Numerical Models
369
elements of the standard layer models as well as aspects of conventional continuous models. In particular, the temperature (i.e., density) of each layer is
allowed to vary horizontally within each layer. It is assumed that the density
remains independent of z within the layer. The horizontal velocity, if it is
geostrophic, therefore varies linearly with z. The advection of the density by the
geostrophic thermal wind is identically zero. If the velocity is geostrophic then,
the depth-dependent advective terms in the density equation for each layer are
zero, and the assumption of purely horizontal variations of temperature within
the layer is consistent with the density equation, assuming the sources of
heating are uniform over each layer's thickness. However, as (6.3.17) shows, it
is unlikely that the meridional velocity is geostrophic in the equatorial region,
and the assumption of a z-independent temperature field is somewhat inconsistent there. Nonlinear advection by the ageostrophic velocity produces a zdependent density field. A similar problem arises in the zonal momentum
equation due to the nonlinear advection of momentum.
Heuristic entrainment and detrainment laws, governing the transfer of
mass between the upper two layers, allow the model to mimic the process of
subduction in midlatitudes. Indeed, when the thermodynamics of the model is
idealized as adiabatic, McCreary and Lu are able to reproduce the ventilated
thermocline solution of Luyten et al. (1983) for the subtropical part of the
circulation. The solution in the equatorial region is found numerically without
the boundary layer assumptions used in previous sections of this chapter. The
model domain is large enough to easily contain both the midlatitude circulations of subpolar and subtropical gyres as well as the equatorial flow.
The model, in spite of its inconsistencies in the equatorial region, is thus an
ingenious heuristic representation of the circulation. Its ability to close the
circulation linking the midlatitudes and the equator, without a priori assumptions of boundary layer matching, makes it a useful vehicle for examining
the connection between the two regions. The reader is referred to the paper for
further details of the model's formulation.
The wind stress in the "standard" model calculations is chosen to have a
negative curl over the region of the subtropical gyre and to become independent of latitude over a broad tropical region straddling the equator. Thus
in the region outside the anticipated equatorial boundary layer we expect a
subsurface geostrophic flow to balance the poleward meridional Ekman flux.
Experiments with more complex stress patterns, in which the stress vanishes in
the equatorial domain, are also considered.
Figure 6. 7.1 shows the velocity fields as calculated for the upper and lower
layers. McCreary and Lu superimposed on the results of the numerical calculation the boundaries of the shadow zone (emanating from the point labeled
Yd on the eastern edge of the basin) and the critical streamline connecting the
outcrop line with the bifurcation latitude of the western boundary current
which occurs at the point labeled Yb in panel b. For the parameters of this
calculation the shadow strikes the western boundary at Ye, well outside the
equatorial boundary layer region but south of the bifurcation latitude. It is
369
elements of the standard layer models as well as aspects of conventional continuous models. In particular, the temperature (i.e., density) of each layer is
allowed to vary horizontally within each layer. It is assumed that the density
remains independent of z within the layer. The horizontal velocity, if it is
geostrophic, therefore varies linearly with z. The advection of the density by the
geostrophic thermal wind is identically zero. If the velocity is geostrophic then,
the depth-dependent advective terms in the density equation for each layer are
zero, and the assumption of purely horizontal variations of temperature within
the layer is consistent with the density equation, assuming the sources of
heating are uniform over each layer's thickness. However, as (6.3.17) shows, it
is unlikely that the meridional velocity is geostrophic in the equatorial region,
and the assumption of a z-independent temperature field is somewhat inconsistent there. Nonlinear advection by the ageostrophic velocity produces a zdependent density field. A similar problem arises in the zonal momentum
equation due to the nonlinear advection of momentum.
Heuristic entrainment and detrainment laws, governing the transfer of
mass between the upper two layers, allow the model to mimic the process of
subduction in midlatitudes. Indeed, when the thermodynamics of the model is
idealized as adiabatic, McCreary and Lu are able to reproduce the ventilated
thermocline solution of Luyten et al. (1983) for the subtropical part of the
circulation. The solution in the equatorial region is found numerically without
the boundary layer assumptions used in previous sections of this chapter. The
model domain is large enough to easily contain both the midlatitude circulations of subpolar and subtropical gyres as well as the equatorial flow.
The model, in spite of its inconsistencies in the equatorial region, is thus an
ingenious heuristic representation of the circulation. Its ability to close the
circulation linking the midlatitudes and the equator, without a priori assumptions of boundary layer matching, makes it a useful vehicle for examining
the connection between the two regions. The reader is referred to the paper for
further details of the model's formulation.
The wind stress in the "standard" model calculations is chosen to have a
negative curl over the region of the subtropical gyre and to become independent of latitude over a broad tropical region straddling the equator. Thus
in the region outside the anticipated equatorial boundary layer we expect a
subsurface geostrophic flow to balance the poleward meridional Ekman flux.
Experiments with more complex stress patterns, in which the stress vanishes in
the equatorial domain, are also considered.
Figure 6. 7.1 shows the velocity fields as calculated for the upper and lower
layers. McCreary and Lu superimposed on the results of the numerical calculation the boundaries of the shadow zone (emanating from the point labeled
Yd on the eastern edge of the basin) and the critical streamline connecting the
outcrop line with the bifurcation latitude of the western boundary current
which occurs at the point labeled Yb in panel b. For the parameters of this
calculation the shadow strikes the western boundary at Ye, well outside the
equatorial boundary layer region but south of the bifurcation latitude. It is