Geology Reference
In-Depth Information
@S
@T
p
D
tomography techniques (Ritsema et al.
2004
).
The velocity of this flow and the typical ther-
mal diffusivity of peridotite (›
Š
0.6 mm
2
s
1
at
T
D
1,400
ı
K,
p
D
7
10 GPa) prevent sub-
stantial transfer of heat to the surrounding man-
tle, because for a path length
L
D
300 km the
dimensionless quantity
vL
/
Š
1.59
10
2
(
Peclet
number
), which measures the relative importance
of advection with respect to conduction, is much
greater than unity. Therefore, we can consider
this as an adiabatic process, which is dominated
by the advection of heat rather than by thermal
conduction.
Let us consider a small rising volume element
of the asthenosphere beneath a mid-ocean ridge.
During its ascent, the temperature
T
and the
ambient pressure
p
change in such a way that
no heat is lost or gained (
dQ
D
0). This process
is generally considered to be reversible to a first
approximation, so that
dQ
D
TdS
and the entropy
is invariant too (
dS
D
0, isentropic process). In
these conditions, it is simple to determine how
the temperature changes as the volume element
moves towards the surface. By the cyclic relations
of thermodynamics, we have that:
@T
@p
T
¡V
T
¡V.@T=@S/
p
c
p
D
(1.5)
where ¡ is the density. Therefore, the rate of
change of the temperature with pressure will be
given by:
@T
@p
S
D
'T
¡c
p
(1.6)
In order to determine the variation of
T
with
depth, we can use the simple relationship express-
ing the variation of the hydrostatic pressure with
depth:
@P
@
z
D
¡g
(1.7)
where
g
is the acceleration of gravity and
z
is the
depth. Therefore, the adiabatic rate of change
of the temperature with depth will be given
by:
@T
@
z
S
D
'Tg
c
p
(1.8)
The coefficient of thermal expansion varies
between '
D
5
10
5
K
1
at
p
D
0.1 MPa,
T
D
1,700
ı
K (an adiabatically decompressed
mantle) and '
D
1
10
5
K
1
at the core-
mantle boundary (Chopelas and Boehler
1992
).
For
T
D
1,700
ı
K,
p
D
10 GPa (
300 km depth)
this quantity assumes the value: '
Š
3.5
10
5
K
1
. The specific heat at constant pressure for
the mantle is
c
p
Š
1.2kJ K
1
kg
1
(Stacey
2010
). Therefore, using these values in Eq. (
1.8
),
the
@T
@S
@S
@p
S
D
(1.1)
p
T
Furthermore, Maxwell's relations require that:
@S
@p
@V
@T
T
D
(1.2)
p
The rate of variation of the volume with tem-
perature can be expressed through the
coefficient
of thermal expansion
. This quantity is given by:
adiabatic
gradient
of
temperature
in
the
uppermost asthenosphere is approximately:
@T
@
z
@V
@T
1
V
S
Š
0:5
ı
Kkm
1
(1.9)
˛
D
(1.3)
p
Equations (
1.6
)and(
1.8
) describe the depen-
dence of temperature from pressure,
T
D
T
(
p
),
and depth,
T
D
T
(
z
), beneath ridges. These
isen-
tropic adiabat geotherms
have great importance
in geodynamics. A key observation is that an
isentropic adiabat has a less steep slope
dT/dp
(or
dT/dz
) than the
peridotite solidus
, which is the
set of (
p
,
T
) pairs where these rocks start melting.
Substituting into Eq. (
1.1
) we obtain:
@T
@p
S
D
˛V
@T
(1.4)
@S
p
The last derivative in Eq. (
1.4
) can be calcu-
lated through the
specific heat at constant pres-
sure
and the usual Maxwell relations:
