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Survival in the Surf Zone Author(s): Mark Denny Reviewed work(s): Source: American Scientist, Vol. 83, No. 2 (MARCH-APRIL 1995), pp. 166-173 Published by: Sigma Xi, The Scientific Research Society Stable URL: http://www.jstor.org/stable/29775404 . Accessed: 29/03/2012 21:23

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http://www.jstor.org Survival in the Surf Zone

Random propertiesof ocean waves combinedwith measurements ofphysical strengthpredict optimal sizesfor wave-swept organisms

Mark Denny

a an ocean a waves are a physicist hoping to establish ed where interacts with the short-period that typical To law of nature, random behavior in shore in the surf zone. As we shall see, of locally produced "seas," and long a a waves in near-shore waters "swells" are retained. The natural system is delightful thing. operate period peri? over ocean The predictive power of gas laws, for sufficiently randomly the short od of swells typically varies a instance, arises from the random mo? term that robust statistical techniques from six to about 20 seconds. As wave tion of gas molecules: Moving at a provide an accurate prediction of their approaches shore and enters effects. the shallow its of travel de? speed dictated by temperature, they long-term Furthermore, water, speed wave creases collide occasionally and rebound in climate profoundly affects the and itsheight grows gradually. a a wave unpredictable directions, like so many distribution and abundance of plants As result, becomes steeper more at some tinybilliard balls. Ifgases acted in even and animals that inhabit the shore. This and peaked, and point it a slightly less random fashion on the combination of predictable "weather" becomes unstable and breaks. Such waves or? molecular level (which they do at very and its effects on organisms presents tall, breaking interactwith a an an on low temperatures), the behavior of opportunity to explore interac? ganisms the shore. gas in bulk would be much more com? tion between biology and a physical plicated, and gas pressures would be environment. The Random Sea hard to predict. The plants and animals ofwave-swept a Outside basic physics, however, ex? Surface Waves shores clearly inhabit physically are on an arise stressful where waves amples of purely random behavior Waves ocean's surface pri? environment, rare. Take for ironi? action of wind. Air break over them half-dozen sec? weather, example; marily from the every a sea some or so. the commu? cally, it cannot be predicted accurately blowing over exchanges of onds Nevertheless, because it is not unpredictable enough. its kinetic energy with thewater below, nities of wave-swept shores represent some The progression ofweather fromday to thereby raisingwaves. Faster wind that of themost diverse and produc? a an ones on ex? day involves both stochastic (random) has longer time to interact with tive earth. The challenge of behavior by atmospheric systems and a ocean produces largerwaves. Typically, plaining such rich biological diversity waves are created over in face of arous? historical component: Yesterday's continuously the physical adversity es weather affects today's. The historical large expanses of ocean surface, and considerable interest, leading inves? and ani? influence, a lack of day-to-day indepen? wave height?the vertical distance be? tigators to explore how plants dence, vexes weather forecasters. tween a wave crest and the preceding mals have been designed through the ex? But there is at least one case inwhich trough?varies gradually fromplace to course of evolution to cope with the more the vari? wind waves. "weather"?or precisely, place because of varying strength. igencies imposed by breaking ation of the environment inwhich or? Once produced, a wave travels freely, Three relations play particularly im? so The roles in an interac? ganisms live?is thoroughly unpre? often forvery long distances. large portant organism's a dictable over the short term that its waves that batter the northwest coast tion with wave-swept environment. are a long-term effects can be predicted ac? of the United States, for instance, First, there is basic consideration of That is the environment creat storms farout in the curately. often produced by population dynamics: Larger organ? northern Pacific Ocean or in the Gulf isms are more likely to produce more of Alaska. In an extreme the a relative con? Mark Denny is an associate professor of biologi? example, young, leaving greater at He received waves that de? tribution to the next Sec? cal sciences Stanford University. exceptionally large generation. in at Duke a his bachelor's degree zoology stroyed the breakwater at Long Beach, ond, larger organisms face greater where he was to the a wave. University, first exposed field California, on an otherwise sunny and risk of being broken by Com? His interests in the ofbiomechanics. engineering calm day in 1930, arose thousands of bining the first and second relation? were aspects of biology given rigor during gradu? miles in a storm in the Southern raises a basic of organis ate at British away ships question studies theUniversity of Columbia, How should a Hemisphere. mal design: large expanded during postdoctoral stints at the As waves travel, their form changes wave-swept plant or animal grow? A University ofWashington and the Smithsonian Waves of different (the smaller faces less chance of Tropical Research Institute, and continue today subtly. periods organism time from crest to travel at dif? wave-induced but itwill in the application of biomechanical principles to crest) breakage, ferent and lose at differ? fewer A the study of ecology. Address: Hopkins Marine speeds energy probably produce offspring. ent rates. In travel removes on the other Station, Pacific Grove, CA 93950-3094. practice, larger organism, hand,

