Shape, Drag, and Power in Ammonoid Swimming...Paleobiology, 18(2), 1992, pp. 203-220 Shape, drag, and power in ammonoid swimming David K. Jacobs Abstract.-This study assesses swimming

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Shape, Drag, and Power in Ammonoid SwimmingAuthor(s): David K. JacobsSource: Paleobiology, Vol. 18, No. 2 (Spring, 1992), pp. 203-220Published by: Paleontological SocietyStable URL: http://www.jstor.org/stable/2400999Accessed: 23/02/2010 16:25

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Paleobiology, 18(2), 1992, pp. 203-220

Shape, drag, and power in ammonoid swimming

David K. Jacobs

Abstract.-This study assesses swimming potential in a variety of ammonoid shell shapes on the basis of coefficients of drag (Cd) and the power needed to maintain a constant velocity. Reynolds numbers (Re) relevant to swimming ammonoids, and lower than those previously studied, were examined. Power consumption was scaled to a range of sizes and swimming velocities. Estimates of power available derived from studies of oxygen consumption in modern cephalopods and fish were used to calculate maximum sustainable swimming velocities (MSV).

Laterally compressed, small thickness ratio (t. r.) ammonoids, previously assumed to be the most efficient swimmers, do not experience the lowest drag or power consumption at all sizes and velocities. At low values of size and velocity associated with Reynolds numbers below 104, less compressed forms have smaller drag coefficients and reduced power requirements. At hatching a roughly spherical shell shape would have minimized drag in ammonoids; with increasing size, hydrodynamic optima shift toward compressed morphologies.

The high energetic cost of ammonoid locomotion may have limited dispersal and excluded ammonoids from high current velocity environments.

David K. Jacobs. Museum of Paleontology, University of California, Berkeley, California 94720. Present address: Department of Invertebrates, American Museum of Natural History, New York, New York 10024

Accepted: August 21, 1991

Introduction

The involute, laterally compressed, oxyconic shell tapers smoothly from its widest point near the center of coiling to the venter. Ammonoid workers have long assumed that the oxyconic shell form conferred the lowest values of drag and hence the greatest swimming ability among the planispiral ammonoids. The question that this study will address is, If oxycones have a distinct hydrodynamic advantage, and if this advantage were important to ammonoids generally, then why weren't all ammonoids oxyconic in shape?

The opinion that the oxyconic morphology was hydrodynamically advantageous has been supported by a number of workers (Schmidt 1930; Kummel and Lloyd 1955; Chamberlain 1976, 1980, 1981). In addition, true oxycones evolved independently at least ten times (Spath 1919; Arkell 1957), suggesting that this shell morphology may have had a particular selective advantage. However, the large majority of ammonoids are less compressed and lack the tapered venter characteristic of oxycones.

If the oxyconic morphology represents a 1 1992 The Paleontological Society. All rights reserved,

truly optimal morphology, the many depar- tures from this form might not be expected. However, explanations for why all ammonoids are not oxycones may lie no further afield than the conditions and assumptions employed in previous examinations of hydrodynamic efficiency. Even if only the hydrodynamic aspects of the landscape are considered, oxycones may not occupy a simple peak in the adaptive landscape of planispiral ammonoid shell design.

Recent studies of drag in ammonoids (Chamberlain 1976, 1980, 1981; Chamberlain and Westermann 1976) were conducted at Reynolds numbers (Re) in excess of 104, and swimming performance for shell shapes were characterized using a single coefficient of drag (Cd) (Kummel and Lloyd 1955; Chamberlain 1976, 1981). Re increase with increasing size and swimming velocity. High Re pertain to large organisms swimming relatively fast. Wells (1987), O'Dor (1988) and Chamberlain (1990) suggested that cephalopod swimming speed may be limited by the high energy de- mands of jet propulsion. Low swimming speed and the small size of the majority of ammonoids correspond to lower Re than examined

0094-8373/92/ 1802-0006/$1.00

204 DAVID K. JACOBS

in previous studies of drag on cephalopod shells. If Cd is not constant over the range of Re pertinent to swimming in ammonoids, then the Cd reported in previous studies may not be sufficient to determine the range and variety of hydrodynamically advantageous ammonoid forms.

This study was designed to determine whether Cd is constant for compressed and less compressed ammonoids over the range of Re representative of biologically reasonable swimming velocities. If Cd were to vary with Re, then a variety of shell shapes might have had advantageously low Cd, depending on the Re at which they swam. In addition, Cd does not scale with the power available to propel the organism. With increasing size, drag increases as a function of area whereas power available increases as a function of volume. Larger volume means more power available and the ability to swim faster even if Cds are similar. Thus, power required per unit of volume provides an alternative, and possibly more relevant, means of comparing swimming ability than does the Cd. Power required can also be compared to estimates of power available, derived from oxygen-consumption rates in modern cephalopods, to estimate maximum sustainable swimming speeds.

In this study, drag forces were measured in a flow tank over a range of Re lower than those previously examined. A small number of shell models of ammonoids ranging in shape from laterally compressed to depressed were examined (see fig. 1). Cd and power required for swimming at various sizes and velocities were determined. Comparison of laterally compressed and less compressed ammonoids should reveal whether the laterally compressed morphs, such as the much- vaunted oxyconic shell shape, were advantageous in terms of drag and resulting power consumption at all sizes and swimming velocities, or whether other shapes might also have had lower drag or power requirements under some circumstances.

Previous Studies

Many paleontologists have commented on the relative swimming abilities of different

ammonoid morphologies (e.g., Hyatt 1889; Spath 1919; Scott 1940; Trueman 1941; Mutvei and Reyment 1973). A number of these workers expressed the opinion that laterally compressed morphs were better swimmers. Fewer workers have empirically examined the drag forces on ammonoid shells (Schmidt 1930; Kummel and Lloyd 1955; Chamberlain 1976, 1980).

According to Kummel and Lloyd (1955), Schmidt (1930) observed the drag on four ammonites and came to the conclusion that involute, laterally compressed morphologies were more "streamlined" than depressed or evolute forms. Kummel and Lloyd (1955) made a more extensive study of drag on cephalopod shell form using a larger number of shell models and came to conclusions similar to those expressed by Schmidt (1930). Later- ally compressed involute forms were more "streamlined" or more efficient swimmers.

In 1976 Chamberlain published a more sys- tematic study in the theoretical morphology tradition of Raup (Raup 1966, 1967; Raup and Chamberlain 1967). This study employed a series of ideal models derived from the Raup- ian coiling parameters for spiral shell ge- ometries, whorl expansion W, the distance of the aperture from the coiling axis D, and the shape of the generating curve S. Chamber- lain's (1976) study was based on 37 planispiral models. Drag forces were measured in a naval test tank. The models involved were in excess of 10 cm in length, and the velocities were above 10 cm/s indicating that the suite of Re examined was in excess of 104.

Chamberlain (1976) intensively investigated the influence of whorl expansion W and distance from the coiling axis D on the Cd in forms with circular whorl sections. The region of morphospace containing the highest diversity of ammonoids with circular whorl sections was found to overlap one of the regions that conferred low Cd (Chamberlain 1981).

