## SUMMARY

Many fish species span two or three orders of magnitude in length during
the growth from larvae to adults, and this change may have dramatic
consequences for locomotor performance. We measured how the performance of
coasting changes over the life history of zebrafish (*Danio rerio*) and
examined the scaling of mechanics underlying this change. Adult zebrafish
coast disproportionately further and faster and maintain their speed for a
longer duration than do larvae and juveniles. Measurements of drag on tethered
dead fish suggest that adult fish operate in an inertial regime by coasting at
relatively high Reynolds numbers (*Re*>1000), and *in vivo*
drag measurements showed adults to operate with a drag coefficient
(*C*_{inert}≈0.024) that was consistent with previously
published estimates. However, drag scaled differently at lower *Re*
values than those assumed in previous studies. We found a viscous regime at
*Re*<300, which corresponds to the routine coasting of larvae and
juveniles. Despite these changes in hydrodynamics over growth, a mathematical
model of coasting mechanics suggests that the disproportionately longer
coasting of adults is caused primarily by their large body mass and high speed
at the beginning of coasting. We therefore propose that changes in coasting
performance with growth are dictated primarily by the scaling of momentum
rather than resulting from hydrodynamic changes. These results provide an
opportunity for new interpretations of function in the growth and evolution of
fish.

## Introduction

The size of a swimming organism has important consequences for the
hydrodynamic resistance to its motion. Microorganisms act against the
viscosity of water, while animals greater than a few centimetres in length
must overcome an uneven distribution in pressure along the surface of the body
(Daniel et al., 1992;
Videler et al., 2002;
Wu, 1977). For the tremendous
diversity of animals at a scale intermediate to these body sizes, it remains
unclear to what degree viscous and inertial forces contribute to drag during
locomotion (Fuiman and Batty,
1997; McHenry et al.,
2003; Muller et al.,
2000). Zebrafish (*Danio rerio*) grow from a length of a
few millimetres as larvae to a few centimetres as adults and therefore provide
a model system for examining the locomotor mechanics of this intermediate
scale. Like many fish species, zebrafish swim intermittently and thus spend at
least half of a locomotory period coasting passively (i.e. maintaining a
stretched straight posture while moving forward) against the surrounding water
(Fuiman and Webb, 1988). The
goals of the present study were to examine how the mechanics of coasting scale
with size and to consider how these changes influence locomotor performance
over the life history of zebrafish.

### The scaling of drag

The ratio of viscous to inertial drag is proportional to the Reynolds
number (*Re*) of an animal, but this proportionality cannot easily be
estimated by hydrodynamic theory and therefore requires measurements to be
understood. In order to measure the relationship between drag and Reynolds
number *Re=ρUL/μ*, where *U* is swimming speed,
*L* is body length, and ρ and μ are, respectively, the density
and dynamic viscosity of water: Lamb,
1945), *Re* may be varied by changing the speed of flow
past the body or altering the density or viscosity of the fluid. At high
*Re*, inertial drag is predicted to vary with water density, the square
of the swimming speed and the wetted area of the body (*S*;
Batchelor, 1967). Therefore,
measured drag, *D*, normalized by these terms yields the inertial drag
coefficient, *C*_{inert} (most commonly denoted as
*C*_{d}; Batchelor,
1967), which is predicted to remain constant with respect to
*Re* when inertia dominates:
1

In contrast, viscous drag varies with body length, the viscosity of water
and speed (Lamb, 1945), so
measurements of drag normalized by these terms provides a viscous drag
coefficient, *C*_{visc}, that is predicted to remain constant
with respect to *Re* in a viscous-dominated regime:
2
Both measurements of non-dimensional drag are expected to vary with
*Re* if drag is generated by a combination of viscous and inertial
forces. Therefore, measurements of these coefficients over a range of
*Re* may be used to identify viscous, inertial and intermediate regimes
and thereby define the scaling of drag for a particular animal. The present
study used this approach to define the scaling of drag in zebrafish.

### Ontogenetic change in behaviour and performance

The scaling of drag provides a means for interpreting ontogenetic changes
in the behaviour of fish. Weihs
(1980) proposed that anchovy
(*Engraulis mordax*) operate in a viscous regime as larvae swimming at
*Re*<10 and an inertial regime as adults at *Re*>200. It
followed that during the growth between these stages, inertial forces
increasingly played a dominant role in the hydrodynamics of swimming. This
scaling was consistent with measurements of drag for bluff bodies (e.g. the
sphere and cylinder) at comparable *Re* values (where few measurements
existed for streamlined bodies; Hoerner,
1965) and it provided the fundamental set of assumptions for a
mathematical model of ontogenetic change in swimming mechanics
(Weihs, 1980). The results of
this model suggested that larval anchovy conserve energy by switching from a
steady to an intermittent swimming behaviour
(Hunter, 1972) upon growing to
a size where they operate outside of the proposed viscous regime
(*Re*>10; Weihs,
1980). It is at this transition that other species of juvenile
fish adopt foraging behaviours that were considered to benefit from the low
energetic cost of intermittent motion in an inertial environment
(Webb and Weihs, 1986). At
this stage in their life history, zebrafish are capable of changing their
direction of swimming to an equal or greater extent than the earlier larval
stage despite using their paired fins less and beating their tails with lower
curvature. These kinematics give juvenile fish the appearance of greater
manoeuvrability than larvae (Budick and
O'Malley, 2000; Fuiman and
Webb, 1988; Muller and van
Leeuwen, 2004; Thorsen et al.,
2004).

