Vertical Structure of the Abyssal Circulation
437
This completes the mathematical formulation of the problem. However,
some general insight into the solution can be obtained by simple scaling
considerations. In particular we can estimate the ratio of the baroclinic
velocity, induced by the density gradients, to the barotropic velocity that must
still satisfy the Stommel-Arons integral relation (7.7.2). Thus we can estimate
the vertical average, or barotropic, component of the flow from (7.7.2) as:
fwo
Vbarotropic = {3H ·
(7.7.17)
The baroclinic velocity can be estimated from the thermal wind relation as:
gH8p
Vbarociinic = f Po ax
g H 11p
f Po x • .
(7.7.18)
However, the density anomaly, 11p, can be related to the vertical velocity
through the density equation, (7.7.9), i.e., assuming that w is O(w0):
11p w0N 2 H 2
(7.7.19)
Po
gK
so that:
(7.7.20)
Therefore, the ratio of the barotropic to baroclinic velocity is of the order of:
Vbaroclinic
f3N 2 H 4
Vbarotropic
Kj 2 Xe
(7.7.21)
If {3 = 2 x to- 13 cm-I, N = l0- 13 s- 1 , H = 1 km (for the scale of the motion),
K = 0.1 cm 2 s- 1 , f = 10-4 s- 1 , and x. = 3000 km, this parameter is about 70. A
larger value of the diffusion coefficient or a weaker background stratification
reduces the ratio, but the implication is clear. The baroclinic velocity is large
compared with the vertical average, and on the basis of the physical argument
given above we can expect the Stommel-Arons barotropic velocity to be a
relatively small residual obtained by vertically averaging the large baroclinic
velocity.
An important constraint on the interior solution which allows us to
determine the density anomaly on the eastern boundary arises from the
inability of the boundary layers to transfer a net vertical mass flux locally. This
inability follows from the balance that occurs in the density equation in the
boundary layer. Each variable in the western boundary layer can be
represented as the sum of the interior solution (subscript I) and a boundary
layer correction denoted by a tilde, e.g.:
Pn =PI+ P·
(7.7.22)
437
This completes the mathematical formulation of the problem. However,
some general insight into the solution can be obtained by simple scaling
considerations. In particular we can estimate the ratio of the baroclinic
velocity, induced by the density gradients, to the barotropic velocity that must
still satisfy the Stommel-Arons integral relation (7.7.2). Thus we can estimate
the vertical average, or barotropic, component of the flow from (7.7.2) as:
fwo
Vbarotropic = {3H ·
(7.7.17)
The baroclinic velocity can be estimated from the thermal wind relation as:
gH8p
Vbarociinic = f Po ax
g H 11p
f Po x • .
(7.7.18)
However, the density anomaly, 11p, can be related to the vertical velocity
through the density equation, (7.7.9), i.e., assuming that w is O(w0):
11p w0N 2 H 2
(7.7.19)
Po
gK
so that:
(7.7.20)
Therefore, the ratio of the barotropic to baroclinic velocity is of the order of:
Vbaroclinic
f3N 2 H 4
Vbarotropic
Kj 2 Xe
(7.7.21)
If {3 = 2 x to- 13 cm-I, N = l0- 13 s- 1 , H = 1 km (for the scale of the motion),
K = 0.1 cm 2 s- 1 , f = 10-4 s- 1 , and x. = 3000 km, this parameter is about 70. A
larger value of the diffusion coefficient or a weaker background stratification
reduces the ratio, but the implication is clear. The baroclinic velocity is large
compared with the vertical average, and on the basis of the physical argument
given above we can expect the Stommel-Arons barotropic velocity to be a
relatively small residual obtained by vertically averaging the large baroclinic
velocity.
An important constraint on the interior solution which allows us to
determine the density anomaly on the eastern boundary arises from the
inability of the boundary layers to transfer a net vertical mass flux locally. This
inability follows from the balance that occurs in the density equation in the
boundary layer. Each variable in the western boundary layer can be
represented as the sum of the interior solution (subscript I) and a boundary
layer correction denoted by a tilde, e.g.:
Pn =PI+ P·
(7.7.22)