Download Ocean waves, nearshore ecology, and natural selection

 Springer 2006
Aquatic Ecology (2006) 40:439–461
DOI 10.1007/s10452-004-5409-8
-1
Ocean waves, nearshore ecology, and natural selection
Mark W. Denny*
Stanford University, Hopkins Marine Station, Pacific Grove, CA 93950, USA; *Author for correspondence
(e-mail: [email protected])
Received 21 October 2004; accepted in revised form 27 October 2004
Key words: Biomechanics, Drag, Intertidal communities, Lift, Wave theory
Abstract
Although they are subjected to one of the most stressful physical environments on earth, wave-swept rocky
shores support a highly diverse community of plants and animals. The surprising presence of such diversity
amidst severe environmental adversity provides a unique opportunity for exploration of the role of extreme
water flows in community ecology and natural selection. Methods are described by which the maximal
water velocity and acceleration can be predicted for a site on the shore, and from these values maximal
hydrodynamic forces are calculated. These forces can limit the range and foraging activity of some species,
and can determine the rate of disturbance in others, but in general, wave-swept organisms have surprisingly
high factors of safety. This apparent over-design can help to explain the diversity of forms present on waveswept shores, and provides examples of how mechanics can limit the ability of natural selection to guide
specialization. Although flow itself may commonly be prohibited from selecting for optima in morphology,
it nonetheless continues to play a potentially important role in evolution by providing a mechanism for
breaking or dislodging individuals that have been selected by other means.
Introduction
Wave-swept rocky shores are among the most
physically stressful habitats on earth. When the
tide is low, areas of the shore are exposed to
terrestrial conditions, and intertidal organisms
(which are typically of marine descent) must cope
with the possibility of desiccation, increased irradiance by ultraviolet light, and drastic shifts in
temperature (Newell 1979; Tomanek and Helmuth
2002). At high tide, ocean waves break on the
shore, imposing large hydrodynamic forces on
intertidal plants and animals (e.g., Koehl 1977;
Denny 1988, 1995; Carrington 1990; Dudgeon and
Johnson 1992; Utter and Denny 1996; Gaylord
2000). This combination of physical stresses is a
unique contrast to other severe environments.
Desert organisms, for example, can be subjected to
desiccation and extreme changes in temperature,
but are not dunked in rapidly flowing saltwater
hundreds of times per day.
Despite the apparent physical harshness of their
environment, wave-swept nearshore and intertidal
communities are among the most diverse and
productive on earth (Connell 1978; Leigh et al.
1987). The presence of such diversity in a severe
environment is surprising, akin to finding a lush
tropical forest in the midst of the Sahara. This
unusual diversity, coupled with the relative accessibility of the habitat and the ease with which
intertidal plants and animals can be manipulated,
has led ecologists to employ wave-swept shores as
a model system for experiments regarding the roles
of competition, predation, succession, recruitment,
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and disturbance in community ecology (e.g.,
Connell 1961, 1972, 1978; Paine 1966, 1974;
Dayton 1971; Menge 1976, 1978; Lubchenco 1978;
Sousa 1979; Paine and Levin 1981). Similarly, the
physical severity and temporal variability of the
nearshore habitat have led to studies of the process
of physiological adaptation in wave-swept organisms (e.g., Newell 1979; Hoffman et al. 2002;
Somero 2002; Stillman 2002; Tomanek 2002).
In this review, I explore this unusual combination of environmental severity and biological
diversity, and assess the insights that this peculiar
habitat can provide regarding the role of extreme
water flows in community ecology and natural
selection. As we will see, in some cases the severity
of the environment may be less stressful than it
initially appears. We begin with a description of
water motion in waves and the hydrodynamic
forces that they cause.
The wave-swept environment
Ocean waves
Ocean waves are among the most spectacular of
natural phenomena. Poets, sailors, and surfers
have each developed a colorful vocabulary to
describe them, but the discussion here will use the
more utilitarian language presented in Figure 1.
Waves are fluctuations of the sea surface away
from its equilibrium, still-water level. Each wave
has a crest and a trough, and the vertical distance
between them is the wave’s height, H. The horizontal distance between two crests (measured in
the direction of wave motion) is the wavelength, k.
The time it takes for a crest to travel one wavelength is the wave period, T. The speed at which
the wave form travels (which is often greater than
Figure 1. The nomenclature of waves.
the speed of the water under the wave) is known as
the phase speed, c.
Wave heights
Wave generation and the random sea
Typically, the waves that arrive at a site on the
shore are produced at sea by the interaction of wind
with the water’s surface (Kinsman 1965). Because
waves are created haphazardly across a wide
expanse of ocean, it is highly unlikely that they will
all proceed in lockstep as they approach a particular shore. Instead, the crests and troughs of various wave trains combine stochastically to produce
a complicated time series of surface elevations
know as the ‘random sea’ (Longuet-Higgins 1952).
On the short term, this series is unpredictable –
information about the height of one wave gives
very little information about the height of the next
or succeeding waves.
Fortunately, this short-term random behavior
provides the basis for a robust statistical calculation of the long-term characteristics of wave
heights. Building on the seminal work of Lord
Rayleigh (1880), Longuet-Higgins (1952, 1980)
calculated the maximal wave height one would
expect to encounter in a given situation. This
height is a function of (1) the ‘waviness’ of the
ocean (traditionally quantified by the significant
wave height, Hs, the average height of the highest
1/3 of waves) and (2) the time, t, over which one
observes the waves:
h i1=2
h t i1=2 t
Hmax ¼ 0:6541Hs ln
þ0:2886 ln
:
s
s
ð1Þ
Here s is the average wave period. The wavier the
ocean and the longer one waits, the higher the
wave that, on average, one will encounter. For
example, if the wave period is 10 s (a typical
value), in the course of an hour, the highest wave is
expected to be approximately 1.7 times the significant wave height. In a day, the highest wave will
be about 2 times Hs.
Longuet-Higgins’ formula assumes that the
ocean’s waviness is constant, and therefore it can
be used to estimate maximum wave heights only
over the course of a few hours to perhaps a day.
Over longer periods, Hs inevitably varies. On
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temperate shores there is commonly a pronounced
seasonal fluctuation in significant wave height
(wavy conditions in winter and calm conditions in
summer) as well as random variation associated
with storms at sea. As a result, the prediction of
maximum wave heights for periods longer than a
day must take into account the temporal variation
in significant wave height.
A variety of methods are available (e.g., Battjes
1970; Denny 1995), but they all follow the same
basic logic. As noted above, the wavier the ocean
and the more time for which a given sea state is
present, the higher the maximal wave produced.
However, above the modal significant wave height,
the higher the sea state, the shorter the time that
state is present in a given period. At some point the
decrease in the time available for the production of
a large wave offsets the increase in maximum wave
height associated with an increase in average
waviness, and it is thus possible to predict the
maximum wave height. For example, on the west
coast of North America, the yearly maximal wave
height is predicted to be approximately 5.9 times
the yearly average significant wave height (Denny
1995). Given that for many exposed stretches of
shoreline the yearly average significant wave
height is 1.5–3 m, this implies that the highest
wave that will approach that shore in a year is
9–18 m high.
In summary, a statistical treatment of the random sea allows us to predict both the short- and
long-term maximal wave height. The accuracy of
these predictions has been verified in the field (for
a brief review, see Denny 1995).
Keep in mind that the significant wave heights
used in these calculations are commonly measured
in deep water, so the predictions made here typically refer to waves well away from the coast.
Before these waves can interact with shoreline
organisms, they must move into shallow water, a
process that can affect wave height in two important ways.
Shoaling
First, wave height might increase. In raising waves,
the wind imparts to the ocean both kinetic energy
(in the form of horizontal and vertical water
velocities) and gravitational potential energy (due
to the change in elevation of the water’s surface
away from its equilibrium position). The energy
contained in waves is noteworthy. For example, a
wave with a height of 2 m contains enough energy
in each square meter of ocean surface to lift a metric
ton (1000 kg) more than half a meter. As a series of
waves approaches shore it carries energy with it.
The flux of total wave energy (per length of wave
crest) is equal to the product of the energy per area
in each wave (a value proportional to the square of
wave height) and the group velocity of the wave
train (see Denny 1988; Denny and Gaines 1999 for
an explanation of group velocity). In the absence of
viscous energy dissipation (a function primarily of
turbulence after the wave has broken), energy flux
remains constant as the wave train moves shoreward (US Army Corps of Engineers 1984).
This constancy of energy flux does not mean
that the form of individual waves is constant,
however. As the water’s depth decreases near the
shoreline, the group velocity changes, first
increasing slightly and then decreasing drastically.
In response, the energy per area initially decreases
slightly and then rises. The rise in energy per area
near the shore entails an increase in wave height.
The net result is that, as a wave approaches shore,
its height increases substantially, a process known
as shoaling. This process suggests that an offshore
wave with the already substantial maximal height
noted above (9–18 m), could increase its height to
truly momentous proportions as it approached the
shore.
Wave breaking
The process of shoaling is limited, however. As a
wave moves into ever-shallower depths, the period
of the wave stays the same, but the speed of the
waveform is reduced. As a result, the wavelength
decreases. Simultaneously, the velocity of the
water at the crest increases, an effect that is augmented by the shoaled increase in wave height. At
some point, the water velocity at the crest exceeds
the velocity of the waveform itself, the now-steep
waveform becomes unstable, and the wave breaks.
On a shore with a gently sloping substratum,
theory predicts that waves break when their height
is equal to approximately 80% of the water’s
depth (US Army Corps of Engineers 1984). On
steeper shores, waves can travel into shallower
water before they break. Given the appropriate
topography, waves can reach average breaking
heights of 140% of the water depth (Galvin 1972).
Visual evidence of the relationship between
water depth and breaking height can be found on
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any sandy beach. When the surf is up, waves break
far from shore where the water depth is relatively
deep. For example, the 9–18 m waves mentioned
above would break at a depth of 11–25 m,
depending on the slope of the shore. On calmer
days, waves break nearer the shore in shallower
water.
As the process of breaking proceeds, the rapidly
moving wave crest pitches forward to impact the
wave on its shoreward face, and the resulting
turbulence is mixed down into the waveform. This
spreading turbulence is associated with rapid viscous transduction of kinetic energy to heat and an
overall reduction in the mechanical energy of the
wave. The reduction in energy results in a reduction in wave height. Thus, after a wave breaks its
height continually decreases as the wave moves
inshore through the surf zone (Figure 2a). At any
point after breaking, the local significant wave
height is a constant fraction a of the local stillwater depth, d (Thornton and Guza 1983):
Hs ¼ ad
ðin the surf zone):
ð2Þ
Measured values of a on sandy beaches are in the
range of 0.5–0.7. Note, however, that Hs is a
measure of the average wave height. There is
substantial variation around this average. For
example, in the course of a year the maximal wave
height at a given depth is approximately 2.7 times
Hs (Eq. (1)). Thus, the expected yearly maximal
height of broken waves is
Hmax ffi 1:6d
ðin the surf zoneÞ:
ð3Þ
This maximum height for broken waves in the surf
zone is approximately the same fraction of depth
as that for the average height of waves that initially break on a steep shore (1.4d).
The situation is more complicated on shores for
which there is a distinct ‘step’ in the sea floor near
the shoreline (Figure 2b). In this case, breaking is
typically initiated at the step, and the steep leading
face of the resulting broken wave (commonly referred to as a turbulent bore) surges up the shore
above still-water level in what is termed the swash
zone. As with waves breaking on a planar beach,
there is an upper limit to wave height for waves
breaking at a step in the seafloor. In this case, the
maximal breaking height is
Hmax ¼ vds ;
ð4Þ
where ds is the water-column depth at the base of
the step and v depends on the wave period and the
slope of the sea floor seaward of the step (US
Army Corps of Engineers 1984; Denny 1995). The
steeper the slope and the longer the wave period,
the larger v is. For typically steep rocky seafloor
slopes (1:10) and typical wave periods (8–15 s), v is
in the range of 1–2. Thus, if the depth of water at
Figure 2. (a) Broken waves in the surf zone. On a gently sloping shore, waves break when their height is approximately 80% of the
water’s depth. (b) A wave breaking at a sudden step in the sea floor. The breaking wave subsequently surges up the shore in the swash
zone.
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the step is 5 m, the maximum height of a wave that
can break directly onto the shore is 5–10 m.
In summary, the process of breaking counteracts the process of shoaling by imposing an upper
limit to the height of the waves that can interact
with plants and animals near the shore. Below this
limit, the height of waves near the shore may vary
in proportion to the offshore waviness. However,
as long as the shoaled height of waves reaches the
breaking limit (which seems likely given the substantial long-term maximal heights predicted for
waves offshore), the long-term maximum height of
waves in the surf and swash zones is set by the
breaking limits imposed by local topography.
Wave-induced water velocity
The variation in wave height as waves move inshore is associated with a predictable pattern of
water velocities, a pattern that can be divided into
two categories separated by the process of breaking. In each case, near the substratum water
velocity perpendicular to the seafloor is inhibited
by the impermeable nature of the seafloor, so for
the sake of simplicity we deal only with the dominant flow, the velocity of water parallel to the
seabed.
Figure 3. The parameter K (Eq. (4)) decreases with an increase
in the ratio of depth to wavelength. K ¼ 1= sinhð2pd=kÞ. See
Denny (1988) for a thorough explanation.
Unbroken waves
Prior to breaking, water motion can be described
with reasonable accuracy by the equations of linear wave theory (see Denny 1988). According to
this theory, the horizontal water velocity at the
seafloor is
uLW ¼
pHK
T
ðbefore waves breakÞ:
ð5Þ
Here H is the height of an individual wave, T is the
period of the waves, and K is a function of the
ratio of the local depth of the water column (d) to
the wavelength (k) (Figure 3). The shallower the
water or the longer the wavelength, the larger K is,
and the faster the velocity imposed on benthic
organisms. Wavelength is, in turn, a function of
wave period and depth, and is best calculated
using an approximation derived by Eckart (1951)
(Figure 4). The shallower the water and the
shorter the wave period, the shorter the wavelength. These relationships have been combined in
Figure 5 to portray water velocity under unbroken
Figure 4. Wavelength increases with water-column depth (data
calculated using the approximation of Eckart 1951). The longer
the wave period T, the longer the wavelength.
waves as a function of depth. Water velocity at the
substratum increases dramatically as a wave
moves toward shore. The longer the wave period,
the higher the water velocity at any given depth.
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in this general review of wave-induced flows we
will not be able explore this interesting detail.
At any instant, the velocity imposed by an
unbroken wave varies with position relative to the
waveform. The magnitude of horizontal velocity is
maximal under the crest and trough, and is zero
where the waveform crosses still-water level.
However, given the large wavelengths typical of
ocean waves (Figure 3), the rate of spatial variation in velocity is small. For example, consider the
following situation. Waves with a period of 10 s
and a height of 2 m travel on a water-column with
a depth of 5 m. Even if the benthic organism is
quite large (a meter in shoreward–seaward length),
the instantaneous velocity at the head of the
organism is maximally only 0.41% different from
the velocity at its toes.
Figure 5. Horizontal water velocity (here normalized to wave
height) increases drastically as wave move into shallow water.
At any given depth, the longer the wave period T, the higher the
velocity.
As noted above, these relationships can be used
(albeit with caution) up to the point at which a
wave breaks. For a more detailed account of flows
near the seafloor outside of the surf zone, consult
Kawamata (1998).
A word of caution is in order at this point.
Linear wave theory (like most wave theories) assumes that water has no viscosity, an unrealistic
simplification that potentially can cause problems
when one attempts to estimate the velocity imposed on benthic organisms. In reality, seawater is
viscous, and this viscosity, coupled with what is
known as the ‘no-slip condition,’ ensures that the
velocity directly at the seafloor is zero (Schlichting
1979; Vogel 1994). As a result, in the benthic
boundary layer adjacent to the substratum there is
a steep, time-varying gradient in velocity, and the
velocity we have calculated here for the substratum actually applies to flow a short distance above
it. The shape of the velocity gradient, and therefore the actual distance above the seafloor at which
our calculations apply, depends in a complicated
fashion on the period of the flow and the roughness of the seafloor. However, for common conditions (wave period=10 s, a flat but rugose
seabed) the actual velocity should match the
velocity calculated from wave theory within 10%
at a height of less than 1 cm. Plants and animals
may well utilize this thin layer of retarded flow, but
Broken waves
The process of wave breaking is complex, and the
accurate prediction of post-breaking flows is
fraught with difficulties. Nonetheless, reasonable
estimates (essentially, rules of thumb) can be calculated based on a combination of empirical
measurement and the theory of flow in bores and
solitary waves. The nature of these estimates depends on the topography of the seafloor.
If the slope of the seafloor is relatively constant,
a distinct surf zone is present (Figure 2a). As turbulent broken waves move through the surf zone,
their phase speed (the speed of the wave form as
distinct from the speed of the water) can be
roughly estimated using a simple theory of bores
(Tricker 1964):
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
ðH þ dÞ
;
c ¼ g H
ð6Þ
þd
2
where g is the acceleration due to gravity
(9.81ms1), and H is the wave height at watercolumn depth d.