The hydrodynamic scaling proposed by Weihs (1980) has also provided a basis for interpretations of ontogenetic change in morphology. For example, the elongated body of larval fish in many species is thought to contribute to their ability to exit precociously from the viscous regime by having a greater body length than if they possessed the compressed body shapes typical of adults (Muller and Videler, 1996; Osse, 1990; Webb and Weihs, 1986). It is subsequent to growing out of the viscous regime that many species assume a streamlined body shape that reduces pressure drag in an inertial hydrodynamic environment.

Fuiman and Batty (1997)
presented one of the few experimental tests of the hydrodynamic scaling used
to interpret the functional consequences of changes in morphology and
behaviour. They found a linear correlation between stride length (i.e. the
distanced traversed by the body in one half tailbeat) and the viscous drag
product (*ULμ*) in larval herring (*Clupea harangus*) at
*Re*<300 but found that the relationship became disrupted at
*Re*>300. These results were interpreted as evidence that the
viscous regime extends into Reynolds numbers that are greater than an order of
magnitude above that proposed by Weihs
(1980). However, Fuiman and
Batty (1997) did not offer
*a priori* predictions for the relationship between stride length and
the viscous drag product or for the effect of inertial forces based on
hydrodynamic theory. It is therefore unclear if the disruption between stride
length and the viscous drag product was due to alterations in hydrodynamics or
the result of other changes, such as the pattern of midline kinematics (which
vary with size and speed in other species;
Muller and van Leeuwen, 2004).
Therefore, it remains unclear if the results of Fuiman and Batty
(1997) present a challenge to
the hydrodynamic scaling proposed by Weihs
(1980).

### Drag measurements

Biologists and engineers remain challenged by the question of how best to
measure the drag experienced by a swimming animal. This area of biomechanics
has shifted from the use of kinematics, direct force measurements of dead
animals and analytical modelling (reviewed by
Blake, 1983;
Videler, 1993;
Webb, 1975) in favour of
digital flow visualization (e.g. Anderson
et al., 2001; Drucker and
Lauder, 1997) and computational fluid dynamics (e.g.
Liu et al., 1996;
Wolfgang et al., 1999) to
estimate drag (Schultz and Webb,
2002). Proponents of these relatively new techniques have
emphasized the need to independently validate novel experimental and
theoretical approaches (e.g. Drucker and
Lauder, 1999; Liu et al.,
1997), yet there are few direct or indirect measurements of drag
at the *Re* scale where larval and juvenile fish operate.

Drag is most directly measured with a force transducer attached to a dead fish that is exposed to flow. Although the force measured in such a `dead drag' experiment (Webb, 1975) is unlike the drag experienced by an undulating body (Anderson et al., 2001; Lighthill, 1971; Schultz and Webb, 2002), it may approximate the drag on a coasting fish (or one propelled by paired fins with low-amplitude oscillations; Drucker and Lauder, 1999) under a few assumptions. Specifically, these measurements require that differences in the surface properties, body posture and motion of the fins between the dead and coasting fish have a relatively small effect on the total drag generated by the body. Concerns about these assumptions, along with the potential for experimental artefacts generated by fin fluttering or tethering the body at an erroneous angle of attack, are cause for scepticism about the accuracy of dead drag measurements (see Blake, 1983; Schultz and Webb, 2002; Videler, 1993; Webb, 1975).

Indirect measurements of drag from kinematics provide an alternative to the
dead drag approach but introduce other assumptions and sources of error.
Because drag is the sum total of hydrodynamic forces acting in opposition to
the body motion of a coasting animal, it should be equivalent to the product
of body mass and the average rate of deceleration and therefore may be
calculated from measurements of mass and the kinematics of coasting
(Clark and Bemis, 1979;
Lang and Daybell, 1963;
Videler, 1981). However,
calculating the drag coefficient in this manner from video recordings is
susceptible to large experimental error. Differentiating position measurements
twice to yield acceleration generates a low signal-to-noise ratio
(Lathi, 1998;
Walker, 1998), which is
compounded by dividing these values by the square of velocity (found by
differentiating position once) to calculate the inertial drag coefficient
(Eqn 1). Furthermore, such
measurements of drag coefficient have generally been based on only a single
pair of velocity estimates from the coasting period. To address these
concerns, Bilo and Nachtigall
(1980) estimated the drag
coefficient for a penguin by fitting the first integral of the equation of
motion to velocity measurements. This method offered an improvement over the
use of acceleration measurements by requiring only a single differentiation of
positional recordings and by using the entire time series of data for a
coasting sequence to calculate a single coefficient value. However, this
approach assumes that the animal maintains a fixed posture and a high enough
*Re* to remain in the inertial regime throughout the coast. Therefore,
applying the Bilo and Nachtigall
(1980) approach to juvenile and
larval fish is tenuous because it is possible that these animals operate in
the viscous or intermediate regimes.

