The Ventilated Thermocline: The Two-Layer Model
189
model are labeled 1 and 2. For now, layer 3 and all lower layers do not
outcrop, i.e., they are unventilated in the subtropical gyre, and we provisionally
assume that they are at rest. The solution so obtained is consistent, but, as we
saw in Chapter 3 the possibility that unventilated layers are actually in motion
depends on the strength of the forcing and the stratification of these layers. We
return to the question of the motion of an unventilated layer 3 and deeper
layers later in this chapter. For now, using the results of Chapter 3, we assume
simply that layer 3 is so thick that the motion of the overlying layers is unable
to distort the potential vorticity in layer 3 sufficiently to close the geostrophic
contours in layer 3, which being unventilated must then be at rest. Only layers 1
and 2 are in motion.
The Ekman pumping vanishes on the line (} = (} 0 on which f = fo. For
now we take this line to be a latitude circle, although this can easily be relaxed.
The specification of the surface density field is equivalent to specifying the
position in which layers deeper than the lightest layer are exposed to the
interface. These are the lines {) = {)n, n = 1, 2 .. . N, on which f = fn· These
lines too may in general be functions of longitude, but the theory rapidly
becomes technically more complex if this occurs. Fortunately, the surface
density and temperature fields in the world oceans are more zonal than not,
and this idealization is an acceptable one. In the present case of the two-layer
model there is a single outcrop line on which the surface density changes from
p 1 to p 2 as we move northward beyond the outcrop line. Layers 1 and 2 are
therefore, each, exposed somewhere to Ekman pumping, and the principal
question of interest is whether a layer that is exposed to Ekman pumping at
high latitudes is in motion even at lower latitudes where it is shielded from
direct forcing by the Ekman pumping. If so, this provides another mechanism,
in addition to the closure of potential vorticity contours and the subsequent
process of potential vorticity homogenization, to set the thermocline in motion
below its uppermost layer.
With the surface density field given, the structure of the thermocline is
determined by the geography of the interfaces after they leave the sea surface.
This is equivalent to the problem of the mapping of the surface density field
onto a vertical line at a fixed latitude and longitude that we referred to in
Section 4.1. The subsurface spacing of the density interfaces determines the
vertical density variation at each geographical location. Adiabatic flow slides
along the density interfaces, bringing fluid of that density to regions below the
surface, but we do not know a priori the configuration of these pathways of the
flow. In distinction to quasi-geostrophic theory where the potential vorticity
contours are known ahead of time from the barotropic transport stream
function (see Chapter 3), here the flow path is entirely a part of the final
solution. It is conceivable, although wrong, to imagine a priori that the flow in
the ventilated layers is nonzero only where the layer is exposed directly to
Ekman pumping, and that the meridional flow would be halted once an
overlying layer interposed itself, shielding the layer from Ekman pumping.
However, only the full solution allows us to settle the issue.
189
model are labeled 1 and 2. For now, layer 3 and all lower layers do not
outcrop, i.e., they are unventilated in the subtropical gyre, and we provisionally
assume that they are at rest. The solution so obtained is consistent, but, as we
saw in Chapter 3 the possibility that unventilated layers are actually in motion
depends on the strength of the forcing and the stratification of these layers. We
return to the question of the motion of an unventilated layer 3 and deeper
layers later in this chapter. For now, using the results of Chapter 3, we assume
simply that layer 3 is so thick that the motion of the overlying layers is unable
to distort the potential vorticity in layer 3 sufficiently to close the geostrophic
contours in layer 3, which being unventilated must then be at rest. Only layers 1
and 2 are in motion.
The Ekman pumping vanishes on the line (} = (} 0 on which f = fo. For
now we take this line to be a latitude circle, although this can easily be relaxed.
The specification of the surface density field is equivalent to specifying the
position in which layers deeper than the lightest layer are exposed to the
interface. These are the lines {) = {)n, n = 1, 2 .. . N, on which f = fn· These
lines too may in general be functions of longitude, but the theory rapidly
becomes technically more complex if this occurs. Fortunately, the surface
density and temperature fields in the world oceans are more zonal than not,
and this idealization is an acceptable one. In the present case of the two-layer
model there is a single outcrop line on which the surface density changes from
p 1 to p 2 as we move northward beyond the outcrop line. Layers 1 and 2 are
therefore, each, exposed somewhere to Ekman pumping, and the principal
question of interest is whether a layer that is exposed to Ekman pumping at
high latitudes is in motion even at lower latitudes where it is shielded from
direct forcing by the Ekman pumping. If so, this provides another mechanism,
in addition to the closure of potential vorticity contours and the subsequent
process of potential vorticity homogenization, to set the thermocline in motion
below its uppermost layer.
With the surface density field given, the structure of the thermocline is
determined by the geography of the interfaces after they leave the sea surface.
This is equivalent to the problem of the mapping of the surface density field
onto a vertical line at a fixed latitude and longitude that we referred to in
Section 4.1. The subsurface spacing of the density interfaces determines the
vertical density variation at each geographical location. Adiabatic flow slides
along the density interfaces, bringing fluid of that density to regions below the
surface, but we do not know a priori the configuration of these pathways of the
flow. In distinction to quasi-geostrophic theory where the potential vorticity
contours are known ahead of time from the barotropic transport stream
function (see Chapter 3), here the flow path is entirely a part of the final
solution. It is conceivable, although wrong, to imagine a priori that the flow in
the ventilated layers is nonzero only where the layer is exposed directly to
Ekman pumping, and that the meridional flow would be halted once an
overlying layer interposed itself, shielding the layer from Ekman pumping.
However, only the full solution allows us to settle the issue.