laboratory study of solitary-wave transformation over bed-form roughness on fringing reefs

7/25/2019 Laboratory Study of Solitary-wave Transformation Over Bed-Form Roughness on Fringing Reefs

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Laboratory study of solitary-wave transformation over bed-formroughness on fringing reefs

Pablo D. Quiroga 1, Kwok Fai Cheung

Department of Ocean and Resources Engineering, University of Hawaii at Manoa, Honolulu, HI, USA

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

Article history:

Received 16 August 2012

Received in revised form 1 May 2013Accepted 6 May 2013

Available online 14 June 2013

Keywords:

Bed-form roughness

Bores

Breaking waves

Fringing reefs

Macro roughness

Surf zone processes

This paper presents the formulation and implementation of a series of two-dimensional ume experiments

to investigate effects of bed-form roughness on coastal wave processes. The experiments were carried out

on a fringing reef model in a 104-m long and 4.6-m high ume at Oregon State University. The reef model

has a 1:12 face slope and a long at section for examination of wave shoaling and breaking as well as bore

formation and propagation. The model is 2.36 m tall and the water depth ranges from 2.36 to 2.66 m to pro-

duce dry and wet reef conditions. The bed-form roughness is modeled by timber beams placed across the

ume in four congurations by varying the height from 0.038 to 0.076 m and the spacing from 0.388 to

0.768 m to provide a range of pitch ratios from 5 to 20. The incident solitary wave height varies from 10 to

50% of the water depth to cover a range of breaking and non-breaking conditions. A series of wire and

sonic gauges measured the wave transformation along the ume and a digital camera recorded images of

the breaking waves on a background grid painted on a ume wall. The bed forms decrease the effective

depth for wave propagation and modify the structure of the free surface ow. In comparison to a control

experiment with plain concrete surface, the solitary wave breaks earlier and dissipates more energy on the

reef slope. The subsequent bore slows down with undulation over the shallow reef at, but speeds up for

more energetic ows in deeper water.

2013 Elsevier B.V. All rights reserved.

1. Introduction

Many tropical andsub-tropical islands in thePacic are susceptible

to ood hazards due to tsunamis, hurricanes, and high-surf events.

Accurate prediction of the near-shore wave conditions is important

in coastal structure design, land-use planning, and hazard assessment.

The presence of fringing reefs along these coastlines results in more

complex near-shore processes than those on gentle slopes and sandy

beaches in non-tropical environments (Gerritsen, 1981).Fig. 1shows

a cross section of the reef at Mokuleia on the north shore of Oahu,

Hawaii. The prole, which references to the mean sea level (MSL),

includes a fore reef and a shallow reefat typical of Pacic island envi-

ronments. The abrupt slope transition at the reef edge introduces ener-

getic breaking waves thatresult in bore formation andpropagation over

the shallow reefat (Roeber et al., 2010). The energy dissipation pro-

cesses are augmented by the irregular reef surface with an abundance

of coral heads and colonies of reef organisms (Hardy and Young,

1996; Lowe et al., 2005; Nelson, 1996).

Wave breaking and dissipation in fringing reef environments have

recently received attention in the research community. Nwogu and

Demirbilek (2010) reported a wave ume experiment on irregular

wave transformation over a scaled model of a fringing reef on Guam.

Roeber (2010) described two series of large-scale ume experiments

with 10 two-dimensional reefs modeled after cross-shore proles in

Hawaii, Guam, and American Samoa. Swigler (2009)conducted basin

experiments for solitary wave transformation over a three-dimensional

reef design. These studies provided data for validation and calibration

of numerical models and understanding of wave processes over reef

geometries (e.g., Bai and Cheung, 2012, 2013; Filipot and Cheung,

2012; Kazolea and Delis, 2013; Roeber and Cheung, 2012; Sheremet et

al., 2011; Shi et al., 2012; Tonelli and Petti, 2013). These experiments,

however, were performed on plexiglass or nished concrete surface

without the bed-form roughness commonly found in reef environments.

Lowe et al. (2005) concluded from a eld experiment at Kaneohe Bay on

the east shore of Oahu, Hawaii that bottom friction may dominate the

energy dissipation over the reefat.

The dissipation mechanism due to free surface ows over rough

beds has been a subject of intense investigation.Sleath (1987),Chen

et al. (2007),Dixen et al. (2008), andLowe et al. (2008)conducted

ume experiments to investigate dissipation over gravel, stone, and

coral beds. Parameterized roughness geometries consisted of regu-

larly placed pipes and triangles provide a systematic approach to

examine energy dissipation in unidirectional and oscillatory ows

Coastal Engineering 80 (2013) 3548

Corresponding author. Tel.: +1 808 956 3485; fax: +1 808 956 3498.

E-mail addresses:[email protected](P.D. Quiroga),[email protected]

(K.F. Cheung).1 Present address: ESA PWA, 550 Kearny Street, Ste 900, San Francisco, California

94108.

0378-3839/$ see front matter 2013 Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.coastaleng.2013.05.002

Contents lists available at SciVerse ScienceDirect

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(e.g., Mirfenderesk and Young, 2003; Ojha and Mazumder, 2010;

Suntoyo et al., 2008). Their results advance the understanding of

laminar and turbulent boundary layers over rough beds and provide

a good resource for validation of computational uid dynamic, wave

propagation, and circulation models (e.g.,Fuhrman et al., 2009; Lowe

et al., 2010; Suntoyo and Tanaka, 2009). These ume experiments use

a small roughness height compared to the water depth allowing

formation of laminar and turbulent layers well below the free surface

for parameterization of the dissipation processes with a wave fric-

tion factor.

Coral reef organismsform some of the most jagged surfaces in coast-

al waters. The roughness height, for example, is typically 0.51.0 m on

Oahu, Hawaii (Nunes and Pawlak, 2008). The reefat is quite shallow,

usually less than a couple of meters deep. The dissipation might depend

on the water depth in addition to the roughnessheight and spacing sug-

gested by Raudkivi (1988). For reef environments with sparse coral

communities, theow from surface waves detaches behinda roughness

element and re-attaches in front of the next. A recirculation region isformed adjacent to the roughness element with an internal boundary

layer. This process plays an important role on the macro-turbulence

structure and dissipation rate from bed-form roughness (Leonardi et

al., 2003). For shallow ows, the wakes behind the roughness elements

may extend to the water surface. Thecoupling between free surface and

bed-form induced ows is sporadic and not well understood with little

information in the technical literature.

Prior experiments on wave breaking over fringing reefs have been

performed over smooth or nished surfaces, while dissipation due to

bed-form roughness has been investigated with water depth much

greater than the roughness height. Due to coupling between bottom

friction and wave breaking over shallow reefs, it is necessary to combine

the two dissipation mechanisms in a single laboratory experiment to

characterize the physical processes. Since scaling is an issue for these

processes, physical experiments in a large ume are preferred.Quiroga

(2012) extended the large-scale experiments of Roeber (2010) by

including bed-form roughness on a fringing reef model. The laboratory

study provided measurements of solitary wave transformation over

roughness elements constructed of timber beams with the height and

spacing varied under a range of

ow conditions. The controlled labora-tory environment enables a systematic investigation of the bed-form

effects on wave shoaling, breaking, and bore propagation. This paper

provides a summary of the experiments and results from Quiroga

(2012)as well as further analysis and interpretation of the data.

2. Laboratory experiments

A seriesofumeexperimentswere carried out at theO.H. Hinsdale

Wave Research Laboratory, Oregon State University in 2009. The test

facility is a National Science Foundation designated site for tsunami

research within the Network for Earthquake Engineering Simulation.

Fig. 2 showsa schematicof theexperiments andthe pertinent physical

variables. The wave ume measures 104 m long, 3.66 m wide, and

4.6 m high. Prefabricated concrete slabs of 0.2 m thickness, 3.66 m

width, and 4.57 m length were mounted at bolt holes in the ume

walls to construct a fore reef with a 1:12 slope and a reefat 2.36 m

above the bottom. A wedge in front of the fore reef provides a smooth

transition between the 0.2-m slab and the oor of the ume. A

piston-type wavemaker with a programmable hydraulicactuator gen-

eratesthe incident solitary wave, which has been commonly used

in laboratory studies of wave transformation and runup (e.g., Briggs

et al., 1995; Grilli et al., 1994; Hsiao et al., 2008; Li and Raichlen,

2002; Roeber, 2010; Swigler, 2009; Synolakis, 1987). The use of soli-

tary waves allows precise measurements of wave transformation

and energy dissipation without interference from return ows, wave

setup, and end-wall reection. The laboratory experiments represent

a simplication of wind generated waves, which resembles a series

of solitary waves in shallow reef environments.

Coral reef roughness is inhomogeneous with varying length scalesand a broad spectral distribution (Nunes and Pawlak, 2008). For gen-

eralization and ease of installation, a geometrical representation of

the bed-form roughness was made using assemblies of 2 4 timber

beams at regular intervals across the reef model.This allows parameter-

ization of theroughnessin terms of theheight k and spacing. The pitch

ratio /kdenes the wake behind a roughness element and the overall

dissipation mechanism(Raudkivi, 1988).We focus on the k-typerough-

ness with /k 5 that exposes the recirculation vortices to the free

surface ow.Fig. 3illustrates the four bed-form congurations in the

experiments. BF1 and BF3 have the same pitch ratio of 10, but rough-

ness heights of 3.8 and 7.6 cm. BF2 and BF3 have the same roughness

Fig. 1.Fringing reef prole at Mokuleia, Hawaii.

Fig. 2.Schematic of wave ume experiments.

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height of 7.6 cm, but pitch ratios of 5 and 10. The friction factor varies

with the pitch ratio and becomes maximum around /k = 7 in the ab-

sence of a free surface (Leonardi et al., 2003). The four congurations

cover an important range of roughness conditions as well as a large

pitch ratio of 20 for BF4 that produces negligible interactions of the

turbulence between roughness elements. This parameter setting allows

examination of the roughness height and pitch ratio with a minimum

number of experiments.Fourteen resistance-typewave gauges(WG1 to WG14)with parallel

wires andsevensonicwave gauges(SW1 to SW7) measured thesurface

elevation along theume at a sampling rate of 50 Hz. Resistance-type

wire gauges give accurate readings of the surface elevation for non-

breaking waves. Since ultrasonic wave gauges can track sheet ows

over dry beds as well as turbulent bores with air entrainment and

spray, they are deployed on thereefat to provide redundant measure-

ments of the ow conditions. Each wave gauge was mounted at a

distance of 0.44 m from a side wall. Because of the relatively steep

fore reef, the wave amplitude increases rapidly and a steep wave front

develops just prior to breaking. The wave gauges might not capture

the breaking wave height. A video camera simultaneously recorded im-

ages of breaking waves at 30 fps over a 0.5 m 0.5 m grid painted on a

ume wall. Incipience of wavebreaking is dened at the moment whenthe wave front becomes near vertical just before spilling or jet forma-

tion at the crest (Grilli et al., 1997; Hsiao et al., 2008). Post-processing

of the video images identies the breaking wave and the breaker type.

Fig. 4 shows a video image of a test from which the breaking wave

heightHband depthhbcan be determined.

Experiments were conducted at four water depths,ho= 2.36, 2.46,

2.56, and 2.66 m, measured from the bottom of the ume. These corre-

spond to water depths ofhf = 0, 0.1, 0.2, and 0.3 m over the reefat for

examination of sheet ow and bore propagation. The wavemaker

generates the input solitary wave with dimensionless heights ofHo/ho= 0.1 to 0.5 at 0.05 increments. Despite formation of trailing

waves, the solitary wave remains stable before reaching the reef

slope.Fig. 5shows good repeatability of the measured wave proles

at WG1 over the range of test conditions (tdenotes time andgaccel-

eration due to gravity). The variation of the recorded wave heights

from each set of tests is less than 2% of the target value. The trailingwaves indicate adjustment of the solitary wave prole during

the generation process. The averaged recorded wave height is 2%

larger than the target value at Ho/ho= 0.5, but that increases to 6%

atHo/ho= 0.1 as the longer wave becomes less reproducible by the

wavemaker. The incident wave height is thus dened by measure-

ments from WG1 at 8.3 m in front of the reef slope. The roughness

elements were initially installed on the reef slope from WG2 to

WG9 for examination of wave shoaling and breaking. In the second

series of experiments, the roughness elements were transferred to

the reefat from WG9 to the end of the ume. The incident solitary

wave transforms over the plain reef slope into a bore on the reefat

for investigation of the roughness effects.

3. Data post-processing and analysis

The laboratory study included 32 series of tests for the four water

depths, four bed-form congurations, and two bed-form placements as

well as four series of control tests without the bed forms. Each test pro-

vides time series of the surface elevation at the gauges along the ume

and videos images of the breaking wave near the reef edge for the given

incident wave height. The quality of the recorded signals is a concern

for breaking waves and turbulent bores, which involve air entrainment

Fig. 3.Roughness element congurations and arrangements.

Fig. 4.Video capture of a breaking wave and its measurements.

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and splashing. Additional post-processing with a weighted-average

lter was necessary to remove spikes and outliers from the wave

gauge signals. Very often these noises are severe enough to obscure

the signals of the breaking waves or turbulent bores. The redundant

measurements by the wire and sonic wave gauges on the reef at

allow cross validation and reconstruction of the wave signals. For qual-

ity assurance, each test was conducted at least twice and the measure-

ments with minimal or correctable anomalies were averaged forsubsequent analysis. This results in a total of 746 tests in the three-

month laboratory study.

The experiments provided a large volume of time series data for

post-processing and analysis. Fig. 6 shows, for example, the time

series of the free surface from the control and bed-form tests with

hf= 0.1 m and Ho/ho= 0.2. The recorded surface elevations with

BF2 on the reef slope and at are shown in the left and right columns,

respectively. The results from the control test in both columns are

identical. The input solitary wave shoals on the plain slope with

noticeable steepening. Reection from the reef slope was observed

during the laboratory experiments and recorded by WG5 and WG6

as an elongated tail of the wave prole. A near vertical front develops

at WG9 and a plunging breaker occurs across the reef edge. WG10 on

the reefat recorded two peaks in the control test corresponding to

the initial splash up from the plunger and the subsequent broken

wave. The ow transforms into a turbulent bore on the shallow

water over the at. The data at WG14 shows the incident bore and

its reection from the end wall. The presence of bed forms on the

fore reef slightly modies the shoaling processes, but does not seem

to increase reection. The wave breaks just before reaching WG9

resulting in a reduced height comparing to the control test data. The

bed forms on the at cause noticeable reection of the input bore at

WG9 that in turn extends the tail of the surface pro

les on the slope.The bed forms reduce the initial splash up at WG10 and increase the

subsequent surface undulation modifying both the bore height and

speed on the at.

The post-processed data denesthe time historiesof solitary wave

and bore propagation over the reef model for analysis of the bed-form

effects. Solitary waves produce net transport of water mass in the

direction of propagation. The horizontal water particle displacement

is of the order of the ow depth, which includes the water depth

and the wave height. An examination of the post-processed data

shows that the ow depth on the reef slope is greater than the

maximum roughness element spacing of 0.768 m for tests withHo/ho 0.2. This is true even at breaking near the reef edge as the

wave height is several times larger than the water depth as seen

inFig. 4. The subsequent surge on the reef slope has a much larger

horizontal length scale associated with the transition from subcrit-

ical to supercritical ow during the breaking process. The supercrit-

icalow in turn generates a bore downstream via a hydraulic jump

near the reef edge. Bore propagation on the reefat is analogous to

open channel ow that the water traverses across the roughness

elements regardless of the bore height. Because the water particle

displacement exceeds the roughness element spacing in most of

the tests, the results enable a comparative study of the effects

of roughness height and pitch ratio on wave propagation and

dissipation.

4. Results and discussion

The post-processed data is divided into three sets, which are evalu-

ated for the bed-form effects on wave shoaling, breaking, and borepropagation. The results from the control tests with plain beds provide

a reference to assess the change in wave properties and dissipation

rates. The primary ow associated with the solitary waves and bores

is unidirectional along the bed. Secondary ow features associated

with the bed forms might extend into the water column to inuence

the free-surface ow. However, the laboratory experiments were not

designed to collect velocity data over the water column for analysis of

the detailed ow structure. We utilize published results and established

theories for steady owsover bed-form roughness,albeit in theabsence

of a free surface, to assist interpretation of the laboratory data.

4.1. Wave shoaling

The bed forms were installed on reef slope during the rst phaseof the experiments to investigate their effects on wave shoaling and

breaking.Fig. 7shows, for example, the normalized surface envelopemax/Ho over the reef slope for the water depth hf= 0.1 m over the

reefat and the full range of input wave conditions from Ho/ho= 0.1

to 0.5. The initial wave transformation on the reef slope is primarily

shoaling. The smaller solitary waves have longer wavelengths and

thus shoal earlier, but do not break on the slope as observed for the

tests withHo/ho= 0.1. The wave height reduction between WG8 and

WG9 is due to reection by the reef slope and energy transmission

from the non-breaking wave to the reefat. The larger solitary waves

shoal later, but break earlier on the slope as observed in the tests with

Ho/ho = 0.2 to 0.5. The breaking wave height is not necessarily the

largest due to reection and transmission of energy. The reduction

of max/Ho at WG9 for the larger values of Ho/ho is due to a

Fig. 5.Solitary wave proles at WG1 from four separate test runs denoted by dark blue,

red, green, and light blue lines.



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combination of reection and breaking. The bed forms show clear

effects on the surface elevation as the wave shoals and breaks on

the slope. Since the slope is uniform, an increase in water depth sim-

ply shifts the shoaling process toward the reefat and vice versa. An

exception occurs at the reef edge, where the shoaling process is

interrupted and the wave height also depends on the water depth

over the reefat. For Ho/ho 0.2, gradual shoaling occurs in front

of WG3, the surface elevation increases rapidly from WG3 to WG7,

and wave breaking occur between WG7 and WG9. Since the solitarywave does not produce a trough during the transformation, the

maximum surface elevation recorded at any location is equivalent

to the local wave height.

The steady increase in wave height from WG3 and WG7 covers the

zone of rapid shoaling as dened bySynolakis and Skjelbreia (1993).

The measured data allows examination of the bed-form effects on the

shoaling process prior to wave breaking. Fig. 8plots the wave height

gradientH/xas a function of the normalized incident wave height

Ho/ho, where x is the distance between WG3 and WG7. The data

obtained with the four water depths, which represent different stages

of the shoaling process, show a similar pattern with the roughness

conguration. The control tests with a smooth concrete surface give

the largest wave height gradient and the scatter of the data about

the blue trend line provides an indication of laboratory errors. The

bed-form roughness increases the friction and dissipates wave energy

during the shoaling process. The wave height gradient decreases

appreciably from the control data beyond the margin of laboratory

errors. Both the pitch ratio and roughnessheight inuence the results.

Over the range of pitch ratios from 5 to 20, BF3 with a value of 10 re-

sults in the largest reduction. The green dotted lines tted through

the data shows a consistent pattern with the range of water depth

considered. This corroborates the nding ofLeonardi et al. (2003)

that the energy dissipation is maximum at an intermediate range of

pitch ratios. The energy dissipation is also a function of the roughnessheight. BF1 has the same pitch ratio of 10, but produces consistently

larger gradients because of having half of the roughness height in

comparison to BF3.

The celerity, which is the same as group velocity for long waves, is

a good indicator of the shoaling process.Fig. 9shows the normalized

average celerity C=

ffiffiffiffiffiffigh

q computed by timing the peaks between WG3

and WG7, wheregis acceleration due to gravity andh is the average

water depth between the two gauges. A larger water depth corre-

sponds to an earlier stage of the shoaling process that results in a

more gradual increase of the celerity with the normalized incident

wave height Ho/ho. The data does not show a clear pattern of the rough-

ness effects on the celerity. This can be explained by the nonlinear

long-wave equations, in which the bottom friction is an external force

with no effects on the characteristic lines. However, the scatter appears

Fig. 6.Time series of surface elevation at wave gauges along the ume forHo/ho= 0.2 and hf= 0.1 m. Blue and black lines denote gauge data from the control and BF2 tests.



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to increase with the roughness height and is beyond the range of labo-

ratory errors as inferred from thet of the control data. Forsteady ows

over bed forms, Jimnez (2004) approximated the roughness layer

thickness as

kR k min 1 =k; 5 : 1

This gives a value of 0.38 m for the largest roughness height ofk=

7.6 cm. The average water depth between WG3 and WG7 ranges from1 to 1.3 m for the four experiments. The turbulence introduced by the

bed forms probably does not extend far enough into the water column

to inuence the momentum balance in the free-surface ow and the

resulting wave properties. The green dotted line tted through the

BF3 data indicates a slight reduction of the celerity probably associated

with the smaller wave height in comparison to the control tests. The

bed-form roughness dissipates energy and decreases the wave height

through bottom friction, but does not signicantly modify the celerity

during the shoaling process.

4.2. Wave breaking

Wave breaking occurs between WG7 and WG9 for most of thetests.

The wave height shows themost rapid decrease across a narrow region

as noted by Synolakis and Skjelbreia (1993). Despite reection from the

steep slope and transmission to the at, the wave height gradient be-

tween these two gauges provides a general indication of the bed-form

effects on the breaking process especially for the larger wave heights

and the tests with shallower water.Fig. 10shows the wave height gra-

dient H/x as a function of the normalized incident wave height Ho/hofor thefourseries of tests with the waterdepth hf= 0 to0.3 mover the

reefat. Thewave height gradient, in general, shows a downward trend

with Ho/ho as energetic breaking of the largerwaves dissipates more en-ergy over a short distance. An increase in water depth effectively shifts

thesmaller breakingwaves toward thereef edge at WG9 and enhances

the effects of the standing water over the reefat on the breaking pro-

cess. In some cases, the incident wave does not break on the reef slope

but transforms into a bore immediately after propagating over the reef

edge. This results in less energy dissipation and even some effects of

shoaling between WG7 and WG9 that accounts for the initial upward

trend and positive values of the gradient for hf= 0.2 and 0.3 m.

The results show a clear inuence of the bed forms on the wave

breaking process. The control tests with a smooth bed yield the highest

gradient or lowest energy dissipation rate as indicated by the blue trend

line. The energy dissipation shows much stronger dependence on the

pitch ratio than theroughnessheight. Thegreendash linetted through

the BF2 data with the lowest pitch ratio of 5 shows the highest

Fig. 7.Maximum surface elevation along thereef slope for hf= 0.1 m. Bluecrosses connected by bluelines denote gauge data andback circles denote video dataof breaking waves from

the control tests; magenta squares, green diamonds, green triangles, and magenta circles denote gauge data from the BF1, BF2, BF3, and BF4 tests.



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dissipation rate among the four bed-form congurations, while BF4

with the largest pitch ratio of 20 produce results close to the control

tests. BF1 and BF3, which have the same pitch ratio of 10, produce

very similar results despite their roughness heights of 3.8 and 7.6 cm.

In contrast to wave shoaling, a smaller pitch ratio of 5 produces the

highest dissipation rate in the surf zone. Bed friction due to roughness

is not the primary mechanism for energy dissipation during wave

breaking. The average water depth between WG7 and WG9 rangesfrom 0.25 to 0.55 m for the fourexperiments. Secondaryows generat-

ed by the bed forms probably extend far enough into the water column

to modify the momentum balance and the subsequent wave breaking

process. It was observed during the experiments that the bed forms ini-

tiate more energetic, plunging breakers in deeper water and increase

the subsequent splashing and air entrainment in the surf zone.

Analysis of the videos taken during the experiments pinpoints the

breaking wave location and height. Only the data with incident wave

heights ofHo/ho 0.2 is considered for their well-dened breaking

incipience up to the reef edge. Fig. 11shows strong correlation be-

tween the normalized breaking depth hb/ho with the four bed-form

congurations. The results fromhf= 0 and 0.1 m, which both corre-

spond to plunging breakers on the reef slope, give similar relations for

cross validation. As the water depth increases to hf= 0.2 and 0.3 m,

the smaller waves develop into bores at the reef edge resulting in a

constant breaking depth. Other than that the trend lines through

the control and BF2 data highlight the increase in breaking depth

with the roughness element height and density. The increase in

breaking depth varies with the incident wave height and reaches an

average value ofhb/ho= 0.034 or hb= 9.2 cm for Ho/ho 0.45.

The roughness elements introduce a displacement height at the bot-

tom that decreases the effective depth for the ow and forces thewaves to break earlier. Jackson (1981) estimated the displacement

height at 0.7 k from a large range of bed-form roughness in steady

ow. This suggests a displacement height of 5.3 cm for BF2 that is

considerably smaller than the increase in breaking depth. A second

mechanism, in which the roughness elements redirect the predomi-

nantly horizontal ow upward, might explain the change in the

free-surface ow. Numerical model results of solitary wave propaga-

tion on bed forms show transfer of momentum from the bottom to

the free surface causing acceleration of the wave crest to produce

more energetic breaking at deeper water (Sambe et al., 2011).

Fig. 12plots the breaking wave heightHbas a function of the inci-

dent wave heightHo, both normalized by the water depthho. The four

series of tests with hf = 0 to 0.3 m show very similar trend lines

through the control and BF2 data. Despite producing a larger breaking

Fig. 8.Wave height gradient on the reef slope between WG3 and WG7 as a function of the solitary wave height at WG1. Blue crosses, magenta squares, green diamonds, green tri-

angles, and magenta circles denote data from the control, BF1, BF2, BF3, and BF4 tests; blue and green dotted trend lines pass through the control and BF3 data.



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depth, the bed forms do not have signicant effects on the breaking

wave height. The larger breaking depth appears to offset the upward

momentum transfer in maintaining the wave height. The breaking,

however, is more energetic contributing to larger wave height reduc-

tions as seen inFig. 10. The breaking index, which is dened as the

ratio of the breaking wave height and depth, is an important param-

eter in surf-zone processes.Fig. 13shows the breaking index Hb/hbas

a function of the normalized incident wave height Ho/ho. Because of

the steep 1:12 slope, the waves break in shallower water and produce

relatively high breaking indices. In comparison, Grilli et al. (1997)reported breaking indices of 1.36 to 1.47 with the same criterion for

numerically generated waves on a gentle beach slope of 1:35. The

breaking index decreases with the roughness height and element

density since the waves break in deeper water while the breaking

height remains essentially unchanged. An exception to this general

pattern arises from the breaking waves at the reef edge in which

the standing water over the at denes the breaking depth of the

smaller waves resulting in drastically different trend lines for hf=

0.2 and 0.3 m.

4.3. Bore propagation

Bore propagation and decay have been studied extensively through

the nonlinear shallow-water equations since the seminal works of

Stoker (1957)and Whitham (1958). The shock-capturing techniques

approximate breaking and broken waves as bores and conserves mo-

mentum across the discontinuities to account for energy dissipation.

Recent Boussinesq and non-hydrostatic models have incorporated

these techniques to describe bore propagation and energy dissipation

in the surf zone with encouraging results (e.g.,Bai and Cheung, 2012,

2013; Kazolea and Delis, 2013; Roeber and Cheung, 2012; Shi et al.,

2012; Tonelli and Petti, 2013; Zijlema et al., 2011). Here, we present a

laboratory dataset that elucidates these processes in the presence of

bed-form roughness.The bed-form roughness was transferred from the reef slope to the

at in the second phase of the laboratory study. The experiments

were repeated with smooth concrete surface on the reef slope to pro-

vide the input wave at the reef edge. The broken or breaking wave

transforms into a bore over the bed forms on the reef at. Fig. 14

plots, for example, the surface elevation envelope from WG9 to WG14

for hf= 0.1 m. The input wave height at WG9 is generally well de-

ned, but might include effects of reection and ow-backup from

the bed forms. Turbulence and air entrainment from the breaking

wave produce modulations of the surface elevation as far as WG12.

