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
© 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. http://creativecommons.org/licenses/by-nc/3.0/us/
This dissertation is online at: http://purl.stanford.edu/fj342cd7577
ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in 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 adequate in 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 adequate in 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 adequate in 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 in University Archives.
iii 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
iv 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.
v
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
vii 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
viii
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
ix
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
x
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
xi
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
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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
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
xiii
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
xiv
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 5-1. 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
xvi
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
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xviii
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].
1
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
2 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
3 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
4
[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
5 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
6 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
7 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 (푓푤 ≈ 1.8) is significantly higher than previously measured on reefs at Kaneohe Bay, Hawaii (푓푤 ≈
0.3) [Lowe et al., 2005], and John Brewer Reef, Australia (푓푤 ≈ 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,
8 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.
9
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
10 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.
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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. Storlazzi3.
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.
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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.
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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.,
14
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
15 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:
휕휂 + ∇ ∙ [(ℎ + 휂)풖] + ∇ ∙ 푀 = 0, (1) 휕푡 푑 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
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휕풖 흉 + 풖 ∙ ∇풖 = −푔∇휂 + 푭 + 푭 − 풃 − 휐 ∇4풖 − 푭 , (2) 휕푡 푑 푏푟 휌(ℎ + 휂) 푏푖 풔 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 4 for the biharmonic friction (∇ u) 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
흉풃 = 휌퐶푑풖|풖|, (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
1 푭 = ∇ ∙ [휐 (ℎ + 휂)∇풖], (4) 푏푟 (ℎ + 휂) 푏푟 where νbr is the eddy viscosity associated with the breaking waves [Kennedy et al.,
2000]. When 휕휂⁄휕푡 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]:
푼푬 = 풖̅, (5)
(̅̅ℎ̅̅+̅̅̅̅휂̅)̅̅풖̅ 푼 = , (6) 푳 ℎ̅̅̅+̅̅̅휂̅
푼푺 = 푼푳 − 푼푬, (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
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Appendix A. The wave height H can be approximated from the variance of the surface [e.g., Svendsen, 2007]:
퐻 = √8(휂̅̅′̅2̅), (8) where 휂 = 휂̅ + 휂′. The mean wave direction θ is given by,
2퐶 tan 2휃 = 푢푣 , (9) 퐶푢푢 − 퐶푣푣 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 휃 = 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:
푈푐 = 푈퐿 cos(휑), (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 (휕풖/휕푡) and dispersive (Fd) terms in the Boussinesq momentum equations (Eq. 2) are effectively zero. Additionally, the velocity u can be decomposed into mean (풖̅) and wave (u’) components, essentially a Reynolds decomposition
풖 = 풖̅ + 풖′, (11) and the phase-averaged nonlinear term of Eq. 2 becomes (with the use of Eq. 5):
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ′ ′ 풖̅̅̅∙̅̅∇̅̅풖̅ = (풖̅ + 풖′) ∙ ∇(풖̅ + 풖′) = 푼푬 ∙ ∇푼푬 + 풖̅̅̅̅∙̅̅∇̅̅풖̅̅. (12)
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The phase averaged momentum equation can then be written as:
̅̅̅̅̅흉̅̅̅̅̅̅ 푼 ∙ ∇푼 + 풖̅̅̅′̅∙̅̅∇̅̅풖̅̅′ = −푔∇휂̅ + 푭̅̅̅̅̅ − 풃 − 휐 ∇4푼 − 푭̅̅̅. (13) 푬 푬 푏푟 휌(ℎ + 휂) 푏푖 푬 풔
The effect of the waves on the mean Eulerian velocity is given by the nonlinear wave term (풖̅̅̅′̅∙̅̅∇̅̅풖̅̅′). 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:
̅̅̅̅̅̅ 흉̅̅풃̅ = 휌퐶푑풖|풖| (14)
2 In a weak current regime, where 푈퐸/휎푢 is small, where 휎푢 is the total velocity variance, and away from the surf zone where 휂 ≪ ℎ, the bottom stress is proportional to the mean velocity, 흉̅̅풃̅ ∝ 푼푬, [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(푥, 푦) = ℎ푏푎푠푒 − ℎ푠푝푟ℎ푥ℎ푦 + 휂푡푖푑푒, (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
−(푥 − 휇)2 ℎ = exp [ ], (16) 푥 2휖2
휋푦 ℎ푦 = max [(1 − 훼) |cos ( )| − 훼, 0], (17) 휆푆퐴퐺 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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휋 푊 |cos [ (1 + 푔푟푣)]| 2 휆 훼 = 푆퐴퐺 . (18) 휋 푊 1 − |cos [ (1 + 푔푟푣)]| 2 휆푆퐴퐺
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
4 -1 (퐼) of 0.2 m s was used, with wave breaking parameters of: 훿푏 = 1.2, 휂푡 = 0.65√푔ℎ, (퐹) ∗ 휂푡 = 0.15√푔ℎ, and 푇 = 5√ℎ/푔 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,
퐶 푼 |푼 |휌 푭 = 흉 ⁄(ℎ휌) = 푑푤 ퟏퟎ ퟏퟎ 푎 , (19) 풔 풘 ℎ휌
-3 where drag Cdw = 0.0015, density of air ρa =1 kg m , 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.
22
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 휂̅, 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). 휂̅ 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 휂̅ between the spur and groove profiles which are much smaller than the cross shore variability in 휂̅ (i.e. |휕휂̅⁄휕푦| ≪ |휕휂̅⁄휕푥|).
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
23 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 휏̅̅푏푥̅̅ 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), 휏̅̅푏푥̅̅ 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 waves, steady-state mean velocities, small alongshore currents, and no surface forcing, the phase-averaged cross-shore (x) momentum equation (Eq. 13) is given by
휕푈 ̅̅̅̅휕̅̅푢̅̅′ ∂휂̅ ̅̅̅̅휏̅̅̅̅̅̅̅ 푈 퐸 + 푢′ = −푔 − 푏푥 , (20) 퐸 휕푥 휕푥 ∂x 휌(ℎ + 휂) where the terms are referenced from left to right as nonlinear advective mean (NLM), nonlinear advective wave (NLW), pressure gradient (PG), and bottom stress (BT). The remaining terms in Eq. 13 are negligible (confirmed through model results). The NLW term is analogous to the radiation stress gradient in wave-averaged models [Longuet-Higgins and Stewart, 1964] (see Appendix A for comparison of the direct radiation stress estimates with those of second-order wave theory).
24
The fore reef (400 m < x < 600 m) has a classic set-down balance [e.g., Bowen, 1969; Kumar et al., 2011] between PG and NLW terms (Figure 2-8a). Closer to where wave-breaking occurs (330 m < x <400 m), BT also becomes important (Figure 2-8a). Within the surf zone (270 m < x < 330 m) and on the reef flat (x < 270 m), the classic surf zone setup (PG-NLW-Fbr) and reef-flat (PG-BT) cross-shore momentum balances were obtained from the model [e.g., Monismith, 2007].
On the SAG formations, (400 m < x < 600 m), the alongshore variation of the cross-shore momentum balance (Eq. 20) shows that the PG and NLW terms do not balance and their difference is largely balanced by BT (Figure 2-9a). The NLM term is very small. The PG and NLW mismatch depends upon alongshore position on the SAG bathymetry (Figure 2-9c). The alongshore variation in NLW is primarily due to the local cross-shore slope, while the alongshore variation in PG is primarily due to the local depth (see Appendix B). On the spurs, the PG and NLW terms are basically in balance as in a classic set-down balance [Bowen, 1969], whereas on the grooves, they are out of balance, and the PG and NLW mismatch is balanced by the BT. The residual forcing accelerates the flow until BT is large enough to balance it which drives an offshore UE. US is very weakly alongshore variable so the alongshore variation in UL, and hence the circulation, is largely due to UE (Figure 2-9b). Note that the fore reef circulation does not depend on wave breaking within the surf zone
(confirmed through separate model runs with smaller Hi that did not have a surf zone).
2.4.3 Effects of Hydrodynamic Conditions and SAG Geometry The base-configuration parameters were varied in the model (denoted variation models,
Table 2-1) to assess their effect on Uc and 휏̅̅푏푥̅̅ at a reference location (xr = 440 m, yr = spur top) as a representative location to assess the hydrodynamics. This location captures the main cross-shore UL circulation cell for a wide range of modeled parameters (e.g., Hi, Cd, hspr, etc.). To evaluate relative changes to Uc and 휏̅̅푏푥̅̅, these are normalized by the base-configuration values at the reference location:
푈̂푐 = 푈푐⁄푈푐푏 , (21)
25
휏̂푏푥 = 휏̅̅푏푥̅̅⁄휏̅̅푏푥푏̅̅̅̅, (22) with Ucb(xr,yr) = -0.0060 m/s and 휏̅̅푏푥푏̅̅̅̅ (xr,yr) = -0.37 Pa, representing the base- configuration. The reference water depth hr is the depth at the reference location h(xr,yr). For the variation in slope βf and cross-shore location μ models, the cross-shore reference location xr was positioned in the same relative cross-shore position on the SAG formation for each geometric configuration (i.e. base-configuration μ = 500 m and xr = 440 m ; for μ = 550 m , xr = 490 m).
The modeled dependence of 푈̂푐 and 휏̂푏푥 on the model variables are shown in Figure 2-10 and Figure 2-11, respectively. From a maximum at a spur-normal wave incidence angle (θi = 0°), 푈̂푐 quickly decreases to nearly zero with oblique incidence (θi = 20°), with 휏̂푏푥 remaining nearly constant (Figure 2-10a and Figure 2-11a). 푈̂푐 and 휏̂푏푥 increase linearly and quadratically, respectively with increasing Hi, (Figure 2-10b and
Figure 2-11b). Increased Ti slightly decreases 푈̂푐 but shows oscillations in 휏̂푏푥 (Figure 2-10c and Figure 2-11c). The effects of refraction/diffraction are likely important here. 푈̂푐and 휏̂푏푥 weakly decrease with increasing ηtide, (Figure 2-10d and Figure
2-11d). 푈̂푐and 휏̂푏푥 show no variation with U10 as expected due to closed cross-shore boundaries (Figure 2-10e and Figure 2-11e). Here, wind and waves are not coupled, so increased wind forcing does not influence wave growth. Increased V10 decreases 푈̂푐 (Figure 2-10f). The circulation cells driven by the SAG bathymetry (Figure 2-7) are essentially overwhelmed by the increasingly stronger alongshore current, which decreases 푈̂푐 proportional to cos(휑) (Eq. 10). This is similar to oblique wave incidence. Increased V10 shows a slight decrease in 휏̂푏푥 (Figure 2-11f).
푈̂푐and 휏̂푏푥 vary inversely with decreasing hspr, (Figure 2-10g and Figure 2-11g).
Similarly, 푈̂푐and 휏̂푏푥 vary inversely with increasing spur cross-shore position μ
(Figure 2-10h and Figure 2-11h). The dependence of 푈̂푐 on λSAG shows small peaks at
80 m and 200m, while 휏̂푏푥 shows a broad, but weak peak centered around 200 m
(Figure 2-10i and Figure 2-11i). The hydrodynamics of different λSAG, will be discussed in more detail in Section 2.5.2. Increased reef Cd shows decreased 푈̂푐 and
26 increased 휏̂푏푥, (Figure 2-10j and Figure 2-11j). Increased Wgrv to λSAG ratio, shows increased 푈̂푐 and nearly constant 휏̂푏푥 (Figure 2-10k and Figure 2-11k). 푈̂푐 and 휏̂푏푥 linearly increase with βf (Figure 2-10l and Figure 2-11l).
The effect of particular model variables (
Table 2-1) on SAG-influenced UE and 휏̅̅푏푥̅̅ on the fore reef can be derived from a simplified scaling (Appendix B) of the dominant cross-shore x momentum balance (Section 2.4.2),
1⁄2 3⁄2 휋√푔훽퐴푆퐻퐴푆 훾푢 훾훽훾퐻 훾휂훾ℎ 훾퐻 푈퐸 ≈ [ 1⁄2 − 1⁄2 ], (23) 16퐶푑√ℎ퐴푆 훾ℎ 훾푢
2 훾 훾 훾2 휌푔훽퐴푆퐻퐴푆 푢 훽 퐻 2 휏̅̅푏푥̅̅ ≈ [ − 훾휂훾ℎ훾퐻], (24) 16ℎ퐴푆 훾ℎ which are Eq. B13 and Eq. B12 in Appendix B, respectively. (_AS) and (_’) denote an alongshore average and alongshore deviation respectively. The local depth factor
훾ℎ = 1 + ℎ′⁄ℎ퐴푆, γβ and γH are similarly defined. γη and γu are kh dependent correction terms defined in Appendix B.
The terms in brackets in Eq. 23 and 24 contain the local alongshore variability in UE and 휏̅̅푏푥̅̅; the dominant factors are local depth (γh) and local slope (γβ). Since the strength of Uc is due to alongshore variations in UE, the nondimensional scaled Uc to first order scales proportionally to 푈퐸⁄푈퐸푏, where UEb is the base condition UE. Thus, for terms that vary independently in Eq. 23 and 24, with normally-incident waves on the spur, and relatively small hspr (훽퐴푆 ≈ 훽푓):
퐻푖 √ℎ푏 퐶푑푏 훽푓 푈̂푐 ∝ 푈̂푐 [ , , , ], (25) 퐻푖푏 √ℎ 퐶푑 훽푓푏
2 퐻푖 ℎ푏 훽푓 휏̂푏푥 ∝ 휏̂푏푥 [ 2 , , ], (26) 퐻푖푏 ℎ 훽푓푏
27 where Hib, hb, Cdb, and βfb are the base condition Hi, h, Cd, and βf respectively. Eq. 25 and 26 capture the first order effects of variables on 푈̂푐 and 휏̂푏푥, but do not capture more complex processes such as wave refraction/diffraction, local alongshore- variability of h, H, and β (Eq. 23 & 24), as well as other second order effects ignored in this scaling (such as correlations between η and u’ in the BT term). The results for
푈̂푐 and 휏̂푏푥 based on the model (Eq. 21 & 22) and scaling approximation (Eq. 25 & 26) are generally similar [Figure 10 (b, d, h, j, l) and Figure 11 (b, d, h, l)], with differences likely due to these more complex processes.
2.4.4 Effect of Spatially Variable Drag Coefficient
The base-configuration model had spatially uniform drag coefficient Cd. However, on typical SAG formations, spurs are covered with hydraulically rough corals (high Cd), while the grooves are often filled with less-rough sediment (low Cd) (example Figure
2-1). The difference in Cd between spur and groove could also have consequences on the net circulation, independent of SAG geometry. To test this idea, a separate model run was performed with SAG formations (hspr = 2.9 m), but with spatially variable Cd between spurs (Cd=0.06) and grooves (Cd=0.01). A Lagrangian circulation pattern similar to the base-configuration (Figure 2-7) was created, but of slightly larger magnitude (≈4%). In another model run assuming no SAG formations (hspr = 0 m) but with spatially variable Cd between spurs (Cd=0.06) and groove (Cd=0.01), a Lagrangian circulation pattern similar to the base-configuration (Figure 2-7) was created, but much smaller, O(10%). Thus, SAG bathymetry is the primary driver of the Lagrangian circulation patterns shown in Figure 2-7, while alongshore differences in Cd between the coral and sediment-filled grooves have a negligible role. Reef-scale
Cd, however, is important to the overall circulation as it sets the magnitude of the circulation (Section 2.4.3).
2.5 Discussion
2.5.1 Relative Effect of Return Flow to SAG-Induced Circulation Many reefs have channels or lagoons onshore of the reef flat with a connection back to the open ocean (open back reef), while other reefs have a closed back reef with no
28 ocean connection [Spalding et al., 2001]. SAG formations are often found on the fore reefs for both open and closed back reef geometries. A net onshore flow over reef flats has been measured in numerous field experiments on reefs with such open ocean back connections [Symonds et al., 1995; Bonneton et al., 2007; Monismith, 2007]. The funwaveC model has a closed onshore boundary at x = 0, which is reasonable for the closed back reef on southern Moloka’i, Hawai’i. A relevant question then is for open back reefs, how strong is the SAG-driven circulation on the fore reef compared to the net onshore flow driven by the open ocean connection?
On the reef flat, neglecting bottom boundary layer wave dissipation, the primary momentum balance on the reef flat is between pressure gradient and bottom stress [e.g., Hearn, 1999],
퐶 푔(ℎ + 휂̅)∇휂̅ = − 푑 푞 |푞 |, (27) (ℎ + 휂̅)2 퐸 퐸 where 푞퐸 = (ℎ + 휂̅)푈퐸 is the mean Eulerian transport. If the overall reef flat depth change is assumed to be small, the reef flat flow can be approximated by,
1 1 푔ℎ 2 휂̅̅푟̅ − 휂̅̅퐿̅ 2 푞퐸 ≅ ( ) ( ) , (28) 퐶푑 퐿푟 where h is the mean depth on the reef, 휂̅̅푟̅ is the setup at the end of breaking, 휂̅̅퐿̅ is the mean surface at the lagoon, and Lr is the length of the reef flat. For the modeled base- configurations, at the end of model domain where all wave energy is dissipated (x = 0 m), the setup 휂̅̅푟̅ is 0.025 m (not shown in Figure 2-5), while the offshore (x = 720 m) setup 휂̅̅퐿̅ is -0.027 m (Figure 2-5). Using Cd = 0.06, varying Lr from 100 m to 2000 m, and varying h from 0.5 m to 1.5 m, the results indicate that the qE has the potential to vary from -0.02 to -0.5 m2s-1 (directed onshore). The funwaveC model results indicate that the SAG formation-induced mean Lagrangian transport [푞퐿 = (ℎ + 휂̅)푈퐿] is -0.06 m2 s-1 and 0.09 m2 s-1 on the spur and groove, respectively (Figure 2-6c).
Although this analysis is qualitative, it indicates that it is possible for transport induced over the reef flat to be of a similar magnitude as the SAG-induced circulation.
29
Under certain conditions, such as strong offshore wave forcing inducing strong transport over the reef flat, the onshore transport on the spur would be strengthened, while the offshore transport on the grooves would be reduced, or potentially reversed. If there is no reef pass or back channel, i.e., a pure fringing reef like at Moloka’i, the SAG induced circulation will likely be the only fore reef exchange. In all cases, at shallow depths the net Lagrangian flow over the spurs is onshore.
2.5.2 SAG Wavelength Waves encountering SAG formations is analogous to the classical problem of waves propagating through a breakwater gap [Penney and Price, 1952]. In the latter case, for a breakwater gap less than one wavelength, the waves in the lee of the breakwater propagate approximately as if from a point source; diffraction is predominant within several wavelengths of the gap and further away, refraction dominated [Penney and Price, 1952; Dean and Dalrymple, 1991]. Although SAG formations are submerged (instead of protruding from the surface), and their alongshore shape is rounded (instead of vertical), wave transformation over SAG formations may have some qualitative similarity to the breakwater gap where λSAG/2 corresponds to an approximate gap scale. Thus for λSAG much less than the surface gravity wavelength, the wave transformation over the SAG formations may be dominated by diffraction, which tends to alongshore “diffuse” wave height, whereas for λSAG much larger than the surface gravity wavelength, refraction dominates the wave transformation. For wavelength larger than the gap scale, the effect of refraction becomes important several wavelengths from the end of the spur or approximately 400 m as shown in the oscillations in θm for x < 350 m (Figure 2-5c).
For the base-configuration, the surface gravity wave wavelength over the SAG formations varied from 115 m (near the front face) to 50 m (near the surf zone). For small λSAG < 100 m, the fore reef H difference between the spur and groove is small and grows slowly with λSAG (Figure 2-12a) and θ is zero at both spurs and grooves
(Figure 2-12b) consistent with diffraction being dominant. At larger λSAG, (>100 m) the spur-groove difference in H grows rapidly and equilibrates at λSAG >200 m (Figure
30
2-12a). Similarly, the alongshore maximum and minimum θ increases similar to the wave height equilibrating to 5° at λSAG >200 m (Figure 2-12b). This large λSAG behavior is consistent with refraction being dominant.
The effect of λSAG variation on the SAG circulation is seen through the x-momentum terms on the groove which has the largest signal (Figure 2-13a). The PG and NLW mismatch balanced by BT (Section 2.4.2) increases with λSAG driving an offshore UE that also generally increases with λSAG (Figure 2-13b). US generally increases with
λSAG, but has opposite sign of UE, resulting in UL (and thus Uc) that has a maximum near λSAG =80 m, with a secondary maximum at larger λSAG (Figure 2-13b). 휏̂푏푥 (Figure 2 2-13c) is a function of H (Section 2.4.3). Thus for λSAG less than 90 m, 휏̂푏푥 remains relatively constant, while for larger λSAG, 휏̂푏푥 increases to a local maxima at λSAG equal to 200 m due to the effects of diffraction/refraction (Figure 2-13c). It appears the maximum circulation and bottom stress occurs when the SAG wavelength allows for the effects of diffraction to create alongshore differences in wave height without changing the mean wave angle, in which case the SAG formations are most efficient at driving Lagrangian circulation cells.
2.5.3 Two-Dimensional SAG Circulation and Potential Three-Dimensional Effects The predominant two-dimensional horizontal Lagrangian circulation pattern induced by the waves is counter-rotating circulation cells. From x ≈ 530 m to the surf zone, transport is onshore over the spur and offshore over the groove; while from the end of the spur to x ≈ 530 m the flow direction is reversed (Figure 2-7). A wide range of hydrodynamic conditions and SAG geometries were modeled (Section 2.4.3). For all modeled conditions except strongly angled waves (high θ) or strong alongshore currents, the waves over SAG formations induce the same basic Lagrangian circulation cells.
The present study focuses on the barotropic (depth-averaged) circulation. The modeled conditions are within the range of values of kh for which the governing equations [Nwogu, 1993] and associated numerical methods [Feddersen, 2007] are
31 valid. Thus the depth-averaged flows in this study should be accurate. Even so, it is reasonable to expect three-dimensional flow effects to become important, especially in the deeper areas of the SAG model domain. For example, while the vertical structure of US is easily calculated, funwave C only calculates depth-averaged mean Eulerian flows (UE) meaning that the vertical structure of the mean Lagrangian flows remain to be determined. Additionally, the model does not represent more complicated three- dimensional flow processes such as separation that might occur. Clearly, these more complicated flow features could have important hydrodynamic and biological implications. Thus, further study of the wave-induced currents over the SAG geometry using fully three-dimensional modeling techniques or field studies would seem warranted.
2.6 Conclusions In summary, a time-dependent Boussinesq wave model, funwaveC that resolves individual waves and parameterizes wave breaking was used to numerically simulate current velocities and sea surface height along SAG formations based on idealized bathymetry from Moloka’i, Hawai’i. The predominant two-dimensional Lagrangian circulation pattern is counter-rotating circulation cells induced by the shoaling wave field over the SAG bathymetry. In shallow depths, transport is directed onshore over the spur and offshore over the groove, while near the end of the spur in deeper water the circulation is reversed. The primary driver of these Lagrangian circulation patterns is the waves interacting with the SAG bathymetry, not alongshore differences in bottom drag due to variation in a drag coefficient. The dominant phase-averaged momentum balance is between pressure gradient and nonlinear wave advection terms on the fore reef. The alongshore variation of the x-momentum terms shows that the pressure gradient and nonlinear wave advection term are not in exact balance and their difference is balanced by bottom stress.
The effect of model variables on circulation Uc and cross-shore average bottom stress,
휏̅̅푏푥̅̅ on the fore reef was approximated using scaling arguments of the dominant cross- shore x momentum balance. The model results show Uc varies approximately
32
-1/2 -1 proportionally to Hi, h , Cd , and βf consistent with a simple scaling. The parameters that created the strongest Uc were spur-normal incident waves (θi = 0°), increased Hi, no alongshore currents, and increased hspr.
The present study focuses on the barotropic (depth-averaged) circulation, but it is reasonable to expect three-dimensional flow effects to become important, especially in the deeper areas of the SAG model domain. Thus, further study of the wave-induced currents over the SAG geometry using fully three-dimensional modeling techniques or field studies are a logical next step.
Many reefs have channels or lagoons onshore of the reef flat with a connection back to the open ocean, while other reefs have a closed back reef with no connection [Spalding et al., 2001]. Using an order of magnitude analysis, results indicate that it is possible for flow induced over the reef flat to be of a similar magnitude as the circulation induced by the SAG formations. Under all onshore reef flows at shallow depths, the net Lagrangian flow over the spurs remains directed onshore.
An investigation was conducted into the hydrodynamic behavior of SAG formations of different SAG wavelength. It appears the maximum circulation and low bottom stress occurs when the SAG wavelength allows for the effects of diffraction to create alongshore differences in wave height without changing the mean wave angle, thus the SAG formations are most efficient at driving Lagrangian circulation cells.
The typical circulation pattern noted in this study likely brings low-sediment, high “food” water from the open ocean up over the corals on the spur; while simultaneously transporting coral debris and sediment from the surf zone and reef flat along the groove sand channels and away from the reef, (assuming low alongshore exchange between spurs and grooves). Average cross-shore bottom shear stress is stronger on the spur than the groove, thus for large wave events that generate shear stress above the capacity of the corals, the corals on the spur would exceed their capacity and break. However, the increased bottom stress on the spur also likely allows for sediment shedding towards the grooves and possibly more nutrient exchange due to increased turbulence on the spur under certain conditions. Thus, while the net effect of
33 bottom shear stress on coral growth remains unclear, increased circulation may favor growth on the coral spur and inhibit coral growth in the groove. Based on variations in the assumed model parameters, some of the strongest factors affecting SAG circulation include spur-normal waves (θ=0°), increased wave height, and increased spur height. If increased circulation is favorable to coral growth, the modeling results are qualitatively consistent with field observations that SAG formations are orthogonal to typical predominant incident wave angle and are largest and most well developed in areas of larger incident waves [Munk and Sargent, 1954; Roberts, 1974; Storlazzi et al., 2003].
2.7 Acknowledgements The authors are grateful to Oliver Fringer for assistance with modeling, Rob Dunbar for constructive discussions, two anonymous reviewers, as well as Robert Arthur, Jamie Dunckley, Joshua Logan, Lianna Samuel, Sean Vitousek, Ryan Walter, Phillip Wolfram, and Simon Wong. This research was made with Government support under and awarded by the U.S. Department of Defense, Office of Naval Research, National Defense Science and Engineering Graduate (NDSEG) Fellowship, 32 CFR 168a. Additional financial support came from the National Science Foundation and the Singapore Stanford Partnership.
2.8 Appendix A – Comparison to Second Order Wave Theory It is of practical interest to compare the Stokes drift and radiation stress obtained from second order wave theory (i.e. nonlinear quantities accurate to second order in ak, whose origins are in linear wave theory) to the results obtained from the nonlinear Boussinesq model. The Stokes drift can be approximated from the wave height, assuming second order wave theory by [e.g., Svendsen, 2007]:
푔푘퐻2 푈 = cos 휃, (A1) 푆 8ℎ휎 where US is in the x direction, H is the wave height, k is the wavenumber, and σ is the wave radian frequency which are related in the usual dispersion relation (휎2 =
34
푔푘 tanh 푘ℎ). The results for Us using H (Eq. A1) and the difference in Eulerian and Lagrangian velocities (Eq. 7) methods have fairly good agreement offshore of wave breaking (x > 350 m), where the waves are weakly nonlinear (Figure 2-14a). Agreement is poor in the surf zone and reef flat due to strong nonlinearity in the wavefield (Figure 2-5a).
3 The radiation stress tensor Sij correct to O(H/L) is given by [e.g. Mei et al, 2005]:
η 휌푔 η ′ ′ ̅̅̅′2̅ ̅̅̅̅′2̅ 푺풊풋 = 휌 ∫ 풖̅̅̅풊̅풖̅̅풋푑푧 + 훿푖푗 { 휂 − 휌 ∫ 푤 푑푧}, (A2) −ℎ 2 −ℎ
′ where L is the wavelength (k/2π) and u’ is based on 푢 = 푢 − 푈퐿. For the nonlinear model dynamics, Sij can be approximated by assuming linear wave theory for the third term for vertical velocity w’ above, and using the instantaneous depth averaged velocities, which can account for weak reflections:
휌푔 2푘ℎ 푺 = 휌(̅̅휂̅̅+̅̅̅ℎ̅̅)̅풖̅̅′̅풖̅̅′ + 훿 휂̅̅̅′2̅ ( ), (A3) 풊풋 풊 풋 푖푗 2 sinh 2푘ℎ
Eq. A3 can be evaluated from the model results assuming the frequency is known. For progressive waves S11 (Sxx) is given by [Longuet-Higgins and Stewart, 1964]:
퐸 2퐶푔 2퐶푔 푆 = 푆 = [ cos2 휃 + ( − 1)], (A4) 11 푥푥 2 퐶 퐶 where E is the energy, C is the phase velocity, Cg is the group velocity, and θ is the wave angle. While the magnitude of Sxx from the two methods is similar (Figure 2-14b), the results from linear wave theory show more local variability in in the cross shore gradient of Sxx (i.e. 휕푆푥푥⁄휕푥) which is the result of small cross-shore oscillations in H (Figure 2-5b).
