128
Vertical Structure: Baroclinic Quasi-Geostrophic Models
(x, y) = (0, r 1). At this point the barotropic streamfunction is zero and
{Jy = {Jr1 so that on this outer most curve ?12 = rJ2 0 = fJr1, and (3.7.14) becomes:
A2
{
y- YJ}
t/12 =
Hi t/ln + fJ~ .
A2 +r2 H
(3.7.15)
Once t/1 2 is known t/1 1 is determined from (3.5.9a).
Figure 3.7.2 shows the streamfunctions in the upper and lower layers for
the example of the disk forcing in the limit of negligible bottom friction and
thus when the form of the circulation is independent of the magnitude of the
frictional parameters. Panel a shows the lower layer streamfunction for the case
r1 / y 0 = 4. The circulation in layer 2 is limited to the zone of closed geostrophic
contours and its streamfunction is given by (3.7.15) with r2 = 0 and with t/Jn
given by (3.6.4). Panel b shows the circulation in the upper layer. Outside the
region of closed q2 contours the upper layer flow coincides with the streamlines
of the Sverdrup transport, and therefore the upper layer streamlines are
portions of circles centered at x = y = 0. Above the region of closed q 2 the upper
layer shares the total Sverdrup transport with the layer 2. The meridional
velocity in the upper layer is therefore less and the zonal velocity somewhat
larger. The streamlines thus "flatten" in this region, as can be seen in the figure.
At the boundary of the region there is a kink in the upper layer streamlines. In
order that the mass flux in the upper layer across the pool region be
continuous, the velocity normal to the pool boundary must be continuous. This
implies that the magnitude of the upper layer streamfunction is continuous at
the pool boundary. The tangential velocity in the upper layer is not continuous.
This follows from the fact that the tangential velocity in the lower layer is
discontinuous at the pool boundary, and that the tangential velocity of the
upper layer is determined by (3.7.8). That is, the Sverdrup velocity is
everywhere continuous so that if any component of the velocity in one layer is
discontinuous, the same component in the other layer must have a
compensating discontinuity. This implies that the first derivative of t/1 1 normal
to the pool boundary is discontinuous, and this shows up as a kink in the
streamlines in Fig. 3.7.2b.
The potential vorticity in the pool region in layer 2 can now be directly
determined from (3.5.14). For the form of the dissipation that we have chosen
(3.7.5) it follows that:
(3.7.16)
In the limit where bottom friction is unimportant h ~ 0), and where the
density jump across the interface between layers 2 and 3 is much larger than the
Vertical Structure: Baroclinic Quasi-Geostrophic Models
(x, y) = (0, r 1). At this point the barotropic streamfunction is zero and
{Jy = {Jr1 so that on this outer most curve ?12 = rJ2 0 = fJr1, and (3.7.14) becomes:
A2
{
y- YJ}
t/12 =
Hi t/ln + fJ~ .
A2 +r2 H
(3.7.15)
Once t/1 2 is known t/1 1 is determined from (3.5.9a).
Figure 3.7.2 shows the streamfunctions in the upper and lower layers for
the example of the disk forcing in the limit of negligible bottom friction and
thus when the form of the circulation is independent of the magnitude of the
frictional parameters. Panel a shows the lower layer streamfunction for the case
r1 / y 0 = 4. The circulation in layer 2 is limited to the zone of closed geostrophic
contours and its streamfunction is given by (3.7.15) with r2 = 0 and with t/Jn
given by (3.6.4). Panel b shows the circulation in the upper layer. Outside the
region of closed q2 contours the upper layer flow coincides with the streamlines
of the Sverdrup transport, and therefore the upper layer streamlines are
portions of circles centered at x = y = 0. Above the region of closed q 2 the upper
layer shares the total Sverdrup transport with the layer 2. The meridional
velocity in the upper layer is therefore less and the zonal velocity somewhat
larger. The streamlines thus "flatten" in this region, as can be seen in the figure.
At the boundary of the region there is a kink in the upper layer streamlines. In
order that the mass flux in the upper layer across the pool region be
continuous, the velocity normal to the pool boundary must be continuous. This
implies that the magnitude of the upper layer streamfunction is continuous at
the pool boundary. The tangential velocity in the upper layer is not continuous.
This follows from the fact that the tangential velocity in the lower layer is
discontinuous at the pool boundary, and that the tangential velocity of the
upper layer is determined by (3.7.8). That is, the Sverdrup velocity is
everywhere continuous so that if any component of the velocity in one layer is
discontinuous, the same component in the other layer must have a
compensating discontinuity. This implies that the first derivative of t/1 1 normal
to the pool boundary is discontinuous, and this shows up as a kink in the
streamlines in Fig. 3.7.2b.
The potential vorticity in the pool region in layer 2 can now be directly
determined from (3.5.14). For the form of the dissipation that we have chosen
(3.7.5) it follows that:
(3.7.16)
In the limit where bottom friction is unimportant h ~ 0), and where the
density jump across the interface between layers 2 and 3 is much larger than the