252
Theory of the Ventilated Thermocline
Since the velocity field in the mixed layer beneath the Ekman layer is
geostrophic, the meridional velocity in the mixed layer is given by:
1
apm
1
{aPs
aps}
Vm = Po f R cos (} a¢ = Po f R cos (} a¢ - gz a¢
( 4.11.29)
whose integral over the mixed layer depth gives the meridional transport of the
mixed layer:
1 °
1
{ a Ps gh 2 aps }
Vm = -hm Vmdz= Po fRcosO h a¢ +2 a¢ .
(4.11.30)
The Sverdrup balance, over the region of moving fluid is:
1 0
1-hm
10
1-hm
J
vdz=
vdz+
Vmdz=
vdz+ Vm =-pWE·
-D
-D
-hm
-D
(4.11.31)
The integral over the region below the mixed layer can be written as an integral
in p over the equivalent interval of density, that is:
1 -hm 1Ps az 1PD az
vdz =
vadp = -
vadp.
-D
PD
p
Ps
p
(4.11.32)
The order of integration ( 4.11.32) is interchanged here so that the limits of
integration in density run from lower to higher values of p.
When (4.11.30) and (4.11.15) are used in (4.11.31) we obtain:
rD 1 an 8z f
Vm- }p, PofRcosOa¢apdp=pwE.
(4.11.33)
If the integral ( 4.11.33) is integrated by parts, and the boundary conditions are
used to evaluate the terms arising from the end points of the integration, we
obtain for the Sverdrup balance:
(4.11.34)
In evaluating the third term in ( 4.11.34) some care must be taken. The
derivative with respect to longitude is taken at constant density. Thus:
an) _ an) _ an ap _ a Ps
h aps
a¢ p - a¢ z ap a¢ - a¢ + g m a¢
at p = Ps where z = -hm.
( 4.11.35)
The derivative with respect to longitude can be brought outside of the
integral in ( 4.11.34) if proper attention is paid to the variation of the limits of