2.9 Appendix B – Scaling of the Boussinesq Equation Here we present an approximate scaling for the SAG circulation developed to help explain the results in Section 2.4.3. The circulation is given by 푈푐 = (푈퐸 +
푈푆) cos(휑) (Eq. 10). The alongshore variation in UL (and thus Uc) is primarily a
35 result of alongshore variation in UE, not Us (which is nearly alongshore uniform)
(Section 2.4.2). Thus, to first order, the strength of Uc is due to alongshore variations in UE.
On the fore reef, away from the surf zone (휂 ≪ ℎ), but not too deep (kh < 1.5), the primary phase-averaged cross-shore (x) momentum balance (Eq. 20) is between NLW, PG and BT (Figure8a).
̅̅̅̅휕̅̅푢̅̅′ ∂휂̅ ̅̅̅̅휏̅̅̅̅̅̅̅ 푢′ ≈ −푔 − 푏푥 , (B1) 휕푥 ∂x 휌(ℎ + 휂)
Linear wave theory is assumed for normally incident (θ = 0) waves of the form
휂 = (퐻⁄2) cos(휔푡), with wave speed C can be expressed as 퐶 = 휔⁄푘 = √훾푐푔ℎ, where employing the dispersion relation 휔2 = 푔푘 tanh 푘ℎ, a correction to the shallow water wave speed is 훾푐 = tanh(푘ℎ)⁄(푘ℎ). Taking the standard form for linear wave velocity u’ [e.g. Dean and Dalrymple, 1991], is evaluated at 푧푟 = −훼ℎ, with α = 0.531 [Nwogu, 1993]. The wave velocity is then
′ 푢 = 푈0 cos(휔푡), (B2) where 푈0 = (√푔훾푢퐻)⁄(2√ℎ), and the kh dependent wave velocity terms are combined
2푘ℎ 훾 = cosh2[(1 − 훼)푘ℎ] (B3) 푢 sinh(푘ℎ)
For small kh, 훾푢 = 1, for kh = 1, 훾푢 = 0.68. The NLW wave term from Eq. B1 can then be evaluated using Eq. B2,
̅̅̅̅휕푢̅̅̅′ 푔훾 퐻 휕퐻 휕ℎ 푢′ = 푢 [2ℎ − 퐻 ], (B4) 휕푥 16ℎ2 휕푥 휕푥
-3 -1 where 휕훾푢⁄휕푥 is very small [O(2푘ℎ휔훽⁄√푔ℎ) ~ O(10 m ) confirmed through model results and first-order scaling]. The mean set-down for alongshore uniform bathymetry in a classic pressure gradient – radiation stress balance offshore of the surf
36 zone is given by 휂̅ = − (푘퐻2)⁄[8 sinh(2푘ℎ)] [Longuett-Higgins and Stewart, 1962; Bowen, 1969]. This solution is based on alongshore uniform bathymetry, therefore is it most appropriate that k and h are taken as the alongshore average denoted by kAS and hAS respectively. The set-down can then be written as,
2 훾휂퐻 휂̅ = − , (B5) 16ℎ퐴푆 where the kh dependent set-down terms are given by 훾휂 = (2푘퐴푆ℎ퐴푆)⁄sinh(2푘퐴푆ℎ퐴푆).
For small kh, 훾휂 = 1, for kh = 1, 훾휂 = 0.55. The modeled mean set-down was well approximated by Eq. B5. The PG term from Eq. B1 is then evaluated using Eq. B5,
휕휂̅ 푔훾휂퐻 휕퐻 휕ℎ퐴푆 푔 = − 2 [2ℎ퐴푆 − 퐻 ], (B6) 휕푥 16ℎ퐴푆 휕푥 휕푥
-3 -1 where 휕훾휂⁄휕푥 is very small [O(2푘퐴푆ℎ퐴푆휔훽퐴푆⁄√푔ℎ퐴푆) ~ O(10 m ) confirmed through model results and first-order scaling]. Let the local slope 휕ℎ⁄휕푥 and alongshore average slope 휕ℎ퐴푆⁄휕푥 be denoted by β and βAS respectively (note for small hspr, 훽퐴푆 ≈ 훽푓). The sum of NLW and PG terms (Eq. B4 and B6) can be −1 −1 rearranged into two terms, ignoring common terms, one with (휕퐻⁄휕푥)(ℎ − ℎ퐴푆 ) −1 −2 and the second with (퐻⁄2ℎ)(−ℎ 훽 + ℎ⁄ℎ퐴푆 훽퐴푆). The first is much smaller than the second (confirmed through model results) since 휕퐻⁄휕푥 is much smaller than 퐻⁄2ℎ and differences in local vs. alongshore depths are linear in the first term, but squared in the second. The sum of NLW and PG terms becomes, ̅̅̅̅̅̅̅̅′ 2 2 휕푢 ∂휂̅ 푔훾푢훽퐻 훾휂훽퐴푆ℎ 푢′ + 푔 ≈ 2 [−1 + ( 2 )] (B7) 휕푥 ∂x 16ℎ 훾푢훽ℎ퐴푆
The modeled NLW+PG was reasonably approximated by Eq. B7. For purposes of scaling, the BT term in Eq. B1 is approximated by
̅̅̅̅휏̅̅̅̅̅̅̅ 휏̅̅̅̅ 푏푥 ≈ 푏푥 , (B8) 휌(ℎ + 휂) 휌ℎ
37 where it is assumed ℎ ≫ 휂. Combining equations B1, B7, and B8 with some rearrangement yields,
2 2 휌푔훾푢훽퐻 훾휂훽퐴푆ℎ 휏̅̅푏푥̅̅ ≈ [1 − ( 2 )], (B9) 16ℎ 훾푢훽ℎ퐴푆 where the terms in the large bracket above come from the NLW and PG terms respectively. In a weak current, small angle regime, where 푢′ ≫ 푈퐸 is small, for monochromatic, unidirectional waves, the mean bottom stress 휏̅푏 is commonly parameterized by [e.g. Feddersen et al., 2000]:
흉̅̅풃̅ ≈ (4⁄휋)휌퐶푑푈0푼푬, (B10)
Combining equations B9 and B10 with some rearrangement yields,
2 휋√푔훾푢훽퐻 훾휂훽퐴푆ℎ 푈퐸 ≈ [1 − ( 2 )] (B11) 16퐶푑√ℎ 훾푢훽ℎ퐴푆 where the terms in the large bracket above come from the NLW and PG terms respectively.
Separating alongshore variable h, H and β into an alongshore average (_AS) and a local ′ ′ alongshore deviation (_’) yields, ℎ = ℎ퐴푆 + ℎ = ℎ퐴푆훾ℎ, 퐻 = 퐻퐴푆 + 퐻 = 퐻퐴푆훾퐻, and ′ 훽 = 훽퐴푆 + 훽 = 훽퐴푆훾훽, where the local depth, local wave height, and local slope factors are given by 훾ℎ = 1 + ℎ′⁄ℎ퐴푆, 훾퐻 = 1 + 퐻′⁄퐻퐴푆, 훾훽 = 1 + 훽′⁄훽퐴푆 respectively. Substituting these expressions into Eq. B9 and Eq. B11 and rearranging so that all alongshore variability (i.e., γh, γH, γβ, γu) is in the parentheses yields,
2 훾 훾 훾2 휌푔훽퐴푆퐻퐴푆 푢 훽 퐻 2 휏̅̅푏푥̅̅ ≈ [ − 훾휂훾ℎ훾퐻], (B12) 16ℎ퐴푆 훾ℎ
1/2 3/2 휋√푔훽퐴푆퐻퐴푆 훾푢 훾훽훾퐻 훾휂훾ℎ 훾퐻 푈퐸 ≈ [ 1/2 − 1/2 ]. (B13) 16퐶푑√ℎ퐴푆 훾ℎ 훾푢
38
The first term in brackets of Eq. B13 originates from the NLW term and is denoted NLW* (note change of sign in NLW from Figure 8a), while the second originates from the PG term and is denoted PG*. The alongshore variability in NLW* is most affected by γβ while γu, γH and γh have a minor effect (Figure 2-15a). The alongshore variability in PG* is most affected by γh with little to no effect from γη, γH and γu
(Figure 2-15b). Thus, the alongshore variability in UE (and thus Uc) is primarily the result of a mismatch between the local slope coefficient γβ and the local depth coefficient γh to the 3/2 power. The alongshore variation in UE shows good agreement between model results and Eq. B13 (Figure 2-15c).
2 Equation B13 is highly approximate to O(H/h) , but explains to first order the Uc dependence on model parameters, as discussed in Section 2.4.3. Note that if the 2 bathymetry is alongshore-uniform (훾휂 = cosh [(1 − 훼)푘ℎ] 훾푢, 훾ℎ = 훾퐻 = 훾훽 = 1) and Eq. B13 will predict alongshore-uniform UE > 0 (directed offshore); in this case, second order effects ignored in this scaling would become important.
39
2.10 Figures and Tables
Figure 2-1. Underwater image of a typical SAG formation off southern Moloka’i, Hawai’i. For scale, the height between the sand-floored groove and the top of the coral spurs is approximately 1.5 m, the width of the groove is approximately 2 m. Wave-generated symmetrical ripples cover the sand bed; view is seaward.
40
Figure 2-2. Morphology of characteristic SAG formations off southern Moloka’i, Hawai’i. Contour lines are 2 m spacing. Location is approximately 21°05’N, 157°10’W.
Figure 2-3. Distribution of SAG wavelength λSAG and spur height hspr of SAG formations at 5, 10, 15, and 20 m depth alongshore bathymetric profiles from southern Moloka’i, Hawai’i.
41
Figure 2-4. Idealized spur and groove model domain. (a) x-z profile, with spur (blue solid), groove (green dash), wave maker (magenta dash-dot), and sponge layers (black dash), (b) x-y contours. Note difference in cross-shore scale.
42
Figure 2-5. Model surface results for base-configuration. (a) instantaneous surface η(t=3600 s), (b) wave height H, (c) mean alongshore wave angle θ (red-solid) and max/min alongshore θ (red-dash), (d) mean setup 휂̅, and (e) cross-shore depth h. Alongshore location for (a), (b), (d), and (e) at spur y = 50 m (blue solid), groove y = 75 m (green dash). Vertical lines (magenta dash), indicate surf zone extent.
43
Figure 2-6. Model velocity and bed shear results for base-configuration. (a) cross-shore Stokes drift Us, (b) Eulerian velocity UE, (c) Lagrangian velocity UL, (d) average cross-shore bed shear stress 휏̅̅푏푥̅̅, and (e) cross-shore depth profile at spur y = 50 m (blue solid) and groove y = 75 m (green dash). Vertical lines (magenta dash), indicate surf zone extent.
44
Figure 2-7. Lagrangian velocity UL vectors from base-configuration and 1 m bathymetric contours zoomed to SAG formations. Maximum velocity vector scale is 0.05m/s, and vertical dashed blue line represents the offshore edge of the surf zone.
Figure 2-8. Phase averaged cross-shore momentum balance for base-configuration at top of spur y = 50 m. (a) Cross-shore phase-averaged x-momentum significant terms, NLM, NLW, PG, and BT terms, residual error is small, (b) cross-shore depth profile for spur (blue solid) and groove (green dash).
45
Figure 2-9. Alongshore variation of x-momentum terms and velocity for base-configuration at x = 440 m. (a) Alongshore phase-averaged x-momentum significant terms, NLM, NLW + PG, and BT terms, residual error is small, (b) UE, US and UL velocities, and (c) depth h.
46
Figure 2-10. Variation of model parameters and their effect on normalized circulation 푈̂푐 for model results (red solid) and scaling approximation (Eq. 25) (black dash) at xr = 440 m, yr = 50 m (spur) as a function of model variables (a) incident wave angle θi, (b) incident wave height Hi, (c) incident wave period Ti, (d) depth as a function of tide level ηtide, (e) cross-shore wind U10, (f) alongshore wind V10,(g) spur height hspr, (h) depth as a function of cross-shore location μ, (i) SAG wavelength λSAG, (j) drag coefficient Cd, (k) fraction groove width Wgrv/ λSAG, and (l) fore reef slope βf. Scaling approximation only shown on (b, d, h, j, and l), blue circle indicates base-configuration.
47
Figure 2-11. Variation of model parameters and their effect on normalized average cross- shore bottom stress 휏̂푏푥 for model results (red solid) and scaling approximation (Eq. 26) (black dash) at xr = 440 m, yr = 50 m (spur) as a function of model variables (a) incident wave angle θ, (b) incident wave height Hi (note larger scale), (c) incident wave period Ti, (d) depth as a function of tide level ηtide, (e) cross-shore wind U10, and (f) alongshore wind V10,(g) spur height hspr, (h) depth as a function of cross-shore location μ, (i) SAG wavelength λSAG, (j) drag coefficient Cd, (k) fraction groove width Wgrv/ λSAG, and (l) fore reef slope βf. Scaling approximation only shown on (b, d, h, and l); blue circle indicates base-configuration.
48
Figure 2-12. Variation of wave height H and wave angle θ with SAG wavelength λSAG at x = 440 m. (a) alongshore mean H (solid) and max/min H (dash), (b) alongshore mean θ (solid) and alongshore max/min θ (dash).
49
Figure 2-13. Variation of x-momentum terms, velocity, circulation and average bottom shear with SAG wavelength λSAG at x = 440 m. (a) phase-averaged x-momentum significant terms NLM, NLW + PG, and BT; residual error is small, at groove y = 1.5λSAG (b) UE, US and UL velocities, at groove y = 1.5λSAG and (c) normalized circulation 푈̂푐 (red) and normalized bottom shear stress 휏̂푏푥 (black), at spur y = λSAG.
50
Figure 2-14. Comparison of cross-shore Stokes drift US and radiation stress Sxx for base- configuration at top of spur y = 50 m; (a) US obtained directly from the model Eq. 7 (blue solid), and using linear wave theory Eq. A1 (blue dash); and (b) Sxx from instantaneous velocities and η Eq. A3 (blue solid), and linear wave theory Eq. A4 (blue dash). Vertical lines (magenta dash) indicate surf zone.
51
Figure 2-15. Comparison of alongshore contribution to NLW* and PG* terms and results for UE from model and scaling at x = 440 m; (a) contribution to NLW* term in Eq. B13 (black solid) from γβ (magenta dash), γH (blue dash), γu (green dash), and γh (red dash); (b) contribution to PG* term in Eq. B13 (black solid) from γη (cyan dash), γH (blue dash), γu (green dash), and γh (red dash); (c) comparison of UE from model (blue solid) and scaling approximation (Eq. B13) (black dash); and (d) depth h.
Table 2-1. Parameters used for base-configuration model, and range of parameters for variation models.
Model Base-Configuration Variation Models Variation Models Variable Model Min Max θi (°) 0 0 32.5 Hi (m) 1 0.25 2.5 Ti (s) 10 8 22 ηtide (m) 0 -0.9 0.9 U10 (m/s) 0 -30 30 V10 (m/s) 0 0 30 hspr (m) 2.9 0 8 μ (m) 550 500 600 λSAG (m) 50 20 240 Cd coral 0.06 0.01 0.12 Wgrv /λSAG 0.06 0 0.82 βf 0.07 0.02 0.13
52
Chapter 3 Field Observations of Wave-Driven Circulation over 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, Robert B. Dunbar2, and David Koweek2.
1. Environmental Fluid Mechanics Laboratory, Stanford University, 473 Via Ortega, Stanford, California, 94305, USA
2. Department of Environmental Earth System Science, Stanford University, Stanford California, 94305, USA
J. Geophys. Res. Oceans, 120, 145–160, doi:10.1002/2014JC010464. © 2015. American Geophysical Union. All Rights Reserved. Used with Permission.
53
Key Points Waves over spur and groove formations induce Lagrangian circulation cells
Horizontal flow is offshore over spurs and weak onshore over grooves
Downward flow over spur is likely due to variation in alongshore bottom stress
Abstract Spur and groove (SAG) formations are found on the forereefs of many coral reefs worldwide. Modeling results have shown that SAG formations together with shoaling waves induce a nearshore Lagrangian circulation pattern of counter-rotating circulation cells, but these have never been observed in the field. 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. Favorable forcing conditions must be maintained on the order of one hour to accelerate and develop the SAG circulation cells. The primary cross- and alongshore depth-averaged momentum balances were between the pressure gradient, radiation stress gradient and nonlinear convective terms, and the bottom drag was similar to values found on other reefs. The vertical structure of these circulation cells was previously unknown and the results show a complex horizontal offshore Lagrangian flow over the spurs near the surface driven by alongshore variability in radiation stress gradients. 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.
54
3.1 Introduction One of the most prominent features of coral reefs worldwide are elevated periodic shore-normal ridges of coral (“spurs”) separated by shore-normal patches of sediment (“grooves”), generally located on the forereef offshore of the surf zone [Storlazzi et al., 2003]. These features, termed “spur and groove” (SAG) formations, have been observed on fringing reefs, barrier reefs, and atolls, and vary in their size and shape [Rogers et al., 2013].
Hydrodynamic processes influence coral growth in several ways [Chappell, 1980]. First, waves and mean flows can suspend and transport sediments and reef debris. 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]. Second, 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]. Third, the rates of nutrient uptake on coral reefs [Atkinson and Bilger, 1992; Thomas and Atkinson, 1997, Atkinson et al., 2001], photosynthetic production and calcification by coral [Dennison and Barnes, 1988] and particulate capture by reef organisms [Genin et al., 2009] increase with increasing water motion.
Although the geometric properties of SAG formations are well documented [Munk and Sargent, 1954; Roberts, 1974; Blanchon and Jones, 1997; Storlazzi et al., 2003], analysis of their hydrodynamic function has been limited [Rogers et al., 2013]. Using a depth-averaged, phase-resolving model (funwaveC), Rogers et al. [2013] showed that SAG formations together with shoaling waves induce a nearshore Lagrangian circulation pattern of counter-rotating circulation cells, confirming suggestions from geologic literature [Munk and Sargent, 1954; Roberts et al., 1977; Storlazzi et al., 2003]. This model also revealed that this type of circulation is enhanced by spur- normal waves, increased wave height, weak alongshore currents, increased spur height, and decreased bottom drag.
55
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]. Specifically for SAG formations, the 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]. The alongshore variation in radiation stress gradient is primarily due to the local cross-shore slope, whereas the alongshore variation in mean pressure gradient is primarily due to the local depth; the residual forcing from this imbalance accelerates the flow until the bottom stress or nonlinear convection is large enough to balance it, resulting in alongshore variable flow and the counter-rotating circulation cells [Rogers et al., 2013].
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 refined by others including Kumar et al. [2012]. The vortex force is the interaction of the Stokes drift with flow vorticity, and is essential in the mechanism for Langmuir circulation. 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.
56
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 predicted by Rogers et al. [2013] have never been observed in the field.
Here we present field observations of wave-induced circulation cells over SAG formations, including their vertical structure, and discuss several mechanisms consistent with the observed circulation. Based on the near-bed observations we discuss why coral growth and development may be enhanced on the spurs. To address these questions, two separate field studies were conducted on Palmyra Atoll in the Central Pacific (Section 3.2.1) and data was processed per accepted methods (Section 3.2.2). Results show wave-induced circulation and their vertical velocity profiles (Section 3.3.1), momentum balances (Section 3.3.2), bottom roughness characteristics (Section 3.3.3), and near-bed velocity and bottom stress (Section 3.3.4). A discussion on waves and circulation (Section 3.4.1), mechanism for circulation (Section 3.4.2), implications for coral health (Section 3.4.3), and conclusions (Section 3.5) follow.
3.2 Methods
3.2.1 Field Experiment Palmyra Atoll (5° 52’N, 162° 05’W) is part of the Northern Line Islands of the central equatorial Pacific (Figure 3-1a) and largely because of the absence of acute anthropogenic stress on the ecosystems, its reefs contain abundant calcifiers, namely hard corals and crustose coralline algae [Williams et al., 2013] and high growth rates [Koweek et al., 2014]. Two separate field experiments were conducted on the atoll to characterize SAG circulation cells (Figure 3-1b). The first experiment, hereafter referred to as SFR12, was conducted from September 16-26th, 2012, on the south forereef in approximately 8-10 m depth (Figure 3-1b, d). The SAG formations at
57
SFR12 had approximately 1.8 m high spurs, 15 m average wavelength, 2 m wide grooves, and the spurs were oriented with an approximately 175° heading offshore, approximately parallel to the larger scale reef slope, which was approximately 15% (Figure 3-1d). The second experiment, hereafter referred to as NFR13, was conducted from September 4-9th, 2013, on the northwest corner of the atoll in approximately 8-11 m depth (Figure 3-1b,c). The SAG formations at NRF13 had approximately 1.9 m high spurs, 14 m average wavelength, 1.4 m wide grooves, and the spurs were oriented with an approximately 0° heading offshore, approximately parallel to the larger scale reef slope, which was approximately 7% (Figure 3-2c). Live coral and coralline algae cover was high on the spurs (Figure 3-2a,d), whereas the bottom of the grooves were typically covered with reef debris, sediment, and fewer live coral colonies (Figure 3-2b,e).
The NFR13 instrument array consisted of two cross-shore transects of instrumentation covering two spurs and two grooves (Figure 3-1c). Transect A consisted of four sites (A1-A4) measuring velocity and pressure, and transect B consisted of four sites (B1- B4) measuring pressure located approximately 15 m offshore from Transect A (Table 3-1). A deep forereef mooring, Site C1, measuring velocity and pressure was located approximately 115 m downslope from transect A in approximately 20 m water depth. The SFR12 instrument array consisted of one cross-shore transect (Transect M) covering one spur and one groove (Figure 3-1d), consisting of two sites (M1 and M2) measuring velocity and pressure (Table 3-1). A weather station was located on the atoll which measured wind speed and direction (RM Young 3002 sensor), and other meteorological variables (Figure 3-1b).
3.2.2 Data Analysis Instantaneous measured velocity data u(u,v,w) were rotated to local bathymetry coordinates of cross-shore (x) and alongshore (y) directions based on the orientation of spurs. The cross-shore coordinate at NFR13 corresponded to a 0° compass heading, directed offshore (Figure 3-1c), and cross-shore coordinate at SFR12 corresponded to a 175° compass heading, directed offshore (Figure 3-1d). The vertical (z) coordinate is
58 taken as upwards from mean sea level (MSL). Time averaging ( ̅ ) was computed over 15 minute intervals for mean velocity 풖̅, average free surface deviation from MSL, 휁,̅ and wave statistics. Only velocity data from the ADCP/APDs between selected depth ranges were used for analysis (Table 3-1). The ADV data was combined with the ADCP data at sites A2 and A3 to create velocity profiles.
Linear wave theory has been shown to be accurate on rough reefs with steep and complex geometry, and large wave amplitudes nearly equal to the mean depth [Monismith et al., 2013]. The use of spectral wave definitions (i.e. the superposition many sine waves over many frequencies) is common practice, which does not assume a constant wave form and there is some flexibility to include nonlinear wave forms in the formulation [Dean and Dalrymple, 1991; Sheremet et al., 2011]. Wave analysis was conducted on pressure p and velocity data by dividing each 15 minute segment into 33 sections of equal length, each with 75% overlap, applying a Hanning window to the segments and computing spectra 푆(푓) of frequency f. The rms wave height Hrms
1/5 1/2 was calculated by 퐻 = [ 8푆 푑푓] , where 푆 (푓) is the power spectral 푟푚푠 ∫1/25 휁휁 휁휁 2 density of the free surface ζ, calculated from, 푆휁휁 = 푆푝푝[cosh 푘ℎ⁄(휌푔 cosh 푘ℎ푔)] , and 푆푝푝(푓) is the power spectral density of p, ρ is density, g is gravitational constant, h is depth of the bottom below MSL, hg is the height of the pressure gauge above the bottom, and wavenumber k is related by the dispersion relation 휎2 = 푔푘 tanh 푘ℎ, and radian frequency 휎 = 2휋푓 = 2휋⁄푇 [Dean and Dalrymple, 1991]. Mean wave period,
Tm was calculated based on the first spectral moment of 푆휁휁(푓). Mean wave direction
θm was computed from the first spectral moment of 휃(푓) calculated by, tan 2휃(푓) =
푆푢푣(푓)⁄[푆푢푢(푓) − 푆푣푣(푓)], where Suu and Svv are the autospectra and Suv is the cospectra of u and v from the ADVs and near bed ADCP/ADPs bins [Herbers et al., 1999].
The mean Lagrangian velocity 풖̅̅푳̅ was calculated by [Andrews and McIntyre,
1978], 풖̅̅푳̅ = 풖̅̅̅푬̅ + 풖̅̅푺̅ , where 풖̅̅̅푬̅ = 풖̅ is the mean measured Eulerian velocity and Stokes drift,
59
휎퐻2 cosh 2푘(ℎ + 푧) 풖̅̅̅ = 푟푚푠 풌, (1) 푺 8 sinh2 푘ℎ was computed spectrally and integrated from 1/5 to 1/25 Hz, and k is the magnitude of wavenumber vector 풌 [Dean and Dalrymple, 1991]. The Lagrangian depth-averaged mean velocity 푼푳(푈퐿, 푉퐿, 푊퐿) was calculated by combining available data at a given location (ADV/ADCP/ADP), assuming 풖̅̅̅푬̅ = 0 at the bottom, linearly interpolating in z and taking the average. To quantify the strength of circulation we define a cross- shore circulation velocity 푈푐 as [Rogers et al., 2013],
푈푐 = (푈퐿 − 〈푈퐿〉) cos 휑, (2) where 〈푈퐿〉 denotes a spatial average in the alongshore direction to remove the average cross-shore reef flow, and φ is the angle between x and y components of 푼푳. In the presence of strong alongshore current (휑 ≈ 휋⁄2) 푈푐 will approach zero; while in the presence of strong cross-shore current (휑 ≈ 0) 푈푐 will approach 푈퐿 − 〈푈퐿〉 .
Error in the velocity measurements was taken as the measured error from the redundant beam of the ADCPs, 1% of the measurement ± 0.5 cm/s for the ADPs, and 0.5% of the measurement ± 0.1 cm/s for the ADVs. The error was propagated through the calculations per standard error analysis methods [e.g. Emery and Thomson, 2004].
The approximate depth integrated momentum equations for horizontal flow are given by [e.g. Mei et al., 2005],
휕푼 1 푳 + 푼 ∙ ∇푼 = −푔∇휁̅ − [∇ ∙ 푺 + 흉̅̅̅ − 흉̅ ], (3) 휕푡 푳 푳 휌(휁̅ + ℎ) 풃 풔 where S is the radiation stress tensor, 흉̅̅풃̅ is the mean bottom stress, and 흉̅풔 is the mean surface stress. The terms in Eq. 3 will be referred to from left to right as unsteady (US), convective nonlinear (NL), mean pressure gradient (PG), radiation stress gradient (RS), bottom stress (BT), and surface stress (ST). Equation 3 was evaluated for the NFR13 experiment in the alongshore direction at the midpoint between stations A1-A2 and A2-A3, but was not evaluated between A3-A4 due to variable local
60 bathymetry and larger distance between sites. Time derivatives were taken using the leapfrog method, while alongshore spatial derivatives used a central difference scheme, and cross-shore spatial derivatives used a forward Euler scheme. A leapfrog scheme was used to compute the alongshore PG due to lack of highly accurate p (RBR) at A2.
1 For linear waves 푆 = 푆 [푛(sin2 휃 + 1) − ], and 푆 = 푆 = 푦푦 휁휁 2 푥푦 푦푥 1 푆 푛 sin 2휃, were computed spectrally and integrated over 1/5 to 1/25 Hz, where 2 휁휁
푛 = 퐶푔⁄퐶, with group velocity Cg and phase speed C [Longet-Higgins and Stewart,
1964]. To evaluate 휕푆푥푦⁄휕푥, 휃푚 at the B stations was assumed equal to the corresponding A station.
The measured velocity can be expressed as 풖 = 풖̅ + 풖′ + 풖̃, where 풖′ is the turbulent velocity and 풖̃ is the wave-induced velocity. Assuming motions that are correlated with the free surface are due to waves, and those that do not are due to ∗ ̅̅′̅̅̅̅′ turbulence, 푆푢̃푤̃ = 푆푢휁푆푤휁⁄푆휁휁, and 푆푢′푤′ = 푆푢푤 − 푆푢̃푤̃ , and 푢 푤 = ∫ 푆푢′푤′푑푓 [Benilov and Filyushkin, 1970; Benilov et al., 1973]. The process is similar for the other components of the Reynolds stress tensor (i.e. 푢̅̅̅′̅푢̅̅′) and wave motions (i.e. 푢̅̃̅̅푢̃̅). 1 The total kinetic energy due to turbulence (TKE) is 푇퐾퐸 = (푢̅̅̅′2̅ + 푣̅̅̅′2̅ + 푤̅̅̅̅′2̅). 2
Mean bottom stress, 흉̅̅풃̅, was computed from the turbulent Reynolds stress, which are assumed constant within the inertial sublayer, [e.g. Reidenbach et al., 2006],
′ ′ 흉̅̅풃̅ = −휌풖̅̅̅푤̅̅̅ (4) using the measured turbulent velocities (풖′) from the ADVs. 흉̅̅풃̅ used in Eq. 3 at the midpoint between A1-A2 was assumed equal to the measured 흉̅̅풃̅ at A2, and at the midpoint A2-A3 was taken as the average of 흉̅̅풃̅ at A2 and A3. The surface stress was approximated by a typical quadratic drag law, 흉̅풔 = 휌푎퐶퐷푎풖ퟏퟎ|풖ퟏퟎ|, where air density -3 휌푎 = 1 kgm , wind drag coefficient 퐶퐷푎 = 0.0015, and wind velocity u10 [Smith, 1988].
