Geology Reference
In-Depth Information
13
Flow and Fluid Behaviour
of the Mantle
Abstract
With Chap. 13, we move from the lithosphere to the underlying mantle. In
fact, this chapter introduces the important theme of mantle dynamics, in
particular thermal convection and asthenosphere currents. Navier-Stokes
and energy balance equations are derived and discussed, along with the
classic Boussinesq approximation.
motion and conservation laws in fluid dynamics.
In the
Eulerian
formulation, the changes of any
intensive variable through time are considered at
arbitrary
fixed
locations
r
. For example, the fluid
velocity
v
(
r
,
t
), which represents the fundamental
kinematic variable, is viewed as the velocity of
the parcel that travels through a location
r
(
x
,
y
,
z
) at time
t
, while its time derivative @
v
/@
t
represents the variation of velocity between the
particles that travel through
r
at times
t
and
t
C
dt
.
Therefore, in this representation the coordinates
of
r
are expressed in an inertial reference frame
and the velocity is a function of four indepen-
dent variables:
v
D
v
(
x
,
y
,
z
,
t
). Conversely, in the
Lagrangian
representation we consider a single
particle
P
, which is assigned a path
r
D
r
(
t
)
from an initial location
r
0
. In this instance, any
change of the intensive variables is referred to
a frame that is moving with
P
. For example,
we can consider the variations of temperature
or density of
P
while it is moving between two
locations. Substituting the parametric equations
r
D
r
(
t
) into the velocity field
v
gives a ve-
locity vector
v
(
t
)
D
d
r
/
dt
D
v
(
r
(
t
),
t
) that depends
only from time. In the context of the Lagrangian
13.1
Continuity Equation
In this chapter, we are going to introduce the
fluid behaviour of the Earth's mantle, in par-
ticular of the asthenosphere, and the influence
of mantle flows on plate tectonics. As we have
described by rheological constitutive equations
that link stresses to strain rates rather than strains.
Therefore, differently from seismology, where
displacements and infinitesimal strains are the
basic kinematic variables, a formulation of the
laws that determine the long-term dynamics of
the mantle will require a kinematic framework
based on velocity fields and strain rates. In fluid
dynamics, just as in the more general context
terial is ideally subdivided into small parcels
(or particles) of volume
dV
and it is assumed
that the intensive variables of the system, such
as velocity, density, temperature, and pressure,
change continuously from point to point through-
out the material. There are two different ap-
proaches to the formulation of the equations of
