Reference: Biol. Bull. 183: 220-232. (October, 1992)
Biological Consequences of Topography on Wave-swept Rocky Shores: I. Enhancement of External Fertilization
MARK DENNY, JEFF DAIRIKI’, AND SANDRA DISTEFANO Department of Biological Sciences, Stanford University, Hopkins Marine Station, Pacific Grove, California, 93950
Abstract. Surge channels on wave-swept rocky shores males are present, but is lower at higher velocities (Pen- are characterized by the violent hydrodynamic mixing nington, 1985; Levitan et al., 1992). Even in fish that that accompanies broken waves. It has been suggested actively pair to spawn, the fraction of eggs fertilized can that this mixing rapidly dilutes gametes shed into the surf be low (40-50% ) when the flow exceeds 0.1 m s-l and zone, thereby severely reducing the fraction of eggs that wave-induced turbulence disperses the gametes (Petersen, can be fertilized externally. Although surge channels are 1991; Petersen, et al. 1992). well mixed within themselves, field experiments show that These studies (and the low rates of fertilization they the exchange of water between these small embayments report) raise important questions about the role of water and the adjacent mainstream is surprisingly slow. Thus, motion in external fertilization and the consequences it surge channels may act as “containment vessels,” limiting may have in the evolution of reproductive strategies. the rate at which gametes are diluted, and thereby en- However, flow during these experiments was relatively hancing the efficacy of external fertilization. Indeed, a benign. How effective can external fertilization be under mathematical model of fertilization in surge channels more extreme hydrodynamic conditions? suggests that given a sufficient population of adult males As ocean waves break upon rocky shores they gen- within a surge channel, 80- 100% of eggs may be fertilized. erate water velocities as high as 15 m SC’ (Denny, 1988; This result must be tempered, however, by the possibility Vogel, 198 1 ), twenty to loo-fold faster than those in that the small-scale shears induced by turbulence interfere the studies cited above. These high velocities are ac- with fertilization. companied by much more violent mixing of water. The “white water” characteristic of the surf zone is visual Introduction evidence of the extreme turbulence in breaking waves, Recent field experiments with marine organisms have and Denny and Shibata ( 1989) proposed that turbu- demonstrated that the fraction of eggs externally fertilized lence in the benthic boundary layer of the surf zone during spawning can be adversely affected by water mo- reduces the effectiveness of external fertilization well tion. For example, when water velocity is 0.1-0.2 m s -’ below even the low values measured subtidally. They and an individual male is separated from a female by estimate that only about 0.0 1-O. 1% of eggs released into more than 5 m, studies of sea urchins (Pennington, 1985; the surf zone are fertilized by a single adult male under Levitan, 199 1) and a hydroid (Yund, 1990) show that typical conditions, and a maximum of only about 3% only 5-15% of eggs are fertilized. This low rate of fertil- are fertilized when multiple males spawn synchro- ization is attributed to the dilution of gametes by turbulent nously. These predicted rates of fertilization are one to mixing in flow. The percentage of eggs fertilized can be two orders of magnitude lower than those measured in somewhat higher (approximately 30-50%) if multiple subtidal flows, and suggest even more dramatic con- sequences for intertidal plants and animals than those Received 1 I April 1992; accepted 27 July 1992. proposed for subtidal organisms. ’ Present address: School of Oceanography WB-10, University of Denny and Shibata ( 1989) note, however, that the Washington, Seattle, Washington, 98195. fraction of eggs fertilized in the surf zone could be sub-
220 FERTILIZATION IN SURGE CHANNELS 221 stantially higher than they predict if the volume into which heights provided an estimate of the mean low water level. gametes can be diluted is limited. Experiments described The number of surges encountered, divided by the 5-min here show that blind-ended surge channels (the small- duration of the recording session, provided an estimate scale embayments typical of rocky shores) can act to limit of the average period of the surge. Periods ranged from the dilution volume for gametes of benthic organisms. about 9- 14 s. The height of each surge was estimated as Shoreline topography may thereby enhance the effective- the maximal height minus the subsequent minimal height. ness of external fertilization in the surf zone, and the frac- Heights recorded in each 5-min period were averaged to tion of eggs fertilized in surge channels may actually be provide an estimate, Havg, of the sea state during each similar to that found in subtidal habitats. trial. Average surge heights varied from 0.14 m to 0.74 m. Materials and Methods The concentration of dye in each sample was measured with an Aminco model SPF 500 spectrofluorometer and Field experiments were conducted on the shoreline ad- a standard series of dilutions from the stock dye solution. jacent to Hopkins Marine Station, Pacific Grove, Cali- Concentrations were corrected for any background lluo- fornia. Four blind-ended surge channels (essentially, rescence present in the control sample, and were expressed small-scale bays) were chosen to span the size typical of as a fraction, C,, of the concentration in the initial ex- this coastline. Each was surveyed with a surveyor’s level, perimental sample. A model of the time series of relative tape measure, and stadial rod to provide the location (in concentrations from each trial was then determined by cylindrical coordinates) of loo-150 points, and these fitting the data to the equation: points were analyzed (Surfer, Golden Software, Inc.) to C,(t) = a exp( -kt) + c; produce a topographic map of each surge channel (Fig. (Eq.1) 1). From these maps, the volume of each channel was a simplex algorithm (Caceci and Cachet-is, 1984) was used estimated as a function of still water level (Fig. 2). Volume to estimate the least-squares fit. The parameter c represents (at approximately the mean still-water level present during the quasi-equilibrium concentration of dye at the end of this study) ranged from 3.5 to 45 m3. each trial when the concentration in the surge channel is The residence time of water in these channels was mea- equal to that in the adjacent mainstream flow. In all cases, sured at high tide as follows. A control sample of water c is small (typically
Benchmark Seagull Channel
8 8
2
0 2 4 6 8 10 0 2 4 Meters Meters
Dogrock Junior
8
6
6
2
0 0 0 2 4 6 0 2 Meters Meters
Figure 1. Topographical maps of the four embayments used in this study. Vertical gradations are 0. I m: hachures denote local depressions. the channel the water level rises, increasing the chan- ‘IV nel’s volume by an amount AV. If the initial volume Probabthty per surge = (v,, + TV) (Eq. 3) (V,) and added volume (AV) are thoroughly mixed. the probability that a particle is carried out of the chan- But the mixing of initial and added volumes may not be nel as the surge subsides and the water returns to its complete; in such a case the probability of escape from original level is the channel is only a fraction. m. (0 I m I 1) of this FERTILIZATION IN SURGE CHANNELS 223
shore that lie shoreward of this average were defined to be surge channels; segments seaward of the average shore- line were promontories. At each scale, the fraction of the average shoreline that comprised the mouths of surge channels, A, was then taken as a measure of relative surge channel abundance. This procedure was repeated for sev- eral starting points and the results averaged. The abun- dance of surge channels was measured at scales ranging from 2.2 to 22.4 m. The shore at Hopkins Marine Station is formed from granite, and is not atypical of rocky shores on the west coast of North America. -2 -1 0 1 2
Still Water Level (m) Theory of external fertilization Figure 2. The relation between still water level and surge channel volume for the four embayments used in this study. These curves were If the exchange between water in surge channels and calculated using the topographical maps of Figure I. that in the adjacent mainstream is slow, then surge chan- nels may function as “containment vessels” for gametes, thereby allowing for the effective fertilization of eggs. This maximal probability. If the average period between surges possibility can be examined mathematically. is T seconds, the probability that a dye particle is carried In this analysis, we follow the model of Denny and out of the channel in one second is thus Shibata ( 1989), based primarily on data from sea urchins. Males are assumed to release sperm into a surge channel at a constant rate Qs per second resulting in a concentra- tion of sperm C, (with units of mv3) that may vary through time. Denny and Shibata ( 1989) also allowed C, to vary where k is again the exchange parameter of Eq. 1. in space, but for present purposes we assume that sperm Vo, AV, T, and k can be determined experimentally, are thoroughly mixed, ensuring that C, is the same every- allowing us to calculate the mixing parameter m: where within the surge channel. Sperm (like dye particles) have a probability k in each second of passively escaping m= k(V, + AV)T AV ’ (Eq.5) from the surge channel into the adjacent mainstream, and we assume that sperm that escape from the surge channel Abundance of surge channels do not re-enter. Sperm can swim and, under certain conditions, can The abundance of surge channels on a shore is strongly move preferentially toward eggs (e.g., Miller, 1985). The dependent on the local topography, and therefore may speed of sperm locomotion is very slow, however, a few vary greatly from site to site. Due to the fractal nature of tens of pm s-l. In contrast, the motion of neutrally buoy- shorelines (Mandelbrot, 1982; Denny, in press), it is dif- ficult to quantify precisely the abundance of surge chan- nels for even a single site, because the exact location of &a the shoreline varies with the scale at which the shore is examined. An estimate of the abundance of surge channels on the shores at Hopkins Marine Station was made, how- ever, using the following technique (Fig. 3). An aerial photograph of the shore was traced onto paper to provide a two-dimensional representation of the shore- line with accurate detail present at a scale smaller than that used in the analysis. A point was chosen at random on this trace, and a caliper placed with one tip at this point. The caliper was then “walked” down the trace, and the distance between the tips of the caliper set the scale at which the shoreline was measured. The series of con- Figure 3. An example showing the traced shoreline (thick line), “average” shoreline at a particular scale (thin line segments), and the tiguous lines joining points where the caliper intersected surge channel and promontory portions for a single segment of the average the trace defined one realization of the “average” shoreline shore. This shoreline was traced from an aerial photograph of the shore at that particular scale (Fig. 3). Segments of the actual at Hopkins Marine Station. 224 M. DENNY ET AL. ant particles in turbulent flow can be characterized by a assumed to be the friction velocity u* ) and the effective velocity u* (the friction velocity), a measure of the in- “collision cross section” of an egg (Denny and Shibata, tensity of turbulence in flow (Schlichting, 1979; Denny, 1989), and has units m3 SK’. 1988; Denny and Shibata, 1989). For the intensely tur- If an egg could be fertilized by the first sperm that con- bulent flow found in the surf zone of wave-swept shores, tacted any point on its surface, the “collision cross section” u* is approximately equal to 0.1 uavg, where u,,,~ is the would be the same as the projected area of the egg. For ensemble average velocity of the water (see Denny, 1988; sea urchin eggs with a diameter of lop4 m this is 7.85 Denny and Shibata, 1989). Water velocities in the surf X 10 -9 m2. Laboratory measurements suggest, however, zone vary from approximately 1- 15 m s-’ (Denny, 1988)) that sea urchin eggs behave as if only a small fraction of suggesting that u* is 0. l- 1.5 m SK’ . Thus, the velocity their surface area is fertilizable. The reason for this reduced imposed by turbulence is four to five orders of magnitude cross section is unclear. Vogel et al. ( 1982) suggest that faster than a sperm’s swimming speed, and we assume apparent fertilizable area is only about 1% of the overall that the swimming speed of sperm can be safely neglected area for Paracentrotus lividus. More recent experiments in the present context. with Strongylocentrotus fransiscanus (Levitan et al., 199 1) Given these assumptions, we see that the rate of change suggest an apparent fraction of 3%. Using this latter value, of sperm concentration is equal to the difference between we find that the rate at which concentration increases due to the release C$= uJ2.4 X lo-” m2) of sperm and the rate at which concentration decreases 3 (Eq. 13) due to the passive escape of sperm to the mainstream: and because u* x 0.1 uavg, -=--G(t) Qs kC (t) 4 e u&2.4 X lo-” m2). (Eq. 14) dt V ” Given that average water velocities in the surf zone vary where V is the still-water volume of the surge channel. from approximately 1 to 15 m s-l, 4 is likely to vary from If we assume that C,(O) = 0, we calculate that 2.4 X lo-” to 3.5 X lo-” m3 s-‘. Denny and Shibata ( 1989) calculated that an individual urchin extrudes sperm at approximately 10’ s-’ or eggs C,(t) = 2 [1 - exp(-kt)]. (Eq.7) at lo4 s-’ , and these values are used here. Due to the vast At infinite time C, reaches a steady-state concentration, overabundance of sperm it seems safe to assume that the c * concentration of sperm is minimally affected by the at- s,co* tachment of sperm to eggs; thus QS C =- C,” = G. (Eq. 15) ‘** kV ’ (Q-8) Given these assumptions, we can describe the flux of virgin We assume that females release eggs at the constant eggs through a surge channel. Virgin eggs are added to rate Q, per second, and that eggs are mixed and exchanged the local population as they are released by females and as for sperm. Thus, are removed either by escaping to the mainstream or by dG(t) Qe fertilization. Thus: - = v - kc,(t), (Eq.9) dt dG,v(t) Qe = v - K,,(t) - Kdt)Ce,dt). (Eq. 16) dt C,(t) = 2 [l - exp(-kt)], (Eq. 10) At this point the mathematics can be simplified if we Qe convert each of the variables of interest to a dimensionless C =- (Eq.11) form. In the present context, time is important primarily ‘** kV ’ as it relates to the residence time of gametes in a surge where C, is the instantaneous concentration of eggs ( rnm3) channel. Thus we may specify a dimensionless time 7, and C,, is the steady-state concentration. 7 = t/T, = kt. We assume that the rate at which sperm fertilize eggs (Eq. 17) is governed by the co-occurring concentrations of both Note that sperm and eggs and by a “reaction” parameter, 4. dg dg dt 1 dg -c--c-- 0%. 18) Rate of fertilization = @C,,(t)C,,Jt). (Eq. 12) dr dt dr k dt Here, C,, and C,, are the concentrations of unattached for any function g. sperm and virgin eggs, respectively. The instantaneous concentration of eggs can be ex- The reaction parameter is the product of the effective pressed in dimensionless form by dividing by the steady- speed at which sperm are mixed through the water (here state concentration of eggs. Thus, FERTILIZATION IN SURGE CHANNELS 225
v([i + 1167) e(7) = y - CAT) (Eq. 19) e,m (QJW * = v(is7) { 1 - (&/2)[1 + 0 - 0 exp - (idr + Y2&)]} Similarly, the instantaneous concentration of virgin eggs { 1 + (&/2)[1 + 0 - 0 exp - (i& + 1/2&)]} is: G,“(~) _ C,,“(7) + {1+(&/2)[1 +G-60ierp-(i&+i/2&)])’ v(7) = (Eq.20) C e,oo (Qe/W * (Eq. 28) The instantaneous concentration of sperm is made di- where 8~ is the time step and i is a counter (i = 0, 1, 2, mensionless by expressing it as a fraction of the steady- 3 , . . .). At 7 = 0, all eggs are assumed to be virgin so that state concentration of sperm: v(0) = 1. We again set f = 1 - v and follow f as a function of G(7) G(T) s(7)=c= (J3.21) dimensionless time. s,co (Qs/kV) * Results Finally, we devise a dimensionless fertilization parameter 0 that incorporates several of the variables that govern Mixing in surge channels the rate of fertilization: The decrease in dye concentration within a surge channel is modeled accurately by the exponential decrease of Equa- tion 1 (Fig. 4). The average coefficient of determination (Eq.22) (r2) for the 70 trials is 0.955 (with a standard deviation of Inserting these dimensionless variables into our previous 0.054), indicating that, to a first approximation, each dye results (Eqs. 7, 8, 10, 11, and 16), we find that: particle can be regarded as having a fixed probability of escaping from the surge channel in a given interval. S(T) = e(T) = 1 - exp(-r), (Eq.23) Exchange between surge channel and mainstream water s, = em = 1, is surprisingly slow. The exchange parameter k varies from (Eq.24) approximately 0.00 1 to 0.04, implying average residence times, T,, of 25 to 1000 s. -MT) = 1 - v(7) - OS(7)V(T). dr (Eq.25) The mixing parameter, m, varies from about 0.05 to 0.9. When data from all four surge channels are combined, Thus, at steady state (that is, when dv(~)/dr = 0), m is found to be an increasing function of average surge VCO= l/(1 +o). (Eq.26) height (Fig. 5 ), but variation in