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A common bottom stress parameterization is given by [e.g. Grant and Madsen, 1979, Feddersen et al., 2000],
̅̅̅̅̅̅ 흉̅̅풃̅ = 휌푪푫풖|풖| (5) where u is evaluated near the bed but above the bottom boundary layer, and CD is a nondimensional drag coefficient which may depend on the flow environment, height above the bed and bottom roughness. Combining Eqns. (4) and (5) gives,
−풖̅̅̅′̅푤̅̅′ 푪푫 = (6) 풖̅̅̅|̅풖̅̅| where in environments with low wave and turbulence energy, the denominator is often simplified to 푢̅|풖̅|, and 푢̅ is either the depth averaged or near bed velocity, see Rosman and Hench [2011] for a complete discussion.
3.3 Results During the NFR13 experiment, the tidal elevation 휁 ̅ varied by 0.8 m (Figure 3-3a). Wave energy was characterized by two main events, the first on 4-5 September with wide frequency spread, and average Hrms of 0.6 m at 9 s period, and the second on 8
September with narrower frequency spread and average Hrms of 0.5 m at 11 s period
(Figure 3-3b,c,d). Hrms was slightly higher at deep forereef station C1, but showed only very slight difference between all stations at A and B (Figure 3-3c). Tm increased with propagation onshore (from C1 to A), but showed very little alongshore variation (Figure 3-3d). Shorter period waves were directed on average at -θ, while the longer period swell were directed at +θ (Figure 3-3e), and average wave direction was nearly incident for the 4-5 September wave event, while angled at about 20° on 8 September.
θm was directed more normally incident with onshore propagation consistent with
Snell’s law, and θm varied by approximately 10° within sites A1-A4 (Figure 3-3f), essentially the accuracy of the instrument reference frame. The wind speed was directed generally in the positive x and y directions (approximately to the NW) and its magnitude varied from 0 to 8 m/s (Figure 3-3g).
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During the SFR12 experiment 휁 ̅ varied by 0.8 m, (Figure 3-4a). The waves were characterized by Hrms between 0.6 and 1.3 m (Figure 3-4b), Tm of 8 to 13 s (Figure
3-4c), and θm of about zero (incident to cross-shore coordinate x) (Figure 3-4d). Site
M1 (spur) had slightly higher Hrms, and lower Tm compared to Site M2 (groove), but
θm was nearly the same.
3.3.1 Circulation and Vertical Structure
For the NFR13 experiment, the cross-shore depth averaged Lagrangian velocity UL varied from -12 to 12 cm/s and was generally directed offshore (+) with velocities on the deep forereef (C1) showing similar trends (Figure 3-5a). UL was generally more positive at A2 (spur) than at A3 (groove) especially on 4, 5 and 7 September. The deep forereef (C1) VL varied from -50 to 30 cm/s and was generally positive (to the west)
(Figure 3-5b). At the SAG site A1-A4, VL was smaller in magnitude but followed a similar trend. Two larger-scale anomalous flow events occurred on 5 and 8 September with strong flow at C1 directed onshore and negative alongshore (southeast). WL varied from -4 to 2 cm/s and was generally negative (down) at A2 and A4 (spurs) and
A1 (groove), and positive at A3 (groove) (Figure 3-5c). Uc varied from -6 to 6 cm/s and was nearly always positive at A2 and A4 (spurs) and negative at A1 and A3 (grooves), and was strongest on 4 and 5 September (Figure 3-5d). The horizontal Stokes drift was O(0.1 cm/s) over the experiment duration, and was thus a small O(10%) fraction of the Lagrangian velocity.
During the SFR12 experiment, UL varied between -6 and 12 cm/s but was generally directed onshore (Figure 3-4e), VL varied between -26 and 28 cm/s (Figure 3-4f), and the circulation Uc varied from -1.2 to 1.2 cm/s, and was generally directed offshore at M1 (spur) and onshore at M2 (groove) (Figure 3-4g). The horizontal Stokes drift was O(1 cm/s) over the experiment duration, and was thus a significant fraction of the cross-shore Lagrangian velocity.
A snapshot of the Lagrangian velocity field interpolated in the alongshore direction
(NFR13 A stations) at a time of weak alongshore flow and strong Uc, shows characteristic offshore flow cells centered over the spurs and weak onshore flow over
63 the groove and at depth (Figure 3-6). The alongshore and vertical velocities show a strong counterclockwise rotation on the side of the spur (y = 12 m, h = 5 m), and there is a suggestion of clockwise rotation on the opposite side of the spur (y = 2 m, h = 6 m; and y = 24 m, h = 5 m).
The characteristic circulation cells shown Figure 3-6 were present under different flow conditions with strong Uc, including weak VL and weak UL (Figure 3-7a), strong VL and weak UL (Figure 3-7b), weak VL and strong offshore UL (Figure 3-7c), strong VL and strong offshore UL (Figure 3-7d). However, for periods of weak Uc, these characteristic circulation cells are not present for different flow conditions including weak VL and weak UL (Figure 3-7e), strong VL and weak UL (Figure 3-7f), weak VL and strong UL (Figure 3-7g), strong VL and strong onshore UL (Figure 3-7h). Where weak and strong refer to below and above the mean, and VL and UL are taken as the average of stations A1-A4, and Uc is taken as the average of the magnitude from stations A2 and A3.
In the cross-shore direction, the profile of the rms of the Eulerian velocity (푢̅̅퐸̅)푟푚푠 is a complex shape, with NFR13 of larger magnitude than during the SFR12 experiment
(Figure 3-8a). The profile of Stokes velocity (푢̅̅푆̅)푟푚푠 (from Eq. 1) is similar for all NFR13 profiles and very small compared to the Eulerian velocities, while during the SFR12 experiment, Stokes drift is higher over the spur than the groove and of a similar magnitude to the Eulerian velocity (Figure 3-8b). The Lagrangian velocity
(푢̅̅퐿̅)푟푚푠 profile shape is different between periods of weak and strong Uc for all sites
(Figure 3-8c). For periods of strong Uc, during the NFR13 experiment 푢̅̅퐿̅푟푚푠 was much stronger on the spurs about 3 m below the surface (A2,A4), while during the SFR12 experiment the trend is similar but weaker.
In the alongshore direction, the profile of (̅푣̅퐸̅)푟푚푠 for the bottom half of the water column is similar to a log-layer shape on the spurs, while on the grooves the shape is also like a log-layer but with an offset of about the spur height (Figure 3-8d). The profiles are similar higher in the water column. The profile of (푣̅̅푆̅)푟푚푠 (from Eq. 1) is similar for all profiles and very small compared to the Eulerian velocities (Figure
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3-8e). For both experiments, the profile of 푣̅̅퐿̅푟푚푠 had a similar log-layer-like shape for both strong and weak Uc conditions, but strong Uc conditions had a larger magnitude (Figure 3-8f).
3.3.2 Momentum Balance The significant terms in the depth averaged cross-shore momentum balance (Eq. 3), were of O(10-4 m/s2) between nonlinear convective, mean pressure gradient, and radiation stress gradient (RSxy) terms (Table 3-2) i.e.,
휕푈 휕휁̅ 1 휕푆푥푦 푉 퐿 ≈ −푔 − , (7) 퐿 휕푦 휕푥 휌(휁̅ + ℎ) 휕푦 while the unsteady, second nonlinear convective (푈퐿 휕푈퐿⁄휕푥), radiation stress -5 2 gradient (RSxx), and bottom stress terms were of secondary importance O(10 m/s ). The significant terms in the depth averaged alongshore momentum balance (Eq. 3), were of O(10-4 m/s2) between nonlinear convective, mean pressure gradient, and radiation stress gradient (RSyy) terms (Table 3-2) i.e.,
휕푉 휕휁̅ 1 휕푆푦푦 푉 퐿 ≈ −푔 − , (8) 퐿 휕푦 휕푦 휌(휁̅ + ℎ) 휕푦 while the unsteady second nonlinear convective (푈퐿 휕푉퐿⁄휕푥), bottom stress, and -5 2 radiation stress gradient (RSyx) terms were of secondary importance O(10 m/s ). Surface stress from wind forcing was insignificant at both sites O(10-6 m/s2) (Table 3-2). Because the relative difference in elevation between pressure gauges is not known, there may be net bias in the calculated pressure gradient, which is constant in time. It is assumed to first order this bias is zero. This assumption is reasonable to first order because the change in wave direction (Figure 3-3), and the periodic reversal of
UL and VL (Figure 3-5) suggests the temporal mean of the mean pressure gradient over the entire measurement period should be near zero. Secondly, the mean pressure gradient arises as a response to the other forcing terms and is expected to be of the same order as the other major terms [Rogers et al., 2013], which is the result in Table 3-2.
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3.3.3 Bottom Roughness
The bottom coefficient CD was computed at A2 and A3 using the Reynolds stress using Eq. 6 from the ADVs (Table 3-3). The CD was similar O(0.01) at both sites in the cross-shore direction and in the alongshore direction over the spur (A3), while it was much larger O(0.1) in the alongshore direction over the groove (A2). Fits to logarithmic profiles were performed for the alongshore Eulerian velocity profiles, [e.g.
Reidenbach et al., 2006] but the results for bottom roughness scale z0 (A3: 0.009 ± 0.023 m, A4: 0.008 ± 0.016 m, C1: 0.02 ± 0.04 m) and offset height d (A3: 0.62 ± 0.12 m, A4: 0.51 ± 0.33 m, C1: 0.57 ± 1.36) had very large scatter. Use of Grant and Madsen [1979] to remove the effect of waves and predict a physical roughness scale kN (A3: 0.06 ± 0.03 m, A4: 0.05 ± 0.04 m, C1: 0.12 ± 0.12 m) also had very high scatter.
3.3.4 Near Bed Results During the NFR13 experiment, the near-bed average cross-shore Lagrangian velocity u̅̅L̅ measured by the ADVs was slightly higher in magnitude over the spur (A2) compared to the groove (A3) (Figure 3-9a), while v̅̅L̅ was significantly higher in magnitude over the spur (Figure 3-9b). w̅ was similar in magnitude at the two sites but was variable over the groove but generally negative (down) over the spur (Figure
3-9c). The rms of the near-bed cross-shore wave velocity (푢̃)푟푚푠 was similar over the spur and the groove for periods of stronger, directly incident wave forcing (5 September), but larger in the groove for weaker wave forcing (Figure 3-9d). The measured wave velocity was smaller at both locations than what would be predicted by linear wave theory (based on the mean Hrms, Tm, and θm). This different was most ′ ′ pronounced over the spur. The 푢̅̅̅푤̅̅̅, TKE and |휏̅푏|computed from the wave-separated Reynolds stress, were significantly higher over the spur than over the groove by up to a factor of four (Figure 3-9e,f,g).
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3.4 Discussion
3.4.1 Waves and Circulation The effect of SAG’s on waves appears to be minimal. The NFR13 results show only very small alongshore differences in wave properties Hrms, Tm, and θm (Figure
3-3c,d,f). For higher wave forcing during the SFR12 experiment, Hrms was slightly higher over the spur (Figure 3-3b). At both field sites, the alongshore SAG wavelength O(10 m) is much smaller than the surface wavelength O(100 m). As shown by Rogers et al. [2013], for small SAG wavelengths, the effects of diffraction are likely strong, and alongshore differences in wave parameters are therefore minimal. Linear wave theory predicts larger horizontal wave motions over the spur, but the measured wave velocities were very similar for strong normally incident waves, and smaller over the spur for weaker incident waves (Figure 3-9d). Thus, the SAG formations appear to be slowing the horizontal wave motion over the spur more than what would be predicted by linear wave theory.
For both experiments, the results show the presence of persistent O(1 cm/s) cross- shore circulation velocity Uc, directed offshore over the spurs and onshore over the grooves (Figure 3-4g, Figure 3-5d) as predicted by the modeling results of Rogers et al
[2013]. The period of strongest measured Uc is on 4 September (NFR13), which for that experiment is also the period of highest Hrms, nearly incident waves, and relatively small alongshore flow VL. Periods of weaker Hrms but directly incident waves (6
September) or high Hrms but angled waves (8 September) show weaker Uc. This is consistent with the modeling results of Rogers et al [2013], which predicted stronger
Uc with high Hrms, directly incident waves (θm = 0), and weak alongshore flow (VL).
The SFR12 experiment had smaller Uc than the NFR13 experiment likely because while Hrms was higher, the SAG formations were smaller, less well-defined (Figure
3-2), and the alongshore VL was stronger. For the events on 4, 5, 6, and 7 September,
(NFR13) there appears to be a similar growth of Uc at approximately 1 cm/s/hr (Figure 3-5d). Similar growth rate is observed during SFR12 (Figure 3-4g). Thus, there appears to be a spin up time for the observed Uc on the order of one hour for which
67 favorable forcing conditions must be maintained to accelerate and develop the SAG circulation cells.
The vertical structure of the circulation cells showed strong horizontal offshore flow over the spurs near the surface, while the grooves had weak horizontal onshore flow
(Figure 3-6). For periods of strong Uc, these cells were persistent features during periods of different cross- and alongshore flow conditions, and were absent during periods of weak Uc (Figure 3-7).
The cross-shore Lagrangian flow shows a significant difference in shape in the mean rms profile between periods of strong and weak Uc, i.e. the circulation cells appear to be modifying the velocity profile shape in the cross-shore direction (Figure 3-8c). This effect was most pronounced during the NFR13 experiment but was also observed in the SFR12 experiment. This is in contrast to the alongshore direction which shows a log-layer like flow profile for both strong and weak Uc, but simply of different magnitudes (Figure 3-8f). The profile shape is similar to a log-layer shape on the spurs, while on the grooves the shape is also like a log-layer but with an offset of about the spur height (Figure 3-8d). We emphasize that because both field site locations were near the deeper end of the spurs, the direction of cross-shore flow was offshore over the spur and onshore over the groove. However, as predicted by Rogers et al. [2013] this is likely reversed at shallower depths, but was not investigated in this study.
3.4.2 Mechanism for Circulation The cross-shore momentum balance indicates the primary terms were nonlinear -4 convective, mean pressure gradient (RSxy), and radiation stress gradient of O(10 m/s2). This was the same primary cross-shore momentum balance obtained by Rogers et al. [2013] for the same relative position on the SAG formations (near the deeper offshore slope). The radiation stress gradient term is the direct result of the shoaling waves, are most affected by the local bathymetric slope [Rogers et al. 2013], which varies in the alongshore direction. The mean pressure gradient arises as a response to wave forcing. The residual forcing from the imbalance between the pressure gradient
68 and radiation stress gradient accelerates the flow until the nonlinear convection is large enough to balance it, resulting in alongshore variable UL flow and the counter- rotating circulation cells (Figure 3-5a, Figure 3-6) consistent with modeling results of Rogers et al. [2013].
The alongshore momentum balance indicates the primary terms were nonlinear -3 convective, mean pressure gradient and radiation stress gradient (RSyy) of O(10 m/s2). Changes to the waves propagating in the alongshore direction from refraction, bathymetric changes or bottom dissipation create a radiation stress gradient (RSyy). As in the cross-shore direction, the pressure gradient arises in response to the forcing and the imbalance between the mean pressure gradient and radiation stress gradient accelerates the flow until the nonlinear convection is enough to balance it, driving an alongshore VL flow (Figure 3-5b, Figure 3-7).
For both cross- and alongshore directions, the nonlinear convective terms are important at this site due to the large variability in bathymetry from the SAG formations and rough reef. The largest errors in the momentum balance are likely from the pressure gradient terms due to instrument accuracy in p measurement and the unknown net constant bias between gauges (assumed to be zero). The second largest errors in the momentum balance are likely the smaller nonlinear convective terms
(cross-shore 푈퐿 휕푈퐿⁄휕푥, alongshore 푈퐿 휕푉퐿⁄휕푥), which were likely larger than the calculated value due to the approximation of 휕 ⁄휕푥 between Transect A and C.
The characteristic shape of the flow field appears analogous to Langmuir circulation cells but of opposite rotation in the yz plane (Figure 3-6). Since the vertical vorticity
휔푧 at h = 4 m, y = 11 m is positive, and 푢푆 is negative, the horizontal vortex force,
푢푆 × 휔푧, [Craik and Leibovich, 1976] is directed toward positive y, contrary to the observed circulation in the yz plane. For waves and currents opposed, the vortex force is stabilizing [Leibovich, 1983], and thus it appears unlikely the vortex force is the mechanism for the observed rotation in the yz plane. However, only limited studies have been conducted on Langmuir circulation with opposing waves and currents and to our knowledge no studies have been conducted on alongshore periodic bathymetry
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[Thorpe, 2004]. Additionally, further onshore where waves and currents are aligned [Rogers et al., 2013], the vortex force would be destabilizing [Leibovich, 1983]. Thus, further study seems warranted.
The near-bed cross-shore Reynolds stress 푢̅̅′̅푤̅̅̅′ is nearly always higher on the spur than on the groove (Figure 3-9e), and the near bed mean vertical motions WL are nearly always directed down over the spur and up over the groove (Figure 3-9c), resulting in the larger secondary flow pattern seen in Figure 3-6, consistent with the lateral stress mechanism and secondary flow pattern proposed by Townsend [1976]. On 7 Sept, 푢̅̅′̅푤̅̅̅′ is nearly the same on the spur and groove corresponding to very little difference in WL between the spur and groove also consistent with this mechanism. Therefore, the lateral stress mechanism of Townsend [1976] seems the most likely mechanism for the observed downward flow over the spur and upward flow over the groove. However, to our knowledge no studies exist to examine this mechanism in a similar flow environment and thus further study seems warranted.
The computed bottom drag coefficient values, CD were similar to values computed for other reefs [Rosman and Hench, 2011]. CD on the spur in both directions and over the groove in the cross-shore direction are of similar magnitude, while over the groove in the alongshore direction is an order of magnitude larger. The larger CD is likely the result of the SAG morphology affecting the flow profile over the groove in the alongshore direction. This effect is seen in the alongshore Eulerian velocity profile which shows low velocity water over the grooves near the bed, while higher up in the water column the profiles are similar over both the spur and grooves (Figure 3-9d). These results suggest that the form drag from the SAG morphology in the alongshore direction may be dominant over differences in the frictional drag from benthic cover
(i.e. coral vs. debris/sand). These results for CD are local measurements, but the net average CD over the larger scale SAG morphology O(100 m) requires further study.
3.4.3 Implications for Coral Health The near bed velocity, turbulence, and bottom shear stress are of particular interest because they directly affect organisms on the bed. Differences in these parameters
70 between the spur and groove may illuminate why corals thrive on the spurs and not in the grooves, beyond the influence of light levels that are generally higher on the spur. While the cross-shore near-bed velocity was slightly higher on the spur (Figure 3-9a), the alongshore near-bed velocity was much higher (Figure 3-9b) during the NFR13 experiment. The vertical velocity was also directed down over the spur, while directed up over the groove (Figure 3-6, Figure 3-9c). There was also increased turbulence kinetic energy over the spur (Figure 3-9f). Corals subjected to stronger water motion have greater mass transfer, including increased nutrient uptake rates [Atkinson and Bilger, 1992; Thomas and Atkinson, 1997; Atkinson et al., 2001], photosynthetic production and calcification [Dennison and Barnes, 1988], and particle capture [Genin et al., 2009]. Since corals living on the spurs are the recipients of much higher alongshore flows and variable vertical flow from the surface providing increased “food” supply, and increased turbulent motions providing increased contact, they may have a net advantage to corals living in the grooves.
The bottom shear stress was also significantly higher and more variable on the spur than the groove with an average of 0.37±0.21 and 0.17±0.07 Pa respectively during the NFR13 experiment (Figure 3-9g). Assuming incipient motion based on the shields parameter, [e.g. Julien, 2010], this would correspond to flow capable of suspending sediments smaller than very fine gravel on the spur and coarse sand on the groove during the relatively small wave conditions observed during the NFR13 experiment. During higher wave events, bottom stress would be much higher. Since the spur has much higher mean flows primarily in the alongshore direction, and bottom slope is toward the grooves, sediment suspended by the bottom stress would be shed from the spurs toward the grooves where it could accumulate. This sediment and debris accumulation was present at the study sites (Figure 3-2b,e), and has been observed as a primary feature of SAG formations around the world [Rogers et al., 2013]. Over time, the sediment in the grooves is carried downslope [Storlazzi et al., 2003]. Lower debris and sediment accumulation on the spurs relative to the grooves would be a significant advantage for recruitment and coral growth [Buddemeier and Hopley, 1988; Acevedo et al., 1989; Rogers, 1990; Fortes 2000; Fabricius, 2005]. During large
71 wave events, the bottom stress would likely be much greater than what was observed during the NFR13 study period (such as during SFR12 experiment); potentially subjecting the coral on the spur to proportionally higher bottom shear stress and perhaps breakage. However, the magnitude of this effect remains unclear and requires further study.
3.5 Conclusions The results from two separate field studies of SAG formations on Palmyra Atoll show the effect of SAG formations on waves was small, and there was a persistent O(1 cm/s) depth-averaged Lagrangian circulation (Uc) of offshore flow over the spurs and onshore flow over the grooves. This circulation was stronger for larger, directly- incident waves and low alongshore flow conditions. There also appeared to be a spin- up time for the observed Uc on the order of one hour for which favorable forcing conditions must be maintained to accelerate and develop the SAG circulation cells. These are the first field observations of SAG hydrodynamics and confirm the modeling results from Rogers et al. [2013]. The primary cross- and alongshore momentum balances were between the pressure gradient, radiation stress gradient and nonlinear convective terms. The vertical structure of these circulation cells was previously unknown and the results show a complex horizontal offshore Lagrangian flow over the spurs near the surface driven by alongshore variability in radiation stress gradients consistent with Rogers et al. [2013]. Vertical flow was downward over the spur and upward over the groove, likely driven by alongshore differences in bottom stress [Townsend, 1976] and not by vortex forcing [Craik and Leibovich, 1976]. The bottom drag coefficients were similar to values found on other reefs; and were enhanced over the groove in the alongshore direction. Beyond the influence of light levels that are generally higher on the spur, 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.
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The present study was conducted near the deeper end of the SAG formations. At shallower depths, the direction of the depth-averaged cross-shore circulation (Uc) is likely reversed [Rogers et al., 2013], but this effect and the three-dimensional Lagrangian velocity structure at these shallower depths remains unknown and requires further study. The observed similarity to Langmuir circulation but with opposite rotation, and the possible importance of the vortex force and lateral variation of stress mechanism would warrant further analytical or modeling work. Additionally, SAG hydrodynamics under large wave conditions, as well as investigation into the sediment transport through the grooves, also warrants further inquiry.
3.6 Acknowledgements Data from this study will be deposited at the NOAA National Oceanographic Data Center once the article is accepted and can be obtained there after February 1, 2015. This project was funded by the Gordon and Betty Moore Foundation, and supported with an NDSEG Fellowship to JSR (U.S. Department of Defense, Office of Naval Research, 32 CFR 168a), and an NSF Graduate Research Fellowship to DK. We wish to acknowledge able field assistance from Mallory Barkdull, Hank Lynch, David Mucciarone, and Lida Teneva. Helpful discussions with Falk Feddersen and Curt Storlazzi and comments from two anonymous reviewers improved this manuscript. Logistical and operational support on Palmyra Atoll was provided by the Nature Conservancy; this research was conducted under US Fish and Wildlife Service Special Use Permit #12533-13030, and this is Palmyra Atoll Research Consortium contribution number PARC-0113.
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3.7 Figures and Tables
Figure 3-1. Palmyra Atoll with field experiment location and layout. (a) Palmyra Atoll location in the Central Pacific, (b) overview of the western side of the atoll including the NFR13 experiment location on the north forereef and SFR12 experiment location on the south forereef, and (c) zoom in of NFR13 field experiment showing regional bathymetry and instrument locations with velocity and pressure sensors A1-A4 and C1 (red), and only pressure B1-B4 (blue), and local coordinate system cross-shore (x) and alongshore (y) in relation to true north, and (d) zoom in of SFR12 field experiment showing regional bathymetry and instrument layout with velocity and pressure sensors locations M1and M2 , and local coordinate system cross-shore (x) and alongshore (y) in relation to true south. Note, bathymetry shown on (c) and (d) is regional scale without SAG formations. Image and bathymetry courtesy of NOAA.
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Figure 3-2. Field experiment images and spur and groove bathymetry, for NFR13 experiment (a)-(c), and SFR12 experiment (d)-(f). (a) typical spur and Station A2 (center) with divers for scale looking onshore, (b) typical groove and Station A3 (center) showing reef debris looking onshore, and (c) NFR13 spur and groove bathymetry in alongshore direction (Stations A1-A4) showing instrument placement (red squares), (d) Station M1 showing spur looking alongshore from M2, (e) Station M2 showing groove and reef debris looking alongshore from M1, and (f) SFR12 spur and groove bathymetry in alongshore direction (Stations M1-M2) showing instrument placement (red squares).
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Figure 3-3. Physical forcing of tide, waves, and wind during NFR13 experiment duration, (a) mean free surface 휁,̅ (b) power spectral density of surface 푆휁휁(푓) as a function of frequency f at A2, (c) rms wave height Hrms, (d) mean wave period Tm, (e) wave angle θ(f) as a function of frequency at A2 (°), (f) mean wave angle θm, and (g) wind velocity U10x (-) and U10y (--). For (a), (c), (d), and (f) blue is A1-A4, red is B1-B4 [not shown for (f)] and black is C1.
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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, (a) mean free surface 휁,̅ (b) rms wave height Hrms, (c) mean wave period Tm, (d) mean wave direction θm, , (e) cross- shore UL, (f) alongshore VL, and (g) circulation Uc. Blue is M1 (spur) and red is M2 (groove). Solid lines (-) are mean values, dotted lines (..) are mean ± the 95% confidence limit are only shown for (e), (f) and (g).
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Figure 3-5. Depth averaged mean Lagrangian velocity UL results and circulation velocity Uc over NFR13 experiment duration, (a) UL, (b) VL, (c) WL, and (d) Uc, Colors are green, blue, red, black, and cyan for A1 (groove), A2 (spur), A3 (groove), A4 (spur), C1 (deep forereef)[not shown for (c) and (d)] respectively. Solid lines (-) are mean values, dotted lines (..) are mean ± the 95% confidence limit.
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Figure 3-6. Mean Lagrangian velocity uL, in the alongshore (y) and vertical (z) direction showing characteristic spur and groove circulation cells during NFR13 experiment. Arrows indicate alongshore (푣̅̅퐿̅, 푤̅̅̅퐿̅) velocity, color shading is cross-shore 푢̅̅퐿̅, + is offshore and – is onshore flow. Dashed black line is free surface 휁,̅ solid black is bottom bathymetry, and black x are Stations A1-A4.
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Figure 3-7. Mean Lagrangian velocity uL in alongshore (y) and vertical (z) direction under different flow conditions during NFR13 experiment. For periods of strong Uc with characteristic circulation cells, (a) weak VL and weak UL, (b) strong VL and weak UL, (c) weak VL and strong UL, (d) strong VL and strong UL; and for periods of weak Uc with little to no circulation cells, (e) weak VL and weak UL, (f) strong VL and weak UL, (g) weak VL and strong UL, (h) strong VL and strong UL. . Color scheme and labels are the same as Figure 6, color shading is cross-shore 푢̅̅퐿̅ (cm/s).
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Figure 3-8. Average profiles over depth for rms of mean Eulerian (풖̅̅̅푬̅)푟푚푠, Stokes drift (풖̅̅̅푺̅)푟푚푠, and Lagrangian (풖̅̅̅푳̅)푟푚푠 velocity during NFR13 and SFR12 experiments. Cross- shore direction (a) (푢̅̅̅퐸̅)푟푚푠, (b) (푢̅̅푆̅)푟푚푠, (c) (푢̅̅퐿̅)푟푚푠, and alongshore direction (d) (푣̅̅퐸̅)푟푚푠, ̅̅̅̅̅ (e) (푣̅̅푆̅)푟푚푠, (f) (푣̅̅퐿̅)푟푚푠. For (c) and (f), periods of above average circulation (|푈푐| > |푈푐|) ̅̅̅̅̅ are (-x), and periods of below average circulation (|푈푐| < |푈푐|) are (--o).
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Figure 3-9. Near-bed mean Lagrangian velocity and bottom stress results from ADVs (h = 0.7 m) during NFR13 experiment duration, (a) cross-shore 푢̅̅퐿̅ (cm/s), (b) alongshore 푣̅̅퐿̅ (cm/s), (c) ′ ′ 2 2 vertical 푤̅̅̅퐿̅ (cm/s), (d) cross-shore (푢̃)푟푚푠 (cm/s), (e) cross-shore Reynolds stress 푢̅̅̅푤̅̅̅(m /s ), 2 2 (f) turbulent kinetic energy TKE (m /s ), and (g) bottom shear stress magnitude |휏̅̅푏̅| (Pa). Colors are blue and red for A2 (spur) and A3 (groove) respectively. For (d), result from linear wave theory is cyan and magenta for A2 (spur) and A3 (groove) respectively.
