Mathematical Models of the Massachusetts Bay

Abstract

Part 1. The vertically integrated conservation of mass and momentum equations for shallow water bodies are reviewed. The equations used in this study are based on only two assumptions: hydrostatic pressure and squares of surface elevation gradients negligible with respect to unity. The finite element method is applied to reduce the governing equations to a system of ordinary non-linear differential equations in time for which two different numerical integration schemes are described. Model results are compared with analytical solutions. Also, numerical predictions of the tidal response for Massachusetts Bay are presented. Part 2. A need for qualitative information concerning the hydrodynamics of Massachusetts Bay has been seen from recent oceanographic measurements and current studies in the Bay area. In response to this, two analytical models have been derived for a simple rectangular configuration which can be applied to the geometry of Massachusetts Bay. A one layer model has been developed to simulate the conditions found during the winter season when the water column is well mixed. A two layer model represents the stratified case generally observed, with the presence of a strong thermocline, during the summer. Both models are derived from the linearized long wave equations in two dimensions and analytical solutions are obtained by neglecting Coriolis force, bottom friction, and wind stress. The models are depth averaged and the geometry of the Bay is represented by a rectangle. The boundary conditions are specified as zero normal velocity along the walls and a constant surface slope across the opening connecting Massachusetts Bay to the ocean. The results of the two models indicate that the surface elevations at high tide are fairly insensitive of the assumed conditions (one or two layer model). However, for the two layer model, relatively large interfacial waves are predicted as well as velocities which at some locations in the upper layer, are directed shoreward on the ebbing tide, rather than seaward. Comparison of available field observations with these results verify, qualitatively, that these conditions do exist and 'shows that if a model capable of predicting velocities in the Bay is desired, it must incorporate the conditions corresponding to a two layer flow. Part 3. A three-dimensional analytical model is proposed for the description of the dispersion of fine suspended sediments in coastal waters. The model basically predicts the quasi-steady state sediment concentration as a function of space and tidal time and the deposition pattern in the region surrounding a continuous vertical line source. It requires that the sediment settling velocities and the hydrodynamic features of the area, the net drift and the tidal velocities as well as the dispersion coefficients be known. Effects of wave action and vertical stratification are not explicitly considered. A separation of variables technique permits a rather independent treatment of the vertical and horizontal distributions; they are linked primarily through the decay factor, which represents the loss of material to the bottom. The model is applied to a hypothetical dredging situation in Massachusetts Bay. Values for the hydrodynamic parameters were obtained from the analysis of field data collected during the past year. Laboratory experiments were carried out for the determination of settling rates of clays in seawater, in view of unknown flocculation factors. Stoke's law was considered adequate for silt and very fine sand. The model results indicated very long and relatively narrow dispersion patterns, under the assumption of constant drift direction. The net drift and the sediment settling velocity seem to be the most important factors controlling the dispersion of fines in coastal waters.