FISHERY BULLETIN: VOL. 83, NO. 3 



^1 

 No 



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O 



z 

 < 

 o 



z 



m 

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0) 



H.; 





DISTANCE - 

 DOWNSTREAM 



X (KM) 



500 



20 

 OFFSHORE 



cm/s 



ONSHORE 

 -20 



Figure 3. -The steady state abundance of fish larvae with 

 distance aion^ the shelf. This abundance is expressed as a frac- 

 tion of the number of larvae continuously being produced at the 

 spawning site, Nq. There is a 80 km wide, stationary eddy at the 

 shelf edge, inducing onshore and offshore flows of 20 cm/s. The 

 longshore velocity Uq of the shelf water is 5 cm/s. Biological 

 losses (f.() are set equal to zero. The dotted line shows the steady 

 abundance of fish larvae with distance down the shelf when 

 there is no eddy present. 



shows only the spatial distribution of larval density in 

 the water moving down the shelf, as affected by flow 

 convergences or divergences associated with the 

 physics of the ring. The flux of larvae off the shelf 

 (not shown) is given by V',, hN,^ in the regions where 

 Vo is greater than zero and amounts in total to 



i 



Vq h N^^ dx ^ A hL N^e 



}k 



66% of the flux into the domain (L^,, A^,, hY) at x = 0. 

 Next, we shall see that the physics and biology ac- 

 tually interact to produce a greater net impact than 

 when either is considered separately. 



For this second model problem, we shall still use a 

 steady onshore and offshore flow pattern, but now 

 include the biological loss term and the time-depen- 

 dence in the source function Nq. When the flow is off- 

 shore or zero, the population distribution is given by 



N{x,t) = NS- T)e' 



(13a) 



whej-e the variable t measures the length of time 

 necessary to reach the point x from the upstream 

 edge of the domain. In general r is given by 



T = 



s 



' rfo:' 

 ., U{x) 



(13b) 



where U{x) can be found by integrating the mass 

 conservation equation 



U{x) = f/o- \ dx —. 



i 



(13c) 



In the absence of ring-induced onshore-offshore 

 flows (V,) = 0), however, t is just equal to xlU^^ and 



N{x,t) = iVo [t 



1 ilL\ 



(13d) 



The population at any downstream point lags that at 

 the origin by the travel time xIUq and has also 

 decayed exponentially during its travel. This solution 

 is an important base case for understanding the 

 distributions in a spawned patch which has not been 

 impacted by rings. 



When water is being drawn off the shelf, the along- 

 shelf decay in concentration is again purely due to 

 travel time, since the effect of losses off the shelf on 

 the density is compensated for by the convergence. 

 However the spatial density of the larvae is still 

 noticeably altered by the offshore flow because the 

 travel time necessary to reach any point is increased. 

 This occurs because U is decreasing with i- as a result 

 of the advection of water off the shelf (as shown in 

 Equation (13c)). Since U is less than Uq, the travel 

 time T in Equation (13b) is necessarily greater than 

 that in the absence of the ring (xlU^^). We have 

 sketched t(x) for the three possible signs of F,, in 

 Figure 4a. These results suggest that there will be an 

 enhanced spatial decay rate of larval density in the 

 regions where the flow is offshore. 



318 