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Table 3-1. Experiment instrumentation for NFR13 and SFR12 experiments, sites, depth, instrumentation and sampling rates.
Site Depth h (m) Instrumentation, Sample Rate A1 10.31 1MHz Nortek Aquadopp 0.5m bins, 4.0-8.0 MAB, 1Hz, (Groove) RBR Virtuoso, 1 Hz A2 (Spur) 8.59 600 kHz RDI ADCP 0.4m bins, 5.3-7.3 MAB, 2 Hz Nortek ADV, 0.7 MAB, 4 Hz A3 10.88 1200 kHz RDI ADCP 0.4m bins, 2.3-9.7 MAB, 2 Hz (Groove) Nortek ADV, 0.7 MAB, 4 Hz RBR Virtuoso, 1 Hz A4 (Spur) 9.10 2MHz Nortek Aquadopp 0.5 m bins, 0.75-7.5 MAB, 1Hz
NFR13 RBR Virtuoso, 1 Hz B1 (Groove) 10.59 RBR 1050, 1 Hz B2 (Spur) 10.22 RBR 1050, 1 Hz B3 (Groove) 10.69 RBR 1050, 1 Hz B4 (Spur) 10.33 RBR 1050, 1 Hz C1 (Deep 19.02 1200 kHz RDI ADCP 0.75 m bins,3.5-17.0 MAB, 2 Hz Forereef) RBR 1050, 1 Hz
M1 (Spur) 8.72 1200 kHz RDI ADCP, 0.5m bins, 2.1-7.5 MAB, 0.67 Hz
RBR 1050, 1 Hz
M2 10.53 1200 kHz RDI ADCP, 0.5m bins, 2.1-9.1 MAB, 0.67 Hz SFR12 (Groove) RBR 1050, 1 Hz
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.
Cross-shore (x) Alongshore (y) Description Term O(m/s2) Term O(m/s2) -5 -5 Unsteady ∂UL⁄∂t 1x10 ∂VL⁄∂t 2x10 -5 -4 Nonlinear convective 1 푈퐿 휕푈퐿⁄휕푥 1x10 푉퐿 휕푉퐿⁄휕푦 2x10 -4 -5 Nonlinear convective 2 푉퐿 휕푈퐿⁄휕푦 2x10 푈퐿 휕푉퐿⁄휕푥 2x10 Mean pressure gradient 푔 휕휁⁄̅ 휕푥 2x10-4 푔 휕휁⁄̅ 휕푦 6x10-4 -5 -4 Radiation stress gradient 1 훽 휕푆푥푥⁄휕푥 4x10 훽 휕푆푦푦⁄휕푦 6x10 -4 -6 Radiation stress gradient 2 훽 휕푆푥푦⁄휕푦 3x10 훽 휕푆푦푥⁄휕푥 6x10 -5 -5 Bottom stress 훽휏̅̅푏푥̅̅ 2x10 훽휏̅̅푏푦̅̅ 1x10 -6 -6 Surface stress 훽휏̅̅푠푥̅̅ 2x10 훽휏̅̅푠푦̅̅ 2x10 Note: 훽 = 1⁄[휌(휁̅ + ℎ)]. Order O is the average of the absolute value of the term over the experiment duration.
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Table 3-3. Bottom drag coefficient CD results from NFR13 experiment from near-bed ADV measurements in cross-shore (x) and alongshore (y) directions.
A2 (spur) A3 (groove)
CD (x) 0.015±0.001 0.011±0.001
CD (y) 0.0072±0.0007 0.10±0.007
Note: Reference height zref = 0.7m, confidence intervals are one standard deviation.
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Chapter 4 Wave Dynamics of a Pacific Atoll with High Frictional Effects
This chapter is prepared as a manuscript for future submittal. As the main author of the work, I made the major contributions to the research and writing. Co-authors 1 2 2 include: Stephen G. Monismith , David A. Koweek , and Robert B. Dunbar .
1. Environmental Fluid Mechanics Laboratory, Stanford University, 473 Via Ortega, Stanford, California, 94305, USA
2. Department of Earth System Science, Stanford University, Stanford California, 94305, USA
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Key points Bottom friction can dominate breaking in high frictional environments
New frictional parameterization in SWAN better predicts wave heights on reefs
Waves exert control on geomorphic structure and coral cover
Abstract 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 the SWAN wave model to characterize the wave dynamics operant on the atoll. Our results indicate that wave energy is primarily from the north during the northern hemisphere winter and from the south in the northern hemisphere summer. Refraction of waves along the reef terraces due to variations in bathymetry leads to focusing of waves in specific locations. 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 modeled wave dissipation rates from bottom friction are on average larger than dissipation rates due to breaking and are an order of magnitude larger than what has been observed on other, less geometrically complex reefs, a result which should be corroborated with future in situ measurements. The SWAN wave model with a modified bottom friction formulation better predicts bulk wave energy properties than the existing formulation at our measurement stations. The near bed squared velocity, a proxy for bottom stress, shows strong spatial variability across the atoll and exerts control over geomorphic structure and benthic community composition.
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4.1 Introduction 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 oceanographic 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 rise [Ferrario et al., 2014; Baldock 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].
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 measurements. Due to the lack of on-island measurements, previous estimates of waves 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 reef with little
87 anthropogenic degradation and abundant calcifiers. Thus, characterizing the wave dynamics in this isolated system with an intact exterior reef structure and highly frictional environment is of interest.
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].
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 (푓푤 ≈ 1.8) is significantly higher than previously measured on reefs at Kaneohe Bay, Hawaii (푓푤 ≈
0.3) [Lowe et al., 2005], and John Brewer Reef, Australia (푓푤 ≈ 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.
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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 parameter [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, 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 parameter has been well- studied on sandy beaches and is typically assumed constant at about 0.8 [Battjes and Jansen, 1978]. 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.
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 this study 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.
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4.2 Study Site Palmyra Atoll (5° 52’N, 162° 05’W) is part of the Northern Line Islands of the central equatorial Pacific (Figure 4-1a). Largely because of the absence of acute anthropogenic stressors, Palmyra’s exposed reef tracts (outside of the lagoons) contain abundant and diverse calcifiers, namely hard corals and crustose coralline algae [Williams et al., 2013] with relatively high community production and calcification rates [Koweek et al., 2015].
The atoll consists of a forereef, reef crest and shallow back reef region on both its northern and southern sides, while the western and eastern edges are dominated by open terraces of 5 to 20 m depth with abundant corals. (Figure 4-1b, Figure 4-2). The forereefs are characterized by abundant live hard coral cover (Figure 4-1c,d; Figure 4-2). Near the reef crest where the surfzone is found, the substrate largely consists of rubble whereas further inshore, larger corals are common on the back reef (Figure 4-1e). The open terraces are typically characterized by high live coral cover with high rugosity and complex bathymetry (Figure 4-2) [Williams et al., 2013].
4.3 Field Measurements
4.3.1 Field Experiments and Data Analysis The field experiment consisted of an array of velocity and pressure sensors deployed between Sept-2011 and Jul-2014 designed to characterize the wave dynamics around the atoll (Figure 4-1b, Table 4-1). During this study period, two high-resolution short- term experiments were also conducted. The first on the north forereef during Sept 2013, hereafter referred to as NFR13, with additional details in Rogers et al., [2015], and the second during July 2014 on the south forereef FR3, hereafter referred to as SFR14 with additional details in Monismith et al., [2015] (Figure 4-1b, Table 4-1). Pressure measurements were made with Richard Brancker Research DR1050 and VirtuosoD sensors, and velocity and pressure measurements were made with RDI Teledyne ADCPs and Nortek ADPs with sampling rates and locations in Table 4-1. Note that sampling rates for long term experiments were constrained by battery power due to the long times (ca. 1 year) between instrument deployment and recovery. Winds
90 were taken from a local weather station (Campbell Scientific with RM Young Wind Sentry, Figure 4-1b) for 2013-2014, and the NASA Modern-Era Retrospective Analysis for Research and Applications (MERRA) global reanalysis model for 2012- 2013. Monitoring stations with ADCPs or ADPs attached to polyethylene plates were secured to the reef in areas of dead coral or sand. Monitoring stations with only a pressure sensor were secured directly to dead corals on the reef substrate.
Instantaneous measured velocity data u(u,v,w) in geographic coordinates (east, north, up) were rotated to local bathymetry coordinates of cross-shore (x) and alongshore (y) directions with positive cross-shore coordinate defined as towards the atoll center. The vertical (z) coordinate is taken as upwards from mean sea level (MSL). Time averaging ( ̅ ) was computed over 30 minute intervals for mean velocity 풖̅, average free surface deviation from MSL, 휁,̅ and wave statistics.
Wave analysis was conducted on pressure p and velocity data by dividing each 30 minute segment into sections of equal length such that the frequency resolution was 0.0017 Hz, each with 75% overlap, applying a Hanning window to the segments and computing spectra 푆(푓) of frequency f. The significant wave height Hs was calculated by,
1/2 퐻푠 = 4 [∫ 푆휁휁(푓)푑푓] , (1)
(root mean square (rms) wave height 퐻푟푚푠 = 퐻푠⁄√2), integrated from 4 to 25 s for the swell band (ss), and 33 to 600 s for the infragravity band (ig), where 푆휁휁(푓) is the power spectral density of the free surface ζ, calculated 2 from, 푆휁휁 = 푆푝푝[cosh 푘ℎ⁄(ρg cosh 푘ℎ푔)] . 푆푝푝(푓) is the power spectral density of p with the mean removed (detrending was not needed as tidal variations are small in the analysis window), ρ is density, g is gravitational acceleration, h is depth of the bottom below MSL, hg is the height of the pressure gauge above the bottom, and wavenumber k is related by the dispersion relation 휎2 = 푔푘 tanh 푘ℎ, and radian frequency 휎 =
2휋푓 = 2휋⁄푇 [Dean and Dalrymple, 1991]. Peak wave period Tp, was the period of
91 peak 푆휁휁(푓), and mean wave period, Tm was calculated based on the first spectral moment of 푆휁휁(푓). At several sites in the interior lagoons (CHAN, BE, DOCK, EL), instrument depths were too deep to accurately measure dynamic pressure from the dominant wind generated waves, with peak periods typically less than 3 s. At one site
(RT13), with low resolution sampling rate (12 s), Hs was obtained by 퐻푠 = 4퐸{[var(휂)]1/2}, where η was obtained from the measured dynamic pressure [Dean an Dalrymple, 1991] with assumed peak period from a nearby gauge (FR9), variance, var(휂) was obtained from a bootstrapping method with 30 iterations to remove potential aliasing effects, and E is the expected value.
4.3.2 Wave Climate The WWIII model results compare reasonably well to the time variability in field measurements of significant swell wave height extrapolated to offshore Hs0 by using conservation of energy flux to the north (from FR9) and to the south (from FR5) correcting for shoaling, refraction and bottom friction (Figure 4-3a). On average, the WWIII results are 25% higher for the average (50%) and 15% higher for the high wave events (98%), (i.e. comparing the larger of Hs0 from north or south to WWIII
Hs0). The trends in offshore wave height from field measurements (2012-2014) are similar to results from the long term WWIII model (1979-2014) (Sup. Inf. Figure S1).
Based on the field measurements of the swell wave Hs on the north forereef of the atoll (FR9) varied from 0.4-1.0 m during the summer months to 3.1 m from wave swell events in the winter months. Hs on the southern side of the atoll (FR3 and FR5) was typically 0.4-1.0 m except for swell events in summer months of up to 2.6 m
(Figure 4-3b). Hs on the western terrace (RT4, RT13) was typically about 0.1 m, but increased to 0.9 m during large swell wave events. The swell wave peak period Tp on the forereef was typically 5-10 s for periods of wind wave forcing, and up to 20 s for strong swell wave events (Figure 4-3c). The peak wave events are associated with generally higher period waves, with decreasing peak frequency with time, consistent with the dispersion relation and waves originating from distant sources (Figure 4-3d,e, and Sup. Inf. Figure S3). On the forereef, there was energy concentrated in short
92 period waves coincident with increased local winds, and consistent with locally generated wind waves (Figure 4-3d,e,f). Thus the power spectra often had a characteristic double peak. Note wind measurements prior to Sept 2013 were from MERRA global reanalysis model, and after from the onsite weather station, thus the increase in variability is likely from a change in measurement method (Figure 4-3f). Additional discussion on swell and infragravity wave climate is shown in Text S1 and S2 in the Supporting Information.
Overall, the wave climate shows dominant waves from the north likely originating from storms in the northern hemisphere winter and dominant waves from the south likely originating from storms in the southern hemisphere winter in both the offshore and on atoll measurements (Figure 4-3). On average, the north forereef receives approximately twice the total swell energy flux as the southern forereef, and at some locations within the interior of the atoll, tides play a significant role in modulating the wave energy (Sup. Inf. Figure S2).
4.3.3 Wave Friction We used techniques similar to Monismith et al. [2015] in order to analyze the effects of bottom friction. Briefly, the total energy flux ℱ was computed as,
1/4
ℱ = 휌푔 ∫ 퐶푔푥(푓)푆휁휁푑푓, (2) 1/25 where Cgx is the group velocity in the cross shore direction. Mean wave direction θm was computed from the first spectral moment of 휃(푓) calculated by, tan 2휃(푓) =
푆푢푣(푓)⁄[푆푢푢(푓) − 푆푣푣(푓)], where Suu and Svv are the autospectra and Suv is the cospectra of u and v from the near bed ADCP/ADPs bins [Herbers et al., 1999]. Average rate of frictional wave dissipation 휀 ̅ between two points, using the simplified energy flux equation [Dean and Dalrymple, 1991],
푑ℱ ∆ℱ = = −휀,̅ (3) 푑푥 ∆푥
93 where Δℱ is the change in energy flux, and Δx is the distance between two points. Wave reflected energy is assumed small, see Monismith et al. [2015]. In most wave models, frictional dissipation is assumed to take the form,
3 휀̅ = 0.6푓푤휌푈푟푚푠, (4) where fw is the wave friction factor, and the near-bottom wave velocity,
1/2 1/4 휎2 푈푟푚푠 = [∫ 2 푆휁휁(푓)푑푓] (5) 1/25 sinh 푘ℎ
[Dean and Dalrymple, 1991]. The maximum bottom stress 휏푏from waves is then,
1 휏 = 휌푓 푈2 (6) 푏 2 푤 푏
2 where 푈푏 = √2푈푟푚푠, and fw is essentially equal in Eq. 4 and 6 [Nielsen, 1996]. 푈푏 is a proxy for bottom stress, which has the advantage of being independent of the local fw
(which is generally approximate or unknown). A parameterization for fw was proposed by Swart [1974],
푎2 푓푤 = exp[푎1(퐴푏⁄푘푁) + 푎3], (7) with a1 = 5.213, a2 = -0.194, and a3 = -5.977, a bottom roughness scale kN,, with the wave excursion distance 퐴푏 = 푈푏⁄휎.
The loss of energy flux due to bottom friction was computed for three study sites, which were located on the forereef such that they receive swell energy from only one predominant direction (Figure 4-1b). The first was during the NFR13 experiment during four days in September 2013, between deep forereef (C1), shallow terrace (B4) and shallow terrace (A4) separated by 100 m and 15 m respectively [Rogers et al., 2015]. The second was between November 2013 and March 2014 between the FR9 and RT4 stations, separated by 1400 m. The third was the six-day SFR14 experiment during July, 2014, between 11.2 m depth and 6.2 m depth on the south forereef near FR3 separated by 56 m [Monismith et al., 2015].
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Onshore propagation wave energy flux decreased with onshore propagation for the three study sites, with the highest total dissipation occurring for sensors with the greatest separation at the Western Terrace site (Figure 4-4a,b,c). Using the average 3 bottom rms velocity (the mean of 푈푟푚푠at each site) (Figure 4-4d,e,f), and Eqs. 3 and 4, we calculated that fw varied from 0.5 to 5 (Figure 4-4g,h,i), and kN was approximated using a least squares fit to Eq.7 which varied from 1.1 to 2.5 m.
From classical laboratory experiments on sediment, kN is 2 to 3 times the characteristic diameter [Nielsen, 1992]. In this case, the characteristic diameter would be 0.37 to 1.25 m, which is consistent with the scale of coral heads observed at the site (Figure
4-1c,d,e). Determination of kN without direct wave dissipation measurements is an area of active research. Lowe et al. [2005] effectively computed kN from the standard deviation σr of a cm scale bathymetric survey with 푘푁 ≈ 4휎푟, however this method has yet to be applied to other reefs. Compared to other remote sites in the pacific, the reefs on Palmyra are about average coral cover [Bruno and Selig, 2007; Knowlton and Jackson, 2008] and coral species richness [Maragos and Williams, 2011], and thus likely about average benthic complexity. Other reef sites exist in the Pacific with much higher complexity. For example, compare the very rugose spur and groove formations on the forereef of Millennium Atoll, Kiribati (Figure 4-1f) and described in Barott et al. [2010] with that of Palmyra (Figure 4-1c,d,e). Thus it is likely that higher wave friction factors than those measured here exist at other reefs.
4.3.4 Wave Breaking Wave breaking is typical along the north and south reef crests depending on incoming wave forcing and is often a clearly visible feature underwater (Figure 4-1e) or in aerial images (Figure 4-1b). We assume a simple one dimensional model for waves and mean setup 휁,̅ assuming a radiation stress gradient and pressure gradient balance, no net flow, and wave dissipation from bottom friction and breaking,
휕ℱ = −휀̅ − 휀̅ , (8) 휕푥 푏
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휕푆 휕휁̅ 푥푥 + 휌푔(휁̅ + ℎ) = 0, (9) 휕푥 휕푥 where Sxx is the radiation stress [Dean and Dalrymple, 1991]. Bottom dissipation 휀 ̅ is taken from Eq. 4, and the breaking dissipation 휀̅푏 is taken to satisfy depth limited ̅̅̅ breaking, where the breaking parameter is 훾푠 = 퐻푏⁄ℎ푏. This is similar to the methods employed by Vetter et al. [2010], but including wave bottom dissipation, which is important at this site. Note, if the rms wave height is referenced 훾푟푚푠 = 훾푠⁄√2.
We apply Eqs. 8 and 9 on the north side of the atoll between the forereef (FR7) and the shallow back reef (NBE) which experiences little net flow, with measured waves from the northwest forereef (FR9). Wave forcing along the respective side of the atoll is typically relatively uniform in space, for example between FR3 and FR5 measured
Hs was very similar (Figure 4-3b). Therefore, it is reasonable to assume that the measured Hs at FR9 approximates the Hs on the forereef offshore of FR7. To determine the constant offset between depth gauges on the forereef and back reef, average setup was regressed against incoming ℱ (Supp. Inf. Figure S6a). A forward
Euler model of Eq. 8 and 9 was employed and 훾푠 was iterated until the setup matched the field observations (Supp. Inf. Figure S6b).
On average, the computed breaking parameter γs, was 0.83±0.21 (Supp. Inf. Figure S6). Previous studies have shown the mean setup due to waves increases with increasing bottom friction, especially on the reef flat [Apostos et al., 2007; Franklin et al., 2013; Vetter et al., 2010].
4.4 Wave Modeling
4.4.1 Wave Model The SWAN module [Booij et al., 1999] of the Coupled-Ocean-Atmosphere-Wave- Sediment Transport (COAWST) program was used to model waves on the atoll [Warner et al., 2010]. The evolution of the wave spectrum is described by the spectral action balance equation first shown by Bretherton and Garrett [1968] and refined by others including Hasselmann et al., [1973]:
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휕푁 휕푐 푁 휕푐 푁 + ∇ [(푐 + 푈)푁] + 휎 + 휃 = 푆 , (10) 휕푡 푥 푔 휕휎 휕휃 푡표푡 where the wave action density 푁(휎, 휃) = 퐸(휎, 휃)/휎, E is wave energy, and U is mean velocity. The terms in Eq. 10 from left to right represent the unsteadiness, propagation in geographical space (푃푥푦), shifting of relative frequency (푃휎), and shifting of wave direction due to refraction (푃휃), and the source/sink term Stot given by,
푆푡표푡 = 푆푖푛 + 푆푛푙3 + 푆푛푙4 + 푆푑푠,푤 + 푆푑푠,푏 + 푆푑푠,푏푟, (11) where Sin is transfer of wind energy to waves, Snl3 is transfer of energy from wave triad interactions, Snl4 is transfer of energy from wave quadruplets, Sds,w is wave dissipation from white capping, Sds,b is wave dissipation from bottom friction, and Sds,br is wave dissipation from breaking and are described in Booij et al. [1999].
Of several methods available to model the bottom dissipation Sds,b , including those of Hasslemann et al. [1973], and Collins [1972], this study uses the method of Madsen et al. [1988] where Sds,b is solved using a method similar to Eq. 4, which is dependent on wave properties and a bottom roughness scale kN. This method uses a parameterization for fw originally proposed by Jonsson [1966] for rough turbulent conditions in implicit form (Figure 4-5a) (limited to 푓푤 < 0.3 for a defined solution),
1 1 퐴푏 + log10 ( ) = 푚푓 + log10 ( ) (12) 4√푓푤 4√푓푤 푘푁 with the wave excursion distance 퐴푏 = 푈푏⁄휎, and mf = -0.08. Swart [1974] recast Eq. 12 as an explicit formulation not limited to this constraint (Eq. 7).
The model grid consists of a rectangular (xy) grid covering 34.1 by 14.1 km at 50 m grid resolution, extending from -162.2387 to -161.9313 °W and 5.8189 to 5.9470 °N (a zoomed in view of the atoll is shown in Figure 4-2a). The data used for the model bathymetry was based on NOAA ship-based multi-beam bathymetry for depths greater than 10 m, and linear regression of 5 m grid IKONOS multispectral data for shallow depths [Pacific Islands Benthic Habitat Mapping Center,
97 http://www.soest.hawaii.edu/pibhmc]. Grid bathymetry was interpolated from data sources and smoothed using a Shapiro filter until the appropriate grid stiffness parameters were met (R<0.4). Additionally, the reef crest was explicitly included in the grid based on field measurements and aerial images, and max depth was trimmed at 200 m. Mapping of the geomorphic structure (Figure 4-2b) and dominant biological cover (Figure 4-2c) was obtained from NOAA NCCOS Benthic Habitat Mapping [http://ccma.nos.noaa.gov/ecosystems/coralreef/palmyra], and in conjunction with computed values of bottom roughness height kN at FR3,FR9, and RT4 (Section 4.3.3) were used to infer kN over the model domain (Figure 4-5b).
Boundary conditions were taken from measured wave height and period on the north (FR9) and south (FR5) of the atoll, corrected for changes to height, travel time, and bottom dissipation from the measured location to the model boundary (Figure 4-3a). The FR9 and FR5 sites are on the outward facing forereefs with nearly parallel alongshore bathymetry (Figure 4-2a), and thus free from focusing effects and representative of the wave dynamics on the respective sides of the atoll. The primary boundary wave angle was taken from the NOAA Wave Watch III Model results [http://polar.ncep.noaa.gov/waves/index2.shtml], while the wave angle for the opposite boundary was taken as due south or due north for the north and south boundaries respectively. The properties for the east and west boundaries were taken from the north or south boundary with the larger Hs. Wave properties are assumed uniform along each boundary for waves coming into the domain. The boundaries employed a JONSWAP spectrum based on wave height and period. Wind forcing was included, see Section 4.3.1 (Figure 4-3f). Model directional resolution was 5°, and there were 40 logarithmically spaced frequency bins between 0.04 and 1 Hz. The enabled SWAN physics modules included: Madsen et al. [1988] friction method with spatially variable kN, zero mean currents, enabled wave triad interaction, enabled wave quadruplets, constant breaking parameter, and constant water level. The model was run at a 1.5 hour time step for Sept-2012 to Jul-2014. Revisions to the SWAN friction parameterization are discussed in the following section.
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4.4.2 Model Modifications and Performance The two primary adjustment factors in the SWAN model are the breaking parameter and bottom friction. A breaking parameter of γs = 0.66 was used in the model.
Measurements of the wave friction factor fw from three sites on Palmya range from 0.4 to 5.7 (Figure 4-4, Figure 4-5a). These results are higher than previous field studies on reefs from Nelson [1996] (N96), and Lowe et al. [2005] (L05), but fall within the range of values measured by lab experiments (Figure 4-5a). These include studies by Mirfeneresk and Young [2003] (M03), Simons et al. [1988] (S88), Kemp and Simons [1982] (KS82), Simons et al. [1992] (S92), Kamphuis [1975] (K75), Sleath [1987] (S87a), Jensen [1989] (J89), Sumer [1987] (S87b), Jonsson and Carlsen [1976](JC76), Brevik [1980] (B80), and Riedel et al. [1972] (R72) .
The existing formulation for fw in SWAN using the Madsen et al. [1988] formulation is based on Jonsson [1966] (J66), and parameterized as a function of 퐴푏⁄푘푁, with a maximum value of 0.3 (Eq. 12), which is well below the measured values on Palmyra (Figure 4-5a). To account for the higher measured frictional effects, we propose a modified friction parameterization in SWAN based on Swart [1974] (S74):
푎2 exp[푎1(퐴푏⁄푘푁) + 푎3], 퐴푏⁄푘푁 ≥ 0.0369 푓푤 = { (13) 50 , 퐴푏⁄푘푁 < 0.0369 with a1 = 5.213, a2 = -0.194, and a3 = -5.977 (Figure 4-5a). This parameterization is equal to the existing SWAN friction parameterization for large 퐴푏⁄푘푁 (Eq.12), but extends the parameterization for low 퐴푏⁄푘푁. The maximum value of 50 was selected based on the highest reported experimental data [Simons et al., 1988]. A fit to all the available data (excluding the present study) gave an R2 of 0.92 for the proposed method. An alternate set of coefficients for Eq. 7 proposed by Nielsen [1992] (a1 = 2 5.5, a2 = -0.2, a3 = -6.3) have a similar fit to the data, with R = 0.90 (Figure 4-5a).
A maximum value for fw must be specified in the model since in this formulation, the 3 3 dissipation is the product of fw and 푈푏 (Eq. 4). With increasing depth, 푈푏 decays like −3 sinh 푘ℎ (Eq. 5), while fw grows exponentially with depth (Eq. 7). For typical waves,
99 the exponential growth of fw starts to dominate the dissipation for 퐴푏⁄푘푁 less than 3 about 0.006 (푓푤 ≈ 3x10 ), and thus the cutoff of 0.0369 is well above this limit.
Additionally, for very low 퐴푏⁄푘푁, inertial forces will dominate over drag forces [Lowe et al., 2005], and thus the parameterization in Eq. 13 may not be correct in this regime.
The SWAN model was run for three scenarios, (1) existing SWAN friction formulation (J66) with γs = 0.66, (2) proposed friction formulation (S74) with γs =
0.66, and (3) proposed friction formulation (S74) with γs = 0.92. To compare model predictions of a given variable of interest Xmodel to the observations Xobs in reef environments [Lowe et al., 2009], we used a quantitative measure of model skill [Willmott et al., 1982],
2 ∑(푋푚표푑푒푙 − 푋표푏푠) Skill = 1 − 2 , (14) ∑(|푋푚표푑푒푙 − 푋̅̅̅표푏푠̅̅̅| + |푋표푏푠 − 푋̅̅̅표푏푠̅̅̅|) where perfect agreement between model results and observations will yield a skill of one and complete disagreement yields a skill of zero. Model skill of modeled vs. measured wave height at the forereef sites (FR3, FR5, FR9, NFR13) have similar skill scores of 0.84-0.94 for all model runs, but model skill at the western terrace sites (RT4, RT13) varies significantly depending on the friction formulation (Figure 4-6, Sup. Inf. Table S1). At the western terrace (RT4,RT13) model skill is significantly improved from 0.33 to 0.85 and 0.21 to 0.43 respectively using the proposed friction formulation. Using a different value of γs = 0.92 gives nearly identical model skill (Sup. Inf. Table S1), and results were insensitive to selected directional and frequency resolution.
With the revised bottom friction formulation (Eq. 13), the SWAN model generally shows good skill in predicting wave height at the measurement locations (Figure 4-6). Model skill is most improved over the existing friction formulation at the terrace sites (RT4, RT13), and the change in wave height between the existing (J66) and proposed (S74) friction methods is most pronounced where cumulative frictional effects are
100 large compared to the forereef where they are smaller (Figure 4-6g). Thus reef width has a large effect on the wave height with the different friction formulations.
One likely factor in residual error is the forcing wave angle in the model which is taken from the WWIII regional model. In addition, variations in the mean water level are not included in the model which shows strong coherency with measured wave energy at some locations (Sup. Inf. Figure S2a). However, there was no significant correlation between residual errors and mean water levels at the sites (max R2 = 0.06), suggesting this may be a minor effect. In addition, SWAN does not include the effect of nonlinear transfer of wave energy to low frequencies, which is observed in the field data (Supp. Inf. Figure S4). However, there was also no significant correlation between residual errors and infragravity wave energy at the sites (max R2 = 0.04) suggesting this may also be a minor effect. Finally, the model does not include the effects of diffraction or mean currents, which could be important in some locations on the atoll. Despite these approximations, the SWAN model well predicts the bulk swell wave properties on the atoll at the gauge locations using the new friction formulation.
