## SUMMARY

Previous studies have hypothesized that wave-induced drag forces may
constrain the size of intertidal organisms by dislodging or breaking organisms
that exceed some critical dimension. In this study, we explored constraints on
the size of the articulated coralline alga *Calliarthron*, which
thrives in wave-exposed intertidal habitats. Its ability to survive depends
critically upon its segmented morphology (calcified segments separated by
flexible joints or `genicula'), which allows otherwise rigid fronds to bend
and thereby reduce drag. However, bending also amplifies stress within
genicula near the base of fronds. We quantified breakage of genicula in
bending by applying known forces to fronds until they broke. Using a
mathematical model, we demonstrate the mitigating effect of neighboring fronds
on breakage and show that fronds growing within dense populations are no more
likely to break in bending than in tension, suggesting that genicular
morphology approaches an engineering optimum, possibly reflecting adaptation
to hydrodynamic stress. We measured drag in a re-circulating water flume
(0.23–3.6 m s^{–1}) and a gravity-accelerated water flume,
which generates jets of water that mimic the impact of breaking waves
(6–10 m s^{–1}). We used frond Reynolds number to
extrapolate drag coefficients in the field and to predict water velocities
necessary to break fronds of given sizes. Laboratory data successfully
predicted frond sizes found in the field, suggesting that, although
*Calliarthron* is well adapted to resist breakage, wave forces may
ultimately limit the size of intertidal fronds.

- adaptation
- biomechanics
- breaking stress
- Calliarthron
- decalcification
- drag
- flexibility
- geniculum
- intertidal
- macroalgae
- material properties

## INTRODUCTION

The intertidal zone of rocky shores is a hydrodynamically stressful
environment, where breaking waves can generate water velocities greater than
25 m s^{–1} (e.g. Denny et
al., 2003) and impose great forces on intertidal inhabitants
(Koehl, 1984;
Koehl, 1986;
Carrington, 1990;
Gaylord et al., 1994;
Denny, 1995;
Gaylord et al., 2001;
Helmuth and Denny, 2003). The
severity of wave-induced forces has been hypothesized to limit the maximum
size to which intertidal organisms can grow (e.g.
Denny et al., 1985;
Gaylord et al., 1994;
Denny, 1999). For example,
unlike whales and giant kelps that live in deeper water, intertidal flora and
fauna rarely exceed 0.5 m in any dimension
(Denny et al., 1985).
Blanchette found that intertidal algae transplanted from sheltered to
wave-exposed locations `tattered' back to a smaller size
(Blanchette, 1997). Such damage
is likely to be the result of drag, the primary wave-induced force applied to
intertidal macroalgae (Denny and Gaylord,
2002).

Several studies have measured drag on seaweeds in an attempt to predict the
size to which various species can grow in the intertidal zone
(Carrington, 1990;
Dudgeon and Johnson, 1992;
Gaylord et al., 1994;
Wolcott, 2007) but have had
mixed success. This may be due in part to the characterization of drag at slow
speeds (<3 m s^{–1}) in re-circulating water flumes and the
need to extrapolate from these data to environmentally relevant water
velocities (20–30 m s^{–1}). Such long-range
extrapolations can be misleading (Vogel,
1994; Bell, 1999).
In particular, drag coefficient (*C*_{d}) decreases as flexible
macroalgae bend and reconfigure with increasing water velocity
(Bell, 1999;
Boller and Carrington, 2006a),
but the extent of this reconfiguration and its effect on
*C*_{d} have never been characterized at the high velocities
found on wave-swept shores. Here, we introduce a gravity-accelerated water
flume, capable of generating jets of water (meant to mimic crashing waves) up
to 10 m s^{–1}. Thus, for the first time, it is possible to
measure drag and re-configuration of seaweeds at high velocities, reducing the
need for extrapolation.

The articulated coralline alga *Calliarthron cheilosporioides* Manza
thrives in wave-swept intertidal habitats along the California coast
(Abbott and Hollenberg, 1976).
Unlike fleshy algae, which are flexible along their entire length,
*Calliarthron* thalli are firmly calcified but have flexible joints
(genicula) that allow fronds to bend when struck by breaking waves. Flexible
genicula also define breakage points along calcified thalli
(Martone, 2006) and are
hypothesized to be especially susceptible to bending stresses, as segmented
bending may locally amplify stress within genicula [see accompanying paper
(Martone and Denny, 2008)].
Nevertheless, *Calliarthron* fronds can grow to a length of 25 cm,
including more than 100 genicula, and can dominate the most wave-exposed
habitats.

When struck by incoming waves, erect articulated fronds bend in the direction of flow parallel to the substratum. Most genicula are stretched in tension by each passing wave, but basal genicula, which are farthest from the free end of any frond, experience the greatest bending moments (see Martone and Denny, 2008) and are hypothesized to be the most prone to breakage (Martone, 2006). Morphological characteristics of bending genicula are significantly different from those of tensile genicula [see table 3 of the accompanying paper (Martone and Denny, 2008)], helping them increase flexibility and decrease stress amplification [see figure 7 of accompanying paper (Martone and Denny, 2008)], and bending angles may be constrained by the close proximity of neighboring fronds, further mitigating the amplification of stress within bending genicula. The mechanical advantages of such traits may be limited, however, if tensile genicula ultimately break first.

In this study, we compared breaking forces of genicula and drag forces on
articulated fronds to explore physical constraints on the size and survival of
this ecologically successful intertidal seaweed. We used empirical and
modeling techniques to quantify forces to break bending genicula, with and
without neighbors, and compared them with forces sufficient to break tensile
genicula. These data allowed us to evaluate the mechanical limits of bending
genicula and to estimate the average drag force required to break fronds in
the field. Drag on articulated fronds was measured in the lab, and the size to
which fronds can grow without breaking in a given water velocity was
predicted. We tested laboratory predictions by measuring maximum water
velocities and frond sizes in the field and propose that, although articulated
fronds are remarkably well adapted to resisting wave-induced drag, breaking
waves may, indeed, be sufficient to constrain the size of intertidal
*Calliarthron*.

## MATERIALS AND METHODS

### Material properties

*Calliarthron* fronds (*N*=15) were collected from the low
intertidal zone in a moderately wave-exposed surge channel at Hopkins Marine
Station (HMS) in Pacific Grove, CA, USA. Stress–strain curves were
generated for one geniculum in each frond loaded in tension using a
custom-made tensometer and a video dimension analyzer (model V94, Living
Systems Instrumentation, Burlington, VT, USA) as described in the accompanying
paper (Martone and Denny,
2008). Tensile moduli (*E*_{t}) were calculated as
the slopes of linear stress–strain regressions forced through the
origin; breaking strains (ϵ_{break}) were assumed to be the
strain measurements immediately preceding frond breakage during mechanical
tests. *E*_{t} and ϵ_{break} were correlated, and
a linear regression was fitted to the *E*_{t} *versus*ϵ
_{break} data. Residuals were calculated for each datapoint
(*N*=15), and the standard deviation of residuals was calculated.

### Bend-to-break tests

In the field, *Calliarthron* fronds often break near the base
(Martone, 2006), where bending
moments and bending stresses are greatest
(Martone and Denny, 2008). To
explore breakage in bending, *Calliarthron* fronds (*N*=7) were
collected from the field site described above. Branches were removed from each
frond by cutting below the first dichotomy, and the remaining straight chains
of segments were tested as follows. Individual fronds were gripped in clamps
by the first few genicula and held horizontal
(Fig. 1A). To quantify the
force to bend genicula to failure, a second clamp was secured near the tenth
genicula (numbered from the clamp) and masses were hung, in 20 g and 50 g
increments, from the clamp until fronds broke
(Fig. 1B). As bending genicula
experienced increasing force, they broke gradually
(Fig. 1C). An image analysis
revealed that unbroken genicular cells approached, but had not yet exceeded,
the average breaking strain (ϵ_{break}) of genicula loaded in
tension (P.T.M. and M.W.D., unpublished data).

Dimensions of broken genicula were quantified as described in
figure 1 of the accompanying
paper (Martone and Denny,
2008). Genicular radii (*r*_{1}*,
r*_{2}) and intergenicular radii (*y*) were measured with
an ocular dial-micrometer. Genicular lengths (ω) and gap lengths
(ω*–*2*x*) were measured in wet, long-sectioned
genicula adjacent to broken genicula and briefly decalcified in HCl. Average
length measurements were assumed for broken genicula. Intergenicular lip
length (*x*) of broken genicula was estimated as half the difference
between mean ω and mean gap length.

### Modeling breakage in bending

The mathematical model that we present in our other study
(Martone and Denny, 2008) was
augmented to allow genicula to break gradually in order to estimate the force
to bend experimental genicula to failure. The distribution of breaking strains
was assumed to be normal with a mean of
*x̄*_{ϵ,break} and standard
deviation of s.d. _{ϵ,break}, and when tensile moduli
(*E*_{t}) were plotted against breaking strains
(ϵ_{break}), residuals were assumed to be normally distributed
around the linear regression with a mean of 0 and standard deviation of
s.d._{residual}. To generate unique bivariate normal pairs of
*E*_{t} and ϵ_{break} for each iteration of the
model, a random value was chosen from the ϵ_{break} distribution
and a corresponding *E*_{t} was calculated from the normal
distribution of values around the regression. As virtual fronds deflected, the
model eliminated any portion of the first geniculum whose strain exceededϵ
_{break} (see Appendix). This reduction of tissue was factored
into the internal moments and neutral axis positions calculated by the model.
In some cases, reduction of genicular cross-section created a positive
feedback, increasing bending angles and further increasing strain. Breakage
occurred when ϵ >ϵ_{break} across the entire geniculum.
All other components of the original bending model were unchanged (see
Martone and Denny, 2008).

To validate the model, forces (means ± s.d.) to break experimentally
broken genicula described above (*N*=7) were estimated mathematically
from 1000 model iterations. Observed and predicted breaking forces were
compared.

### Breaking force predictions

To estimate breakage of fronds in the field, morphological dimensions of 10
fronds were used from our other analysis [see
table 3 in Martone and Denny
(Martone and Denny, 2008)].
Forces to break first genicula (nearest the base) of these experimental fronds
in bending were calculated from 1000 model iterations (means ± s.d.;
*N*=10).

*Calliarthron* grow in dense clusters in the field, and fronds
emerging from individual bases are tightly packed together
(Fig. 2A). The spatial density
of fronds probably limits bending angles of basal genicula. To evaluate this
`neighbor effect', the spatial density of fronds was measured in 15
*Calliarthron* individuals growing in the low intertidal zone at HMS.
Average distance between fronds (*D*) was calculated from the average
diameter of basal intergenicula (2*y*=1.334 mm) [see
table 3 in Martone and Denny
(Martone and Denny, 2008)] and
the number of fronds growing within 1 cm of one another (*N*=15;
Fig. 2B). Given average
intergenicular diameter and spacing, the maximum bending angle
(ϕ* _{i}*) of each frond depends upon the bending angle
(ϕ

_{i}_{–1}) of the neighboring frond: (1) According to this equation, bending angles of articulated fronds equilibrate a few fronds within the periphery, assuming that fronds at the edge can bend 90 deg. (Fig. 2C). To evaluate the effect of this constraint on breaking force, the mathematical model was adjusted to prevent bending angles of first genicula from exceeding the mean bending angle of central fronds (i.e. positioned five neighbors within the periphery). Forces to break first genicula in the 10 experimental fronds (mean± s.d.) with neighbors were calculated from 1000 model iterations. Forces to break fronds with and without neighbors were compared.

When fronds bend over, most genicula experience drag force in tension [see
right panels of figure 4 in
Martone and Denny (Martone and Denny,
2008)]. Tensile forces required to break tenth genicula in the 10
experimental fronds were estimated using genicular cross-sectional area and
breaking stresses sampled from a normalized distribution with
*x̄*_{σ,break}=25.9 MN
m^{–2} and s.d._{ϵ,break}=2.3 MN
m^{–2} [determined by Martone
(Martone, 2006)]. Mean and
standard deviation of 1000 breaking force estimates were calculated for each
geniculum. Forces to break tenth genicula in tension and to break first
genicula in bending were compared, assuming fronds would break at the lesser
force. Mean and standard deviation of forces to break fronds were calculated,
and these values were used to predict breakage of articulated fronds in the
field.

### Drag force measurements

When intertidal algae are struck by breaking waves, drag can be calculated
from the following equation:
(2)
where ρ is seawater density (approximately 1025 kg m^{–3}),
*U* is water velocity, *A* is algal planform area, and
*C*_{d} is the drag coefficient, a dimensionless index of shape
change and reconfiguration of flexible fronds
(Carrington, 1990;
Dudgeon and Johnson, 1992;
Gaylord et al., 1994;
Bell, 1999) (see also
Boller and Carrington,
2006a).

To quantify the effect of frond size and growth on drag force,
*Calliarthron* fronds (*N*=24) were collected from the low
intertidal zone at HMS and were tested in re-circulating and
gravity-accelerated water flumes. In both flume types, fronds were attached
with cyano-acrylate glue to custom-made force transducers. In the
re-circulating flume, drag force was measured on fronds (*N*=8) at
0.23, 0.46, 0.69, 0.92, 2.0 and 3.6 m s^{–1}. In the
gravity-accelerated water flume, drag force was measured as fronds were struck
with jets of water (Fig. 3).
Flow was fully turbulent as it fell through the 10 cm diameter pipe, and
velocity was adjusted by varying the distance through which the water fell.
Fronds were tested at 6.8 m s^{–1} (*N*=6) and 10.0 m
s^{–1} (*N*=10).

To explore changes in drag force over the lifetime of
*Calliarthron*, fronds were `de-grown' by sequentially removing apical
branches, and the resulting effect on drag force was quantified. This method
reasonably approximated *Calliarthron* ontogeny (in reverse), since
most growth occurs at the apical meristem
(Johansen, 1981) and genicula
are approximately the same size in young and old fronds
(Martone, 2007). Drag on whole
fronds was measured, apical branches of fronds were removed, and drag force
was re-measured. Then sub-apical branches of fronds were removed, and drag
force was re-measured. This process was repeated until all branches had been
removed. Severed branches were digitally photographed and planform areas were
measured using an image analysis program (ImageJ, NIH Image,
http://rsb.info.nih.gov/ij/).
The correlation between frond planform area and drag force was plotted for
fronds at all eight water velocities.

To compare the performance of *Calliarthron* fronds with that of
streamlined bodies and fleshy algae in flow, Vogel's *E*
(Vogel, 1984) was calculated
as the slope of a linear regression fitted to a log–log plot of
speed-specific drag (*F _{drag}/U*

^{2})

*versus*velocity (

*U*).

### Calculating drag coefficients, *C*_{d}

Given drag force measurements, drag coefficients were calculated for every
combination of frond planform area (*A*) and water velocity
(*U*) by re-arranging Eqn
2:
(3)
Data revealed that drag coefficient decreased with both increasing velocity
and increasing frond planform area, making extrapolations based on one
parameter alone inaccurate. Instead, drag coefficient was plotted against
frond Reynolds number, Re_{f}, a function of both velocity and area:
(4)
where *L* is characteristic frond dimension (represented here by the
square-root of *A*) and ν is the kinematic viscosity of water
(1×10^{–6} m^{2} s^{–1} at
20°C). Drag coefficients decreased with increasing Re_{f}
according to the power curve:
(5)
Parameters *a, b* and *c* were estimated for the data using
Matlab (v7.0.1, The Mathworks, Natick, MA, USA), and 95% confidence intervals
(CI) around the fitted curve were calculated from 1000 bootstrapped datasets
created by sampling with replacement of the original data
(Efron and Tibshirani,
1993).

### Drag force extrapolation

Re-arranging Eqn 4 and
squaring both sides yields:
(6)
By substituting Eqns 5 and
6 into the drag equation
(Eqn 2), we obtain drag in terms
of Re_{f}:
(7)
*F*_{drag} was plotted against Re_{f} for all frond
areas and velocities. Parameters *a, b* and *c* (calculated in
Eqn 5) were used to plot a
*F*_{drag} *versus* Re_{f} curve; 95% CI around
this curve were calculated in Matlab by estimating parameters for the same
1000 bootstrapped log*C*_{d} *versus* logRe_{f}
datasets and then calculating the range of *F*_{drag}.

### Predicting breakage in the field

Fronds are expected to break in the field when drag force experienced by
genicula exceeds breaking force. This expectation can be represented as
follows:
(8)
Critical frond Reynolds numbers (Re_{f,crit}) predicted to break
genicula were calculated iteratively using Matlab. For the mean breakage
prediction, Re_{f,crit} was estimated using mean
*F*_{break} and the mean *F*_{drag}
*versus* Re_{f} curve. For the best case scenario,
Re_{f,crit} was estimated using mean *F*_{break}+s.d.
and the mean curve –95% CI. For the worst case scenario,
Re_{f,crit} was estimated using mean
*F*_{break}–s.d. and the mean curve +95% CI. Then, using
Eqn 4, we explored the
combinations of velocity (*U*) and frond area (*A*) that would
yield each critical Re_{f}. *U versus A* curves were plotted
for each Re_{f,crit} and were used to predict the maximum area to
which fronds could grow in a given water velocity or, conversely, the minimum
water velocity necessary to break fronds of a given size:
(9)

### Field measurements

From November 2003 to November 2006, *Calliarthron* fronds were
collected every few months, totaling eight collections, from the intertidal
field site described above. During each collection we searched for the largest
available fronds. Collections typically consisted of 10–20 fronds.
Fronds were digitally photographed and frond planform areas were measured
using image analysis (ImageJ). Maximum frond area was noted on each date over
the 3 year span.

From November 2005 to August 2006, maximum water velocities were measured. On 2 November 2005, three dynamometers (Bell and Denny, 1994; Helmuth and Denny, 2003) were installed at mean lower low water approximately 0.75 m apart, spanning the field site. Dynamometers were first checked and reset on 4 November 2005 and were checked and reset during sufficiently low tides (13 times) until 10 August 2006. The maximum water velocity recorded by any dynamometer was noted on each date over the 9 month span.

Field measurements were compared with breakage predictions to determine
whether water velocities in the field were sufficient to generate drag forces
that would equal forces experimentally determined to break
*Calliarthron* fronds.

## RESULTS

### Material properties

Mean ϵ_{break} of genicula was 1.18±0.44 (mean ±
s.d.), and mean *E*_{t} of genicula was 27.7±12.4 MN
m^{–2} (mean ± s.d.). ϵ_{break} and
*E*_{t} were significantly negatively correlated
(Fig. 4;
*R*^{2}=0.62, *P*<0.001), such that:
(10)
Standard deviation of residuals around the regression was 7.7 MN
m^{–2}.

### Bend-to-break tests and model validation

Predicted and observed breaking forces were similar
(Table 1) and, on average, were
not significantly different (*P*=0.47, Student's paired
*t*-test).

### Breaking force predictions

Without neighbors, bending genicula were predicted to break before tensile
genicula because, on average, tensile genicula resisted significantly more
force (*P*<0.01, paired *t*-test;
Fig. 5). However, fronds were
predicted to resist greater forces in bending when supported by neighboring
fronds (Fig. 5). Neighboring
fronds were spaced 0.4 mm apart, on average, limiting bending angles to
approximately 54 deg. With neighbors, bending genicula and tensile genicula
were predicted to resist similar forces
(Fig. 5), which, on average,
were not significantly different (*P*=0.30, paired *t*-test).
Mean force to break first genicula in bending with neighbors was 26.3 N, and
the mean force to break tenth genicula in tension was 22.7 N
(Table 2). Assuming fronds
would break at the lesser of the two breaking forces for each frond, mean
force to break *Calliarthron* fronds was 20.0±3.8 N (mean±
s.d.; Table 2).

### Drag force measurements and drag coefficient estimates

For all water velocities, drag force increased with frond planform area,
and fronds of any given area experienced more drag force at greater velocities
(Fig. 6). Vogel's *E*
was calculated to be –0.68 (*R*^{2}=0.70;
Fig. 7).

Drag coefficients decreased with increasing water velocity and increasing
frond area (Fig. 8). Given its
dependence on both frond area and water velocity, drag coefficient decreased
with increasing frond Reynolds number (Fig.
9). The following curve captured nearly all of the variation in
the data (*R*^{2}=0.95):
(11)

### Predicting drag and breakage in the field

Drag was plotted against frond Reynolds number
(Fig. 10):
(12)
Using Eqn 12 and associated 95%
CI (see Materials and methods), Re_{f,crit} was calculated for
*F*_{drag}=*F*_{break} (determined to be
20.0±3.8 N, mean ± s.d.). For *F*_{break}=16.2 N
(mean – 1 s.d.), Re_{f,crit}=1.15×10^{6}; for
*F*_{break}=20.0 N (mean),
Re_{f,crit}=1.45×10^{6}; for
*F*_{break}=23.8 N (mean + 1 s.d.),
Re_{f,crit}=1.89×10^{6}
(Fig. 10).

Substituting these values for Re_{f,crit} in
Eqn 9, the following three
equations were generated to predict water velocities that would break fronds
of given planform area in the field (Fig.
11):
(13)
corresponding to *F*_{break}=16.2, 20.0 and 23.8 N,
respectively. Larger fronds were predicted to break at slower water
velocities. Fronds smaller than 10 cm^{2} were predicted to resist
water velocities greater than 40 m s^{–1}.

The greatest water velocity recorded at the field site was 22.1 m
s^{–1} (Table 3).
On average, the largest frond collected from the field site was
40.9±7.8 cm^{2} (mean ± s.d.), and the largest frond
ever collected from the site was 51.9 cm^{2}
(Table 4). These field
measurements corresponded well to breakage predictions
(Fig. 11).

## DISCUSSION

### Optimized to resist breakage

Despite the amplification of bending stresses within genicular tissue
(Martone and Denny, 2008),
*Calliarthron* genicula are clearly well adapted to resist mechanical
failure in wave-swept habitats. Even without the support of neighbors, fronds
located near the periphery of aggregations are able to bend to 90 deg., and
several experimental fronds supported more than 1 kg of weight (>9.8 N) in
this position (Table 1). When
neighboring fronds are considered, bending angles of central fronds are
constrained, allowing first genicula to resist approximately twice the force
of fronds near the periphery. Ultimately, fronds growing within dense
populations are just as likely to break at tenth genicula in tension as they
are to break at first genicula in bending.

These data suggest that genicula are not `over-designed' in an evolutionary
sense. The tensile strength of calcified intergenicula in
*Calliarthron* (28.5 MN m^{–2})
(Martone, 2006) is similar to
that of coral skeleton (25.6 MN m^{–2})
(Vosburgh, 1982) and to that
of several bivalve and gastropod shells that appear similar to
*Calliarthron* cell walls (`homogeneous' type, 30 MN
m^{–2}; `foliated' type, 38.3 MN m^{–2})
(Currey, 1980). This suggests
an upper limit to the tensile strength of biologically deposited calcium
carbonate within *Calliarthron* cell walls. This mechanical constraint
may be biologically linked to genicula, whose tissue is equally strong (25.9
MN m^{–2}) (Martone,
2006). Indeed, genicular tissue is far stronger than other algal
tissues (up to an order of magnitude)
(Martone, 2006) but may be
biologically constrained from growing any stronger. Given this putative
maximum tensile strength of genicular tissue, there may not be a selective
advantage to adjusting the spatial density of fronds or the dimensions of
bending genicula, if tensile genicula would ultimately break first. In other
words, growing more densely packed clusters of fronds or decreasing the length
of intergenicula (see Martone and Denny,
2008) may indeed increase the breaking force of bending genicula,
but such fronds would probably break at tensile genicula anyway; and breakage
of either bending or tensile genicula near the base is likely to be
disadvantageous, resulting in the loss of an entire frond that may have been
reproductive and several years old
(Johansen and Austin,
1970).

