Vertical Structure of the Abyssal Circulation
433
Wo=O
wo
N2.~
CR
8 w
X= Xe
~------- oR -------<~
Fig. 7.7.1. Schematic diagram showing a zonal cross section of a stratified abyssal region driven by
an upwelling velocity, w0, which is limited to the eastern portion of the basin.The abyss has a
background buoyancy frequency, N. The thermal anomaly produced by the upwelling propagates
westward with the Rossby wave speed cR and extends a distance DR westward as determined by a
balance between propagation and dissipation of the anomaly. In the western region the anomaly
arrives at a location where w0 is zero, but where the local vertical velocity, w, in the body of the fluid
produced by the local density anomaly is different from zero
The physical essentials of the idea can be grasped by considering a simple,
extreme conceptual example. Suppose, as in Fig. 7. 7.1, the upwelling into the
abyss is concentrated entirely on the eastern boundary as shown in the figure.
Suppose there is a background stratification measured by the buoyancy
frequency N. The upwelling in the eastern portion of the basin produces a
density anomaly locally which propagates westward as an ensemble of
baroclinic Rossby waves each with a different vertical structure. Each Rossby
wave extends westward a distance !JR which is appropriate to its vertical mode
structure. This distance arises as a balance between the propagation westward
as a Rossby wave and the decay of the anomaly through thermal dissipation.
The westward extent of the anomaly reaches into a region in which w0 is zero
(this can always be arranged conceptually by limiting the region in which w 0 is
different from zero to a narrow zone in the east). In this region the density
anomaly carried by the wave is nonzero and forces, locally, an accompanying
vertical velocity in consequence of the distortion of the density surfaces. The
vertical velocity in turn implies a meridional velocity as a consequence of the
Sverdrup vorticity equation:
f3v = 1 aw
az
(7.7.1)
whose validity is independent of the stratification. When (7. 7.1) is integrated
over the entire depth of the abyss:
433
Wo=O
wo
N2.~
CR
8 w
X= Xe
~------- oR -------<~
Fig. 7.7.1. Schematic diagram showing a zonal cross section of a stratified abyssal region driven by
an upwelling velocity, w0, which is limited to the eastern portion of the basin.The abyss has a
background buoyancy frequency, N. The thermal anomaly produced by the upwelling propagates
westward with the Rossby wave speed cR and extends a distance DR westward as determined by a
balance between propagation and dissipation of the anomaly. In the western region the anomaly
arrives at a location where w0 is zero, but where the local vertical velocity, w, in the body of the fluid
produced by the local density anomaly is different from zero
The physical essentials of the idea can be grasped by considering a simple,
extreme conceptual example. Suppose, as in Fig. 7. 7.1, the upwelling into the
abyss is concentrated entirely on the eastern boundary as shown in the figure.
Suppose there is a background stratification measured by the buoyancy
frequency N. The upwelling in the eastern portion of the basin produces a
density anomaly locally which propagates westward as an ensemble of
baroclinic Rossby waves each with a different vertical structure. Each Rossby
wave extends westward a distance !JR which is appropriate to its vertical mode
structure. This distance arises as a balance between the propagation westward
as a Rossby wave and the decay of the anomaly through thermal dissipation.
The westward extent of the anomaly reaches into a region in which w0 is zero
(this can always be arranged conceptually by limiting the region in which w 0 is
different from zero to a narrow zone in the east). In this region the density
anomaly carried by the wave is nonzero and forces, locally, an accompanying
vertical velocity in consequence of the distortion of the density surfaces. The
vertical velocity in turn implies a meridional velocity as a consequence of the
Sverdrup vorticity equation:
f3v = 1 aw
az
(7.7.1)
whose validity is independent of the stratification. When (7. 7.1) is integrated
over the entire depth of the abyss: