ofFig. 6) suggests

that the velocity distribution along the depth is close to uniform.

Therefore, the approximations made in the derivation of shallow water

equationsare reasonably goodin thescenarioof theexperimental setup.The measured and predicted pore pressure time series are

presented in Fig. 7. In spite of the sophistication of the numerical

model for the soil bed, the agreement was not excellent. This was dueto unavoidable spatial variations of physical soil bed properties. In

most experiments involving sediment, this one included, there areusually large uncertainties in the physical properties of the sediment

due tothedifculties in obtaining thein situ parameters and the large

spatial variations of soil properties.

Fig. 5.Time series of water surface elevation at four representative locations. Comparison of numerical simulation and experimental results. Top left: WG8 ( x =23 m). Top right:

WG10 (x =25 m). Bottom left: WG12 (x =27 m, nominal shoreline). Bottom right: DS12 (x =29 m). Locations of these sensors are shown in Fig. 4. The time origin is shifted by 4.4 s

compared to the gures in (Young et al., 2010b-this issue).



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In theexperimental validationstudies, we usedone setof results fromPPS7 to calibrate the uid compressibility, and then used this value to

compute the results for all other pore pressure sensors. The permeability

of the soil was obtained from laboratory measurements of the sediment

samples taken from the bed. Young's modulus and Poisson's ratio were

obtained from values reported in the literature (Army Corps, 1990). The

results from the simulation and the experiments were then compared.

Fig.7 shows reasonable agreement betweenthe predicted and measured

pore pressure responses of PPS58. On the other hand, the agreement for

PPS14 results arenot as good. Theagreement can be improved by usingdifferent material properties at this location (specically,uid compress-

ibility, soil porosity, Young's modulus, and Poisson's ratio), but ad hoc

adjustments are not particularly meaningful. Nevertheless, the qualita-tive features of the pore pressure response are well captured by the

numerical model. Specically, the following aspects are noted:

1. The decay of the pore pressure signal along the depth. Note the

blurring of the peaks and the decrease in magnitude from top to

bottom. This shows the diffusive nature of pore pressure evolution

in the sediment bed.

2. The initial sudden increase of the pore pressure (a kink) at t4 s.

This is particularly obvious for the bottom PPSs (PPS12 and PPS5

6). This kink was due to the compression of the soil skeleton whichreduced the pore space and led to pore pressure increases. The

pressure increase due to skeleton compression arrived earlier thanthat caused by the pressure diffusion.

The morphological changes predicted by the simulation and those

measured from the experiments are compared in Fig. 8. The same

coefcientscalibrated from the30 cm wave experiments were used in

this prediction. The numerical results shown here were obtained with

the original Shields parameter given in Eq. (10). The Shieldsparameter (Eq. (10)) and modied Shields parameter (Eq. (11)),

together with the wave andbed proles, in the computational domain

during the initial period of the wave breaking (at t=2.94 s) are

shown inFig. 9. Note the steep front of the wave prole atx22 m,

which represents wave breaking, and the resulting larger Shieldsparameter values in this region. This correlation suggests that the

sediment suspension due to wave breaking is partly accounted for inan implicit way by the erosion ux formula in Eq. (9). The role of

inltration and exltration on the bed erosion ux will be presented

inSection 5. It is noted here that the Shields parameters computed

from Eqs. (10) and (11) are almost identical except in a region behind

the wave front (x =18 to 20 m). This is a region with upward seepage

Fig. 6.Time series of cross-shore velocities at three representative locations. Comparison of numerical simulation and experimental results. Top left: ADV5 (velocity at x =29 m and3 cm above bed surface). Top right: ADV4 (velocity atx =28 m and 3 cm above bed surface). Bottom left: ADV8 (velocity at x =23 m and 9 cm above surface). Bottom right: OBS4

(concentrationx =23 m; 9 cm above bed surface. Co-located with ADV8). Locations of these sensors are shown in Fig. 4. The time origin is shifted by 4.4 s compared to the gures in

(Young et al., 2010b-this issue). Note that the numerical results were the depth-averaged values at the corresponding cross-shore locations, while the experimental results were

obtained from specic measurement points.



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caused by the passage of the wave (which induced soil compression

immediately followed by decompression).

The erosion model was able to capture the bulk of the physical

process, which is evident from the good agreement of the erosion zoneon the shore face (x27.535 m), as shown in Fig. 8. The physics

contributed to the erosion in this region were high drawdown velocity

(upto 3 m/s) andshallow water depth(a fewcentimeters), andpossibly

interactions with the permeable bed and uid acceleration, among

others. From the erosionux formulations (Eqs. (5), (9), and (10)), it

can be seen that bothshallow water depth and high velocity contributeto high bed shear stress and consequently large erosion ux. Therefore,

the good agreement in the predicted erosion/deposition patters by the

numerical model and those measured from the experiments was

expected.As explainedin (Young et al.,2010b-thisissue), the deposition

in the wave-breaking region was caused by the large recirculation

region caused by the hydraulic jump. Although the depth-averaged

modelcouldnot accuratelysimulate the recirculatingow,it didpredict

a hydraulic jump at the end of the drawdown and sediment deposition

in this region due to the signicant decrease of the ow velocity.The thin layer of deposition at the runup tip(betweenx =35 m and

x =38.5 m) was also captured by the numerical model, consistent with

the experimental observations. The factors contributed to this deposi-tion region included: shallow water depth, high sediment concentra-

tion, and most importantly, a brief period ofow stagnation during thetransition between runup and drawdown phases. These features were

all well represented by the numerical model.

In summary, the numerical simulation compared well with the

experimental measurements. Although the depth-averaged hydrody-

namic model and the sediment transport model are relatively simple,

the global physical behavior was well captured, particularly during

the runup and the initial stage of the drawdown. In the later stage of

the drawdown, the effects of wavewave interactions and the

recirculation ow caused by hydraulic jump became signicant,

which could notbe captured using a depth-averagedmodel. However,

the overall prediction performance of the model is satisfactory for thespatial andtemporal domains of interest.As to thecomputationof bed

responses, the usp formulation is adequate, and the uidskeleton

coupling is necessary to correctly predict the observed pore pressure

evolution. However, the uncertainties and spatial variations of thematerial properties, which are difcult to obtain in situ, became the

bottleneck of accurate predictions.

5. Discussion

The effects ofltration ows are examined using the formulation of

Nielsenet al. (2001), where theShields parameter is modiedas shown

in Eq. (11). The simulated morphological change caused by a 60-cm

solitary wave with and without ltration effects (Eqs. (11) and (10),

respectively), as well as the experimental measurements, are shown in

Fig. 10. It can be seen that both formulations gave satisfactoryagreements with the experimental results. At x2123 m and

x2527 m, the model with ltration effects yield slightly better

comparison with experimental measurements. The slight favorable

comparison with experiments does not necessarily prove or disapprove

the superiority of the model accounting for the ltration effects due to

themany uncertainties presentin boththe model parameters andthe insitu properties. Therefore, this issue still needs further investigations.

6. Conclusions

In this work, a comprehensive numerical model is developed with

the capability of modeling breaking solitary waves runup and

Fig. 7.Time history of the pore pressure variation at PPS14 (shown in the four subplots in the left column; located at x =25 m; Co-located with WG10) and PPS58 (shown in the

four subplots in the right column; located at the nominal shoreline, x =27 m; Co-located with WG12.). The time origin is shifted by 4.4 s compared to the gures in (Young et al.,

2010b-this issue).



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drawdown over a mobile bed, including the effects of sediment

transport, morphological changes, and bed responses. The individual

components of the numerical model are rst validated against

previous analytical, numerical, and experimental studies. The simu-

lation results are compared to the data obtained from a set of large-

scale experimental studies with breaking solitary waves runup and

drawdown over a ne sand beach presented in (Young et al., 2010b-

this issue).

The comparisons between numerical predictions and experimen-tal measurements suggest that the current comprehensive numerical

model is able to capture the bulk of the physics during breaking

solitary wave runup and drawdown processes over a sand beach.However, there are some discrepancies between the simulation

results and experimental measurements due to the limitations ofthe depth-integrated wave simulator, the empirical nature of the

sediment transport model, as well as the uncertainties in the physical

properties of the bed material.

The study also suggests that including the ltration effects in the

numerical model resulted in slightly favorable comparison to the

experimental results in some cases. However, due to the many

uncertainties in the model parameters, soil properties, and the limited

experimental results, denite conclusions cannot be made regarding

the role ofltrationows, which warrants further research.

Acknowledgements

The authors would like to acknowledge funding by the NationalScience Foundation through the NSF George E. Brown, Jr. Network for

Earthquake Engineering Simulation (grant nos. 0530759 and0653772) and through the NSF CMMI grant no. 0649155. Discussions

with Mr. Volker Roeber, and the data provided by Dr. Yong Wei are

gratefully acknowledged.

References

Army Corps, 1990. Engineering and Design: Settlement Analysis. Department of theArmy, U.S. Army Corps of Engineers, Washington DC. EM 1110-1-1904.

Biot, M.A., 1941. General theory of three-dimensional consolidation. Journal of AppliedPhysics 12, 155164.

Fig. 8. Morphological evolution caused by three consecutive 60-cm solitary wavesrunning onto a nominally 1:15 beach (with 15 min between the wave to allow the

water to return to a calm state). (a) The initial bed pro le and that after three waves

(numerical and experimental). (b) The amount of morphological change by three

waves (numerical and experimental). The envelopes in thin dash lines are the total

deposition and erosion from the numerical simulation, the sum of which being net

change (solid line).

Fig. 9. Variation of the Shields parameter according to Eq. (10) and modied Shields

parameter according to Eq. (11) during initial period of the wave breaking at t=2.94 s.

The wave and bed proles are also shown as references.

Fig. 10. Inuence of exltration and inltration on sediment transport and

morphological changes.



13/13

Borthwick, A.G.L., Ford, M., Weston, B.P., Taylor, P.H., Stansby, P.K., 2006. Solitary wavetransformation, breaking and run-up at a beach. Proceedings Institute of CivilEngineering: Journal of Maritime Engineering 159 (3), 97105.

Cao, Z., Carling, P., 2002. Mathematical modelling of alluvial rivers: reality and myth.Part I: general overview. Water and Maritime Engineering 154, 207219.

Cao, Z., Pender, G., Wallis, S., Carling, P., 2004. Computational dam-break hydraulicsover erodible sediment bed. Journal of Hydraulic Engineering 130 (7), 689703.

Carrier, G., Greenspan, H., 1958. Water waves ofnite amplitude on a sloping beach.Journal of Fluid Mechanics 4, 97109.

Carrier, G., Wu, T., Yeh, H., 2003. Tsunami run-up and draw-down on a plane beach.Journal of Fluid Mechanics 475, 7999.

Conley, D.C., Douglas, L.I., 1994. Ventilated oscillatory boundary layers. Journal of Fluid

Mechanics 273, 261284.Coussy, O., 2004. Poromechanics. John & Wiley, Hoboken, NJ.Drake, T.G., Calantoni, J., 2001. Discrete particle model for sheet ow sediment

transport in the nearshore. Journal of Geophysical Research 106 (C), 1985919868.Fraccarollo, L., Toro, E., 1995. Experimental and numerical assessment of the shallow

water model for two-dimensional dam-break type problems. Journal of HydraulicResearch 33 (6), 843864.

Harten, A., Lax, P., van Leer, B., 1983. On upstream differencing and Godunov-typeschemes for hyperbolic conservation laws. SIAM Review 15 (1), 3561.

Hoefel, F., Elgar, S., 2003. Wave-induced sediment transport and sandbar migration.Science 299, 18851887.

Jeng, D.S., 2003. Wave-induced seaoor dynamics. Applied Mechanics Review 56 (4),407429.

Kim, D., Cho, Y., Kim, W., 2004. Weighted averagedux-type scheme for shallow waterequations with fractional step method. Journal of Engineering Mechanics 130 (2),152160.

Kobayashi, N., Lawrence, A., 2004. Cross-shore sediment transport under breakingsolitary waves. Journal of Geophysical Research 109, C030047.

Lin, P., Liu, P., 2000. A numerical study of breaking waves in the surf zone. Journal of

Fluid Mechanics 259, 239264.Madsen, O.S., Durham, W.M., 2007. Pressure-induced subsurface sediment transport in

the surf zone. Proceeding of Coastal Sediments '07. ASCE, New Orleans, LA, pp.8295.

Madsen, P.A., Srensen, O.R., 1992. A new form of the Boussinesq equations withimproved linear dispersion characteristics. Part II: a slowly-varying bathymetry.Coastal Engineering 18 (3), 183204.

Madsen, P.A., Murray, R., Srensen, O.R., 1991. A new form of the Boussinesq equationswith improved linear dispersion characteristics. Part I. Coastal Engineering 15 (4),371388.

Meyer-Peter, E., Mller, R., 1948. Formulas of bed-load transport. Proc. 2nd meeting,IAHR. Stockholm, Sweden, pp. 3964.

Nielsen, P., 1997. Coastal groundwater dynamics. Proceeding of Coastal Dynamics '97.ASCE, Plymouth, pp. 546555.

Nielsen, P., Robert, S., Mller-Christiansen, B., Oliva, P., 2001. Inltration effects onsediment mobility under waves. Coastal Engineering 42, 105114.

Prvost, J.H., 1983a. DYNAFLOW: A Nonlinear Transient Finite Element AnalysisProgramme. Dept. of Civil Engineering, Princeton University, Princeton, NJ. (lastupdated 2008).

Prvost, J.H., 1983b. Implicitexplicit schemes for nonlinear consolidation. ComputerMethods in Applied Mechanics and Engineering 39, 225239.

Prvost, J.H., 1997. Partitioned solution procedure for simultaneous integration ofcoupled-eld problems. Communications in Numerical Methods in Engineering13,239247.

Pritchard, D.,Hogg, A.J.,2005.On thetransport ofsuspended sedimentby a swasheventon a plane beach. Coastal Engineering 52, 123.

Puleo, J., Holland, K., Plant, N.G., D.S., Hanes, D., 2003. Fluid acceleration effects onsuspended sediment transport in the swash zone. Journal of Geophysical Research

108 (3350).doi:10.1029/2003JC001943.Sassa, S., Sekiguchi, H., 1999. Wave-induced liquefaction of beds of sand in a centrifuge.

Gotechnique 49 (5), 621638.Sassa, S., Sekiguchi, H., 2001. Analysis of wave-induced liquefaction of sand beds.

Gotechnique 51 (2), 115126.Sassa, S., Sekiguchi, H., Miyamoto, J., 2001. Analysis of progressive liquefaction as a

moving-boundary problem. Gotechnique 51 (10), 847857.Schoonees, J.S., Theron, A.K., 1995. Evaluation of 10 cross-shore sediment transport/

morphological models. Coastal Engineering 25 (1), 141.Simpson, G., Castelltort, S., 2006. Coupled model of surface water ow, sediment

transport and morphological evolution. Computers & Geosciences 32, 16001614.Synolakis,C., 1987. Therunup of solitarywaves. Journalof Fluid Mechanics 185,523545.Tivoli, V., Synolakis, C., 1995. Modeling of breaking and nonbreaking long wave

evolution and runup using VTSC-2. Journal of Waterway, Port, Coastal, and OceanEngineering 121, 308316.

Toro, E., 2000. Shock-Capturing Methods for Free-Surface Shallow Flows. John Wiley &Sons, Ltd.

Tuck, E., Hwang, L., 1972. Long wave genertion on a sloping beach. Journal of FluidMechanics 51, 449461.

Turner, I.L., Masselink, G., 1998. Swash inltrationexltration and sediment transport.Journal of Geophysical Research 103 (C), 3081330824.

USACE, 1984. Shore Protection Manual, 4th Edition. Department of the Army, U.S.Corpsof Engineers, Washington, DC.

Verruijt, A., 1969. Elastic storage of aquifers. In: De-Wiest, R.J.M. (Ed.), Flow throughPorous Media. Academic Press, New York, pp. 331376. Ch. 8.

Wei, Y., Mao, X., Cheung, K., 2006. Well-balanced nite-volume model for long waverunup. Journal of Waterway, Port, Coastal, and Ocean Engineering 132 (2),114124.

Young, Y.L., White, J.A., Xiao, H., Borja, R.I., 2009a. Liquefaction potential of coastalslopes induced by solitary waves. Acta Geotechnica 4 (1), 1734.

Young, Y.L., Xiao, H., Maddux, T., 2010b. Hydro- and morpho-dynamic modeling ofbreaking solitary waves over a ne sand beach. Part I: Experimental study. MarineGeology 269 (3-4), 107118 (this issue).

Zoppou, C., Roberts, S., 2000. Numerical solution of the two-dimensional unsteady dambreak. Applied Mathematical Modelling 24, 457475.


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