CHAPTER 8 



result from the loading of an area, the determination of 

 mixing volumes and transport rates is critical. Cases of 

 point loadings are difficult to approach because we must 

 know the details of advection and diffusion processes on 

 smaller time and space scales than is usually possible. In 

 addition, large concentration gradients complicate com- 

 putation procedures and introduce errors in the results. 

 When a substance is well distributed over a region, dif- 

 fusion processes can be neglected by averaging over a 

 large volume. Then simpler models of mass flow rates can 

 be assumed and longer time and space scales can be used, 

 which are usually a better match to the scales of obser- 

 vations. 



Previous studies of transport in the Bight followed salt 

 balance methods initiated by Ketchum et al. (1951). Their 

 studies of the distribution of physical properties produced 

 estimates of flushing (residence) times based upon dilution 

 of seawater by the Hudson River outflow. Estimates of 

 this sort may be valid for the Apex, where salinities are 

 often low enough to make an accurate calculation. When 

 considering the larger area of the continental shelf, how- 

 ever, the alongshore component of circulation, which 

 usually parallels the contours of the salinity distributions, 

 produces far greater flushing in a small area of the shelf 

 than a dilution ratio calculation would indicate. 



MODEL DESCRIPTION 



Theory and Averaging Schemes 



The basis for transport modeling is the conservation of 

 material equation 



dp/dl + V-(uc) = 0. 



(3) 



Dc/Dt = Is 



(i; 



where c is the concentration of the material, S.? is the sum 

 of sources and sinks, and D/Dt is the total or material 

 derivative. Molecular diffusion effects are ignored here. 

 When working in a Lagrangian coordinate system (fol- 

 lowing a parcel of water), eq. (1) describes the balance 

 between the time rate of change and the net production 

 (l5>0) or net utilization (Ss<0) of the constituent. This 

 strategy is used in laboratory experiments and sometimes 

 attempted in field studies. In most field studies, however, 

 , measurements are made in an Eulerian coordinate system 

 I at fixed points in space so that material is advected past 

 the points of observation. In this system the material de- 

 rivative must be expanded and eq. (1) is written as 



dc/dt + V-(uc) = Is 



(2) 



where dc/dt is the locally observed rate of change of con- 

 centration, u = (m, V, w) is the velocity vector, and V 

 = d/dx,d/dy,d/dz) is the divergence operator. The equiv- 

 alent relationship for conservation of water mass is 



Equation (2) shows that the material derivative is com- 

 posed of the local time rate of change of c and the diver- 

 gence of the flux of c; it is the sum of these which balances 

 the sum of production and utilization. All these quantities 

 are evaluated at a fixed position. 



Since we cannot accurately estimate these gradients on 

 infinitesimal time and space scales, we must approximate 

 the derivatives so that the observed time and space gra- 

 dients are consistently treated. When observations in time 

 are made at a spacing of T days, the resolution in space 

 must be of the order of L = UT, where L, T, U are the 

 characteristic length, time, and velocity. Thus, if T is of 

 the order of 10 days and U is 5 cm/s or 5 km/d then L 

 must be of the order of 50 km. If the temporal rates of 

 certain processes are inferred from observations at time 

 interval 7, then L is the smallest length scale that can be 

 resolved. 



One way to approximate spatial gradients is to smooth 

 the data over a selected volume segment. Transports be- 

 tween segments are then treated as advective fluxes. Dif- 

 fusion effects are introduced by assuming that each volume 

 is well mixed. Any material entering the volume is "in- 

 stantaneously" mixed over time 7" throughout the volume. 

 The effective diffusion thus introduced contributes to the 

 transport on scales smaller than those resolved. 



Because New York Bight waters were strongly stratified 

 during the period of this study, we will average over an 

 upper layer and over a lower layer, separated at the pyc- 

 nochne. Vertical diffusion between layers is neglected 

 because the strong stratification implies inhibited vertical 

 mixing; vertical advective fluxes are included. 



If observations are available at two discrete times, / 

 = /, and / = ?,, we must decide how to approximate the 

 time derivative. We know the concentrations at /, and /,, 

 but there is no way of knowing dc/dt |, , „. If we approx- 

 imate 5c/ar by Ac/Ar where Ac = (cj-c,) and Ar = (^-fl), 

 then we should use concentration and velocity data inter- 

 polated to f = (/, -h r,)/2. 



After performing appropriate volume and time aver- 

 ages, the discrete version of eq. (2) becomes 



AVC 



^t 



+ I.QC + QX = Is, 



and assuming constant density eq. (3) becomes 

 ^V 



A/ 



+ IQ + g, = 0, 



(4) 



(5) 



where the capitalized letters are now volume averaged 

 quantities and Q is the outward horizontal volume flux, 

 Q, the vertical flux, IQC the sum of divergence of con- 

 centration flux around the boundaries of each segment. 



167 



