Heat and Convection in the Earth
We have seen
how seismology, etc. enables us to determine density, seismic velocity etc. as
a function of depth in the Earth.
Observation
also tells us that the Earth is active - volcanoes, earthquakes, mountain belts
and magnetic fields.
These must be
due to an internal energy or heat source.
What is the
thermal structure of the Earth?
Temperature is
the most poorly constrained planetary parameter.
It cannot be
directly obtained from seismology - need to carry out other studies, e.g.
heat-flow, etc.
We know - Earth
has a hot interior:
- volcanoes;
- hot springs, etc.;
- hot mines at depth.
Heat flows from
centre of Earth to surface via:-
conduction - thermal vibrations: every atom is
physically bonded to its neighbours in some way. If heat energy is supplied to
one part of a solid, the atoms vibrate faster. As they vibrate more, the bonds
between atoms are shaken more. This passes vibrations on to the next atom, and
so on:
convection - mass transfer
radiation - photon transfer
Silicates are
poor conductors and therefore conductive heat transfer is only important in
cold lithosphere.
We will see
that in the Earth, convective transfer
occurs.
Radiative heat
transport (e.g. heating like an electric bar fire) may play a minor role deep
in Earth where T > 2000 K. But minerals are probable too opaque for this to
me very significant.
Two important
questions:-
(1) How
does heat-flow from Earth compare with other energy sources?
(2) Where
does Earth's heat energy come from?
Energy Sources on Earth
Energy sources
responsible for Earth Processes are:-
(a) External
sources
- solar energy
-
gravitational energy from Sun and
Moon
(b) Internal
sources
- Earth's internal heat
- Earth's rotational and gravity field
The external
sources tend to dominate surface processes such as ocean circulation and tides,
atmospheric processes, biological activity.
Internal
sources responsible for volcanism, earthquakes, metamorphism, mountain
building, etc.
Despite massive
effects of internal sources (e.g. creating the Alps), they are much less than
external energy:
Sun -> 1.7 x
1017 W to Earth, of which 60% reaches the surface.
Earth heat ->
4 x 1013 W to surface.
i.e. ~4500
times more energy from Sun than from Earth's interior!
Below the
surface, however, things are different.
No seasonal
variation of T felt below 20 m:
Therefore Sun's
effect not felt at depth - here all processes result from internal sources (20
m maximum depth because silicates are poor thermal conductors).
The energy
reaching the surface of the Earth from within can be measured to get heat flux, q.
q = - k dT/dz
Units of heat
flux = Wm-2 which is
equivalent to Js-1m-2,
and k is the thermal conductivity (Wm-1K-1).
The average
heat flow from the Earth gives a q of approximately 0.08Wm-2
(equivalent to 80mWm-2).
But the flow is
very uneven. Some areas, such as volcanoes and mid-ocean ridges have very high
q ~400 mWm-2.
Could we use
the average heat flow as an
industrial energy source?
If we collect all the energy that flows through
a football pitch (100 x 70 m2), then:
Total power
= 7 x 103 x
0.08 = 560 W
=
5.5 light bulbs!!
Not a generally
viable power source. But locally…
Grand Prismatic
Geothermal Pool in Yellowstone National Park
Geothermal
power only possible in areas of anomalous heat flow (eg Iceland or Japan).
Nevertheless
the internal energy in the Earth can generate large scale changes on geological
time scale.
What are the
sources of the Earth's internal heat processes?
Sources of the Internal Heat of
the Earth
Two
categories:-
(i) primordial
heat - generated during Earth formation.
(ii) radioactive
heat - generated by long-term radioactive decay
Primordial Heat Sources
These are somewhat
speculative as they depend on the hypotheses of Earth formation.
(a) Accretion energy - conversion of K.E.
of smaller planetary objects into heat as they collided on accretion. Collision -> seismic shock -> internal
heating.
(b) Adiabatic compression - as compresses
something cause it to heat up (c.f. bicycle pump) -> adiabatic heating.
As more particles accreted in
planet those at centre squashed by growing gravitational load -> adiabatic
heating.
(c) Core formation Energy - settling of Fe
to centre of Earth converts P.E. of iron to heat energy.
(d) Decay of short-lived radio-isotopes -
in early-solar system have isotopes such as Al26, Cl36,
Fe60, with half-lives of approximately 0.3 Ma.
Heat released
early in Earth's history. How important
these were depends on how rapidly the Earth accreted.
If <20 Ma,
Al26, etc. would have been present and given heat, but if > 100
Ma little effect on heat from Al26 etc.
Relative
importance of these four depends on formation models. Prior to Rutherford, Curie, etc. the
non-radiogenic primordial heat sources were the only known source of heat
energy in the Earth.
This led to calculations by Lord Kelvin of the
age of the Earth, based on cooling rates, of <100 Ma - not 4.5 Ga!
Long-lived radio-isotopes
All radioactive
decay -> heat, but only break-down of isotopes with large half-life will have made a continuing contribution to heat
source over geological time.
Four long-lived
isotopes occur in sufficient abundances as to be important heat sources:-
Isotope Half-life
(x 109 y) Heat generation (mWkg-1)
K40 1.3 2.8 x 10-2
Th232 13.9 2.6 x 10-2
U235 0.7 56.0 x 10-2
U238 4.5 9.6 x 10-1
The total
contribution to global heat production depends on the abundance of the
isotope.
That abundance
has varied throughout geological time because of half-life.
Abundance Now 109 years ago 4.5 x
109 yrs ago
K40 1.0 1.7 10.9
Th232 1.0 1.05 1.25
U235 1.0 2.64 80.00
U238 1.0 1.17 2.0
Heating effect
of U235 much more important at beginning of Earth's history than
today as a result of being x80 more abundant.
To assess
importance of radio-active heating need to know true abundances and
distribution of isotopes.
Cannot sample core and lower mantle, therefore
some uncertainty.
Variation of K, Th and U in rocks means that
some rocks generate more heat from radioactive decay than others, e.g.
Granodiorite 3.5 ppm K40
(Continental Crust) 18.0 ppm Th232 -> 96.4 x 10-8mWkg-1
3.97
ppm U238
0.03
ppm U235
Peridotite 1.2 x 10-3
ppm K40
(Mantle) 0.06 ppm Th232 -> 0.26 x 10-8
mWkg-1
0.01
ppm U238
7 x
10-5 ppm U235
Gabbro -> 18.63 x 10-8
mWkg-1
(Oceanic Crust)
Continental
crust has concentration of radioisotopes (they happen to be incompatible
elements), therefore heat generation in continental crust is more
concentrated.
Mantle has less heat
producing isotopes per kg, but has a much
larger volume than crust.
What about
global heat production from these elements?
Geochemical
models of Earth suggest that the Earth has same chemistry as a chondritic meteorite. Can this be used to estimate radio-isotope
heating effect?
In chondrite
have K40 0.1 ppm
Th232 0.04 ppm
U238 0.01 ppm
U235 0.00007 ppm
This gives
total heat generation of 0.48 x 10-8
mWkg-1 of chondrite, of which K40 and Th232
contribute major part.
If the Earth (mass = 5.97 x 1024 kg) was chondritic
this would give a heat flow of 28 TW.
In Earth we
have:-
Heat
generation Mass
Upper Cont.
Crust 96.4 x 10-8 mWkg-1 8 x 1021 kg
Lower Cont.
Crust 40.0 " 8 "
Oceanic Crust 18.6 " 7 "
Mantle 0.26 " 4080
"
Core ? 1880 "
-> 0.38 x 10-8 mWkg-1
of silicate in Earth -> 23 TW
This suggests possible lost K in
core??
Recall global
heatflow is ~40 TW, so we can conclude that heat-flow in Earth is dominated by radio-active decay heat energy.
Estimates 60 – 70% of heat flow is due to
radioactive heat, and so 30-40% is contributed from loss of primordial heat.
If losing primordial heat the Earth must be
cooling slowly. Estimates range from:
5 to 10 K per 100 Ma -> 230 to 460 K over life span of Earth.
Explains the
formation of the inner core - crystallising as the Earth cools.
But how does
heat escape and what how does it affect the nature of the Earth???
The Lithosphere
Ridge
When the plate
forms, it makes a ridge which is a topographic high because it is hot, and so
has lower density and 'floats' higher.
The heat flow
from oceanic crust is well studied since it was one of fundamental inputs in
development of Plate-Tectonic model.
In a conducting
system heat flow (q) and thermal conductivity (k) are related to temperature
gradient (dT/dZ) since:
q = - k.dT/dZ
Find high q at
ridge - variable, but up to 400+ mWm-2, dropping to about 50-55 mWm-2
at age 80 Ma.
The ridge is a
topographic high, because of the hot underlying asthenosphere.
As the mantle
cools, the lithosphere thickens and the depth of the oceans increases as the
ridge subsides:
From such a
cooling model would expect ocean depth (d in m) to depend on the age (t in Ma)
of the crust thus:
d = 2500 +350t0.5
Also have a
heat flow which depends on t0.5.
q ~ t-0.5
Seen in this
log q v. log t plot:
At t < 5 Ma
the heat flow due to conduction is less than expected because heat is also
transferred by hydrothermal circulation at ridge.
At t > 100
Ma, heat flow is greater than expect from a simple cooling model, because of
effect of heat escape from underlying mantle.
Subduction Zone
Heat flow is
altered by subduction, low at trench due to cold slab, high at island arc
because of volcanics.
Temperature Gradients in the
Lithosphere
Important to
know if we are to construct full T profile of Earth.
Can measure
near-surface (top 5-10 km) gradients.
Find values of
between 10 Kkm-1 and 40 Kkm-1.
Ave value of
20-25 Kkm-1
This must
decrease with depth because otherwise it would produce an inner core T of
approx. 180,000 K - impossibly hot! (surface of sun approx. 6000K).
We will see
below that at depth in the Earth convection
in the asthenosphere is the main mechanism for heat transfer, and the
extrapolation of conduction models
below the lithosphere are not valid.
Convective Instability and Mantle
Dynamics
Plate tectonics is a convective
processes.
That is heat is transferred by
the motion of matter - cold slabs sink and hot magma is emplaced near the
surface at ridges.
To accommodate this, there must
be motion below surface too.
What is the nature of the
subsurface motions in the Earth's interior, what drives the convection and what
is the cause of the dynamics of the Earth's mantle?
We can envisage two ways in
which plate tectonics occurs:
(1) Mantle
convects actively ("like a pan of soup") and plates are driven by
this motion.
(2) Plate
motion is determined by lithospherical forces, with the mantle motion not
necessarily being strongly correlated to plate motion.
In this section, we will look at
model (1), while model (2) will be discussed later.
In order to address the problem
of mantle convection, we need to look at some basic fluid dynamics.
Most results have been derived
from the analysis of simple systems and the general principles hold.
What is physics of convection? It is concerned
with flow.
We know the mantle flows but we need to quantify
this flow. Need to consider:
viscosity - A measure of how easily flow
occurs.
Materials flow when subjected to stress.
η =
σ / (dε/dt)
where, η = Viscosity (Pa.s);
σ = Stress (Pa = Nm-2);
Strain rate = dε / dt (s-1).
Viscosity of
liquids that we know are low –
ηH2O = 10-3
Pas,
ηhoney = 102
Pas:
The viscosity
of the mantle depends on the mechanism of flow.
In H2O
have a liquid - no long range atomic order, but the mantle is crystalline.
Apply stress to crystal - elastic response.
Elastic limit -
Plastic - Permanent Strain
Creep can occur by
two mechanisms
Dislocation
Glide + Climb
Diffusional
Flow - Atomic Vacancy Movement
By diffusing to surface can give shape
change.
Both processes
need atomic motion so they are thermally
activated
Rate α exp (-H/RT)
H = activation energy; T =
temperature
Viscosity of
Rock/Crystals is therefore very temperature
dependent.
Cold lithosphere with η -> ¥ is not
plastic.
Hot mantle has large but finite η and
so can be plastic.
Behaviour is
also a function of dε/dt.
Small stress applied for long t gives
plastic behaviour.
Large stress over small t -> brittle
behaviour.
Mantle
Viscosity
Revealed by
observation of isostatic rebound due
to loss of ice-cap, or post-glacial
uplift. Ice-cap loads lithosphere. Deflected downwards - elastically.
Fennoscandian ice cap lost approx. 104y ago.
Max. measured
uplift approx. 200 m; therefore rate approx. 2 cmy-1 , and gravity
studies indicate there will be further uplift.
To calculate
η need to analyze system.
Two possible
models -
DEEP FLOW CHANNEL FLOW
Channel flow
-> peripheral bulge to accommodate displaced material.
No such bulge in Fennoscandia, therefore
deep flow.
In model, can
calculate η to a depth = half diameter of ice cap.
η = t g
d ρ / 4 π
where,
t = time for uplift approx. 104 years
(approx. 3 x 1011s);
g = 10 ms-2;
ρ = 3300 kgm-3;
d = radius of old ice sheet approx. 1500 km.
This gives a
estimate of mantle viscosity of:
η= 1 x 1021
Pas
More detailed
results indicate that the LVZ has η
approx. 4 x 1019 Pas, while the rest of the whole mantle has
approx. constant η with range 1021 - 1022 Pas.
Having found
vital data of viscosity, can model mantle like a fluid system.
But in reality
remember that its is not just like a soup-pan, because:
Spherical earth
Radioactive internal heating
T dependent η
Non-Newtonian
Fluid Dynamics
The behaviour of a uniform
liquid heated uniformly from below was studied experimentally by Benard - found
three stages in such process:
(1) Small ΔT- No convection. Heat transfer by conduction.
(2) Larger
ΔT - Stable convection occurs - regular hexagonal mesh of cells: aspect
Ratio approx. 1 : 1
(3) Very
large ΔT - turbulent convection regular cell pattern broken up.
Theory developed by Rayleigh:
He introduced a term now known
as the Rayleigh Number (Ra) to
describe convective systems.
Ra shows balance between
buoyancy forces which promote convection, and η and conduction effects
which inhibit it.
Ra = α ΔT g z3 ρ / K η
α = Coefficient vol. expansion (K-1)
)
ΔT = Temperature
gradient )
ρ = Density ) Buoyancy Terms
g = 10ms-2 )
z = Depth of cell )
K = Thermal diffusivity (m2s-1)
η = Viscosity (Pas)
Theory and expt. shows that
systems will convect only if Ra approx. > 1700
Calculations show that the
mantle can indeed convect, and has:
Ra ~ 105 to 107.
This would suggest that the
convection is turbulent or time dependent.
What Fluid Dynamics says about
Mantle Convection
In a convective system have upper and lower thermal boundary layers
- where heat flow is dominated by conduction of thickness δ, and an
adiabatic gradient in core region.
Gradient in boundary layer is ~
ΔT / 2δ . From fluid dynamics know that :
δ ~ z/2
(Racritical/Ra)1/3
(if η constant!)
For z ~ 1800 km, Ra ~ 107,
Rac ~ 103, then δ
~ 100 km.
Thus from the mantle convection
model, can consider oceanic lithosphere as the upper boundary layer.
Such an analysis would further
suggest that the thickness of the oceanic lithosphere would varies as the
square root of the age of the sea floor, which is indeed the case for sea floor
older than 5 Ma and less than about 80 Ma.
Predict gradient to be approx.
15 K km-1, which is in good accord with average global values.
Stability analysis of the lower
thermal boundary layer suggests that it may be unstable, and perhaps the
D" zone is the source of hot spot plumes.
Also from fluid dynamics can
calculate flow rates or plate velocities:
u = 0.15 Ra2/3 K/Z ~
10 cm y-1
Thus mantle convection model gives excellent predictions of plate processes:
There is still great uncertainty
about the nature of mantle convection:
·
is convection layered, or does
the mantle convect as a whole?
·
what effect does P,T dependent
viscosity have on convection?
·
what effect does the
spherical nature of the Earth have on
the style of convection?
Layered or Whole Mantle Convection
Major question because of
influence on evolution of Earth: not only thermal but also chemical.
What points to
layered convection?
(1) Geochemistry: need many different
mantle sources to explain, Sr, Pb, Nd isotope data found in MORB, OIB, etc.
rock types. The argument being that if we had whole mantle mixing would have a
homogeneous mantle and no isotopic distinct sources.
(2) Seismology: no undisputed evidence for
slab passing 670 km discontinuity on subduction.
Also the deep
seismic focal mechanism solutions are compressive, suggesting the resistance to
subduction of the 670 km discontinuity.
Implications
of layered model
(1) No
mass transfer across 670 km boundary, plates accumulate at 670km – “megalith”
model of Ringwood.
(2) U.M.
and L.M. are each well mixed, but chemically distinct.
(3) Heat
transfer at 670 km boundary is conductive.
(4) Must
have a double thermal boundary layer. Temperature increase at 670 km could be
500 to 1000 K. This higher T in the
L.M. might suggest a lower η in the lower mantle.
(5) Convection
in U.M. unlikely to be uncorrelated with plate size/motion. As we have already
seen, the aspect ratio for cells are approx. 1:1. In the U.M. the cell size is approx. 600 km.,
but plates atr approx. 4000 km., so could not have simple relationship between
plate motion and convection.
[But see for
example Craig and McKenzie (Earth
Planet. Sci. Letters, 78, p 420-426, 1986), who show that the existence
of a low viscosity layer under the lithosphere is likely in any case to
decouple plate kinematics and deeper mantle processes.]
Points for
Whole Mantle Model
(1) Simple!
(2) Fluid dynamics
model gives good prediction of lithosphere thickness, etc.
(3) Convective
cell size approx. size of plates.
(4) probably
no kink in the geotherm. If L.M. minerals behave in same way to U.M. we would
expect a 1000 K kink to give major drop in η, as viscosity is so dependent
on T. No evidence for low η in lower mantle - in fact analyses of the
geoid suggest that the lower mantle may have a slightly higher viscosity that
the upper mantle (Hager and Richards, Phil Trans Roy Soc Lond A328, p 309-327, 1989).
How does the
model allow for different Isotope sources?
Whole mantle convection and
isotopic heterogeneity are not incompatible because convection does not
necessarily produce perfect mixing in mantle.
Calcs. show heterogeneity in a
convective system can be long-lived, giving a "Marble-cake mantle"
or regions of segregation of
denser material in, for instance, the D" zone (e.g. Kellogg, Ann Rev Earth Planet
Sci, 20, 365-388, 1992; Christensen, Phil Trans Roy Soc Lond, A328, p 417-424,
1989).
Why should 670
km boundary be a barrier to convection?
(1) Our knowledge of mineral
physics is still imprecise, and, from seismic data, we cannot exclude the
possibility that the lower mantle could be chemically distinct from the upper
mantle and transition zone - possibly being more Fe rich.
This would mean that given
isothermal volumes of U.M. + L.M. would have different ρ
- a so called chemical density effect.
Chemical density effects could
outweigh thermal expansion effects which would normally drive hot buoyant lower
mantle material upwards.
For a given volume the buoyancy
which drives convection is given by:
α ΔT ρ g
if ρ is constant
If we have two units, the lower
of which has a higher chemical density, and try to mix them, then the
difference in force acting on a given volume is:
(ρ2 - ρ1)g =
Δ ρc g.
If:
Δ ρc g >
α ΔT ρ1 g
the chemical density effect will
counteract the negative buoyancy and so boundary will act as a barrier to
convection.
(2) There is a phase change with
a negative slope (endothermic) at the 670 km boundary associated with the
disproportionation reaction:
Spinel -> Perovskite + (Mg,Fe)O
The negative slope means that
spinel is stable in a cold slab even
below 670 km.
Spinel is less structurally dense than perovskite, and
so this preservation of spinel at high P would inhibit convective mixing.
If
dP/dT approx. 0 then small effect
If dP/dT -> - ∞ then
get larger effect.
So what
happens in the Earth?
Analysis of these factors of
structural and chemical density effects are not easily treated, and full
computational models are needed. An early study is described by Christensen, Phil Trans Roy Soc Lond, A328, p
417-424, 1989.
He found that if Δρ /
ρ and slope
-> 0, then no barrier, but if Δρ / ρ ->
3% OR slope ~ -60 bar/K, then the boundary is a Convection Barrier.
Not just simple barrier could
have leaky state, with some mass
change, and also deep slab penetration.
For Earth, find data close to boundary of single or layered convection:
so from this analysis, it is not
possible to rule out either process.
A more recent study by
Tackley et al. (Nature, 361, p 699-704, 1993) was carried out using a
spherical shell model.
They found that a 3D flow
pattern was produced containing cylindrical plumes and linear sheets.
The dynamics were dominated by
the accumulation of down welling cold material above the 670 km boundary, which
periodically avalanches into the
lower mantle.
Similar results have been
presented by Solheim and Peltier (JGR, 99, p 6997-7018), who found that
their simulation was Ra number dependent, and that the periodicity of the
avalanches was controlled by instabilities which developed in the internal
thermal boundary layer that develops when the convection is layered.
Are these models however
supported by direct observation of the mantle?
Seismic Tomography
In recent years, seismic
tomography (see for example Woodhouse and Dziewonski, Phil Trans Roy
Soc Lond, A328, p 291-308, 1989; Romanowicz, Ann Rev Earth Planet Sci, 19,
77-99, 1991) has given an increasingly resolved view of the internal
thermal structure of the mantle, and it is now possible to correlate the
observations with fluid dynamical models.
Seismic tomography gives 3D
image of seismic velocity of the Earth's interior, which can reasonably be
interpreted in terms of the thermal structure of the Earth.
At shallow depths, the mantle
beneath ridges is hot and under continental shield areas it is cold, but these
anomalies do not necessarily persists below about 300 km.
It appears that the distribution
of hot spots correlates strongly with anomalously hot regions at the core
mantle boundary (CMB), supporting the suggestion that at least a significant
number of hot spots are the result of plume initiation in the unstable D"
zone.
There is a ring of high
velocities extending through the lower mantle around the rim of the Pacific,
apparently correlated with the circum-Pacific
subduction zones.
The geoid anomalies correlate
with the general distribution of velocity highs and lows in the mantle. All of
these support the view that tomography images mantle convection.
Current picture of convective flow
in the Earth's mantle in which whole mantle style flow is dominant at present
but in which phase transition induced localised layering also exists.
Thus fluid dynamics and seismic
tomography seem now to suggest that the upper and lower mantle do mix, but
perhaps aspects of the circulation are layered.
Also note that geochemistry samples whole past history,
but seismology shows structure today
– could have had a change of convection
style in the past billion years.
Mantle Thermal
Structure
To calculate T
in mantle, we cannot use conductivity measurements. Instead we must use:-
(1) Known experimental data to constrain P and T;
(2) Knowledge of convective systems to model
average T.
Under
lithosphere containing oceanic and young continent, have the L.V.Z. at depth of
approx. 100 km.
The L.V.Z. is
believed to be region of partial melting of peridotite. Depending on H2O content, this
corresponds to a T of approx. 1100-1200oC.
Thus this gives
a fixed point for lower lithosphere temperature (N.B. also gives dT/dZ of
approx. 10 Kkm-1, which is heat-flow inferred gradient for old
oceans).
At greater
depth melting T of peridotite increases rapidly (increasing P) and so rest of
mantle is not melted and has a high seismic velocity.
No L.V.Z. under
cratons. This means a lower average geotherm - in accord with lower q for these
areas. Cold roots to cratons seen by
seismic tomography too.
What are mantle
geotherms?
Have seen that
have shallow geotherm in average mantle, why?
In a convective system, have upper and lower
thermal boundary layers with high gradients (e.g. lithosphere and D").
Also have an "isothermal" average core
region which would correspond to the bulk mantle.
In reality, it is not "isothermal",
but has T increase via adiabatic heating.
For this:
dT/dZ = -αgT/Cp
where Cp
is the heat capacity, α is thermal expansion coefficient.
For mantle
material dT/dZ adiabatic approx. 0.3 Kkm-1, this is 1 to 2 orders of
magnitude less than the conductive geotherm in lithosphere.
A further T - Z
constraint is the olivine -> beta-phase phase transformation at the 400 km
seismic discontinuity. Experimental studies indicate that this occurs at 1700 K
at approx. 10 kbars, the P at 400 kms.
Base of Transition Zone (from
phase diagram of Mg2SiO4) ~1600 C, 1873 K
Base of mantle ~2700 K at D”.
From work of Alfe et al, top of
core ~4000K:
IOB ~ 5500 K.