208
Theory of the Ventilated Thermocline
line south of the first outcrop line. The first outcrop line is at 8 = 83 where
f = /3. The addition of another layer has required a slight alteration of
notation and the new, more southern outcrop line is a latitude circle along
which f = fz . This is the latitude in which layer 2, which was the upper
layer of the two-layer model, subducts. South of this latitude there are now two
ventilated layers each of which is assumed to conserve its potential vorticity.
The solution north of this outcrop line is exactly the same as that obtained for
the two-layer model. This is an important point for it means that as we increase
the resolution by adding more layers, the previous solution still holds north of
the new outcrop lines and remains undisturbed by the addition of new layers.
Information in the ventilated region travels southward along the streamlines as
long as potential vorticity is conserved. We need not iterate the solution
already found as new layers are added. Thus merely letting 2 --+ 3 and 1 --+ 2,
we can rewrite the solution in the region north of 8 = 82. That is, for
83 ~ 8 ~ 80 the solution in which only layer 3 is moving is given by (4.4.4) or:
( 4.7.1)
where his the total depth of the moving fluid (equal to -z4 in this region) while
D5 is now defined as:
2/21e
D~ = --p
wE(',8)Rcos8 d'
Y3
(4.7.2)
and H 3 is the constant depth of the lowest layer on the eastern boundary.
In the region 82 ~ 8 ~ 83 layers 2 and 3 are in motion and contain all the
structure described above. In particular, the boundary between the shadow
zone and the ventilated region in layer 3 follows the curve given by:
D2(<1> 8) = H2 Y2 (1 _1_)2
0
.,
3
f
Y3
3
(4.7.3)
while the relation in the ventilated zone between the thickness of the lowest
layer and the total depth is again given by the potential vorticity relation
determined at the outcrop line, (4.4.14) i.e.:
f /3
q3 = - = -
h3
h
so that
(4.7.4)
The solution itself in each of the ventilated, shadow zone and pool regions
is given by the formulae in Sections 4.4--4.6, with an obvious change in
Theory of the Ventilated Thermocline
line south of the first outcrop line. The first outcrop line is at 8 = 83 where
f = /3. The addition of another layer has required a slight alteration of
notation and the new, more southern outcrop line is a latitude circle along
which f = fz . This is the latitude in which layer 2, which was the upper
layer of the two-layer model, subducts. South of this latitude there are now two
ventilated layers each of which is assumed to conserve its potential vorticity.
The solution north of this outcrop line is exactly the same as that obtained for
the two-layer model. This is an important point for it means that as we increase
the resolution by adding more layers, the previous solution still holds north of
the new outcrop lines and remains undisturbed by the addition of new layers.
Information in the ventilated region travels southward along the streamlines as
long as potential vorticity is conserved. We need not iterate the solution
already found as new layers are added. Thus merely letting 2 --+ 3 and 1 --+ 2,
we can rewrite the solution in the region north of 8 = 82. That is, for
83 ~ 8 ~ 80 the solution in which only layer 3 is moving is given by (4.4.4) or:
( 4.7.1)
where his the total depth of the moving fluid (equal to -z4 in this region) while
D5 is now defined as:
2/21e
D~ = --p
wE(',8)Rcos8 d
Y3
(4.7.2)
and H 3 is the constant depth of the lowest layer on the eastern boundary.
In the region 82 ~ 8 ~ 83 layers 2 and 3 are in motion and contain all the
structure described above. In particular, the boundary between the shadow
zone and the ventilated region in layer 3 follows the curve given by:
D2(<1> 8) = H2 Y2 (1 _1_)2
0
.,
3
f
Y3
3
(4.7.3)
while the relation in the ventilated zone between the thickness of the lowest
layer and the total depth is again given by the potential vorticity relation
determined at the outcrop line, (4.4.14) i.e.:
f /3
q3 = - = -
h3
h
so that
(4.7.4)
The solution itself in each of the ventilated, shadow zone and pool regions
is given by the formulae in Sections 4.4--4.6, with an obvious change in