420
Abyssal Circulation
Kawase presents a discussion of a dynamics in which the upwelling depends
only on the motion of the abyss. This represents an opposite extreme from the
classical theory. The Stommel-Arons model imagines the upwelling as a
response to the wind- and buoyancy-driven thermocline motion. The
thermocline dynamics provides an upwelling velocity, w0, at the base of the
thermocline, which is considered given. In Kawase's model there is no wind
driven circulation and the forcing for the interior circulation must be generated
entirely internally, i.e., there is no forcing from the thermocline. As we see
below, this places a stronger burden on the abyssal dynamics than is perhaps
realistic dynamically, but the theory is very revealing in its delineation of the
way in which the circulation is established and controlled by baroclinic wave
processes.
The fundamental model is shown in Fig. 7.6.1. The ocean is represented as
a two-layer model and, as in the Stommel-Arons theory, the lower layer
represents the abyss. The layers in the absence of motion have thicknesses H1
and H2, and the subscript 1 for all variables refers to the upper layer and 2
refers to the lower layer. The departure of the interface from its rest position is
h. A source of deep water is introduced at the northwest boundary of the basin
with a strength Q, and this represents a flux of water from the upper to the
lower layer occurring outside the region to which the dynamics directly applies.
The total upwelling through the interface between the two layers must equal Q
in the steady state.
.
Although it is possible to write the equations in spherical coordinates, it is
useful to simplify the discussion to its essentials by employing cartesian
coordinates. The basin, however, straddles the equator. In fact the basin itself
(although not the placement of the sources) is symmetrical about the equator
and the equatorial P-plane approximation is used in which the . Corio lis
parameter is approximated as f = f3y.
If p: is the total pressure in the nth layer, then the linear, time-dependent
horizontal momentum equations for the two-layer model can be written as:
OUn
A
_
p: _
~+ jk X Un = -'i;J-- KUn·
ut
Po
(7.6.1)
In (7.6.1) the dynamics is linearized as in the Stommel-Arons theory, and
the Boussinesq approximation is made so that the density, in the horizontal
equations of motion, is replaced by the mean density p 0 • A simple drag law,
proportional to the velocity, with coefficient " is introduced to allow the
representation of the western and equatorial boundary layers where needed.
Otherwise this term is considered negligible.
The hydrostatic approximation to the vertical equation of motion implies
that:
P~ = -pngz + Pn(x, y, t)
(7.6.2)
while continuity of total pressure at the interface implies that:
Abyssal Circulation
Kawase presents a discussion of a dynamics in which the upwelling depends
only on the motion of the abyss. This represents an opposite extreme from the
classical theory. The Stommel-Arons model imagines the upwelling as a
response to the wind- and buoyancy-driven thermocline motion. The
thermocline dynamics provides an upwelling velocity, w0, at the base of the
thermocline, which is considered given. In Kawase's model there is no wind
driven circulation and the forcing for the interior circulation must be generated
entirely internally, i.e., there is no forcing from the thermocline. As we see
below, this places a stronger burden on the abyssal dynamics than is perhaps
realistic dynamically, but the theory is very revealing in its delineation of the
way in which the circulation is established and controlled by baroclinic wave
processes.
The fundamental model is shown in Fig. 7.6.1. The ocean is represented as
a two-layer model and, as in the Stommel-Arons theory, the lower layer
represents the abyss. The layers in the absence of motion have thicknesses H1
and H2, and the subscript 1 for all variables refers to the upper layer and 2
refers to the lower layer. The departure of the interface from its rest position is
h. A source of deep water is introduced at the northwest boundary of the basin
with a strength Q, and this represents a flux of water from the upper to the
lower layer occurring outside the region to which the dynamics directly applies.
The total upwelling through the interface between the two layers must equal Q
in the steady state.
.
Although it is possible to write the equations in spherical coordinates, it is
useful to simplify the discussion to its essentials by employing cartesian
coordinates. The basin, however, straddles the equator. In fact the basin itself
(although not the placement of the sources) is symmetrical about the equator
and the equatorial P-plane approximation is used in which the . Corio lis
parameter is approximated as f = f3y.
If p: is the total pressure in the nth layer, then the linear, time-dependent
horizontal momentum equations for the two-layer model can be written as:
OUn
A
_
p: _
~+ jk X Un = -'i;J-- KUn·
ut
Po
(7.6.1)
In (7.6.1) the dynamics is linearized as in the Stommel-Arons theory, and
the Boussinesq approximation is made so that the density, in the horizontal
equations of motion, is replaced by the mean density p 0 • A simple drag law,
proportional to the velocity, with coefficient " is introduced to allow the
representation of the western and equatorial boundary layers where needed.
Otherwise this term is considered negligible.
The hydrostatic approximation to the vertical equation of motion implies
that:
P~ = -pngz + Pn(x, y, t)
(7.6.2)
while continuity of total pressure at the interface implies that: