4 A Generalization of a Sigma
Coordinate Ocean Model and an
Intercomparison of Model Vertical Grids
GEORGE L. MELLOR 1 , SIRPA M. HAKKlNEN 2 , TAL EZER 1 AND RICHARD C. PATCHEN 3
1 Atmospheric and Oceanic Sciences Program Princeton University, Princeton, NJ
2NASA Goddard Space Flight Center, Greenbelt, MD
3Dynalysis of Princeton, Princeton N.J.
4.1 Introduction
Numerical ocean models increasingly make use of cr - coordinate systems. A
paper by Gerdes (1993) shows that these coordinate systems can be more general;
he termed the generalized form an "s - coordinate" system. The main advantage of
the cr or s - system is that, when cast in a finite difference form, a smooth representation ofthe bottom topography is obtained; one can also easily incorporate a bottom boundary layer as well as a surf ace boundary layer in those coordinate
systems. This is intuitively appealing and Gerdes has shown that superior numerical results are obtained relative to a z - level system. However, in regions of steep
topography and crude resolution - a limiting case would be a seamount represented
by a single grid point surrounded by a tlat bottom - the so-called sigma coordinate
pressure gradient error exists (Haney 1991, Mellor et al. 1994, 1998) and at least
locally a z - level coordinate system might be preferred. On the other hand, in a
recent study, Bell (1997) has shown that the step structure of z - level models lead
to vorticity errors and consequent errors in the barotropic component of the flow
which, he reports, cause rather large temperature errors (3 to 4 0 C) on a 1 0 x 1 0 grid
of an Atlantic Ocean model after 3 months of integration. And it is difficult to
model bottom boundary layers in a z - level model (Winton et al. 1998).
The pioneer Bryan-Cox model (Bryan 1969, Cox 1984) is a z - level model. A
modification by Spall and Robinson (1990) is termed a "hybrid" coordinate system;
they describe it as a z - level system in the region, O > Z > - Zc = constant, and a cr
system when Zc > Z > - H(x, y) where the transformed sigma equations apply. Presumably, this system is adopted so that surf ace mixed layers, which do not scale on
depth, may be best represented. However, the hybrid system would appear to
require separate numerical implementations for the two regions; their objectives
can be realized more simply with the s - coordinate system described here.
For the sigma coordinate system of the Princeton Ocean Model (henceforth
POM), the top numericallevel, k = 1, follows the free sea surface and the lowest
numeric al level, k = kb, follows the bottom depth; for 1 < k < kb, the distance
between levels are in fixed proportion to each other independent of elevation or
Coordinate Ocean Model and an
Intercomparison of Model Vertical Grids
GEORGE L. MELLOR 1 , SIRPA M. HAKKlNEN 2 , TAL EZER 1 AND RICHARD C. PATCHEN 3
1 Atmospheric and Oceanic Sciences Program Princeton University, Princeton, NJ
2NASA Goddard Space Flight Center, Greenbelt, MD
3Dynalysis of Princeton, Princeton N.J.
4.1 Introduction
Numerical ocean models increasingly make use of cr - coordinate systems. A
paper by Gerdes (1993) shows that these coordinate systems can be more general;
he termed the generalized form an "s - coordinate" system. The main advantage of
the cr or s - system is that, when cast in a finite difference form, a smooth representation ofthe bottom topography is obtained; one can also easily incorporate a bottom boundary layer as well as a surf ace boundary layer in those coordinate
systems. This is intuitively appealing and Gerdes has shown that superior numerical results are obtained relative to a z - level system. However, in regions of steep
topography and crude resolution - a limiting case would be a seamount represented
by a single grid point surrounded by a tlat bottom - the so-called sigma coordinate
pressure gradient error exists (Haney 1991, Mellor et al. 1994, 1998) and at least
locally a z - level coordinate system might be preferred. On the other hand, in a
recent study, Bell (1997) has shown that the step structure of z - level models lead
to vorticity errors and consequent errors in the barotropic component of the flow
which, he reports, cause rather large temperature errors (3 to 4 0 C) on a 1 0 x 1 0 grid
of an Atlantic Ocean model after 3 months of integration. And it is difficult to
model bottom boundary layers in a z - level model (Winton et al. 1998).
The pioneer Bryan-Cox model (Bryan 1969, Cox 1984) is a z - level model. A
modification by Spall and Robinson (1990) is termed a "hybrid" coordinate system;
they describe it as a z - level system in the region, O > Z > - Zc = constant, and a cr
system when Zc > Z > - H(x, y) where the transformed sigma equations apply. Presumably, this system is adopted so that surf ace mixed layers, which do not scale on
depth, may be best represented. However, the hybrid system would appear to
require separate numerical implementations for the two regions; their objectives
can be realized more simply with the s - coordinate system described here.
For the sigma coordinate system of the Princeton Ocean Model (henceforth
POM), the top numericallevel, k = 1, follows the free sea surface and the lowest
numeric al level, k = kb, follows the bottom depth; for 1 < k < kb, the distance
between levels are in fixed proportion to each other independent of elevation or