The Linear Problem
41
In deriving (2.7.4) we have used the fact that the integral across the basin
of v must vanish to conserve mass and that the gradient of vorticity at the
eastern boundary is utterly negligible in comparison with its value on the
western boundary. The result (2.7.4a) shows us that in the linear model the flux
of vorticity into the western boundary layer at each latitude exactly balances
the vorticity input by the wind across the basin at the same latitude. The
vorticity balance is attained locally at each latitude. It is important to emphasize that this is a consequence of ignoring advective effects. Note that the
viscous flux required is proportional to the discrepancy between the value of
the interior streamfunction at the western boundary and the value (zero) the
actual streamfunction must attain there.
If instead of using (2.4.5) as the second boundary condition to determine
C(y) in (2.7.2) we were to use the no-stress, or slip, condition (2.4.6), the
solutions for ljJ, v, and ( would be:
(2.7.5)
The northward velocity and the vorticity are shown in Fig. 2.7.3. Now the
northward velocity has its maximum at x = 0. The vorticity is zero there and
negative for x > 0. The vorticity gradient is again negative at x = 0, and there
is, as in the case of the no-slip condition, a flux of positive relative vorticity into
1.0 ,---o;::--,----,----r--,----,----,---r------,
0.8
v
0.6
0.4
0.2
0
-o.2
rei. vort.
Fig. 2.7.3. As in Fig. 2.7.2 but -0.4
now with slip (no-stress) conditions
-0.6 ':---:c'-::--...,.-'-:---:-'c:-----:"-::--7-::---='c::---::-7-----:"
0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
x;sm