Inertial Runaway
79
2.14 Inertial Runaway
The original hypothesis in our approach to constructing the dynamic picture of
the ocean circulation started with the notion of a Sverdrup interior. The
incomplete character of the Sverdrup theory means that its validity depends on
the existence of a western boundary current to accept the Sverdrup transport in
the south and return it in the north after having made sufficient alterations of
the total vorticity of the fluid in its travel through the boundary layer so it can
smoothly rejoin the interior. Furthermore, our interest is in a western
boundary layer, whose existence, basic structure, and ability to perform the
tasks just described are not sensitive to the strength or type of the explicit
dissipative mixing or the boundary condition applied at the western boundary.
These hopes, really, are based on an understanding that neither the
parameterization of the mixing or the applied boundary condition are known
or deducible from first principles. We would prefer that the global structure of
the ocean circulation did not depend on our arbitrary representations of smallscale mixing.
If dissipation of the large-scale motion is important, we might hope, as has
often been expressed, that with sufficiently powerful computers and sufficiently
fine spatial and temporal resolution the naturally occurring eddy field due to
instabilities of the large-scale flow would provide the necessary brake on the
circulation and allow us, as the resolution is continuously improved, to simultaneously reduce the level of explicit dissipation, AH, say, until the calculated solutions become independent of the explicit mixing parameter. The
eddies would take over the role of the explicit dissipation and produce a deductive relation between the large-scale motion and its own dissipation. Finally, in such a case our circulation would saturate at a level and structure
independent of AH in a realistic circulation structure independent of arbitrary
parameterizations.
There are several reasons why, after some thought, we should anticipate
difficulty with such an asymptotic scenario. First of all, we have seen that the
boundary conditions associated with the explicit dissipation, no matter how
small, are vital in determining the ability of the flow to transfer vorticity into or
out of the basin. Regardless of the advection of vorticity by the eddies within
the basin, the flux of vorticity through either the side wall or the bottom
depends on the level of the explicit dissipation. As we have seen, whether this
flux occurs through the side walls or the bottom depends on the boundary
condition at the side wall.
The motion of the fluid, even on the eddy scale of oceanic turbulence, is
fundamentally two-dimensional. Oceanic eddies that we might hope to resolve
have horizontal scales of tens to hundreds of kilometers and vertical scales of
only 1-2 km. A fundamental attribute of such two-dimensional turbulence is
the transfer of energy from small to large scales (Batchelor 1969). The effect of
the eddies in two-dimensional motion is to energize larger scales rather than
79
2.14 Inertial Runaway
The original hypothesis in our approach to constructing the dynamic picture of
the ocean circulation started with the notion of a Sverdrup interior. The
incomplete character of the Sverdrup theory means that its validity depends on
the existence of a western boundary current to accept the Sverdrup transport in
the south and return it in the north after having made sufficient alterations of
the total vorticity of the fluid in its travel through the boundary layer so it can
smoothly rejoin the interior. Furthermore, our interest is in a western
boundary layer, whose existence, basic structure, and ability to perform the
tasks just described are not sensitive to the strength or type of the explicit
dissipative mixing or the boundary condition applied at the western boundary.
These hopes, really, are based on an understanding that neither the
parameterization of the mixing or the applied boundary condition are known
or deducible from first principles. We would prefer that the global structure of
the ocean circulation did not depend on our arbitrary representations of smallscale mixing.
If dissipation of the large-scale motion is important, we might hope, as has
often been expressed, that with sufficiently powerful computers and sufficiently
fine spatial and temporal resolution the naturally occurring eddy field due to
instabilities of the large-scale flow would provide the necessary brake on the
circulation and allow us, as the resolution is continuously improved, to simultaneously reduce the level of explicit dissipation, AH, say, until the calculated solutions become independent of the explicit mixing parameter. The
eddies would take over the role of the explicit dissipation and produce a deductive relation between the large-scale motion and its own dissipation. Finally, in such a case our circulation would saturate at a level and structure
independent of AH in a realistic circulation structure independent of arbitrary
parameterizations.
There are several reasons why, after some thought, we should anticipate
difficulty with such an asymptotic scenario. First of all, we have seen that the
boundary conditions associated with the explicit dissipation, no matter how
small, are vital in determining the ability of the flow to transfer vorticity into or
out of the basin. Regardless of the advection of vorticity by the eddies within
the basin, the flux of vorticity through either the side wall or the bottom
depends on the level of the explicit dissipation. As we have seen, whether this
flux occurs through the side walls or the bottom depends on the boundary
condition at the side wall.
The motion of the fluid, even on the eddy scale of oceanic turbulence, is
fundamentally two-dimensional. Oceanic eddies that we might hope to resolve
have horizontal scales of tens to hundreds of kilometers and vertical scales of
only 1-2 km. A fundamental attribute of such two-dimensional turbulence is
the transfer of energy from small to large scales (Batchelor 1969). The effect of
the eddies in two-dimensional motion is to energize larger scales rather than