The present study used a combination of dead drag and *in vivo* drag
measurements to examine changes in the hydrodynamics of coasting over
ontogeny. Dead drag measurements have the advantage of being applicable at any
*Re* value. We therefore used this method to define viscous, inertial
and intermediate regimes. We did not expect experimental errors in these
measurements (Videler, 1993;
Webb, 1975) to approach the
magnitude of the effect of hydrodynamic scaling
(Hoerner, 1965). However, we
tested the accuracy of the dead drag approach with measurements of *in
vivo* drag on coasting adult fish that could safely be assumed to operate
in the inertial regime (Muller et al.,
2000).

### The present study

Intermittent swimming may be considered in terms of a periodic building up of momentum during propulsion that is followed by its loss during coasting. The ability of a fish to cover distance and maintain speed during a coast depends not only on its rate of momentum loss (i.e. drag) but also on the momentum gained during the propulsive phase. Therefore, the present study considered the scaling of both drag and momentum on the coasting performance of zebrafish. This work focused on addressing the following questions: (1) how does coasting performance change with size; (2) how do the hydrodynamics of coasting change with size and (3) how does the scaling of drag and momentum affect performance?

## Materials and methods

The body dimensions, motion and biomechanics of zebrafish were measured at
different stages of growth. These fish ranged in size from larvae with newly
inflated swim bladders (∼5 days post-fertilization and ∼4.2 mm in body
length) to mature adults (>90 days post-fertilization, ∼45 mm in
length). All wild-type *Danio rerio* Hamilton 1922 were maintained
according to standard protocols
(Westerfield, 1995) on a 14
h:10 h light:dark cycle at 26°C.

### Coasting kinematics

We measured the kinematics of coasting from video recordings of routine
swimming in fish over the full range of growth stages. In order to automate
measurements of body position, high-contrast images of the body's silhouette
were produced by illuminating the translucent acrylic floor of the aquarium
with a compact florescent light bulb (40 W;Greenlite, Irvine, CA, USA). The
water temperature during experiments was maintained between 25°C and
26°C by cooling the light source with a computer fan (12 V;
Fig. 1A). Coasting was recorded
with high-resolution, high-speed digital video (500 frames s^{-1} at
1280×1024 pixels) with the camera (NAC Hi-Dcam II) focused on the centre
of the aquarium using a macro lens (50 mm Nikkor, Nikon, Tokyo, Japan) with a
field of view ranging from ∼4 mm to ∼50 mm in width. The eyes of the
zebrafish appeared as the darkest pixels in video frames and thereby provided
a consistent pair of landmarks that were tracked by a custom image-processing
program (written in Matlab 6.0; Mathworks). The position of the caudal
peduncle was acquired by manual tracking within this Matlab program.

Kinematic variables were calculated from measurements of cranial position and orientation and body posture. The cranial position was taken as the centre point between a fish's eyes, and its orientation, θ, was perpendicular to the axis between the centres of the eyes and directed anteriorly. The body posture was measured by the caudal position, φ, which is the angle between the orientation of the cranium and the position of the caudal peduncle. The coast phase was considered to be the period when φ maintained a constant zero value, but ended if the body ceased to move forward. Coasting sequences were rejected for analysis if fish visibly moved their fins.

Kinematic parameters describing coasting were found by non-linear curve
fitting kinematic equations to cranial position measurements. The temporal
changes in speed, *U*, were approximated as an asymptotic decay towards
zero:
3
where *U*_{0} is the initial speed and τ is a time constant
that expresses the ability of a fish to maintain speed. Integrating this
equation with respect to time yields an equation describing cranial position,
*x*, with a zero initial value:
4
Values for *U*_{0} and τ were found by a non-linear
least-squares curve fit of this equation to position measurements
(Gauss-Newton method in Matlab). In order to correct for behavioural variation
in coast duration, the coast distance, *d*, was calculated with
Eqn 4 as the distance traversed
in the mean coast duration for all fish. Therefore, the performance of each
coast was characterized by the kinematic parameters *d*, τ and
*U*_{0}.

