A      Model Formulation on C- and A-grids

A.1  The Governing Equations

We consider the three dimensional (3D) primitive equations for an incompressible, stratified fluid using the rigid-lid, Boussinesq, and hydrostatic approximations. They are essentially the same as those considered by Dietrich [1992], except that spherical polar coordinates are used here.


where u,v,w are the east page47_lambda.gif, north page47_phi.gif, and vertical (z) components of the velocity, p is pressure, page47_rho.gif; is density, T and S are potential temperature and salinity, Km and Kh are vertical eddy viscosity and diffusivity coefficients, f is the Coriolis parameter,page47_rho.gifo is a reference density, R and g are the Earth's radius and gravitational acceleration, page47_curlyL.gif is an advection operator defined as


and page47_curlyD.gifm and page47_curlyD.gifh are diffusion operators defined as


where page47_CurlyA.gifm and page47_CurlyA.gifh are horizontal eddy viscosity and diffusivity coefficients, respectively.

Under the hydrostatic approximation (3), the pressure field at depth z can be expressed in terms of the pressure at the rigid upper surface, ps, plus that due to the fluid between the rigid lid and the depth z:


Lateral boundary conditions are required to close the above governing equations. No normal flow across solid boundaries is imposed and the component of horizontal velocity tangent to solid walls satisfies either free-slip or no-slip boundary conditions. In the case of free-slip boundary conditions the tangential stress at vertical boundaries is set to zero, and for the no-slip boundary condition, the tangential velocity at the boundary is set to zero. Note that since the real bathymetry is represented by a series of steps in the Z-level model, lateral boundary conditions are required not only at the coast, but also at all submerged vertical boundaries.

The boundary conditions for the vertical velocity component w are based on the rigid-lid approximation and the condition of no normal flow at the bottom. For general topography these conditions can be expressed as


where the subscripts 0 and -h indicate evaluation at the sea surface z=0 and at the sea bottom z=-h, respectively.

Integrating the continuity equation (4), and using (11) gives


and substituting (12) into (13), we obtain the depth-integrated continuity equation:


Equation (14) holds for general topography and is valid in both z and sigma coordinate formulations. Note that for the step-like topography of Z-coordinate models, h is uniform over each individual grid cell and there is no flow through solid boundaries; hence page48_4.gif.

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