422
Abyssal Circulation
surface of the model, and therefore the mass conservation equation for the
total transport, yields:
V' · Un = 0
(7.6.5)
where:
Un = H1 ii1 + H2ii2
(7.6.6)
so that the total transport is horizontally nondivergent and derivable from a
streamfunction, i.e.:
(7.6.7)
If the momentum equations for each layer are each multiplied by their
respective Hn and added together to form an equation for the barotropic
transport, we obtain:
(7.6.8)
Aside from a solution exponentially decaying in time, the only solution of
(7.6.8) which satisfies the condition that '1'0 vanish on the basin boundary is the
solution in which '1'0 and all the barotropic velocities are zero. In the Kawase
model there is no barotropic mode, and the only circulation pattern is the
internal mode driven by the source-sink forcing. There is no forcing of the
abyss by the thermocline's wind-driven motion, and in this sense the physics of
the model is quite distinct from the Stommel-Arons model.
The vanishing of the barotropic transport implies that:
_
H2_
UI = --u2.
H1
(7.6.9)
Subtraction of the two momentum equations yields the governing equation for
the baroclinic mode:
[:t + K + /kx] (iii- ii2) = g'V'h
(7.6.10)
where (7 .6.3) is used to express the difference of the pressure gradients between
the two layers. If (7.6.9) is used, (7.6.10) can be expressed entirely in terms of
the velocity of the flow in the lower layer. Thus:
8ii2 fkA -
""'h
-
fu + X U2 = -g v - KU2
(7.6.11)
where g" = g' HI/(HI +H2).
The key assumption of Kawase's model is the connection suggested
between the cross-interface velocity w and the displacement of the interface.
Kawase takes:
Abyssal Circulation
surface of the model, and therefore the mass conservation equation for the
total transport, yields:
V' · Un = 0
(7.6.5)
where:
Un = H1 ii1 + H2ii2
(7.6.6)
so that the total transport is horizontally nondivergent and derivable from a
streamfunction, i.e.:
(7.6.7)
If the momentum equations for each layer are each multiplied by their
respective Hn and added together to form an equation for the barotropic
transport, we obtain:
(7.6.8)
Aside from a solution exponentially decaying in time, the only solution of
(7.6.8) which satisfies the condition that '1'0 vanish on the basin boundary is the
solution in which '1'0 and all the barotropic velocities are zero. In the Kawase
model there is no barotropic mode, and the only circulation pattern is the
internal mode driven by the source-sink forcing. There is no forcing of the
abyss by the thermocline's wind-driven motion, and in this sense the physics of
the model is quite distinct from the Stommel-Arons model.
The vanishing of the barotropic transport implies that:
_
H2_
UI = --u2.
H1
(7.6.9)
Subtraction of the two momentum equations yields the governing equation for
the baroclinic mode:
[:t + K + /kx] (iii- ii2) = g'V'h
(7.6.10)
where (7 .6.3) is used to express the difference of the pressure gradients between
the two layers. If (7.6.9) is used, (7.6.10) can be expressed entirely in terms of
the velocity of the flow in the lower layer. Thus:
8ii2 fkA -
""'h
-
fu + X U2 = -g v - KU2
(7.6.11)
where g" = g' HI/(HI +H2).
The key assumption of Kawase's model is the connection suggested
between the cross-interface velocity w and the displacement of the interface.
Kawase takes: