Vertical Structure of the Abyssal Circulation
p 0 fu = -8pj8y
pHg = -8pj8z.
435
(7.7.6a,b,c)
The density appearing in (7.7.6c) is the density anomaly PH since prefers to the
horizontally varying pressure field. The thermal wind relation implies that:
(7.7.7a,b)
In (7.7.6) and (7.7.7) f = fo + f3y.
The condition of incompressibility of the abyssal flow is:
(7.7.8)
from which (7.7.1) follows by eliminating the pressure in the momentum
equations.
The density equation is a balance between advection and the diffusion of
density (or equivalently temperature and salinity) by small-scale turbulence. If
the assumption that the laterally varying density anomaly is small with respect
to Pv is used, the density equation reduces to the linear balance:
(7.7.9)
Here the assumption is made that it is vertical mixing of the density anomaly
which is important in the interior of the basin rather than horizontal mixing.
The density equation can be rewritten in terms of the pressure as:
" iJ3p
wN 2 = - -
Po oz3 .
(7.7.10)
On the lower boundary, i.e., at z = -H, the vertical velocity is zero, while at the
top of the abyss, at z = 0, the vertical velocity is given by w0 (x, y), thus, with
(7.7.10):
Z=-H
(7.7.11a,b)
z= 0.
The density anomaly on both the bottom at z = -Hand at the top of the
abyss are taken to be zero, although the latter condition can easily be relaxed.
However, the simplest model is one in which, as in the Kawase model, the
temperature anomalies are driven internally by the response to the upwelling
and the sources rather than externally imposed density forcing. Thus: