Geology Reference
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tion to the net areal flux, which coincides with
the first term at the right-hand side of ( 13.48 ), is
2.50 10 10 ms 1
D 8 mm year 1 . With a plate
velocity v 0 D 50 mm year 1 , approximately ¼ of
the total areal flux would come from pressure
gradients, but for @ p /@ x D 1 kPa this contribution
would be reduced to 3 %. Therefore, granted
that the average viscosity assumed for the as-
thenosphere is correct, the pressure-driven flow
becomes important only when the horizontal gra-
dient of pressure is significant. Substantial pres-
sure gradients in the asthenosphere have been hy-
pothesized since the 1990s (e.g., Phipps Morgan
et al. 1995 ). More recently, lateral pressure varia-
tions of 7,500-8,000 Pa km 1 have been reported
for the East Pacific Rise region (Conder et al.
2002 ), and possibly in excess of 100 kPa km 1
along the Tonga trench (Conder and Wiens 2007 ).
An interesting consequence of the asthenospheric
flow is represented by the drag, £ D ,exertedonthe
overlying tectonic plates. This is easily calculated
from ( 13.42 )and( 13.46 ):
Fig. 13.4 Poiseuille-Couette flow and counterflow (this
is an example of passive drag)
is the velocity (relative to the transition zone
reference frame) at which a tectonic plate,
initially at rest, would be accelerated by an
asthenospheric pressure-driven flow. When v 0
and @ p /@ x have the same sign, the pressure-
driven drag reinforces the viscous stress and the
flow assumes the shape illustrated in Fig. 13.4 .
This is called the asthenospheric counterflow
(Schubert and Turcotte 1972 ;Chase 1979 )and
represents a hypothetical situation that does not
find confirmation in the observed pattern of
gravity anomalies and dynamic topography (e.g.,
Turcotte and Schubert 2002 ).
Another interesting feature of the simple
model discussed above is the predicted amount of
shear heating. From ( 13.20 )and( 13.42 )wehave
that the viscous dissipation function is given by:
LJ LJ LJ LJ z D0 D
£ D D £ xy .x;0/ D ǜ @ v x
@ z
v 0 ǜ
h
@p
@x
(13.49)
h
2
Using the same parameters of the example
above, we see that the viscous drag term v 0 ǜ/ h
gives a contribution of 0.53 MPa to the total
drag, while with a pressure gradient of 10 kPa
km 1 the pressure-driven component is three
times larger: ( h /2)@ p /@ x 1.5 MPa. Therefore, for
ǜ D 10 20 Pa s, substantial variations of pressure in
the asthenosphere generate active drag of tectonic
plates. The expression ( 13.49 ) shows that when v 0
and @ p /@ x have opposite sign (as in Fig. 13.2 ), the
active pressure-driven drag opposes the passive
viscous stress, thereby an equilibrium velocity
exists such that £ D D 0. This is given by:
@x j D ǜ @ v x
2
@ v i
ˆ.x; z / D £ 0 ij
@ z
D ǜ 1
ǜ
z
@p
@x
2
h
2
v 0
h
(13.51)
In the case of a Couette flow this expression
reduces to:
ˆ D ǜ v 0
h 2
(13.52)
h 2
@p
@x
v eq D
(13.50)
Now let us consider the law of conservation
of energy ( 13.41 ), which now assumes the simple
form:
When @ p /@ x D 10 kPa km 1 and using the
same asthenosphere thickness and average
viscosity of the examples above, we have that
v eq D 4.5 10 9
ˆ C k @ 2 T
@ z 2 D 0
(13.53)
ms 1
D 142 mm year 1 .This
 
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