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Thermal biology of rocky intertidal mussels: quantifying body temperatures using climatological data.

INTRODUCTION

The role of the physical environment in determining animal body temperatures and its subsequent effects on their physiological performance has received considerable attention in terrestrial habitats (e.g., Porter and Gates 1969, Porter et al. 1973, Riechert and Tracy 1975), and these studies have led to a heightened awareness of the importance of considering an organism's physiology when examining population- and community-level interactions (Schoener 1986, Kingsolver 1989, Huey 1991). Using thermal engineering (heat balance) approaches, studies of terrestrial organisms such as vertebrates (Porter and Gates 1969, Porter et al. 1973, Grant and Porter 1992) and insects (Kingsolver 1979, 1983a, b) have shown that accurate predictions of body temperatures can be generated when given sufficient information about small-scale environmental conditions (i.e., the organism's microclimate). Such approaches have more recently been applied in the rocky intertidal (Johnson 1975, Thomas 1987, Bell 1992, 1995, Helmuth 1997, 1998), a habitat where heterogeneity over a wide range of spatial and temporal scales (Menge 1976, Menge and Olson 1990, Berlow 1997) has in part contributed to its tractability as a model system for examining general ecological principles (Paine 1994). However, these methods have seldom been extended to population-level processes in this habitat (but see Bell 1992).

Intertidal invertebrates are marine ectotherms that must regularly contend with a terrestrial environment, and as such provide a unique perspective on examining the effects of fluctuating temperatures on organismal physiology and ecology (e.g., Hofmann and Somero 1995, 1996, Stillman and Somero 1996, Roberts et al. 1997). During daytime exposure at low tide, the temperature of an invertebrate's body can rapidly rise from that of the ambient seawater to well above air temperature, an increase of 15 [degrees] C or more on even moderately warm days (Fig 1; Southward 1958, Carefoot 1977). Following re-immersion, the temperature of the animal's body is rapidly reset to that of the surrounding water, but only a few hours later can decrease to several degrees below air temperature during aerial exposure on clear nights. Not surprisingly, these rapid changes in body temperature can profoundly affect an organism's physiology, reproductive output and survival (Bayne et al. 1976a, Widdows 1976, Johnson and Shick 1977, Suchanek 1978, Almada-Villela et al. 1982, Tsuchiya 1983, Wethey 1984, Branch et al. 1988, Bertness 1989, Seed and Suchanek 1992, Liu and Morton 1994, Hofmann and Somero 1995, Stillman and Somero 1996, Roberts et al. 1997), which can in turn have cascading effects on the structure, diversity, and productivity of the intertidal ecosystem (Suchanek 1978, Leigh et al. 1987, Seed and Suchanek 1992). In fact, "physical factors" (temperature and desiccation) are generally considered to be one of the most important factors in setting the upper distributional limits of species within the intertidal community (Doty 1946, Carefoot 1977, Levinton 1982, Swinbanks 1982). However, despite literally decades of research demonstrating the effects of body temperature on organismal performance and survival, we know very little about what constitutes a "typical" intertidal invertebrate body temperature during aerial exposure in the field (but see specific examples in Davies 1970, Elvin and Gonor 1979, Liu and Morton 1994, Stillman and Somero 1996, Roberts et al. 1997), or of how body temperatures change temporally and spatially. Subsequently, the frequencies with which intertidal organisms experience physiologically optimal, stressful, and lethal body temperatures remain virtually unexplored. This lack of information not only limits our ability to address physiological questions within an ecologically relevant context, but also reduces our ability to address the potential roles of physical factors and physiological performance in driving the ecology of intertidal communities. Methods for predicting the temperatures of sessile invertebrates can therefore serve as useful tools for investigating their physiology and ecology, and perhaps more importantly provide a link between studies conducted at these two scales. In this paper I use thermal engineering approaches to predict the body temperature of a common intertidal invertebrate, the mussel Mytilus californianus, at a site in the northeastern Pacific (Crescent Bay, Olympic Peninsula, Washington State, USA). This approach is then employed to explicitly and quantitatively dissect the role of the tidal cycle in determining the timing of aerial exposure and hence body temperatures.

During low tide, the body temperatures of intertidal organisms are driven by the interactions of solar radiation, cloud cover, wind speed, and air and ground temperatures. Given accurate measurements of these parameters and estimates of the organism's mass and size, body temperatures can be predicted to within a few degrees (Bell 1992, Helmuth 1998; [ILLUSTRATION FOR FIGURE 1 OMITTED]). Predicting body temperatures without continuous measurements of the microclimate, however, can be more problematic. Thus, although deterministic heat budget models based on continuous microclimatological records (i.e., conditions immediately adjacent to an organism) are useful for examining the consequences of organismal characteristics such as size, shape, and spatial position (e.g., Bell 1992, 1995, Helmuth 1997, 1998), a more generalized approach is necessary to estimate body temperatures over longer temporal and larger spatial scales. For example, Bell (1992) extrapolated from climate data recorded at a small weather station to the microclimate of a population of intertidal algae located within several hundred meters of the weather station. However, while such larger scale approaches have been explored to varying extents using terrestrial plants and animals (e.g., Kingsolver 1979) and marine algae (Bell 1992), these methods have not been previously applied to rocky intertidal invertebrates.

One of the primary strengths of mechanistic approaches to predicting organism temperatures in the terrestrial environment has been the ability to reconstruct patterns over large spatial and temporal scales using standard meteorological data (e.g., Kingsolver 1979, 1981, Huey et al. 1989). By necessity, these predictions are generally based on environmental conditions averaged over periods of hours to days (the time scales usually reported by federal meteorological data bases), but the potential consequences of using time-averaged climate data have not been fully explored (but see Buatois and Croze 1978). In this paper I examine some of the limits inherent in using environmental averages to predict the temperatures of intertidal organisms. Using a simple deterministic, individual-based model developed for intertidal mussels (Helmuth 1998) and microclimatological records collected from two semi-exposed rocky intertidal beaches, I explore the degree of error introduced by averaging environmental conditions over a range of time scales. Realistically, however, some degree of environmental averaging must be used, particularly when examining long time scales using historical climate data. Given this limitation and a better understanding of the errors involved, I then use a slightly more complicated model to predict the body temperatures of aggregations of Mytilus californianus at a semi-exposed site on the northern coast of the Olympic Peninsula during aerial exposure under "typical" climatological conditions representative of a 30-yr time period (1961-1990). Although this study is designed to develop methods for predicting the body temperatures of intertidal mussels, in principle the approach is applicable to a wide range of organisms and habitats.

Specific goals of this study were to: (1) assess the consequences of using time-averaged climate data to predict the body temperatures of intertidal mussels; (2) develop methods for extrapolating from regional-scale climatological databases to predictions of intertidal mussel (Mytilus californianus) body temperatures; (3) examine the temporal variability in mussel body temperatures over a "typical" climatological year in the northeastern Pacific; and (4) examine quantitatively the role of the tidal cycle in driving the body temperatures of mussels during aerial exposure.

Thermal modeling background

The flux of heat between a mussel and its environment determines the total heat contained in the animal and, subsequently, its temperature. In general, the mechanisms of heat exchange that drive body temperature are divided into six categories: short-wave solar radiation, long-wave (infrared) radiation to and from the sky and ground, conduction to and from the ground, heat convected between the animal and the surrounding air, and heat lost through the evaporation of water (Porter and Gates 1969, Campbell 1977, Monteith and Unsworth 1990). Because energy is neither created nor destroyed, the sources, sinks, and stores of heat in the system must, at all times, be balanced. The net sum of these inputs and outputs of heat determines the change in heat stored within the mussel, which, given its mass and specific heat, drives the temperature of the body ([T.sub.b]). Equations describing each component of the heat balance model have the general form of the product of a driving "force" (e.g., heat from the sun or a temperature differential), some area over which the heat exchange occurs (either a projected area or a surface area) and a series of coefficients that often depend on body size, morphology, material properties, and the animal's spatial arrangement (see Helmuth [1997, 1998] for a thorough description of the application of this model to intertidal mussels). As such, an organism's morphological characteristics can significantly affect its heat balance and thus its temperature (Kingsolver 1981, Thomas 1987, Bell 1995, Helmuth 1998). For example, during the day mussels in aggregations are predicted to experience lower body temperatures than solitary mussels under identical "weather" conditions due to their decreased areas subject to solar radiation (Helmuth 1998). Similarly, larger solitary mussels (in the absence of evaporative cooling) heat to greater temperatures than smaller mussels, but do so at a much slower rate (Helmuth 1998).

Potential pitfalls of using environmental averages

This simple, individual-based heat budget model (Helmuth 1998) provides a deterministic method for generating predictions of body temperature using the characteristics of the organism (size, morphology, and mass) and of the microclimate (wind speed, air and ground temperature, cloud cover, and solar radiation). However, two potential complications arise when predictions are based on environmental data that are averaged over long periods of time rather than reported continuously. First, many components of the heat budget equation vary nonlinearly with environmental parameters. For example, a doubling of wind speed does not double the rate of convective heat flux, and, as such, estimates of convection and hence body temperatures based on average measures of wind speed are subsequently flawed. Second, the temperatures of organisms such as a mussels vary in their response time to changes in heat flux. An important distinction here is the difference between heat and temperature. Larger mussels possess greater areas over which heat flux can occur, and consequently absorb a greater amount of heat energy per unit time. However, the flux of heat (energy) into any system increases the temperature (average kinetic energy) of the object at a rate proportional to the product of the object's mass and specific heat. Thus, a larger, more massive object, or one composed of a material with a high specific heat, requires substantially more heat energy to raise its temperature to the same degree as a smaller object (in this case, mussel). The capacity to dampen the response in body temperature to fluctuations in the environment is termed "thermal inertia" and is quantified as a time constant ([Tau]), a ratio of factors that resist changes in temperature (mass and specific heat) to those that promote them (such as areas of exposure and the coefficient of heat transfer; Helmuth 1997, 1998). Typical values for mussels range from a few minutes for a small solitary individual to [greater than or equal to]40 min for a large mussel in a bed (Helmuth 1997, 1998, and unpublished data). Thus, for example, although a larger mussel (e.g., 10 cm length) can experience an average heat flux four to five times that of a smaller (5 cm) mussel, its mass may be ten times greater, resulting in an increase in body temperature that is only half that of the smaller individual. Spatial position can also lead to a greater buffering effect against environmental change, and mussels living in aggregations display a larger thermal inertia than solitary mussels of identical size in gaps (= patches created by disturbance: Paine and Levin 1981). Thus, compared to a mussel of identical size and initial heat content, a mussel living in an aggregation will heat (or cool) much more slowly when environmental conditions change (Helmuth 1998).

Clearly, depending on their size and spatial position, mussels respond to changes in the environment differently. In general, environmental fluctuations that occur much faster than the time constant of the mussel (i.e., high frequency components) will tend to be "filtered" by the mussel's thermal inertia. Thus, although a small mussel may experience a relatively quick change in body temperature from higher frequency changes in the environment, larger mussels or mussels in beds may exhibit changes only in response to lower frequency components, such as the gradual increase in air temperature from morning to noon. This differential response to varying frequencies in environmental conditions, coupled with the nonlinear response of body temperature to environmental conditions, forms a potential pitfall for the use of environmental averages for the prediction of average body temperatures in the intertidal. Even larger errors can be incurred when predicting measures of body temperature such as maxima. For example, using simple models of fluctuating environmental conditions I have previously shown that predictions of maximum body temperature that are based on means or extremes of environmental conditions can theoretically introduce errors of [greater than or equal to]6 [degrees] C (Helmuth 1998). Clearly, high-frequency changes in the physical environment can occur, primarily in the parameter of wind speed, but how important are these high frequency components to the overall heat budget of an intertidal mussel under normal conditions in the field? Using real environmental records obtained in the intertidal, I attempt to estimate the range of error that the use of environmental averages can produce by "binning" environmental data over time widths of 560 min and explore the consequences of using such data for predictions of average and maximum body temperatures of intertidal mussels on an hourly basis.

Ideally, in order to avoid the potential problems associated with time-averaged data, studies of body temperature in the intertidal should utilize environmental data that are collected continuously and at a location very close to the organism in question, or else should simply measure these temperatures directly. For most studies these approaches are obviously untenable, particularly when examining trends over large geographic regions or over long time scales. However, mechanistic heat budget models can be still used to predict these larger scale trends when coupled with standard meteorological data (e.g., Beckman et al. 1972, Porter et al. 1973, Kingsolver 1979, Bell 1992). I used a series of steps to extrapolate from long-term, regional climatic data sets to predictions of average body temperatures of mussels over a "typical" year [ILLUSTRATION FOR FIGURE 2 OMITTED]. Some degree of inaccuracy is unavoidably introduced in extrapolating between such diverse spatial scales (regional to local to microclimatological), and at each step I describe the major sources of error. Thus, the goal is not to extrapolate from weather data to the precise physical conditions in the intertidal on any given hour or day, but rather to use a mechanistic approach to estimate what constitutes a "typical" distribution of mussel body temperatures during low tide in this region over an entire season. The approach that I describe is thus similar to the one developed by Denny (1995) for predicting rates of physical disturbance due to wave forces, in that large-scale processes are used to derive successively smaller scale events using a combination of statistical correlations and physical principles [ILLUSTRATION FOR FIGURE 2 OMITTED].

METHODS

Estimating error from the use of environmental averages

Hour-long environmental records were collected at semi-exposed rocky benches during the springs and summers of 1995 and 1996 on the northern coast of the Olympic Peninsula (Tongue Point, Crescent Bay, 48 [degrees] 10 [minutes] N, 123 [degrees] 40[minutes] W) and during summer 1995 at Cattle Point, San Juan Island, WA (48 [degrees] 35[minutes] N, 123 [degrees] 10[minutes] W). Solar radiation, wind speed, air temperature and ground temperature were monitored at the level of the mussel every 5 s. Wind speed was measured using a Kurz 490 mini-anemometer (Kurz Instruments, Monterey, California), and solar flux density was measured with a LI-COR (LI-COR, Lincoln, Nebraska) flat (2[Pi]) pyranometer (for these trials only direct solar radiation was considered, as it is usually the dominant source of solar heat flux). Air and ground temperatures immediately adjacent to the mussel were measured with copper-constantan thermocouples. Thermocouples were shielded from solar radiation using aluminum foil and were cold-junction-compensated using a Campbell 21X data logger (Campbell Scientific, Logan, Utah). Output from the instruments were channeled through the data logger into a laptop computer (Macintosh Powerbook 180), and were recorded using a program designed in Labview (Version 3.0, National Instruments, Austin, Texas). A total of 12 hourly records were obtained on 11 different days: 6 from gaps (adjacent to solitary mussels) and 6 from large mussel beds (3 of the records for solitary mussels were included in the original tests of model accuracy; Helmuth 1998). For two of these records, occasional spurious data points ([less than] 1 min total) due to instrument error were replaced with interpolations from the time interval immediately preceding the error. Similarly, in one trial interpolated data totalling 1 min were used to splice two 30-min records into one hour-long record. Environmental parameters were then used to generate predictions of body temperature for 5-cm and 10-cm solitary mussels (gap data) and 10-cm bed mussels (bed data), using both unsteady and steady-state heat balance approaches (Helmuth 1998). For these simulations, evaporative cooling was considered to be zero, as can occur when the mussel clamps its valves shut or is otherwise prevented from gaping (e.g., due to the presence of predators). Initial conditions of body temperature in the unsteady model were estimated to equal air temperature ([T.sub.a]) for mussels in beds and [T.sub.a] + 2 [degrees] C for solitary mussels (e.g., Helmuth 1998). This choice of initial conditions therefore does not apply to conditions immediately following emersion (when the body temperature is equivalent to that of the water), but instead represents a "ballpark" first approximation of conditions more than an hour after initial aerial exposure (Helmuth 1997, 1998; [ILLUSTRATION FOR FIGURE 3 OMITTED]).

