Numerical Examples
71
balance (2.10.2) applied to a streamline which is coincident with the basin
boundary.
The flux of vorticity through the side wall in the western boundary layer is
proportional to the eastward derivative of the vorticity. The flux occurs
through the sublayer with a thickness given by (2.8.9) while the tangential
velocity in the sublayer is the same order as that in the inertial layer. Thus in
the sublayer our estimate for the vorticity flux over a meridional length L is:
(2.12.6)
where use has again been made of mass conservation to relate the boundary
layer velocity, v;, to the interior velocity and of the definition of the inertial
layer thickness. The last step in the estimate chain of (2.12.6) uses the Sverdrup
relation. We see from (2.12.6) that for the basin as a whole there appears always
to be a possible balance between the vorticity input by the wind and the
frictional flux through the western boundary sublayer. Recall that in the noslip case the dissipation balance takes place latitude by latitude, and that there
can be no advective export of vorticity anomaly which would produce a large
recirculation as in the slip case. This makes at least plausible the relatively
small region of flow in Fig. 2.12.3c that departs from the pattern of a western
boundary layer/Sverdrup interior picture of the total solution. The perceptive
reader will have noted, however, that this still leaves a problem for those
streamlines that do not pass through the sublayer. In the limit l>J/l>M » 1 this
involves most of the Sverdrup streamlines. On these streamlines the vorticity
gradient is much smaller than in the sublayer because the length scale is now
lJ1 » l>., and the vorticity flux is therefore much less. How the vorticity balance
is achieved for such streamlines is left for Sections 2.13 and 2.14.
For the case of slip boundary conditions the sublayer has the same length
scale as in the no-slip case, but the vorticity gradient is weaker in the sublayer
by a factor of l>.jl>1 than in the no-slip case because now the sublayer acts only
to bring the velocity gradient to zero at the wall rather than the velocity. This
reduces the flux of vorticity through the side wall by the same factor, and the
local balance (2.12.6) is unattainable. Instead the balance is locally met in the
western boundary layer by the northward export of the anomalous vorticity
where it produces a large region of recirculation. The flow which rejoins the
interior recirculates around this small-scale gyre near the northern boundary.
This augments the dissipation of vorticity along the path of the streamline until
the balance is achieved.
A very interesting series of experiments with slip boundary conditions were
carried out by Boning (1986). Both lateral mixing and bottom friction were used.
In these experiments both the ratio l>J/l>M and the ratio l>s/l>M were varied.
71
balance (2.10.2) applied to a streamline which is coincident with the basin
boundary.
The flux of vorticity through the side wall in the western boundary layer is
proportional to the eastward derivative of the vorticity. The flux occurs
through the sublayer with a thickness given by (2.8.9) while the tangential
velocity in the sublayer is the same order as that in the inertial layer. Thus in
the sublayer our estimate for the vorticity flux over a meridional length L is:
(2.12.6)
where use has again been made of mass conservation to relate the boundary
layer velocity, v;, to the interior velocity and of the definition of the inertial
layer thickness. The last step in the estimate chain of (2.12.6) uses the Sverdrup
relation. We see from (2.12.6) that for the basin as a whole there appears always
to be a possible balance between the vorticity input by the wind and the
frictional flux through the western boundary sublayer. Recall that in the noslip case the dissipation balance takes place latitude by latitude, and that there
can be no advective export of vorticity anomaly which would produce a large
recirculation as in the slip case. This makes at least plausible the relatively
small region of flow in Fig. 2.12.3c that departs from the pattern of a western
boundary layer/Sverdrup interior picture of the total solution. The perceptive
reader will have noted, however, that this still leaves a problem for those
streamlines that do not pass through the sublayer. In the limit l>J/l>M » 1 this
involves most of the Sverdrup streamlines. On these streamlines the vorticity
gradient is much smaller than in the sublayer because the length scale is now
lJ1 » l>., and the vorticity flux is therefore much less. How the vorticity balance
is achieved for such streamlines is left for Sections 2.13 and 2.14.
For the case of slip boundary conditions the sublayer has the same length
scale as in the no-slip case, but the vorticity gradient is weaker in the sublayer
by a factor of l>.jl>1 than in the no-slip case because now the sublayer acts only
to bring the velocity gradient to zero at the wall rather than the velocity. This
reduces the flux of vorticity through the side wall by the same factor, and the
local balance (2.12.6) is unattainable. Instead the balance is locally met in the
western boundary layer by the northward export of the anomalous vorticity
where it produces a large region of recirculation. The flow which rejoins the
interior recirculates around this small-scale gyre near the northern boundary.
This augments the dissipation of vorticity along the path of the streamline until
the balance is achieved.
A very interesting series of experiments with slip boundary conditions were
carried out by Boning (1986). Both lateral mixing and bottom friction were used.
In these experiments both the ratio l>J/l>M and the ratio l>s/l>M were varied.