Chamberlain's (1976) examination of the variation in Cd with changing whorl shape was less exhaustive. Three models in which whorl shape, S, departed from 1 (a circle) were examined. These results indicated that re- ductions in S (to 0.3) produced the lowest Cd

SHAPE, DRAG, AND POWER IN AMMONOIDS 205

Less Compressed

hnodised as Microceras dia-S.13 wid=i.o0 Lr.=o.19 dia-4.23 wid=1.35 t.r.-=0.32 W-3.02 D=0.03 S=0.32 W-1.77 D=0,47 S=1.05 Jl5.0 V=6.39 A--3.44 1..4.5 v--9.26 A=4.41

eras dicras

dk-s4.54 wid=1.00 Lr.-0.21 W-1.85 D=0.35 S=0.53 wid=1.91 tr.=0.33

1- 4.5 v=10.01 A=4.64 P4 D-0A47 S=0.95 .65 v=25.00 A=8.55

Axahoplites dia=4.80 wid=1.15 t.r.=0.24 i plitua W=1.83 D=0.28 S=0.52 1=5.0 v=8.88 A=4.29 _488 wids1,89 tr.=0.39

_ _ _::S--M D-0.37 S=1.00 _15.5 v=20.20 A=7.42

Oppelia

dia=4.32 wid=1.10 tr.=-0.25 W=2.17 D1=0.15 S=0.462 1=4.0 v=7.82 A=3 94

dm-5.32 wid=2.25 t.r=0.42 W-2.10 D=0.16 S=0.80

_ -- S1.8 v=26.70 A=8.93 pressed

Scaphites

_Stephanoceras di=3.50 wid=1.60 t.r.=0.46 W=2.53 D=0.09 8--0.80

dia-4.11 wid=2.15 t.r.=0.52 1=3.7 v=7.00 A=3.66 W=2.04 D=0.33 S= 1.20 1-5.0 v=17.17 A=6.66

e_eras dia=5.02 wid=3.10 t.r.=0.62 W=1.60 D1=0.35 S= I.55

_1- I.5 v=30.20 A=9.70

FIGURE 1. Specimens in the first column in each photograph show the venter of the specimen that would be presented to flow. In the second column, casts of the specimens including body protheses are displayed in profile. Forms with a thickness ratio less than 0.3 are referred to as compressed, between 0.3 and 0.5 they are "less compressed" and with thickness ratios greater than 0.5 "depressed." Specimens and cast are accompanied by generic names, dimensions, and Raup's coiling parameters: dia, diameter in centimeters; wid, width in centimeters; t. r., thickness ratio (width/diameter); W, whorl expansion rate; D, departure of the whorl from the coiling axis (evoluteness); S, shape of the whorl. Information on the models is included: 1, length in direction of flow in centimeters; v, volume in cubic centimeters; and A, the characteristic area.

of any of the models. Thus whorl shape in- fluences drag to a greater extent than the coiling parameters W and D. This is not surpris- ing; reduction in whorl width (small S) contributes directly to lateral compression (smaller t. r.) of the coil as a whole. Previous studies (Schmidt 1930; Kummel and Lloyd 1955) identified lateral compression as a ma- jor contributor to drag reduction.

The hydrodynamic properties of ammo-

noid shells may be influenced by details of whorl form in addition to Raup's parameter, S. The widest point on the whorl can range from near the venter to the umbilicus; the venter of the whorl can vary from square or quadrate, to round or tapered to a point; the venter can be multiply keeled or sulcate; and the whorl can be ornamented. These additional variables are likely to influence the drag of the shell. Previous studies (Schmidt 1930;

206 DAVID K. JACOBS

Kummel and Lloyd 1955), and the conven- tional wisdom, specify oxycones as the form likely to be lowest in drag, not only because it is laterally compressed and involute, but also because the whorl tapers smoothly from its maximal width near the coiling axis to the venter. This tapered form is similar in cross section to the trailing portion of teardrop- shaped streamlined objects. These details of shape led many workers to conclude that the oxyconic ammonites experienced lower drag than other compressed forms.

In 1980, Chamberlain demonstrated that when models of soft parts were attached to cephalopod shell shapes, drag was reduced by as much as 23%. Without an extended body, the abrupt edge of the shell at the aperture produces flow separation resulting in a large drag-producing wake. Consequently, the addition of some proxy for the extended soft parts seems to be a reasonable and necessary step when attempting to assess the drag production and swimming ability of extinct shelled cephalopods.

One intriguing observation in Chamber- lain's study is that Cd actually decreases with increasing velocity for laterally compressed forms equipped with a body (Chamberlain 1980: fig. 11, p. 457). This suggests that, at least for models of ammonoids where flow separation has been reduced by the addition of a model of the soft parts, Cd will not nec- essarily be constant for all Re.

Stability. -Upper limits on propulsive force and swimming speed have been inferred on the basis of shell stability (Chamberlain 1981; Ebel 1990). In Nautilus the center of mass is well below the center of buoyancy, and gravity returns the shell to its original position if it is perturbed. In essence, the body chamber containing the dense soft parts hangs below the buoyant chambered portion of the shell. In Nautilus the restoring force plays an inte- gral role in the swimming mechanism. The jet exits near the venter, below the center of mass, imparting a rotational force on the shell; the restoring force opposes this rotation.

Stability is a function of the distance between the center of mass and the center of buoyancy. Calculations of stability have been performed for a range of fossil and theoretical

shell morphologies (Trueman 1941; Raup 1967; Raup and Chamberlain 1967; Saunders and Shapiro 1986). These calculations assume that uniformly dense soft parts just fill the body chamber. Chamberlain (1981) argued that the restoring force resulting from stability provides an upper limit on swimming velocity in coiled shelled cephalopods including ammonoids. Chamberlain's analysis appears to assume a ventral position and horizontal orientation of the jet. Using a different set of assumptions as to the orientation of the jet, but similarly incorporated calculations of stability, Ebel (1990) also attempted to determine maximum swimming velocities.

There are both phylogenetic and mechan- ical reasons why an analogy to the role of stability in Nautilus may not apply to swimming in ammonoids (Jacobs 1990; Jacobs and Landman 1991). Ammonoids are more closely related to coleoids than nautiloids (Berthold and Engeser 1987). The coleoid mechanism of mantle contraction differs markedly from that of Nautilus; therefore, there is little phylogenetic basis for assuming identical function in ammonoid and nautiloid swimming. In addition, Saunders and Shapiro (1986) observed that the longer body chamber of many ammonoids places the aperture higher up on the body where a horizontal jet could act more directly through the center of mass. Trueman (1 941) argued that ammonoids controlled their orientation by minor extension or retraction of the body. If either Saunders and Shapiro's or Trueman's arguments are important factors, calculations of swimming velocity based on shell stability may not be applicable across a broad range of cephalopod taxa. Here maximum sustainable swimming velocity (MSV) is assessed by comparison of power available with the power required to overcome drag. This assessment is independent of the stability argument.

Acceleration. -In addition to the drag or power consumption associated with constant swimming, it has been documented that acceleration has an influence on optimal shape in swimming forms. During acceleration an additional mass of fluid must be accelerated along with the mass of the organism (Daniel 1984). Although it is not considered in this


work, added mass is minimized by slender morphology. An examination of this potential influence on ammonoid morphology may be possible once the range of advantageous form for continuous swimming has been determined.

Should Coefficient of Drag (Cd) Be Constant with Changing Reynolds Number (Re)?

Re is a ratio characterizing the relative importance of inertia and viscosity for a given flow condition:

Re = Ul/lv, (1)

where length is 1 and velocity is U and the kinematic viscosity is v. The Re is often taken to indicate flow similarity; flow will be similar in character around objects of the same shape as long as the Re is the same. Because the objects compared in this study are not identical, flow need not be identical at comparable Re.

In estimating the range of Re pertinent to ammonoid swimming, both linear dimension and swimming velocity must be addressed. In water at 20?C, kinematic viscosity, v, is approximately 0.01 cm2/s; it varies only by a factor of about two between 0?C and 40?C (Vogel 1981). The modal diameter of adult ammonoids is less than 8 cm (Raup 1967). Fewer than 1% of ammonoids exceed 25 cm in length. It is hard to know a priori exactly what swimming speeds are relevant. Maxi- mum swimming velocities in ammonoids have been calculated (Chamberlain 1981) based on an analogy to swimming in modern Nautilus. These estimates are on the order of 10 cm/s for an ammonoid 10 cm in length. However, it is not clear that Nautilus is an appropriate model, or that selective advantage will always lie with maximum velocity. Cost of transportation is often minimized at velocities well below the maximum swimming speed (Schmidt-Neilson 1972; Wells 1987; O'Dor 1988). Given these considera- tions, a range of Re, from 103 to over 105, can be envisioned for adult ammonoids.

Pressure Drag, Skin Friction Drag, and Con- stancy of Cd. -There are two components to

drag, "pressure drag" related to cross-sectional area and "skin-friction drag" related to surface area. Pressure drag is essentially the inertial component of drag; consequently, it is important at high Re where inertial terms (the numerator of the Re) dominate. With increasing Re, laminar flow will separate from the object forming vortices in the wake. In the region where the flow has separated from the object, pressure acting on the object is actually lower than on comparable points on the object facing into the flow. It is this asym- metry of pressures that results in pressure drag. Pressure drag is related to frontal area, the cross-sectional area perpendicular to flow, and to details of shape that alter flow separation.

Reduction of cross-sectional area, and tapered trailing edges that reduce flow separation, tend to increase surface area. Skin- friction drag is a function of surface area. Thus, there is a trade-off; shapes that minimize pressure drag at higher Re may not be advantageous with regard to skin-friction drag at lower Re.

Cd is a nondimensional number that iso- lates a factor of drag associated with shape rather than size, velocity, or any other particular variable. The usual formula for Cd is

Cd = Df/(0.5ApU2), (2)

where Df is drag force, A is an area term, p is density of the fluid, and U is fluid velocity. The area term A is often chosen to scale with important variables in the question under in- vestigation. An organism's volume, rather than linear dimension or area, provides a more realistic basis of comparison (Vogel 1981). Consequently, in this and other studies (volume)213 was used for A.

In the Cd, the denominator is the equation for inertial force, 0.5ApU2. Therefore, the Cd is drag force corrected for the expected increase in intertial, or pressure, drag with velocity. Skin-friction drag is not compensated for; it will be important at low Re where vis- cous forces dominate. Thus, at low enough Re, the Cd should vary as a function of Re. In addition, in streamlined forms, pressure drag increases with velocity at a lower power function of velocity than the squared term in

208 DAVID K. JACOBS

100

0) ^ > I . . I @ w Sphere

.01

1 10 102 103 104 105

R e

FIGURE 2. Log x log plot of Reynolds number, Re, and Coeffient of drag, Cd, for a sphere. Note that Cd declines with Re up to about 103. This is a consequence of the declining effects of skin-friction drag with increasing Re. Between Re 103 and 105, Cd for a sphere does not vary. In this region, pressure drag is dominant and the effects of skin-friction drag are no longer evident. Also plotted are the Re-versus-Cd curves for streamlined forms of thickness ratio 0.25 and 0.12 (after Hoerner 1965). Note that, unlike the sphere, the Cd declines between Re 103 and 105.

0.5ApU2 (fig. 2). If the tapered oxyconic shell can also be considered a streamlined form, then oxycones could have declining Cd through Re - 105. The combined effects of skin-friction drag and the potential for streamlining in some compressed ammonoid shell shapes suggest that the Cd may not be constant over the range of Re from 103 to 105.

It is important to keep in mind that these changes are a consequence of the formulation of Cd employed. Other coefficients, with different power functions of velocity can be used to eliminate the change in Cd associated with streamlined forms (fig. 2). Similarly, skin- friction drag at low Re can be modeled or accounted for by additional functions of the Re (Hoerner 1965). In order to compare the results within this study, and between this study and previous work, the standard formulation of Cd was used.

In ammonoids of different shape, the effects of surface area and streamlining should vary resulting in different Cd-versus-Re curves. In a Cd-versus-Re plot for a sphere, skin-friction drag becomes important below

a Re of about 103, and the Cd begins to increase with decreasing Re (fig. 2). However, ammonoid shells are not spherical; they have larger surface areas than spheres of comparable volume. In disks of constant volume, a change in thickness ratio (t. r.) from 1 to 0.1 almost doubles the surface area (fig. 3). With the same change in t. r., frontal area declines by about half. The greatest change in surface area (important for skin-friction drag) relative to frontal area (important for pressure drag) occurs in a range of thickness ratios from 0.2 to 0.6. (fig. 3). These thickness ratios encompass the range exhibited by most planispiral ammonoid shells. It follows that the Cd for thin disks, or compressed ammonoids, will be influenced by skin-friction drag to a greater extent and to higher Re (above 103) than the Cd for spheres or thicker disks.

The Hypothesis. -The hypothesis under in- vestigation is that Cd (or, alternatively, power required) should vary with Re for compressed ammonoids because of the greater importance of skin-friction drag in these forms. In less compressed ammonoids, skin-friction


drag should not play as large a role. Of greatest interest is whether changes in Cd with Re number result in changes in the relative efficiency of ammonoid shapes. In other words, do the Cd, or power-required, curves for different morphologies actually cross. Crossing points will reveal the position where one morphology becomes "better" in terms of lower drag or power consumption than another. If crossing points occur in the range of Re important for swimming, a variety of hydrodynamically advantageous ammonoid shapes may exist.

Drag Measurement

In order to determine Cd and power required, drag forces on models of ammonoid shells were measured over a range of velocities in a Vogel and LaBarbera (1978) type flow tank constructed for this purpose. The plexiglas tank is 182.9 cm long, 25.4 cm tall, and 20.3 cm wide. Flows from 0 cm/s to 50 cm/s are produced using a Minarik mm 21001 motor speed control. Velocities are measured in the tank by injecting dye and timing its movement over 1 meter with a stopwatch. The range of Re examined in this study was from 2 x 103 to 2.3 x 104.

Each model was attached to a thin rod or "sting" and immersed in the flow. An alu- minum beam attached to the sting was used as a force transducer. Four strain gauges attached to the beam responded to the drag forces acting on the model. A Wheatstone bridge circuit and amplifier converted the strain experienced by the strain gauges into a voltage difference.

Millivolts were measured using a volt meter. In order to eliminate noise in the ampli- fication system, ten measurements were av- eraged for each velocity examined. This should render the data comparable to that taken with a chart recorder or an amplifica- tion system with greater capacitance (Vogel personal communication). Frequent readings at zero velocity were taken to guard against drift in the electronics. Force was determined at 20 to 30 velocities between 3 and 45 cm/s. Millivolt readings were corrected for the drag force on the portion of the sting exposed to the flow in the tank. The force transducer

Constant Volume Disk 11 1.6

e Surface A E 10 Projected A 1.4

E~~~~~~~~~~~~ > 9 1.2 >7

8 - ~~~~~~~1.0

57- 0.8

to 6 - ~~~~~0.62&O

5. - 0.4 0.00 0.50 1.00 1.50

Thickness Ratio

FIGURE 3. For a dish of constant volume, cross-sectional or frontal area decreases dramatically with increasing thickness ratio (width/diameter) over the range of thickness ratios pertinent to ammonoids (0.15 to 1.5). In addition, over this same range of thickness ratios, surface area increases dramatically.

behaved linearly and could measure mass differences as little as 0.005 g. The instrument was calibrated using weights, and millivolts were converted to units of force.

The variance in force determination during operation of the equipment appeared to be larger than the error in force calibration and is attributed to vibration and the additional error associated with measuring velocity by timing dye movement. Drag force varies as a square of velocity, and velocity appears as a squared function in the calculation of Cd. Thus, error in velocity measurement is es- pecially important. To compensate for these sources of error, force measurements were made at 20-30 independently determined velocities for each model examined.

Velocity was measured in cm/s, and the unit of force employed, the dyne (10-5 joules), was also based on centimeters and grams (1 dyne = g x cm s-2). The temperature in the tank never exceeded 20?C and was never below 19?C. Consequently, a kinematic viscosity of 0.010 cm2/s pertaining to water at this temperature was used in calculation of the Re. For calculation of Re, length in the direction of flow, including the added body, was measured (fig. 1). Volume of the models including the attached body was determined using Ar-

210 DAVID K. JACOBS

Compressed Morphologies 0.9

* Sphenodiscus 0.8- * Oppelia

* Cardioceras

0.7-

0.6

0.5

0.4

0.3 - U

0.2 0 10000 20000 30000

Re

Less Compressed Morphologies 0.6-

* Lytoceras chimedes' principleandusdGastroplites

t Scaphites 0.50

0 0%

0.4

0.3 U

0.2 0 10000 20000 30000

Re FIGURE 4. Coefficient of drag (Cd) versus Reynolds number (Re) curves for compressed and less compressed ammonoid morphologies. Compressed forms show a steep decline in Cd at Re of 1000-10,000. Less compressed forms show a more constant Cd over the range of Re observed.

chimedes' principle and used to determine the characteristic area, A. Cd and Re were then calculated for all data.

The primary concern in selecting specimens for study was that they be planispiral and vary in thickness ratio (t. r.). Eleven specimens were chosen (for additional information, including Raupian parameters and il- lustrations of each of the specimens, refer to fig. 1). They ranged in t. r. from 0.19 for Sphe- nodiscus, an oxycone, to 0.62 for Cadoceras, a depressed form. Four of the ammonoid specimens were compressed (t. r. < 0.25), five

were less compressed (0.3 < t. r. < 0.5), and two were depressed (t. r. > 0.5). In addition, an attempt was made to select pairs of specimens similar in shape but varying in t. r. For example, the Scaphite specimen (t. r. of 0.46) is comparable to Oppelia (t. r. of 0.25) in that both are involute and have rounded venters. Several evolute forms were selected to determine the effects of the umbilicus on drag over a range of t. r. Cadoceras, Stephanoceras, and Cardioceras are all comparably evolute (D of 0.33-0.35), but decline dramatically in t. r. (0.612, 0.52, and 0.21, respectively) as a consequence of narrowing of the whorl (declining S). The even more evolute Lytoceras (t. r. of 0.33) can also be compared to these forms. Lastly, two highly ornamented forms, Micro- ceras (t. r. of 0.32) and Otohoplites (t. r. of 0.39) were included. The specimens ranged in diameter from 3.5 to 5.8 cm. Molds of all specimens were made using Dow Corning RTV 3110 molding compound. Models were then cast using fiberglass resin.

Previous studies of drag on cephalopod shells (Kummel and Lloyd 1955; Chamberlain 1976, 1980, 1981), assumed that the planispiral coil was oriented parallel to flow with the aperture pointed backward. Backward direction of the jet is the norm for all modern cephalopods engaged in rapid jet propulsion. However, Nautilus does swim forward when engaged in slow search for food (Tschudy 1989), and squid can swim both forward and backward. Despite these limited observations of forward swimming, it is assumed in this study that the aperture faces backward.

All specimens were oriented with the aperture at approximately 550 from the vertical, that is, 350 below the horizontal. Extended bodies were constructed of modeling clay and attached to the aperture. The bodies were tapered from the aperture margin and oriented in the direction of flow. Constant taper should insure that the added bodies have approximately the same hydrodynamic effect. These models are comparable to the model D used by Chamberlain (1980), which, due to its orientation in the direction of flow, produced his greatest reduction in Cd. Because of the differences in size of the model and shape of the aperture, different amounts of clay were


Crossing Points

0.6 0.9* 0.6-

\ * Sphenodiscus \ * Cardioceras * Oppelia *Gastroplites 0.8 * Lytoceras *Scaphites

0.5 0.5

* ~~~~~~~~0.7

0.4 = f - 0.6 0.4 *

00 ~~~~~~0.5- U 0

0.3 *- 0.3 *o .

0.4-

0.2 'A0.3 A0.2 Al 0 10000 20000 30000 0 10000 20000 30000 0 10000 20000

Re

FIGURE 5. Comparisons of Cd-versus-Re curves for pairs of compressed and less compressed ammonoids of similar form reveal a crossing point (shown by the arrow on the Re axes) at Re of 5000-8000. At Re below the crossing point, the less compressed form has a lower Cd than the more streamlined compressed form.

required ranging from 5.7% to 18.5% of the volume of the specimen. Regression analyses show no obvious relationship between the volume percentage of clay added and thickness ratio or any of Raup's parameters, suggesting that this variation is unlikely to bias the results.

Cd versus Re

The Cd-versus-Re data for all specimens were graphed. The compressed forms, Spheno- discus, Cardioceras, and Oppelia, had descend- ing Cd, forming a hollow curve over the range of Re observed (fig. 4). A semilog regression produced a good fit to these data. This semilog approach was chosen because of the larger range Re (2 x 103-2.3 x 104) relative to Cd (0.2-0.7). In contrast to the compressed forms, and three less compressed specimens that lack ornament (Gastroplites, Scaphites, and Lytocer- as) had relatively constant Cd (in the 0.35 to 0.48 range) over the Re examined (fig. 4). Or- namented forms and depressed forms had higher Cd, ranging from 0.5 to 0.6.; they did not have the declining trend in Cd observed in the compressed forms.

Regression Analysis. -To determine whether declining Cd was significantly related to t. r., a semilog curve was fitted to the Cd- versus-Re data for each specimen using least squares. Slopes of these curves were then re-

gressed using the t. r. of the specimens as the independent variable. This regression had a positive slope of 0.995 and was highly significant (p < 0.001). The use of slopes as data in a regression analysis is no cause for concern; regression slopes are normally distrib- uted (Draper and Smith 1981). To allay possible concern about the use of t. r., a fraction, in the analysis, a log transformation of t. r. was regressed against the slopes of the Cd- versus-Re curves. This procedure also produced a highly significant positive slope.

These results indicate that for planispiral ammonoids there is a strong relationship between the slope of the Cd-versus-Re curve and the t. r. in the Re 103-104 region. This confirms a prediction based on the relationship between surface area and cross-sectional area. Compressed forms have declining Cd with increasing Re; less compressed and depressed forms do not.

Crossing Points. -Cd-versus-Re plots of compressed versus less compressed morphs intersect each other in the Re region investigated. The Cd-versus-Re curves of paired forms that differ primarily in t. r. (Sphenodiscus vs. Gastroplites, Oppelia vs. Scaphites, and Car- dioceras vs. Lytoceras) all cross in the region between Re of 5000-8000 (fig. 5). Below Re

5000, less compressed forms have lower Cds than compressed forms. Between Re

212 DAVID K. JACOBS

TABLE 1A. Power required, ergs/s/cm3, scaled to a set of sizes and velocities for Sphenodicus, Cardioceras, and Oppelia. Velocity is in cm/s; length is in cm.

Velocity Length

cm/s 1 cm 2.5 cm 5 cm 10 cm 15 cm 25 cm 50 cm 100 cm

Sphenodiscus 2.5 1.68 x 101 5.71 x 100 2.49 x 100 1.06 x 100 6.37 x 10-1 3.27 x 10-1 1.27 x 10-1 4.50 x 10-2 5.0 1.20 x 102 3.98 x 101 1.70 x 101 7.02 x 100 4.11 x 100 2.03 x 100 7.20 x 10-1 2.62 x 10-1

10.0 8.39 x 102 2.71 X 102 1.12 x 102 4.44 x 101 2.51 x 101 1.16 x 101 4.19 x 10? 1.70 x 100 15.0 2.60 x 103 8.23 x 102 3.32 x 102 1.28 x 102 6.91 x 101 3.09 x 101 1.25 x 101 5.09 x 100 25.0 1.07 x 104 3.27 x 103 1.27 x 103 4.50 x 102 2.39 x 102 1.23 x 102 4.97 x 102 2.02 x 101 50.0 7.05 x 104 2.03 x 104 7.20 x 103 2.62 x 103 1.55 X 103 7.96 x 102 3.23 x 102 1.31 X 102

100.0 4.46 x 105 1.15 x 105 4.19 x 104 1.70 x 104 1.01 X 104 5.17 x 103 2.10 x 103 8.51 X 102

Cardioceras 2.5 1.73 x 101 5.82 x 100 2.53 x 100 1.08 x 100 6.44 x 10-1 3.26 x 10-1 1.28 x 10-1 4.49 x 10-2

5.0 1.23 x 102 4.05 x 101 1.72 x 101 7.10 x 100 4.15 x 100 2.01 x 100 7.18 x 10-1 3.19 x 10-1 10.0 8.59 x 102 2.76 x 102 1.14 x 102 4.47 x 101 2.51 x 101 1.13 x 101 5.10 x 10? 2.55 x 100 15.0 2.67 x 103 8.35 x 102 3.35 x 102 1.27 x 102 6.88 x 101 3.40 x 101 1.72 x 101 8.61 x 10? 25.0 1.09 X 104 3.31 x 103 1.28 x 103 4.49 x 102 2.66 x 102 1.57 x 102 7.97 x 101 3.99 x 101 50.0 7.19 x 104 2.04 x 104 7.18 x 103 3.19 x 103 2.13 x 103 1.26 x 103 6.38 x 102 3.19 x 102

100.0 4.53 x 105 1.15 X 105 5.11 X 104 2.56 x 104 1.70 X 104 1.01 X 104 5.10 X 103 2.55 X 103

Oppelid 2.5 1.32 x 101 4.52 x 10? 1.98 x 100 8.52 x 10-1 5.14 x 10-1 2.67 x 10-1 1.06 x 10-1 3.92 x 10-2 5.0 9.41 x 101 3.17 x 101 1.36 x 101 5.71 x 100 3.37 x 10? 1.69 x 100 6.26 x 10-1 2.83 x 10-1

10.0 6.63 x 102 2.18 x 102 9.14 x 101 3.68 x 101 2.10 x 101 1.00 x 101 4.53 x 100 2.27 x 100 15.0 2.06 x 103 6.66 x 102 2.73 x 102 1.07 x 102 5.93 x 101 3.06 X 101 1.53 X 101 7.65 x 100 25.0 8.53 x 103 2.68 x 103 1.06 x 103 3.91 X 102 2.36 x 102 1.42 x 102 7.09 x 101 3.54 x 101 50.0 5.71 x 104 1.70 x 104 6.26 x 103 2.83 x 103 1.89 x 103 1.13 x 103 5.66 x 102 2.83 x 102

100.0 3.68 x 105 1.00 x 105 4.53 x 104 2.27 x 104 1.51 X 104 9.07 x 103 4.53 x 103 2.27 x 103

5000 and 8000, Cds in compressed and less compressed forms are comparable. Above Re

8000, compressed forms have lower Cds. Within the range of Re investigated, depressed morphologies have higher Cds than less compressed forms, and only lower Cds than compressed forms below Re 103. How- ever, depressed forms may have an advantage relative to all other forms at yet lower Re.

Power Scaling

As discussed in the introduction, the amount of power required per unit of volume for an organism of a given size to swim at a given speed may be a better comparative measure of swimming ability than Cd. Assuming isometric size change, power required can be determined over a range of sizes and velocities from the Cd-Re curves. Given the drag force of an object and its velocity, power consumed in drag will be the multiple of the two (power = force x distance/time or force x velocity). To calculate drag force (Df) an al- gebraic manipulation of the equation for Cd (Df = CdO.5ApU2) was used. From isometry

we can calculate a characteristic area A' at a new length from the area A of the original model:

(1' /I)2 x A = A', (3)

where 1 is the original length and 1' is the new length. Similarly, we can determine a new volume v' of our ammonite of length 1' using the cube of the ratio 1'/1. Given a new length and velocity and assuming constant viscosity, the Re can be determined. With the Re, Cd can be interpolated from the Cd-Re curve. (The semilog regressions fit to the Cd- Re data were used to determine Cd, up to Re of 30,000. Above 30,000, the Cd was assumed to be constant. This extrapolation is supported by the relatively constant Cd observed by Chamberlain at these Re (1976, 1980); however, this constancy may not be applicable in compressed forms if streamlining is important. Using this method, power per unit of volume was calculated over a range of sizes (1 cm to 100 cm in length), and swimming velocities (2.5 cm/s to 100 cm/s) for 7 of the 11 morphologies (table 1A,B). Ornamented


TABLE 1B. Power required, ergs/S/Cm3, scaled to a set of sizes and velocities for Lytoceras, Gastroplites, Scaphites, and Cadoceras. Velocity is in cm/s; length is in cm.

Velocity Length

cm/s 1 cm 2.5 cm 5 cm 10 cm 15 cm 25 cm 50 cm 100 cm

Lytoceras 2.5 8.19 x 100 3.29 x 100 1.63 x 100 8.07 x 10-1 5.35 x 10-1 3.19 x 10-1 1.58 x 10-1 7.83 x 10-2

5.0 6.49 x 101 2.61 x 101 1.29 x 101 6.40 x 100 4.25 x 100 2.53 x 100 1.25 x 100 6.25 x 10- 10.0 5.15 x 102 2.07 x 102 1.02 x 102 5.07 x 101 3.37 x 101 2.01 X 101 1.00 X 101 5.00 x 10? 15.0 1.73 x 103 6.94 x 102 3.44 x 102 1.70 x 102 1.13 x 102 6.75 X 101 3.38 X 101 1.69 x 101 25.0 7.95 x 103 3.20 x 103 1.58 x 103 7.83 x 102 5.21 X 102 3.13 x 102 1.56 x 102 7.82 x 101 50.0 6.31 X 104 2.53 x 104 1.25 x 104 6.25 x 103 4.17 x 103 2.50 x 103 1.25 x 103 6.25 x 102

100.0 5.00 x 105 2.01 x 105 1.00 x 105 5.00 x 104 3.33 x 104 2.00 x 104 1.00 X 104 5.00 x 103

Gastroplites 2.5 5.71 x 10? 2.59 x 10? 1.18 x 10? 6.26 x 10-1 4.13 x 10-1 2.45 x 10-1 1.20 x 10-1 5.89 X 10-2 5.0 4.94 x 101 2.04 x 101 9.30 x 10? 4.92 x 10? 3.25 x 10? 1.92 x 10? 9.43 x 10-1 4.70 x 10-1

10.0 3.53 x 102 1.60 x 102 7.31 x 101 3.87 x 101 2.55 x 101 1.51 x 101 7.51 x 100 3.75 x 100 15.0 1.18 x 103 5.35 x 102 2.44 x 102 1.29 x 102 8.51 x 101 5.07 x 101 2.53 x 101 1.27 x 101 25.0 5.40 x 103 2.45 x 103 1.11 X 103 5.89 x 102 3.91 X 102 2.35 x 102 1.17 x 102 5.86 X 101 50.0 4.24 x 104 1.92 x 104 8.75 x 103 4.69 x 103 3.13 x 103 1.88 x 103 9.39 x 102 4.69 x 102

100.0 3.33 x 105 1.51 x 105 6.97 x 104 3.75 x 104 2.50 x 104 1.50 x 104 7.51 X 103 3.75 x 103

Scaphites 2.5 5.67 x 10? 2.15 x 10? 1.13 x 100 5.67 x 10-1 3.78 x 10-1 2.27 x 10-1 1.13 x 10-1 5.67 x 10-2

5.0 4.53 x 101 1.72 x 101 9.07 x 100 4.53 x 10? 3.02 x 10? 1.81 x 10? 9.07 x 10-1 4.53 x 10-1 10.0 3.63 x 102 1.38 x 102 7.26 x 101 3.63 x 101 2.42 x 101 1.45 x 101 7.25 x 100 3.63 x 100 15.0 1.22 x 103 4.64 x 102 2.45 x 102 1.22 x 102 8.16 x 101 4.90 x 101 2.45 x 101 1.22 x 101 25.0 5.67 x 103 2.15 x 103 1.13 x 103 5.67 x 102 3.78 x 102 2.27 x 102 1.13 x 102 5.67 X 101 50.0 4.53 x 104 1.72 x 104 9.07 x 103 4.53 x 103 3.02 x 103 1.81 X 103 9.07 x 102 4.53 x 102

