Numerical Hydrodynamics of Estuaries
Classically, estuaries have been classified dimensionally on the basis of the dominant salinity gradients. Following Pritchard (1958) the general classifications based on spatial averaging of the constituent transport relationship are: (1) three-dimensional; (2) laterally homogeneous with longitudinal and vertical spatial gradients dominant; (3) vertically homogeneous with longitudinal and lateral spatial gradients dominant, and (4) sectionally homogeneous with longitudinal gradients dominant. Development of the hydrodynamic (momentum transport) relationships follow similar spatial averaging and classification.
In general, the momentum balances determine the flow field by which the constituent is transported. The momentum and constituent transport are interrelated in estuaries through the horizontal density gradient as determined from the constituent distribution. Only the fourth case, sectional homogeneity, is solvable for a few limiting situations without use of the hydrodynamic relationships, and are situations for which the advective flow field can be inferred from fresh water inflow.
The development of numerical hydrodynamics for estuaries begins with a presentation of the equations of motion and constituent transport in three dimensions. The basic equations are: (1) the u-velocity or longitudinal momentum balance; (2) the v-velocity or lateral momentum balance; (3) the pressure distribution, p, as determined from the vertical momentum balance as the hydrostatic approximation; (4) the w or vertical velocity as determined from local continuity; (5) the salinity, S, constituent transport; (6) the equation of state relating density, p, to the constituent concentration, and (7) the free water surface elevation, n, as determined from vertically integrated continuity. The general numerical problem is, therefore, to spatially integrate numerically over time seven equations for the seven unknowns of u, v, w, P, S, p, n given appropriate geometry and time-varying boundary data. The seven equations are interrelated with the constituent distribution, S, determining density, p; with density and the free water surface elevation, n, determining pressure, P, and with the pressure distribution entering the momentum balance.
The two-dimensional and one-dimensional cases are derived from the three-dimensional relationships by spatial averaging. The laterally homogeneous estuary dynamics include a majority of the interrelationships of density, pressure and surface elevation incorporated in the three-dimensional equations. Explicit and implicit solution procedures can be illustrated for the laterally homogeneous relationships as they depend upon the inclusion of vertically integrated-velocities in the surface elevation computations. Laterally averaged hydrodynamic solution procedures that utilize simplifying assumptions for the longitudinal density gradient are also examined. The sectionally homogeneous hydrodynamics is shown to be a reduced case of the laterally homogeneous relationships.
The two-dimensional vertically homogeneous dynamics is presented as a reduced form of the vertically integrated three-dimensional case. The vertically homogeneous case has spatially explicit and implicit solution procedures, the properties of which can be illustrated from the basic equations. It will be shown that a surface elevation relationship exists for this case that has a variational statement leading to spatial finite element description.
KeywordsRichardson Number Momentum Balance Water Surface Elevation Free Water Surface Estuarine Circulation
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