80
Homogeneous Models of the Ocean Circulation
draining away the energy of the large-scale motion. This would leave the explicit parameterization of small-scale dissipation, acting directly on the largescale, as the only way to balance the large-scale energy and vorticity input by
the wind. It would be futile in this case to expect the resolved eddy field to
inhibit the amplitude of the large-scale motion.
In the presence of the f3 effect Rhines (1975) showed that there is a scale of
the order of /fiTf[ above which the eddy energy is unable to cascade. However, the value of U is determined by the motion field itself. If the circulation is
limited to the order of the Sverdrup velocity, we know the above parameter
would be of the order of the inertial boundary-layer width, which is a modest
100 km. On the other hand, if the explicit dissipation is unable to put a brake
on the large-scale field, U might be much larger, and the Rhines arrest scale
would be larger as well. Hence the barrier to the cascade to larger scales is part
of the answer to the problem and cannot be deduced a priori.
We have already seen in the calculations of Boning (1986) the tendency for
the circulation, in the case with slip boundary conditions, to become much
stronger than the Sverdrup theory would predict as the boundary-layer Reynolds number becomes large. This is not entirely surprising either, given our
global estimate in Section 2.12, of the inadequate flux of vorticity in the sublayer in the slip case when bJ/bM » 1. We discuss the no-slip case below;
however, the numerical demands placed on calculations with the no-slip condition (because of the rapid variation of the velocity in the sublayer) make it
difficult to examine the limit fJ I/ fJ M » 1 for the no-slip condition.
Slip Conditions
The most extensive and careful calculations of the steady, slip condition
problem have been carried out recently by Ierley and Sheremet (1995).
Calculations by Boning previously showed that for large fJ I/ fJ M allowing time
dependence nevertheless naturally yields a steady solution for reasons that we
elaborate below. Hence, indeed especially, for large bJ/bM steady solutions are
the relevant solutions to the full problem.
Figure 2.14.1 shows a schematic diagram of the solution curve of the
steady solutions obtained numerically by Ierley and Sheremet. On the abscissa
is plotted the boundary-layer Reynolds number R = (fJI/fJM) 3 . On the ordinate
is the maximum value of the streamfunction. For small values of R solutions
resembling the Munk solution are found. As R increases, the circulation becomes more nonlinear, as we saw in Boning's solutions. The remarkable feature discovered by Ierley and Sheremet, and shown in the figure, is the existence
in a range of R of multiple solutions of the vorticity equation for the same
forcing and dissipation. Furthermore, this multiplicity of solutions occurs in a
very moderate range of R. For example, for fJM/ L = 0.04 , the multiplicity
occurs in the range RH = 1.0377:S:R:S:l.3203 = RL. As might be expected, solutions in the middle branch are found to be unstable with respect to the
Homogeneous Models of the Ocean Circulation
draining away the energy of the large-scale motion. This would leave the explicit parameterization of small-scale dissipation, acting directly on the largescale, as the only way to balance the large-scale energy and vorticity input by
the wind. It would be futile in this case to expect the resolved eddy field to
inhibit the amplitude of the large-scale motion.
In the presence of the f3 effect Rhines (1975) showed that there is a scale of
the order of /fiTf[ above which the eddy energy is unable to cascade. However, the value of U is determined by the motion field itself. If the circulation is
limited to the order of the Sverdrup velocity, we know the above parameter
would be of the order of the inertial boundary-layer width, which is a modest
100 km. On the other hand, if the explicit dissipation is unable to put a brake
on the large-scale field, U might be much larger, and the Rhines arrest scale
would be larger as well. Hence the barrier to the cascade to larger scales is part
of the answer to the problem and cannot be deduced a priori.
We have already seen in the calculations of Boning (1986) the tendency for
the circulation, in the case with slip boundary conditions, to become much
stronger than the Sverdrup theory would predict as the boundary-layer Reynolds number becomes large. This is not entirely surprising either, given our
global estimate in Section 2.12, of the inadequate flux of vorticity in the sublayer in the slip case when bJ/bM » 1. We discuss the no-slip case below;
however, the numerical demands placed on calculations with the no-slip condition (because of the rapid variation of the velocity in the sublayer) make it
difficult to examine the limit fJ I/ fJ M » 1 for the no-slip condition.
Slip Conditions
The most extensive and careful calculations of the steady, slip condition
problem have been carried out recently by Ierley and Sheremet (1995).
Calculations by Boning previously showed that for large fJ I/ fJ M allowing time
dependence nevertheless naturally yields a steady solution for reasons that we
elaborate below. Hence, indeed especially, for large bJ/bM steady solutions are
the relevant solutions to the full problem.
Figure 2.14.1 shows a schematic diagram of the solution curve of the
steady solutions obtained numerically by Ierley and Sheremet. On the abscissa
is plotted the boundary-layer Reynolds number R = (fJI/fJM) 3 . On the ordinate
is the maximum value of the streamfunction. For small values of R solutions
resembling the Munk solution are found. As R increases, the circulation becomes more nonlinear, as we saw in Boning's solutions. The remarkable feature discovered by Ierley and Sheremet, and shown in the figure, is the existence
in a range of R of multiple solutions of the vorticity equation for the same
forcing and dissipation. Furthermore, this multiplicity of solutions occurs in a
very moderate range of R. For example, for fJM/ L = 0.04 , the multiplicity
occurs in the range RH = 1.0377:S:R:S:l.3203 = RL. As might be expected, solutions in the middle branch are found to be unstable with respect to the