Continuous Models of the Ventilated Thermocline
With the mass conservation equation ( 4.11.8) this becomes:
dq- (-q-) aw*
dt- 8zj8p 8p
where:
f
q = - 8zj8p ·
249
(4.11.17)
(4.11.18)
The minus sign is introduced into the definition ( 4.11.18) since az I 8p is
negative for a stably stratified ocean. The minus sign then defines the potential
vorticity as positive in the northern hemisphere as in the layer model's
definition (4.3.1) which again could be interpreted as a finite difference version
of (4.11.18). Indeed, (4.11.17) is identical, from this point of view, with the
layer model's potential vorticity equation (4.3.26) when midocean friction or
eddy momentum mixing is ignored.
For steady, adiabatic motion the function i1 8zj8p is horizontally
nondivergent and is derivable from a streamfunction, i.e.:
(4.11.19)
It follows that both q and n are constant on streamlines in each density surface
when the motion is adiabatic and so both q and n are functions only of t/J and p.
Eliminating t/1 between q and n yields:
f
q = - 8zj8p = Q(n, p).
(4.11.20)
This is the same statement as (4.3.18) where now the value of the density, p,
substitutes for the density layer index, n.
The physical model is shown in Fig. 4.11.1. This is similar to the model
discussed in Section 4.10. Now the density is a continuous function of position
everywhere. A mixed layer of variable thickness hm lies over a region in which
the thermocline motion is assumed to conserve potential vorticity. The model is
steady, and the density layers fall into three classes. There is a surface
z = -D( ¢, 8) below which the fluid is at rest, and where the density is a
function only of z and is given by the relation:
z = Zn(P)
(4.11.21)
which expresses the stratification of the resting ocean. Above this surface are
density surfaces which do not outcrop but are set into motion, as discussed in
Section 4.8, as recirculating fluid. These layers have homogenized potential
vorticity. Above these layers are layers which outcrop in the gyre and are