Together, these data are consistent with the engineering theory of optimal design [Maxwell's Lemma (see Wainwright et al., 1982)], which states that all components in a mechanically stressed system should be equally strong to avoid wasting resources in their construction. The fact that bending and tensile genicula are morphologically distinct (Martone and Denny, 2008) but resist similar drag forces suggests that these structures may have been shaped by selective pressures imposed by breaking waves. Morphological differences among tensile and bending genicula, therefore, may represent adaptations to hydrodynamic stress.

### Environmentally relevant drag coefficient

To our knowledge, this is the first study to report drag coefficients for
an intertidal seaweed at high, environmentally relevant water velocities. Drag
coefficients reported here for *Calliarthron* are up to an order of
magnitude lower than those reported for several other algae at slow water
velocities (Carrington, 1990;
Dudgeon and Johnson, 1992;
Gaylord et al., 1994). Our
data suggest that drag coefficient continues to decrease as water velocity
increases, at least up to 6 m s^{–1}
(Fig. 8), contrary to the
assumption that reconfiguration is solely a low-velocity phenomenon (see
Bell, 1999). Vogel's *E*
(–0.68) suggests that the drag coefficient of reconfiguring
*Calliarthron* fronds drops faster than that of a typical streamlined
body (–0.50) (Vogel,
1984) and faster than those of several branched red algae,
including a congeneric species (–0.35±0.13, mean ± s.d.,
*N*=7 species), although slower than those of flat bladed algae
(–1.11±0.10, *N*=3 species;
Fig. 7) [data compiled from
table 4 in Gaylord et al.
(Gaylord et al., 1994)].
Without high-speed data for other seaweeds, it is unknown at this time whether
such low drag coefficients are a distinct characteristic of
*Calliarthron* or a shared feature of intertidal algae. Our data
emphasize the importance of measuring drag at high water velocities to avoid,
or at least improve, extrapolation. For example, Gaylord and colleagues
extrapolated fivefold beyond their data to generate drag predictions
(Gaylord et al., 1994); in
contrast, we would need to extrapolate less than twofold beyond our high-speed
flume velocities (i.e. a short distance along the logRe_{f} axis of
Fig. 9) to generate the same
predictions. The accuracy of past predictions awaits verification in the
high-speed flume.

Our data demonstrate effects of both planform area and water velocity on drag coefficient, suggesting yet another source of error in previous studies that treated drag coefficient strictly as a function of velocity and tested only a narrow size range of fronds (e.g. Carrington, 1990; Dudgeon and Johnson, 1992; Gaylord et al., 1994; Bell, 1999). Flexible algal fronds have lower drag coefficients in faster water because of the increased reconfiguration that occurs as fronds bend. Similarly, larger fronds are likely to have lower drag coefficients because of their capacity to re-arrange their branches and collapse to be more streamlined, unlike smaller fronds whose sparse branches are perhaps less capable of reconfiguration (Fig. 8).

