FISHERY BULLETIN: VOL. 83, NO. 3 



and z positive upwards. We shall assume that there is 

 a source of larvae somewhere upstream of x = 0, 

 e.g., the spawning grounds on Georges Bank, 

 leading to a specified flux of organisms into the 

 domain at the x = boundary. Alternatively, and 

 more conveniently, the upstream spawning can be 

 thought of as leading to a specified abundance of 



organisms A^q (^) ^^ ^ = ^• 



Although we could, in principle, specify the three- 

 dimensional currents and the source function Aq 

 from either measurements or models, the available 

 data and models are not really adequate for this to be 

 possible. We have chosen, therefore, to simplify the 

 model further by averaging the larval fish density 

 vertically and across the shelf 



A(x,0 = 



hY 3„ 3_ 



nix,y,z,t)dzdy. 



(2) 



Here Y is the width of the shelf and h is the depth 

 (both assumed independent of x). We can find the 

 equation governing this N by averaging Equation (1) 

 over the shelf width and depth, applying boundary 

 conditions of zero flux through the upper and lower 

 surfaces and the continent side of the domain. This 

 gives 



— A + 

 dt 



— \ I w^ dz dy I 



hY 3„ 3_, J 



hY 3- 



+ _ I v(x,Y,z,t)n{x,Y,z,t)dz= -t^N. (S) 



hY y, 



The downstream variation of Y and h has been 

 neglected although it is not too difficult to include it. 

 The term representing downstream advection of lar- 

 vae will be simplified by assuming that the cross- 

 shelf mixing of larvae is sufficiently intense that n is 

 uniform in y. The along-shelf advection term then 

 becomes 



d 1 

 dz hY 



n -a 



un dz dy = — UN; 

 , dx 



U{x,t) = — 

 hY 



— \ \ u{x,y,z, 

 ^Y 3o ) k 



t) dz dy. 



(4) 



Similarly, we shall ignore vertical variations in n at 

 the shelf edge so that the flux off the shelf becomes 



hY ) , 



— \ v(x,Y,z,t) n(x,F,z,0 dz = 



hY y . Y 



V,{x,t) = 



h: 



v{x,Y,z,t) dz. 



(5) 



Here VQ{x,t) is the depth-averaged onshore-offshore 

 flow (positive offshore) and A, is the depth-averaged 

 larval fish density in the water which is moving onto 

 or off of the shelf. (Vertical migratory behavior 

 which is somehow correlated with vertical shears 

 would alter this parameterization of the outflowing 

 flux of larvae.) 



When there is onshore or offshore flow, the aver- 

 aged velocity along the shelf cannot be constant. The 

 variations in U can be calculated from the conserva- 

 tion of fluid volume integrated across the shelf 



dx 



— I I u dz dy\ 



hY 3o 3_, J 



^0 



u dz dy\ + — =0 



which implies — 

 dx 



V, 



(6) 



Finally we must introduce a parameterization for 

 the density of larvae carried on or off the shelf at the 

 edge A, in terms of the average density N{x,t). It is 

 assumed that the Slope Water pushed onto the shelf 

 by the ring is devoid of shelf fish larvae. If we 

 presume that this Slope Water mixes completely 

 with the shelf water, then the water leaving the shelf 

 carries larvae with density A, as sketched in Figure 

 2. These considerations suggest that the entrain- 

 ment term can be modelled by 



N.= 



for Vn < 



A for Fn > 



(7) 



(Again, we must remark upon the limitations of the 

 present calculation; certainly the shelf water is not 

 thoroughly mixed and the density of the outflowing 

 larvae is much more complicated and perhaps 

 smaller on the whole than this formula would sug- 

 gest. We hope that our results will spur further 

 modelling and observational efforts to assess the pro- 

 cesses we have been forced to represent so crudely.) 

 When all of these simplifications are gathered 

 together, the approximate equations for the average 

 density of larvae N{x,t) become 



10 for Fo < 

 VoN ] = -1^ 

 for Fo > 



316 



