THE INFLUENCE OF DIFFERENT WATER LEVELS AND WATER … · Pearce 1999). The reef also supports a...

THE INFLUENCE OF DIFFERENT WATER

LEVELS AND WATER HEIGHT IN WAVE

DRIVEN CIRCULATION OVER AN

IDEALIZED NINGALOO REEF

Thesis by

HELEN REYNOLDS

Centre for Water Reserch

The University of Western Australia

2001

ii

Abstract

Ningaloo Reef is located off the western coast of Australia, stretching from NorthWest

Cape to Gnaraloo Bay (Environment Australia 2000). Coral reefs are complex

ecological systems closely connected to their physical environment. Therefore, effective

management of a reef system requires an understanding of the physical oceanographic

processes controlling the movement of water over and around the reef. This is

increasingly important in the Ningaloo region as the tourism industry grows and

population pressures, such as waste disposal and boating, increase.

Studies undertaken to date have provided a basic description of the general

oceanographic characteristics of the Ningaloo Reef system (D’Adamo & Simpson 2001)

They have also included an analytical assessment of possible forcings on water

circulation within the backreef lagoons. These studies have concluded circulation within

coral reef lagoons is largely driven by wave pumping of water across the reef (Hearn et

al. 1986, Hearn 1999). This project conducted a preliminary investigation of wave driven

circulation over an idealized version of Ningaloo Reef. The work considered the effects

of different water levels and wave conditions using the 2D vertically integrated numerical

model FUNWAVE.

The results of the numerical modelling were used to describe wave setup and the

magnitude and direction of flow over the reef in an idealized Ningaloo lagoon.

Preliminary estimates were made of flushing times for the idealized lagoon under wind,

tidal and wave forcing.

Numerical modelling produced results that agreed well with observed and theoretical

values of current speed and wave setup published in the literature. The relative

importance of wave-driven flushing was reconfirmed. However, the project was highly

idealized and field data for model verification was not available. This limited the

conclusions that could be drawn using the magnitude of modelled setup and currents.

iii

Table of Contents

Acknowledgements _______________________________ Error! Bookmark not defined.

Abstract _______________________________________________________________ ii

1 Introduction _______________________________________________________ 1

2 Literature Review ___________________________________________________ 4

2.1 Regional Oceanography and Climate of Ningaloo Reef_________________________42.1.1 Climate and Meteorology ___________________________________________________ 42.1.2 Large Scale Currents _______________________________________________________ 52.1.3 Tides ___________________________________________________________________ 72.1.4 Waves __________________________________________________________________ 82.1.5 Temperature and Salinity __________________________________________________ 10

2.2 Reef Geomorphology ___________________________________________________112.2.1 General Coral Reef Morphology _____________________________________________ 112.2.2 Geomorphology of Ningaloo Reef ___________________________________________ 13

2.3 Circulation within Coral Reef Lagoons_____________________________________172.3.1 Wind Driven Circulation ___________________________________________________ 182.3.2 Tidally Driven Circulation__________________________________________________ 192.3.3 Wave Driven Circulation___________________________________________________ 20

2.4 Numerical Modelling of Wave Driven Flow in Coral Reefs_____________________30

3 Methodology ______________________________________________________ 33

3.1 Numerical Model ______________________________________________________333.1.1 Wave Generation and Breaking______________________________________________ 333.1.2 Bottom Friction __________________________________________________________ 34

3.2 Model Inputs __________________________________________________________343.2.1 Bathymetry _____________________________________________________________ 353.2.2 Water Levels ____________________________________________________________ 373.2.3 Wave Forcing ___________________________________________________________ 38

3.3 Simulations ___________________________________________________________39

3.4 Model Outputs ________________________________________________________393.4.1 Surface Elevation ________________________________________________________ 403.4.2 Velocity Vectors _________________________________________________________ 403.4.3 Wave Gauges ___________________________________________________________ 41

4 Results and Analysis________________________________________________ 43

4.1 Modelled Results_______________________________________________________43

4.2 Comparisons with Experimental Results ___________________________________43

4.3 Surface Elevation ______________________________________________________454.3.1 Wave Setup _____________________________________________________________ 454.3.2 Comparison with Experimental Results________________________________________ 494.3.3 Wave Measurements at the Gauges ___________________________________________ 50

iv

4.4 Velocity ______________________________________________________________534.4.1 Velocity Direction________________________________________________________ 544.4.2 Velocity Magnitude_______________________________________________________ 584.4.3 Cross-Reef Velocity ______________________________________________________ 604.4.4 Velocity at the Wave Gauges _______________________________________________ 63

4.5 Discharge_____________________________________________________________654.5.1 Cross Reef Discharge _____________________________________________________ 654.5.2 Total Discharge __________________________________________________________ 67

4.6 Flushing Times ________________________________________________________694.6.1 Wind Driven Flushing _____________________________________________________ 704.6.2 Tidal Flushing ___________________________________________________________ 714.6.3 Wave Driven Flushing_____________________________________________________ 71

5 Discussion________________________________________________________ 73

5.1 Wave Setup ___________________________________________________________73

5.2 Wave Induced Currents and Discharge ____________________________________75

5.3 Flushing Times ________________________________________________________78

5.4 Influence of Other Factors _______________________________________________795.4.1 Wave Period ____________________________________________________________ 795.4.2 Bottom Friction __________________________________________________________ 805.4.3 Irregular Waves__________________________________________________________ 80

6 Conclusions ______________________________________________________ 81

7 Recommendations _________________________________________________ 83

7.1 FUNWAVE and Wave-driven circulation __________________________________83

7.2 Other Forcings ________________________________________________________84

8 Bibliography ______________________________________________________ 85

9 Appendix A _______________________________________________________ 89

10 Appendix B _______________________________________________________ 91

11 Appendix C _______________________________________________________ 96

12 Appendix D ______________________________________________________ 100

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List of Figures

Figure 1.1 Location of Ningaloo Reef on Western Australian Coastline showing location of the continentalshelf (200m isobath) (Taylor & Pearce 1999) ________________________________________________ 1

Figure 2.1 Large-Scale Current Regime at Ningaloo (redrawn from Taylor & Pearce 1999) ___________ 6

Figure 2.2 : Timeseries of Wave Height and Current Speed at Ningaloo Reef. For location of wave riderand current meter, refer to Figure 2.4. (Hearn 1999) _________________________________________ 9

Figure 2.3 Main Geomorphological Features of a Coral Reef __________________________________ 12

Figure 2.4: Schematic map of a section of Ningaloo Reef (Hearn 1999)___________________________ 14

Figure 2.5 Division of Ningaloo Reef into sectors based on topographical features (Hearn et al. 1986) __ 16

Figure 2.6 Transect taken in the northern sector of Ningaloo Reef, (Hearn et al. 1986)_______________ 17

Figure 2.7 Definition Diagram for Wave Setup (Massel & Gourlay 2000) _________________________ 21

Figure 2.8 Setup on a Berm or a Reef _____________________________________________________ 22

Figure 2.9 Idealized reef defining theoretical model parameters (redrawn from Symonds et al. 1995)____ 23

Figure 2.10 Correlation between Hs and Current Speed, (Hearn 1999) ___________________________ 28

Figure 2.11 Current speeds indicated by drogue tracking under prevailing southerly-south easterly windsat Turquoise Bay, (Sanderson 1996) ______________________________________________________ 29

Figure 3.1 Transect of Turquoise Bay (Sanderson 1996) ______________________________________ 35

Figure 3.2 Idealized Bathymetry of Ningaloo Reef Used in FUNWAVE ___________________________ 36

Figure 3.3 Location of Wave Gauges relative to bottom contours________________________________ 42

Figure 4.1 Modelled Wave Setup ________________________________________________________ 46

Figure 4.2 Change in Maximum Setup with water depth and wave height _________________________ 48

Figure 4.3 Non-dimensional Comparison of Setup Results _____________________________________ 48

Figure 4.4 Comparison of Experimental and Modelled Results (after Gourlay 1996a)________________ 50

Figure 4.5 Surface Elevation over Time at Gauges, H=1.34____________________________________ 51

Figure 4.6 Change in Surface Elevation with Time at Gauges, H=1.55m __________________________ 52

Figure 4.7 Velocity Field at Lowest Water Level, Wave Height =1.55m___________________________ 55

Figure 4.8 Velocity Field at Mean Water Level, Wave Height=1.55m ____________________________ 56

Figure 4.9 Velocity Field at the Highest Water Level, Wave Height=1.55m ________________________ 57

vi

Figure 4.10 Velocity Contours at Mean Water Level _________________________________________ 58

Figure 4.11 Velocity Contours at the Highest Water Level _____________________________________ 59

Figure 4.12 Velocity Profile Along the Reef ________________________________________________ 61

Figure 4.13 Velocity Measurements at Gauges, Mean Water Level ______________________________ 64

Figure 4.14 Discharge per meter through gaps and over reef at a wave height of 1.34m ______________ 66

Figure 4.15 Per Meter Discharge Across Reef and Through Gaps at a wave height of 1.55m __________ 67

Figure 4.16 Total Discharge Across the Reef Top____________________________________________ 68

List of Tables

Table 2.1 Tidal Height at Selected Locations along Ningaloo Reef (Tide Tables 1996) ________________ 8

Table 2.2 Tidal Constituent Amplitudes and Form Factor, (after Hearn 1999)_______________________ 8

Table 3.1 Summary of Water Depths______________________________________________________ 38

Table 3.2 Significant Wave Height and Wave Period _________________________________________ 38

Table 4.1 Wave Heights at Wave Gauges __________________________________________________ 53

Table 4.2 Volume Calculations at Low Water Level __________________________________________ 69

Table 4.3 Volume Calculations at Mean Water Level _________________________________________ 70

Table 4.4 Volume Calculations at High Water Level__________________________________________ 70

Table 4.5 Discharge Over the Reef at each Water Level _______________________________________ 72

Introduction

1

1 Introduction

Ningaloo reef lies along the west coast of Australia, stretching from NorthWest Cape

down to Gnaraloo Bay. It is the largest fringing reef system in Australia and the only reef

system of its kind located off the western coast of a continent. The Ningaloo Marine

Park, which includes most of the main reef line, is under the jurisdiction of both state and

federal agencies (Environment Australia, 2000).

Figure 1.1 Location of Ningaloo Reef on Western Australian Coastline showing locationof the continental shelf (200m isobath) (Taylor & Pearce 1999)

Introduction

2

Ningaloo reef is significant for both its size and ecological composition. It is becoming

an internationally famous tourist destination, based on easy access to the reef from shore

and unique wildlife. For example, whale sharks, the largest fish in the world, are in the

region of the marine park between April and June each year. It is believed they are

attracted to the reef after the mass spawning of corals in March and April (Taylor &

Pearce 1999). The reef also supports a diverse community of more than 500 fish species,

over 200 species of coral, and 600 species of mollusc. It is home to dugongs, marine

turtles, whales and dolphins (Environment Australia 2000).

The geomorphology of Ningaloo reef is significantly different to many other reef

systems. The sedimentary lagoon backing the reef is shallow, with a mean depth of only

two meters. It is also located very close to shore, the maximum offshore distance to the

reef line is 7km. This proximity to shore is in direct contrast to other major reef systems

such as the Great Barrier Reef off Queensland, which is separated from the mainland by

an up to 100km wide expanse of lagoon. The only large reef system similar to Ningaloo

in its’ proximity to shore is found off the west-coast of Madagascar (Environment

Australia 2000).

Coral reefs are productive, biochemically complex systems that exist in an oligotrophic

environment. The ecology of any reef system is thus closely connected to the circulation

of water, which transports nutrients and disperses animal larvae. Effective management

of a reef system requires an understanding of the physical oceanographic processes

controlling the movement of water over and around the reef. In addition, the prediction of

water movement is vital for risk analysis of contaminant dispersal. This may become

increasingly important in the Ningaloo region as the tourism industry grows and

population pressures, such as waste-disposal and boating, increase.

Limited studies of the oceanographic conditions within the Ningaloo region have been

completed to date (D’Adamo & Simpson 2001). AIMS conducted intensive

oceanographic investigations in the lagoons and adjacent ocean area near Vlamingh Head

during 1997, including inner and outer lagoon wave measurements. These data are

Introduction

3

currently being analyzed and were not available to this project (D’Adamo & Simpson

2001). The studies that have been undertaken so far have provided a basic description of

the general oceanographic and geomorphic characteristics of the Ningaloo Reef system.

They have also provided an analytical assessment of the possible forcings on water

circulation within backreef lagoons. These studies have concluded that circulation within

the lagoons is largely driven by wave pumping of water across the reef. However, these

analytical assessments were not predictive and there is considerable uncertainty about the

effects of natural variability in water level and swell conditions on wave driven flows.

The aim of this study was to conduct a preliminary investigation of wave driven

circulation over an idealized version of a Ningaloo Reef lagoon. In particular, it

examined differences in wave setup and wave driven velocities at a range of water levels

and wave conditions using a numerical model.

Literature Review and Background Information

4

2 Literature Review

Although this study is an investigation of modelled circulation in an idealized

environment, the bathymetry and forcings aim to reflect the Ningaloo Reef environment

as closely as possible. In this context, the regional oceanography and climate of

Ningaloo are discussed to provide a description of potential forcings on circulation. The

features that make Ningaloo a unique coral reef environment are described in terms of the

general geomorphology of coral reefs. A summary of the current state of knowledge on

wave driven circulation and in particular, circulation around coral reefs is presented.

Finally, numerical modeling of wave driven circulation is discussed.

2.1 Regional Oceanography and Climate of Ningaloo Reef

The regional oceanography and climate of Ningaloo Reef can be described in terms of the

climate of the region, large-scale currents, water temperature and salinity, the tidal

regime and wave conditions.

2.1.1 Climate and Meteorology

The mid-west of Western Australia, where Ningaloo Reef is located, is a very arid and

windy region. Evaporation exceeds precipitation by more than 2 meters per year

(D’Adamo & Simpson 2001). Most of the annual rainfall occurs during summer storms

and cyclones associated with the southerly movement of the belt of anti-cyclonic high-

pressure systems (D’Adamo & Simpson 2001). During summer, when the belt moves to

its most southerly extent, monsoonal wind systems dominate the weather (D’Adamo &

Simpson 2001). Summer is also cyclone season; on average, two cyclones cross the

Pilbara coast each year, accompanied by intense winds of up to 300kmh-1 and heavy

rainfall (D’Adamo & Simpson 2001).

The sporadic nature of rainfall in the Ningaloo region means there is no regular flow of

terrestrial run-off into the marine park (Hearn et al. 1986). Storm water flows into the

ocean through seasonal creeks, which are generally associated with breaks in the reef

(Hearn et al. 1986). An important implication of the lack of regular freshwater flow is


5

the absence of density stratification due to river outflow, which might inhibit a vertically

well-mixed water column. In addition, influxes of low salinity water limit coral growth

as coral has a very narrow range of salinity tolerance (Mann 2000). The association of

breaks in the reef with seasonal creeks may be attributable to occasional but severe coral

stress caused by high influxes of low salinity water.

The wind regime around Ningaloo shows seasonal variations, which like the rainfall

patterns, change with the movement of the anti-cyclonic high pressure belt. It also shows

a year-round pattern of diurnal variability, with strong afternoon sea breezes replacing

weaker morning offshore trade winds. In summer, wind records at Cape Cuvier (45 km

south of Gnaraloo) indicate a mean wind speed of 7-9 ms-1, but this falls to about 3 ms-1

in winter due to the more variable wind directions (Taylor & Pearce 1999). Peak wind

speeds exceed 14ms-1 throughout the year (Taylor & Pearce 1999). The wind patterns

described by Taylor and Pearce (1999) are similar to those described by Hearn et al.

(1986) at Carnarvon and Learmonth. This implies that, allowing for topographic effects

such as sheltering, wind regimes are probably similar throughout the Ningaloo region.

2.1.2 Large Scale Currents

Large ocean current systems can influence the advection of water through a region. It has

been suggested (Taylor & Pearce 1999) that there are two large-scale current systems, the

Leeuwin and the Ningaloo, operating in the Ningaloo region.

The Leeuwin Current is a southward flowing current of low salinity, warm tropical water.

It contributes to maintaining the temperature of the water off the Western Australian

coast at temperatures suitable for coral growth (Taylor & Pearce 1999). The Leeuwin is

driven by an along-shore pressure gradient and flows most strongly in autumn, winter and

early spring when pressure head outweighs the prevailing winds (Taylor & Pearce 1999).

Satellite imagery indicates the Leeuwin current is narrow and close to the shelf-break in

the vicinity of Ningaloo reef. Hearn et al. (1986) reasoned that bottom friction in the

shallow reef system probably prevented direct flow of tropical water into the lagoons by


6

out-balancing the influence of regional pressure head. However, tropical water advected

onto the shelf by the current may enter the lagoons via ocean/lagoon exchange processes

(Hearn et al. 1986).

Figure 2.1 Large-Scale Current Regime at Ningaloo (redrawn from Taylor & Pearce1999)

During summer, the prevailing southerly winds are much stronger and the increased wind

stress forces the Leeuwin current further offshore. When the Leeuwin current is further

offshore, a northward counter-current has been observed (Taylor & Pearce 1999). Taylor

and Pearce (1999) proposed the counter current be called the Ningaloo Current. It is a

relatively cool counter current driven by strong south-south westerly breezes during

summer. It flows equator-ward, with a major perturbation to its flow between Point

Cloates and Coral Bay (Taylor & Pearce 1999). This eddy appears to re-direct some of

the flow back southwards. This has important ecological significance, as the re-

circulation of water means coral spawning in March and April may remain within the

NingalooCounterCurrent

LeeuwinCurrent


7

Ningaloo area itself and not be advected out of the region as previously thought (Taylor

& Pearce, 1999)

2.1.3 Tides

Water level variability in the coastal environment is due to the combined effects of

astronomical tides, atmospheric pressure variations and wind direction. Astronomical

tides cause cyclic changes in water level and meteorological changes are superimposed

on to the tidal record. The magnitude of meteorological effects can be approximated by

the inverse pressure effect, where for every 1hPa drop in atmospheric pressure there is a

sea level rise of about 1cm (Pond & Pickard 1983). For example, extreme water levels

may be induced by the passage of low-pressure cyclones. On-shore wind forcing during

Cyclone Vance elevated water levels at Exmouth by nearly 3m (D’Adamo & Simpson

2001).

Ningaloo reef is located just north of the tidal transition area between the southwestern

and northwestern Australian zones (D’Adamo & Simpson 2001). Southwestern tides are

diurnal micro-tidal while northwestern tides are semi-diurnal macro-tidal. The form

factor, which describes whether tides are diurnal or semidiurnal, varies considerably

along the coast between Carnarvon and Point Murat (Tide Tables 1996). It is important

to point out that although Hearn et al. (1986) stated that tides in the Ningaloo area are

mixed, predominately semi-diurnal with a form factor near 0.8 (Hearn et al. 1986),

according to Table 2.2, at Point Murat the form factor is 0.4. This implies the tide is

more semi-diurnal at the northern end of the reef than at the southern end.

The mean tidal amplitude within Ningaloo is about 0.55m, which is very close to the

mean sea level over the reef flat (Hearn 1999). The tidal range at springs is between one

and two meters, increasing towards the northern end of Ningaloo (Table 2.1, Tide Tables

1996, Hearn 1999). Much of the reef is exposed for several hours during the lower of the

two low waters for the four days either side of the spring tides. Cumulatively, this means

the reef is exposed for about 10% of each year (Hearn 1999).


8

Table 2.1 Tidal Height at Selected Locations along Ningaloo Reef (Tide Tables 1996)

Tidal Levels, meters ref. To LAT1

Port Name LAT MHHW MSL MLLW Range

Carnarvon 1.03 1.5 1.0 0.6 0.9

Coral Bay 0.84 1.4 0.8 0.2 1.2

Exmouth 1.43 2.3 1.4 0.5 1.8

Point Murat 1.22 2.0 1.2 0.5 1.5

Learmonth 1.55 2.6 1.5 0.5 2.1

Table 2.2 Tidal Constituent Amplitudes and Form Factor, (after Hearn 1999)

Amplitude (h) meters Form Factor Tide Type

Location M2 S2 K1 O1

Carnarvon 0.3 0.14 0.2 0.13 0.8 Mixed, mainly semi-diurnal

Coral Bay 0.29 0.14 0.19 0.13 0.7 Mixed, mainly semi-diurnal

Point Murat 0.49 0.27 0.18 0.13 0.4 Mixed, mainly semi-diurnal

Learmonth 0.66 0.36 0.19 0.14 0.3 Mainly semi-diurnal

2.1.4 Waves

The wave climate around Ningaloo and the Northwest Cape has been described by wave-

rider data and shipping information (Hearn 1999, WNI Science and Engineering 2000,

Sanderson 1996). As expected, the significant wave height appears to show a strong

dependence on weather conditions. For example, cyclonic conditions can generate very

large sea and swell waves. Extreme cyclone wave conditions typically have significant

wave heights of around 10meters, wave periods of 8 to 13 seconds and arrive at the

Northwest Cape from the north-northeast (D’Adamo & Simpson 2001).

Extreme conditions are significantly different to the mean wave climate. According to

wave-rider data collected off the Northwest Cape (21° 36’ 27’’S, 114° 2’ 5’’E), the swell

1 Explanation of abbreviation in Table 2.1, LAT- Lowest Astronomical Tide, MHHW- Mean Higher High Water, MSL

– Mean Sea Level, MLLW – Mean Lower Low Water


9

direction is predominately from the southwest in both summer and winter (WNI Science

and Engineering 2000). Long period swell (T = 12-22s) with a mean significant wave

heights (Hs) of 1.5 meters is generated in the Southern Ocean. There is a slight seasonal

variation in wave height; the mean Hs in summer is 1.34m, while in winter it increases to

1.55m (WNI Science and Engineering 2000). The southwesterly swell refracts as it

passes over the shelf, causing an increase in the western component of swell direction

(D’Adamo & Simpson, 2001).

The WNI wave-rider data appears to agree with wave rider data (Figure 2.2) collected off

Ningaloo between the 12th of August and the 12th of September 1987 (Hearn 1999). A

wave rider buoy was deployed in the ocean outside the main reef in 47 meters of water.

This time series recorded significant wave heights that were generally between 1 and 2

meters, with a peak value of 3 meters (Hearn 1999).

Figure 2.2 : Timeseries of Wave Height and Current Speed at Ningaloo Reef. Forlocation of wave rider and current meter, refer to Figure 2.4. (Hearn 1999)

Local winds cause sea waves with periods of between 2 and 8 seconds and heights of 1 to

2 meters to be superimposed on the swell (D’Adamo & Simpson 2001). Even under non-

cyclonic conditions sea-wave heights can reach 3.0 to 3.5 meters. Within embayments

and lagoons on the reef it appears swell is blocked by the reef and sea-waves predominate

(Hearn et al. 1986, D’Adamo & Simpson 2001). According to D’Adamo and Simpson


10

(2001), sea-waves would be limited to about heights of about 1 meter under typical sea

breeze conditions.

2.1.5 Temperature and Salinity

The combination of strong winds, shallow lagoons and lack of freshwater inflow suggests

the water column would be vertically well mixed. A well-mixed water column has been

observed both in deeper lagoons and in regions featuring weaker winds (Prager 1991,

Kraines et al. 1998). The well-mixed assumption has been confirmed at Ningaloo by

observations that appear to show there is no usual vertical temperature or salinity

stratification (Hearn et al. 1986).

Although there is no normal vertical density stratification, unpublished observations have

been made of horizontal density stratification in the Bills Bay area (D’Adamo & Simpson

2001). The temperature difference between the lagoon and oceanic water masses

suggests large-scale intrusion of oceanic waters into lagoons in Ningaloo. This is

supported by the fact flushing of a water-body is never instantaneous. Incomplete mixing

across the interface between the two water masses would then tend to maintain a

horizontal density gradient. Neap tides were suggested as the optimal time for oceanic

incursions as the reef crest is covered by water over the entire tidal cycle (Hearn et al.

1986).

Localized areas of poor flushing are suggested by patches of high salinity/low

temperature water in Bills Bay recorded by D’Adamo in 1999 (D’Adamo & Simpson

2001). Evaporative salinity change is a slower process than heat transfer in terms of

equivalent density change (D’Adamo & Simpson 2001). This implies these patches of

water have been undisturbed for a relatively long time. Given these high salinity/low

temperature patches were found within embayments, their existence suggests mean

lagoonal circulation fields bypass these areas under particular environmental conditions

(D’Adamo & Simpson 2001).


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2.2 Reef Geomorphology

Coral reefs are a unique marine environment, flourishing in apparently nutrient poor

waters (Mann 2000). The physical morphology of a coral reef has a significant influence

on wave breaking and attenuation (Gourlay 1996a, Gourlay 1996b, Lugo-Fernandez 1998

& others). Thus, it will have a significant influence on wave driven flow and circulation.

It is therefore worth describing key reef geomorphological features and their effect on

wave breaking and energy attenuation.

2.2.1 General Coral Reef Morphology

Coral reefs are typically described as platform, barrier or fringing reefs. The morphology

of each type of reef is shaped by the interaction of ecological and physical factors such as

biological growth and the pre-existing substrates (Gourlay 1996a). Platform reefs, also

known as atolls, are flat-topped and island-like. The reef tends to form a ring around a

central lagoon, which may be very deep (Mann 2000). Barrier reefs, such as the Great

Barrier Reef, are associated with a landmass. However, the reef line occurs some

distance out to sea. For example, the main line of the Great Barrier Reef is generally

located about 100km offshore (Gourlay, 1996a). Fringing reefs, such as Ningaloo, are

also associated with a landmass, but they are much narrower and closer to shore than

barrier reefs (Mann, 2000).

The different zones of a reef are named in Figure 2.3. The reef slope rises rapidly from

depths of about 16 to 18 meters. It may have a slope as steep as 1:1 or even be nearly

vertical (Massel & Gourlay 2000). The reef crest may also be known as the reef front.

The reef front is the zone of the most active growth of corals and coralline algae (Mann

2000). It is exposed to the maximum wave energy, which encourages coral growth

through a constant renewal of water (Andrews & Pickard 1990, Mann 2000). The reef

flat is located just behind the reef crest, which may be exposed at low tide. If the crest is

exposed, the reef flat may be kept moist by water and spray from the waves breaking on

the reef front (Mann 2000)


12

Figure 2.3 Main Geomorphological Features of a Coral Reef

The seaward reef slope effectively acts as a breakwater and dissipates the energy of

incident waves. It is estimated that 70 to 95% of the wave energy impinging on a reef is

dissipated through frictional processes and wave breaking (Prager 1991, Lugo-Fernandez

1998). Most coral reefs have groove and spur structures on the reef slope and top that

dissipate a major fraction of the wave energy through frictional processes (Hearn et al.

1986, Gourlay 1996b). Munk and Sargent (1954) recorded the occurrence of a groove

and spur system in their work at Bikini Atoll. These grooves can be described as a

natural energy dissipating device tuned to the average wave characteristics (Gourlay

1996b) beginning at the depth where wave action becomes significant (Hearn et al.

1986).

Deeper channels are common in coastal lagoons behind reefs; these channels are also

known as moats, gutters or drainage channels (Hearn et al. 1986). The currents within

these channels are generally substantial enough to be visible to the eye (0.1 to 0.5 ms-1).


13

Volume flux within fringing reef lagoons is usually concentrated in the moat. Outflow

channels through the reef line are often fed by the flux through the moat (Andrews &

Pickard 1990).

2.2.2 Geomorphology of Ningaloo Reef

Ningaloo Reef is a unique environment; although it is not as long as the Great Barrier

Reef, it is much closer to shore. Few large reef systems are so easily accessible from the

shoreline. The total length of the reef is approximately 280km (Hearn et al. 1986). The

average distance from shore to the reef flat is 2.5 km, although this distance varies from

just hundreds of meters up to 7km (D’Adamo & Simpson 2001). In contrast, the Great

Barrier Reef is generally around 100km offshore (Hearn et al. 1986).

Groove and spur structures occur on the seaward slope of Ningaloo Reef. The reef line

has an outward normal, orientated from northwest to southwest. Grooves appear to be

absent from north-facing reef sections, expected as the swell direction is predominately

southerly (Hearn et al. 1986). Grooves occur approximately every 10 meters and are

normally between 10 and 20 meters long. The water depth at which wave action

becomes significant and grooves can be expected to occur is approximately 20 meters

under west-coast wind and swell conditions (Hearn et al. 1986).

The fringing reef is broken up into sections, where elongated sections of reef are

separated by relatively deep channels. Aerial photography has been used to calculate that

under low swell conditions, gaps comprise approximately 15% of the total reef length

(Hearn et al. 1986). The size of the gaps in the reef varies along the reef. For example, at

Turquoise Bay (Figure 2.11) the main gap in the reef is about 800 meters wide, however

there are smaller gaps of about 200 meters width about 3 kilometers south of the main

gap (Sanderson 1996). Figure 2.4 shows a schematic map of a section of Ningaloo Reef,

including features such as the deeper channel, wide reef flat and sharp drop from the reef

crest to the 20m isobath.


14

Figure 2.4: Schematic map of a section of Ningaloo Reef (Hearn 1999)

Although Ningaloo Reef does display common features all the way along its full 280km

length, it is not topographically uniform. It can be divided into smaller sections, within

which the reef shows fairly homogenous geomorphological features. Hearn et al. (1986)

divided the reef into three sectors, based on topographic features (Figure 2.5).

Northern sector:

This sector runs about 120 kilometers from NorthWest Cape to Point Cloates. The

lagoon is less than three kilometers wide and the reef runs parallel a straight coast. The

shelf break is also parallel to the shore and is located approximately ten kilometers

offshore. Lengths of straight barrier reef so close to shore are comparatively rare (Hearn

et al. 1986).

Central sector:

The 50-kilometer central sector runs from Point Cloates to Point Maud. In this section

the lagoon is about six kilometers wide and has the structure of a long embayment with a

major break in the reef at its southern end near Point Maud.


15

Southern sector

The most southerly part of Ningaloo reef consists of 90 kilometers of scattered reef

between Point Maud and Gnaraloo Bay. The reef at Amherst Point is very scattered and

a definite structure is only evident some 35 kilometers south of Point Maud, at Pelican

Point. In this sector the lagoon is about 1 kilometer wide (Hearn et al. 1986).

Transects across the reef to determine reef bathymetry have been taken at Turquoise Bay

and Sandy Bay (northern sector) have been used to describe the bathymetry as much of

the region is unmapped. They show the reef crest is usually at the mean sea level (MSL),

the reef flat is less than 2 meters below MSL and the depth of the lagoon as a whole is

about 2 meters (Sanderson 1996, Hearn et al. 1986). The gaps in the reef are deeper than

the lagoon as a whole. For example, the reef break at the Northern Embayment at

Turquoise Bay is about 4 meters below MSL, while the lagoon is between one and three

meters deep (Sanderson 1996).


16

Figure 2.5 Division of Ningaloo Reef into sectors based on topographical features(Hearn et al. 1986)


17

Figure 2.6 Transect taken in the northern sector of Ningaloo Reef, (Hearn et al. 1986)

2.3 Circulation within Coral Reef Lagoons

Circulation within coral reef lagoons could have a variety of different driving forces,

including wind, tides, buoyancy and waves (Andrews & Pickard 1990, Kraines et al.

1998, Prager 1991). The dominant forcing will vary with the reefs’ physical and

oceanographic environment. Circulation patterns determine the residence time of the

water within a backreef lagoon. This makes the residence time of water within the

lagoon a function of lagoon geometry, depth and bathymetric complexity, as well as

circulation near gaps in the reef, mixing and the currents flowing over the reef (Andrews

& Pickard 1990, Prager 1991).

Residence times are often determined using calculations of flushing. There are a variety

of ways to define flushing, but generally flushing time can be defined as the time taken to

replace a volume of water at a particular rate of replacement. So, flushing time can be

calculated as

Q

V=τ , Equation 2-1

where τ is the flushing time, V is the volume and Q is the rate of discharge either into or

out of the volume.


18

2.3.1 Wind Driven Circulation

Wind driven forcing may be an important component of overall circulation within a coral

reef. Wind imparts a surface stress on the ocean surface, adding momentum to the water

body and creating a current. The importance of wind forcing varies depends on the

typical wind speed, direction and water depth within a particular lagoon. For example,

wind speed and direction apparently have a significant effect on current strength in some

lagoons (Prager 1991, Yamamoto et al. 1998), but at other locations modeled results

including wind forcing are indistinguishable from those that do not include wind forcing

(Kraines et al. 1998). The variation in these results is probably due to differences in wind

speed and direction as well as topographical differences between locations.

Within Ningaloo, wind stress is predominately from the south with easterly and westerly

components at different times of the year. This creates a steady southerly wind pattern

with occasional more energetic storm gales from the north (Hearn et al. 1986). The

direction of wind-driven circulation is function of both the wind direction and

topographical effects. Wind stress within a long, shallow reef lagoon tends to set up a

flow in the direction of the wind (Hearn et al. 1986). Given the morphology of Ningaloo

reef, wind stress probably creates a gyre that moves water north and out of the lagoon

through breaks in the reef (Hearn et al. 1986).

To determine the importance of wind-driven flow in lagoons behind Ningaloo reef, Hearn

et al. (1986) made an approximate calculation of the magnitude of the current in the

lagoon due to wind stress. They assumed the water depth in the lagoon was shallow

enough to allow a force balance between wind stress and bottom friction (Hearn et al.

1986). This yielded the relationship

uC U

C uwindA W

D f

= ρρ

2

Equation 2-2


19

where ρA is the density of air, CW is the surface drag coefficient, U is the wind speed, ρ is

the density of the water and uf is a background water velocity (Hearn et al. 1986). Under

typical conditions for Ningaloo, Hearn et al. (1986) calculated a uwind of 0.15 m s-1, but

stated that this was probably an over-estimate as the calculation neglected set-up forces

within the lagoon. The conclusion drawn from these calculations was that wind driven

circulation was probably only significant close to shore (D’Adamo & Simpson 2001)

Wind-driven flushing was estimated using the calculated value of uwind. This gave an

order of magnitude estimate of water velocity of 0.1 ms-1 (Hearn et al. 1986).

Disregarding additional flow over or through the reef from the starting point of the

particle, it was estimated a particle would exit the reef 28 hours after release for a travel

distance of 10 kilometers (Hearn et al. 1986). This yielded a wind driven flushing time in

the order of about a day (Hearn et al. 1986).

2.3.2 Tidally Driven Circulation

The tidal cycle causes changes in lagoonal circulation due to fluctuations in water level

over the reef and within the lagoon. The difference in water level between the ocean and

the lagoon at different stages of a tidal cycle creates a pressure gradient (Prager 1991).

This drives water exchange through flow over the reef and through any gaps in the reef.

The back-reef lagoon can be considered analogous to a semi-enclosed water body

because it is assumed the lagoon is a bounded region. The simplest calculation of tidal

flushing for a semi-enclosed water body uses the tidal prism method (Hearn et al. 1986,

Kraines et al. 1998, Prager 1991). It uses only the mean lagoon volume (V), tidal period

(T) and volumetric difference between high and low water (∆V) to obtain a residence

time, τ.

TV

Vtide ∆

=τ Equation 2-3


20

For a typical Ningaloo lagoon this gives a flushing time of 1 or 2 days (Hearn et al.

1986). However, in shallow water systems such as coral reef lagoons, the incoming tidal

prism has a different salinity and temperature to the water within the lagoon. The

incoming oceanic water may not mix completely with the water remaining in the lagoon.

This could result in the formation of a vertical front that moves in on the flood tide and

out on the ebb (Kraines et al. 1998, Prager 1991). According to Hearn et al. (1986) this

would increase the tidal flushing time to about five days, depending on the strength of

mixing forced by waves, wind and the density difference between the two water masses.

2.3.3 Wave Driven Circulation

Wave-pumping by waves breaking on the reef flat is a third forcing that may have a

major impact on circulation around a coral reef. In many environments it has been

observed to be the dominant forcing, controlling transport of water in and around a coral

reef (Hearn et al. 1986, Kraines et al. 1986, Prager 1991, Pickard & Andrews 1990 &

others).

2.3.3.1 Theory

When a wind-wave shoals, its celerity decreases and height increases as the wave feels

the effect of the sea floor. Wave steepening can only occur until a critical point, after

which the wave breaks. Essentially, a wave breaks when the crest of the wave is

travelling faster than the base celerity of the wave. The critical point can be described

either in terms of a crest angle of 120° or a ratio of water depth to wave height.

b

i

h

H=γ Equation 2-4

The ratio, γ, has been found to vary across the surf zone (Hearn 1999). In most coastal

engineering applications, it is assumed γ equals 0.78 or 0.8 (Horikawa 1978). This value

is suitable for the initiation of breaking of monochromatic waves, but it has been shown


21

the breaking ratio decreases as waves move through the surf zone (Hearn 1999). A range

of values of γ, all less than 0.55, have been suggested by several authors. The most

relevant to this discussion is the result of Hardy et al. (1991) who found γ reduces to 0.4

over a coral reef.

Figure 2.7 Definition Diagram for Wave Setup (Massel & Gourlay 2000)

The excess momentum flux induced by wave breaking is called radiation stress. The

concept of radiation stress can be briefly explained in terms of a momentum argument.

Surface waves induce a momentum, M, in the direction of wave propagation. When a

wave train hits an obstacle the momentum direction is changed and wave reflection

occurs at the surface of the obstacle. A force on the obstacle equal to the rate of

momentum change is created (Horikawa, 1978). In shallow water, the cross-shore

component of radiation stress is

2

16

3gHS xx ρ= Equation 2-5

Wave Setup and Setdown

Waves exert a net time averaged force on the fluid mass in which they propagate (van

Rijn 1990). This creates a net momentum and net mass flux, which contributes to

variations in local mean water depth (van Rijn 1990). The radiation stress gradient


22

(excess momentum) is balanced by a hydrostatic pressure gradient due to a mean water

level variation (van Rijn 1990), which is expressed for waves shoaling normal to the

shore in the equation below.

0)( =ƒ++ƒ

ƒdx

hgx

Sxx ηηρ Equation 2-6

So, when waves move into shallow water towards the shore on a plane beach the decrease

in momentum is balanced by an increase in water height over the still water level (Figure

2.8).

Wave setup is preceded by wave setdown at the breakpoint (Figure 2.8). Wave setdown

may also be associated with the passage of non-breaking waves. Assuming no energy

dissipation (i.e. no breaking) and η<<h, this can be seen when Equation 2-6 is integrated

for waves normal to the coast. This yields a negative water elevation,

)2sinh(8

2

kh

kH−=η Equation 2-7

where k is the wave number and H is the local wave height (van Rijn 1990).

Figure 2.8 Setup on a Berm or a Reef

The above theory describes setup on a plane beach. One of the first analytical

descriptions of wave set up over a coral reef was made by Tait (1972) who applied

Bowens’ (1968) setup on a plane beach theory to observations made at Bikini Atoll.

Essentially, this showed the magnitude of the set up, ηr, was determined by the depth of

Ho Set down new SWL


23

water at the reef top (hr), the depth at breaking (hb) and the ratio of wave height to

breaker depth (γ).

η γγr

bb r

hh h= −

+

−( )2

2161

1 83

Equation 2-8

Wave set up was at a maximum when hr = 0 and minimized when hb=hr. That is, setup

was lowest when the water depth over the reef was equal to the critical depth and waves

did not break on the reef (Gourlay 1996a). Although this theory was developed using

observations of setup under relatively large swell conditions, observations of wave set up

in a micro-tidal environment also agree reasonably well with Taits’ (1972) equation.

Cross Reef Flow

The magnitude of wave induced currents depends on both the geometry of the reef and

magnitude of the forcing (Symonds et al. 1995). A linear, one-dimensional model, which

includes wave forcing over an idealized reef has been developed by Symonds et al.

(1995). This theoretical model includes both pressure driven flow and bottom friction.

Although this model does not account for the three-dimensional nature of a coral reef, it

does provide an explanation of the force balance driving flow over a reef.

The theoretical model was based on an idealized one-dimensional reef, shown in Figure

2.9

Figure 2.9 Idealized reef defining theoretical model parameters (redrawn from Symondset al. 1995)

β

surf zonex=0

x

hhb


24

Conservation of momentum is expressed as

h

fu

x

S

hxg xx −

ƒƒ−=

ƒƒ

ρη 1

Equation 2-9

where g is gravitational acceleration, h is the depth, f is a linear friction coefficient, η is

sea surface elevation and u is the cross reef current. Sxx is the cross-shore component of

the radiation stress. Equation 2-9 shows the change of momentum across the surf zone is

balanced by cross reef flow and a pressure gradient. The offshore pressure gradient is

increased by the high values of friction associated with flow over a rough, shallow reef

(Symonds et al 1995). Conservation of mass also applies to the flow, so

0)( =

ƒƒ

x

hu Equation 2-10

A change in water depth over the reef alters the across reef current through two physical

effects. First, if the wave set up is considered to be unaltered, an increase in water depth

increases the total force because of the resultant pressure gradient relative to bottom

friction (Hearn 1999). This tends to increase the current. Second, the increase in water

depth over the reef results in a decline in wave breaking, which reduces the wave setup

and tends to reduce the current (Hearn 1999).

Symonds et al. (1995) applied this one-dimensional model to observations of wave driven

currents at John Brewer Atoll on the Great Barrier Reef. They managed to get good

agreement with the observed cross reef currents, however the solutions were non-unique.

Different combinations of scaled friction factor and surf zone width could be used to

arrive at the same solution (Symonds et al. 1995).

A further limitation of this one-dimensional model was neglect of the along shore

component of radiation stress and the existence of long-shore currents. This was justified

in the context of their work by noting they found little correlation between offshore wave

height and along reef currents (Symonds et al. 1995). In the context of an atoll, where

flow is directed over the top of the reef and is not constrained by a shoreline, the

omission of variability in the long-shore direction is probably not a major concern.


25

However, neglect of this is a problem for the conversion of the problem two dimensions,

where variability is permitted in the y-direction. For example, in shallow water there is

also radiation stress in the y-direction, Syy.

2

16

1gHSyy ρ= Equation 2-11

The problem becomes even more complicated when the assumption that wave crests

shoal normal to the reef is discarded. This means radiation stress has both a cross-shore

and a long shore component, which is resolved into Sxy.

ααρ cossin8

1 2gHSxy = Equation 2-12

On a barred beach, this non-normal component of radiation stress drives long-shore

currents and potentially creates rip currents (van Rijn 1990).

For conservation of mass, water entering over the reef must also exit the lagoon. The

morphology of the reef will govern how the discharge exits the reef. In a fringing reef

discharge is often constrained through relatively narrow outflow gaps. The force driving

discharge is the water elevation within the lagoon caused by wave setup. Wave energy is

transformed from kinetic energy into potential energy within the lagoon as a higher water

level (Hearn et al. 1986). Then, water exits the lagoon through gaps in the reef as

potential energy is transformed back into kinetic energy (Hearn et al. 1986).

2.3.3.2 Experiments and Observations

Experimental work and theoretical modelling of waves over coral reefs has been limited

due to the complexity of reef hydrodynamics. Steep slopes, the variable roughness of the

reef bottom and a complicated bottom slope make it difficult to parameterize work

successfully (Massel & Gourlay, 2000). Most of the work that has been done has been

limited to studies of wave setup and cross reef flow on two-dimensional reefs (Gourlay


26

1996a). This neglects the three dimensional nature of a real coral reef which shows

bathymetric variability in both the long-shore and cross-shore directions.

Wave Setup

The earliest published observations of wave setup over a coral reef were at Bikini Atoll in

the early 1950’s. Munk and Sargent (1954) observed swell waves caused the water level

to be between 0.45 and 0.6 meters higher over the reef-top than in the surrounding ocean.

They also observed wave pumping caused an inflow of ocean water into the lagoon

(Gourlay 1996a). While measurements of setup over coral reefs in the Pacific have

ranged from 0.10m to 0.6m, setup has been measured at only 0.8cm to 1.5cm over a reef

in the Caribbean (Lugo-Fernandez 1998). The difference in magnitude can be explained

by the micro-tidal regime and low wave energy environment in the Caribbean (Lugo-

Fernandez 1998).

The magnitude of meso-scale processes, such as wave currents and wave set up depends

on the geometry of the reef and the magnitude of the wave forcing. Despite different reef

topographies, the results of most experimental studies, summarized in Gourlay (1996a),

tend to agree that wave set up increases both with increasing wave height and period and

decreasing water depth over the reef.

Cross Reef Flow

Observations of wave induced flow over reefs have been made at several locations over

different types of reefs. Currents over coral reefs have almost universally been described

as strong, with speeds of up to 0.8ms-1 over the reef and speeds of more than 1.5ms-1

through outflow channels (Andrews & Pickard 1990).

However, there is severe shortage of long-term observations of wave-driven currents that

can be correlated to variables such as wave height and water level. The longest data series

collected to date has been described by Symonds et al. (1995). It was a one-month long

set of observations taken at John Brewer Reef, an atoll 70 kilometers northwest of

Townsville. The data were used to observe variability in cross-reef currents due to tidal


27

variations in sea level and variations in wave height. Symonds et al. (1995) found cross-

reef currents at sub-tidal frequencies were highly correlated with offshore rms wave

height and that offshore-directed currents were associated with small waves. They

theorized a forcing other than wave pumping drove the offshore-directed currents.

John Brewer Reef is a coral atoll, which has a completely different morphology to

Ningaloo Reef. A study of wave driven currents in a fringing reef lagoon in Guam by

Marsh et al. (1981) is may be more comparable to Ningaloo. This study showed water

entered the lagoon via wave pumping over the reef. The water was then entrained into an

along-shore current in a drainage channel and flowed out of the lagoon through a large

break in the reef. Current speeds through the break in the reef reached speeds of up to

1ms-1 (Marsh et al. 1981).

At the small scale, the direction of flow over and around a reef is complicated by the

irregular surface created by coral growth. The roughness of a coral reef varies over the

reef profile, according to both the morphology and ecology of the reef (Gourlay 1996b).

Coral growth also affects the porosity of the reef. The coral framework may contain

significant cavities that permit flow within the reef matrix as well as over the top of the

reef (Gourlay 1996b). Very little work has addressed this aspect of flow within a coral

reef (Andrews & Pickard 1990, Gourlay 1996a).

2.3.3.3 Wave Driven Circulation in Ningaloo Reef

Hearn et al. (1986) made a series of observations at Ningaloo reef that led them to

conclude wave generated flow was an extremely important component of lagoonal

circulation. These observations included examination of aerial photos, current meters

and are supported by drogue tracking by Sanderson (1996).

Aerial photos showed lines in the seabed that run across the reef towards shore. These

grooves were probably created by erosion and scour caused by transport of biological

material originating on the reef crest (Hearn et al. 1986). They are likely to be specific to


28

wave-pumped currents because high current speeds are required to transport particles the

size of coral rubble (Hearn et al. 1986). These lines are roughly normal to the direction of

the reef and terminate in a deeper channel that runs along-shore. A northward along-shore

motion in the deep channels is suggested by the way the lines swing north as they

approach the shore. This probably reflects the predominately southerly swell and wind

waves (Hearn et al. 1986). Near breaks in the reef, the lines make a 180-degree turn and

exit back out the reef (Hearn et al. 1986). The seabed grooves provide a long-term

average picture of wave induced currents and cover almost the entire bottom of the

lagoon (Hearn et al. 1986).

A further observation suggesting a strong linkage between wave overtopping and

circulation within the lagoon was a correlation between offshore significant wave height

and current speeds within the along-shore channel (Hearn et al. 1986). However, this

data does not isolate how flow over the reef changes with wave height. Other forcings,

particularly wind, may also affect current speed in the near shore deep channel.

Figure 2.10 Correlation between Hs and Current Speed, (Hearn 1999)


29

Observations by Sanderson (1996) of surface currents at Turquoise Bay reinforced the

conclusions drawn by Hearn et al. (1986) from the aerial photographs. Drogue tracking of

surface currents under southeasterly to southerly wind conditions showed current vectors

directed shoreward behind the reef and then turning northward to run along-shore. The

most rapid movement of water occurs through the gap in the reef (Sanderson 1996). The

strength of the exit current has been observed to increase as the surf-state becomes

heavier (Hearn et al. 1986) but this has not been measured directly.

Figure 2.11 Current speeds indicated by drogue tracking under prevailing southerly-south easterly winds at Turquoise Bay, (Sanderson 1996)

<5cm/s5-10 cm/s15-20 cm/s20-25 cm/s>25 cm/s


30

2.4 Numerical Modelling of Wave Driven Flow in Coral Reefs

Numerical modelling can be a useful tool to represent physical processes that occur in the

natural environment, and for making predictions about those processes. However, like

any problem there are a variety of ways to approach a solution in modelling. For

example, there have been two different approaches to numerical modelling of wave

driven flow. One approach incorporates radiation stress as a forcing in the conservation

of momentum equations, yielding an overview of meso-scale circulation in the lagoon.

The other focuses on the propagation of waves in shallow water, giving better resolution

of the dynamics forcing circulation.

Modelling of wave induced flow in coral reef lagoons, using a vertically integrated two-

dimensional model has been carried out in three locations (Prager 1991, Kraines et al.

1998, Wolanski et al. 1993). These models have incorporated a numerical algorithm for

radiation stress based on the work of Longuett-Higgins and Stewart (1964) into the

conservation of momentum equation. This accounts for momentum transfer due to

breaking waves, localized in reef containing model grids (Kraines et al. 1998, Prager

1991, Wolanski et al. 1993). They also accounted for bottom friction in shallow water

using a quadratic friction law (Kraines et al. 1998, Prager 1991, Wolanski et al. 1993)

that relates friction to water depth over the reef (Hearn 1999)

Using this approach, changing water levels associated with the tidal cycle have been

found to affect wave driven flow across coral reefs. According to radiation stress theory,

when the water depth has increased past the breaking depth, waves will pass over the reef

without breaking. Thus, there will be no mass transport and no wave pumping (Prager,

1991, Kraines et al. 1998, Hearn, 1999). One study found the magnitude of across reef

flow depended more on the water depth over the reef than on whether the tide was ebbing

or flooding (Kraines et al 1998). However, the Prager (1991) study concluded that flow

over the reef was strongest in the early to mid-flood tide and weakest at the ebb. The

difference between these conclusions demonstrates the variability of the natural


31

environment. For example, the differences could be attributed to different tidal ranges,

different water depths over the reef or swell conditions in Japan and the Caribbean.

While previous 2D vertically integrated modelling approaches have incorporated the

effect of changing wave heights on wave driven circulation, they have also made a range

of simplifying assumptions about the nature of the wave field approaching the reef. The

effect of wave direction and non-monochromatic wave fields is a component of coral reef

circulation that has not been addressed in any of the published studies to date. The only

reference to a relationship between wave direction and current direction was made in the

Prager (1991) paper. According to Prager (1991), wave induced back-reef currents tend

to flow roughly parallel to the reef trend, independent of the direction of wave approach.

In all cases, the simplifying assumption has been made that all wave shoal normal to the

shore and radiation stress is considered only normal to the reef flat (Kraines et al 1998,

Wolanski et al. 1993, Prager 1991).

These models appear to describe the general circulation features in a coral reef lagoon.

However, they do not resolve smaller scale circulation features that occur on the reef. The

key features required to model wave shoaling over a coral reef include the ability to deal

with a relatively large model domain, resolution of wave breaking processes and the

ability to model non-linear interactions between waves and currents. Models based on

the Boussinesq equations can predict the propagation and shoaling of shallow water

nonlinear waves in the nearshore region (Naval Postgraduate School 2000). Models of

this type can be used to accurately predict the wave height decay and shape changes of

waves propagating across the surf zone (Chen et al. 1999).

Only one study so far has attempted to use a Boussinesq model to describe wave set

down and setup on a coral reef (Skotner & Apelt 1999). This study compared

experimental measurements and the results of numerical modelling using a weakly non-

linear model. Skotner and Apelt (1999) concluded their model accurately computed the

set down and setup of regular waves of small incident wave height, but there was a

tendency to underestimate wave setup as the incident wave height increased. Their


32

model was not fully non-linear; but they predicted that using a fully non-linear

Boussinesq model would improve the agreement between modelled and experimental

results (Skotner & Apelt 1999).

FUNWAVE2D is a fully non-linear Boussinesq wave model, available in the public

domain. It was developed by Kirby et al. (1998) at the Center for Applied Coastal

Research at the University of Delaware. It has been used in nearshore circulation studies,

such as wave shoaling, rips, and wave run-up on planar beaches (Kirby et al. 1998, Chen

et al. 1999). It allows prediction of mean flows, including long-shore and rip-currents

and the interaction of waves and currents (Chen et al. 1999). This is particularly

important for a coral reef as strong currents exiting outflow gaps may block incoming

waves.

The use of FUNWAVE to model large nearshore regions has been made possible by

recent advances in computer technology (Chen et al. 1999), however model runtime can

still be very long. In addition, FUNWAVE was not developed for use on the extremely

steep slopes characteristic of coral reefs. However, Chen et al. (1999) stated adjustments

to shore permeability and localized filtering may be used to avoid numerical instability.

This means it is probably suitable for investigation of the fundamental characteristics of

wave setup and wave driven flow across a coral reef.

Methodology

33

3 Methodology

The methodology details FUNWAVE2D, the numerical model used to describe

circulation around the idealized reef. It also describes the inputs to the model and forms

of data generated by the model.

3.1 Numerical Model

FUNWAVE2D is a publicly available fully non-linear Boussinesq wave model,

developed by Kirby et al. (1998). The model simulates the nearshore propagation of

nonlinear surface gravity waves and predicts the underlying unsteady flow generated by

wave breaking (Kirby et al. 1998). FUNWAVE provides simulation of a range of

dynamic information including velocity vectors and surface elevation. It has been used in

nearshore circulation studies, such as wave shoaling and wave run-up on planar beaches

(Kirby et al. 1998). It has also been used to model rip currents off a barred beach (Chen

et al. 1999). The barred beach profile used to model rip currents is analogous to the

profile of the idealized reef used in this study.

3.1.1 Wave Generation and Breaking

FUNWAVE calls input files for the initial wave field and either a time-series of wave

amplitude or a source function for wave input. Waves are generated using an internal

source mechanism, where water mass is added or subtracted along a source line within

the computational domain (Kirby et al. 1998). The index line used in all runs was x=31.

FUNWAVE uses a spatially distributed source function f(x,y,t) where f(x,y,t)=g(x)s(y,t).

g(x) is a Gaussian shape function and s(y,t)=Dsin(λy-ωt) describes the wave form, where

D is the magnitude of the source function and λ is the component of the wave number in

the y-direction (i.e. λ=ksinθ) (Kirby et al. 1998). Calculation of the source function

requires information about the frequency, direction and power of the wave field (Kirby et

al. 1998). The wave field may be monochromatic or directional.

Methodology

34

Sponge layers are placed at the ends of the domain to damp the energy of outgoing waves

with different frequencies and directions (Kirby et al. 1998). The usual values (Kirby et

al. 1998) were used for the coefficients of the three different types of sponge layers.

The start and finish of wave breaking is determined by the parameter η t* (Kirby et al.

1998). For bar/trough beaches this parameter is defined as

<−≤−−

+

?

=g

httghgh

g

h

ttgh

g

htgh

t50)35.015.0(

5

35.0

515.0

00*η Equation 3-1

where h is the water depth, g is gravitational acceleration, t0 is the time when wave

breaking occurs, and t-t0 is the age of the breaking event (Kirby et al. 1998).

3.1.2 Bottom Friction

Bottom friction is modelled in FUNWAVE using the quadratic law (Chen et al. 1999),

ααηuu

h

fR f +

= Equation 3-2

The friction coefficient was chosen as f=4.0 x 10-3. The choice of f was taken at the upper

end of the range of typical values suggested by Kirby et al. (1998). This is still likely to

be an underestimate of the actual friction coefficient over a coral reef (Gourlay 1996a).

3.2 Model Inputs

FUNWAVE calls input files for the water depths within the model domain and the wave

field. The contents of the input files are described in the following sections. Details of

the data files used, including relevant parameter values, are provided in Appendix A.

Methodology

35

3.2.1 Bathymetry

Ningaloo Reef is not well mapped, and a bathymetric map suitable for digitization was

not available. Instead, an idealized bathymetry was generated based on the general

characteristics of the Ningaloo lagoons described by Sanderson (1996), Hearn et al.

(1986) and Hearn (1999). The bathymetry was generated by adapting MATLAB code

developed by Johnson (2001). The reef bathymetry included the usual geomorphic

features described in Section 2.2, such as a steep reef face, a slightly elevated reef rim

and a broad reef flat (Figure 3.1). It does not include a deeper channel within the lagoon.

This feature was omitted as flow was not constrained in the along-shore direction due to

the open boundary condition of the model. Other small-scale bathymetric irregularities,

such as the spur and groove system, could not be resolved within the spatial scales used

in the model domain.

Figure 3.1 Transect of Turquoise Bay (Sanderson 1996)

Methodology

36

Figure 3.2 Idealized Bathymetry of Ningaloo Reef Used in FUNWAVE

In the idealized bathymetry (Figure 3.2), the reef ran parallel to the shoreline with a

north-south orientation, approximately 1.8km offshore. The reef crest was 100 meters

wide and was backed by a reef flat 350 meters wide. The reef line was broken by two

gaps. A wide range of gap widths has been observed along the Ningaloo reef line

(Sanderson 1996, Hearn et al. 1986). To investigate the differences in circulation caused

by differences in gap width and depth, the gaps were asymmetric. Gap 1 was 600 meters

wide and 4 meters deep. Gap 2 was wider (800 m) and deeper (6m) than Gap 1. The

stretch of unbroken reef between the two gaps was 2000m. The backreef lagoon is

generally shallow at Ningaloo (Sanderson 1996, Hearn et al. 1986). Depths typically

range between 1 and 2 meters (Figure 3.1). The lagoon depth in the idealized bathymetry

was set at -2 meters datum level. The beach had a slope of 1:15 and a width of 200

meters. The water depths at the datum level over each part of the bathymetry are

summarized in the first column of Table 3.1.

Methodology

37

The model-grid was a Cartesian domain divided into 301 10-meter wide grids in the

cross-shore direction and 201 20-meter long grids in the along-shore direction. This gave

a total model domain of 4km by 3km. The grids were non-square as a compromise

between the size of the lagoon and resolution of wave processes. Greater resolution in

the cross-shore direction than the along shore direction was required to accurately capture

the processes occurring on the face of the waves as they break. Offshore width from the

reef crest to the boundary was originally 1000m, however a preliminary model run

suggested greater width was required to prevent outflow being pushed back into the reef

gaps. The offshore width was increased to 1500 meters; this appeared to allow sufficient

space for wave generation in the ocean.

3.2.2 Water Levels

The water levels used in the model were based on a combination of depth observations

(Hearn et al. 1986, Sanderson 1996) and tidal information from points along the Mid-

west coast (Tide Tables 1996).

Observations of water depth (Hearn et al. 1986, Sanderson 1996) were used to establish

datum water levels within the lagoon. The reported mean sea level (MSL) over Ningaloo

reef is 0.53m (Hearn, 1986). For convenience, this was rounded up to 0.55m. The level

of the lowest water, 15cm below the reef crest, was established by the observation that

the reef was exposed at low water for four days either side of spring tide (Hearn 1999).

The magnitude of the water level range was established using tidal records. Tide tables

indicate the range of MLLW to MHHW varies along the coast from approximately 0.9

meters at Carnarvon to 2 meters at Point Murat (Table 2.1). However, Hearn et al (1986)

found the predicted tides at Carnarvon correlated better in phase and amplitude to the

tides in their study area than the tides at Point Murat (Hearn et al. 1986). For this reason,

the final choice of a 1.15m range in MLLW to MHHW was based on the tidal range at

Coral Bay, at the south end of the reef rather than Exmouth, at the northern end of the

reef. Water levels used in the model runs are summarized in Table 3.1.

Methodology

38

Table 3.1 Summary of Water Depths

Location AHD MLLW Mid 1 MSL Mid 2 MHHW

Depth from 0 AHD 0 0.05 0.25 0.55 0.85 1.15

Off reef -16 16.05 16.25 16.55 16.85 17.15

Reef crest 0.2 -0.15 0.05 0.35 0.65 0.95

Reef flat 0 0.05 0.25 0.55 0.85 1.15

Lagoon -2 2.05 2.25 2.55 2.85 3.15

Gap 1 -4 4.05 4.25 4.55 4.85 5.15

Gap 2 -6 6.05 6.25 6.55 6.85 7.15

3.2.3 Wave Forcing

The model was run using a seasonally divided monochromatic wave regime. WNI

Science and Engineering (2001) provided percentage occurrence data from their wave

rider off the Northwest Cape, which was used to determine the wave regime at Ningaloo.

The data was recorded at 21° 36’ 27’’S, 114° 2’ 5’’E between June 1999 and July 2000.

The water depth at the recording location was 200m MSL (WNI Science and Engineering

2001).

The percentage occurrence matrices were divided into seasonal sea and swell. Summer

was defined as October to March, while winter was defined as April to September. Swell

waves are considered more significant in wave-pumping than sea waves, consequently

the wave field used in the model was based on swell conditions. The significant wave

height and peak period for seasonally divided swell are summarized in Table 3.2.

Table 3.2 Significant Wave Height and Wave Period

Summer Winter

Max Min Mean Max Min MeanWave Period (T) s 23.08 9 12.95 21.68 9 14

Significant Height (Hs) m 4.087 0.348 1.335 4.098 0.32 1.55

Methodology

39

According to the WNI data, swell arrives at the Northwest Cape from the southwest more

than 90% of the time in both summer and winter. This was incorporated into the model

by setting the wave direction at 45 degrees to the reef line.

3.3 Simulations

The number of model runs was severely limited by the very long run-time required by

each simulation. Producing an “hour” of model time required a run-time in the order of

days. Minimum coverage was provided by runs simulating circulation patterns at five

discrete tidal water levels with two different wave regimes for a total of 10 runs.

The size of the time step dt was chosen using

max

5.0gh

dxdt < Equation 3-3

where dx was the grid size in meters in the cross-shore direction and hmax was maximum

water depth (Kirby et al. 1998). This gave a value of 0.43s, which was rounded up to 0.5s

to reduce run time slightly for no obvious loss of stability.

One of the runs, at hr=0.05m, Ho=1.55m, crashed due to numerical instability. The

instability could have been fixed by decreasing the size the time-step or increasing the

porosity of the reef to reduce reflection (Johnson pers. comm. 2001). Increasing the

porosity of the reef, while not unreasonable in terms of reef morphology, may have

reduced the comparability of the runs. However, reducing the time-step was not an

option due to the extended run-time required by small time-steps. Consequently, the

hr=0.05m and Ho=1.55m run was omitted from the results.

3.4 Model Outputs

FUNWAVE outputs a range of dynamic information, including velocity and surface

elevation. The data is output for each grid point and averaged over a defined number of

Methodology

40

wave periods to condense the size of output files. The length of the time average depends

on the number of waves over which the average is taken and the period of those waves.

For all runs, a five-wave average was taken. During the summer runs, the wave period

was 13 seconds, so averaged results were output every 65 seconds. The wave period in

winter was one second longer, so results were output every 70 seconds. However, a time-

series of velocity and surface elevation at each time-step was recorded at five points

within the model grid using “wave gauges”.

It took some time for the simulation to ramp up and approach a fully developed,

relatively steady velocity pattern. The exact length of time varied between simulations

but was generally between 45 and 50 mean time-steps. Comparisons between the outputs

for different water levels were made on the 50th mean time-step. This ensured

comparability between steady-state model simulations.

3.4.1 Surface Elevation

The initial surface elevation was defined as the mean water level at every point it the

model domain. The time averaged output of η was used to determine the mean water

elevation within the model domain. This was then plotted to show wave setup and

setdown over the reef.

3.4.2 Velocity Vectors

The initial velocity of all points within the domain was zero. The depth integrated

velocity vectors for the entire model domain could be visualized as an animation cycling

through the time averaged results or as single frames taken at each mean time-step.

Velocity vectors could also be extracted and displayed as transects in the along-shore or

cross-shore direction. The velocity along the reef line was isolated and displayed to

highlight how velocity changes with depth along the reef line.

The u components of the velocity vectors were used to calculate the mass flux of water

across the reef. The values were used to interpolate a velocity profile along the front of

Methodology

41

the reef. The mass flux was calculated through grids 10 meters wide and with a height

equal to the water depth.

3.4.3 Wave Gauges

In FUNWAVE, “wave gauges” record u, v and η at a point without time averaging the

variables. This allows visualization of the variation in time within the mean time-step, in

this case five waves. Wave gauges were positioned at five points along the reef line.

There were five gauges positioned along the reef crest on the same along-shore line. Two

of the gauges were positioned 200 meters apart within the shallower gap. There were

another two gauges positioned within the deeper gap, also 200 meters apart. The last

gauge was positioned over the reef flat. Gauge position relative to the reef flat and gaps

is indicated in Figure 3.3.

The velocity data were used to construct plots showing the magnitude and direction of the

velocity at the each gauge for each combination of water level and wave height.

Comparisons between the simulations were made using the last six waves before the

model run finished.

Methodology

42

0 500 1000 1500 2000 2500 3000 3500 4000

0

1

2

3

4

5

6

7

Distance Along the Reef, meters

Depth, meters

Location of Gauges Along the Reef

1 23

4 5

Figure 3.3 Location of Wave Gauges relative to bottom contours

The water elevation η recorded at the gauges was used to determine the wave height at

each location. Wave height was determined using

H=max(η)-min(η) Equation 3-4

Results and Analysis

43

4 Results and Analysis

The results of the nine successful model simulations were data sets of surface elevation

and velocity. These results were used to calculate discharge at different water levels.

Preliminary estimates of wind, tidal and wave driven flushing times were calculated for

the idealized lagoon.

4.1 Modelled Results

The results of the numerical modeling were used to describe the dynamics of circulation

over the idealized Ningaloo reef. The results obtained can be summarized as follows.

Surface elevation data were used to describe wave breaking and wave set-up over the

reef. Surface elevation data taken at the wave gauges were used to identify changes in

waveform and height. Velocity data were used to describe the magnitude and direction

of flow within the lagoon. The still water level and velocity data were used to calculate

discharge at different water levels. Where appropriate, modelled set-up was compared

with experimental results from Gourlay (1996a). Finally, estimates were made of the

flushing time of the idealized lagoon under wind, tidal and wave forcing.

4.2 Comparisons with Experimental Results

One of the difficulties in working with a numerical model is determining if the results

give a good approximation of the environmental processes being represented. In the study

of coral reef circulation it is difficult to compare results to the real world directly, as there

is a scarcity of current meter or setup measurements taken over real reefs. This is

especially true for currents through outflow gaps. In addition, several authors (Gourlay

1996b, Pickard & Andrews 1990) have pointed out that reef profiles are very variable and

it cannot be expected that results in one location will give a good description of the

conditions at other locations. In particular, Gourlay’s (1996b) comparison of several sets

of experimental data showed clearly that reef profile will have a considerable effect on

wave setup.


44

Despite these problems, comparisons of wave setup values have been attempted with

Gourlay’s (1996a) experimental results. There are enough similarities between the

experimental and modelled scenarios to potentially allow a comparison. Gourlays’

(1996a) experimental set-up was a “relatively smooth, impermeable horizontal reef top,

with a steep, rough reef face subjected to steady, regular waves.” The modelled scenario

could also be described as a relatively smooth, impermeable horizontal reef top.

However, there was a step down of 20 centimeters from the reef crest to the reef flat in

the modelled bathymetry. This was probably less important than the similarity between

the model and experiment of the reef face slope. Gourlay (1996b) identified one of the

main controls on set up being the reef face slope, because it controls attenuation of wave

energy as the waves approach the reef. The modelled reef face had a slope of 1:1.2,

while the experimental reef had a slope of 1:1. The modelled reef face was not as rough

as the experimental reef face. Both reefs were subjected to steady, regular waves,

although the direction of wave approach differed. In the experiment waves approached

normal to the reef, while in the numerical model the waves had a 45° angle of approach.

Dimensional analysis was required to compare the modelled and experimental results.

The dimensionless parameters used follow from the argument of Gourlay (1996a). He

stated that for a given reef geometry, including roughness, the maximum setup, ηmax, and

the unit discharge q, are functions of wave height H, period T, water depth on the reef hr,

and gravitational acceleration g, i.e.

(ηmax, q) = f(H, T, hr, g) Equation 4-1

He showed a suitable set of parameters for describing non-dimensional wave setup were

√√↵

=

oo

r

o gHTH

hf

H

ηη,max Equation 4-2


45

4.3 Surface Elevation

Surface elevation data were used to determine wave setup and wave set down. Values of

maximum wave setup were compared with Gourlay’s (1996a) experimental values.

4.3.1 Wave Setup

Wave setup was calculated by taking the mean of the surface elevation across the reef flat

and the gaps. Bottom contours were used to define the boundaries between the gaps and

the reef flat.

Wave set down began before the usual surf-zone wave breaking condition of hb/H = 0.78

was reached. This condition occurred in the same grid square, grid 90, for both the wave

heights of 1.34m and 1.55m. The location of grid 90 is indicated by a straight-line

through 900m offshore on Figure 4.1. The end of the reef flat was also indicated by a

straight-line on Figure 4.1. These lines highlight the rapid changes in mean surface

elevation across the reef flat and the more gradual changes through gaps in the reef.

The low water surface profile was distinct from the other surface profiles at both wave

heights. Setup peaked much closer to the front of the reef than at the other water levels

and it was then constant across the reef into the lagoon. There was no change in surface

profile that might indicate a change in bottom contours at the leeward edge of the reef. At

both wave heights the final set-up within the lagoon was lowest at the low water level.

There appeared to be some wave set down in the gaps in the reef. The size of the set

down varied little with changes in water depth or gap width. The surface elevation did

increase from the front of the gap to the back, and once the gap met the lagoon, the water

level increased to converge with the final setup in the lagoon.


46

500 700 900 1100 1300 1500 1700 1900 2100 2300 2500-0.4

-0.3

-0.2

-0.1

0

0.1

Mean Surface Elevation from Offshore to Shore Over the Reef Fla t, Ho

=1.34m

Surface Elevation,

500 700 900 1100 1300 1500 1700 1900 2100 2300 2500-0.4

-0.3

-0.2

-0.1

0

0.1Mean Surface Elevation from Offshore to Shore Through Gap 1, H

o=1.34m

Surface Elevation, meters

h r=0.95

hr=0.65

h r=0.35

hr=0.05

h r=-0.15

500 700 900 1100 1300 1500 1700 1900 2100 2300 2500-0.4

-0.3

-0.2

-0.1

0

0.1Mean Surface Eleva tion from Offshore to Shore Through Gap 2, H

o=1.34m

Distance from Offshore, meters


hr/H

o=0.78 Lagoon star ts

(A)

(B)

(C)

500 700 900 1100 1300 1500 1700 1900 2100 2300 2500-0.4

-0.3

-0.2

-0.1

0

0.1

Mean Surface Elevation from Offshore to Shore Over the Reef Fla t, Ho=1.55m

Surface Elevation,

500 700 900 1100 1300 1500 1700 1900 2100 2300 2500-0.4

-0.3

-0.2

-0.1

0


o=1.55m


hr=0.95

hr=0.65

hr=0.35

hr=-0.15

500 700 900 1100 1300 1500 1700 1900 2100 2300 2500-0.4

-0.3

-0.2

-0.1

0


o=1.55m

Distance from Offshore, meters


(A)

(B)

(C)

hr/H=0.78

Lagoon star ts

Figure 4.1 Modelled Wave Setup


47

The magnitude of the wave set down was greater for larger wave heights, but there did

not appear to be a corresponding increase in setup. The size of the final mean setup

within the lagoon did not appear to vary greatly with water depth either. The final setup

experienced in the lagoon was a few centimeters different despite a water depth range of

one meter.

Generally, set-down on the reef increased as the water level decreased. The greatest set-

down occurred at the second lowest water level. The point of maximum set-down also

moved shoreward as water depth decreased. However, the point of maximum set-up

converged at the leeward edge of the reef.

The maximum surface elevation at each water level over the reef flat was plotted against

the still water depth at the reef crest in Figure 4.2. This clearly shows a trend for the

Ho=1.34m run where set up decreases as water depth increases. For the Ho=1.55m run,

the 0.35m and 0.65m runs were on the same line as the smaller wave heights. However,

the setup at the highest water level was more than a centimeter greater when the run was

repeated with the larger wave-height. The setup on the reef flat when the water level was

below the reef crest was low, and not on the line of the other results.


48

-0.2 -0.05 0.1 0.25 0.4 0.55 0.7 0.85 1 1.150.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

Still Water Depth on Reef Crest, meters

Maximum Mean Water Surface Ele Change in Maximum Mean Water Surface Elevation with SWL Depth at Reef Crest

Ho=1.55

Ho=1.34

Figure 4.2 Change in Maximum Setup with water depth and wave height

Figure 4.3 Non-dimensional Comparison of Setup Results

Non-dimensional Comparison of Set-up Results

y = -0.0006x + 0.0011R2 = 0.7535

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8hr/Ho

ηηηη/(T

sq

rt(g

Ho

))

ModelledData

Linear(ModelledData)


49

As the model runs were not controlled for wave period, set up results were non-

dimensionalized for set up, offshore wave height, period and water depth at the reef crest

(Figure 4.3). This was done using the non-dimensional parameters

√√↵

o

r

H

h and √

√↵

ogHTmaxη

, where ηmax was the maximum set-up on the reef top, hr was the

still water level depth at the reef crest and Ho was the offshore wave height,. As

suggested by Gourlay (1996a) and described in Section 4.2. The set up values when the

water level was below the reef top were omitted. The remaining data fit a linear

relationship with an R2 value of 0.75. This was only slightly different to the R2 value of

0.77 calculated for the plot of water depth versus setup (Figure 4.2).

4.3.2 Comparison with Experimental Results

Gourlay’s (1996a) experimental results for set-up were compared with modelled results

using the same parameters as in the previous section. The two series of experimental data

compared against represented a “fringing” reef and a “platform” reef. In the fringing reef

flow was constrained to remain within the lagoon and it could only flow out of the lagoon

during the backwash phase of wave breaking. In the platform reef scenario, flow moved

across the reef and exited the lagoon at the back of wave tank. However, flow was still

constrained laterally.

Including the modelled results initially suggested values of wave setup had been seriously

underestimated (Figure 4.4). However, it should be noted that data for various “natural”

reef profiles were consistently below that for an idealized horizontal reef (Gourlay

1996b). Probably more importantly, lateral variability was not permitted in either

experimental scenario. In contrast, lateral flow was permitted and clearly occurred in the

modeled scenario.


50

Figure 4.4 Comparison of Experimental and Modelled Results (after Gourlay 1996a)

4.3.3 Wave Measurements at the Gauges

The wave gauges recorded a time-series of water elevation at each time-step. This was

used to describe the waveform as it passed each location.

A time-series of measurements was produced for a 75second period as this amount of

time exceeded the number of time-steps over which the mean results were produced.

That is, mean results were produced over 5 wave periods, which was equivalent to

65seconds in summer and 70seconds in winter. The pattern was steady and cyclic so for

visual clarity a 25second portion of the time-series was used in

Figure 4.5. Incident wave height was determined at the five wave gauges by subtracting

the minimum surface elevation from the maximum surface elevation. These values are

summarized in Table 2.1.

Comparison of Experimental and Modelled Results

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 0.5 1 1.5 2 2.5 3

hr/Ho

ηηηη/T

sqrt

(gH

o)

Platform

Modelled

Fringing


51

0 5 10 15 20 25

-2

-1

0

1

Surface Elevation, Me

0 5 10 15 20 25

-2

-1

0

1

0 5 10 15 20 25

-2

-1

0

1

Gauge 1

Gauge 2

Gauge 3

Change in Water Elevation with Time at Gauges

0 5 10 15 20 25

-2

-1

0

1

data1data2data3data4data5

0 5 10 15 20 25

-2

-1

0

1

Time, Seconds

Surface Elevation

0.95m 0.65m 0.35m 0.05m -0.15m

Gauge 4

Gauge 5

Figure 4.5 Surface Elevation over Time at Gauges, H=1.34


52

0 5 10 15 20 25

-2

-1

0

1

0 5 10 15 20 25

-2

-1

0

1

Surface Elevation, Me

0 5 10 15 20 25

-2

-1

0

1

Gauge 1

Gauge 2

Gauge 3

Change in Surface Elevation with Time at Gauges

0 5 10 15 20 25

-2

-1

0

1

0 5 10 15 20 25

-2

-1

0

1

Time, Seconds

0.95m 0.65m 0.35m -0.15m

Surface Elevation

Gauge 4

Gauge 5

Figure 4.6 Change in Surface Elevation with Time at Gauges, H=1.55m


53

Table 4.1 Wave Heights at Wave Gauges

SWL @ Crest -0.15 0.05 0.35 0.65 0.95

Ho 1.34 1.55 1.34 1.55 1.34 1.55 1.34 1.55 1.34 1.55

Gauge 1 0.22 0.32 1.26 - 1.31 1.42 1.29 1.39 1.25 1.34

Gauge 2 0.22 0.29 1.20 - 1.25 1.30 1.30 1.35 1.29 1.41

Gauge 3 0.19 0.29 2.50 - 1.99 2.47 1.65 1.89 1.37 1.56

Gauge 4 0.27 0.37 1.18 - 1.22 1.27 1.08 1.21 1.08 1.26

Gauge 5 0.26 0.36 1.12 - 1.37 1.17 1.32 1.25 1.08 1.11

The waveform recorded at the gauges in Gap 1 was steady at each water level. The wave

was peaked, with a broader and slightly asymmetric trough. There was very little

difference between the recordings at Gauges 1 and 2, which were located 200 meters

apart in the narrower gap.

