Quasi-Geostrophic Model with Continuous Stratification
149
0
Fig. 3.10.1. The continuously stratified model. The motion is confined to a region Z >-D(x,y). On
the boundary of the region the streamfunction and density are continuous with the surrounding
resting fluid
condition (3.10.13). This then is the situation we discussed in the introduction,
in which linear theory compresses the Sverdrup transport into an absurdly thin,
delta function zone at the upper boundary. The closing of geostrophic contours
and the homogenization of potential vorticity gives the fluid a way to establish
deep motion beneath the layers directly influenced by Ekman pumping and to
distribute the Sverdrup transport over a nonzero depth, D(x, y). We must find
the vertical and lateral extent of the field of motion and the streamfunction
within that zone.
For small dissipation the steady form of (3.10.5) implies that:
q = Q(l/f,z).
(3.10.14)
Here Q (not to be confused with the heating function, which is zero in this
example) is a function of streamfunction and vertical coordinate. This is similar
to the relation (3.8.4) where now z is a continuous counterpart to the layer label
n in the layer model. The relation (3.10.14) applies in the bowl region
-D :<:: z :<:: 0. If the potential vorticity is homogenized in this region in the sense
that for each z we have 8Qj 81/f = 0, then we may write, with the approximation
that the relative vorticity is negligible:
a [No
2
81/f]
q = y+- - - = Q(z) = yo(z)
8z N 2 8z
(3.10.15)
for-D:<:: z :<:: 0. Once y 0 is specified, we can integrate (3.10.15) and apply the
boundary conditions (3.10.12) and (3.10.13) to determine the motion in the
bowl-shaped pool. Since (3.10.15) is a second-order ordinary differential