The bore height decreases along the at due to reection by the

bed forms, dissipation from turbulence and bottom friction, and at-

tenuation of the input ow at the reef edge over time. The bores typ-

ically stabilize between WG13 and WG14 to dene the output

Fig. 9.Average celerity on the reef slope between WG3 and WG7 as a function of the solitary wave height at WG1. Blue crosses, magenta squares, green diamonds, green triangles,

and magenta circles denote data from the control, BF1, BF2, BF3, and BF4 tests; blue and green dotted trend lines pass through the control and BF3 data.



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condition of the transformation process. The control tests with a

smooth bed give the largest surface elevation. The bed-form rough-

ness increases the turbulence in theow and the dissipation causes

signicant reduction of the bore height. For the dry bed case, the

bore completely collapses and turns into a sheet ow on the reef

at conforming to the theoretical analysis and numerical results of

Hibberd and Peregrine (1979). The bed forms create extensive local

splashing that is not amenable to either the wire or sonic gauges.

We only consider the results obtained from the water depth hf=

0.1, 0.2, and 0.3 m in the subsequent analysis.Fig. 15 plots the wave height gradient H/x between WG9 and

WG14 as a function of the normalized wave height H/hf, where H is

the average between the two gauges. The use ofHinstead of the input

wave height at WG9 reduces the data scatter and improves the correla-

tion. The gradientshows a clearrelation with thewaveheight,roughness

height, and water depth. The dissipation rate increases with the wave

height due to more energetic wave breaking and turbulence at the

front. The shorter wavelength associated withthe larger incidentsolitary

wave also contributes to the attenuation of the wave height along the

reefat. The results show strong dependence on the roughness height,

because of the direct relation between the form drag on the roughness

element and the bottom friction. A higher pitch ratio for the same

roughness height contributes to slight reduction of the dissipation rate

within the range of laboratory errors. Since the roughness layer

thickness is of the same order as the water depth, the wake extends

to the downstream free surface regardless of the element spacing. The

wave height reduction due to the bed forms also depends on the

water depth. A smaller water depth amplies the effects of the bed

forms in dissipating the wave energy. These ndings have important

implications for numerical modeling of wave transformation over fring-

ing reefs. The pitch ratio, which is often poorly dened and highly var-

iable,plays a secondary rolein energydissipation. Theroughnessheight

and water depthare key parameters that dene the wave friction factor

over the reefat.The bore speed is an important characteristic parameter in shock

propagation that can be inuenced by the turbulence generated

from the bed forms. Fig. 16 shows the normalized average celerityC=

ffiffiffiffiffiffiffiffighf

q between WG9 and WG14 as a function of the dimensionless

wave heightH/hf. The results show that the bed-form roughness plays a

signicant role in borepropagation onthe reefat.When thewave height

is small, the bed forms tend to decrease the bore speed. Observations

during the experiments show that the roughness elements generate an

undulating bore, which has a smaller propagation speed. As the ow

depth increases with the wave height, the undulation over the bed

forms diminishes. The increase in propagation speed can be attributed

in part to reduction of the effective depth. The bed forms also enhance

vertical advection that increases the ow speed near the surface as

shown in the numerical results ofSambe et al. (2011). A higher pitch

Fig. 10.Wave height gradient across the wave breaker on the reef slope between WG7 and WG9 as a function of the solitary wave height at WG1. Blue crosses, magenta squares,

green diamonds, green triangles, and magenta circles denote data from the control, BF1, BF2, BF3, and BF4 tests; blue and green dash trend lines pass through the control and BF2

data.

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10/14

ratio with more widely spaced roughness elements slightly reduces

their effect on the free surface ow. The overall effect of the bed forms

on the bore speed diminishes with increasing water depth. The rough-

ness layers from all four bed forms likely extend to the water surface

to inuence the free surface ow forhf= 0.1 m. The results from BF1

and BF4 with an estimated 0.18-m roughness layer from Eq.(1)show

convergence to the control test data athf= 0.2 m, while BF2 and BF3

with an estimated 0.36-m roughness layer still have inuence on the

bore speed athf= 0.3 m.

5. Conclusions and recommendations

A series of large-scalelaboratory experimentshave provideda unique

dataset to elucidate wave shoaling, breaking, and bore propagation over

bed-form roughness on an idealized fringing reef. The use of solitary

waves in the experiments allows precise measurements of wave trans-

formation across the reef without interference from return ows, wave

setup, and end-wall reection. Construction of the bed forms by timber

beams facilitates a systematic approach to investigate their effects

in terms of the roughness height and pitch ratio. The experiments

utilized the k-type roughness, which generates turbulence into the

water column, to examine bed-form effects on wave propagation and

dissipation. A set of control tests with smooth concrete surface on the

reef provides a reference dataset for comparison. The wave transforma-

tion processes were recorded by resistance and sonic wave gauges

along the ume as well as a video system focusing on the wave breaker.

The bed-form roughness increasesthe bottomfriction and dissipates

wave energy during the shoaling process. Since the roughness height

is small compared to the water depth, the bed forms do not modify

the wave propagation characteristics appreciably. The measurements

reafrm a previous nding that the highest dissipation rate occurs atan intermediate range of pitch ratios. As the water depth decreases

relative to the roughness height, a smaller pitch ratio produces the

highest dissipation rate at wave breaking. The bed forms reduce the

effective water depth for wave propagation and increase the vertical

advection. The breaking depth increases with the roughness height

and density, while the breaking wave height remainslargely unaffected.

The roughness height and water depth, instead of the pitch ratio,

dene the bore propagation and dissipation rate on the reef at. The

celerity of the smaller waves decreases because of formation of undular

bores over the bed forms, while the larger bores speed up due to reduc-

tion in the effective depth and enhancement of vertical advection.

The experiments also produced a dataset to validate and calibrate

numerical free-surface ow models for tropical coastal environments.

The recorded dissipation on the reef slope and at can be readily

Fig. 11. Wave breaking depth on the reef slope as a function of the solitary wave heightat WG1. Blue crosses, magenta squares, green diamonds, green triangles, and magenta circles

denote data from the control, BF1, BF2, BF3, and BF4 tests; blue and green dash trend lines pass through the control and BF2 data.

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used to calibrate the wave friction factor and the eddy viscosity for

phase-averaged and phase-resolving models. Existing parameteriza-

tions, however, are insufcient to account for the bed form effects on

wave propagation in shallow water. These sub-grid features modify

intrinsicwave properties in addition to thedissipationrate.It is neces-

sary to develop new algorithms that enhance vertical advection in a

numerical wave model to describe acceleration of the surface ow,

formation of undular bore, and modication of the bore speed. Futurelaboratory studies might try to model more realistic wave conditions.

Additional experiments with a wider range of roughness heights and

pitch ratios will certainly help produce a comprehensive dataset for

development of numerical models.

Acknowledgment

This study was funded in part by the National Science Foundation

Grant No. 0530759 through the Network for Earthquake Engineering

Simulation. The National Tsunami Hazard Mitigation Program provid-

ed additional support via Hawaii State Civil Defense through Grant

No. NA09NWS4670016. The authors would like to thank Dan Cox,

Jason Killian, Tim Maddox, Ian Robertson, Abdulla Mohamed, Volker

Roeber, and Kim Quesnel for the assistance with the laboratory

experiments as well as the two anonymous reviewers, whose com-

ments and suggestions have greatly improved this paper. SOEST Contri-

bution No. 8770.

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588.

Fig. 13.Wave breaking index on the reef slope as a function of the solitary wave height at WG1. Blue crosses, magenta squares, green diamonds, greentriangles, and magenta circles


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List of symbols

BF: bed formC: celerityhb: wave breaking depth determined from video imagesho: water depth in the wave umehf: water depth over the reefat

Hb: breaking wave height determined from video imagesHo: incident wave height recorded at WG1h: average water depth between WG7 and WG9H: average bore height between WG9 and WG14k: roughness heightSG: sonic gaugeWG: wire gaugeH: wave height difference between WG3 and WG7 for wave shoaling, WG7 and WG9for wave breaking, and WG9 and WG14 for bore propagationhb: increase in breaking depth due to bed formsx: distance between WG3 and WG7 for wave shoaling, WG7 and WG9 for wavebreaking, and WG9 and WG14 for bore propagation: roughness element spacing: surface elevation from the still water levelmax: maximum surface elevation from the still water level

Fig. 14.Maximum surface elevation along the reefat for hf= 0.1 m. Blue crosses, magenta squares, green diamonds, green triangles, and magenta circles denote data from the

control, BF1, BF2, BF3, and BF4 tests; the control data is connected by blue lines.

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Fig. 16.Average celerity on the reefat between WG9 and WG14 as a function of the

average wave height. Blue crosses, magenta squares, green diamonds, green triangles, and

magenta circles denote data from the control, BF1, BF2, BF3, and BF4 tests; blue and green

dash trendlinespassthrough thecontrol andBF2 data.Note that thevertical andhorizontal

axes of the three panels have different ranges for presentation of the data.

Fig. 15.Waveheight gradienton thereefat between WG9 andWG14as a functionof the

average wave height. Blue crosses, magenta squares, green diamonds, green triangles, and

magenta circles denote data from the control, BF1, BF2, BF3, and BF4 tests; blue and green

dash trend lines pass through the control and BF2 data. Note that the horizontal axes of the

three panels have different ranges for presentation of the data.

http://localhost/var/www/apps/conversion/tmp/scratch_7/image%20of%20Fig.%E0%B1%B5http://localhost/var/www/apps/conversion/tmp/scratch_7/image%20of%20Fig.%E0%B1%B6

laboratory study of solitary-wave transformation over bed-form roughness on fringing reefs

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