FISHERY BULLETIN: VOL. 83, NO. 3 



dx 



[(U-c)N] = 



dU _ 

 dx 



-Fn 



V, 



N 



for Vo > 



for V. < 



(19) 



and the population change caused by the eddy when 

 the downstream flow is sufficiently faster than the 

 ring's translation rate is 



N(-oo) 



-;i 



V,{x,0) 



y„>o ^0 - c 



v.. dx 



dx, 



c<a 









(20) 



>o 



When the flow stagnates relative to the ring at some 

 point, we have 



N(oo) = U, 



Y 



i 



V,,dx<c<Uo (21) 



X) 



and when the ring is moving faster than the shelf 

 waters 



Ni-°o) - N(oo) 



N{oo) 



il 



Vo dx 



^ Jv„>o 



c - [/,, + - 



^ Jv.r->o 



OUa. 



\i 



Vq dx 



These are plotted in Figure 7. Notice that the ring 

 may cause substantial losses in the population, 

 especially when its speed is roughly matched to the 

 mean flow on the shelf. 



In principle, one could write down an analytical 

 solution to the full problem (Equations (8) and (i))), 

 including a translating ring and a time-dependent 

 source at the upstream edge of the domain. 

 However, this is a rather cumbersome calculation, 

 and we have chosen instead to solve these equations 

 numerically and simply present representative pic- 

 tures here of the ring induced effects when a cohort 

 -a single patch of larvae spawned roughly simul- 

 taneously - moves down the shelf and is disturbed by 

 a ring. Some care is necessary in selecting a 

 numerical scheme, since centered differencing is 



100 



\ loss ol 

 population 



^a--b\ Vo di 



X,>0 



PHASE SPEED C 



Figure 7. -Percent of the total number of larvae produced at the 

 spawning site which are ultimately lost when a moving eddy is pres- 

 ent at the shelf edge. When the speed c of the eddy is greater than 

 Uq - 1/Y /" Vq, all the larvae advected along the shelf are drawn 

 offshore. When the ring is moving faster than the down-shelf cur- 

 rent Uq, the ring catches up to the larvae, which are then diluted 

 before being drawn offshore. 



unstable while upstream differencing introduces a 

 numerical diffusivity (Roache 1972). This may not be 

 undesirable, since in reality one would expect some 

 mixing to occur along the shelf; but unfortunately 

 the diffusivity is spatially variable, being lowest in 

 the vicinity of the eddy, and this we do not want in 

 the model. We compromised by choosing a very 

 small grid scale (5 km) so that the effective diffu- 

 sivity is only 1.2 x 10*^ cm^/s. This is not completely 

 negligible, as shown in Figure 8a which plots suc- 

 cessive snapshots of the larval fish density at 15-d in- 

 tervals in the absence of any rings (the time step was 

 one-half day). The population at the beginning of the 

 domain is assumed to enter in a pulse 



(22) A^o(0 = exp 



(23) 



The gradual decrease in the peak abundance and the 

 spread in width is caused by the numerical diffusion. 

 Also included in this figure are the simple cases add- 

 ing biological decay (Fig. 8b), a stationary ring (Fig. 

 8c), or both simultaneously (Fig. 8d). Note the large 

 decreases in density induced as the population passes 

 the ring and also the slower advection of the popula- 

 tion down the shelf so that the organisms down- 

 stream are half a month older than they would other- 

 wise have been (compare Figure 8b and 8d). 



Finally, we show in Figure 9 two cases when a 

 moving ring interacts with a patch of larvae. In the 

 first case, the eddy is moving at 7 cm/s (faster than 

 the 5 cm/s drift rate of the shelf water), so that the 

 eddy catches up to the population and passes by it. 



322 