100.0 3.63 x 105 1.38 X 105 7.26 X 104 3.63 X 104 2.42 X 104 1.45 X 104 7.25 x 103 3.63 x 103

Cadoceras 2.5 8.30 x 100 3.32 x 100 1.66 x 100 8.30 x 10-1 5.53 x 10-1 3.32 x 10-1 1.66 x 10-1 8.30 X 10-2 5.0 6.65 x 101 2.66 x 101 1.33 x 101 6.56 x 100 4.43 x 100 2.66 x 100 1.33 x 100 6.65 x 10-1

10.0 5.32 x 102 2.13 x 102 1.06 x 102 5.32 x 101 3.54 x 101 2.13 x 101 1.06 x 101 5.32 x 100 15.0 1.79 x 103 7.18 x 102 3.59 x 102 1.79 x 102 1.20 x 102 7.18 x 101 3.59 x 101 1.79 x 101 25.0 8.31 X 103 3.32 x 103 1.66 x 103 8.31 X 102 5.54 x 102 3.32 x 102 1.66 x 102 8.31 x 101 50.0 6.65 x 104 2.66 x 104 1.33 x 104 6.65 x 103 4.43 x 103 2.66 x 103 1.33 x 103 6.65 x 102

100.0 5.32 x 105 2.13 x 105 1.06 x 105 5.32 x 104 3.54 x 104 2.13 x 104 1.60 x 104 5.32 x 103

forms, and forms where the Cd-versus-Re curve is not well fit by the semilog regression, were excluded from the analysis.

Comparison of Power Required. -The power required for the compressed forms (Oppelia, Sphenodiscus, and Cardioceras) are nearly identical; Sphenodiscus does not have a clear advantage over the other compressed forms. Be- low Re 30,000, this result is supported by the data; above Re 30,000, it may be a consequence of the extrapolation employed. Thus, if oxyconic forms have a profound advantage in power required for constant swimming, it must be in the Re region above 30,000, which is poorly constrained by these data.

A Sphenodiscus 10 cm in length requires the same amount of power as a Gastroplites 10 cm in length when they are both swimming at 15 cm/s; below this speed, the Gastroplites re-

quires less power per unit of volume; above this speed, the Sphenodiscus requires less power. At this size and velocity, the Gastroplites and Sphenodiscus have a Re 15,000, about twice the Re of the crossing point of their Cd- Re curves. If we look at the diagonals in table 1A and 1B we find that the relative efficiency of these forms crosses over at about this same multiple of velocity and size, that is at about the same Re of 15,000. In the upper left-hand portion of table 1B, Gastroplites is more efficient; in the lower left of table 1A, at larger multiples of velocity and size, Sphenodiscus is more efficient (see fig. 6). The Re-versus-Cd crossing points of the Cardioceras versus Ly- toceras and the Oppelia versus Scaphites are similarly elevated. Crossing points for power required occur in the region of Re of 10,000- 15,000 rather than in the region of 5000-8000

214 DAVID K. JACOBS

Gastroplites-Sphenodiscus

100 100

- ,j> ,>, ,,r,, >l, X _ 50 00C

501 |= i5 50 e's ,,j>,,, j, ? >X-3X_~5 0 XX ~~0

0

-50 E -50~~~~~~~5 00

-100 au~~~~~~~~~~0 100 25

Velocty cm/s 1052 25 Size cm

FIGURE 6. This surface represents the power per unit of volume required for swimming in Gastroplites minus the power per unit of volume required for swimming in Sphenodiscus. When values are negative, as they are at low multiples of velocity and size, then Gastroplites has a lower power required per unit of volume; at high multiples of size and velocity, values are positive and Sphenodiscus is preferred. The declining power required per unit of volume at large size and low velocity gives a saddle shape to the surface. At combinations of large size and low velocity, the power required per unit of volume is small and so are power differences. At small sizes and at high velocity, values are extreme and the surface was trun- cated at a power difference of 100 ergs/s/cm3.

for Cd crossing points. Thus, if one looks at efficiency in terms of power required per unit of volume, the larger relative volume of less compressed forms has an effect; the crossing points occur at a higher Re.

Cadoceras, a depressed form with a deep umbilicus, requires less power than the oxycone, Sphenodiscus, at Re z 5000. Thus, at modest sizes and swimming speeds, even ful- ly depressed forms have lower power requirements than do compressed forms. Inter- mediate less compressed forms with involute whorls, Gastroplites and Scaphites, require substantially less power than Cadoceras. The evolute Lytoceras, the intermediate t. r. form most comparable to Cadoceras, approaches it in power required at 1 cm in size and 2.5 cm/s. Thus, depressed forms may become advantageous at Re < 103. Less compressed forms with a t. r. of 0.3-0.5 will have the lowest power required in a Re region of 1000-15,000. At Re above 15,000, compressed forms are preferred.

Power Available. -In order to calculate an actual swimming speed using the power-required curves, some notion of power available is needed. The power available for swimming has been empirically determined for a few squid (O'Dor 1982; Webber and O'Dor 1986) and for Nautilus (Redmond 1987; O'Dor and Wells 1990) by examining oxygen consumption rates. The use of oxygen consumption to calculate power available requires a number of assumptions. However, the curves for power required rise rapidly with increasing velocity. Thus, large variation in the estimate of power available will produce much smaller differences in the velocity determined.

The proportion of metabolic activity available for swimming (metabolic scope) is calculated by subtracting the rate of oxygen consumption at rest from the rate of oxygen consumed during maximum sustainable ex- ertion. Metabolic scope relates to a maximum aerobic or sustainable power supply; anaerobic metabolism may transiently produce greater power resulting in brief "bursts" of even faster swimming. In modern cephalopods, metabolic scope varies greatly. A value of 95 ml/ kg/h of oxygen is quoted for Nautilus (O'Dor and Wells 1990; lower values have also been published by Redmond [1987]). Es- timates of metabolic scope for coleoids are around 410 ml 02/kg/h (O'Dor 1982) or higher (Webber and O'Dor 1986). Coleoids are more closely related to ammonoids than Nau- tilus (Berthold and Engeser 1987), and all aspects of squid life history are speeded up (e.g., they have extremely high metabolic rates and short life spans relative to other marine taxa). Ammonoids appear to have longevities intermediate between Nautilus and squid (Wes- termann 1971). In this work, an ammonoid metabolic scope of 200 ml 02/kg/h, about twice that of Nautilus and half that of squid, is assumed. Conversion of oxygen consumption to power/unit volume (0.56 ml O2/kg/s x 20 joules/ml 02 X 107 ergs per joule x 1000 g/kg = 11,000 ergs/s/cm3) results in 11,000 ergs/s/cm3.

Froude and muscular efficiency must also be factored in. Froude efficiency is a consequence of the relationship of propulsive force


Maximum Sustained Swimming Velocity

so so

A B C

40 40

30 30 E

20 -20 o Cadoceras

. if 0 Gastroplites 0 Scaphites 0 Lytoceras

10- A Sphenodiscus A Oppellia Cardioceras 10

0. .....I10 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 50

Size (cm)

FIGURE 7. Maximum sustainable swimming velocity (MSV) is presented as a function of size for the pairs of less compressed and compressed ammonoids: Gastroplites and Sphenodiscus (A), and Scaphite and Oppelia (B). The umbilicate forms Cadoceras, Lytoceras, and Cardioceras (C) are also graphed together. MSV was determined by interpolation of the power-required data using a maximum power available figure of 400 ergs/s/cm3. Note that differences among the velocities of all the forms are not large and that curves for compressed and less compressed forms cross between 5 and 10 cm in size. Multiples of size and velocity above 300 represent extrapolations beyond the Re range of drag measurements made.

to energy. In swimming, momentum is produced by the propulsion of a mass (m) of water at some velocity (U) in the opposite direction. Energy = m/2MU2; to minimize energy use or power consumption while maximizing the production of momentum, the squared velocity term must be minimized; the largest mass of water possible must be accelerated to just slightly faster than the swimming speed. In cephalopods, the mantle cavity size limits the mass of water that can be propelled, lim- iting Froude efficiency. The mantle cavity of Nautilus is the smallest, relative to body size, of any living cephalopod (Chamberlain 1990). Other cephalopods have larger mantle cavi- ties and Froude efficiencies. A Froude efficiency of 30%, in the middle of the range of values calculated for cephalopods, was chosen (O'Dor 1982; Wells 1987; Chamberlain 1990). Estimates of muscular efficiency in cephalopod swimming range from about 11% to 25% (O'Dor 1982; Wells 1987); a value of 20% was used here. Incorporating these factors reduces the power available to 660 ergs/ s /cm3.

Ammonoid soft parts were contained in the body chamber of a larger-volume shell. Pow- er required was calculated for the gross volume of the shell and body; consequently, power available must be scaled to the pro-

portion of soft parts in the whole ammonoid. Volume equations for the body chambers of planispires (Raup 1966; Raup and Chamber- lain 1967) can be used in conjunction with shell thickness estimates (Westermann 1971) to estimate the ratio of body mass to total volumes. This varies in a range of approximately 0.5 g/cm3-0.7 g/cm3 total volumes. The 660 ergs /s/ g was corrected by an average factor of 0.6 g/cm3 to generate the value of 396 or 400 ergs /s/cm3.

