Numerical and Observational Evidence
157
circulation and represent them adequately. One important reason for this is
that although the horizontal velocities may be discontinuous at the layer
interfaces, the vertical velocities are continuous, and it is w that drives the fluid
by stretching planetary vorticity filaments. One important difference between
the layer model and the continuous model is in the nature of the velocity at the
lower boundary of the bowl. In both the layer model and the continuous model
the boundary of the bowl is a locus of positions where the isopycnal surfaces
have a discontinuity in their slope, as seen in Fig. 3.10.6. In the layer model,
because of the 0(1) jump in the density across each interface, this slope is
accompanied by a finite jump in the velocity across the pool boundary. We
noted this above in our discussion of the streamline patterns in Sections 3.8 and
3.9. In the continuous model the sloping isopycnals, by the thermal wind
equation, support a jump in the shear of the velocity rather than the velocity
itself. The velocity across the bowl boundary is continuous with the resting
fluid around it and is thus zero at z = -D. As the number of layers increase,
and the density difference decreases in proportion, the velocity jumps of the
layer model diminish, and the velocity structure smoothly approaches that of
the continuous model.
The principal question that we have avoided so far is the validity of the
homogenization hypothesis for situations such as the gyre circulation where the
geostrophic contours "close" through the western boundary current.
3.11 Numerical and Observational Evidence
We have accepted the hypothesis of the homogenization of potential vorticity
and applied it in the previous two sections in the pool regions of the gyre where
the geostrophic contours or the isolines of potential vorticity are so bent by the
overlying wind-driven flow that they no longer strike the eastern boundary.
This escape of the contours from striking the eastern boundary really only
makes it plausible that motion occurs in such regions. Whether the potential
vorticity should be expected to be homogeneous within those regions depends
on a good deal more.
The discussion in Section 3.8 demonstrated that there are several physical
conditions that must be satisfied for homogenization. To summarize briefly,
these are:
1. The isolines of potential vorticity must close.
2. The dissipation of potential vorticity must be equivalent to a diffusion of
potential vorticity.
3. There must be nearly no dissipation of potential vorticity at the lowest
order, at least on the bounding contour of the "pool."
4. There must be no other sources of potential vorticity within the region
enclosed in the pool.
157
circulation and represent them adequately. One important reason for this is
that although the horizontal velocities may be discontinuous at the layer
interfaces, the vertical velocities are continuous, and it is w that drives the fluid
by stretching planetary vorticity filaments. One important difference between
the layer model and the continuous model is in the nature of the velocity at the
lower boundary of the bowl. In both the layer model and the continuous model
the boundary of the bowl is a locus of positions where the isopycnal surfaces
have a discontinuity in their slope, as seen in Fig. 3.10.6. In the layer model,
because of the 0(1) jump in the density across each interface, this slope is
accompanied by a finite jump in the velocity across the pool boundary. We
noted this above in our discussion of the streamline patterns in Sections 3.8 and
3.9. In the continuous model the sloping isopycnals, by the thermal wind
equation, support a jump in the shear of the velocity rather than the velocity
itself. The velocity across the bowl boundary is continuous with the resting
fluid around it and is thus zero at z = -D. As the number of layers increase,
and the density difference decreases in proportion, the velocity jumps of the
layer model diminish, and the velocity structure smoothly approaches that of
the continuous model.
The principal question that we have avoided so far is the validity of the
homogenization hypothesis for situations such as the gyre circulation where the
geostrophic contours "close" through the western boundary current.
3.11 Numerical and Observational Evidence
We have accepted the hypothesis of the homogenization of potential vorticity
and applied it in the previous two sections in the pool regions of the gyre where
the geostrophic contours or the isolines of potential vorticity are so bent by the
overlying wind-driven flow that they no longer strike the eastern boundary.
This escape of the contours from striking the eastern boundary really only
makes it plausible that motion occurs in such regions. Whether the potential
vorticity should be expected to be homogeneous within those regions depends
on a good deal more.
The discussion in Section 3.8 demonstrated that there are several physical
conditions that must be satisfied for homogenization. To summarize briefly,
these are:
1. The isolines of potential vorticity must close.
2. The dissipation of potential vorticity must be equivalent to a diffusion of
potential vorticity.
3. There must be nearly no dissipation of potential vorticity at the lowest
order, at least on the bounding contour of the "pool."
4. There must be no other sources of potential vorticity within the region
enclosed in the pool.