Continuous Models of the Ventilated Thermocline
253
the integral. When that is done, and ( 4.11.35) is also used in ( 4.11.34) we
obtain:
~1PD :l-d -:?( )aPn=2f 2 p0Rcos()
a¢ p,
P
PD a¢
g{3
W£.
( 4.11.36)
This can be integrated in longitude from the eastern boundary at¢= cPe to the
point at ¢ to obtain:
1
PD
1PD(,)
r"'
ap
:l-dp:l-(¢e)dp- Jm :l- (Pn) a: d¢'
Ps
p,(,)
,
'+'
2f2p r"' ·
= - f3g
0 } wERcos()dcjJ'.
( 4.11.37)
The third integral is given in terms only of the resting fluid density and its
variation along the slanting base of the thermocline and can be rewritten as an
integral over the density itself:
r"'
ap
1PD()
1PD()
lm z'l(pn) a: dc/J' =
z'l(pn)dpD =
z;(p)dp.
,
PD(,)
PD(,)
( 4.11.38)
The last step in ( 4.11.38) recognizes that the isopycnals are flat between the
eastern wall and their intersection of the base of the thermocline. Hence the
integral can be carried out in terms of the isopycnal depths on the eastern wall.
The density interval for the integration is given by the range of abyssal density
spanned along the thermocline base from its intersection with the eastern wall
to the point under consideration at ¢, e. When ( 4.11.38) is combined with
(4.11.37) we obtain as the statement of the Sverdrup balance:
1
PD
2f2p f"'•
1PD()
:l-dp = -T lm wERcos() d¢' + ZJ;dp.
Ps
g
p,(,)
( 4.11.39)
Although the derivation is complicated it is apparent that ( 4.11.39) is simply the
continuous limit of the statement ( 4.3.15) of the Sverdrup balance for the layer
model.
The statement of the proper boundary conditions on the eastern wall of the
basin presents the same difficulty in the continuous case as in the layer model
discussed in the previous section. A mixed layer in which both the layer depth
and the mixed layer density are functions of latitude on the eastern boundary
gives rise to a flow normal to the boundary. In the layer model we balanced this
flow with an opposite zonal flux in the layer just beneath the mixed layer
without describing the physics of the boundary layer that would effect the
required vertical mass flux. In the continuous model we could similarly specify
a mass flux in the region below the mixed layer so that its zonal mass flux
balances that of the mixed layer. This would require an arbitrary specification
of the distribution of this balancing flow with depth (or density). Several
253
the integral. When that is done, and ( 4.11.35) is also used in ( 4.11.34) we
obtain:
~1PD :l-d -:?( )aPn=2f 2 p0Rcos()
a¢ p,
P
PD a¢
g{3
W£.
( 4.11.36)
This can be integrated in longitude from the eastern boundary at¢= cPe to the
point at ¢ to obtain:
1
PD
1PD(,)
r"'
ap
:l-dp:l-(¢e)dp- Jm :l- (Pn) a: d¢'
Ps
p,(,)
,
'+'
2f2p r"' ·
= - f3g
0 } wERcos()dcjJ'.
( 4.11.37)
The third integral is given in terms only of the resting fluid density and its
variation along the slanting base of the thermocline and can be rewritten as an
integral over the density itself:
r"'
ap
1PD()
1PD()
lm z'l(pn) a: dc/J' =
z'l(pn)dpD =
z;(p)dp.
,
PD(,)
PD(,)
( 4.11.38)
The last step in ( 4.11.38) recognizes that the isopycnals are flat between the
eastern wall and their intersection of the base of the thermocline. Hence the
integral can be carried out in terms of the isopycnal depths on the eastern wall.
The density interval for the integration is given by the range of abyssal density
spanned along the thermocline base from its intersection with the eastern wall
to the point under consideration at ¢, e. When ( 4.11.38) is combined with
(4.11.37) we obtain as the statement of the Sverdrup balance:
1
PD
2f2p f"'•
1PD()
:l-dp = -T lm wERcos() d¢' + ZJ;dp.
Ps
g
p,(,)
( 4.11.39)
Although the derivation is complicated it is apparent that ( 4.11.39) is simply the
continuous limit of the statement ( 4.3.15) of the Sverdrup balance for the layer
model.
The statement of the proper boundary conditions on the eastern wall of the
basin presents the same difficulty in the continuous case as in the layer model
discussed in the previous section. A mixed layer in which both the layer depth
and the mixed layer density are functions of latitude on the eastern boundary
gives rise to a flow normal to the boundary. In the layer model we balanced this
flow with an opposite zonal flux in the layer just beneath the mixed layer
without describing the physics of the boundary layer that would effect the
required vertical mass flux. In the continuous model we could similarly specify
a mass flux in the region below the mixed layer so that its zonal mass flux
balances that of the mixed layer. This would require an arbitrary specification
of the distribution of this balancing flow with depth (or density). Several