CHARACTERISTICS AND IMPLICATIONS OF
BRIEF DURATION HYDRODYNAMIC FORCES
IN THE INTERTIDAL ZONE
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF BIOLOGY AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
Megan Morrison Jensen June 2014
© 2014 by Megan Morrison Jensen. 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/mt991tp5954
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.
Mark Denny, 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.
Derek Fong
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
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.
George Somero
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.
James Watanabe
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
iv Abstract
The hydrodynamic forces produced by breaking waves make the rocky intertidal zone one of the most physical stressful environments on Earth. Although the classical in- line hydrodynamic forces, drag and the acceleration reaction, are well studied and characterized, a third in-line force – the impingement force – may be the largest hydrodynamic force in the intertidal zone. Impingement is characterized by a sharp, transient spike in force at the instant of wave arrival, and very few measurements of it exist. I ask several questions about impingement: what variables affect its magnitude and frequency of occurrence? Most importantly, how do brief hydrodynamic loads of this sort affect intertidal organisms? I measured impingement in the laboratory, using a gravity-driven water cannon to simulate waves and tested which variables affect impingement magnitude. I show that impingement is likely drag, rather than an undescribed hydrodynamic force; it likely occurs due to brief increases in flow velocity at the front of a wave. To assess the frequency and magnitudes of impingement events in the field, I built a custom force transducer to record high- frequency forces in the intertidal zone. My data reveal that impingement events occur in 2-7% of waves in the field, but surprisingly, they are not the largest forces: the average magnitude of impingement events is lower than the average maximum force produced by all recorded waves in this study. This suggests that impingement is not the largest hydrodynamic risk to organisms, and future work measuring flows in the intertidal zone should focus on identifying large transient forces regardless of where they occur in a wave. Using these measurements, I explore the likely effects of brief forces on intertidal organisms in two ways: first, I model a limpet as a mechanical spring-mass-damper system and show that its foot has the capacity to reduce the effective force on the animal. I also predict dislodgment rates of three species of gastropod using the maximum forces recorded in my field study. I find that at my site, snails are very susceptible to wave-induced dislodgment. The chapters comprising this dissertation illustrate the necessity of recording high-frequency flows when studying intertidal hydrodynamics.
v
vi Acknowledgments
This thesis is dedicated to my parents, David and Mary Jensen.
It takes a village: I could not have written this dissertation and completed this scientific journey from naval architect to biomechanics Ph.D. without an incredible amount of help from many amazing people along the way. These few pages are not enough to express my gratitude for all who have helped me over the last several years, but I hope that the brief thanks here will suffice in lieu of a second dissertation composed solely of acknowledgments.
First and foremost, I would like to thank my advisor, Mark Denny. It is impossible to summarize everything I have learned from Mark, but I will attempt to list a few of the things I’ve valued most about the last few years. Mark’s enthusiasm and curiosity are inspiring and contagious, and I view the world differently now – in the best possible way. Mark’s excitement about learning and his expansive knowledge about an astonishingly broad range of topics make him an incredible mentor, and I feel incredibly fortunate to have had the opportunity to work with him. Mark is always willing to answer questions big or small, and more valuably, teach his students to answer those questions for themselves. And for questions without easy answers, Mark taught me that half the fun is figuring out how to ask the question, and then building something to answer it. Mark gave me the tools to succeed, the room to fail when necessary, and helped shape my projects, questions, and ideas into a dissertation. He taught me to be a scientist, and I cannot thank him enough for that.
My thesis committee has been exceptionally helpful and a source of many great ideas for my work. George Somero is one of the kindest, most insightful teachers I’ve ever had the pleasure of working with, and even made me tea during a meeting when a coughing fit broke out. Jim Watanabe’s door is always open, and he is perpetually ready to discuss ideas, limpets, algae, statistics, and education, and I appreciate every one of these topics more after sharing perspectives with him. Derek Fong taught one
vii of the best classes I have ever taken, and if I ever find myself teaching, I hope to model my lessons on his. Derek asked incredibly insightful questions which always cut straight to the point, and my thesis is greatly improved for his comments along the way. Stephen Monismith lent his expertise at blending hydrodynamics and biology to my committee, and graciously provided me a home away from Hopkins in Y2E2, which allowed me to get to know some of the wonderful people in the EFML.
I’ve been fortunate to share the lab with amazing labmates. I especially want to thank our two postdocs, Luke Miller and Kerry Nichols, for all of their help. They have been incredible role models, and have become great friends as well as coworkers. Nearly every aspect of this thesis was helped in some way by one of them: Luke helped with field work, machining, photographing, coding problems, more field work, and even gave car advice on numerous occasions. Kerry was always supportive, offering figure- making tips, fieldwork help, lab and life advice, and a friendly ear. My fellow graduate students, Tom Hata, Diana LaScala-Gruenewald, and Paul Leary, have all helped with field work, lunch chats, and great ideas, and I thank them for everything they’ve done. Kevin Miklasz and Sarah Tepler made the lab a fun place to work, even (especially) when an escalating prank war got slightly out of hand. Anton Staaf imparted circuitry and machining wisdom that continues to be valuable. I am also grateful to a number of visitors to the lab for their ideas, help, and guidance: talking through ideas with Chris Harley, Josh Madin and Bengt Allen during their tenures in the lab improved my work immensely, as well as made days in the lab even more enjoyable.
I would like to thank my cohort: Judit Pungor, Carolyn Tepolt, and Chelsea Wood were an incredible resource and support system over the last five-plus years. I count all three of these women among my close friends, and I cannot thank them enough for their examples of the kind of scientist and friend I want to be. From Boston Legal craft nights with Judit to Say Yes to the Dress and wine nights with Carolyn to weekly breakfasts with Chelsea, I owe these women a tremendous amount of thanks for their help, love, and friendship. I am grateful for all of the meals, editing, practice talk help,
viii stress relief, and laughter that they brought into my life. Extra thanks must go to Carolyn for supplying baked goods, wine, a beautiful wedding cake, and for telling me over and over and over, “It’ll be ok. Have a margarita.”
At Hopkins, we are extremely fortunate to have an amazing staff, who have helped me in innumerable ways over the years. Captaining the Hopkins ship is Judy Thompson, who has saved me from myself more times than there are pages in this thesis. From every rushed order to question about reimbursements, Judy has solved every logistical problem I’ve ever brought her. In the front office, Doreen Zelles cheerfully re-taught me how to send a fax after every committee meeting, and handled more McMaster- Carr orders than I should admit to placing. More importantly, Doreen always offered a smile and a place to relax and chat. John Lee patiently helped me with all of my machining, and I could not have made all of the equipment I did without his skilled eye and suggestions. I also need to thank John for helping me make my own engagement ring from a stainless steel bolt on the Hopkins lathe. Chris Patton helped with printing, video manipulation, and recordings that enriched my presentations. Joe Wible is an incredibly valuable resource, and tracked down a 1929 paper in German – and the only known copy of its English translation. I would also like to thank Don Kohrs, Freya Sommer, Nik Myers, Barbara Compton, Bob Doudna, and the late Peter Ferrante.
I am so lucky to have gotten to know so many people at Hopkins and Monterey over the years: to thank everyone properly would require a second volume, so I will cheat and simply list them: Stacy Aguilera, Bryan Barney, Rachael Bay, Ernest Daghir, Alyssa Gehman, Ana Guerra, Alison Haupt, Kristin Hunter-Thompson, Hannah Jaris, Nishad Jayasundara, Laurie Kost, Judith Levine, Brent Lockwood, Michelle Ow, Melissa Pespeni, Malin Pinsky, Lupita Ruiz-Jones, Danna Staaf, and Julie Stewart. Thank you all for your friendship, lunches, dinners, ski days, movie nights, and for being people I am proud to know.
ix I had the joy of being on two teams while living in Monterey. My lovable losing softball team, the Ekman Pumpers, introduced me to some of my most cherished friendships. The Pumpers also taught me a valuable lesson about sportsmanship: it’s not how often you win or lose, it’s how often you go to the pub after the game with your teammates that counts. The Pumpers are champions of the post-game beer, and I thank Shawn Wagoner, Jon Detka, Erin Barnes, and John and Kelley Petersen. A group of amazing women on the team have adopted me into their fold, and I thank Cara Wilson, June Babilla, MB Robertson, and Anke Richter for the monthly dinners, career advice, hugs when projects failed, and even delivering me dinner when writing this thesis came between me and an evening with their wonderful group. My trivia team, the Borons, made Tuesday nights a highlight of my weekly social calendar, and I thank Sam Wilding, Luke Miller, and Elliott Hazen. I also thank Tony for asking such great trivia questions.
Several friends who have gone above and beyond the call of friendship-through-a- dissertation deserve thanks: Pam Holt, for instituting “Thesis Watch 2014” and checking in on me via text every day for the bulk of thesis writing. Dan Eling, for his unwavering friendship, engineering advice, and teaching me to love the sport of baseball. Jane Louie, for her incredibly sage advice regarding productivity, working with others, and for being an inspiring role model.
I am incredibly lucky to have valuable friendships with people I consider family. Jon Sanders deserves a thesis of his own were I to describe what his friendship means to me properly, but I’ll attempt to be brief: Jon has been my best friend since high school, and I have learned so much from him about what truly matters in life. It is also entirely his fault that I am writing this acknowledgements section: I first visited Hopkins with Jon his freshman year at Stanford, and several more times over the years that Jon spent here. Jon called me in 2006 and told me that if I was considering graduate school, I needed to talk to Mark Denny before doing anything else.
x Elliott and Lucie Hazen (and Phoebe and Griffin) deserve thanks for making their home a second home for me, and for all of their love, support, advice, editing, and hugs. I also thank Chris, Lisa, and A.J. Lenaway for their love, support, beer, food, wine, holiday celebrations, ski trips, and for teaching Rich and me over and over again that family doesn’t have to be biological.
Toni Mizerek dubbed herself my “Monterey Spouse” and declared that we would take care of one another during the week when Rich and I weren’t together. Toni is someone who instantly makes everyone feel at ease, and I have learned so much about love, life, and happiness from her. Toni delivered me meals in the lab late at night, and would often work on the lab couch so I wouldn’t be sitting in the lab alone. After she moved to Australia to pursue her own Ph.D., she suggested virtual work sessions via webcam, so I wouldn’t be working alone long after the rest of the lab had gone home. Her encouragement and support had an enormous effect on my ability to complete this dissertation, and I cannot thank her enough.
I thank my Aunt Care, Uncle Dave and cousins Colin and Kate for their love, support, and for being such fun-loving, hilarious people. I would be remiss if I didn’t also thank them for the original choreography of the dance now known as the “The Jellyfish,” which premiered at our wedding reception.
My family members have always been supportive of my decisions, regardless of how far away from the nest they took me. I am so lucky to have such a loving and supportive family, and I thank my brothers Matt and Mike for everything they’ve done to help me through graduate school – be it limpet wrangling advice, Star Wars Facebook posts, or Legos in the mail. My parents are incredible people who, though surprised when I announced I was heading back to school for biology, were always supportive and who took enormous care to describe my work to their coworkers better than I ever could. I cannot thank them enough for all of their love and sacrifices over the years to get me to where I am today.
xi Finally, I thank Rich Shuma for both his technical help and emotional support: he designed my field transducer and formatted this thesis, both of which I am incredibly grateful for. More importantly, Rich is an supportive partner who graciously dealt with years of living apart so that I could follow and achieve my goals. I am not exaggerating when I say I could not have completed this journey without him. Rich, thank you for everything. I love you.
xii
TABLE OF CONTENTS
Abstract ...... v!
Acknowledgments ...... vii!
List of Figures ...... xvii!
List of Tables ...... xx!
Introduction ...... 1!
Chapter 1 ...... 5!
What a drag: experimental determination of the hydrodynamic forces responsible for wave impact events 5!
1.1! Abstract ...... 5!
1.2! Introduction ...... 5!
1.3! Methods and Materials ...... 10!
1.3.1! Wave simulation ...... 10!
1.3.2! Wave velocity measurements ...... 12!
1.3.3! Force measurements ...... 13!
1.3.4! Test objects ...... 16!
1.3.5! Signal post-processing ...... 19!
1.3.6! Data analysis ...... 19!
1.4! Results ...... 23!
1.4.1! Impingement occurrence ...... 23!
1.4.2! Impingement duration ...... 24!
1.4.3! Force as a function of area ...... 25!
xiii 1.4.4! Force as a function of object volume ...... 27!
1.4.5! Force as a function of object drag coefficient ...... 28!
1.4.6! Water velocity ...... 30!
1.4.7! Predicted vs. measured drag ...... 31!
1.5! Discussion ...... 34!
1.5.1! A new look at impingement ...... 34!
1.5.2! Impingement magnitude scaling behavior ...... 35!
1.5.3! Predicted vs. measured drag ...... 36!
1.5.4! Are organisms capable of responding to brief loads? ...... 37!
1.5.5! Does impingement limit intertidal organism size? ...... 38!
1.5.6! Conclusions ...... 39!
1.6! Appendix ...... 40!
Chapter 2 ...... 43!
Characterizing wave impact forces in the field using high-frequency hydrodynamic measurements 43!
2.1! Abstract ...... 43!
2.2! Introduction ...... 43!
2.3! Methods and Materials ...... 47!
2.3.1! Drag elements ...... 47!
2.3.2! Transducer ...... 48!
2.3.3! Field site ...... 50!
2.3.4! Data collection ...... 52!
2.3.5! Post processing...... 53!
xiv 2.3.6! Environmental data ...... 54!
2.3.7! Wave analysis ...... 55!
2.4! Results ...... 58!
2.4.1! Drag sphere measurements ...... 58!
2.4.2! Limpet model measurements ...... 64!
2.5! Discussion ...... 69!
2.5.1! Implications for studies using dynamometers ...... 69!
2.5.2! Frequency of impingement ...... 69!
2.5.3! Comparison to earlier studies ...... 70!
2.5.4! Do biological shapes experience impingement events? ...... 72!
2.5.5! Conclusions ...... 73!
Chapter 3 ...... 75!
Predicting the mechanical response of Lottia gigantea to transient hydrodynamic forces 75!
3.1! Abstract ...... 75!
3.2! Introduction ...... 75!
3.3! Theory ...... 80!
3.4! Methods and Materials ...... 91!
3.4.1! Limpet collection ...... 91!
3.4.2! Limpet stiffness and damping measurements ...... 91!
3.4.3! Displacement and force predictions ...... 92!
3.5! Results ...... 93!
3.5.1! Force-displacement curves ...... 93!
xv 3.5.2! Stiffness and damping ...... 95!
3.5.3! Effects of pulse frequency ...... 100!
3.5.4! Response to measured waves ...... 102!
3.6! Discussion ...... 104!
Chapter 4 ...... 107!
Predicting gastropod dislodgment risk using hydrodynamic force measurements 107!
4.1! Abstract ...... 107!
4.2! Introduction ...... 107!
4.3! Methods and Materials ...... 108!
4.3.1! Determining Wave Velocity ...... 108!
4.3.2! Drag Predictions ...... 112!
4.3.3! Tenacity Estimates ...... 114!
4.3.4! Predicting Dislodgment Probabilities ...... 115!
4.4! Results ...... 118!
4.4.1! Maximum Velocities ...... 118!
4.4.2! Drag Distribution ...... 118!
4.4.3! Probability of Dislodgment ...... 119!
4.5! Discussion ...... 120!
References ...... 123!
xvi LIST OF FIGURES
Figure 1.1: Force from one wave over time, including impingement event ...... 7
Figure 1.2: Side and front photos of water cannon ...... 11
Figure 1.3: Camera frame showing sphere track points over time ...... 13
Figure 1.4: Diagram of force transducer ...... 14
Figure 1.5: Transducer mount in front of water cannon ...... 15
Figure 1.6: Shape difference between front- and side-facing rectangular solids ...... 17
Figure 1.7: Data ensemble for objects under high velocity jet ...... 20
Figure 1.8: Calculating impingement duration ...... 22
Figure 1.9: Data ensembles for three jet velocities ...... 23
Figure 1.10: Impingement duration vs. cylinder length ...... 25
Figure 1.11: Impingement magnitude vs. object area ...... 26
Figure 1.12: Force per area vs. object volume ...... 28
Figure 1.13: Impingement magnitudes normalized by drag coefficient ...... 30
Figure 1.14: Water velocity time-series of the high velocity jet ...... 31
Figure 1.15: Predicted impingement magnitudes vs. measured drag ...... 32
Figure 1.16: Measured impingement & predicted drag (front-facing rect. solid) ...... 33
Figure 1.17: Measured impingement & predicted drag (side-facing rect. solid) ...... 33
Figure 1.18: Measured impingement & predicted drag (sphere) ...... 34
Figure 1.19: Measured impingement & predicted drag (all front rect. solids) ...... 40
Figure 1.20: Measured impingement & predicted drag (all side rect. solids) ...... 41
Figure 1.21: Measured impingement & predicted drag (all spheres) ...... 42
xvii Figure 2.1: Force transducer used for field measurements ...... 49
Figure 2.2: Photo of field site at low tide ...... 51
Figure 2.3: Examples of impingement waves (definite and probable) ...... 57
Figure 2.4: Density plot, sampled environmental conditions ...... 59
Figure 2.5: Density plot, waves per tidal and significant wave heights ...... 60
Figure 2.6: Force ratio distributions for impingement events ...... 61
Figure 2.7: Force ratio distribution for all waves ...... 62
Figure 2.8: Maximum force distributions for impingement events ...... 63
Figure 2.9: Maximum force distribution for all waves ...... 63
Figure 2.10: Maximum force per wave vs. Hs ...... 64
Figure 2.11: Density plot, sampled environmental conditions, limpet ...... 65
Figure 2.12: Density plot, waves per tidal and significant wave heights, limpet ...... 65
Figure 2.13: Force ratio distributions for impingement events, limpet ...... 67
Figure 2.14: Force ratio distribution for all waves, limpet ...... 67
Figure 2.15: Maximum force distributions for impingement events, limpet ...... 68
Figure 2.16: Maximum force distribution for all waves, limpet ...... 68
Figure 2.17: Force ratio distribution, Gaylord’s method ...... 71
Figure 3.1: Example of a mass-spring-damper system ...... 76
Figure 3.2: Response of a mass-spring-damper system ...... 77
Figure 3.3: Anatomy of L. gigantea ...... 79
Figure 3.4: Limpet body modeled as a Kelvin-Voigt element ...... 81
Figure 3.5: Critically damped, underdamped, overdamped, and undamped motion . 85
xviii Figure 3.6: Ratio of effective force to applied force under forced oscillation ...... 86
Figure 3.7: Calculating stiffness from a force-displacement curve ...... 88
Figure 3.8: Force displacement curves under loading and unloading ...... 88
Figure 3.9: Ideal loading / unloading force time-series ...... 89
Figure 3.10: Force-displacement curves for representative stationary limpet ...... 94
Figure 3.11: Force-displacement curves for representative escaping limpet ...... 95
Figure 3.12: Estimated stiffness values for stationary and escaping limpets ...... 96
Figure 3.13: Estimated damping coefficients for stationary and escaping limpets .... 97
Figure 3.14: Estimated damping ratios for stationary and escaping limpets ...... 98
Figure 3.15: Limpet stiffness vs. limpet mass ...... 99
Figure 3.16: Limpet damping coefficient vs. limpet mass ...... 100
Figure 3.17: Mean ratio of max / input force for varying durations, stationary ...... 101
Figure 3.18: Mean ratio of max / input force for varying durations, escaping ...... 101
Figure 3.19: Predicted force on limpet from a half-sine pulse input ...... 102
Figure 3.20: Ratio of max / applied wave forces, stationary limpet ...... 103
Figure 3.21: Ratio of max / applied wave forces, escaping limpets ...... 104
Figure 4.1: Drag coefficient vs. Reynolds number ...... 111
Figure 4.2: Velocity vs. force curve ...... 112
Figure 4.3: Predicting snail dislodgment, simple case ...... 116
Figure 4.4: Predicting snail dislodgment, actual waves ...... 117
Figure 4.5: Histogram of maximum wave velocities ...... 118
Figure 4.6: Distributions of drag normalized by foot area ...... 119
xix LIST OF TABLES
Table 1.1: Test object dimensions, areas, and volumes ...... 18
Table 1.2: Impingement magnitude vs. area ...... 27
Table 1.3: Drag coefficients calculated for rectangular solids and spheres ...... 29
Table 4.1: Ratios of organism dimensions and areas ...... 113
Table 4.2: Drag coefficients for gastropods ...... 114
Table 4.3: Shear tenacities for gastropods ...... 115
Table 4.4: Probabilities of dislodgment for gastropods ...... 120
xx Introduction
“…you must not confound statics with dynamics, or you will be exposed to grave errors.” -Captain Nemo to the naturalist Pierre Aronnax 20,000 Leagues Under the Sea
On her first visit to the tidepools of the rocky intertidal zone, a rational person can be forgiven for assuming that the immense power of waves crashing upon the shore must strip away anything unlucky enough to settle on the rocks, leaving the shoreline bare. Learning that, instead, this is one of the most diverse and productive ecosystems to grace our planet might understandably come as a shock. Watching waves impinge on these intertidal communities, she might reasonably wonder why anything – let alone so many things – may choose to call this stressful and chaotic region of the world home. An intertidal biomechanic, however, wonders how these organisms survive and thrive in such a harsh environment.
