Homogenization of Potential Vorticity
135
the midocean. Therefore the Sverdrup vorticity balance for the barotropic field
is unlikely to be upset by the eddies. On the other hand, the fact that the largescale field is large compared to the Rossby deformation radius encourages
baroclinic instability (Pedlosky 1987) if the vertical shear exceeds a critical
value. This critical value for instability is exactly the value that would make the
potential vorticity gradient in the meridional direction vanish at the northern
boundary of the gyre where the velocity is zonal. If the motion in the lower
layer is zero, use of(3.5.14) shows that this condition would be identical to the
condition that closed geostrophic contours elude the eastern boundary, as is
shown in detail in the next section. The instability produces a rectified
transport of potential vorticity, as has been noted in many numerical
experiments (see, for example, Holland 1978). Indeed, the equivalence of the
condition for closed geostrophic contours and the criterion for instability
satisfyingly identifies the mechanism for the production of motion with the
necessary condition for the existence of that motion. When the motion is
allowed, the advent of baroclinic instability is capable of producing it.
The potential vorticity equation for the time averaged mean flow in layer 2,
temporarily denoted by an overbar, can be written, if the average effect of the
eddies is included:
(3.8.9)
where the primed variables represent the fluctuations in the eddy field.
Although the average of any linear fluctuation variable is zero by definition,
the average of the quadratic terms is generally not zero, giving rise to nonlinear transports of potential vorticity on the large scale.
If the fluctuations were small, linear perturbation theory would relate the
fluctuation of the potential vorticity to the Lagrangian displacements of the
fluctuation field. If e: is the fluctuation displacement in the ith coordinate
direction (i = 1, 2), linear theory suggests that:
I - _J!I8tjz
qz- '-ia ·
Xi
(3.8.10)
The convention of implied summation over the repeated index i = 1, 2, has
been used in (3.8.10) and x, = x while xz = y. The content of (3.8.9) is that a
parcel arrives at a distance e: from its initial position carrying with it the
potential vorticity of the mean field. This is a small amplitude, conservative
dynamics approximation for the perturbation field. If (3.8.10) is used to
evaluate the potential vorticity flux by the eddies:
135
the midocean. Therefore the Sverdrup vorticity balance for the barotropic field
is unlikely to be upset by the eddies. On the other hand, the fact that the largescale field is large compared to the Rossby deformation radius encourages
baroclinic instability (Pedlosky 1987) if the vertical shear exceeds a critical
value. This critical value for instability is exactly the value that would make the
potential vorticity gradient in the meridional direction vanish at the northern
boundary of the gyre where the velocity is zonal. If the motion in the lower
layer is zero, use of(3.5.14) shows that this condition would be identical to the
condition that closed geostrophic contours elude the eastern boundary, as is
shown in detail in the next section. The instability produces a rectified
transport of potential vorticity, as has been noted in many numerical
experiments (see, for example, Holland 1978). Indeed, the equivalence of the
condition for closed geostrophic contours and the criterion for instability
satisfyingly identifies the mechanism for the production of motion with the
necessary condition for the existence of that motion. When the motion is
allowed, the advent of baroclinic instability is capable of producing it.
The potential vorticity equation for the time averaged mean flow in layer 2,
temporarily denoted by an overbar, can be written, if the average effect of the
eddies is included:
(3.8.9)
where the primed variables represent the fluctuations in the eddy field.
Although the average of any linear fluctuation variable is zero by definition,
the average of the quadratic terms is generally not zero, giving rise to nonlinear transports of potential vorticity on the large scale.
If the fluctuations were small, linear perturbation theory would relate the
fluctuation of the potential vorticity to the Lagrangian displacements of the
fluctuation field. If e: is the fluctuation displacement in the ith coordinate
direction (i = 1, 2), linear theory suggests that:
I - _J!I8tjz
qz- '-ia ·
Xi
(3.8.10)
The convention of implied summation over the repeated index i = 1, 2, has
been used in (3.8.10) and x, = x while xz = y. The content of (3.8.9) is that a
parcel arrives at a distance e: from its initial position carrying with it the
potential vorticity of the mean field. This is a small amplitude, conservative
dynamics approximation for the perturbation field. If (3.8.10) is used to
evaluate the potential vorticity flux by the eddies: