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
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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.
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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
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.
Rob Dunbar
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.
Oliver Fringer
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.
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.
iii
iv
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.
vii
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
ix
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
4.7 Figures and Tables ............................................................................................ 107
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
5.5 Acknowledgements .......................................................................................... 136
5.6 Figures and Tables ............................................................................................ 138
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
xiii
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
6.5 Acknowledgements .......................................................................................... 171
6.6 Figures and Tables ............................................................................................ 173
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
xiv
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
xv
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
xvi
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 5-1. Palmyra Atoll location, site layout and experiment instrumentation ...... 138
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 6-1. Palmyra Atoll location, site layout and experiment instrumentation ...... 173
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
xviii
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