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Numerical Scheme

The model used in this study is called ECOM-si, a semi-implicit variant of the three-dimensional Estuary, Coastal and Ocean Model (ECOM) described by Blumberg and Mellor (1987). ECOM-si was selected because it includes a free surface, nonlinear advective terms, coupled density and velocity fields, river runoff, heating and cooling of the sea surface, a 2.5 level turbulence closure scheme to represent vertical mixing (Mellor and Yamada, 1982), and is designed to easily allow ``realistic'' simulations. In addition, the combination of orthogonal curvilinear coordinates in the horizontal plane and sigma-coordinates in the vertical dimension allows grid refinement in regions of interest without sacrificing the well-known characteristics of Cartesian grid schemes.

The basic equations are expressed in a sigma coordinate system

where is the bottom topography and is the surface elevation. The basic governing equations are presented here in Cartesian coordinates to facilitate discussion. The equations as expressed in curvilinear coordinates may be found in Blumberg and Mellor (1987).

The continuity equation is

the x momentum equation is

and the y momentum equation is

where is the surface elevation, u and v are the x and y components of velocity, D is the total water depth , is the transformed vertical velocity (normal to sigma surfaces), is the vertical eddy viscosity, is the water density, is a reference water density, and and are the horizontal viscous terms defined by

and

where is the horizontal viscosity. The parameterizations of and are discussed in chapters 2.3 and 2.4, respectively. The model also solves prognostically for temperature, salt, concentration (used here to model sewage effluent), turbulence kinetic energy and turbulence macroscale.

ECOM-si differs from the Blumberg and Mellor (1987) ECOM model in that it uses a semi-implicit scheme for calculating the free surface, therefore avoiding the gravity wave CFL condition required by explicit schemes (eg Casulli, 1990). This has the advantage that larger time steps may be taken (on the order of minutes, rather than tens of seconds). A potential disadvantage of implicit schemes is that they more readily damp free wave motions, but in strongly forced and damped shallow regions such as Massachusetts Bay, the effect is small. This was determined by halving the time step and observing negligible differences in simulation results. Another disadvantage is that because the calculation of surface elevation requires solving a large matrix equation at each time step and efficient solution of this equation requires positive definiteness; boundary conditions for elevation must be formulated in matrix form and must not destroy the positive definiteness of the matrix. We use a combination of clamped and gravity wave radiation conditions on the open boundary, made possible by the implementation of the partially-clamped formulation of Blumberg and Kantha (1985) discussed in chapter 2.8.