4.4.3 Wave Transformation and Dissipation Model results for representative wave conditions from dominant north waves on 21
Dec 2013 (Figure 4-7 a, c, e, g) show high Hs on the north side of the atoll, and shielding effect on the south. The terraces have localized areas of highly focused waves and Hs generally decreases inwards to the lagoons which had very low Hs. Wave period mostly corresponds to the dominant source waves, and wave direction reflects refraction patterns around the atoll. Wave breaking is strong on the north side of the atoll and weak on the south side. Model results for representative wave conditions from dominant south waves on 4 Jul 2013 (Figure 4-7b,d,f,h) show similar patterns as the dominant north waves condition but with reversed direction. Strong wave focusing occurred on the terraces, and wave breaking is similar on both sides of the atoll.
In general, the waves on the east and western edges of the atoll (and the model domain) have higher wave heights because they are outside the shadow effect of the
101 atoll and receive energy from both the north and south directions. SWAN solves the action density equation using the full frequency-directional spectrum. Thus, the model should be able to simulate double-peaked spectra. The results for Tm and θ (Figure 4-7c,d,e,f) are mean properties and indicate the properties corresponding to the dominant energy. Thus in the locations of sharp transitions in Tm and θ, the spectra are double peaked and transitions are smoother than indicated by the bulk properties.
Offshore in deep water, and within the interior deeper lagoons, the dominant terms in the wave action equation are propagation, wind input and dissipation by whitecapping (Figure 4- 8). This is consistent with long period waves propagating from the boundary and local generation of wind waves.
As the waves approach the atoll they shoal and refract approaching normal to the local bathymetry (Figure 4-7e,f), such that the leading terms in the wave action equation are generally between geographical propagation and wave refraction (Figure 4- 8a,b), while dissipation from bottom friction also becomes important (Figure 4- 8e). Dissipation by wave breaking is only important in the surf zone, along the north and south forereefs (Figure 4- 8f). The fraction of total dissipation (푆푑푠 = 푆푑푠,푏 + 푆푑푠,푏푟 +
푆푑푠,푤) due to bottom friction shows bottom friction is the dominant dissipation mechanism except for a few locations on very shallow terrace (h < 1 m) where dissipation due to whitecapping is larger (Figure 4- 8g). Within the surf zone, breaking accounts for about 25-50% of the total dissipation (Figure 4- 8h).
Thus, bottom friction is the dominant average energy dissipation mechanism on the atoll, even within the surf zone. The estimated wave friction factors from three separate measurements on the north and south forereef range from 0.4 to 5 (Figure 4-4). These estimates are an order of magnitude higher than previous measurements on reefs [Nelson, 1996; Lowe et al., 2005], and indicative of the importance of a robust healthy reef with complex geometry (Figure 4-1). While wave friction is the largest source of dissipation, the effects of breaking are also important. Wave breaking generally occurs only on the north and south forereef where the reef crest is well
102 defined (Figure 4-7g,h). For large wave events, or at some locations at scales smaller than the grid resolution, breaking may locally dissipate more wave energy.
4.4.4 Ecological Implications The importance of bottom friction in reducing wave energy over a relatively healthy and diverse reef is demonstrated in both the field experiments and model results. Our field and modeling results demonstrate the enormous potential of hard coral- dominated reef systems to dissipate wave energy through bottom friction. Because of the vibrant coral reef ecosystem on the atoll’s terraces and forereefs, the spatial variation in wave regime is of particular ecological relevance.
Over the entire modeled time period (Sep 2012 – Jul 2014), the average wave energy flux was largest at the eastern and western terraces of the atoll, and the northern forereef received about twice as much incoming offshore energy flux as the southern forereef (Figure 4-9a). The average wave friction factor was largest in the interior of the atoll, ranged from 1 to 10 on the terraces and was near 1 along the shallow forereef and increased with depth (Figure 4-9b). The mean value of the top 2% of near-bed bottom velocities squared, a proxy for bottom stress independent of the assumed bottom friction (Eq. 6), was largest along the northern forereef at shallow depths, and weaker but still significant along the southern forereef and terraces (Figure 4-9c).
Figure 4-10 shows over the entire model grid, the cumulative probability of bottom stress (mean of the top 2% of near-bed bottom velocities squared) (Figure 4-9c) and depth (Figure 4-2a) for each of the geomorphic structures and biological cover based on benthic mapping (Figure 4-2b,c). The spatial variation in bottom stress exerts control on both the geomorphic structure and biological cover on the atoll (Figure 4-10). In terms of the geomorphic structure (Figure 4-10a,b), reef rubble, patch reefs, and pavement (low-relief solid carbonate rock) occur along a wide range of bottom stress, suggesting waves have less importance on these structures; and mud and sand occur at low bottom stress as expected. However, aggregate reef (continuous, high- relief coral formation) occurs within a narrow band of moderate bottom stress (0.08 < ̅̅̅2̅ 2 2 푈푏 < 0.48 m /s ) and over a wide range of depth, suggesting that bottom stress, not
103 depth, is the primary control on aggregate reef formation. Spur and groove formations, as well as pavement with sand channels, occur in a narrow band of high bottom stress ̅̅̅2̅ 2 2 (0.28 < 푈푏 < 1.1 m /s ) at shallow depth, suggesting co-controls of waves and depth. Altogether, our results suggest that waves can be a significant factor in formation of these geomorphic structures, a result also noted in previous studies [Storlazzi et al., 2003; Rogers et al., 2015].
Waves also exert control on the ecological community composition (Figure 4-10c,d). Very low coral cover (0<10%) occur over a wide range of bottom stress, but primarily in shallow depths. Moderate-to-high coral cover (10-50%, 50-90%, and >90%), ̅̅̅2̅ 2 2 however, occur in a narrow band of moderate wave stress (0.10 < 푈푏 < 0.41 m /s ) over a wide range of depth. These results suggest that for high coral cover, waves exert a primary control, while depth (and likely light) appears to have a limited effect. Nonetheless, there are a few localized places on the atoll where high coral cover exists under very low wave conditions, for example on the eastern side of the atoll just onshore of the reef crest (North Barren - NB) (Figure 4-2c). Hard coral at these locations may thrive based on other processes such as enhanced infragravity wave energy through increased low-frequency wave motions (Sup. Inf. Figure S2b) and/or enhanced mean flow. In contrast to the wave controls on coral distribution, algae occur in a wide range of bottom stress and depth, suggesting waves may have a limited influence on algal cover. We note, however, that since the NOAA benthic surveys aggregate fleshy and calcifying macroalgae, which are known to thrive in very different wave environments [Williams et al., 2015; Gove et al., 2015], our interpretive power for understanding wave controls on algal cover is diminished. Coupling high- resolution ecological community composition information along with detailed information about wave dynamics is necessary for improving our understanding of biophysical coupling on coral reef ecosystems.
4.5 Conclusions We characterize the wave dynamics of Palmyra Atoll in the Central Pacific, using field measurements from Sept-2011 to Jul-2014 along with a calibrated and validated
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SWAN wave model. Results indicate that dominant wave energy changes seasonally, with greater energy from the north during the northern hemisphere winter and from the south during the southern hemisphere winter. Storm wave events originating from distant locations are the most significant energy source, but locally generated wind waves are also present.
With the modification of the bottom friction formulation, the SWAN wave model better predicts the bulk wave energy properties at the measured locations than with the existing formulation. Refraction of waves along the reef terraces due to variations in bathymetry leads to focusing of waves in specific locations and varies in time based on wave conditions. Bottom friction is the significant source of wave energy dissipation on the atoll, and wave friction factor results are significantly larger than results from other locations with less geometric complexity. Modeled wave dissipation rates from bottom friction are on average larger than dissipation rates due to breaking, a result which should be corroborated with future in situ measurements. The model could be improved through the inclusion of a coupled hydrodynamic model, specifically the effect of water level due to tides and wave-induced setup. Additionally, future work could include connecting bathymetric roughness and complexity data with prediction of wave bottom dissipation, and quantification of wave friction factors at locations with higher bottom friction than measured here.
The near bed squared velocity, a proxy for bottom stress independent of local frictional effects, shows strong spatial variability across the atoll and is shown to exert control on presence of aggregate reef and spur & groove formations geomorphic structures. We show that bottom stress also exerts control over moderate-to-high percent coral cover (>10%), which occurs within a narrow band of stress, while depth seems to have a limited effect. While these relationships may extend to other reefs with low anthropogenic effects, additional work should be conducted to verify their applicability to other reefs. Previous studies have shown the importance of wave stress in shifting benthic regimes between species and coral morphology. Future work
105 should aim to couple high resolution ecological and wave studies to better understand biophysical coupling on coral reefs.
4.6 Acknowledgements The data from this study will be deposited at the NOAA NCEI data repository after the manuscript is accepted for publication. The authors wish to acknowledge the field team: Mallory Barkdull, Ron Harrell, Joel Leavitt, Hank Lynch, David Mucciarone, Lida Teneva, and Gareth Williams. We thank two anonymous reviewers, as well as Oliver Fringer, Matthew Rayson, Sean Vitousek, John Warner, Alex Sheremet, and Nirnimesh Kumar for their comments and suggestions. Bathymetry and other information were provided by Jamison Gove. Research funding was provided by Stanford University along with two grants from the Gordon and Betty Moore Foundation (“Observations and modeling of the C system dynamics at Palmyra Atoll: In support of the development of management strategies for ocean acidification impacts in the tropics,” to RBD and, “Understanding coral reef resilience to advance science and conservation,” to RBD and SGM). This research was made with Government support under and awarded by the U.S. Department of Defense, Office of Naval Research, NDSEG Fellowship, 32 CFR 168a to JSR. DAK was funded by an NSF Graduate Research Fellowship. This is Palmyra Atoll Research Consortium contribution number PARC- 0120.
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4.7 Figures and Tables
Figure 4-1. Palmyra Atoll location, site layout and experiment instrumentation (a) location of Palmyra Atoll, (b) layout of atoll and instrument locations for long term measurement (high frequency, magenta squares; and low frequency only, magenta circles), NFR13 and SFR14 short-term experiments (yellow circles), and weather station (green star), image courtesy of NOAA. (c) typical northern forereef with spur and groove formations near NFR13, (d) typical southern forereef near FR3, (e) breaking wave on reef crest near PSM courtesy of Brian Zgliczynski, and (f) Millennium Atoll, Kiribati with very rugose spur and groove formations on the forereef, courtesy of Stanford@SEA.
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Figure 4-2. SWAN model grid bathymetry zoomed to atoll; total grid extent is about double view shown. (a) depth grid h (m) with location of selected field sites, (b) geomorphic structure, and (c) dominant biological cover. Gray lines are (5, 10, 30, 60, 200 m) depth contours for (a), 5 and 60 m for (b,c); white shading is land mask.
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Figure 4-3. Wave and wind observations on Palmyra Atoll, Sept 2012 to July 2014. (a) Offshore significant swell wave height Hs0, north and south indicate direction of wave origin, (b) significant swell wave height Hs, (c) peak swell wave period Tp, (d) power spectral density Sζζ at north forereef FR9, (e) power spectral density Sζζ at south forereef FR5, and (f) wind U10 2 (direction going). For (d,e) Sζζ is shown as log10 (m /Hz), and white indicates no available data.
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Figure 4-4. Wave friction factor calculation from field observed energy flux and dissipation for North Forereef NFR13 (left), Western Terrace RT4-FR9 (middle), and South Forereef SFR14 (right). (a,b,c) Onshore energy flux, (d,e,f) near bottom rms velocity assumed average for model, and (g,h,i) calculated wave friction factor and roughness scale, using 6 hour averaging. Colors for (a,b,c) are blue, red, and orange from offshore to onshore. Colors for (d) and (g) are blue, red, and orange for B4-C1, A4-C1, and A4-B4 respectively.
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Figure 4-5. Wave friction parameterizations and model bottom roughness grid. (a) wave friction factor fw as a function of wave excursion distance Ab to roughness scale kN for previous lab (gray) and field (black) studies and present study (blue, green, orange), and existing SWAN (J66) friction parameterization with maximum of 0.3, and proposed friction parameterization (S74) with maximum of 50, and (b) bottom roughness model grid kN, gray lines are 5 and 60 m bathymetric contours and white shading is land mask.
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Figure 4-6. Observed and modeled significant wave height, and average change in wave height with friction method. Observed and modeled significant wave height Hs at (a) FR3, (b) FR5, (c) FR9, (d) RT4, (e) RT13, and (f) NFR13, with SWAN (J66) friction formulation (blue) and the proposed (S74) friction formulation (red). (g) Average change in wave height between original friction method (J66) to proposed friction method (S74), ∆̅̅퐻̅̅̅푠 for 2011-2013, with location of validation sites.
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Figure 4-7. SWAN model results for dominant north waves on 21-Dec-2013 (left), and dominant south waves on 4-Jul-2013 (right). (a)(b) Significant wave height Hs (m), (c)(d) mean wave period Tm (s), (e)(f) mean wave angle θ (°), and (g)(h) breaking fraction. Gray lines are 5 and 60 m depth contours, and white shading is land mask.
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Figure 4- 8. Average wave action terms and dissipation from SWAN model. (a) geographic propagation Pxy, (b) refraction Pθ, (c) wind input Sin, (d) dissipation from whitecapping Sds,w, (e) dissipation from bottom friction Sds,b, (f) dissipation from breaking Sds,br, (g) fraction of total dissipation from bottom friction (Sds,b / Sds), and (h) fraction of total dissipation from breaking (Sds,br / Sds), from daily output, 2012-2014. (a) through (f) show log10 of terms (W/m2). Gray lines are 5 and 60 m depth contours, and white shading is land mask.
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Figure 4-9. Wave energy flux, wave friction factor and high near bed velocity squared from ̅̅̅ SWAN model, (a) average wave energy flux ℱ̅, (b) average wave friction factor 푓푤 in log10 ̅̅̅2̅ scale and (c) mean of top 2% near-bed bottom velocity squared 푈푏 , a proxy for bottom stress, from daily output, 2012-2014. Gray lines are 5 and 60 m depth contours, and white shading is land mask.
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Figure 4-10. Cumulative probability of geomorphic structure and biological cover as a function of modeled near bottom velocity squared, a proxy for bottom stress and depth. Geomorphic structure with (a) bottom stress and (b) depth; and biological cover with (c) ̅̅̅2̅ bottom stress and (d) depth. 푈푏 is the top 2% near-bed bottom velocity squared, a proxy for bottom stress, from daily output, 2012-2014.
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Table 4-1. Field experiment instrumentation, depth, deployment time, and sampling at each site.
Site h (m) Dates1 Instrument and sampling rate Oct-11 – Oct-12 RBR 1050, 10 s FR3 12.1 Oct-12 – Sep-13 RBR Virtuoso, 2 s Sep-13 – Jul-14 RBR Virtuoso, 2 s Nortek ADP (burst mode, 1024 samples Oct-12 – Sep-13 at 1 Hz, every 25 min) Forereef FR5 20.1 Nortek ADP (burst mode, 1024 samples Sep-13 – Jul-14 at 1 Hz, every 25 min) FR7 11 Oct-11 – Oct-12 RBR 1050, 10 s Oct-12 – Sep-13 RBR Virtuoso, 2 s FR9 11.5 Sep-13 – Jul-14 RBR Virtuoso, 2 s PSM 5.9 Sep-13 – Jul-14 RBR 1050, 14 s Western RT4 4.9 Sep-13 – Jul-14 RBR Virtuoso, 2 s Terrace RT13 3.7 Oct-12 – Sep-13 RBR 1050, 12 s Oct-11 – Oct-12 RBR 1050, 10 s NB 3.9 Oct-12 – Sep-13 RBR 1050, 12 s Reef Flat Sep-13 – Jul-14 RBR 1050, 14 s NBE 0.55 Sep-13 – Jul-14 RBR 1050, 14 s BE 2.4 Oct-12 – Sep-13 RBR 1050, 12 s Nortek ADP (burst mode, 1024 samples CHAN 7.7 Oct-12 – Sep-13 at 1 Hz, every 25 min) Lagoons DOCK 3.5 Sep-13 – Jul-14 RBR 1050, 14 s Oct-12 – Sep-13 RBR 1050, 14 s EL 1.9 Sep-13 – Jul-14 RBR 1050, 14 s C1 19.0 3-8 Sep-13 RDI ADCP (0.5 s), RBR 1050 1 s North Forereef B4 10.6 3-8 Sep-13 RBR 1050, 1 s (NRF13) 2 A4 10.3 3-8 Sep-13 Nortek ADP (0.5 s), RBR 1050 1 s South Forereef B 11.2 16-23 Jul-14 Nortek ADP (2 s), RBR 1050 2s 3 (SFR14) A 6.2 16-23 Jul-14 RBR 1050, 2 s 1. Deployment and recovery were in Oct 2012, Sept 2013, and July 2014 with typically two week maintenance downtime. 2. See Rogers et al. [2015] for additional details. 3. See Monismith et al.,
[2015] for additional details.
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Chapter 5 Field Observations of Hydrodynamics and Thermal Dynamics in an Atoll System: Mechanisms and Ecological Implications
This chapter is prepared as a manuscript for future submittal. As the main author of the work, I made the major contributions to the research and writing. Co-authors include: Stephen G. Monismith1, David A. Koweek2, Walter I. Torres1, and Robert B. Dunbar2.
1. Environmental Fluid Mechanics Laboratory, Stanford University, 473 Via Ortega, Stanford, California, 94305, USA
2. Department of Earth System Science, Stanford University, Stanford California, 94305, USA
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Key Points Atoll scale currents are forced primarily by tides and waves
Sites with high coral cover have lower weekly temperature similar to offshore
Low weekly temperature is governed by wave-driven mean advection
Abstract We present results of the hydrodynamics and temperature dynamics of an atoll system based on field measurements from 2012 to 2014 at Palmyra Atoll in the Central Pacific. 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 between the atoll exterior and the interior lagoons. Wave driven flow is significant at most of the field sites, and is a strong function of the dominant wave direction. The typical condition of strong waves from the north drives flow from north to south across the atoll, and from east to west through the lagoon system and out the channel. Wind driven flow is generally weak, except on the shallow terraces.
Bottom roughness values z0 were computed and were similar to values found on other reefs (0.2 – 5 cm). The sites with high coral cover have high diurnal temperature variability, but their average weekly temperature variability is similar to offshore waters, compared to sites with little coral cover which had higher weekly temperatures. The mechanism for maintaining this low weekly temperature is high mean advection, which occurs on timescales of a week, and is primarily governed by wave driven flows.
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5.1 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 and recycling nutrients, thereby supporting both phytoplankton and zooplankton [Odum and Odum, 1955; Yahel et al., 1998]. Water motion appears to be beneficial to coral reefs through increasing the rates of nutrient uptake [Atkinson and Bilger, 1992; Thomas and Atkinson, 1997], photosynthetic production and nitrogen fixation by both coral symbionts and algae [Dennison and Barnes, 1988; Carpenter et al., 1991], and particulate capture by coral [Genin et al., 2009]. Reef-building corals have experienced global declines resulting from bleaching events caused week to month- long warm-water exposure [Hughes et al., 2003; Hoegh-Guldberg et al., 2007; Carpenter et al., 2008]. However, corals in naturally warm environments can exhibit enhanced 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]. Corals can often resist high temperature variability at hourly time scales, but corals may experience mortality with elevated temperatures at time scales of several days to weeks [Williams et al., 2011; Palumbi et al., 2014].
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 reef scale, typically ten to hundreds of meters, 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 dissipation from breaking or bottom friction increases the mean water level, known as wave setup, establishing a pressure gradient that drives flow across the reef in the direction of wave propagation[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
120 relatively narrow, and the constricted exchange of water between these lagoons and the open ocean can cause significant phase lags between lagoon and offshore water levels [e.g., Dumas et al., 2012; Lowe and Falter, 2015]. Wind 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].
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 reef flat with connecting channel systems 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.
Numerous small islands and atolls dot the Central Pacific, including Palmyra Atoll, in the Northern Line Islands. To our knowledge, none of the Northern Line Islands including Palmyra, have been the location of any published long-term hydrodynamic measurements. Due to the lack of on-island measurements, previous estimates of hydrodynamics 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]; Palmyra in particular because of its status as a National Wildlife Refuge, is thought to represent a reef with little anthropogenic degradation and abundant calcifiers. Thus, characterizing the hydrodynamics in this isolated atoll system with an intact exterior reef structure and highly frictional environment is of interest.
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The classical dynamical basis by which waves drive flow is by 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 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. A typical method of parameterization is a log-layer roughness height including the effects of waves [Madsen, 1994]. 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].
Reef thermal variability (i.e., spatial and temporal deviations from climatology) can be driven by atmospheric forcing (e.g., solar radiation, wind, and surface heat fluxes) [Smith, 2001; Wells et al., 2012; Zhang et al., 2013], as well as advective processes from waves, tides and other processes [Davis et al.; 2011; Herdman et al., 2015]. While diurnal temperature variability can be quite high on shallow backreefs [Herdman et al., 2015], the longer term weekly or monthly averaged temperatures appear to be more stressful to corals [Williams et al., 2011; Palumbi et al., 2014]. These long term temperatures are the basis for reef bleaching predictors such as degree heating week, a cumulative measure [Strong et al., 2011]. Little work has been conducted on atolls investigating the mechanisms for this weekly temperature variability, and its relation to observed benthic community.
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While the hydrodynamic forcing and thermal dynamics on fringing and barrier reef systems has been well investigated, little work has been done on atoll systems to quantify the effect of different forcing mechanisms on driving flow and temperature. Additionally, little work has been conducted on atolls connecting the reef ecology to the temperature within the atoll system. The aim of this study is to address this knowledge gap by characterizing the hydrodynamics of Palmyra Atoll through field measurements made from 2011-2014. We examine the effects of different forcing mechanisms in driving flow and present results of thermal dynamics relevant to the reef. We then address the role of temperature in shaping the ecological community structure of Palmyra.
5.2 Methods
5.2.1 Study Site Palmyra Atoll (5° 52’N, 162° 05’W) is part of the Northern Line Islands of the central equatorial Pacific (Figure 5-1a). Largely because of the absence of acute anthropogenic stressors, Palmyra’s exposed reef tracts (outside of the lagoons) contain abundant and diverse calcifiers, namely hard corals and crustose coralline algae [Williams et al., 2013] with relatively high community production and calcification rates [Koweek et al., 2015].
The atoll consists of a forereef, reef crest and shallow back reef region on both its northern and southern sides, while the western and eastern edges are dominated by open terraces of 5 to 20 m depth with abundant corals. (Figure 5-1b,g). The forereefs are characterized by high percentages of live stony coral cover (Figure 5-1c). Near the reef crest where the surfzone is found, the substrate largely consists of rubble whereas further inshore, larger corals are common on the back reef (Figure 5-1d). The open terraces are typically characterized by high live coral cover with high rugosity and complex bathymetry, while lagoonal areas typically exhibit seabeds of sediment (Figure 5-1e,f) [Williams et al., 2013].
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The atoll is generally within the North Equatorial Counter Current (NECC), which flows primarily to the east at typically 0.2 to 0.8 m/s from August to January, and is weak the rest of the year [Hsin and Qiu, 2012; Maragos et al., 2008a]. The wave climate is seasonal, dominated by strong storm waves (1-3 m) from the north in the northern hemisphere winter, and strong storm waves (1-2 m) from the south in the southern hemisphere winter, similar to the Hawaiian Islands [Rogers et al., Chapter 4]. The winds on the atoll are dominated by the northeasterly trade winds for much of the year, and are strongest in February through May, typically near 20-30 km/hr [Maragos et al., 2008a]. Average daily air temperature is fairly constant near 26-28 °C [Maragos et al., 2008a].
5.2.1 Field Experiment and Data Analysis The field experiment consisted of an array of velocity and pressure sensors deployed between Sept-2011 and Jul-2014 designed to characterize the waves and hydrodynamics around the atoll (Figure 6-1, Table 5-1). The wave dynamics of the atoll are described in Rogers et al. (Chapter 4). Additionally, several short term experiments were conducted; the NFR13 and SFR12 experiments are described in Rogers et al., [2015], the south forereef FR3 experiment are described in Monismith et al., [2015], and the SIB experiment is described in Koweek et al., [2015]. Pressure measurements were made with Richard Brancker Research DR1050 and VirtuosoD sensors; velocity and pressure measurements were made with RDI Teledyne ADCPs, Nortek ADPs, and Nortek ADVs; salinity was inferred from Sea-Bird Electronics SBE-37 CTDs; and temperature measurements were made with Sea-Bird Electronics SBE-56 thermistors with sampling rates and locations in Table 5-1. Note that sampling rates for long term experiments were constrained by battery power due to the long times (ca. 1 year) between instrument deployment and recovery. Monitoring stations with ADCPs or ADPs attached to polyethylene plates were secured to the reef in areas of dead coral or sand. Monitoring stations with only a pressure and/or temperature sensors were secured directly to dead corals on the reef substrate. Most of the sites were outfitted with a single bottom mounted thermistor, but at the forereef sites (FR3, FR5, FR7, FR9) bottom mounted thermistors were placed at 10, 20, and 30m depth,
124 and vertical moorings were placed at the channel inlet and outlet (CHAN and OCM, 3 SBE56s each) and within the west lagoon (WL, 5-SBE56s).
Instantaneous measured velocity data 풖(푢, 푣, 푤) were taken in (푥, 푦, 푧) coordinates, with y corresponding to geographic north, and the vertical (z) coordinate is taken as upwards from mean sea level (MSL). Time averaging ( ̅ ) was computed over 30 minute intervals for mean velocity, average free surface deviation from MSL, ζ, temperature T, salinity S, and wave statistics. The mean Lagrangian velocity 풖̅ was calculated by [Andrews and McIntyre, 1978], 풖̅ = 풖̅̅̅푬̅ + 풖̅̅푺̅ , where 풖̅̅̅푬̅ is the mean measured Eulerian velocity and 풖̅̅푺̅ is the Stokes drift computed spectrally from the wave data. Details of wave computations of significant wave height Hs, and other wave properties are discussed in Chapter 4. The Lagrangian depth-averaged mean velocity 푼(푈, 푉, 푊) was calculated by combining available data at a given location
(ADV/ADCP/ADP), assuming 풖̅̅̅푬̅ = 0 at the bottom, linearly interpolating in z and taking the average. For some portions of the analysis, mean velocities were rotated into a cross-shore (CS) and alongshore (AS) components, i.e., (푢퐶푆, 푢퐴푆) and
(푈퐶푆, 푈퐴푆), with positive cross-shore coordinate defined as towards the atoll center.
5.3 Results and Discussion
5.3.1 Circulation and Tides The primary field results used for this analysis are from September 2012 to July 2014
(Figure 5-2). Wave height Hs varies seasonally with strongest waves on the north of the atoll up to 3.2 m in the northern hemisphere winter, and stronger waves on the south of the atoll up to 2.2 m during the northern hemisphere summer (Figure 5-2a). The mean free surface ζ on the forereef varies primarily with tidal fluctuations of up to
0.8 m, but also with long-term sea level fluctuations ζlp of 0.2 m (Figure 5-2b). Winds
U10 are directed primarily in the southwest to northwest directions with speeds up to 12 m/s (Figure 5-2c). Note prior to September 2013, wind measurements are from the global MERRA model, and after from on island measurements, and thus the change in variance is likely from the difference in instrumentation. The regional currents derived from the average upper 50 m in the vicinity of Palmyra from the global HYCOM
125 model show generally easterly flows up to 0.8 m/s, which are strongest in the northern hemisphere winter consistent with the north equatorial counter current (Figure 5-2d).
Mean low pass filtered temperature Tlp across all instruments varies during the two- year period by about 1 ºC, with 95% of measurements (2σT spatially and temporally) typically within 1 ºC of the mean (Figure 5-2e). The alongshore depth averaged velocities UAS are largest on the forereef, up to 0.6 m/s (Figure 5-2f), and weaker in the cross-shore direction UCS up to 0.2 m/s (Figure 5-2g).
The amplitude of free surface of the M2 tide is 0.30 m on the exterior of the atoll and with propagation into the lagoon system, the amplitude decreases to 0.26 m, and the phase lags by up to 4.7 hours in the East Lagoon (Figure 5-3a). The results are similar for the K1 tide, with 0.08 m amplitude on the reef exterior, decreasing to 0.07 m with 6.0 hr phase lag in the East Lagoon. The current ellipses show flow generally aligned with bathymetric contours, except on the terrace; and the contribution of the M2 tide to flow is relatively small on the forereef (< 20% of overall velocity), while on the western terrace and channel the contribution is large (> 50% of overall velocity) (Figure 5-3b). The net mean flow over the measurement period is to the west on the south forereef, out of the lagoon in the channel, near zero on the western terrace, and to the east on the north forereef (Figure 5-3b).
5.3.2 Forcing Mechanisms Flow on reefs is governed by the depth integrated momentum equations for horizontal flow given by [e.g. Mei et al., 2005],
휕푼 1 푳 + 푼 ∙ ∇푼 = −푔∇ζ − [∇ ∙ 푺 + 흉̅̅̅ − 흉̅ ], (1) 휕푡 푳 푳 휌(휁̅ + ℎ) 풃 풔
2 where S is the radiation stress tensor (proportional to 퐻푟푚푠), 흉̅̅풃̅ is the mean bottom stress, and 흉̅풔 is the mean surface stress here approximated by a typical quadratic drag law, 흉̅풔 = 휌푎퐶퐷푎풖ퟏퟎ|풖ퟏퟎ|, where air density 휌푎, wind drag coefficient 퐶퐷푎, and wind velocity u10 [Smith, 1988].
