intertidal mussel microclimates: predicting the …california 93950-3094 usa. are aerially exposed...

51

Ecological Monographs, 68(1), 1998, pp. 51–74q 1998 by the Ecological Society of America

INTERTIDAL MUSSEL MICROCLIMATES: PREDICTING THE BODYTEMPERATURE OF A SESSILE INVERTEBRATE

BRIAN S. T. HELMUTH1

University of Washington, Department of Zoology, Box 351800, Seattle, Washington 98195-1800 USA

Abstract. To elucidate the determinants of intertidal invertebrate body temperaturesduring aerial exposure, I developed deterministic models using the environmental inputsof solar radiation, air temperature, ground temperature, and wind speed to predict the bodytemperatures of intertidal mussels (Mytilus spp.). Combined with field studies, these modelswere used to determine the effects of body size on body temperature, and to compare theheat budgets of mussels living as solitary individuals vs. those living in aggregations (beds).On average, the model accurately predicted the body temperatures of solitary mussels inthe field to within ;18C. Steady-state simulations (using constant environmental conditions)predicted that, under conditions where evaporative water loss is limited, smaller (5 cm)mussels experience lower body temperatures than larger (10 cm) mussels exposed to iden-tical environmental parameters. When evaporative cooling is limited only by intoleranceto desiccation, the trend in body size reversed due to a disproportionately greater amountof tissue (per unit length) in larger mussels, which provides them with a greater reservoirof water available for evaporative cooling. In both scenarios, larger mussels display a greater‘‘thermal inertia’’ (time constant of change), which buffers them against rapid changes inenvironmental conditions. No one environmental factor controls body temperature, and thusmeasurements of single environmental parameters such as air temperature are very unlikelyto serve as accurate indicators of mussel body temperature. Results of unsteady simulations(using fluctuating environmental conditions) further indicated a significant effect of thespectral characteristics of the physical environment on body temperature. In many casespredictions of body temperature based only on daily means or extremes of environmentalparameters are off by 68C or more due to the time dependence of the system. Models ofbody temperature must therefore be based upon repeated measurements of multiple envi-ronmental parameters, rather than simple statistical measures such as daily mean, maximum,or range. Significantly, several parameters in the model presented here are modified by theproximity of neighboring organisms, including predators and competitors. During extremeenvironmental conditions (using steady-state conditions), mussels living in beds are pre-dicted to experience substantially lower (48–58C) body temperatures than those living ingaps. Furthermore, living within an aggregation also augments a mussel’s thermal inertia,which dampens the effects of rapid temporal changes in the physical environment. Incontrast to most previous studies in rocky intertidal habitats, results thus suggest that‘‘physical factors’’ are not immutable boundaries imposed by the environment, but may besignificantly altered by the organism itself through its size, morphology, and interactionswith neighbors, which may create feedback loops between abiotic and biotic controls.

Key words: body temperature; climate change; heat budget; intertidal; microclimate; mussel;Mytilus; physical factors; thermal inertia; time history.

INTRODUCTION

Sessile organisms living in the marine intertidal mustcontend with highly variable and often extreme envi-ronmental conditions. Water motion from crashingwaves exerts enormous forces upon organisms, and therisk of dislodgement from the substrate can be an im-portant selective factor (e.g., Witman and Suchanek1984, Denny et al. 1985, Gaylord et al. 1994, Denny1995). Furthermore, most intertidal animals and algae

Manuscript received 20 January 1997; revised 16 April1997; accepted 5 May 1997.

1 Present address: Hopkins Marine Station, Stanford Uni-versity, Department of Biological Sciences, Pacific Grove,California 93950-3094 USA.

are aerially exposed on a daily or twice-daily basis. Asa result, intertidal organisms essentially live on the‘‘edges’’ of both the terrestrial and marine environ-ments, and must contend with the physical conditionsof both. To ectotherms such as marine invertebrates,two important consequences of aerial exposure at lowtide are the effects of the environment on body tem-perature and water loss rates. While submerged, anorganism’s body temperature approximates the tem-perature of the surrounding water. In contrast, solarradiation, evaporative cooling, and convective heat ex-change can all strongly affect body temperature duringexposure to air. Consequently, the body temperature ofan intertidal invertebrate can increase by .158C above

52 BRIAN S. T. HELMUTH Ecological MonographsVol. 68, No. 1

water temperature (Carefoot 1977) or can decrease tofreezing after exposure during low tide (Seed and Su-chanek 1992). Either extreme can profoundly affect anorganism’s physiological performance and subsequentfitness, and under extreme conditions its probability ofsurvival (e.g., Fyhn et al. 1972, Suchanek 1978, Elvinand Gonor 1979, Sebens 1982, Tsuchiya 1983, Wethey1984, Branch et al. 1988, Bertness 1989, Wilbur andHilbish 1989, Liu and Morton 1994, Gosselin and Chia1995, Hofmann and Somero 1995, Williams and Mor-ritt 1995). For example, Tsuchiya (1983) found thatthermal stress led to the deaths of .50% of a largepopulation of intertidal mussels (Mytilus spp.) withinone hour of exposure. In a less extreme but perhapsfar more common scenario, Hofmann and Somero(1995) have found that cellular damage (heat shock)due to extreme temperatures is commonly observed inintertidal mussels, which can in turn have significantconsequences to their energy allocations. Furthermore,evidence suggests that intraspecific variability in sen-sitivity to thermal stress may contribute to patterns inthe latitudinal distribution of mussels within this genus(Hofmann and Somero 1996). Body temperature anddesiccation are also thought to set both the upper in-tertidal and geographic limits of many other individualintertidal invertebrates (e.g., barnacles—Connell 1961,Wethey 1984; mussels—reviewed in Seed and Sucha-nek 1992). Finally, explicit, mechanistic approaches togaining an understanding of the role of the physicalenvironment in controlling body temperatures may becritically important if we are to predict the impact offorecasted changes in the earth’s climate (e.g., Kareivaet al. 1993, Asrar and Dozier 1994). Surprisingly littleis known, however, of the relative importance of thevarious environmental determinants of invertebratebody temperature during aerial exposure (e.g., sun,wind, air temperature), or of how the effects of thesefactors are influenced by the size, spatial position, andgeometry of the organism itself (e.g., Peters 1983, Dun-ham 1993, Bell 1995). Thus mechanistic, quantitativemethods for predicting body temperatures under abroad suite of environmental conditions can aid in un-derstanding the role of the physical environment ininfluencing the physiology and community dynamicsof intertidal animals.

Mechanistic approaches to predicting body temper-ature have met with considerable success in studies ofterrestrial organisms such as insects (e.g., Buatois andCroze 1978, Kingsolver 1979, 1983a, b, Kingsolverand Watt 1983), arachnids (e.g., Riechert and Tracy1975), and vertebrates (e.g., Porter and Gates 1969,Porter et al. 1973, Grant and Porter 1992, Dunham1993) as well as marine algae (Bell 1992, 1995) andterrestrial plant communities (e.g., Nobel 1983). Forexample, using thermal energy budgets Kingsolver(1979) showed that small-scale spatial and temporalheterogeneity in the physical environment of a popu-lation of pitcher plant mosquitoes led to significant

differences in larval mortality and development rate,which in turn translated into population-level effects.Such mechanistic approaches thus serve as a bridgebetween the behavioral and physiological ecology ofindividual organisms and the ecological dynamics ofpopulations and communities (e.g., Porter et al. 1973,Schoener 1986, Kingsolver 1989, Huey 1991). How-ever, these methods have seldom been fully applied tointertidal invertebrates (but see Thomas 1987), animalsthat, although aquatic in origin, are periodically ex-posed to a terrestrial environment. Unlike most (adult)terrestrial animals (but see Huey et al. 1989), intertidalinvertebrates are often almost completely immobile,and therefore cannot quickly alter their position in re-sponse to changing environmental conditions. Here Idescribe a mechanistic model that quantifies the inter-active effects of organism size, behavior, spatial ar-rangement, and characteristics of the physical environ-ment on body temperature. I use it to examine the de-terminants of body temperature of marine mussels (My-tilus spp.), and to explicitly address the hypotheses thatthe heat budgets, and thus operative body temperatures,of mussels significantly change with body size andproximity to conspecifics (i.e., aggregated vs. solitarymussels).

Mytilus californianus and M. trossulus (formerlygrouped with M. galloprovincialis as M. edulis,McDonald et al. 1991) are two of the most commonconstituents of hard-substrate intertidal communities inthe northeastern Pacific, and range geographically fromsouthern Alaska to central California (Gosling 1992,Seed and Suchanek 1992). Both species typically livein aggregations (beds) of conspecifics and congeners,but following disturbance events solitary individualsinhabit gaps (5patches) within the beds (Levin andPaine 1974, Paine and Levin 1981, Sousa 1984). M.californianus is the competitive dominant over a widerange of intertidal habitats and, as such, determines thestructure and diversity of the intertidal community(Paine 1966, 1974, 1984, Suchanek 1981, 1985, Seedand Suchanek 1992). In addition, mussel beds havebeen shown to exhibit very high levels of productivity,and may potentially support a large proportion of thetotal productivity of the near-shore benthic environ-ment (Leigh et al. 1987, Seed and Suchanek 1992). Asa result, attempts to predict the determinants of survivaland productivity of Mytilus spp. are particularly im-portant due to their extensive impact on other specieswithin the community. While species within this genusvary in their physiological responses to temperature(e.g., Hofmann and Somero 1996) as well as their sizeand morphology (e.g., Seed 1968, Seed and Suchanek1992), models of heat flux incorporate these factors incalculating body temperature. Thus, while I focus onthe mussel M. californianus, the model applies to allspecies within the genus.

The goal of this paper is to develop a mechanisticapproach for predicting the body temperatures of in-

February 1998 53INTERTIDAL INVERTEBRATE MICROCLIMATES

FIG. 1. Approach to predicting steady-state body temperature under constant environmental conditions. After a predictabletime lag (time to equilibrium) a mussel achieves a steady-state body temperature where the effect of ‘‘thermal inertia’’ (timeconstant of heating) disappears.

tertidal mussels, which may then be used to addressquestions pertaining to the effects of body temperatureon a mussel’s ecology and physiology. Elsewhere I de-scribe methods for extrapolating from climatic recordsand other large-scale data sets to the body temperaturesof intertidal invertebrates (B. S. T. Helmuth, unpub-lished manuscript). Thus, while the ultimate goal ofthis approach is to address questions at an ecologicalscale, the specific objective of this paper is to lay thegroundwork for examining the thermal interactions ofa sessile invertebrate with its physical and biotic en-vironment.

METHODS

The model system described here pertains directlyto mussels, but the principles entailed may be readilyadapted to other sessile intertidal organisms (e.g., Tho-mas 1987, Bell 1992, 1995). I first present the generalform of a heat balance equation and then provide detailsof the various components of the model. Two generalforms of this model are presented. First, an unsteadyheat budget model is used to track the body temperatureof mussels (Mytilus californianus and M. trossulus)under fluctuating environmental conditions in the fieldas a test of the model parameters. Then, to examine

the effects of body size on the body temperature of M.californianus, I use a simpler steady-state version ofthe model, which assumes ideal, unchanging environ-mental conditions (Fig. 1; Monteith and Unsworth1990). I also use this approach to examine the thermalconsequences of living in a bed of conspecifics vs.solitarily in a gap. Lastly, returning to the unsteadymodel, I explore the consequences of a temporally fluc-tuating environment on intertidal invertebrate heat bud-gets.

Theoretical approach

The instantaneous heat flux between an animal andits environment determines the total heat stored in theanimal and, therefore, its temperature (Fig. 2). At alltimes, the sources, sinks, and stores of heat in the sys-tem must be balanced (e.g., Porter and Gates 1969,Campbell 1977, Monteith and Unsworth 1990). In gen-eral, six factors determine the rate of change in theamount of heat stored within a body (Qstored): short-wave solar radiation (Qsol), radiation to and from fromthe sky (Qrad, sky), radiation to and from from the ground(Qrad, ground), conduction to and from the ground (Qcond),heat convected between the animal and the surrounding


FIG. 2. Determinants of heat flux in a sol-itary mussel. At all times the sources, sinks, andstores of heat in the system must be in balance.In general, six categories of heat transfer areused to describe the system: short-wave solarradiation to the mussel (Qsol), infrared radiationbetween the mussel and the sky (Qrad, sky) andsurroundings (Qrad, ground), convective heat ex-change (Qconv), conduction to and from theground (Qcond), and evaporative cooling fromwater loss (Qevap). The net sum of all the inputsand outputs of heat determines the rate ofchange in heat stored within the mussel (Qstored),which in turn determines the temperature of thebody (Tbody).

air (Qconv), and heat lost through the evaporation ofwater (Qevap). That is,

Q 5 Q 6 Q 6 Q 6 Qstored sol rad,sky rad,ground cond

6 Q 2 Q . (1)conv evap

Model development.—While a thorough understand-ing of each parameter used in the equations is not nec-essary to interpret the results presented in this paper,a brief explanation of several components of the heatbalance model will aid in interpreting the effects offactors such as body size and aggregation (Fig. 2), andwill shed light on the means by which organisms in-teract with the characteristics of their physical envi-ronment. I have therefore collapsed all ‘‘nonessential’’components into constants (k1–5), which are presentedin an Appendix along with a list of symbols, parametervalues, and methods for obtaining empirically derivedcoefficients specific to mussels.

Although the entire unsteady heat balance equation(Eq. 1 under nonconstant environmental conditions) issolved numerically, each component may be thoughtof as a source or sink of heat to the system and hasthe units of watts (W), a measure of energy flux perunit time (joules per second [J/s]). Because heat energyis neither created nor destroyed, accounting for all ofthe inputs and outputs of heat into and out of the musselpermits the calculation of the change in heat storedwithin the mussel (Qstored), which, given its mass andspecific heat, determines the temperature of the body(Tb). Equations describing each heat source and/or sinkhave the general form of the product of a driving‘‘force’’ (e.g., heat from the sun or the difference intemperature between the mussel and its surroundings),some area over which the heat is exchanged (either aprojected or surface area), and a series of coefficients(physical constants and factors that depend on bodysize, morphology, material properties, and the animal’sspatial arrangement).

1. Change in stored heat (Qstored).—The total heat

stored in an organism (or any other system) is the prod-uct of its mass (m), specific heat (c), and body tem-perature (Tb, in units of Kelvin [K]). Thus, the rate ofchange in stored heat is

Qstored 5 d(mcTb)/dt. (2a)

Since the specific heat of shell is likely to be differentfrom that of the largely water-filled body, this term isbest separated into two components with the testableassumption that the temperature is relatively uniformthroughout (see Appendix). Additionally, with a con-stant value for specific heat and shell mass, the storedheat term becomes:

Qstored 5 (mshellcshell)/dTb/dt 1 cbody/d(mbodyTb)/dt (2b)

where mshell and mbody are the masses of shell and body,respectively, and cshell and cbody are their specific heats.Under conditions in which no evaporation occurs (andthus mbody remains constant), the last term on the right-hand side of Eq. 2b is simplified as (mbodycbody)/dTb/dt.

2. Short-wave solar flux (Qsol).—Solar heat flux istraditionally (and arbitrarily) divided into long andshort wavelengths due to differences in their interac-tions with the atmosphere and the objects upon whichthey impinge. The total short-wave solar heat flux hasdirect (sun only), diffuse (sky-scattered sunlight) andalbedo (reflected) components (Campbell 1977):

Qsol 5 a(AsolS 1 AdSd) (3a)

where a is the solar absorptivity (the percentage of totalimpinging radiation absorbed by the body, a functionof the properties of the shell and any associated epi-fauna), Asol is the projected area of the animal in thedirection of the sun, S is the direct solar flux densityacting upon that area, Sd is the sum of the diffuse andalbedo flux densities, and Ad is the total (not projected)surface area exposed to Sd. Since, however, extremeconditions are the central interest here, I focus on thedominant term—direct solar radiative flux, which, un-


der cloudless sunny skies, dominates the total short-wave heat flux (Monteith and Unsworth 1990):

Qsol 5 aAsolS. (3b)

3. Long-wave radiation (Qrad, sky and Qrad, ground).—Long wavelength (infrared) radiation is emitted fromthe animal and its surroundings (the sky and theground) with a dependence on temperature to the fourthpower:

4 4Q 5 sA (« T 2 « T ) (4a)rad,sky rad,sky org b sky a

4 4Q 5 sA (« T 2 « T ) (5a)rad,ground rad,ground org b surr g

where s is the Stefan-Boltzman constant, Ta and Tg areair and ground temperatures, respectively, and the areasArad, sky and Arad, ground are each taken to be half of thetotal surface area (half facing the sky and half facingthe ground). The coefficients «org and «surr are the long-wave thermal emissivities (and subsequently the ab-sorptivities as well) of the organism and its surround-ings, and both have values of ;1.0 (Campbell 1977,Nobel 1983). The term «sky represents a ‘‘functional’’infrared emissivity of the sky, and is dependent uponair temperature and cloud cover (Campbell 1977, Mon-teith and Unsworth 1990). Values range from ;0.75(on perfectly clear skies on cold nights) to near unity(on very cloudy, warm days; Campbell 1977). Becausethe difference in temperature between the air and theanimal’s body is small relative to their values raised tothe fourth power, Eqs. 4a and 5a may be linearized witha Taylor series approximation:

3/4 3 1/4Q 5 4.0« « sA T (T 2 « T ) (4b)rad,sky org sky rad,sky a b sky a

3Q 5 4.0« sA T (T 2 T ). (5b)rad,ground org rad,ground g b g

(In Eq. 5b, «org 5 «surr 5 1.0). While this simplificationdoes introduce some error into the infrared radiationterms, under normal daylight conditions this effect isnegligible to the entire heat budget equation, and per-mits the formulation of a simple analytic solution tothe heat balance equation under steady-state condi-tions.

4. Conduction (Qcond)—The rate of heat conductionbetween a body and a surface in contact with the bodyis the product of the temperature differential betweenthe body and the surface, the conductivity of heatthrough the object, and the area over which the contactoccurs:

Qcond 5 k5 Acond(Tb 2 Tg) (6)

where Acond is the area of contact between mussel bodyand substrate, and k5 represents several lumped coef-ficients and is expanded and defined in the Appendix.

5. Convection (Qconv)—The flux of heat between thesurface of an animal or plant and the surrounding fluid(air) is affected by the pattern and rate of fluid move-ment around the body of the organism, the amount ofsurface area in contact with the fluid, and the gradient

in heat between the organism’s body and that of thesurrounding fluid:

Qconv 5 hc Aconv(Tb 2 Ta) (7)

where hc is the coefficient for forced convection, andAconv is total surface area exposed to convective heatloss (approximately equal to total surface area, A). Val-ues of hc are measures of the interaction between theorganism’s size and shape with the fluid moving aroundthe organism and must be extracted empirically, andaccount for any effects of organism or group mor-phology on local shear stress. This pattern of fluid mov-ing around a body is generally represented by the Reyn-olds number (Re), a measure of the relative importanceof inertial and viscous stresses in a moving fluid (Vogel1981, Denny 1988, 1993):

Re 5 UL9y21 (8)

where U 5 fluid velocity, L9 5 characteristic length ofthe organism, and y is the kinematic viscosity of thefluid. For irregularly shaped organisms such as mus-sels, L9 can be considered as an average dimension ofthe body, in this case the mean of the mussel’s length,width, and height. Re increases both with fluid velocityand organism size, and conservation of Re implies dy-namic similarity in the pattern of fluid movement (Vo-gel 1981). The characteristics of the local convectiveregime similarly affect the flux of heat away from thesurface, which is represented by the Nusselt number,Nu, a heat analog for the momentum-based Re:

21Nu 5 h L9k (9)C a

where ka 5 thermal conductivity of air. Nu is tradi-tionally presented as a power function of Re (Nu/Replots, Fig. 3) as a means of relating the effect of theconvective regime on heat vs. momentum flux. Valuesof the heat transfer coefficient (hc) are then extractedfrom Nu (methods described in Appendix).

6. Evaporative cooling (Qevap).—The rate of heatloss due to the evaporation of water at an organism’ssurface is the rate of mass (water) loss, , multiplied·mby the latent heat of vaporization of water (l):

Qevap 5 l .·m (10a)

The rate of mass flux is, in turn, affected by the char-acteristics of the local convective regime, much in thesame way that heat is convected away from a surface.Like convective heat exchange, the interaction of thelocal fluid flow with the size and morphology of theorganism is represented by a coefficient of mass trans-fer, hm, which like its heat analog, hc, is empiricallyderived. The flux of mass is furthermore determinedby the concentration gradient in the mass item, in thiscase water vapor, between the surface of the organismand the surrounding fluid, as well as the area over whichexchange occurs:

Qevap 5 lhmAevap (rn,body 2 rn,air) (10b)


FIG. 3. Nusselt/Reynolds (Nu/Re) plots forsolitary mussels in turbulent flow, posterior endupwind, and in growth position. Note that Reand hence Nu increase with both body size (L9)and wind speed (U), but that increases in heatloss rate (Nu) begin to level off at higher Re.

where Aevap 5 area of mass exchange, as determinedby mussel gape, and rn,body and rn,air are the vapor den-sities of the body and air, respectively, and are nonlin-ear functions of body temperature, air temperature, andrelative humidity (Campbell 1977). Under conditionsof excessively high relative humidity, the differencebetween rn,body and rn,air is small so that evaporativecooling cannot occur. Similarly, very low wind veloc-ities severely reduce the magnitude of the coefficientof mass exchange, hm, again limiting the rate of evap-orative water loss. Furthermore, bivalves can effec-tively prevent evaporative water loss by closing theirvalves (e.g., Shick et al. 1986, McMahon 1988, Dav-enport and Wong 1992). Because the degree of gape(Aevap) is under the behavioral control of the mussel andthus difficult to predict, for the purposes of the presentstudy the average rate of water loss, , was considered·mto be either zero (no gaping) or continuous and constant(the limits to which are set by desiccation stress); hm

was therefore not measured directly. However, valuesof hm are generally very similar to those of hc due tosimilarities in the diffusivities of heat and water vaporin air (e.g., Monteith and Unsworth 1990).

Unsteady heat balance equation.—Combining theexpanded (and in some cases simplified) form of eachterm in Eq. 1 we arrive at the full unsteady heat balancefor fluctuating environmental conditions:

d(mcT )/dtb

35 k A S 2 k A T (T 2 k T )1 sol 2 rad,sky a b 3 a

32 k A T (T 2 T ) 2 k A (T 2 T )4 rad,ground g b g 5 cond b g

·2 h A (T 2 T ) 2 lm (11)C conv b a

where the product mcTb is the total heat content of themussel (shell and body combined, Eq. 2b) and k1–5 arelumped physical constants and measured coefficients

(expanded in Appendix). Again, each component in themodel represents a flux of heat to or from the animal,and is determined by the magnitude of the driving‘‘force’’ (e.g., solar radiation or air temperature), thearea over which the exchange occurs, and some coef-ficient describing the interactive effects of the organismand the environment on the rate of exchange (e.g., hc,a measure of the interactive effects of body size, mor-phology, wind speed, and turbulence on convectiveheat exchange).

Steady-state heat balance model.—Under naturalfield conditions each of the above environmental pa-rameters can fluctuate over relatively short temporalscales. However, a simpler, less realistic but more trac-table version of the above full unsteady model is usefulfor examining the steady-state (equilibrium) body tem-perature of an organism exposed over long periods oftime to constant environmental conditions (Fig. 1;Monteith and Unsworth 1990). In this scenario, thebody temperature remains constant, and the change ofstored heat (Qstored) equals the body temperature(Tb, steady) times the average rate of mass (water) loss( ) times the mass specific heat of water (c):·m

·T mc 5 Q 6 Q 6 Q 6 Qb,steady sol rad,sky rad,ground cond

6 Q 2 Q . (12)conv evap

Rearranging to solve for the steady-state body tem-perature (Tb, steady, the equilibrium body temperature thata mussel achieves after long periods of time, Fig. 1)thus gives the following analytic solution to the heatbudget model:

4 4T 5 [k A S 1 k A k T 1 k A Tb,steady 1 sol 2 rad,sky 3 a 4 rad,ground g

·1 k A T 1 h A T 2 lm]5 cond g C conv a

3 33 [k A T 1 k A T2 rad,sky a 4 rad,ground g

· 211 k A 1 h A 2 mc] (13)5 cond C conv


where is a constant but constrained rate of mass loss·mfrom evaporation, and can be set to zero to representconditions when the mussel does not gape or otherwisecannot evaporatively cool.

Field trials

The accuracy of the heat budget model (eq. 11) wastested with live Mytilus californianus and M. trossulusin May–August 1995, both under semicontrolled con-ditions (flat pavement with natural sunlight and wind)and during low tide in the field at semi-exposed rockybenches on the northern coast of the Olympic Peninsula(Crescent Bay, 488109 N, 1238409 W) and at CattlePoint, San Juan Island, Washington (488359 N, 1238109W). Lowest summer tides in this region can occur nearmidday, and as such mussels are exposed to extremeaerial conditions for several hours. In addition, mod-erate exposure to wave action, especially during wintermonths, frequently creates gaps (5patches) in the mus-sel beds so that mussels can be found as solitary in-dividuals (e.g., Levin and Paine 1974, Paine and Levin1981, Sousa 1984). I monitored the body temperatureof solitary mussels of varying sizes in gaps, as well asenvironmental parameters of solar flux, local windspeed, air temperature, and ground temperature, overperiods of 30–70 min. Mussels were clamped shut withrubber bands to eliminate gaping (and hence to preventevaporative cooling), but were otherwise left in growthposition (dorsal side up). Similar methods were usedto track the body temperatures of mussels placed onconcrete pavement under natural conditions of light andwind speed for comparison to results from the moreheterogeneous intertidal.

Local wind speed was measured with a Kurz mini-anemometer by placing the probe ;15 cm upwind ofand at a height comparable to that of the mussel. Windvelocities immediately adjacent to the mussel were thusmonitored directly rather than extrapolating local ve-locities from free stream velocities and boundary layerprofiles. Solar flux density (S, Eq. 3) was measuredusing a LI-COR pyranometer (LI-COR, Lincoln, Ne-braska) facing directly away from the substratum. Airtemperature and ground temperature were measuredimmediately adjacent to the mussel with copper-con-stantan thermocouple wire, as was body temperature,in which the tip of the wire was inserted through theposterior end of the mussel into the middle of its body.Output from the thermocouples, wind meter, and pyr-anometer were channeled through a Campbell 21X datalogger (Campbell Scientific, Logan, Utah) into a laptopcomputer (Macintosh Powerbook 180) and recordedevery 5 s. Live input was monitored using a programdesigned in Labview (Version 3.0, 1993, National In-struments, Austin, Texas). In three instances occasionalspurious data points due to instrument misfiring (,105-s readings per trial) were replaced with interpolationsfrom the time interval immediately preceding the error.At the end of each trial mussels were kept for mor-

phometric measurements in the laboratory. Shell length(posterior/anterior axis), width (left/right axis), andheight (dorsal/ventral axis) were measured to the near-est 0.05 mm with calipers, and shell and tissue massmeasured to the nearest 0.01 g. Any water retained inthe mussel was considered as body tissue for purposesof heat content. Total surface area of each mussel wasmeasured by covering the surface of each valve withaluminum foil (with no overlap), which was thenweighed and compared to a regression of foil mass vs.surface area.

Environmental inputs from the field trials were usedto generate predictions of body temperature with a nu-merical (fourth-order Runge-Kutta) solution to Eq. 11.Results were then compared to body temperatures ac-tually measured in the field. Accuracy of the modelwas quantified by calculating the root mean square(rms) of the difference between the predictions and themeasured values (mean difference 5 rms[obs 2 pred]averaged over the entire run, as well as a similar rel-ative error term (relative error 5 rms[(obs 2 pred)/obs] to correct for the magnitude of the forcing func-tion, body temperature (Tb).

To generalize the model for a wide range of musselsizes, estimates of body and shell mass and total shellsurface area as a function of length were obtained frommeasurements of 25 Mytilus californianus, ranging inlength from 50 to 150 mm, collected from a semi-exposed location on the northern coast of the OlympicPeninsula, Washington (see Appendix). All of theseparameters are known to change significantly in re-sponse to the environment (e.g., Seed 1968), but arehere used to represent ‘‘average’’ mussels in the ex-posed rocky intertidal. Model predictions were againgenerated using environmental data collected in thefield but using allometry equations instead of measuredvalues for body and shell mass and surface area. Resultsof the simulations were again compared to measuredbody temperatures using the previously described errorterms.

Model simulations

Once the completeness of the model had been ver-ified using actual field data, general predictions of theheat budget model were applied to ‘‘average’’ solitarymussels using allometry equations under simulatedconditions of both a constant (steady state, Eq. 13, Fig.1) and an unsteady (Eq. 11) environment. Because theallometry equations were calculated from Mytilus cal-ifornianus only, these simulations therefore only applydirectly to this species. Steady-state simulations werecompleted at two extremes of evaporative cooling:none (mass loss rate, , 5 0), and at constant levels·mset by maximum limits to desiccation ( 5 constant·mand non zero), and were conducted using Mathematica(Version 2.2, 1993, Stephen Wolfram, Wolfram Re-search, Champaign, Illinois). I also completed a seriesof unsteady (fluctuating environment) simulations, in


FIG. 4. Results of model test under semicontrolled conditions. Environmental parameters and mussel body temperatureswere monitored over periods of 25–85 min both in the field and over homogeneous surfaces (pavement). Environmentalinputs were then used to generate predictions of body temperature using the unsteady heat balance equation. The meandeviation between predicted and measured body temperature, when compared over all runs, was ;18C. In the example shownhere, mean deviation was ;0.508C.

which all environmental parameters but one were heldconstant. The changing parameter, wind speed, was sin-usoidally varied at several frequencies. In this simu-lation wind speed was chosen as the fluctuating variablebecause it changes so rapidly in the field, but the sameapproach may be applied to any environmental param-eter or combinations of parameters, such as solar fluxdensity and wind speed on a day with patchy cloudcover. No evaporative cooling was permitted. In allcases the mean, range, and extrema in wind speed re-mained constant, but the cycle time of the signal (timefrom peak to peak) varied between simulations from afew minutes to several hours, a reasonable approxi-mation of the range of conditions on a gusty day in theintertidal. Results of these simulations were then com-pared to steady-state predictions using the mean andthe extremes of wind speed. Thus, for example, un-steady simulations in which wind cycled between 0.1and 3.1 m/s with varying frequencies were comparedto steady-state simulations that used wind speeds of0.1, 1.6 (mean wind speed), and 3.1 m/s. Finally, pa-rameter values obtained from artificial and natural mus-sel beds were used to quantify the effect of living inan aggregation on the steady-state body temperature of

Mytilus californianus as compared to solitary musselsliving in adjacent gaps.

RESULTS

Field tests of model accuracy

Tests of the unsteady version of the heat budget mod-el (fluctuating environmental conditions, Eq. 11) in-dicated a fairly accurate fit between predicted and ob-served body temperatures. On average, the mean de-viation between observed and predicted values waswithin ;18C of body temperatures measured in the field(e.g., Fig. 4, Table 1). Results based on allometry equa-tions of Mytilus californianus shells (Allometry col-umn, Table 1) showed a similar degree of accuracy toresults based on actual measurements of body and shellmass and surface area (Measured parameters column,Table 1) (Mean difference, Table 1). Relative error was;5% for predictions based both on measured morpho-metrics and allometry equations (Table 1). There wasno apparent effect of either body size or run length onmodel accuracy.

Steady-state model results

Model simulations in which environmental condi-tions are held constant (steady-state model) indicate a


TABLE 1. Results of trials comparing model predictions to body temperatures measured both in the field (Crescent Bay[Olympic Peninsula] and Cattle Point [San Juan Island], Washington State) and under seminatural conditions (concrete‘‘pavement’’ with natural light and wind) in the summer of 1995.

Date LocationBody L(mm)

Measured parameters

Mean diff. Rel. error

Allometry

Mean diff. Rel. error

18 May18 May18 May9 Aug9 Aug9 Aug9 Aug

17 May17 May17 May13 July13 Aug

PavementPavementPavementPavementPavementPavementPavementCattle PointCattle PointCattle PointOlympic Pen.Olympic Pen.

120108

8282696682

15055866855

1.310.901.070.541.210.741.361.110.980.460.172.35

0.0650.0410.0420.0150.0350.0210.0440.0590.0610.0270.0110.082

2.031.230.850.50

0.531.520.800.450.89

0.0990.0580.0350.015

0.0290.0950.0460.0200.031

Mean pavement 87 1.02 6 0.31 0.038 6 0.016 1.15 6 0.66 0.052 6 0.036Mean fieldAll combined

8385

1.01 6 0.841.02 6 0.55

0.048 6 0.0290.042 6 0.022

0.84 6 0.420.98 6 0.41

0.044 6 0.0300.048 6 0.031

Notes: Model comparisons were completed using both measured values of shell and body mass and shell surface area anddimensions (Measured parameters columns), as well as allometry equations (Allometry columns) of these parameters basedon Mytilus californianus collected in the field. Three M. trossulus used on 9 August were not included in comparisons usingallometry equations; all other trials used M. californianus. Model accuracy was quantified as the root mean square (rms) ofthe mean difference between observed (obs.) and predicted (model) values (Mean diff. 5 rms[obs. 2 model]) and as a relativeerror value (Rel. error 5 rms[(obs. 2 model)/obs.]).

FIG. 5. Interaction of wind speed and bodysize in determining the equilibrium (steadystate) body temperature of mussels in the ab-sence of gaping (i.e., no evaporative cooling).All other environmental parameters were heldconstant (Ta 5 258C, Tg 5 308C, solar flux 5500 W/m2, «sky 5 0.90). Mussel body size hasa significant effect on steady-state (equilibrium)body temperature (Tb, steady), primarily due to al-lometries in surface area and the Nu/Re rela-tionship (Fig. 3). Likewise, wind speed has anonlinear effect on convective cooling such thatsmall differences in wind speed at lower windvelocities (e.g., 0.5 vs. 1.0 m/s) have a largereffect on Tb, steady than similar changes at highervelocities (e.g., 2.5 vs. 3.0 m/s).

marked effect of body size on equilibrium body tem-perature, and suggest that during the day, evaporativewater loss, convective cooling from wind, and short-wave solar inputs represent the dominant sources andsinks of heat to the mussel’s thermal budget (in termsof percentage total heat flux). The interaction of bodysize and wind speed in determining Nu and hence con-vective exchange (Fig. 3) results in fairly large changesin body temperature (Tb) with small changes in windspeed, particularly at low velocities (Fig. 5). However,no one environmental parameter controls body tem-perature. For example, results of the model understeady-state conditions indicate that mussel body tem-perature increases with air temperature, but due to thenonlinear dependence of the long-wave radiative term(Eq. 4) on air temperature, the increase does not con-form to a simple 1:1 relationship (Fig. 6).

Simulations in which evaporative cooling is not per-mitted, as may occur in the presence of predators orexcessively high relative humidity, suggest that underidentical environmental conditions steady-state bodytemperature (Tb, steady, the equilibrium temperature thata mussel experiences under hypothetical, unchangingenvironmental conditions) increases with body size(Figs. 5 and 7a). For example, at a solar flux of 600W/m2 and a local wind speed of 1 m/s, the equilibriumbody temperature of a 5-cm mussel is ;298C, comparedto ;318C in a 10-cm mussel (Fig. 7a). Thus, in theabsence of gaping, for example, as occurs in some trop-ical oyster species (Davenport and Wong 1992), largermussels are predicted to achieve higher equilibriumbody temperatures when compared to smaller individ-uals exposed to the same constant environmental con-ditions (Figs. 5 and 7a). Significantly, in the absence


FIG. 6. Effect of air temperature (Ta) onsteady-state body temperature (Tb, steady) in theabsence of evaporative cooling. Values are re-ported as the change in body temperature (Tb)vs. change in Ta relative to initial conditions ofTa 5 158C and Tb 5 228C for a 5-cm musseland Tb 5 238C for a 10-cm mussel. Other en-vironmental parameters are fixed as indicated inthe figure. Under this set of conditions, the rateof change of body temperature is lower than thatof air temperature (compare to 1:1 line 5 pro-portional response). Thus, under this set of con-ditions a 18C increase in average air temperaturecan result in a less than 18C change in musselbody temperature.

→

FIG. 7. Body temperature as a function of wind speed and solar flux for mussels of three sizes and three rates of evaporativecooling. (a) When the mussel does not gape, or cannot evaporatively cool due to extremely high relative humidity or extremelylow wind speed, exposure time is not limited by desiccation, but body temperatures are much higher than those in whichmussels are able to lose water through evaporation. Under these conditions larger mussels achieve higher body temperatures.(b, c) When mussels do gape, the limit to exposure can be set either by extreme body temperatures or by death fromdesiccation, which is predicted to occur when the animal has lost an amount of water equivalent to ;37% of its body mass.

of evaporative cooling, during most daytime conditions(i.e., when the long-wave radiative component of theheat budget model is small) the lower range of bodytemperature is effectively set by the temperature of theair, that is, convective heat exchange by wind can onlyserve to bring the temperature of the mussel closer tothat of the air. However, this lower limit is unlikely tooccur at night when long-wave (infrared) radiation isimportant, as is evident from the formation of frost atnight even when air temperature is above freezing.Thus, only in cases of very low sun coupled with highwind (that is, only when the dominant mode of heatexchange is convection) is air temperature an even re-motely reasonable estimate of the body temperature ofintertidal mussels.

The trend in equilibrium body temperature with bodysize reverses when evaporation is permitted (Figs. 7b,c). Indeed, a trade-off exists between the coolingachieved through evaporative water loss and the phys-iological costs associated with desiccation, as has beenshown for intertidal algae (e.g., Bell 1995). Thus, deathfrom desiccation sets a limit to the amount of waterthat can be made available for evaporative cooling, andis under some measure of control by the mussel viathe amount of gape between the valves (5Aevap, Eq.10). Evaporative water loss may also be limited by therelative humidity of the air, which determines the con-centration gradient of the solute (water vapor) to beconvected, as well as by the local wind speed and themussel’s body size, both of which drive the coefficientof mass transfer, hm (Eq. 10b). Previous studies withthe mussel Modiolus sp. have shown that this bivalve

can tolerate a water loss of 36–38% of its total bodymass before death occurs (Lent 1968). Although hm wasnot measured directly in this study, values are typicallyconsistent with those of the coefficient of heat transfer,hc, due to similarities in the diffusivities of heat andwater vapor in air (Monteith and Unsworth 1990). Runsof the steady-state model using values of hc as a proxyfor hm showed that mussels can readily lose 37% oftheir body mass over a period of 2–3 h of exposurewith only a small amount of gape (#0.5% of totalsurface area), even under conditions of high relativehumidity (#95%) and low wind velocities (#0.5 m/s).Thus, it appears reasonable that under average envi-ronmental conditions the amount of water lost throughevaporation can be controlled by the mussel via theamount of gape between its valves. Over any givenexposure time, the average maximum rate of water loss(37% of animal mass/exposure time) thus sets the upperlimit to the average rate of evaporative cooling that themussel can achieve before death due to desiccationoccurs. As a result, the equilibrium body temperatureof a mussel undergoing a range of evaporative coolingrates may be calculated using Eq. 13. Fig. 7 displaysthe equilibrium body temperature of mussels of threesizes undergoing evaporative cooling at three rates: (a)with no evaporation (not limited by desiccation), (b)when the maximum permissible amount of water is lostover 6 h (i.e., a fairly low rate of evaporative waterloss), and (c) when the same amount of water is evap-orated over 3 h (i.e., rate of water loss twice that of[b]). When evaporative cooling is permitted, largermussels can maintain significantly lower body tem-


When this amount of water is lost over a period of (b) 6 h, the lowered average rate of water loss leads to higher bodytemperatures than exposure times of (c) 3 h (i.e., same amount of water lost over a longer period of time). Here air andground temperatures are fixed at 258C and 308C, respectively, and «sky 5 0.90 (;50% cloud cover). In the presence ofevaporative cooling (b, c), and in contrast to scenarios where evaporation does not occur (a), larger mussels are able toremain at lower body temperatures due to a disproportionately greater amount of water available for evaporative cooling.


FIG. 8. Time constant of heating as a function of bodysize for solitary mussels. This time constant (approximatelytwo-thirds of the total time to equilibrium under steady-stateconditions) increases from a few minutes for a 5-cm musselto tens of minutes for larger mussels, particularly under con-ditions of low wind. Larger mussels thus require a longertime to achieve steady-state body temperature, although thetemperature to which they heat can in some situations behigher.

peratures than smaller individuals due to dispropor-tionately greater amounts of tissue relative to bodylength and hence more water available for evaporativecooling (Fig. 7b, c). Not surprisingly, mussels thatevaporatively lose all of their available water overshorter periods (higher rates of mass loss and henceevaporative cooling) remain cooler than mussels thatlose the same amount of water over a longer period oftime (Fig. 7b, c). Thus, while the behavioral determi-nants of mussel gape and subsequent evaporative cool-ing are uncertain, larger mussels could potentially farebetter than small mussels due to a disproportionatelygreater water reservoir and thus potentially faster ratesof evaporative cooling under identical exposure times.Alternatively, these mussels could withstand longer ex-posure times while undergoing the same rate of evap-orative water loss as smaller mussels.

In the steady-state simulations just described (Figs.5 and 7) environmental parameters of air and groundtemperature were held constant, but in reality these maybe thought of as additional dimensions of the two-di-mensional plots presented above (Fig. 7). For example,Fig. 7 represents extremes in a range of water loss ratesbetween none and that permitted before death fromdesiccation occurs. Intermediate to these two extremesis a continuous range of water loss rates that stronglyaffect body temperature, and are functions of the degreeof mussel gape, relative humidity, and wind speed. Un-der conditions that limit evaporative cooling, smallermussels may be at a thermal advantage. At the oppositeend of this spectrum where the rate of evaporation islimited only by the degree of gape, and is thus undercontrol of the mussel, larger mussels are expected toachieve lower body temperatures than smaller individ-uals. Thus, the addition of a third (water loss), fourth(air temperature), or even fifth (ground temperature)axis to the two-dimensional plots presented here wouldproduce a multidimensional ‘‘climate space’’ (Porterand Gates 1969) in which body temperature is deter-mined by unique sets of each of these parameters.

Fluctuating (unsteady) environmental conditions andthe effect of environmental time history

The above steady-state models predict that largermussels experience higher temperatures in the absenceof evaporation. However, larger mussels also requireconsiderably longer times to achieve equilibrium bodytemperature (Fig. 8), which may have significant ef-fects in a fluctuating environment. The time course ofheating under steady-state conditions is generally quan-tified as a time constant, a measure roughly equivalentto two-thirds of the time required to reach equilibriumbody temperature (shown in Fig. 1). Because of thedependence of several components of the heat budgeton body size, the time constant contains both morpho-logical and environmental parameters. While smallermussels require only a few minutes to achieve equi-librium, larger mussels do not do so for tens of minutes

(Fig. 8). Thus, although larger mussels may ultimatelyheat to higher average temperatures (in a constant en-vironment), their greater mass results in a greater ‘‘ther-mal inertia,’’ which may to some degree buffer themagainst rapid environmental changes during more re-alistic nonequilibrium dynamics in nature. Conversely,however, this result also means that larger mussels willcool more slowly after reaching high body tempera-tures. In other words, the dependence of the time con-stant of heating on body size suggests that the timehistory of the environment may be very important (e.g.,Buatois and Croze 1978). Simulations in which windspeed fluctuated (Fig. 9) as a simple sine wave sug-gested a marked effect of the time history of the en-vironment on the body temperatures of mussels, andemphasize the potential inadequacy of using steady-state approaches to predicting body temperature max-ima in a fluctuating environment. In this scenario,steady-state predictions of average body temperature(using mean wind speed) were similar (within 0.58C)to those predicted by the fluctuating, unsteady model(Fig. 9). In contrast, however, steady-state predictionsof maximum body temperatures (based on minimumwind speeds, i.e., extreme conditions) overestimatedmaximum body temperature predicted by the unsteadymodel by as much as 68C (Fig. 9). Thus, predictionsof extremes (and perhaps even averages) in body tem-perature require measures of the time history of theenvironment.

Mussels in aggregations

As previously mentioned, many of the parameters inthe above heat balance equations are altered by the


FIG. 9. Predicted average (mean Tb) and maximum body temperatures (max. Tb) experienced by solitary mussels of twobody sizes (L9 5 5, 10 cm) when exposed to simple fluctuating environmental conditions of varying cycle time. In thissimulation, mussels were exposed to constant solar irradiance, air temperature, and ground temperature, but wind speedcycled sinusoidally between 0.1 and 3.1 m/s. No evaporative cooling was permitted. In all cases the mean, range, and extremain wind speed remained constant, but the cycle time of the signal (time from peak to peak) varied. Mussels in rapidlyfluctuating wind (20-min cycle time) experienced slightly lower mean and significantly lower maximum body temperaturesthan those exposed to conditions where wind changed more slowly (120-min cycle time). In this simple, sinusoidally varyingexample mean body temperatures are similar to those predicted by the equilibrium (steady-state) model using mean windspeed (indicated by arrows). In contrast, steady-state predictions of maximum body temperatures based on the lowest windspeed (extreme conditions) greatly overestimated (.68C) the maximum temperatures predicted for a fluctuating environmentusing an unsteady heat balance equation.

presence of neighboring organisms. Living in an ag-gregation, for example, decreases the projected area ofthe mussel subject to solar radiation by up to 40%, butalso substantially reduces the rate of convective heatexchange with the environment (Appendix). Thus,there is an apparent trade-off in that, while mussels ina bed receive less energy from solar radiation, theyalso are less able to convectively lose (or gain) heat tothe surrounding air. Steady-state simulations of an ‘‘av-erage’’ (7.5 cm) bed mussel under conditions of noevaporative cooling show that, relative to an identicalmussel inhabiting an adjacent gap, mussels living in

beds can experience markedly lower body tempera-tures, particularly when exposed to high rates of solarflux (Fig. 10). Under conditions of low solar flux, thatis, when the relative advantage of presenting a smallarea to the sun is small, solitary mussels are slightly(,0.58C) cooler than mussels in aggregations (Fig.10b). In contrast, under extreme conditions of highsolar flux and low wind, mussels in aggregations canbe .4.08C cooler than solitary mussels exposed toidentical physical conditions. Furthermore, mussels inaggregations display a higher time constant of changethan do solitary mussels (Fig. 11), suggesting that they


FIG. 10. (a) Steady-state temperature of an ‘‘average’’7.5-cm mussel living within a bed, and (b) a comparison ofequilibrium body temperatures of the same mussel in a gapvs. in a bed. At low solar flux, solitary mussels are slightlycooler than bed mussels. At extreme conditions (high solarflux), however, solitary mussels are several degrees hotterthan mussels in aggregations exposed to the same environ-mental conditions.

FIG. 11. Time constant of heating for three intermediate-sized (L 5 7.5 cm) mussels, two living as solitary individualsin a gap, but with different orientations to the wind, and thethird living in an aggregation (bed). Under identical envi-ronmental conditions, mussels living in an aggregation havean effectively much larger time constant, which in turn buffersthem against rapid changes in environmental conditions.

are less susceptible to rapid changes in the environ-ment. These results, therefore, strongly suggest a pos-itive thermal interaction between conspecific Mytilusspp. living in beds, both in terms of lower equilibriumbody temperature and a greater ability to buffer bodytemperature against rapid spikes in environmental con-ditions.

DISCUSSION

The physical ecology of Mytilus spp.

Body temperatures and desiccation rates can signif-icantly affect the physiology and survival of intertidalinvertebrates. Although previous studies have shownthat the body temperature of mussels can deviate sig-nificantly from air or ground temperature (e.g., South-ward 1958, Carefoot 1977, Elvin and Gonor 1979, Tsu-chiya 1983), the models presented here represent aquantitative, first-principles approach to predicting thebody temperature of intertidal mussels in the field.Clearly, a number of environmental parameters deter-mine the daytime body temperatures of mussels, mostimportantly (in terms of percentage total heat flux) so-lar radiation, wind speed, and air temperature. Again,however, the relative importance of each factor willchange during nighttime conditions, when infrared ra-diation is likely to become much more important. Bodysize plays a significant role in the heat balance, as doesthe presence of neighboring conspecifics. Fig. 7 maythus be thought of as two-dimensional ‘‘slices’’ of aclimate space (Porter and Gates 1969) of survival andperformance, the upper limits to which are set by bothtolerance to desiccation and high body temperature aswell as by sublethal effects of these two factors.

For example, Tsuchiya (1983) found that body tem-peratures of $408C led to the mortality of over 50%of the individuals in an intertidal population of Mytilusedulis within 1 h of exposure. Model results suggestthat body temperatures of this magnitude are likely tooccur only when both solar flux density and air tem-perature are unusually high, and generally only whenevaporative cooling is severely limited. Using a steady-state approximation of body temperature, Fig. 12 showsthe maximum solar flux density (S, Eq. 3) that a 10-


FIG. 12. Maximum solar flux density (S) that can be tol-erated by a 10-cm solitary mussel before death due to heatstress occurs (here assumed to occur at a body temperatureof 408C). Ground temperature is fixed at 308C, wind speedis 1 m/s, and air temperature varies from 258 to 358C. Underconditions where the evaporative loss of water by gaping isnot limited by environmental conditions, limits to evaporationare set by the maximum amount of water that can be lostbefore death due to desiccation occurs (37% of total bodymass, here lost at a constant rate over a 6-h exposure), anddeath due to excessive body temperature is unlikely. Underconditions where evaporative cooling is prohibited, criticaltemperatures are achieved at more realistic lower solar irra-diances. While mussels are unlikely to experience these ex-tremes under normal maximal summer conditions in thenortheastern Pacific (S # 1000 W/m2, 238–258C air temper-ature), records of air temperatures in excess of 348C havebeen reported for this region historically.

cm solitary mussel can tolerate before its body tem-perature reaches a critical value of 408C, both underconditions of maximum evaporative cooling (the limitsto which are set by desiccation) and of no evaporativewater loss. The amount of solar energy that can betolerated declines with increasing air temperature, andis higher for mussels that can evaporatively cool. Forexample, a mussel that is permitted to gape in an en-vironment with an air temperature of 308C would haveto experience a solar flux of .1400 W/m2 to achievea body temperature of 408C (Fig. 12). The same musselunder identical physical conditions but in the absenceof evaporative cooling (which can occur when relativehumidity approaches 100% or the action of predatorsprevents the mussel from gaping) would only requirean exposure to a solar flux of 1050 W/m2, a high butnot unrealistic value (Fig. 12; Monteith and Unsworth1990). On the northern coast of the Olympic Peninsula(Washington State) where this study was conducted,average maximum values of solar flux density seldomexceed 1000 W/m2 (Fig. 12). Solar data for this andmany other regions of the United States are availableonline from the National Solar Radiation Data Base[NSRDB].2 While air temperatures of up to 358C have

2 URL 5 http://rredc.nrel.gov

occurred historically, average maximum summer airtemperatures in this region are generally on the orderof 238–258C (30-yr average, NOAA Western RegionalClimate Center Data).3 Death due to such extreme bodytemperatures (408C) is therefore likely to be a rare butnot impossible occurrence in the Northeastern Pacific,and mussels in beds are predicted to be even less sus-ceptible than solitary mussels given their lower equi-librium body temperatures (Fig. 10) and greater timeconstants (Fig. 11).

More difficult to predict are the potentially additiveeffects of exposure to sublethal temperatures and des-iccation rates for longer periods of time, which, whilenot as catastrophic as lethal body temperatures, none-theless strongly affect an invertebrate’s fitness. For ex-ample, Hofmann and Somero (1995, 1996) found thatcellular damage from excessive heat commonly occursin intertidal mussels. Bayne et al. (1976) showed thatthe physiological performance (‘‘scope for growth’’)of Mytilus californianus is maximized at a body tem-perature (Tb) of 178–228C, and declines markedly at Tb

. 268C. Interestingly, these optimum temperatures arewell above those of seawater during most of the yearin the northeastern Pacific, suggesting that mussels inthese regions may actually benefit from periodic ex-posures at low tide. Under the steady-state conditionsof the model shown in Fig. 7 (air temperature of 258C),a 10-cm mussel undergoing a maximum average rateof water loss (as set by desiccation) over an exposureperiod of 6 h (Fig. 7b) is expected to maintain its bodytemperature within this range at a solar radiance of;300 W/m2 or less, a fairly low value likely to occuronly on cloudy days or early in the morning in tem-perate latitudes (this study, Monteith and Unsworth1990, NSRDB data). In contrast, if the same musselevaporatively loses this maximum amount of waterover a period of 3 h (Fig. 7c), optimal conditions maybe achieved under solar irradiances as high as 600–700 W/m2, a common range of values for midday inlate spring and early summer in the northeastern Pacific(NSRDB data).

Similarly, mussel physiological performance andgrowth are affected not only by absolute body tem-perature, but also by fluctuations in body temperature(e.g., Widdows 1976). For example, Almada-Villela etal. (1982) found that mussels exposed to fluctuatingbody temperatures (cycled between 38 and 208C) ex-hibited faster rates of shell growth than mussels ex-posed to constant temperatures at either extreme. Theability of mussels to maintain their body temperatureswithin some optimal range or even to fluctuate at someoptimal rate varies not only with environmental con-ditions but also with limits to desiccation set by ex-posure time, wind speed, relative humidity, and otherfactors affecting the degree of mussel gape (e.g., aerialrespiration: Bayne et al. 1976, Shick et al. 1986, Mc-

3 URL 5 http://wrcc.sage.dri.edu


FIG. 13. Limits to evaporative cooling set by relative hu-midity, air temperature, and body temperature. The verticalaxis, Ta 2 Tb, represents the maximum number of degreesbelow air temperature (dew point) that a mussel can achievebefore evaporative cooling becomes completely ineffective.

Mahon 1988), and has important consequences to mus-sel fitness that await further exploration.

Clearly, evaporative cooling can significantly affectthe heat budget of an intertidal mussel. In the absenceof gaping, the lower body temperature under daytimeconditions is effectively set by air temperature (Figs.5 and 7a). In contrast, a mussel that cools throughevaporative water loss can achieve a temperature sig-nificantly lower than that of the surrounding air (Fig.7b, c). However, there are limits to the cooling that canbe maintained through evaporation. Specifically, thedew point temperature of the environment, which isdetermined by air temperature and relative humidity,sets the lower limit to which evaporation can cool amussel (Fig. 13; Campbell 1977). The dew point isdefined as the temperature to which air, at any givenrelative humidity, must be cooled in order to be com-pletely saturated with water vapor (Monteith and Un-sworth 1990). Because warm air can hold more watervapor, air parcels with high relative humidity requirehigher temperatures to maintain saturation and thushave a higher dew point. The exchange of mass (watervapor) between the surface of the mussel and the airis in turn determined to a large extent by the differencein the concentration of water vapor between the surfaceof the mussel and the air (Eq. 10b). When surfaces incontact with the air cool to temperatures equivalent tothat of the dew point of the air, water vapor on thesurface condenses such that mass continues to be lostbut the surface experiences no further cooling (Camp-bell 1977, Monteith and Unsworth 1990). When thesurface temperature (i.e., the body temperature of themussel) reaches this critical limit, usually a few degreesbelow air temperature depending on the relative hu-midity, no further cooling from evaporation is possible.Thus, although mussels may evaporatively lose heatquite readily when body temperature exceeds air tem-

perature, evaporative cooling may be severely curtailedwhen the temperature of the body falls more than afew degrees below that of the air (Fig. 13), even whenlarge amounts of water are still available for cooling.For example, while a mussel exposed to 90% relativehumidity and 258C air temperature can continue toevaporatively cool until a body temperature of only238–248C is reached, the same mussel exposed to 70%relative humidity can continue to cool via evaporativewater loss at body temperatures above 198–208C (Fig.13). Thus, while in theory large mussels undergoinghigh rates of mass loss and exposed to low solar ir-radiances and wind speeds (Fig. 7c; 10-cm mussel) canexperience equilibrium body temperatures 108C ormore below air temperature, this can only be achievedunder rather low values of relative humidity, whichmay not be realistic under normal intertidal conditions.

The relative importance of thermal fluxes to an in-tertidal invertebrate’s physiology and ecological inter-actions is almost certain to be highly variable. Themarine intertidal is both spatially and temporally het-erogeneous (e.g., Menge 1976, Menge and Olson1990), and it is likely that the body temperatures oforganisms may change over relatively small scales (Su-chanek 1985). Environmental conditions vary with lat-itude, and the subsequent effects on body temperatureplay a role in determining species distribution patternsand dominance hierarchies (e.g., Hofmann and Somero1996). Perhaps even more importantly, and in contrastto studies of terrestrial systems, the thermal biology ofintertidal organisms is set not only by conditions ofweather but also by tidal rhythms. Thus, organismsinhabiting littoral zones that are typically exposed dur-ing early morning or late afternoon rather than nearmidday may be less susceptible to thermal damage thanthose living in locations with less extreme environ-mental conditions but less desirable tidal cycles. Forexample, tidal cycles in the Strait of Juan de Fuca(northern coast of the Olympic Peninsula, WashingtonState) generally result in low tides, which occur duringmorning and afternoon hours on the exposed outercoast. As one moves eastward into Puget Sound, how-ever, low tides during summer months occur closer tosolar noon, when environmental conditions are themost hazardous. Thus, at any given tidal height, inter-tidal organisms living closer to Puget Sound not onlyexperience cumulatively longer aerial exposures duringmidday hours, but also are more commonly exposedduring times when solar radiative flux is greatest (Fig.14). Similarly, organisms inhabiting lower zones of theintertidal are exposed to a terrestrial environment lessfrequently than are those higher up (Fig. 14). However,it should be noted that zonation patterns can also shiftmoving into the sound. For example, mussels subjectto wave splash on Tatoosh Island can inhabit MLLW(mean lower low water) 13m (Suchanek 1978, Leighet al. 1987), much higher than mussels in the Sound.

It must be strongly emphasized that while the results


FIG. 14. Estimated cumulative aerial exposure time (inhours) at different times of day during July 1996. Exposuretimes are divided by time from solar noon (here, ;1210 localtime): within 1 h, 1–2 h, 2–3 h, and 3–4 h before or aftersolar noon. Total exposure time within 4 h of solar noon wasalso calculated. Determining the body temperature of inter-tidal organisms requires estimates not only of local physicalconditions but also of time and duration of aerial exposure.Estimates of aerial exposure were made for three sites andthree tidal heights (cm above or below Mean Lower LowWater [MLLW] along the northern coast of the Olympic Pen-insula (WA) by modeling tidal height as a simple cosine curvewith varying amplitude. When compared to mussels from theeastern section of the peninsula (Port Townsend, at the mouthof Puget Sound) living at the same tidal height, mussels onthe exposed outer coast of the peninsula (Tatoosh Island) arenot only less aerially exposed during these times, but are alsoless exposed during hours closest to solar noon. For any giventidal height, the risk of death and sublethal effects due toexcessive body temperatures would thus be expected to behigher in Puget Sound, which may account for observed pat-terns in the distributions of Mytilus californianus (outer coast)and M. trossulus (Puget Sound). Also note that small differ-ences in tidal height can lead to large changes in cumulativeaerial exposure time.

of the steady-state model (constant environmental con-ditions) represented in Figs. 5, 7, 10, and 12 do providea means of assessing the relative impacts of environ-mental parameters, body size, and aggregation, theyneglect the mussel’s ‘‘thermal inertia,’’ and thus com-pletely ignore the time-history dependence of the sys-tem (e.g., Figs. 8, 9, 11). Significantly, the results ofthe unsteady model in which the environment is per-mitted to fluctuate (Fig. 9) suggest that approachesbased solely on simple statistical measures of the en-vironment such as daily means, ranges, or even ex-tremes are highly unlikely to accurately predict thebody temperature of mussels over short time scales.These estimates may, however, prove more useful whenexamining patterns over longer periods of time. Likeother sessile invertebrates (e.g., Thomas 1987), musselbody temperatures are a product of several environ-mental inputs, as well as the rates at which these inputsfluctuate. Accurate predictions of mussel body tem-perature must therefore be based on multiple environ-mental parameters, and whenever possible should in-corporate measurements of the physical environmentcollected over short time scales.

Applications to global climate change

Mechanistic approaches such as the one describedhere may prove promising for predicting the effects offorecasted changes in local and global climate. Un-derstanding and predicting the ecological consequencesof environmental change necessarily demand an ex-plicit understanding of the interaction between thosephysical processes and their effect on the organisms inquestion. While some organisms may exist very neartheir limits of tolerance to abiotic factors, others mayperform well over a range of environmental conditionsand may be less affected by environmental change. Tothis end, the literature is replete with studies examiningthe survival and physiological performance of individ-ual organisms experimentally subjected to a range oftemperatures. Several workers have suggested individ-ual-based models to predict the effects of global changeon biotic communities, in which the changing char-acteristics of the physical environment affect the bodytemperatures and subsequently the performance andsurvival of the animals within that community. Thedynamics of the community or population may then beextrapolated given differential susceptibility to changesin local or global climate (e.g., Dunham 1993, Murdoch1993). However, the accurate prediction of intertidalorganismal body temperatures demands a quantitative,mechanistic approach to determining the relative im-portance of environmental parameters such as solarflux, wind speed, and air temperature, all of which arepredicted to change globally in coming decades (Asrarand Dozier 1994). As has been shown here, estimatesof the effects of global climate change that are basedon any one parameter alone (such as air temperature)are very unlikely to prove accurate for larger inver-


tebrates such as mussels, and thus require examinationof the full heat budget model including all environ-mental parameters. For example, a 38C increase in airtemperature (e.g., Schneider 1993), all other factorsremaining constant, can produce a less than 38C in-crease in equilibrium body temperature (Fig. 6) due tothe nonlinear nature of the complete heat balance (andspecifically the radiative terms). However, changes inone environmental parameter, such as air temperature,are likely to be linked to others such as solar flux (S).Thus, for example, (under the same set of conditionspresented in Fig. 6) a 38C increase in average air tem-perature coupled with a 20% increase in solar flux doeslead to an increase of 38C in average body temperature.

Limitations to the model

Individual-based approaches such as the methods de-scribed here are powerful in that in a true reductionistmanner they are ultimately based upon well-formulat-ed, first-principles models of heat exchange (e.g.,Kreith 1976). Given sufficient information about en-vironmental parameters, initial conditions, and accu-rate measurements of an organism’s physical charac-teristics, it is thus possible to not only accurately pre-dict the effect of changes in any one factor, but moreimportantly to mechanistically gain an understandingof the means by which organisms interact with theirphysical environment. As with any modeling approach,the methods presented here are limited by several as-sumptions that must be taken into consideration whenextrapolating to actual conditions in the field, or whenexpanding the model to other applications. For ex-ample, during daytime conditions my accounting ofshort-wave solar radiation (Eq. 3b) focuses on directsolar flux at the expense of diffuse and radiative (al-bedo) solar flux components. While this assumption iscertainly valid when examining the thermal effects ofextreme environmental conditions, as is the focus here,under conditions of heavy cloud cover, when diffusesolar flux dominates, the model is less apt to accuratelypredict the effects of total solar flux. Models of bodytemperature under these conditions should therefore in-corporate the full short-wave solar flux term (Eq. 3a).

While I have shown that it is possible to scale upfrom individual mussels to aggregations and popula-tions, confidence in the ability of the model to accountfor all physical factors is necessarily reduced. For ex-ample, as has been shown for aggregations of organ-isms ranging in size from forests of trees (Raupach andThom 1981) to aggregations of scleractinian corals(Shashar et al. 1996, Helmuth et al. 1997) and subtidalmussels (Frechette et al. 1989), the spatial arrangementof organisms can significantly modify the character-istics of the boundary layer overlying their surfaces.In this study, the empirically derived coefficient of heattransfer (hc) effectively lumps the effects of organismsize, morphology, and even the effects of neighboringconspecifics on the local flow regime and hence the

rate of convective heat transfer. When considering in-dividual organisms or even small aggregations of or-ganisms, it thus serves as an effective metric of con-vection. However, any larger scale, community-wideeffects of organisms on the local convective regimeand patterns of shear stress are not considered. Whilethe structure of the boundary layer in the intertidal ismost likely influenced more by the heterogeneity of thesubstrate, large aggregations of mussels could poten-tially affect the local patterns of shear stress and windvelocity relative to organisms living in adjacent gaps.For example, Frechette et al. (1989) have shown thatthe roughness of mussel beds can enhance the verticalflux of particles from the water column overlying thebed, and analogous processes could potentially occurfor convective heat exchange during aerial exposure.Similarly, the local convective regime may differ be-tween the edge and the center of a large bed. Whilethe coefficient of heat transfer reported here does ef-fectively account for the roughness effects of smallaggregations on convection in turbulent air flow,boundary layers overlying extremely large mussel bedscould potentially modify local flow velocity and shearstress immediately above the bed. Extensions of themodel to larger scale regional processes of boundarylayer formation in the intertidal (and in particular, ameans for adjusting the local wind velocity and heattransfer coefficient terms in the equation for convectiveheat exchange, Eq. 7) thus await further study.

Feedback loops between biotic and abiotic factors?

Despite the limitations of an individual-based ap-proach, the above methods, when combined with phys-iological studies of organism performance, provide afirst step to predicting the influence of the physicalenvironment on the intertidal, latitudinal, and habitat-dependent distribution of a wide range of intertidalinvertebrates, and suggest the existence of feedbackloops between abiotic and biotic factors in the marineintertidal. The concept that sessile organisms, and inparticular aggregations, modify the characteristics ofthe physical environment is not new. For example,Frechette et al. (1989) have shown that the physicalstructure of Mytilus beds can significantly affect theflux of phytoplankton to the mussels. Similarly, thespacing between the ramets of colonial cnidarians (e.g.,McFadden 1986, Lesser et al. 1994, Helmuth et al.1997, Sebens et al. 1997) and bryozoans (e.g., Okamura1984, 1988) can drive rates of feeding and gas ex-change, as can the size and morphology of the indi-vidual units within the aggregation (e.g., Thomas1994). Despite the results of these predominantly sub-tidal studies, however, current ecological paradigms ofintertidal communities generally consider the aerial en-vironment as an uncontrollable, albeit fluctuating,backdrop against which biotic interactions controlcommunity structure, and abiotic factors limit the sur-vival, performance, and interactions of the players.


Most studies of the marine intertidal thus operate underthe premise that, while the dynamics of the physicalenvironment during low tide may ultimately drive thedynamics of the biotic community, seldom do the an-imals themselves affect the characteristics of the aerialphysical environment (but see Hay 1981 and Bell 1992[algae], Thomas 1987 and Bertness 1989 [barnacles]).At best, organisms large enough to affect their localphysical environment are considered as spatially het-erogeneous habitat (similar to forests of trees) withinwhich other organisms interact. The approach takenwith many intertidal animal communities thus variesmarkedly from those applied to many plant commu-nities (e.g., Nobel 1983) where the physical structuresof forest and crop systems have been shown to affectenvironmental parameters such as wind speed, solarradiative flux and relative humidity. The results of thisstudy indicate that similar mechanisms may also occurin communities of sessile animals during aerial expo-sure, where shading by neighbors and effect on theconvective regime result in a significantly different mi-croclimate for mussels living in beds as compared tothose living in gaps.

Clearly, mussel aggregation behavior has significant,positive effects on the thermal dynamics of musselbody temperature, although other negative effects ofaggregation have been shown to occur subtidally (e.g.,lowered food availability and growth rates: Okamura1986, Frechette et al. 1989). Mussels in beds not onlyexperience a lowered body temperature during ex-tremes in environmental conditions (Fig. 10), but alsodisplay a much larger time constant of change due tothe lowered rate of convective exchange (Fig. 11). Thisgreater ‘‘thermal inertia’’ in turn almost certainly buf-fers these animals against rapid spikes in environmentalparameters, just as body size dampens the thermal re-sponse of large solitary mussels (Figs. 8 and 9). Thisenhanced capacity to buffer body temperature againstrapid fluctuations in the environment is moreover likelyto be enhanced by several factors not included in thismodel. For example, the ground temperature beneathmussel beds is almost always several degrees coolerthan the surface of rocks in adjacent gaps (B. S. T.Helmuth, unpublished data). Furthermore, many bedsretain large amounts of water as the tide goes out. Thisreservoir probably serves as a source for evaporativecooling, and perhaps even more importantly providesa huge sink for heat conducted from the mussels sittingwithin it. Both of these factors would not only increasethe mussels’ time constant of change but also lowertheir equilibrium body temperature (Fig. 10). Finally,mussels living in beds may directly buffer one anotheragainst extremes in temperature. Although direct con-tact along the curved surfaces of mussels is rathersmall, mussels reradiate heat (long-wave radiation) toone another within the matrix of the bed. As a result,differences in body temperature between, for example,small and large mussels, may be reduced via this mech-

anism. Thus, not only does group living lower the bodytemperature of the aggregation’s constituents (Fig. 10),but it is also possible that heat stress is to some extent‘‘shared’’ between its members. This interaction is un-fortunately difficult to model using the individual-based approach presented here, but may play a signif-icant role in the heat budgets of mussel aggregations.

Similarly, differences in mussel spatial arrangementcould potentially drive the body temperature of smallpredators (e.g., the gastropod Nucella) feeding on mus-sels within a bed. For example, living in an aggregationreduces the amount of heat that is lost through con-vection (Appendix), and a gastropod feeding on mus-sels within a bed would similarly experience a loweredrate of convective heat exchange. For the mussel, thisdetriment is offset by a reduced rate of solar heat fluxdue to mutual shading by the mussels. In contrast, whenNucella feed on the thin (upright) posterior margin ofthe mussel shell, they are perched on top of the bed,where they are subject to full solar radiation. Similarly,while larger body size can afford some protectionagainst rapid fluctuations in the physical environment(thermal inertia), smaller predators would be expectedto be much less buffered against these rapid changes.Under identical ‘‘extrinsic’’ physical conditions, there-fore, predator and prey in close proximity could po-tentially experience very different body temperaturesand desiccation rates from one another due to differ-ences in body size, exposure to solar irradiance, andlocation within the boundary layer overlying the bed.As has been suggested previously (Connell 1961, Menge1978), predators may be more restricted in their dis-tribution and foraging ability by the physical environ-ment than are their prey. The results of this study sug-gest the possibility that this restriction may in part befurther affected by the behavior and spatial arrange-ment of the prey. Thus, in marked contrast to mostcurrent ecological approaches, it is likely that manysessile animals may to some extent actually controltheir physical environment, as well as that of their com-petitors and predators. In other words, the local phys-ical environment (microclimate) may not merely pro-vide a backdrop in which organisms interact, but in-stead may itself be dynamically controlled by the or-ganisms through their morphology, behavior, andspatial arrangement. The utilization of a microclimateapproach to community dynamics thus indicates thepossibility of feedback loops among organisms and thelocal physical environments of themselves, their pred-ators and their competitors, and suggests that ‘‘abioticfactors’’ are not independent of the biological inter-actions that they themselves modify.

ACKNOWLEDGMENTS

Financial support for this research was provided in part byNSF Mathematical Biology Training Grant BIR9256532 toT. Daniel and G. Odell, and by a grant from the AmericanMuseum of Natural History Lerner-Gray Fund for MarineResearch to the author. B. Timmerman-Helmuth was indis-


pensable for her help in the field, as was T. Daniel for hisconstant support and advice. Helpful comments on the re-search and manuscript were provided by E. C. Bell, S. Baird-Daniel, T. Daniel, M. Frye, A. Kohn, G. Odell, J. Kingsolver,B. Menge, R. T. Paine, K. P. Sebens, J. Sherman, L. Stockwell,B. Timmerman-Helmuth, A. Trimble, members of the Uni-versity of Washington Mathematical Biology Training Grantprogram, and two anonymous reviewers. This research wassubmitted in partial fulfillment of the requirements for a Ph.D.in the Department of Zoology, University of Washington.

LITERATURE CITED

Almada-Villela, P. C., J. Davenport, and L. D. Gruffydd.1982. The effects of temperature on the shell growth ofyoung Mytilus edulis L. Journal of Experimental MarineBiology and Ecology 59:275–288.

Asrar, G., and J. Dozier. 1994. EOS: science strategy for theearth observing system. AIP Press (American Institute ofPhysics), New York, New York, USA.

Bayne, B. L., C. J. Bayne, T. J. Carefoot, and R. J. Thompson.1976. The physiological ecology of Mytilus californianusConrad. Oecologia 22:211–228.

Bell, E. C. 1992. Consequences of morphological variationin an intertidal macroalga: physical constraints on growthand survival of Mastocarpus papillatus Kutzing. Disser-tation. Stanford University, Stanford, California, USA.

. 1995. Environmental and morphological influenceson thallus temperature and desiccation of the intertidal algaMastocarpus papillatus Kutzing. Journal of ExperimentalMarine Biology and Ecology 191:29–55.

Bertness, M. D. 1989. Intraspecific competition and facili-tation in a northern acorn barnacle population. Ecology 70:257–268.

Branch, G. M., P. Borchers, C. R. Brown, and D. Donnelly.1988. Temperature and food as factors influencing oxygenconsumption of intertidal organisms, particularly limpets.American Zoologist 28:137–146.

Buatois, A., and J. P. Croze. 1978. Thermal responses of aninsect subjected to temperature variations. Journal of Ther-mal Biology 3:51–56.

Campbell, G. S. 1977. An introduction to environmentalbiophysics. Springer-Verlag, New York, New York, USA.

Carefoot, T. 1977. Pacific seashores: a guide to intertidalecology. University of Washington Press, Seattle, Wash-ington, USA.

Connell, J. H. 1961. The influence of interspecific compe-tition and other factors on the distribution of the barnacleChthamalus stellatus. Ecology 42:710–723.

Davenport, J., and T. M. Wong. 1992. Effects of temperatureand aerial exposure on three tropical oyster species, Cras-sostrea belcheri, Crassostrea iradelei and Saccostrea cu-cullata. Journal of Thermal Biology 17:135–139.

Denny, M. W. 1988. Biology and the mechanics of the wave-swept environment. Princeton University Press, Princeton,New Jersey, USA.

. 1993. Air and water: the biology and physics oflife’s media. Princeton University Press, Princeton, NewJersey, USA.

. 1995. Predicting physical disturbance: mechanisticapproaches to the study of survivorship on wave-sweptshores. Ecological Monographs 65:371–418.

Denny, M. W., T. L. Daniel, and M. A. R. Koehl. 1985.Mechanical limits to size in wave-swept organisms. Eco-logical Monographs 55:69–102.

Dunham, A. E. 1993. Population responses to environmentalchange: operative environments, physiologically structuredmodels, and population dynamics. Pages 95–119 in P. M.Kareiva, J. G. Kingsolver, and R. B. Huey, editors. Bioticinteractions and global change. Sinauer, Sunderland, Mas-sachusetts, USA.

Elvin, D. W., and J. J. Gonor. 1979. The thermal regime of

an intertidal Mytilus californianus Conrad population onthe central Oregon coast. Journal of Experimental MarineBiology and Ecology 39:265–279.

Frechette, M., C. A. Butman, and W. R. Geyer 1989. Theimportance of boundary-layer flows in supplying phyto-plankton to the benthic suspension feeder, Mytilus edulisL. Limnology and Oceanography 34:19–36.

Fyhn, H. J., J. A. Petersen, and K. Johansen. 1972. Eco-physiological studies of an intertidal crustacean, Pollicipespolymerus (Cirripedia, Lepadomorpha). I. Tolerance tobody temperature change, desiccation and osmotic stress.Journal of Experimental Biology 57:83–102.

Gaylord, B., C. A. Blanchette, and M. W. Denny. 1994. Me-chanical consequences of size in wave-swept algae. Eco-logical Monographs 64:287–313.

Gosling, E. M. 1992. Systematics and geographic distribu-tion of Mytilus. Pages 1–20 in E. M. Gosling, editor. Themussel Mytilus: ecology, physiology, genetics and culture.Elsevier Science, Amsterdam, The Netherlands.

Gosselin, L. A., and F. S. Chia. 1995. Characterizing tem-perate rocky shores from the perspective of an early ju-venile snail: the main threats to survival of newly hatchedNucella emarginata. Marine Biology 122:625–635.

Grant, B. W., and W. P. Porter. 1992. Modeling global mac-roclimatic constraints on ectotherm energy budgets. Amer-ican Zoologist 32:154–178.

Hay, M. E. 1981. The functional morphology of turf-formingseaweeds: persistence in stressful marine habitats. Ecology62:739–750.

Helmuth, B. S. T., K. P. Sebens, and T. L. Daniel. 1997.Morphological variation in coral aggregations: branchspacing and mass flux to coral tissues. Journal of Experi-mental Marine Biology and Ecology 209:233–259.

Hofmann, G. E., and G. N. Somero. 1995. Evidence forprotein damage at environmental temperature: seasonalchanges in levels of ubiquitin conjugates and hsp70 in theintertidal mussel Mytilus trossulus. Journal of ExperimentalBiology 198:1509–1518.

Hofmann, G. E., and G. N. Somero. 1996. Interspecific vari-ation in thermal denaturation of proteins in the congenericmussels in Mytilus trossulus and M. galloprovincialis: ev-idence from the heat-shock response and protein ubiqui-tination. Marine Biology 126:65–75.

Huey, R. B. 1991. Physiological consequences of habitatselection. American Naturalist 137:S91–S115.

Huey, R. B., C. R. Peterson, S. J. Arnold, and W. P. Porter.1989. Hot rocks and not-so-hot rocks: retreat-site selectionby garter snakes and its thermal consequences. Ecology 70:931–944.

Kareiva, P. M., J. G. Kingsolver, and R. B. Huey. 1993. Bioticinteractions and global change. Sinauer, Sunderland, Mas-sachusetts, USA.

Kingsolver, J. G. 1979. Thermal and hydric aspects of en-vironmental heterogeneity in the pitcher plant mosquito.Ecological Monographs 49:357–376.

. 1983a. Thermoregulation and flight in Colias but-terflies: elevational patterns and mechanistic limitations.Ecology 64:534–545.

. 1983b. Ecological significance of flight activity inColias butterflies: implications for reproductive strategyand population structure. Ecology 64:546–551.

. 1989. Weather and the population dynamics of in-sects: integrating physiological and population ecology.Physiological Zoology 62:314–334.

Kingsolver, J. G., and W. B. Watt. 1983. Thermoregulatorystrategies in Colias butterflies: thermal stress and the limitsto adaptation in temporally varying environments. Amer-ican Naturalist 121:32–55.

Kreith, F. 1976. Principles of heat transfer. Third edition.


Intext Educational Publishers (IEP), New York, New York,USA.

Leigh, E. G., R. T. Paine, J. F. Quinn, and T. H. Suchanek.1987. Wave energy and intertidal productivity. Proceedingsof the National Academy of Sciences (USA) 84:1314–1318.

Lent, C. M. 1968. Air-gaping by the ribbed mussel, Modiolusdemissus (Dillwyn): effects and adaptive significance. Bi-ological Bulletin 134:60–73.

Lesser, M. P., V. M. Weis, M. R. Patterson, and P. L. Jokiel.1994. Effects of morphology and water motion on carbondelivery and productivity in the reef coral, Pocilloporadamicornis (Linnaeus): diffusion barriers, inorganic carbonlimitation, and biochemical plasticity. Journal of Experi-mental Marine Biology and Ecology 178:153–179.

Levin, S. A., and R. T. Paine. 1974. Disturbance, patch for-mation and community structure. Proceedings of the Na-tional Academy of Sciences (USA) 71:2744–2747.

Liu, J. H., and B. Morton. 1994. The temperature tolerancesof Tetraclita squamosa (Crustacea: Cirripedia) and Septifervirgatus (Bivalvia: Mytilidae) on a sub-tropical rocky shorein Hong Kong. Journal of Zoology, London 234:325–339.

McDonald, J. H., R. Seed, and R. K. Koehn. 1991. Allo-zymes and morphometric characters of three species of My-tilus in the Northern and Southern Hemispheres. MarineBiology 111:323–333.

McFadden, C. S. 1986. Colony fission increases particle cap-ture rates of a soft coral: advantages of being a small col-ony. Journal of Experimental Marine Biology and Ecology103:1–20.

McMahon, R. F. 1988. Respiratory response to periodicemergence in intertidal molluscs. American Zoologist 28:97–114.

Menge, B. A. 1976. Organization of the New England rockyintertidal community: role of predation, competition, andenvironmental heterogeneity. Ecological Monographs 46:355–393.

. 1978. Predation intensity in a rocky intertidal com-munity. Oecologia 34:1–16.

Menge, B. A., and A. M. Olson. 1990. Role of scale andenvironmental factors in regulation of community struc-ture. Trends in Ecology and Evolution 5:52–57.

Monteith, J. L., and M. H. Unsworth. 1990. Principles ofenvironmental physics, Second edition. Edward Arnold,London, UK.

Murdoch, W. W. 1993. Individual-based models for pre-dicting the effects of global change. Pages 147–162 in P.M. Kareiva, J. G. Kingsolver, and R. B. Huey, editors.Biotic interactions and global change. Sinauer, Sunderland,Massachusetts, USA.

Nobel, P. S. 1983. Biophysical plant physiology and ecology.W. H. Freeman, New York, New York, USA.

Okamura, B. 1984. The effects of ambient flow velocity,colony size, and upstream colonies on the feeding successof bryozoa. I. Bugula stolonifera Ryland, an arborescentspecies. Journal of Experimental Marine Biology and Ecol-ogy 83:179–193.

. 1986. Group living and the effects of spatial positionin aggregations of Mytilus edulis. Oecologia (Berlin) 69:341–347.

. 1988. The influence of neighbors on the feeding ofan epifaunal bryozoan. Journal of Experimental Marine Bi-ology and Ecology 120:105–123.

Paine, R. T. 1966. Food web complexity and species diver-sity. American Naturalist 100:65–75.

. 1974. Intertidal community structure: experimentalstudies on the relationship between a dominant competitorand its principal predator. Oecologia 15:93–120.

. 1984. Ecological determinism in the competition forspace. Ecology 65:1339–1348.

Paine, R. T., and S. A. Levin. 1981. Intertidal landscapes:disturbance and the dynamics of pattern. Ecological Mono-graphs 51:145–178.

Peters, R. H. 1983. The ecological implications of body size.Cambridge University Press, Cambridge, UK.

Porter, W. P., and D. M. Gates. 1969. Thermodynamic equi-libria of animals with environment. Ecological Monographs39:245–270.

Porter, W. P., J. W. Mitchell, W. A. Beckman, and C. B.DeWitt. 1973. Behavioral implications of mechanisticecology: thermal and behavioral modeling of desert ecto-therms and their microenvironment. Oecologia 13:1–54.

Raupach, M. R., and A. S. Thom. 1981. Turbulence in andabove plant canopies. Annual Review of Fluid Mechanics13:97–129.

Riechert, S. E., and C. R. Tracy. 1975. Thermal balance andprey availability: bases for a model relating web-site char-acteristics to spider reproductive success. Ecology 56:265–284.

Schneider, S. S. 1993. Scenarios of global warming. Pages9–23 in P. M. Kareiva, J. G. Kingsolver, and R. B. Huey,editors. Biotic interactions and global change. Sinauer, Sun-derland, Massachusetts, USA.

Schoener, T. W. 1986. Mechanistic approaches to communityecology: a new reductionism? American Zoologist 26:81–106.

Sebens, K. P. 1982. The limits to indeterminate growth: anoptimal size model applied to passive suspension feeders.Ecology 63:209–222.

Sebens, K. P., J. Witting, and B. Helmuth. 1997. Effects ofwater flow and aggregation on particle capture by the reefcoral Madracis mirabilis. Journal of Experimental MarineBiology and Ecology 211:1–28.

Seed, R. 1968. Factors influencing shell shape in the musselMytilus edulis. Journal of the Marine Biological Associa-tion of the United Kingdom 48:561–584.

Seed, R., and T. H. Suchanek. 1992. Population and com-munity ecology of Mytilus. Pages 87–169 in E. M. Gosling,editor. The mussel Mytilus: ecology, physiology, geneticsand culture. Elsevier Science, Amsterdam, The Nether-lands.

Shashar, N., S. Kinane, P. L. Jokiel, and M. R. Patterson.1996. Hydromechanical boundary layers over a coral reef.Journal of Experimental Marine Biology and Ecology 199:17–28.

Shick, J. M., E. Gnaiger, J. Widdows, B. L. Bayne, and A.de Zwaan. 1986. Activity and metabolism in the musselMytilus edulis L. during intertidal hypoxia and aerobic re-covery. Physiological Zoology 59:627–642.

Sousa, W. P. 1984. Disturbance and patch dynamics on rockyintertidal shores. Pages 101–125 in S. T. A. Pickett and P.S. White, editors. Disturbance and the patch dynamics per-spective. Academic Press, New York, New York, USA.

Southward, A. J. 1958. Notes on the temperature toleranceof some intertidal animals in relation to environmental tem-peratures and geographic distribution. Journal of the Ma-rine Biological Association of the United Kingdom 37:49–66.

Stephens, E. G., and M. D. Bertness. 1991. Mussel facili-tation of barnacle survival in a sheltered bay habitat. Jour-nal of Experimental Marine Biology and Ecology 145:33–48.

Suchanek, T. H. 1978. The ecology of Mytilus edulis L. inexposed rocky intertidal communities. Journal of Experi-mental Marine Biology and Ecology 31:105–120.

. 1981. The role of disturbance in the evolution oflife history strategies in the intertidal mussels Mytilus ed-ulis and Mytilus californianus. Oecologia 50:143–152.

. 1995. Mussels and their role in littoral and sublit-toral community dynamics. Pages 70–96 in P. G. Moore


and R. Seed, editors. The ecology of rocky coasts. Colum-bia University Press, New York, New York, USA.

Thomas, F. I. M. 1987. The hot and cold of life on the rocks:determinants of body temperature of the northern rock bar-nacle Semibalanus balanoides. Thesis. Brown University,Providence, Rhode Island, USA.

. 1994. Morphology and orientation of tube exten-sions on aggregations of the polychaete annelid Phrag-matopoma californica. Marine Biology 119:525–534.

Touloukian, Y. S., and E. H. Buyco. 1970. Thermophysicalproperties of matter. Volume 5. Specific heat: nonmetallicsolids. IFI/Plenum, New York, New York, USA.

Tsuchiya, M. 1983. Mass mortality in a population of themussel Mytilus edulis L. caused by high temperature onrocky shores. Journal of Experimental Marine Biology andEcology 66:101–111.

Vogel, S. 1981. Life in moving fluids. Princeton UniversityPress, Princeton, New Jersey, USA.

Wethey, D. S. 1984. Sun and shade mediate competition inthe barnacles Chthamalus and Semibalanus: a field exper-iment. Biological Bulletin 167:176–185.

Widdows, J. 1976. Physiological adaptation of Mytilus edulisto cyclic temperatures. Journal of Comparative Physiology,Series B 105:115–128.

Wilbur, A. E., and T. J. Hilbish. 1989. Physiological ener-getics of the ribbed mussel Geukensia demissa (Dillwyn)in response to increased temperature. Journal of Experi-mental Marine Biology and Ecology 131:161–170.

Williams, G. A., and D. Morritt. 1995. Habitat partitioningand thermal tolerance in a tropical limpet, Cellana grata.Marine Ecology Progress Series 124:89–103.

Witman, J. D., and T. H. Suchanek. 1984. Mussels in flow:drag and dislodgement by epizoans. Marine Ecology Prog-ress Series 16:259–268.

APPENDIX

PARAMETER MEASUREMENTS

The model presented in this study may be easily adaptedto studies of other intertidal invertebrates, but in many casesparameter values used here are specific to mussels. The fol-lowing are descriptions of the parameter measurements fromseveral components of the model, and the methods by whichthey were collected. A complete list of symbols and valuesof constants used in the heat budget model is presented inTable A1.

Stored heat

The specific heat of mussel tissue (cbody) was considered toapproximate that of water (4180 J·kg21·K21). The precise spe-cific heat of bivalve shell (cshell) has not been reported, butwas calculated as 815 J·kg21·K21, the average of values re-ported for calcite (Touloukian and Buyco 1970). This as-sumption was verified by measuring the convective loss ofheat from small pieces of fresh shell with known values ofhc (plates) under conditions of uniform flow and then back-calculating values of specific heat (see Convection sectionbelow). Results converged on values of 700–900 J·kg21·K21.

Solar heat flux

Although values of a (short-wave solar absorptivity) couldpotentially change with shell characteristics (such as erodedperiostracum or heavy diatom cover) a value of 0.75 was usedfor the purposes of the model, an intermediate estimate ofmany other biological structures (Campbell 1977, Monteithand Unsworth 1990). Asol (projected surface area exposed toshort-wave solar radiation) for use in field trials was calcu-lated as the projected area of a prolate spheroid, where shelllength (L) is the major axis and shell width and height arethe minor axes. Values of Asol for both solitary mussels (dorsalside up) and mussels in aggregations (posterior end up) wereestimated by measuring projected surface areas of actual mus-sel shells at a range of solar angles. For the purposes of thesteady-state models and for descriptions of ‘‘generic’’ mus-sels I used values of Asol 5 0.25A for solitary mussels andAsol 5 0.15A for mussels in aggregations. These parameterestimates were verified with measurements taken from mus-sels in the field. Values of S in the model thus reflect the totalflux density of a beam traveling normal to Asol. Because solarflux was measured with a flat (2p) pyranometer, measuredvalues of S were divided by sin (u) to yield the equivalentsolar energy intercepted by projected surface area Asol. Thesolar elevation angle u is a function of latitude, time of year,and time of day, and was derived from a mathematical fit ofdata given in Campbell (1977).

ConductionIf the average body temperature is considered to be that of

the center of mass of the mussel (probably a valid assumptiongiven the active and passive circulation of fluids within themussel), the distance over which this temperature gradientoccurs is that between the center of the mussel and the ground,approximately one-half its shell height:

Qcond 5 kb Acond(Tb 2 Tg)(0.5 H)21 (A.1)

where kb is the thermal conductivity of heat in body, Acond isthe area of contact between mussel body and substrate, andH is the maximum height of the mussel (dorsal–ventral axis).

Because most of the body of a mussel is composed of water,the thermal conductivity of heat in the body, kb, was consid-ered to be equal to that of water, 0.60 W·m21·K21 (Denny1993). Empirical tests were conducted, which indicated thatconductivity from the body through the shell is very rapidand may thus be neglected as a significant barrier to con-duction. Furthermore, the conduction of heat between solitaryepifaunal mussels and the ground is, under most circum-stances, unlikely to be very important given the small surfacearea of contact. The ventral surface of a solitary mussel,including the byssal notch through which byssus threads at-tach to the ground, is in general relatively small in comparisonto the total surface area of the mussel, and the posterior endof a mussel in a bed even smaller. For the purposes of thisstudy, therefore, Acond was approximated as 5% of the mussel’stotal surface area.

It should be noted, however, that this assumption is almostcertainly violated when the mussel is infaunal (e.g., Stephensand Bertness 1991), or is positioned within a depression orany other area that contains water during low tide. In such acase, the contact area of the mussel with the substrate or watercan be considerably larger, which in turn would be expectedto have significant effects on the mussel’s heat budget: insituations where the volume of retained water is large, andthus maintains a large ‘‘thermal inertia,’’ the pool would beexpected to serve as a significant sink of heat from the mussel,particularly when water evaporates from the surface of theshell. In contrast, when the pool of water is warmer than Tb,it would serve as a major source of heat. Due to the widerange of variations in water volume and the geomorphologyof depressions, these cases will not be considered here, butshould be kept in mind as potentially important modifiers ofan animal’s heat budget.

Convection and the heat transfer coefficientIn order to extract values of the convective heat transfer

coefficient (hc; Eqs. 7 and 9), heat loss via convection was


TABLE A1. Parameters used in the heat balance equations, including values (of constants), their units, and equations (inthe text and Appendix) where they are used. Units include Joules (J), Watts (W 5 1.0 J/s), and Kelvin (K 5 273 1 8C).

Parameter Description Value UnitsEqua-tions Source†

AAcond

Aconv

Ad

Aevap

Arad, ground

total mussel shell surface areaarea of contact between mussel and groundsurface area exposed to convective heat losssurface area exposed to diffuse and albedo solar fluxarea over which evaporation occurssurface area subject to long-wave radiation

from ground

0.05AA0.5A

0.5A

m2

m2

m2

m2

m2

m2

A667, 1.13105

112

1Arad, sky

Asol

cbody

cshell

Hhc

hm

ka

kb

surface area subject to long-wave radiation from skyprojected area in direction of sunspecific heat of mussel bodyspecific heat of mussel shellmussel height (dorsal/ventral axis)proportionality constant for forced convectionmass transfer coefficientthermal conductivity of airthermal conductivity of heat in body

0.5A

4180815

0.0260.60

m2

m2

J·kg21·K21

J·kg21·K21

mW·m22·K21

m/sW·m22·K21

W·m22·K21

4322A17, A2109A1

1123

4555

LL9·mmbody

mshell

NuQstored

Qcond

Qconv

Qevap

Qrad, sky

mussel length (anterior/posterior axis)average mussel dimension (mean of L, H, and width)mass flux ratebody massshell massNusselt numberchange in total heat within musselheat conducted between mussel and groundheat convected between mussel and airheat lost through evaporation of waterheat exchanged via long-wave radiation

between mussel and sky

mmkg/skgkg

J/sJ/sJ/sJ/sJ/s

A4–A68, 9102, A52, A4926, A17104

4

Qrad, ground heat exchanged via long-wave radiationbetween mussel and ground

J/s 5

Qsol

ReSSd

tTa

heat received from short-wave solar irradianceReynolds numbershort-wave direct solar radiative heat flux densitysum of diffuse and albedo short-wave flux densitiestimeair temperature

J/s

W/m2

W/m2

sK

383311, A34, 7

6

2, 7

Tb

Tb, steady

Tg

U

body temperaturesteady-state body temperatureground temperatureair velocity

KKKm/s

2, 4–7, 11135, 6, A18

aeorg

esky

esurr

solar absorptivity of mussel shellinfrared emissivity of organismfunctional infrared emissivity of skyinfrared emissivity of surroundings (ground)

0.751.00.70–0.991.0

34, 54, 54, 5

2222

lyrv, air

rv, body

su

latent heat of vaporization of waterkinematic viscosity of airvapor density of airvapor density at surface of the mussel tissueStephan-Boltzman constantsolar elevation angle

2.4816 3 1026

5.67 3 1028

J/kgm2/skg/m3

kg/m3

W·m22·K24

radians

10810104, 53

265521, 2

† 1 This study; 2 Campbell 1977; 3 Touloukian and Buyco 1970; 4 Kreith 1976; 5 Denny 1993; 6 Vogel 1981; 7 Monteith andUnsworth 1990.

measured independently of all other heat fluxes. Empty My-tilus californianus shells were sealed with epoxy and thenfilled with warm water (;108 above air temperature) andplaced in a unidirectional wind tunnel with turbulent flow.Mussels were supported above the bottom boundary layer onan airfoil-shaped support to reduce boundary layer effects.Temperature was monitored with a thermocouple placed inthe center of the mussel and the end sealed with clay toeliminate any evaporative cooling. Wind speed was measuredusing a Kurz mini-anemometer immediately upstream of themussel. Both the thermocouple and the wind meter sampledat a rate of 1 Hz and were connected to a Campbell 21X datalogger, which was in turn connected to a Macintosh computer.Outputs were monitored using a program designed in Lab-

view. Mussel shell lengths ranged from 60 to 100 mm, andwind speeds ranging from 0.1 to 3.5 m/s were used to yielda range of Reynolds numbers (Re). Mussels were allowed tocool convectively until body temperature was within 28–38Cof air temperature in order to generate cooling curves. Be-cause mussels were also losing heat radiatively (Qrad, sky; Eq.4), heat loss via this mechanism was subtracted from theexponentially declining cooling curve. The resulting coolingcurve therefore represented heat loss via convection alone;given values for all other parameters, values of hc could thusbe extracted. Cooling curves were fit to both a one-lump (shelland body mass weight-averaged) and a two-lump model (shelland body considered separately), as described in Kreith(1976). Conduction between the body and the shell was found


to be sufficiently rapid that no difference was apparent be-tween the two approaches; the simpler one-lump model wasthus used to calculate hc:

hc 5 [slope mc 2 Qrad, sky]Aconv21 (A.2)

where the quantity mc is the weighted average of the shelland body heat content (Eq. 2b). ‘‘Slope’’ is the slope of thenatural-log transform of the exponentially declining coolingcurve:

21slope 5 ln[(T (t) 2 T )/(T 2 T )]t (A.3)b a b a0

where Tb(t) is body temperature at time t, Ta is air temperature,and is initial body temperature (at t 5 0). In this exper-Tb0

iment both the area of convective heat exchange, Aconv, andthe area of radiative exchange, Arad, sky were considered toequal the total surface area of the mussel, A.

Values of Nu as a function of Re were calculated for singlemussels in growth position (dorsal side up) oriented eitherwith the anterior/posterior axis oriented parallel to the windor the left/right valve (side) facing into the wind. Nu (andthus convective heat exchange) was found to vary markedlywith orientation, and as expected, conformed well to a powerfunction of Re (Fig. 3). Nu/Re relationships varied slightlybetween mussels, possibly reflecting morphological differ-ences, although no consistent trends were evident. Fits of thegrouped data (Levenberg-Marquardt curve-fitting algorithm,Kaleidagraph) yielded relationships of Nu 5 0.38 Re0.51 formussels with the anterior or posterior end facing upwind andNu 5 0.63 Re0.47 for mussels with the valve facing upwind.For the purposes of the model and field tests mussels wereassumed to be in the former orientation, although in realityorientation to the wind obviously varies both spatially andtemporally.

Similar methods were used to calculate the rate of con-vective heat exchange from mussels living within beds. Asmall aggregation (20 3 20 cm) of mussels ranging in sizefrom 5 to 10 cm was constructed for use in the wind tunnel.Mussels were glued in approximate growth position (posteriorend attached to substrate) on a plexiglass plate, which wasin turn mounted on airfoil-shaped supports to reduce any walleffects in the wind tunnel. Heat flux was measured from thesame mussel shells used above, which were placed in thecenter of the artificial aggregation. Despite the tight packing

of the bed, direct contact between the experimental shell andother mussels was minimal. Compared to results from solitarymussels, bed mussels experienced a significant decrease inthe rate of convection: Nu 5 0.67Re0.42. Here, as in mea-surements of solitary mussels, the area subject to convection(Aconv) was considered as the entire surface area of the mussel,A. While in reality Aconv is smaller than the entire area of theshell A, the issue here is the product of the area of convectionAconv times the coefficient of heat transfer, hc. Because valuesof hc are empirically derived, the area used for Aconv is irrel-evant as long as values of hc are adjusted accordingly andthe product of these two variables remains constant.

Lumped coefficients

For simplicity many of the above parameters and constants(as defined in Table A1) were combined for use in the heatbalance equations in the text. Expanded values of these co-efficients and the equations from which they are first derivedare as follows:

k 5 a /sin(u) (Eq. 3)1

3/4k 5 4.0s« « (Eq. 4b)2 org sky

1/4k 5 « (Eq. 4b)3 sky

k 5 4.0s« (Eq. 5b)4 org

21k 5 k (0.5H) . (Eq. A.1)5 b

Mussel allometries

Both shell and body (including enclosed water) mass in-creased allometrically as functions of length (mass in kilo-grams and length in meters):

2.92m 5 57.6L (A.4)shell

3.53m 5 191L (A.5)body

(R 5 0.95 and 0.99, respectively). Total surface area (A) wasdescribed by a second-order polynomial:

2A 5 1.08L 1 0.0461L 2 0.0016 (A.6)

(R 5 0.99). Average body dimensions (for hC calculations;Eqs. 7, 9, and A.2) was considered to equal 2/3L (i.e., wherewidth 5 height 5 1/2L).

intertidal mussel microclimates: predicting the …california 93950-3094 usa. are aerially exposed...

Documents