CHAPTER 8 



otropic-geostrophic, baroclinic-geostrophic, wind stress 

 curl, and bottom friction components. 



To aid in visualizing the equations, consider a simplified 

 system with constant density, no wind stress curl, and no 

 bottom friction. Then only the barotropic term remains 

 and eq. (6) becomes 



J(i. H) = 0. 



(7) 



A property of the operator, J, or the Jacobian, is that it 

 is zero whenever one variable in the argument list is a 

 function of the other variable. Equation (7) is satisfied if 

 the surface elevation contours parallel the depth contours. 

 Thus, in this simplified case, the geostrophic velocity field 

 at all depths is described by the C, contours as streamlines. 

 This flow is, of course, everywhere parallel to the bottom 

 contours as well. 



Once eq. (6) is solved for the ^ field, the geostrophic 

 velocity profile can be calculated at any depth z= o to z 

 = -//by 



kx^i^i 





(8) 



where / is the Coriolis parameter. The current profiles 

 derived from eq. (8) are somewhat simplified in that the 

 vertical shear associated with the density gradients is as- 

 sumed uniform. The complete velocity profile can be 

 formed by superimposing the proper profiles for a surface 

 and bottom Ekman layer. 



MODEL APPLICATION AND RESULTS 



Water Fluxes 



Results from the diagnostic model are used to calculate 

 the water flux in the layer below the pycnocline. To apply 

 the diagnostic model, the data required are depth {//), 

 vertically integrated density (a), wind stress field (t„), and 

 appropriate boundary conditions. To calculate the bound- 

 ary conditions on t,, current meter data are needed on the 

 two cross-shelf boundaries. 



Model equation (6) is solved for the surface elevation 

 field using a finite element technique, developed by Gait 

 (1975), on a grid shown in figure 8-2, where each triangle 

 vertex is an STD station location. Values of //, a, and 

 T^ are specified at these vertices. The station grid and 

 density data were used from MESA cruise XWCC-9 (May 

 17-24, 1976). Several stations were added near the Hud- 

 son Shelf Valley to better approximate the sharp depth 

 gradients. Values of a were smoothly interpolated to these 

 new stations. Figures 8-3 and 8-4 show the depth and a 

 fields used in the calculations. Wind stress was calculated 

 from EB34 wind velocity data; EB34 is located at midshelf 



just northeast of Hudson Shelf Valley (ch. 7, fig. 7-1). 

 Wind stress was taken as constant over the domain (grid). 

 Current meter records from the MESA 1976 survey 

 were averaged over four separate time periods spanning 

 May 18 to June 29, 1976. The four periods were selected 

 so that wind and current conditions were relatively uri- 

 form over each interval. The averaged currents and wind 

 stresses are shown in figure 8-5. 



To solve the model equation, the surface elevation fidd 

 must be specified around the entire boundary because (♦) 

 is an elliptic equation. By calculating V^ between ea<h 

 boundary point, ^ is then calculated relative to the nortl- 

 east corner elevation, fixed arbitrarily at 10 cm became 

 only V^ enters into the calculation. There are three types 

 of boundaries: 1) solid boundaries on the New Jersey av6 

 Long Island coasts for which no-flux condition is impose*?; 

 2) cross-shelf open boundaries for which elevations are 

 calculated from velocity data on those boundaries; and 3) 

 shelf-break open boundary for which the elevations are 

 interpolated between the seaward points on the cross-shelf 

 boundaries. 



In calculating the cross-shelf elevation profile, data are 

 used from the deepest current meter on each array that 

 is not a bottom-mounted meter. These meters, located 

 approximately 8 m above the bottom, are used because 

 we are attempting to model the lower layer velocities and 

 wish to avoid the influence of bottom friction. The two 

 velocity data points on each boundary, as shown in figure 

 8-5, are used to first construct a smooth profile across the 

 shelf of velocity perpendicular to the boundaries. Then 

 the geostrophic velocity profile (eq. 8) is used to solve for 

 V^ along the boundary. This procedure produces values 

 of ^ along the solid and northeastern open boundary, but 

 on the southern open boundary only the values of V^ are 

 known. The final I, field is determined by evaluating sev- 

 eral solutions of eq. (6) with varying slopes on the shelf- 

 break boundary until a smooth flow at the southern bound- 

 ary is produced. This procedure is equivalent to fine-tun- 

 ing by adjusting the imposed alongshore pressure gra- 

 dient. However, the nature of the equations causes the 

 northeastern cross-shelf boundary (the major inflow 

 boundary of the circulation) to have the greatest influence 

 on the flow because topographic vorticity waves propagate 

 alongshelf from that boundary. The nature of the equa- 

 tions is such that changing C, on the southern boundary 

 influences the flow only in a narrow zone adjacent to that 

 boundary. 



Once the t, field is found from the solution of eq. (6), 

 the velocity profile and then the transport in the lower 

 layer are calculated by integrating eq. (8) from the pyc- 

 nocline depth to the bottom and adding the transport in 

 the bottom Ekman layer. The lower layer transport, T 

 becomes 



169 



