Transcript Introduction of the Solid Earh Geophysics

Earth as a Planet
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Mass M = 6 x 1027 g.
Radius R = 6371 km.
Mean density = M/(4/3
p R3) = 5.5 g/cm3
Moment of inertia I of
the Earth:
• I =  r2 dm
• I/(MR2) = 0.331.
• for a uniform sphere
I/(MR2) = 0.4.
Differentiation
early in Earth’s
history
Interior of the Earth
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Crust: variable thickness with an average value of 35
km in the continents and 7-8 km in the oceanic regions.
Volume ~1019 m3 Mass 2.8 x 1022 kg.
Mantle: between the Moho discontinuity (crustmantle) and the core-mantle boundary (R = 3480 km).
Volume 9 x 1020 m3 Mass 4 x 1024 kg.
Core: from the center of the Earth to the core-mantle
boundary. Volume 1.77 x 1020 m3 Mass 1.94 x 1024 kg.
More Details about Layering…
What is Earth made of?
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Why do we need to look outside the Earth
to learn about what is inside the Earth?
• We know the Earth is layered and that what we
can sample on the outside is not typical.
• By studying members of the Solar System, it is
possible to estimate its original composition
and the physical and chemical processes that
have led to its present state.
The Solar System: A highly diverse zoo!
Overview of the Solar System
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Sun
Mercury
Venus
Earth
• Moon
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Mars
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Asteroids
Jupiter
Saturn
Uranus
Neptune
Pluto
The Origin of the Solar System
Frank Crary, CU Boulder
Here is a brief outline of the current theory of the events
in the early history of the solar system:
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A cloud of interstellar gas and/or dust (the "solar nebula") is disturbed and
collapses under its own gravity. The disturbance could be, for example, the
shock wave from a nearby supernova.
As the cloud collapses, it heats up and compresses in the center. It heats
enough for the dust to vaporize. The initial collapse is supposed to take less
than 100,000 years.
The center compresses enough to become a protostar and the rest of the gas
orbits/flows around it. Most of that gas flows inward and adds to the mass
of the forming star, but the gas is rotating. The centrifugal force from that
prevents some of the gas from reaching the forming star. Instead, it forms
an "accretion disk" around the star. The disk radiates away its energy and
cools off.
The Origin of the Solar System
Frank Crary, CU Boulder
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First break point. Depending on the details, the gas orbiting
star/protostar may be unstable and start to compress under its
own gravity. That produces a double star. If it doesn't ...
The gas cools off enough for the metal, rock and (far enough
from the forming star) ice to condense out into tiny particles.
(i.e. some of the gas turns back into dust). The metals
condense almost as soon as the accretion disk forms (4.55-4.56
billion years ago according to isotope measurements of certain
meteors); the rock condenses a bit later (between 4.4 and 4.55
billion years ago).
The dust particles collide with each other and form into larger
particles. This goes on until the particles get to the size of
boulders or small asteroids.
The Origin of the Solar System
Frank Crary, CU Boulder
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Run away growth. Once the larger of these particles get big enough to have
a nontrivial gravity, their growth accelerates. Their gravity (even if it's very
small) gives them an edge over smaller particles; it pulls in more, smaller
particles, and very quickly, the large objects have accumulated all of the
solid matter close to their own orbit. How big they get depends on their
distance from the star and the density and composition of the
protoplanetary nebula. In the solar system, the theories say that this is large
asteroid to lunar size in the inner solar system, and one to fifteen times the
Earth's size in the outer solar system. There would have been a big jump in
size somewhere between the current orbits of Mars and Jupiter: the energy
from the Sun would have kept ice a vapor at closer distances, so the solid,
accretable matter would become much more common beyond a critical
distance from the Sun. The accretion of these "planetesimals" is believed to
take a few hundred thousand to about twenty million years, with the
outermost taking the longest to form.
The Origin of the Solar System
Frank Crary, CU Boulder
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Two things and the second break point. How big were those
protoplanets and how quickly did they form? At about this
time, about 1 million years after the nebula cooled, the star
would generate a very strong solar wind, which would sweep
away all of the gas left in the protoplanetary nebula. If a
protoplanet was large enough, soon enough, its gravity would
pull in the nebular gas, and it would become a gas giant. If not,
it would remain a rocky or icy body.
At this point, the solar system is composed only of solid,
protoplanetary bodies and gas giants. The "planetesimals"
would slowly collide with each other and become more
massive.
The Origin of the Solar System
Frank Crary, CU Boulder
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Eventually, after ten to a hundred million years, you end up
