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:


rooms           convection     -    mass transfer








current_eit_304small           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



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



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




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:

2 - ρ1)g  =  Δ ρc g.



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.


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