and (3) vertical diffusion is Fickian with a constant vertical diffusivity (Okubo, 

 1974), how large a volume will the eggs occupy 24 hours later? Calculations 

 (Okubo, pers. comm.) are based on diffusion diagrams that plot horizontal variance 

 a^ (i.e., the mean square distance from the center of mass) of dye distribution as i 

 function of time since point release of dye in the ocean. Horizontal diameter of 

 the egg patch is considered to be 2a x , i.e., twice the standard deviation (Okubo, 

 1971), and vertical height of the patch is 2a z . After 24 hours, eggs would be 

 spread 3-9 m vertically and 400-1000 m horizontally. Mean separation between 

 neighboring eggs would be 24-71 cm vertically and 28-70 m horizontally for a 

 spawn of 2,000 eggs with zero mortality, and 9-26 cm vertically, 10-26 m horizon- 

 tally for 40,000 eggs. The average density of 40,000 eggs spread horizontally in a 

 circle 700 m in diameter, and neglecting vertical spread, would be 0.10 eggs/m 2 . 

 Assuming the eggs are dispersed at random within the circle according to a Poisson 

 distribution, the probability of two or more eggs occupying a 1 m? portion of the 

 circle is 0.004679. We could expect 1800 units, each 1 m? in size, within this 

 circle each to contain >_ 2 eggs. 



What factors operate to make aggregation of sibling eggs more or less likely 

 than this simple model suggests? Firstly, eggs probably do not behave like dye in 

 the ocean. While the chorionic surface of fish eggs may often be smooth (Hemple, 

 1979), 27 tropical species (e.g., Robertson, 1981) produce eggs with external 

 chorionic hooks, tendrils, spikes, or raised hexagonal ridges. In some cases 

 these structures hold fertilized eggs in compact sheets. In others, chorionic 

 sculpturing increases drag (Robertson, 1981). Female lionfishes of the family 

 Pteroidae produce transparent mucous balls containing 2,000-15,000 eggs that remain 

 in the mass until hatching 36 hours later (Fishelson, 1975). 



Turbulent motion is customarily broken into Fourier components with different 

 length scales. Components with length scales smaller than or equal to the egg 

 patch tend to break the patch and mix and spread the eggs. Components with length 

 scale larger than the patch move the patch as a whole (Okubo, 1971). "As a result, 

 the apparent power of the mixing which acts on a diffusing patch increases with 

 patch size or, to put it another way, with the time elapsed since the patch first 

 began to mix." (Okubo, 1971: 91). Empirically, horizontal variance is a function 

 of diffusion time raised to the power 2.34 (Okubo, 1974). Thus, any tendency for 

 newly spawned eggs to adhere to each other or otherwise remain in an aggregated 

 cloud would delay the time at which any particular eddy length became capable of 

 dispersing it and produce an inordinately large reduction of egg patch dimensions 

 24 hours later. 



Species in many families lay demersal eggs that have adhesive structures or 

 properties that maintain their benthic attachments (Thresher, 1980). Dispersion 

 does not begin for demersal spawners, for mouthbrooders such as cardinal fish and 

 jawfish, or for livebearers such as some clinids, until after hatching or live 

 birth. The period of completely passive egg dispersion is by-passed in these 

 species. 



The simple model predicted an egg density 24 hours after spawning of 0.10 eggs 

 per m?. Much higher egg densities have been recorded: up to 46,000 eggs/m 2 in 

 temperate fishes and up to 12 eggs/m? in tropical fishes (Hunter, 1980; Watson & 

 Leis, 1974). Inter-egg distances may be as low as 1-2 cm at spawning and 15-20 cm 

 several days later (Hunter, 1980). These distances are an order of magnitude 

 lower than those in the above model. While these field observations suggest that 

 egg clumping may be more likely than expected by the above model, very high egg 

 densities almost certainly result in part from superposition of egg products from 

 multiple spawns. I will return to the effect of multiple spawns on sibling 

 aggregation later. 



Unlike an assumption of the model, dispersion is not uniform throughout the egg 

 patch, but is Gaussian with densities greater at the center than at the periphery 



40 



