FISHERY BULLETIN: VOL. 78, NO. 1 



of water, even though that mass is convected on a 

 much larger scale. 



The motionless larva and its surrounding water 

 mass may therefore be analyzed separately, as a 

 distinct system in the thermodynamic sense. 

 Within this system, oxygen transport to the larva 

 is controlled by molecular diffusion because the 

 gill system is not developed at this stage. This 

 process is time dependent, beginning when the 

 larva arrives in a certain location (by swimming) 

 and rests, ending when swimming begins again. 

 Initially the oxygen concentration in the water 

 mass surrounding the larva is uniform, but the 

 larva now starts acting as an oxygen sink, gradu- 

 ally depleting the oxygen content of the water 

 surrounding it. This concept of the larva as an 

 oxygen sink simplifies the calculations, as knowl- 

 edge of the exact distribution of oxygen diffusivity 

 on the animal's surface is not required. The sink 

 model also is useful here as it averages out the 

 direction of local transport and the body of the 

 larva into which the oxygen diffuses can be taken 

 as an equivalent sphere of equal surface area 

 (Figure 1). 



Diffusion into a sphere is most conveniently 

 analyzed in the spherical coordinate system. The 

 governing conservation of mass equation can be 

 written (Crank 1975) as 



dt 



= D 



c 2 dc 



+ 



dr 



(1) 



where c is the mass fraction of oxygen (a function 

 of the distance and time); r is the radial distance, 

 measured for the center of the equivalent spheri- 

 cal body (the sink); t is the time; and D is the 

 diffusion coefficient of oxygen in seawater. 



The temporal boundary condition is the initial, 

 uniform state 



c(r,0) = c^ 

 while the spatial conditions are 

 c(^,0 =Co 



(2) 



(3) 



which states that far from the animal the oxygen 

 concentration stays unchanged at all times. 

 Strictly, the condition should be defined at some 

 finite distance but as that distance is much larger 

 than the animal equivalent radius, it can be ap- 



proximated by ^. Next, the oxygen concentration 

 boundary condition at the surface of the equiva- 

 lent sphere r = a is obtained. Muscles and vas- 

 cularized tissues have much higher oxygen trans- 

 port rates than seawater, due to internal uptake 

 augmented by active transport. Thus, oxygen will 

 be absorbed at the surface of the larva as fast as it 

 arrives by diffusion from the surrounding water. 

 The oxygen concentration c at the larva's surface 

 (r = a) is thus constant, and very low, i.e.. 



c (a, t) = Cj 

 where Cj->0 and ^>0. 



(4) 



Equations (2)-(4) enable solving equation (1) 

 analytically, by classical methods. The solution 

 can be written in nondimensional form for the 

 concentration as 



C-Cq 



ci -Co 



± erfc ^~° 



'' 2Vd7 



(5) 



where the complementary error function, erfc, is 

 defined as 



erfc(2) 



2 " — 22 



f e dz 



\/lT 



(6) 



Numerical values of the complementary error 

 function are found in most mathematical tables 

 (e.g., Abramowitz and Stegun 1965). 



The rate of mass transfer (flux) J to the animal 

 is now obtained from 



A O^ 



dA (surface A) 



(7) 



where p is the density and A the surface area of the 

 body. For the equivalent sphere of radius a,A- 

 Aira"^. dc/dr is assumed spherically symmetric so 

 that 



J = -pDA 



dc_ 

 dr 



(8) 



where the concentration derivative is obtained 

 from Equation (5). Substituting this, and the 

 value for the surface area, and setting c^ = as in 

 Equation (4), the total mass flux per unit time J^ is 



110 



