Numerical Examples
73
forcing over the area of the recirculation gyre to which (2.11.11) is applied, we
must have dQ/ dljl ~ 0 to achieve a balance in (2.11.11 ). That is, the total
vorticity tends to become uniform in this limit. At this level of moderate
nonlinearity the maximum transport in the basin, i.e., the maximum value of
the streamfunction exceeds the maximum of linear theory by a factor of 2.33.
As the ratio ch / (J M is increased, the zone of recirculation extends eastward
until it touches the eastern boundary and then extends southward for further
increases in bJ/bM, filling an ever larger part of the basin.
A very interesting heuristic model of the recirculation in this parameter
regime has been described by Cessi et al. (1987). They suggest that the recirculation is driven by the anomaly of total vorticity produced at the northern
boundary of the gyre by the nonlinear advection of vorticity in the western
boundary current. If this anomaly is specified, the recirculation can be calculated and is driven directly by the anomaly. The wind forcing enters only
indirectly into the problem to the extent that it produced the inertial boundary
current which has transported the anomalous vorticity to the northern
boundary. In the case where the recirculation is limited in the north-south
direction to less than the full extent of the basin, Cessi et al. provide an estimate
of the southward extent of the recirculation. If Qn is the total vorticity of the
field on the northern boundary, and fn is the planetary vorticity there, the
southward extent, A, of the recirculation gyre from the northern boundary is
given by:
A= 3(fn- Qn)/2{1.
(2.12.8)
Note that the total vorticity on the northern boundary, Qn, must be less than
the planetary valuefn· This is expected since the inertial boundary layer carries
fluid which has entered the boundary layer with total vorticity equal to fs,
which is the planetary vorticity that it has when it leaves the Sverdrup interior
at the southern latitude where f =Is < fn. As the circulation intensifies, and the
anomaly increases in strength, the north-south scale increases (see also Ierley
and Young 1988). Unfortunately, there is no simple deductive way to link the
anomaly which drives the recirculation to the external forcing and external
parameters, but the discussion of Cessi et al. is illuminating because it describes
a clear physical mechanism for the production of the recirculation related to
the nonlinear production of anomalous vorticity near the northern boundary.
Figure 2.12.5 shows another calculation in the sequence by Boning, this
time for bJ/bM = 1.5. The recirculation zone now fills most of the basin. The
Sverdrup interior and the western intensification have both disappeared. The
transport is an order of magnitude greater than would be found in linear
theory. This is quantitatively similar to the Veronis model with only bottom
friction when a resonance with the Fofonoff mode occurs. In the present case,
in the absence of bottom friction, the same use of (2.11.11) for a gyre that fills
the basin and is therefore in balance with an 0(1) forcing, tells us that contrary
to the Fofonoff mode, dQ/dl/1 < 0. We can in fact see from Fig. 2.12.5b the
coincidence of streamlines and isolines of total vorticity in the gyre.
73
forcing over the area of the recirculation gyre to which (2.11.11) is applied, we
must have dQ/ dljl ~ 0 to achieve a balance in (2.11.11 ). That is, the total
vorticity tends to become uniform in this limit. At this level of moderate
nonlinearity the maximum transport in the basin, i.e., the maximum value of
the streamfunction exceeds the maximum of linear theory by a factor of 2.33.
As the ratio ch / (J M is increased, the zone of recirculation extends eastward
until it touches the eastern boundary and then extends southward for further
increases in bJ/bM, filling an ever larger part of the basin.
A very interesting heuristic model of the recirculation in this parameter
regime has been described by Cessi et al. (1987). They suggest that the recirculation is driven by the anomaly of total vorticity produced at the northern
boundary of the gyre by the nonlinear advection of vorticity in the western
boundary current. If this anomaly is specified, the recirculation can be calculated and is driven directly by the anomaly. The wind forcing enters only
indirectly into the problem to the extent that it produced the inertial boundary
current which has transported the anomalous vorticity to the northern
boundary. In the case where the recirculation is limited in the north-south
direction to less than the full extent of the basin, Cessi et al. provide an estimate
of the southward extent of the recirculation. If Qn is the total vorticity of the
field on the northern boundary, and fn is the planetary vorticity there, the
southward extent, A, of the recirculation gyre from the northern boundary is
given by:
A= 3(fn- Qn)/2{1.
(2.12.8)
Note that the total vorticity on the northern boundary, Qn, must be less than
the planetary valuefn· This is expected since the inertial boundary layer carries
fluid which has entered the boundary layer with total vorticity equal to fs,
which is the planetary vorticity that it has when it leaves the Sverdrup interior
at the southern latitude where f =Is < fn. As the circulation intensifies, and the
anomaly increases in strength, the north-south scale increases (see also Ierley
and Young 1988). Unfortunately, there is no simple deductive way to link the
anomaly which drives the recirculation to the external forcing and external
parameters, but the discussion of Cessi et al. is illuminating because it describes
a clear physical mechanism for the production of the recirculation related to
the nonlinear production of anomalous vorticity near the northern boundary.
Figure 2.12.5 shows another calculation in the sequence by Boning, this
time for bJ/bM = 1.5. The recirculation zone now fills most of the basin. The
Sverdrup interior and the western intensification have both disappeared. The
transport is an order of magnitude greater than would be found in linear
theory. This is quantitatively similar to the Veronis model with only bottom
friction when a resonance with the Fofonoff mode occurs. In the present case,
in the absence of bottom friction, the same use of (2.11.11) for a gyre that fills
the basin and is therefore in balance with an 0(1) forcing, tells us that contrary
to the Fofonoff mode, dQ/dl/1 < 0. We can in fact see from Fig. 2.12.5b the
coincidence of streamlines and isolines of total vorticity in the gyre.