A harsh environment it is: wave-swept rocky shores are among the most physically stressful habitats on Earth. The stresses imposed on organisms are almost as diverse as the plants and animals themselves. When the tide recedes, marine organisms must survive in a terrestrial environment for large portions of the day, subjecting them to large temperature swings and forcing them to avoid lethal desiccation. When the tide comes in, these organisms are suddenly thrust back into the maelstrom of chaotic and turbulent flows that result from breaking waves.
In light of the obvious hydrodynamic stresses these waves impose on organisms, it seems counterintuitive to consider how essential moving fluids are to the success of intertidal communities. Indeed, the flows resulting from breaking waves are critical to every stage of life in the intertidal zone: from fertilization to settlement to dislodgment and death, as well as the necessary steps for survival in between.
Introduction 1 To understand the hydrodynamic mechanisms that drive these processes, we need quantitative measurements of surf zone flows. However, these data are hard to come by: for one, the harshness of the very processes we are trying to study makes the intertidal zone a brutal place to deploy equipment. Second, the complex topography that characterizes rocky shores is extremely difficult to reconcile with existing wave theories: while the literature contains theories predicting breaking wave heights and velocities on gently sloping beaches, these theories break down when applied to rocky shores – shores composed of cliff faces, crevices, surge channels, rock outcroppings, and variations at every scale from the texture of granite to kilometers of jagged coastline. Finally, fluid dynamics is neither an inviting nor attractive tool for many biologists.
Despite these obstacles, many studies have been devoted to quantifying flows in the intertidal zone with empirical measurements. As technology advances, building equipment with robust, small, and affordable electronics has become easier, as have solutions to the problem of storing massive amounts of data. Tiny electronics and large hard drives have allowed me to collect an enormous dataset in the service of exploring an intriguing aspect of intertidal flows: “impingement forces” are the briefest, yet seemingly largest, hydrodynamic forces in the surf zone. What causes them? How often do they occur? And most importantly, what do these forces mean for intertidal organisms?
Impingement events are sharp, transient spikes in force occurring at the instant of wave impact, first discovered by Brian Gaylord during his graduate work in the Denny Lab. In the most complete empirical and quantitative measurements of intertidal flows to date, his work suggested that these are the largest hydrodynamic forces imposed on intertidal plants and animals. They appeared to be rare and potentially controlling events, and traditional hydrodynamic models do not take them into account. However, his study had comparably few waves with impingement events, and used a less-than- ideal force transducer, leaving several important questions regarding impingement unresolved.
Introduction 2
How important are impingement events and other transient loads in the intertidal zone? My dissertation focuses on exploring this question in several ways, and explores the effects of transient forces on intertidal inhabitants through quantifying high- frequency forces. To that end, I present the following four chapters.
In Chapter 1, I explore the variables of impingement in the laboratory. Using a gravity-driven water cannon, I was able to recreate impingement events and determine which variables control the magnitude of these forces. Contrary to previous thought, impingement is drag rather than an undescribed hydrodynamic force, most likely due to high velocities at the front of a wave.
In Chapter 2, I measure impingement events in the field using a custom transducer built specifically to make extremely high-frequency measurements in the intertidal zone. By recording at 40 kHz for weeks at a time, I was able to record hundreds of these events. With these data, I was able to estimate how frequently these events occur and determine if, as previously thought, impingement events are the largest hydrodynamic forces acting on organisms. They are not: waves with impingement events have a smaller average maximum force than that of all waves in this study. My data suggest that impingement events are not any more dangerous to organisms than other forces occurring in a wave. As a result, future studies need not concern themselves with searching for impingement events. Rather, it appears that high- frequency fluctuations in force – which can occur anywhere in a wave – generate maximum forces in the surf zone.
Armed with the characterization of high-frequency forces in the intertidal zone, in Chapter 3 I set out to determine how transient forces might affect intertidal organisms. Specifically, are gastropods able to mechanically respond to brief forces, or does the rubbery foot characterizing these creatures act as a shock absorber of sorts? By measuring the structural properties of Lottia gigantea, I modeled a limpet as a mass- spring-damper system, and investigated the response to 500 waves measured in
Introduction 3 Chapter 2. My results showed that on average, the foot reduces the effective force on the limpet by 30% – though this is not always the case. Over 500 waves, force reductions varied from 1-92%.
In Chapter 4, I explore the possible ecological consequences of the forces I measured in Chapter 2. How many of those 9,311 waves were likely to dislodge organisms? Using the maximum forces of each wave, I predicted the likely forces acting on three species of gastropod. By comparing these to published tenacity values, I was able to predict probabilities of dislodgment for each species. Snails face high risks of dislodgment by wave forces, and thus must take behavioral precautions to avoid becoming dinner for intertidal predators.
Together, these studies paint a picture of how hydrodynamic forces in the intertidal zone vary over extremely small temporal scales. The motivation behind this work was to determine if impingement events are an important contributor to hydrodynamic forces in the surf zone, because traditional hydrodynamic and ecological models do not take them into account. It turns out that’s okay: impingement events are not the largest forces acting on intertidal plants and animals.
This dissertation underscores the necessity of high-frequency hydrodynamic measurements, and explores a few of the ways that dynamic forces in the surf zone might affect intertidal organisms. Perhaps my favorite result of this work, however, is evident every time I watch a wave crashing against the shore: learning to quantify how these organisms survive has only added to my wonder when I ask why.
Introduction 4 Chapter 1
What a drag: experimental determination of the hydrodynamic forces responsible for wave impact events
1.1 ABSTRACT Breaking waves can impose enormous forces on intertidal plants and animals. While hydrodynamic forces such as drag, lift, and the acceleration reaction are well-studied, the factors affecting the magnitude of a fourth force – the impingement force – remain unknown. Characterized by a sharp, transient spike in force at the instant of wave arrival, impingement has been thought to be the largest hydrodynamic force in the intertidal zone – yet the variables affecting its magnitude are unstudied. To determine which factors influence the magnitude of impingement forces, I tested a variety of objects encompassing a range of areas, volumes, aspect ratios, and drag coefficients in simulated waves using a gravity-driven “water cannon.” Additionally, I measured the velocity of the jet produced by the water cannon. I found the force of impingement events to be proportional to object frontal area and drag coefficient, and to occur concomitantly with spikes in water velocity at the front of waves. I conclude that impingement is a brief spike in drag caused by an increase in velocity at the wave front, rather than an undescribed hydrodynamic force. As a consequence, previous hypotheses regarding impingement’s ability to limit organism size in the intertidal zone are rejected.
1.2 INTRODUCTION The rocky intertidal zone is one of the most physically stressful environments on the planet, in large part due to breaking waves crashing ashore. The hydrodynamic forces imposed by these turbulent bores can be enormous, and can have a cascade of effects that shape and structure surf zone communities. Wave intensity influences species assemblages (Lewis 1964, 1968; Paine 1979; Paine and Levin 1981; McQuaid and Branch 1985; Ricketts et al. 1985; Rius and McQuaid 2006, 2009; Branch et al. 2010;
Chapter 1 5 Tam and Scrosati 2014), and contributes to species turnover by dislodging organisms (Dayton 1971; Paine and Levin 1981; Koehl 1984; Sousa 1984; Denny et al. 1985; Denny 1987b; Trussell et al. 1993; Denny 1995; Bell and Gosline 1997; Denny and Blanchette 2000; Lau and Martinez 2003). Dislodged organisms free space for others to settle, offering valuable real-estate for colonization. Hydrodynamic forces also impact the intertidal food chain: dislodged organisms are a significant food resource in the surf zone (Gaylord 2007), and large waves discourage predator and grazer movement (e.g., Menge 1978a, 1978b; Jenkins and Hartnoll 2001). Because of the critical role hydrodynamics plays in shaping intertidal communities, understanding the fluid-dynamic mechanisms that drive these processes is essential. While the parameters controlling the magnitudes of the classical hydrodynamic forces – lift, drag, and the acceleration reaction – have been understood for decades, the variables that determine the magnitude of impingement remain unknown.
Impingement is the sharp, transient spike in force occurring within the first fraction of a second of wave impact on an emersed object. Impingement has been assumed to be related to either the impact physics of a wave striking an exposed object or an undescribed hydrodynamic force, possibly resulting from an emersed organism being hit by a wave with no surrounding fluid to absorb some of the impact (Gaylord 1999, 2000; Gaylord et al. 2001). In either case, impingement events have been thought to be the largest forces imposed on intertidal organisms, with an average magnitude twice that of drag (Gaylord 2000). Figure 1.1 shows a representative impingement event, measured in the field in Chapter 2, followed by more-or-less steady classical drag as the wave advects past the object in flow.
Chapter 1 6
Figure 1.1: Magnitude of hydrodynamic force over time as a wave flows past an emersed object, including an impingement event.
The goal of this study is to elucidate the fluid-dynamic mechanisms of impingement. To that end, I begin by discussing the classical in-line hydrodynamic forces, which frequently depend on Reynolds number. The Reynolds number Re is a dimensional index of flow conditions that describes the ratio of inertial to viscous forces in flow. Reynolds number is a function of object size L, flow velocity u, and the kinematic viscosity of the fluid ν, which for seawater at 10°C is 1.17x10-6 m2/s:
!" !" = Equation 1.1 !
In the surf zone, u is typically greater than 1 m/s, so for organisms 1 cm in length and longer, Reynolds numbers are greater than 8500. Thus, this chapter deals with high Reynolds number flows (Re ≫ 1) and associated hydrodynamic forces.
The in-line forces in high Reynolds numbers flows can be divided into two groups: those whose magnitude is proportional to fluid velocity and those that depend on fluid acceleration. Drag is a velocity-scaling force, while the components of the acceleration reaction are, appropriately, acceleration-scaling.
In high Reynolds number flows, the magnitude of drag FD on an organism is a function of the fluid density ϱ, organism shape (indexed by the drag coefficient CD),
Chapter 1 7 the velocity of the fluid relative to the organism ur, and the projected area of the organism exposed to flow, Apr. (Projected area can be thought of as the “frontal area” of the object exposed to flow, equivalent to the area of the shadow created by shining light in the direction of flow.) Drag can be modeled as:
1 ! = ρ! !!! Equation 1.2 ! 2 ! ! !"
The drag coefficient can be thought of as a dimensionless index describing how streamlined an object is: a blunt object (such as a semi-truck) will have a much higher drag coefficient than an aerodynamic, smooth object (such as an Indianapolis 500 race car). For many shapes, drag coefficients are dependent on the Reynolds number. Drag results both from skin friction drag (the force produced by the shearing of fluid as it moves past the surface of an object) and pressure drag, caused by the difference in pressure between the upstream face of an object in flow (high pressure) and the downstream wake (low pressure). In the surf zone, pressure drag dominates for all but the most streamlined objects.
At the instant of wave arrival at an emersed object, there is no water in the wake, and therefore only ambient air pressure downstream. As a result, until the wake is fully established, the net pressure difference, and therefore drag, acting on the object could exceed that during the remainder of the wave. In this case, impingement might be explained by this “delayed wake” effect.
In contrast to drag, the acceleration reaction FA scales with the acceleration of the fluid a and object volume V, rather than fluid velocity and object area. The acceleration reaction is also a function of fluid density and organism shape, and can be modeled as:
!! = !!!!" Equation 1.3
Chapter 1 8 where Cm is the inertia coefficient, which takes into account the object’s added mass and the virtual buoyancy associated with accelerating flow. Although large accelerations have been measured in the intertidal zone, the spatial scales of these accelerations are too small to result in large forces on organisms (Gaylord 2000). Because of the variations in acceleration at a centimeter spatial scale, the lack of uniform acceleration prevents large acceleration forces for large organisms. Small organisms, on the other hand, lack the necessary volume to create large accelerational forces.
For a more complete treatment of the acceleration reaction and drag, see Denny (1988), Vogel (1994), and Batchelor (1967).
These classical forces – drag and the acceleration reaction – are predictable for an organism if one can measure the appropriate variables and flow conditions (either velocity or acceleration, appropriately). For impingement, though, the variables of importance remain unknown. Does flow velocity or acceleration control impingement? Does the magnitude of impingement depend on the area or volume of an organism in flow, or something else entirely?
Whether impingement is proportional to organism area or volume may have important consequences regarding organism size limitation in the intertidal zone. Denny et al. (1985) hypothesized that hydrodynamic forces due to breaking waves are responsible for limiting organism size. Indeed, organisms living in the intertidal zone are, typically, smaller than their subtidal or terrestrial counterparts, and individuals within some species vary in size with degree of wave exposure. For these groups, such as seaweeds and barnacles, larger individuals tend to be found in more protected environments, while smaller individuals are often found in more exposed places.
However, only hydrodynamic forces that scale with volume are intrinsically capable of limiting size. An organism’s ability to resist hydrodynamic force is dependent on its attachment strength, which is assumed to vary with area (Denny et al. 1985).
Chapter 1 9 Assuming an organism grows isometrically (which is a dangerous assumption for many taxa: see the Discussion for an elaboration), a hydrodynamic force varying with area (such as drag) has no capacity to limit organism size: as the organism gets larger, both the force acting on it and its ability to resist that force increase at the same rate. For volume-scaling forces (such as the acceleration reaction), however, force increases with the volume of the organism, while attachment strength still scales with area. If the organism grows isometrically, its volume increases at a faster rate than its attachment area. Because force increases faster than the organism’s ability to resist dislodgment, there might exist a critical length at which the force acting on the organism exceeds its attachment strength. If impingement scales with volume, it could potentially limit organism size, and it is therefore important to determine the scaling behavior of impingement.
Does impingement scale with area or volume? If impingement scales with area, is it due to the delayed wake effect? To answer these questions, I tested a variety of objects comprising a range of areas, volumes, aspect ratios, and drag coefficients using simulated waves from a gravity-driven “water cannon.” By exploring the relationships between these variables and impingement magnitude, I sought to describe the mechanics behind what may possibly be the controlling hydrodynamic events in the intertidal zone.
1.3 METHODS AND MATERIALS
1.3.1 Wave simulation
Controlled, repeatable waves were simulated using a gravity-driven “water cannon.” This apparatus consists of a large (approximately 25 feet tall) vertical pipe with a 10.2 cm internal diameter. The bottom of the pipe releases water, simulating a crashing wave (Figure 1.2). To control the volume of water in the pipe and the height at which it is released (and thus, the velocity of the jet upon exit), the pipe has two pneumatically-operated gate valves. Overflow holes in the pipe allow it to be filled to the same height each trial. When the corresponding gate valve is abruptly released, the
Chapter 1 10 water falls down the pipe into a wide-radius 90 degree elbow that directs the flow out of the pipe horizontally. Using this apparatus, repeatable simulated waves can be generated (Martone and Denny 2008; Martone et al. 2012).
Figure 1.2: a) Side photo of water cannon. b) Front photos of water cannon, showing valve locations and relative volumes of water released at each of the three velocities.
Experiments were conducted using three water velocities. The high velocity jet is controlled by a valve 3.1 m above the cannon outlet, which supports a 1.6-m column of water in the pipe above the valve. A second, lower valve 0.8 m above the jet outlet produces both the medium and low jets. The columns of water held are 2.2 m and 0.7 m above the valve for the medium and low jets, respectively. Maximum velocities are estimated to be 9.8 m/s for the high velocity jet, 6.8 m/s for the middle velocity jet, and 5.1 m/s for the low velocity jet, and were measured by Boller and Miller (unpublished) for all three jets using a method similar to the one I describe in the following section.
Chapter 1 11 1.3.2 Wave velocity measurements
For the highest velocity flow, the water velocity of the jet was measured using a combination of neutrally buoyant spheres and a high speed video camera (FASTCAM- 512PCI, Photron, San Diego, CA). Spheres were constructed by taking hollow, 1.9 cm diameter polypropylene spheres and using a syringe to fill them with water. Because these spheres were still slightly positively buoyant, a small, flat thumbtack was pushed into each injection hole to add additional weight. Spheres were not perfectly neutrally buoyant, but spanned a small range of densities very near neutral buoyancy.
By varying the time between adding the spheres to the water cannon pipe and firing the water cannon, I could control the spacing of spheres through the jet. Longer wait times (25-30 seconds) put the spheres at the front of the jet because they settled to the bottom of the chamber before release, while short wait times (5-10 seconds) provided higher sphere concentrations at the tail of the jet. The slight variation in density allowed the spheres to spread out slightly through the water column in the chamber, preventing sphere “clumping.”
The spheres were painted to maximize contrast with the water, enabling the camera to see them against the jet. By analyzing the horizontal distance traveled by the spheres between frames, the velocity of each sphere was estimated as a function of both time and distance from the jet outlet. The horizontal position of each sphere was identified manually, using the tracking software Photron Motion Tools (Photron, San Diego, CA) (Figure 1.3).
Chapter 1 12
Figure 1.3: Camera frame showing sphere track points over time. Frames are 0.0164 s apart.
Velocity measurements were made only for the highest velocity jet; the flows produced by the two slower jets were assumed to follow the same trends (for example, how velocity varied with time) as the highest velocity jet.
1.3.3 Force measurements
Force measurements were made using a waterproof double-cantilever force transducer (Figure 1.4). Two semi-conductor strain gauges were glued to the base of each aluminum beam and wired into a full Wheatstone bridge. As a horizontal force was applied to the test object, the beams bent, producing strain in the beam and therefore a voltage proportional to applied force. This signal was fed through an anti-aliasing, low-pass filter at 15 kHz, amplified 1000 times, and recorded by a computer running LabView (National Instruments, Austin, TX). The primary natural frequency of the transducer was 8200 Hz when submersed.
Chapter 1 13
Figure 1.4: Diagram of the force transducer used for measurements.
Prior to experiments, the transducer was calibrated by hanging known weights from it to obtain static displacement voltages for known forces. A linear regression was performed to determine a calibration curve for the transducer (R2 = 0.9999).
To isolate the transducer from any forces aside from those produced by impinging waves, an aluminum mounting block was designed to hold the transducer in front of the water cannon outlet. The mounting block was bolted to the concrete sidewalk in front of the water cannon to maximize the stiffness of the mount and minimize extraneous noise (Figure 1.5).
Chapter 1 14 Jet outlet
Test object
Transducer (inside water bath)
Transducer mount
Figure 1.5: Transducer mount in front of water cannon.
Within the mounting block, the transducer was surrounded by a plastic housing that was filled by a steady stream of water at the same temperature as that used in the cannon. As the strain gauges are temperature sensitive, this feature kept the strain gauges at the same temperature as the water used for waves, minimizing error from rapid temperature changes.
A piece of acrylic with a small hole for the test object’s mounting rod separated the transducer beams from direct flow, ensuring the jet hit only the test object.
Chapter 1 15
Force was sampled at 40 kHz, and four to five seconds of data were recorded per wave.
1.3.4 Test objects
Twenty-one objects were tested in front of the water cannon: ten rectangular prisms, five spheres, and six cylinders. Each object was attached to a ¼”-20 threaded rod, which was bolted to the transducer through a clearance hole in the center of the top beam (Figure 1.4). All of the objects were smaller than the diameter of the water cannon jet.
Rectangular prisms were fabricated from PVC tubes with square cross-sections. Prism dimensions were chosen to encompass a range of areas, aspect ratios, and volumes. Prism faces ranged in width from 20 mm to 50 mm, and prism lengths varied from 16 mm to 45 mm (Table 1.1). The prisms were tested in two different orientations – first with the square face oriented towards the water cannon outlet, and then with that face rotated ninety degrees about the mounting rod. By testing both sides of the prism, measurements were thus taken for two different frontal areas for each volume.
The tubing used to construct the rectangular prisms was not precisely square – plastic tubing of this sort is manufactured with rounded edges (Figure 1.6). As a result, “front” facing rectangular solids had sharp edges but rounded corners, while “side” facing ones had rounded edges and sharp corners. Slight deviations in drag coefficient might be expected due to the differences in the shape of the edges and corners. The difference in area between a square face with sharp corners and the rounded corners in my objects was taken into account, and those areas are reflected in Table 1.1. Rectangular solid 3 was not tested in the sideways condition due to damage sustained after front-facing trials.
Chapter 1 16
Figure 1.6: Diagram showing difference in shape between front- and side-facing rectangular solids.
Test spheres were constructed by inserting threaded rods into polypropylene spheres ranging in diameter from 19 mm to 75 mm. All of the plastic spheres were hollow except for the 25.4 mm sphere, which was solid; threaded rods were anchored using silicone sealant in the hollow spheres and epoxy in the solid sphere.
Cylinders were constructed of solid PVC pipe 25.8 mm in diameter; cylinder lengths varied from 13 mm to 43.4 mm. Threaded rods were screwed into tapped holes in the center side of the cylinder, and the flat ends were oriented perpendicular to flow.
Chapter 1 17 Rectangular Width Height Length Area - front Area - side Volume Solid (mm) (mm) (mm) (mm2) (mm2) (mm3)
1 19.98 20.26 25.52 384.7 517.0 10,330.4
2 20.02 20.26 32.09 385 650.1 13,015.9
3 20.10 20.20 38.44 393.5 776.5 15,607.4
4 20.05 20.23 44.59 400.3 902.1 18,086.2
5 25.99 26.10 25.73 650.7 671.6 17,453.7
6 26.05 26.18 52.94 652.7 1,386.0 36,104.5
7 35.06 35.42 25.84 1200.7 915.3 32,088.8
8 35.07 35.35 32.68 1184.1 1,155.2 40,514.2
9 50.07 49.97 16.01 2515.9 800.0 40,057.0
10 50.18 49.97 22.74 2586.5 1,136.3 57,020.4
Diameter Area Volume Sphere (mm) (mm2) (mm3)
1 19.2 28.9 3,692.7
2 25.4 50.7 8,580.2
3 35.6 99.3 23,544.2
4 50.0 196.4 65,499.3
5 75.0 441.9 220,948.0
Diameter Length Area Volume Cylinder (mm) (mm) (mm2) (mm3)
1 25.8 13.0 522.0 6,788.8
2 25.8 18.6 522.0 9,719.1
3 25.8 22.0 522.0 11,495.9
4 25.8 27.8 522.0 14,532.3
5 25.8 35.4 522.0 18,496.8
6 25.8 43.4 522.0 22,660.3
Table 1.1: Test object dimensions, areas, and volumes.
Chapter 1 18 25 trials were conducted for each object at each of the three flow velocities, with the exception of front-facing rectangular solid 10 for the low velocity jet: in this case, only 20 trials were conducted.
1.3.5 Signal post-processing
All structures, and therefore all components of the experimental setup, have resonant frequencies of vibration when an external force is applied. To accurately measure brief forces such as impingement, these vibrations must be minimized. To identify the first (and therefore lowest) natural frequencies of the transducer mount, the transducer itself, and the object mounted to the transducer, “tap tests” were performed by bouncing metal spheres off the objects prior to each set of experiments. The tap tests identified how the experimental setup responded to impulses, and the resonant frequencies of vibration that would need to be minimized in post-processing.
Power spectra were calculated for each object’s tap tests to identify frequency ranges with resonance. Additionally, power spectra were calculated for every object and velocity combination. By comparing the corresponding tap test and wave power spectra for each object, frequency ranges with energy due to resonance were identified. To minimize the effects of resonance from the experimental set-up, resonant frequency ranges were 4th-order band-stop filtered using Butterworth filters in MATLAB (MathWorks, Natick, MA, 2008).
1.3.6 Data analysis
Due to the turbulent nature of the water cannon jet (which mimics the turbulent flow following breaking ocean waves), measured forces were not identical from run to run. To manage this variation, I visually inspected all runs, and manually selected the best estimate of the impingement peak, defined as the local maximum during the initial rise of the jet before force decreased prior to the steady-state drag characterizing the bulk of the wave. After aligning all 25 runs at this peak, they were ensemble averaged at
Chapter 1 19 each time point to provide an estimate of the average force for each object/velocity combination over the course of the wave among the 20 - 25 replicates. (Figure 1.7).
The force magnitude at the aligned impingement peak was compared to the areas and volumes of each tested object. For volume calculations, the force per projected area (normalized force) was calculated and plotted against object volume, because for many of the objects tested, there is an implicit relationship between projected area and volume. That is, the objects with larger faces (and projected areas) exposed to flow also tend to have larger volumes (Table 1.1), confounding the relationship between volume and impingement magnitude. Thus, normalized forces were used.
Figure 1.7: 25 runs (light gray) and ensemble average (black) for one of the objects under the high velocity jet.
Forces measured by the transducer after the impingement peak of each wave were assumed to be due to drag: Gaylord’s (2000) work showed that for fully breaking waves, drag is the dominant hydrodynamic force and fairly accurately predicts measured forces. Given the similarity between waves produced by the water cannon jet and surf-zone waves, I assumed the force produced by the jet in the steady-state
Chapter 1 20 drag region of the wave was indeed drag. For each object and velocity, drag coefficients were determined by identifying the maximum steady-state drag following each impingement peak (Figure 1.8), and then solving Equation 1.2 for CD two ways.
First, a single representative CD was calculated for each object using average maximum steady-state drag of the ensemble. Second, drag coefficients were calculated for each object using the maximum drag measured in each experimental run. The variance among these drag coefficients allowed me to calculate confidence intervals for the average CD estimates. The drag coefficient calculated from the ensemble average was used in all analyses.
Impingement peak durations were calculated for cylinders. Because I had six cylinders of varying length with the same frontal area, these test objects served as the best test for existence of a delayed wake effect. I calculated peak durations by first finding the local minimum following the impingement peak (Figure 1.8). The width of the peak at half-height between this minimum and the impingement peak was then calculated for each run for each jet velocity for each cylinder, and this is used as an index of peak duration.
Chapter 1 21 40
35
30 width at half-height 25
20
Force (N) Force height 15
10 min
5
0 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 Time (s)
Figure 1.8: Impingement spike and following minimum point illustrating how the half- height used to calculate impingement durations was calculated.
As I will show in the following section, impingement varies with area, drag coefficient, and water velocity – the three variables that determine the magnitude of drag. This strongly suggests that what we measure as impingement is in fact an increase in drag caused by brief spikes in water velocity.
To test this prediction, drag was predicted as a function of time for all of the tested objects over the measured velocity time-series using Equation 1.2, measured areas, calculated drag coefficients, and the measured water velocity time-series. Maximum predicted forces could then be compared to maximum measured forces.
Chapter 1 22 1.4 RESULTS
1.4.1 Impingement occurrence
Impingement events were observed in 98% of water cannon runs, across all three water velocities. Impingement peaks were not necessarily the maximum forces in the wave: the middle (6.8 m/s) velocity jet produced a wave with a pronounced peak at wave impact, but the subsequent drag was greater in magnitude than this peak (Figure 1.9). For the high (9.8 m/s) and low (5.1 m/s) velocity jets, impingement peaks were the largest forces in each ensemble average.
Figure 1.9: All runs (gray) and ensemble averages (black) for an object under the a) high velocity, b) medium velocity, and c) low velocity jets.
The medium velocity jet’s odd time course is presumed to be due to the way the wave was generated in the water cannon. As shown in Figure 1.2, the medium velocity jet is produced by a large volume of water (representing a large height of water in the pipe) released from the lower valve. The initial impact from this jet is from water that has traveled a small vertical distance relative to the water initially stored in the top of the
Chapter 1 23 pipe, corresponding to later regions of the jet. Water from the top of the pipe has a larger vertical drop to the pipe elbow, which accelerates that water and produces higher water velocities and therefore, higher drag forces later in the wave. This effect is not as pronounced in the high and low velocity jets, because the height of water in the pipe above the valves is much shorter than that in the medium velocity jet. This likely explains why the drag region of the medium velocity jet produces a greater force magnitude than the impingement spike.
1.4.2 Impingement duration
Impingement durations for cylinders under all three jets are shown in Figure 1.10. Neither the high nor medium velocity jets have a significant relationship between duration and cylinder length (p > 0.5), in contrast to what would be expected if delayed wake effects caused impingement events. However, the low velocity jet does have a significant relationship: impingement duration increases with increasing cylinder length (p = 0.012).
Chapter 1 24 6 a) high velocity jet 6 b) medium velocity jet
4 4 Time (ms) Time (ms) Time 2 2 2 R = 0.07 R2 = 0.02 p > 0.5 p > 0.5 0 0 0 0.02 0.04 0 0.02 0.04 Cylinder length (m) Cylinder length (m)
c) low velocity jet 6
4
Time (ms) Time 2 R2 = 0.83 p = 0.012 0 0 0.02 0.04 Cylinder length (m)
Figure 1.10: Impingement duration vs. cylinder length for all three jets. Only the low- velocity jet is significant. Error bars represent 95% confidence intervals of the mean.
The low velocity jet velocity is 5.1 m/s. If delayed wake effects are operative, we would expect the duration of impingement to be at least the time it takes the jet to traverse the length of the cylinder (flow velocity divided by the cylinder length). For cylinders ranging from 13 mm to 43.4 mm, expected durations range from at least 2.5 ms to 8.5 ms. Measured impingement durations at the low velocity, however, range only from 3.6 ms to 5.2 ms. This suggests that the observed relationship between impingement duration and cylinder length is not easily explained by a delayed wake effect.
1.4.3 Force as a function of area
I plotted impingement magnitudes against projected area for rectangular prisms and spheres for each of the three velocities (Figure 1.11), and performed linear regressions (Table 1.2). There is a very strong linear relationship between area and impingement magnitude for rectangular solids, both front- and side-facing (R2 > 0.93, p < 0.00005). The five spheres also displayed what appear to be strong linear relationships between
Chapter 1 25 area and magnitude (R2 > 0.994 for each valve, p < 0.0005), at least for areas greater than 5 cm2. However, best-fit lines have non-zero y-intercepts (Table 1.2), suggesting that at least for small areas, the relationship between area and impingement magnitude may be slightly curved and downwardly concave. Cylinders were excluded from this analysis, as they all have the same frontal area.
100 100 a) high velocity b) medium velocity
50 50 Force (N) Force (N)
0 0 0 10 20 30 40 50 0 10 20 30 40 50 Area (cm2) Area (cm2) 100 c) low velocity Rectangular Solids (Front) 50 Rectangular Solids (Side) Spheres Force (N) Force
0 0 10 20 30 40 50 2 Area (cm )
Figure 1.11: Plot of impingement magnitude vs. area for each jet velocity. Error bars represent 95% confidence intervals of the mean.
Chapter 1 26 Shape Jet Velocity n R2 Slope Intercept p-value
High 10 0.991 4.449 ± 0.346 0.374 ± 4.556 1.82E-09 Rectangular solids Medium 10 0.994 2.253 ± 0.138 -0.520 ± 1.813 2.68E-10 (front) Low 10 0.983 1.868 ± 0.197 -0.009 ± 2.597 2.05E-08
High 9 0.950 3.980 ± 0.813 -3.682 ± 7.659 8.12E-06 Rectangular solids Medium 9 0.934 2.001 ± 0.479 -1.978 ± 4.481 2.23E-05 (side) Low 9 0.944 1.778 ± 0.385 -2.267 ± 3.627 1.20E-05
High 5 0.998 0.990 ± 0.090 3.754 ± 1.995 5.09E-05
Spheres Medium 5 0.994 0.533 ± 0.075 2.152 ± 1.676 1.92E-04
Low 5 0.996 0.380 ± 0.046 1.747 ± 1.031 1.24E-04
Table 1.2: Impingement magnitude vs. area regressions and p-values for each object/valve. Error estimates for slopes and intercepts represent 95% confidence intervals of the mean.
1.4.4 Force as a function of object volume
Normalized force versus volume relationships for all shapes are presented in Figure 1.12. For front- and side-facing rectangular solids, there was no significant relationship between normalized force and volume (p > 0.05 for all three velocities). Spheres and cylinders did show a significant negative relationship between force per area and volume, although not at every velocity. Spheres had a weakly significant negative relationship (p = 0.049) only for the 6.8 m/s jet. Similar correlations were found for cylinders for the 6.8 m/s and 5.1 m/s jets (p = 0.003 and p = 0.036, respectively).
In cases where normalized force did vary with volume, the trends were negative. This is the opposite of what we would expect if accelerational forces were influencing impingement magnitudes.
Chapter 1 27 6 3 ) ) 2 9.8 m/s 2 6.8 m/s
4 2
2 1 Force/area (N/m Force/area (N/m Force/area 0 0 0 50 200 250 0 50 200 250 Volume (cm3) Volume (cm3)
) 3 2 5.1 m/s
2 Rectangular Solids (Front) Rectangular Solids (Side) Spheres 1 Cylinders Force/area (N/m Force/area 0 0 50 200 250 3 Volume (cm )
Figure 1.12: Force per area vs. volume for all three water velocities. Error bars represent 95% confidence intervals of the mean.
1.4.5 Force as a function of object drag coefficient
Drag coefficients for the high velocity (9.8 m/s) jet are given in Table 1.3. Front facing rectangular solids have larger drag coefficients than side facing ones, and both are greater than those observed for spheres. Drag coefficients for spheres decrease with sphere size (sphere diameters increase with sphere number), which is to be expected: for Reynolds numbers on the order of 105, smooth spheres undergo a boundary-layer transition that decreases drag coefficient as Re increases (Vogel 1994). As spheres with increasing diameters have larger Reynolds numbers, calculated drag coefficients match the expected decreasing drag coefficient pattern.
Chapter 1 28 Rectangular solids Rectangular solids Spheres (front) (Side)
Avg. Max. Avg. Max. Avg. Max.
maximum average maximum average maximum average
1 0.90 0.92 ± 0.04 0.74 0.76 ± 0.02 0.35 0.43 ± 0.03
2 0.87 0.91 ± 0.02 0.64 0.65 ± 0.01 0.35 0.38 ± 0.02
3 0.75 0.79 ± 0.01 - - 0.22 0.27 ± 0.01
4 0.84 0.86 ± 0.02 0.69 0.70 ± 0.01 0.20 0.22 ± 0.01
5 0.97 1.00 ± 0.02 0.74 0.75 ± 0.01 0.17 0.19 ± 0.01
6 0.88 0.89 ± 0.01 0.73 0.74 ± 0.01 - -
7 0.85 0.86 ± 0.02 0.71 0.72 ± 0.02 - -
8 0.97 0.97 ± 0.01 0.76 0.77 ± 0.01 - -
9 0.88 0.89 ± 0.01 0.67 0.70 ± 0.01 - -
10 0.87 0.88 ± 0.01 0.75 0.77 ± 0.01 - -
Table 1.3: Drag coefficients calculated for rectangular solids (front- and side- facing) and spheres, both for the ensemble averages (avg. maximum) and the average and 95% confidence intervals of the mean for all 25 runs composing the ensemble average (max. average).
For rectangular prisms and spheres, area accounted for over 93% of the variation of impingement magnitude. However, the slopes of the lines relating force to area differed for front-facing rectangular solids, side-facing rectangular solids, and spheres, which all had different shapes. Because CD accounts for shape, could the differences in drag coefficients between these objects be responsible for this variation?
To remove the effect of shape, I normalized impingement magnitude by drag coefficient. When this was done, the front and side rectangular solids plotted against area became indistinguishable, suggesting that the difference in magnitude between these two groups of objects was due solely to their shape (Figure 1.13). It is clear from
Chapter 1 29 Figure 1.13 that drag coefficient is also responsible for much (but not all) of the difference in magnitude between the spheres and rectangular solids. Cylinders were excluded from this analysis due to their invariant frontal areas.
400 Rectangular Solids (Front) 350 Rectangular Solids (Side) Spheres 300
250
200
150
100 Force/Drag Coefficient (N) 50
0
0 5 10 15 20 25 30 35 40 45 50 2 Area (cm )
Figure 1.13: Impingement magnitudes for the high velocity jet normalized by drag coefficient, plotted against area. Error bars represent 95% confidence intervals of the mean.
In essence, Figure 1.13 suggests that Equation 1.2 is an accurate model of impingement.
1.4.6 Water velocity
The horizontal velocities of 217 spheres over 35 waves produced by the 9.8 m/s jet are shown in Figure 1.14. There is a brief period of high water velocities at the first part of the wave, concomitant with observed impingement spikes.
Chapter 1 30
Figure 1.14: Water velocity time-series of the water cannon jet. Each point represents the measured velocity of one neutrally-buoyant sphere. 217 spheres from 35 water cannon jets are shown.
1.4.7 Predicted vs. measured drag
Over the first 0.3 s of the wave, predicted drag was approximately equal to measured force, although predicted impingement forces (using a constant drag coefficient and the measured water velocities) generally underestimate measured values (Figure 1.15). Predicted impingement magnitudes are, on average, 5% lower than measured values for rectangular solids, and 19% smaller for spheres. Representative predicted and measured drag values are shown for a front-facing rectangular prism, a side-facing rectangular prism, and a sphere in Figure 1.16 through Figure 1.18. (Plots for all objects can be found in the Appendix on Page 40.) Predicted values from measured water velocities were aligned as closely as possible with the start of the jet, but a few predicted values lie before the impingement peak. In the 0.1 s following the impingement peak, predicted drag values generally overestimated measured drag
Chapter 1 31 forces for side-facing rectangular prisms and spheres, but were within observed ranges of individual runs (Figure 1.17 and Figure 1.18).
150 Rectangular Solids (Front) Rectangular Solids (Side) Spheres
100
50 Predicted Impingement Magnitude (N)
0 0 50 100 150 Measured Impingement Magnitude (N)
Figure 1.15: Predicted impingement magnitudes vs. measured drag for all rectangular solids and spheres. Overall, predicted magnitudes are slightly underestimated relative to measured forces.
Chapter 1 32 20
18
16
14
12
10
Force (N) 8
6
4
2
0 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 Time (s)
Figure 1.16: Measured ensemble average (dark gray line) and predicted drag (black points) for a front-facing rectangular solid. Individual runs are shown in light gray to illustrate variability.
50
45
40
35
30
25
Force (N) 20
15
10
5
0 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 Time (s)
Figure 1.17: Measured ensemble average and predicted drag (black points) for a side- facing rectangular solid. Individual runs are shown in light gray to illustrate variability.
Chapter 1 33 15
10 Force (N)
5
0 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 Time (s)
Figure 1.18: Measured ensemble average and predicted drag (black points) for a sphere. Individual runs are shown in light gray to illustrate variability.
1.5 DISCUSSION My results suggest that, in contrast to previous assumptions, impingement is simply drag. The reasons for this are as follows:
● Impingement force scales linearly with area. ● When object drag coefficients are accounted for, differences between force per area are explained. ● Impingement magnitudes can be predicted using drag (Equation 1.2) as a model: predicted drags closely match measured impingement values.
1.5.1 A new look at impingement
Drag is arguably the most well-known hydrodynamic force, and is well-characterized. As a result, drag can be predicted for nearly any object by measuring its drag
Chapter 1 34 coefficient and area. Provided the density of the fluid is known, drag can be predicted at any velocity using Equation 1.2. Thus, for waves with known velocities, such as those generated by the water cannon, impingement magnitudes can be predicted (within the limits of experimentally measured velocity and drag coefficient). Flow velocities generated by breaking waves are difficult but not impossible to measure (Gaylord 1999, 2000; Chapter 2 of this thesis), and improving intertidal flow velocity measurements in the future will allow researchers to predict impingement magnitudes on organisms in the field.
If impingement is a transient spike in drag, there is no reason to expect that these spikes may be limited to the front of a wave. Sharp velocity increases may occur at any point throughout a wave in the field if velocity momentarily increases, and these spikes may be just as important to organisms as those occurring at the moment of wave impact. Gaylord reported that the largest forces observed in his studies were impingement events. Is there some phenomenon associated with velocity transients at impact that cause impingement events to be larger there than in the rest of the wave? Are spikes in drag occurring at wave impact indeed the largest hydrodynamic forces in the field? I explore these questions using high-frequency field measurements of breaking waves in Chapter 2.
1.5.2 Impingement magnitude scaling behavior
For the rectangular solids in both orientations, there was a positive and highly significant relationship between impingement magnitude and object area, but no significant relationship between force per area and object volume. However, some results for spheres are unexpected: spheres displayed a slightly curved and downwardly concave relationship between impingement magnitude and area, and for one velocity, have a decreasing relationship between normalized force and volume for one jet. However, these results are not incompatible with my conclusion that impingement is drag: if impingement were an accelerational force, normalized force would increase with volume, which is opposite of the pattern observed.
Chapter 1 35 Why cylinders display a relationship between magnitude and volume but rectangular solids do not remains unknown.
1.5.3 Predicted vs. measured drag
In general, predicted estimates of drag matched the force time-series measured for each rectangular solid and sphere. Most variation in predicted measurements was within the range of force measurements for the turbulent jet used in these experiments; due to the turbulence of the jet, such variation is expected. Impingement magnitudes were not as well predicted as the rest of the wave, although values were all between -4 to 12% of measured impingement spikes for rectangular solids in both orientations. Spheres impingement magnitudes were underestimated by 10 to 30%.
What might be causing the discrepancies in impingement prediction? First, the jet produced by the water cannon is extremely turbulent, and velocities vary both temporally and spatially within the jet, particularly at the very front of the wave. As a result, my velocity measurements of this region of the wave are variable and may not accurately describe the rapid changes in velocity occurring within the jet. With more tests to measure jet velocity, the discrepancies between predicted and measured impingement forces may be partially closed.
Additionally, my estimates of drag coefficient rely on measured velocities at one time in each “wave,” and are therefore likely inexact. Drag coefficients are dependent on the flow regime and velocity, and my experimentally determined drag coefficients (calculated at maximum steady-state drag) likely do not capture the rapidly changing flow conditions in the first 0.3 s of the wave. If, as a result, my drag coefficient estimates are off by 15%, all of the variation in the rectangular solids (and much of the spheres) may be explained. This may also account for the apparent over-prediction of drag in the 0.1 s of the wave immediately following the impingement peak (Figure 1.16 through Figure 1.18).
Chapter 1 36 One additional explanation for the discrepancy between predicted and measured impingement forces may be imperfect filtering of inertial responses of the test objects. Because filtering does not remove all energy within the filtered range, the measured impingement magnitudes may be overestimated: that is, the inertia of the test object may be contributing force to my measurements that was not entirely removed by filtering. This effect is discussed further in Chapter 3.
1.5.4 Are organisms capable of responding to brief loads?
There is some question as to whether organisms that are not perfectly rigid are capable of responding to impingement events, which occur over brief time periods. An object’s mechanical response to a transient force depends on its structural properties: a tuning fork made out of soft rubber would not vibrate at the same frequency as one made from steel. Intertidal organisms span a wide range of materials, from stiff barnacle shells to rubber-like gastropod feet to extremely pliable seaweeds. We can reasonably expect organisms with such divergent structural properties to respond differently to brief hydrodynamic loads.
Indeed, Gaylord (2000) noted a difference in magnitude between compliant and rigid organisms – measured impingement magnitudes for flexible seaweeds were much lower than for rigid urchins. He proposed that the difference was due to the ability of seaweeds to reorient with flow, and thus expose a smaller projected area to oncoming waves. Another possible explanation for the smaller impingement forces seen in seaweeds is a longer organism response time. Because seaweeds are flexible, they deform when loaded. This deformation takes time, and loads of brief duration (like impingement) may be over before an organism reaches maximum deformation. In addition, any viscous components of biological materials may cause a lag between applied force and a deformation response, which may reduce the effective force and maximum deformation. (For a more detailed description of how compliant organisms may ameliorate transient loads, see Gaylord et al. (2001)). While Gaylord et al. explored the effects of transient loads on seaweeds, no work has been done on how the visco-elastic foot of a limpet may be affected by transient loads. Whether the
Chapter 1 37 structural properties of limpet feet ameliorate or enhance brief forces is explored in Chapter 3.
1.5.5 Does impingement limit intertidal organism size?
This study suggests that impingement spikes are actually brief drag spikes, and therefore scale with area. As a result, impingement is not likely to limit organism size in the intertidal zone, and the mechanism that does remains unknown – regardless of whether it is biological or mechanical. However, two major assumptions go into this conclusion, and these may be worth examining in the future. First, the argument that velocity-scaling forces cannot limit size assumes that tenacity varies linearly with attachment area. Multiple studies have found linear relationships between attachment strength and foot area in gastropods (Miller 1974; Branch and March 1978; Grenon and Walker 1981), though some of these relationships may in fact be nonlinear. Branch and Marsh (1978) report an obvious nonlinear fit for one species of Patella, and Grenon and Walker (1981) noted that all but one of their relationships were slightly nonlinear. However, force must increase faster than tenacity to provide a mechanism for size limitation; this requires the nonlinear fit to have a negative second derivative. This is not true of Branch and Walker’s limpet, nor several of those in Grenon and Walker’s study. However, in these studies, not all tenacity measurements were made in shear, which is how these animals are likely to be loaded in the field. If the relationship between shear tenacity and cross-sectional area has a negative second derivative, size may be limited as force increases faster than tenacity.
Second, the scale-independence argument assumes that organisms grow isometrically. This is not true for some marine species, including some seaweeds (coralline algae do not grow isometrically, for example), but this seems to vary by taxon: within the European Patella genus of limpets, two of four species studied displayed isometric growth while the other two did not (Cabral 2007). At any rate, assuming isometric growth across taxa is not a safe assumption, and may provide a mechanism by which organism size is limited in the intertidal zone. For a more thorough discussion of the
Chapter 1 38 possible size-limiting mechanisms in the intertidal zone, including failed and promising predictive models, see Denny (1999).
1.5.6 Conclusions
This study produced the first laboratory measurements of impingement. Contrary to previous assumptions, I demonstrate that impingement events are likely to be sharp increases in drag, caused by brief spikes in water velocity at the instant of wave arrival. The drag equation can thus be used to predict impingement forces under flows of known velocity. In the next chapter, I explore the implications of this study on high- frequency hydrodynamic measurements taken in the surf zone.
Chapter 1 39 1.6 APPENDIX
Figure 1.19: Measured ensemble average (dark gray line) and predicted drag (black points) for front-facing rectangular solids. Individual runs are shown in light gray to illustrate variability.
Chapter 1 40
Figure 1.20: Measured ensemble average and predicted drag (black points) for side- facing rectangular solids. Individual runs are shown in light gray to illustrate variability.
Chapter 1 41
Figure 1.21: Measured ensemble average and predicted drag (black points) for spheres. Individual runs are shown in light gray to illustrate variability.
Chapter 1 42 Chapter 2
Characterizing wave impact forces in the field using high- frequency hydrodynamic measurements
2.1 ABSTRACT Hydrodynamic forces produced by breaking waves make the rocky intertidal zone one of the most physically stressful environments on the planet. Despite the recognized importance of quantifying intertidal flows, hydrodynamic measurements of high enough frequency to resolve “impingement events” – the large, transient spikes in force occurring at wave impact, and hypothesized to be the largest hydrodynamic force in the intertidal zone – have not been made. I built a custom force transducer to measure high-frequency flows in the intertidal zone, and took measurements of force acting upon both a drag sphere and a limpet shell. I estimate impingement events to occur in 2-7% of waves. Mean ratios of the maximum force to the second largest force within a wave are approximately 1.5 for impingement waves, and 1.2 for all waves. Despite the increased variability in force they inject into a wave, impingement events are not the largest forces imposed on intertidal organisms. At my site, impingement events had a mean maximum force of 0.77 N, while the mean maximum force of all waves in this study was 1.21 N. These results suggest that impingement is not in and of itself dangerous to organisms. Future studies should focus on high-frequency measurements of surf zone flows to identify the largest transient forces regardless of where they occur in the wave.
2.2 INTRODUCTION The hydrodynamic forces produced by breaking waves define many aspects of life on wave-swept rocky shores, shaping intertidal communities in a number of ways. Perhaps the most obvious consequence is the breakage and dislodgment of organisms (Dayton 1971; Paine and Levin 1981; Koehl 1984; Denny et al. 1985; Denny 1987b; Trussell et al. 1993; Denny 1995; Bell and Gosline 1997; Denny and Blanchette 2000;
Chapter 2 43 Lau and Martinez 2003). Dislodgment occurs from single waves that are greater than an organism can withstand, or from repeated sublethal loads if organisms fail by fatigue (Denny et al. 1989; Hale 2001; Mach et al. 2007a; Mach et al. 2007b; Mach 2009; Mach et al. 2011). Wave action has also been shown to limit species assemblages in the rocky intertidal zone (Lewis 1964, 1968; Paine 1979; Paine and Levin 1981; McQuaid and Branch 1985; Rickets et al. 1985; Rius and McQuaid 2006, 2009; Branch et al. 2010; Tam and Scrosati 2014). In many species, growth and life histories are at least partially dictated by flow regime: sea stars alter body shapes depending on the degree of wave exposure (Hayne and Palmer 2013), anemones display evidence of hydrodynamic forces limiting growth (Wolcott and Gaylord 2002), and several snail species have been shown to have growth rates or sizes tied to wave exposure (Grahame and Mill 1986; Brown and Quinn 1988; Etter 1989; Etter 1996; Trussell 1997; Trussell 2002). Additionally, reproductive success is tied to fluid forces for many organisms. Some snails have higher reproductive efforts on exposed coasts versus protected areas (Etter 1989), and flow conditions and topography have been shown to greatly influence external fertilization success (Denny et al. 1992; Mead and Denny 1995; Berndt et al. 2002).
Although quantifying flow conditions in the intertidal zone is critical to understanding and predicting these processes, accurately assessing hydrodynamics on wave-swept rocky shores is a difficult proposition. First, flows are stochastic and both spatially and temporally variable due to the complex topography that characterizes rocky shorelines. Consequently, measurements made using dynamometers that record maximum forces in a given time interval (Jones and Demetropolous 1968; Palumbi 1984; Bell and Denny 1994; O’Donnell and Denny 2008) do not provide the temporal detail needed to understand an organism’s wave-to-wave flow environment. Time-series measurements of flow in the intertidal zone using force transducers (such as those used by Koehl (1977), Denny (1985), Denny et al. (1985), Gaylord (1999, 2000), and O’Donnell (2008)) paint a more accurate picture of instantaneous forces and flow conditions. However, the higher cost of these instruments and associated electronic, data collection, and data storage requirements have historically made long-term
Chapter 2 44 measurements of this kind impractical. Advances in computing power and data storage have made these types of measurements more feasible in recent years, and Gaylord’s (1999, 2000) measurements have provided the most in-depth look at instantaneous flow conditions on rocky shores to date.
Among the insights Gaylord’s work provided was the presence of what he termed “impingement forces” (Gaylord 2000). Impingement events are brief spikes in force occasionally present at the instant of wave impact, and they appear to be the largest hydrodynamic forces acting on intertidal organisms. These events were observed when waves struck the emersed (out of water) transducer, and Gaylord hypothesized that they occur due to a lack of surrounding fluid to cushion the impact of a wave against the object (Gaylord 1999). Gaylord described several notable patterns regarding impingement: first, ratios of impingement force to subsequent force within a wave differed between rigid objects and flexible seaweeds. Maximum ratios were 10 for rigid samples and 4 for flexible ones. Second, impingement events occur in only a subset of waves. Third, measured impingement durations were brief: Gaylord et al.’s (2001) estimates range from 0.02 s to 0.24 s. In these studies, an imprecise definition of “impingement” was used, which, as I will discuss, leaves room for misinterpretation of what constitutes an impingement event. Finally, Gaylord concluded impingement spikes are a stochastic phenomenon whose maximum forces cannot be predicted using traditional hydrodynamic approaches.
Gaylord’s measurements remain the only observations of impingement events in the field: no other studies have recorded brief forces upon wave impact, although they have been hypothesized to occur (Denny 1988). Here, absence of evidence is not necessarily evidence of absence: it is unlikely the transducer used by Denny (1985) and Denny et al. (1985) to record breaking waves was able to respond quickly enough to record such brief forces. And although Gaylord’s measurements provided a leap forward for high-frequency, empirical flow measurements in the rocky intertidal zone, similar response-time limitations prevented several questions about impingement events from being resolved.
Chapter 2 45 One of the primary questions about impingent is why these events occur. In Chapter 1, I detailed the first laboratory measurements of these brief forces. I showed that impingement is likely a brief spike in drag caused by a short-term increase in water velocity. As a result, there is no reason to expect that sharp velocity increases occur only at the front of a wave. Whether velocity transients (and therefore, hydrodynamic forces) occurring at wave impact are larger than those in subsequent portions of the waves remains unknown; transient forces occurring at any point in a wave may be equally as important to an organism as those at impact.
With regard to the ecology of wave-swept shores, perhaps the most important question raised by Gaylord’s field data is how the presence of large, transient forces affects studies performed using dynamometer flow measurements. Because of the difficulty involved in making high-frequency time-series measurements of hydrodynamic forces in the surf zone, the majority of studies quantitatively exploring intertidal flow have used dynamometers such as those described by Bell and Denny (1994). Dynamometers of this sort have been deployed in oceans around the world, measuring hydrodynamic force for a variety of applications. These measurements have been used to assess breakage and survival of organisms such as seaweeds (Blanchette 1997; Mach et al. 2011; Demes et al. 2013) and limpets (Denny and Blanchette 2000). Dynamometers have also been used to explore sea star movement in relation to wave environments (Barahona and Navarette 2010), investigate species richness and diversity with environmental stress models (Zwerschke et al. 2013), and describe patterns of abundance, size structure and microhabitat patterns of urchins (Jacinto et al. 2013).
However, dynamometers do not have the capability of responding to brief loads (Miller unpublished), and Gaylord’s measurements of impingement raised an important question: are studies using dynamometers inadvertently missing the largest hydrodynamic force in the intertidal zone? Gaylord reported impingement forces up to ten times greater than subsequent drag. If the measurements made in studies using dynamometers underestimate maximum forces by an average factor of 2.5 (Gaylord
Chapter 2 46 2000), previous conclusions regarding surf-zone hydrodynamics are questionable. Therefore, there is a need to investigate – with a dedicated, high-frequency transducer – how much larger impingement events are than subsequent forces in a wave.
In response to this need, I constructed a force transducer capable of making high- frequency measurements of hydrodynamic force. In this chapter, I use this transducer to answer several questions regarding impingement events. First: are they a general phenomenon, or were they unique to Gaylord’s site and time? Second: how frequently do impingement events occur? Third, how much larger are impingement events than the next largest force within a wave? Finally, are impingement events the largest forces acting on intertidal organisms?
2.3 METHODS AND MATERIALS
2.3.1 Drag elements
Hydrodynamic forces were measured on two objects: a 2.54 cm sphere and a rigid model of a limpet. The sphere was chosen for its rotational symmetry, as forces from any direction are measured equally. The limpet model was used to determine if patterns observed with spheres hold for forces acting on a common intertidal organism.
A 2.54 cm sphere was chosen because its size is on the order of larger intertidal organisms, such as limpets, mussels, and large snails. The drag sphere was roughened with a file; the ratio of roughness elements to the diameter of the sphere was approximately 1/100. By roughening the sphere, I hoped to avoid the variable drag coefficients experienced by spheres due to the turbulent boundary layer transition (Schlicting 1979). This “drag crisis” typically occurs for Reynolds numbers on the order of 105 – an expected range of Reynolds numbers for many intertidal organisms (see Chapter 1). The sphere was mounted to the transducer by a ¼”-20 stainless steel threaded rod, which tightly fastened into the body of the cross-shaped aluminum transducer (Figure 2.1).
Chapter 2 47 In addition to the drag sphere, I constructed a rigid limpet model by filling an empty Lottia gigantea shell with epoxy. Before the epoxy hardened, a ¼”-20 bolt was positioned in the epoxy near the limpet’s center of mass so it could be attached to the transducer. When the limpet model was used, it was positioned so the anterior of the shell faced the dominant direction of flow.
2.3.2 Transducer
To search for impingement events in the intertidal zone, Shuma (2012) designed a specialized force transducer. This transducer is a 2-axis, cantilever-style transducer similar in principle to those of Gaylord (1999) and O’Donnell (2008). Two matched semiconductor strain gauges (Micron Instruments, SS-060-033-500PU) were mounted to the four transducer faces (Figure 2.1) and configured as Wheatstone bridges (Horowitz and Hill, 1989), providing a voltage proportional to the bending of the transducer in each axis as a load was applied to the drag sphere. The shape of the transducer was chosen to maximize the first natural frequency of the instrument (10,000 Hz in air) to facilitate high-frequency measurements in the surf zone (Shuma 2012). To maximize the first natural frequency of the drag sphere, the threaded rod connecting the drag sphere to the body of the transducer was made as short as possible (approximately 3 mm). The drag sphere/connecting rod element had a first natural frequency of approximately 1500 Hz (roughly 3 times higher than that of Gaylord’s (1999) transducer). The heavier mass of the limpet drag element relative to the drag sphere lowered its first natural frequency to approximately 500 Hz.
Chapter 2 48
Figure 2.1: The force transducer used for field measurements.
The strain gauges were waterproofed using a coating originally developed for sealing aircraft fuel tanks (PR-1422 B-2, PPG Aerospace-PRC-DeSoto), which has proven effective in protecting electronics in the intertidal zone. The interior electronics cavity was made waterproof with a lubricated O-ring.
Signals from both axes of the transducer were amplified 100x by an amplifier located in the waterproof cavity of the transducer. After amplification, a referenced single- ended signal was transmitted to a computer via 76 m of shielded instrumentation cable. To protect the cable from intertidal debris, hydrodynamic forces, and other stresses, the 15 m of cable nearest the transducer cable was encased in plastic tubing. Eye bolts were anchored into rocks near the field site, and the cable was attached to these eye bolts with zip ties. Eye bolt spacing varied: eye bolts nearest the transducer (subject to the highest wave forces) were spaced approximately 20 cm apart, while those much higher on the shore were spaced 1-2 m apart.
