Section 8Thermal state of the Earth – Part 1 EPS-320 / Fall-2007 – Professor Michael Riedel...

Section 8 Thermal state of the Earth – Part 1

EPS-320 / Fall-2007 – Professor Michael Riedel ([email protected]) Slide S8-1

Contents:

Heat within the Earth: What are the sources and sinks?

Heat transfer by conduction, convection, radiation

Basic equations of conductive heat transfer and general thermal diffusion

Oceanic and continental heat flux

Temperature-gradient inside the Earth

Consequences of temperature distribution on mineralogy, rheology, physical properties of rocks and ultimately on global mantle convection

Models of plate tectonics: subduction zones and hot-spot plumes

Is the Earth cooling?


Heat in the Earth – What are possible sources?

(1) Accretion process of early Earth generated heat – amount Q0 is unknown

(2) Core-formation and differentiation – heat is estimated to 1.0 ∙1012 W

(3) Radioactivity – depends on amount of radioactive material within rocksU, Th, 40KGranite ~1000 ∙ 10-12 W/kg

Basalt ~20 – 200 ∙ 10-12 W/kg Ultra-mafic rocks ~1 – 10 ∙ 10-12 W/kg

radioactive material is enriched in crust; amount in crust can be calculated, amount within mantle is uncertain but meteorites may provide some estimates, but there is no information about core!

(4) Chemical/mineralogical reactions – heat flux is estimated to 2.0 ∙ 1012 W

The total sum is estimated to30 – 35 ∙ 1012 W


Heat in the Earth – What are possible losses?

(1) Loss of heat from core to mantle

(2) Loss of heat from mantle to crust

(3) Loss of heat from crust into atmosphere

Measured heat flow at surface of Earth:

Continental crust Precambrian shields ~40 mW/m2

young rifts/Volcanoes ~100 mW/m2

Oceanic crust trenches ~50 mW/m2

mid-ocean ridges ~80-120 mW/m2

Total heat flow (1/3 continental, 2/3 oceanic) ~30-40 ∙ 1012 W

Production and loss of heat are about equal!




Heat transfer mechanisms:

(1) Conduction – requires contact of material; thermal conductivity of rocks is temperature dependent

Heat energy flows from regions of high T to regions of low Theat flux occurs if and only if there is a T-gradient

(2) Radiative heat – energy is radiated in form of electromagnetic waves; Infrared (IR)-radiation invisible – but you can feel it

Amount of radiated energy is proportional to T4 and depends on transparency of medium

(3) Convection – exchange of heat by macroscopic motion material of high T is more buoyant, motion is dependent on viscosity of material



Heat conduction

Let’s start by considering the heat transfer across a slab.

Laboratory measurements showed us that:

(a) Heat flows from the hotter to the cooler end,

(b) The heat flux (j) is linearly dependent on the temperature difference between the two end

(c) The heat flux is proportional to the thermal conductivity (K) of the material

(d) The heat flux is inversely proportional to the distance between both ends.

Thus:

dx

dTK

xx

TTKj xxx

12

12


EPS-320 / Fall-2007– Professor Michael Riedel ([email protected]) Slide S8-6

Heat conduction

We have so far considered only the x-component of the heat flux. If we expand this into 3D and also allow anisotropy (K is dependent on the direction of the heat flux), the former equation becomes:

Anisotropy mainly occurs in crystalline material which has preference for thermal conduction in a particular direction. In multi-crystalline rocks the anisotropic effect averages out due to the more or less random distribution of the individual crystals.

Thus we can simplify the above equation:

z

y

x

K

K

K

K

TKj

00

00

00

TKj



Equation for thermal diffusion

Lets consider a small volume element (dV);

If there is more heat coming out of this volume element than what is going in, the total heat (Q) of the volume element is changing over time. Thus the outgoing heat flux (jout) is larger than the incoming heat flux (jin). In this case the volume element is cooling over time (e.g. like a cup of coffee or tea).

Mathematically this is described by the divergence (div) operator:

jdVdivt

Q



We can relate the change in heat with the change in temperature.

The heat capacity of the volume element is: c dV.

jdVdivt

Q

Tt

T 2

is also called the thermal diffusion coefficient.

TTc

KTgraddiv

c

K

t

T 22))((



Heat conduction in a 1-D medium – Oceanic crust

A 1-D medium is a first-order, but reasonable approximation for heat conduction in the oceanic lithosphere or for extended sills in continents. The physical properties are only depth (z) dependent. We further have neglected heat generation (h) in the previous case as we just looked into the change in flux over time. Thus the diffusion equation becomes:

In a further simplification we can assume that thermal conductivity K is independent of depth z:

hz

TK

zt

Tc

c

h

z

T

t

T

2

2



The vertical temperature gradient is related to heat flow. Measurements of heat flow on oceanic crust show that heat flow is reduced the further away you are from the mid-ocean ridge, i.e. the older the crust. [Sclater et al., 1980]

Simplified, heat flow (and water depth) is proportional to age

From Stacey (1992), page 295 But this holds only until an age of 60 Ma!



EPS-320 / Fall-2007 – Professor Michael Riedel ([email protected]) Slide S8-2-1

Adiabatic processes

A key question in geophysics is still whether mantle convection appears in layers, or if it is continuous across the depth range from outer core to the upper mantle.

The temperature distribution inside the Earth could yield significant clues to answer this question. In order to understand the argumentation made by some scientists, we need to first introduce the concept of adiabatic processes.

An important equation in this discussion is the Thomson-equation, as well as the Grüneisen-parameter.

The temperature-distribution inside the Earth is important to decide whether rocks can cross the solidus, i.e. if melt or partial melt can occur.

From seismic work we know several key discontinuities inside the mantle linked to know phase-changes in the mineralogy. Those sudden phase-changes are fix-points in our temperature-depth curve.



Observe the mass-element with constant mass M in the diagram on the right.

If this element rises it also expands. The work used to increase the volume is taken from the heat-energy of the mass-element, thus it cools while rising.

In the case of our Earth the thermal conductivity of the mantle material is relatively low, thus there is no (or only negligible) exchange of heat between the mass-element and its surroundings. This is called an adiabatic process.

After Strobach (1991)



If over longer periods of time some mass-elements rise (and cool) and others sink (and compress – thus increase their heat) a temperature field will be established that is such that at any depth the moving mass element is always in equilibrium with its surrounding temperature.

With = thermal expansion coefficient, = density, cp = specific heat at constant pressure, T absolute temperature , dp = change in pressure.

This is called an adiabatic temperature distribution. It can be described by the Thomson equation: ∙cp∙dT = ∙T∙dp

After Strobach (1991)



If we exchange in the Thomson equation our expression of pressure (p) at a depth z by using the general hydrostatic equation:

dp/dz = g ∙ ,

then we get the following expression:

dT = ( dz ∙ T ∙ g ) ∙ ( / cp).

In theory we could now simply integrate this equation with respect to depth (z) if we would know g and the term (/cp) as function of depth.

Now a few more substitutions are made to simplify the integration: Instead of the expression (/cp), we would like to use the better known Grüneisen-parameter:

Here kS is the adiabatic incompressibility coefficient.

P

S

c

k



Thus Thomson’s equation can now be written as follows:

In this equation we notice the expression / kS, or the reciprocal value:

This new value is well known, as it relates to seismic P-wave (Vp) and S-wave (Vs) velocities:

= Vp2 – 4/3 Vs2.

Thus, we can finally get an expression that is easy to integrate:

dzk

TgdT

S

Sk

dzTg

dT


EPS-320 / Fall-2007– Professor Michael Riedel ([email protected]) Slide S8-2-6

As so often in geophysics, early hopes of defining the Grüneisen-parameter did not turn out to be an easy task, as it appears that this parameter is heavily dependent on not just the material itself, but more so on temperature and pressure, which makes the Thomson equation not the best tool to calculate temperature with depth absolutely.

