Abstract

To counter gravitational settling, planktonic foraminifera adjust their buoyancies, in part by manufacturing low-density lipids or gases. The biochemical energy that a foraminifer expends in this way is a function of the speed at which it would otherwise settle if it did not expend this energy. In turn, the settling speed varies with foraminifer shape. We consider here foraminifera that have acicular spines, for example Orbulina universa and Globigerinoides sacculifer. Growing spines produces two counteractive effects: spines increase the weight of a foraminifer, and therefore tend to increase its settling speed; they also increase the fluid drag on the foraminifer, and therefore tend to decrease its settling speed. If growing spines is part of an evolutionary strategy to impede settling, then it is reasonable to expect that the advantage of increasing drag by growing spines outweighs the disadvantage of increasing weight.

The complexity of foraminiferal shapes precludes directly solving the equations of fluid motion for drag and settling speed. We therefore appeal to the efficacy of dimensional analysis to define a coefficient of drag CD and a Reynolds number Re for spinose foraminifera. Experiments that involve settling scaled models of foraminifera (constructed from beeswax and pins) in viscous liquids are then used to confirm the forms of generalized dimensional formulae relating the settling speed W to test radius R, spine number n, spine length l, and spine radius r. Geometrically similar foraminifera whose spine arrangements possess quasispherical symmetry settle according to an inverse relation between CD and Re, homologous to Stokes's law for spheres. Fluid drag systematically increases with both n and l. For given R, l, and r, a minimum settling speed occurs at an intermediate spine number n0. Similarly, for given R, n, and r, a maximum settling speed W0 occurs at an intermediate spine length l0. Results suggest that insofar as there is disadvantage in settling rapidly, there is advantage in remaining small; or, if growth of tests occurs, there is advantage in manufacturing many long thin spines. Investments of mass and energy associated with this strategy must be weighed against those involved in achieving neutral (or positive) buoyancy by other mechanisms, and limitations on lengths of spines imposed by their finite strength. A comparison of the theory with modern foraminifera suggests that the geometries of adult Orbulina universa and Globigerinoides sacculifer, in the absence of external protoplasm, are well suited to impede settling. With external protoplasm, however, l is effectively decreased and the drag associated with spines is not sufficient to provide viscous settling unless the protoplasm possesses positive buoyancy. For an individual at or near a state of neutral buoyancy, drag associated with spines decreases the sensitivity with which its settling speed responds to unavoidable changes in the buoyancy of its protoplasm related to metabolic activity, and to changes in the density and viscosity of sea water related to external factors. The effect is to hydrodynamically dampen vertical motions that would otherwise occur if the individual did not possess spines. In contrast, the small drag associated with few short spines is advantageous to juveniles that must ascend from deep to shallow waters during their ontogenies.

A partitioning of finite spine mass into many moderate to short spines is less effective in producing drag than one involving fewer long spines. Long spines, however, are more susceptible to mechanical breakage due to the torque that viscous forces apply to them. Foraminifera with approximately 10 (or fewer) spines that possess mechanical properties equivalent to those of spines of adult Orbulina universa and Globigerinoides sacculifer can withstand motions at speeds of only a few tenths of a centimeter per second (or more, depending on spine strength) without breakage. With increasing numbers of spines, the resulting hydrodynamic interaction among them has the effect of significantly reducing the chance of spine breakage related to momentarily rapid motions; this is attributable to a decrease in the proportion of the spine length l exposed to significant viscous forces, whereby the torque on individual spines is decreased. External protoplasm also reduces the chance of breakage by decreasing the length of spines exposed to surrounding fluid.

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