PHYSICAL OCEANOGRAPHY IN CORAL REEF …jsrogers/Dissertation...scales of order ten to a hundred...

PHYSICAL OCEANOGRAPHY IN CORAL REEF ENVIRONMENTS:

WAVE AND MEAN FLOW DYNAMICS AT SMALL AND LARGE

SCALES, AND RESULTING ECOLOGICAL IMPLICATIONS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF CIVIL AND

ENVIRONMENTAL ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Justin Scott Rogers

December 2015

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/fj342cd7577

© 2015 by Justin S Rogers. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii

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I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Stephen Monismith, Primary Adviser


Rob Dunbar


Oliver Fringer


Curt Storlazzi

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

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Abstract

This dissertation investigates the physical oceanography of coral reef environments,

specifically focusing on waves and mean flows at small and large scales. At small

scales of order ten to a hundred meters, the role of spur and groove formations and

their interaction with surface waves and mean flow is examined. Spur-and-groove

formations are found on the fore reefs of many coral reefs worldwide. Although these

formations are primarily present in wave-dominated environments, their effect on

wave-driven hydrodynamics is not well understood. A two-dimensional, depth-

averaged, phase-resolving non-linear Boussinesq model (funwaveC) was used to

model hydrodynamics on a simplified spur-and-groove system. The modeling results

show that the spur-and-groove formations together with shoaling waves induce a

nearshore Lagrangian circulation pattern of counter-rotating circulation cells. We

present results from two separate field studies of SAG formations on Palmyra Atoll

which show their effect on waves to be small, but reveal a persistent order 1 cm/s

depth-averaged Lagrangian offshore flow over the spur and onshore flow over the

grooves. This circulation was stronger for larger, directly-incident waves and low

alongshore flow conditions, consistent with predictions from modeling. Vertical flow

was downward over the spur and upward over the groove, likely driven by alongshore

differences in bottom stress and not by vortex forcing. We suggest that the conditions

for coral recruitment and growth appear to be more favorable on the spur than the

groove due to (1) higher “food” supply from higher mean alongshore velocity,

downward vertical velocity, and higher turbulence, and (2) lower sediment

accumulation due to higher and more variable bottom shear stress.

At large scales of order hundreds of meters to kilometers, the wave and mean flow

dynamics of a pacific atoll are investigated. We report field measurements of waves

and currents made from Sept-2011 to Jul-2014 on Palmyra Atoll in the Central Pacific

that were used in conjunction with a coupled wave and three-dimensional

v

hydrodynamic model (COAWST) to characterize the waves and hydrodynamics

operant on the atoll. Bottom friction, modeled with a modified bottom roughness

formulation, is the significant source of wave energy dissipation on the atoll, a result

that is consistent with available observations of wave damping on Palmyra. Indeed

observed and modeled dissipation rates are an order of magnitude larger than what has

been observed on other, less geometrically complex reefs. At the scale of the atoll

itself, strong regional flows create flow separation and a well-defined wake, similar to

the classic fluid mechanics problem of flow past a cylinder. Circulation within the

atoll is primarily governed by tides and waves, and secondarily by wind and regional

currents. Tidally driven flow is important at all field sites, and the tidal phasing

experiences significant delay with travel into the interior lagoons. Wave driven flow is

significant at most of the field sites, and is a strong function of the dominant wave

direction. Wind driven flow is generally weak, except on the shallow terraces. The

near bed squared wave velocity, a proxy for bottom stress, shows strong spatial

variability across the atoll and exerts control over geomorphic structure and high coral

cover. Based on Lagrangian float tracks, the mean age was the best predictor of

geomorphic structure and appears to clearly differentiate the geomorphic structures.

While high mean flow appears to differentiate very productive coral regions, low

water age and low temperature appear to be the most important variables for

distinguishing between biological cover types at this site. The sites with high coral

cover can have high diurnal temperature variability, but their average weekly

temperature variability is similar to offshore waters. The mechanism for maintaining

this low mean temperature is high mean advection, which occurs at timescales of a

week, and is primarily governed by wave driven flows. The resulting connectivity

within the atoll system shows that the general trends follow the mean flow paths;

however, some connectivity exists between all regions of the atoll system.

vi

Acknowledgments

It is impossible to eloquently and succinctly summarize a journey that has taken the

last five years, and to adequately acknowledge all the people who have supported and

guided me through this process. Nevertheless, here is an attempt.

First I would like to acknowledge my advisor, Stephen, for his work in training me to

think like an oceanographer and giving me the opportunities and tools to succeed in

academia. I sincerely appreciate not only his scientific brilliance when discussing

perplexing questions, but also his loyalty and care for me as a person. I am very

grateful for his dedication to connecting me with other influential people in the field,

and for supporting me through this process.

Secondly, I would like to thank my committee members. Rob Dunbar has been very

influential in my time here at Stanford, and I sincerely appreciate the opportunities he

has given me to collaborate with others outside of EFML. Oliver Fringer has been my

favorite teacher at Stanford, as well as an incredible mentor in modeling and life. Curt

Storlazzi has been influential in training me in physical oceanography, and helping me

get my first paper published.

I sincerely appreciate the collaboration and friendship of Dave Koweek, I could not

have done this without him. Spending many long days working on a remote tropical

atoll together either makes you sincere friends or bitter enemies, and I am happy to say

we are the former!

I would also like to acknowledge the following colleagues: Jeff Koseff, Falk

Feddersen, Derek Fong, Dave Mucciarone, Brock Woodson, Fiorenza Micheli, Steve

Litvin, Nirnimesh Kumar, Alex Sheremet, Amatzia Genin, and everyone from the

Reefs Tomorrow Initiative. I am indebted to my master’s advisor, Ken Potter, as well

as Chin Wu and John Hoopes at UW - Madison for starting this whole thing by

imparting their love of research to me.

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I am very grateful to be a part of the wonderful community of Stanford EFML; I

certainly could not have completed this journey without all of them. A few people who

have had special influence on this dissertation include Ivy Huang, Maha AlNajjar,

Bobby Arthur, Kara Scheu, Mallory Barkdull, Simon Wong, Matt Rayson, Phil

Wolfram, Sean Vitousek, Ryan Walter, Franco Zarama, Walter Torres, Mike Squibb,

Jamie Dunckley and Sarah Giddings. I also acknowledge the administrative support of

Jill Filice, Yusong Rogers, and Marguerite Skogstrom.

I am grateful to the US Department of Defense NDSEG fellowship for funding me for

the first three years. I was also funded by a grant from the Gordon and Betty Moore

Foundation, “Understanding coral reef resilience to advance science and

conservation,” and teaching support from the Stanford Department of Civil and

Environmental Engineering.

On a personal level, this journey has been the most difficult five years of my life.

There were many times I did not know if I would be able to complete my degree. I am

grateful to have been surrounded by many colleagues mentioned above, but also a

community of friends and family who encouraged me to keep pushing forward and

pursue my passion even in the face of difficulty and at times outright despair. Coming

through difficult times has made me appreciate even more that which is good, true,

and beautiful in life, a few of which are love, passion, friendship, and faith. A few

people who have been especially influential to me include my parents, my brother

Grant, sister Heather, my aunt Nellie and uncle Chuck, my grandmother Lorraine,

Minna, Tim and Helga, Fatima, my Christian church community at PBC, especially

Matt and Laurice Vitalone, Nii and Jana Dodoo, Brad Powley and Lisa Cram. Finally,

my daughter Maya continues to provide such joy, inspiration and fun to life; she

makes all of this worthwhile.

I would like to dedicate this dissertation to my beloved grandmother, Carol, who

always believed the best in me. As a lifelong learner, she always valued education

having received her master’s in psychology at a time when that was uncommon for

women. She was so excited for me to be at Stanford because she knew that was my

viii

dream. But beyond that, and more importantly, she loved me and believed in who I

am. Her confidence in me changed the course of my life.

Be kind, for everyone you meet is fighting a harder battle. ― Plato

As I grow older, I appreciate more and more the people in my life who take the time to

look outside themselves and help another. I hope that I have learned to do the same. I

also have grown to understand that this life is a sacred gift and our time here is short,

and therefore it is my obligation to make the very best of the opportunities in front of

me. Soli Deo Gloria! While this PhD journey has been difficult, I love the work I do. I

am so glad I made the decision to change my career direction, and am thankful for all

the people who have helped me on the way. I am excited to see what the future brings!

Justin Rogers

Stanford, California

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Table of Contents

Abstract .......................................................................................................................... iv

Acknowledgments ......................................................................................................... vi

Table of Contents .......................................................................................................... ix

List of Tables ............................................................................................................... xiv

List of Figures ............................................................................................................... xv

Chapter 1 Introduction .................................................................................................... 1

1.1 Background and Motivation ................................................................................. 1

1.2 Small Scales – Spur and Groove Formations ....................................................... 4

1.3 Large Scales – A Pacific Atoll System ................................................................. 7

1.4 Dissertation Outline ............................................................................................ 10

Chapter 2 Hydrodynamics of Spur and Groove Formations on a Coral Reef .............. 12

Abstract ..................................................................................................................... 13

2.1 Introduction ........................................................................................................ 14

2.2 . The Boussinesq Wave and Current Model ....................................................... 16

2.3 Model Setup and Conditions .............................................................................. 19

2.3.1 Model SAG Bathymetry .............................................................................. 19

2.3.2 Model Parameters and Processing ............................................................... 21

2.4 Results ................................................................................................................ 23

2.4.1 Base-Configuration Model Results ............................................................. 23

2.4.2 Mechanism for Circulation .......................................................................... 24

2.4.3 Effects of Hydrodynamic Conditions and SAG Geometry ......................... 25

x

2.4.4 Effect of Spatially Variable Drag Coefficient ............................................. 28

2.5 Discussion ........................................................................................................... 28

2.5.1 Relative Effect of Return Flow to SAG-Induced Circulation ..................... 28

2.5.2 SAG Wavelength ......................................................................................... 30

2.5.3 Two-Dimensional SAG Circulation and Potential Three-Dimensional

Effects ................................................................................................................... 31

2.6 Conclusions ........................................................................................................ 32

2.7 Acknowledgements ............................................................................................ 34

2.8 Appendix A – Comparison to Second Order Wave Theory ............................... 34

2.9 Appendix B – Scaling of the Boussinesq Equation ............................................ 35

2.10 Figures and Tables ............................................................................................ 40

Chapter 3 Field Observations of Wave-Driven Circulation over Spur and Groove

Formations on a Coral Reef .......................................................................................... 53

Key Points ................................................................................................................ 54

Abstract ..................................................................................................................... 54

3.1 Introduction ........................................................................................................ 55

3.2 Methods .............................................................................................................. 57

3.2.1 Field Experiment ......................................................................................... 57

3.2.2 Data Analysis ............................................................................................... 58

3.3 Results ................................................................................................................ 62

3.3.1 Circulation and Vertical Structure ............................................................... 63

3.3.2 Momentum Balance ..................................................................................... 65

3.3.3 Bottom Roughness ....................................................................................... 66

3.3.4 Near Bed Results ......................................................................................... 66

3.4 Discussion ........................................................................................................... 67

xi

3.4.1 Waves and Circulation ................................................................................ 67

3.4.2 Mechanism for Circulation .......................................................................... 68

3.4.3 Implications for Coral Health ...................................................................... 70

3.5 Conclusions ........................................................................................................ 72

3.6 Acknowledgements ............................................................................................ 73

3.7 Figures and Tables .............................................................................................. 74

Chapter 4 Wave Dynamics of a Pacific Atoll with High Frictional Effects ................ 85

Key points ................................................................................................................. 86

Abstract ..................................................................................................................... 86

4.1 Introduction ........................................................................................................ 87

4.2 Study Site ............................................................................................................ 90

4.3 Field Measurements ............................................................................................ 90

4.3.1 Field Experiments and Data Analysis ......................................................... 90

4.3.2 Wave Climate .............................................................................................. 92

4.3.3 Wave Friction .............................................................................................. 93

4.3.4 Wave Breaking ............................................................................................ 95

4.4 Wave Modeling .................................................................................................. 96

4.4.1 Wave Model ................................................................................................ 96

4.4.2 Model Modifications and Performance ....................................................... 99

4.4.3 Wave Transformation and Dissipation ...................................................... 101

4.4.4 Ecological Implications ............................................................................. 103

4.5 Conclusions ...................................................................................................... 104

4.6 Acknowledgements .......................................................................................... 106


xii

Chapter 5 Field Observations of Hydrodynamics and Thermal Dynamics in an Atoll

System: Mechanisms and Ecological Implications .................................................... 118

Key Points .............................................................................................................. 119

Abstract ................................................................................................................... 119

5.1 Introduction ...................................................................................................... 120

5.2 Methods ............................................................................................................ 123

5.2.1 Study Site ................................................................................................... 123

5.2.1 Field Experiment and Data Analysis ......................................................... 124

5.3 Results and Discussion ..................................................................................... 125

5.3.1 Circulation and Tides ................................................................................ 125

5.3.2 Forcing Mechanisms ................................................................................. 126

5.3.3 Vertical Structure and Bottom Roughness ................................................ 129

5.3.4 Thermal Dynamics and Ecological Implications ...................................... 132

5.4 Conclusions ...................................................................................................... 136



Chapter 6 Modeling the Hydrodynamics of an Atoll System: Mechanisms for Flow,

Ecological Implications, and Connectivity ................................................................. 151

Key Points .............................................................................................................. 152

Abstract ................................................................................................................... 152

6.1 Introduction ...................................................................................................... 153

6.2 Methods ............................................................................................................ 157

6.2.1 Study Site ................................................................................................... 157

6.2.2 Hydrodynamic Model ................................................................................ 158

6.3 Results and Discussion ..................................................................................... 161

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6.3.1 Model Validation and Performance ........................................................... 161

6.3.2 Interaction of Atoll with Regional Flow ................................................... 162

6.3.3 Circulation within Atoll ............................................................................. 164

6.3.4 Ecological Implications ............................................................................. 166

6.3.5 Connectivity .............................................................................................. 169

6.4 Conclusions ...................................................................................................... 170



Chapter 7 Conclusion ................................................................................................. 185

7.1 Summary of Key Findings ................................................................................ 185

7.1.1 Small Scales – Spur and Groove Formations ............................................ 185

7.1.2 Large Scale – Waves and Hydrodynamics of a Pacific Atoll System ....... 186

7.2 Future Research ................................................................................................ 188

Appendix A - Supporting Information for Wave Dynamics of a Pacific Atoll with

High Frictional Effects ............................................................................................... 190

Appendix B – Supporting information for Hydrodynamics of a Pacific Atoll System –

Mechanisms for Flow, Ecological Implications and Connectivity ............................ 201

List of References ....................................................................................................... 208

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List of Tables

Table 2-1. Parameters used for base-configuration model, and range of parameters for

variation models. .......................................................................................................... 52

Table 3-1. Experiment instrumentation for NFR13 and SFR12 experiments, sites,

depth, instrumentation and sampling rates. .................................................................. 83

Table 3-2. Order of terms in depth-averaged momentum equations (Eq. 3) from

NFR13 experiment in the cross-shore (x) and alongshore (y) directions. .................... 83

Table 3-3. Bottom drag coefficient CD results from NFR13 experiment from near-bed

ADV measurements in cross-shore (x) and alongshore (y) directions. ........................ 84

Table 4-1. Field experiment instrumentation, depth, deployment time, and sampling at

each site. ..................................................................................................................... 117

Table 5-1. Field experiment instrumentation, depth, deployment time, and sampling at

each site. ..................................................................................................................... 148

Table 5-2. Bottom roughness and drag results from field measurements at various sites

using fits to velocity profiles, and Reynolds stress. ................................................... 150

Table 6-1. Model run details and computed efolding flushing time for each zone. ... 184

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List of Figures

Figure 2-1. Underwater image of a typical SAG formation off southern Moloka’i,

Hawai’i. ........................................................................................................................ 40

Figure 2-2. Morphology of characteristic SAG formations off southern Moloka’i,

Hawai’i. ........................................................................................................................ 41

Figure 2-3. Distribution of SAG wavelength λSAG and spur height hspr of SAG

formations ..................................................................................................................... 41

Figure 2-4. Idealized spur and groove model domain. ................................................. 42

Figure 2-5. Model surface results for base-configuration. ........................................... 43

Figure 2-6. Model velocity and bed shear results for base-configuration. ................... 44

Figure 2-7. Lagrangian velocity UL vectors from base-configuration ......................... 45

Figure 2-8. Phase averaged cross-shore momentum balance for base-configuration .. 45

Figure 2-9. Alongshore variation of x-momentum terms and velocity for base-

configuration ................................................................................................................. 46

Figure 2-10. Variation of model parameters and their effect on normalized circulation

...................................................................................................................................... 47

Figure 2-11. Variation of model parameters and their effect on normalized average

cross-shore bottom stress .............................................................................................. 48

Figure 2-12. Variation of wave height H and wave angle θ with SAG wavelength .... 49

Figure 2-13. Variation of x-momentum terms, velocity, circulation and average bottom

shear with SAG wavelength ......................................................................................... 50

Figure 2-14. Comparison of cross-shore Stokes drift US and radiation stress Sxx for

base-configuration ........................................................................................................ 51

Figure 2-15. Comparison of alongshore contribution to NLW* and PG* terms and

results for UE from model and scaling .......................................................................... 52

Figure 3-1. Palmyra Atoll with field experiment location and layout. ......................... 74

Figure 3-2. Field experiment images and spur and groove bathymetry ....................... 75

Figure 3-3. Physical forcing of tide, waves, and wind during NFR13 experiment

duration ......................................................................................................................... 76

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Figure 3-4. Physical forcing of tide, waves, depth averaged mean Lagrangian velocity

UL results and circulation velocity Uc during SFR12 experiment duration ................. 77

Figure 3-5. Depth averaged mean Lagrangian velocity UL results and circulation

velocity Uc over NFR13 experiment duration .............................................................. 78

Figure 3-6. Mean Lagrangian velocity uL, in the alongshore (y) and vertical (z)

direction showing characteristic spur and groove circulation cells .............................. 79

Figure 3-7. Mean Lagrangian velocity uL in alongshore (y) and vertical (z) direction

under different flow conditions during NFR13 experiment. ........................................ 80

Figure 3-8. Average profiles over depth ....................................................................... 81

Figure 3-9. Near-bed mean Lagrangian velocity and bottom stress results ................. 82

Figure 4-1. Palmyra Atoll location, site layout and experiment instrumentation ...... 107

Figure 4-2. SWAN model grid bathymetry zoomed to atoll ...................................... 108

Figure 4-3. Wave and wind observations on Palmyra Atoll ...................................... 109

Figure 4-4. Wave friction factor calculation from field observed energy flux and

dissipation ................................................................................................................... 110

Figure 4-5. Wave friction parameterizations and model bottom roughness grid. ...... 111

Figure 4-6. Observed and modeled significant wave height, and average change in

wave height with friction method. .............................................................................. 112

Figure 4-7. SWAN model results ............................................................................... 113

Figure 4- 8. Average wave action terms and dissipation from SWAN model. ......... 114

Figure 4-9. Wave energy flux, wave friction factor and high near bed velocity squared

from SWAN model, .................................................................................................... 115

Figure 4-10. Cumulative probability of geomorphic structure and biological cover . 116


Figure 5-2. Field measurements, Sept 2012 to July 2014. ......................................... 139

Figure 5-3. Measured tidal amplitude, flow averages and current ellipses. ............... 140

Figure 5-4. Wave driven flow through lagoon system measured in the channel, Dec

2013. ........................................................................................................................... 141

Figure 5-5. Coherence between forcing mechanisms (tides, waves, and wind) with

measured depth-averaged flow ................................................................................... 142

xvii

Figure 5-6. First empirical orthogonal function (EOF) of measured Eulerian velocity

profiles ........................................................................................................................ 143

Figure 5-7. Bottom roughness results on north forereef (FR9) as a function of wave

height. ......................................................................................................................... 144

Figure 5-8. Cumulative probability of temperature at sites with varying biological

cover compared to offshore, ....................................................................................... 145

Figure 5-9. Thermal dynamics at Channel site (left) and Terrace RT4 site (right) in

Nov 2013. ................................................................................................................... 146

Figure 5-10. Effect of mean advection, nonlinear advection, and surface heating in

driving high mean temperatures at sites with different biological cover. .................. 147


Figure 6-2. Model grid bathymetry and bottom roughness zoomed to atoll .............. 174

Figure 6-3. Selected model validation data for four sites ........................................... 175

Figure 6-4. Regional flow interaction with the atoll, for 24 hour sequence ............... 176

Figure 6-5. Model results average magnitude of significant momentum terms ......... 177

Figure 6-6. Atoll scale model results snapshot of waves, free surface and surface

velocity ....................................................................................................................... 178

Figure 6-7. North-south profile of atoll showing waves, free surface, velocity, and

significant momentum terms during Run 1, 5-Oct-2012 ............................................ 179

Figure 6-8. Lagrangian float tracks ............................................................................ 180

Figure 6-9. Model results average velocity, age and high temperature. ..................... 181

Figure 6-10. Cumulative probability of geomorphic structure and biological cover as a

function of average near bottom velocity, water age, and high diurnal temperature. 182

Figure 6-11. Connectivity between hydrodynamic zones. ......................................... 183

1

Chapter 1

Introduction

To myself I am only a child playing on the beach, while vast oceans of truth lie

undiscovered before me. – Isaac Newton

This dissertation investigates the physical oceanography of coral reef environments,

specifically focusing on waves and mean flows at small and large scales and some

resulting ecological implications. It is hoped it is a small contribution of knowledge

from the vast ocean of what is yet unknown.

1.1 Background and Motivation

Coral reefs provide a wide and varied habitat that supports some of the most diverse

assemblages of living organisms found anywhere on earth [Darwin, 1842]. Reefs are

areas of high productivity because they are efficient at trapping nutrients, zooplankton,

and phytoplankton from surrounding waters [Odum and Odum, 1955; Yahel et al.,

1998; Genin et al., 2009]. The hydrodynamics of coral reefs involve a wide range of

scales of fluid motions, but for reef scales on the order of hundreds to thousands of

meters, surface wave-driven flows often dominate [e.g., Monismith, 2007].

Hydrodynamic flows are the primary mechanism for dispersal and thus connectivity

for small larval species such as corals and are thus of ecological significance [Cowen

and Sponaugle, 2009].

Hydrodynamic processes influence reef growth in several ways [Chappell, 1980].

First, increased water motion from waves or mean flows appears to be beneficial to

reefs through increasing the rates of nutrient uptake on coral reefs [Atkinson and

Bilger, 1992; Thomas and Atkinson, 1997], photosynthetic production and nitrogen

fixation by both coral and algae [Dennison and Barnes, 1988; Carpenter et al., 1991],

and particulate capture by coral [Genin et al., 2009].

2

Second, terrestrial systems appear to generally negatively impact reefs through

increased nutrient loading and sedimentation, among other factors [Buddemeier and

Hopley, 1988; Acevedo et al., 1989; Rogers, 1990; Fortes 2000; Fabricius, 2005]; and

their retention and removal of terrigenous sediment depends on hydrodynamic

processes (flushing rates, dilution, resuspension), hydrology (e.g., accumulation and

slow discharge via groundwater) as well as biological processes [Fabricuis, 2005].

Often, suspended sediment concentrations are highest near the shore, and are much

lower in offshore ocean water [Ogston et al., 2004; Storlazzi et al., 2004; Storlazzi and

Jaffe, 2008].

Third, forces imposed by waves can subject corals to breakage, resulting in trimming

or reconfiguration of the reef [Masselink and Hughes, 2003; Storlazzi et al., 2005].

Finally, reef-building corals have experienced global declines resulting from bleaching

events sparked by pulses of warm-water exposure [Hughes et al., 2003; Hoegh-

Guldberg et al., 2007; Carpenter et al., 2008]. However, corals in naturally warm

environments can have increased resistance to bleaching at high temperatures, and

results show both short-term acclimatory and longer-term adaptive acquisition of

climate resistance [Palumbi et al., 2014].

Surface waves are often the primary forcing mechanism which drives flow on coral

reefs [Monismith, 2007]. At shallow depths, surface waves create oscillatory motion

and bottom stresses, which have important effects on the reef ecosystem such as

modulating substrate type and benthic community structure and morphology [Gove et

al., 2015; Williams et al., 2015]. Wave regime also influences coral growth rates

[Dennison and Barnes, 1988] as well as local bathymetric features such as spur and

groove formation [Rogers et al., 2013; Rogers et al., 2015], and ultimately impacting

the morphology of reef platforms [Chappell, 1980].

Waves serve as a connector between basin-scale winds and reefs through their transfer

of energy [Lowe and Falter, 2015]. Waves often serve as a strong control on the

hydrodynamics and geomorphology of reef systems, and as such, are deserving of

increased attention in a future climate of potential greater storm intensity and sea level

3

rise [Ferrario et al., 2014; Storlazzi et al., 2011]. Despite their importance for

understanding the fate of reefs in a changing climate, we know very little about the

wave activity across many of the most vulnerable atolls and low-lying islands of the

Pacific [Riegl and Dodge, 2008; Woodroffe, 2008].

Classically, waves have been studied through linear wave theory and represented as a

time average over many waves, with real seas approximated as the spectral sum over

many frequencies [Dean and Dalrymple, 1991]. While reef environments are often

characterized by steep slopes and by rough and uneven topography, features that

violate assumptions used to derive linear wave theory, field studies have shown

excellent agreement with many aspects of theory [Monismith et al., 2013].

The hydrodynamics of reef systems are governed primarily by the forcing mechanisms

that drive flow, typically waves, tides, regional flow, wind, and buoyancy effects.

These mechanisms have different importance depending on the scale [Monismith,

2007]. At the island scale, typically kilometers, flow is primarily governed by the

interaction of the island with the large scale regional flow, tides, Coriolis, and

buoyancy effects [Monismith, 2007]. Depending on flow conditions, vortices can be

shed from local bathymetric features such as headlands, or from the island itself

[Aristegui et al., 1994; Wolanski, 1996].

At the reef scale, typically ten to hundreds of meters, breaking waves have long been

recognized as the dominant forcing mechanism on many reefs [Munk and Sargent,

1954; Symonds et al., 1995; Kraines et al., 1998; Lugo-Fernandez et al., 2004;

Callaghan et al., 2006; Lowe et al., 2009]. Conceptually, wave breaking increases the

mean water level in the surf zone, wave setup, establishing a pressure gradient that

drives flow across the reef and into a lagoon [Munk & Sargent, 1954, Young, 1989;

Lowe et al., 2009]. In addition, tides can play a more direct role in driving circulation

in larger and more enclosed lagoons where the channels connecting the lagoon with

the open ocean are relatively narrow, and the constricted exchange of water between

these lagoons and the open ocean can cause significant phase lags between a lagoon

and offshore water levels [e.g., Dumas et al., 2012; Lowe and Falter, 2015]. Wind

4

stresses often play only a minor role in driving the circulation of shallow reefs;

however, wind forcing can be important or even dominant in the circulation of deeper

and more isolated lagoons [Atkinson et al. 1981, Delesalle & Sournia, 1992, Douillet

et al., 2001, Lowe et al., 2009]. Finally, buoyancy forcing can drive reef circulation

through either temperature- or salinity-driven stratification which may also be

important in certain reef systems [Hoeke et al. 2013, Monismith et al. 2006].

The classical dynamical basis by which waves drive flow is through changes to the

waves from physical processes such as shoaling, refraction, dissipation, etc., which

create spatial gradients in radiation stresses and impart a force in the momentum

equation [Longuet-Higgins and Stewart, 1964]. The radiation stress gradient can be

recast as a vortex force in the full three-dimensional momentum equations, first

proposed by Craik and Leibovich [1976] and developed more fully by Uchiyama et al.

[2010]. The vortex force is the interaction of the Stokes drift with flow vorticity, and is

essential in the mechanism for Langmuir circulation.

Corals have irregular, branching morphologies and reef topography varies at scales

ranging from centimeters to kilometers, therefore flow within these systems is

complex [Rosman and Hench, 2011]. In wave and circulation models, variability in

reef geometry occurs at scales smaller than the resolution of the computational grid;

thus, drag due to the small scale geometry must be parameterized. On reefs, bottom

friction is often a significant term in the momentum balance and the primary

dissipation loss; and thus correct parameterization of the bottom drag is essential

[Monismith, 2007].

1.2 Small Scales – Spur and Groove Formations

At scales of ten to one hundred meters, one of the most prominent features of many

forereefs are elevated periodic shore-normal ridges of coral (“spurs”) separated by

shore-normal patches of sediment (“grooves”), generally located offshore of the surf

zone [Storlazzi et al., 2003]. These features, termed “spur-and-groove” (SAG)

formations, have been observed in the Pacific Ocean [Munk and Sargent, 1954; Cloud,

1959; Kan et al., 1997, Storlazzi et al., 2003; Field et al., 2007], the Atlantic Ocean

5

[Shinn et al., 1977, 1981], the Indian Ocean [Weydert, 1979], the Caribbean Sea

[Goreau, 1959; Roberts, 1974; Geister, 1977; Roberts et al., 1980; Blanchon and

Jones, 1995, 1997], the Red Sea [Sneh and Friedman, 1980], and other locations

worldwide. SAG formations are present on fringing reefs, barrier reefs, and atolls.

The alongshore shape of the SAG formations varies from smoothly varying rounded

spurs [Storlazzi et al., 2003], to nearly flat spurs with shallow rectangular channel

grooves [Shinn et al., 1963, Cloud, 1959], or deeply cut rectangular or overhanging

channels often called buttresses [Goreau, 1959]. The scales of SAG formations vary

worldwide, but in general spur height (hspr) is 0.5 m to 10 m, SAG alongshore

wavelength (λSAG) is 5 m to 150 m, the width of the groove (Wgrv) is 1 m to 100 m, and

SAG formations are found in depths (h) from 0 m to 30 m below mean sea level,

[Munk and Sargent, 1954; Roberts, 1974; Blanchon and Jones, 1997; Storlazzi et al.,

2003].

Although the geometric properties of SAG formations are well documented, analysis

of their hydrodynamic function has been limited. On Grand Cayman [Roberts, 1974]

and Bikini Atoll [Munk and Sargent, 1954], SAG formations were shown to be related

to incoming wave energy: high incident wave energy areas have well-developed SAG

formations, whereas those with low incident wave energy have little to no SAG

formations. The spur and groove formations of southern Moloka’i, Hawai’i, have been

well-characterized; and incident surface waves appear to exert a primary control on

the SAG morphology of the reef. [Storlazzi et al., 2003; Storlazzi et al., 2004; and

Storlazzi et al., 2011]. Spurs are oriented orthogonal to the direction of predominant

incoming refracted wave crests, and λSAG is related to wave energy [Munk and

Sargent, 1954; Emry et al., 1949; Weydert, 1979; Sneh and Friedman, 1980; Blanchon

and Jones, 1995]. SAG formations are proposed to induce a cellular circulation

serving to transport debris away from the reef along the groove [Munk and Sargent,

1954; Roberts et al., 1977; Storlazzi et al., 2003]; however, no field or modeling

studies have been carried out to assess this circulation. Although the relationship

between SAG orientation and incoming wave orientation, and the relationship between

6

hspr, λSAG, and incoming wave energy are qualitatively known, the mechanism for

these relationships has not been investigated.

A possible mechanism for circulation is an imbalance between the cross-shore

radiation stress gradient and cross-shore pressure gradient terms in the depth-averaged

momentum equations [Rogers et al., 2013]. For SAG formations, the three-

dimensional velocity structure is unknown but it is hypothesized that due to the

coincident Stokes drift and horizontal vorticity in the mean flow, the vortex force may

be important in driving secondary flow.

Another important mechanism capable of creating secondary flow is from lateral

(normal to the main flow direction) periodic variations of bottom stress first proposed

by Townsend [1976]. The mechanism of instability is the induction by the normal

Reynolds stresses of a pattern of secondary flow, directed from regions of large stress

to ones of small stress; this also induces, by continuity, downward flow over the

regions of high stress and upward flow over those of small stress [Townsend, 1976]. It

is hypothesized this may be an important mechanism influencing the secondary flow

circulation on SAG formations due to the periodic large alongshore differences in

bottom roughness between the spur and groove.

Roberts et al. [1977] present, to our knowledge, the only known field measurements of

currents on SAG bathymetry based on a single dye release in strong alongshore flow

conditions at Grand Cayman, (Cayman Islands). They measured 31 cm/s onshore near-

bed velocity in the groove which carried the dye plume onshore and up and over the

spur before being advected alongshore. Beyond the limited data in Roberts et al.

[1977] which did not resolve the three-dimensional velocity structure, the wave-

induced circulation cells suggested by geologic literature [Munk and Sargent, 1954;

Roberts et al., 1977; Storlazzi et al., 2003] have never been observed in the field.

The primary purpose of Chapter 2 is to examine the hydrodynamics of a typical fore

reef system (seaward of the surf zone) with SAG formations to determine the effects

of the SAG formations on the shoaling waves and circulation. To address this

question, a phase resolving nonlinear Boussinesq model was used with idealized SAG

7

bathymetry and site conditions from Moloka’i, Hawai’i. The results show that SAG

formations together with shoaling waves induce a nearshore Lagrangian circulation

pattern of counter-rotating circulation cells.

The primary purpose of Chapter 3 is to present field observations of wave-induced

circulation cells over SAG formations, including their vertical structure, and discuss

several mechanisms consistent with the observed circulation and model predictions

from Chapter 2. Based on the near-bed observations we discuss why coral growth and

development may be enhanced on the spurs.

1.3 Large Scales – A Pacific Atoll System

At scales of hundreds of meters to kilometers, atolls represent a geologic end member

for reefs, and are a common feature throughout the world’s tropical oceans [Riegl and

Dodge, 2008]. The distinctive geometry of exterior reefs and interior lagoon system

separated by a reef crest and channel system is a unique feature which creates different

hydrodynamic regimes. Previous studies on atolls have focused on portions of the

system [Andréfouët et al., 2006; Andréfouët et al, 2012; Kench, 1998; Dumas et al.,

2012], but to our knowledge no studies exist to examine the atoll system as a whole

using combined field data and three-dimensional wave and hydrodynamic modeling

studies.

Numerous small islands and atolls dot the Central Pacific, including Palmyra Atoll, in

the Northern Line Islands. Due to its location within the trade wind belts, Palmyra was

chosen as a major field site for Walter Munk’s three-month study of wave propagation

across the Pacific [Snodgrass et al., 1966]. To our knowledge, since that time, none of

the Northern Line Islands including Palmyra, have been the location of any published

long-term wave or flow measurements. Due to the lack of on-island measurements,

previous estimates of waves and currents at Palmyra have used results from remote

sensing or models [Riegl and Dodge, 2008; Gove et al., 2015; Williams et al., 2015],

which have not been locally validated. The Northern Line Islands are of significant

ecological interest [Stevenson et al., 2006; Sandin et al., 2008]; and Palmyra in

particular because of its status as a National Wildlife Refuge, is thought to represent a

8

reef with little anthropogenic degradation and abundant calcifiers. Thus, characterizing

the wave and mean flow dynamics in this isolated system with an intact exterior reef

structure and highly frictional environment is of interest.

An important feature of waves on reefs is the fact that the high rugosity of reefs

creates relatively high rates of frictional wave energy flux dissipation [Young, 1998;

Lowe et al., 2005]. Dissipation by features much smaller than the wavelength are

typically approximated using a wave roughness friction factor fw [Kamphuis, 1975;

Grant and Madsen, 1979]. For sediment beds fw is well described in extensive

literature using classical concepts of sand grain roughness [Dean and Dalrymple,

1991]. In contrast, wave friction on reefs can be more complicated and has only been

the subject of a handful of studies. Recent work by Monismith et al. [2015] indicates

wave friction on the structurally complex forereef at Palmyra (𝑓𝑤 ≈ 1.8) is

significantly higher than previously measured on reefs at Kaneohe Bay, Hawaii (𝑓𝑤 ≈

0.3) [Lowe et al., 2005], and John Brewer Reef, Australia (𝑓𝑤 ≈ 0.1) [Nelson, 1996].

Waves on reefs are commonly modeled using a phase-averaged wave action approach,

in which bottom dissipation is parameterized as a function of wave excursion to

bottom roughness scale with a maximum fw of 0.3 [Jonsson, 1966; Madsen et al.,

1988]. For reefs with fw below 0.3, this approach has shown good model skill when

compared with field data [Lowe et al., 2005]. However, this approach has not been

tested in high friction environments. Since the measured fw on Palmyra is well above

0.3 in some locations [Monismith et al., 2015], we anticipate that models using this

friction parameterization [e.g. Simulating WAves in the Nearshore (SWAN)] will

perform poorly and thus require revision.

Wave breaking, another important source of energy dissipation on reefs, occurs where

the depth is on the order of the wave height, and is typically approximated as a

constant breaking fraction [Symonds et al., 1995; Becker et al., 2014]. The breaking of

waves creates a net increase in the water level behind the surf zone, typically a reef

flat or lagoon, an effect that depends on the breaking fraction [Symonds et al., 1995;

Vetter et al 2010]. Given that this setup usually drives flow through the reef system,

9

wave breaking is seen to be an important influence on the hydrodynamics of interior

reefs and lagoons, and thus on residence time and mean currents, both of which are

important for ecological and biogeochemical processes [Baird and Atkinson, 1997;

Atkinson et al., 2001; Falter et al., 2013]. The wave breaking fraction has been well-

studied on sandy beaches and is typically assumed constant at about 0.8 [Battjes and

Jansen, 1978], although it has been shown to be a function of beach slope

[Raubenheimer et al., 1996]. Beyond the studies of Vetter et al. [2010] and Monismith

et al., [2013], the breaking fraction has not been well characterized on reefs for steep

bathymetry with high friction.

The vortex force formalism has recently been implemented in numerical models, and

has shown increased skill over traditional radiation stress methods in predicting

velocity profiles in conditions of coincident waves and currents [Kumar et al., 2012,

2015]. While the vortex force formalism has shown good results in certain field

conditions, it has not yet been implemented on coral reefs with high bottom drag and

steep slopes.

To the best of our knowledge, the wave dynamics of a reef with the high frictional

effects observed on Palmyra Atoll have not been characterized previously.

Additionally, a phase-averaged wave model has not been applied in high frictional

environments with coincident field data to parameterize frictional effects and wave

breaking. Finally, the effect of wave induced bottom stress on geomorphic structure

and biological cover in this environment is of significant ecological interest.

The aim of Chapter 4 is to address this knowledge gap by characterizing the wave

dynamics of Palmyra Atoll through field measurements made from 2011-2014 and

modeling studies. We examine the effects of high friction on the wave dynamics of the

atoll and suggest modifications to the SWAN model to account for the exceptionally

high bottom friction of the reef. We then address the role of waves in shaping the

geomorphic and ecological community structure of Palmyra and address the

extensibility of these findings to other reef systems.

10

While the hydrodynamic forcing on fringing and barrier reef systems has been well

investigated, little work has been done specifically on atoll systems in quantifying the

effect of different forcing mechanisms. No studies on reefs have implemented the

vortex force formalism to predict flows. Additionally, little work has been conducted

on atolls connecting the reef ecology to the hydrodynamics parameters of mean water

age, or connectivity within the atoll system.

The aim of Chapters 5 and 6 to address this knowledge gap by characterizing the

hydrodynamics of Palmyra Atoll through field measurements made from 2011-2014

and modeling studies. We examine the effects of different forcing mechanisms in

driving flow and thermal dynamics, and present results using the vortex force

modeling framework. We then address the role of hydrodynamics in shaping the

geomorphic and ecological community structure of Palmyra and investigate the

interconnectivity of the atoll.

1.4 Dissertation Outline

This dissertation is organized into seven chapters and two supplementary Appendices.

Chapter 1 provides a background and motivation for the research, and outlines the

major research objectives. Chapters 2 through 6 are presented as early and/or

completed drafts for peer-reviewed journals. Accordingly, each of these chapters

contains a separate introduction and review of the relevant literature, experimental

setup and methods, results, discussion, and conclusion section. Chapter 2 examines the

effect of spur and groove formations on a coral reef on waves and resulting

hydrodynamics using modeling simulations. This chapter was published in the Journal

of Geophysical Research – Oceans [Rogers et al., 2013]. Chapter 3 investigates the

hydrodynamics of a spur and groove system with field experiments from Palmyra

Atoll. This chapter was published in the Journal of Geophysical Research – Oceans

[Rogers et al., 2015]. Chapter 4 investigates the wave dynamics of a pacific atoll using

field measurements and modeling. Chapter 5 explores the hydrodynamics of a pacific

atoll system from field observations, focusing on mechanisms for flow, thermal

dynamics and ecological implications. Chapter 6 explores the hydrodynamics of a

11

pacific atoll system based on modeling results, including the mechanisms for flow,

ecological implications and inter-atoll connectivity. Chapters 4, 5, and 6 are prepared

as a draft for future journal submission. Chapter 7 highlights the findings and

discusses avenues for future research. Appendix A contains additional wave data from

Palmyra Atoll and is supporting information for Chapter 4. Appendix B contains

additional validation data from the COAWST modeling results of Palmyra Atoll and is

supporting information for Chapter 5.

12

Chapter 2

Hydrodynamics of Spur and Groove Formations

on a Coral Reef

This chapter is a reproduction of the work published in the Journal of Geophysical

Research – Oceans. As the main author of the work, I made the major contributions to

the research and writing. Co-authors include: Stephen G. Monismith1, Falk

Feddersen2, and Curt D. Storlazzi

3.

1. Environmental Fluid Mechanics Laboratory, Stanford University, 473 Via Ortega,

Stanford, California, 94305, USA

2. Scripps Institution of Oceanography, 9500 Gilman Dr., #0209, La Jolla, California,

92093, USA

3. US Geological Survey, Pacific Coastal and Marine Science Center, 400 Natural

Bridges Dr., Santa Cruz, California, 95060,USA

J. Geophys. Res. Oceans, 118, 3059–3073, doi:10.1002/jgrc.20225.

© 2013. American Geophysical Union. All Rights Reserved. Used with Permission.

13

Abstract

Spur-and-groove formations are found on the fore reefs of many coral reefs

worldwide. Although these formations are primarily present in wave-dominated

environments, their effect on wave-driven hydrodynamics is not well understood. A

two-dimensional, depth-averaged, phase-resolving non-linear Boussinesq model

(funwaveC) was used to model hydrodynamics on a simplified spur-and-groove

system. The modeling results show that the spur-and-groove formations together with

shoaling waves induce a nearshore Lagrangian circulation pattern of counter-rotating

circulation cells. The mechanism driving the modeled flow is an alongshore

imbalance between the pressure gradient and nonlinear wave (NLW) terms in the

momentum balance. Variations in model parameters suggest the strongest factors

affecting circulation include spur-normal waves, increased wave height, weak

alongshore currents, increased spur height, and decreased bottom drag. The modeled

circulation is consistent with a simple scaling analysis based upon the dynamical

balance of the NLW, PG and bottom stress terms. Model results indicate that the spur-

and-groove formations efficiently drive circulation cells when the alongshore spur-

and-groove wavelength allows for the effects of diffraction to create alongshore

differences in wave height without changing the mean wave angle.

14

2.1 Introduction

Coral reefs provide a wide and varied habitat that supports some of the most diverse

assemblages of living organisms found anywhere on earth [Darwin, 1842]. Reefs are

areas of high productivity because they are efficient at trapping nutrients, zooplankton,

and possibly phytoplankton from surrounding waters [Odum and Odum, 1955; Yahel

et al., 1998]. The hydrodynamics of coral reefs involve a wide range of scales of fluid

motions, but for reef scales of order 100 m to 1000 m, surface wave-driven flows often

dominate [e.g., Monismith, 2007].

Hydrodynamic processes can influence coral growth in several ways [Chappell, 1980].

Firstly, waves and mean flows can suspend and transport sediments. This is important

because suspended sediment is generally recognized as an important factor that can

negatively affect coral health [Buddemeier and Hopley, 1988; Acevedo et al., 1989;

Rogers, 1990; Fortes 2000; Fabricius, 2005]. Often, suspended sediment

concentrations are highest along the reef flat, and are much lower in offshore ocean

water [Ogston et al., 2004; Storlazzi et al., 2004; Storlazzi and Jaffe, 2008]. Secondly,

forces imposed by waves can subject corals to high drag forces breaking them,

resulting in trimming or reconfiguration of the reef [Masselink and Hughes, 2003;

Storlazzi et al., 2005]. Thirdly, the rates of nutrient uptake on coral reefs [Atkinson

and Bilger, 1992; Thomas and Atkinson, 1997], photosynthetic production and

nitrogen fixation by both coral and algae [Dennison and Barnes, 1988; Carpenter et

al., 1991], and particulate capture by coral [Genin et al., 2009] increase with

increasing water motion.

One of the most prominent features of fore reefs are elevated periodic shore-normal

ridges of coral (“spurs”) separated by shore-normal patches of sediment (“grooves”),

generally located offshore of the surf zone [Storlazzi et al., 2003]. These features,

termed “spur-and-groove” (SAG) formations, have been observed in the Pacific Ocean

[Munk and Sargent, 1954; Cloud, 1959; Kan et al., 1997, Storlazzi et al., 2003; Field

et al., 2007], the Atlantic Ocean [Shinn et al., 1977, 1981], the Indian Ocean [Weydert,

1979], the Caribbean Sea [Goreau, 1959; Roberts, 1974; Geister, 1977; Roberts et al.,

15

1980; Blanchon and Jones, 1995, 1997], the Red Sea [Sneh and Friedman, 1980], and

other locations worldwide. SAG formations are present on fringing reefs, barrier reefs,

and atolls. Typical SAG formations off the fringing reef of southern Moloka’i,

Hawai’i, are shown in Figure 2-1 and Figure 2-2.

The alongshore shape of the SAG formations varies from smoothly varying rounded

spurs [Storlazzi et al., 2003], to nearly flat spurs with shallow rectangular channel

grooves [Shinn et al., 1963, Cloud, 1959], or deeply cut rectangular or overhanging

channels often called buttresses [Goreau, 1959]. The scales of SAG formations vary

worldwide, but in general spur height (hspr) is of order 0.5 m to 10 m, SAG alongshore

wavelength (λSAG) is of order 5 m to 150 m, the width of the groove (Wgrv) is of order 1

m to 100 m, and SAG formations are found in depths (h) from 0 m to 30 m below

mean sea level, [Munk and Sargent, 1954; Roberts, 1974; Blanchon and Jones, 1997;

Storlazzi et al., 2003].

Although the geometric properties of SAG formations are well documented, analysis

of their hydrodynamic function has been limited. On Grand Cayman [Roberts, 1974]

and Bikini Atoll [Munk and Sargent, 1954], SAG formations were shown to be related

to incoming wave energy: high incident wave energy areas have well-developed SAG

formations, whereas those with low incident wave energy have little to no SAG

formations. The spur and groove formations of southern Moloka’i, Hawai’i, have been

well-characterized; and incident surface waves appear to exert a primary control on

the SAG morphology of the reef. [Storlazzi et al., 2003; Storlazzi et al., 2004; and

Storlazzi et al., 2011]. Spurs are oriented orthogonal to the direction of predominant

incoming refracted wave crests, and λSAG is related to wave energy [Munk and

Sargent, 1954; Emry et al., 1949; Weydert, 1979; Sneh and Friedman, 1980; Blanchon

and Jones, 1995]. SAG formations are proposed to induce a cellular circulation

serving to transport debris away from the reef along the groove [Munk and Sargent,

1954]; however, no field or modeling studies have been carried out to assess this

circulation. Although the relationship between SAG orientation and incoming wave

16

orientation, and the relationship between hspr, λSAG, and incoming wave energy are

qualitatively known, the mechanism for these relationships has not been investigated.

The primary purpose of the present work is to examine the hydrodynamics of a typical

fore reef system (seaward of the surf zone) with SAG formations to determine the

effects of the SAG formations on the shoaling waves and circulation. To address this

question, a phase resolving nonlinear Boussinesq model (Section 2.2) was used with

idealized SAG bathymetry and site conditions from Moloka’i, Hawai’i (Section 2.3).

The model shows that SAG formations induce Lagrangian circulation cells (Section

2.4.1). A mechanism for this circulation in terms of the momentum balance (Section

2.4.2), the role of various hydrodynamic and geometric parameters (Section 2.4.3),

and the effect of spatially variable drag coefficient (Section 2.4.4), are investigated. A

discussion follows on the relative effect of an open back reef on the SAG-induced

circulation (Section 2.5.1), the hydrodynamics of different SAG wavelengths (Section

2.5.2), and the SAG induced-circulation and potential three-dimensional effects

(Section 2.5.3), with conclusions in Section 2.6.

2.2. The Boussinesq Wave and Current Model

A time-dependent Boussinesq wave model, funwaveC, which resolves individual

waves and parameterizes wave breaking, is used to numerically simulate velocities

and sea surface height on the reef, [Feddersen, 2007; Spydell and Feddersen, 2009;

and Feddersen et al. 2011]. The model Boussinesq equations [Nwogu, 1993] are

similar to the nonlinear shallow water equations but include higher order dispersive

terms. The equation for mass (or volume) conservation is:

𝜕𝜂

𝜕𝑡+ ∇ ∙ [(ℎ + 𝜂)𝒖] + ∇ ∙ 𝑀𝑑 = 0, (1)

where η is the instantaneous free surface elevation, t is time, h is the still water depth,

Md is the dispersive term, and u(u,v) is the instantaneous horizontal velocity at the

reference depth zr = -0.531h (approximately equal to the depth averaged velocity for

small kh), where z = 0 at the still water surface. The momentum equation is

17

𝜕𝒖

𝜕𝑡+ 𝒖 ∙ ∇𝒖 = −𝑔∇𝜂 + 𝑭𝑑 + 𝑭𝑏𝑟 −

𝝉𝒃𝜌(ℎ + 𝜂)

− 𝜐𝑏𝑖∇4𝒖 − 𝑭𝒔, (2)

where g is the gravitational constant, Fd are the higher-order dispersive terms, Fbr are

the breaking terms, τb is the instantaneous bottom stress, and υbi is the hyperviscosity

for the biharmonic friction (∇4u) term, and Fs is the surface forcing. The dispersive

terms Md and Fd are given by equations 25a and 25b in Nwogu [1993]. The bottom

stress is parameterized with a quadratic drag law

𝝉𝒃 = 𝜌𝐶𝑑𝒖|𝒖|, (3)

with the nondimensional drag coefficient Cd and density ρ. The effect of wave

breaking on the momentum equations is parameterized as a Newtonian damping where

𝑭𝑏𝑟 =1

(ℎ + 𝜂)∇ ∙ [𝜐𝑏𝑟(ℎ + 𝜂)∇𝒖], (4)

where νbr is the eddy viscosity associated with the breaking waves [Kennedy et al.,

2000]. When 𝜕𝜂 𝜕𝑡⁄ is sufficiently large (i.e., the front face of a steep breaking wave),

νbr becomes non-zero. Additional details of the funwaveC model are described by

[Feddersen, 2007; Spydell and Feddersen, 2009; and Feddersen et al., 2011].

Post processing of the instantaneous model velocity and sea-surface elevation output

were conducted to separate the Eulerian, Lagrangian and Stokes drift velocities [e.g.,

Longuet Higgins 1969; Andrews & McIntyre, 1978]:

𝑼𝑬 = �̅�, (5)

𝑼𝑳 =(ℎ + 𝜂)𝒖̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅

ℎ + 𝜂̅̅ ̅̅ ̅̅ ̅, (6)

𝑼𝑺 = 𝑼𝑳 − 𝑼𝑬, (7)

where, an over bar ( ̅ ) indicates phase (time) averaging, UE(UE,VE) is the mean

Eulerian velocity, UL(UL,VL) is the mean Lagrangian velocity, and US(US,VS) is the

Stokes drift. This form for US is compared to the linear wave theory form in

18

Appendix A. The wave height H can be approximated from the variance of the surface

[e.g., Svendsen, 2007]:

𝐻 = √8(𝜂′2̅̅ ̅̅ ), (8)

where 𝜂 = �̅� + 𝜂′. The mean wave direction θ is given by,

tan 2𝜃 =2𝐶𝑢𝑣

𝐶𝑢𝑢 − 𝐶𝑣𝑣, (9)

where the variance (Cuu, Cvv) and covariance (Cuv), are used with a monochromatic

wave field, and are equivalent to the spectral definition given by Herbers et al.,

[1999], and 𝜃 = 0 corresponds to normally incident waves. Although realistic ocean

waves are random, monochromatic waves are used here for simplicity and to highlight

the linkage of the wave shoaling on SAG bathymetry with the resulting circulation. A

cross-shore Lagrangian circulation velocity Uc is defined as:

𝑈𝑐 = 𝑈𝐿 cos(𝜑), (10)

where φ is the angle between the x and y components of UL. In the presence of a

strong alongshore current, cross-shore circulation is negligible (φ ≈ π/2) and Uc will

approach zero; while in the presence of strong cross-shore current (φ ≈ 0), Uc will

approach UL.

Under steady-state mean current conditions, the phase averaged unsteady (𝜕𝒖/𝜕𝑡) and

dispersive (Fd) terms in the Boussinesq momentum equations (Eq. 2) are effectively

zero. Additionally, the velocity u can be decomposed into mean (�̅�) and wave (u’)

components, essentially a Reynolds decomposition

𝒖 = �̅� + 𝒖′, (11)

and the phase-averaged nonlinear term of Eq. 2 becomes (with the use of Eq. 5):

𝒖 ∙ ∇𝒖̅̅ ̅̅ ̅̅ ̅̅ = (�̅� + 𝒖′) ∙ ∇(�̅� + 𝒖′)̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ = 𝑼𝑬 ∙ ∇𝑼𝑬 + 𝒖′ ∙ ∇𝒖′̅̅ ̅̅ ̅̅ ̅̅ ̅̅ . (12)

19

The phase averaged momentum equation can then be written as:

𝑼𝑬 ∙ ∇𝑼𝑬 + 𝒖′ ∙ ∇𝒖′̅̅ ̅̅ ̅̅ ̅̅ ̅̅ = −𝑔∇�̅� + 𝑭𝑏𝑟̅̅ ̅̅ ̅ −𝝉𝒃

𝜌(ℎ + 𝜂)

̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅− 𝜐𝑏𝑖∇

4𝑼𝑬 − 𝑭𝒔̅̅ ̅. (13)

The effect of the waves on the mean Eulerian velocity is given by the nonlinear wave

term (𝒖′ ∙ ∇𝒖′̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ). This is analogous to a radiation stress gradient on the mean

Lagrangian velocity, but without the effect of the free surface. The phase averaged

bottom stress follows from Eq. 3:

𝝉𝒃̅̅ ̅ = 𝜌𝐶𝑑𝒖|𝒖|̅̅ ̅̅ ̅̅ (14)

In a weak current regime, where 𝑈𝐸/𝜎𝑢 is small, where 𝜎𝑢2 is the total velocity

variance, and away from the surf zone where 𝜂 ≪ ℎ, the bottom stress is proportional

to the mean velocity, 𝝉𝒃̅̅ ̅ ∝ 𝑼𝑬, [Feddersen et al., 2000].

2.3 Model Setup and Conditions

2.3.1 Model SAG Bathymetry

An idealized and configurable SAG bathymetry for use in numerical experiments was

developed based on well-studied SAG formations on the southwestern coast of

Moloka’i, Hawai’i (approximately 21°N, 157°W). High-resolution Scanning

Hydrographic Operational Airborne Lidar Survey (SHOALS) laser-determined

bathymetry data were utilized in combination with previous studies in the area [Field

et al., 2007]. The reef flat, with an approximate 0.3% slope and water depths ranging

from 0.3 to 2.0 m, extends seaward from the shoreline to the reef crest (Figure 2-2, x <

400m) [Storlazzi et al., 2011]. Shore-normal ridge-and-runnel structure characterizes

the outer reef flat. Offshore of the reef crest, from depths of 3 to 30 m lies the fore reef

that is generally characterized by an approximately 7% average slope (βf) and shore-

normal SAG structures covered by highly variable percentages of live coral (Figure

2-2) [Storlazzi et al., 2011]. Note the SAG formations have a roughly coherent λSAG

and cross-shore position, yet with natural variability.

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Analysis of the SHOALS bathymetric data used in Storlazzi et al. [2003] was

conducted, of the fringing reef of southern Moloka’i from Kaunakakai west

approximately 18.5 km to the western extent of the island. Alongshore bathymetric

profiles taken at the 5, 10, 15, and 20 m depth isobaths were analyzed using a zero

crossing method (similar to wave height routines). Of a total 784 measured SAG

formations across all depths, the results show a mean λSAG of 91 m, and a mean hspr of

3.0 m, (Figure 2-3). SAG formations generally had larger λSAG and hspr at deeper

depths, a conclusion also noted in Storlazzi et al. [2003].

A selection of 10 prominent SAG formations from this same area of southern

Moloka’i, from areas with documented active coral growth in Field et al. [2007] was

used to further characterize λSAG, h, Wgrv, and hspr using alongshore and cross-shore

profiles. The geometric shape of the SAG formations was variable, but in general an

absolute value of a cosine function well-represented the planform alongshore

geometry and a skewed Gaussian function well-represented the shore-normal profile

shape. Adopting a coordinate system of x being positive offshore from the coast, and y

being alongshore, the functional form of the idealized depth h(x,y) is given by:

ℎ(𝑥, 𝑦) = ℎ𝑏𝑎𝑠𝑒 − ℎ𝑠𝑝𝑟ℎ𝑥ℎ𝑦 + 𝜂𝑡𝑖𝑑𝑒 , (15)

where hbase(x) is the cross-shore reef form with reef flat and fore reef, ηtide is the tidal

level, and the cross-shore SAG variability hx(x) and alongshore SAG variability hy(y)

are given by

ℎ𝑥 = exp [−(𝑥 − 𝜇)2

2𝜖2], (16)

ℎ𝑦 = max [(1 − 𝛼) |cos (𝜋𝑦

𝜆𝑆𝐴𝐺)| − 𝛼, 0], (17)

where μ is the x location of peak SAG height, ε is a spreading parameter with ε = ε1

for x ≥ μ and ε = ε2 for x < μ to create the skewed Gaussian form, and α is a coefficient

depending on Wgrv and λSAG given by:

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𝛼 =|cos [

𝜋2 (1 +

𝑊𝑔𝑟𝑣𝜆𝑆𝐴𝐺

)]|

1 − |cos [𝜋2 (1 +

𝑊𝑔𝑟𝑣𝜆𝑆𝐴𝐺

)]|

. (18)

These equations were used with the typical SAG parameters of: λSAG = 50 m, hspr = 2.9

m, μ = 550 m, ε1 = 77 m, ε2 = 20 m, Wgrv = 3 m, ηtide = 0, (Figure 2-4). Maximum depth

was limited to 22 m based on kh model constraints. Qualitatively, this form is similar

to SAG formations in Figure 2-2 thus giving some confidence that this idealized

model bathymetry is representative of SAG formations.