### Drag measurements

The scaling of drag was characterized from measurements of drag
coefficients for adult fish over a range of *Re*. For these
experiments, fish were euthanized by an overdose of MS-222 (Argent, Chemical
Laboratories, Redmond, WA, USA) and fixed in a 10% formalin solution overnight
to stiffen the body and fins. During fixation, the body was turned upside down
and supported on the dorsal surface with wires that did not touch the fins.
This arrangement allowed the fins to be fixed in their resting posture, with
the pectoral fins approximately aligned with the frontal plane of the body and
the median fins flattened along the sagittal plane. During experiments, fish
were attached by their ventral surface to a glass micropipette (50 μl
microcaps; Drummond Scientific, Broomall, PA, USA), using light suction (as in
McHenry and Strother, 2003),
and oriented toward flow in a flow tank
(Fig. 1B) by adjusting the
pitch and yaw of the body by eye. The fixed bodies of fish were sufficiently
rigid that no fluttering by the fins was observed. Deflection of the
micropipette was recorded with an optical strain gauge (11-04-001; UDT
Sensors, Hawthorne, CA, USA) and calibrated for force after each drag
measurement. This calibration consisted of finding the relationship between
the output voltage for the strain gauge and force by rotating the apparatus to
a horizontal orientation and hanging weights at the tip of the tether. The
relationship between voltage and force was linear within the range of strains
observed during experiments. We subtracted the drag acting on the tether from
the total drag from measurements taken after releasing the fish's body from
the tether. The accuracy of our experimental approach was confirmed by
comparing drag measurements for a sphere with published values
(White, 1991). Reynolds number
was varied in these experiments by altering the viscosity of water (as in
Johnston et al., 1998) through
the addition of up to 8% Dextran 500 (Amersham Biosciences, Piscataway, NJ,
USA) and controlling the speed of flow in our gravity-fed flow tank (from 0.9
mm s^{-1} to 60.7 mm s^{-1}). Drag on each fish was measured
at six *Re* values ranging from 1 to 10,000. For each measurement, we
calculated the non-dimensional drag coefficients using Eqns
1,
2. The inertial hydrodynamic
regime was defined as the range of *Re* values where the inertial drag
coefficient, *C*_{inert} (Eqn
1), remained constant with respect to *Re*. The viscous
regime was defined as the range of *Re* values where the viscous drag
coefficient, *C*_{visc} (Eqn
2), maintained a constant value, and the intermediate regime was
taken as the range between viscous and inertial regimes.

The accuracy of dead drag measurements was tested with *in vivo*
drag measurements of coasting fish. We found the inertial drag coefficient
using the equation for the inverse of the first integral of the equation of
motion for a gliding animal (Bilo and
Nachtigall, 1980):
5
The slope of this line, *h*, was found by a linear least-squares curve
fit to the inverse of instantaneous measurements of speed throughout a
coasting sequence, with an intercept equal to the inverse of the initial
speed. The inertial drag coefficient was found from this slope with the
following relationship (Bilo and
Nachtigall, 1980):
6
where *m* is body mass. Since this method requires that the fish
operate in an inertial regime, we used only sequences of coasting adult fish
at *Re*>1000.

### Morphometrics

The size of zebrafish bodies was measured from high-contrast photographs of dorsal and lateral views. From these photographs, the peripheral shape of the body was traced with an automated program (in Matlab), and the body was reconstructed in three dimensions under the assumption of an elliptical shape in the transverse plane. The wetted surface area was calculated as the sum of area of polygonal elements describing the peripheral body shape, and the volume was calculated as the space enclosed by this surface (as in McHenry, 2001; McHenry et al., 2003).

### Mathematical modelling

The equations of motion for fish coasting in the viscous and inertial
regimes were used to examine the effects of scaling in hydrodynamics and
momentum on coasting performance. We used the following equation of motion to
model coasting in the inertial regime:
7
where *a* is body acceleration, *V* is body volume, and
*A* is the added mass coefficient. It was assumed that a zebrafish's
body possesses an added mass coefficient equal to a 1:6 ellipsoid (where
*A*=0.045; Johansson,
2003; Kochin,
1964; Munk, 1922).
This equation may be solved analytically to generate a prediction of changes
in body position, *x*, as a function of time:
8
where
9
Coasting in the viscous regime has a similar form to
Eqn 7 but includes a term for
viscous instead of inertial drag:
10
The analytical solution to this equation is equivalent to
Eqn 4, where:
11
Mathematical simulations consisted of calculating positional changes with Eqns
4 and
8 using the measured drag
coefficients from tethering experiments and the mean morphometric measurements
for the small larvae, large larvae, juveniles and adults
(Table 1). By using drag
measurements for adults only, this approach assumed that morphological
differences influenced drag coefficient to a smaller degree than did
differences in *Re*. Simulations including acceleration reaction
(*AρVa*) were compared with simulations that excluded that term in
order to examine the effects of added mass. We focused on the transition
between juvenile and adult stages by running a series of simulations that
varied individual parameters for body size (i.e. *L* or *S*),
hydrodynamic regime (i.e. Eqns 4
or 8), body mass and initial
speed. These simulations were calculated for durations equal to the mean of
measured coast periods, the mean + 1 s.d. and the mean - 1
s.d. in order to examine the effect of coast duration.

## Results

### The scaling of coasting kinematics

Zebrafish at all stages of growth routinely swam with intermittent beat and
coast phases. Caudal position and cranial orientation oscillated rapidly
during tail beating but maintained relatively constant values during the coast
phase (Fig. 2A-C). The
coefficient of determination was high (*r*^{2}>0.98) for
curve fitting Eqn 4 to position
measurements at all stages of growth (Fig.
2D), which suggests that this equation characterizes the
kinematics of coasting well.

The duration of coasts varied greatly at all stages of growth and did not
correlate with body length. For example, one small larva (*L*=4.25 mm)
coasted for durations of 0.30 s, 0.42 s and 0.54 s and one adult
(*L*=37.2 mm) coasted for durations of 0.20 s, 0.49 s and 0.60 s
(Fig. 3A). This apparent lack
of a size effect was supported by a linear regression between body length and
mean coast duration that was not statistically significant (*P*=0.42,
reduced major axis regression; Sokal and
Rohlf, 1995). The distribution of coast duration was therefore
approximated by the mean (μ_{T}=0.37 with 95% confidence intervals
of *L*_{1}=0.33 and *L*_{2}=0.42) and the
standard deviation (σ_{T}=0.10 with 95% confidence intervals of
*L*_{1}=0.08 and *L*_{2}=0.15;
Fig. 3B) for all individuals
(*N*=21).

We found that the size of a fish strongly influenced its coasting
kinematics. For example, one representative larva traversed a mean coasting
distance (*d*) of less than 20% of its small body length
(*L*=4.38 mm) whereas an adult (*L*=36.9 mm) coasted for more
than 50% of its much larger body (Fig.
4A-C). Furthermore, adults sometimes achieved coasting distances
in excess of two body lengths, and the mean coasting distance of fish larger
than 23.0 mm (*d*=0.97*L*; *N*=9) was more than three
times the mean distance of smaller fish (*d*=0.31*L*;
*N*=12; Fig. 4D). This
variation in coasting distance may be attributed to the scaling of initial
speed (Fig. 4E) and the time
constant (Fig. 4F) because all
distances were calculated with the same period (*t*=0.37 s) using
Eqn 4. The mechanics that account
for the ability of adult fish to coast disproportionately further than larvae
and juveniles were investigated in the remainder of this study.

### The scaling of hydrodynamics and momentum

Viscous and inertial hydrodynamic regimes were evaluated from dead drag
measurements over a range of *Re* values between 1 and 10,000
(Table 2). The inertial drag
coefficient varied across the 1<*Re*<1000 range, but remained
relatively constant (*C*_{inert}=0.067±0.006;
*N*=6; five *Re* values per fish) at higher *Re* values
(Fig. 5A). By our criteria for
hydrodynamic regimes (see Introduction), the range of *Re* where the
inertial drag coefficient remained constant (*Re*>1000) defines the
inertial regime. The viscous regime was defined at *Re*<300 because
it is in this range that the viscous drag coefficient maintained a constant
value (*C*_{visc}=5.72±0.41; *N*=6; three
*Re* values per fish; Fig.
5B). The intermediate hydrodynamic regime was therefore found to
be 300<*Re*<1000 (Fig.
5C). Differences in values for the drag coefficients at
*Re*≈300 (*C*_{visc}=5.37±0.84, *N*=6;
*C*_{inert}=0.064±0.019, *N*=6) and at
*Re*≈1000 (*C*_{visc}=16.83±5.42,
*N*=6; *C*_{inert}=0.076±0.011, *N*=6)
were small (<0.1%) relative to the range of measured coefficient values
(Fig. 5A,B).

We found differences between indirect and direct measurements of drag. The
mean value (± 1 s.d.) for the inertial drag coefficient from
*in vivo* measurements (*C*_{inert}=0.024±0.011;
*N*=4; three glides per fish; Fig.
6) was nearly one-third the dead drag value for adult fish at
*Re*>1000 (Fig. 5A).
This is a highly significant difference (*P<*0.001), according to a
two-tailed, unpaired Student's *t*-test
(Sokal and Rohlf, 1995).
Although significant, this discrepancy between methods of drag measurement is
small (<0.2%) relative to the range of *C*_{inert} spanned
across the *Re* values examined
(Fig. 5A).

Drag measurements were related to the routine locomotion of fish by
measuring the range of instantaneous *Re* values exhibited during
coasting. The upper and lower limits of this range were progressively greater
at larval, juvenile and adult stages (Fig.
7A). Small larvae generally ranged from *Re*≈1 to
*Re*≈100 during coasting, whereas large adults spanned from
*Re*≈1000 to *Re*≈10,000
(Fig. 7B). Larval and juvenile
fish primarily operated at *Re* values within the viscous regime
defined by our drag measurements (Fig.
5). By contrast, adult fish (*L*>23 mm) primarily
operated in the inertial regime.

Mathematical models were used to examine the influence of scaling in hydrodynamics and momentum on coasting kinematics. Model fish operating in the same hydrodynamic regime travelled further if they had greater momentum at the start of the coast (Fig. 8). For example, when we created a model with the body size of a larva and the momentum of an adult, it traversed more than 100 times the length-specific distance in 0.5 s of a model with the momentum of a larva (Fig. 8A). A similar pattern was predicted for model fish the size of juveniles and adults (Fig. 8B-D). When momentum was held constant, model fish of larger body size encountered greater drag and therefore coasted less far. For instance, a model having the momentum and size of a large juvenile (green lines in Fig. 8C) coasted about half the length-specific distance of the same model reduced to the size of a larva (green lines in Fig. 8A).

