Inertial Runaway
87
solution Reynolds number is too large, the eddies which naturally develop
would produce a resolved turbulent flux (equivalent to a larger viscous flux)
and allow the inertial flow to deliver a sufficient flux of vorticity across mean
streamlines to the sublayer on the whole western boundary. The sublayer could
then flux that vorticity across the boundary in the manner that we have noted
above and allow a balance to be struck for each time-mean streamline. The
evidence for this conceptually attractive scenario is still problematic. Very few
numerical, full-basin calculations come close to the asymptotic limit of large
(j 1/ (j M in the no-slip case.
A few years ago Ierley, Young, and this author initiated a series of highresolution, low-dissipation computations of two and five-layer geostrophic
models which are dynamically similar, insofar as the recirculation is concerned,
to the single-layer model discussed in this chapter. The goal was to establish a
baseline circulation whose naturally resolved eddy field would saturate and
render the solutions independent of further changes in the explicit dissipation
AH. We were dismayed to find that every reduction in AH led to further gross
changes in the circulation pattern, and that convergence or saturation as a
function of Reynolds number was never achieved. Figure 2.14.5 shows two
images of the calculated circulation. Panel a shows the time-averaged upperlayer circulation of the two-layer model, which has a boundary-layer Reynolds
number equal to 10, corresponding to a value of fJJ/(jM = 2.2. The calculation
employs no-slip boundary conditions on the western and eastern boundaries
and slip conditions on the northern and southern boundaries. In this case we
see a large recirculation reaching nearly three-quarters of the way across the
basin. Further reduction in AH, leading to a value of bJ/bM = 4.7, yields the
circulation shown in Fig. 2.14. 5b. The recirculation reaches right across the
basin and the interior is increasingly disturbed by the recirculation. This is the
circulation with the largest bJ/bM that we were able to achieve numerically.
However, the development sequence is similar to that which we saw above for
the slip case in which the recirculation first extends across the northern
boundary until the eastern boundary is struck and then extends southward,
increasingly filling the basin. That last development has so far not been observed in no-slip calculations but may plausibly be expected.
If so, this would imply that for every boundary condition and for every
form of dissipation the very existence of the Sverdrup interior depends vitally
on the existence of a sufficiently large explicit frictional dissipation in the
western boundary current. If the explicit dissipation representing very smallscale turbulent motions is not sufficient, the solution achieves inertial runaway
and circulates ever faster, reaching speeds far in excess of those observed.
Internal Compensation in a Two-Gyre Model
There is one rather singular case in which the inertial runaway can be avoided.
This occurs when no net vorticity is put into the basin. This can be achieved in
87
solution Reynolds number is too large, the eddies which naturally develop
would produce a resolved turbulent flux (equivalent to a larger viscous flux)
and allow the inertial flow to deliver a sufficient flux of vorticity across mean
streamlines to the sublayer on the whole western boundary. The sublayer could
then flux that vorticity across the boundary in the manner that we have noted
above and allow a balance to be struck for each time-mean streamline. The
evidence for this conceptually attractive scenario is still problematic. Very few
numerical, full-basin calculations come close to the asymptotic limit of large
(j 1/ (j M in the no-slip case.
A few years ago Ierley, Young, and this author initiated a series of highresolution, low-dissipation computations of two and five-layer geostrophic
models which are dynamically similar, insofar as the recirculation is concerned,
to the single-layer model discussed in this chapter. The goal was to establish a
baseline circulation whose naturally resolved eddy field would saturate and
render the solutions independent of further changes in the explicit dissipation
AH. We were dismayed to find that every reduction in AH led to further gross
changes in the circulation pattern, and that convergence or saturation as a
function of Reynolds number was never achieved. Figure 2.14.5 shows two
images of the calculated circulation. Panel a shows the time-averaged upperlayer circulation of the two-layer model, which has a boundary-layer Reynolds
number equal to 10, corresponding to a value of fJJ/(jM = 2.2. The calculation
employs no-slip boundary conditions on the western and eastern boundaries
and slip conditions on the northern and southern boundaries. In this case we
see a large recirculation reaching nearly three-quarters of the way across the
basin. Further reduction in AH, leading to a value of bJ/bM = 4.7, yields the
circulation shown in Fig. 2.14. 5b. The recirculation reaches right across the
basin and the interior is increasingly disturbed by the recirculation. This is the
circulation with the largest bJ/bM that we were able to achieve numerically.
However, the development sequence is similar to that which we saw above for
the slip case in which the recirculation first extends across the northern
boundary until the eastern boundary is struck and then extends southward,
increasingly filling the basin. That last development has so far not been observed in no-slip calculations but may plausibly be expected.
If so, this would imply that for every boundary condition and for every
form of dissipation the very existence of the Sverdrup interior depends vitally
on the existence of a sufficiently large explicit frictional dissipation in the
western boundary current. If the explicit dissipation representing very smallscale turbulent motions is not sufficient, the solution achieves inertial runaway
and circulates ever faster, reaching speeds far in excess of those observed.
Internal Compensation in a Two-Gyre Model
There is one rather singular case in which the inertial runaway can be avoided.
This occurs when no net vorticity is put into the basin. This can be achieved in