Download heat and convection in the earth

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? [see Lowerie 4.2 and Fowler Chapter 7 ]
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
convection
-
mass transfer
radiation
-
photon transfer
Silicates are poor conductors and therefore conductive heat transfer 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) also plays a role deep in Earth where T > 2000 K.
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
-
(b)
Internal sources
-
gravitational energy from Sun and Moon
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. 2500 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.
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.
Accretion energy - conversion of K.E. of smaller planetary objects into heat as they collided on accretion.
Collision -> seismic shock -> internal heating.
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.
Core formation Energy - settling of Fe to centre of Earth converts P.E. of iron to heat energy.
Decay of short-lived radio-isotopes - in early-solar system have isotopes such as Al26, Cl36, Fe60, with halflives 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
Th232
U235
U238
2.8 x 10-2
2.6 x 10-2
56.0 x 10-2
9.6 x 10-1
1.3
13.9
0.7
4.5
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
109 years ago
Now
K40
Th232
U235
U238
1.0
1.0
1.0
1.0
4.5
x 109 yrs ago
1.7
1.05
2.64
1.17
10.9
1.25
80.00
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
(Continental Crust)
3.5 ppm K40
18.0 ppm Th232
3.97 ppm U238
0.03 ppm U235
Peridotite
(Mantle)
1.2 x 10-3 ppm K40
0.06 ppm Th232 -> 0.26 x 10-8 mWkg-1
0.01 ppm U238
7 x 10-5 ppm U235
Gabbro
(Oceanic Crust)
-> 96.4 x 10-8mWkg-1
-> 18.63 x 10-8 mWkg-1
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
Th232
U238
U235
0.1 ppm
0.04 ppm
0.01 ppm
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.
In Earth we have:-
Upper Cont. Crust
Lower Cont. Crust
Oceanic Crust
Mantle
Core
Heat generation
Mass
96.4 x 10-8 mWkg-1
40.0 "
18.6 "
0.26 "
?
8 x 1021 kg
8 "
7 "
4080 "
1880 "
-> 0.38 x 10-8 mWkg-1 of silicate in Earth
(N.B. Crust + Mantle make about equal contribution overall)
Generally good agreement between expected heat production from chondrite model and that calculated for Earth,
given uncertainty of role of core.
This suggests possible lost K in core??
Thus heat production in Earth is approximately explained by chondritic Earth model (in detail other problems to
worry about -see later courses in Geochemistry).
Can conclude that heat-flow in Earth is dominated by radio-active decay heat energy. However, estimates of
chemistry are approx.
Best estimates for origins of heat flow from Earth give 80-50% radioactive origin, 20-50% primordial.
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 - crystallizing as the Earth cools.
But how does heat escape and what how does it effect 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 PlateTectonic model.
In a conducting system if know 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 would produce 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.
Recall from our lectures on Plate Tectonics:
What is physics of convection?
Concerned with flow.
We know the mantle flows but need to quantify. 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 = tgdρ/4π
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 connection, 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
g = 10ms-2
z = Depth of cell
)
)
) Buoyancy Terms
)
)
K = Thermal diffusivity (m2s-1)
η = Viscosity (Pas)
Theory and expt. shows that systems will convect only if Ra approx. > 0(103)
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 (see Fowler p 249-254).
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)
(3)
U.M. and L.M. are each well mixed, but chemically distinct.
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 L.M. might suggest a lower η in the lower mantle.
Convection in U.M. unlikely to be uncorrelated with plate size/motion.
As seen, aspect ratio for cells are approx. 1 : 1. In U.M. cell size approx. 600 km., but
plates approx. 4000 km., so could not have simple relationship between plate motion and
convection.
(5)
[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 geotherm.
If L.M. minerals behave in same way to U.M. 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) Recall that 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 have two units, the lower of which has a higher chemical density and try to mix, 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
disproportination:
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 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 localized 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)
(2)
Known experimental data to constrain P and T;
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 TOPOGRAPHY 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 (see Price).
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