In addition to the other assumptions, it is assumed that the maximum power available, or metabolic scope, does not vary with changing volume. This appears to be the case for the few fish so far examined (Brett 1965; Pe- ters 1983). Too few data are available to sub- stantiate or refute this point in cephalopods.

Results.-Curves of power required versus velocity were interpolated at 400 ergs/s/cm3 for each size for the seven ammonoid forms examined, yielding the maximum sustainable swimming velocity (MSV). Maximum sustainable swimming velocity varies dramatically with size (fig. 7). At 1 cm in length, compressed forms had a MSV of 7 to 8 cm/s; less compressed forms had a MSV of 10 cm/s. At 100 cm in length, compressed forms had a MSV of 55-58 cm/s; less compressed forms 41 to 46 cm/s.

216 DAVID K. JACOBS

There are multiple uncertainties in the es- timation of power available. However, the steepness of the power-required curves makes the calculation of MSV relatively insensitive to error in the estimate of power available. Doubling or halving power available changes MSV by 15%-20%. If the coleoid metabolic scope were correct for ammonoids, MSV would be 15%-20% greater. If the Nautilus metabolic scope is appropriate, MSV would be 15%-20% lower than those recorded in figure 7. Thus, these calculations provide a rough estimate of maximum power available. It is unlikely that the uncertainties in the estimate would covary to produce power available many times larger than that calculated here. However, organisms can always devote energy to other activities than swimming and the tissues required for propulsion; consequently, the power available could always have been lower than the calculated values.

Discussion

Maximum sustainable velocity (MSV) increases rapidly with size, and curves for less compressed forms cross those for compressed forms between 5 and 10 cm in length (fig. 7). Despite the trends and crossing patterns in the data, there is not a great deal of difference between the MSV of different forms. Cham- berlain's (1981) work, based on drag observation and analogy to swimming in Nautilus, concluded that there were much larger differences in maximum velocity between the best and worst swimmers and did not predict that maximum swimming velocity for different shapes would cross with changing size. This work supports Chamberlain's (1981) conclusion that the swimming velocities attained by ectocochliate cephalopods are only a fraction of those attained for fish of comparable size.

Relative Efficiency. -Small differences in MSV distract one from the large differences in relative efficiency at somewhat lower velocities (table 1A,B; fig. 6). Thus, at a length of 15 cm and a speed of 15 cm/s, a less compressed form like Gastroplites required 23% more power per unit of volume, and at a length of 25 cm and a speed of 25 cm/s nearly twice as much power per unit of volume as

a compressed form such as Sphenodiscus. As a consequence of its larger volume, the Gastro- plites would have expended a much larger total amount of energy. On the other hand, at 5 cm in length, the less compressed Gas- troplites could have sustained swimming speeds of 17 cm/s, just a bit faster than Spheno- discus; and at velocities of 5 and 10 cm/s, the 5-cm Gastroplites would have required only 55% to 65% of the power per unit of volume.

In the range of sizes (5-15 cm in diameter) that encompasses the majority of adult ammonoids, the t. r. that confers the lowest power requirements will range from about 0.2 to 0.5. The precise optimum depends greatly on the particular range of swimming behaviors that are selectively important to the ammonoid. Different species of ammonoids may have had different modes of life. Some ammonoids may have swum at high, but sustainable, velocities for periods of time. Other ammonoids may have indulged in brief periods of even higher-velocity swimming where they relied on anaerobic resources. Such bursts might have occurred during prey capture or avoidance of predators. O'Dor (1988) has recently emphasized the cost of transportation in squids that must migrate to spawning areas on limited energy resources. The need to minimize the cost of transportation may have confronted forms that have to search out widely-spaced ephemeral resources. Wells (1987) has argued that Nautilus may be adapted to low-energy transport per- haps for this purpose. Thus, a wide range of Re specialization may have been possible. Some forms could have specialized for a particular Re. Others may have been generalists, moderately efficient over a wide range of sizes and velocities. Given this multiplicity of possible modes of life, the range of forms observed with t. r. of 0.2 to 0.5 is expected from an adaptive scenario. It is not evidence for a lack of hydrodynamic adaptation.

Oxycones.-Within the range of Re examined in this study, oxycones are not superior in terms of lower drag or power required than other moderately compressed forms. How- ever, at the higher Re associated with larger sizes, compressed ammonoids in general have an increasing advantage in swimming effi-


ciency relative to other forms. Oxycones could have an additional hydrodynamic advantage over other forms at Re < 30,000 if they behave as streamlined forms. In streamlined forms with t. r. similar to that for oxycones (fig. 2), Cd declines about 50% for each order of magnitude increase in Re over 104 (Hoerner 1965). Oxycones in excess of 30 cm in length may have attained Re of 105 where their tapered "streamlined" form could contribute to lower power required. In addition, if their slender shape minimizes the added mass of water that hinders acceleration (Daniel 1984), they may have had the advantage of more efficient acceleration.

Effect of Umbilicus. -Evolute forms, such as Lytoceras, had a slightly lower MSV than the comparable less compressed forms. This is not the case for compressed forms with open coiling. The evolute Cardioceras is virtually identical in Cd, power required, and MSV to the similarly compressed Sphenodiscus or Oppelia. Thus, the shallow umbilicus of evolute but compressed forms does not produce the drag of evolute forms with round whorl sections such as Lytoceras or the evolute models with round whorl sections documented by Cham- berlain (1976, 1981). This implies that many evolute and even serpenticonic forms will be able to swim with moderate efficiency as long as their umbilicus is not deep or well defined.

Early Ontogeny. -Immediately after hatching, ammonoids must have swum at low Re. Ammonoids hatched at a size range of 0.6- 1.5 mm (Landman 1988). At low Re, drag is a linear function of both area and velocity (Stokes law). Power available and propulsive force should scale with volume. As a consequence of the relationship between area and volume, MSV in small ammonoids should decline in proportion to linear dimension. Thus, at 1 mm, an ammonoid is likely to have had a MSV of one-tenth that of its MSV at 1 cm. The analysis of power required versus power available suggests that MSV at 1 cm must have been less than 10 cm/!s; consequently, a MSV of approximately 1 cm/s pertained at hatching in ammonoids. This corresponds to a Re of 10, where spherical shapes that minimize surface area and maximize volume will be optimal.

Ontogenetic Change in Thickness Ratio 1.6

1.4 * Placenticeras

_ Lytoceras 0 t 1.2-

0 1.0 0

X 0.8 t 0.6

0.46

0.2 0 5 10 15

Shell diameter (mm)

FIGURE 8. Ontogenetic change in thickness ratio for two ammonoid genera, Placenticeras and Lytoceras (data from Smith 1898, 1900). Ammonoids hatch at about 1 mm in size. Consequently, immediately after hatching, they must have swum at low Reynolds numbers, where the optimal thickness ratio approaches 1.0. With growth to 10 mm, optimal thickness ratio declines to 0.3-0.5. These plots of thickness ratio in early ontogeny suggest that ammonoids conformed to this changing optimum during post-hatching growth.

During growth from 1 mm (hatching) to 10 mm, Re should have increased by approximately two orders of magnitude to approximately 103. With this change in Re, optimal shape should change from spherical to a t. r. of 0.3-0.5 documented in this study. Thus, if there were selection for hydrodynamic efficiency, t. r. should change dramatically in early ontogeny as diameter increases from 1 mm to 10 mm. In Placenticeras pacificum and Lytoceras, t. r. falls from about 1 (spherical) at hatching, to values of 0.4 and 0.3, respectively, at a diameter of 6 mm (fig. 8).

Despite the similar change in t. r., different combinations of coiling parameters are involved. In Placenticeras, which has large whorl expansion, increasing compression of the whorl (smaller S) results in a decline in t. r. through early ontogeny. In an evolute serpenticonic form such as Lytoceras, small whorl expansion results in a declining t. r. through ontogeny as the coil increases in diameter faster than in thickness. This ontogenetic change in shape is a consequence of coiling parameters alone, rather than a change in whorl shape. Thus, evolute forms with a small whorl expansion W and slight whorl com-