The transducer was bolted to a custom thick-walled cylindrical housing at the field site. The bottom face of the housing contained threaded holes which matched the bolt-
Chapter 2 49 hole pattern in the base of the transducer. The housing was cemented into a 4-inch diameter hole drilled with a standard core drill, and the lip of the mount was approximately flush with the surrounding rock. To ensure the drag sphere or limpet model was the only element of the device exposed to flow, a lid was constructed to shield the body of the transducer from waves. The lid had a hole in the center slightly larger than the bolt attaching the drag elements to the transducer, permitting the drag elements to be fastened without any contact from the lid.
2.3.3 Field site
The transducer was deployed at a site located in a surge channel on the rocky shoreline of Hopkins Marine Station in Pacific Grove, California (36.622°N, 121.906°W). This site is typical of the rocky coastlines in this region, with variable topography over small spatial scales (for a more thorough discussion of variability along spatial and temporal scales in the intertidal zone, see Denny et al. (2004)).
This site faces the shore, and is located on a gently sloping rock. During high tides, the tidepool in front of the site fills with seawater, submerging the transducer. The dominant direction of water flow is along the seaward-facing surge channel, but during periods of large waves, waves occasionally break over the top of the rock (Figure 2.2).
Chapter 2 50
Figure 2.2: Photo of site at low tide, with arrows showing typical flow directions.
The site was located at 1.52 meters above mean lower low water (MLLW), an elevation that offers three distinct advantages. First, the primary interest in this study was waves striking emersed objects: therefore the sensor was placed at a tidal height where it would be out of the water as much as possible, but still frequently hit by waves. Second, having a site this high above MLLW facilitated easy access to the site; that is, I did not have to wait for extremely low tides to service the transducer. Third, this site roughly corresponded to the tops of the nearby mussel beds, an important boundary of zonation patterning in the intertidal zone demarcating middle and high intertidal species. Referencing my measurements to an “ecological tidal height” allowed hydrodynamic effects on organisms living at or near this boundary to be considered.
Chapter 2 51
Owing to the relatively high position and location within a surge channel, the transducer experienced primarily postbreaking flows (see Denny (1988) for a more thorough discussion). These flows resulted from waves that broke seaward of the sensor and propagated up the shore to the transducer as turbulent bores, and are described by Gaylord (1999, 2000) as “fully breaking” waves – waves whose crest had already broken and whose wave face had degenerated into turbulent flow.
Although ocean waves are typically considered oscillatory flows, flows at the site were treated as unidirectional due to both the topography of the site and the period parameter of the drag sphere. When the emersed transducer was struck by a wave, flow typically did not reverse direction and flow back over the transducer; generally the fluid went around the site, providing more of a pulsating than oscillating flow. Second, the period number K, an index relating the length of an object in flow to the distance travelled by the flow in each oscillation, shows that the drag sphere at this site experiences flows that can be approximated as unidirectional (K ≫!30, Sarpkaya and Storm (1985)). Denny (1988) provides a more thorough explanation of the period number.
2.3.4 Data collection
The transducer was connected to a computer running LabView (National Instruments Austin, TX). Both signals were low-pass filtered at 20 kHz for anti-aliasing purposes, and amplified an additional 10x. The signals were sampled at 40 kHz by LabView, and saved as data files in 60-second segments. Sampling with the drag sphere took place across 41 days in August, September and October of 2013. Limpet data were collected over two weeks in November 2013.
Chapter 2 52 2.3.5 Post processing
2.3.5.1 Calibrating To process the raw data, I first converted the measured voltages to force in Newtons using calibration curves developed for each axis of the transducer, developed by hanging known weights from a threaded rod fastened to the body of the device. The transducer was positioned horizontally, with one face of the transducer parallel to the ground. A series of weights were suspended from the rod at a distance corresponding to the center of area of the drag sphere, and multiple voltages per weight were recorded. This was repeated for all four transducer legs, and regression lines fitting voltage per force were calculated for each transducer axis. Additionally, calibrations were carried out at varying heights from the transducer base to determine corrections for objects either taller or shorter than the drag sphere. All data recorded with the transducer were converted to the appropriate force in Newtons using these calibration values.
2.3.5.2 Filtering The power spectra for representative portions of recorded force signals were computed using MATLAB (MathWorks, Natick, MA, 2008), which allowed me to see the approximate frequencies that contained electrical noise and inertial resonance from both the transducer and the measured object. Electrical noise was omnipresent through the entire sampling period at 60 Hz (and harmonics thereof), and Butterworth band- stop filters were used to filter these out. An unknown, nearly constant noise source between 8 and 13 Hz was also present throughout the entire sampling period, and was filtered out. This noise was assumed to be inherent to the power source used for the recording computer because a separate intertidal experiment running nearby (with a different cable, computer, and electronics) recorded noise at a similar frequency.
The power spectrum also provided an estimate of the first natural frequency of the transducer when bolted to its mount in the rock. This was approximately 10 kHz (9998 Hz for one axis and 10005 Hz for the other). This varied slightly depending on how
Chapter 2 53 tightly the transducer was bolted in, although care was taken to fasten the transducer consistently and tightly each deployment. The drag sphere on the threaded rod had an inertial response from approximately 1500-2200 Hz, while the limpet drag element had an inertial response from approximately 500-1500 Hz (due to its greater mass – see Chapter 3). Although I attempted to fasten drag elements consistently, the frequency of responses varied slightly from deployment to deployment – but by no more than a few hundred Hz in either direction. For each deployment, a Butterworth band-stop filter was designed to filter out the inertial response of the drag elements.
2.3.5.3 Zero balancing As the strain gauges on the transducer are somewhat temperature sensitive and prolonged data recording therefore involved some DC drift, the zero for each axis needed to be determined and periodically subtracted from the signal. This was straightforward at low tide, when the transducer was clearly emersed. For these periods of time, identifying regions with no waves (and therefore, no force) allowed a reliable zero to be determined and subtracted from force measurements. However, at high tide, determining “zero” was far more difficult as regions without any force could not be positively identified. For this reason, all measurements made at tidal heights greater than the height of the transducer (see Environmental data, below) were not used. For intermediate tides, a reliable zero could not be established for some periods of time due to the temperature fluctuation between warm air temperature and relatively cold seawater. This was especially true when the transducer was being hit by the first waves following a daytime low tide. To minimize uncertainty in my measurements due to poor zero-balancing, waves for which a clear zero could not be established were ignored.
2.3.6 Environmental data
The NOAA tidal station located in Monterey Harbor (Station 9413450, 36.605°N, 121.886°W) records tidal height every 6 minutes, referenced to mean lower low water (MLLW). Verified tide data were downloaded for sampling periods.
Chapter 2 54 Significant wave height Hs was measured by the Coastal Data Information Program (CDIP) buoy located 0.6 kilometers offshore of Hopkins Marine Station (Buoy 158, 36.6263°N, 121.9071°W). The buoy calculates significant wave height (the average height of the largest one-third of waves) every 30 minutes, and I downloaded data corresponding to sampling periods. Each 6-minute period of force data (to line up with the tide data) in the 30 minutes preceding each data point was assumed to have a significant wave height equal to that reported by the buoy (since Hs is calculated over a 30 minute period).
The CDIP buoy also provides estimates of wave period every 30 minutes, but these measurements were not used in this analysis, as waves of all periods broke well seaward of my field site.
2.3.7 Wave analysis
Following calibration, filtering, and zero-balancing for signals from both transducer axes, the overall magnitude of force was calculated and used for all further analysis. Small fluctuations about zero in each axis produced a “noise floor” of positive forces ranging from 0 to 0.05 N when magnitudes were calculated. Maximum forces reported here, then, may be overestimated by 0.05 N.
2.3.7.1 Wave selection To maximize the environmental conditions covered by my analysis, periods of time representing the most diverse array of environmental conditions were selected for analysis. Additionally, data were subsampled to every other minute for analysis (though for some periods of time early in the analysis, every minute was analyzed). Subsampling every other minute (as opposed to larger periods of time, such as every other hour) allowed a reduction in waves identified while still sampling evenly across both time and environmental gradients.
I visually inspected the data to identify waves, and selected waves for analysis only if they met each of two criteria. First, maximum force needed to be at least 0.25 N to
Chapter 2 55 ensure each wave was sufficiently above the noise floor of the transducer. Second, I only counted a wave if the transducer was clearly out of the water prior to impact. This was done for two reasons. (1) The original impingement data (Gaylord 1999, 2000) were measured on emersed force transducers, and I wanted to be able to compare my data to those. (2) As described in the previous section, determining a reliable “zero” value was difficult during periods where the transducer was submerged. If the magnitude of the force acting on the sphere wasn’t clearly zero in the immediate half- second leading up to a wave’s arrival at the sensor, I did not include the wave in my analyses.
The starting point of each wave was determined by visual selection, and the time of occurrence for each wave was recorded. The data for the half-second before the wave and 2.5 seconds after the wave were saved as a separate file. Using each wave’s time stamp, the tidal height and significant wave height at the time of each wave were determined.
2.3.7.2 Impingement and wave identification No specific criteria for reliably identifying impingement events exist, apart from waves whose maximum forces occurred at wave impact, or those that superficially resembled the wave Gaylord et al. (2001) described as an impingement wave.
Attempts to implement a set of criteria that could accurately identify waves with impingement events were found to be subjective: how long, exactly, can elapse between the identified start point of a wave (itself somewhat subjective) and the maximum force? If any local maxima or minima exist between the start and maximum forces in a wave, is it still an impingement event? To deal with this uncertainty, I created two rules: first, the maximum force needed to occur within 0.5 seconds of wave start (as in Gaylord (2000)). Second, the maximum forces calculated each 0.01 second between wave start and the wave maximum needed to be below either the minimum cutoff for a wave (0.25 N) or one fourth the wave maximum, whichever was greater. (The one-fourth wave maximum was implemented because this algorithm was
Chapter 2 56 biased to smaller waves without it: larger waves tended to have larger fluctuations immediately preceding the maximum force, and these were frequently larger than 0.25 N while still being small relative to the maximum force.) Limiting the maxima of any forces leading up to the maximum force in a wave ensured that in waves with an impingement event, the first large force to strike the transducer was the impingement spike itself.
Waves that passed through these initial filters were grouped into three classes: first, waves which inarguably contain impingement spikes, second, waves which might have an impingement event if the criteria are more broadly defined, and third, waves with no impingement event. Waves in the first group (referred to as definite impingement events) have a near instantaneous rise from wave start to force maximum, with very little (if any) noise in the signal between those two points (Figure 2.3a). Waves in the second group (referred to as probable impingement events) have an overall shape that superficially resembles Gaylord’s impingement example, but may have a noisier signal between wave start and the wave maximum (Figure 2.3b). Finally, the third group of waves passed through the initial filters but did not have an impingement event by any reasonable definition; these waves were not considered impingement events.
Figure 2.3: a) Example of a definite impingement wave (characterized by a near instantaneous rise time to maximum force) and b) a probable impingement wave (characterized by a slightly longer rise time).
Chapter 2 57 2.3.7.3 Maximum force and force ratio calculation To estimate how much larger impingement peaks are relative to the rest of a wave, the maximum force was calculated for each 0.01 second period within the wave. Maximum force was compared to the second largest force and a ratio obtained.
The 0.01 second binning was chosen due to the high sampling rate used to collect wave data. Because each wave was measured at 40 kHz, it is very likely that the maximum and second largest forces will inevitably result in two values only 1/40,000 of a second apart. To avoid this issue and accurately assess maxima from reasonably distinct time periods of the wave, maxima were calculated within each 0.01 second bin. In cases where the two maxima occurred near bin divisions – e.g., one at the end of one 0.01 segment, and the other at the very beginning of the next 0.01 segment – maxima were only calculated if there existed at least 0.01 seconds of separation between values. If this requirement was not met, the next maximum was chosen to ensure maxima that did not functionally co-occur. Visual inspection of many waves showed that this method and bin size did a reliable job of picking force maxima occurring in different regions of the wave.
Distributions of this force ratio as well as the distribution of maximum forces were calculated for waves with definite impingement events, probable impingement events, and the larger set of all waves.
2.4 RESULTS
2.4.1 Drag sphere measurements
9,311 waves were identified that met the minimum magnitude requirement and were clearly emersed prior to wave impact.
Over the study period, I sampled over a range of tidal conditions from 0.3 m below MLLW to approximately 2 meters above MLLW. However, due to the variability of the tides themselves, not every day of sampling saw these extremes. Significant wave
Chapter 2 58 heights ranged from 0.35 meters to 2.5 meters during sampled periods. A plot showing the distribution of sampled conditions is shown in Figure 2.4.
Figure 2.4: Density plot showing number of six minute sampling periods analyzed per tidal height (above MLLW) and significant wave height (m).
I attempted to measure the widest range of wave and tidal conditions possible, but was limited by environmental conditions occurring during sampling periods. However, sampling periods did include large waves between Hs = 2 and 2.5 meters, though those waves only occurred at tides higher than 0.7 m above MLLW (less than 2% of significant wave heights recorded by the buoy in 2013 were greater than 2.0 meters).
Wave and tidal conditions under which waves were recorded are shown in Figure 2.5. There are two primary differences between my wave occurrence plot and the plot representing all environmental conditions. First, the lower left portion of the plot is not well represented, due to the placement of the transducer on the shore. Because the transducer was located at 1.52 m above MLLW, low tides that coincided with small waves did not produce any waves that reached the transducer’s position. Second, the
Chapter 2 59 largest waves present during the sampling period did not result in waves included in this study. Many of these waves filled up the tidepool surrounding the transducer, keeping it effectively submerged between waves and causing the transducer to record small forces. These waves were not included because the transducer was not clearly emersed in the half-second prior to wave impact.
Figure 2.5: Density plot showing number waves recorded per tidal height (above MLLW) and significant wave height (m). Outline of sampled environmental conditions is shown in orange.
2.4.1.1 Impingement frequency One hundred thirty seven waves (out of 9311 total) were identified as having a definite impingement peak, while an additional 518 waves were classified as probable impingement events. These results suggest that 1.5% to 7.0% of waves contain an impingement event, depending upon the criteria used to define impingement.
Chapter 2 60 2.4.1.2 Force ratios For definite impingement events, the mean force ratio (maximum force to second maximum force) is 1.43, and the median is 1.30 (Figure 2.6a). Probable impingement events have a mean force ratio of 1.49, and a median of 1.30 (Figure 2.6b). For all waves measured in this study (including both impingement and non-impingement waves), the mean force ratio is 1.26, while the median force ratio is 1.13 (Figure 2.7). Therefore, on average, the maximum force in impingement waves is nearly 50% larger than the second-highest force occurring in the rest of the wave. In a wave chosen at random, the mean maximum force is only 26% larger. A t-test calculated using logarithmic transformations of mean force ratios indicates this difference is significant (Welch two-sample t-test, p << 0.01).
Figure 2.6: Force ratio distributions for a) definite impingement events and b) probable impingement events.
Chapter 2 61
Figure 2.7: Force ratio distribution for all waves.
2.4.1.3 Maximum forces While impingement events are, on average, larger than forces in the rest of the wave, the absolute forces produced by impingement events are, on average, smaller than the mean maximum force in all waves. The mean maximum force produced by a wave was 0.68 N for definite impingement events, and 0.77 N for probable impingement events. For all waves, the mean maximum force is 1.21 N – 78% larger than that of the mean impingement event. This difference in mean force is statistically significant (Welch two-sample t-test on log transformed data, p << 0.01). Distributions are shown in Figure 2.8 for impingement events and Figure 2.9 for all waves.
Chapter 2 62
Figure 2.8: Maximum force distributions for a) definite impingement events and b) probable impingement events.
Figure 2.9: Maximum force distribution for all waves.
Although there is a trend between maximum force per wave and corresponding significant wave height (p < 0.0001), this relationship predicts only 6% of variation in maximum force recorded during a wave (Figure 2.10). This is not unexpected: Helmuth and Denny (2003) found that increased significant wave heights do not necessarily lead to higher maximum forces due to variations in topography, especially if waves break far offshore (as at my field site).
Chapter 2 63 16
F = 0.49 H + 0.61 14 max s R2 = 0.06
12
10
8
6 Maximum Force (N)
4
2
0 0 0.5 1 1.5 2 2.5 H (m) s
Figure 2.10: Maximum force per wave vs. corresponding Hs.
2.4.2 Limpet model measurements
In the limpet data, 3,165 waves were identified. The sampling period for the limpet model was only two weeks, compared to nearly six weeks for the drag sphere measurements. Due to the shorter sampling period, a narrower range of environmental conditions was experienced during limpet data sampling. Measurements spanned 0.4 meters below to 1.8 meters above MLLW, covering significant wave heights from 0.50 meters to 2.35 meters. The environmental coverage map is shown in Figure 2.11.
Conditions under which waves were recorded are shown in Figure 2.12. Similar patterning between the drag sphere and the limpet model can be seen: very few small waves are recorded at low tides (due to transducer placement as discussed earlier), and conditions with large significant wave heights produced fewer waves with the clear zeros required by this study.
Chapter 2 64
Figure 2.11: Density plot showing number of six minute sampling periods included per tidal height (m above MLLW) and significant wave height (m) for the limpet element.
Figure 2.12: Density plot showing number waves recorded per tidal height (above MLLW) and significant wave height (m) for the limpet element. Outline of sampled environmental conditions is shown in orange.
Chapter 2 65 Individual waves recorded on the limpet model tended to have “noisier” signals than those recorded on the drag sphere, perhaps because the limpet model is heavier than the drag sphere, and has a first natural frequency approximately one-third that of the drag sphere. Although filters were designed to remove inertial resonance of the limpet model, filtering was imperfect, and some energy at these frequencies remains.
2.4.2.1 Impingement frequency Of the 3,165 limpet waves, 86 had definite impingement events, and an additional 359 waves had probable impingement events. This suggests impingement frequencies of 2.7 to 14.1%
2.4.2.2 Force ratios As with the drag sphere data, the ratios between the largest and second largest forces within each wave were calculated and distributions plotted (Figure 2.13 and Figure 2.14). The mean force ratio for limpet impingement events was 1.55 for both definite and probable impingement events, with medians of 1.35 and 1.36 respectively. For all waves, the mean force ratio was 1.38 and the median was 1.21. As with drag sphere force ratios, the difference in mean force ratio was significant (Welch two sample t- test on log transformed data, p << 0.01).
These force ratios are all greater than those for the drag sphere, possibly due to the inertial effects from the increased mass of the limpet model (the inertial effects of mass are explored in more detail in the Discussion and Chapter 3.) However, the difference in impingement wave force ratios is not significant between sphere and limpet measurements (Welch two sample t-test, p = 0.12). For all waves, the larger force ratios for the limpet model are significantly greater than those for the sphere (Welch two sample t-test, p << 0.01).
Chapter 2 66
Figure 2.13: Force ratio distributions on limpet drag element for a) definite impingement events and b) probable impingement events.
Figure 2.14: Force ratio distributions on limpet model for all waves on limpet drag element.
2.4.2.3 Maximum forces For limpet impingement events, the absolute values of the impingement force are smaller than the average maximum forces in all waves, as observed with the drag sphere (Figure 2.15 and Figure 2.16). The mean maximum impingement forces were 0.67 N for definite and 0.73 N for probable impingement events, while the mean
Chapter 2 67 maximum force for all waves was 0.82 N. Medians for the same three distributions were 0.58, 0.60, and 0.65 N, respectively. The difference in mean maximum force for all impingement forces was significantly lower than mean maximum force of all waves (Welch two sample t-test, p << 0.01).
Figure 2.15: Maximum force distributions on limpet drag element for a) definite impingement events and b) probable impingement events.
Figure 2.16: Maximum force distribution for all waves on the limpet drag element.
Chapter 2 68 2.5 DISCUSSION Impingement is not the largest force acting on rigid intertidal organisms: the average magnitude for all waves measured on the drag sphere was 78% larger than the average magnitude of impingement events. This result suggests that contrary to previous suggestions, impingement events are not the largest hydrodynamic risk factor for intertidal organisms. Rather, fluctuations in force occurring elsewhere in the wave appear to be the largest risk.
