Numerical and Observational Evidence
161
on the edge of the pool where the outer potential vorticity contour must be
wrapped around the pool by inertial processes, subsequently enclosing the pool
with a constant boundary value for potential vorticity.
It is very important to note that in this experiment, as in most of the
detailed numerical investigations of the homogenization process, the wind field
that is used is antisymmetric about a midlatitude, yielding a double-gyre
circulation. There is no net input of vorticity. As in the Marshall double-gyre
calculation described in Chapter 2, this implies that the fluid need not export
an order 1 amount of potential vorticity through the western boundary. Eddies
can allow the flow to reach equilibration through fluxes of vorticity from the
subpolar to subtropical gyres where the vorticity input in one gyre is exactly
balanced in the other. This favors the homogenization process by obviating the
need for dissipation in the western boundary current.
One of the first numerical demonstrations, of the process of potential
vorticity homogenization was that of Rhines and Young (1982) who presented
the result of a calculation by Holland shown in Fig. 3.11.2. The figure, again,
shows the middle layer of a three-layer model in which the layer evolves to a
state in which the central zone possesses uniform potential vorticity. Panel b
shows a three-dimensional rendering of q2 over the gyre showing the plateau of
constant q2 . What is particularly interesting is that in addition to the
homogenization of the large-scale, time-averaged potential vorticity, the
potential vorticity fluctuations are also expelled from the pool region. Panel
c shows the instantaneous field of q2, and it is clear that there are also no
fluctuations of q2 in the pool region. This is consistent with a law such as (3.8.9)
in which the fluctuations of the potential vorticity are due to the eddy
advection of large-scale potential vorticity. If the potential vorticity is uniform
on the large scale, the turbulent motion of a fluid element carrying its potential
vorticity from one position to another leaves the local value of the potential
vorticity unchanged so that the eddying motion is not accompanied by a
fluctuation in the potential vorticity.
Thus, in those controlled cases where the physics is otherwise a priori
favorable to potential vorticity homogenization the process is seen to proceed
to completion even when the fluid must complete its circuit in a western
boundary current for which there is no adequate theory, and even when the
dissipation of large-scale potential vorticity is accomplished by an eddy field
whose effect on the mean flow is not specified a priori as a diffusion process.
This is a real conceptual triumph for the theory.
Fig. 3.11.1. Streamline and potential vorticity patterns for the time-averaged flow in a three-layer
quasi-geostrophic model. Left panels, streamfunction in layers 1-3; right panels, potential vorticity
in each layer. Layer 1 is in contact with the surface forcing. Layer 3 is very deep (4000 m) and has
no closed contours. Layer 2 has geostrophic contours which close through the western boundary
layer. Note the homogenization of potential vorticity in this region. (From Rhines and Schopp
1991)
161
on the edge of the pool where the outer potential vorticity contour must be
wrapped around the pool by inertial processes, subsequently enclosing the pool
with a constant boundary value for potential vorticity.
It is very important to note that in this experiment, as in most of the
detailed numerical investigations of the homogenization process, the wind field
that is used is antisymmetric about a midlatitude, yielding a double-gyre
circulation. There is no net input of vorticity. As in the Marshall double-gyre
calculation described in Chapter 2, this implies that the fluid need not export
an order 1 amount of potential vorticity through the western boundary. Eddies
can allow the flow to reach equilibration through fluxes of vorticity from the
subpolar to subtropical gyres where the vorticity input in one gyre is exactly
balanced in the other. This favors the homogenization process by obviating the
need for dissipation in the western boundary current.
One of the first numerical demonstrations, of the process of potential
vorticity homogenization was that of Rhines and Young (1982) who presented
the result of a calculation by Holland shown in Fig. 3.11.2. The figure, again,
shows the middle layer of a three-layer model in which the layer evolves to a
state in which the central zone possesses uniform potential vorticity. Panel b
shows a three-dimensional rendering of q2 over the gyre showing the plateau of
constant q2 . What is particularly interesting is that in addition to the
homogenization of the large-scale, time-averaged potential vorticity, the
potential vorticity fluctuations are also expelled from the pool region. Panel
c shows the instantaneous field of q2, and it is clear that there are also no
fluctuations of q2 in the pool region. This is consistent with a law such as (3.8.9)
in which the fluctuations of the potential vorticity are due to the eddy
advection of large-scale potential vorticity. If the potential vorticity is uniform
on the large scale, the turbulent motion of a fluid element carrying its potential
vorticity from one position to another leaves the local value of the potential
vorticity unchanged so that the eddying motion is not accompanied by a
fluctuation in the potential vorticity.
Thus, in those controlled cases where the physics is otherwise a priori
favorable to potential vorticity homogenization the process is seen to proceed
to completion even when the fluid must complete its circuit in a western
boundary current for which there is no adequate theory, and even when the
dissipation of large-scale potential vorticity is accomplished by an eddy field
whose effect on the mean flow is not specified a priori as a diffusion process.
This is a real conceptual triumph for the theory.
Fig. 3.11.1. Streamline and potential vorticity patterns for the time-averaged flow in a three-layer
quasi-geostrophic model. Left panels, streamfunction in layers 1-3; right panels, potential vorticity
in each layer. Layer 1 is in contact with the surface forcing. Layer 3 is very deep (4000 m) and has
no closed contours. Layer 2 has geostrophic contours which close through the western boundary
layer. Note the homogenization of potential vorticity in this region. (From Rhines and Schopp
1991)