However, it can be used to interpolate between several fix-points in the temperature-depth curve, which we can determine from somewhere else independently.

From mineralogy-experiments we know quite well the behavior of the olivine-spinel system and how it behaves with temperature and pressure.



Olivine is an Fe-Mg silicate, also described as -(Mg,Fe)2SiO4.We further distinguish a -(Mg,Fe)2SiO4 (modified spinel structure) and -(Mg,Fe)2SiO4 (spinel-structure).

The phase-changes from - to - and to -(Mg,Fe)2SiO4 occur at distinctive pressures (depth) and temperatures. They are also associated with sudden jumps in density and thus in seismic velocities.

The 1st phase change from - to -(Mg,Fe)2SiO4 occurs at ~410 km depth with a temperature of ~1700ºK.

The 2nd phase change from - to -(Mg,Fe)2SiO4 occurs at ~520 km depth with a temperature of ~1820ºK.

The 3rd phase change from -(Mg,Fe)2SiO4 to perovscite occurs at ~670 km depth and a temperature of ~1950ºK.

The olivine-spinel-perovscite system:



Do we have other fix-points at greater depth to control our temperature gradient?

The core-mantle boundary (CMB) and the thin layer just above it (known as D’’ – D-double-prime) offers such a fix-point. The CMB is the transition from the mantle into the liquid outer core. Knowing the melting point (liquidus) of the core-material - mostly iron (Fe) – we can define the temperature at the CMB.

The D’’ layer is marked by a jump of 500ºK as defined from seismic velocities. The CMB is at 2900 km depth, marking a temperature of 3500ºK.

Another fix-point is given at the boundary between inner (solid) and outer (liquid) core (ICB = inner core boundary) again by looking at the solidus/liquidus of the core material. The ICB at 5120 km depth marks a temperature of 4900ºK.

In between all these temperature fix-points we could interpolate using the Thomson equation and making assumptions on the Grüneisen-parameter .



Before continuing to the actual temperature distribution inside the Earth, let’s consider what is required for melting to occur in the core.

Rocks made out of several individual minerals do not have a uniform melting temperature. The individual melting points define a solidus-curve, above which those minerals start to melt that have the lowest melting point.

The temperature at which all material is melted is called the liquidus.

The melting point of minerals is highly dependent on pressure and can be approximated by: (Law by Lindemann)

Where Tm = melting temperature, Tm0 = melting temperature at atmospheric pressure, = density, 0 = density at atmospheric pressure, = Grüneisen-parameter.

3/12

00

mm TT



Modified after Strobach (1991)



A couple of observations and comments about this temperature-depth curve:

A Grüneisen-parameter of ~1.3-1.4 was used to calculate the gradient inside the inner and outer core.

In the upper mantle was set to ~2 to interpolate between the individual Olivine-spinell-perovscit phase changes.

The lower mantle was modeled using an iso-viscous interpolation from 670 km depth to the D’’ layer and not an adiabatic gradient! The difference is only ~ 100K, but makes a huge difference in the overall question if convection in the mantle can occur. This method by Poirier and Liebermann (1984) results in a uniform viscosity of ≈ 1021 Pa∙s at all depths of the lower mantle.

In order to allow convection, the T-grad has to be above the adiabatic T-grad. If T-grad is above the adiabatic gradient turnover of the material (instability) results automatically and convection tries to re-establish an adiabatic gradient.

The thermal boundary of the D’’-layer (T jumps by 500K over less than 150 km) is origin of large mantle plumes. At the base of the D’’ layer viscosity is reduced by 4 orders of magnitude compared to the lower mantle above!

Section 8Thermal state of the Earth – Part 1 EPS-320 / Fall-2007 – Professor Michael Riedel...

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