Fl.lKRLAN'U\VKOKLh;\VSKl:\VAKiVU'OKK(;ri.FSIKKAMKlN(;s 



dU _ ^0 



dx Y 



with the boundary condition 



iV(0,0 = iVo(0. 



STATIONARY EDDIES 



(8) 



(9) 



In nature, there are pulses of larvae entering the 

 domain as the fish spawn. In addition, the shelf-edge 

 velocities V^){x,t) are changing as mesoscale eddies 

 and Gulf Stream warm core rings impinge upon the 

 shelf. We shall present in the section on moving ed- 

 dies several numerical solutions of Equations (8) and 

 (9), simulating this complex situation. However, in 

 order to fully understand the importance of the rings 

 and eddies in determining the fish larvae's spatial 

 distribution, it is first useful to consider some 

 simpler, analytically tractable cases. We shall begin 

 by discussing the distributions which occur when the 

 shelf-edge flows are not changing with time, i.e., the 

 eddies are stationary. This problem also has bearing 

 on the real situation south of Long Island, where 

 rings may often stop for considerable lengths of 

 time. 



As a first example, consider the larval fish distribu- 

 tion which would occur in the absence of any biolo- 

 gical loss processes (f^ = 0) and when the source term 

 A^Q is independent of time. The resulting equations 



dx 



iUN) = 



N 

 Fo — foryo>0| 

 Y 



for F„ < 



au 



dx 



Y 



(10) 



can be solved readily 



where f/,, is the longshore velocity and N,, is the 

 (time-independent) population density at the 

 upstream boundary x = 0. We can now see explicitly 

 the effects of the physics alone upon the larval fish 

 distribution. In the regions where the flow is onto the 

 shelf (Equation (11a)), the shelf break boundary con- 

 tribution to Equation (10) is zero. But the effects of 

 the tlow field are still felt in that the along-shelf flow 

 is divergent. U increases downstream as water 

 comes onto the shelf, spreading out the larvae and 

 reducing their average density. In contrast, when 

 the flow is offshore (Equations (lib) or (lie)), there 

 are direct loss terms due to larvae being carried off 

 the shelf. Some of the water flowing into a section 

 are diverted offshore while some continues down the 

 shelf, with the larvae separating in the same propor- 

 tions. Thus, although there is a decreased flux down 

 the shelf, this does not affect the density since there 

 are no biological losses which need to be balanced by 

 this flux. The net effect is that the physics by itself 

 does not change the population density in regions of 

 offshore flow (Equation (lib)). The only exception 

 would occur when the offshore transport (/ F,, dx) is 

 sufficiently strong so that all of the normal 

 alongshore flow (f/,, Y) is diverted off the shelf. In 

 this case (Equation (lie)), the flow in regions farther 

 down the shelf is reversed and the water moves up 

 the shelf. Since this water is from regions without 

 . sources of larvae, the population density is zero. 



By putting together these two results, we can con- 

 struct a picture of the density of larvae in continental 

 shelf water flowing past a stationary ring centered 

 a.tx = D. This is shown in Figure 3. For these calcu- 

 lations we have used 



n = 



,x- D 

 ■A exp 



[2 2 L' ] 



(12) 



with A = 20 cm/s the peak offshore velocity) and L 

 = 20 km (so that roughly 80 km along the shelf is 

 strongly influenced by the ring currents). This figure 



A^ = 



'A^o 



Nn 



U 



N,a 



o"-^o 



C/n 



S 



Va 



Y 



if Fo < 



if Fo > and U^ > 



if F„ > and Uo < 



Va 



Y 



(11a) 



(lib) 

 (lie) 



317 