Microclimate data were collected every 5 s, which for the purposes of this study were considered as essentially continuous. These data thus served as a baseline for comparison to body temperature predictions generated using time averages of these parameters. To test for the degree of information lost from the use of environmental averages, I binned data from the microclimate records into means based on intervals of 5, 10, 20, and 60 min [ILLUSTRATION FOR FIGURE 3 OMITTED], the latter representing the shortest time average generally available from standard weather bureau records. I then re-ran simulations using environmental data of these different bin widths. For each simulation, I estimated the average and maximum body temperatures of each mussel, as well as the average rate of change in body temperature (in degrees Celsius per minute), a potentially important measure for ectotherms whose body temperature fluctuates over short time scales (Newell 1969, Widdows 1976, Johnson and Shick 1977). Predictions were also generated using a simpler steady-state approach, which assumes that all environmental changes are slower than the mussel's time constant, and thus eliminates all aspects of thermal inertia over the time period in which it is measured (1 h). Under this latter scenario, a single measure [TABULAR DATA FOR TABLE 1 OMITTED] of body temperature was calculated for each hourly period, rather than a time series of body temperatures (Monteith and Unsworth 1990; [ILLUSTRATION FOR FIGURES 1, 3C OMITTED]).

Two types of error were calculated for measures of body temperature: (1) the mean difference between the time courses of body temperatures predicted using time-averaged environmental data and those predicted using (baseline) continuous environmental data (i.e., the average absolute difference between curves of temperature vs. time in [ILLUSTRATION FOR FIGURE 3B, C OMITTED]; unsteady model predictions only), and (2) the error incurred from using time-averaged environmental data to predict the metrics of mean and maximum body temperature and the average rate of temperature change. Whereas the first form of error (mean difference) describes discrepancies at any given point in time (as was used to test the accuracy of the heat balance equation, Helmuth 1998), the second describes an average performance over the entire hour-long record and is perhaps more relevant to studies that require estimates of mean or maximum body temperature over these time periods.

Long-term predictions of mussel temperature in the field

Sources of meteorological and solar data. - From 1961 to 1990 the National Renewable Energy Laboratory (NREL, Table 1) measured or calculated hourly measurements of direct and diffuse surface (horizontal) solar radiation from a network of stations throughout the United States (National Solar Radiation Data Base [NSRDB]; Marion and Urban 1995). These data are available in several formats. First, hourly solar data for the full 30-yr period have been combined with measurements of wind speed, air temperature, cloud cover, relative humidity and other meteorological information from the National Climatic Data Center (NCDC) to form the Solar and Meteorological Surface Observational Network (SAMSON) data set (Table 1). Second, NREL has constructed a "typical meteorological year" ("TMY2") for each site, in which hourly data from months most representative of a 30-yr period are concatenated to form a composite record for a complete year of representative climatological conditions (see Marion and Urban [1995] for a description of statistical methods). I used this latter resource as a means of extrapolating from "typical" climatological conditions to estimates of mussel body temperature in the rocky intertidal of the Olympic Peninsula, Washington State.

Extrapolating from regional- to microclimate-scale processes. - Using regional data reported for a site near the northern coast of the Olympic Peninsula (Quillayute; 47 [degrees] 57[minutes] N, 124 [degrees] 33[minutes] W), I scaled down to environmental data at the level of a mussel bed at Crescent Bay ([approximately]50 km distant, [ILLUSTRATION FOR FIGURE 4 OMITTED]). I then used a steady-state heat budget model, slightly more complicated than the individual-based model described in Helmuth (1998), to calculate body temperatures that should be representative of the 30-yr period of time for which environmental data were reported.

1. Regional-local scale climate correlations. - Although conditions reported at sites used to construct the "TMY2" data sets represent regional climate patterns, conditions of wind speed and air temperature at local scales more relevant to the intertidal are available from weather buoys and unmanned weather stations, which are scattered along the coast of the United States (NDBC, Table 1). In some cases data are available for considerable lengths of time and may be used directly, but, in several cases, conditions at the regional site must be extrapolated to those at a local site by correlating patterns in available data collected over identical time periods. From December 1986 to October 1988 a navigational buoy (number 46039, U.S. Coast Guard Platform type) was moored directly off the coast of Crescent Bay, providing a "window" of local climate conditions at Crescent Bay for comparison to regional conditions at Quillayute. Air temperature and wind speed were recorded at a height of [approximately]4 m, and were reported on an hourly basis throughout much of the buoy's 22-mo deployment. Air temperatures were reported to the nearest 0.1 [degrees] C; wind speeds were reported with resolutions of 1 m/s in 1987 and 0.1 m/s in 1988. Using cross-correlation analysis (cross-covariance), I compared the time lags in air temperature and wind speed between Quillayute and the Crescent Bay buoy as a means of extrapolating from regional conditions to smaller-scale, local climate patterns directly offshore from the intertidal of Crescent Bay [ILLUSTRATION FOR FIGURES 2 and 4 OMITTED].

Wind speed and air temperature data were divided into four "seasons" for analysis of time lags in wind and air temperature: winter (January-March), spring (April-June), summer (July-September) and autumn (October-December). Because of large gaps in wind data at the buoy site, I used the longest nearly continuous records available for each season of both 1987 and 1988. I used linear interpolation to fill in missing data points for [approximately] 1% of the data used for this comparison (one instance of three consecutive hours, the remainder all [less than or equal to]2 consecutive hours of missing data). Analyses of air temperature data indicated that conditions at the two sites were, on average, exactly in phase (no time lag) for all records examined. In contrast, wind speeds at Crescent Bay consistently lagged those at Quillayute by a period of 6 h. Both wind speed and air temperature showed a strong diurnal periodicity. During winter months, mean air temperature at Crescent Bay was slightly higher than at Quillayute, but during the remainder of the year was relatively cooler (Appendix A). Fluctuations in air temperature were also consistently damped at Crescent Bay relative to Quillayute. In contrast, mean wind speeds were consistently higher at Crescent Bay than at Quillayute, particularly during summer months, and fluctuated with a greater amplitude (Appendix A).

Data on solar radiation and cloud cover collected at Quillayute were used directly, as no local data on these parameters were available. Like wind speed and air temperature, solar radiation can change over relatively small spatial scales (Marion et al. 1992), especially in coastal environments. This assumption thus represents a potential source of error in this approach, and is one reason why such methods probably cannot be used to extract accurate predictions of body temperature over limited temporal scales (e.g., on any given day).

2. Extrapolating from local conditions to mussel bed microclimates. - Environmental conditions at the location of the buoy must differ from those of a mussel's microclimate. Specifically, conditions of wind speed and air temperature at the level of a mussel are strongly influenced by the mixing characteristics of the velocity gradient (boundary layer) overlying the bed (Vogel 1981, Denny 1993). I therefore used a fuller, surface-area based version of the heat budget model to incorporate these effects, which is described in Appendix B. In contrast to the individual-based model presented in Helmuth (1998), the heat budget model used for these predictions implicitly solved for mussel body temperatures given "free-stream" values (i.e., those at the height of the buoy) of air temperature and wind speed, coupled with measurements of horizontal direct and diffuse solar radiation. Conduction to the ground below the mussels was considered to be negligible, as suggested by the individual-based model (Helmuth 1998).

a) Wind speed. I measured boundary layer profiles in April 1997 at Crescent Bay in the centers of large (10-15 m diameter) mussel aggregations as a representative example of mussel beds at this site (Helmuth 1997). While several models are available for describing turbulent boundary layer profiles, I used a simple version for estimating the velocity at any given height (e.g., Middleton and Southard 1978, Eckman et al. 1981, Denny 1988):

