Geology Reference
In-Depth Information
which furnishes the following minimum for
Ra
cr
:
5. Determine the corner flows of a subduction
zone assuming that the subducting plate is at
rest with respect to the transition zone and that
the overriding plate moves with velocity
v
0
in
the positive
x
direction;
6. Prove the transformation Eq. (
13.72
);
7. Write the continuity equation for an incom-
pressible fluid in polar coordinates;
8. Rewrite the equations of motion (
13.69
)in
polar coordinates;
9. Find the velocity field (
u
,
v
) for a fluid that is
moving in the positive (downward)
z
direction
relative to a spherical object fixed at the ori-
gin of the reference frame. This is a Stokes'
flow. To solve the problem, you must solve
the equations found in exercises (7) and (8).
Assume that the velocity field is a uniform
field in the
z
direction with magnitude
v
as
r
!1
.
min.Ra
cr
/
Š
657:5
(13.149)
An important feature of thermal convection is
represented by the
aspect ratio
of the convective
cells, which is the ratio of the horizontal width
of the cells,
w
, to the vertical thickness
H
of the
fluid layer. Linear stability analysis shows that the
aspect ratio of the most rapidly growing fluctu-
ations is
w
/
H
D
œ/(2
H
)
D
p
2. Therefore, upper
mantle convective rolls should have a horizontal
width of
950 km.
Problems
1. Write and solve the equations of motion for
a one-dimensional steady flow in the astheno-
sphere, considered as a two-layers Newtonian
incompressible fluid. It is assumed that the up-
per asthenosphere has viscosity ǜ
1
D
10
20
Pa s
and thickness
h
1
D
200 km, while the lower
layer has viscosity ǜ
2
D
10
21
Pa s and thick-
ness
h
2
D
200 km.
v
0
D
100 mm year
1
is the
velocity of the overlying lithosphere. Deter-
mine the depth of maximum velocity for a
horizontal pressure gradient @
p
/@
x
D
10 kPa
km
1
;
2. Determine the horizontal pressure gradient
that was necessary to accelerate India to
v
0
D
180 mm year
1
in the early Paleocene,
assuming a 400 km thick asthenosphere and
an average viscosity ǜ
D
10
20
Pa s. Determine
the time required to attain such a velocity
v
D
100
References
Batchelor GK (2000) An introduction to fluid dynamics.
Cambridge University Press, Cambridge, UK, 615 pp
Becker TW, Faccenna C (2011) Mantle conveyor beneath
the Tethyan collisional belt. Earth Planet Sci Lett
310(3):453-461
Bercovici D, Schubert G, Glatzmaier GA (1989) Three-
dimensional spherical models of convection in the
Earth's mantle. Science 244(4907):950-955
Cande SC, Stegman DR (2011) Indian and African plate
motions driven by the push force of the Réunion plume
head. Nature 475:47-52. doi:
10.1038/nature10174
Carslaw HS, Jaeger JC (1959) Conduction of heat in
solids, 2nd edn. Oxford University Press, Oxford,
510 pp
Chase CG (1979) Asthenospheric counterflow: a kine-
matic model. Geophys J R Astr Soc 56:1-18
Chorin AJ (1968) Numerical solution of the Navier-Stokes
equations. Math Comput 22(104):745-762
Chung
mm/yr
starting
from
v
D
0mm
year
1
;
3. Let us assume that the lower boundary of a
tectonic plate can be represented by a function
z
D
f
(
x
). Assuming that no streamline has
a cusp along this boundary, use the no-slip
boundary condition to determine the boundary
values of velocity for the asthenosphere along
the LAB;
4. Repeat the previous exercise assuming that the
lower boundary of a tectonic plate is repre-
sented by a surface
z
D
f
(
x
,
y
);
TJ
(2002)
Computational
fluid
dynamics.
Cambridge
University
Press,
Cambridge,
UK,
1012 pp
Conder JA, Wiens DA (2007) Rapid mantle flow be-
neath the Tonga volcanic arc. Earth Planet Sci Lett
264(1):299-307
Conder JA, Forsyth DW, Parmentier EM (2002) Astheno-
spheric flow and the asymmetry of the East Pacific
Rise, MELT area. J Geophys Res 107(B12):2344.
doi:
10.1029/2001JB000807
Conrad CP, Behn MD (2010) Constraints on litho-
sphere net rotation and asthenospheric viscosity from
global mantle flow models and seismic anisotropy.
