A Generalization of a Sigma Coordinate Ocean Model
63
Fig. 4.2 shows the three vertical grids in a north-south cross section at the center
ofthe domain shown in Fig. 4.1. For the z-level grid (Fig. 4.2a), we have a conventional grid with the characteristic step structure at the bottom. For the conventional
sigma grid (Fig. 4.2b), the spacing between the different layers have the same proportion independent of depth, and the lowest layer smoothly folIows the bottom
topography. For the generalized sigma grid (Fig. 4.2c)~ we have devised a grid
which is mostly a z - level grid, but departs from the z - leve1 constraint near the
bottom boundary such that the lowest layer folIows the Qottom topography. Layers
therefore converge near the bottom and alIow for good resolution of the bottom
boundary layer in most of the domain. The minimum layer thickness is constrained
to be 5 cm at the bottom.
In the first set of experiments the three models were integrated for one year using
viscosity and diffusivity values of AM = 10 5 m 2 s-1 and AH = 0.2 AM , respective1y.
This level ofhorizontal viscosity is that used in the GFDL, Bryan-Cox applications
for this resolution (Cox 1975, Bryan and Lewis 1979, Rosati and Myakoda 1988,
Derber and Rosati 1989, BeU 1997) and a reduced diffusivity is also typical; we
have found that calculations with AH = AM do not differ significant1y from those
discussed below. However, the reduced diffusivity is generalIy favorable to the
sigma systems to reduce a vertical component of the along - sigma diffusion.
Another way to counter along-sigma diffusion is to subtract the (initial in the
present case) c1imatological temperature fields before reckoning along sigma gradients oftemperature and salinity and fluxes, and this has been invoked in alI three
model runs (but is inconsequential for the z - level run). The ultimate solution is to
set the along-sigma diffusivity equal to zero which we find possible in many applications of the sigma grid (Mellor et al. 1998, Ezer and MeUor 2000).
Surface e1evations, representing the surf ace velocities, are shown in the left panels and transport streamfunctions, representing the depth integrated velocities, are
shown in the right panels in Fig. 4.3 for alI three grids. The two sigma models are
almost identical. The z - level ca1culations are also quite similar to the two sigma
models. However, the z - level surf ace velocities in the shalIow northem region are
much stronger. In Fig. 4.4, the temperature cross sections of the three experiments
appear to be virtualIy identical in most of the domain, producing similar upper
ocean mixed layers, except near the continental shelf, where only the two sigma
models appear to produce also bottom boundary layers. In Fig. 4.5, velocity cross
sections of the three experiments are fairly similar in the deep ocean, but significant1y differ from each other near the continental slope and shelfwhere the z -level
grid produces noisy velocity fields.
If the horizontal viscosity and diffusivity are reduced by an order of magnitude,
AM = 10 3 m 2 s- 1 andA H = 0.2 AM' the situation changes. The sigma models tolerate
the reduced values; the z -level model does not as seen in Figs. 4.6, 4.7 and 4.8. At
this reduced viscosity and diffusivity, the sigma models exhibit Rossby waves near
the equator (Figs. 4.6d and 4.6f) that are absent from the high viscosity calculations
(Fig. 4.3); additional plots (not shown) reveal that the waves propagate westward at
a speed of about 20 km d- 1 , about 30% higher than the speed ofthe simplest baro-
63
Fig. 4.2 shows the three vertical grids in a north-south cross section at the center
ofthe domain shown in Fig. 4.1. For the z-level grid (Fig. 4.2a), we have a conventional grid with the characteristic step structure at the bottom. For the conventional
sigma grid (Fig. 4.2b), the spacing between the different layers have the same proportion independent of depth, and the lowest layer smoothly folIows the bottom
topography. For the generalized sigma grid (Fig. 4.2c)~ we have devised a grid
which is mostly a z - level grid, but departs from the z - leve1 constraint near the
bottom boundary such that the lowest layer folIows the Qottom topography. Layers
therefore converge near the bottom and alIow for good resolution of the bottom
boundary layer in most of the domain. The minimum layer thickness is constrained
to be 5 cm at the bottom.
In the first set of experiments the three models were integrated for one year using
viscosity and diffusivity values of AM = 10 5 m 2 s-1 and AH = 0.2 AM , respective1y.
This level ofhorizontal viscosity is that used in the GFDL, Bryan-Cox applications
for this resolution (Cox 1975, Bryan and Lewis 1979, Rosati and Myakoda 1988,
Derber and Rosati 1989, BeU 1997) and a reduced diffusivity is also typical; we
have found that calculations with AH = AM do not differ significant1y from those
discussed below. However, the reduced diffusivity is generalIy favorable to the
sigma systems to reduce a vertical component of the along - sigma diffusion.
Another way to counter along-sigma diffusion is to subtract the (initial in the
present case) c1imatological temperature fields before reckoning along sigma gradients oftemperature and salinity and fluxes, and this has been invoked in alI three
model runs (but is inconsequential for the z - level run). The ultimate solution is to
set the along-sigma diffusivity equal to zero which we find possible in many applications of the sigma grid (Mellor et al. 1998, Ezer and MeUor 2000).
Surface e1evations, representing the surf ace velocities, are shown in the left panels and transport streamfunctions, representing the depth integrated velocities, are
shown in the right panels in Fig. 4.3 for alI three grids. The two sigma models are
almost identical. The z - level ca1culations are also quite similar to the two sigma
models. However, the z - level surf ace velocities in the shalIow northem region are
much stronger. In Fig. 4.4, the temperature cross sections of the three experiments
appear to be virtualIy identical in most of the domain, producing similar upper
ocean mixed layers, except near the continental shelf, where only the two sigma
models appear to produce also bottom boundary layers. In Fig. 4.5, velocity cross
sections of the three experiments are fairly similar in the deep ocean, but significant1y differ from each other near the continental slope and shelfwhere the z -level
grid produces noisy velocity fields.
If the horizontal viscosity and diffusivity are reduced by an order of magnitude,
AM = 10 3 m 2 s- 1 andA H = 0.2 AM' the situation changes. The sigma models tolerate
the reduced values; the z -level model does not as seen in Figs. 4.6, 4.7 and 4.8. At
this reduced viscosity and diffusivity, the sigma models exhibit Rossby waves near
the equator (Figs. 4.6d and 4.6f) that are absent from the high viscosity calculations
(Fig. 4.3); additional plots (not shown) reveal that the waves propagate westward at
a speed of about 20 km d- 1 , about 30% higher than the speed ofthe simplest baro-