Download Chapter 3: heat flow

CHAPTER 3
Heat flow and energetics of the Earth
3.1 Introduction . The seismic activity observed in the lithosphere suggests that the surface of the
Earth is in motion. The energy required for this motion must come from the Earth’s interior and we now
ask: what are the energy sources and how is the energy transmitted to the surface? Potential sources
of energy include: radioactive decay, global cooling, and gravitational redistribution of matter such as
core formation or planetary accretion. Since energy is lost by the Earth mainly by heat flow through the
surface, we will estimate this energy loss by examination of global heat flow measurements.
3.2 Global Energy Sources . The discovery of radioactivity radically changed the view of the global
energy budget and consequent estimates of the age of the Earth. Kelvin required a very young age
for the Earth (a few million years) in order to explain the observed heat flow. With the discovery of
radioactivity at the turn of this century, conductive cooling no longer had to be the primary source of
heat within the Earth. In fact, radioactivity could be a very important source of heat and calculation of
the global energy budget requires a good knowledge of the composition of the Earth.
The composition of the bulk Earth is usually assumed to be close to that of the carbonaceous chondrites
as discussed in Chapter 1. If this is true, then the major heat producing elements in the present Earth are
238
U , 235 U , 232 T h and 40 K. Other radioactive elements either decay too slowly or are present in such
small abundances that they are relatively unimportant. The concentrations of the major heat producers
form consistent ratios in a wide range of terrestrial and meteoritic materials. The ratio of potassium
to uranium and thorium to uranium in the sun and the carbonaceous chondrites are about 104 and 4
respectively and probably are inherited from the composition of the original solar nebula. Owing to the
volatility of potassium it is quite possible that the potassium/uranium ratio in the bulk Earth is somewhat
lower, although how much lower is not well constrained. If we assume a chondritic abundance as a
starting point, we can make a rough guess at the heat production within the Earth due to radioactivity
by:
[T h]
[K]
HT h +
HK
Hr = [U ] HU +
[U ]
[U ]
where
and
HU =9.71 × 10−5
HK =3.58 × 10
−9
HT h =2.69 × 10
−5
W/kg
W/kg
W/kg
Taking a concentration of Uranium [U] of 1.3×10−8 kg/kg (∼13ppb) consistent with that in carbonaceous
chondrites and the ratios of the other heat producing elements to be the same as in the sun, we get Hr to
be ∼ 3 × 10−12 W/kg or about 1 → 1.5 × 1013 W for the whole Earth (mass of mantle is ∼ 4 × 1024 kg
– it is currently unclear how much, if any, radioactivity is in the core).
Owing to the greater abundance of radioactive elements early in the history of the Earth, radioactive
heat production was higher in the past than now. In Figure 3.1, we show the heat production from
radioactive sources as a function of time. 26 Al is a short lived isotope which would have been present a
few million years after a supernova explosion. Whether or not this energy source was available depends
on how quickly the Earth formed after the last supernova. Some models for the formation of the solar
system actually use a supernova explosion to initiate the collapse. In such a case, 26 Al probably served
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as a major energy source for a few million years. In any case, 3 billion years ago, the total radioactive
heat production was about twice the present value.
Fig. 3.1 Heat production from radiocative sources
with chondritic abundances
One consequence of a greater radioactive heat production in the past is that the Earth has probably
cooled. Comparing the geothermal gradient during the Archean with the present geotherm yields an
estimate of the cooling rate of about 100◦ /billion years. When a body cools, energy is released according
to the equation:
Q(t) = −M Cp
dT
dt
where t is time, Cp is the specific heat at constant pressure and M is the mass of the body. [The definition
of the specific heat Cp is, the energy required to raise the temperature of a unit mass of material by
1K (at constant pressure)]. Taking the appropriate values for the mantle, we calculate 1.5 × 1013 W,
comparable to the radioactive sources. Cooling of the core at a similar rate would contribute another
0.5 × 1013 W.