with ten or so planets, in stable orbits, and that's a solar
system. These planets and their surfaces may be heavily
modified by the last, big collision they experience (e.g. the
largely metal composition of Mercury or the Moon).
Note: this was the theory of planetary formation as it stood
before the discovery of extrasolar planets. The discoveries
don't match what the theory predicted. That could be an
observational bias (odd solar systems may be easier to detect
from Earth) or problems with the theory (probably with subtle
points, not the basic outline.)
The Big Questions
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What is the origin of the solar system? It is generally agreed that
it condensed from a nebula of dust and gas. But the details are far
from clear.
How common are planetary systems around other stars? There is
now good evidence of Jupiter-sized objects orbiting several nearby
stars. What conditions allow the formation of terrestrial planets? It
seems unlikely that the Earth is totally unique but we still have no
direct evidence one way or the other.
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Is there life elsewhere in the solar system? If not, why is Earth
special?
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Is there life beyond the solar system? Intelligent life?
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Is life a rare and unusual or even unique event in the evolution of
the universe or is it adaptable, widespread and common?
Solar abundance of elements
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Determined from spectral absorption
lines
• Light from visible surface of sun passing
through cooler gases above the surface
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This is thought to represent total
solar abundance because nuclear
reactions powering the star take
place deep inside and there is little
convection there to mix modified
material up with original material.
Meteorites
Meteorites: Summary
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The fabric of chondrites is quite unlike that of any terrestrial
rock and required very different conditions in which to
form. These are identified with early stages in the
development of the Solar nebula.
The carbonaceous chondrites are a close approximation to
the material of the Solar Nebula, having lost only the most
volatile elements. It is, therefore, plausible to regard them
as a starting point from which the composition of the Earth
has evolved. This leads to the Chondritic Earth Model.
The meteorites derive from the asteroids by collision.
The differentiated meteorites were formed within minor
planets, or asteroids, which heated sufficiently to segregate
into layers, forming an iron core, silicate mantle and
transitional region between. Subsequent break-up due to
collisions produced iron, achondrite, and stony-iron
meteorites.
Next Question:
What is the origin of the
distribution of elements in the
Solar System?
Hydrogen, the simplest element, is
the basic building block.
•Hydrogen burning – 4 protons become alpha particle (helium nucleus)
•Helium burning - 3 alpha particles become 12C (which can absorb another
to become 16O
•Carbon burning and oxygen burning produce 28Si (very stable), 24Mg , 32S,
and other elements
•Each of these requires more heat than the fusion reaction before it.
•Silicon burning involves breaking of pieces of other nuclei and adding them to
others. This produces many stable nuclei heavier than Si.
•As temperature rises, this becomes the equilibrium e-process which is like shaking
and breaking up all the existing nuclei and recombining them randomly to make all
possible stable nuclei up to the iron group elements.
•Everything bigger than the iron group is less stable and the e-process would
rearrange them into iron group elements.
•Neutron capture becomes the method that builds larger nuclei.
•Slow-neutron or the s-process. Add a neutron to a nucleus. If nucleus becomes
unstable because its neutron/proton ratio is too high, it has time to “fix” itself before
another neutron arrives. It “fixes” things by a beta decay. A neutron converts to a
proton and an electron is emitted. The nucleus has moved one element up the
periodic table.
•The s-process can build elements up to 209Bi. At that point there is no
neutron/proton ratio stable enough to allow the one by one conversion of neutrons to
protons.
•The rapid-neutron or r-process involves adding neutrons to a nucleus faster than
things can be “fixed.” Much heavier nuclei can be built up. Once the bombardment is
over, the neutron-rich nucleus will undergo repeated beta decays to produce nuclei
that are relatively more stable but which, in turn, are unstable to alpha decay and so
break down into lighter nuclei. These include 238U, 235U, and 232Th which have halflives comparable to the age of the Earth, and so have not yet decayed to negligible
amounts.
•The s-process also helps fill in gaps between some of the lighter elements such as between 12C
and 16O. This mechanism can only produce neutron-rich nuclei, so other processes must
account for known nuclei with lower than average neutron/proton ratios.
•The p-process resolves this by adding protons to nuclei.
•Light elements Li, Be, and B are not produced by any of the above processes. In fact, they are
destroyed at the temperatures required for hydrogen burning. They are probably formed as
fragments when heavy nuclei in interstellar dust are struck by cosmic rays. This is a very slow
process, but interstellar dust spends a lot of time in space!
•The most likely place for these reactions to take place is in the interior of a large star.
•The Sun is not large enough to ever get beyond hydrogen burning, and therefore will not
generate the distribution of elements found in the Sun or meteorites.
•One or many larger stars were needed to produce these elements that formed the nebula that
became the Solar System.
•The matter from these stars would have been disseminated by supernova explosions at the
end of their existence as stars.
•There is time between the formation of our galaxy, 15x109 years ago and the formation of the
Solar System 4.6x109 years ago for many generations of stars to form, explode and slowly
enrich the interstellar medium. This heavy material is the 2% of the Solar System.
•One additional observation suggests that the last of these supernovae must have occurred just
2-3 million years before the initiation of the formation of the Solar System. It must have
occurred close to the dust cloud that became the Solar System.
•There is evidence that certain elements like 26Al with very short half-lives were present in the
material that formed the Solar System. If these had been formed gradually by many stars, these
elements would have decayed away. They could only have been formed and disseminated by a
very recent supernova.
•It is possible that this supernova not only contributed material to the cloud, but also initiated
its collapse.
Differentiation of the Earth
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Differentiation is the process by
which random chunks of primordial
matter were transformed into a body
whose interior is divided into
concentric layers that differ from one
another both physically and
chemically.
This occurred early in Earth’s history,
when the planet got hot enough to
melt.
What was the starting point for
differentiation?
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Heterogeneous/Hot starting model
• Initial layering as Earth solidified from gas
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Homogeneous/Cold starting model
• Little or no initial layering because Earth
formed from the agglutination of cold, uniform
particles
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Neither model seems to work completely
When did differentiation happen?
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About 4.5 billion years ago
After beginning of Earth’s accretion
at 4.56 billion years ago
Before the formation of the Moon’s
oldest known rocks, 4.47 billion
years ago
Sources of heat to melt Earth
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Frequent and violent impacts
• There was likely one particularly large
impact
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Moon aggregated from the ejected debris
Earth’s spin axis was tilted
Decay of radioactive elements
• This heat generation was greater in the
past than today
Basic processes of differentiation
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In a liquid or soft solid sphere, denser
material sinks to the center and less
dense material floats to the top.
When rock is partially melted, the melt
and the remaining solid generally have
different chemistry and density. The melt
is usually less dense than the “residue.”
The melt is enriched in “incompatible”
elements. The residue is enriched in
“compatible” elements.
Earth’s Core
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Iron, nickel, and other heavy elements
were the densest material and formed the
core. Core radius is 2900 km.
They are about 1/3 of the planet’s mass
Inner core is solid. Inner core
radius=1200 km. Inner core is solid
because pressure is too great for iron to
melt at Earth’s current temperature.
Outer core is liquid. Some of the iron in
the outer core is iron sulfide.
The Iron, Oxygen, Sulfur,
Magnesium, and Silicon
story
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There were large amounts of these five elements in the early Earth
The fate of the iron was controlled by its affinity for bonding with oxygen
and sulfur.
Iron bonds preferentially with sulfur. All available sulfur is consumed. Iron
remains.
Oxygen bonds preferentially with magnesium and silicon. This uses up the
magnesium and silicon. Oxygen remains.
Iron then combines with oxygen. Oxygen is now used up. Iron remains as
elemental iron.
The iron, magnesium, and silicon oxides are light and form the Earth’s
crust and mantle.
The iron sufide is dense, but less dense than iron, so it forms the outer
part of the core of Earth.
The elemental iron is densest of all, so it forms the inner core of the
Earth.
Note: The amount of oxygen in the starting material plays a key role in
determining the size of the core of a planet. What does adding oxygen do
to the core radius? What does adding sulfur do to the core radius?
Earth’s Crust
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Lighter rocks floated to the surface of the magma
ocean.
The crust is formed of light materials with low
melting temperature and is up to 40 km thick.
These are generally compounds of silicon,
aluminum, iron, calcium, magnesium, sodium,
and potassium, mixed with oxygen.
Fragments of crustal rocks (zircons) of age 4.34.4 billion years were found recently in western
Australia. If this is confirmed, we can conclude
that Earth cooled enough for a solid crust to form
only 100 million years after the large impact.
Earth’s Mantle