Gauge 3, on the reef flat, recorded different waveforms at each water level. The

amplitude of the wave was greatest at the second lowest water level. However, the wave

amplitude decreased as the water depth increased. In general, the wave crests were

broader and the wave troughs narrower than recorded at the gauges situated in the gaps.

The waveform recorded by the gauges in the broader gap (Gap 2) was the same general

shape as that recorded by the gauges in the first gap. However, there were differences

between the water elevations recorded at the two gauges within the gap. The amplitude

of the wave decreased as the water depth increased at the fifth Gauge. At the fourth gauge

however, the wave amplitude was roughly constant.

4.4 Velocity

The velocity field produced by wave breaking during each model run was visualized in a

variety of ways to fully describe the spatial variability of velocity within the model

domain.


54

4.4.1 Velocity Direction

The velocity vector field was used to visualize changes in the direction and magnitude of

currents over and around the reef. Selected graphs are presented in the results section,

the complete set is provided in Appendix B.

The velocity field varied along the reef, showing along-shore and cross-shore variability

that may be attributed to the presence of the gaps in the reef. Generally, water was

directed into the reef across the reef top and flowed back out the gaps. This pattern was

obvious during all the model runs, except for the two simulations where the water level

was below the reef crest.

At the lowest water level, flow was directed parallel to the reef front (Figure 4.7). Flow

was also parallel to the reef line in the narrower gap. However, in the wider gap, flow

was slightly more outward directed, which appeared to be drawn from within the

backreef lagoon. There was very little difference in either direction or magnitude in the

low water model simulations at the different wave heights.


55

0

500

1000

1500

2000

2500

3000

05001000150020002500300035004000Distance Along the Reef Line, meters

Mean Velocity Field at t=58min, hr=-0.15m, H

o=1.55m

0 0002

2 2

2 2 2

2

22

1.0 m/s

Figure 4.7 Velocity Field at Lowest Water Level, Wave Height =1.55m

When the water level was above the reef crest, the velocity field was far more complex.

It showed evidence of eddying, deflection, re-circulation and asymmetry. Changes in

water level and wave height affected the occurrence and location of these features.

Flow into the lagoon occurred across the reef. There were strong inward directed vectors

at the south side of both gaps. This feature occurred at all water levels, however the

vectors were most intense at higher water levels. As the water level decreased, inward

directed vectors also started to occur at the north side of the outflow gaps. This feature

started to develop at the north side of the wider gap when hr was 0.65m and the wave


56

height was 1.55m. However, when the water level was decreased to 0.35m at the reef

crest (Figure 4.8), there was strong unidirectional inward flow at both the northern and

southern edges of both gaps.

0

500

1000

1500

2000

2500

3000

05001000150020002500300035004000

Distance Along the Reef Line, meters

Mean Velocity Field at t=58min, hr=0.35m, H

o=1.55m

2

2 2

2 2 2

2

2

2

1.0 m/s

Figure 4.8 Velocity Field at Mean Water Level, Wave Height=1.55m

Re-circulation of water from the outflow currents back into the lagoon was evident at

higher water levels. This feature appeared to be most intense at the northern edge of both

gaps. Water exiting the lagoon from the gap seemed to be moving back into the lagoon

across the reef almost immediately when the water level was greater than 0.65m (Figure

4.9)


57

The two gaps tend to pull water towards themselves from the south. At most water

levels, the broader gap appeared to draw inflow from about 1500m south along the reef

line. The narrower gap pulled water from about 400m north and from the south to the

model domain boundary. As the water level decreased the direction of flow became

more parallel to the reef.

0

500

1000

1500

2000

2500

3000

05001000150020002500300035004000

W


Mean Velocity Field at t=58min, h r=0.95m, H o=1.55m

2

2 2

2 2 2

2

2

2

1.0 m/s

Figure 4.9 Velocity Field at the Highest Water Level, Wave Height=1.55m

Eddying features in the outflow current were more pronounced at higher water levels and

larger wave heights. At the two highest water levels the outflow currents seemed to

terminate in an eddy (Figure 4.9). In contrast, at lower water levels the currents

maintained their direction and only lost magnitude as they progressed into deeper water

(Figure 4.8).


58

4.4.2 Velocity Magnitude

Velocity contour plots were used to identify areas of maximum inflow and outflow. It

was easier to identify these areas using filled contour plots of the u component of velocity

than in the directional vector field plots. Selected plots are presented here, the complete

set is presented in Appendix C. Despite the lack of bottom contour information in these

plots, it is usually obvious from the velocity profiles where the gaps and reef crest are

located. However, it is less clear where the reef flat starts to fall away into the backreef

lagoon.

-1

-0.5

0

0.5

1

500 1000 1500 2000 2500 3000

500

1000

1500

2000

2500

3000

3500

4000

Magnitude of Velocity Field, Ho=1.34m, hr=0.35s

Distance, meters

Distance, meters

-1

-0.5

0

0.5

1

500 1000 1500 2000 2500 3000

500

1000

1500

2000

2500

3000

3500

4000


Distance, meters

Distance, meters

(A)

(B)

Figure 4.10 Velocity Contours at Mean Water Level


59

The fastest velocities were localized at the front of the reef and the edges of the gaps.

That is, the most rapid inward directed velocities were associated with rapid changes in

bottom contours. Also, current velocity tended to increase as the wave height increased.

This was particularly noticeable in Figure 4.11 where the water depth was 0.95m. A

large patch of high velocity appeared on the reef flat at Ho=1.55m that was not present

when Ho=1.34m. An increase in velocity with increased wave height was also obvious in

Figure 4.10, which shows velocity at mean water level.

-1

-0.5

0

0.5

1

500 1000 1500 2000 2500 3000

1000

2000

3000

4000

Magnitude of Velocity Field, Ho=1.34m, h

r=0.95s

Distance, meters

Distance, meters

-1

-0.5

0

0.5

1

500 1000 1500 2000 2500 3000

1000

2000

3000

4000


r=0.95s

Distance, meters

Distance, meters

(A)

(B)

Figure 4.11 Velocity Contours at the Highest Water Level


60

The location of the strongest outflow currents was highlighted in dark blue in Fig. 4.10

and 4.11. Outflow currents were strongest at higher water levels, and the peak currents

tended to be elongated and narrow within the gaps. At the highest water level, there were

patches of high velocities at the leeward edge of the broader gap. These patches

increased in size at the higher wave height (Figure 4.11). There was also a patch of

outward directed velocities at the rear of the reef in the mean water level winter

simulation (Figure 4.8). This feature was not present in any of the other contour plots.

4.4.3 Cross-Reef Velocity

The results of the velocity contour plots suggested the best place to directly compare

wave driven velocities at different water levels and wave heights was at the front of the

reef. Direct comparison of the velocity at the reef crest and through the gaps was carried

out by taking the mean of the u-component of velocity across 5 grids (50meters) along

the line of the reef crest. This was then plotted along the transect of the reef line for each

water level. The different wave heights were presented on separate graphs (Figure

4.12A,B)


61

0 500 1000 1500 2000 2500 3000 3500 4000-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Velocity profile along the reef crest, Ho=1.34m

Distance along the reef, meters

Velocity, m/s

hr=0.95

hr=0.65

hr=0.35

hr=0.05

hr=-0.15

(A)

0 500 1000 1500 2000 2500 3000 3500 4000-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1Velocity profile along the reef crest, Ho=1.55m

Distance along the reef, meters

Velocity, m/s

hr=0.95

hr=0.65

hr=0.35

hr=-0.15

(B)

Figure 4.12 Velocity Profile Along the Reef


62

The velocity profile along the front of the reef line indicated the location of the gaps and

reef flat at all water levels. Velocity directed into the reef was positive and negative out

through the gaps. Figure 4.12 clearly showed the difference in wave induced velocities at

different water levels.

As seen in the contour plots, the velocity across the reef crest tended to increase with

water depth. The velocity over the reef at low water was much less than the velocity

experienced at the front of the reef at the other water levels. However, the maximum

wave driven velocity did not occur at the highest water level when the wave height was

1.34m. The wave driven velocity was actually greater at the second highest (hr=0.65m)

water level. In contrast, the greatest wave driven velocity occurred at the highest water

level when the wave height increased to 1.55m. This suggested the increase in wave

height was enough to continue to make the waves break at the front of the reef and drive

water across the reef.

The velocity profile through the gaps also showed variability with water level. In

general, the greater the velocity across the reef, the greater the velocity through the gaps.

The velocity of the current through the narrower gap was greater than the velocity of

currents through the wider gap. They were also more strongly localized down the

northern edge of the gap. In contrast, the wider gap often showed a two or three peaked-

velocity profile. The high water, second highest and mean water level all showed a three

peaked pattern with inflow on the northern edge of the gap, strong outflow in the center

of the gap and weaker outflow at the northern edge of the gap. The magnitude of the

velocity was similar at mean and high water, however the peak was shifted slightly north

at the mean water level. At low water, the velocity out of the gaps was much lower than

at higher water levels.


63

4.4.4 Velocity at the Wave Gauges

A time-series of velocity was recorded at each of the five gauges along the reef. This

showed the oscillation in current direction in time during each wave cycle. The time-

series showed the swash and backwash phase of wave movement. At the lowest water

level, the velocity profile was positive throughout the time-series. However, it still

showed an oscillation in magnitude. In general, at each water level, the velocity was

stronger when the wave height was 1.55m than when it was 1.34m.

Gauge 3 was located on the reef flat. The time-series of velocity shows the swash and

backwash phases as the wave moves past the gauge. When compared to the swash and

backwash at the other gauges, the strongest backwash clearly occurred at Gauge 3.

The gauges also showed that waves tend to arrive at Gauges 1 and 2 before they arrive at

Gauges 3,4 and 5. This can be seen in Figure 4.13, where the peak current speed

occurred at t=8s (16*0.5s) at Gauges 1 and 2, but only reached Gauge 3 at t=12s. This

effect was more pronounced at particular water levels (refer to Appendix C).


64

0 10 20 30 40 50-1

0

1

2

Velocity at the Wave Gauges, hr=0.35m, Ho=1.34m

0 10 20 30 40 50-1

0

1

2Gauge 2

0 10 20 30 40 50-1

0

1

2

Gauge 3

0 10 20 30 40 50-1

0

1

2

Gauge 4

0 10 20 30 40 50-1

0

1

2Gauge 5

Time, half secondsVelocity, ms

-1

ms-

1

ms-

1

ms-

1

ms-

1

Gauge 1

Figure 4.13 Velocity Measurements at Gauges, Mean Water Level


65

4.5 Discharge

The total discharge was calculated using velocity data generated by FUNWAVE at the

50th mean time-step for all simulations. The u-components of the vectors were

interpolated from a 20meter wide grid to a 10meter grid. The discharge through each

grid was calculated using the simple formula Q=V*A for each grid, where A=10 * hr.

4.5.1 Cross Reef Discharge

The discharge across the reef was expressed on a per-meter basis. The discharge across

the reef crest into the lagoon was summed over each section of reef between the limits

defined by bottom contours. The total cross-reef flow was divided by the total length of

the reef sections to get a per meter value. Discharge through each gap was calculated

separately. The total discharge through Gap 1 was divided by 600 meters, while the total

discharge through Gap 2 was divided by 800m to yield discharge per-meter (m3s-1m-1).

The discharge at each water level was plotted against the still water level at the reef crest

with each wave height as a different series. Following the definitions in Section 4.4.3

flow into the reef was designated positive and flow out through the gaps was negative.

Like velocity, discharge across the reef tended to increase with both increasing water

depth and wave height. However, the discharge during the lower wave height

simulations peaked at less than the simulated maximum water depth. This was followed

by a decrease in discharge across the reef and through the gaps to almost zero at the

highest water level. In contrast, the discharge over the reef continued to increase with

water depth for the run when H=1.55m.

At the lowest water level for both wave heights, flow through the wider Gap 2 was about

0.5m3s-1m-1, while the flow out of gap 1 was almost zero. Flow through the gaps

increased as water depth increased. The per-meter flow through the gaps, at Ho=1.34m,

was approximately the same at the still water depths of 0.05 and 0.35m. It was also

approximately the same volume for the Ho = 1.55m and hr = 0.35m water depth run.


66

However, a successful simulation was not run at hr = 0.05m and Ho=1.55m and thus no

comparisons could be made with this scenario.

As the water level increased, the flow through Gap 1 increased and was eventually

greater than the flow through Gap 2 at water levels greater than about 0.5m. This was the

reverse of the situation at the lowest water level.

-0.2 0 0.2 0.4 0.6 0.8 1 1.2-2

-1.5

-1

-0.5

0

0.5

1

Still Water Depth at Reef Crest, meters

Discharge per m (m

3s-1m-1)

Discharge per m over the reef and through the gaps, Ho=1.34m

Gap 1Gap 2Reef

Figure 4.14 Discharge per meter through gaps and over reef at a wave height of 1.34m


67

-0.2 0 0.2 0.4 0.6 0.8 1 1.2-2

-1.5

-1

-0.5

0

0.5

1


Discharge per m (m

3s-1m-1)

Discharge per m over the reef and through the gaps, Ho=1.55m

Gap 1Gap 2Reef

Figure 4.15 Per Meter Discharge Across Reef and Through Gaps at a wave height of1.55m

4.5.2 Total Discharge

The total discharge across the reef at each water level was plotted in Figure 4.16. The

two wave heights were plotted as different series. Total discharge was calculated by

summing the interpolated values of discharge along the entire reef line. This did not

include the discharge through the gaps.

The total discharge increased less per unit depth at shallower depths than deeper depths

when the wave height was 1.55m. As expected from the velocity results for the summer

case (Ho =1.34m), the maximum discharge occurred at 0.65m. It was slightly greater

(~150m3s-1) than the discharge at 0.65m for the winter simulation. The maximum


68

discharge for the wave height of 1.55m for the simulated water depths occurred at the

highest water level. At the same water depth, the total discharge for the Ho = 1.34m was

very slightly negative. Greater discharge occurred during the 1.55m wave height

simulation at either end of the range of water depths. In between however, the greater

wave height had slightly less flow than the 1.34m wave height.

-0.2 0 0.2 0.4 0.6 0.8 1 1.2-500

0

500

1000

1500

2000

2500Total Discharge Across the Reef Top


Discharge, m

3s-1 Ho=1.55m

Ho=1.34m

Figure 4.16 Total Discharge Across the Reef Top


69

4.6 Flushing Times

One of the observations this study was based on was the relative importance Hearn et al.

(1986) ascribed to wave forcing on flushing at Ningaloo. For the purpose of their

calculations, flushing was defined as “the average time taken for a particle released

within one of the channel regions to exit through the reef line” (Hearn et al. 1986). The

validity of the assumption that wave flushing was more important than wind or tidal

flushing was checked using the modelled wave driven circulation. Hearn et al.’s (1986)

methods were used to recalculate wind and tidal flushing times for the idealized lagoon.

Then the wave driven flushing time was calculated using the modelled rate of discharge

across the reef. This was carried out to determine if the relative importance of each type

of forcing was maintained.

The volume of the lagoon was calculated at the high, mean and low water depths. The

lagoon was divided into three sections, the backreef lagoon, reef flat and the reef crest,

based on the bottom bathymetry. The volume of each section was calculated as depth x

width x length. The results of the calculations are summarized in Table 4.2 to Table 4.4.

