Numerical Examples
67
ment about the impossibility of eastern boundary currents becomes irrelevant.
Finally, panel d shows the solution for fJI/fJs = 8. In this limit the east-west
asymmetry of the circulation is completely lost. As the nonlinearity increases,
the transport in the circulation also increases until, in the last panel, the
maximum transport of the anticyclonic cell is of the order of ten times larger
than would be anticipated by the Sverdrup theory! Indeed, in this limit the
Sverdrup theory for the interior is no longer relevant. The solution has resonated with a Fofonoff mode which, as we have seen, is consistent with a
circulation balanced only by bottom friction. Niiler (1966) exploited this fact
and used the circulation integral and the form of the Fofonoffmode to estimate
the large amplitude limit of the Veronis calculation with considerable success.
On the other hand, Bryan's (1963) calculations showed a completely different transformation as the degree of nonlinearity increases. In his model he
used no-slip conditions on the eastern and western boundaries and no-stress
conditions on the northern and southern boundaries. Figure 2.12.2 shows a
sequence of increasing fJI/fJM from 0.4 to about 0.9, after which the calculated
solution became unsteady due to instabilities in the western boundary layer.
The similarity to Veronis' calculation for weak nonlinearity is evident. The
advection of vorticity northward produces a similar north-south asymmetry.
However, for larger values of 6/ 6M the circulation develops a gyre of recirculation in the northwest corner and a damped wave train of one or two
oscillations in the region of eastward interior flow.
Blandford (1971) then constructed a model with both bottom friction and
lateral friction and tested the effects of using either no-slip or no-stress
boundary conditions (or neither, in the case of AH = 0). In Blandford's calculation 6/6 8 is 4, while 6/6M is approximately -3.7. Figure 2.12.3 shows the
results of three calculations. Each of the three solutions very nearly achieves a
steady state although the no-slip case possesses a small residual fluctuation in
the energy level due to eddies produced by instabilities in the western
boundary layer. In the first panel is the calculation with only bottom friction; that is, in this calculation alone, AH = 0. The result is qualitatively
similar to Veronis' calculation with the same value of 6/6 8 • A strong northern
and eastern boundary current feeds a large interior flow which departs
substantially from the Sverdrup solution. In the second panel, in which
the slip condition is applied, the effect of the horizontal diffusion eliminates the
tendency to form a basin wide Fofonoff mode as our previous discussion
anticipated. Instead, a Sverdrup interior is reached by the fluid after it circumvents a large zone of recirculation near the northern boundary, reminiscent
of the above calculations of Ierley (1987). The third panel shows the no-slip
case. The Sverdrup interior is again maintained by the presence of lateral
friction, and now the recirculation shrinks to a relatively small region near the
northwestern corner of the basin.
Recall that all three models have about the same overall dissipation time
scales, i.e., the Reynolds number and bottom friction level are the same in each
case. Only the type of dissipation or the boundary conditions differ, and yet the
67
ment about the impossibility of eastern boundary currents becomes irrelevant.
Finally, panel d shows the solution for fJI/fJs = 8. In this limit the east-west
asymmetry of the circulation is completely lost. As the nonlinearity increases,
the transport in the circulation also increases until, in the last panel, the
maximum transport of the anticyclonic cell is of the order of ten times larger
than would be anticipated by the Sverdrup theory! Indeed, in this limit the
Sverdrup theory for the interior is no longer relevant. The solution has resonated with a Fofonoff mode which, as we have seen, is consistent with a
circulation balanced only by bottom friction. Niiler (1966) exploited this fact
and used the circulation integral and the form of the Fofonoffmode to estimate
the large amplitude limit of the Veronis calculation with considerable success.
On the other hand, Bryan's (1963) calculations showed a completely different transformation as the degree of nonlinearity increases. In his model he
used no-slip conditions on the eastern and western boundaries and no-stress
conditions on the northern and southern boundaries. Figure 2.12.2 shows a
sequence of increasing fJI/fJM from 0.4 to about 0.9, after which the calculated
solution became unsteady due to instabilities in the western boundary layer.
The similarity to Veronis' calculation for weak nonlinearity is evident. The
advection of vorticity northward produces a similar north-south asymmetry.
However, for larger values of 6/ 6M the circulation develops a gyre of recirculation in the northwest corner and a damped wave train of one or two
oscillations in the region of eastward interior flow.
Blandford (1971) then constructed a model with both bottom friction and
lateral friction and tested the effects of using either no-slip or no-stress
boundary conditions (or neither, in the case of AH = 0). In Blandford's calculation 6/6 8 is 4, while 6/6M is approximately -3.7. Figure 2.12.3 shows the
results of three calculations. Each of the three solutions very nearly achieves a
steady state although the no-slip case possesses a small residual fluctuation in
the energy level due to eddies produced by instabilities in the western
boundary layer. In the first panel is the calculation with only bottom friction; that is, in this calculation alone, AH = 0. The result is qualitatively
similar to Veronis' calculation with the same value of 6/6 8 • A strong northern
and eastern boundary current feeds a large interior flow which departs
substantially from the Sverdrup solution. In the second panel, in which
the slip condition is applied, the effect of the horizontal diffusion eliminates the
tendency to form a basin wide Fofonoff mode as our previous discussion
anticipated. Instead, a Sverdrup interior is reached by the fluid after it circumvents a large zone of recirculation near the northern boundary, reminiscent
of the above calculations of Ierley (1987). The third panel shows the no-slip
case. The Sverdrup interior is again maintained by the presence of lateral
friction, and now the recirculation shrinks to a relatively small region near the
northwestern corner of the basin.
Recall that all three models have about the same overall dissipation time
scales, i.e., the Reynolds number and bottom friction level are the same in each
case. Only the type of dissipation or the boundary conditions differ, and yet the