184
Theory of the Ventilated Thermocline
The smallness of the frictional terms in the vorticity equation and the slow
time scale of the large-scale motion allow ( 4.2.13) to be approximated by:
( 4.3.4)
which also follows from the divergence of (4.3.3). Note that in ignoring the
contribution of the frictional terms in the interior we are also ignoring the effect
of the cross-isopycnal momentum flux which is included in S'n ( 4.2.6). Assuming
that W* I hn is of the order of U I L from continuity of mass considerations, the
neglected term is of the same order as the horizontal advection of momentum
and is therefore negligible in the momentum and vorticity balances as long as
Ro and U I f3L 2 « 1.
Expanding (4.3.4) and using the continuity equation lead to the more
familiar form of the planetary vorticity relation:
awn
f3vn = f az .
( 4.3.5)
As in Chapter 3, the continuity of the cross-isopycnal velocity from one
density layer to the next across an isopycnal surface follows from elementary
considerations of mass conservation. The continuity of the vertical velocity
itself follows from the geostrophic balance for the horizontal velocity, so that
the vertical shear of the horizontal velocity across each density interface is
perpendicular to the direction of slope of the interface. The horizontal velocity
in the direction of the interface slope is therefore continuous. Thus, as shown in
Section 3.2, the vertical velocity itself is continuous across each density
interface. Just as in quasi-geostrophic theory we write the overall integral
constraint of the Sverdrup balance for the layer model as the sum:
(4.3.6)
n
where the sum is taken over all the moving layers.
We have discussed above the requirements for the validity of the Sverdrup
balance, most fundamental of which is the assumption that there is negligible
interaction with the ocean bottom. Indeed, in all the models described in this
chapter we assume either that there is a layer below which no motion occurs, or
if motion exists down to the bottom, that the bottom is flat and exerts a
negligible drag on the interior flow.
The hydrostatic relation (3.2.12) of course remains valid on the planetary
scale, as also does the condition on the dynamic part of the pressure field that is
obtained by matching the total pressure across each density interface. Thus,
(3.2.13 a,b) applies and can be written as a difference equation for the dynamic
pressure, nn, namely:
1rn - 'lrn-1 = Yn-JZn,
PJ+i- PJ
l'j =
g.
Po
(4.3.7a,b)
Theory of the Ventilated Thermocline
The smallness of the frictional terms in the vorticity equation and the slow
time scale of the large-scale motion allow ( 4.2.13) to be approximated by:
( 4.3.4)
which also follows from the divergence of (4.3.3). Note that in ignoring the
contribution of the frictional terms in the interior we are also ignoring the effect
of the cross-isopycnal momentum flux which is included in S'n ( 4.2.6). Assuming
that W* I hn is of the order of U I L from continuity of mass considerations, the
neglected term is of the same order as the horizontal advection of momentum
and is therefore negligible in the momentum and vorticity balances as long as
Ro and U I f3L 2 « 1.
Expanding (4.3.4) and using the continuity equation lead to the more
familiar form of the planetary vorticity relation:
awn
f3vn = f az .
( 4.3.5)
As in Chapter 3, the continuity of the cross-isopycnal velocity from one
density layer to the next across an isopycnal surface follows from elementary
considerations of mass conservation. The continuity of the vertical velocity
itself follows from the geostrophic balance for the horizontal velocity, so that
the vertical shear of the horizontal velocity across each density interface is
perpendicular to the direction of slope of the interface. The horizontal velocity
in the direction of the interface slope is therefore continuous. Thus, as shown in
Section 3.2, the vertical velocity itself is continuous across each density
interface. Just as in quasi-geostrophic theory we write the overall integral
constraint of the Sverdrup balance for the layer model as the sum:
(4.3.6)
n
where the sum is taken over all the moving layers.
We have discussed above the requirements for the validity of the Sverdrup
balance, most fundamental of which is the assumption that there is negligible
interaction with the ocean bottom. Indeed, in all the models described in this
chapter we assume either that there is a layer below which no motion occurs, or
if motion exists down to the bottom, that the bottom is flat and exerts a
negligible drag on the interior flow.
The hydrostatic relation (3.2.12) of course remains valid on the planetary
scale, as also does the condition on the dynamic part of the pressure field that is
obtained by matching the total pressure across each density interface. Thus,
(3.2.13 a,b) applies and can be written as a difference equation for the dynamic
pressure, nn, namely:
1rn - 'lrn-1 = Yn-JZn,
PJ+i- PJ
l'j =
g.
Po
(4.3.7a,b)