U(z) = [u.sub.*][k.sup.-1] ln(z/[z.sub.0]) (1)

where U(z) is the mean horizontal (x) velocity at height z above the surface of the mussel bed (i.e., the tops of the mussels), [u.sub.*] is friction (shear) velocity (a measure of mixing within the boundary layer), k is von Karman's constant (0.4, dimensionless) and [z.sub.0] is roughness height (a measure of the "effective" height of objects protruding from the substrate into the overlying boundary layer; Denny 1988). Values of [u.sub.*] and [z.sub.0] were estimated using linear regressions of U(z) vs. ln(z) (Denny 1988). Velocity measurements and extrapolations of boundary layer profiles indicated that velocity changed very little (5-10%) above a height of [approximately]2 m, approximating the thickness of the small-scale boundary layer overlying the bed. Velocities recorded or predicted at the buoy anemometer (4 m) were therefore considered as [U.sub.[infinity]], the "free stream" velocity of the small-scale boundary layer overlying the bed (e.g., Vogel 1981, Denny 1993). Because shear velocity depends on wind velocity, values of [u.sub.*] were calculated as a linear function of [U.sub.[infinity]], which was extrapolated from the boundary layer profile between the ground and 2 m. As predicted by previous estimates (Denny 1993), shear velocity ([u.sub.*]) values varied linearly with [U.sub.[infinity]] ([u.sub.*] = 0.03[U.sub.[infinity]]), but were lower than the generic estimate of [u.sub.*] = 0.05[U.sub.[infinity]] (Middleton and Southard 1978, Denny 1993). Values of [U.sub.[infinity]] extrapolated from measurements of boundary layer profiles ranged from 2 to 5 m/s, but the linear relationship between [u.sub.*] and [U.sub.[infinity]] was assumed to extend to the full range of values reported for the buoy site (0-18 m/s). Roughness height ([z.sub.0]) was fairly insensitive to flow, with mean values ([+ or -]1 SD) of 1.7 [+ or -] 1.1 mm. Thus, given an estimated flow velocity ([U.sub.[infinity]]) at 4 m (local environment) and knowledge of the velocity profile ([u.sub.*]), I extrapolated from velocities at Quillayute (regional weather) to conditions at the level of the mussel (microclimate).

b) Air temperature. Like wind speed, air temperatures immediately adjacent to a mussel bed can differ from those a few meters above the substrate. Indeed, gradients in temperature above the substrate are determined to a large extent by mixing within the boundary layer, as measured by the shear velocity parameter [u.sub.*]. For example, in the terrestrial environment, the ground often serves as a heat sink during early morning hours so that air temperatures close to the ground are lower than those of the overlying air; in contrast, as the ground warms throughout the day from solar radiation, this heat is transferred to the layers of air above the substrate (Campbell 1977). Modeling a rocky substrate as inhomogeneous as the rocky intertidal is difficult (but see methods of Rejmanek 1971), and is a limitation for using the methods described here to predict the body temperatures of animals whose temperatures are closely linked to that of the substrate (e.g., barnacle species lacking basal plates [Thomas 1987]; rupestrine crabs [Stillman and Somero 1996]). However, because tightly packed mussel beds almost always completely cover the underlying rock, mussel aggregations can be considered to effectively act as the substrate, creating a feedback between the body temperature of the mussels and the rate of convection to the overlying air. The heat budget model (Appendix B) thus incorporates not only the effect of wind speed on the rate of convective exchange, but also the interactive effects of body temperature, air temperature, wind speed, and shear velocity on the local air temperature immediately adjacent to the mussel bed.

c) Evaporation. The rate of evaporative cooling by mussels under natural field conditions is, unfortunately, one of the most difficult portions of the heat budget model to predict and can significantly affect the thermal energy balance (e.g., Helmuth 1998). Unlike organisms such as algae, which almost always have wet surfaces exposed to air and can only influence their rates of desiccation though their morphology and water content (Bell 1995), mussels have very tight behavioral control over the rate of water loss through the degree of gape between their shell valves. The determinants of gape by bivalves in air remain unclear and relate not only to thermal and desiccation stresses, but also to aerial respiration (Lent 1968, 1969, Bayne et al. 1976b, Widdows et al. 1979, Guderley et al. 1994). However, Bayne et al. (1976b) have suggested that aerial respiration by Mytilus californianus is only effective when relative humidities approach 100%. Under these conditions, mussels are unable to evaporatively cool due to the lack of a gradient between the vapor pressure density at the surface of their tissues and that of the air (Helmuth 1998). While this result does not preclude gaping as a means of evaporative cooling at relative humidities lower than 100%, it does suggest that M. californianus do not continuously gape during aerial exposure as a means of respiring. Thus, for these simulations, I considered evaporative cooling to be either zero (no gape) or at a fixed mass loss rate of 5% of initial body mass per hour of exposure, both within the range of values reported for other species of mussels (e.g., Liu and Morton 1994) while generally not exceeding lethal levels of desiccation (e.g., Lent 1968). Thus, while the interactive relationship among desiccation, thermal stress, and aerial respiration in determining mussel gape remains enigmatic, these two scenarios (no gape or 5% per hour) provide a bracket of expected values of body temperatures during average periods of aerial exposure.

3. Time of exposure. - The last remaining step, which is unique to intertidal systems, is to calculate the timing of exposure to the aerial environment as a function of the tidal cycle. Denny and Paine (1998) have recently shown that tidal cycles vary in amplitude over an [approximately]18.6-yr cycle due to fluctuations in lunar inclination, which can result in variability in exposure times of up to 30-40 min, particularly in higher regions of the intertidal ([greater than] 1 m + Mean Lowest Low Water [MLLW]). Therefore, although this analysis represents "typical" climatological conditions, tidal amplitudes used here are not necessarily representative of the entire 30-yr period for which environmental data were collected. In general, however, within any given season the timing of low tides remains relatively consistent from year to year (NOS tide tables, Table 1). Because variability in tidal cycle estimated by Denny and Paine (1998) is within the hourly interval used here, as a default I estimated tidal heights using the tide series of 1996 for Crescent Bay by modeling the tidal cycle as a simple cosine function with varying amplitude (details given in Helmuth 1997). I calculated exposure times (based on still-water tidal levels) for mussel beds at two tidal heights: MLLW + 1 m and MLLW + 0 m. By this scheme, mean exposure duration for each emersion event was [approximately]2-3 h at MLLW + 0 m and 6-6.5 h at MLLW + 1 m (Table 2). Exposure durations varied slightly by season and were longest during spring months. On average, mussels at + 1 m were exposed more than once per day, but mussels at the lower tidal height were aerially exposed only once every 3 or 4 d (Table 2).
TABLE 2. Durations and frequencies of aerial exposure at tidal
heights of Mean Lowest Low Water (MLLW) + 0 m and + 1 m predicted
using a simple model of tidal fluctuations at Crescent Bay (1996
tidal series).

                                           Average exposure
                  Average exposure             frequency
                    duration (h)           (no. exposures/d)
Season           + 0 m      + 1 m          + 0 m        + 1 m

Winter            2.44       6.11           0.26        1.21
Spring            3.10       6.51           0.47        1.24
Summer            2.79       6.31           0.38        1.36
Autumn            2.81       6.43           0.48        1.17
Year              2.83       6.34           0.40        1.17

Note: Tidal exposures are both more frequent and for longer periods
of time during the spring tidal series.


4. Calculating mussel body temperatures. - Using the above steps, data from a "typical meteorological year" (TMY2) at Quillayute (Marion and Urban 1995) were used to generate body temperatures of mussels during aerial exposures over an entire representative year using the environmental parameters of wind speed, air temperature, solar radiation (horizontal diffuse and direct), cloud cover, and relative humidity (Appendix B). Using a steady-state approximation, estimated body temperatures were generated for each hour of exposure for aggregated mussels at tidal heights of MLLW + 1 m and MLLW + 0 m and with two rates of mass loss: none and a constant rate of 5% initial body mass per hour.

Decoupling the effects of climate and tidal cycle

Whereas the characteristics of the terrestrial environment ultimately drive the body temperatures of mussels during aerial exposure, the tidal cycle determines the timing of this exposure. As a means of independently assessing the role of the tidal cycle independent of climate, I estimated body temperatures at Crescent Bay at a tidal height of MLLW + 1 m using spring climate conditions but with exposures determined by summer tidal cycles, and likewise coupled summer climate conditions with spring tidal cycles.

