s. Afr. J. m£lr. Sci. 10: 353-363
353
1991
BIOLOGY, NATURAL SELECTION AND THE PREDICTION OF
MAXIMAL WAVE-INDUCED FORCES
M. W.DENNY*
Appropriate interpretation of body form often requires knowledge regarding the maximal force that an organism
is likely to encounter in its Iifetime. In many situations. prediction of maximal forces is nearly impossible, but the
statistical distribution of ocean wa\es allows prediction of the maximal lift and drag to which a benthic plant or
animal is subjected. These predictions can, in tum. be used to provide quantitative estimates of the severity of natural
selection on specific attributes of body form.
Toepaslike venolking van liggaamsvorm vereis dikwels ke~nis aangaande die maksimale krag wat 'n organisme
waarskynlik in sy leeftyd salteekom. fn vele omstandlghede IS dll haas onmoonthk om makslmale kragte te voorspel, maar die statistiese verspreiding van oseaangoJwe maak voorspelling moontlik van die maksimale hef- en
sleurkragte waara an 'n bentiese plant of dier blootgestel word. Op hul beun kan hierdie voorspellings aangewend
word om kwantitatiewe ramings van die hefligheid van natuurlike seleksie ten opsigte van bepaalde kenmerke van
liggaamsvorm te verskaf.
When an engineer designs a structure, it is necessary
that he or she has an estimate of the maximal force
that the structure is expected to endure. For instance,
when designing a house, an architectural engineer
must take into account the maximal load to be caused
by storm winds, the possibility of snow on the roof
and the likelihood of earthquakes. Only then can the
structure be designed appropriately. For a house, an
"appropriate" design is one strong enough to ensure
that the house will not collapse, but not so strong that
inordinate expense is incurred in the building process.
Scientists face an analogous problem when examining
the design of plants and animals. In such a case, however,
the biologist is presented with the finished product, a
living organism, and the task is to understand how the
organism works. An important component of this task
is to assess which of the myriad possible criteria of
mechanical design has actually been influential in the
COurse of a particular organism's evolution. For example, the question may be posed as to why the wing
bones of a pigeon are the size and shape that they are,
or why kelps are small and flexible whereas trees are
large and stiff? Questions of this nature, which concern the direction and severity of natural selection on
mechanical design, can be answered only when estimates are available of the maximal forces to which the
organism has been subjected.
T~erein lies one of the fundamental problems in
studIes of evolution: it is often very difficult to estimate
the maximal force a living structure encounters
(Alexander 1981, 1984). For instance, it may be supposed
that the wings of a pigeon must be strong enough to
*
resist lift forces without snapping, but that they cannot
be too strong, lest the additional mass required to
strengthen the bones be an impediment to efficient
flight. The "appropriate" trade-off between strength
and weight is therefore set by the maximal lift force
likely to be encountered when, for example, the bird
makes a rapid turn (pennycuick 1967). However, how
rapidly the bird turns is, in large part, under the bird's
control. Because the maximal force can thus be affected
by behaviour, it is nearly impossible to quantify the
trade-off between strength and weight. As a result,
arguments regarding the evolved mechanical design of
wing bones (and many other biological structures) are
doomed to be qualitative.
There are, however, situations in which maximal
lifetime forces may be open to quantitative estimation
(see, for example, Lowell 1985, 1987). One such case
is examined in this paper; the hydrodynamic forces
imposed on nearshore benthic plants and animals by
wave-induced water motion.
THE WAVE-SWEPT ENVIRONMENT
As ocean waves approach a shore, they cause the
water at the sea bed to move. Outside the surf zone,
these oscillatory motions can be described with
reasonable accuracy by linear wave theory, which
predicts that the maximal velocity at the substratum is
proportional to wave height H (the vertical distance
between trough and crest) and is inversely proportional
Hopkins Marine Station of Stanford University, Pacific Grove, California 93950, U.S.A.
Manuscript received : August 1990
354
South African Journal of Marine Science 10
Table I: Symbols used in the text, and the equation in which
they first appear
Symbol
Definition
Apr
Cd
C,
d
H
Hm
H"
Hr",,'i
H,
L
M
M"
M
A1::.max
II
T
ltcreSI
lima);
~
p
1:
6
5
5
6
=
(3)
6
I
I
II
\I
15
7
I
7
9
13
16
7
I
Ucrest = (g[H
+ d])2
(4)
I
4
I
15
5
8
(rrH/D (I/sinh[27td/L]) ,
(1)
where sinh is the hyperbolic sine function [sinh(x) =
e-X)] and L is the wave length at depth d. As a
practical matter, L can be estimated as :
t (eX -
I
L
I
u max "" 0,3 (g[H + d])2
Thus, for a wave. I m high in a water depth of 2 m
the maXimal veloCity at t.he substratum is about I
m' S-I. Note that the velocity caused by these bores:
the surf zone is independent of wave period, a ()"ener~~
attribute of waves in very shallow water.
b
a
Eq~ation I applies before a wave brea ks and
Equatfon 3 thereafter. Unfortunately, max imal Water
velocity is much less easi~y defined for breaking
waves themselves. At the lOstant of breaki ng, the
water at the crest of a wave moves at a velocity
(Denny 1988)
to wave period T. (A bst of alJ the symbols used in the
text, and the equation in which they first appear, is
given in Table I.) Velocity also depends on the depth
of the water column d, decreasing with increasing
depth (Denny 1988):
U max
1949). For steeply peaked waves such as those f
in the surf zone:
oUlld
Equation
Planfonn area
Projected area
Drag coefficient
Lift coefficient
Depth of water column
Wave height
Median significant wave height
Normalized significant wave height
Root-mean-square wave height
Significant wave height
Wavelength
Max.imal wave height
M/Hs
M/H m
Yearly maximum M"
Number of waves
Wave period
Crest velocity
Maximum velocity
Depth parameter
Water density
Interval period
ApI
199]
= (gT2/[2rcJ) (tanh[4rc 2d/(T 2g)])2,
(2)
where tanh is the hyperbolic tangent function ([eX e-X]/[eX + e-X ]) and g is the acceleration due to gravity
(9,81 m's-2 ) - Eckart (1952) .
For example, when the wave period is 10 s (a typical value for ocean waves), a wave I m high in 15 m
of water imparts a maximal velocity of approximately
0,3 m' S-I to the fluid at the sea bed.
As waves move into shallow water, a point is
reached where they become unstable and break.
Breaking usually occurs when the wave height is
80-140 per cent of the water depth (Denny 1988).
After breaking, waves may continue to move inshore
as bores, the height of which gradually decreases as
wave energy is expended in turbulence. The velocity
at the substratum under one of these turbulent bores
can be estimated from solitary wave theory (Munk
If the wave crest arches forward and impacts directly
on the substratum, organisms in its p ath may be
subjected to very rapid water velocities. For example,
a wave 2 m high, breaking in 2 m of water and subsequently striking the substratum, would impose a velocity of 6,3 m·s- I . Water velocities as high as 14-16
m ' S-I have been measured on exposed rocky shores
(Vogel 1981, Denny el al. 1985).
These extreme velocities are imposed , however,
only when the crest itself strikes the shore, a si tuation
that most often occurs on shores with steep slopes;
under other circumstances, the velocity at the substratum is likely to be considerably smaller. Nev rthe l~ss,
because of the complexity of motion in a brea ktng
wave, the precise maximal velocity is diffic ul t to predict from present theory. Empirical results presen ted
by Denny and Gaines (1990) suggest tha t maX imal
velocities at the substratum near the break po mt of
waves increase in proportion to wave height, r ather
than the square root of wave height, as in~ ~lted by
Equation 4. The question of maximal velOC Ities near
the break point will be raised again later.
HYDRODYNAMIC FORCES
The wave-induced water motions outlined above
impose three types of force on benthic organ Isms
(Denny 1988). These are di scussed in the follow tng
three subsections.
Drag
As water moves past an object on the substratuo:,
the pattern of flow is such that a high pressure IS pres-
Denny: Maximal Wave-Induced Forces
1991
ent on the upstr~am face of the object and a lower
pressure is assocIated wIth the turbulent wake downstream. This upstream-downstream difference in pressure exerts a force, pressure drag, on the area of the
object that extends into the flow (the proftle area Apr)'
Drag tends to push the object downstream.
.
Maximal drag can be descnbed for most CIrcumstances by the equation
Maximal drag = t pu 2max Apr Cd '
(5)
where p is the density of the water (about I 025 kg'm- 3
for seawater), and Cd. the drag coefficient, is a parameter determined primarily by the shape of the object.
For streamlined objects, Cd decreases with increasing
water velocity, but for bluff, i.e. non-streamlined, objects (which includes most benthic organisms), Cd is
nearly constant over a wide range of water velocities.
Here it is assumed that Cd is constant, leading to the
conclusion that, for a plant or animal of a given size
and shape, drag increases with the square of maximal
water velocity.
Acceleration reaction
In addition to drag, there is a second force that acts
in the direction of flow. This force, the acceleration
reaction, is proportional to the acceleration of the
water rather than to its velocity, and to the volume of
the object rather than to its profile area. Although
acceleration reaction can have interesting consequences regarding limits to size for benthic organisms
(Denny et 01. 1985, Denny 1988), its magnitude is
generally much smaller than that of drag, and for this
reason its consequences will be considered only
briefly at the end of this discussion.
Lift
. The same flow-induced pressure distribution that
unpo.ses drag also can result in a force, lift, acting perpendIcular to the direction of flow. Maximal lift can
be described by an equation similar to that for drag:
Maximal lift = ~ P u 2max Ap,CI
'
(6)
~here ApI is the planform area of the object (the pro~ected area one would see looking down on the object
~e direction of lift) and C, is the lift coefficient,
agam a parameter primarily determined by the shape
of the object. Lift coefficients have been measured for
?nly a few benthic organisms (Denny 1988, 1989), but
~these cases CI is independent of velocity. Thus, like
g, maximal lift is expected to increase in propor-
In
---
355
tion to the square of maximal water velocity.
The combined action of lift and drag can place substantial forces on benthic organisms, and there are
many cases in which these forces are suspected both
of controlling the distribution and abundance of
species and of contributing to the ecological dynamics
of benthic communities (for reviews see Sousa 1985,
Denny 1988). Because these forces are proportional to
the square of water velocity, and because maximal
water velocity is controlled by wave height, it behoves
one to search for some method of predicting the maximal
wave height that a particular organism will encounter
in its lifetime. If accurate predictions of maximal
wave height can be made, they will form a useful tool
for exploring the role of hydrodynamic forces in the
evolution of body form in benthic organisms.
THE THEORY OF MAXIMAL WAVE HEIGHTS
On an individual basis, wave heights are nearly
impossible to predict, a fact to which any surfer or
intertidal ecologist can attest. The heights of waves
striking a shore in any period are quite variable and, in
general, knowledge regarding the height of one wave
provides no information regarding the height of subsequent waves. In other words, individual waves act independently. This stochastic nature of waves, however,
opens the door for a robust statistical approach to the
prediction of maximal wave heights.
On any given day, waves approaching a shore have.
a mean wave period T. Individual waves may vary
from this mean, but as long as the variation is slight,
the wave field is said to be narrow-banded. Given this
prerequisite, the heights of individual waves are expected to conform to a particular distribution, the
Rayleigh distribution, as shown in Figure I (LonguetHiggins 1952).
The Rayleigh distribution is described completely
by a single parameter proportional to the local "waviness" of the sea. One can, for instance, measure the
elevation of the sea surface at one point through time
and express "waviness" as the standard deviation of
surface elevation. The Rayleigh distribution can then
be described in terms of this value. More often, how~ver, the root-rpean-square wave height H rms , which
IS equal to 2(2)2 times the standard deviation, is used.
Alternatively, the Rayleigh distribution can be described in terms of a significant wave height Hs . This
parameter, which corresponds roughly to visual estimates of wave height, is defined as the average height
of the 1- highest waves pres\ent. Longuet-Higgins
(1952) has shown that Hs '" 2 i H rms , or four times the
standard deviation of surface elevation.
According to the Rayleigh distribution (Fig. 1),
356
South African Journal of Marine Science 10
199/
1.°-;::-------------------------------~ 1,0
0,8
0,8
I
I
I
I
I
>-
I
0,6
I
I
I-
::::;
I
a;
I
~
I
I
Q
I
I
a.. 0,4
I
I
I
I
,,
,,
,,
\
\
,
\
>0 ,6 IVi
Z
w
,
Q
>-
.....
\
'H
, s
004
,,
,
,,
I
I
\
I
0,2
I
I
I
I
I
I
I
°
~
«
0""
'""-
,..
....
0,2
'"
' ..
.... ..... .. _-
2
3
H/H rms
Fig. 1: The Rayleigh distribution of wave heights. The probability that a wave chosen at random has a height H such
that HI Hrms is greater than a given value is shown by the solid line and the left-hand ordinate. The derivative
of this exceedance curve gives the probability density curve shown by the dashed line and the right-hand
ordinate
there are few waves with small heights, the most frequently observed heights are slightly less than the
root-mean-square height, and there is a long tai I to the
distribution that extends to waves much greater in
height than either Hmrs or Hs. It is this tail of infre quent, high waves that is of particular interest here:
these are the "rogue" or "sneaker" waves that cause
maximal damage to benthic organisms.
Several empirical studies have shown that, under
many circumstances. ocean waves indeed conform approximately to a Rayleigh distribution (Sarpkaya and
Isaacson 1981). Thus, the statistical propel1ies of the
local wave heights can, to a reasonable approximation,
be described at any time as functions of Hrm.1 or Hs.
For the purposes of the present discussion, the major
exception to this rule applies to waves at the point of
breaking, whi c h can deviate substantially from a
Rayleigh distribution (Thornton and Guza 1983) .
Other exceptions will be discussed later.
For those circumstances in which individual wave
heights conform to a Rayleigh distribution, Longu~t­
Higgins (1952) has shown that it is possible to predict
the maximal height M one can expect from a collec-
4,------~--~--~---~-,
l-
I
C)
iIi
I3
w
I
~
/
5:
o
~
--'
«
~2
«
~
ow
N
::::;
«
' - - T=5s
----- T=lOs
_._-- T=20s
~
0<
oZ
3
4
5
6
7
8
9
LOG TIME (seconds)
Fig. 2: The largest wave encountered for a sea state characterized by a constant Hs depends on elapsed time and
wave period T. Curves calculated from Equation 10
Denny: Maximal Wave-Induced Forces
1991
357
I.Or---'OO!!!!!!!III;~:::;-----------------------~
0.9
0.8
wO. 7
u
Z
~0.6
w
w
~
wO.S
u...
o
~0.4
::::;
iii
~0.3
...""
0.2
0.1
2
°
3
NORMALIZED SIGNIFICANT WAVE HEIGHT
Fig. 3: Exceedance curves for significant wave height at five sites on the west coast of North America. 1982-1987
tion of n waves:
I
M = (~!n(n) l' Hs
(7)
for n > 50.
Alternatively, maximal wave height can be expressed
in terms of time. If the average period of waves present
is T seconds, T.IT waves move over a given point on
the substratum in a period T. . If it is assumed that T. » T,
the maximal height of these waves is
I D
!
M = !2rn(T./T) I' Hs .
(8)
It is useful to normalize this maximal wave height to
a parameter that describes the time-averaged "waviness"
of a particular site. By this strategem, a generalized
description of maximal wave height can be obtained.
For example, a normalized maximal wave neight can
be defined by
(9)
in which case
Mn
= H£n(T./n}~
(10)
To convert from Mn to M for a particular situation,
it is necessary simply to multiply by the operative Hs.
This relationship is shown in Figure 2 for average
wave periods ranging from 5 s (characteristic of locally
created seas) to 20 s (characteristic of swell from distant storms). For example, in the course of an hour it
can be expected that a wave will be encountered with
a height 1,5 times that of the significant wave height.
In a day, a wave approximately twice the significant
height would be encountered. It would require more
than a decade, however, before a benthic organism
would encounter a wave three times Hs.
The predictions shown in Figure 2 assume that Hs is
constant, but on any real shore this is very unlikely.
During storms Hs increases, and during periods of
calm, it decreases . If, however, the variation of Hs can
be taken into account, it is still possible to predict a
maximal wave height.
This prediction involves two steps. First, information must be obtained on the long-term variation in
Hs. As an example, the literature (U.S. Army Corps of
Engineers 1982-1987) gives the yearly distribution of
significant wave heights at five sites on the west coast
of North America. To facilitate comparison of the
pattern of variation in H s , significant wave heights at a
particular site in a particular year are normalized to the
358
Soulh African Journal of Marine Science /0
199/
9
_1000
@B
~
c
0
~7
v
~
u
~6
BOO
<f)
'"::l
<f)
~
~5
0
600
~
I4
0
~3
;;j'
:5
400
::l
200
::;
6I 2
u
>-1
0
1
2
0
3
Fig. 4: The least-squares fit to the exceedance curves of
Figure 3, drawn using Equation 12
median significant wave height Hm for that site and
year. Thus, by definition,
(II)
For the particular data set used in this example, the
fraction of the year for which H~ exceeded a given
normalized value was similar among sites and years
(Fig. 3), and an average distribution was calculated
by fitting a Weibull distribution to the pooled data,
as described by Denny and Gaines (1990). In this
case, the Weibull distribution closely approximates a
Gumbel type 1 distribution (Jacocks and Kneile
1975), such that
p(HIl
> H)
=I-
expl-exp[-3,532(H - 0,881)]). (12)
where p(H n > H) is the probability (or fraction of
time) that the observed Hn exceeds a given value H.
The results are shown in Figure 4. TtUs curve is presented here primarily for the sake of argument. It may
or may not be a valid representation of how significant
wave heights vary on shores in general.
By dividing the probability curve of Figure 4 into a
series of contiguous ranges in HIl , and taking the difference in expected exceedance time across each
range, the cumulative hours during a year fo~ whic.h
each range of normalized significant wave heights IS
present can be calculated (Fig. 5). The most comm~nly
occurring Hn (the mode) is about 0,88, and the tIme
spent at a certain HII decreases rapidly fO.r Hn > I.
.
This distribution of cumulative hours IS then used In
the second step of the calculation of yearly maximal
wave height. As before, it is useful to express this
maximal height relative to the "waviness". of the sit~,
thereby to obtain a generalized deSCription. In this
2
NORMALIZED SIGNIFICANT WAVE HEIGHT
NORMALIZED SIGNIFICANT WAVE HEIGHT
3
Fig. 5: Yearly cumulative hours for which Hn falls within the
ranges centred at the dots. Calculated from data presented in Figure 4
case it is the long-telm waviness that is o f interest. a
value for which Hili can be used as an index . Thus. the
normalized yearly maximal wave he ight can be defined by
M\"=M/HIII
( 13)
Dividing both sides of Equation R by the median
yearly significant wave height.
( 14)
where L is the cumulative time spent at a partic ular HII
during the year. Carrying out this calcul ation for the
cumulative times shown in Figure 5, an est imate of
how M" varies with HII can be arrived at (Fig. 6).
Owincr to the short period of time spent at large HII
(Fig. 5),cthe overall maximal yearly normali zed ~ave
height MI'.max is associated with a Jess-than-ma xl l11 al
Hn. The magnitude of the yearly maximu m Val les
slightly depending on the average wave pen od (t.h~
shorter the wave period, the more waves encountele
in a year), but for the range ?f expected wave pe nod~l:
M" lies between 5 and 6 (FIg. 6). In other words.' ~ nt
exposed shore where the yearly median slglllflca d
wave height is 2 m, the maximum wave height ex pecte
in a year lies between 10 and 12 m .
e
If the pattern in variation of Hs docume nted h~r
(Fig. 4) is representative, maximal wave heIghts t~r
periods longer th.an .a y~ar can be esttmated slmpl~ ~
multiplyincr the distributIOn of cumulative tunes (Flo· .
by the appbroptiate constant and calculating as befdo~e.
o pre ICHowever, there are four caveats regard ·tng th IS
tion:
an
(i) The prediction assumes that the Rayleigh dis tri-
bution is an accurate description for even ve ry
Denny: Maximal Wave-Induced Forces
1991
359
and any storm-induced set-up.
(iv) As a result of the hydrodynamic complexities associated with waves breaking, it is difficult to
predict from theory the maximum breaking wave
height on steeply sloping shores. Denny and
Gaines (1990) have shown, however, that the
maximal hydrodynamic forces imposed by
waves on steep rocky shores are, to a first approximation, distributed as if the wave heights
causing them conform to a Rayleigh distribution.
MAXIMAL FORCES
o
1
2
3
4
5
NORMALIZED SIGNIFICANT WAVE HEIGHT
Fig. 6: Expected normalized maximal yearly wave height My
as a function of Hn
large waves. This assumption will be discussed
later.
(ii) It should be noted that the prediction of Figure 6
applies only for waves before they have broken.
At the point of breaking, waves are known not to
conform precisely to a Rayleigh distribution, and
therefore they cannot be predicted accurately by
this method.
(iii) After waves have broken, their significant wave
heights are governed more by the local water
depth than by the height of incoming waves.
Thornton and Guza (1982) have shown that, on
gently sloping shores, the average height of
waves in the surf zone (here expressed as Hrms)
is:
H rms
= ~d
,
(15)
where ~ is a parameter characteristic of a particular shore. For the shore studied by Thornton
and Guza (op. cit.), ~ = 0,42. As a result of the
dependence of wave height on depth, maximal
water velocities within the surf zone are expected
to decrease with decreasing depth of the water
column. Further, as long as the prevailing sea
conditions are sufficient to provide waves whose
broken height exceeds ~d, Hs in the surf zone on
gently sloping shores is likely to vary little. In
such a case, the maximal wave height at a particular spot in the surf zone will be governed by
those factors that set the local depth: the tides
The predictability of maximal wave heights discussed above allows prediction of the maximal hydrodynamic forces that a benthic organism is likely to
encounter in a year. In each case, the link between
maximal wave height and maximal force is via a
particular theory for how water moves in waves, and
the resulting predictions are only as good as the theory
on which they are based.
In the case of organisms outside the surf zone, linear
wave theory is used to predict water velocity (Equation I) and, under these conditions, it is likely to provide
estimates of acceptable accuracy.
M5'
Maximal lift = ~
max H;, (rr"/l T" sinh" [2rrd/Lll)
pC/Api ,
(16)
Maximal drag
=~ M5',max H;,
pCdApr
,
(rr"/l T" sinh" [2n:d/Lll)
(17)
Noting, from Figure 6, that My,max is approximately
5,5, the maximal yearly lift and drag on a benthic
organism under these conditions is expected to be:
Maximal lift "" 15 H;, (rr"/l T" sinh" [2n:d/Lll)
pC/Apt ,
(18)
Maximal drag"" 15 HJ, (rr2/1T" sinh" [2n:d/LJl)
P Cd Apr
(19)
Consider, for example, a stony coral such as Acropora
cerviconis living at a depth of 15 m. A. cervicornis
resembles a circular cylinder with a length about 10
times its diameter. As such, its lift coefficient is likely
to be negligible (Denny 1988), but its drag coefficient
is approximately I. If the median significant wave
height at the site is 2 m, and the coral is 3 cm in diameter
and 30 cm long, the maximal expected wave-induced
drag force expected in a year is about 65 N (Equation
19). This force tends to bend the specimen, and exerts
a force per area (a tensile stress) on the coral's skele-
360
South African lournal of Marine Science 10
ton of about 3,7 x 10 Pa. Now, the average breaking
stress for A. cervicornis is 1,23 X 107 Pa, with a standard deviation of 0,43 x 107 Pa (Denny op. cit.). The
maximal applied stress is 2 standard deviations below
the mean and therefore is sufficient to break only
about 2,3 per cent of this species. One can conclude
that drag force alone is unlikely to result in substantial
disruption of a population of A. cervicornis.
A similar calculation can be made for organisms in
the surf zone on a gently sloping shore. If it is assumed
that the depth at tre site is a constant d, the significant
wave height is 2 i ~d. This value can be inserted into
Equation 8 to calculate the maximum wave height
corresponding to a 1: of 1 year (3,15 x 107 s if the organism is continuously submerged), and that value
can be inserted into Equation 3 to estimate the maximum force:
6
Maximal lift = 0,045 g (Hin(3,15 x 107/Dl~ ~d + d)
p CI ApI
(20)
Maximal drag = 0,045 g ([~in(3,15 x 10 7/Dll ~d + d)
P Cd Apr
(21)
Consider, for instance, an urchin (Strongylocentrotus purpuratus) living at a depth of I m on a
shore where ~ = 0,42. A typical specimen (Denny el
01. 1985) might have a profile area of 1,9 x 10- 3 m2 , a
planform area of 3,9 x 10- 3 m2, and require 148 N (SD
90 N) to dislodge. Denny et 01. (1985) estimated that
the drag coefficient for S. purpura/us is I and the lift
coefficient 0,55. Inserting these values into Equations
20 and 21, the urchin is clearly expected to encounter
a maximal hft force of 2, I N and a maximal drag of
1,9 N. Adding these forces as vectors, the overall
force on the urchin will be 2,8 N, 1,6 standard deviations below the mean dislodgement force. Thus, only
about 5 per cent of the urchins present can be expected
to be dislodged in a year. This somewhat surprising
result is due to the fact that, in the surf zone on gently
sloping shores, wave heights are severely limited by
water depth.
It should be noted that the prediction for maximal
wave forces in the surf zone on gently sloping shores
scales differently with time than that for organisms
outside the surf zone. Because solitary wave theory
predicts that water velocity is approximately proportional to the square root of wave height, the maximal
force encountered in the surf zone (proportional to £(2)
should increase in proportion to M, i.e. in proportion
to 1£n(1:) ]1. Outside the surf zone, where velocity is
proportional directly to wave height, maximal force
increases in proportion to M2, or £n(1:). The significance of this difference in temporal scaling remains to
be determined.
199/
Although they can perhaps provide guidance
the manner in which estimates of maxi mal as to
. ht can b.e use d to pre d"ICt maxImal forces, thewave
helg
tw
exa.mples cited above are hypothetical. No case i~
whIch actual long-term measurements ha ve b
. h
' .
een
rna de In t ese envIronments LS known to the auth
and, therefore, these predictions cannot yet be testedor,
In contrast, Denny and Gaines (1990) report on'
series of empirical force measurements made in th:
surf zone on steeply sloping shores. These measurements show that the probability of encounterincr a
force of a given magnitude relative to the lona-tebon
mean maximal force typical of a site is simi lar : mono
a variety of sites and surf conditions. Their results ar~
consistent with the approach presented here if the
maximal water velocity on steep shores is assumed to
vary directly with wave height (analogous to linear
wave theory) rather than with the square root of wave
height (as suggested by solitary wave theory ). The
empirical data of Denny and Gaines (op. cit.) allow a
demonstration of how the prediction of maximal wave
forces can be used in an exploration of the ro le of
hydrodynamics in natural selection of body fOlm .