2.3.2 Model Parameters and Processing

Bottom roughness for the reef was evaluated using the methods of Lowe et al. [2009],

assuming an average coral size of 14 cm, and thus a drag coefficient Cd = 0.06.

Similar values of drag coefficients for coral reefs are reported in Rosman and Hench

[2011]. The base-configuration model had a spatially uniform Cd = 0.06, with no Cd

variation between spurs and grooves. As grooves often do not have coral but are

instead filled with sediment (see Figure 1, and Storlazzi et al., 2003), some additional

runs were conducted with a spatially variable Cd that was lower (Cd = 0.01) in the

grooves to determine the potential effect of variable bottom roughness (Section 2.4.4).

The Cd = 0.01 used for the sand channels was assumed to have higher roughness than

for flat sand due to likely sand waves and coral debris.

Typical wind and wave conditions on Moloka’i have been summarized in Storlazzi et

al. [2011]. In general, wind speed varies from 0 to 20 m/s, and direction is variable

depending on the season. Average incident wave conditions are also variable and

dependent on the season, but in general from offshore buoy data the average deep-

water wave height varies from 0.5 to 1.5 m, average deep-water wave period varies

from 6 to 14 s, and average observed deep-water wave angle varies from 0 to 80° (0°

corresponds to normally incident waves). The wave angle was assumed to follow

Snell’s law in propagating from deep-water offshore to the model wave maker at 22 m

depth, thus limiting the range of possible θi. Tidal variation for southern Moloka’i is

0.4 m to 1.0 m.

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A grid size of Δx = Δy = 1m was used with bathymetry, as shown in Figure 2-4

Sponge layers were located at 60 m and 800 m offshore (Figure 2-4a). The wave

maker center was located at 752 m (Figure 2-4a), with forcing incident wave height

Hi, period Ti and angle θi. The computational time step was 0.01 s, and instantaneous

values of u, v, η, and νbr were output at 0.2 s intervals. The maximum value of kh in

the model domain was 1.1 for the base-configuration (offshore) and 1.5 for all runs,

and is within the limits suggested by Nwogu [1993]. A biharmonic eddy viscosity νbi

of 0.2 m4s

-1 was used, with wave breaking parameters of: 𝛿𝑏 = 1.2, 𝜂𝑡

(𝐼)= 0.65√𝑔ℎ,

𝜂𝑡(𝐹)

= 0.15√𝑔ℎ, and 𝑇∗ = 5√ℎ/𝑔 as defined by Kennedy et al. [2000]. Surface

forcing due to wind was input to the model assuming a typical drag law in the

momentum equation,

𝑭𝒔 = 𝝉𝒘 (ℎ𝜌)⁄ =𝐶𝑑𝑤𝑼𝟏𝟎|𝑼𝟏𝟎|𝜌𝑎

ℎ𝜌, (19)

where drag Cdw = 0.0015, density of air ρa =1 kg m-3

, and the wind velocity

U10(U10,V10) at a reference level of 10 m.

The model was first run in a base-configuration with model parameters typical of

average conditions on Moloka’i (Table 2-1) to diagnose the SAG-induced circulation.

Subsequently the model parameters were varied (denoted variation models –Table

2-1). The variation models configuration is similar to that described previously.

However, for θi variation, the alongshore length was extended to 700 m to allow the

oblique waves to fit into the alongshore domain with periodic boundary conditions.

Additionally, for βf variation the cross-shore dimension was adjusted so that the wave

maker and sponge layers were the same distance from the SAG formations. For

example, for βf = 2%, the cross-shore domain length was 1692 m, the wave maker was

located at x = 1466 m, and the sponge at x = 1512 m. For variation in Ti, the cross-

shore width of the wave maker was held constant at approximately 60 m. For variation

in λSAG, the alongshore model length was adjusted to model 2λSAG.

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Model run time was 3600 s, with 3240 s of model spin-up and the last 360 s for post

processing analysis. At the model spin-up time, the mean Eulerian currents at all

locations in the model domain had equilibrated. Simulations conducted with variable

alongshore domains that are multiples of λSAG gave identical results, thus an

alongshore domain that spanned 2λSAG was used here.

2.4 Results

2.4.1 Base-Configuration Model Results

This section describes the idealized base-configuration model based on typical

parameters for southern Moloka’i, Hawai’i (Table 2-1). Results are shown for the

model domain from the edge of the onshore sponge layer (x = 60 m) to the onshore

side of the wave maker (x = 720 m). The cross-shore variation of η at the end of the

model run (t = 3600 s), H, θ, and �̅�, for both the spur and groove profiles are shown in

Figure 2-5. As the waves approach the fore reef they steepen and increase in height

from 1.0 m to 1.8 m (trough-to-crest) (Figure 2-5a), and from 1.0 m to 1.3 m (based on

surface variance H) (Figure 2-5b). Within the surf zone (demarked by the vertical

dotted lines), the waves were actively breaking, reducing H (Figure 2-5b). H

continues to decay with onshore propagation along the reef flat. H is slightly higher

along the spur, due to the effects of diffraction and refraction. The alongshore mean θ

is nearly zero along the model domain, but the alongshore maximum and minimum θ

show small oscillations induced along the reef flat due to effects of diffraction and

refraction (Figure 2-5c). �̅� is slightly set down just before wave breaking, is set-up

through the surf zone, and is fairly constant on the reef flat (Figure 2-5d). This cross-

shore reef setup profile is qualitatively in agreement with field observations [e.g.,

Taebi et al., 2011; Monismith, 2007]. There are very small O(1%) differences in �̅�

between the spur and groove profiles which are much smaller than the cross shore

variability in �̅� (i.e. |𝜕�̅� 𝜕𝑦⁄ | ≪ |𝜕�̅� 𝜕𝑥⁄ |).

The cross-shore variation of US, UE, and UL for both spur and groove profiles are

shown in Figure 2-6. Positive velocities are oriented offshore and negative velocities

are oriented onshore. US (computed from Eq. 7) increases from offshore to wave

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breaking, and decreases within the surf zone and on the reef flat. Along the SAG

system, there is a small O(20%) difference in US between the spur and groove profiles.

Model derived US (Eq. 7) and US based on second order wave theory (i.e. a nonlinear

quantity accurate to second order in ak, whose origins are based in linear wave theory,

Eq. A1) are similar in the shoaling fore reef region (Appendix A). Along the majority

of the fore reef (350 < x < 520 m), UE is O(50%) larger over the groove than over a

spur (Figure 2-6b). The circulation Uc is nearly identical to UL in Figure 2-6 (c), due to

weak alongshore currents along the spur and groove profiles in this model. The

predominant two-dimensional UL circulation pattern is onshore flow over the spur and

offshore flow over the groove along the majority of the SAG formation up to the surf

zone (330 m < x < 520 m) (Figure 2-7). Near the offshore end of the spur (x ≈ 550 m),

this UL circulation pattern is reversed, see Section 2.5.3 for further discussion on

potential three-dimensional effects.

From offshore, the magnitude of 𝜏𝑏𝑥̅̅ ̅̅ generally increases up to wave breaking, and

slowly decreases on the reef flat Figure 2-6(d). Along the majority of the SAG

formation up to the surf zone (330 m < x < 520 m), 𝜏𝑏𝑥̅̅ ̅̅ is stronger on the spur than the

groove and is oriented onshore on the spur, while oscillating sign on the groove.

2.4.2 Mechanism for Circulation

Outside the surf zone, assuming normally-incident wave

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