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Wave-driven flow through the lagoon system is forced by strong waves from the north and with heights Hs up to 3 m, and relatively weak southern waves of 1.0 m height during December 2013 (Figure 5-4a). The free surface with tides removed, ζdt shows a large increase in water levels on the northern shallow reef flat (NBE), and a slight increase in the eastern lagoon (EL), compared to the channel (CHAN) (Figure 5-4b). The velocity in the channel varies with the tide and depth up to 0.38 m/s (Figure
5-4c,d), but the low pass filtered depth averaged velocity ULlp is directed out of the channel (negative), and is largest during the period of strong waves up to 0.2 m/s (Figure 5-4d). Wind forcing at this site does not show significant effect on the flow.
To evaluate the relative effect of each of the mechanisms in Eq. 1 at each site we use the squared coherency γ2 (Coh) between two signals which is given by,
2 2 훾 (푓) = |푆12(푓)| ⁄푆11(푓)푆22(푓), (2) where 푆푖푗(푓) is the cospectra of signal i with signal j as a function of frequency f. The 2 [1/(퐸퐷푂퐹−1)] limiting value for 95% confidence, 훾95 = 1 − 0.05 , where EDOF is number of independent realizations [Emery and Thomson, 2004]. The relative effect of each forcing from tides, waves and wind on accelerating flow in Eq. 1 is then,
2 푼푳 ∝ 푼푳(휁푂푆, 퐻푟푚푠, 푢10|풖ퟏퟎ|), (3) where 휁푂푆 is obtained from an average of all forereef sites, and filtered to retain only 2 tidal frequencies. Preconditioning of the 휁,̅ 퐻푟푚푠, 푢10|풖ퟏퟎ| signals was conducted so that coherency between them was not significant.
The coherence between tidal height and observed flow is significant for all sites in both the alongshore and cross-shore directions, and the maximum coherence has a period of either the M2, K1 or M4 tides (Figure 5-5a,b). Coherence between north waves and observed flow is significant for most sites on the northern side of the atoll and the channel (Figure 5-5c,d), while for south waves the coherence is weakly significant on the terrace and southeast forereef (FR5) (Figure 5-5e,f). Coherence between wind and observed flow is significant only on the western terrace sites and
127 northeast forereef (FR7) (Figure 5-5g,h). Overall, tides and north waves show the strongest coherence, while south waves and wind show weak coherence.
The relative importance of density driven flows are evaluated, by approximating a two layer system, where ΔU and ΔT are the average velocity and temperature of the upper half of the water column minus the average of the lower half. The results show buoyancy driven flow is important at all the forereef and channel sites, but especially in the cross-shore direction (Figure 5-5i,j).
Mean regional currents derived from the upper 50 m of the HYCOM global model, showed significant coherence at RT4 in the alongshore direction, but not at any other sites. Velocities measured on the forereef showed significant coherence at the M2 tidal frequency (FR3-FR5, FR5-FR7, FR7-FR9), the K1 tidal frequency (FR3-FR5, FR3- FR7, FR5-FR7), but also at longer periods near 50 to 200 h, (FR3-FR7, FR5-FR7, and FR7-FR9).
In summary, circulation within the atoll is primarily governed by tides and waves, and secondarily by wind and regional flows. Tidal forcing is significant at all field sites in both cross- and alongshore directions, primarily at the M2 frequency, and the oscillations generally follow the alongshore bathymetric contours. The free surface tidal amplitude is only slightly reduced with propagation into the lagoon system, but is significantly delayed by up to 4.3 hours in the interior East Lagoon due to the constricted inlets.
Wave-driven flow is significant at most of the field sites. Waves drive flow through breaking and dissipation from bottom friction which create a radiation stress gradient. The resulting pressure gradients drive flow from the side of the atoll with larger waves to the side of smaller waves. This setup effect is largest when one side of the atoll has much larger waves than the other. For larger waves on the north side of the atoll (which are the most common conditions), this drives flow generally from north to south. Flow is enhanced through the lagoon system from east to west and flow exits through the channel, a response clearly seen in the field data and modeling float results. For larger waves on the south of the atoll, this generally drives flow from
128 south to north across the atoll. Flow is not increased in the channel, since there is not a pressure gradient in that direction, but is increased across the terraces. Over the field record, waves are most often larger on the north of the atoll, and thus the dominant flow path is from east to west through the lagoon system and out the channel, explaining the net long-term outflow from the channel.
Wind-driven flow is weakly coherent at a few of the field sites, primarily on the western terrace. Winds drive flow by imparting a surface stress (Sstr), which is generally weak, except in shallow regions, where it can be a significant term. Wind driven flow in the terrace and northeast forereef are present but weak.
Density driven flows are also likely important on the atoll. The lagoon system is stratified, and the channel velocity profile shows likely evidence of classic baroclinic exchange flow (Figure 5-6f). Additionally, surface jets of flow in the direction away from the atoll are observed in the ADCP/ADV field data at the diurnal frequency, likely from heated shallow reef flat water exiting over the reef crest. Shallow water on the reef flats is likely saltier due to differential evaporation; and with cooling at night, this water could form bottom density currents which propagate into the interior lagoons or along the exterior forereefs.
The measurement stations on the forereef (FR3, FR5, FR7, FR9) are largely sheltered from the large scale regional flow due to their location on the forereef (Chapter 6), and showed no significant coherence with the large scale HYCOM flows. However, flows on the western terrace (RT4), were coherent with the regional flow in the alongshore direction (approximately north-south) as the regional flows can drive flow across the exposed terrace, as seen in the model results (Chapter 6).
5.3.3 Vertical Structure and Bottom Roughness We employed an empirical orthogonal function (EOF) analysis on the measured velocity profiles at each site to decompose the signal into its principal components, or dominant statistical modes. The EOF analysis provides a description of the spatial variability of the velocity field through the modal shapes (eigenfunctions), as well as
129 the temporally variability through the modal amplitude time series [Emery and Thompson, 2004].
The first mode of the empirical orthogonal functions (EOFs) of the velocity profiles in the alongshore directions explains (97, 97, 97, 98, 93, and 41%) of the variance and in the cross-shore directions explains (71, 56, 84, 83, 90, and 96%) of the variance for sites FR3, FR5, FR7, FR9, RT4, and CHAN respectively (Figure 2-7). At most of the sites, the deeper part of the profile is roughly linear, (note log scale), indicating a log- layer like flow. Near the surface on the forereef and channel sites, the profile deviates from the linear profile especially in the cross-shore direction.
For several short term experiments (Table 5-1), mean bottom stress, 흉̅̅풃̅, was computed from the turbulent Reynolds stress, which are assumed constant within the inertial sublayer, [e.g. Reidenbach et al., 2006],
′ ′ 흉̅̅풃̅ = −휌풖̅̅̅푤̅̅̅ (4) using the measured turbulent velocities (풖′) from the ADVs. A common bottom stress parameterization is given by [e.g. Grant and Madsen, 1979, Feddersen et al., 2000],
̅̅̅̅̅̅ 흉̅̅풃̅ = 휌푪푫풖|풖| (5) where u is evaluated near the bed but above the bottom boundary layer, and CD is a nondimensional drag coefficient which may depend on the flow environment, height above the bed and bottom roughness. Combining Eqns. (4) and (5) gives,
−풖̅̅̅′̅푤̅̅′ 푪 = (6) 푫 풖̅̅̅|̅풖̅̅| where in environments with low wave and turbulence energy, the denominator is often simplified to 풖̅|풖̅|, and 풖̅ is either the depth averaged or near bed velocity, see Rosman and Hench [2011] for a complete discussion.
For a well-developed turbulent boundary layer, an inertial sublayer region exists where mean velocities exhibit a logarithmic profile. Within this region, the mean
130 velocity profile is related to the generation of turbulence by shear at the bed, and the law of the wall takes the form [e.g., Reidenbach et al., 2006; Kundu, 1990]:
풖∗(푡) 푧 − 풅 풖̅̅̅푬̅(푧, 푡) = ln ( ) (7) 휅 풛ퟎ풂
Where 풖∗ is the friction velocity, κ is von Karman’s constant (0.41), d is the displacement height (MAB), and z0a is the apparent bottom roughness length scale. In very rough boundary layers, it is not clear where z = 0 should be defined, and thus d is typically included in the fitting function [Rosman and Hench, 2011]. Values of u*, d, and z0a were adjusted to obtain a best fit in the least squares sense from measurement points from the ADV/ADCP/ADP measurements between the bottom and ½ of the depth. To ensure a well-defined log region over the measurement points used in the fit, only alongshore profiles having strong mean flow (푢̅̅퐸̅ > 5 cm/s), physically realistic 2 roughness 0.001 < 30푧0푎 < 0.5ℎ, offset 0.01 < 푑 < 1.25 m, and good fit (r > 0.8) were used in the analysis.
When waves are present, they act to increase the bottom roughness felt by the mean current, therefore the apparent roughness scale (z0a) is larger than the physical roughness scale (z0) [Grant and Madsen, 1979]. Using the methods of Grant and
Madsen (1994) with the wave and near bed velocity data, z0a and computed reference height (푧푟푒푓 − 푑) from the log fit, the physical roughness 푘푁 = 30푧0 was iterated until z0a from both methods matched within 1%. Only physically realistic values of kN were kept, 0.01 < kN < 2 m. The mean z0a, d, and kN were computed as the log mean of the ̅̅̅̅̅̅̅̅̅̅̅ results from each segment, i.e. 푧0푎 = 푒푥푝(ln 푧0푎(푡)) [Reidenbach et al., 2006].
Using a log-layer fit to the measured velocity profiles (Eq. 7) the two-day low pass filtered 푧̅̅0̅푎̅ at the north forereef (FR9) increases with increased wave height, as expected (Figure 5-7a). The results for d, zoa, and zo are highly variable, and their distributions are typically exponential thus confirming the use of the log mean for statistics (Figure 5-7b,c,d). Over all the sites, offset height d varies from 0.01 to 0.95 m, the apparent z0 varies from 0.010 m to 0.10 m, and the physical roughness height
131 z0, (removing the effects of waves where applicable), varies from 0.002 to 0.05 m
(Table 5-2). Results for bottom drag coefficient CD using Reynolds stress (Eq. 6) varied from 0.0037 (CHAN) to 0.033 (NFR13) (Table 5-2).
Bottom stress is a significant term in the momentum balance for shallow depths less than about 10 m (Chapter 6). Thus for these shallow regions which cover a significant fraction of the atoll, correct parameterization of the roughness is essential to accurately modeling the flow. For this study, extensive field data at multiple sites allowed for calculation of roughness values, which are consistent with results from other reefs [Rosman and Hench, 2011; Lentz et al., in prep]. However, the theory is currently not well developed for selection of roughness values a priori from high resolution bathymetric data or other site mapping, and should be considered for future work.
5.3.4 Thermal Dynamics and Ecological Implications Because of the vibrant coral reef ecosystem on the atoll’s terraces and forereefs, the spatial variation in hydrodynamic properties is of particular ecological relevance. The physical properties of interest include wave stress, light, mean flows, offshore travel time (age), and temperatures. On Palmyra, wave stress has been shown to exert a significant control on both geomorphic structure and biological cover, while depth (a proxy for light) had only a limited effect (Chapter 4). Mean flows, offshore travel time (age), and diurnal temperature variability also had a significant effect (Chapter 6). This section takes a more detailed look at the long term temperature dynamics with the extensive field records available.
The timescale most relevant to coral reef mortality from high temperatures is over several days to weeks [Palumbi et al., 2014]. Therefore, while diurnal temperature fluctuations can be quite large, corals can often resists these changes, but long term average temperatures can stress corals beyond their capacity leading to mortality [Williams et al., 2011; Palumbi et al., 2014]. The cumulative probability of measured temperature 푇30푚푖푛 shows high variability in temperature, with sites of low coral cover (<10%) and algae experiencing temperatures up to 36°C, while sites with high coral cover (>50%) experiencing lower temperature (Figure 5-8a). For weekly
132 averaged temperature 푇̅7푑, sites with high coral cover (>50%) have generally lower temperatures which are similar to offshore temperatures, while sites with low coral cover (<10%) and algae experiencing up to 0.6°C higher temperatures than offshore (Figure 5-8b). Note that the median temperature is similar for all sites at both time scales. These results are consistent with previous laboratory studies noting that sites of high coral cover can experience high diurnal temperature variability, but on a weekly timescale experience low temperatures similar to offshore waters. The spatial distribution of the top 80% of weekly averaged temperature 푇̅7푑,80, shows the locations of lower temperatures are located in more open areas connected to the offshore, while areas of high temperature (north backreef, and east lagoon) are within relatively closed systems (Figure 5-8c). We now investigate the mechanism for temperature variability on a weekly timescale.
Assuming depth averaged instantaneous temperatures T, no horizontal diffusion, and no heat flux from the sea floor, the heat conservation equation is [e.g. Herdman et al., 2015],
휕푇 퐻 + 푼푳 ∙ ∇푇 = , (8) 휕푡 휌퐶푝ℎ
Where 푼푳 is depth averaged velocity vector, ρ is density, Cp is specific heat of water, h is depth, and H is the surface heat flux, the sum of short wave radiation (HSW), long wave radiation (HLW), and the sensible (HS) and latent heat (HL) fluxes,
퐻 = 퐻푆푊 + 퐻퐿푊 + 퐻푆 + 퐻퐿, (9)
And H>0 implies heat flux into the water column. Direct measurements of long and short wave radiation along with variables necessary to estimate sensible and latent heat flux using bulk formulae [Pawlowicz et al., 2001] were made at the weather station from Sep 2013 to May 2014 (Figure 5-1b). As we are interested in temperature variability of time averaged temperature 푇̅, we take a time average of Eq. 8, and decompose 푇 = 푇̅ + 푇′ and 푈퐿 = 푈̅ + 푈′, and assume alongshore temperature gradients are weak,
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휕푇̅ 휕푇̅ ̅̅̅̅휕푇̅̅̅̅′ ̅̅̅퐻̅̅̅̅ + 푈̅ + 푈′ = , (10) 휕푡 휕푥 휕푥 휌퐶푝ℎ where the terms from left to right will be referred to as unsteady, mean advection, nonlinear advection, and surface heating. We evaluate Eq. 10 between each site and offshore, and take the offshore temperature TOS as the mean of all forereef sites, with a three day low pass filter to remove diurnal effects and internal waves.
We compare the heat conservation at the Channel and Terrace (RT4) sites for three weeks during November 2013, noting that while diurnal temperature variations in T are about 1°C, mean temperatures 푇̅ are within ±0.2°C of the offshore temperature TOS (Figure 5-9a,b). Net heating is similar for both sites, varying from 800 M/m2 in midday to -200 W/m2 at night (Figure 5-9c,d). Measured cross shore velocity at the channel site has strong tidal variability (±0.3 m/s) with a net negative mean, while at the terrace site, variability is much smaller (±0.1 m/s) with a small net negative mean (Figure 5-9e,f). The nonlinear transport of temperature 푈̅̅̅′̅푇̅̅′ results from the correlation of temperature deviations (primarily diurnal) and velocity deviations (primarily tidal), with net positive at the channel sites and varyable at the terrace site (Figure 5-9g,h). The resulting heat balance in the channel site is primarily between mean advective and nonlinear advective terms, while at the shallower terrace site, surface heating also becomes important (Figure 5-9i,j). Note that while the heat budget does not close exactly, the residual error is of the same order as the terms.
To statistically evaluate the importance of different terms at all the sites, we employ coherency γ2 (see Forcing Mechanisms) between the mean of the top 80th percentile of weekly average temperatures 푇̅80, which differentiates biological cover types (Figure 5-8b), and each term in Eq. 10. To effectively use coherency in a linear system, each of the inputs must not be coherent with each other [Emery and Thomson, 2004], thus a proxy is employed for each term.
The mean advection term is approximated by 휕푇̅ ≈ 푇̅ − 푇푂푆, which should be 휕푇̅ proportional to TOS, or 푈̅ ∝ 푇 when 푈̅ is significant. This is confirmed by 휕푥 푂푆
134 comparing TOS (Figure 5-9a,b) to the computed mean advective term (Figure 5-9i,j). ′ To evaluate the nonlinear advective term, since TOS is a long term mean,푇푂푆 = 0, and ̅̅̅̅휕푇̅̅̅′ thus 푈′ ∝ 푈̅̅̅′̅푇̅̅′. This is confirmed by comparing 푈̅̅̅′̅푇̅̅′ (Figure 5-9g,h) to the 휕푥 computed nonlinear advective term (Figure 5-9i,j). For sites where velocity measurements were not available, U’ was assumed the same as nearby gauges in similar regimes (i.e., RT4 is similar to RT1). For sites where no nearby observations existed, (all shallow backreef sites), a least squares regression was performed using the ROMS simulations (Chapter 6), at each site assuming U’ is primarily tidally driven
휕휁(푡−훽) by pressure gradients with some time offset, 푈′ ≈ 훼 to determine α and β. To 휕푡 evaluate the surface heating term, measured meteorological parameters were assumed spatially constant across the atoll, while local water T and h were used to compute H at each site [Pawlowicz et al., 2001]. Thus 푇̅ is assumed proportional to
′ ′ ̅̅̅̅̅̅ 푇̅ ∝ 푇̅(푇푂푆, 푈̅̅̅̅푇̅̅, 퐻⁄ℎ), (11)
The results of the coherency analysis show the relative effect of mean advection, nonlinear advection and surface heating on the average high temperatures (Figure 5-10). Sites with stronger mean advection experience lower temperatures, while sites with strong nonlinear advection or surface heating experience higher temperatures. Additionally, sites with high coral cover (>50%) are generally associated with high mean advection and low nonlinear advection compared to sites with no low (<10%) to no cover.
These results suggest that the sites with high coral cover can have high diurnal temperature variability, but their average weekly temperature is similar to offshore waters. The mechanism for maintaining this low temperature is high mean advection, which occurs at timescales of a week, and is primarily governed by wave driven flows. This suggests that lower mean temperatures may be primarily governed by residence time or offshore travel time (age), a result also noted by Zhang et al. [2013] and explored in Chapter 6.
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5.4 Conclusions We present results of the hydrodynamics of an atoll system based on field measurements from 2012 to 2014 on Palmyra Atoll in the Central Pacific. 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 by up to 4.3 hr. Wave driven flow is significant at most of the field sites, and is a strong function of the dominant wave direction. The typical condition of strong waves from the north drives flow from north to south across the atoll, and from east to west through the lagoon system and out the channel. When wave direction is reversed with stronger waves from the south, flow is generally from south to north across the terraces, but flow within the lagoons is unaffected. Wind driven flow is generally weak, except on the shallow terraces. Density driven flows are important in the channel and forereefs.
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. This suggests that lower mean temperatures may be primarily governed by residence time or offshore travel time (age), a result also noted by Zhang et al. [2013] and explored in Chapter 6.
Future work could further examine the effects of density driven flow, especially focusing on the interior lagoons. Additionally, for this study, extensive field data at multiple sites allowed for calculation of roughness values to parameterize the model. Development of methods for selection of roughness values a priori from high resolution bathymetric data or other site mapping would warrant further inquiry.
5.5 Acknowledgements The data from this study will be deposited at the NOAA NCEI data repository after the manuscript is accepted for publication. The authors wish to acknowledge the field team: Mallory Barkdull, Ron Harrell, Joel Leavitt, Hank Lynch, David Mucciarone,
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Lida Teneva, and Gareth Williams. Bathymetry and other information were provided by Jamison Gove. Research funding was provided by Stanford University along with two grants from the Gordon and Betty Moore Foundation (“Observations and modeling of the C system dynamics at Palmyra Atoll: In support of the development of management strategies for ocean acidification impacts in the tropics,” to RBD and, “Understanding coral reef resilience to advance science and conservation,” to RBD and SGM). This research was made with Government support under and awarded by the U.S. Department of Defense, Office of Naval Research, NDSEG Fellowship, 32 CFR 168a to JSR. DAK was funded by an NSF Graduate Research Fellowship. This is Palmyra Atoll Research Consortium contribution number PARC-XXXX.
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5.6 Figures and Tables
Figure 5-1. Palmyra Atoll location, site layout and experiment instrumentation (a) location of Palmyra Atoll, (b) layout of atoll and instrument locations for long term measurement [(U, ζ, T) magenta squares; (ζ, T) magenta circles; (T) small magenta circles), short term hydrodynamics experiments (yellow circles), and weather station (green star), image courtesy of NOAA. (c) typical northern forereef near FR9 with abundant live coral and fish, (d) typical terrace with blacktip shark near PSM courtesy of Brian Zgliczynski, (e) typical shallow back reef with live coral near NB, (f) typical lagoon rubble bottom with low visibility water near CHAN with diver and ADV for scale, and (g) dominant biological cover mapping courtesy of NOAA.
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Figure 5-2. Field measurements, Sept 2012 to July 2014. (a) measured significant wave height Hs at north and south forereef, (b) forereef average surface ζ and low pass filtered surface ζlp, (c) wind velocity U10, (d) regional currents Ureg from upper 50m of HYCOM model, (e) average low pass filtered temperature Tlp and standard deviation σT of all measurements, (f) measured depth averaged alongshore velocity UAS, and (g) measured depth averaged cross- shore velocity UCS. Positive cross-shore coordinate is towards atoll center.
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Figure 5-3. Measured tidal amplitude, flow averages and current ellipses. (a) Amplitude (seize) and phase lag (color) of M2 tide from surface ζ, (b) depth averaged Lagrangian velocity U ellipses, M2 tidal velocity UM2 ellipse, average flow 푈̅ vector. Black lines are 5 and 60 m depth contours, white shading is land mask.
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Figure 5-4. Wave driven flow through lagoon system measured in the channel, Dec 2013. (a) significant wave height Hs at north and south sides of atoll, (b) detided free surface ζdt at north back reef (NBE), eastern lagoon (EL), and channel (CHAN), (c) free surface (blue) and along- channel velocity uCS with depth at channel, (d) depth averaged velocity UCS and 36-hr low pass filtered velocity UCS,lp at channel. Positive velocity is towards lagoon.
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Figure 5-5. Coherence between forcing mechanisms (tides, waves, and wind) with measured depth-averaged flow in alongshore UAS (left) and cross-shore UCS (right) directions. Forcing with (a,b) tides, (c,d) north waves, (e,f) south waves, (g,h) wind. (i,j) Coherence between exchange flow ∆푈 and temperature stratification ∆푇. Size of markers represent maximum squared coherency over all frequencies with largest marker in plot (a) equal to 0.98, and smallest marker equal to 95% significance (0.18). Values below 95% significance are black x. Colors represent period of maximum coherency. Black lines are 5 and 60 m depth contours, and dark gray shading is land.
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Figure 5-6. First empirical orthogonal function (EOF) of measured Eulerian velocity profiles 푢̅̅̅퐸̅ with height above bottom at six sites, (a) FR3, (b) FR5, (c) FR7, (d) FR9, (e) RT4, and (f) CHAN, in cross-shore (CS) and alongshore (AS) directions. Note log scale with depth.
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Figure 5-7. Bottom roughness results on north forereef (FR9) as a function of wave height. (a) Computed apparent bottom roughness 푧̅̅0̅푎̅ and rms wave height 퐻̅̅̅푟푚푠̅̅̅̅, (two day low pass filtered), (b) distribution of offset height d, (c) distribution of apparent roughness height z0a, and (d) distribution of roughness height z0. Black dashed line in (b-d) is the log-mean of each variable.
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Figure 5-8. Cumulative probability of temperature at sites with varying biological cover compared to offshore, (a) with measured 30 min data 푇̅30푚푖푛, and (b) 7 day time average 푇̅7푑,80, and (c) spatial distribution of top 80% of weekly average temperatures 푇̅7푑,80. Results are over all available data from 2012-2014, note only high cumulative probability shown (0.5- 1) for (a) and (b).
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Figure 5-9. Thermal dynamics at Channel site (left) and Terrace RT4 site (right) in Nov 2013. (a,b) Measured instantaneous T, time averaged 푇̅, and offshore average temperature TOS, (c,d) net solar heat flux H, (e,f) measured instantaneous UCS and time averaged velocity 푈̅̅̅퐶푆̅̅, (g,h) nonlinear transport 푈̅̅̅′̅푇̅̅′, and (i,j) terms in heat equation.
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Figure 5-10. Effect of mean advection, nonlinear advection, and surface heating in driving 2 high mean temperatures at sites with different biological cover. Coherency 훾 of 푇̅80 with (a) ̅̅̅̅̅ mean advection Tos, (b) nonlinear advection 푇̅̅′̅푈̅̅′, and (c) surface heating 퐻⁄ℎ. Analysis is for September 2013 to April 2014, only results with significant 훾2 are shown.
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Table 5-1. Field experiment instrumentation, depth, deployment time, and sampling at each site.
Site h (m) Dates1 Instrument (sampling rate)2,3 FR3 30.5 Oct-11 – Oct-12 RDI ADCP (15 min), RBR 1050 (10 s) Oct-12 – Sep-13 RDI ADCP (5 min), RBR Virtuoso (2 s), 3-SBE 56, SBE 37 (2 min) Sep-13 – Jul-14 RBR Virtuoso (2 s), 3-SBE 56, SBE 37 (12 min) 16-23 Jul-14 Nortek ADP (2 s), RBR 1050 (2 s) SFR12 9.0 16-26 Sep 12 2-RDI ADCP (1.5 s), 2-RBR 1050 (1 s) FR5 20.1 Oct-12 – Sep-13 Nortek ADP (5 min, & burst: 1 s for 1024 s), RBR Virtuoso (2 s), 3-SBE 56 Sep-13 – Jul-14 Nortek ADP (5 min, & burst: 1 s for 1024 s), 3-SBE 56 Forereef FR7 22.5 Oct-11 – Oct-12 RBR 1050, (10 s), SBE 37 (2 min) Oct-12 – Sep-13 RDI ADCP (5 min), 3-SBE 56, SBE 37 (2 min) Sep-13 – Jul-14 RDI ADCP (2 min), 3-SBE 56 FR9 11.5 Oct-12 – Sep-13 RDI ADCP (5 min), RBR Virtuoso (2 s), 3-SBE 56 Sep-13 – Jul-14 RDI ADCP (2 min), RBR Virtuoso (2 s), 3-SBE 56 NFR13 9.0 3-8 Sep-13 2-Nortek ADP (1s), 2-Nortek ADV (0.25 s), 2-RDI ADCP (1 s), 3-RBR 1050 (1 s) SIB 4.2 20-25 Sep-12 Nortek ADP (1 s) PSM 5.9 Oct-12 – Sep-13 RBR 1050 (12 s), SBE 56 Sep-13 – Jul-14 RBR 1050 (14 s), SBE 56 RT4 4.9 Oct-11 – Oct-12 RDI ADCP (15 min)
Western Oct-12 – Sep-13 Nortek ADV (10 min), SBE 56, SBE 37 (2 min) Terrace Sep-13 – Jul-14 Nortek ADV (10 min), RBR Virtuoso, (2 s), SBE 56 RT10 3.0 Oct-12 – Sep-13 Nortek ADV (10 min), SBE 56 Sep-13 – Jul-14 Nortek ADV (10 min), RBR Virtuoso, (2 s), SBE 56 RT13 3.7 Oct-12 – Sep-13 RBR 1050 (12 s), SBE 56 Sep-13 – Jul-14 SBE 56 Reef Flat NB 3.9 Oct-11 – Oct-12 RBR 1050, (10 s), SBE 56
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Oct-12 – Sep-13 RBR 1050, (12 s) , SBE 56 Sep-13 – Jul-14 RBR 1050, (14 s) , SBE 56 NBE 0.55 Sep-13 – Jul-14 RBR 1050, (14 s) , SBE 56 BE 2.4 Oct-12 – Sep-13 RBR 1050, (12 s) , SBE 56 Sep-13 – Jul-14 SBE 56 CHAN 7.7 Oct-11 – Oct-12 RDI ADCP (15 min), 4-SBE 56 Oct-12 – Sep-13 Nortek ADP (5 min, & burst: 1 s for 1024 s), 3-SBE 56 Sep-13 – Jul-14 Nortek ADP (15 min), 3-SBE 56, SBE 37 (12 min) Lagoons 19-24 Jul-14 Nortek ADV (0.125 s) DOCK 3.5 Oct-11 – Oct-12 RBR 1050 (14 s) Oct-12 – Sep-13 SBE 56 Sep-13 – Jul-14 RBR 1050 (14 s), SBE 56 EL 1.9 Oct-12 – Sep-13 RBR 1050, (14 s), SBE 56 Sep-13 – Jul-14 RBR 1050, (14 s), SBE 56 1. Deployment and recovery were in Oct 2011, Oct 2012, Sept 2013, and July 2014 with typically two week maintenance downtime. 2. All SBE 56 instruments sampled at 10 s. 3. Additional sites with only SBE 56 thermistors, a. Oct-11 – Oct-12: RT1(5), b. Oct-12 – Sep-13: CG, CC, LL, OCM, PSI, PSO, RT1, SIB, TG, WCG, WL(5);
c. Sep-13 – Jul-14: CG, CC, OCM(3),PSI, PSO, RT1, RP, TG, WL(5)
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Table 5-2. Bottom roughness and drag results from field measurements at various sites using fits to velocity profiles, and Reynolds stress.