MAXIMAL WAVE FORCES AND NATURAL
SELECTION
The empirical relationship between maximal fo rce
and time found by Denny and Gaines (1990) can be
approximated if maximal force is described by:
Maximal lift = d) M~ H} P CI Api,
(22)
Maximaldrag=({)M,;H} pCdApr,
(23 )
for a period in which Hs is constant. If H.\ varies
through the year as described above,
Maximal lift == ({) M ~,max. H,;1 p CI Api,
(24)
Maximal drag= ({) M~"max H~ P Cd Apr.
(25)
These empirical approximations are similar to the expressions obtained from linear wave theory, but they
replace the term 11:2/1 yz sinh 2 [2rcdILl] with a constant
equal to 0,5.
From this appr,oach, the yearly maximal force on a
typical intertidal animal can be predicted. Co.nslder,
for instance, the limpet Lottia pelta, a common mh abl-.
tant of exposed rocky shores on the Pacific coast of
North America. A representative individual of thIS
species (based on data presented by Denny 1989) has
a profile area of 2,05 x 10-4 m\ and a planform area oj'
Denny: Maximal Wave-induced Forces
i991
361
0,7
>-
0,6
Z
::E
w
/
00,5
Cl
9
/
If)
/
Ci 0.4
o
>-
,
>-
g 0,3
<{
<C
~
<l.
,,
,
0,2
,,
,
,,
/
/
/
... -
°
2
.... ....
...... .....
3
YEARLY MEDIAN SIGNIFICANT WAVE HEIGHT
~"",
-----_ --
/
/
/
/
/
;,,'
0,1
/
/
... ... .. ..
..... .
....
4
5
(m)
Fig, 7: Probability of dislodgement in the course of a year for a crawling limpet (L. pella) as a function of the local
wave exposure as indexed by Hm , The solid lines are estimates made using measured force coefficients, The
dashed lines are estimates made using force CDefiicients that have been decreased by 10 per cent
5,77 X 10-4 m2 , The drag coefficient is 0,55 and the Lift
coefficient is 0,25, The mean force required to dislodge
this individual while it is crawling on the substratum
is 23 N, if applied in the direction of lift (SD 9 N), and
35 N if the force is applied in the direction of drag
(SD 12 N),
With these values, the probability of dislodging an
individual limpet, chosen at random, by the maximal
water velocity encountered in a year can be calculated,
Given the above data, the calculation depends only on
thl': yearly median significant wave height and the
mean wave period, For simplicity, a mean wave period
of 10 s has been selected, and data are presented solely
as a function of Hm (Fig, 7).
As expected, the probability of being dislodged increases with increased Hm. In this respect, the annual
median significant wave height serves as an appropriate
index of "wave exposure". Further, the probability of
being dislodged by a lift force is considerably greater
than that of being dislodged by drag. This effect is
discussed in some detail by Denny (1989) and Denny
and Gaines (1990).
Of more interest here is the effect of changing the
lift or drag coefficient. What happens, for instance, if
a minor modification is made to the shape oCthe typi-
cal L. pella shell such that C, is reduced by \0 per
cent? The answer is shown as the upper dashed line in
Figure 7. A 10 per cent reduction in lift coefficient results in a decrease in the probability of dislodgement,
and the amount of decrease depends on H m.
This effect is shown explicitly in Figure 8. For a
site where the yearly median significant wave height
is 1 m, a 10 per cent reduction in lift coefficient results
in a mere 0,02 per cent reduction in the probability of
dislodgement. In other words, limpets with the lowerlift shell would survive the year only 0,02 per cent
better than those with the higher-lift shell. This level
of natural selection might well be negligible. In contrast, if Hm is 3 m, the difference in probability of dislodgement is seventyfold greater, i.e. 1,4 per cent. In
other words, on a highly exposed shore, the selective
advantage of a decrease in lift is much greater than on
a protected shore.
Dislodgement attributable to drag follows a similar
pattern, but for this limpet the change in probability of
being dislodged (i.e. the selection coefficient for a
particular change in shell shape) is much lower for
drag than for lift.
As qualitative predictions, these are hardly surprising.
Most marine biologists would know as a matter of
362
South African Journal of Marine Science 10
0,12,-------------------,
>-
0,10
>:::;
~ 0,08
Lift
~
i 0,06
w
u
a; 0,04
~
Z5 0,02
(ii)
°
2
3
4
YEARLY MEDIAN SIGNIFICANT WAVE HEIGHT 1m)
5
Fig, 8: The difference in yearly probability of dislodgement for
a limpet with high- and low-lift (or high- and low-drag)
shell morphologies. Calculated from data shown in
Figure 7
intuition that selection for reduced hydrodynamic
force is much more stringent on exposed than on protected shores, The real utility of this exercise lies in its
ability to make quantitative predictions. Few marine
biologists could trust their intuition to predict that
selection for a decrease in lift in limpets is greater than
selection for a decrease in drag, and fewer still would
be able to say by how much,
Further, the quantitative predictions made by this
method can be used to assess the likelihood that a
particular hydrodynamic force (rather than some other
force or another physical stress, such as predation) has
played the primary role in evolution of body form. For
instance, the case developed here was chosen primarily
because it predicts that lift can act as a substantial
selective factor, but it also serves to highlight the fact
that drag in this limpet species is likely not to have
been a potent agent of selection,
For other animals (corals, for instance), the method
proposed here might predict that neither lift nor drag
form a risk, In such a case, it would be futile to attempt
to relate body form to hydrodynamic force, and much
time and effort can be saved by dismissing the possibility at the outset.
(iii)
(iv)
FUTURE STUDIES
The approach presented here is still in its infancy,
and much work remains to be done before it can be
used as a practical tool for exploring the process of
natural selection. In particular, five areas require further
study and are listed below.
(i) It is necessary to test the predictions made by this
approach through long-term monitoring of maxi-
(v)
199}
mal wave heights and forces at a variety of wave.
swept sites. Only when these empirical tests have
been conducted should the results presented her
be accepted with confidence, For instance, severa~
authors (reviewed by Sarpkay~ and, Is aaCSon
1981) have shown that the RayleIgh dIstribution
overpredicts the heights of very large waves in
deep water. If this is true for waves in intem1edi_
ate and shallow depths as well, the method pro.
posed here will overestimate maximal forces .
It is necessary to understand better the mechanis_
tic link between maximal wave height and maximal water velocity at the substratum. Th is is
especially true in and near the surf zone on rocky
shores where steeply sloping sea beds and topographical complexity at the substratum may
cause velocities to deviate substantially from
those predicted by theory and possibly even from
those predicted from empirical work on sandy
beaches,
It may be necessary to take acceleration reac tion
into account. If the water accelerations presen t in
storm waves are as large as proposed by Denny
et ai, (1985), i.e, > 400 m's-2, the acceleration reaction can impose a substantial force in addition
to lift and drag. As Denny et aI, (op. cit.) ha ve
shown, this additional force is relatively larger
in bigger organisms, and can thereby act as a
mechanical limit to the size of plants and anin1als.
Practical use of this idea, however, will req uire
knowledge regarding the maximal acceleration
an organism will encounter and the timing of this
acceleration relative to maximal water velocity.
For instance, the effects of maximal acceleration
reaction may differ depending on whether or not
it is applied by the same wave that applies the
maximal lift and drag,
There can be complications in interpreting the
probability that an organism will be dislodged by
a force of known magnitude. Here, this probability
has been estimated on the basis of strength distributions measured for individuals at a particu lar
site. By definition, these individuals were strong
enough to have survived to the time of measurement; any weak individuals that did not survive
have not been taken into account. As a result, the
mean strengths used in the calculations in this
paper may overestimate the actual population
mean, and thereby underestimate the probability
of dislodgement (Denny 1988), Careful, long-term
measurements of mean strengths will be required
to determine how in1portant this effect might be
for each species in question.
More should be known about how individual
variation in the shape and the flexibility of organisms affects Cd, C, and reproductive output.
Denny: Maxinwl Wave-Induced Forces
1991
These factors interact with the probability of
dislodgement to determine one component of the
organis~s' ~it~ess, and a full understanding of
their intncaCles IS needed before a complete understanding of the evolutionary implications of
biological safety factors is possible (Alexander
1981, 1984, Lowell 1985, 1987).
SUMMARY
• Appropriate interpretation of natural selection on
body form requires a knowledge of the maximal
forces to which an organism is subjected.
• The Rayleigh distribution of ocean wave heights
makes possible the prediction of long-term maximal
water velocities at the substratum.
• These predicted velocities can, in tum, be used to
predict the maximal hydrodynamic forces imposed
on benthic organisms.
• These predicted forces form a tool for quantitatively
determining the severity of natural selection upon
particular morphological or mechanical traits.
• Although enticing, the results described here require
testing and verification.
ACKNOWLEDGEMENTS
The empirical data on which this discussion is
based were collected during the course of NSF grants
OCE83-1459l and 87-11688. I thank two anonymous reviewers for helpful comments.
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