Greater body mass and initial speed endows adults with more momentum at the beginning of a coast than juveniles, but adults also have a larger body size for drag to act upon. Direct comparisons of the mechanics of juvenile and adult stages are complicated by our finding that they operate in different hydrodynamic regimes (Figs 5, 7). Therefore, mathematical simulations were useful for differentiating the mechanical changes occurring during growth from juvenile to adult stages. Model fish coasted less far with the body size and inertial drag of adults (Fig. 8D) than with the size and viscous drag of juveniles having the same initial momentum (Fig. 8C). Simulations run within the viscous regime predicted that model fish the size of juveniles (Fig. 9A) coasted further than models the size of adults (Fig. 9B), and this effect was more pronounced in coasts of longer durations. A shift from the viscous to inertial hydrodynamics resulted in even shorter predicted coast distance (Fig. 9C), which suggests that the ontogenetic increase in coast distance (Fig. 4D) is not generated by changes in hydrodynamics. However, increasing body mass and initial speed to adult values (Fig. 9D,E) greatly increased predicted coast distance, which suggests that differences in momentum are dominant in determining the change in coast distance between juveniles and adults (Fig. 4D). Regardless of parameter values in these simulations, the differences between predictions that included the acceleration reaction were negligibly different from simulations that did not (Figs 8, 9).

## Discussion

### How does coasting performance change with size?

Our measurements of gliding kinematics over the full range of growth stages revealed nonlinear changes in performance over ontogeny. The most surprising of these changes was the large increase in coast distance and initial speed in the growth between juvenile and adult stages (Fig. 4D,E). On average, juveniles begin gliding at one-third the length-specific speed and travel less than half the length-specific distance of adults. Furthermore, the time constant gradually increased during growth to four times the value of larvae (Fig. 4F).

Our findings that adult zebrafish coast disproportionately further and faster, and maintain their speed for a longer duration than larvae and juveniles (Fig. 4D-F) are consistent with prior studies (e.g. Fuiman and Webb, 1988; Hunter, 1972; Muller et al., 2000). Our method of calculating coasting distance at an equivalent coast period (using Eqn 4) accounts for the small discrepancies between the values that we report and those of Muller et al. (2000) and Fuiman and Webb (1988). Furthermore, the `resting period' data used in prior studies included time that fish spent motionless whereas our measure of coast duration included only the period of deceleration and therefore was of smaller value.

### How do the hydrodynamics of coasting change with size?

The hydrodynamic scaling suggested by our measurements of dead drag does
not agree with some previous studies on fish locomotion. We found that viscous
forces dominate the generation of drag at *Re*<300, which has an
upper limit that is more than an order of magnitude greater than that proposed
by Weihs (1980) and considered
by others (e.g. Fuiman and Webb,
1988; Muller and Videler,
1996; Osse, 1990;
Osse and Drost, 1989;
Videler, 1993;
Webb and Weihs, 1986).
Furthermore, the lower limit of our measured inertial regime
(*Re*≈1000) is greater than what many have suggested (e.g.
Osse and Drost, 1989;
Videler, 1993;
Webb and Weihs, 1986;
Weihs, 1980), which leaves an
intermediate transition over a relatively narrow range of Reynolds numbers
(300<*Re*<1000) and drag coefficient values
(Fig. 5). The scaling proposed
by Weihs (1980) was reasonable
because it was consistent with drag measurements of rigid bodies in the
1<*Re*<1000 range from the engineering literature (e.g.
Hoerner, 1965). However,
measurements at this scale have primarily focused on bluff bodies of spherical
or cylindrical shape, with relatively little attention paid to streamlined
bodies (although an abundance of research has examined the hydrodynamics of
streamlined bodies at higher *Re* values;
Hoerner, 1965). Our results
suggest that the scaling of drag for a coasting fish is not well approximated
by measurements of bluff bodies. This conclusion is perhaps not surprising
given that bluff bodies exhibit separation and a turbulent wake within this
scale, while airfoils at higher *Re* with a zero angle of attack
generate little turbulence (Granger,
1995).

The hydrodynamic regimes defined by our drag measurements are entirely
consistent with the results of Fuiman and Batty
(1997). That study found that
a linear relationship between stride length and the viscous drag product
(*ULμ*) in anchovy was disrupted at *Re*>300, which was
interpreted as evidence of inertial drag and lead to the conclusion that fish
are viscous-dominated at *Re*<300. Our results support this
conclusion and raise the possibility that the scaling of drag examined
presently may apply to the propulsive phase as well. Although the drag
coefficient of a swimming fish is different from that of a coasting fish
(Anderson et al., 2001;
Blake, 1983;
Drucker and Lauder, 1999;
Schultz and Webb, 2002;
Videler, 1981,
1993;
Webb, 1975), drag may vary
with *Re* in a similar manner for both phases. This hypothesis is at
odds with the mathematical model of propulsion proposed for ascidian larvae by
McHenry et al. (2003), which
suggested a negligible contribution of viscous drag at *Re*≈100
during body undulation. However, it is expected that the spherical trunk of an
ascidian larva should more closely resemble the hydrodynamics of a bluff body
and therefore have a lower limit to its viscous regime than the relatively
streamlined body of a fish (McHenry,
2005). Computational fluid dynamics modelling verified with
experimental measurements (e.g. Liu et
al., 1996; Wolfgang et al.,
1999) holds promise for examining differences in the scaling of
drag between coasting and propulsive phases.

The present approach of expressing hydrodynamic scaling in terms of regimes
is useful for interpreting large-scale ontogenetic change but has the danger
of concealing physically important phenomena. For example, it is informative
that drag varies directly with speed and not to its second power in juvenile
fish. However, it would be inaccurate to predict that larvae and juvenile fish
should generate identical flow fields because they both operate in the viscous
regime. Inertia is present and dynamically relevant at low *Re*, and
viscosity is required for generating turbulence at high *Re*.
Furthermore, drag coefficients may vary with *Re* within regimes,
although to a lesser degree than observed across regimes
(Fig. 5). Hydrodynamic regime
categorization provides a useful first-order approximation of dominant
hydrodynamic forces but is limited in its ability to predict the flow fields
that generate those forces or higher-order effects. Nonetheless, our results
provide an empirical basis for verifying new experimental and theoretical
approaches for understanding hydrodynamics at this scale.

Our measurements of drag coefficient are comparable to values reported in
prior studies. Numerous studies have examined the inertial drag coefficients
(usually denoted *C*_{d}) of the rigid bodies of adult fish and
marine mammals at *Re*>1000. Employing a variety of experimental
techniques, this literature reports values for *C*_{inert}
ranging from 0.0034 to 5.59 (reviewed by
Blake, 1983;
Videler, 1993;
Webb, 1975). Webb's dead drag
measurements of *Salmo gairdneri* (*L*=300 mm,
3000<*Re*<20,000; Webb,
1970) come close in body morphology to those of zebrafish. Our
mean *in vivo* measurement of *C*_{inert}=0.024 is of
comparable magnitude to Webb's mean of *C*_{inert}=0.036
(Webb, 1970). Despite a
difference in body shape, computational simulations of dead drag in frog
tadpoles (*Re*=7200) by Lui et al. (1997) also predicted a similar
value (*C*_{inert}=0.0287).

Although measurements of *in vivo* drag have been restricted to the
inertial regime in the present study, a method for measuring drag in larval
and juvenile fish is apparent from our results. The time constant found by a
non-linear curve fit to position data (using
Eqn 4) or velocity data (using
Eqn 3) is inversely proportional
to the viscous drag coefficient. As long as a coasting fish is operating in
the viscous regime, measurements of time constant, body length, mass and body
volume may be used to calculate the viscous drag coefficient with
Eqn 11. Although no comparable
equations exist for fish operating in the intermediate regime
(300<*Re*<1000), our finding that the *C*_{inert}
varies little in this range suggests that methods for measuring
*C*_{inert} *in vivo* (e.g.
Bilo and Nachtigall, 1980)
provide a reasonable approximation for coasting in this regime.

Discrepancies between dead drag and *in vivo* measurements of drag
coefficient demonstrate the limitations on what may be interpreted from dead
drag experiments. The range of *C*_{inert} values spanned in
the 1<*Re*<10,000 range (Fig.
5A) far exceeded (by more than a factor of 5000) the difference
between *in vivo* and dead drag measurements. However, we did find that
tethered adult fish generated nearly three times the drag of freely coasting
fish. Possible sources for this difference include chemical fixation (which
presumably alters the mechanics of mucus and surface compliance) or
differences in the posture of the fins or angle of attack between the bodies
of dead and live fish. Although this discrepancy appears minor in the context
of hydrodynamic scaling, a threefold difference in drag may be substantial for
considerations of the energetics of locomotion. Our results should therefore
contribute to scepticism about the application of dead drag as an absolute
measurement of force coefficients for physiological studies
(Schultz and Webb, 2002).

Our mathematical modelling suggests that the added mass of the surrounding water does not make a meaningful contribution to the dynamics of coasting zebrafish. The coasting distances predicted in models that included the added mass were negligibly different from those that did not (Figs 8, 9). This result is not surprising, given that the added mass was assumed to be equal to that of a 1:6 ellipsoid, which is a geometry that accelerates a relatively small volume of fluid (Kochin et al., 1964; Munk, 1922). Furthermore, a coasting body does not generate the large oscillatory accelerations created during the propulsive phase, where acceleration reaction is expected to play a much larger role in hydrodynamics (Lighthill, 1975; Wu, 1971). However, it remains possible that the added mass could contribute to coasting dynamics if the added mass coefficient was at a comparable magnitude to what Webb (1982) found in the fast-starts of trout.