Log Layer Fit Reynolds Stress 1 Site Waves? h (m) d (m) z0 app (m) z0 (m) CD FR3 Yes 29 0.40 ±0.21 0.034 ±0.036 0.0033 ±0.0032 FR5 Yes 20 0.66 ±0.18 0.035 ±0.019 0.0026 ±0.0010 Forereef FR7 Yes 22 0.62 ±0.25 0.043 ±0.020 0.0024 ±0.0010 FR9 Yes 21 0.86 ±0.19 0.055 ±0.019 0.0032 ±0.0013 NFR13 C1 Yes 19 0.17 ±0.28 0.022 ±0.016 0.0040 ±0.0033 NFR13 Spur Yes 9 0.40 ±0.16 0.023 ±0.011 0.0017 ±0.0014 0.0072 ±0.0007 Spur and NFR13 Groove Yes 11 0.63 ±0.07 0.050 ±0.019 0.0019 ±0.0008 0.10 ±0.007 Groove SFR12 Spur Yes 9 0.24 ±0.19 0.033 ±0.014 0.0017 ±0.0016 SFR12 Groove Yes 11 1.34 ±0.23 0.029 ±0.011 0.0018 ±0.0013 Terrace SIB No 4 0.35 ±0.12 0.037 ±0.010 0.037 ±0.010 Lagoon CHAN No 8 0.24 ±0.09 0.012 ±0.008 0.012 ±0.008 0.0037 ±0.0004 1. Using Grant and Madsen [1979] to remove wave effects, 푧0 = 푘푁⁄30. 2. Results are averaged over all available data at each site, and uncertainties are reported as one standard deviation.
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Chapter 6 Modeling the Hydrodynamics of an Atoll System: Mechanisms for Flow, Ecological Implications, and Connectivity
This chapter is prepared as a manuscript for future submittal. As the main author of the work, I made the major contributions to the research and writing. Co-authors include: Stephen G. Monismith1, David A. Koweek2, and Robert B. Dunbar2.
1. Environmental Fluid Mechanics Laboratory, Stanford University, 473 Via Ortega, Stanford, California, 94305, USA
2. Department of Earth System Science, Stanford University, Stanford California, 94305, USA
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Key Points Large scale currents are forced by regional flow interacting with atoll geometry
Atoll scale currents are forced primarily by tides and waves
Low travel time from offshore and temperature are important for high coral
cover
Abstract We present results of the hydrodynamics of an atoll system modeling simulations using a coupled waves and three-dimensional hydrodynamic model (COAWST) applied to Palmyra Atoll. 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. Wave driven flow is significant over most of the atoll, and is a strong function of the dominant wave direction. The typical condition of strong waves from the north drives flow from north to south across the atoll, and from east to west through the lagoon system and out the channel. Wind driven flow is generally weak, except on the shallow terraces. Bottom stress is significant for depths less than about 10 m. Based on Lagrangian float tracks, the average travel time from offshore (age) appears to clearly differentiate the geomorphic structures. The biological cover shows significant trends with mean flow, age and temperature. While high mean flow is associated with very productive coral regions, low water age and low temperature appear to be the most important variables for distinguishing between biological cover types at Palmyra. 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.
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6.1 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 and recycling nutrients, thereby supporting both phytoplankton and zooplankton [Odum and Odum, 1955; Yahel et al., 1998]. Water motion appears to be beneficial to coral reefs through increasing the rates of nutrient uptake [Atkinson and Bilger, 1992; Thomas and Atkinson, 1997], photosynthetic production and nitrogen fixation by both coral symbionts and algae [Dennison and Barnes, 1988; Carpenter et al., 1991], and particulate capture by coral [Genin et al., 2009]. Reef-building corals have experienced global declines resulting from bleaching events caused week to month- long warm-water exposure [Hughes et al., 2003; Hoegh-Guldberg et al., 2007; Carpenter et al., 2008]. However, corals in naturally warm environments can exhibit enhanced 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]. Corals can often resist high temperature variability at hourly time scales, but corals may experience mortality with elevated temperatures at time scales of several days to weeks [Williams et al., 2011; Palumbi et al., 2014]
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 the retention and removal of terrigenous sediment depends on hydrodynamic processes (flushing rates, dilution), hydrology (e.g., accumulation and groundwaterdischarge) as well as biological processes [Fabricuis, 2005]. Travel time for a parcel of water moving from a boundary to some point within a region is often termed “water age” in literature [Monsen et al., 2002]. For isolated tropical atolls, the relevant boundary is the atoll exterior, and age is thus a measure of the atoll influence on the offshore waters for quantities such as temperature and nutrients.
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Hydrodynamic flows are the primary mechanism for dispersal and thus connectivity for small larval species such as corals [Cowen and Sponaugle, 2009]. Studies of the connectivity between reef systems at large scales using hydrodynamic models have been completed [Roberts, 1997; Jones et al., 2009; Andréfouët et al., 2002], but to our knowledge, no studies exist studying interconnectivity within an atoll system.
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, the flow is primarily governed by the interaction of the island with the large-scale regional flow, tides, Coriolis, and buoyancy [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, 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 dissipation from breaking or bottom friction increases the mean water level, known as wave setup, establishing a pressure gradient that drives flow across the reef in the direction of wave propagation[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 lagoon and offshore water levels [e.g., Dumas et al., 2012; Lowe and Falter, 2015]. Wind 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
154 temperature- or salinity-driven stratification, which may also be important in certain reef systems [Hoeke et al. 2013, Monismith et al. 2006].
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 reef flat with connecting channel systems 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 modeling studies.
Numerous small islands and atolls dot the Central Pacific, including Palmyra Atoll, in the Northern Line Islands. To our knowledge, none of the Northern Line Islands including Palmyra, have been the location of any published long-term hydrodynamic measurements. Due to the lack of on-island measurements, previous estimates of hydrodynamics 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]; Palmyra in particular because of its status as a National Wildlife Refuge, is thought to represent a reef with little anthropogenic degradation and abundant calcifiers. Thus, characterizing the hydrodynamics in this isolated atoll system with an intact exterior reef structure and highly frictional environment is of interest.
The classical dynamical basis by which waves drive flow is by 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
155 the mechanism for Langmuir circulation. 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.
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 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. A typical method of parameterization is a log-layer roughness height including the effects of waves [Madsen, 1994]. 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].
While the hydrodynamic forcing on fringing and barrier reef systems has been well investigated, little work has been done on atoll systems to quantify 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 this study is to address this knowledge gap by characterizing the hydrodynamics of Palmyra Atoll through modeling studies, validated through extensive field measurements (Chapter 5). We examine the effects of different forcing mechanisms in driving flow 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.
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6.2Methods
6.2.1 Study Site Palmyra Atoll (5° 52’N, 162° 05’W) is part of the Northern Line Islands of the central equatorial Pacific (Figure 6-1a). Largely because of the absence of acute anthropogenic stressors, Palmyra’s exposed reef tracts (outside of the lagoons) contain abundant and diverse calcifiers, namely hard corals and crustose coralline algae [Williams et al., 2013] with relatively high community production and calcification rates [Koweek et al., 2015].
The atoll consists of a forereef, reef crest and shallow back reef region on both its northern and southern sides, while the western and eastern edges are dominated by open terraces of 5 to 20 m depth with abundant corals. (Figure 6-1b, Figure 6-2). The forereefs are characterized by high percentages of live stony coral cover (Figure 6-1c, Figure 6-2). Near the reef crest where the surfzone is found, the substrate largely consists of rubble whereas further inshore, larger corals are common on the back reef (Figure 6-1e). The open terraces are typically characterized by high live coral cover with high rugosity and complex bathymetry, while lagoonal areas typically exhibit seabeds of sediment (Figure 6-1d,f; Figure 6-2) [Williams et al., 2013]. We grouped the atoll into different hydrodynamic zones with similar geologic and ecologic characteristics and similar area (Figure 6-2e).
Detailed observations of waves and mean flows are discussed in detail in Chapter 4 and 5 respectively. The atoll is generally within the North Equatorial Counter Current (NECC), which flows primarily to the east at typically 0.2 to 0.8 m/s from August to January, and is weak the rest of the year [Hsin and Qiu, 2012; Maragos et al., 2008a]. The wave climate is seasonal, dominated by strong storm waves (1-3 m) from the north in the northern hemisphere winter, and strong storm waves (1-2 m) from the south in the summer hemisphere winter, similar to the Hawaiian Islands (Chapter 4). The winds on the atoll are dominated by the northeasterly trade winds for much of the year, and are strongest in February through May, typically near 20-30 km/hr [Maragos
157 et al., 2008a]. Average daily air temperature is fairly constant near 26-28 °C [Maragos et al., 2008a].
6.2.2 Hydrodynamic Model The Coupled-Ocean-Atmosphere-Wave-Sediment Transport (COAWST) program [Warner et al., 2010] was used to model the atoll. Only the ocean (Regional Ocean Modeling System – ROMS) and wave (Simulating Waves in the Nearshore – SWAN) modules were used in this study. The SWAN wave component of the model is discussed in detail in Rogers et al., [Chapter 4]. ROMS is a three-dimensional, free surface, topography following numerical model, which solves finite difference approximation of Reynolds Averaged Navier Stokes equations using hydrostatic and Boussinesq approximation with a split explicit time stepping algorithm [Shchepetkin and McWilliams, 2005; Haidvogel et al., 2008; Shchepetkin and McWilliams, 2009]. The vortex force formalism expresses the Navier Stokes equations to include the effects of waves [Uchiyama et al., 2012],
휕풖 휕풖 휕 휕풖 + (풖 ∙ ∇ )풖 + 푤 + 푓푧̂ × 풖 + ∇ 휑 − 푭 − 푫 + (푢̅̅′̅푤̅̅̅′ − 휈 ) 휕푡 ⊥ 휕푧 ⊥ 휕푧 휕푧 풘 = −∇⊥κ + 푱 + 푭 , (1) where (u,w) are the Eulerian mean velocities, f is the Coriolis parameter, φ is the dynamic pressure (normalized by density), F is the non-wave, non-conservative force, D is the diffusive terms (viscosity and diffusion), κ is the lower order Bernoulli head, Fw is the sum of momentum flux due to all non-conservative wave forces, and J is the horizontal vortex force,
휕풖 푱 = −푧̂ × 풖풔풕[(푧̂ ∙ ∇ × 풖) + 푓] − 푤푠푡 , (2) ⊥ 휕푧
The vortex force formalism is implemented in ROMS, which solves for the Lagrangian velocities rearranging Eq. 8, with the significant terms in the three dimensional momentum equation [Kumar et al., 2012], being acceleration (ACC), horizontal advection (HA), vertical advection (VA), Coriolis (COR), Stokes-Coriolis (StCOR), pressure gradient (PG), horizontal vortex force (HVF), non-wave body force
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(BF), wave breaking acceleration (BA), wave roller acceleration (RA), bottom streaming (BtSt), surface streaming (SuSt), horizontal mixing (HM), vertical mixing (VM), and curvilinear grid (FCurv). For complete discussion and definition of terms see Kumar et al. [2012]. Vertical integration of the three dimensional equations, applying surface and bottom boundary conditions, assuming a rectangular grid, and ignoring wave streaming and Stokes-Coriolis effects gives the two-dimensional momentum equation [Kumar et al., 2012],
ACC + HA + COR = PG + HVF + Bstr + Sstr + [BA + RA] + HM, (3) where bottom stress (Bstr) and surface stress (Sst) arise from the boundary conditions.
The model grid consists of a rectangular (xy) grid covering 34.1 by 14.1 km at 50 m grid resolution, extending from -162.2387 to -161.9313 °W and 5.8189 to 5.9470 °N (a zoomed in view of the atoll is shown in Figure 6-2a). Vertical resolution consists of 20 vertical layers in sigma-coordinates and directional wave resolution is 5°. The data used for the model bathymetry is based on NOAA ship-based multi-beam bathymetry for depths greater than 10 m, and linear regression of 5 m grid IKONOS multispectral data for shallow depths [Pacific Islands Benthic Habitat Mapping Center, http://www.soest.hawaii.edu/pibhmc]. Grid bathymetry is interpolated from data sources and smoothed using a Shapiro filter until the appropriate grid stiffness parameters were met [Shchepetkin and McWilliams, 2003]. Additionally, the reef crest is explicitly included in the grid based on field measurements and aerial images, and to reduce the complexity of the model, max depth was limited to 200 m, essentially the upper ocean mixed layer. Mapping of the geomorphic structure (Figure 6-2b) and dominant biological cover (Figure 6-2c) was obtained from NOAA NCCOS Benthic Habitat Mapping (http://ccma.nos.noaa.gov/ecosystems/coralreef/palmyra), and in conjunction with computed values of bottom roughness heights z0 across the atoll
(Chapter 5) are used to infer z0 over the model domain (Figure 6-2d).
Initial conditions and lateral boundary conditions (u, T, S, ζ) are interpolated from the National Ocean Partnership Program (NOPP)'s HYbrid Coordinate Ocean Model (HYCOM) global ocean model (http://hycom.org). Boundary conditions for the
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SWAN model are taken from measured wave height and period on the north (FR9) and south (FR5) of the atoll, corrected for changes to height and travel time from the measured location to the model boundary conditions. Wave angle is taken from the NOAA Wave Watch III Hawaii Model results (http://polar.ncep.noaa.gov/waves/index2.shtml). Tidal constituents are derived from the 2011 Oregon State University Pacific Ocean 1/12° Tidal Atlas (http://volkov.oce.orst.edu/tides/PO.html ). Initial conditions for T and S within the lagoon system are taken from the WL temperature mooring, and annual CTD casts within the lagoon.
Atmospheric winds, shortwave and longwave radiation, rainfall, air temperature, relative humidity, and air pressure were included in the model and taken from the weather station on the atoll (Figure 6-1b). For periods of time when data was not available from on island measurements, atmospheric forcing was interpolated from NASA's Modern-Era Retrospective Analysis for Research and Applications (MERRA) global atmospheric model (http://disc.gsfc.nasa.gov). Notably, these periods include wind (prior to Sep-2013), rainfall (after Dec-2013), and longwave radiation (prior to Sep-2013).
Lagrangian floats were released for each simulation starting approximately 2 days after model initialization, with some simulations varying this time to vary the tidal cycle of release. 22,447 floats were released within the model domain in and around the atoll, spaced evenly between the bottom, mid and top of the water column. The
Lagrangian residence time, (휏퐿) is defined as a time taken for a parcel of fluid to leave a boundary [Monsen et al., 2002], here taken as the 100 m depth contour. 휏퐿 was computed for each particle and then averaged over the three initial floats at each location. The offshore travel time, or age (휏퐴) represents the amount of time an individual fluid parcel or particle has spent inside of a domain since it entered a boundary [Monsen et al., 2002], here taken as the 100 m bathymetric contour. 휏퐴 was computed for each float as the time since the float crossed the 100 m bathymetric contour into the atoll. Results of 휏퐴 for all particle tracks were interpolated to a fine
160 scale 10 m grid and then down sampled to the model grid using a median filter. The
Lagrangian flushing time 휏푒 (efolding time) was computed as a best fit to, 푁(푡) =
−푡/휏푒 푁0푒 , where N is the number of particles at any time t, and N0 is the initial number of particles within a region of interest [Monsen et al., 2002].
6.3 Results and Discussion
6.3.1 Model Validation and Performance Two separate periods of simulation were modeled using the COAWST model. Due to processing time limitations, runs were limited to 14 day periods. The first, referred to as Run 1, simulates 3-Nov to 17-Nov 2012, and is a period of stronger northern waves
(1.19 m Hs from the north), and eastern regional currents (Table 6-1). The second, referred to as Run 2, simulates 26-Sep to 10-Oct 2013, and is a period of stronger southern waves (1.05 m Hs from the south) and northeastern regional currents (Table 6-1). Run 1 represents approximately average conditions for the 2012-2014 field measurements, while Run 2 represents less frequent (although not uncommon) forcing scenario (Figure 5-2).
Run 1 is used for validation because it is the period of average conditions with substantial coincident field data (Chapter 5). The model results compared to measured field observations for ζ, Hs, U, V, and T at five selected sites generally show good agreement (Figure 6-3). Overall available field data sites, model skill for ζ is 0.94 to
0.99 at all sites (Figure 6-3a-e), for Hs is 0.90 to 0.98 (Figure 6-3f-h), for U is 0.18 to 0.86 (Figure 6-3k-m), for V is 0.12 to 0.45 (Figure 6-3p-s), and for T is 0.61 to 0.84 (Figure 6-3u-y). More detailed validation information and model skill are shown in Appendix B to this dissertation. Overall, after a period of about 2 days for model spin up, the model results for ζ, Hs, U, V, and T reasonably represent the mean and variability at each site.
The COAWST model performs well in simulating the field scale flows observed on the atoll (Figure 6-3). The model skill is excellent for the free surface and waves, and
161 the model reasonably reproduces the mean and overall variability of velocities and temperature, which is typical of field scale models [Kumar et al., 2015].
Bottom roughness values computed for various sites (Chapter 5) in conjunction with benthic mapping (Figure 6-2b,c) were used to infer roughness values for the entire model grid (Figure 6-2d). The model uses a revised friction formulation in the SWAN wave model which correctly parameterizes the high wave friction on the atoll, as well as breaking coefficient values obtained from site data (Chapter 4). Considering the model complexity needed to achieve realistic results in bathymetry, bottom roughness, forcing (waves, wind, metrological, tides, regional boundary conditions), the results are quite good. For predicting local flows on the atoll, the model bathymetry is the most important input parameter, especially in areas of hydraulic controls such as the reef crest, channel, and lagoon inlets. Additionally, a critical factor is smoothing the grid sufficiently to achieve model stability, while still maintaining realistic geometry. Finally, because bottom stress is a significant term in the momentum balance for areas of shallow depth on the atoll, correct parameterization of the bottom roughness is essential.
Initial implementation of wave forcing using the radiation stress method of Mellor [2008, 2011], yielded unrealistically high velocities near the surface, an issue also noted by [Kumar et al., 2012]. Implementation of the vortex force method for this model produces realistic vertical velocity profiles and mean flows, and thus appears to be the preferred method for modeling the effect of waves on currents in three- dimensions.
6.3.2 Interaction of Atoll with Regional Flow In the presence of a strong easterly regional current on 06-Nov-2012, the atoll creates flow separation and a large wake, which extends a significant distance offshore from the forereef, with shed vortices clearly visible, while in the presence of a northerly regional current on 29-Sep-2013, the wake is less clearly defined but with vortices still present (Figure 6-4).
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At the scale of the atoll itself, strong regional flows create flow separation and a well- defined wake in the model results (Figure 6-4). Vortices are shed from leading edge of the wake which propagate downstream in the mean flow. At times, these vortices migrate towards the atoll and interact with the forereef, or alternately may spin off into the open ocean. There are also some locations of persistent circulation forced by irregularities in the atoll shape, most notably on the southwest corner of the atoll.
The primary two-dimensional momentum balance at this scale is between acceleration (ACC), pressure gradient (PG), and horizontal advection (HA) (Figure 6-5). The model includes only the upper 200 m, and is essentially modeling the upper mixed layer, the depth of which is set by the regional stratification. Thus the flow dynamics are primarily governed by the forcing from the regional flow and its interaction with the geometry of the atoll.
These processes are similar in nature to the fundamental fluid mechanics problem of two-dimensional flow past a cylinder, where the Reynolds number governs the flow. For island wakes, Wolanski [1996] proposed an island wake parameter P, essentially the Reynolds number times the similarity ratio,
푢퐿 퐻 2 푃 = ( ) ( ) , (4) 퐾푧 퐿 where u is the free stream velocity, L is the island horizontal scale, H is the island vertical scale and Kz is the vertical mixing viscosity. Based on laboratory data, for P < 1, there is no wake, for 1< P < 10 the wake is stable, and for P > 10 vortices are shed from the island [Wolanski, 1996]. For Palmyra, we assume L varies from 4 to 19 km depending on flow direction, u varies from 0.1 to 0.8 m/s, the upper mixed layer depth -5 -7 H is typically 100 to 200 m, and Kz is order 10 to 10 . With these assumptions, P would be order 105 to 107, well within the unstable vortex shedding regime and consistent with modeling results. For certain conditions such as very low free stream velocity, high mixing, or shallow mixed layer depth, the wake could become stable, an effect which could be studied further. In addition, this simplified model ignores
163 stratification within the upper mixed layer, Coriolis effects, and bottom stress, all of which could be considered.
6.3.3 Circulation within Atoll At the smaller scale of the atoll, the waves are sheltered on the lee of atoll and focused on the exterior edges (Figure 6-6a,b). Mean free surface setup is highest along the shallow reef flats on the side of the atoll with highest wave height (Figure 6-6c,d). (Figure 6-6f). Surface temperature is lowest during the night (Figure 6-6e), and highest during the day (Figure 6-6f) on the shallow interior areas. The wave driven flow is in the direction of the dominant waves over the reef crest, easterly through the lagoon system, and variable on the eastern and western terraces (Figure 6-6g,h).
Model results for Run 1 average magnitude of the momentum terms are shown in Figure 6-5. Offshore of the atoll, the significant terms are ACC, HA, and PG (Figure 6-5a,b,d), while on the atoll the significant terms are ACC, HA, PG, HVF, Bstr, Str, and RA depending on the location (Figure 6-5a,b,d,e,f,g). The COR and HM terms are weak everywhere (Figure 6-5c, HM not shown).
Waves primarily coming from the north (Figure 6-7a), create a classic set down in the shoaling region and then setup within the surf zone, and decrease in mean surface across the western terrace (Figure 6-7b). In the x direction the PG term is important, which on the forereef and surf zone is balanced by RA, HA, BT, and HVF, and within the shallow reef terrace onshore of the surf zone is balanced by HA, BT, and Sstr (Figure 6-7c), and thus driving a flow in the negative x direction (Figure 6-7e). In the y direction the pressure gradient PG (from the gradient of ζ, Figure 6-7b), on the forereef and surf zone is balanced by RA, HA, and BT (in shallow depth), and within the shallow reef terrace onshore of the surf zone is balanced by HA, BT, and Sstr (Figure 6-7d), overall driving a flow in the negative y direction into the atoll interior (Figure 6-7e).
Circulation within the atoll is primarily governed by tides and waves, and secondarily by wind and regional flows. Tidal forcing is significant at all field sites in both cross- and alongshore directions, primarily at the M2 frequency, and the oscillations
164 generally follow the alongshore bathymetric contours (Chapter 5). The regional tidal oscillations drive local pressure gradients (PG), which are balanced by horizontal advection (HA) and bottom stress (Bstr). The free surface tidal amplitude is only slightly reduced with propagation into the lagoon system, but is significantly delayed by up to 4.3 hours in the interior East Lagoon due to the constricted inlets (Chapter 5).
Waves drive flow through two processes (1) breaking and dissipation from bottom friction which create a roller acceleration (RA), and (2) Stokes drift from waves interacting with the mean vorticity to produce a horizontal vortex force (HVF) (Figure 6-6). In the cross shore direction, RA is the dominant forcing mechanism from waves, while in the alongshore direction, HVF may also be important. Both forces are opposed by a pressure gradient (PG), bottom stress (Bstr) at shallow depths, and horizontal advection (HA), create an elevated mean surface onshore of the surfzone, and also drive flow. The resulting pressure gradients drive flow from the side of the atoll with larger waves to the side of smaller waves. This setup effect is largest when one side of the atoll has much larger waves than the other. For larger waves on the north side of the atoll (which are the most common conditions), this drives flow generally from north to south. Flow is enhanced through the lagoon system from east to west and flow exits through the channel, a response clearly seen in the field data (Chapter 5) and modeling float results. For larger waves on the south of the atoll, this generally drives flow from south to north across the atoll. Flow is not increased in the channel, since there is not a pressure gradient in that direction, but is increased across the terraces. Over the field record, waves are most often larger on the north of the atoll, and thus the dominant flow path is from east to west through the lagoon system and out the channel, explaining the net long-term outflow from the channel (Chapter 5).
Wind-driven flow is weakly coherent at a few of the field sites, primarily on the western terrace (Chapter 5). Winds drive flow by imparting a surface stress (Sstr), which is generally weak, except in shallow regions, where it can be a significant term. Wind driven flow in the cross shore direction at the channel site (likely from wind
165 effects on the west lagoon) and northeast forereef (likely from wind effects on the shallow back reef) are present but weak.
The measurement stations on the forereef (FR3, FR5, FR7, FR9) are largely sheltered from the large scale regional flow due to their location on the forereef, and showed no significant coherence with the large scale HYCOM flows. However, flows on the western terrace (RT4), were coherent with the regional flow in the alongshore direction (approximately north-south) as the regional flows can drive flow across the exposed terrace, as seen in the model results (Figure 6-6g,h).
Bottom stress is a significant term in the momentum balance for shallow depths less than about 10 m (Figure 6-7). Thus for these shallow regions which cover a significant fraction of the atoll, correct parameterization of the roughness is essential to accurately modeling the flow. For this study, extensive field data at multiple sites allowed for calculation of roughness values, which are consistent with results from other reefs [Rosman and Hench, 2011; Lentz et al., personal communication]. However, the theory is currently not well developed for selection of roughness values a priori from high resolution bathymetric data or other site mapping, and should be considered for future work.
While the analysis of circulation likely captures the primary mechanisms for flow, density driven flows are also likely important on the atoll. The lagoon system is stratified, and the channel velocity profile shows likely evidence of classic baroclinic exchange flow (Figure 5-6f). Shallow water on the reef flats is likely saltier due to differential evaporation; and with cooling at night, this water could form bottom density currents which propagate into the interior lagoons or along the exterior forereefs. Thus, further inquiry into density driven flows at the site is warranted.
6.3.4 Ecological Implications Because of the vibrant coral reef ecosystem on the atoll’s terraces and forereefs, the spatial variation in hydrodynamic properties is of particular ecological relevance. The physical properties of interest include wave stress, light, mean flows, offshore travel time (age), and temperatures. On Palmyra, wave stress has been shown to exert a
166 significant control on both geomorphic structure and biological cover, while depth (a proxy for light) had only a limited effect (Chapter 4).
Results for particle tracks for Run 1 show particles generally going east once offshore in the strong regional flow (Figure 6-8). Largrangian flow paths within the interior lagoons are nearly always to the west and out the main channel or shallow reef flats (Figure 6-8d,g,h). Despite these mean trends, individual particles can follow alternate routes due to tidal phasing, vortices and other nonlinear interactions.
The flushing time for the different hydrologic regions varies from 1.0 to 9.8 hr for the terraces forereef and reef flat locations, while the interior lagoons averaged over all depths vary from 9.6 to 174 hr (Table 6-1). The deep lagoon areas greater than 20 m depth likely have flushing times of 1 to 3 years based on dissolved oxygen profiles [Gardner et al., 2014]. The steep slopes of the lagoon sides and exterior forereefs in the model likely create spurious vertical currents, which while of minor effect on the energetic forereef, may significantly alter mixing in the deeper lagoons. Thus model results for flushing times in the deep lagoons are likely too high. Stratification within the lagoons likely has a strong effect on these results.
Average flow properties were computed from the results of both runs. The mean near- bottom velocity magnitude is highest along the western forereef, and shallow reef crest, up to 0.5 m/s (Figure 6-9a). Mean depth-averaged water age was less than 20 hours on the exterior reefs and terraces and up to 130 hours in the interior lagoons (Figure 6-9b). The average 95th percentile of temperature was relatively uniform on the exterior reefs and terraces near 29.5º C, but varied up to 33º C on the shallow lagoon flats (Figure 6-9c). Mean flows are highest on the western forereef and on the reef crest. Mean age is less than 15 hours on the terraces, and varies up to 140 hours in the west lagoon. These results are consistent with bulk flushing time results, while the deep lagoon areas have significantly higher flushing times up to 369 hr. These times appear to be most dependent on the general flow conditions (Run 1 vs. Run 2) than on the starting tidal phase (Table 6-1). Finally, the average 95th percentile of temperature
167 is relatively consistent on the terraces, and maximum on the shallow flats within the lagoons and backreef areas.
Figure 6-10 shows over the entire model grid, the cumulative probability of mean flow ̅̅̅̅̅ th |푢푏| (Figure 6-9a), mean age (Figure 6-9a), and mean of 95 percentile of temperature
푇̅̅푏̅95̅̅ (Figure 6-9c) for each of the geomorphic structures and biological cover based on benthic mapping (Figure 6-2b,c). For geomorphic structure, while mean flow and temperature show some trends, age appears to be the largest differentiator between types with Aggregate Reef and Spur & Groove having low age, and Sand and Mud having high age (Figure 6-10a,b,c). For biological cover, sites with coral cover greater than 50% show high mean flow, low water age and low temperature compared to sites of algae and low coral cover (Figure 6-10d,e,f).
The geomorphic structure of the atoll shows some expected trends with mean flow and temperature. Notably, mud areas have low mean flow and low temperature, while sand areas have moderate mean flow and high temperature. Aggregate reef generally has higher mean flows and lower temperature than the other geomorphic types, but the trends are generally weak. Mean offshore travel time age, however, appears to clearly differentiate the geomorphic structures with types of with high biological cover (aggregate reef, spur and groove) having a low mean age less than 10 hours, while geomorphic structures of with little biological cover(reef rubble, sand, mud) having much higher mean offshore travel time.