2.5.1 Implications for studies using dynamometers
Across all waves, the maximum forces (almost invariably due to brief force spikes) were estimated to be 1.2 times greater than the second largest forces. This result has important implications for studies performed using dynamometers: it suggests that forces measured by these devices are likely to be underpredicted by an average of 20%. While not optimal, this factor of 20% is much less worrisome than the factor of 2.5 suggested by Gaylord’s (2000) work.
The efficacy of dynamometer measurements remains an open question, however. The 20% value measured here assumes that the second highest force is of long enough duration to be recorded by the dynamometer. If this is not the case, the error of dynamometers might be greater than 20%. To test this, future work should include a simultaneous deployment of a fast-response transducer and a dynamometer to determine the differences between forces measured with a dynamometer and the actual maximum hydrodynamic forces.
2.5.2 Frequency of impingement
This study found impingement events in approximately 2-7% of waves measured in a rocky intertidal site on a drag sphere. However, this frequency is sensitive to the criteria used to define these events. In the literature, no criteria defining impingement exist: as a result, the definition of impingement (beyond “a large force occurring at wave impact”) is subjective. If different criteria than those used here are employed, the
Chapter 2 69 estimates of impingement frequency will change. The ambiguity in estimating the frequency of impingement events is unlikely to cause a problem, however. Because impingement events are not the largest hydrodynamic forces, the frequency of their occurrence probably has little ecological significance.
2.5.3 Comparison to earlier studies
Although Gaylord (1999, 2000) and Gaylord et al. (2001) offered the first intriguing insights into impingement, this study was the first to record measurements with sufficiently high frequency to explicitly search for these brief force spikes. Therefore, it is worth comparing my estimates of ratios between impingement and subsequent drag to those presented in earlier studies.
Gaylord’s studies likely measured larger impingement events due to differences in the transducers used. The drag element of Gaylord’s transducer had a lower first natural frequency. For reasons detailed in Chapter 3, this likely caused an overestimation of force due to inertial effects of the drag element. These overestimates are more likely to happen at the very beginning of a force or for very brief forces: as impingement events are both brief and at the beginning of a wave, it is possible that the events observed by Gaylord were due to inertial effects of the transducer.
Gaylord (2000) reported the mean ratio of impingement to drag force under all fully breaking waves as 2.5, where “impingement” was the largest force observed in the first 0.5 second “impact” region of a wave, and “drag” was the largest force after that initial period. A ratio of 2.5 is clearly much higher than the force ratio for drag spheres of 1.4 that I determined, but this is largely explained by the difference in force ratio calculation. Force ratios reported here were determined by comparing the largest forces in each 0.01-second period of a wave, while Gaylord had comparatively long impact and drag regions of a wave. For waves in this study, my method is likely more accurate, because many of the waves I measured lasted approximately one second. Simply treating the first 0.5 seconds of each of these waves as an “impact” region
Chapter 2 70 unfairly categorizes the makeup of these waves, particularly given the very short durations of impingement events.
To accurately compare my force ratios to Gaylord’s, I replicated his force ratio analysis using my data by determining the maximum force in the first 0.5 s (impact region) and the maximum force (assumed to be drag) after the initial half second. When calculated in this way, force ratios were similar to Gaylord’s: the mean force ratio of my dataset was 2.28 (compared to 2.5 for Gaylord) (Figure 2.17). However, my maximum ratios were much higher than the maximum of 10 reported by Gaylord – my maximum was approximately 27 when calculated using his method. This is likely because many of the waves I measured were shorter than 0.5 s in duration: in these cases, the maximum value was essentially being compared to zero because the entire wave passed within Gaylord’s 0.5 s impact region, leaving negligible force in the drag region. This leads to artificially high force ratios comprising the long tail shown in Figure 2.17.
Figure 2.17: Force ratios for all waves on the drag sphere, calculated using Gaylord’s method.
Chapter 2 71 My method of calculating force ratios, using 0.01-second bins, may have the opposite problem as Gaylord’s approach: it may occasionally compare regions very close together in the wave if there isn’t a great deal of force variability. A minimum separation of 0.01 seconds was instituted to minimize this effect, and a visual inspection of a subsampling of waves show that the majority of them appear to have accurately calculated force ratios.
2.5.4 Do biological shapes experience impingement events?
The results of sampling using an epoxy-filled limpet shell instead of a drag sphere showed that a biological shape experiences impingement forces similar to those imposed on spheres. In general, many of the same patterns I described for the drag sphere hold true for the limpet model: ratios of maximum to second-largest force are larger within impingement waves than all waves, yet the force in an average wave is greater in magnitude than that of a typical impingement wave. Notably, though, both impingement frequencies and force ratios for the limpet model were higher than for the drag sphere.
Both the higher frequency of events and increased force ratios are likely due to the higher mass and resultant lower first natural frequency of the limpet model. As discussed in the previous section, and detailed in Chapter 3, the increased mass results in greater inertia of the drag element as it is excited by a wave, likely causing force to be overestimated. Though inertial forces were filtered out, they cannot be completely erased: indeed, even after filtering, the force signals from the limpet model were “noisier” than drag sphere data. This likely caused both higher impingement frequencies and higher force ratios for the limpet model. It is possible that a less dense material than epoxy may have mitigated this effect, but it may also have been less stiff: if so, any gains in natural frequency would be offset by the decrease in stiffness (see Chapter 3).
The limpet drag element clearly shows that impingement events occur on biological shapes as well as geometric ones, but the limitations of the data should be kept in
Chapter 2 72 mind. The drag sphere’s advantages regarding non-directionality and higher first natural frequency make its estimates of force ratios and maximum forces more reliable, and in Chapter 4, I use these results to extrapolate velocities (and therefore drag) on several species of gastropod.
2.5.5 Conclusions
Impingement events occur in roughly 7% of waves measured at my field site – yet they are not, as previously thought, the largest forces acting on intertidal organisms. Despite their lower absolute force magnitudes, waves with impingement events display more variability over the course of the wave: ratios between the maximum and second largest force are 1.5 times for impingement waves and 1.2 for all waves. This result underscores the need for high-frequency measurements in the surf zone: dynamometers cannot capture the dynamic changes in velocity and drag acting on intertidal organisms and are likely to underpredict maximum forces by approximately 20% for a rigid organism.
Because impingement events are not, on average, larger than the maximum forces in all waves, these results suggest that impingement events are not any more dangerous to an organism than any other transient forces in a wave. Future studies, then, should not be concerned with searching for and quantifying impingement events. Rather, high-frequency measurements should focus on the largest transient forces, regardless of where in the wave they occur, and explore how these brief loads affect intertidal organisms. To that end, I investigate the mechanical response of gastropods to transient loads in the following chapter.
Chapter 2 73
Chapter 2 74 Chapter 3
Predicting the mechanical response of Lottia gigantea to transient hydrodynamic forces
3.1 ABSTRACT Previous studies have explored how transient forces affect rigid organisms and flexible seaweeds, but how the rubbery foot of a limpet responds mechanically to brief forces has never been tested. Does the foot act as a shock absorber, or is the full magnitude of hydrodynamic force from a wave transmitted to the limpet’s attachment surface? I modeled a limpet as a mass-spring-damper system, and numerically tested its response to half-sine pulses of varying durations for both stationary limpets glued to the substrate with mucus, as well as those undergoing an escape response from a predatory sea star. In addition, I predicted likely responses to 500 randomly selected waves from those recorded in Chapter 2. The degree of force reduction varied with pulse width, and I found that, on average, both stationary and escaping limpets reduced the forces imposed on them by waves by approximately 30%. However, force reductions were not assured: in response to 500 waves, force reductions varied from 1- 92%.
3.2 INTRODUCTION How any object or system responds to a force depends on its structural properties. One of the simplest ways to model structures is as a mass-spring-damper system, which consists of an elastic element (such as a spring), a damping element, and the mass of the system (Figure 3.1).
Chapter 3 75
Figure 3.1: Example of a mass-spring-damper system.
How the system responds to a load – and thus, the effective force applied to the attachment surface of the structure – depends on the properties of these elements. One of the most important parameters dictating the structure’s response is the natural period at which it oscillates when an externally applied force is released. This is determined by the system’s mass and the stiffness of the elastic element. For forces applied over periods of time much longer than the natural period, the elastic force dominates: the effective force felt by the system is equal to that of the applied force, and is only dependent upon the stiffness of the elastic element. Forces applied very rapidly (relative to the natural period) are resisted by the mass’s inertia, which reduces the effective force on the attachment surface. For durations between these two extremes, response is modulated by the degree of damping in the system. If the external force is applied at the natural period and the system is poorly damped, resonance occurs, and responses may become large. Figure 3.2 illustrates these general properties.
Chapter 3 76
Figure 3.2: Response of a mass-spring-damper system.
How an intertidal organism responds mechanically to fluctuating hydrodynamic forces, then, depends upon its structural properties – its mass, stiffness, and damping – and thus, its natural period. Many studies have hypothesized that the time it takes for an organism to reach maximum deflection from an applied force may be longer than the duration of a brief force, and that this may reduce net force on an organism’s attachment to the substratum. Because deformation takes time, the force may have passed before the organism’s inertia is overcome (Koehl 1984; Denny et al. 1985; Koehl 1986; Denny 1987a; Johnson and Koehl 1994; Gaylord 2000; Gaylord et al. 2001). In addition, a viscous component can contribute a “lag” in the deformation of an organism in flow (Gaylord et al. 2001), and affect the organism’s structural rigidity (Koehl 1977; Koehl 1984; Gaylord et al. 2001), and thus, its response to a force.
Very rigid organisms – such as barnacles, which are cemented to the substrate – behave like rigid engineered structures. Very flexible organisms – seaweeds in particular – have the capacity to reduce the effective force on an organism
Chapter 3 77 substantially (Gaylord et al. 2001). Gaylord and colleagues explored the degree of force amelioration based on several variables, including seaweed shape, stiffness, and how the load was applied (e.g., tension or bending). By modeling the response of seaweeds numerically, the authors showed that in many cases, the loads experienced by the seaweeds were fractions of the forces applied. Yet, this was not always the case: long, compliant organisms in tension experienced higher forces than those applied.
How do organisms with stiffnesses somewhere between those of rigid barnacles and very compliant seaweeds respond to transient forces? Specifically, is there any force amelioration for brief loads by limpets, and possibly by extension, other gastropods, which attach to the substratum using a rubbery foot? In this study, I numerically explore the mechanical response of a limpet to brief forces.
The giant owl limpet Lottia gigantea (Sowerby) is a common denizen of the mid-to- upper rocky intertidal zone on the west coast of North America, reaching 9 cm in length (Morris et al. 1980) (Figure 3.3). L. gigantea is territorial, and uses its shell to clear its territory of competitors so it can graze algal film from the rock. The muscular foot is used in combination with secreted mucus to attach to intertidal rocks, and is itself attached to the shell with a horseshoe-shaped tendon. The foot takes up nearly all of the aperture area of the shell. For more detailed anatomy, see Fisher (1904).
Chapter 3 78 Ventral View
Mouth Shell Muscle attachment Visceral mass tendon
Foot Transverse View
Visceral mass
Shell
Foot Lateral View Tentacles 1 cm
Figure 3.3: Anatomy of L. gigantea.
When not foraging, L. gigantea tends to sit with the shell slightly elevated above the substratum, presumably to facilitate respiration (Abbott 1956). During high tides, limpets remain stationary and glue themselves in place using a secreted mucus (Smith 1992). When disturbed, individuals will quickly clamp the shell directly to the rock and “hunker down” (Abbott 1956; Cook et al. 1969; Ellem et al. 2002). This allows the edge of the shell to take advantage of the textural variation in the rock surface, using it to increase friction and thereby resistance to dislodgment. This is thought to be a defense mechanism against predators (Ellem et. al 2002), but may also lower drag as waves sweep over the animal (Denny 1988). Limpet survivorship is high year-to-year, and due to their size, high adhesive strength, and position on the shoreline, they are likely to be immune to predation by birds. (Denny and Blanchette 2000). Large individuals are estimated to be 10-15 years old (Morris et al. 1980). Nonetheless, they may be subject to dislodgment by hydrodynamic forces.
Limpets have very high tenacities – up to hundreds of Newtons for large organisms – although adhesive strength varies with the animal’s behavior: stationary limpets are better able to resist dislodgment than moving ones (Denny and Blanchette 2000). Because limpets must move to forage, they modulate their behavior to reduce the risk
Chapter 3 79 of dislodgment during rough seas: when waves are high, limpets reduce their foraging time, thus lowering the risk of dislodgment (Wright 1978; Judge 1988; Wright and Nybakken 2007).
L. gigantea individuals exhibit an escape response when they encounter the predatory sea star Pisaster ochraceus. When the margin of a limpet is touched by the tube foot of a sea star, the limpet makes its fastest known motions: the shell raises up, rocks back and forth for several seconds, and the individual then rotates and flees (Bullock 1953). While fleeing, its attachment strength is at its lowest (Denny and Blanchette 2000).
If I treat a limpet as a purely mechanical system, can I predict its response to the forces produced by breaking waves? In Chapter 2, I showed that the largest hydrodynamic forces are due to very brief increases in force throughout the wave. How do these transient forces affect a limpet? Do L. gigantea individuals feel the full force of a wave, or is there some mitigation due to the structural properties of the foot? If the latter is true, does the behavior of a limpet at the time of impact – namely whether it is firmly attached with mucus or undergoing an escape response – affect this force amelioration?
To answer these questions, I measured the structural properties of six L. gigantea individuals under both stationary (glued with mucus, but not clamped down) and escape response conditions, which represented what is likely to be the largest range of structural properties exhibited by limpets. I used the results to numerically predict responses to applied forces.
3.3 THEORY I will treat a limpet as a simplified mass-spring-damper system, known as a Kelvin- Voigt element (Figure 3.4).
Chapter 3 80
Figure 3.4: Limpet body modeled as a Kelvin-Voigt element.
The spring resists displacement according to Hooke’s Law:
! = −!" Equation 3.1 where F (in Newtons) is the restorative elastic force exerted by the structure, k is the structure’s stiffness (Newtons per meter), and y is its displacement (in meters) measured from the spring’s undeformed (equilibrium) position. The negative sign indicates that force acts opposite the direction of displacement.
If a static external force F is applied, the system will deform to y and come to rest. If that force is then removed (so F = 0), the elastic restoring force accelerates the mass:
−!" = !" Equation 3.2
As the mass accelerates, velocity v increases such that it is maximal when y = 0, at the equilibrium position. However, because the momentum of the mass is maximal at that
Chapter 3 81 position, it overshoots and comes to a halt at position -y. The process then repeats and the mass oscillates about its equilibrium position.
The frequency with which this oscillation occurs, or the natural frequency of the oscillating system, ωn, is a function of both stiffness and mass:
! ! = Equation 3.3 ! !
measured in radians per second. As there are 2π radians in a cycle, the natural period, or time between oscillations, is:
2! ! !! = ! = 2! Equation 3.4 !! !
As the system oscillates, the elastic element stores potential energy:
! 1 potential!energy! = ! !"#" = !!! Equation 3.5 ! 2
As the structure is released from its deflected position and moves toward its equilibrium point, potential energy is converted to kinetic energy:
1 kinetic!energy! = !!! Equation 3.6 2
As the mass oscillates, potential energy is traded for kinetic energy, and vice-versa.
In a frictionless world, this structure would oscillate forever, with nothing to slow it down. In the real world, however, motion is always damped; energy is lost to friction.
Chapter 3 82 In many cases (including biological materials), damping is due to viscous friction, a force that varies not with a structure’s displacement, but rather with the rate of displacement, v. Thus, in the absence of an externally applied force,
!"! + !!"! + !!" = 0! Equation 3.7
where c is the damping coefficient with units Ns/m. In damped oscillation, in the absence of external force, displacement decreases with each cycle as energy is lost to friction.
If this spring-mass system is attached to a surface, the effective force experienced by that surface as the structure responds to the external force is the sum of the spring and damping terms of Equation 3.7:
!!"" = !" + !" Equation 3.8
Equation 3.7 is a differential equation, with a solution of the form
!" ! = ! Equation 3.9
When substituted into Equation 3.7, we find:
! !" (!! + !" + !)! = 0 Equation 3.10
which is true for all values of t only if
! ! !! + ! + = 0 Equation 3.11 ! !
Chapter 3 83 Equation 3.11 is a quadratic equation, for which there exist two roots, S1 and S2:
c c S = − + ( )! − k/m ! 2m 2m
c c S = − − ( )! − k/m Equation 3.12 ! 2m 2m
The nature of S1 and S2 determine the system’s response. If the roots are zero, then
Equation 3.12 can be solved for c, yielding the critical damping coefficient cc:
!! = 2 !" Equation 3.13
A given damping coefficient can be expressed as a fraction of cc, giving the damping ratio ζ:
! ζ= Equation 3.14 !!
If ζ is 1, the system is critically damped. If ζ is less than one, it is underdamped, and oscillations decay in amplitude. If ζ is greater than one, the system is overdamped, and no oscillations occur. In all cases, the behavior of the system depends on the starting y and v. Examples of critically damped, underdamped, overdamped, and undamped motion are shown in Figure 3.5.
Chapter 3 84
Figure 3.5: Illustration of critically damped, underdamped, overdamped, and undamped motion.
If the system is damped, its damped natural frequency, ωd, is modified from the undamped natural frequency, ωn:
! !! = 1 − ζ !! Equation 3.15
So far, these equations have assumed a system that is released from position y and is in free oscillation. What if, however, there is an external force acting on the system?
If there is an external force, the right hand side of Equation 3.7 is no longer zero. In the case of a sinusoidally varying force,
!"! + !!"! + !!" = !!!"#(!") Equation 3.16
Chapter 3 85 where the external excitation force has amplitude F0 and frequency ω radians/sec
(note that this frequency is independent of ωn).
Oscillations resulting from forced harmonic motion are dependent on the ratio of the frequency of the applied force, ω, to the natural frequency of the system ωn (Figure 3.6). When this ratio is very low, the excitation frequency is much lower than the natural frequency. In this case, the system responds as if a static force is deflecting it: the effective force felt (ky) is equal to the external force. As the frequency of the excitation force increases, the system’s inertia comes into play, and ky increases until the structure resonates – when the excitation frequency equals the natural frequency of the system. At this point, the system’s displacements grow large (infinitely so, with no damping). Resonance is modulated by damping: as shown in Figure 3.6, maximum displacement decreases as damping ratios increase.
3 ζ = 0
ζ = .15
applied 2 ζ = .25 ) / Force ky ζ = .5
1
ζ = 1 Elastic Force ( Force Elastic
0 0 1 2 3 4 Frequency Ratio ω / ω n
Figure 3.6: The ratio of effective force on an object to force applied under force oscillation, as a function of damping and frequency. For underdamped systems (ζ < 1), effective force is increased when forcing frequencies are closer to the natural frequency.
Chapter 3 86 Applied force need not be harmonic, however: any applied force will cause the system to deform, and through the principles outlined above, that displacement can be predicted. In response to an external force F (varying with time ε), the displacement of a structure at time t is:
1 ! ! ! = !"# −!!! ! − ! ! ! sin!(!!(! − !))!" Equation 3.17 !!! !
That is, the displacement of the structure at time t depends on not just force F(ε) at time t, but F(ε) at all time steps ε between 0 and t.
The brief explanations presented to this point are based on derivations in Thomson (1981) and Denny (1988), which discuss these concepts in more detail. (I have also found the systems dynamics text by Ogata (1998) to be particularly useful.)
Equation 3.17 gives displacement as a function of time, and its derivative yields the velocity of the mass v with respect to time. Given y and v, the force on the attachment structure’s base can be predicted with Equation 3.8.
For simple structures, k and c can be estimated: the stiffness of a structure is the ratio of a static applied load to the structure’s resultant displacement. When force is plotted against displacement, the slope of the resulting curve is a measure of stiffness, and the area under the curve (the product of force and distance) is the energy put into the system (that is, the work done to deform it) (Figure 3.7).