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Lies between the crust and the core.
Depth range is 40 km to 2900 km.
The mantle consists of rocks of
intermediate density, mostly
compounds of oxygen with
magnesium, iron, and silicon
New continental crust may be
produced during partial melting of
mantle material.
Radiometric Dating: General Theory
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The radioactive decay of any radioactive
atom is an entirely random event,
independent of neighboring atoms,
physical conditions, and the chemical state
of the atom.
It depends only on the structure of the
nucleus.
λ, the decay constant, is the probability of
an atom decaying in unit time. It is
different for each isotope.
Suppose that at time t there are N atoms and that at time t+δt, δN of those
have decayed, then δN can be expressed as
δN = -λ N δt
In the limit as δN and δt go to 0, this becomes
dN/dt = -λ N
Thus, the rate of decay is proportional to the number of atoms present.
Rearrangement and integration gives:
loge N = -λ t + c
If at t=0 there are N0 atoms present, then c = loge N0
N = N0 e-λt
The half-life, T½, is the length of time required for half of the original
atoms to decay.
N0/2 = N0 e-λT½
or
T½ = (loge 2) / λ
Consider the case of a radioactive Parent atom decaying to an atom called
the Daughter. After time t, N = N0 – D parent atoms remain and
N0 – D = N0 e-λt
Where D is the number of daughter atoms (all of which have come from
decay of the parent) present at time t. Thus
D = N0 (1 – e-λt)
However, it is not possible to measure N0, but only N
Use the previous equation and
N = N0 e
–λt
yields
D = N (eλt – 1)
This equation expresses the number of daughter atoms in terms of the number
of parent atoms, both measured at time t, and it means that t can be
calculated by taking the natural log
t = loge (1 + D/N) / λ
In practice, measurements of D/N are made using a mass spectrometer.
http://www.chemguide.co.uk/analysis/masspec/howitworks.html
Major radioactive elements used in radiometric dating
Parent
Isotope
Daughter
Isotope
Half Life of
Parent (years)
Effective dating
range (years)
Materials that
can be dated
238U
206Pb
4.5 billion
10 million –
4.6 billion
Zircon
Apatite
235U
207Pb
0.7 billion
10 million –
4.6 billion
Zircon
Apatite
1.3 billion
50,000 –
4.6 billion
Muscovite
Biotite
Hornblende
10 million –
4.6 billion
Muscovite
Biotite
Potassium
Feldspar
100 70,000
Wood, charcoal, peat,
bone and tissue, shell
and other calcium
carbonate,
groundwater, ocean
water, and glacier ice
containing dissolved
CO2
40K
87Rb
14C
40A
87Sr
14N
47 billion
5730
Radiometric dating is not always that simple!
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There may have been an initial concentration of the
daughter in the sample
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Not all systems are closed. There may have been exchange
of parent and/or daughter with surrounding material.
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If dates from different isotope systems match within
analytical error, we say the ages are concordant. If they are
not, then we say they are discordant.
When discordant, we suspect problems like those above
with one or all of the systems.
The date t obtained is not always the date of formation of
the rock. It may be the date the rock crystallized, or the
date of a metamorphic event which heated the rock to the
degree that chemical changes took place.
Radioactive decay schemes are not all as simple as a parent
and exactly one daughter. 87Rb to 87Sr is a simple one step
decay. The two U to Pb series have a number of
intermediate daughter products.
Fission Track Dating
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As well as decaying to 206Pb as described before,
238U is also subject to spontaneous fission.
It disintegrates into two large pieces and several
neutrons. This is a very rare event, occurring just
once per 2 million α decays.
Each event is recorded as a trail of destruction
about 10 m long through the mineral structure.
These “fission tracks” can be observed by etching
the polished surface of certain minerals. The
tracks become visible under a microscope.
Spontaneous
Fission Tracks
Consider a small polished sample of a mineral. Assume that it has
[238U]now atoms of 238U distributed throughout its volume
The number of decays of 238U, Dr, during time t is:
t
Dr  [ U ]now (e 1)
238
The number of decays of 238U by spontaneous fission, Ds, which occur in
time t is:
s 238
Ds  [ U ]now (e t  1)
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Where s is the decay constant for spontaneous fission of 238U.
To determine an age, we must count the visible fission tracks, estimate
the proportion of the tracks visible (crossing) the surface, and measure
[238U]now.
Fortunately, we do not need to do this in an absolute manner, because
another isotope of Uranium, 235U, can be made to fission artificially. This is
done by putting our sample in a nuclear reactor and bombarding it with slow
neutrons for a specified time (hours). This provides us with a standard against
which to calibrate the number of tracks per unit area (track density). The
number of induced fissions is:
DI [235U ]nown
Where σ is the known neutron capture cross-section and n is the neutron
dose in the reactor.
We assume that if the two isotopes of U are equally distributed in the sample,
then the proportion of tracks that cross the surface will be the same. We can
combine equations to get:
s [ 238U ]now (et  1) Ds N s