### How does the scaling of drag and momentum affect performance?

Although the hydrodynamic changes occurring over the life history of zebrafish are dramatic, our results suggest that changes in coasting performance are largely driven by the scaling of momentum (Fig. 10). Our mathematical model predicted that the increase in relative coast distance from juvenile to adult stages is actually hindered by the transition from viscous (Fig. 9B) to inertial (Fig. 9C) regimes. This detrimental consequence of hydrodynamics is overcome by an increase in momentum between these stages. The greater mass and, especially, the faster initial speed propel the body of an adult disproportionately further (Fig. 9). High initial speed may be the result of reduced drag production from the more streamlined body shape of adults (Fig. 7A), or adults may be more effective at generating thrust. The end result is that the scaling of performance in the coasting phase is largely dictated by the performance of the propulsive phase.

### Mechanical scaling and ontogenetic change

Our results present an opportunity for new interpretations of ontogenetic
changes in the behaviour of fish. The newly hatched larvae of zebrafish
(Buss and Drapeau, 2001) and
other species of comparable size (Hunter,
1972; Osse and van den
Boogaart, 2000) routinely swim in relatively long bouts at steady
speed but shorten the duration of the propulsive phase and swim more
intermittently as they grow to a size that routinely operates at
*Re*>30. During the transition towards more intermittent swimming,
some fish appear to develop greater manoeuvrability, as the curvature of tail
beating decreases while maintaining or enhancing their ability to change
direction (Fuiman and Webb,
1988; Muller and van Leeuwen,
2004; Osse and van den
Boogaart, 2000). A number of studies have suggested that
intermittent swimming and greater manoeuvrability are consequences of
operating in a more inertial hydrodynamic environment (e.g.
Fuiman and Webb, 1988;
Webb and Weihs, 1986;
Weihs, 1980). However, our
results indicate that the viscous regime extends into *Re* of more than
an order of magnitude greater than assumed by these analyses, which suggests
that these behavioural transitions may not be attributed to a hydrodynamic
regime change. Furthermore, we found that coasting performance is better
predicted by ontogenetic changes in momentum than the scaling of drag. This
suggests that even a revised consideration of hydrodynamic scaling must be
coupled with the inertial dynamics of the body in order to accurately evaluate
the functional significance of ontogenetic change in behaviour, especially in
fish that swim intermittently.

Patterns of morphological change during growth may be more a reflection of
developmental or historical constraints than an adaptation to changing
hydrodynamic conditions. Zebrafish, like many fish species, have an elongated
body at hatching but gradually develop a more compressed and streamlined shape
during larval and juvenile growth (Osse,
1990; Webb and Weihs,
1986). Investigators have posited that this elongated shape allows
larvae to precociously exit the viscous regime by having a greater body
length, and therefore higher *Re*, than if they distributed their mass
in a streamlined form. According to this hypothesis, juveniles gradually
abandon an elongated shape after exiting the viscous regime in order to reduce
drag by streamlining as they grow into an increasingly inertial environment
(Fuiman and Webb, 1988;
Muller and Videler, 1996;
Osse, 1990;
Osse and Drost, 1989;
Sagnes et al., 2000;
Webb and Weihs, 1986).
However, our results suggest that these changes in body shape occur well
within the viscous regime (*Re*<300). The fusiform body shape of
adults is probably an adaptation for locomotion, but it develops in juvenile
zebrafish before coasting can benefit from the inertial drag reduction that
comes from streamlining (Fig.
3). Given that an elongated larval body is the plesiomorphic
condition in the Chordata (Berrill,
1955; McHenry and Patek,
2004; Stokes,
1997), it is more plausible that this morphology is a
developmentally or historically constrained trait rather than an adaptation to
locomotion (as in jellyfish; McHenry and
Jed, 2003). Furthermore, an elongated shape may facilitate gas
exchange or other physiological functions
(Liem, 1981;
Webb and Weihs, 1986), which
suggests that stabilizing selection may have acted in the evolution of aquatic
vertebrates to retain this ancestral condition.

**List of symbols**

- A
- added mass coefficient
- a
- body acceleration
*C*_{inert}- inertial drag coefficient
*C*_{visc}- viscous drag coefficient
- d
- coast distance
- D
- measured drag
- h
- slope of the inverse of speed
*vs*time - L
- body length
- m
- body mass
- p
- momentum
- Re
- Reynolds number
- S
- wetted area of the body
- t
- time
- U
- speed
*U*_{0}- initial speed
- V
- body volume
- φ
- angular fin position
- μ
- water dynamic viscosity
- θ
- cranial orientation
- ρ
- water density
- τ
- time constant

## ACKNOWLEDGEMENTS

We thank E. Tytell, T. Hsieh, J. Liao, J. Roman, A. Summers, W. Korff, and S. Patek for their input on the manuscript. W. Korff provided critical technical advice. This work was supported by a NSF Postdoctoral Research Fellowship to M.J.M. and NSF grant IBN0316675 to G.V.L. P. Webb and an anonymous reviewer provided valuable feedback.

- © The Company of Biologists Limited 2005