A Generalization of a Sigma Coordinate Ocean Mode!
6!
This formulation accommodates an orthogonal, curvilinear, horizontal coordinate system; the elements of each cell have the horizontal dimensions, ex, ey and
the area, A == oxoy. The vertical dimension is os. FormalIy, there are additional
horizontal viscous terms in (22) and (23) due to curvature, but these have been
neglected since the terms themselves are highly empirical and we prefer the values,
AM and AH, be as small as possible. In the calculations discussed below, the horizontal coordinates will be spherical, a special case of more general orthogonal curvilinear coordinates accommodated by the model.
4.5 Model Strategy
The algorithmic structure of the generalized model does not differ greatly from
the previous POM (Blumberg and Mellor 1987, Mellor 1996) in that it has a free
surface, split internal, external time steps and second order horizontal differencing
on a staggered C grid (although work on other differencing schemes is in progress)
and an imbedded turbulence closure scheme which primarily governs surface and
bottom boundary layers. The new features are that the more general vertical increment, os(i,j,k,n), where i,j,k are the spatial indices and n is the time index, has
replaced the sigma increment, os(k)(H(i,j) + T\(i,j,n» and the previous two-dimensionalland mask is now three-dimensional to accommodate the z - level grid when
that system is invoked. Ali of the code except for one subroutine is universal. The
exceptional subroutine is crafted to produce whichever coordinate system is
desired. In the examples discussed below, there are three such subroutines (SUBROUTINE MAKSIG, for example), one for each ofthe three grids we have chosen
to test in this paper. The active subroutine is called at each interna! mode time step.
4.6 Numerical Simulations Comparing Three Vertical
Coordinate Systems
We will examine calculations using three vertical grids, a z - leve1 grid, a cr grid
and a generalized cr grid. Fig. 4.1 illustrates the domain used in the calculations
which includes a bowl shaped deep bas in in the north, east and west and a vertical
wall at the equator; the basin connects to a shallow continental shelf in the north.
Fig. 4.la represents both the sigma and generalized sigma grids. The horizontal
resolution of alI the experiments is about 1 °xl o on a 51 x 76 horizontal grid. There
are 21 vertical grid levels with spacing decreasing near the surface; of course, for
the z - level grid, levels are truncated for all but the deepest depths (Fig. 4.1 b). AII
vertical walls including those of the z - level steps are adiabatic and frictionless.
6!
This formulation accommodates an orthogonal, curvilinear, horizontal coordinate system; the elements of each cell have the horizontal dimensions, ex, ey and
the area, A == oxoy. The vertical dimension is os. FormalIy, there are additional
horizontal viscous terms in (22) and (23) due to curvature, but these have been
neglected since the terms themselves are highly empirical and we prefer the values,
AM and AH, be as small as possible. In the calculations discussed below, the horizontal coordinates will be spherical, a special case of more general orthogonal curvilinear coordinates accommodated by the model.
4.5 Model Strategy
The algorithmic structure of the generalized model does not differ greatly from
the previous POM (Blumberg and Mellor 1987, Mellor 1996) in that it has a free
surface, split internal, external time steps and second order horizontal differencing
on a staggered C grid (although work on other differencing schemes is in progress)
and an imbedded turbulence closure scheme which primarily governs surface and
bottom boundary layers. The new features are that the more general vertical increment, os(i,j,k,n), where i,j,k are the spatial indices and n is the time index, has
replaced the sigma increment, os(k)(H(i,j) + T\(i,j,n» and the previous two-dimensionalland mask is now three-dimensional to accommodate the z - level grid when
that system is invoked. Ali of the code except for one subroutine is universal. The
exceptional subroutine is crafted to produce whichever coordinate system is
desired. In the examples discussed below, there are three such subroutines (SUBROUTINE MAKSIG, for example), one for each ofthe three grids we have chosen
to test in this paper. The active subroutine is called at each interna! mode time step.
4.6 Numerical Simulations Comparing Three Vertical
Coordinate Systems
We will examine calculations using three vertical grids, a z - leve1 grid, a cr grid
and a generalized cr grid. Fig. 4.1 illustrates the domain used in the calculations
which includes a bowl shaped deep bas in in the north, east and west and a vertical
wall at the equator; the basin connects to a shallow continental shelf in the north.
Fig. 4.la represents both the sigma and generalized sigma grids. The horizontal
resolution of alI the experiments is about 1 °xl o on a 51 x 76 horizontal grid. There
are 21 vertical grid levels with spacing decreasing near the surface; of course, for
the z - level grid, levels are truncated for all but the deepest depths (Fig. 4.1 b). AII
vertical walls including those of the z - level steps are adiabatic and frictionless.