52
Homogeneous Models of the Ocean Circulation
infinity were satisfied, the calculation would be over. In the more likely case
that the initial guess is not correct, the value of the initial guess is iterated until
the boundary condition at x = oo is satisfied. In the case of the no-slip condition the missing initial value that must be guessed is the second derivative,
while in the case of slip boundary conditions it is the first derivative at the
origin that must be guessed. In each case Ierley and Ruehr called the guessed,
missing derivative at the origin /3. This is an unfortunate terminology from the
oceanographic point of view for it has nothing to do with the planetary vorticity gradient, but we retain their terminology for the discussion of this section
(in which the dimensional planetary vorticity gradient does not explicitly appear).
The results of careful numerical calculations are rather surprising.
Figure 2.9.1 shows the relation between A. and f3 for which solutions to the
boundary-layer problem could be found in the case of no-slip boundary conditions. The solid line is the numerical result (which agrees closely with the
earlier numerical work of Il'in and Kamenkovich 1964), and the dotted curve is
the result of an analytical approximation of Ierley and Ruehr based on an
expansion and continued fraction solution for small A.. For A. > 0 there is a
single solution represented by a single value of f3 for each A.. This is the branch
labeled II in the figure. For A. < 0 (eastward interior flow) there is a range of A.
for which two solutions are found, i.e., two values of f3 for each negative A.
greater than a critical value Ac. In the region where two solutions are possible
Ierley and Ruehr showed that the solutions on the upper branch are unstable,
and therefore only the solutions on the lower branch which continues smoothly
from A. > 0 are relevant. More important from the point of view of the circulation problem is that for A < Ac there are no boundary-layer solutions at all.
For the no-slip case this critical value of A.= -0.79130.
If this threshold were to occur for large negative values of A. where inertial
effects are dominant, the absence of solutions would call to mind Greenspan's
theorem on the impossibility of a purely inertial boundary layer for outflow
conditions at x = oo. However, the value of the critical value found by Ierley
and Ruehr and Il'in and Kamenkovich corresponds to a ratio of ch/1JM of
about 0.89. That is, even when the boundary layer is largely dissipative, so that
the inertial and viscous terms are of the same order, no boundary-layer solution can be found for outflow conditions. (Note that with Ierley and Ruehr's
slightly different definition of the inertial layer thickness the critical A would
correspond to an even smaller value of b db M.)
Figure 2.9.2 shows the profiles of ¢(x) (whose derivative yields v) for the
realizable solutions. The solid line shows the solution for A> 0 (A.= 2). The
solution is not essentially different than the inertial solution (2.8.6) except that
v vanishes at x = 0. The dotted line shows the solution near A= 0 for which
nonlinearity is relatively unimportant. Here, the solution is similar to the
Munk solution. Note the slight overshoot of¢ before it asymptotes to its final
value of unity as x tends to infinity. This overshoot describes the weak recirculation present in the Munk solution that we mentioned above. The dashed
Homogeneous Models of the Ocean Circulation
infinity were satisfied, the calculation would be over. In the more likely case
that the initial guess is not correct, the value of the initial guess is iterated until
the boundary condition at x = oo is satisfied. In the case of the no-slip condition the missing initial value that must be guessed is the second derivative,
while in the case of slip boundary conditions it is the first derivative at the
origin that must be guessed. In each case Ierley and Ruehr called the guessed,
missing derivative at the origin /3. This is an unfortunate terminology from the
oceanographic point of view for it has nothing to do with the planetary vorticity gradient, but we retain their terminology for the discussion of this section
(in which the dimensional planetary vorticity gradient does not explicitly appear).
The results of careful numerical calculations are rather surprising.
Figure 2.9.1 shows the relation between A. and f3 for which solutions to the
boundary-layer problem could be found in the case of no-slip boundary conditions. The solid line is the numerical result (which agrees closely with the
earlier numerical work of Il'in and Kamenkovich 1964), and the dotted curve is
the result of an analytical approximation of Ierley and Ruehr based on an
expansion and continued fraction solution for small A.. For A. > 0 there is a
single solution represented by a single value of f3 for each A.. This is the branch
labeled II in the figure. For A. < 0 (eastward interior flow) there is a range of A.
for which two solutions are found, i.e., two values of f3 for each negative A.
greater than a critical value Ac. In the region where two solutions are possible
Ierley and Ruehr showed that the solutions on the upper branch are unstable,
and therefore only the solutions on the lower branch which continues smoothly
from A. > 0 are relevant. More important from the point of view of the circulation problem is that for A < Ac there are no boundary-layer solutions at all.
For the no-slip case this critical value of A.= -0.79130.
If this threshold were to occur for large negative values of A. where inertial
effects are dominant, the absence of solutions would call to mind Greenspan's
theorem on the impossibility of a purely inertial boundary layer for outflow
conditions at x = oo. However, the value of the critical value found by Ierley
and Ruehr and Il'in and Kamenkovich corresponds to a ratio of ch/1JM of
about 0.89. That is, even when the boundary layer is largely dissipative, so that
the inertial and viscous terms are of the same order, no boundary-layer solution can be found for outflow conditions. (Note that with Ierley and Ruehr's
slightly different definition of the inertial layer thickness the critical A would
correspond to an even smaller value of b db M.)
Figure 2.9.2 shows the profiles of ¢(x) (whose derivative yields v) for the
realizable solutions. The solid line shows the solution for A> 0 (A.= 2). The
solution is not essentially different than the inertial solution (2.8.6) except that
v vanishes at x = 0. The dotted line shows the solution near A= 0 for which
nonlinearity is relatively unimportant. Here, the solution is similar to the
Munk solution. Note the slight overshoot of¢ before it asymptotes to its final
value of unity as x tends to infinity. This overshoot describes the weak recirculation present in the Munk solution that we mentioned above. The dashed