The water velocity at the substratum in the surf
zone, uSZ, is a fraction / of the phase speed:
uSZ ¼ /c
(in the surf zone):
ð7Þ
According to solitary wave theory, the coefficient
/ varies with the ratio of H/d (Munk 1949), but
for the steep waveforms typical of the surf zone, /
is approximately 0.3–0.4 (see the review in Gaylord 1999). Given our assumption that Hmax@ 1.6d
(Eq. (3)), the maximum water velocity at the
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substratum is
uSZ
pffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi 0:6 gHmax
(in the surf zone):
ð8Þ
The situation is more complicated on a stepped
shore (as shown in Figure 2b), where a distinct
swash zone exists. In this zone, waves do not break
until they reach the shoreline. As a result, organisms are typically out of the water prior to the
arrival of the wave, and are subsequently impacted
by the leading edge of a bore as it surges up the
shore. Laboratory measurements (van Dorn 1976,
1978) and theory (Hibberd and Peregrine 1979)
suggest that the ‘surge’ speed in the swash zone
(the phase speed of the bore), cSW, is proportional
to the phase speed of the wave at breaking
cSW ¼ wc
(in the swash zone):
ð9Þ
where w can be as high as approximately 2. Now,
the maximum velocity of water in this surge (at its
leading edge) is equal to the phase speed of the
bore. Thus, the maximum velocity imposed on a
benthic organism in the swash is predicted to be
uSW ¼ cSW ¼ wc
(in the swash zone):
ð10Þ
In this expression, c can be estimated from Eq. (6)
using Hmax and ds. As we have seen (Eq. (4)), the
upper limit to wave height as a wave breaks on a
stepped shore is set by the depth of the step and
the parameter v, which in turn is set by the slope of
the bottom offshore and the wave period. Using a
conservative value of v=1, we arrive at a rough
rule of thumb – the maximal velocity imposed on
intertidal organisms by the leading face of a turbulent bore in the swash is
pffiffiffiffiffiffiffiffiffiffiffiffiffi
uSW ffi 1:6w gHmax (in the swash zone): ð11Þ
When w=2, the velocities imposed in the swash on
stepped shores are about 5 times those found in the
surf zone on gently sloping shores.
Note that this calculation applies only to water
at the leading edge of a bore in the swash. After the
crest of the bore has passed by, the velocity is likely
to be reduced. If the water velocity at the substratum behind the crest of a bore behaves similarly to that in a solitary wave, the velocity trailing
the crest is a fraction of that of the wave form:
u ¼ /uSW ¼ /wc (in the swash zone):
ð12Þ
If / in the swash is similar
to that
in p
the
surfffi
ffi
pffiffiffiffiffiffiffiffiffiffiffiffi
ffiffiffiffiffiffiffiffiffiffiffiffi
(0.3–0.4), u ranges from 0:5 gHmax to 1:3 gHmax ,
depending on the value of w. These swash-zone
values are in fact close to those measured in the
field (Gaylord 2000).
Note that Eqs. (11) and (12) apply only to
shores with a well-defined step in the sea floor,
adjacent to the shoreline. However; this caveat is
not overly limiting – many rocky shores fit the bill.
Note also that the values calculated here are the
maximum velocities, those imposed by waves of
maximal height. Given the scenario shown in
Figure 2b, waves with a height less than v ds also
break at the step in the shore, but impose lower
velocities on intertidal organisms. In contrast,
waves higher than v ds break before they reach the
step in the shore, and the velocity they impose is
predicted to be no greater than that calculated via
Eq. (11).
Summary
The maximal wave-induced water velocity that
can potentially be imposed on a benthic organism is, in theory, a predictable function of where
the organism lives. On gently sloping shores, the
water velocity at the substratum increases as a
wave moves toward shore and is maximal under
the break point of the waves. On stepped shores,
the water velocities are maximal in the swash
zone, and are larger than those imposed subtidally. In both cases, however, there is an upper
limit to velocity, set by wave breaking. As we
will see, it is this limit that is of particular
interest in the application of biomechanics to the
study of ecology and evolution in wave-swept
organisms.
A caveat
A major note of caution is necessary at this
point. Each of the calculations above assumes
that the topography of the shoreline varies in
only two dimensions. That is, each assumes that
a wave approaching the shore will encounter a
certain depth gradient as the seabed rises to the
shore, but that this pattern is constant along the
coast. In contrast, real rocky shores have substantial alongshore variation: there are headlands
and bays across a wide variety of spatial scales,
surge channels, tide pools, crevices, and caves.
This three-dimensionality can have dramatic
effect on water velocities in the surf and swash
zones. As a consequence, the rules of thumb
drawn above should be used to predict the flow
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imposed on a stretch of shore with an understanding that the rules apply as an average
across a spatial scale of meters to tens of meters.
At smaller scales the site-specific nature of the
substratum must be taken into account. For
example, Denny et al. (2003) have shown that
when a wave breaks on a prow-like rock, the
interaction of the wave with itself as it refracts
onto the beach can result in the formation of a
jet of water with a local velocity 30–40% higher
than the phase velocity of the bore. This velocity
can be further amplified if the jet impacts a
vertical wall at the back of the beach. Overall,
Denny et al. found that for their experimental
topography the water velocity could locally exceed twice the velocity of the wave as it travels in
the swash. Other processes of velocity amplification are quite possible, as are situations in
which water velocity is likely to be slower than
that predicted here. For example, velocity may
be increased as a wave is ‘squeezed’ by a surge
channel, and velocities in a tide pool or the wake
of a boulder are reduced relative to the bulk flow
in a wave. These topographical effects inevitably
complicate the interpretation of the consequences
of wave-induced water velocity.
Prediction of the water accelerations that accompany ocean waves follows a pattern similar to that
for velocity – accelerations prior to breaking can
be accurately predicted using linear wave theory,
those after breaking are less easily (and less accurately) predicted.
Unbroken waves
According to linear wave theory, the maximal
horizontal acceleration at the substratum is
2p2 HK
;
T2
Broken waves
Accelerations in the turbulent flow of broken
waves can be exceedingly high. For example,
Denny et al. (1985) and Gaylord (1999) measured
values in excess of 400 m s2 in the swash zone on
exposed shores. The very rapid accelerations that
occur in the swash are primarily a result of turbulent eddies being carried past an organism by
the bulk water motion of the wave. Because the
size and orientation of these eddies are predictable
only statistically, the wave-by-wave prediction of
water accelerations is problematic. However, a
theoretical treatment of the problem (Appendix A)
suggests that
amax ffi
Wave-induced water accelerations
amax ¼
As with velocity, the acceleration imposed by an
unbroken wave varies with position relative to the
waveform. The magnitude of horizontal acceleration is maximal where the waveform crosses stillwater level, and is zero under the crest and trough.
However, given the large wavelengths typical of
ocean waves (Figure 4), the spatial variation in
acceleration is relatively gradual. For example, as
with velocity, the instantaneous acceleration varies
negligibly from one end to the other in an organism a meter in shoreward–seaward length.
ð13Þ
where H is again the height of an individual wave,
T is wave period, and K is the depth-dependent
term shown in Figure 3. For example, with a
typical period of 10 s, a 2-m-high wave near its
breaking point (d=2.5 m) imposes a maximal
acceleration of 1.3 m s2 at the substratum, small
compared to the hundreds of m s2 common in the
surf (see below).
2phu2
/y
(in the surf and swash zones)
ð14Þ
where y is the distance above the substratum at
which the acceleration is measured, u is the
velocity at y, and h is an empirically measured
coefficient that ranges from about 0.015 to 0.074.
Note that there is a lower limit to the appropriate
value of y that can be used in this equation. In
theory, this is set by the thickness of the viscous
sublayer in the turbulent boundary layer (a value
that may be a fraction of a millimeter under waveinduced flows, Schlichting 1979), but in practice,
the lower limit of y is set by the height of the
roughness elements on the substratum. In the case
of rocky shores, this typically constrains y to a
lower limit of about 1–2
cm.
pffiffiffiffiffiffi
ffi Using values of
y=2 cm, /=0.3, u ¼ 0:5 gH,
amax ffi 39H to 190H m s2
ð15Þ
depending on the value of h. This prediction is in
reasonable accord with empirical measurements
made by Gaylord (1999). He found that the
447
accelerations imposed on a small sphere 2 cm
above the substratum were on average equal to
approximately 114H m s2 for swash-zone waves
in which turbulence was well-developed, with a
broad distribution of accelerations extending from
approximately 50H to 230H m s2. Although this
theoretical approach is promising, the considerable variation in measured values of h is problematic, and further work is necessary before
reliable and well-understood predictions can be
made for surf-zone accelerations.