The timing of exposure can also vary over relatively short spatial scales. For example, as one moves eastward from the outer coast of the Olympic Peninsula to the inner Sound, tides occur later in the day so that low tides at Port Townsend occur [approximately]2 h later than at Crescent Bay, [approximately]50 km distant (Fig. 4; Helmuth 1998). In order to dissect the relative impact of this shift in emersion time on mussel body temperatures, I calculated temperatures at MLLW + 1 m using climatological conditions predicted for spring and summer at Crescent Bay, but imposed tidal conditions for Port Townsend.

RESULTS

Errors incurred through the use of time-averaged environmental data

Mean difference error. - Not surprisingly, mean difference error (the average error incurred in predicting temperature at any given point in time during each hour-long trial) varied markedly with the degree to which environmental conditions fluctuated; thus, the errors presented here are intended only as examples rather than absolute measures. Days on which data were collected, however, were not atypical of other days at these sites in 1995-1997 (Helmuth 1998 and unpublished data). During periods of time when environmental conditions fluctuated very slowly, estimates based on long time averages (wide time bins) of the environment introduced little error when compared to estimates based on continuous environmental measurements. In contrast, on days with gusty wind or patchy cloud cover (and thus erratic solar radiation) mean deviations exceeded 1 [degree] C (Helmuth 1997). On average, simulations using smaller (5 cm) mussels displayed greater error than those used to predict the temperatures of larger (10 cm) mussels, primarily due to the faster response time of smaller mussels (Fig. 5). Because environmental records in gaps (= patches) were collected on different days from those collected over beds, error measurements calculated from these two groups could not be directly compared, as could differences between 5-cm and 10-cm mussels, which were based on the same environmental data. However, that errors were always lowest in aggregated mussels (Fig. 5) is consistent with their higher thermal inertia relative to solitary mussels. Mean differences increased with the width of the time bin, but were surprisingly low, ranging from 0.02 [degrees] C to just over 1 [degrees] C (Helmuth 1997). In fact, this degree of error is similar to that inherent in the model itself (due to errors in mussel allometry equations, small-scale spatial heterogeneity, etc.; Helmuth 1998).

Errors in mean and maximum body temperatures and rate of change. - Error in predicting average body temperatures during each hour-long record also increased with time bin width [ILLUSTRATION FOR FIGURE 6A OMITTED], on some days up to 1 [degrees] C (Helmuth 1997), and generally reflected overestimates. In contrast, increasing time bin width usually underestimated maximum body temperature [ILLUSTRATION FOR FIGURE 6B OMITTED], on some days by [greater than]1.5 [degrees] C (Helmuth 1997). Measures of the rate of change in body temperature were very sensitive to the time over which environmental data were averaged, and the degree of error incurred increased with larger time bins [ILLUSTRATION FOR FIGURE 6C OMITTED] due to significant decreases in the predicted magnitude of fluctuations (Helmuth 1997). Whereas errors in estimating average body temperature of larger solitary mussels increased with time bin width more rapidly than for smaller mussels [ILLUSTRATION FOR FIGURE 6A OMITTED], errors in predicting maximum body temperatures increased more slowly for larger mussels, presumably due to their greater capacity to filter high-frequency fluctuations in environmental parameters [ILLUSTRATION FOR FIGURE 6B OMITTED]. Similarly, error in predicting average rates of change in body temperature were consistently higher in smaller, solitary mussels, again due to a lower thermal inertia [ILLUSTRATION FOR FIGURE 6C OMITTED].

Steady-state predictions of average hourly body temperature consistently overestimated those generated using continuous data, but were generally within 2 [degrees] C (5-cm solitary mussels: 0.61 [+ or -] 0.36 [degrees] C; 10-cm solitary mussels: 1.44 [+ or -] 0.61 [degrees] C; 10-cm bed mussels: 0.71 [+ or -] 0.40 [degrees] C). Note that because estimates of body temperature based on 1-h averages of environmental conditions asymptote to those predicted by steady-state models, maximum body temperatures (and hence errors) are identical for these two simulations. Similarly, because the steady-state model generates only a single value of body temperature (the equilibrium temperature under constant environmental conditions), mean rates of change in body temperature cannot be generated.

Overall, therefore, estimates of average and maximum body temperature using unsteady heat budget models based on environmental data binned over periods of time as long as an hour displayed a surprisingly small error when compared to estimates obtained using continuous data. Estimates using the simpler steady-state approach were less accurate than those using an unsteady model, but were still not more than a few degrees off. In contrast, predictions of small-scale rates of change in body temperature were greatly underestimated with longer time-scale averages. These results thus suggest that while hourly averages of climate data are insufficient to predict rapid rates of change in body temperature they are adequate for predicting average and maximum temperatures at a resolution of 1 h (e.g., [ILLUSTRATION FOR FIGURE 1 OMITTED]).

Patterns in mussel body temperatures and effects of the tidal cycle

Mussels at a tidal height of MLLW + 1 m were predicted to experience a much wider range of temperatures than mussels lower in the intertidal, and the average daily range of temperature fluctuations during summer months at this height was estimated as 15 [degrees] 20 [degrees] C [ILLUSTRATION FOR FIGURE 7 OMITTED]. Daily mean temperatures during aerial exposure at the two heights were similar, ranging from 5 [degrees] C in December and January to 20 [degrees] C in May [ILLUSTRATION FOR FIGURE 7 OMITTED]. These strong seasonal trends in body temperature resulted not only from variability in climate but also from differences in the timing of exposure to the aerial environment as set by the tidal cycle. Mean body temperature during low tide at Crescent Bay (i.e., not including submerged temperatures) was predicted to be slightly higher in spring (April-June) than in summer (July-September), although maximum temperatures were predicted to occur during summer months [ILLUSTRATION FOR FIGURES 7B and 8 OMITTED]. Similarly, mean body temperature at this site was predicted to be slightly lower in autumn than in winter [ILLUSTRATION FOR FIGURES 7A and 8 OMITTED].

Significantly, these patterns did not strictly correlate with average measures of environmental conditions. For example, while total solar radiative flux was higher during spring than summer, air temperatures were slightly warmer in the summer than in the spring (11.8 [degrees] vs. 8.7 [degrees] C). At Crescent Bay, however, tidal cycles tend to expose mussels closer to noon more frequently in the spring than in the summer. Results of simulations where the effects of seasonal variations in climate and tidal cycles were decoupled indicated that, with the influence of tidal cycle removed, climatic conditions in the summer could theoretically lead to higher (2 [degrees] -3 [degrees] C) body temperatures than in the spring [ILLUSTRATION FOR FIGURE 9 OMITTED]. However, the effects of emersion time exceeded those of seasonal changes in climate. For example, mussels aerially exposed by a spring tidal cycle to springtime climate conditions were predicted to experience a mean body temperature of 15.94 [degrees] C (upper left of [ILLUSTRATION FOR FIGURE 9 OMITTED]). Exposing the mussels using an identical tidal cycle but superimposing summertime climate conditions led to a hypothetical increase of nearly 2 [degrees] C in mean body temperature, with a similar variance (upper right of [ILLUSTRATION FOR FIGURE 9 OMITTED]). However, when springtime climate conditions were held constant but summertime tidal cycles were used to determine the timing of exposure, a decrease in body temperature of nearly 4.5 [degrees] C, with a slightly lower variance, was predicted (lower left of [ILLUSTRATION FOR FIGURE 9 OMITTED]). Thus, while terrestrial climatic conditions drive the body temperatures of sessile intertidal invertebrates during low tide, the timing of aerial exposure determines when organisms are exposed to those conditions (e.g., closer to noon in the spring than in the summer at Crescent Bay). In other words, climatic conditions near noon in spring are "hotter" than those in early morning or late afternoon in summer.

Similarly, differences in the timing of tidal exposure which occur at a spatial scale of only 50 km (Crescent Bay vs. Port Townsend) raised predictions of mean body temperature by [approximately]3 [degrees] C during both spring (no evaporative cooling, Crescent Bay tide (mean [+ or -] 1 SD): 15.94 [degrees] [+ or -] 7.77 [degrees] C; Port Townsend tide: 18.86 [degrees] + 7.31 [degrees] C) and summer (Crescent Bay: 14.40 [degrees] [+ or -] 7.75 [degrees] C; Port Townsend: 17.67 [degrees] [+ or -] 8.88 [degrees] C). Again, small geographical differences in tidal emersion time had effects on body temperature comparable to predicted differences between spring and summer.

DISCUSSION

Physical ecology of the rocky intertidal

The thermal biology of intertidal invertebrates fundamentally differs from that of terrestrial animals for several reasons. First, many intertidal organisms are sessile, and thus, once settled, have limited behavioral control over their local physical environment. However, despite large-scale heterogeneity in the surrounding substrate, many invertebrates, and particularly those that aggregate, can affect the characteristics of their microclimate (e.g., Bertness and Grosholz 1985, Lively and Raimondi 1987, Bertness 1989, Stephens and Bertness 1991, Bertness and Leonard 1997, Helmuth 1998). Thus, an invertebrate living in an aggregation can experience markedly different body temperatures than an adjacent solitary individual (Helmuth 1998). In this regard intertidal animal communities are perhaps most similar to terrestrial plant communities, where such "switches" have been found to occur regularly (Wilson and Agnew 1992). This potential for organisms to modify the microclimates of other organisms thus strongly suggests the potential for feedback loops between abiotic and biotic factors (Bertness 1989, Stephens and Bertness 1991, Wilson and Agnew 1992, Hacker and Bertness 1995, Jones et al. 1997, Helmuth 1998), and mechanistic approaches to predicting body temperatures can serve as valuable tools for investigating these processes.

Second, intertidal organisms are unique in that they must contend with two very different thermal environments where the timing of exposure to each is driven not by a day/night cycle but by a tidal cycle. This interaction of tidal cycle and environmental factors thus leads to unusual temporal patterns in body temperatures that are not strictly coupled with diurnal rhythms. Along the Pacific Coast of North America intertidal invertebrates are emersed twice daily (mixed semidiurnal tide), so that organisms high in the intertidal spend more time in air than in water (Carefoot 1977). In between this upper extreme and the subtidal is a range of exposure regimes that can influence the upper distributional limits of species within the intertidal community (Doty 1946, Carefoot 1977, Swinbanks 1982). While lower distributional limits generally depend on competition and predation pressures (Connell 1961, Paine 1969, 1974), aerial exposure and its effects on temperature and desiccation can play a role by setting the excursion limits of predators (Menge 1978, Moran 1985, Seed and Suchanek 1992, Tokeshi and Romero 1995) and the performance of competitors (Wethey 1984, Lively and Raimondi 1987, Bertness 1989). Because of these steep gradients and the irregular alternation between terrestrial and aquatic environments, the rocky intertidal provides a unique habitat for examining the role of the physical environment in driving the ecology and physiological performance of ectothermic organisms.

The role of tidal cycle in determining body temperatures

The results of this study demonstrate that variation in the timing of aerial exposure due to differences in the tidal cycle, which can occur over rather small spatial scales, can be more important than larger scale patterns of the local climate. That is, sites with "hotter" climate conditions do not necessarily lead to higher body temperatures. Significantly, the relative importance of climate and tidal cycle can only be deciphered through a mechanistic approach such as the one presented here. This pattern also suggests the presence of a "vernier effect" in the interactions of tidal cycle and climate conditions such that stressful or lethal body temperatures may not always occur when climate conditions exceed some critical maximum, but rather only when low tides occur during these events. Thus, in regions of the Olympic Peninsula (e.g., Tatoosh Island, [ILLUSTRATION FOR FIGURE 4 OMITTED]) where lowest low tides seldom occur during mid-day in summer (i.e., when climatic conditions are "hottest"), lethally high temperatures are less likely to occur than at comparable tidal heights in regions where the timing of low tides near noon in summer is common (Puget Sound). Therefore, mussels at similar intertidal heights but at locations separated by only tens of kilometers on the Olympic Peninsula are likely to differ substantially in their thermal regime, and hence in their physiology.

Conversely, these results at least tentatively suggest that there may be effects of the tidal cycle on latitudinal patterns in aerial body temperature and that these temperatures may not necessarily increase with decreasing latitude (also see Bell 1992). For example, mussel populations inhabiting a northern site where low tides tend to occur closer to noon during spring and summer may experience higher temperatures than a more southern site where low tides consistently occur either earlier in the day or later in the afternoon. While the possibility of such patterns remains to be investigated, if correct such heterogeneity in the aerial thermal regime could partially explain the patchy distribution of Mytilus species that has been observed over a range of spatial scales (Suchanek et al. 1997).

Results of this study also confirm that mussels in higher sections of the intertidal are much more likely to be exposed to extreme conditions than those lower in the intertidal [ILLUSTRATION FOR FIGURE 7 OMITTED], as exposure at these levels can occur at noon even when low tide is much earlier in the day due to longer exposure periods. These results are consistent, for example, with the findings of Roberts et al. (1997), who have recently shown that the incidence of heat stress effects increases with tidal height, as well as other reports of mortality that were restricted to upper tidal heights (e.g., Suchanek 1978).

Extrapolations to mussel performance and survival

How important are fluctuations in body temperature to the physiology and survival of intertidal invertebrates? During the last several decades numerous studies have investigated the role of body temperature in determining the performance and risk of mortality of Mytilus spp., but most have focused on measurements conducted over exposure to constant conditions, and many have tended to examine the effects of body temperature on physiological processes during submersion (but see Lent 1968, 1969, Newell 1969, Bayne et al. 1976b, Elvin and Gonor 1979, Hofmann and Somero 1995, 1996, Roberts et al. 1997). Clearly, mussels and other sessile intertidal invertebrates spend a substantial portion of their existence in the terrestrial environment, and exposure to critical temperatures and levels of desiccation during low tide is generally thought to set the upper distributional limits of Mytilus spp. in the rocky intertidal (Seed and Suchanek 1992). These limits can be set both by excessively high body temperatures as well as by freezing. Suchanek (1978) reported a massive summer mortality of M. edulis (= M. trossulus, Suchanek et al. 1997) high in the intertidal for three consecutive years at a protected site on Tatoosh Island, and Tsuchiya (1983) recorded a similar event in northern Japan. In the latter study, mussels reached body temperatures [greater than] 40 [degrees] C in a period of an hour. Craciun (1980) has suggested upper tolerable temperatures as low as 28 [degrees] C for M. galloprovincialis, but exposure times in these experiments were conducted using submerged individuals exposed to constant temperatures over periods of 30-60 h. Perhaps even more importantly, Hofmann and Somero (1995, 1996) and Roberts et al. (1997) have shown that evidence of cellular damage (both reversible and irreversible) from extreme temperatures is common in intertidal mussels.

Mortality has also been shown to occur at extremely low temperatures, and while Williams (1970) has shown that M. edulis can tolerate body temperatures as low as -10 [degrees] C, M. californianus is thought to be less able to tolerate freezing (Seed and Suchanek 1992). My results suggest, perhaps not surprisingly, that in "typical" years body temperatures above [approximately] 30 [degrees] C or below approximately -4 [degrees] C are uncommon at a tidal height of + 1 m at Crescent Bay, provided evaporative cooling is permitted [ILLUSTRATION FOR FIGURE 8 OMITTED]. Estimates of the risks of mortality derived from these or similar simulations must, however, be considered with caution. For these predictions an average rate of mass loss of 5% initial body mass per hour was permitted, but under extreme conditions mussels may be able to tolerate extremely hot conditions by evaporatively cooling at a faster rate but over a shorter period of time, the limits to which are set by desiccation stress and respiratory requirements. Furthermore, other environmental factors such as wave splash can affect the incidence of mortality due to both freezing and overheating. For example, R. T. Paine (personal communication) has noted that heavy ice cover resulting from wave splash can protect mussel beds in high exposure regimes during winter freezes.

Significantly, although evaporative water loss lowers the average body temperature of mussels by several degrees, perhaps more importantly it greatly lowers the incidence of extreme body temperatures. Thus, whereas the maximum body temperature that a mussel would experience during "typical" summer months in the absence of evaporative cooling is [approximately] 45 [degrees] [ILLUSTRATION FOR FIGURE 8E OMITTED], with a reasonable rate of mass loss maximum temperature is reduced to [approximately] 34 [degrees] C [ILLUSTRATION FOR FIGURE 8F OMITTED]. Because aerial respiration is thought to be most effective when relative humidities approach 100% (Bayne et al. 1976b) where evaporative cooling cannot occur (Helmuth 1998), from a thermal standpoint gaping is probably most effective during extreme conditions rather than as a constant mechanism for cooling. By storing water during marginal times, mussels would be expected to incur a substantial benefit during otherwise lethal exposures. This result also suggests that, because mussels must evaporatively cool to prevent excessive heat, desiccation stress in mussels is indirectly controlled by body temperature.

Limitations to thermal engineering approaches

The methods presented here provide a mechanism for predicting "typical" body temperatures of intertidal mussels over a range of spatial and temporal scales, as well as a means of quantitatively assessing the potential impacts of climate change on these patterns. However, how well do these predictions represent reality? The answer partially depends upon the scale of question being asked. Clearly, estimates of body temperature based on hourly environmental averages collected several kilometers away have little hope of providing insight into rates of temperature change on the order of minutes, and even predictions of fluctuations on the order of hours are suspect. Similarly, the intertidal is sufficiently heterogeneous (e.g., the presence of boulders and surge channels, or any feature that provides shade or blocks wind flow) that predictions at small spatial scales are beyond the scope of this approach. Indeed, measurements of the body temperatures of mussels within the same bed show that these temperatures can vary by several degrees due to differences in mussel morphology and orientation to sun and wind (e.g., [ILLUSTRATION FOR FIGURE 1 OMITTED]; Helmuth, unpublished data). Finally, organisms vary in their morphologies. For example, changes in shell coloration (e.g., Etter 1988), the presence or absence of periostracum on molluscs (larger Mytilus sp. often lose portions of this layer), and the growth of algae on the shell are all likely to modify a mussel's reflectivity and hence its body temperature. Questions at these scales are best addressed using individual-based models where all factors but the one of interest are controlled or accounted for (e.g., Thomas 1987, Bell 1995, Helmuth 1998).

In contrast, methods based on large-scale climatological data are probably most useful for generating broad-scale comparisons, for example seasonal differences, changes in global climate or geographic patterns. However, it must be emphasized that the approach described here, despite its mechanistic components, is still simply a model and has yet to be fully tested as to its accuracy and limitations. However, preliminary evidence based on measurements at Crescent Bay suggests that, at any given point in time (i.e., at any specific hour), body temperatures predicted from wind speed and air temperature data recorded at Quillayute (but using solar data collected at Crescent Bay) are, on average, within [approximately] 2.5 [degrees] C of the means of body temperatures actually measured in the field (Appendix B). Thus, while there are many potential sources of error in the approach described here, the evidence to date suggests a fairly reasonable fit to real temperatures in the intertidal.

Summary

The results of this study show that, first, the use of environmental averages for predicting body temperatures will produce errors due to the nonlinear nature of the heat budget equation and the "thermal inertia" of the organism's body. The acceptable level of error, and hence the minimum sampling interval, will vary with the nature of the study. Studies that require high accuracy over small temporal or spatial scales, or those that require measurements in the rates of change in body temperature, must use data collected adjacent to the organism and sampled at high frequencies (unsteady heat budget model), or else should measure body temperatures directly. Studies of long-term trends and averages, such as the study presented here, can be based on hourly averages of environmental parameters, but ignore the time history of body temperature within each hourly period (steady-state model). Secondly, given the limits to environmental averaging, climate data from long-term data sets can be used to extrapolate to "typical" conditions at nearby shorelines. However, the limitations and accuracy of this approach are still being explored. Thirdly, at Crescent Bay (Olympic Peninsula, Washington), aerially exposed mussels are, on average, coldest during autumn (October-December) and warmest during spring (April-June) due to the interaction of the tidal cycle with the local climatological conditions in driving body temperatures. Warmer temperatures during spring are not strictly the result of "hotter" average climate conditions, per se, but of tides that tend to occur closer to noon at this location. In other words, conditions near solar noon in spring lead to higher body temperatures than conditions in early morning or late afternoon in the summer. Significantly, this result suggests that the importance of variability in the timing of aerial exposure due to differences in the tidal cycle, which can occur over relatively small spatial scales ([less than] 100 km), can exceed the effects of larger scale (seasonal and perhaps latitudinal) variability in climatic conditions. Finally, in addition to behaviors related to respiratory requirements, mussels must gape periodically during aerial exposure as a means of preventing lethal body temperatures. However, while gaping lowers average body temperatures to varying degrees, perhaps more importantly it serves as a very effective means of substantially reducing the incidence of excessively high body temperatures during fairly rare events, but only at the cost of desiccation. While the interactions between biotic and abiotic factors in the intertidal still remain to be fully explored, the merger of quantitative approaches to sessile invertebrate microclimates, studies of organismal physiology and biochemistry, and experimental studies of competition, predation, and positive interactions form a potentially strong basis for explorations of the physiological ecology of rocky intertidal systems.

ACKNOWLEDGMENTS

Financial support for this research was provided in part by NSF Mathematical Biology Training Grant BIR9256532 to T. Daniel and G. Odell, and by a grant from the American Museum of Natural History Lerner-Gray Fund for Marine Research to the author. B. E. H. Timmerman and J. von Klingelberg were indispensable for their help in the field, as was T. Daniel for his unwavering support and advice. Helpful comments on the research and manuscript were provided by T. Daniel, M. Denny, G. Gilchrist, C. Harley, R. Huey, J. Kingsolver, A. Kohn, B. Menge, R. T. Paine, B. E. H. Timmerman, P. Ward, two anonymous reviewers, and members of the University of Washington Mathematical Biology Training Grant program. I thank N. Terwilliger for unknowingly providing the impetus for this study. I am particularly grateful to the staffs of the National Renewable Energy Laboratory and the Western Regional Climate Center for their assistance in accessing climate data. This research was submitted in partial fulfillment of the requirements for a Ph.D. in the Department of Zoology, University of Washington.

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APPENDIX A

PREDICTING LOCAL WIND SPEEDS AND AIR TEMPERATURES FROM REGIONAL DATA

Records of air temperature and wind speed at local scales are available from NOAA weather buoys and unmanned (C-MAN) stations for many sites in the northeastern Pacific as well as other coastal regions of the United States (Table 1). Data from Crescent Bay were only collected from 19871988, but nonetheless serve as a means of correlating regional conditions reported at Quillayute with those at a scale more local to the site of interest. For each season, I calculated the difference in the seasonal means of air temperature and wind speeds (i.e., a "D.C." offset) between these two sites and then calculated the relative magnitude (gain) of fluctuations away from these means. Amplitude was defined as the average (absolute) deviation from the mean within each season, and the gain in amplitude calculated as the ratio of the amplitude at Crescent Bay (CB) to that at Quillayute (Q). All reported data were used and no attempt was made to interpolate missing data points. Given the mean value of air temperature at Quillayute (Q, mean) over an entire season, one can extrapolate from any given temperature at hour t to air temperature at the buoy at Crescent Bay (height of 4 m):

[T.sub.a](CB, t) = gain[[T.sub.a](Q, t) - [T.sub.a](Q, mean)] t [T.sub.a](Q, mean) + offset. (A.1)

Similar methods were used to predict wind speed (U), only predictions also incorporated a 6-h time lag:

U(CB, t) = gain[U(Q, t - 6) - U(Q, mean)] + U(Q, mean) + offset. (A.2)

All predictions that generated a negative wind speed were assigned a value of 0.1 m/s (to prevent a zero denominator in the heat balance equation, Appendix B). Values of offsets and gains used in these models are presented in Table A1.
TABLE A1. Parameter estimates used to predict wind speeds and air
temperatures at Crescent Bay given conditions at Quillayute.

