Geology Reference
In-Depth Information
1 C
2 Pr E
h 2 1
2 Pr E
(13.59)
Substituting ( 13.52 )gives:
z 2
z
h
1
™. z / D
ǜ v 0
h 2 C k @ 2 T
@ z 2 D 0
(13.54)
Plots of the expected temperatures in the
asthenosphere for various plate velocities are
shown in Fig. 13.5 .When v 0 D 0(Pr E D 0),
there is no frictional heating and the geotherm
is a straight line. The temperatures in excess
of this linear trend are associated with viscous
dissipation. In terms of dimensionless quantities,
the excess temperature due to frictional heating
is given by:
This equation can be integrated immediately
assigning the boundary conditions: T (0) D T 0 and
T ( h ) D T a . The result is:
T a T 0 C
ǜ v 0
2k
ǜ v 0
2k
z 2
h 2
(13.55)
z
h
T. z / D T 0 C
It is useful to rewrite this expression in terms
of dimensionless temperature ratio. We have:
1
1
2 Pr E z
z
h
•™. z / D
(13.60)
1 C
h
ǜ v 0 =.2k/
T a T 0
T. z / T 0
T a T 0 D
z
h
™. z /
The excess temperature has a maximum
•™ max D (1/8)Pr E for z / h D ½. For example, for
v 0 D 50 mm year 1 (Pr E D 0.21) we would
have • T max D 7.86 K at z D 150 km below the
LAB. In this example, the maximum increase of
temperature due to shear heating would be only
2.6 % of the temperature difference between
the base and the top of the asthenospheric
channel. However, if the average viscosity in
the asthenosphere were one order of magnitude
greater, say ǜ D 10 21 Pa s, a fast moving plate
that travels at v 0 D 100 mm year 1 would
trigger an asthenospheric flow with Pr E D 8.38.
Accordingly, the maximum excess temperature
would be 314 K and we would have downward
heat loss at the base of the asthenosphere! This
simple thermodynamic consideration suggests
that an appropriate value of the average viscosity
in the asthenosphere should not exceed 10 20 Pa
s. This value is confirmed by recent accurate
estimates based on surface wave tomography and
seismic anisotropy (Conrad et al. 2007 ; Conrad
and Behn 2010 ), which give an average upper
mantle viscosity ǜ um D 0.5-1.0 10 21 Pa s, and
a value of asthenosphere viscosity ǜ D 0.5-
1 10 20 Pa s. Consequently, there is strong
evidence that active asthenospheric drag of
tectonic plates associated with pressure-driven
flows represents a real mechanism explaining
the
ǜ v 0 =.2k/
T a T 0
z 2
h 2
(13.56)
Therefore, in the case of a pure Couette flow,
the temperature distribution in the asthenosphere
is governed by the single dimensionless pa-
rameter (ǜ v 0 /(2 k ))/( T a T 0 ). The dimensionless
quantity:
ǜ
¡›
Pr
(13.57)
is a characteristic of the fluid and represents the
ratio of the momentum diffusivity (or kinematic
viscosity ) ǜ/¡ to the thermal diffusivity ›.It
is called Prandtl number and says how much
rapidly a fluid diffuses its momentum relative
to the heat diffusion rate. Another dimensionless
parameter that describes a fluid property is the
Eckert number , which represents the ratio of
kinetic to thermal energy:
v 0
c p .T a T 0 /
E
(13.58)
Combining the definitions ( 13.57 )and( 13.58 ),
we see that the temperature distribution in the
asthenosphere is governed by the dimensionless
parameter ½Pr E :
non-equilibrium states
discussed
in
Sect.
6.7 .
The
lateral
variations
of
pressure
that
 
 
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