Although theory suggests (and measurements
confirm) that extreme accelerations are present in
the surf zone, it is important to note that the
duration of these accelerations is very short and
their spatial scale is very small. Field measurements of the temporal scale of surf zone accelerations (Gaylord 2000) show that at a point near the
substratum, accelerations remain high for less than
0.1 s and are relatively constant across a spatial
scale of less than approximately 1 cm. The biological consequences of these scales are discussed
below.
coefficient (Vogel 1994). From Eqs. (5), (8), and
(11), we know that us varies with wave height,
potentially allowing us to predict drag or lift from
a knowledge of the wave climate and shoreline
topography. For example, in subtidal flows outside of the surf zone, us,max=p HmaxK/T (Eq. (5)).
Inserting this expression into Eq. (16), we see
that
2
1 pHmax K
FD;max ¼ q
Af CD :
ð18Þ
2
T
At the front
offfi a bore in the swash
pffiffiffiffiffiffiffiffiffiffiffiffi
us;max ffi 1:6w gHmax (Eq. (11)), an expression we
can again insert into Eq. (16). When the dust settles, we see that in this case
FD;max ffi 1:3qgw2 Hmax Af CD
(front of the bore):
ð19Þ
By a similar process utilizing Eq. (12) we can calculate that after the bore’s crest has passed
FD;max ¼ 0:5qgHmax Af CD
(behind the crest):
ð20Þ
Wave-induced hydrodynamic forces
Drag, lift, and the impingement force
The wave-induced water motions described above
impose hydrodynamic forces on nearshore and
intertidal organisms that must be resisted if these
organisms are to remain intact. For example, the
drag imposed on a stationary benthic organism (a
force in the direction of flow) is
1
FD ¼ qu2s Af CD ;
2
ð16Þ
where q is the density of seawater (nominally
1025 kg m3), Af is the frontal area of the organism, and CD is the dimensionless drag coefficient
(Vogel 1994). Here us is the velocity at the substratum (and parallel to the substratum), calculated as appropriate for a given location. A similar
relationship exists between lift (a force perpendicular to flow) and water velocity:
1
FL ¼ qu2s Ap CL ;
2
ð17Þ
where Ap is traditionally taken to be the object’s
platform area and CL is a dimensionless lift
Analogous expressions can be written for lift.
Note that both drag and lift are proportional to
the area exposed to flow. As a result, the stress
(force per area) placed on the organism’s structure
is largely independent of size (Denny et al. 1985;
Denny 1988). This conclusion must be modified,
however, if the organism changes shape as it grows
– an effect we explore below (Section ‘Biological
determinants of strength’).
Gaylord (2000) has noted that when the leading
edge of a bore impinges on a hitherto immersed
organism, the resulting force is often larger than
the subsequent drag force imposed later after the
organism is fully submerged. In other words, there
is an augmentation of force associated with
movement through the air–water interface at the
leading edge of a broken wave. Gaylord suggests
that this ‘impingement force’ can be modeled by
an equation similar to that for drag:
1
FI ¼ qu2s Af CI ;
2
ð21Þ
where CI is an impingement coefficient. Gaylord’s
field measurements suggest that CI is approximately 1.5–3.0 times CD for sea urchins and
rockweeds, but further measurements are required
448
before these measurements can be extrapolated to
other organisms.
Note that drag, lift, and the impingement force
are all proportional to the density of the fluid.
Near the leading edge of breaking waves and turbulent bores, substantial amounts of air may be
entrained, and the density of the seawater may
thus be less than that calculated using nominal
values.
Accelerational force
In addition to the hydrodynamic forces that are
determined by water velocity (drag, lift, impingement force), there is a hydrodynamic force that
results from the water’s acceleration:
FA ¼ qVCM a;
ð22Þ
where V is the volume of water displaced by the
organism and CM is a dimensionless inertia coefficient, dependent on shape (Batchelor 1967;
Denny 1988). Because acceleration is (at least
loosely) dependent on wave height (Eqs. (13) and
(15)), this hydrodynamic force can be related to
the wave climate. For example, in the subtidal
zone:
FA;max ¼
2qVCM p2 Hmax K
:
T2
ð23Þ
The accelerations found in subtidal habitats,
while low in magnitude, are large enough in
spatial scale to enclose an entire organism. The
resulting accelerational forces may be the dominant forces on large animals (such as massive
coral heads) in this habitat (Massel and Done
1993; Massel 1996).
Note that FA is proportional to the volume of an
organism (Eq. (22)). This raises an issue of size in
determining the force per area (the stress) imposed
on an organism. If the volume of the organism is
proportional to the cube of a characteristic length
L, and area is proportional to the square of L, the
stress imposed by the accelerational force is
rA / qCM La:
ð24Þ
The larger the organism, the larger the stress. This
could lead to a physical limit to the size of waveswept organisms (Denny et al 1985; Gaylord et al.
1994). In the surf zone, the combined stress due to
lift and drag can be large, but it is typically not
large enough to dislodge limpets (Denny 2000;
Denny and Blanchette 2000). If, however, an
additional stress is imposed by the accelerational
force, it might serve as the straw that breaks the
camel’s back. Consider for example, a hypothetical
case in which the stress required to dislodge a
limpet (in excess of that imposed by lift and drag)
is 1000 N m2, CM is 2 (a typical value, Denny
and Blanchette 2000), and a=100 m s2. Given
this situation, if L is larger than approximately
5 cm, the organism is in danger of being broken.
There is a catch, however. Recall that in the surf
zone the spatial scale of acceleration is very small
(<1 cm). Thus, if an organism is large enough to
experience a dangerous accelerational force in the
surf (>5 cm), it is likely to be too large to be
completely enclosed by the highest accelerations in
the flow. As a result, although surf- and swashzone accelerations are extreme in magnitude, they
are probably not capable of exerting large accelerational forces (Gaylord 2000), and therefore may
not be of great biological importance.
The flexibility of many wave-swept plants and
animals may also limit the effects of accelerational
hydrodynamic forces. If an organism is capable of
moving with the flow during a period of high water
acceleration, it can potentially avoid the accelerational force (Koehl 1984, 1986, 1999). The very
short duration of surf-zone accelerations (<0.1 s)
may contribute to this effect. Although flexible
organisms may be able to minimize accelerational
hydrodynamic forces, the motion of the body can
lead to another form of accelerational loading. By
going with the flow, the mass of the body attains
some velocity, and the organism may thus develop
a substantial momentum. If the organism subsequently comes to the end of its tether, the resulting
jerk rapidly reduces this momentum, thereby
placing an inertial load on the organism. The effect
is only indirectly related to water motion, so I will
not discuss it in detail here. A thorough discussion
can be found in Denny et al. (1998).
In the subtidal zone, accelerations are two orders
of magnitude smaller than those in the surf, but
they may nonetheless pose a limit to size. For
example, if a is 1 m s2 (a common subtidal value)
and the excess stress required to break an organism
is 1000 N m2, an organism will be broken if its
characteristic length exceeds 50 cm. Although this
size exceeds that of most wave-swept organisms, it
is well within the range of massive coral heads, and
449
Massel and Done (1993) report that loosely attached corals are indeed overturned by cyclone
waves, presumably due to accelerational effects.
We are now in a position to examine the ecological and evolutionary consequences of waveinduced hydrodynamic forces. We begin by briefly
exploring the concept of wave exposure.
Wave exposure and community ecology
Exposure defined
‘Wave exposure’ is a loosely defined index that
combines the effects of all environmental factors
that vary as a function of the interaction of a
particular organism with wave-induced water
motion. For example, the higher the incident
waves, the larger the hydrodynamic forces imposed on an organism (a factor affecting the risk of
breakage or dislodgment), and the more wave
splash the organism might receive at low tide (a
factor affecting desiccation, osmotic stress, and
temperature). The higher the waves, the more
effectively the water is stirred, potentially affecting
nutrient uptake and light-flecking, and thereby the
photosynthetic rate of benthic algae (e.g., Leigh
et al. 1987; Wing and Patterson 1993). Wave-induced water motion can control the amount and
type of sediment that is delivered to the shore,
including particles whose impact is large enough to
cause bodily harm (Shanks and Wright 1986). A
wide variety of other environmental variables can
also be tied directly or indirectly to wave-induced
flow (Lewis 1964; Newell 1979).
Community effects
As loosely defined as it is, wave exposure has
nonetheless been a unifying concept in nearshore
ecology. For instance, intertidal species are traditionally classified by the exposure of the habitat in
which they are characteristically found. Indeed, if
one tells an ecologist the general location of a site
(thereby establishing the potential pool of species)
and the site’s wave exposure, he or she can predict
with surprising accuracy what species will be
present (e.g., Ricketts et al. 1985).
Among the host of factors that comprise wave
exposure, wave-induced hydrodynamic forces are
thought to play an important (often dominant)
role in community ecology. In this context, the
forces imposed by waves have three primary
effects.
Limits to species distribution
If, at a particular site on the substratum, the imposed forces exceed the resistive ability of a particular species, that organism cannot survive at
that site. As a result, if there is a gradient in wave
exposure, hydrodynamics may form a controlling
factor in the distribution of nearshore and intertidal plants and animals (e.g., Shaughnessy et al.
1996; Graham 1997).
Rate of disturbance
Hydrodynamic forces play a central role in determining the rate at which individuals are disturbed.
This rate of disturbance can in turn have important ecological consequences. For example, on the
west coast of North America, the California
mussel (Mytilus californianus) is the dominant
competitor for space in the mid- to upper-intertidal zone. That is, in the absence of physical disturbance, primary space on the rock in this zone
on the shore would be completely taken up by
mussels to the exclusion of other species. In reality,
wave forces occasionally rip patches of mussels
from the beds, opening space on the rock that can
be colonized by less competitive species. The more
frequent and more severe the disturbance, the
larger the fraction of the substratum that can be
occupied by species other than mussels. In this
fashion, the mid- to upper-intertidal zone is typically a dynamic patchwork of plants and animals,
and the species diversity and community interactions are sensitive functions of the rate of disturbance (Dayton 1971; Paine and Levin 1981). For
example, the sea palm, Postelsia palmaeformis is
found only at intertidal sites that are exposed to
surf conditions sufficiently extreme to provide bare
patches for colonization (Paine 1979; Blanchette
1996), and the species composition in offshore kelp
beds is governed in large part by the frequency of
disturbance of the dominant canopy-forming alga
(Dayton et al. 1984, 1992; Dayton 1985; Ebling
et al. 1985)
Effects on foraging
Lastly, wave-induced hydrodynamic forces can
affect the ability of herbivores and predators to
450
forage for food. For example, limpets graze small
algae on wave-exposed rocks, but are more susceptible to being dislodged by waves when they are
feeding than when they are not. As a result, the
feeding time available to a limpet is set in part by
the severity of the waves impinging on the shore
(e.g., Judge 1988). A similar constraint is imposed
on subtidal sea urchins (Kawamata 1998). Rates of
foraging by one species can in turn have effects on
rate of disturbance in other species. For example,
off the coast of Maine, when urchins are free to
forage they prohibit kelp from growing on mussels. Where urchins are prohibited from foraging
by wave action, kelps readily grow on mussels, and
the resulting increase in drag renders mussels
highly susceptible to dislodgment by waves
(Witman 1987).
Note that water motion can have biological
consequences separate from the forces it imposes.
For example, flow can be effective at delivering
nutrients and gasses to plants and animals, and
can remove harmful sediment from sessile species.
Predicting the effects
Following the logic outlined earlier, Denny (1995)
proposed that it is feasible to calculate the maximal hydrodynamic force imposed on nearshore
organisms given basic information regarding
yearly average offshore wave conditions and the
topography of the shore. Furthermore, these calculations suggest that this approach can explain
both the physical limits to the distribution of some
species of algae and the rate of disturbance in
mussel beds. Thus, in the rocky intertidal zone
these results hold forth the promise that the primary effects of water motion on community ecology can be predicted from the principles of fluid
mechanics and a knowledge of the local shoreline
topography.
However, considerable additional field work is
necessary before the accuracy of this approach can
be fully evaluated. I have noted above the difficulties posed by the interaction of waves with local
topography. Biological factors may also complicate matters. For example, preliminary data
(Helmuth and Denny unpublished) suggest that it
is often not the largest waves that do the most
damage in mussel beds. In this case, fluctuations in
the organisms’ adhesive strength (rather that
fluctuations in ocean waviness) may determine
when mussels are disturbed (Carrington 2002). As
we have seen above, the rate of disturbance on
subtidal mussels can depend less on the imposed
water velocity than on whether the mussels are
overgrown by kelps. Similar complications are
found in other organisms, and several examples
are discussed below (Section ‘Biological determinants of strength’). Ultimately, the success of a
purely mechanical approach to the prediction of
the ecological consequences of water motion will
depend on how common these biological ‘complications’ are and how well they are understood.
Wave exposure and functional morphology
The concept of wave exposure is of potential
interest to functional morphologists in that it
provides evidence of a quantifiable and demonstrably important selective factor. If we accept the
evidence provided by ecologists that wave exposure has predictable effects on the distribution and
abundance of organisms, we may reasonably
hypothesize that wave-induced hydrodynamic
forces have played (and are playing) a role in the
structural evolution of nearshore plants and animals. Our ability to predict the maximal hydrodynamic forces to which organisms are subjected
then becomes a potentially valuable tool with
which to examine the evolved morphology of
nearshore macrophytes and animals.
But can this potential be realized? Have waveinduced water motions indeed served as a potent
selective factor in the evolved design of nearshore
algae and animals? Does the nearshore environment provide clear examples of morphological
adaptation that are demonstrably important in
community ecology? The answer to these questions is both yes and no, and therein lies an
interesting story.
Factors of safety
Let us begin by quantifying the risk that waveinduced water motions impose on organisms.
Traditionally, the question of risk has been addressed in terms of a ‘factor of safety’ (Alexander
1981; Lowell 1985, 1987), defined as the ratio of
the expected failure strength of an organism to the
451
expected maximal stress to which that organism is
subjected. In the wave-swept environment, this
approach has been applied to the giant bull kelp,
Nereocystis luetkeana, by Johnson and Koehl
(1994). In the absence of long-term measurements
of flow at their sites, they estimated maximal
velocities from short-term records and used these
to estimate maximal stress imposed on the stipe of
N. luetkeana. The ratio of the mean breaking stress
to this estimated maximal stress is the ‘environmental stress factor,’ an index similar to a factor of
safety. They found that the environmental stress
factor was high (3.2–11.6) and independent of the
habitat in which the plants grew. They suggest that
N. luetkeana has evolved a plastic phenotype that
maintains a constant risk by adjusting the shape
and material properties of the kelp.
Unfortunately, there is a complication when
applying the concept of a safety (or stress) factor
directly to questions of risk. This is best seen
through an example. Consider the situation shown
in Figure 6. Here probability density distributions
are shown for both the maximal stress applied to
an organism, p, and the organism’s strength (its
ability to resist applied stress), h. In this case, the
ratio of expected (= mean) strength to expected
maximal applied stress is 3, a seemingly ‘safe’
value. But the variances in applied stress and
strength are such that there is considerable overlap
between the two distributions. The high-stress
events in the right-hand tail of the applied stress
distribution are capable of breaking the relatively
weak individuals in the left-hand tail of the distribution of strength. In other words, despite the
fact that on average this organism is three times as
strong as the maximal stress imposed on it, there is
still a substantial risk that an organism chosen at
random will have a strength less than the maximum applied by chance, and the organism will
break. The larger the variation in either the applied stress or the strength of the organism, the
larger the overlap between distributions, and the
greater the risk (Alexander 1981; Lowell 1985,
1987). Unless both these variations are known, the
risk cannot be calculated from the safety factor.
Furthermore, different safety factors are required
to achieve the same risk of failure in systems
subject to different variations in strength and applied load. Indeed, building codes require different
safety factors for different types of manmade
structures depending on the variability of their
strengths (from 1.7 to 2.05 for steel buildings all
the way to 11.25 for the wire ropes that suspend
elevators, Alexander 1981).
Calculating risk
The actual risk imposed on an individual is calculated through a process shown schematically in
Figure 7. First one calculates P, the probability of
exceeding a given applied stress, r. This is done by
integrating p, the probability density distribution
of applied stress, and subtracting the integral from
1:
PðrÞ ¼ 1 Zr
pðxÞ dx:
ð25Þ
0
Figure 6. In this hypothetical example, the average strength of
an organism is three times the average applied stress. Nonetheless, the overlap between the two distributions ensures that
some organisms will be broken.
This exceedance probability is then combined with
the probability density function of strength, h. For
each value of r, h(r)dr is (by definition) the
probability that the organism’s strength falls in the
range between r and r+dr. Thus, P(r)h(r)dr is
the probability that the organism will be broken by
452
Figure 7. The exceedance probability of applied stress quantifies the probability that a stress chosen at random exceeds stress
r. The product of this value and the probability that an
organism has strength r is the probability that an organism
with strength r will be broken. Integrated over all values of r,
this product yields the probability of breakage.
an applied stress in the range r to r+dr. Integrating over all values of r yields the overall risk of
breakage:
risk ¼
Z1
PðrÞhðrÞ dr:
ð26Þ
0
While straightforward in theory, the application of
this approach is difficult in practice. It is relatively
simple to measure the distribution of strengths in
organisms at a given time. In many cases, however,
measuring the distribution of maximal forces is
problematic. For instance, predatory snails may be
able to sense an incoming tide. To avoid wave
forces, they might then respond by hiding in crevices or the interstices of a mussel bed. If so, the
maximal hydrodynamic stress (and thereby risk)
imposed on these organisms by breaking waves
would be set as much by sensory physiology and
behavior as by fluid mechanics.
The situation is simplified, however, if the
mechanics of the situation accurately define rmax,
the maximal force that will be applied. In this case,
the problems with variation in maximal stress is
avoided: P=1 for r £ rmax and P=0 for r>rmax.
As a result, the overall risk can be estimated by
simply calculating
risk ¼
rZ
max
hðrÞ dr:
ð27Þ
0
This approach was used by Lowell (1985, 1987) to
examine the risk of shell breakage in limpets. In
that case, the adhesive strength of the limpet sets
an upper limit to the prying force that can be applied to the shell by a predatory crab or bird, and
this limit allowed Lowell to examine the evolution
of shell strength. It is in this context that the waveswept environment shines. As we have seen, in
many cases the maximal wave height is set by wave
breaking. The maximal wave-induced hydrodynamic forces can therefore be estimated with reasonable accuracy, providing a straightforward
estimate of the risk of failure. (Note that this
procedure is substantially complicated if the distribution of strength varies through time.)
Assessing the risk from exposure
This approach has been used to examine the risk of
breakage or dislodgment in a variety of waveswept algae and animals. For example, Denny
(1995) calculated maximal inshore wave height as
a function of bottom topography and yearly
average offshore wave height, and applied these
data to the risk of dislodgment in barnacles,
mussels, and several algae. In general, the risk of
dislodgment among species found on wave
exposed shores was low (<5.2% per year).
Massel and Done (1993) and Massel (1996) used
historical records of the occurrence of cyclones
and theory regarding the shoaling of waves to
estimate the risk of overturning in massive corals
on the Great Barrier reef. They conclude that this
risk is negligible as long as a coral head is even
slightly attached to a stable substratum. Only if the
head is lying loose on the sea floor, is there an
appreciable risk of overturning.
Calculations from several studies suggest that
large kelps are subjected to only a small risk of
breakage by wave-induced hydrodynamic forces.
For example, only 0.1–26% of individual fronds of
the giant kelp Macroscystis pyrifera are predicted to
be broken by even the exceptionally large waves of
storms (Utter and Denny 1996), and similar values
are predicted for N. luetkeana (Denny et al. 1997).
Other algae may be at even lower risk. Only 0.16–
0.32% of large individual fronds of the feather boa
kelp are predicted to be broken by extreme waves in
the surf zone (Friedland and Denny 1995).
Gaylord et al. (1994) estimated the risk of dislodgment for a variety of intertidal macroalgae,
and concluded that all were at substantial risk of
453
breakage by the accelerational forces accompanying breaking waves. However, subsequent work by
Gaylord (2000) suggests that by mistakenly
assuming a large spatial scale for accelerations,
this study substantially overestimated the accelerational forces. If only drag forces are taken into
account, the risk of dislodgment for most algal
species appears to be low.
Limpets appear to be similarly resistant to dislodgment by waves (Judge 1988; Denny 1989;
Denny and Blanchette 2000), and a sea urchin
typically found exposed on intertidal shores in
Hawaii (the shingle urchin, Colobocentrotus atratus) has only a very low risk of being dislodged
(Denny and Gaylord 1996).
In summary, the physics of breaking waves allows one to predict for wave-swept organisms a
well-defined maximal applied stress. When this
approach is used to calculate the risk of breakage,
the predicted risk to intact organisms is typically
low. In other words, with a few exceptions, waveswept shores appear not to be as intrinsically
hydrodynamically stressful as is generally
assumed.
At this point it is important to raise a voice of
caution. These calculations have used the rules of
thumb developed above. As we have seen, local
topography can augment velocity, and in these
locales risk may be higher than that discussed here.
Note also that this conclusion applies only to intact organisms: algae that have not been weakened
by herbivory, senescence, or abrasion, and animals
that are healthy and clinging to the rock as best
they can. Many biological factors can compromise
the strength of organisms, and some of these are
explored below (Section ‘Biological determinants
of strength’).
tentative conclusion we have just reached has a
certain appeal. If the hydrodynamic environment
is less stressful than previously thought, there is
less of a reason to suppose that diversity has been
adversely affected by environmental severity.
It is unlikely that the answer can be this
straightforward, however. Water velocities in
excess of 20 m s1 have indeed been measured on
wave-swept rocky shores, and velocities in excess
of 10 m s1 are common (Bell and Denny 1994;
Denny et al. 2003). Thus the hydrodynamic conditions on wave-swept rocky shores are undeniably
extreme. If intact intertidal algae and animals on
exposed shores are indeed at low risk, it seems
reasonable to attribute this fact not to any lack of
environmental severity, but rather to an effectively
evolved design. That is, we may hypothesize that
the flow environment has been a sufficiently potent
selective factor in the past that only those organisms that have evolved an effective, finely ‘tuned’
resistance to flow are currently present. The
question then becomes, how can we account for
the diversity of form if all morphologies have been
finely ‘tuned’ by natural selection?
I would like to suggest two answers to this
question. First, wave-swept organisms provide
examples of how mechanics can limit the ability of
natural selection to guide the specialization of
form – if many different forms have the same
probability of survival, it is less surprising that a
variety of morphologies exist. Second, although
nearshore hydrodynamics may provide the proximate mechanism for breakage and dislodgment in
wave-swept algae and animals, the ultimate cause
of selection may rest with biological (rather than
environmental) effects.
Limits to specialization
Environmental stress and the diversity of form
Diversity amidst adversity
Let us now return to the peculiar contrast noted in
the Section ‘Introduction’ – the wave-swept environment appears to be one of the most physically
stressful on earth, but it is inhabited by an
unusually diverse set of morphologies and taxa. If
wave-swept shores are as physically stressful as
they initially appear, how do all those diverse life
forms manage to live there? In this context, the
In the wave-swept environment, several factors
conspire to limit the evolution of morphologies
specialized to the flow environment. Consider an
example. When wandering around rocky intertidal
shores, one is often struck by the lack of convergence in form of benthic macroalgae. At a given
site exposed to large waves one can find species
with unbranched, smooth, planar blades living
next to species with three-dimensional, highly
branched blades. Several species exhibit within
themselves such a range of forms that it is difficult
454
to believe that individuals are not from separate
genera. Similar morphological diversity is common among wave-swept animals. How is this
diversity of form possible when all the individuals
are exposed to the same, potentially harmful flow?
Macroalgae
In an attempt to answer this question, Carrington
(1990) measured the drag characteristics of a wide
variety of intertidal algae. The species used in her
tests differed greatly in their morphology, but
showed surprising similarity in their behavior in
flow. As velocity increased, the plants bent
downstream and passively reconfigured to a more
streamlined shape. As a result, the drag coefficient
decreased with increasing velocity, and at high
velocities the drag coefficient was similar across
species despite the dissimilarity in still-water
morphologies.
This similarity in drag coefficient suggests that
flexibility rather than still-water shape governs the
response of algae to flow. As long as the stipe and
blades of an alga are sufficiently flexible, the
organism will assume an appropriately streamlined
shape in flow. If this is true, the evolution of
flexibility has pre-empted the ability of drag to
serve as a selective factor on plant morphology. In
effect, sufficient flexibility renders shape neutral to
selection by hydrodynamic forces, granting the
‘permission’ of the flow environment for shape to
evolve in response to other factors. For example,
the different shapes among algae may be a response to factors related to self-shading, desiccation, resistance to herbivores, nutrient uptake, or
mode of reproduction (Koehl 1986; Koehl and
Alberte 1988; Bell 1993).
Limpets
Another example concerns limpets. Denny (1989)
measured the lift and drag characteristics of the
limpet Lottia pelta, and found one individual with
a surprisingly low drag coefficient. When facing
upstream at a certain water velocity, this individual’s drag was suddenly reduced by about 40%
relative to its conspecifics, an effect that was
attributed to an abrupt transition from a laminar
to a turbulent boundary layer. The gross morphology of this unusual individual was not
noticeably different from its brethren, suggesting
that a slight adjustment of shell shape in this
species could (under appropriate circumstances)
result in a drastic decrease in drag. Given the
demonstration that this effect is possible, the
question then arises – why have most L. pelta not
acquired this morphological modification?
One possible answer involves the linkage between drag and lift. The shell with the reduced
drag coefficient did not exhibit a similar reduction
in its lift coefficient, and measurements of lift and
the adhesive tenacity of L. pelta show that this
species is at much greater risk of being dislodged
by lift than by drag. Thus, although a slight
change in shape could drastically reduce the drag
coefficient, the consequent reduction in hydrodynamic force results in a negligible reduction in risk.
Without a reduction in risk, the shell shape with
reduced drag cannot be selected. In this fashion,
the existence of a high lift coefficient (a factor that
is probably unavoidable due to the fact that limpet
shells lie next to a solid substratum) pre-empts the
evolution of a shell shape specialized for the
reduction of drag.
The pre-emptive interaction among the various
factors that contribute to the risk of dislodgment
in limpets is also affected by the strength of their
adhesive. Denny (2000) found that the shape of
limpet shells is typically far from the optimum that
would allow the shell to minimize the overall
hydrodynamic force on a body of a given volume.
However, the risk of dislodgment in these animals
is generally very low, due to a highly tenacious
adhesive system. The presence of a strong adhesive
(perhaps in response to predation by birds and
crabs) appears to have given limpets the ‘permission’ to evolve shells that are not notably welldesigned to interact with flow (Lowell 1987).
The ability to adjust
At this point we are again faced with a basic
question. If (as proposed above) most intact waveswept organisms are much stronger than they need
be to resist wave-induced hydrodynamic forces
and at the same time have broad ‘permission’ from
the flow environment to evolve diverse morphologies, what governs the shapes that we actually
see? It seems likely that there is no single answer to
this question; a broad spectrum of biological factors (and the interactions among them) may be
responsible (Koehl 1986). Despite the apparent
lack of a single guiding principle, it is informative
455
to explore individual examples in which the
mechanics of flow may have conspired with biological factors to affect the functional morphology
of wave-swept organisms. The first of these
involves phenotypic plasticity.
Plastic growth in algae
The maintenance of a low risk of dislodgment by
wave-swept plants can in some cases be attributed
to the ability of these organisms to alter their
morphology plastically as they grow. For example,
the kelp Laminaria saccharina typically has a different morphology at wave-exposed sites than at
sheltered sites. The exposed blades are thinner and
longer, and presumably have a lower drag coefficient. In a series of laboratory experiments, Gerard
(1987) elegantly demonstrated that this difference
in form could be elicited by growing blades with
different amounts of tension applied along their
length. Plants under increased tension grew to
resemble the wave-exposed morphology, suggesting that the increased drag on blades in the field is
a sufficient cue for the plants to adjust their morphology appropriately.
Similar phenotypic plasticity has been noted
among other species of kelps, although in many
cases it is unclear what the exact cue is that governs
the pattern of growth. For example, the ruffled
edges of the blades of the bull kelp, N. luetkeana,
could be an adaptation to increase turbulent flow
over the blades in slow water velocities. Alternatively, ruffles could be a mechanism to keep blades
from superimposing themselves, and thus an
adaptation to increase light availability (e.g., Koehl
and Alberte 1988). It is quite possible that both
effects operate simultaneously.
Intertidal macroalgae are capable of passively
adjusting their morphology in response to flow
even when fully grown. Blanchette (1997) performed a series of reciprocal transplants of the
rockweed Fucus gardneri. Large individuals
transplanted from protected to exposed sites ‘tattered,’ a process that reduced their size to match
that of individuals that had been raised under exposed conditions. This ability to reduce their size
maintains a constant rate of survival of this species
across a wide range of flow conditions. Note that
effective tattering in an alga involves a considerable element of mechanical design. The alga must
be structured such that applied hydrodynamic
forces break the distal portions of the plant first.
Plastic growth in animals
The ability to adjust morphology to flow is not
confined to algae. For example, Trussell (1997)
showed that the intertidal snail Littorina obtusata
adjusts the size and shape of its shell and the
adhesive area of its foot during growth to maintain
a constant ratio between drag at a given velocity
and the strength of adhesion to the substratum.
Both Trussell (1997) and Etter (1988) note that
phenotypic plasticity in intertidal snails is ‘asymmetric.’ Snails born in protected sites readily
adjust their shape if exposed to increased water
velocities either in the field or in the lab. In contrast, snails initially living in exposed sites show a
reduced tendency to acquire a ‘low-flow’ morphology.
Biological determinants of strength
Ecological effects
We have noted above that a sufficiently high
strength can limit the ability of natural selection to
guide the evolution of form. There are situations,
however, in which interactions among species
constrain the ratio of applied force to structural
strength, and in these cases hydrodynamic forces
may still be of importance in natural selection. We
have seen how overgrowth by kelps can affect the
rate of dislodgment in subtidal mussels (Witman
1987). Presumably, if a mussel evolved an antifouling coating effective against kelps, it would be
at a selective advantage. This advantage would be
set by an ecological interaction, however, not by
the intrinsic strength of the mussel itself. In a
similar fashion, Koehl and Wainwright (1977)
found that a large fraction of dislodged bull kelps
had broken where they had been gnawed upon by
urchins. In this case, the ultimate strength of the
alga is set by herbivory rather than factors intrinsic
to the organism, and the typically high safety
factors in algae may be in response to the need to
cope with damage from herbivory. Given the
brittle nature of the materials from which marine
algae are constructed (Denny et al. 1989),
456
reductions in strength due to herbivory are likely
to be common in wave-swept macrophytes.
Thus, on wave swept shores there is a certain
symmetry to the roles of mechanics and biology in
the evolution of form. Wave exposure, interacting
with the strength and structure of organisms, can
limit which species co-occur. At the same time,
damage incurred by the interaction of co-occurring species (e.g., through herbivory) can play a
role in the evolution of strength and form.
Allometry
Other examples exist in which constraints on
strength may have affected the morphology of
intertidal algae. Carrington (1990) showed that
Mastocarpus papillatus grows with an unusual
allometry. The blades of the alga continue to grow
through time, but the cross-sectional area of the
stipe does not. As a result, the ratio of blade area
to stipe cross-sectional area increases through
ontogeny, and the drag applied to the alga increases relative to the organism’s ability to resist.
This pattern of growth is likely to set an upper
limit to the size of blades that can survive in a
given flow regime, a limit that can vary seasonally
as the ocean’s waviness fluctuates (Denny and
Wethey 2000). Because the reproductive output in
this species is proportional to the size of blades,
the fixed cross-sectional area of the stipe limits the
reproductive output that can be maintained in a
given habitat. Conversely, for a blade of a given
size, the allometry of the alga determines the
maximum velocity it can withstand. For blades of
typical size, this velocity falls well within the range
seen on rocky shores, and this alga is often broken.
In teleological terms, why would an alga
potentially constrain its reproductive output or
survival by limiting the cross-sectional area of its
stipe? The life history of the species provides a
possible answer. In M. papillatus blades grow from
a perennial crustose holdfast, and the persistence
of this holdfast as a means for maintaining space
on the substratum may be of paramount importance in the long-term fitness of an individual. If
this is true, it may be advantageous to have blades
with a built-in safety link such that if an unusually
large force is applied to the blade, the stipe breaks
before the holdfast is dislodged. Indeed, Carrington (1990) found that when she pulled on these
algae, in 88% of the cases it was the stipe that
broke rather than the holdfast. If this scenario is
valid, the life history strategy of M. papillatus may
be as follows. In a year characterized by mild flow
conditions, blades can grow to a large size, and
reproductive output is large. In years when flow
conditions are severe, the reproductive blades may
be broken, but the holdfast will survive to try
again in the future (Denny and Wethey 2000). The
effectiveness of this strategy is augmented by the
fact that dislodged blades may retain their viability
for a time. In this case, broken blades may serve as
a means of dispersal. The strategy outlined here is
in contrast to that proposed for Fucus (Blanchette
1997), in which the plant allows the size of a frond
to be reduced by tattering. It remains to be seen
where the tradeoffs lie between ‘tattering back’ and
breakage at the base of the stipe.
Other examples of the interaction between algal
allometry and fluid dynamics can be found in
Gaylord and Denny (1997) and Denny et al.
(1997).
Irresistible forces
In light of the great diversity of forms among
wave-swept algae and animals, it is tempting to
suppose that designs can evolve to resist any waveinduced force that the environment can produce.
