NO A A PROFESSIONAL PAPER 11 



and V the segment volume. Assuming incompressibility, 

 water density has been taken equal to a constant density, 

 Po, in eq. (5). To estimate these fluxes, we will use a 

 diagnostic model of shelf circulation. Once the advective 

 terms of eq. (4) are known, the time rate of change of 

 concentration term can be related to the source and sink 

 term. 



Diagnostic Model 



Computing water transport on the shelf requires spatial 

 resolution of a velocity field greater than is normally avail- 

 able from current meter observations alone. Calculation 

 of the velocity field requires understanding the dynamics 

 of shelf circulation; that is, understanding how water 

 moves in response to the imposed forces, as well as having 

 adequate measurements of all the various forcing and in- 

 dicator fields. 



One major forcing field is wind stress. Only a portion 

 of water transport on the shelf, however, is in the frictional 

 layer driven directly by wind. Simple Ekman theory ex- 

 plains how a wind stress along a coastline will transport 

 surface water perpendicular to the coast, resulting in a 

 cross-shelf gradient in surface elevation. The response to 

 this force is a geostrophic velocity along the shelf. Hori- 

 zontal density gradients and bottom friction will modify 

 the vertical profile of horizontal velocity, but it is mainly 

 the forcing by seasurface elevation gradients (set up by 

 the wind stress) that determines the gross characteristics 

 of the flow. 



At present we cannot calculate the barotropic compo- 

 nent of flow from wind data alone. Winds acting on dif- 

 ferent sections of the Middle Atlantic Bight produce var- 

 ied responses, particularly at a bend in the coastline, such 

 as in the New Yoric Bight. A "prognostic" dynamic model 

 should be able to calculate flow directions, given wind 

 stress, river discharge, bottom topography, and other 

 boundary conditions. Such a model would probably have 

 to include the entire Middle Atlantic Bight and be able 

 to approximate many processes we now understand only 

 poorly. 



A more limited approach, but one that can yield the 

 required flow field, is to construct a "diagnostic" model 

 as is done here. The model is a steady-state representation 

 of the flow for which it is assumed that the structure of 

 the density field is known and is not being changed by the 

 velocity field. The general tlow condition must be known 

 from current measurements at appropriate points to cal- 

 culate boundary conditions which strongly influence the 

 flow. This type of modeling might alternatively be con- 

 sidered a formalism for interpolating or synthesizing over 

 the field of available data. The result is tightly bound to 

 the observed data. Wind stress does not enter the for- 

 mulation explicitly but is reflected in the boundary con- 

 ditions, calculated from current velocity data for the open 



boundaries and from a no-flux condition for the solid 

 boundaries. Only the curl of the wind stress enters the 

 model, and this is much less important than the other 

 terms for the cases considered. 



The model addresses a condition which is not truly 

 steady state but is actually a time average over a certain 

 period. The period should be long enough to span any 

 storm events but short enough to allow the approximation 

 of an unchanging density field. The diagnostic model con- 

 cept is supported by recent work of Csanady (1976), who 

 hypothesized that short-term storms generate flows which 

 tend to organize the density field into patterns such that 

 the time-average flows can be analyzed from the density 

 patterns with a simple linearized equation of motion. 

 Thus, a reasonable averaging period is of the order of 5 

 to 30 days. 



The fundamental physical problem reduces to solving 

 a differential equation for the shape of the sea surface. 

 The model equation is; 



p„gJ{lH) + gJiaM) + k-(VxTj 



+ -^^Poj— + V-^q = 0, (6) 



where C, is the elevation of the sea surface, H bottom 

 depth, T„ surface wind stress, y linear bottom friction pa- 

 rameter taken as 1600 cm following Csanady (1976), p„ 

 reference density, k unit vector in the vertical direction, 



r" 



g gravitational acceleration, a = p dz. where z is the 



vertical coordinate, zero at the surface and positive up- 

 ward, and 



m-n) = 



dx dy 



dy dx 



A complete description of this equation is not possible m 

 here. Alternative forms of the model have been devel- ^ 

 oped and described by Hsueh et al. (1976) and by Gait 

 (1975). Gait's vorticity equation, used here, results from 

 the linearized equations of motion on a rotating Earth and 

 is derived by summing the transports in the surface Ekman 

 layer, the geostrophic interior, and the bottom Ekman 

 layer, and then imposing the continuity condition that the 

 divergence of the transport must be zero. This is an ex- 

 tension of the classical geostrophic current calculation to 

 include frictional layers at the sea surface and over a slop- 

 ing bottom. The diagnostic variable, ^, is used as the sur- 

 face boundary condition for a calculation of the geos- 

 trophic velocity profile. The equivalent condition in the 

 deep ocean is the assumption of a depth of no motion. 

 The terms in eq. (6) are interpreted consecutively as: bar- 



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