Inertial Runaway
83
a) R = 1 maxt/J = 5.37
b) R = 1 maxt/J = 14.2
c) R = 1 maxt/J = 382
Fig. 2.14.3a--c. Solutions on the lower, middle, and upper branch of the solution curve in the region
of multiple solutions for the case R =I and 15M/L = 0.02. (From Ierley and Sheremet 1995)
These are extraordinary results. At least for these boundary conditions the
solution to the general circulation problem is neither unique nor necessarily in
Sverdrup balance, even when it is unique, if R exceeds R£. The fact that these
critical values of R are order 1 and therefore in a range that cannot be dismissed as artificially high and unrealistic emphasizes the problematic character,
from a theoretical point of view of simple Sverdrup theory. If we believe that
Sverdrup theory is observationally plausible for the interior, the implication is
(if the present models and the slip condition are relevant) that the small-scale
mixing in the western boundary current is strong enough to keep the boundarylayer Reynolds number sufficiently low to prevent inertial runaway.
No-Slip Conditions
The numerical task of finding solutions for large ?h j (J M in the case of no-slip
conditions is made particularly difficult by the existence of the strong viscous
sublayer in which the tangential velocity is brought to rest. This is a source of
time-dependent motions, and the solution for Reynolds numbers » 1 is
typically time dependent. In the case of no-slip boundary conditions we saw
above in (2.12.6) that an apparent vorticity flux balance for the basin as a whole
can exist even for large boundary-layer Reynolds numbers. We might therefore
imagine that at least in this case the Sverdrup theory for the interior remains
valid as the value of the explicit dissipation parameter AH is continuously
reduced.
However, this consideration is relevant only to the streamline which coincides with the basin boundary. In the limit (J I/ (J M » 1, the boundary layer for
the steady problem splits into an outer inertial layer and the viscous sublayer.
Most of the streamlines flow in the outer layer and only a small fraction,
( (J I/ (J M) -I, are in the sublayer. For the streamlines that traverse the western
boundary-layer region in the outer layer there is an insufficiency crisis for the
flux of vorticity across the boundary of the area encircled by the streamline. If
83
a) R = 1 maxt/J = 5.37
b) R = 1 maxt/J = 14.2
c) R = 1 maxt/J = 382
Fig. 2.14.3a--c. Solutions on the lower, middle, and upper branch of the solution curve in the region
of multiple solutions for the case R =I and 15M/L = 0.02. (From Ierley and Sheremet 1995)
These are extraordinary results. At least for these boundary conditions the
solution to the general circulation problem is neither unique nor necessarily in
Sverdrup balance, even when it is unique, if R exceeds R£. The fact that these
critical values of R are order 1 and therefore in a range that cannot be dismissed as artificially high and unrealistic emphasizes the problematic character,
from a theoretical point of view of simple Sverdrup theory. If we believe that
Sverdrup theory is observationally plausible for the interior, the implication is
(if the present models and the slip condition are relevant) that the small-scale
mixing in the western boundary current is strong enough to keep the boundarylayer Reynolds number sufficiently low to prevent inertial runaway.
No-Slip Conditions
The numerical task of finding solutions for large ?h j (J M in the case of no-slip
conditions is made particularly difficult by the existence of the strong viscous
sublayer in which the tangential velocity is brought to rest. This is a source of
time-dependent motions, and the solution for Reynolds numbers » 1 is
typically time dependent. In the case of no-slip boundary conditions we saw
above in (2.12.6) that an apparent vorticity flux balance for the basin as a whole
can exist even for large boundary-layer Reynolds numbers. We might therefore
imagine that at least in this case the Sverdrup theory for the interior remains
valid as the value of the explicit dissipation parameter AH is continuously
reduced.
However, this consideration is relevant only to the streamline which coincides with the basin boundary. In the limit (J I/ (J M » 1, the boundary layer for
the steady problem splits into an outer inertial layer and the viscous sublayer.
Most of the streamlines flow in the outer layer and only a small fraction,
( (J I/ (J M) -I, are in the sublayer. For the streamlines that traverse the western
boundary-layer region in the outer layer there is an insufficiency crisis for the
flux of vorticity across the boundary of the area encircled by the streamline. If