Geology Reference
In-Depth Information
10
4
N/s (newtons per second) for
Using the values quoted above yields
b
=
7
×
Hawaii.
7.2.2 Heat transported by plumes
If the plume buoyancy is thermal, it can be related to the rate at which heat is
transported by the plume, since both depend on the excess temperature,
T
=
T
p
-
T
m
, of the plume (Figure 7.4), as we will now see.
The difference between the plume density,
ρ
p
, and the mantle density,
ρ
m
, is due
to thermal expansion, which we introduced in Chapter 5. Thus from Eq. (5.3)
ρ
p
−
ρ
m
=−
ρ
m
αT.
(7.5)
The rate of flow of thermal buoyancy is then, from Eq. (7.2),
π
r
2
ugρ
m
αT.
b
=
(7.6)
We can relate the rate of heat flow to the volumetric flow rate
φ
(Eq. (7.1)). In
Chapter 5 we introduced the specific heat,
C
P
, which is the amount of heat required
to raise the temperature of a unit mass of a material by one degree Celsius. The
mass of the cylinder section in Figure 7.4 is
ρφt
, so its excess heat content, due
to its excess temperature, is
H
=
ρφtC
P
T .
The rate at which heat flows up the cylinder is
Q
=
H/t
,so
Q
=
ρ
m
φC
P
T .
(7.7)
(We can approximate the density in this expression with the mantle density,
ρ
m
.)
Now if we take the ratio of
Q
and
b
, remembering that
φ
π
r
2
u
from Eq. (7.1),
=
then
Q
=
C
P
b/gα.
(7.8)
This is a remarkably simple relationship between
Q
and
b
. It involves only
g
and
two material properties,
C
P
and
α.
It does
not
depend on the excess temperature
of the plume, which cancels out. Nor does it depend on the flow velocity,
u
,orthe
plume radius,
r
.
Thus we can calculate the heat flowing up the Hawaiian plume without having
to know
T
or
u
or
r
.Using
C
P
=
1000 J/kg
◦
Cand
α
=
10
−
5
◦
C
−
1
,Eq.(7.8)
3
×
10
11
W
yields about
Q
=
2
×
=
0.2 TW. This is about 0.5% of the global heat flow
(41 TW, Table 6.1).
The total rate of heat transport by all known plumes was estimated roughly by
Davies [54], and more carefully by Sleep [55], with similar results. Although there