Table 4.2 Volume Calculations at Low Water Level

Depth (m) Width (m) Length (m) Volume (m3)

Lagoon 2.05 1400 4000 114.8x105

Reef Flat 0.05 250 4000 0.5 x105

Reef Crest -0.15 100 4000 0

Total 115.3x105


70

Table 4.3 Volume Calculations at Mean Water Level


Lagoon 2.55 1400 4000 143x105

Reef Flat 0.55 250 4000 5.5x105

Reef Crest 0.35 100 4000 1.4x105

Total 149.9x105

Table 4.4 Volume Calculations at High Water Level


Lagoon 3.15 1400 4000 176.4x105

Reef Flat 1.15 250 4000 11.5 x105

Reef Crest 0.95 100 4000 3.80 x105

Total 191.7x105

4.6.1 Wind Driven Flushing

Hearn et al. (1986) estimated a mean annual wind-driven current of 0.15ms -1, which they

considered likely to be an overestimate. Therefore, in their wind-driven flushing

estimates they used an order of magnitude current speed estimate of 0.1ms-1 (Hearn et al.

1986). From the description of their methods, it appears they simply chose a typical

travel distance from within a lagoon to a gap and assumed the particle would travel at a

constant speed along that travel path. Although this is a questionable method of

calculating wind driven flushing, for comparison the same method has been used here.

So, assuming a travel path of 4 kilometers, from one end of the lagoon to the other, gives

a wind driven flushing time of about 12 hours. This calculation probably under estimates

the wind driven flushing time.


71

4.6.2 Tidal Flushing

A standard calculation for the tidal flushing of an enclosed water body is the volumetric

ratio time,

TV

Vvolume ∆

=τ Equation 4-3

Where ∆V is the volume difference between high and low water, V is the mean volume

of the water body and T is the tidal period. So, for the values given in Table 4.3, this

yields a τvolume of 24 hours for the idealized lagoon.

However, in shallow water systems, the incoming tidal prism does not mix completely

due to the formation of a front between the incoming tide and the water inside the lagoon.

Mixing across this front depends on the density difference between the water bodies and

the strength of any forcing driving mixing. The dispersion coefficient for this process is

poorly known and so an accurate mixing time is difficult to calculate (Hearn et al. 1986).

In addition, there may be re-circulation of water back into the lagoon. Empirically, a

return coefficient, ‘r’ varying from zero (no re-entry) to one (total return) can be included

to represent this aspect of the flushing process. This gives a τtide that depends on τvolume

and r. Hearn et al. (1986) used a typical value of r=0.5.

rvolume

tide −=

1

ττ Equation 4-4

This increases τvolume to about two days.

4.6.3 Wave Driven Flushing

Hearn et al. (1986) calculated a wave driven flushing time by assuming that flow occurs

inwards across the reef and into deep channels where it is then guided out through breaks

in the reef. They calculated a range of final exit velocities through the reef break and

used these to determine how long the lagoon would take to empty (Hearn et al. 1986).

This yielded a range of flushing times from 5 to 23 hours.


72

Table 4.5 Discharge Over the Reef at each Water Level

Water Depth at Reef Crest Summer m3s-1 Winter m3s-1

0.95 -45.3 2073

0.65 1467.3 1322

0.35 821.7 703

0.05 311 -

-0.15 90.5 210

Mean 530 1077

The modelled data was used to recalculate the flushing time. This was based on the

amount of water coming into the lagoon over the reef (Table 4.5). Flushing time was

simply calculated as

Volume

eDischwave

arg=τ Equation 4-5

Using the mean values of discharge across the reef into the lagoon and the mean water

volume of the lagoon (Table 4.3) yielded a flushing time of about 4 hours in winter. This

increased to about 8 hours in summer.

Discussion

73

5 Discussion

5.1 Wave Setup

The magnitude of the maximum wave-setup was affected by changes in water level and

wave height. There was approximately 3cm difference between the modelled maximum

and minimum setup. The maximum setup was about 5cm and the minimum setup was

about 2cm. In general, the magnitude of the setup increased as the water depth

decreased. This gave good agreement with other experimental and modelled results. For

example, according to Gourlay (1996a), the magnitude of wave setup increases as the

water depth over the reef decreases for a given wave height and period.

At a given water depth and wave period, the wave setup increases as the incident wave

height increases (Gourlay 1996a). The divergence between setup values at different

wave heights appears to be greatest at the lowest and highest water levels. At the lowest

water depth, when the water level was below the reef crest, waves broke on the reef front.

Effectively, the waves were breaking on a very steep beach. Consequently, the force

balance producing wave setup was not identical to the force balance for waves breaking

on the reef top. When the water level is below the reef crest, setup on the reef flat occurs

from overtopping by wave run-up. Relative run-up will depend on the roughness and the

permeability of the reef face (Gourlay 1996a). This implies adjusting the permeability

and roughness of the model reef could change the magnitude of the setup produced at low

water levels.

At the upper end of the depth range, the difference in setup with wave height can

probably be attributed to a reduction in wave breaking intensity. At the highest water

level, when the water was almost a meter over the top of the reef, the 1.34m wave was

not breaking as strongly as the 1.55m wave. This changed the force balance governing

the conservation of momentum, reducing the increase in pressure required to balance the

gradient in radiation stress.

Discussion

74

Although not simulated in this study, when the reef is submerged, there is a threshold

value of the offshore wave height Ho below which there is no setup. Gourlay (1996a)

suggested this value was ro hH 4.0∪ . For the range of water depths considered in the

model simulations, this relationship yields an offshore wave height of approximately

0.46m. Wave heights less than this threshold value would not produce set-up. This is a

very small swell height for the Northwest Cape region. According to the WNI

percentage occurrence data (WNI 2000), a significant swell wave height of 0.4 is

exceeded more than 99% of the time in both summer and winter. However, if the water

depth over the reef was increased to 2m, the threshold wave height required to produce

setup would increase to 0.8m. This threshold is still exceeded more than 85% of the time

(WNI 2000). This implies a situation where no setup over the reef occurs is

comparatively rare at Ningaloo.

When the modelled results were compared to Gourlay’s (1996a) experimental results, it

appeared FUNWAVE could be seriously underestimating wave setup. While this should

not be ruled out, it is also possible that differences between the constraints placed on flow

in his experiments and the modelled scenario might be significant. Gourlay’s (1996a)

two scenarios involved different flow restrictions. The so-called “fringing reef” scenario

only allowed water to exit the lagoon during the backwash phase of wave breaking. The

“platform reef” scenario allowed flow to occur out of the lagoon at the back of the

lagoon. It was then re-directed back to the “ocean” in the wave flume through side

channels that were separated from the central flume by solid walls. In contrast, the return

flow out of the lagoon in the model was not constrained. This led to an uneven setup

profile along the reef-line in the long-shore direction. Gourlay (1996a) found setup was

reduced when water was not trapped in the lagoon, by an amount at least equal to the

velocity head of the wave generated flow across the reef. This implies reducing

constraints on flow by including breaks in the reef would continue to reduce wave setup.

However, despite variability in the magnitude of setup, the approximate location of

maximum setup was the same for each simulation. The maximum wave setup always

occurred at the leeward end of the surf zone. This agrees with the conservation of

Discussion

75

momentum theory and experimental observations made by Gourlay (1996b). That is, the

maximum wave set-up should occur at the back of the reef flat where the wave breaking

process was complete (Gourlay 1996a). However, the point of maximum set down

shifted progressively towards the lagoon as the water depth decreased. This may be

explained by differences in magnitude of radiation stress and depth gradients across the

reef flat.

5.2 Wave Induced Currents and Discharge

Modelled current speeds reached a maximum of 0.5ms-1across the reef and 1ms-1 through

the outflow channels. However, it is difficult to compare these values directly to currents

recorded at other locations around the world as reef profile can have a significant affect

on wave dynamics (Gourlay 1996b, Lugo-Fernandez 1998). However, as order of

magnitude estimates, the modelled values seem to match reported current speeds quite

well. For example, Hearn et al. (1986) reported that current speeds of up to 1ms-1 in

outflow channels were recorded in Guam. Landward flow over the reef was measured at

0.3ms-1 (Hearn et al 1986). It is possible that the velocity measurements may be

underestimates given the particular reef profile given Gourlay’s (1996a) reported

experimental setup values were larger than the setup values modelled by FUNWAVE. If

setup is underestimated then it is likely velocity will also be underestimated.

The modelled velocity profiles showed that the velocity at the front of the reef and

through the outflow channels increased with water depth and wave height. This agreed

with the trend identified in the experimental results of Gourlay (1996a), who observed

that velocity increased as incident wave height increased at a given water depth. He also

observed that wave generated flow increased as wave period increased. Unfortunately,

the wave period used in the two sets of simulations was not the same so the effects of

increasing wave height could not be completely isolated from the effect of increasing

period. However, the two wave periods used in the model simulations differed by only

one second. When a linear fit was applied to the raw results and non-dimensionalized

results, the R2 values of 0.75 and 0.77 were approximately the same. This suggests that

Discussion

76

relative differences in modelled results are probably more likely to be due to different

wave height than wave period.

Discharge was calculated using the modelled velocities and still water depth at the reef

crest. Discharge increased as the water depth increased for a given wave height, in

accordance with the experimental results of Gourlay (1996a). However, it increases to a

maximum value before starting to decrease (Gourlay 1996a). This implies there would

be tidal modulation of the rate of discharge at frequencies of particular tidal constituents.

This has been observed at other locations, including John Brewer Reef (Symonds et al.

1995) and in Japan (Kraines et al. 1998).

The results of modelling were analyzed after the simulation had reached a steady state

condition for a discrete water level. The simulated time for this to occur was about one

hour, which is the same the length of time quoted by van Rijn (1990) to establish

equilibrium conditions for wave setup. However, water levels are not static in the natural

environment. The tidal cycle causes a cyclic change in water level over a 12 or 6 hour

time period. These changes in water level and non-monochromatic sea-states introduce

considerable variability into the forcing on wave driven flow. This aspect of wave driven

flow dynamics was not examined. However, these effects are probably significant, given

observations such as those by Roberts (1980), who reported variations of 50% around the

mean speed of surge currents into the lagoon at timescales of 1 to 2 minutes (Lugo-

Fernandez 1998).

Currents over the reef showed considerable spatial variability both across and along the

reef. A swell direction that is not normal to the reef line creates asymmetry in outflows.

Initial runs, not discussed in this report, were carried out with wave crests approaching

perpendicular to the reef. The currents produced in these simulations were considerably

more symmetrical than those produced when the waves approached at an angle. This

implies that in a real coral reef gradients in radiation stress are probably significant in the

Sxy direction, rather than purely the Sxx direction. This implies theoretical models such as

Discussion

77

that proposed by Symonds et al. (1995) may need further development if they are to be

applied to reef environments that show variability in both the x and y direction.

Interaction between waves and currents in shallow water near the coast may affect wave

characteristics (van Rijn 1990). A current opposing the waves yields increased wave

heights and reduced wave lengths effectively steepening the wave, possibly to the point

of breaking (van Rijn 1990). This situation occurs at the ocean side of the gaps where the

outflow currents exit the lagoon. This could explain the set down in the gaps and the

occurrence of breaking at the front of the gaps.

There is refraction of waves over the reef line. Refraction occurs when waves approach

bottom contours at an angle. One end of the wave experiences the bottom and is slowed,

the other end of the wave curves to become parallel to the reef more slowly. This was

shown by waves reaching gauge locations at different times along the reef line. Waves

arrive at Gauges 1 and 2 a few seconds before they reach Gauges 3, 4 and 5. This implies

waves are not normal to the reef when they reach the gauges on the reef crest. This

reinforces the earlier point that the assumption that waves break normal to the reef made

by other modelling approaches (Prager 1991, Kraines et al. 1998) may not be valid.

Inspection of aerial photographs suggests that only inflow occurring within about one

lagoon width of a break takes the shorter route of a direct arc out through the break

without reaching the inshore channel (Hearn et al. 1986). In the modelled scenario, the

distance where inflow took the shorter route was longer, almost double the gap width.

This is probably due to a combination of omitted topographical effects and forcings. As

the water moves into the lagoon, the wave induced velocity decreases. Hearn et al.

(1986) suggested by that wind forcing becomes more important closer to shore. The

omission of wind forcing could explain why the water does not flow further into the

lagoon before entering an outflow current.

Discussion

78

5.3 Flushing Times

The relative magnitudes of the re-calculated values of flushing agree with the values

originally calculated analytically by Hearn et al. (1986). That is, the most rapid flushing

would occur under a purely wave-driven flushing regime. Tidally driven flushing was

the slowest flushing mechanism and purely wind driven flushing was estimated to take

roughly double the wave driven flushing time. However, the method used to calculate

wind driven flushing was highly questionable. A better method would have been to use

the calculated wind driven velocity to calculate a discharge rate. However, this is only

likely to increase the estimated wind driven flushing time. So, the modelling of wave

driven flow has confirmed the importance of wave driven flushing at Ningaloo Reef.

Although wave pumping is almost certainly the most important factor in flushing of

Ningaloo lagoons, it can not be considered alone if accurate predictions of flushing time

are required. For example, tidal currents may modulate flushing times. Wave driven

outflow currents may be blocked at flood tide or strengthened at ebb tide (Prager 1991).

Wind driven currents may strengthen wave driven currents if they act in the same

direction or weaken them if the wind driven currents act in the opposite direction. Swell

waves of less than a meter would also slow wave driven velocities. This might then

reduce the importance of wave driven flushing relative to tidal and wind forcing.

The effective flushing time for a reef at Ningaloo could be slowed by re-circulation of

water from outflow currents back over the reef top. This question is currently

unquantified, and is usually approximated using an empirical re-circulation coefficient

(Hearn et al. 1986). The velocity vector field results (Appendix B) show the occurrence

of re-circulation. It might be possible to use this type of modelling approach to quantify

the increase in effective flushing time caused by re-circulation.

The formation of density gradients due to tidal intrusion of water with different

temperature and salinity characteristics might also slow mixing between oceanic and

lagoonal waters and consequently increase flushing times. The rate of mixing across the

front will depend on the strength of the density gradient and wave conditions. As

Discussion

79

FUNWAVE is vertically integrated, and assumes uniform water body characteristics,

density gradients can not be included in the model.

Determining accurate flushing times is important for risk analysis of contaminant

dispersal. This is becoming increasingly important at Ningaloo due to proposed

developments, such as the Coral Coast Resort development (EPA 1995). Increased

boating pressure might damage coral, not just by direct physical damage caused by

moorings, but also through biological imbalance caused by discharge of sullage. More

accurate flushing calculations could be developed using a coupled model, incorporating

realistic wave forcing and the effect of density gradients, wind and tides.

5.4 Influence of Other Factors

Numerical modelling of a problem, especially one that is not fully parameterized,

requires assumptions and approximations be made. The results of numerical modelling

should be appraised in view of the factors that have been left out, as well as those that

have been controlled for. In this case, these variables include bottom friction, real sea-

states and wave period.

5.4.1 Wave Period

The model runs were not controlled for the effect of wave period on setup or velocity.

The winter and summer simulations used different wave periods, but as they also used

different wave heights, inferences on the effect of wave period alone on setup or velocity

cannot be drawn. This is unfortunate, as Gourlay (1996a) has shown that wave period

affects wave setup. In his experimental work, it appeared that wave setup increased with

increasing wave period until a limiting condition was reached. It was not possible to

determine conclusively if this limiting condition was reached at wave periods of 13 or

14s.

Discussion

80

5.4.2 Bottom Friction

Bottom friction can have a significant effect on wave-breaking processes. It can affect

the type of breaker by reducing the wave height or changing breaking location (Lugo-

Fernandez 1998). It is relatively straightforward to measure friction in the absence of

swell. Under these conditions, it has been determined that coral has a very high drag

coefficient, up to two orders of magnitude higher than normal ocean shelves. However,

there is currently no information about frictional stresses under large wave conditions

(Hearn 1999). In addition, bottom friction is likely to be spatially variable over a coral

reef. This complicates the incorporation of friction into a two dimensional model.

However, given that a friction factor is related to current speed in Symonds et al.’s (1995)

theoretical model, it is probably an important variable and requires more calibration.

