Geology Reference
In-Depth Information
across plate circuits, test triple junction velocity
triangles, etc. What we cannot do with these mod-
els, is to represent the
absolute
velocities of the
tectonic plates with respect to a reference frame
fixed to the deep mantle, for example fixed to the
top of the transition zone as in Fig.
2.9
.However,
this is a necessary step if we want to consider
the kinematics of tectonic plates in relation to
the asthenospheric flows, and give a complete
geodynamic formulation of plate tectonics. Such
approach represents one the fundamental tasks
of this topic, thereby now we shall illustrate
an approximate method to determine the Euler
vectors in a reference frame fixed to the deep
mantle.
The method was proposed 40 years ago by
Solomon and Sleep (
1974
) and applies equally
well to the modern plates and to a paleotectonic
context (Solomon et al.
1977
). These authors
started from the assumption that the total torque
N
exerted on the lithosphere (Eq.
2.3
) is zero, and
that the asthenosphere is dragged passively by
the overlying lithosphere. The first assumption is
compatible with the fact that, apart from the case
of space geodesy studies, we always represent
plate motions through the
geological time
, not the
physical time, even when we study the present
day plate motions. When we consider physical
processes that occur at the time scale of the last
2-3 Myrs, it is always necessary to neglect any
motion related to the Earth's rotation, including
variations in eccentricity of the orbit, axial tilt,
the evidence that the total angular momentum
L
(Eq.
2.5
) of the lithosphere is constant over time
intervals of several Myrs, which implies that in
equilibrium conditions
N
D
0
at the time scale of
the geological processes. We shall prove that also
the second assumption is correct in conditions of
geodynamic equilibrium, but
not
during episodes
of plate acceleration, such as the northward accel-
eration of India during the Cretaceous to Eocene
time interval (Cande and Stegman
2011
).
The method of Solomon and Sleep is based
on a balance of the torques exerted on the whole
lithosphere. The torques applied on individual
plates are associated with the viscous resistive
drag force that the asthenosphere exerts on the
base of the overlying lithosphere, and with plate
boundary forces, such as the gravitational forces
exerted by slabs. However, it is not necessary
to include symmetric features such as mid-ocean
ridges in the torque balance, because in this
instance the corresponding torques cancel out.
Therefore, the two fundamental torques that must
be included in the torque balance equation are
those associated with drag forces and those aris-
ing from downward pull forces exerted by slabs.
Let us assume that the passive drag applied at the
base of the lithosphere follows a simple viscous
law, so that it depends linearly from the velocity
of the lithosphere relative to the base of the fluid
asthenosphere. It is also reasonable to assume
that the slab pull force exerted along an active
margin does not depend from the plate velocity.
Let ¨
i
be the Euler vector of
i
th plate relative
to the top transition zone, and
v
i
(
r
)
D
¨
i
r
the
corresponding linear velocity field at each point
r
along its surface. The simplest law describing
the resistive drag force per unit area (or
traction
)
at the base of the lithosphere,
T
i
D
T
i
(
r
), is the
following one:
T
i
.r/
D
D
i
v
i
.r/
D
D
i
¨
i
r
(2.58)
In this expression
D
i
is a drag coefficient
which may depend from position. To obtain the
total torque exerted on the
i
thplatewemust
integrate the local torque per unit area,
r
T
i
(
r
),
over the surface
S
i
of the plate:
Z
N
i
D
ŒD
i
r
.¨
i
r/dS
(2.59)
S
i
From this expression, it is easy to calculate
the total torque exerted on the lithosphere by the
underlying asthenosphere:
D
i
Z
S
i
N
D
X
i
N
i
D
X
i
Œr
.¨
i
r/dS
(2.60)
where for simplicity we have assumed that
D
i
is
constant along the surface of a plate. If we expand