In fact, there are a variety of forces common on
rocky shores that are likely to be irresistible. For
example, logs can be carried by waves, acting as
battering rams that smash anything in their path.
On shores where logs are common, the resulting
disturbance can play an important role in community dynamics (e.g., Dayton 1971). In a similar
fashion, waves can propel rocks at the shore
(Shanks and Wright 1986), and these may have
sufficient momentum to be potent sources of
destruction. Bascomb (1980) relates the story of a
rock weighing 130 pounds being thrown over a
lighthouse with its top 139 feet above sea level.
Dayton et al. (1984) and Seymour et al. (1989)
note that much of the disturbance in forests of the
giant kelp M. pyrifera is due to ‘rafts’ of previously
dislodged individuals. These rafts become entangled with intact fronds, vastly multiplying the drag
on these individuals. When a frond gives way under the stress applied by the raft, it becomes part of
the raft, which then drifts on to its next victim. The
457
‘snowball effect’ of this process can render rafts
virtually irresistible, and can cause rates of disturbance far beyond those predicted for individual, intact fronds (Utter and Denny 1997). The
effect of these irresistible forces on the evolution of
shape and size remains to be explored.
Conclusions
The ability to predict maximal wave heights and
maximal wave-induced hydrodynamic forces
provides a valuable tool in the exploration of the
role that biomechanics plays in community ecology and the evolution of shape on wave-swept
shores. To date, the results gained from the use of
this tool highlight the ability of certain physical
attributes of algae and animals (e.g., flexibility,
strong adhesion) to pre-empt the evolution of
forms specialized to cope with flow. Thereby given the ‘permission’ of the flow environment to
evolve in response to other factors, wave-swept
organisms have developed a bewildering variety
of sizes and shapes. However, flow continues to
play a potentially important role in evolution by
providing a mechanism for breaking or dislodging those individual that have been selected by
other means (e.g., damaged by herbivory). The
interactions of flow with other aspects of nearshore biology (e.g., external fertilization: Levitan
1995; Serrão et al. 1996; Pearson and Brawley
1998 and larval settlement: Johnson 1994; Abelson and Denny 1997) provide other areas for
productive research.
Caveats
This review has presented a brief, and in many
ways superficial, overview of wave-induced flows
and the hydrodynamic forces they impose on
organisms. In attempting to draw general conclusions I have given short shrift to many of the
details. The interpretation of the role of hydrodynamics in nearshore biology is very much a
work in progress, and the reader is advised to
take the information presented here with a large
grain of salt. Better yet, the reader is urged to
delve into the literature cited, and draw his or her
own conclusions. Best of all, the reader is
encouraged to visit a wave-swept shore and to
play with the ideas presented here. Let me know
what you find.
Acknowledgements
Much of the work cited here was supported by
NSF Grants OCE-9115688, OCE-9313891, OCE9633070, and OCE-9985946. I thank C. Harley,
P. Martone and two anonymous reviewers for
constructive suggestions.
Appendix A: Water accelerations in the swash
Field recordings (George et al. 1994) and laboratory tests Svendsen (1987) show that in a frame of
reference moving with a wave crest, the amplitude
of turbulent velocity fluctuations, u¢, produced by
waves in the surf and swash zone can be related to
the phase speed of the wave, measured relative to
the stationary substratum. Using Eq. (6) for the
phase speed, c:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
ðH þ dÞ
0
;
u ¼ hwc ¼ hw g H
ð28Þ
þd
2
where h is a constant for which empirical values
range from 0.015 to 0.074 (George et al. 1994). w
(a coefficient accounting for topographical amplification of phase speed) is 1 in the surf zone and
may be as high as 2 in the swash.
Here u¢ is the fluctuation in velocity. If we assume that velocity varies sinusoidally with an
average period s, the instantaneous water velocity
due to turbulent eddies is (in this Lagrangian reference frame):
2pt
2pt
uturb ffi u0 sin
¼ hwc sin
:
ð29Þ
s
s
In the stationary frame of reference of the substratum, the period of fluctuations in velocity is
equal to the length scale of turbulent eddies divided by the rate at which eddies are advected past
a point on the substratum. Because eddies move at
approximately the same velocity as the bulk of the
water near the substratum (/wc)
s¼
‘
:
/wc
ð30Þ
458
Here ‘ is the turbulent length scale and / is (for a
position well behind the wave crest) the fraction of
the phase speed at which water moves (see Eqs. (7)
and (12)). Using this definition of s, we find that
the overall velocity, u, is
2p/wct
þ /wc:
ð31Þ
u ¼ hwc sin
‘
With this expression in hand, we can calculate the
acceleration of water relative to a stationary object
on the substratum:
@u 2p/hw2 c2
2p/ct
¼
a¼
cos
:
ð32Þ
@t
‘
‘
and the maximal acceleration is
amax ¼
2p/hw2 c2
:
‘
ð33Þ
This relationship is very sensitive to the choice of
an appropriate length scale, ‘. The choice of length
scale in turn depends on whether the object experiencing the acceleration is in the benthic boundary
layer or not. Outside the boundary layer, Svendsen
(1987) suggests that ‘=0.2d to 0.3d. Inside the
boundary layer, standard practice (Schlichting
1979) is to set the length scale equal to the distance
from the substratum, that is, ‘=y.
Now, the thickness of the boundary layer in
oscillating flow can be estimated as (Grant and
Madsen 1986):
d¼
kTu
;
2p
ð34Þ
where k is von Kármán’s constant (0.41), T is the
wave period, and u* (the friction velocity) can be
approximated by u¢ as given in Eq. (28). For
example, given a swash-zone bore 1 m high (for
which c=4.4 m s1) and a period of 10 s, and
setting w=2, the boundary-layer thickness is
approximately 9–43 cm depending on the value of
h. Most wave-swept organisms lie closer to the
rock than this when subjected to flow, so we can
reasonably assume that surf-zone organisms lie in
the turbulent boundary layer. We thus set ‘=y. In
other words, if an organism is 2 cm high, ‘ at its
apex is 2 cm.
Inserting y for ‘ in Eq. (33), we see that:
amax ¼
2p/hw2 c2
:
y
ð35Þ
Now, from
2 Eqs. (10) and (12) we know that
u
, thus
c2 ¼ /w
amax ¼
2phu2
:
/y
ð36Þ
Note that that there is a lower limit to the
appropriate magnitude of y. Very near the substratum (in what is known as the viscous sublayer)
the effects of viscosity are sufficient to damp out
turbulent eddies (Schlichting 1979). Consequently
the turbulent length scale is meaningless for y less
than the thickness of the viscous sublayer. In
practice, measuring ‘ at y less than the height of
the local roughness elements is problematic, and
the minimum ‘ is likely to be on the order of this
roughness height.
Alternatively (and equivalently), we can base
our analysis on the notion that a turbulent eddy
has a structure that varies little in the time it takes
for the eddy to be advected past a point on the
substratum. In this ‘frozen turbulence’ approach,
the velocity at a given point in space is the sum of
the bulk velocity (/wc) and the velocity due to the
rotation of a turbulent eddy
2px
0
u ¼ /wc þ u sin
;
ð37Þ
‘
where x is the location of our measurement and ‘
is the diameter of the eddy.
The acceleration at point x can then be estimated
by taking the total derivative of u:
du
du
þu
dt dx
du
2p
2pu0
2px
þ /wc þ u0 sin
¼
cos
dt
‘
‘
‘
a¼
ð38Þ
If we assume that flow is approximately steady (so
that du/dt is negligibly small) and if we ignore
terms in u¢2 (they are small) this reduces to
amax ¼
2p/wcu0
‘
ð39Þ
Recalling (Eq. (28)) that u¢=hw c, we again obtain
Eq. (35):
2p/hw2 c2
:
ð40Þ
y
One final note. We calculate above that the
turbulent boundary layer under waves is likely to
amax ¼
459
be 9–43 cm thick, which may seem at odds with a
statement made earlier in the text, where it is implied that velocities close to those of the mainstream penetrate to within a cm of the substratum.
This apparent discrepancy is an artifact of the
definition of boundary-layer thickness. The total
thickness of a turbulent boundary is the distance
from the substratum at which the average velocity
is equal to 99% of mainstream (Denny 1988;
Vogel 1994). This distance can be quite large.
However, most of the actual velocity gradient is
confined to a very thin sublayer adjacent to the
substratum (Schlicting 1979). Thus, a velocity 90%
of mainstream might well be reached only a fraction of a cm away from the substratum, while the
next 8–42 cm is required to accrue the remaining
9%.
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