The biological cover shows significant trends with mean flow, offshore travel time and temperature. Areas of high coral cover (>50%) have higher mean flows, low age and low temperature. Areas of very low coral cover (<10%) have low water age and high temperature, but about average mean flow. Areas of algae similarly have moderate age and temperature, but about average mean flow. Thus, 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.
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Offshore travel time is defined as the average time taken for offshore water to come to a particular point on the atoll. This metric likely includes the effect of multiple processes on the reef, notably water quality and temperature. In terms of water quality, the longer a parcel of water spends in the atoll system; it experiences increased effects from changes to nutrients and phytoplankton compared to offshore waters. In terms of temperature, the longer a parcel spends in the atoll system, the more it is subjected to the local heating effects, as compared to the relatively stable offshore water temperatures. While the present model results do not incorporate nutrients or phytoplankton, the model results for age and temperature exhibit very similar qualitative trends. Thus, it appears at this site, the effect of age on biological cover is likely mostly a function of temperature and to a lesser extent water quality.
Future work could incorporate a nutrient, sediment, and NPZD components into the model to explicitly examine these effects. It should also be noted, that the temperature effects in the modeling results are based on two-week simulations and are therefore representative of the diurnal heating cycle, but not long-term temperature trends, an effect which is investigated in Chapter 5 using the long-term on-island measurements.
6.3.5 Connectivity The Lagrangian connectivity is obtained from the float track model results and summarized in Figure 6-11. The dominant regions serving as a source are the West Lagoon, Channel and nearby reefs. Within the atoll interior, primary connectivity is east to west, while on the exterior of the atoll primary connectivity is west to east, consistent with mean flow paths. However, limited connectivity occurs between all regions due to tidal phasing, large scale vortices, and other nonlinear effects where individual floats can follow independent tracks which are quite different than the mean flow. Thus it appears possible for a parcel of water to travel between any region of the atoll for a given set of flow conditions.
It should be noted that this study only considered the connectivity of water parcels passing over each region. Thus, while a parcel of water may pass over a region, it may not be in contact with the bottom, which would be of ecological importance.
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Additionally, the Lagrangian floats in these simulations are neutrally carried by the mean velocities, and do not exhibit any swimming patterns, such as exhibited by some larvae. Additional work could include explicitly modeling larval connectivity within the atoll system.
6.4 Conclusions We present results of the hydrodynamics of an atoll system based on modeling simulations using a coupled waves and three-dimensional hydrodynamic model (COAWST) applied to Palmyra Atoll. Using the vortex force formulation, the COAWST model performed well in simulating the field scale flows. 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, and consistent with previous work on island wakes. Vortices are shed from the leading edge of the wake, which can interact with the atoll. The effects from the regional flow are largest on the exposed exterior sides of the atoll, and the field monitoring locations were generally sheltered and did not experience these effects.
Circulation within the atoll is primarily governed by tides and waves, and secondarily by wind and regional currents. Wave driven flow is significant over much of the atoll, and is a strong function of the dominant wave direction. The typical condition of strong waves from the north drives flow from north to south across the atoll, and from east to west through the lagoon system and out the channel. When wave direction is reversed with stronger waves from the south, flow is generally from south to north across the terraces, but flow within the lagoons is unaffected. The vortex force term appears significant only in the alongshore flows near the forereef and surf zone, while in the cross-shore direction the roller acceleration is the primary wave term. Wind driven flow is generally weak, except on the shallow terraces. Bottom stress is a significant term in the momentum balance for shallow depths less than about 10 m. Thus for these shallow regions which cover a significant fraction of the atoll, correct parameterization of the roughness is essential to accurately modeling the flow.
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Based on Lagrangian float tracks, the mean offshore travel time (age) was the best predictor of geomorphic structure and appears to clearly differentiate the geomorphic structures with types of high biological cover (aggregate reef, spur and groove) having a low mean age less than 10 hours, while geomorphic structures with low biological cover (reef rubble, sand, mud) having much higher mean age. This is likely due to higher mean temperatures in the interior lagoons. The biological cover shows significant trends with mean flow, age and temperature. 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 model results for age and temperature exhibit very similar qualitative trends, and thus, it appears at this site, the effect of age on biological cover is likely mostly a function of temperature and to a lesser extent water quality.
The Lagrangian connectivity is obtained from the float track model results. The resulting connectivity within the atoll system shows that the general trends follow the mean flow paths, i.e. from west to east on the atoll exterior reefs and from east to west within the interior lagoon system. However, some connectivity exists between all regions of the atoll system, due to tidal phasing, vortices and other nonlinear interactions.
Future work could examine the effects of density driven flow, which likely important in some regions of the atoll such as the lagoon outlet via the channel. Secondly, future work could explicitly model larval connectivity, considering vertical location in the water column and larval swimming patterns. Finally, for this study, extensive field data at multiple sites allowed for calculation of roughness values to parameterize the model. Development of methods for selection of roughness values a priori from high resolution bathymetric data or other site mapping would warrant further inquiry.
6.5 Acknowledgements The data from this study will be deposited at the NOAA NCEI data repository after the manuscript is accepted for publication. We thank Oliver Fringer, Matthew Rayson,
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Sean Vitousek, John Warner, and Nirnimesh Kumar for assistance with modeling. Bathymetry and other information were provided by Jamison Gove. Research funding was provided by Stanford University along with two grants from the Gordon and Betty Moore Foundation (“Observations and modeling of the C system dynamics at Palmyra Atoll: In support of the development of management strategies for ocean acidification impacts in the tropics,” to RBD and, “Understanding coral reef resilience to advance science and conservation,” to RBD and SGM). This research was made with Government support under and awarded by the U.S. Department of Defense, Office of Naval Research, NDSEG Fellowship, 32 CFR 168a to JSR. DAK was funded by an NSF Graduate Research Fellowship. This is Palmyra Atoll Research Consortium contribution number PARC-XXXX.
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6.6 Figures and Tables
Figure 6-1. Palmyra Atoll location, site layout and experiment instrumentation (a) location of Palmyra Atoll, (b) layout of atoll and instrument locations for long term measurement [(U, ζ, T) magenta squares; (ζ, T) magenta circles; (T) small magenta circles), short term hydrodynamics experiments (yellow circles), and weather station (green star), image courtesy of NOAA. (c) typical northern forereef near FR9 with abundant live coral and fish, (d) typical terrace with blacktip shark near PSM courtesy of Brian Zgliczynski, (e) typical shallow back reef with live coral near NB, and (f) typical lagoon rubble bottom with low visibility water near CHAN with diver and ADV for scale.
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Figure 6-2. Model grid bathymetry and bottom roughness zoomed to atoll; total grid extent is about double view shown. (a) depth grid h (m), (b) geomorphic structure, (c) dominant biological cover, (d) assumed bottom roughness scale z0 (m), and (e) classified zones for flushing time and connectivity computation. Gray lines are (2, 10, 30, 60, 200 m) depth contours for (a), 60 m for (b,c,d,e); white shading is land mask.
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Figure 6-3. Selected model validation data for four sites (FR5, FR9, RT4, CHAN, and NB) for Run 1 (Nov-2012) for (a-d) mean surface 휁, (e-h) significant wave height Hs, (i-l) x velocity U, (m-p) y velocity V¸ and (u-y) temperature T. Blue is field measurement, red is model result. Time is since start of model run. Velocity is depth-averaged, except at RT4 site, which is near bottom. No field measurements available for (h, j, o, t).
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Figure 6-4. Regional flow interaction with the atoll, for 24 hour sequence starting 06-Nov- 2012 (right) and 29-Sep-2013 (left). Black arrows are surface velocity us, coloring is surface vorticity ω (10-4 1/s), gray shading is atoll with black lines as 5 and 60 m bathymetric contours. Results only shown for depths greater than 10 m.
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Figure 6-5. Model results average magnitude of significant momentum terms during Run 1, 5- Oct-2012. (a) acceleration (ACC), (b) horizontal advection (HA), (c) Coriolis (COR), (d) pressure gradient (PG), (e) horizontal vortex force (HVF), (f) bottom stress (Bstr), (g) surface stress (Sstr), (h) wave roller acceleration (RA). Values are log10 of magnitude of terms over two hour period, (m/s2). Gray lines are 5 and 60 m depth contours, white shading is land mask.
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Figure 6-6. Atoll scale model results snapshot of waves, free surface and surface velocity for time period of dominant northern waves (left) and dominant southern waves (right). (a,b) significant wave height Hs, (c,d) free surface 휁, (e,f) surface temperature T, and (g,h) surface velocity 푢푠푢푟푓. Gray lines are 5 and 60 m depth contours; white shading is land mask.
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Figure 6-7. North-south profile of atoll showing waves, free surface, velocity, and significant momentum terms during Run 1, 5-Oct-2012. (a) significant wave height Hs, (b) free surface ζ, (c) x momentum terms, (d) y momentum terms, (e) depth-averaged Lagrangian xy velocities (U,V) and (f) shallow bathymetry (black) with free surface (blue). Location of profile shown in Figure 6-6a.
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Figure 6-8. Lagrangian float tracks for Run 1 (Nov 2012), grouped by starting zone (magenta). Colors are time from float initialization on day 2 (blue), to end of run on day 14 (red). Only 20 floats per zone are shown for clarity. Black lines are 5 and 60 m depth contours, white shading is land.
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Figure 6-9. Model results average velocity, age and high temperature. (a) average near bottom ̅̅̅̅̅ th velocity |푢푏|, (b) mean water age, and (c) average of 80 percentile two day average near bottom temperature 푇̅푏80. Averages taken over all model runs. Gray lines are 5 and 60 m depth contours; white shading is land mask.
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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. ̅̅̅̅̅ th Geomorphic structure with (a) average near bottom velocity |푢푏|, (b) age, and (c) 80 percentile of two day average near bottom temperature 푇̅푏80; and biological cover with (d) ̅̅̅̅̅ |푢푏|, (d) age, and (f) 푇̅푏80. Results are average over all model runs.
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Figure 6-11. Connectivity between hydrodynamic zones. (a) connectivity matrix showing probability a water parcel starting in source zone will pass through destination zone, and (b) geographic connectivity, where size of node is relative importance as an overall source, width of line is relative strength of connection for westward movement (blue) and eastward movement (red). Results based on average of all model runs; gray line is 100 m contour.
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Table 6-1. Model run details and computed efolding flushing time for each zone. Hydrodynamic properties are average over run from float release to end of run.
Run 1 Run 2 Dates 3 to 17-Nov 2012 26-Sep to 10-Oct 2013 Wave Hs (North, South) (m) 1.19, 0.90 0.84, 1.05 NECC Ureg,Vreg (m/s) 0.61, 0.00 0.48, 0.11 Wind U10, V10 (m/s) -3.2, 0.4 -3.7, 1.6 Tidal phase at float start Low High Low WT FR NW 0.7 0.6 1.1 WT FR SW 0.8 1.0 1.4 WT W 1.3 1.3 4.1 FR NW FR9 1.3 0.8 3.2 WT NE RTx 1.3 1.0 2.8 WT SE 1.8 2.1 2.0 CHAN 1.3 2.3 1.3 FR SW FR3 5.6 5.6 14.7 BR SW 1.7 3.0 1.0 West Lagoon (WL) 94.7 104.7 159.7 Flushing BR NW 3.3 2.1 2.1 Time for FR N 2.4 1.6 7.1 Zone (hr) FR NE FR7 1.9 1.0 6.3 BR NE 3.3 4.3 4.4 Central Lagoon (CL) 66.6 50.9 149.4 East Lagoon (EL) 76.4 79.0 174.2 Eastern Lagoon (EL2) 9.6 9.6 15.7 BR SE 0.9 1.4 0.9 FR SE FR5 4.6 8.9 4.9 ET FR S 2.5 2.2 5.9 ET 3.4 1.7 10.8 ET FR N 0.9 1.0 7.4
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Chapter 7 Conclusion
7.1 Summary of Key Findings This dissertation investigates the physical oceanography of coral reef environments, specifically focusing on waves and mean flows at two scales. At small scales on the order of ten to a hundred meters, the role of spur and groove formations and their interaction with surface waves and mean flow is examined. At large scales on the order of hundreds of meters to kilometers, the wave and mean flow dynamics of a pacific atoll are investigated. The effect of hydrodynamic properties on the ecology is discussed throughout.
7.1.1 Small Scales – Spur and Groove Formations 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-
185 and-groove wavelength allows for the effects of diffraction to create alongshore differences in wave height without changing the mean wave angle.
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 depth- averaged Lagrangian offshore flow over the spur and onshore flow over the grooves on the order of 1 cm/s. This circulation was stronger for larger, directly-incident waves and low alongshore flow conditions, consistent with predictions from modeling. Favorable forcing conditions must be maintained on the order of one hour to accelerate and develop the SAG circulation cells. The primary cross- and alongshore depth-averaged momentum balances were between the pressure gradient, radiation stress gradient and nonlinear convective terms, and the bottom drag was similar to values found on other reefs. The vertical structure of these circulation cells was previously unknown and the results show a complex horizontal offshore Lagrangian flow over the spurs near the surface driven by alongshore variability in radiation stress gradients. 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. Beyond the influence of light levels that are generally higher on the spur, 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.
7.1.2 Large Scale – Waves and Hydrodynamics of a Pacific Atoll System 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 waves and three-dimensional hydrodynamic model (COAWST) to characterize the waves and hydrodynamics operant on the atoll. Our results indicate that wave energy is primarily from the north during the northern hemisphere winter and from the south in the northern hemisphere summer. Refraction of waves along the reef terraces due to variations in bathymetry leads to focusing of waves in specific
186 locations. We observed transfer of energy to low frequency infragravity waves strongly coherent with the tides. 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. Mean water level setup around the atoll can be characterized by a constant breaking fraction, and this breaking fraction is dependent on slope, consistent with other results for reefs and sandy beaches. The SWAN wave model with a modified bottom friction formulation predicts bulk wave energy properties at our measurement stations with 0.86-0.94 model skill. The near bed squared velocity, a proxy for bottom stress, shows strong spatial variability across the atoll and exerts control over geomorphic structure and high coral cover.
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. The typical condition of strong waves from the north drives flow from north to south across the atoll, and from east to west through the lagoon system and out the channel. Wind driven flow is generally weak, except on the shallow terraces. Based on Lagrangian float tracks, the mean offshore travel time was the best predictor of geomorphic structure and appears to clearly differentiate the geomorphic structures, likely due to increased average temperatures in the interior lagoons. The biological cover shows significant trends with mean flow, age and temperature. 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
187 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. This suggests that lower mean temperatures may be primarily governed by residence time or offshore travel time (age). The resulting connectivity within the atoll system shows that the general trends follow the mean flow paths, however, some connectivity exists between all ten regions of the atoll system.
7.2 Future Research At small scales of spur and groove formations (Chapters 2 and 3), future work could include extending the modeling to be fully three-dimensional, and extend field measurements beyond the one field measured cross-shore location. At shallower depths, the direction of the depth-averaged cross-shore circulation (Uc) is likely reversed [Rogers et al., 2013], but this effect and the three-dimensional Lagrangian velocity structure at these shallower depths remains unknown and requires further study. The observed similarity to Langmuir circulation but with opposite rotation, and the possible importance of the vortex force and lateral variation of stress mechanism would warrant further analytical or modeling work. Thus a fully three dimensional model incorporating the vortex forcing mechanism, such as COAWST, could yield important results. Additionally, SAG hydrodynamics under large wave conditions, as well as investigation into the sediment transport through the grooves, also warrants further inquiry.
At larger scales of the atoll (Chapters 4, 5 and 6), future work could include connecting bathymetric roughness and complexity data with prediction of wave and mean flow bottom dissipation and quantification of wave friction factors and drag at locations with higher bottom friction than measured on Palmyra Atoll. Additionally, previous studies have shown the importance of physical parameters such as wave stress and mean flow in shifting benthic regimes between species and coral morphology. Future work could aim to couple high resolution ecological and hydrodynamic studies to better understand biophysical coupling on coral reefs.
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Future hydrodynamic work on Palmyra could examine the effects of density driven flow, which is likely important in some regions of the atoll such as the lagoon outlet via the channel. Finally, future work could explicitly model larval connectivity, considering vertical location in the water column and larval swimming patterns.
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Appendix A - Supporting Information for Wave Dynamics of a Pacific Atoll with High Frictional Effects
This appendix contains supporting information for Chapter 4 of this dissertation.
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Supporting Information for
Wave Dynamics of a Pacific Atoll with High Frictional Effects
Justin S. Rogers1, Stephen G. Monismith1, David A. Koweek2, Robert B. Dunbar2
1. Environmental Fluid Mechanics Laboratory, Stanford University, 473 Via Ortega, Stanford, California, 94305, USA
2. Department of Earth System Science, Stanford University, Stanford California, 94305, USA
Contents of this file
Text S1 Text S2 References Figures S1 to S6 Table S1
Introduction
This document contains additional supporting information relating to seasonal wave properties, infragravity wave climate, spectral energy transformation, breaking fraction estimation, and model calibration.
Text S1. Swell Wave Climate
The average wave height well offshore of the atoll Hs0, varied from 2.8±0.6 m in
January to 1.8±0.3 m in September (Figure S1a). Mean wave period Tm was nearly constant seasonally at approximately 11±3 s, and mean wave direction θm in November through February was primarily from the north with high variability (±130°), while in July through September was from the southeast with lower variability (±50°) (Sup. Inf. Figure S1b,c). On average, the north forereef receives
191 approximately twice the total swell energy flux as the southern forereef, and at some locations within the interior of the atoll, tides play a significant role in modulating the wave energy (Figure S2a).
The wave spectra show short period waves of 0.1-0.15 Hz that were strongest on the forereefs (FR9 and FR3) and weaker on the western terrace (RT4), (Figure S3 a,b,c), and coincident with stronger wind speeds (Figure S3 d), thus consistent with locally generated wind waves. Longer period swell waves of 0.04-0.1 Hz were the dominant wave events on the forereef and western terrace, and with a mean frequency increasing in time, consistent with remotely generated swell events and the dispersion relation (Figure S3 a,b,c).
The average energy flux on the north forereef (FR9) was approximately double that on the southern forereef (FR3 and FR5), and an order of magnitude greater than on the western terrace (RT4), during the measurement period (Figure S2a). The squared coherency γ2 (Coh) between two signals is given by, 2 2 2 훾 (푓) = |푆12(푓)| ⁄푆11(푓)푆22(푓) with limiting value for 95% confidence, 훾95 = 1 − 0.05[1/(퐸퐷푂퐹−1)], where EDOF is number of independent realizations [Emery and
Thomson, 2004]. Squared coherence of Hs with the M2 tide is not correlated on the western forereefs (FR3 and FR9), is correlated on the southeastern forereef (FR5), and is strongly correlated on the terrace (RT4) and in the channel (CHAN) (Figure S2 a). Swell wave energy flux on the western terrace (RT4) is strongly correlated with the energy flux on the northern side of the atoll (FR9) for periods of stronger northern waves (Figure S4a) and weakly correlated with the energy flux on the southern side of the atoll for periods of stronger southern waves (FR3) (Figure S4e).
Text S2. Infragravity Wave Climate
Wave energy in reef environments is often concentrated in swell frequency bands with periods less than 25 s; however, wave energy can also be significant at very long periods of 33 to 600 s, also known as infragravity waves (IG). In shallow depths of the surf zone or shoaling zone, the nonlinear character of wave transformation is
192 expressed through energy transfer from the main spectral peak toward higher frequencies (the generation of peak harmonics) as well as lower IG frequencies. [Guza and Davis, 1974; Sheremet et al., 2002; Sheremet et al., 2011]. The transfer of wave energy to higher frequencies is often dissipated quickly, but the energy transferred to low frequency infragravity waves can persist, and is often the dominant wave energy in shallow reef flat or terrace areas [Symonds et al., 1982; Lowe et al., 2005].
Over the measurement period, the average infragravity wave energy flux was highest on the western terrace sites (PSM, RT4, RT13); and shallow reef flats (NB, NBE), weaker on the forereef (FR3, FR5, FR9), and very weak in the lagoons (BE, CHAN, DOCK, EL) (Figure S2 b). The infragravity wave height showed very strong coherence (>0.9) with the M2 tide on the shallow reef flats (NBE, NB), and strong coherence at all the other sites except for the forereef. At FR5, the coherence was significant, but not at FR3 or FR9 sites (Figure S2 b). The ratio of wave energy flux in the swell band was O(1000) times that in the infragravity band on the forereef sites (FR3, FR5, FR9), while on the terrace the ratio was O(10) (RT4), and in the channel O(0.1) (Figure S2 a,b).
The infragravity wave energy on the western terrace shows strong correlation with incoming swell energy at RT4 from the north and south (Figure S4b,f), and at PSM site from the north and south (Figure S4c,g). On the shallow reef flat (NB) the correlation is strong for swell energy from the north, but not significant for swell energy from the south (Figure S4d,h).
The transfer of wave energy flux between frequencies is an important process on the atoll. On the north side of the atoll, the spectral energy flux on the forereef (FR9) has a peak frequency of 0.059 Hz (17 s) corresponding to remote swell, and a second peak at 0.11 Hz (9 s) corresponding to local wind-generated waves (Figure S5a). Onshore on the reef terrace (RT4), the peak frequency is 0.057 Hz, while the higher frequencies (> 0.07 Hz) are significantly reduced whereas the infragravity energy flux is enhanced with a peak near 0.010 Hz (100 s) (Figure S5a). Further to the east on the reef flats where the surf zone is strong, infragravity energy flux is enhanced with nearly flat
193 response at NBE. It is assumed that the forereef energy flux is relatively similar along the northern forereef under these conditions (due to sampling interval, the higher frequency waves were not captured at NBE).
For the NFR13 experiment on the north side of the atoll, the spectral energy flux has two peaks at 0.12 and 0.083 Hz at C, and with onshore propagation at B and A the energy flux is reduced for frequencies greater than about 0.1 Hz, is increased for frequencies greater than about 0.1 Hz (Figure S5b). There is also an increase in infragravity energy flux with a broad peak near 0.012 Hz.
On the south side of the atoll, the spectral energy flux on the forereef (FR3) has a peak frequency of 0.062 Hz (16 s) with a second peak of 0.12 Hz (8 s) (Figure S5c). Onshore of the surf zone and reef crest at PSM, there was significantly enhanced infragravity wave energy flux with a peak near 0.009 Hz (110 s) (due to sampling interval, the higher frequency waves were not captured at PSM). Onshore on the reef terrace (RT4), the peak frequency is 0.060 Hz, while the higher frequencies (> 0.07 Hz) are significantly reduced whereas the infragravity energy flux is enhanced with a peak near 0.010 Hz (100 s).
The net effect of this transfer to lower frequencies is an increase in infragravity wave energy flux on the terrace and reef flat areas (Figure S2b) which are correlated with incoming swell energy flux (Figure S4). The infragravity wave flux has very high coherence with the tidal level at all sites onshore of the surf zone, a process also noted by Lugo-Fernández, [1998], and Péquignet et al. [2009]. Transfer of energy flux to higher frequencies due to wave steepening is not readily apparent in the measurements, likely due to difficulty in separating the effects of friction and nonlinear transfer and the placement of the instruments outside the shallow shoaling zone.
References:
Emery, W. J., & R. E. Thomson (2004), Data Analysis Methods in Physical Oceanography, 2nd ed., 638 pp., Elsevier, Amsterdam.
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Guza, R. T., & R. E. Davis (1974), Excitation of edge waves by waves incident on a beach, J. Geophys. Res., 79(9), 1285–1291, doi:10.1029/JC079i009p01285.
Sheremet, A., J. Kaihatu, S. Su, E. Smith, & J. Smith (2011), Modeling of nonlinear wave propagation over fringing reefs, Coastal Eng., 58(12), 1125–1137, doi:10.1016/j.coastaleng.2011.06.007.
Sheremet, A., S. Jaramillo, S.-F. Su, M. A. Allison, & K. T. Holland (2011), Wave‐ mud interaction over the muddy Atchafalaya subaqueous clinoform, Louisiana, United States: Wave processes, J. Geophys. Res., 116, C06005, doi:10.1029/2010JC006644.
Sheremet, A., T. Guza, S. Elgar, & T. H. C. Herbers, Observations of nearshore infragravity waves: Seaward and shoreward propagating components, J. Geophys. Res., 107(C8), doi:10.1029/2001JC000970, 2002.
Symonds, G., D. A. Huntley, & A. J. Bowen (1982), Two-dimensional surf beat: Long wave generation by a time-varying breakpoint, J. Geophys. Res., 87(C1), 492– 498, doi:10.1029/JC087iC01p00492.
Lowe, R. J., J. L. Falter, M. D. Bandet, G. Pawlak, M. J. Atkinson, S. G. Monismith, & J. R. Koseff (2005), Spectral wave dissipation over a barrier reef, J. Geophys. Res., 110, C04001, doi:10.1029/2004JC002711.
Péquignet, A. C. N., J. M. Becker, M. A. Merrifield, & J. Aucan (2009), Forcing of resonant modes on a fringing reef during tropical storm Man-Yi, Geophys. Res. Lett., 36, L03607, doi:10.1029/2008GL036259.
Lugo-Fernandez, A., Roberts, H. H., Wiseman Jr, W. J., & Carter, B. L. (1998), Water level and currents of tidal and infragravity periods at Tague Reef, St. Croix (USVI). Coral Reefs, 17(4), 343-349, doi: 10.1007/s003380050137.
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Figure S1. Monthly variation in offshore wave parameters from WWIII model data (1979- 2014). (a) significant offshore wave height, (b) mean wave period, and (c) mean wave direction waves are coming from. Error bars are ± one standard deviation.
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Figure S2. Measured average swell and infragravity wave properties across the atoll. (a) swell waves, and (b) infragravity waves, with size of symbols proportional to average measured wave energy flux F (W/m), and squared coherence of wave height with M2 tide, 퐂퐨퐡 = ퟐ 휸 (푯풔, 휻) at 12.42 hr, and 95% confidence level is ±0.08. Results are mean over all available data at each respective site (2012-2014).
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Figure S3. Wave power spectral density 푆휁휁 and wind vectors for Nov 2013 to Jan 2014. 2 log10 푆휁휁 (m /Hz) at (a) North Forereef FR9, (b) Western Terrace RT4, and (c) South Forereef FR3. (d) east west (black) and north south (red) wind vectors.
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Figure S4. Histogram of swell wave energy flux 퓕퐬퐬, from north (top) and south (bottom), with swell 퓕퐬퐬 and infragravity energy flux 퓕퐢퐠 at different locations. North swell flux FR9 vs (a) swell flux at RT4, (b) infragravity flux at RT4, (c) infragravity flux at PSM, and (d) low pass infragravity flux at NB. South swell flux at FR3 vs (e) swell flux at RT4, (f) infragravity flux at RT4, (g) infragravity flux at PSM, and (h) low pass infragravity flux at NB. Color is from highest count (red) to lowest (blue). For top, data only shown for 퓕 larger on north side of atoll, and for bottom data only shown for 퓕 larger on south side of atoll. 퓕 is in kW/m. The low pass filtered infragravity flux is used for the NB site due to the very strong tidal influence.
Figure S5. Spectral energy flux transformation, (a) along northern side of the atoll for strong northern waves on 21 Dec 2013, (b) north forereef NFR13 experiment offshore of the surfzone on 05 Sep 2013, and (c) along southern side of the atoll under strong southern waves on 22 Sep 2013. Results are 6 hour averages.
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Figure S6. Estimation of breaking parameter 휸풔 from incoming wave energy flux. (a) offset height 훼 between forereef (FR7) and back reef (NBE) gauges from wave energy flux ℱ and reef mean setup, and (b) histogram of computed 휸풔between Sept-2013 and Jul-2014.
Table S1. Skill scores (SS) of measured vs model results for significant wave height Hs for 2012-2014 at different locations for different friction parameterizations and breaking parameter γs. Friction Method Existing Proposed Proposed SWAN (J66) (S74) (S74) γs 0.66 0.66 0.92 FR3 0.84 0.85 0.85 FR5 0.92 0.92 0.92 FR9 0.94 0.91 0.91 Skill Score (SS) RT4 0.33 0.85 0.86 RT13 0.21 0.43 0.43 NFR13 0.80 0.79 0.79
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Appendix B – Supporting information for Hydrodynamics of a Pacific Atoll System – Mechanisms for Flow, Ecological Implications and Connectivity
This appendix contains supporting information for Chapter 6 of this dissertation.
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COWAST Model Extended Validation Data
The following figures (B1-B6) contain extended validation data from the COWAST model of Palmyra Atoll. The validation data is for Run 1 (Nov, 2012), and compares all available field data from that time period to the model results. A skill score (SS) is presented for each plot. A skill score of NaN indicates no field data was available for this model run.
Figure B1. Free surface (ζ) validation results.
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Figure B2. Wave validation results of (top) significant wave height Hs, and (bottom) mean wave period Tm.
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Figure B3. Velocity validation results in EW direction u, which is depth-averaged velocity at all sites except RT10 and RT4 where it is near-bottom velocity.
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Figure B4. Velocity validation results in NS direction v, which is depth-averaged velocity at all sites except RT10 and RT4 where it is near-bottom velocity.
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Figure B5. Temperature validation results of near bottom temperature T.
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Figure B6. Atmospheric forcing data, wind (U10), rainfall, net shortwave radiation (SW) and net longwave radiation (LW).
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