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13.3
Energy Balance
Fig. 13.1
Shearing of a fluid bounded by two rigid plates
in relative motion with velocity
v. Arrows
show the veloc-
ity field in the fluid, which varies between
v
and zero
The Navier-Stokes equations (or other more com-
plex versions of the equations of motion for
Newtonian fluids) and the continuity Eq. (
13.6
)
form a system of four differential equations in
the five unknowns
p
(thermodynamic pressure),
(
v
x
,
v
y
,
v
z
) (velocity), and ¡ (density). Therefore, at
least an additional equation is needed to solve the
system (either numerically or analytically). We
note that thermodynamic pressure and density are
not independent each other, because they must
satisfy an
equation of state
together with the
temperature
T
:
mantle rocks are carried to shallow depth. In this
case, the rocks experience a decrease of pressure
and cool adiabatically in so far as they travel
towards the LAB.
The adiabatic temperature gradient in the up-
per mantle has been determined in Sect.
1.3
(Eq.
the mantle material is considered incompress-
ible, variations of hydrostatic pressure cannot
change the volume of a small patch of convecting
asthenosphere. Consequently, in this hypothesis
the adiabatic temperature gradient is zero. To
have an idea of the error associated with this
approximation, we recall that the temperature
gradient determined in Sect.
1.3
for the upper-
most asthenosphere was (@
T
/@
z
)
S
Š
0.5
ı
Kkm
1
.
At depth exceeding 300 km, a more appropriate
values is 0.3
ı
Kkm
1
, but this is still a significant
variation of temperature with depth. Therefore,
the approximation appears justified only if we
limit our attention on upper mantle small-scale
convection.
The most general equation describing the local
energy balance in the fluid mantle must account
for the transport of heat both by conduction
and by advection, as well as for the effect of
frictional heating associated with deformation.
Therefore, we are looking for an equation that
only describes non-steady conduction of heat. To
understand the contribution of frictional heating
to the energy balance, let us consider a fluid sheet
of thickness
h
and area
A
, which is sheared by
the relative motion of two rigid plates as shown
in Fig.
13.1
. The volume of the fluid sheet is
V
D
Ah
. The fluid at contact with the fixed plate
has velocity zero, while at the upper boundary
the fluid velocity coincides with the velocity
v
f.
¡
;p;T/
D
0
(13.19)
This introduces an additional equation but also
an extra unknown, the temperature
T
. With a total
of six unknowns, we now need to solve a system
of six differential equations. However, in addition
to mass conservation, momentum conservation,
and the state Eq. (
13.19
), it must be satisfied the
law of
conservation of energy
, which then com-
pletes the set of equations that are needed to solve
any geodynamic problem. Now we are going to
consider in detail the equation corresponding to
this conservation law.
The thermal structure of the mantle is essen-
tially determined by the convective transport of
heat. This concept is easily proved noting that
if the continental or oceanic geotherms contin-
ued downwards below the LAB, a large part of
the asthenosphere would be molten. Therefore,
the temperature in the sub-lithospheric mantle
must increase with depth approximately along an
adiabat. Assuming that convection is sufficiently
vigorous, mantle rocks that are carried to higher
depth experience larger hydrostatic pressure by
the overlying material and are compressed adi-
abatically, that is, without substantial conductive
heat transfer to or from the surrounding mantle.
Accordingly, their temperature increases only by
adiabatic heating. The opposite occurs when hot