166 American Scientist, Volume 83 M.irtinH. Cluster (l'hotoKi-scirchcrs, Inc.)

conditions. A wave comes ashore dozen or so and wave size Figure 1.Wave-swept organisms endure harsh environmental crashing every seconds, wave allows for the interaction between or? varies randomly from moment tomoment. In fact, the randomness of heights predicting long-term a ocean here at California's Sur. ganisms at shore and the physics of waves, shown Big

a wave That leads to the third relation: The allow of the sizes might be broken by before it predictions optimal waves that crash on a and animals. But how is has a chance to reproduce, in which larger the shore, of plants large the utili? case its fitness is nil. Some size offers the larger the hydrodynamic forces im? the largestwave? Therein lies on an a behavior. an optimum balance between survival posed organism. As result, the ty of random wave sets size that balances waves in a surf and reproductive output. At that size, climate the The crashing given a zone in a wide va? an organism justwithstands the largest survival and reproductive output for probably originated maximum of locations. most waves force imposed during its reproductive wave-swept organism. The riety Although a a to 20 sec lifetime. size ofwaves at particular site should have similar period (from six

1995 March-April 167 organisms on a shore. If individual wave heights are unpredictable, how can a biologist predict the height of the largestwave? The answer lies in a cal? culation made over a century ago by JohnWilliam Strutt,Lord Rayleigh.

The Cocktail-Party Paradox As part of an extensive theoretical treatment of the physics of sound, Lord Rayleigh explored what might be called the "cocktail-party paradox." Imagine being at a party with 50 other revelers, each chatting away at the top or of his her lungs. It seems entirely possible that the peak pressure of each person's sound wave could reach your ear at the same time. The resulting pressure?a sound 50 times as loud as the average party sound?could po? tentially rupture your eardrum. How is it that cocktail parties do not result in widespread hearing damage? Rayleigh solved this paradox by considering the superposition (alge? a braic addition) of large number of sound waves of the same period but random phase. Like ocean waves, sound waves combine and modulate, and Rayleigh's calculations led him to propose the probability distribution as known appropriately the Rayleigh 2. distribution. to the Figure Diverse and productive communities of organisms inhabit shorelines, despite the en? According Rayleigh vironmental sea most common waves constraints. These mussels and stars, for instance, must be designed to survive distribution, the the of waves. a repeated impact breaking reaching partygoer's ear have a slightly less-than-average amplitude. at a onds), they rarely arrive shore in Waves with amplitudes above average lockstep. The arrival of twowave crests do exist, but theirprobability decreases on at same a shore the time is matter of rapidly with increasing amplitude. In pure chance. It is just as likely that a fact,you would have to attend cocktail one wave crest from will arrivewith the parties continuously forover two years trough of another. In fact,waves arriv? to encounter a sound even five times at a are to ing shore likely be assorted louder than average. in terms of same randomly theirphase. The mathematics applies to To a firstapproximation, ocean waves ocean-surface waves. Waves of ran? are wave co? additive. Where two crests dom phase reaching shore modulate the of the wave a incide, height resulting each other,which produces Rayleigh the sum of the of wave equals heights the indi? distribution of heights. In other waves. a vidual Likewise, smaller wave words, an organism on shore must a wave crest a a results when overlaps wait long time, on average, to be sub? size wave Waves of similar an trough. period jected to the forces of exceptionally 3. Size a but random combine at a shore? wave. Figure of wave-swept organism repre? phase large Oceanographers quantify sents a balance between factors. line such The result? competing through processes. waviness with the so-called "signifi? First, as an its chance of modulation of wave wave organism grows larger, ing heights pro? cant height," essentially an aver? because it is more surviving decreases, likely duces a condition called the "random age of the one-third tallest ocean-sur? to be broken a wave. by Second, larger animals sea," and the of an individual facewaves. A wave tend to more height typical significant produce offspring. A successful wave arises a from random interaction height on theWest Coast of North shore balances these factors, organism growing waves. As a for is about 2.3 me? to as among result, predicting America, example, large enough produce as many offspring the of individual waves proves ters. the wave possible, but remaining small enough towith? height By knowing significant wave in one can a stand the largest that it is likely to en? impossible practice. height, generate Rayleigh counter. a The short-term stochastic behavior distribution that the This illustration provides only predicts largest of wave seem wave on a schematic representation of the relations; actual heights might problem? that is likely to crash given curves be more atic a may complex. for studying design constraints for shore over specific period of time.

168 American Scientist, Volume 83 waves Kcomponent nearlyout of phase

? large resultingwave

||f^^^^^^^^^^^H^^^^^^^^Kxv componentwaves /ym ^^^^^^^^^^^^^^^^^m^^^m^^^^^^^^^^^^^^^^^^nh ^^^^^^^^^Bn^ nearlyinphaseXX^ ^^^^^^^^^^^^l^^^^^^^^^^^^^^^^^l

4. Ocean waves interact If two waves a Figure (coral lines) from different places arrive at shore nearly in phase (left)?crests and troughs aligned? the wave sum resulting (blue line) is larger, essentially the algebraic of the two component waves. The arrival of two waves out of phase, however, to a wave leads smaller hitting shore, because the components cancel each other (right).

An ocean's waviness, however, used to calculate the hydrodynamic should travel at a speed of 6.7 meters varies with time. During stormy peri? forces that an organism experiences. per second (about 15miles per hour), ods, the near-shore significant wave Wave height correlateswith theveloc? and water at its crest would move at can height be quite large. Significant ity imparted to the water beneath its the same speed. In a breaking wave heights greater than 10 meters have crest. In deep water, water beneath a with a height 5.9 times the yearly aver? on moves a been recorded the Oregon coast. crest at small fractionof theve? age significant wave height of 2.3 me? Fortunately, the Army Corps of Engi? locity atwhich thewave travels.As wa? ters, thewave and thewater in its crest in neers, collaboration with theCalifor? terdepth decreases during a wave's run would travel at about 16 meters per nia Department of Boating and Water? toward shore, the speed of a wave form second (about 36 miles per hour). ways and the Scripps Institution of decreases, but the speed ofwater moved Waves in a surf zone are also accom? a wave Oceanography, maintains series of by the increases. At the breaking panied by rapid water accelerations. wave sensors along the coasts of Cali? point,water in a wave crestmoves at the As a wave breaks, its ordered motion same as wave. fornia,Oregon and Washington. Mea? speed the Shortly after degenerates into turbulence, and ed? surements wave a wave of significant height breaking, when formhas slowed dies flow across the seabed. A rough are taken four times each day, provid? still further,water in the crest can move consideration of the size of these ed? even ing the basis for calculating the annual faster than thewave, which forms dies and the rate at which they are From these one can cal? wave a average. data, the plunging crest commonly de? moved by wave leads to predicted a culate probability function that de? picted in surfingmagazines. accelerations that increase with wave scribes the variation in significant Crest velocities of breaking waves height, and may reach 1,000meters per wave or a can height along the entire coast, be surprisingly high and depend seconds squared in some areas. Values so-called wave-exposure distribution, on the height of the wave and the in excess of 400 meters per seconds a which is statistical description of how depth ofwater. According towave the? squared have been measured on wave waviness varies over time. ory, a breaking wave 2.3 meters tall swept rocky shores. These are extreme The wave-exposure distribution found on theWest Coast suggests that the largest wave likely in one year is approximately 5.9 times the yearly av? erage significantwave height. So for a yearly average significantwave height of 2.3 meters, the largestwave likely to strike the shore in a year is approxi? mately 13.6 meters high, or the height a of four-storybuilding.

Wave Forces Being able to predict the maximum wave height makes it possible to pro? ceed with predictions of survival and optimal size forwave-swept organ? isms. The firststep consists of connect? wave ing height to hydrodynamic a time(seconds) forces. Knowing wave's height al? lows the calculation of its and velocity Figure 5. The "random sea," random variations in the height of an ocean's surface at a given point acceleration. These values can then be over time, arises from interactions of ocean waves.

1995 March-April 169 tum. Both drag and liftare proportion? al to the dynamic pressure of the flow, or half the density of water times the square of thewater velocity. Being pro? portional to the square of velocity makes these forces sensitive to any in? crease in velocity, and thereby to any increase inwave height. A wave with twice the average velocity, for instance, imposes four times the average force. The forces of lift and drag can be me? quite large. A water velocity of 16 ters per second, that associated with a yearly maximum wave off theWest Coast, produces a dynamic pressure of same 130,000 pascals. Applying the pressure over the projected area of a human body would be equivalent to about 13 metric tons. This ex? 0.1 0.011 10 100 1,000 10,000 100,000 helps survivors of a wrecked togtime (days) plain why ship on a rocky coast do not decide lightly 6. of the wave to hit a shore increases with time, accord? Figure Height largest likely predictably to swim ashore. to a theoretical extension of the distribution and the that the ing Rayleigh assumption significant Size also a fundamental role in wave remains constant waviness on the basis of plays height Oceanographers quantify significant forces. the all waves. The ratio of maximum wave hydrodynamic Doubling wave height, the average height of the largest third of wave with shown here on a linear dimensions of an for height to significant height increases gradually time, logarithmic organism, a its area, scale. In any hour, for instance, the maximum wave hitting shore will probably be less than example, quadruples exposed as wave a before a wave three times the and lift twice as big the significant height And about century passes thereby quadrupling drag than the wave hits a shore. a of higher significant height experienced from given velocity water. The ability of an organism to re? lift. sist in? ly high accelerations (40 to 100 times two types 'of force: drag and Drag hydrodynamic forces,however, on area a or ex? creases with size. The the acceleration of gravity),much high? acts the of plant animal similarly strength an a or er than those found in almost any oth? posed to the flow and tends to push of animal plant increases approxi? er environment. Lift also acts on with the cross-sectional area of aquatic organism downstream. mately an it orwith the area of its adhe? Both the velocity ofwater and its ac? organism's exposed area, but its skeleton across as an celeration impose forces on objects em? tends to pull the organism the sive apparatus. So organism its to resist forces bedded in the flow. Velocity generates flow, typically away from the substra grows larger, ability

water with constant two forces: and lift an Figure 7. Several forces affect organisms in the surf.Moving velocity generates drag (left). Drag pushes to the an from the substrate. Both of these organism in the direction of the flow, and lift acts perpendicular flow, usually pulling organism away area an water an force Newton's second law of motion dictates forces act on the exposed of organism. Accelerating generates accelerational (right). a of water's acceleration and the mass of water the that an organism in accelerating water experiences force that equals the product the displaced by exceed that Newton's second law alone organism. Actual measurements of accelerational force (gray arrow), however, predicted by (white arrow). around an and the rate of a force that enhances the overall ac? The accelerating water changes the pattern of water flow object, change develops on an rather than its area. celerational force. The magnitude of an accelerational force depends organism's volume,

170 American Scientist, Volume 83 drag

8. Size determines the effect of wave-induced forces. an a Figure Imagine organism that is essentially cube-shaped. If cube's side is /units long (left), then it has sides with area I2 and an overall of P. an a volume Increasing organism's linear dimensions by factor of two (right) produces sides with area 4/2 and an overall volume of 8/3.Given that and increase with an area an drag lift organism's frontal and that organism's strength is probably related to its attachment which increases an or or area, similarly organism's risk of drag- lift-induced dislodgment breakage does not increase with size. Accelerational on the other on an force, hand, depends organism's volume, which increases faster than strength as an organism grows. This scaling problem could determine the size at which an organism breaks.

increases inproportion to the drag and lies the potential for an adaptive re? instance, can be induced to grow with lift it are an a feels, because both propor? sponse by organism to its environ? streamlined shape by hanging small tional to area. ment. In many cases, that response weights from their fronds tomimic the Nevertheless, the force due to accel? takes place through the course of evo? force of drag. eration upsets the balance between ap? lution. Hydrodynamic forces have ap? Theoretical predictions of the rela? plied force and strength. Unlike drag parently affected the evolution of rigid tion between biological shape and and lift,the accelerational force is pro? skeletons of some sea urchins, includ? flow-induced forces remain difficult, so an portional to organism's volume, not ing the shingle urchin (Colobocentrotus it is common practice tomeasure di? area. its exposed Consequently, dou? atratus), which possesses reduced rectly the proportionalities between ve? an a bling organism's linear dimension spines and streamlined shape. It in? locity or acceleration and forcewhen results in an increase in eightfold accel? habits wave-swept shores in Hawaii, dealing with wave-swept organisms. A erational force, causing it to increase where it is commonly found fully ex? plant or animal can be attached to a than an to faster organism's strength, posed the breaking swell. Another force transducer and exposed to a care? a more which makes larger organism urchin (Echinometra mathaei) inhabits fully controlled flow in a flume or a to accounts same more likely be dislodged. This the shores, but it has the wind tunnel. By varying the speed and a sea for the size-specific risk proposed typical pincushion appearance of acceleration of the flow, the resulting in can above considering the optimal size urchin that leads to higher drag, per? forces be measured. Using these ofwave-swept plants and animals. haps explaining why it typically lives values, one can calculate the force that in cracks and crevices that are shielded a given organism experiences when Side Effect of Shape from thewaves. In other cases, adapta? subjected to a given water velocity and an a as? Beyond size, organism's shape af? tion develops from physiological re? acceleration, and thereby the force a fectshydrodynamic forces, and therein sponse during growth. Some algae, for sociated with a wave of given height.

1995 March-April 171 produces a curve that quantifies the probability that a mussel, chosen at ran? dom, will be dislodged when a particu? lar force per area is applied. For drag and lift,force per area is a simple func? tion of water velocity and therefore of wave height,which makes itpossible to relatewave climate to theprobability of dislodgment.

Optimal Size and Dominance Preliminary studies suggest that organ? isms may strike a balance between the counteracting effects of increasing size: increasing risk of wave-induced dam? age and increasing reproductive poten? tial. Tom Daniel of the University of Washington, Mimi Koehl of theUniver? sity of California, Berkeley, and I found that the common purple sea urchin forceper area (kilopascals) (Strongylocentrotuspurpuratus) and sev? eral of are 9. the force area boosts the odds that a California mussel will be species limpets approximate? Figure Increasing applied per the size that one would for or force unit area) can be determined a ly predict dislodged. Dislodgment strength (pascals, per by using an? a Other spring scale tomeasure the force required to dislodge a mussel occupying given area of a bed. optimal reproductive output. a imals, acorn barnacles, are Collecting data formany mussels reveals the likelihood that randomly selected mussel will be including a or or a wave much smaller than their would dislodged by given force per area, the lift drag generated by specific height strength and (Adapted from Denny 1993.) allow, the size of these organisms must depend on other factors. One species of coral, firecoral (Millepora cotn a we So from the yearly average significant result, seldom specify thestrength planata), grows substantially larger than wave a an height for particular site, one of organism. Instead, we quantify we predicted, but broken fragments of can predict themaximum wave height, the probability that an organism cho? this species can recement themselves to ac? a themaximum water velocity and sen at random has less than particu? the reef and grow. So their large size celeration and themaximum force im? lar strength. could be a strategy for dispersal and on an posed organism. Consider the strength of attachment vegetative reproduction. What risk does this force impose on inCalifornia mussels (Mytilus californi A more compelling case for the effect an organism? Here again, theory pro? anus). At Tatoosh Island, a site on the of hydrodynamics on optimal size may vides littlepractical utility, so we resort coast ofWashington, 178mussels were exist in algae. Although the abundant al? to empirical measurements. Organisms dislodged from the substratum, and the gae ofwave-swept shores are blade-like of a range of sizes can be broken using forcewas measured with a spring scale. and bend in flow to assume a stream? a force transducer, and the relations These force values can be normalized lined shape, Brian Gaylord of Stanford a among species' size, shape and by dividing them by the size of the dis? University and Carol Blanchette ofOre? can strength be ascertained. In general, lodged mussel, expressed in this case as gon State University showed recently ex? area considerable variation in strength the that themussel occupies in the that the shapes of several macroalgae ists among individuals of a species. As bed. Ranking these normalized values tend to trapwater within a frond's inter? stices, therebymagnifying the apparent volume of a plant. As a result, the accel erational force on an alga may be an or? der ofmagnitude larger than that on an animal of equal mass. Hydrodynamic based predictions of the size at which three species ofwave-swept macroalgae should achieve theirmaximal reproduc? tive output closelymatched the sizes ob? served in nature. Beyond reproductive output, knowl? edge ofmaximal wave height may also explain the competitive allocation of on 10. can space the substratum. The California Figure Organisms adapt to forces by changing shape. The shorter, more streamlined a sea for instance, dominates mid-in spines of shingle urchin (left), for example, allow it to inhabit wave-swept shores inHawaii. mussel, A bears the shores from Alaska to neighboring species (right) standard pincushion of spines that generate more drag, tertidal, rocky which that to cracks and same central relegates species crevices along the shores. (Photographs courtesy California. Although starfish of the author.) prey on thesemussels in the low-inter

172 American Scientist, Volume 83 be dislodged in a year, opening ap? proximately four percent of a mussel bed for colonization. If, on the other hand, the yearly average significant wave height is threemeters, thenmy measurements predict that 14 percent of the mussels will be dislodged. Therefore, a one-meter variation in yearly average significantwave height would result innearly a fourfold varia? tion in the rate at which bare space is created inmussel beds. In fact, data over a 37-year period in theNorth Sea reveal that yearly average significant wave height there varied from 1.0 to 1.8meters around a long-termmean of about 1.4. Such a variation of more than 50 percent of themean could have drastic consequences for intertidal communities. Fundamental principles of ecology lie beneath the seeming disarray of waves crashing against a shore. A com? bination of statistical analyses and em? pirical measurements suggests that an organism in the surf zone often grows large to enhance its reproductive po? tential, but not so large that it cannot survive the largestwave that it is likely to encounter during its life. So the 11.Mussel to short-term stochastic of Figure dislodgment helps determine themakeup of ecological communities along properties ocean waves an excellent are? rocky mid-tidal shores from Alaska to central California. When not disturbed, California-mussel provide communities wall-to-wall can na for the interac? display coverage. Waves uproot sections of mussels, producing exploring long-term bare can spots that be colonized by invading species, including algae, as shown here. (Photo? tion between the physics of the surf of C. A. graph courtesy Blanchette.) zone and the size and strength of wave-swept organisms. tidal zone and a desiccation prevents tive subordinates and relatively long them a from inhabiting thehigh-intertidal period inwhich given patch is likely Acknowledgments form at to a zone, they densely packed beds remain mussel-free. In such case, Much of thework described herewas sup? mid-tidal levels. This dominance poten? subordinate species interact freely, ported byNational Science Foundation tially excludes other species, thereby re? much as theywould in the absence of grants OCE-8711688 and OCE-9115688. ducing the species richness ofmid-inter mussels. A low rate of dislodgment pro? tidal shores. Knowledge of themaximal duces short-lived bare patches, and only Bibliography wave on W. heights these shoresmay help rapidly colonizing species can survive. Bascom, 1981. Waves and Beaches. Second Edition. New York: explain the competitive allocation of the What rate of mussel dislodgment Doubleday. M. W. 1988. and theMechanics substratum. does a statistical approach predict? Us? Denny, Biology of theWave-Swept Environment. Princeton, NJ: Although not prone to biological dis? ing themussel strengths determined at Princeton University Press. turbance,mussels in themid-intertidal Tatoosh Island and the wave significant M. W. 1993. Disturbance, natural selec? zone can be waves. Break? of 2.3 meters from I Denny, dislodged by height nearby sites, tion, and the prediction of maximal wave waves winter for in? that ing during storms, predict about 6.4 percent of the induced forces. Contemporary Mathematics out areas 141:65-90. stance, rip ofmussel beds. A mussels will be dislodged on average. of an M. W., T. L. Daniel and M. A. R. Koehl. variety algae and invertebratesquick? In 11-year study of the dynamics of Denny, 1985. Mechanical limits to size in wave ly colonize the patches of open substra? open patches at this site,Robert Paine of tum. As mussels reclaim the of swept organisms. Ecological Monographs open patches, University Washington and Si? 55:69-102. other patches are formed more mon Levin of Princeton by University Gaylord, B., C. Blanchette and M. Denny. 1994. in an in? recorded rates of waves, resulting ever-shifting average dislodgment Mechanical consequences of size in wave teraction rates 64:287-313. among the of patch for? from 6.3 to 12.3 percent, comfortingly swept algae. Ecological Monographs mation, colonization and reclamation. close to the value predicted by theory. Paine, R. T., and S. A. Levin. 1981. Intertidal and the of The resulting community depends The rate of is landscapes dynamics pattern. predicted dislodgment 51:145-178. on the rate of mussel A sensitive towaviness at a site. If Ecological Monographs dislodgment. quite W. S. 1880. On the resultant of a rate of results in a the wave Rayleigh, J. high dislodgment yearly average significant number of vibrations of the same fraction of the substratum is 2.0 I that large pitch large being height only meters, predict of arbitrary phase. Philosophical Magazine available for colonization by competi only four percent of themussels will 10:73-78.

1995 March-April 173