Geology Reference
In-Depth Information
convective flows in the lower mantle. Therefore,
both the elevation of the mid-ocean ridges and
the deep bathymetry of the trench zones must
be considered as consequences of the dynamic
topography associated with mantle convection.
However, this phenomenon can also explain the
apparent paradox of negative correlation between
long-wavelength geoid undulations and density
anomalies in the mantle mentioned above, which
was first observed by Dziewonski et al. (
1977
).
In fact, as illustrated in Fig.
14.8
, dimples and
swells at the Earth's surface and along mantle dis-
continuities (e.g., the 670 km discontinuity or the
CMB) give respectively negative or positive con-
tributions to the local geoid undulation, and even-
tually can overcome the component associated
with the static density distribution. Hager (
1984
)
and Richards and Hager (
1984
) proved that the
amplitude of dimples and swells is determined by
the magnitude and depth of the density contrasts,
from the depth of the layer, and from the presence
of viscosity stratification in the fluid layer. They
also showed that the process of formation of
dynamic topography could attain a steady state,
because the time interval required to reach sta-
tionary conditions has the same order of magni-
tude of the time employed by a density anomaly
to travel a distance of a few kilometers. There-
fore, a steady state is reached on a postglacial
rebound time interval, essentially instantaneously
on a geological time scale. In general, the effect
of dynamic topography on gravity is quite large.
For example, the topographic low of subduction
zones may reach 10 km depth. Although dynamic
topography is traditionally associated with the
vertical motions of thermal (Rayleigh-Bénard)
convection, horizontal mantle flows also support
that horizontal pressure-driven flows (Poiseuille-
Couette flows) are possible and arise from hor-
izontal pressure gradients in the asthenosphere.
The existence of these flows is testified both by
events of non-equilibrium plate kinematics in the
non-isostatic topographic slope observed in some
regions (e.g., Conder and Wiens
2007
). The re-
lation between topographic slope and horizontal
pressure gradient comes from the observation
that an excess of thermodynamic pressure with
respect to the hydrostatic value at the compensa-
tion depth must be equivalent to uncompensated
topography at the Earth's surface. Therefore, if
h
is the elevation exceeding the normal isostatic
height, we can write the following fundamental
equation (Schubert and Turcotte
1972
; Schubert
et al.
1978
):
@p
@x
D
¡
a
g
@h
(14.67)
@x
where ¡
a
is the density of the asthenosphere.
Therefore, assuming ¡
a
D
3,450 kg m
3
we
have that a pressure gradient of 100 kPa km
1
would generate a dynamic topographic slope of
2.96 m km
1
.
Problems
1. Find the gravity anomaly generated by a
spherical object at 1 km depth with density
contrast
¡
D
200 kg m
3
;
2. Find the gravity anomaly generated by
an infinitely long horizontal dike at 1 km
depth,
with
cylindrical
cross-section
of
R
D
5
radius
m
and
density
contrast
¡
D
100 kg m
3
;
3. Find an expression for the thickness
h
r
of the
crustal root generated by a mountain belt with
average altitude
h
m
in excess of a normal con-
tinental crust, assuming that the lithospheric
mantle has constant thickness;
4. A gravimeter based on measurements of
falling body trajectory has uncertainty
˙
10
6
s on reading the arrival time at distance
d
D
0.5
˙
0.0001 m. Estimate the uncertainty
on gravity.
References
Biancale R, Balmino G, Lemoine JM, Marty JC, Moynot
B, Barlier F, Exertier P, Laurain O, Gegout P,
Schwintzer P, Reigber C, Bode A, König R, Massmann
FH, Raimondo J-C, Schmid R, Yuan Zhu S (2000) A
new global Earth's gravity field model from satellite
orbit perturbations:
GRIM5-S1.
Geophys Res Lett
27(22):3611-3614