                 Air temperature                Wind speed
Season         Offset         Gain         Offset         Gain

Winter         +0.75           0.55        +1.01           1.74
Spring         -1.53           0.33        +2.13           1.71
Summer         -2.40           0.37        +2.66           2.03
Autumn         -0.22           0.43        +0.30           1.40

Notes: Offset is defined as the difference between the means at
the two sites within a given season (Crescent Bay-Quillayute,
[degrees] C or m/s). Gain is the ratio of the average amplitude of
fluctuations at the Crescent Bay buoy to that of the regional
station at Quillayute. Parameters were estimated for four
"seasons": winter (January-March), spring (April-June), summer
(July-September), and autumn (October-December).


APPENDIX B

STEADY-STATE MODEL OF MUSSEL BED MICROCLIMATES

In contrast to the individual-based model (Helmuth 1997, 1998), the steady-state heat flux balance used to predict long-term averages in mussel body temperature ([T.sub.bed]) is based on a generic bed of mussels and is calculated per unit area (-square meters) of substrate. Under this scenario, all mussels are assumed to experience approximately the same body temperature through similar size and behavior, as well as a large degree of infrared radiation between the mussels of the bed. Here I assume a uniform bed containing tightly packed mussels with a body size of 7.5 cm and a density of 700 individuals/[m.sup.2] substrate. Because the rocky intertidal is so spatially heterogeneous, it is difficult to predict the temperature of rock surfaces throughout the time course of aerial exposure (e.g., as in Rejmanek 1971, Mitchell et al. 1975). However, in the case of a tightly packed aggregation, as is typical for Mytilus species, the bed of mussels may be considered as the exchange surface itself. This approach is thus more complete than the individual-based model in that it implicitly includes the presence of a gradient in air temperature above the bed. Specifically, given knowledge of the characteristics of the boundary layer overlying the bed (shear velocity, [u.sub.*]), cloud cover (through its effect [[Epsilon].sub.sky]) and predictions of wind speed ([U.sub.x]) and air temperature ([T.sub.[a.sup.x]])) at some free stream height above the bed (here, 4 m, the height of the buoy instruments), the individual-based model (Helmuth 1997, 1998) may be modified to predict the temperatures of mussels in an aggregation ([T.sub.bed]) as

[Mathematical Expression Omitted] (B.1)

where [A.sub.p] = projected area of a mussel (= [A.sub.sol,dir]) and C = [([A.sub.conv]/[A.sub.p]) x ([u.sub.*][c.sub.p][Rho]k)] x [[ln(z/[z.sub.0])].sup.-1]. Conduction is considered negligible due to the small area of contact, and ground temperature immediately below the bed ([T.sub.g]) is estimated as that of air temperature at 4 m ([T.sub.[a.sup.x]]). The parameter [Mathematical Expression Omitted] is the average rate of water loss per square meter of substrate (amount per mussel times total number of mussels per square meter). The shear velocity parameter [u.sub.*] is computed as a function of [U.sub.x] ([u.sub.*] = 0.03[U.sub.x]), k is von Karman's constant (= 0.4), and [c.sub.p] [Rho] are the specific heat and density of air, respectively ([c.sub.p][Rho] = 1200 J [multiplied by] [m.sup.-3] [multiplied by] [K.sup.-1]; Campbell 1977). The area ([A.sub.sol,diff]) subject to diffuse solar radiation ([S.sub.diff]) is estimated as 0.4 A, and the area subject to convection, [A.sub.conv], = 0.5A (where A = total surface area of one mussel). All temperatures are in units of Kelvin, and constants are as defined for the individual-based model (Helmuth 1997, 1998). In this model, convective heat flux is determined by the shear velocity above the bed rather than by a coefficient of heat transfer (Helmuth 1998). Corrections of areas of exchange are thus necessary to compute the average body temperature of mussels within a bed. For example, the ratio [A.sub.rad,sky]/[A.sub.p] represents the total surface area exposed to long-wave radiation per unit of mussel bed; in a tightly packed bed Ap equals the projected area of a mussel and is estimated as 0.15A, as is the projected area ([A.sub.sol,dir]) subject to direct solar flux ([S.sub.dir]). Similarly, [A.sub.sol,diff]/[A.sub.p], and [A.sub.conv]/[A.sub.p] account for the total areas exposed to diffuse solar radiation and convection per unit of mussel bed projected area, respectively. For this model, mussels in aggregations are assumed to have 40% of their total surface area (A) exposed to long-wave radiation to the sky ([A.sub.rad,sky] = 0.4A), 40% to the ground ([A.sub.rad,ground] = 0.4A) and 20% to other mussels within the bed. Because all mussels within the bed are assumed to be at approximately the same body temperature, this latter term is ignored. The "effective emissivity" of the sky, [[Epsilon].sub.sky], is calculated as a function of air temperature and cloud cover according to empirical fits presented by Idso and Jackson (1969) and Campbell (1977). Under perfectly clear skies, the emissivity ([[Epsilon].sub.clear]) is defined by

[[Epsilon].sub.clear] = 0.72 + 0.005 ([T.sub.[a.sup.x]] - 273). (B.2)

The equation is then adjusted for cloud cover:

[[Epsilon].sub.sky] = [[Epsilon].sub.clear] + [C.sub.sky](1 - [[Epsilon].sub.clear] - 8/[T.sub.[a.sup.x]]) (B.3)

where [C.sub.sky] is the fraction of the sky covered by clouds and is reported in the TMY2 data set.

Rearranging to solve for [T.sub.bed] and substituting [T.sub.[a.sup.x]] for [T.sub.g], the steady-state solution to the heat balance equation for a mussel bed is:

[Mathematical Expression Omitted] (B.4)

where [Mathematical Expression Omitted]; [Mathematical Expression Omitted] and [Mathematical Expression Omitted].

The equation uses environmental inputs of air temperature and wind speed at buoy height (z = 4 m), and direct and diffuse solar radiation and cloud cover at Quillayute to generate predictions of body temperature every hour throughout an entire year. Specific values from this "look-up table" are then chosen using estimated times of aerial exposure, as determined by predictions of the tidal cycle. Only hours in which mussels were aerially exposed for 45 minutes or greater (the approximate time constant of a mussel in a bed) are included. For these simulations, the average rate of mass (water) lost through evaporation, [Mathematical Expression Omitted], was considered to either be zero (conditions of no gape) or a constant mass loss of 5% initial body mass per mussel per hour (assuming a density of 700 mussels/[m.sup.2] and a body length of 7.5 cm this equals 1.99 [10.sup.-4] kg [multiplied by] [s.sup.-1] [multiplied by] [m.sup.-2]; see Helmuth [1998] for allometry equations used). Evaporative cooling was not permitted in either scenario under conditions of relative humidities [greater than or equal to] 95% (assumed to equal those reported for Quillayute). Similarly, mussels were not permitted to evaporatively cool when body temperature was lower than the dew point temperature (Campbell 1977, Helmuth 1998).

Model tests

Unfortunately, because the National Solar Radiation Data Base stopped reporting data in 1990, predictions of body temperature generated using this data base can no longer be compared to those directly measured in the field. However, hourly measurements of wind speed and air temperature at Quillayute are still being reported by the Western Regional Climate Center (Table 1). As a preliminary test of the accuracy of the model presented here, I compared the means of measurements in body temperature from bed mussels at Crescent Bay made during the summer of 1995 and the spring and summer of 1996 to predictions based on data reported from Quillayute. Body temperatures were tracked in sets of 1-4 mussels for periods ranging from 25 to 60 min for each of 15 hourly intervals. Free stream wind speeds and air temperatures were calculated using the methods described in Appendix A. Because solar data were not available, I measured this parameter directly using a pyranometer. This test, therefore, does not include any errors that may have been introduced by differences in solar radiation between Quillayute and Crescent Bay. Data on cloud cover were also unavailable, so I used a generic estimate of 0.90 for [[Epsilon].sub.sky] (Helmuth 1998).

Results indicated that within any given hour-long interval, the error between predicted and measured body temperatures ranged from 0.1 [degrees] to 6.3 [degrees] C. On average, however, estimates of steady-state body temperatures calculated using weather data from Quillayute were within 2.3 [degrees] [+ or -] 1.6 [degrees] C of the mean body temperature measured during that time interval in the field.
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Date:Jan 1, 1999
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