70
Homogeneous Models of the Ocean Circulation
layer. By mass continuity the order of magnitude of the velocity in the
boundary layer is:
L
I/2
Vi = UI (">J = [ UIP]
. L
(2.12.2)
where L is a characteristic length of the basin and hence of the extent along the
boundary of the boundary layer. From the dissipation integral (2.10.2), the
balance requires that, in order of magnitude:
fo
- rUI- [U p]I/2
- W E - - - T
I
D
{JI
(2.12.3)
or that:
(2.12.4)
where Us is the characteristic Sverdrup velocity,J0wE/ np, and {)1 is the inertial
boundary layer thickness based on that velocity (which is the a priori scaling
that we would use to estimate the inertial layer thickness). When the inertial
layer thickness, based on the Sverdrup estimate, exceeds the Stommel bottom
friction thickness, the interior velocity must rise to a level which exceeds the
Sverdrup value by the ratio of the inertial to bottom friction thicknesses. This
is an example of how the Sverdrup solution can be vitiated even though the
wind stress remains small enough that the Sverdrup theory appears to be selfconsistent in terms of its predictions of the size of the interior velocity and the
smallness, on that basis, of the nonlinear and frictional effects in the interior.
The point is that the Sverdrup solution is consistent in every way except in its
inability, in this limit, to close the circulation in a western boundary current
with enough dissipation to satisfy the global constraints of vorticity balance.
That inconsistency is then fatal to the Sverdrup solution's validity.
As r is decreased even further, the solution loses its boundary layer
character completely. In this limit the interior velocity is so large that the p
term in the vorticity equation becomes unimportant, i.e., U1 » PL 2 • If the p
term is ignored:
( L
2 )
fi o . nx . ny
1p= - - Wo-sm-sm2rn2
D
L
L
(2.12.5)
becomes an exact solution since the nonlinear terms cancel exactly and the
balance of the bottom friction term and the forcing leads directly to (2.12.5). A
more complete analysis has been given by Barcilon (unpubl, see also
Zimmerman 1993).
Note that in this limit dQ/ drjl = '\1 2 1/1 N < 0, in distinction to the Fofonoff
mode.
On the other hand, for the no-slip boundary condition we can examine the
overall vorticity balance with quite different results. Consider the vorticity
Homogeneous Models of the Ocean Circulation
layer. By mass continuity the order of magnitude of the velocity in the
boundary layer is:
L
I/2
Vi = UI (">J = [ UIP]
. L
(2.12.2)
where L is a characteristic length of the basin and hence of the extent along the
boundary of the boundary layer. From the dissipation integral (2.10.2), the
balance requires that, in order of magnitude:
fo
- rUI- [U p]I/2
- W E - - - T
I
D
{JI
(2.12.3)
or that:
(2.12.4)
where Us is the characteristic Sverdrup velocity,J0wE/ np, and {)1 is the inertial
boundary layer thickness based on that velocity (which is the a priori scaling
that we would use to estimate the inertial layer thickness). When the inertial
layer thickness, based on the Sverdrup estimate, exceeds the Stommel bottom
friction thickness, the interior velocity must rise to a level which exceeds the
Sverdrup value by the ratio of the inertial to bottom friction thicknesses. This
is an example of how the Sverdrup solution can be vitiated even though the
wind stress remains small enough that the Sverdrup theory appears to be selfconsistent in terms of its predictions of the size of the interior velocity and the
smallness, on that basis, of the nonlinear and frictional effects in the interior.
The point is that the Sverdrup solution is consistent in every way except in its
inability, in this limit, to close the circulation in a western boundary current
with enough dissipation to satisfy the global constraints of vorticity balance.
That inconsistency is then fatal to the Sverdrup solution's validity.
As r is decreased even further, the solution loses its boundary layer
character completely. In this limit the interior velocity is so large that the p
term in the vorticity equation becomes unimportant, i.e., U1 » PL 2 • If the p
term is ignored:
( L
2 )
fi o . nx . ny
1p= - - Wo-sm-sm2rn2
D
L
L
(2.12.5)
becomes an exact solution since the nonlinear terms cancel exactly and the
balance of the bottom friction term and the forcing leads directly to (2.12.5). A
more complete analysis has been given by Barcilon (unpubl, see also
Zimmerman 1993).
Note that in this limit dQ/ drjl = '\1 2 1/1 N < 0, in distinction to the Fofonoff
mode.
On the other hand, for the no-slip boundary condition we can examine the
overall vorticity balance with quite different results. Consider the vorticity