It is important to note that, in this and previous studies of algal reconfiguration, drag coefficient is calculated using frond planform area (i.e. flattened and photographed from above). This methodology assumes a constant area term, allowing the drag coefficient to absorb any change in shape due to reconfiguration; all else being equal, larger fronds will tend to have lower drag coefficients (see Eqn 3). This contrasts sharply with a recent study (Boller and Carrington, 2006a) that calculated drag coefficients using frond projected area (i.e. photographed from upstream in flow) and tracked the independent decline of both projected area and drag coefficient as macroalgae reconfigured with increasing water velocity – a method that had previously been applied to reconfiguring gorgonians (Sponaugle and LaBarbera, 1991). Their data show that as flow increases, projected areas and drag coefficients both decline; fronds with larger projected areas have higher drag coefficients. Unlike our method, theirs generates drag coefficients that are directly comparable to other engineering shapes. However, because projected area cannot be visualized or measured in turbulent water at high speeds (e.g. in the gravity-accelerated water flume or under breaking waves), predictions of projected area, like those for drag coefficient, will inevitably rely upon long-range extrapolations. Thus, even though the drag coefficients presented here cannot be compared with those of standard shapes, our method reduces extrapolation and so seems suited to exploring the maximum size to which wave-swept fronds can grow.

### Limits to frond size in the intertidal zone

Forces estimated to break *Calliarthron* fronds in the field are
consistent with forces previously measured to break genicula in tension
(Martone, 2006). An average
*Calliarthron* frond can resist approximately 20 N of force before
breaking – a substantial amount of force. For example, one large
experimental frond (30 cm^{2}) experienced only 5 N of drag force at
10 m s^{–1} (Fig.
6) – far below the threshold breaking force. This suggests
that *Calliarthron* may be well adapted to resist drag imposed by
intertidal water velocities. However, velocities as high as 35 m
s^{–1} have been recorded at HMS (M.W.D., unpublished data), and
articulated fronds may ultimately be size limited when water velocities
approach this extreme.

Indeed, data presented here suggest that the size of *Calliarthron*
fronds may be limited by drag forces imposed by intertidal water velocities.
According to our breakage model, the maximum water velocity measured at the
field site (22.1 m s^{–1}) closely predicts the mean maximum
size (40.9±7.8 cm^{2}, mean ± s.d.) of
*Calliarthron* fronds expected to survive there
(Fig. 11). These data suggest
that the broad range of frond sizes observed at the field site may, at least
in part, be a consequence of variation in forces to break genicula. Our
simplified breakage model ignores any possible drag-reducing effects of
neighboring algae (Boller and Carrington,
2006b) and, because intertidal water velocities vary widely in
both space and time (Denny and Wethey,
2001; Helmuth and Denny,
2003; O'Donnell,
2005), characterizing years of wave-forces with only a few
dynamometer measurements is a broad generalization. For example, if breaking
waves during some storm event actually generated 28 m s^{–1}
water velocities at the field site – a distinct possibility – then
the size of the largest frond predicted to survive, including 95% model error,
would closely match the observed size of the largest frond (51.9
cm^{2}; Fig. 11). The
close correlation between maximum velocity and frond size across the field
site suggests that, although *Calliarthron* is well adapted to
resisting breakage, growth may ultimately be limited by wave-induced drag
forces. Observations of larger fronds growing subtidally (K. A. Miller,
personal communication), where drag is lower, support this conclusion but have
yet to be properly quantified.

## APPENDIX

Each iteration of the mathematical model accounted for breakage in the
first geniculum by subtracting the portion of genicular cross-sectional area
that had exceeded ϵ_{break}. After calculating frond deflection,
the model determined the position within the geniculum, θ_{max},
where ϵ=ϵ_{break} and subtracted the genicular area whereθ>θ
_{max} (and thereforeϵ>ϵ
_{break}; Fig.
A1):
(A1)

## ACKNOWLEDGEMENTS

This manuscript benefited from comments made by M. Boller, K. Mach, L. Miller, K. Miklasz, R. Martone, and two anonymous reviewers. Research was supported by the Phycological Society of America, the Earl and Ethyl Myers Oceanographic and Marine Biology Trust, and NSF grant no. IOS-0641068 to M.W.D. This is contribution number 309 from PISCO, the Partnership for Interdisciplinary Studies of Coastal Oceans, funded primarily by the Gordon and Betty Moore Foundation and the David and Lucile Packard Foundation.

- © The Company of Biologists Limited 2008