When matter redistributes to a lower energy configuration (denser material on the bottom), energy is
released to the environment as heat. Thus, the formation of an iron core from an initially homogeneous
Earth would be an energetic event of huge proportions. Computing this gravitational energy release at
first may seem like an impossible task. Fortunately, gravity is a conservative force field which means
that we do not need to know the exact mechanism of core formation but can compute the energy release
by knowing the potential energy before and after core formation and then subtracting the two. If we take
into account the fact that g, ρ, P and T change as the Earth forms we get ∼ 1031 J for the energy release
due to core formation. If all the energy were kept inside the Earth we can estimate the temperature rise
it would cause. The change in energy of the Earth by raising its temperature by ∆T is given by
∆E = M Cp ∆T
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where M is the mass of the Earth. For Cp 1000J kg−1 K−1 we get ∆T 3500K. We conclude
that core formation could raise the temperature of the Earth by several thousand degrees, enough to
completely melt it and drive off most of the more volatile materials as well. It is therefore likely that
much of the gravitational energy release was lost during the accretion process by radiation into space.
Gravitational energy release by rearrangement of matter inside the Earth could be an important energy
source today but the problem is knowing the rate of release of gravitational energy. Remember that power
is the rate of change of energy with respect to time (1 Watt = 1 Joule s−1 ) so the power available at any
particular time depends upon the rate at which gravitational energy is being released. Is core formation
still occurring? We simply do not know and it is difficult to assess the importance of this energy source
for the problems we seek to address. It is likely that gravitational energy release occurs during growth
of the inner core and this may be an important power source for the geodynamo.
Another source of energy is the heat generated during planetary accretion. Collision of planetessimals
can generate a huge amount of energy but, again, much of this energy was probably radiated away during
the accretion process, The details of how the planets formed and how much heat was retained as “original”
heat are unknown. If a Mars-size impact generated the Moon, we can be reasonably confident that the
Earth started off hot.
We shall show in the next section that the total energy released through the surface of the Earth is
about 4.4 × 1013 W. Up to perhaps half of this can be accounted for by radioactive decay alone. Taking
into account planetary cooling, core formation and original heat leads us to the conclusion that the
possible supply of energy is almost embarrassingly large and we should not be surprised that the Earth
is a geologically vigorous body.
3.3 Heat Loss . Heat transport can be by radiation, conduction and convection. The latter is a very
efficient means of transporting heat but can only operate if material can flow. The lithosphere is brittle
and can not convect; conduction must be the major form of heat transport through it. Conduction is a
diffusion process described by Fourier’s law. In three dimensions this is
q = −k∇T
(3.1)
where
∇=
∂ ∂ ∂
,
,
∂x ∂y ∂z
and k is the thermal conductivity, a property of the material, q is the heat flow and T is the temperature.
The minus sign reminds us that heat flows down a temperature gradient.
For most problems of heat conduction in the Earth, horizontal heat flow is not important and we can
reduce the problem to a one-dimensional one, i.e., q = −kdT /dz. Consider a slab of material:
Fig 3.2
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We can only access a thin region near the surface of the Earth so the temperature gradient can be
assumed to be roughly linear, i.e., dT /dz ≈ −δT /h. To measure heat flow, all we have to do is measure
a temperature difference over some depth range in the Earth, measure the thermal conductivity of the
material and use the formula:
q = −k
dT
δT
≈ −k
dz
h
(3.2)
This approximation is valid only for a thin shell.
Fig 3.3 Histograms of continental, oceanic and global heat flow
1) Land Measurements
The temperature gradient in the Earth’s crust (the geotherm) can be measured in mines or, to get a
good areal coverage, in boreholes. A borehole must be deep (≈ 300m) to avoid surface effects on the
temperature gradient. An example of such an effect would be the previous presence of an ice sheet.
Conduction is a very slow process and, if the surface temperature is changed, it takes a long time for the
temperature gradient to settle down to its new value. If we want to avoid these climate-induced variations
we must drill below the affected layer. The actual drilling of a borehole perturbs the temperature gradient
and it is often necessary to wait for ∼ 2 years to get a good measurement of the gradient. Boreholes
invariably fill up with water and the temperature gradient is measured by measuring the temperature of
the water at different depths in the hole. We have to be sure that the water is not circulating as this will
mean there will be convective heat transport in the borehole.
Notwithstanding all these difficulties, thousands of measurements have been made giving a mean
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continental heat flow of 65 mWm−2 .
Fig 3.4 Using a corer to do oceanic heatflow. The corer is released
when the trigger wieght hits the ocean floor
2) Sea Measurements
These are done by dropping a “corer” (a tube about 3m long with thermistors attached) into the ocean
floor sediments. The core sample is used to determine the thermal conductivity of the material later in
the laboratory. Ocean floor sediments are often porous or permeable and heat transport by convection
of the sea water in the sediments can bias the measurement of the temperature gradient. If this happens
the sediments are cooled by the water circulation leading to a low estimate of the temperature gradient.
The mean heat flow in the oceans is 101 mWm−2 (after correcting for water circulation in the oceanic
crust). This is very similar to but higher than the continental value which was initially puzzling as the
heat generation by radioactive elements is much higher in continental rocks than in oceanic rocks. We
shall discuss the reason for this later. The following table shows heatflow averages as a function of
the age of both oceanic and continental lithosphere. Note that young ocean floor can have a significant
amount of water circulation leading to low measured heat flows – the table has been corrected for this
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effect.
Fig 3.5
Clearly, there is a strong tendency for oceanic heatflow to be a function of the age of the ocean floor
and we shall come back to this. The continental heat flow is a weak function of the age of the continental
crust (particularly given the scatter in the data – see fig 3.6) but this trend has been used to extrapolate
heat flow into regions where there are no measurements. In fig 3.7, we show the locations of actual
heatflow measurements (this is a new compilation of almost 38,000 measurements worldwide) and fig
3.8 shows the measured heat flux (uncorrected for water circulation effects) averaged into 2X2 cells.
The mean values we are quoting for ocean and continental regions use the age dependence to extrapolate
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globally and also are corrected for convective effects.
Fig 3.6 Two different ways of averaging continental heat low data. The first (a) is by the age of
the last major tectonic event and (b) is by the radiometric date of crustal age. The width of each
box represents one standard deviation
Fig 3.7 Locations of heat flow measurements in both oceans and continents
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Fig 3.8 Map of measured heat flow uncorrected for convective effects
After interpolating the data (using crustal age as a guide), we can compute the total power lost
by the Earth by multiplying the mean oceanic and continental heat flow values by the areas of
the continents and oceans and adding. The area of the continents is about 2 × 108 km2 giving a
continental power loss of 1.3 × 1013 watts. Similarly, the area of the oceans is about 3.1 × 108
km2 giving an oceanic power loss of 3.1 × 1013 watts. The total power loss of the Earth, Q is
therefore about 4.4 × 1013 watts.
3.4 Mechanisms of heat transfer . The lithosphere is by definition too strong to behave in a ductile
fashion and the dominant mode of heat transfer through the lithosphere is by conduction. Because
of the concentration of radioactive elements in the continental crust, the geophysical modelling
of oceanic and continental heat flow must be treated separately. The creation and destruction of
oceanic lithosphere is part of a grand convective process involving at least the upper mantle. In
this section, we will discuss heat flow in continental and oceanic lithosphere.
1) Continental Heat flow
The radioactive elements Uranium, Thorium and Potassium have large ionic radii that do not fit
well into the high density phases that are stable in the mantle and so they are separated preferentially
into melts. These elements have therefore been concentrated into continental crust and contribute
significantly to the observed heat flow. Since radioactive heat produced within the crust is not
available to do the work of moving plates around, this effect should be subtracted out. What we
wish to know is the “reduced heat flow” which is the heat flowing from the mantle into the base
of the crust. We need first to understand heat conduction in a material which also has heat sources
within it.
Consider a thin slab:
Fig 3.9
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For simplicity we shall assume that the temperature distribution in the slab stays constant with
time, that is, it is steady state. The net heat flow per unit time per unit area of slab face is
q(z + δz) − q(z)
If δz is taken to be small we can expand q in a Taylor series:
q(z + δz) = q(z) + δz
dq
+ ···
dz
Thus
q(z + δz) − q(z) = δz
dq
d2 T
= −δzk 2
dz
dz
and we have assumed that k is not a function of z. If ρ is the density of the slab and H is the heat
production per unit mass then the heat generated in the slab per unit area of slab face (per unit
time) is
ρHδz
If there is a steady state we have
ρHδz = −δzk
d2 T
dz 2
d2 T
+ ρH = 0
dz 2
This equation can be integrated with suitable boundary conditions.