5.4.3 Irregular Waves

Irregular waves, particularly if accompanied by wave groups may produce setup

conditions significantly different to those produced by regular waves (Gourlay 1996a).

For example, Seelig (1983) found irregular waves of a given significant wave height

created less setup than monochromatic waves with the same wave height. FUNWAVE

does have the capability to model a real-sea wave spectrum. It could potentially be used

to determine if using a monochromatic wave field is an invalid assumption in modelling

wave driven dynamics over a coral reef.

Conclusions

81

6 Conclusions

This study conducted a preliminary investigation of wave driven circulation over an

idealized version of a Ningaloo Reef lagoon. In particular, it examined differences in

wave setup and wave driven velocities at a range of water levels and wave conditions

using a numerical model.

The magnitude of the maximum wave-setup was affected by changes in water level and

wave height. In general, the magnitude of the setup increased as the water depth

decreased. From comparison with experimental work, it appears that FUNWAVE

predicts the location of setup and setdown accurately, but it may underestimate setup.

Alternatively, it is possible that permitting lateral flow decreases the maximum wave

setup over the reef top when compared to the experimental scenarios, where flow was

constrained in the y-direction.

Modelled current speeds reached a maximum of 0.5ms-1across the reef and 1ms-1 through

the outflow channels. It was difficult to assess the accuracy of these values due to a lack

of field data and realistic bathymetry. However, as estimates of velocities over an

idealized reef, the modelled values seem to match current speeds reported at various

locations around the world quite well.

The relative magnitudes of the re-calculated values of flushing agree with the values

originally calculated analytically by Hearn et al. (1986). That is, the most rapid flushing

occurred under a purely wave-driven flushing regime. Modelling showed the water level

and wave conditions had a significant effect on the rate of wave driven flushing. In

particular, the relative submergence of the reef when compared to wave height is

important in determining the velocity and rate of flushing. The rate of discharge across

the reef slows when the breaking ratio of water depth to wave height is exceeded. This

contributed to flushing being twice as fast in the winter as the summer scenario, despite a

wave height difference of just 20cm.

Conclusions

82

Other outcomes included the possibility that FUNWAVE might be useful in quantifying

re-circulation from outflow gaps back into the reef. This could improve flushing

estimates by quantifying the proportion of lagoonal water that re-enters the lagoon over

the reef. However, to calculate flushing properly, the moderating effects of wind and

tides need to be incorporated. This would require coupling a wave model such as

FUNWAVE to a 3-D hydrodynamic model.

The spatial variability shown by the model along the reef line demonstrates observations

of current speed and direction will be affected by the location of measurements. This

makes it important to understand spatial variability in current speed when planning

fieldwork for model verification.

In conclusion, FUNWAVE has provided a useful tool for investigating the dynamics of

wave driven flow over a coral reef. However, as the model has not been validated against

experimental results or real observations, it is impossible to evaluate the error involved in

the modelled current speeds and wave setup.

Recommendations

83

7 Recommendations

The scope of this study was limited and highly idealized. It focused on only one aspect

of the forces driving water circulation around a simplified coral reef. Although wave

pumping has been identified as the dominant forcing mechanism at Ningaloo, other

factors such as wind and horizontal density gradients may also contribute to the overall

rate of flushing and circulation. The recommendations for further work can be divided

into two parts. First, further investigation of FUNWAVEs’ usefulness in modeling

circulation around a coral reef is discussed. Second, incorporation of other forcings to

develop better predictions of flushing time is suggested.

7.1 FUNWAVE and Wave-driven circulation

A more realistic bathymetry reflecting the actual bottom contours of particular lagoon

would improve the accuracy of modelled results. The surface structure of the reef might

also influence the energy dissipation across the reef through the groove and spur

structures. To resolve these structures the size of the model grids would have to be

reduced. This would cause either an increase in computational time, which is already

long, or a decrease in the maximum size of the domain. A realistic bathymetry would

also include the deep channels that typically exist in back-reef lagoons. This is an aspect

of circulation that was neglected in the study of flow across the reef, although it has been

identified as the location of the greatest mass flux within the reef (Hearn et al 1986).

A major strength of FUNWAVE is its ability to take a directional wave field as an input.

This allows it to better reflect the complexity of a real sea-state. So, the wave field

should be changed from a monochromatic field to a more realistic directional spectrum to

show if variability in the wave field creates variability in wave driven currents.

Recommendations

84

The friction factor used in the modeled runs was at the high end of the typical range

suggested in the FUNWAVE 1.0 Manual (Kirby et al 1998). However, this was probably an

underestimate by at least one order of magnitude (Gourlay 1996a). Bottom friction might

have a significant effect on wave flow at shallow depths. However, the friction factor should

not be increased throughout the model domain. Increased friction would need to be localized

over the top of the reef and on the reef face. The rest of the domain should be left with a lower

friction factor typical of a sandy sea floor. This would provide a more accurate representation of

the influence of friction of wave driven flow.

7.2 Other Forcings

A wave model can only describe one aspect of the forces driving circulation around a

coral reef. Tidal direction and wind strength, as well as wave height and water depth will

moderate the strength of wave driven currents within the lagoon. These forcings could be

incorporated into modeling of circulation around a coral reef by coupling the wave model

to 3-D hydrodynamic model.

Calibration and validation of any circulation model is essential if the coupled model is to

be used for practical applications. This would make it useful for risk assessment or

ecological modeling of neutrally buoyant particles such as coral larvae.

Bibliography

85

8 Bibliography

Andrews, J.C., & Pickard, G.L. (1990) The Physical Oceanography of Coral Reef Systems, In Coral Reefs

ed. Z. Dubinsky, Ecosystems of the World, Vol. 25, Elsevier

Chen, Q., Dalrymple, R.A., Kirby J.T., Kennedy, A.B., Haller, M.C. (1999) Boussinesq modeling of a rip

current system, Journal of Geophysical Research, Vol. 104 NoC9, pp20,617-20,637

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D’Adamo, N. and Simpson, C.J. (2001). Review of the oceanography of Ningaloo Reef and adjacent waters.

Technical Report: MMS/NIN/NIN-38/2001. March 2001. Marine Conservation Branch, Department of

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Gourlay, M.R. (1993) Wave Set-up and Wave-generated Currents on Coral Reefs, Proc. 11th Australasian

Conference on Coastal and Ocean Engineering, pp479-484

Gourlay, M.R. (1996a) Wave set-up on coral reefs. 1. Set-up and wave generated flow on an idealized two

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Gourlay, M.R. (1996b) Wave set-up on coral reefs 2. Set-up on reefs with various profiles. Coastal

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Hardy, T.A., Mason L.B., McConochie J.D., (2000) A wave model for the Great Barrier Reef, Ocean

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Hearn, C.J., (1999), Wave-breaking hydrodynamics within coral reef systems and the effect of changing

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Johnson, David (2001) Center for Water Research, University of Western Australia, personal

communication

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Kraines, S.B., Yanagi, T., Isobe, M., Komiyama, H. (1998) Wind-wave driven circulation on the coral reef

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Appendices

88

Appendix A

89

Appendix AFUNWAVE2D datafile, FUNWAVE2dpt.data

Details of coefficients used in modelling

c*** if idout is set to 1 (eta and mean currents) then itdel is reset internally to be 2*dominant wave period

$data1 ibe = 2 imch = 2 a0 = 0.5 h0 = 15.0 tpd = 12.0 dx = 10.0 dy = 20.0 dt = 0.5 mx = 301 ny =201 nt = 7201 itbgn = 3000 itend = 6001 itdel = 20 itscr = 10 itftr = 200 theta = 45.0 cbkv = 0.35 delta = 0.02 slmda = 20.0 isltb = 60 islte = 301 $end

$data2 isrc = 31 jsrc = 1 cspg = 10.0 cspg2 = 0.0 cspg3 = 0.0 ispg = 21 10 1 1 ngage = 5 ixg = 35 45 180 270 290 iyg = 123 123 123 123 123 itg = 5001 cbrk = 1.2 ck_bt = 0.004 c_dm = 0.05 isld = 1 idout = 2 idft = 0 $end

$data3 f1n = 'dpdata.cacr' f2n = 'inwdata.cacr' f3n = 'specmat.spec' f4n = 'gauges.out' f5n = 'end.out' f6n = 'means.out' f7n = 'timeseries.out' $end

$data4 ihotsave=0 errorcrit=0.0001 ipb=1 iaverno=5 shorecf=1.8 shorefilt=0.8 $end

$ptrack npar = 0 npstart=1000 pstrtx= 110 110.5 110 110.5

Appendix A

90

pstrty= 10 10 10.5 10.5 $end

Appendix B

91

Appendix B

0

500

1000

1500

2000

2500

3000

05001000150020002500300035004000



o=1.34m

2

2

2

2 2 2

2

22

1.0 m/s

0

500

1000

1500

2000

2500

3000



2

2 2

2 2 2

2

2

2

1.0 m/s

Appendix B

92

0

500

1000

1500

2000

2500

3000

05001000150020002500300035004000



o=1.34m

2

2

2

2 2 2

2

2

2

1.0 m/s

0

500

1000

1500

2000

2500

3000



2

22

2 2 2

2

22

1.0 m/s

Appendix B

93

0

500

1000

1500

2000

2500

3000

05001000150020002500300035004000



o=1.34m

2

22

2 2 2

2

22

1.0 m/s

0

500

1000

1500

2000

2500

3000

05001000150020002500300035004000



o=1.55m

2

2 2

2 2 2

2

2

2

1.0 m/s

Appendix B

94

0

500

1000

1500

2000

2500

3000

05001000150020002500300035004000



2

22

2 2 2

2

22

1.0 m/s

Appendix B

95

0

500

1000

1500

2000

2500

3000


Mean Velocity Field at t=56min, h r=-0.15m, H o=1.34m

0 0002

2 2

2 2 2

2

2

2

1.0 m/s

0

500

1000

1500

2000

2500

3000


Mean Velocity Field at t=58min, hr=-0.15m, H

o=1.55m

0 0002

2 2

2 2 2

2

22

1.0 m/s

Appendix C

96

Appendix C

Velocity Contours

-1

-0.5

0

0.5

1

500 1000 1500 2000 2500 3000

1000

2000

3000

4000


r=0.95s

Distance, meters

Distance, meters

-1

-0.5

0

0.5

1

500 1000 1500 2000 2500 3000

1000

2000

3000

4000


r=0.95s

Distance, meters

Distance, meters

(A)

(B)

Appendix C

97

-1

-0.5

0

0.5

1

500 1000 1500 2000 2500 3000

500

1000

1500

2000

2500

3000

3500

4000


r=0.65s

Distance, meters

Distance, meters

-1

-0.5

0

0.5

1

500 1000 1500 2000 2500 3000

500

1000

1500

2000

2500

3000

3500

4000


r=0.65s

Distance, meters

Distance, meters

(A)

(B)

Appendix C

98

-1

-0.5

0

0.5

1

500 1000 1500 2000 2500 3000

500

1000

1500

2000

2500

3000

3500

4000


Distance, meters

Distance, meters

-1

-0.5

0

0.5

1

500 1000 1500 2000 2500 3000

500

1000

1500

2000

2500

3000

3500

4000


Distance, meters

Distance, meters

(A)

(B)

Appendix C

99

-1

-0.5

0

0.5

1

500 1000 1500 2000 2500 3000

1000

2000

3000

4000

Magnitude of Velocity Field, Ho=1.34m, hr=-0.15s

Distance, meters

Distance, meters

-1

-0.5

0

0.5

1

500 1000 1500 2000 2500 3000

1000

2000

3000

4000

Magnitude of Velocity Field, Ho=1.55m, hr=-0.15s

Distance, meters

Distance, meters (A)

(B)

Appendix D

100

Appendix D

Velocity at the Wave Gauges

0 10 20 30 40 50-1

0

1

2 Gauge 1

0 10 20 30 40 50-1

0

1

2 Gauge 2

0 10 20 30 40 50-1

0

1

2 Gauge 3

0 10 20 30 40 50-1

0

1

2


0 10 20 30 40 50-1

0

1

2 Gauge 5

Velocity, ms

-1

Time, half seconds

ms-

1

ms-

1

ms-

1

ms-

1

Gauge 4

Appendix D

101

0 10 20 30 40 50-1

0

1

2Gauge 1

0 10 20 30 40 50-1

0

1

2Gauge 2

0 10 20 30 40 50-1

0

1

2Gauge 3

0 10 20 30 40 50-1

0

1

2

Gauge 4

0 10 20 30 40 50-1

0

1

2



-1 Gauge 5

ms-

1

ms-

1

ms-

1

ms-

1

Appendix D

102

0 10 20 30 40 50-1

0

1

2Gauge 1

0 10 20 30 40 50-1

0

1

2Gauge 2

0 10 20 30 40 50-1

0

1

2Gauge 3

0 10 20 30 40 50-1

0

1

2Gauge 4

0 10 20 30 40 50-1

0

1

2



-1

Gauge 5

ms-

1

ms-

1

ms-

1

ms-

1

Appendix D

103

0 10 20 30 40 50-1

0

1

2Gauge 1

0 10 20 30 40 50-1

0

1

2Gauge 2

0 10 20 30 40 50-1

0

1

2Gauge 3

0 10 20 30 40 50-1

0

1

2 Gauge 4

0 10 20 30 40 50-1

0

1

2



-1

Gauge 5

ms-

1

ms-

1

ms-

1

ms-

1

Appendix D

104

0 10 20 30 40 50-1

0

1

2


0 10 20 30 40 50-1

0

1

2Gauge 2

0 10 20 30 40 50-1

0

1

2

Gauge 3

0 10 20 30 40 50-1

0

1

2

Gauge 4

0 10 20 30 40 50-1

0

1

2Gauge 5


-1

ms-

1

ms-

1

ms-

1

ms-

1

Gauge 1

Appendix D

105

0 10 20 30 40 50-1

0

1

2 Gauge 1

0 10 20 30 40 50-1

0

1

2Gauge 2

0 10 20 30 40 50-1

0

1

2


0 10 20 30 40 50-1

0

1

2Gauge 4

0 10 20 30 40 50-1

0

1

2

Gauge 5


-1

ms-

1

ms-

1

ms-

1

ms-

1

Gauge 3

Appendix D

106

0 10 20 30 40 50-1

0

1

2Velocity at the Wave Gauges, hr=0.05m, Ho=1.34m

0 10 20 30 40 50-1

0

1

2Gauge 2

0 10 20 30 40 50-1

0

1

2Gauge 3

0 10 20 30 40 50-1

0

1

2Gauge 4

0 10 20 30 40 50-1

0

1

2

Gauge 5


-1

ms-

1

ms-

1

ms-

1

ms-

1

Gauge 1

Appendix D

107

0 10 20 30 40 50-1

0

1

2Velocity at the Wave Gauges, h r=-0.15m, H o=1.34m

0 10 20 30 40 50-1

0

1

2

Gauge 2

0 10 20 30 40 50-1

0

1

2 Gauge 3

0 10 20 30 40 50-1

0

1

2Gauge 4

0 10 20 30 40 50-1

0

1

2

Gauge 5

Time, half seconds

Velocity, ms

-1

ms-

1

ms-

1

ms-

1

ms-

1

Gauge 1

Appendix D

108

0 10 20 30 40 50-1

0

1

2

Gauge 1

0 10 20 30 40 50-1

0

1

2

Gauge 2

0 10 20 30 40 50-1

0

1

2 Gauge 3

0 10 20 30 40 50-1

0

1

2

Velocity at the Wave Gauges, hr=-0.15m, H

o=1.55m

0 10 20 30 40 50-1

0

1

2

Gauge 5

Time, half seconds

Velocity, ms

-1

ms-

1

ms-

1

ms-

1

ms-

1

Gauge 4

Appendix A

109

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