218 DAVID K. JACOBS

pression may track the hydrodynamic optimum through ontogeny with minimal ad- justment in coiling parameters. This may explain the large peak in Mesozoic ammonoid forms with small whorl expansion and open coiling observed in W versus D morphospace plots (Raup 1967).

Because of their larger size at hatching (1 cm in modern Nautilus and over 2.5 cm for fossil forms; Landman 1988) nautilids start life swimming at Re 103 with an optimal t. r. of 0.3 or 0.4. Nautilids do not have to track a hydrodynamic optimum through early ontogeny. This may account for the involute coiling and larger whorl expansion of Jurassic to Recent nautilids relative to Mesozoic ammonoids (Ward 1980; Chamberlain 1981).

In some goniatites, such as Goniatites or Paracravenoceras, there is early reduction in t. r. during ontogeny as a consequence of the openness of the coil (Smith 1897; Raup 1967). Subsequently, the whorl widens and becomes involute, producing the more depressed adult shape. This ontogenetic pattern of shape change suggests that these goniatites had an early ontogenetic stage where they responded to the optimal shape dictated by hydro- dynamics. After this period, they ceased to follow this hydrodynamic optimum and be- came depressed and involute. This departure suggests that swimming efficiency ceased to be a dominant selective force at this point in their ontogeny. Thus, the relationship between hydrodynamic optima and actual shape change in ontogeny may provide one line of evidence for changes in the mode of life.

Environmental Evidence. -Independent support for the relationship among large size, lateral compression, and higher sustained swimming velocities may be available from paleoenvironmental evidence. Coarse-grained deposits suggest higher current velocities than fine-grained deposits such as shales. Higher current velocities may have required more rapid swimming on the part of the ammonoids present. In shallowing, coarsening-up- ward cycles in the Cretaceous Seaway of North America, planispiral ammonoids preserved in the coarser-grained sediment tend to be larger and more compressed than those found lower down in the finer-grained portion of the cycle. In the Pierre Shale, scaphites in-

crease in size and lateral compression as one goes up section into the silty Elk Butte Mem- ber; and in the overlying sandy Fox Hills For- mation, large Sphenodiscus are present. In addition, between the Trail City member of the Fox Hills Formation and the overlying coarser-grained Timber Lake member, both the Cosmoscaphites and the Hoploscaphites lineages become more compressed (Landman personal communication). In the underlying Green- horn Cyclothem, Batt (1989) observed that laterally compressed forms were dominant in near-shore coarse-grained settings. Repeated cycles of evolution from smaller to larger more compressed forms are also reported from lineages in the Jurassic of the German Basin during periods of shallow water (Bayer and McGhee 1984). Lineage studies minimize variation from differing modes of life. Thus, lineages that transcend environmental boundaries during the evolution of basins may provide the best evidence for a correlation between shell shape and environmental energy. More studies of such lineages are needed.

Restriction to Epeiric Seas. -Lytoceratids and phyloceratids were long-ranging taxa and ev- olutionarily less volatile than other ammonoids. They are found primarily in deep-water deposits and are presumed to have lived in the water column. Other ammonoids are often associated with particular facies, suggesting that they descended to the bottom to forage or were otherwise closely associated with the benthic environment. It is these groups of ammonoids that exhibit great evolutionary volatility. Ammonoids that lived near, and depended on, benthic resources would have had to maintain their position relative to the bottom. Ammonoids were neu- trally buoyant; consequently, remaining stationary in a current required active swimming. Remaining stationary would have been extremely difficult for small adult and juvenile ammonoids with diameters of less than 5 cm. At 2.5 cm, none of the ammonoid shapes could have sustained speeds of more than 15 cm/s. Even maintaining swimming speeds of 10 cm/s would have been very taxing. On most shelves today, there are tidal or geo- strophic flows in excess of 10 cm/s. These would tend to transport ammonoids away


from their benthic habitats, often off the shelves. Nektobenthic behavior in ammonoids may depend on the presence of suffi- ciently large regions characterized by shallow water and low current velocity where transport out of the general environment would be unlikely. These conditions are most likely to have occurred in epeiric seas. Epeiric seas were subject to frequent anoxic events that testify to their relatively low current velocities. Epeiric seas were ephemeral environments over geologic time; eustatic, sea- level changes emptied them, and sedimen- tation filled them in. This frequent elimina- tion of preferred benthic habitat may have contributed to the evolutionary volatility of the nektobenthic ammonoid groups that tended to occupy shallow basins. The great evolutionary declines of ammonoids, at the end of the Devonian, Permo-Triassic, Rhaetic, and Maastrictian, are all associated with low sea-level stands and progradational filling of epeiric seas.

Conclusions

There is a statistically significant relationship between the slope of the Cd-versus-Re curve and the thickness ratio (t. r.) of ammonoids at Re between 103 and 104. With increasing lateral compression, there is an in- creasingly steep negative slope of the Cd curve with increasing Re. At Re above 8000, compressed forms (t.r. < 0.3) will be preferred; at Re below 5000 less compressed forms will have lower Cd. Because of their lower volume, compressed forms did not have substantially lower power requirements per unit of volume until they achieved a Re of 15,000. At lower Re, thicker, less compressed forms have substantially lower power requirements per unit of volume.

In ammonoids 5-15 cm in diameter, a large range of Re could have selective importance depending on the swimming velocity of the ammonoids in question. A range of optimal t. r. of 0.2-0.5 is possible. Larger forms traveling at higher velocities are more efficient if they are more compressed. Smaller ammonoids traveling at lower velocities are at an advantage at intermediate thickness ratios. Thus, one ammonoid shape is not universally better adapted for swimming than another.

Other information regarding the swimming behavior of the ammonoid would be required to determine an optimal morphology. Only depressed forms, forms with large, deep um- bilici, or heavily ornamented forms, were uniformly at a disadvantage relative to other forms in the range of Re examined in this study. Maximum sustainable swimming velocities, estimated using oxygen consumption rates of modern cephalopods, indicate that swimming speed increases rapidly with size and confirm that shelled cephalopods swam at substantially lower speeds than fish. Ox- ycones do not have a substantial advantage in drag or power requirements over other laterally compressed forms in the range of Re examined.

When ammonoids hatched, they operated at very low Re because of their small size. At this point in ontogeny, the optimal shape ap- proached a sphere. As ammonoids grew, their optimal t. r. changed from 1 to a more compressed morphology. Many ammonoids show increasing compression of the shell during early ontogeny, suggesting that they track this optimal shape. Nautilids hatch at larger sizes avoiding the large changes in optimal shape associated with swimming in smaller juvenile ammonoids. This may account for differences observed in the coiling parameters of ammonoids and nautilids.

Larger, slightly more compressed forms would have been able to sustain higher swimming speeds for longer periods of time. Con- sequently, larger more compressed forms should have been preferred in high-energy environments where an ability to swim faster than the ambient current was critical. Smaller and juvenile ammonoids could not have overcome even moderate current velocities and may have been restricted to low-energy epeiric seas. The evolutionary volatility of some ammonoid groups may be a consequence of the ephemeral nature of conditions in epeiric seas.

Acknowledgments

This research was conducted at Virginia Polytechnic Institute and State University and represents a portion of my Ph.D. dissertation. I would like to thank my committee, R. Bam- bach, E. Benfield, N. Gilinsky, N. Landman,

220 DAVID K. JACOBS

and D. Porter for their support. I thank T. Baumiller, M. LaBarbera, and S. Vogel for suggestions regarding flow-tank construction and operation. I am also indebted to B. Ben- nington, M. Langer, D. Lindberg, and J. Mi- yashiro for assistance with the manuscript, and to T. Baumiller, J. Chamberlain, and 0. Ellers for their detailed reviews.

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Documents

Shape, Drag, and Power in Ammonoid Swimming...Paleobiology, 18(2), 1992, pp. 203-220 Shape, drag, and power in ammonoid swimming David K. Jacobs Abstract.-This study assesses swimming