Midocean Approximations
187
( 4.3.16)
This is a Lagrangian description of the variation of the potential vorticity
evolution following an individual fluid element. It is complementary to the
Eulerian view which is emphasized in the flux formulation, i.e., when the
friction term can be neglected in the midocean:
(4.3.17)
where hnqn = f in the mid ocean approximation.
Note again that (4.3.17) applies regardless of whether there are crossisopycnal fluxes. There can be a flux of large-scale potential vorticity into a
layer in the midocean only if the layer has a boundary in the horizontal plane
across which there can be a normal velocity. One such boundary is an
outcropping line in which the edge of the layer, adjacent to the sea surface (or
mixed layer), allows potential vorticity to enter the layer and so distort the
potential vorticity contours from latitude circles which would be the isolines
for a resting fluid.
In steady, frictionless, and adiabatic flow the potential vorticity is a
constant on streamlines in all layers not locally forced by the Ekman pumping.
For the geostrophic approximation the streamlines coincide with isobars (or
isolines of the Bernoulli function). Thus for adiabatic, frictionless flow the
potential vorticity satisfies:
f
qn = hn = Qn(nn)·
(4.3.18)
Equivalently:
(4.3.19)
where IIn is the inverse function to Qn.
If the isolines of qn = f /hn strike the eastern boundary along which the
pressure anomaly is constant in each layer (to avoid flow through the eastern
wall) that constant value is then carried by (4.3.19) throughout the whole
region reached by those isolines. The pressure is constant in these regions, and
the layer is therefore at rest. Again, in order that the layers beneath the upper
layer can be in motion, the potential vorticity isolines in these layers must be
sufficiently curved to avoid the eastern boundary. Now, if the layer outcrops in
the subtropical gyre, its layer thickness varies from zero at the outcrop line to
order 1 farther south, induced by the filling of the layer by Ekman downwelling. This is sufficient, as we see below, to introduce a family of potential
vorticity isolines in each layer which start at the outcrop line instead of at the
eastern boundary. Thus the process of outcropping, i.e., ventilation, yields
another mechanism by which the potential vorticity isolines, which are the
highways of the flow in the nondissipative limit, can escape contact with the
187
( 4.3.16)
This is a Lagrangian description of the variation of the potential vorticity
evolution following an individual fluid element. It is complementary to the
Eulerian view which is emphasized in the flux formulation, i.e., when the
friction term can be neglected in the midocean:
(4.3.17)
where hnqn = f in the mid ocean approximation.
Note again that (4.3.17) applies regardless of whether there are crossisopycnal fluxes. There can be a flux of large-scale potential vorticity into a
layer in the midocean only if the layer has a boundary in the horizontal plane
across which there can be a normal velocity. One such boundary is an
outcropping line in which the edge of the layer, adjacent to the sea surface (or
mixed layer), allows potential vorticity to enter the layer and so distort the
potential vorticity contours from latitude circles which would be the isolines
for a resting fluid.
In steady, frictionless, and adiabatic flow the potential vorticity is a
constant on streamlines in all layers not locally forced by the Ekman pumping.
For the geostrophic approximation the streamlines coincide with isobars (or
isolines of the Bernoulli function). Thus for adiabatic, frictionless flow the
potential vorticity satisfies:
f
qn = hn = Qn(nn)·
(4.3.18)
Equivalently:
(4.3.19)
where IIn is the inverse function to Qn.
If the isolines of qn = f /hn strike the eastern boundary along which the
pressure anomaly is constant in each layer (to avoid flow through the eastern
wall) that constant value is then carried by (4.3.19) throughout the whole
region reached by those isolines. The pressure is constant in these regions, and
the layer is therefore at rest. Again, in order that the layers beneath the upper
layer can be in motion, the potential vorticity isolines in these layers must be
sufficiently curved to avoid the eastern boundary. Now, if the layer outcrops in
the subtropical gyre, its layer thickness varies from zero at the outcrop line to
order 1 farther south, induced by the filling of the layer by Ekman downwelling. This is sufficient, as we see below, to introduce a family of potential
vorticity isolines in each layer which start at the outcrop line instead of at the
eastern boundary. Thus the process of outcropping, i.e., ventilation, yields
another mechanism by which the potential vorticity isolines, which are the
highways of the flow in the nondissipative limit, can escape contact with the