H,, explains only 17% of the variation in m. The mixing parameter is not sig- In other words, the larger 0 is, the lower the steady-state nificantly correlated with the average fractional change in fraction of virgin eggs, v,. A large 0 can result from a volume during a downsurge, AV/( V0 + AV) (r = 0.06 19, large rate of sperm release (for example, due to the pres- 68 df, P > 0.5). ence of multiple males), a small surge channel volume, When data from all four surge channels are combined, an intense level of turbulence, or a long residence time the exchange parameter k is found to be an increasing (Eq. 22). Thus, any combination of these factors produces function of average surge height (Fig. 6), and the rela- a lowered fraction of virgin eggs. Note, however, that 0 tionship explains 5 1% of the variation in k. The exchange scales with the square of residence time, so v, is especially parameter is also positively correlated with the fractional sensitive to T, (or, equivalently, to k). change in volume in the surge channel, which is itself a Now, the concentration of fertilized eggs is equal to the direct function of Havg, but the correlation (k = 0.179 difference between the total concentration of eggs and the H,, + 0.0023, r = 0.431, 68 df) explains much less of concentration of virgin eggs. By analogy, the fraction of the variation than does surge height alone ( 19 vs. 5 1%). all eggs that are fertilized, f( T ) , is equal to 1 - v ( 7 ) . From Equation 26 we see that at steady-state Abundance of surge channels f, = O/( 1 + 0). (Eq.27) Surge channels form about 46% of the shoreline at Hopkins Marine Station (Table I). The fractional abun- Non-steady-state conditions dance of surge channels is not correlated with the scale at which the shoreline is measured (r = 0.134, 3 df, P This result applies to surge channels once they have > 0.5), at least for the range of scales examined here. reached a steady-state concentration of gametes~Is it realistic to apply these steady-state results to the real world? To an- Predicted fraction of eggs fertilized swerthis question, we solved Equation 25 for v as a function At a realistic value of 0 (that is, a realistic combination of 7 using an implicit tmpazoidal finite difference technique: of k, V, 4, and Q,), a substantial fraction of eggs is expected 226 M. DENNY ET AL.
1.0 0.9 Seagull Channel November 6, 1989 0.8 0.8 g c5 H .“, - 0.32 m ‘zi 0.7 I 0.7 5 k - 0.0102 & k - 0.0071 e 0.6 c 0.6 :: r2 = 0.996 :: r2 = 0.988 E! 0.5 2 0.5 u u : 0.4 :! 0.4 ‘2 P 4 0.3 ; 0.3 rr: d 0.2 0.2 0.1 0.1 0.0 0.0 0 100 200 300 400 500 600 0 150 300 450 600 750
Time (s) Time (s)
1.0 n I 1.0 9 Dogrock 0.9 October 27, 1988 0.8 H ..I - 0.40 m 8 \ 0 NovemberHk .,,r2Junior - - 0.00990.984 0.2528, 1989m k - 0.150 ‘S 0.7 6 r’ = 0.983 5 0.6 ” E 0.5 q u ,” 0.4 ‘S ; 0.3 w 0.2 0.1 - 0.1 0.0 0 100 200 300 400 So0 0 150 300 450 600 750
Time (s) Time (s) Figure 4. Representative examples of the temporal decay in dye concentration in the experimental embayments. The solid curves are least-squares fits to the data using the model of Equation 1. to be fertilized (Fig. 7). For example, if 4 is lo-” m3 s-’ , density of male urchins, smaller channels will have a and the residence time is 50 s (k = 0.02 s-l) for a surge higher fraction of eggs fertilized. channel with a volume of 10 m3 (typical values for the shore at Hopkins Marine Station), about 20% of eggs can be fer- Time dependence offertilization tilized by a single male. This is two to three orders of mag- The time-dependent fraction of eggs fertilized is shown nitude larger than the fraction predicted by Denny and Shi- in Figure 9a. For realistic values of 0, values off within 10% bata ( 1989) for flow outside of surge channels. of steady-state values are obtained within a period of 5 T,. For a typical residence time of 50-120 s, this implies that Multiple males the steady-state estimate of fertilization fraction is reached within 4-10 minutes. Given that an individual urchin in Increasing the number of males present in a surge channel the laboratory releases gametes for approximately 1 h, equi- increases Qs, with a concomitant increase in 0, and thereby librium conditions are likely to apply to most gametes re- an increase in the h-action of eggsfertilized (Fig. 8). Consider, leased. for instance, a surge channel with the same residence time Note that Figure 9a shows the fraction of eggs fertilized as that cited above (k = 0.02 s-l), but with five times the relative to the steady-state fraction. The absolute fraction volume (50 m3). If this channel has an average still water fertilized is shown as a function of dimensionless time in depth of 1 m, it has approximately 70 m2 of substratum Figure 9b. The fraction fertilized decreases with decreasing area. Even given the fivefold increase in volume, if only 3 0 as expected. males are present per square meter of substratum ( 2 10 males Discussion total, a realistic value for wave-swept shores), 90% of eggs are expected to be fertilized. Mixing in small surge channels Smaller channels will have a larger ratio of substratum The rate of exchange between small, well-mixed surge area to channel volume. Thus, given the same spatial channels and mainstream waters is both a simple and a FERTILIZATION IN SURGE CHANNELS 227
Table I Fraction of the average shoreline that comprises the mouths of surge channels at Hopkins Marine Station
Scale (m) Fraction, A
2.2 0.446 4.5 0.440 9.0 0.508 22.4 0.455
only to a relatively narrow range of channel sizes. There I-L (ml is a lower limit to the size at which a local indentation in Figure 5. The relation between average surge height (H& and the the shoreline can be considered a surge channel for our dimensionless mixing parameter, m. The linear regression (solid line) purposes. If the still-water depth in the indentation is less explains approximately 17% of the variation in m: m = 0.424 X H,, than the amplitude of the surge (=H,,/2), the indenta- + 0.094 (r2 = 0.173), where H,, is measured in m. tion is completely emptied by an average downsurge, and the probability of a particle re-entering the channel on complex process. Our data indicate that the probability per the subsequent upsurge is set more by mainstream mixing time that a small, inanimate, neutrally buoyant particle will and advection than by mixing within the indentation itself. escape from a surge channel is constant for at least a few In this case, the indentation is not a surge channel in the tens of minutes, and that this probability can be predicted sense that we have used the term here. with reasonable accuracy from the height of the surge present The upper limit to the size of this relationship is set by in the surge channel. In this respect the results of the process the size at which a surge channel is no longer well mixed are simple to predict as shown in Figure 6. within itself. Only if the channel is well mixed can Equa- It is more difficult to account for these predictions on tion 1 accurately model residence time. We have noticed a mechanistic basis, primarily because of the variability distinct advective patterns in bays in excess of the size of the mixing parameter, m. At present, m cannot be pre- studied here-particles that enter a larger bay at one por- dicted accurately, from either average surge height (Fig. tion of its mouth predictably exit at another specific re- 4) or fractional change in volume. Probably, m is affected gion. In this case, the probability of exchange for a particle by the local topography of each surge channel and by depends in part on its history, and the simple model of chance variation in the shape of each breaking wave, ren- Equation 1 cannot be expected to hold. Thus, the pre- dering it of little value as a general predictive tool. dictions implicit in Figure 6 should not be applied to surge Note that the relationship measured here between surge channels larger than approximately 100 m3, height and exchange parameter (Fig. 6) can be applied
0.04 , 1.00 I
a 0.02
0.01
0.00 0.00 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80
KS Cm) k Figure 6. The relation between average surge height (H,) and the Figure 7. As the probability ofgametes escaping from a surge channel exchange parameter k. The linear regression (solid line) explains ap increases, the equilibrium fraction of eggs fertilized decreases. The larger proximately 51% of the variation in k: k = 0.036 X H,, + 0.002 ( r2 the volume of the surge channel (V), the lower the fraction of eggs = 0.5 14), where H,, is measured in m, and k has the units s-i. fertilized. 228 M. DENNY ET AL. and sperm are mixed, dilution of gametes is so rapid that those few eggs that are fertilized are likely to be fertilized w 0.8 by sperm from the nearest males. In this fashion, rapid 8 z 0.7 dilution insures the localization of fertilization. t 0.6 As a corollary to the effectiveness of fertilization in surge Q w” 0.5 channels, we note the implication that the long residence time of water in channels can again lead to localization of fertilization. Once gametes escape from a channel they k = 0.02 s” are subject to the infinite dilution described by Denny Qs - 10’ S’ mald’ and Shibata ( 1989)) and subsequent fertilization becomes #$ 5 l(pm’ s.l highly unlikely. Eggs released in a surge channel are therefore very likely to be fertilized by males in the same 0 20 40 60 80 100 120 140 160 180 200 channel. Thus both infinite dilution and the limitation of Number of Males dilution result in localized fertilization, leading us to sup- Figure 8. The fraction of eggs fertilized increases with the number pose that, as a general rule, external fertilization in the of males present in a surge channel. surf zone can be effective only over short distances. The effect of localized fertilization due to surge channels The fraction of eggs fertilized on the population genetics of a species is unclear. Although the residence time of small surge channels is surprisingly The predictions made here suggest that external fertil- long, it is still short compared to the larval lifetime of ization in surge channels may be quite effective, with fer- most species, which vary from a few hours to several tilization fractions comparable to or exceeding those measured for benthic invertebrates in relatively benign I subtidal flows (Pennington, 1985; Yund, 1990; Levitan, 199 1; Levitan et al., 1992). If these predictions are valid, fertilization may be much less of a limiting process than suggested by the model of Denny and Shibata ( 1989), at least within surge channels. Furthermore, about 46% of the shoreline at Hopkins Ma- rine Station is formed of surge channels (Table II). If we assume that the fraction of eggs fertilized within the surf zone, but outside of surge channels, is equal to the maximal fraction predicted by Denny and Shibata ( 1989) (3%), and that individuals are randomly distributed along the shore, then the overall fraction of eggs fertilized is approximately 0 1 2 3 4 5 Overall fraction = (f - 0.03)A + 0.03, (h-29) T O/q) where f is again the steady-state fraction of eggs fertilized 1.0 in surge channels, and A is the fraction of shoreline that comprises the mouths of channels. For f = 0.9 and A ‘CI 0.8 I ” = 0.46, the overall fraction of eggs fertilized is thus 43’70, ” suggesting that external fertilization in the surf zone, taken 2 0.6 as a whole, may be considerably higher than that suggested 0 2 by Denny and Shibata ( 1989). 2 0.4 The role of shoreline topography in controlling the overall .s rate of external fertilization is clear in Equation 29. If the s c 0.2 shore is constructed of rocks that are not susceptible to the formation of surge channels, A will be small, and the overall 0.0 fraction of eggs fertilized will be similarly small. Conversely, 0 1 2 3 4 5 on a shore where surge channels are abundant, external fer- tilization may be quite effective. 7 (t/T,) Figure 9. (A) Steady-state conditions are approached within a period Localized fertilization equal to 5 times the retention time (r = 5). The fraction ofeggs fertilized is expressed as a fraction of the steady-state value. (B) The actual fraction Denny and Shibata ( 1989) concluded that, as long as of eggs fertilized is plotted without Being normalized to the fraction at there is no practical limit to the volume into which eggs steady state. FERTILIZATION IN SURGE CHANNELS 229 months. As a result, although fertilization may be local- P’ x 0.177pg3’2H5’2. (h-31 1 ized, the resulting larvae are free to participate in the larger-scale dynamics of the population. We recognize that In making this calculation we have assumed that wave the long residence time of surge channels may have im- height at breaking is equal to water depth (Denny, 1988), portant implications for the spatial pattern of settlement and that waves at breaking move at the speed predicted of larvae, as well as for the probability that newly fertilized by solitary wave theory (in this case, ( 2gH)‘j2 [Denny, 19881). embryos will be consumed by benthic suspension feeders, Some of this incident wave energy may be carried back but these implications will not be developed here. out of the surge channel if the wave reflects from the shore. Instantaneous mixing, shear stress,and a note of If the height of the reflected wave is a fraction R of the caution incoming wave height (0 I R < 1)) the net power deliv- ered to the channel (per meter of channel mouth) is We foresee two factors that could affect the results pre- P:, x 0.177pg3’2Hs’2( 1 - Rs’2). sented here. 0%. 32) First, the model of external fertilization assumes that On rocky shores, R (the reflection coefficient) is typically gametes are instantaneously mixed throughout a surge small. For example, the R for a rubble breakwater (similar channel. As noted, however, thorough mixing within a in topography to a rocky shore) is less than 0.29 when channel requires the entry of one to three breaking waves. the slope of the breakwater is 1:5 (U.S. Army Corps of In the brief period prior to the arrival of the first wave, Engineers, 1984). If the slope is 1: 10, the reflection coef- local high concentrations of gametes may be present, and ficient is less than 0.13. If energy is transported inshore if eggs encounter an area of high sperm concentration, by waves but only a small fraction is carried back out by the probability of fertilization may be very high. Note, waves, most of the incident wave power must either be however, that any momentary reduction in mixing that dissipated in the surf zone or transported out by other allows a high concentration of sperm to be maintained means (e.g., kinetic energy associated with an undertow, also reduces the probability that an egg will be delivered, acoustic energy). Effective transport by these other mech- by chance, to that area of high concentration. Quantitative anisms appears unlikely, and we assume here that most analysis of the dynamics of fertilization over times shorter wave energy is dissipated in the surf zone. than the period of the surge will require a more complete The magnitude of this rate of energy dissipation can be understanding of the structure of turbulence within surge estimated from a simple example. The bottom of a surge channels than is currently available. channel has a slope (Y, as shown in Figure 10, and the wave Second, the calculations made here are based on the height at the mouth of the channel is equal to the depth at assumption that sperm and eggs interact in a fashion akin the mouth of the channel. Given these conditions, waves to molecules in a gas. If a sperm happens to be in the break near the channel mouth, and the entire net energy of vicinity of an egg and aimed in the right direction, it im- the waves is dissipated within the channel. pacts on the egg, sticks, and (perhaps) fertilization follows. The still-water volume of the surge channel per width However, this simple view ignores the small-scale hydro- of mouth is dynamics relevant to sperm and eggs, and as a result may overestimate the probability of fertilization. (Eq.33) To see why, we explore flow in the surf zone at the spatial scale of a gamete. This exploration is carried out As a breaking wave moves into the channel, its height in two steps. First, we examine the rate at which turbulent decreases as wave energy is first converted to tubulent kinetic energy is dissipated in the surf zone. Next, we use kinetic energy and then dissipated. Thornton and Guza these results to estimate the average velocity gradient to ( 1983 ) have shown that the average (root mean square) which gametes are subjected. The biological consequences of this gradient can then be discussed. Ocean waves carry energy with them as they approach Wave Height, H the shore. The rate, P, at which wave energy is transported across the mouth of a surge channel M meters wide is P x ( 1/8)pgH2M(2gH)“2 x 0.1 77pg3’2H5’2M, (Eq. 30) where p is the density of seawater (approximately 1025 kg mm3), g is the acceleration due to gravity (9.8 1 m s-’ ), Figure 10. A schematic cross section through a surge channel illus- and H is wave height (Denny, 1988). Thus, the power trating the relation between bottom slope (a) and water depth, and per meter of surge channel mouth, P’, is showing the typical shape of a broken wave. 230 M. DENNY ET AL. height of waves after they break on a given shore is a d = /d/3. (Eq. 37) constant fraction of the local water depth. For a sandy beach, this fraction was 0.41 and it is likely to be similar If the cube has sides of length dx, the force required to on rocky shores. For ease of computation, however, we shear the cube is thus use a value of 0.5. As the broken wave propagates into F = dx2/.& (Eq.38) the channel, we assume that its shape approximates that of a turbulent bore; that is, it has a steep leading face, and The power, P, dissipated by this force is equal to the prod- the water level behind the face is nearly equal to that of uct of force and velocity. The velocity at the top of the the wave crest (Fig. 10). Given these assumptions, the cube differs from that at the bottom by dx/3. Therefore, broken wave increases the volume of water in the surge P = dx3/.Q2. (Eq.39) channel by maximally i/4 the channel’s still-water value. Thus. the maximal effective volume of water in which The power dissipated per volume ( dx3) is thus the wave energy is dissipated is W’ = pFcp2, (Eq.40) and the power dissipated per mass is (Eq.34) t = d2/P. (Eq.41) The minimal rate at which wave energy is dissipated per Equating Equations 35 and 40 and solving for /3, we find unit of effective water volume ( W’) in the surge channel that the average velocity gradient in the fluid in a surge is thus, channel is: W’ = P’,/V’ a 0.283pg3j2H *12( 1 - R512) tan (Y. P a% e ()~32~‘/2~3/4H1/4( 1 _ R5/2)‘/2(bn ,)1/$-‘0, Wi. 35) m. 42) Now, the turbulent kinetic energy due to wave breaking and (from Eq. 37) the average shear stress in the fluid is is ultimately dissipated as heat via the action of viscosity. M ()532pl/2g3/4H1/4( 1 - R5/2)‘/2(tan a)1/2p1/2a This mechanism is elegantly described by Lazier and ~a, Mann ( 1989): at small spatial scales, turbulent eddies in 0%. 43) the broken wave are damped by the water’s viscosity, Note that these are average values for shear and shear forming in their stead a linear velocity gradient (or shear) stress. At any point in the fluid, instantaneous values likely within the fluid that varies randomly in direction and vary through time. For each interval where the shear stress strength. It is the interaction of this shear with viscosity is lower than average, however, there must be a compen- that converts turbulent kinetic energy to heat. sating interval in which stress is larger than average. The scale at which the chaotic motion of turbulent ed- We are now in a position to evaluate directly the small- dies is effectively damped can be estimated from the Kol- scale flow regime in surge channels and to explore its bio- mogorov length scale, L,,: logical consequences. / ..3 \ l/4 For a slope of 1:5, a breaking wave height of 1 m, a re- Wq. 36) flection coefficient of 0.29, and a dynamic viscosity of 1 X 10m3 Pa s, the average rate of energy dissipation within a where p is the dynamic viscosity of seawater ( 1 to 2 surge channel is 1.66 W kg-’ (Eqs. 40 and 4 1). From this X 10d3 Pa s, depending on temperature), p is again the value we estimate that the eddies associated with maximal water’s density, and E is the rate at which turbulent kinetic shear stresses are approximately 1094 pm in diameter, 1 l- energy is dissipated per mass of fluid (W kg-‘). The largest 14 times the diameter of sea urchin eggs ( 80- 100 pm). The shears are found in eddies with a diameter equal to approx- smallest eddies likely to be found in surf-zone flow are about imately 40 L. Smaller eddies are much less energetic, and 137-273 pm in diameter. Even these small eddies are larger few eddies have a diameter less than 5- 10 L,, (Lazier and than typical gametes, suggesting that gametes in the surf Mann, 1989). zone experience a flow regime characterized primarily by The fact that wave energy is dissipated via small-scale linear shears rather than by turbulent eddies. velocity gradients provides a method for examining the Note that the heat capacity of water is about 4200 J characteristics of flow on the spatial scale of eggs and sperm. kg-’ K-’ (Weast, 1977). At the dissipation rate in this Consider a small cubic volume of water with its top face example, water is heated by only 0.0004”C s-’ . Thus, the moving relative to its bottom face such that a linear velocity energy dissipation within a surge channel is not sufficient gradient /3 (with units of s-l) is established within the cube. to raise the temperature substantially, even for retention As a result, the water in the cube is sheared. From Newton’s times of 1OO- 1000 s. law of viscosity (Vogel, 198 1) , we know that the shear stress For the typical conditions cited above, the average shear (force per area) required to deform water at this rate is stress in the water is about 1.30 Pa, and the average mi- FERTILIZATION IN SURGE CHANNELS 231 croscale velocity gradient is 1304 s-‘. The presence of Literature Cited this gradient means that the velocity at one side of an egg Caceci, M. S., and W. P. Cacheris. 1984. Fitting curves to data. Byte is quite different from that at the opposite side, which 9: 340-362. causes the egg to rotate. The rate of rotation (in cycles Denny, M. W. 1988. Biology and the Mechanics of the Wave-Swept per second) is (Happel and Brenner, 1983; Kessler, 1986) Environment. Princeton University Press, Princeton, NJ. 329 pp. Denny, M. W. in press. Roles of hydrodynamics in the study of life P (Eq.44) on wave-swept shores, in Ecomorphology, P. C. Wainwright and S. 4,. Reilly, eds. University of Chicago Press, Chicago, IL. Denny, M. W., and M. F. Shibata. 1989. Consequences of surf-zone In other words, an egg in a typical surf zone rotates on turbulence for settlement and external fertilization. Am. Nat. 134: average about 104 times per second! Sperm oriented parallel 859-889. to the velocity gradient (i.e., the velocity at the tip of their Happel, J., and H. Brenner. 1983. Low Reynolds Number Hydrody- flagellum is maximally different from that at their head) will namics. Martinus Nijhoff Publishers, Dordrecht. 553 pp. The external dynamics of swimming micro-or- also be rotated by the shear, but because sperm are elongated Kessler, J. 0. 1986. ganisms. Pp. 257-307 in Progress in Phycological Research, Vol. 4, rather than spherical, rotation is likely to cease once the F. E. Round and D. J. Chapman, eds. Biopress Ltd., Bristol. sperm lies perpendicular to the instantaneous gradient (i.e., Lazier, J. R. N., and K. H. Mann. 1989. Turbulence and diffusive with its head and tail in flow at the same speed). layers around small organisms. Deep-Sea Res. 36: 172 I- 1733. The practical results of the rapid rotation of eggs, the Levitan, D. R. 1991. Influence of body size and population density on rapid re-orientation of sperm, and the large shear stresses fertilization successand reproductive output in a free-spawning in- vertebrate. Biol. Bull. 181: 261-268. in the fluid are difficult to estimate. Certainly the egg, as Levitan, D. R., M. A. Sewell, and F.-S. Chia. 1991. Kinetics of fertil- a solid object, will entrain fluid with it as it rotates, creating ization in the sea urchin Strongylocentrotus franciscanus: interaction a complex, 3-dimensional, temporally variable boundary of gamete dilution, age, and contact time. Biol. Bull. 181: 37 l-378. layer that must be crossed by sperm. Whether the rotation Levitan, D. R., M. A. Sewell, and F.-S. Chia. 1992. How distribution itself can affect contact between egg and sperm is less clear. and abundance influence successin the sea urchin Strongylocentrotus Also unclear is whether surf-zone shear stresses are suf- franciscanus. Ecology 73: 248-254. Mandelbrot, B. B. 1982. The Fractal Geometry ofNature. W. H. Free- ficient to dislodge sperm from the egg once they have man and Co, New York. 460 pp. attached, or to damage the sperm or egg directly, although Miller, R. 1985. Sperm chemo-orientation in the metazoa. Pp. 275- Denny and Shibata ( 1989) note preliminary evidence that 337 in The Biology of Fertilization, Vol. 2, C. B. Metz and A. Mon- even low shear stresses can reduce the fraction of eggs ray, eds. Academic Press, New York. fertilized in the laboratory. Pennington, J. T. 1985. The ecology of fertilization of echinoid eggs: While the precise effects of the small-scale flow regime the consequences of sperm dilution, adult aggregation, and syn- chronous spawning. Biol. Bull. 169: 4 17-430. are unclear, the motions of sperm and egg will surely be Petersen, C. W. 1991. Variation in fertilization rate in the tropical reef complex and highly dynamic in turbulent flow which may fish, Halichores bivattus: correlates and implications. Biol. Bull. 181: also impede effective contact and subsequent fertilization 232-237. and development. As long as the effects of shear stress Petersen, C. W., R. R. Warner, S. Cohen, H. C. Hess, and A. T. Sewelf. and egg rotation remain unknown, the calculations made 1992. Variable pelagic fertilization success:implications for mate choice and spatial patterns of mating. Ecology 73: 39 l-40 1. here concerning the effectiveness of fertilization in surge Schlichting, H. 1979. Boundary Layer Theory. 7th ed. McGraw Hill, channels must be taken with a grain of salt. If turbulence New York. 8 17 pp. inhibits the contact and subsequent attachment of sperm Thornton, E. B., and R. T. Gum. 1983. Transformation of wave height to eggs, or if the rapid rotation of eggs is detrimental to distributions. J. Geophys. Res. 88: 5925-5938. their survival, the calculations made here may overesti- U. S. Army Corps of Engineers. 1984. Shore Protection Manual. mate the effectiveness of fertilization. U. S. Government Printing Office, Washington, DC. Vogel, H., G. Czihak, P. Chang, and W. Wolf. 1982. Fertilization ki- netics of sea urchin eggs. Math. Biosci. 58: 189-2 16. Acknowledgments Vogel, S. 1981. Life in Moving Fluids. Willard Grant Press, Boston, We thank E. C. Bell, B. Gaylord, K. Mead, H. Lasker, Massachusetts. 352 pp. Weast, R. C. 1977. Handbook of Chemistry and Physics. Chemical C. Petersen, and an anonymous reviewer for helpful sug- Rubber Company, Cleveland, Ohio. gestions. This study was funded by NSF grant OCE-87- Yund, P. 0. 1990. An in situ measurement of sperm dispersal in a 11688 to M. Denny. colonial marine hydroid. J. Exp. 2001. 253: 102-106. Appendix I
Symbols used in the text and the equation where each is defined orjrst used
Symbol Definition Units Eauation
A Relative abundance of channels 29 G 0) Concentration of eggs at time t m-3 9 G,” (t) Concentration of virgin eggs mm3 16 C Steady-state egg concentration me3 11 d’;, Relative dye concentration 1 G @) Concentration of sperm at time t rnw3 6 C s,v Concentration of unattached sperm mm3 15 C Steady-state sperm concentration me3 8 e T:) Dimensionless egg concentration 19 em Steady-state e 24 f(7) Fraction of eggs fertilized fm Steady-state fraction of fertilized eggs 27 F Force N 38 g Acceleration due to gravity m s-* 30 H Surge height m 31 k Exchange parameter S-’ 1 L Kolmogorov length scale m 35 Mixing parameter 4 F Energy flux into a surge channel W 30 P Energy flux per meter of channel mouth W m-’ 31 ei Net P’ W m-’ 32 Rate of egg release S-l 9 : Rate of sperm release S--l 6 RS Reflection coefficient 32 s (7) Dimensionless sperm concentration 21 SC9 Equilibrium s 24 t Time S 1 T Period of the surge S 4 T, Expected residence time (l/k) S 2 bg Mean water velocity m s-’ 14 U* Friction velocity m s-’ 13 ” (4 Dimensionless concentration of virgin eggs 20 “CO Equilibrium v 26 V Channel volume at still water level 3 vo Surge channel volume at mean low water ;: 3 V’ Maximal volume per meter of channel mouth m* 34 VkVl Volume per meter channel mouth at SWL 33 AV Surge-induced increase in channel volume z: 3 W’ Cl/V’ W m-3 35 Bottom slope 33 Velocity gradient S-’ 37 Average velocity gradient SC’ 42 Rate of energy dissipation W kg-’ 36 Dimensionless interaction parameter 22 Dynamic viscosity of water Pas 36 Water density kg rnp3 30 Shear stress Pa 37 Average shear stress Pa 43 Dimensionless time 17 Reaction parameter m3 s-’ 12 232