235
[ U ]now n
DI
NI
Where Ns and NI are the numbers of spontaneous and induced fission tracks
counted in an area.
The equation can be rearranged and the known present ratio of the two
isotopes of Uranium, [238U]now/[235U]now=137.88,can be inserted to give:
 N s  n 
t  loge 1 


N

137
.
88
I
s


1
In practice, after the number of spontaneous fission tracks Ns has been
counted, the sample is placed in the reactor and then etched again. The
spontaneous tracks are enlarged and the induced tracks are exposed. The
number of induced tracks NI are counted and the age calculated.
Spontaneous
Induced
Fission Tracks
Fission Tracks
There is an additional (and very powerful) way to use
fission tracks.
Fission tracks in a mineral crystal are stable at room
temperature, but can “heal” if the temperature of the
crystal is high enough.
At very high temperature, the tracks heal completely very
quickly. This means that the “age” of a rock can be
completely “reset” by heating.
The rate at which tracks are healed varies with
temperature and mineral type. Therefore there is a
“closure” temperature that is a function of mineral type
and rate of cooling.
For example, fission track
ages determined from
sphene are always greater
than ages determined from
apatite. This is because
healing tracks in sphene
(~300C) requires much
greater temperatures than
healing tracks in apatite
(~90C).
Imagine that rocks are being uplifted and eroded during the creation of a mountain
range. The individual rocks are cooling as they are brought closer to the surface. A
progression of fission track ages in different minerals record the uplift/cooling history
of the rock.
There are newer, even more sophisticated methods, that use the rate at which
tracks heal, they actually shorten before disappearing, to determine more
complicated temperature history curves from each mineral.
http://www.geotrack.com.au/ttinterp.htm
Heat in the Earth
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Volcanoes, magmatic intrusions,
earthquakes, mountain building and
metamorphism are all controlled by the
generation and transfer of heat in the
Earth.
The Earth’s thermal budget controls the
activity of the lithosphere and
asthenosphere and the development of
the basic structure of the Earth.
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Heat arrives at the surface of the Earth from its interior and from
the Sun.
The heat arriving from the Sun is by far the greater of the two
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Heat from the Sun arriving at the Earth is 2x1017 W
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Averaged over the surface this is 4x102 W/m2
The heat from the interior is 4x1013 W and 8x10-2 W/m2
However, most of the heat from the Sun is radiated back into space.
It is important because it drives the surface water cycle, rainfall, and
hence erosion. The Sun and the biosphere keep the average surface
temperature in the range of stability of liquid water.
The heat from the interior of the Earth has governed the geological
evolution of the Earth, controlling plate tectonics, igneous activity,
metamorphism, the evolution of the core, and hence the Earth’s
magnetic field.
Heat Transfer Mechanisms
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Conduction
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Radiation
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Transfer of heat through a material by atomic or molecular
interaction within the material
Direct transfer of heat as electromagnetic radiation
Convection
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Transfer of heat by the movement of the molecules themselves
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Advection is a special case of convection
Conductive Heat Flow
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Heat flows from hot things to cold things.
The rate at which heat flows is
proportional to the temperature gradient
in a material
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Large temperature gradient – higher heat flow
Small temperature gradient – lower heat flow
Imagine an infinitely wide and long solid
plate with thickness δz .
Temperature above is T + δT
Temperature below is T
Heat flowing down is proportional to:
(T  T )  T
z
The rate of flow of heat per unit area up
through the plate, Q, is:
 T  T  T 
Q  k 

z


T
Q( z )  k
z
In the limit as δz goes to
zero:
T
Q( z )  k
z
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Heat flow (or flux) Q is rate of flow of heat per unit area.
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The units are watts per meter squared, W m-2
Watt is a unit of power (amount of work done per unit time)
A watt is a joule per second
Old heat flow units, 1 hfu = 10-6 cal cm-2 s-1
1 hfu = 4.2 x 10-2 W m-2
Typical continental surface heat flow is 40-80 mW m-2
Thermal conductivity k
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The units are watts per meter per degree centigrade, W m-1 °C-1
Old thermal conductivity units, cal cm-1 s-1 °C-1
0.006 cal cm-1 s-1°C-1 = 2.52 W m-1 °C-1
Typical conductivity values in W m-1 °C-1 :
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Silver
Magnesium
Glass
Rock
Wood
420
160
1.2
1.7-3.3
0.1
Let’s derive a differential equation describing the
conductive flow of heat
Consider a small volume element
of height δz and area a.
Any change in the temperature
of this volume in time δt
depends on:
1. Net flow of heat across the
element’s surface (can be in
or out or both)
2. Heat generated in the element
3. Thermal capacity (specific
heat) of the material
The heat per unit time entering
the element across its face at
z is aQ(z) .
The heat per unit time leaving
the element across its face at
z+δz is aQ(z+δz) .
Expand
Q(z+δz) as Taylor series:
Q z   2Q z   3Q
Q( z  z )  Q( z )  z


 ...
z
2! z 2
3! z 3
2
3
The terms in (δz)2 and above are
small and can be neglected
The net change in heat in the
element is (heat entering across z)
minus (heat leaving across Z+δz):
 aQ( z )  aQ( z  z )
Q
  az
z
Suppose heat is generated in the volume element at a rate A per unit
volume per unit time. The total amount of heat generated per unit time is
then
A a δz
Radioactivity is the prime source of heat in rocks, but other possibilities
include shear heating, latent heat, and endothermic/exothermic chemical
reactions.
Combining this heating with the heating due to changes in heat flow in and
out of the element gives us the total gain in heat per unit time (to first
order in δz as:
Q
Aa z  az
z
This tells us how the amount of heat in the element changes, but not how
much the temperature of the element changes.
The specific heat cp of the material in the element determines the
temperature increase due to a gain in heat.
Specific heat is defined as the amount of heat required to raise 1 kg of
material by 1C.
Specific heat is measured in units of J kg-1 C-1 .
If material has density ρ and specific heat cp, and undergoes a
temperature increase of δT in time δt, the rate at which heat is gained is:
T
c p az
t
We can equate this to the rate at which heat is gained by the element:
T
Q
c p az
 Aa z  az
t
z
T
Q
c p az
 Aa z  az
t
z
Simplifies to:
T
Q
cp
 A
t
z
In the limit as δt goes to zero:
T
Q
cp
 A
t
z
T
 2T
cp 
 A k 2
t
z
Several slides back we defined Q as:
T
Q( z )  k
z
T
k  2T
A


2
t c p  z
cp 
This is the one-dimensional heat
conduction equation.
The term k/ρcp is known as the thermal diffusivity κ. The thermal diffusivity
expresses the ability of a material to diffuse heat by conduction.
The heat conduction equation can be generalized to 3 dimensions:
T
k   2T  2T  2T  A

 2  2  2 
t c p   x
y
z  c p 
T
k
A
2

T
t c p 
cp 
The symbol in the center is the gradient operator squared, aka the Laplacian
operator. It is the dot product of the gradient with itself.
  
   , , 
 x y z 
2
2
2



2      2  2  2
x y
z
 T T T 

T  
,
,
 x y z 
 2T  2T  2T
T 2  2  2
x
y
z
2
T
k
A
2

T
t c p 
cp 
This simplifies in many special situations.
For a steady-state situation, there is no change in temperature with time.
Therefore:
A
 T 
k
2
In the absence of heat generation, A=0:
T
k

 2T
t c p 
Scientists in many fields recognize this as the classic “diffusion”
equation.
Talk at board about the
qualitative behavior of the Heat
Conduction equation
Equilibrium Geotherms



The temperature vs. depth profile in the
Earth is called the geotherm.
An equilibrium geotherm is a steady state
geotherm.
2
Therefore:
T
T
A
 0, and 2  
t
z
k
Boundary conditions


Since this is a second order differential
equation, we should expect to need 2
boundary conditions to obtain a solution.
A possible pair of bc’s is:



T=0 at z=0
Q=Q0 at z=0
Note: Q is being treated as positive upward and z is positive downward in this derivation.
Solution


Integrate the
differential equation
once:
Use the second bc to
constrain c1


T
Az

 c1
z
k
Q0
c1 
k
Note: Q is being treated as positive upward
and z is positive downward in this derivation.
Substitute for c1:
T
Az Q0


z
k
k
Solution

Integrate the differential
equation again:
Az 2 Q0 z
T 

 c2
2k
k
Use the first bc to
constrain c2
c2  0

Substitute for c2:
Az 2 Q0 z
T 

2k
k

Link to spreadsheet

Oceanic Heat Flow
Heat flow is higher over young
oceanic crust
Heat flow is more scattered
over young oceanic crust
Oceanic crust is formed by
intrusion of basaltic magma
from below
The fresh basalt is very
permeable and the heat drives
water convection
Ocean crust is gradually
covered by impermeable
sediment and water convection
ceases.
Ocean crust ages as it moves
away from the spreading
center. It cools and it contracts.
These data have been empirically
modeled in two ways:
d=2.5 + 0.35t2
(0-70 my)
and
d=6.4 – 3.2e-t/62.8 (35-200 my)
Half Space Model
Specified temperature at top
boundary.
No bottom boundary condition.
Cooling and subsidence are predicted
to follow square root of time.
Plate Model
Specified temperature at top and
bottom boundaries.
Cooling and subsidence are predicted
to follow an exponential function
of time.
Roughly matches Half Space Model
for first 70 my.
The model of plate
cooling with age
generally works for
continental lithosphere,
but is not very useful.
Variations in heat flow in
continents is controlled
largely by changes in the
distribution of heat
generating elements and
recent tectonic activity.
Range of Continental and Oceanic
Geotherms in the crust and upper
mantle
Convection
Conductive Geotherm
~10-20 C per km
Adiabatic Geotherm
~0.5-1.0 C per km
Convective Geotherm
Adiabatic “middle”
Thermal boundary layer
at top and bottom
Solid and liquid in the Earth
Illustration of mantle
melting during
decompression
Rayleigh-Benard Convection


Newtonian viscous fluid – stress is proportional to strain rate
A tank of fluid is heated from below and cooled from above







Initially heat is transported by conduction and there is no lateral
variation
Fluid on the bottom warms and becomes less dense
When density difference becomes large enough, lateral variations
appear and convection begins
The cells are 2-D cylinders that rotate about their horizontal axes
With more heating, these cells become unstable by themselves and a
second, perpendicular set forms
With more heating this planform changes to a vertical hexagonal
pattern with hot material rising in the center and cool material
descending around the edges
Finally, with extreme heating, the pattern becomes irregular with hot
material rising randomly and vigorously.
Rayleigh-Benard Convection







The stages of convection have
been modeled mathematically
and are characterized by a
“non-dimensional” number
called the Rayleigh number
a is the volume coefficient of
thermal expansion
g gravity
d the thickness of the layer
Q heat flow through lower
boundary
A, κ, k you know
n is kinematic viscosity
a gd  Q  Ad 
Ra 
kn
4
The critical value of Ra for gentle
convection is about 103.
The aspect ratio for R-B convection cells is
about 2-3 to 1
Ra above 105 will produce vigorous
convection
Ra above 106 will produce irregular
convection



Ra for both the upper and lower mantle seems to be
consistent with vigorous convection
While R-B convection models are very useful, they do not
approximate the Earth very well. The biggest problem is
that they model “uniform viscosity” materials. The
mantle is not uniform viscosity!
Reynold’s number – indicates whether flow is laminar or
turbulent





All mantle convection in the Earth is predicted to be laminar
Mantle convection movies from Caltech
More mantle convection movies
More
More
Studies like you did in lab,
seemed to show that
subduction stopped at about
670 km depth. This was
interpreted to mean there
was mantle convection
operating in the upper
mantle that was separate
from convection in the lower
mantle.
Two-layer vs. Whole
Mantle Convection
Modern tomographic
images give a very
different picture!
Plate Driving Forces
Illustration of slab pull
and ridge push
Plate boundaries are marked in several ways:
Names of the plates:
(Lowrie, 1997) Arrows indicate relative velocities (mm/yr) from
NUVEL-1 model of DeMets et al., 1990
Types of plate boundaries:
Assumptions of Plate Tectonics




The generation of new plate material occurs by seafloor
spreading; that is, new oceanic lithosphere is generated
along the active mid-ocean ridges.
The new oceanic lithosphere, once created, forms part of
a rigid plate; this plate may or may not include
continental material.
The Earth’s surface area remains constant; therefore,
seafloor spreading must be balanced by consumption of
plate elsewhere.
The lithospheric plates are capable of transmitting
stresses over great horizontal distances without
buckling; in other words, the relative motion between
plates is taken up only along plate boundaries.
Plate motions can be determined in several ways.
The traditional way is using marine magnetic anomalies:
Magnetic anomalies allow the identification of
isochrons in the worlds oceans.
Very Long Baseline Interferometry (VLBI)
VLBI measures the time difference between the arrival at the
Earth of a radio signals emitted by quasars. The time
difference between arrivals at two satellites is proportional to
the distance between the two satellites and the direction of
the source. These satellites may be separated by some 10,000
km. Using large numbers of time difference measurements
from many quasars observed with a global network of
antennas, VLBI determines the inertial reference frame
defined by the quasars and simultaneously the precise
positions of the antennas. Because the time difference
measurements are precise to a few picoseconds, VLBI
determines the relative positions of the antennas to a few
millimeters and the quasar positions to fractions of a
milliarcsecond. Since the antennas are fixed to the Earth,
their locations track the instantaneous orientation of the
Earth in the inertial reference frame.
Satellite and Lunar Laser ranging (SLR & LLR)
SLR targets are satellites equipped with corner cubes
or retro-reflectors. Currently, the global SLR network
tracks over forty such satellites. The observable is the
round-trip pulse time-of-flight to the satellite.
SLR systems are equipped with short-pulse laser
transmitters that can range to orbiting satellites.
Lunar Laser Ranging (LLR) systems can range to
retro-reflectors located on the moon.
Global positioning system (GPS)
There are 24 GPS satellites currently in circular orbits some 20,200
kilometers above the Earth. At any one time in most places six can be
"seen" by GPS receivers that get and process signals.
GPS receivers calculate current position (latitude, longitude, altitude) with
varying degrees of precision. There are many thousand permanent GPS
receivers located world wide. These provide data for modeling plate
motions on yearly time scales.