Chapter 3 87 Force Slope = stiffness
Energy in
Displacement
Figure 3.7: Calculating stiffness from a force-displacement curve.
Damping is a measure of how much energy is dissipated, or lost, due to viscous forces within a structure. A system with no damping can be loaded multiple times, following the same path on the curve shown in Figure 3.8a. All the energy that goes into the structure is returned as the load is released. Viscosity causes some loss of energy, however (Figure 3.8b-c). In this case, as the load is removed, the force-displacement curve follows a different path. The area between the loading and unloading curves represents this loss, known as hysteresis.
a) b) c) Hysteresis Energy Energy In Out Loading Loading Loading Force Force Force
Unloading Unloading
Displacement Displacement Displacement
Figure 3.8: Force displacement curves under loading and unloading illustrating a) energy going into the system, b) energy going out of the system, and c) energy lost (hysteresis).
Chapter 3 88
These principles can be used to measure k and c for limpets. I used a force transducer to displace a limpet’s shell, approximately as shown in Figure 3.9.
Figure 3.9: Ideal loading/unloading force time-series.
An energy argument can be used to separate the spring and damping components of limpet response. Because deformation increases linearly as the system is ideally loaded (Figure 3.9), the velocity v is constant. In that case, the total energy required to deflect the shell can be expressed as:
!!"# Energy!in! = ! !"! + !!" !"! !
!!"# !!"# ! ! = !!" !" + ! !"! = !"!!"# + !!"# ! ! 2 Equation 3.18 )
Of this total, some energy is lost to viscosity:
Chapter 3 89 !!"# Energy!lost = !"! !"# Equation 3.19 !
Thus, the hysteresis H is:
energy!lost !"!!"# ! = ! ! = Equation 3.20 energy!in ! ! !"!!"# + 2 !!"#
With some algebra, Equation 3.20 can be solved for k:
2(1 − !)!" ! = Equation 3.21 !!!"#
Inserting this value into Equation 3.8, we can solve for c:
! ! = ! !"#$%& 2(1 − !)! Equation 3.22 !{1 + [ ]} !!!"#
where Flimpet is the maximum force measured by the transducer. In this fashion, force- displacement curves can be used to estimate stiffness and damping for limpets, from which their response to brief loads can be predicted.
Other models treating viscoelastic materials as a mechanical analog exist, and may better explain some aspects of a limpet’s structural behavior than the Kelvin-Voigt model I used. However, limpets are not made of simple viscoelastic materials. Limpet feet are muscles, and their stiffness can be controlled by the animal. As a result, the output of a complicated model to predict structural response of a limpet foot is not likely to justify the required development and computational time. For this reason, I opted to use a computationally simple model.
Chapter 3 90 3.4 METHODS AND MATERIALS
3.4.1 Limpet collection
Limpets were collected from Hopkins Marine Station in Pacific Grove, and placed on experimental plates in a seawater tank. The plates were made of PVC covered with Safety Walk tape (3M), and kept in the seawater tank for at least 48 hours prior to experiments.
For trials with the limpet glued but not clamped down (what I will refer to as stationary trials), limpets were moved out of any splashes from the tank, to simulate low tide and encourage them to glue themselves in place with mucus (Smith 1992). Escaping limpets were wetted immediately prior to experiments to simulate a rising tide, when predation from sea stars is a threat. In both cases, care was taken not to tap the shell and provoke a clamping response. For all trials, the limpets appeared to be resting with the shells slightly above the substratum, rather than “hunkered down.”
3.4.2 Limpet stiffness and damping measurements
The experimental plates were clamped to a bench so a high-speed camera (FASTCAM-512PCI, Photron, San Diego, CA) could measure the displacement of the limpet shells. The tip of a force transducer (FT 1000, World Precision Instruments, Sarasota, FL) was used to impose a force lasting approximately 2 seconds on the anterior end of each limpet, approximating the pattern of Figure 3.9. This caused a displacement as the limpet body sheared, and auto-tracking software (Photron Motion Tools, San Diego, CA) was used to measure this displacement over time. The force signal was amplified (Transbridge 4M, World Precision Instruments, Sarasota, FL), and a computer running LabView (National Instruments, Austin, TX) recorded the force imposed on each limpet over time. Each limpet was subjected to nine trials in both stationary and escape response conditions, except for limpet 1 in the stationary condition, which had ten trials. Thus, there were 55 total trials over 6 limpets in the stationary case, and 54 trials for the same 6 limpets undergoing an escape response.
Chapter 3 91
For stationary limpets, no treatment was done to the limpets beyond removing them from water several hours before the experiment. To provoke an escape response, however, tube feet were removed from a Pisaster ochraceus specimen. Using forceps, the margin of the limpet’s mantle was touched with a tube foot until the limpet began to react. At this point, the tube foot was removed, and the limpet was displaced as described above. At least 48 hours elapsed between stationary and escape response experiments.
In MATLAB, the displacement and force data were plotted as force-displacement curves, and polynomials fitted to the loading and unloading regions of the curve (Figure 3.8b). Using Equation 3.21 and Equation 3.22, k and c were then calculated for each trial. These values were averaged for each of the six limpets under both stationary and escape response conditions.
After the experiments, I estimated limpet masses by filling the empty shells with water and recording the masses, which ranged from 9 to 23 grams. Using the measured values of mass, k, and c, natural frequencies (both damped and undamped) were calculated.
3.4.3 Displacement and force predictions
The stiffness and damping data were used to model the limpet’s response to several external forces. First, a half-sine pulse was applied at a range of durations to model limpet response to a brief load (essentially, an isolated impingement event). I calculated displacements using the mean k and c values for each of the six limpets under both stationary and escape response conditions. Equation 3.17 was solved using 0.0001 second time steps. The magnitude of response is a function of the ratio of the applied force’s duration to the natural period of the limpet; I varied half-sine pulse durations from 0.001 to 1 s. In addition, effective force on the limpet was calculated for a 0.005 second half-sine pulse (representative of an impingement event) for all 55 stationary trials and 54 escaping trials to explore the variability in limpet response.
Chapter 3 92
How do limpets respond to brief force fluctuations in actual waves? The exact durations of high-frequency force fluctuations in waves are difficult to quantify, so I calculated a distribution of limpet responses to a variety of measured wave forces to estimate the likely range of responses. 500 waves were randomly chosen from the 9311 measured in Chapter 2. For each wave, limpet displacements were calculated using Equation 3.17 at time steps of 0.0002 s. For both stationary and escaping limpets, I chose three pairs of k and c estimates using the responses to a 0.005 s half- sine pulse: (1) I identified the k - c pair that produced the minimum effective force in response to the 0.005 s half-sine pulse. (2) I used the mean k and c values for all 55 stationary trials and 54 escaping trials. (3) I identified the k - c pair that produced the maximum effective force in response to the 0.005 s half-sine pulse.
For each of the three k - c pairs in each condition (stationary and escaping), I computed the ratio of maximum force felt by the limpet to maximum force imparted by each wave, and recorded the resultant force ratio distributions.
3.5 RESULTS
3.5.1 Force-displacement curves
Force-displacement curves for a representative limpet are shown in Figure 3.10 for the stationary condition, and in Figure 3.11 for the escape response condition. All limpets demonstrated hysteresis, in that loading and unloading paths were not the same. In general, displacements for escape response limpets were greater than those for stationary limpets. In many cases, the limpets exhibited elastic-plastic behavior: that is, the limpet did not return to its starting position by the end of the trial. However, these trials were not able to distinguish if this was true elastic-plasticity, if it was due to active muscle control by the limpet, or whether the starting point would have been reached if measurements continued for a longer period of time after removing the force.
Chapter 3 93
Figure 3.10: Force-displacement curves for a representative stationary limpet. Each panel represents one trial, with gray dots representing measured points. Solid lines are fitted polynomials to the loading part of the curve, while the dashed line is a fitted polynomial to the unloading portion of the curve.
Chapter 3 94
Figure 3.11: Force-displacement curves for a representative escaping limpet. Each panel represents one trial, with gray dots representing measured points. Solid lines are fitted polynomials to the loading part of the curve, while the dashed line is a fitted polynomial to the unloading portion of the curve.
3.5.2 Stiffness and damping
Both stationary and escaping limpets had stiffness values that varied among trials, but stationary limpets had greater predicted stiffnesses than escaping limpets, as shown in Figure 3.12 (paired t-test, p < 0.05).
Chapter 3 95 3 Stationary Escaping 2.5
2 (N/m) k 1.5
Stiffness 1
0.5
0 1 2 3 4 5 6 Limpet
Figure 3.12: Estimated stiffness values k (N/m) for stationary and escaping limpets. Stationary limpets have greater stiffnesses (p < 0.05). Error bars represent 95% confidence intervals of the mean.
Predicted damping coefficients are also variable both among limpets and among trials within limpets (Figure 3.13). However, damping coefficients show no significant difference between stationary and escaping limpets (paired t-test, p = 0.23). Limpets 2 and 5 have larger average damping coefficients during escaping behavior than while stationary, while limpets 1, 3, 4, and 6 show the opposite.
Chapter 3 96 7 Stationary Escaping 6
5 (Ns/m) c 4
3
2 Damping Coefficient
1
0 1 2 3 4 5 6 Limpet
Figure 3.13: Estimated damping coefficients c (Ns/m) for stationary and escaping limpets. There is no difference between stationary and escaping limpets (p = 0.23). Error bars represent 95% confidence intervals of the mean.
All 6 limpets in both stationary and escaping conditions exhibit damping ratios above 1, showing they are overdamped (Figure 3.14). As with the damping coefficient, there is no significant difference between damping ratios for stationary and escaping limpets (paired t-test, p = 0.18).
Chapter 3 97 50 Stationary 45 Escaping
40
35 ζ 30
25
20 Damping Ratio 15
10
5
0 1 2 3 4 5 6 Limpet
Figure 3.14: Estimated damping ratios ζ for stationary and escaping limpets. There is no difference between stationary and escaping limpets (p = 0.18). Error bars represent 95% confidence intervals of the mean.
Because the limpets are overdamped, they cannot exhibit resonance (Figure 3.6). While the limpets may exhibit force reduction based their properties at the time, it is not possible for these limpets to amplify applied forces and feel a higher effective force.
Neither stiffness nor damping coefficients had a statistically significant relationship with limpet mass in either stationary or escaping conditions (Figure 3.15 and Figure 3.16, all p > 0.5).
Chapter 3 98 3 Stationary Escaping 2.5
2 (N/m)
k 1.5
1 Stiffness
0.5
0
8 10 12 14 16 18 20 22 24 Limpet Mass (g)
Figure 3.15: Limpet stiffnesses (N/m) plotted against limpet mass (g). Neither stationary nor escaping limpets change significantly with mass (p = 0.66 for stationary limpets, p = 0.55 for escaping limpets). Error bars represent 95% confidence intervals of the mean.
Chapter 3 99 7 Stationary Escaping 6
5 (Ns/m) c 4
3
2 Damping Coefficient
1
0 8 10 12 14 16 18 20 22 24 Limpet Mass (g)
Figure 3.16: Limpet damping coefficients (kg/s) plotted against limpet mass (g). Neither stationary nor escaping limpets change significantly with mass (p = 0.88 for stationary limpets, p = 0.74 for escaping limpets). Error bars represent 95% confidence intervals of the mean.
3.5.3 Effects of pulse frequency
The ratio of maximum force felt by the limpet’s foot to the magnitude of the applied force varied with pulse duration (Figure 3.17 and Figure 3.18). Both stationary and escaping limpets exhibit the same general patterns: for pulse durations of 0.001 to 0.3 seconds, the force ratios increase dramatically from a minimum of 0.04-0.25 for stationary limpets and 0.03 to 0.17 for escaping limpets, approaching values of 1. From roughly 0.3 s to 1 s, the force ratio is steady at values from 1.000 to 1.004 for both stationary and escape response cases (values slightly above 1 are assumed to be rounding errors). There is variation in response between individuals, as is expected for the varying stiffness and damping values observed.
Chapter 3 100
1 1
0.8 a) 0.8 b) pulse pulse
0.6 0.6 / Max F / Max F Limpet 1 limpet 0.4 limpet 0.4 Limpet 2 Limpet 3 Max F Max F 0.2 0.2 Limpet 4 Limpet 5 Limpet 6 0 0 0 0.5 1 0 0.1 0.2 0.3 Pulse duration (s) Pulse duration (s)
Figure 3.17: Mean ratio of maximum force felt by stationary limpets to input force for a) all pulse durations b) zoomed in to region between 0 and 0.3 s.
1 1
0.8 a) 0.8 b) pulse pulse
0.6 0.6 / Max F / Max F Limpet 1 limpet 0.4 limpet 0.4 Limpet 2 Limpet 3 Max F Max F 0.2 0.2 Limpet 4 Limpet 5 Limpet 6 0 0 0 0.5 1 0 0.1 0.2 0.3 Pulse duration (s) Pulse duration (s)
Figure 3.18: Mean ratio of maximum force felt by escaping limpets to input force for a) various pulse durations b) zoomed in to region between 0 and 0.3 s.
The results shown in Figure 3.17 and Figure 3.18 are for the means of each limpet’s stiffness and damping coefficients, however. Among and within individual trials, stiffness and damping varied, causing large differences in predicted force responses for both stationary and escaping limpets (Figure 3.19). For a 1-N amplitude half-sine pulse of duration 0.005 s, the mean response for all limpets was approximately 0.4 N
Chapter 3 101 for stationary limpets and 0.3 N for escaping ones. However, over all trials, predicted forces ranged from 0.03 to 0.85 N for stationary limpets and 0.01 to 0.79 for escaping limpets.
1 1 a) b) 0.8 0.8 0.6 0.6 0.4 0.4
Total Force (N) 0.2 Total Force (N) 0.2 0 0 0 0.01 0.02 0.03 0 0.01 0.02 0.03 Time (s) Time (s)
Applied force Individual response Mean response (all limpets) 95% confidence interval
Figure 3.19: Predicted force on limpet from a 1-N 0.005 second half-sine pulse for all trials of all a) stationary and b) escaping limpets.
3.5.4 Response to measured waves
Figure 3.20 shows the distributions of the ratio of maximum effective force on stationary limpets to maximum force in 500 random waves, for three pairs of k and c values: those that produced the minimum and maximum responses to a 0.005 s half- sine pulse, and the mean k and c values. Figure 3.21 shows the calculated force ratios for escaping limpets.
In both stationary and escaping cases, the k and c values that produced the smallest response to the half-sine pulse have the lowest mean force ratio, and those that produced the largest response to the pulse have the largest mean force ratio. The mean response calculated from mean k and c values fall between those predicted for minimum and maximum responses.
Chapter 3 102 The mean force ratios for escape response limpets are less than those estimated for stationary limpets (Wilcoxon rank sum test, p << 0.001).
a) b) 0.1 0.1 mean = 0.49 mean = 0.73
0.05 0.05 Fraction of waves Fraction of waves 0 0 0 0.5 1 0 0.5 1 Max F / Max F Max F / Max F limpet wave limpet wave
c) 0.1 mean = 0.82
0.05 Fraction of waves 0 0 0.5 1 Max F / Max F limpet wave
Figure 3.20: Distributions of ratios between maximum effective force on stationary limpets to maximum force in a wave for a) minimum response k - c pair, b) mean k and c values for all limpets, and c) maximum response k - c pair.
Chapter 3 103 0.1 a) 0.1 b) mean = 0.36 mean = 0.69
0.05 0.05 Fraction of waves Fraction of waves 0 0 0 0.5 1 0 0.5 1 Max F / Max F Max F / Max F limpet wave limpet wave
0.1 c) mean = 0.81
0.05 Fraction of waves 0 0 0.5 1 Max F / Max F limpet wave
Figure 3.21: Distributions of ratios between maximum effective force on escaping limpets to maximum force in a wave for a) minimum response k - c pair, b) mean k and c values for all limpets, and c) maximum response k - c pair.
3.6 DISCUSSION
Regardless of whether a limpet was stationary or escaping, all predicted responses suggest that, due to viscous damping of the limpet foot, limpets do not feel the full brunt of a wave’s maximum force. Under both modeled half-sine pulses and measured wave forces, none of the limpets showed the capacity to respond to the full magnitude of brief forces. Half-sine pulse results suggest that for forces lasting longer than several tenths of a second, the limpet may feel a force approaching the full magnitude of the imparted force. However, the limpet responses to 500 random waves suggest that the high-frequency force variations in measured field waves (Chapter 2) are of short enough duration for the forces imparted to limpets to be reduced by 27-31%, on average (Figure 3.20b and Figure 3.21b).
Chapter 3 104 A word of caution is necessary, though: these reductions are for average stiffness and damping values. As a limpet can control these properties, they are variable, and this variation manifests itself in force variation even within an individual. While mean force reductions to a wave are approximately 30%, when the full ranges of k and c estimated in this study are applied, force reduction may vary anywhere from 1-92%.
The high levels of variability observed in force reduction suggest that while limpets may, on average, feel forces smaller than those applied, this is by no means assured: both stationary and escaping limpets experienced force ratios approaching 1. Unless a limpet can anticipate the nature of the waves approaching, and the likely timescales of maximum force pulses, there is unlikely to be a significant advantage to this mechanism. If the limpet can anticipate an advantageous muscle stiffness, however, there may be a significant ecological benefit.
Several studies have shown that limpet tenacity is high enough that they are at little risk of dislodgment when stationary or “hunkered down” (Judge 1988; Denny 1989; Denny and Blanchette 2000). However, as Wright (1978), Judge (1988), and Wright and Nybakken (2007) have shown, L. gigantea reduces its foraging time when waves are large, suggesting that it perceives a risk of dislodgment while moving. My results suggest that effective forces on the limpet are lower due to the structural properties of the foot: in essence, the foot acts as a shock absorber. This effect may allow L. gigantea to increase its foraging time in rough seas if a limpet can control its stiffness, or may serve as a protective buffer if a limpet miscalculates when it should clamp itself to the rock.
There are many possible avenues for continuing this work: the most obvious next steps are to expand measurements to other gastropod species, increase sample sizes, and measure k and c directly at a variety of strain rates. With those measurements, a more complete model of limpet response could potentially be made. However, it is difficult to treat a living limpet with active muscle control as a simple mechanical system; even a more complete model of viscoelastic behavior would need to make gross
Chapter 3 105 simplifications. As stiffness k is under an individual’s control, do limpets modulate their muscular response to wave conditions to further resist hydrodynamic dislodgment, in addition to the “clamping” response observed when limpets are disturbed?
This study was the first to explore the effects of transient loads on limpets, and showed that the material properties of the foot are likely to substantially reduce the largest hydrodynamic forces acting on these creatures. Assuming structural properties of the gastropod foot are similar in other species, this force reduction may be used across many gastropod taxa, suggesting a mechanism of survival for surviving in the wave-swept rocky intertidal zone.
Chapter 3 106 Chapter 4
Predicting gastropod dislodgment risk using hydrodynamic force measurements
4.1 ABSTRACT Dislodgment by waves is a risk for gastropods living in the rocky intertidal zone. For a snail or limpet living at my field site, what is the likelihood of being dislodged by any given wave? Using the maximum forces measured in 9,311 waves in Chapter 2, I estimated drag acting on three species of gastropod and compared them to published tenacity values to calculate the probability of dislodgment for these species at my field site. The limpet species Lottia digitalis has a relatively low risk of dislodgment, but the two snail species Chlorostoma funebralis and Nucella emarginata have much higher risks: a C. funebralis individual moving about the rock faces a 28.8% chance of being dislodged by a wave chosen at random. N. emarginata has an even higher risk of becoming dislodged, at 63.5% per wave. These high risks illustrate the necessity of these organisms taking behavioral action to avoid dislodgment.
4.2 INTRODUCTION For organisms living on wave-swept rocky shores, the risk of dislodgment by waves is omnipresent. Studies have quantified the risks of dislodgment for a variety of species, such as macroalgae (Carrington 1990, Gaylord et al. 1994, Blanchette 1997, Mach et al. 2011), crustaceans (Lau and Martinez 2003), and molluscs (Denny 1987b, Trussell et al. 1993, Denny 1995, Bell and Gosline 1997, Denny and Blanchette 2000). For many gastropods, dislodgment is assumed to have a high risk of mortality (Boulding and Van Alstyne 1993, Denny 1995, Trussell 1997), due to the increased risk of encountering predators such as anemones (Kent 1981).
An organism’s risk of dislodgment is governed by the ratio of force acting on the organism to its attachment strength. For a gastropod resisting hydrodynamic forces,
Chapter 4 107 this is the ratio of combined drag and lift to the strength of its foot on the rock. If the forces acting on an organism exceed its ability to resist that force, the organism will be dislodged. To calculate the probability of dislodgment for an organism, then, we must know both the organism’s attachment strength as well as the likely forces it will face. Tenacity, attachment strength normalized by attachment area, has been measured in a large number of gastropods, including snails (Miller 1974, Trussell et al. 1993, Davies and Case 1997, Trussell 1997, Hohenlohe 2003) and limpets (Branch and Marsh 1978, Grenon and Walker 1981, Denny et al. 1985, Denny and Blanchette 2000), but corresponding hydrodynamic forces have seldom been measured.
In Chapter 2, I recorded the maximum hydrodynamic forces acting on a small sphere in 9,311 waves. In this chapter, I combine these measurements with published gastropod tenacity values. Using Denny’s (1995) mechanistic approach, I estimate the probability of dislodgment for snails and limpets at my field site.
4.3 METHODS AND MATERIALS
4.3.1 Determining Wave Velocity
To determine the maximum velocities at my field site, I used the maximum forces of the 9,311 waves measured in Chapter 2. This site experiences primarily postbreaking flows, resulting from waves that broke seaward of the sensor and whose swash was measured after propagating up the shore as turbulent bores (see Denny (1988) for a more thorough discussion). These conditions are described by Gaylord (1999, 2000) as “fully breaking” waves; he showed that drag is the dominant in-line hydrodynamic force under these conditions (2000). Thus, I assume the force acting on the transducer is due entirely to drag, FD, modeled in Equation 4.1.
1 ! = ρ! !!! Equation 4.1 ! 2 ! ! !"
Chapter 4 108 where ρ is the density of the seawater, CD is the drag coefficient of the object in flow, u is the velocity of the fluid, and Apr is the projected area of the object (see Chapter 1 for a more thorough explanation of these terms). Lift – which acts perpendicular to the substratum and is described by an analogous equation using the lift coefficient CL and the planform area Apl – is not taken into account in this chapter, as the forces measured in Chapter 2 were in the plane of the substrate, rather than normal to it. Lift coefficients for gastropods in this study are lower than drag coefficients (Denny 1995). Because drag acts parallel to the substratum, it induces primarily shear forces on organisms, while lift induces primarily tensile forces. Miller (1974) showed that tenacities for snails in shear are lower than tensile tenacities by a factor of 2-5: as a result, the risk of dislodgment from drag is likely to be higher than that from lift. Additionally, even if the resultant force vector from both lift and drag is calculated, tenacity measurements have only been made in either shear or tension, and not at the angle of the combined force from lift and drag. For these reasons, this analysis considers only drag.
I used Equation 4.1 to predict maximum velocities of each wave by solving for velocity:
2! ! = !,!"# Equation 4.2 !!!"!!
3 where ρ is assumed to be 1025 kg/m , and the projected area (Apr) of the drag sphere is -4 2 5.1x10 m . FD,max is the maximum force identified in each of the 9,311 waves, and
CD is the drag coefficient of the sphere.
The drag coefficient of a sphere is dependent on Reynolds number, a dimensional index of flow conditions that describes the ratio of inertial to viscous forces in flow. Reynolds number Re is a function of object size L, flow velocity u, and the kinematic viscosity of the fluid ν, which for seawater at 10°C is 1.17x10-6 m2/s:
Chapter 4 109
!" Re = Equation 4.3 !
The expected order of Reynolds numbers for the drag sphere I used is 105 in intertidal flows. Spheres undergo a “drag crisis” at Reynolds numbers of this order, characterized by variable drag coefficients as the boundary layer transitions from laminar to turbulent flow (Schlicting 1979). Additionally, drag coefficients vary with the steadiness of flow – typically drag is measured under steady conditions, such as those found in a wind tunnel or flume tank. However, these measurements may not be applicable to the unsteady and chaotic flows that characterize the intertidal zone.
To estimate the drag coefficient of the transducer sphere in intertidal flow, I combined two approaches. First, I used estimates of the drag coefficient of rough spheres in unsteady flows (O’Donnell and Denny 2008). These measurements were made with the water cannon described in Chapter 1, and approximated unsteady flow conditions of intertidal waves. Drag coefficient estimates span Reynolds numbers of 105 to 106. To estimate drag coefficients at Reynolds numbers below 105, I used Achenbach’s (1974) data measuring drag coefficients of spheres of different roughnesses across Reynolds numbers of 4x104 to 6x106 in a high-pressure wind tunnel. (Achenbach did not describe flow conditions more specifically, so I assume that the wind tunnel is a low-turbulence tunnel that provides steady flow.) All of his values asymptote to 0.48 at the low end of the range, while O’Donnell and Denny’s estimates at Re = 105 are approximately 0.42. All of Achenbach’s curves have a similar shape, which decreases sharply from 0.48 to a minimum value, but spheres of different roughnesses exhibit this transition at varying Reynolds numbers. The estimated roughness to diameter ratio ε of my sphere is approximately 1/100, and I used Achenbach’s drag coefficient values for a sphere with a similar roughness. However, the Reynolds numbers where O’Donnell and Denny’s and Achenbach’s estimates converge differ by 3.65x104: I assumed that the relationship between my sphere’s drag coefficient and Reynolds number follows the same general pattern as Achenbach’s spheres, but that the
Chapter 4 110 differences in sphere type and roughness and experimental conditions between O’Donnell and Denny’s measurements and those of Achenbach caused this transition to occur at different Reynolds numbers. To adjust for this, and to extend O’Donnell and Denny’s drag coefficient estimates into lower Reynolds numbers ranges, I shifted Achenbach’s drag coefficient curve to meet O’Donnell and Denny’s curve at a Reynolds number of 104. Below a Reynolds number of 8.90x104, I assumed a drag coefficient of 0.48, in keeping with Achenbach’s asymptote for low Reynolds numbers. The component parts of my drag coefficient estimates, as well as my composite estimate of drag coefficient, are shown in Figure 4.1.
0.5 a) O’Donnell and Denny 0.45 Achenbach Achenbach, shifted 0.4
0.35
0.3 Drag coefficient
0.25
0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Re x 10
0.5 Composite Curve 0.45 b)
0.4
0.35
0.3 Drag coefficient
0.25
0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Re 5 x 10
Figure 4.1: Drag coefficient versus Reynolds number for a) O’Donnell and Denny’s measurements in unsteady flow, Achenbach’s sphere with ε = 1/100, and Achenbach’s measurements translated and b) the composite graph used to estimate drag coefficient as a function of Reynolds number in this study.
Chapter 4 111 Using Figure 4.1, I transformed Reynolds number to velocity using the length of my drag sphere, giving me the relationship between drag coefficient and water velocity. For velocities between 0 and 20 m/s, I calculated force at each velocity using Equation 4.1 and the appropriate drag coefficient for each velocity. The resultant relationship between water velocity and drag is shown in Figure 4.2. With this relationship, I calculated the maximum velocity under each wave using the maximum force recorded under each of the 9,311 waves from Chapter 2.
10
9
8
7
6
5
4 Velocity (m/s)
3
2
1
0 0 1 2 3 4 5 6 7 8 Force (N)
Figure 4.2: Velocity vs. force curve.
4.3.2 Drag Predictions
Once the maximum velocity of each wave was determined, I estimated the drag likely to act on various gastropods using Equation 4.1, normalized by the organism’s foot area. By normalizing drag by area, these estimates are valid for any size organism (assuming isometric growth). Foot area was chosen for normalization because tenacity measurements (discussed below) are measured as a force per foot area; normalizing
Chapter 4 112 drag by foot area allows comparisons between normalized drag and tenacity to be made to predict dislodgment (see Section 4.3.4). Drag measurements were normalized
by foot area by substituting the ratio of projected area to foot area (Apr/Afoot) for Apr in Equation 4.1, yielding Equation 4.4.
! !! 1 ! !" = ρ!!!! Equation 4.4 !!""# 2 !!""#
For L. digitalis, Denny (1995) reported mean ratios between organism lengths, widths and heights; these ratios were used to calculate the relationship between foot area and projected area. I approximated the projected area of L. digitalis as a triangle using the animal’s length and height. Foot area was approximated as an ellipse with axes equal to the length and width of the animal. For the snails, the relationship between foot area and projected area was measured directly for several individuals and averaged. For all three species, the ratios of projected area to foot area are shown in Table 4.1.
Species Length/Width Length/Height Apr/Afoot
Lottia digitalis 1.094* 2.746* 0.282 # # Nucella emarginata 1.472# 1.961 1.929 # # Chlorostoma funebralis 1.090# 1.626 0.216
* from Denny 1995 # determined experimentally
Table 4.1: Ratios of organism length to width, length to height, and projected area to foot area.
I used published drag coefficient values from Denny (1995) (Table 4.2). Where different drag coefficients were reported, I chose the higher value, which in all cases corresponded to animals oriented broadside. Although there are some reports of drag coefficients being dependent on velocity (Branch and Marsh 1978), I assumed a constant drag coefficient across my estimated range of velocities, as Denny (1994) found no significant relationship between drag coefficients of a limpet shell and velocity.
Chapter 4 113
Species Drag Coefficient
Lottia digitalis 0.6999
Nucella emarginata 0.8240
Chlorostoma funebralis 0.4932
Table 4.2: Drag coefficients for gastropods (Denny 1995).
Using Equation 4.1, the estimated maximum velocity of each wave, and the drag coefficients and projected areas in Table 4.2, I estimated the maximum normalized drag acting on each of the three gastropod species for each wave measured in Chapter 2.
4.3.3 Tenacity Estimates
To compare the predicted drags to organism breaking strengths, I used published shear tenacity values for the three gastropod species. Where available, tenacities for both moving and stationary organisms were included. I assumed tenacity is normally distributed for each species, and converted each tenacity mean to N/m2 and standard errors (where necessary) to standard deviations, also N/m2 (Table 4.3).
Denny et al. (1985) measured tenacity for both moving and stationary L. digitalis individuals in shear. For snails, Miller (1974) published shear tenacity values for N. emarginata for both moving and stationary individuals, and for moving C. funebralis individuals.
Only shear tenacities were included in this analysis because drag is likely to produce primarily shear forces on these organisms. (Lift, which is not taken into account in this analysis, is more likely to produce forces normal to the substrate as a wave passes over an organism.)
Chapter 4 114 Tenacity x̅ + SD Species Case (105 N/m2) Source
stationary 3.85 ± 1.75 Lottia digitalis Denny et al. 1985 moving 1.29 ± 0.91 moving 0.05 ± 0.03 Nucella emarginata Miller 1974 stationary 0.21 ± 0.13 Chlorostoma funebralis moving 0.06 ± 0.04 Miller 1874
Table 4.3: Shear tenacities for gastropods.
4.3.4 Predicting Dislodgment Probabilities
Once the distributions for both drag and breaking strength were determined for each species, I calculated the dislodgment probability density function for each organism
and condition. To do this, I first determined the probability of exceedance Pex for the predicted drag magnitudes. For each organism, I empirically determined the
cumulative distribution function for normalized drag, CDFdrag, using the distribution of normalized drag calculated with Equation 4.4. I then calculated the probability of exceedance with Equation 4.5:
!!" = 1 − !"!!"#$ Equation 4.5
The probability of exceedance represents the probability of a given wave exceeding a certain value; since drag values used in this analysis were normalized by foot area, this curve is in units of N/m2.
The tenacities shown in Table 4.3 were assumed to be normally distributed. From these distributions, I calculated the probability density function (PDF) of tenacity for each organism.
Chapter 4 115 The probability of dislodgment is the area under the curve determined by multiplying the wave exceedance function Pex and the tenacity PDF PDFten:
!!"#$%!&' = !!" !"#!"#!" Equation 4.6
As an example, consider a wave exceedance function where the probability of exceedance is one until some value of x, and the probability of exceeding x is zero, as shown in Figure 4.3. That is, all values up to x are certain, and anything above x will never occur.
1 Prob. of wave exceedance PDF of tenacity PDF of dislodgement
0.5 Probability / Density Function
0 x Force (N) / Foot Area (m 2 )
Figure 4.3: Probability of wave exceedance and PDF of tenacity, illustrating how the PDF of snail dislodgment is calculated for a simple wave exceedance function. The shaded area represents the total probability of dislodgment for this case.
In this case, the dislodgment PDF is denoted by the shaded area to the left of x. Because the wave exceedance function is 1 up to x and 0 after, the product of the wave
Chapter 4 116 exceedance distribution and the tenacity distribution is simply the PDF of tenacity up to x.
When the simple exceedance function of Figure 4.3 is replaced by the wave exceedance function calculated for waves at my field site, the resultant PDF represents the net probability of dislodgment for a given snail for waves recorded at my site, provided two assumptions are met. First, I assume that a gastropod randomly samples its tenacity from the tenacity PDF as it moves about the rock. Second, I assume that measured waves at my field site are an accurate representation of all waves at this site (see the Discussion for an elaboration of these assumptions).
1 Prob. of wave exceedance PDF of tenacity PDF of dislodgement
0.5 Probability / Density Function
0 2 Force (N) / Foot Area (m )
Figure 4.4: Probability of wave exceedance and PDF of tenacity, illustrating how the PDF of snail dislodgment is calculated. The shaded area represents the total probability of dislodgment.
Chapter 4 117 4.4 RESULTS
4.4.1 Maximum Velocities
The distribution of calculated maximum wave velocities is shown in Figure 4.5. The mean velocity is 2.9 m/s, with a maximum of 12.8 m/s.
0.12
0.1
0.08
0.06
Fraction of Waves 0.04
0.02
0 0 2 4 6 8 10 12 14 Velocity (m/s)
Figure 4.5: Histogram of maximum wave velocities.
4.4.2 Drag Distribution
Drag values (normalized by organism foot area, N/m2) are shown in Figure 4.6.
Chapter 4 118 0.4 0.4
0.3 0.3
0.2 Lottia digitalis 0.2 Nucella emarginata
0.1 0.1 Fraction of Waves of Fraction Waves of Fraction
0 4 4 0 5 5 02x 10 401x 10 x 10 2x 10 Drag/Foot Area, N/m2 Drag/Foot Area, N/m2 0.4
0.3
0.2 Chlorostoma funebralis
0.1 Fraction of Waves of Fraction
0 4 4 05x 10 10x 10 Drag/Foot Area, N/m2
Figure 4.6: Distributions of drag normalized by foot area (N/m2) on one limpet and two snail species.
4.4.3 Probability of Dislodgment
Dislodgment probabilities for all behavioral cases are shown in Table 4.4. Dislodgment probabilities are higher for moving gastropods than stationary ones, owing to higher tenacities when stationary.
Chapter 4 119 Probability of Species Stationary/Moving Dislodgment
stationary 1.98x10-4 Lottia digitalis moving 1.57x10-3
moving 0.635 Nucella emarginata stationary 0.161
Chlorostoma funebralis moving 0.288
Table 4.4: Probabilities of dislodgment for limpet and snails while stationary or moving.
L. digitalis has a low risk of dislodgment at this site: 0.16% per wave while moving. Snails, however, have much higher risks of dislodgment: I predict N. emarginata faces a 63.5% chance of dislodgment from any given wave if it is moving (assuming that the waves recorded in Chapter 2 are an accurate representation of all waves). C. funebralis fares better: on average, an individual has a 28.8% chance of becoming dislodged by a wave while moving.
4.5 DISCUSSION At this site, L. digitalis faces a dislodgment risk of 0.16% per randomly chosen wave, suggesting that 1 in 637 waves may dislodge a moving L. digitalis individual. Limpets face far lower risks of dislodgment than snails: a given wave has up to a 63.5% chance of dislodging N. emarginata at this site, and each passing wave may pose up to a 28.8% risk of dislodging a C. funebralis individual.
These risks may be overestimates, and represent the worst-case scenario for these snails. First, I am assuming that the snail always takes the full brunt of the force with its maximum projected area, and that the full velocity of the wave acts over the entire projected area. Boundary layers and any associated reductions in velocity (and therefore drag) are not taken into account. Nor are any behavioral strategies for minimizing risk taken into consideration: this analysis assumes no behavior by the
Chapter 4 120 snails, such as seeking shelter in protected microhabitats or behind other organisms. In Chapter 3, although I showed that limpets may reduce forces acting on the foot through its structural properties, force reductions cannot be assumed and so are neglected here. Additionally, the snail tenacities used may be underestimates: Miller (1974) measured snail tenacities on Plexiglas, which she observed tends to produce lower shear tenacity values than those measured on a coarse substrate. This analysis only considers drag, however: by not including lift, I may in fact be underestimating the risk. This underestimate is likely to be slight: drag coefficients are higher than lift coefficients for gastropods in this study (Denny 1995), and shear tenacities for snails are lower than tensile tenacities (Miller 1974). Because the risk of dislodgment from drag is therefore much higher than that of lift, the resultant risk for the animal is likely to be very close to that of drag alone.
Two major assumptions drive this analysis: first, I assume snail breaking strengths are normally distributed, and that each snail randomly samples its breaking strength from this distribution. Second, I assume that the waves recorded in Chapter 2 are an accurate representation of wave velocities (and thus, imposed drag on each species) at this site. These data are limited to waves striking the transducer when the drag sphere was emersed, but it is likely that these represent close to the largest forces at the site. Examination of waves that struck the transducer when the sensor was submerged, while not included in either chapter, suggested that these waves tended to be smaller than those that struck the emersed transducer.
If a snail or limpet is dislodged, how dire are the consequences of being removed from the rock? Dislodgment is frequently assumed to be fatal: not one of 100 artificially dislodged Littorina sitkana on Tatoosh Island in Washington State was subsequently observed at a tidal height above the anemone beds in the low intertidal zone, suggesting no snails were able to return to the high intertidal zone; individuals were presumed eaten by predators (Boulding and Van Alstyne 1993). However, Miller et al. (2007) found Littorina keenae experimentally dislodged from the high intertidal (approximately 3 m above MLLW) had relatively high survival rates: between 54%
Chapter 4 121 and 90% of snails returned to the rocks of the high intertidal zone. These competing results suggest that the density and location of predators at the site are important factors: if a dislodged snail falls directly into an anemone, the consequences are more likely to be dire than for a snail who lands in a predator-free tidepool. At my site, the tidepools surrounding the rocks have varying numbers of anemones. As such, survival is likely to depend on where the dislodged gastropod is ultimately deposited.
Even if every dislodgment event does not prove fatal to a snail, it represents a significant risk. This analysis suggests that N. emarginata may face a dislodgment risk of over 60% per wave. However, since these snails are relatively abundant organisms on the shore, they must obviously take behavioral action to minimize the chances of dislodgment by seeking shelter in crevices, other microhabitats, or behind other organisms. Indeed, gastropods do change their behavior to minimize the risk of dislodgment: limpets demonstrate reduced foraging times during periods of high wave action and “hunker down” against the substratum, taking advantage of larger adhesive strengths when stationary (Denny et al. 1985, Judge 1988). The numbers calculated here serve to illustrate the hydrodynamic risks faced by intertidal gastropods, and underscore the necessity of protection from waves to survive on wave-swept rocky shores.
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