Now what about our problem of heat generation in the continental crust?
or
k
(3.3)
Fig 3.10
We have qs = qm + heat production in the crust; we want qm . Suppose we take a uniform
distribution of heat producing elements and use a value for H typical for granites (H = 9.6 ×
10−10 W kg−1 ). The contribution to qs from heat producing elements in the crust is ρHh = 91mW
m−2 using ρ = 2700Kg m−3 and h = 35km (the thickness of the continental crust.) Something
is wrong as this is higher than the surface heat flux (65 mW m2 ). We can conclude that the
concentration of heat producing elements is not uniform.
In order to choose a more realistic form for the distribution of heat sources, we take a clue from
the correlation of heat flow with crustal age. It is generally true that heat flow is lowest in the
oldest crust and highest in the youngest crust. Since the oldest crust was also the deepest (and has
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since been eroded), this observation suggests that the concentration of heat producing elements
decreases with depth. Let us suppose that it drops off exponentially. This is cast into mathematical
form by:
H(z) = Hs e−z/hr
This looks like:
Fig. 3.11
Now we can rewrite 3.3 for our problem:
d2 T
+ ρHs e−z/hr = 0
(3.4)
dz 2
Integrating this equation once gives us an equation for the temperature gradient which can be
recast into an equation for the heat flux using Fourier’s Law. The integration introduces a constant
which requires a boundary conditon for its evaluation. We shall use the boundary condition that
q(z) = −qs at z = 0 (the minus sign indicates that z is pointing in the opposite direction to q).
Integrating gives
k
dT
− ρHs hr e−z/hr = −q(z) − ρHs hr e−z/hr
dz
where C is a constant to be determined. At z = 0, this equation reduces to
C=k
(3.5)
C = qs − ρHs hr
so now 3.5 can be written
−q(z) = qs − ρHs hr 1 − e−z/hr
(3.6)
This is an equation for the heat flow as a function of depth in the continental crust. Heat flow
must be a continuous function inside the Earth, in particular it will be the same on both sides of
the boundary separating the crust from the mantle. Hs can be measured as can qs , so if we knew
hr , we could use this equation to evaluate the heat flow into the bottom of the continental crust (at
z 35km).
To make further progress, we shall make an assumption that will be verifiable later. We
shall assume that hr is much less than the thickness of the continental crust. This means that the
exponential term in 3.6 will be much smaller than 1 at the base of the crust and so can be neglected.
Now let qm = −q(35) denote the heat flow into the base of the crust and evaluate 3.6 here giving
qs = qm + ρHs hr
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(3.7)
Remember that we can measure ρ, Hs and qs and this equation tells us that if we plot qs as a
function of ρHs , the data should lie on a straight line with intercept qm and slope hr . In fact, for
many regions of the continental crust, this is true (Figure 3.12) and by using fits of straight lines
to the data we estimate that about 6 TW (terrawatts) of the surface heat flow (total 44 TW) are
generated in the oceanic crust so the total coming from the mantle is about 38 TW.
Fig. 3.12 Dependence of surface heat flux on surface radioactive heat
generation for two ”heat flow provinces”.
[If you are worried by the assertion that 35km is much greater than hr we note that
e−z/hr = e−35/10 = .03
which is certainly much smaller than 1]
The actual distribution of radioactive elements in the crust is unclear and, indeed, it is not
always possible to fit a straight line to q/ρHs data though this appears to work well in regions
where crustal radioactivity is high (e.g. granite batholiths). About all that can be said is that it is
likely that radioactive elements are more concentrated to the top of the continental crust.
3.5 The oceanic lithosphere and oceanic heat flow . Modelling heat flow measurements in
oceanic lithosphere requires a different approach from the one we adopted for continental lithosphere. The observation that heat flow depends upon the age of the crust holds for oceanic
lithosphere as well as for the continents and there is an overall similarity in average heat flow values ( 65 mW m−2 in continents versus 101 mW m−2 in the oceans). However, the explanations for
the two are quite different. As discusssed above, heat flow in the continents is controlled partly by
the heat supplied into the base of the lithosphere (the reduced heat flow) and partly by radioactive
sources in the crust. Oceanic lithosphere is not enriched in radioactive elements and radioactive
sources within the oceanic crust can be neglected. The key to oceanic heat flow is the process of
sea-floor spreading whereby oceanic crust is formed at an ocean ridge and then spreads away until
it is eventually subducted back into the mantle at a trench. The concept of seafloor spreading leads
us to try and make a thermal model for the cooling of a lithospheric plate. Since old oceanic crust
has had longer to cool than young oceanic crust, we can explain the dependence of heat flow upon
the age of the crust. We also might expect (and indeed we do observe) a dependence of depth of
the ocean on age (because of the principle of isostasy and the fact that old oceanic lithosphere is
colder and therefore denser than young oceanic lithosphere).
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As a preliminary to developing a model for oceanic heat flow, we must first understand time
dependent heat conduction. Consider a slab as in Fig 3.9. As in the last section we can find an
expression for the net heat flow out of the slab
q(z + δz) − q(z) = δz
dq
d2 T
= −δzk 2
dz
dz
where, in the last step, we have again used the 1-D form of Fourier’s Law (q = −kdT /dz) and we
have assumed that k is independent of position. This net heat flow is due to the fact that the slab
is cooling (and not due to the presence of radioactive heat producing elements as for continental
crust). We have also encountered the energy release, ∆E, due to a temperature change, ∆T
∆E = −M Cp ∆T
(this equation is essentially the definition of the specific heat, Cp which is a property of the
material). What we need is the energy flow per unit cross sectional area of the slab per unit time.
The mass per unit cross sectional area of the slab is ρδz where ρ is the density so the energy flow
per unit cross sectional area per unit time is
−ρδzCp
∂T
∂t
where ∂T /∂t is the rate of change of temperature with time.
By conservation of energy
−δzk
∂2T
∂T
= −δzρCp
2
∂z
∂t
or
∂T
∂2T
=κ 2
∂t
∂z
where κ =
k
ρCp
κ is called the thermal diffusivity and has dimensions of length2 /time. We will use the thermal
diffusivity again in Chapter 5.
Returning to the problem of the cooling of the oceanic lithosphere, we shall consider the
following model:
Fig 3.13
The oceanic plates are created at the ridge (x = 0) and move outward with a velocity u. The
surface is at a temperature Ts because of cooling by seawater and we model the injection of magma
at the ridge crest with a high temperature vertical isotherm at Tm . As the material moves away
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from the ridge, cooling takes place and isotherms become more horizontal. We can think of this as
cooling as a function of distance from the ridge or cooling as a function of age of the lithosphere.
For a plate velocity u, the age, t, is obviously related to the distance from the ridge, x, by t = x/u.
At the ridge a column of mantle material is at a temperature Tm and its surface is suddenly
brought to a temperature Ts . As the column of material moves away from the ridge the surface
temperature is maintained at Ts and the column gradually cools. If we consider a sequence of
such columns:
Fig. 3.14
we note that, away from the ridge the isotherms are nearly horizontal, this implies that the temperature gradient is just a function of depth so that heat conduction is basically one-dimensional.
In what follows we shall neglect the conduction of heat laterally which is a good approximation
for the thin oceanic lithosphere.
The equation governing the behavior of temperature in this model is
∂2T
∂T
=
2
∂z
∂t
The solution is a function of position, z, and time, t, i.e., T = T (z, t) and must fit the boundary
conditions:
κ
T (z, 0) = Tm
and
T (0, t) = Ts
(If we measure temperature in ◦ C it turns out that Ts 0◦ C.)
3.6 Similarity solutions . The boundary conditions are simple constants and have no length or
time scales in them. In this case the solution to the equation is not a function of t and z separately
but can be regarded as a function of a dimensionless parameter which incorporates the length and
time behavior. Bearing in mind the dimensions of κ, one possible choice of dimensionless variable
is
V =
z2
κt
and we can write the solution
T (z, t) = T (V )
We now want to write the diffusion equation in terms of T (V ). We have
∂T
dT V
dT ∂V
dT z 2
=−
=
=−
2
∂t
dV ∂t
dV κt
dV t
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and similarly
∂T
dT ∂V
dT 2z
dT 2V
=
=
=
∂z
dV ∂z
dV κt
dV z
so
∂2T
d
=
2
∂z
dV
dT 2V
dV z
∂V
∂z
2 dT
2V dT dz
d2 T 2V
=
+
− 2
2
dV z
z dV
z dV dV
=
∂V
∂z
d2 T 4V 2
4V dT
2V dT
+ 2
− 2
2
2
dV z
z dV
z dV
=
d2 T 4V 2
2V dT
+ 2
2
2
dV z
z dV
Writing dT /dV as T and d2 T /dV 2 as T , our original equation:
κ
∂T
∂2T
=
2
∂z
∂t
becomes
κ
V
2V
4V 2 T + κ 2 T = −T z2
z
t
Substituting into the expression for V and rearranging gives
2[2V T + T ] = −T V
or finally
T 1
1
d
=− −
=
( ln T )
T
4 2V
dV
Integrating this equation gives
ln T = −
1
V
− ln V + ln A
4
2
where ln A is an integration constant. Thus
ln T = −
V
− ln V 1/2 + ln A
4
so
T V 1/2
V
= exp [− ]
A
4
or
V
A
dT
= 1/2 exp [− ]
dV
4
V
Integrating again gives
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V
A
T (V ) = T (0) +
V
1/2
exp [−
V
]dV
4
0
The integral cannot be done analytically but can be put in the form of a standard tabulated integral
– the error function (so called because it appears in statistics). The error function is defined by
2
erf (η) = √
π
η
exp (−η )dη 2
0
and has the following behavior:
Fig. 3.15. Note that erfc (z) = 1 − erf (z)
√
If we make the identification η = V 1/2 /2 = z/2 κt so that dη = 14 V −1/2 dV , we have
η
T (η) = T (0) + C
exp (−η )dη 2
where C is a constant
0
and so
√
z
C π
√
erf
T (z, t) = T (η) = T (0) +
2
2 κt
The boundary conditions give
at z = 0, η = 0, erf (η) = 0 so T (0, t) = T (0) = Ts = 0◦ C
√
at t = 0, η = ∞, erf (η) = 1 so T (z, 0) = Tm and C π/2 = Tm − Ts .
Our final solution is
z
√
T (z, t) = Ts + (Tm − Ts ) erf
2 κt
We shall now interpret our solution. One question is the thermal structure, i.e., what are the
shapes of the isotherms in the lithosphere? Isotherms are lines of constant temperature and,√because
T = T (η), an isotherm must correspond to lines on which η is a constant. Thus z/2 κt = a
constant so isotherms are parabolas i.e.,
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z∝
√
t
Fig. 3.16
Since x = ut, where u is the spreading velocity, the isotherms are parabolas in space too.
The model predicts the heat flow too which is given by
q = −k
dT
dz
where dT /dz is to be evaluated at the surface (to get the surface heat flow!) at a given time, t. If
we fix t then η ∝ z so we can differentiate the expression for the error function with respect to z.
The result is
2(Tm − Ts )
∂T
= √ √
exp
∂z
π2 κt
z2
−
4κt
At the surface, z = 0 so
dT
Tm − Ts
= √
dz
πκt
Therefore
q=−
k(Tm − Ts ) −1/2
√
t
= −Ct−1/2
πκ
where C is a constant. Using k = 3 Wm−1 K−1 , Tm = 1500K, Ts = 273K and κ = 1.2 × 10−6
m2 s−1 gives C = 330 mWm−2 (Myr)1/2 .
√
If you inspect Figure 3.17, you will find that the t dependence of q is well modeled; however,
the best values of C fitting the data is higher. The values of the constants we have chosen may be
wrong or, more likely, there is a contribution to the surface heat flux from below the lithosphere
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(see below) – in our model, surface heat flux arises only from cooling effects.
Fig. 3.17. Heat flow versus age of ocean floor. The measured values lie close to the straight
line, which shows that the heat flow is inversely proportional to the square root of the age
of the ocean floor. Note that both scales are logarithmic.
Our solution predicts that q → ∞ as t → 0. This is clearly absurd and our assumption that heat
conduction is only in the z direction is poorest when t is small, i.e., this is when the isotherms are
not nearly horizontal. (A more complete analysis still shows the singular behavior of q at t = 0
which is really an artifact of the bondary conditions we have used). Heat flow measurements
tend to actually be depressed near t = 0 and the difference between the model and the data led
people to suspect that water circulation might be an important form of heat transport near the
ocean ridges. This has now been verified by the discovery of vents of very hot water near ridges
and large amounts of heat are being transported by this convection or hydrothermal circulation as
it is called.
As t gets very large, the model predicts a q tending to zero. The heat flow measured in the oldest
parts of the oceans is about 50 mWm−2 (the abyssal value). Our model predicts this value at an
age of about 100 Myr which is consistent with most of the data because only a small fraction of
ocean crust is older than about 120 Myr.
We can use the principle of isostasy to compute the elevation (or bathymetry) of the ocean
floor as a function of age. Consider the following model shown in fig 3.18. The principle of
isostasy states that the lithostatic pressure of various columns of rock and water will be the same at
some depth of compensation. We take this depth to be very deep. An equivalent statement is that
there is the same mass/unit area between the surface and the depth of compensation for different
vertical columns. Consider the two columns in the diagram and let ρ be a function of z and t i.e.,
ρ = ρ(z, t). We shall, for convenience, take the depth of compensation to be infinity. We have
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∞
ρw h 0 +
∞
ρ(z, 0)dz + ∆h(t)ρ(z, 0) = ρw h0 + ρw ∆h(t) +
0
i.e.
ρ(z, t)dz
0
∞
∆h(t)(ρ(z, 0) − ρw ) = [ρ(z, t) − ρ(z, 0)]dz
0
Fig. 3.18
We assume the density differences in the lithosphere are due to the temperature differences predicted by our thermal model. They are related by the coefficient of thermal expansion, α;
δρ = −αρδT
For small density differences we can write, correct to first order in δρ
δρ = ρ(T2 ) − ρ(T1 ) = −α[T2 − T1 ]ρ(T1 )
We shall now assume that ρ(z, 0) = a constant = ρ0 = 3300 kg m−3 (this is consistent with our
thermal model which had a vertical isotherm at the ridge). Now
ρ(z, t) − ρ(z, 0) −αρ0 [T (z, t) − T (z, 0)]
so
ρ(z, t) − ρ(z, 0) = −αρ0 [T (z, t) − Tm ]
z
√
= αρ0 (Tm − Ts ) 1 − erf
2 κt
Substituting into the equation derived from the principle of isostasy gives
∞ z
√
1 − erf
∆h(t)(ρ0 − ρw ) = αρ0 (Tm − Ts )
dz
2 κt
0
This last integral can be done by integrating by parts and the result is
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αρ0 (Tm − Ts )
∆h(t) =
2
ρ0 − ρw
κt
π
1/2
= Ct1/2
where C is another constant which can be evaluated. Using the numbers given above we get C
= 360 m (Myr)−1/2 . The square root dependence of the difference in elevation of the sea floor
with age is seen in the data and a fit to the data gives C = 350 m (Myr)−1/2 for the Pacific. The
distance from the seafloor to the surface should follow the form
H = h0 + Ct1/2
and Figure 3.19 shows how well the data obey this relationship. (The illustration is plotted upside
down as people think in terms of the depth of the seafloor from the ocean surface).
Fig 3.19. The depth of the ocean floor plotted versus the square root of age. Depth has been
corrected for the effect of sediments.
Our model is undoubtedly a crude one but does a remarkably good job of predicting both the heat
flow/age and bathymetry/age observations. One apparent failure of the model is that bathymetry
is overpredicted for ages greater than about 100Ma. A model which does a better job includes
some heating from below. This model (the ”plate” model labelled GDH1 in fig 3.19) differs by
fixing temperature at some depth inside the earth chosen to best fit the observations. This depth,
which defines the thickness of the thermal lithosphere in old oceans is about 100km.
We conclude this chapter by comparing representative temperature profiles (”geotherms”) for
continental and oceanic regions (fig 3.20). Because continental heat flow is lower than oceanic
heat flow, the geotherm tends to be colder under continents down to a depth of about 200 km.
Under very cold shields (such as the Canadian Shield), there may be temperature differences down
to 350km or more. Eventually, we get to a depth where the material is too warm to resist flow and
the mantle experiences convection. Heat transport by convection is much more efficient than heat
transport by conduction and all of the machinery we have developed for looking at conductive heat
flow is irrelevant in a convecting region. As we shall demonstrate in later chapters, the temperature
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profile in a convecting region is close to ”adiabatic” and temperature gradients are much smaller
than they are in conductive boundary layers.
Fig 3.20
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