E4 Energy Cost of Swimming (COT)

Topic

The cost of transport quantifies the metabolic energy a fish must expend to move a unit of its own mass over a unit of distance, and it serves as the central thermodynamic indicator of the fish's locomotor efficiency. Unlike terrestrial mammals—for whom the cost of transport includes the work done against gravity—in fish, this cost primarily reflects the work performed against hydrodynamic drag and the energy losses incurred while generating the propulsive wake described in an earlier model in this series.

Tucker’s physical model describes this cost of transport as the sum of two terms: the fish's basal metabolism (which is independent of swimming speed) and a mechanical power term that increases with the cube of that speed. This second term rises so rapidly because the power expended against drag is the product of drag force and swimming speed; since drag force itself increases with the square of the speed, the product of the two—force and speed—ultimately increases with the cube of the swimming speed.

When this total power is divided by the swimming speed to obtain the cost of transport per unit distance, a minimum appears in the resulting curve, defining the fish's optimal cruising speed; geometrically, this optimal speed corresponds to the point where a straight line drawn from the origin is tangent to the total power curve. At speeds below this optimum, the cost of transport is dominated by basal metabolism, which remains constant while the distance covered per unit of time is low; at higher speeds, however, drag dominates, as its quadratic increase with speed eventually outweighs the constant basal metabolic rate. Furthermore, the cost of transport scales with the fish's body mass according to a power law: larger fish have a lower specific cost of transport—that is, cost per unit of mass—than smaller fish, and this energetic advantage is precisely what enables larger species to sustain long-distance migrations.

The overall performance of the system converting chemical energy into swimming—the product of the efficiency with which fish muscle converts chemical energy into mechanical work and the hydrodynamic propulsive efficiency described in the vortex-propulsion model in this collection—ultimately determines what fraction of the chemical energy stored in muscle adenosine triphosphate is converted into useful propulsive work, rather than being lost as heat or kinetic energy deposited in the wake.

The expected behavior is that the cost of transport reaches its minimum value—at the optimal cruising speed mentioned earlier—at a rate of roughly half a body length to one body length per second; typical cost-of-transport values ​​range from one-tenth of a joule to five joules per kilogram of displaced mass per meter traveled, depending on the fish species and the water temperature.

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