Download Heat (inside the Earth)

Corso di Laurea in Fisica - UNITS
ISTITUZIONI DI FISICA
PER IL SISTEMA TERRA
Heat
(inside the Earth)
FABIO ROMANELLI
Department of Mathematics & Geosciences
University of Trieste
[email protected]
http://moodle2.units.it/course/view.php?id=887
Heat and Earth
•
Earth can be considered as a heat engine, where the flow of
energy out of the system into the environment, space, controls
the internal dynamics of the system, this is manifested in the
long term existence of a magnetic field and in surface
processes which are directly or indirectly observable in the
geological record.
•
From the eighteenth century on, the thermal state of the
Earth’s interior has been the subject of investigation. In the
nineteenth century, the discrepancy between the age of the
Earth estimated from simple physical, conductive cooling
models and the time scales apparently involved in geological
processes was a source of great controversy that lasted until
the discovery of natural radioactive decay and the realisation
that this process could be a significant source for internal
heating of the Earth.
Fabio Romanelli
Heat
•
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
•
•
Heat from the Sun arriving at the Earth is 2x1017 W
Averaged over the surface this is 4x102 W/m2
•
•
The heat from the interior is 4x1013 W and 8x10-2 W/m2
•
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.
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.
Fabio Romanelli
Heat
Earth and Energy budget
221
Table 4.3 Estimates of notable contributions to the
Earth’s annual energy budget
Energy source
Reflection and re-radiation
of solar energy
Geothermal flux from
Earth’s interior
Rotational deceleration by
tidal friction
Elastic energy in earthquakes
Annual
energy [J]
Normalized
[geothermal
flux # 1]
5.4"1024
! 4000
1.4"1021
1
! 1020
! 0.1
! 1019
! 0.01
portional to the temperature of the gas. If we add up the
kinetic energies of all molecules in the container we
obtain the amount of heat it contains. If heat is added to
the container from an external source, the gas molecules
speed up, their mean kinetic energy increases and the temperature of the gas rises.
The change of temperature of a gas is accompanied by
changes of pressure and volume. If a solid or liquid is
heated, the pressure remains constant but the volume
increases. Thermal expansion of a suitable solid or liquid
forms the principle of the thermometer for measuring
temperature. Although Galileo reputedly invented an
early and inaccurate “thermoscope,” the first accurate
thermometers – and corresponding temperature scales –
were developed in the early eighteenth century by Gabriel
Fahrenheit (1686–1736), Ferchaut de Réaumur (1683–
1757) and Anders Celsius (1701–1744). Their instruments
utilized the thermal expansion of liquids and were calibrated at fixed points such as the melting point of ice and
the boiling point of water. The Celsius scale is the most
commonly used for general purposes, and it is closely
related to the scientific temperature scale.
Fabio Romanelli
Heat
Accretion and Initial Temperatures
• If accretion occurs by lots of small impacts, a lot of the
energy may be lost to space
• If accretion occurs by a few big impacts, all the energy
will be deposited in the planet’s interior
• Additional energy is released as differentiation occurs –
dense iron sinks to centre of planet and releases potential
energy as it does so
• What about radioactive isotopes? Short-lived radioisotopes (26Al, 60Fe) can give out a lot of heat if bodies
form while they are still active (~1 Myr after solar system
formation)
• A big primordial atmosphere can also keep a planet hot
• So the rate and style of accretion (big vs. small impacts) is
important, as well as how big the planet ends up
Heat Generation in Planets
• Most bodies start out hot (because of gravitational
energy released during accretion)
• But there are also internal sources of heat
• For some bodies (e.g. Io) the principle heat source is
tidal deformation (friction)
• For silicate planets, the principle heat source is
radioactive decay (K,U,Th at present day)
• Radioactive heat production declines with time
• Present-day terrestrial value ~5x10-12 W kg-1 (or
~1.5x10-8 W m-3)
• Radioactive decay accounts for only about half of the
Earth’s present-day heat loss
Cooling a planet
• Large silicate planets (Earth,
Venus) probably started out
molten – magma ocean
• Magma ocean may have been
helped by thick early atmosphere
(high surface temperatures)
• Once atmosphere dissipated, surface will have cooled rapidly
and formed a solid crust over molten interior
• If solid crust floats (e.g. plagioclase on the Moon) then it
will insulate the interior, which will cool slowly (~ Myrs)
• If the crust sinks, then cooling is rapid (~ kyrs)
Cooling a planet (cont’d)
• Planets which are small or cold will lose heat entirely
by conduction
• For planets which are large or warm, the interior
(mantle) will be convecting beneath a (conductive)
stagnant lid (also known as the lithosphere)
• Whether convection occurs depends if the Rayleigh
number Ra exceeds a critical value, ~1000
Temp.
ρgα(T1 − T 0 )d3
κη
Here ρ is density, g is gravity, α is thermal
expansivity, ΔT is the temperature contrast, d is the
layer thickness, κ is the thermal diffusivity and η is
the viscosity. Note that η is strongly temperaturedependent.
Depth.
Ra =
Stagnant
(conductive) lid
Convecting
interior
Heat Transfer Mechanisms
•
Radiation
•
•
Conduction
•
•
Direct transfer of heat as electromagnetic radiation
Transfer of heat through a material by atomic or molecular
interaction within the material
Convection
Fabio Romanelli
•
Transfer of heat by the movement of the molecules
themselves
•
Advection is a special case of convection
Heat
Conductive Heat Flow
• Heat
flows from hot things to cold
things.
• The
rate at which heat flows is
proportional to the temperature gradient
in a material
• Large temperature gradient – higher heat flow
• Small temperature gradient – lower heat flow
Fabio Romanelli
Heat
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 + δT) − T
δz
The rate of flow of heat per unit area up
through the plate, Q, is:
⎛ T + δT − T ⎞
⎟
Q = −k ⎜⎜
⎟
δz
⎝
⎠
δT
Q(z) = −k
δz
Fabio Romanelli
In the limit as δz goes to
zero:
Q(z) = −k
∂T
∂z
Heat
•
Heat flow (or flux) Q is rate of flow of heat per unit area.
• 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
• Typical continental surface heat flow is 40-80 mW m-2
•
Thermal conductivity k
• The units are watts per meter per degree centigrade, W m-1 °C-1
• Typical conductivity values in W m-1 °C-1 :
•
•
•
•
•
Fabio Romanelli
Silver
Magnesium
Glass
Rock
Wood
420
160
1.2
1.7-3.3
0.1
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
Fabio Romanelli
Heat
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:
2
Q(z + δz) = Q(z) + δz
∂Q
∂z
δz )
(
+
2!
∂2 Q
∂z2
δz )
(
+
3!
3
∂3 Q
∂z3
+ ...
The terms in (δz)2 and above are
small and can be neglected
Fabio Romanelli
The net change in heat in the
element is (heat entering across z)
minus (heat leaving across z+δz):
= aQ(z) − aQ(z + δz)
= −aδz
∂Q
∂z
Heat
Suppose heat is generated in the volume element at a rate H per unit
volume per unit time. The total amount of heat generated per unit time is
then
H 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:
Haδz − aδz
∂Q
∂z
This tells us how the amount of heat in the element changes, but not
how much the temperature of the element changes.
Fabio Romanelli
Heat
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:
c paδzρ
δT
δt
We can equate this to the rate at which heat is gained by the element:
c paδzρ
Fabio Romanelli
δT
δt
= Haδz − aδz
∂Q
∂z
Heat
c paδzρ
δT
δt
= Haδz − aδz
∂Q
∂z
Simplifies to:
c pρ
In the limit as δt goes to zero:
c pρ
c pρ
δT
δt
∂T
∂t
= H−
= H+k
∂Q
δT
δt
= H−
∂Q
∂z
Several slides back we defined Q as:
Q(z) = −k
∂T
∂z
∂2 T
∂T
∂z2
∂t
=
∂z
k ∂2 T
c pρ ∂z
2
+
H
c pρ
1D heat conduction equation
Fabio Romanelli
Heat
Heat 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
∂t
=
k
c pρ
∇ T+
2
H
c pρ
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 ⎠
⎛ ∂T ∂T ∂T ⎞
∇T = ⎜⎜ ,
, ⎟⎟
⎝ ∂x ∂y ∂z ⎠
Fabio Romanelli
2
∇ = ∇•∇ =
∇T=
2
∂2 T
∂x
2
+
∂2
∂x
2
∂2 T
∂y
2
+
+
∂2
∂y
2
+
∂2
∂z2
∂2 T
∂z2
Heat
∂T
∂t
=
k
c pρ
∇ T+
2
H
c pρ
This simplifies in many special situations.
For a steady-state situation, there is no change in temperature with
time. Therefore:
∇ T =−
2
H
k
In the absence of heat generation, H=0:
∂T
∂t
=
k
c pρ
∇2T
Scientists in many fields recognize this as the classic “diffusion” equation.
Fabio Romanelli
Heat
Diffusion length scale
• How long does it take a change in temperature to
propagate a given distance?
• Consider an isothermal body suddenly cooled at the top
• The temperature change will propagate downwards a
distance d in time t
• After time t, Q~k(T1-T0)/d
Temp.
T0
T1
Depth
Initial
profile
d
Profile at
time t
• The cooling of the near surface
layer involves an energy change
per unit area ΔE~d(T1-T0)Cpρ
• We also have Qt~ΔE
• This gives us
d ∼ κt
2
Equilibrium Geotherms
•
The temperature vs. depth profile in the Earth is
called the geotherm.
•
An equilibrium geotherm is a steady state
geotherm. Therefore:
∂T
∂t
Fabio Romanelli
=0
∂T
2
∂z
2
=-
H
k
Heat
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 BCs is:
• T=0 at z=0
• Q=Q0 at z=0
•
Fabio Romanelli
Note: Q is being treated as positive upward and z is positive downward in this
derivation.
Heat
Solution
•
•
•
Integrate the
differential equation
once:
∂T
∂z
Use the second BC to
constrain c1
Substitute for c1:
c1 =
∂T
∂z
Fabio Romanelli
=−
=−
Hz
k
+ c1
Q0
k
Hz
k
+
Q0
k
Heat
Solution
•
•
•
Integrate the differential
equation again:
Use the first BC to
constrain c2
Substitute for c2:
T =−
Hz2
2k
+
Q 0z
k
c2 = 0
T =−
Hz2
2k
+
+ c2
Q 0z
k
According to this model, the temperatures at
only a few hundred km deep are so high that
all the rocks would have melted. This CAN’T
be right! We know that shear waves go
through the solid upper mantle. Our equation
for a geotherm without any heat
"sources" (similar to Kelvin’s calculation of the
age of the earth) would melt the mantle at
approximately 100km! Adding heat sources
improves the situation, but not much
The explanation for this erroneous conclusion
is that when rock gets hot it convects, it
behaves somewhat like a liquid.
Fabio Romanelli
Heat
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.
Fabio Romanelli
Heat
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.
Fabio Romanelli
Heat
Heat
Age (Ma)
0
50
RIGID
100
150
200
plate motion
Mechanical boundary layer
Thermal boundary layer
Onset of
instability
VISCOUS
Thermal structure of plate
and small-scale convection
approach equilibrium
Figure 7.10. A schematic diagram of the oceanic lithosphere, showing the proposed
division of the lithospheric plate. The base of the mechanical boundary layer is the
isotherm chosen to represent the transition between rigid and viscous behaviour.
The base of the thermal boundary layer is another isotherm, chosen to represent
correctly the temperature gradient immediately beneath the base of the rigid plate.
In the upper mantle beneath these boundary layers, the temperature gradient is
approximately adiabatic. At about 60–70 Ma the thermal boundary layer becomes
unstable, and small-scale convection starts to occur. With a mantle heat flow of
about 38 × 10−3 W m−2 the equilibrium thickness of the mechanical boundary layer
is approximately 90 km. (From Parsons and McKenzie (1978).)
Fabio Romanelli
Heat
if the medium is moving
• Two ways of looking at the problem:
– Following an individual particle – Lagrangian
– In the laboratory frame - Eulerian
Particle frame
u
Laboratory frame
Temperature contours
In the Eulerian frame, we write
•In the particle frame, there is no change in
temperature with time
•In the laboratory frame, the temperature at
a fixed point is changing with time
•But there would be no change if the
temperature gradient was perpendicular to
the velocity
∂T
∂t
= −u ⋅ ∇T
The right-hand side is known as the advected term
Material Derivative
• So if the medium is moving, the heat flow equation is
DT
Dt
=κ
∂2 T
∂z2
+
H
C pρ
• Here we are using the material derivative D/Dt, where
⎛∂
⎞
= ⎜⎜ + u ⋅ ∇ ⎟⎟
Dt ⎝ ∂t
⎠
D
• It is really just a shorthand for including both the
local rate of change, and the advective term
• It applies to the Eulerian (laboratory) reference frame
• Not just used in heat transfer (T). Also fluid flow (u),
magnetic induction etc.
Conductive heat transfer
• Diffusion equation (1D, Cartesian)
∂T
∂t
+u
∂T
∂z
=κ
Advected
component
∂2 T
∂z
2
+
Conductive
component
H
C pρ
Heat
production
• Thermal diffusivity 𝛋=k/ρCp (m2s-1)
• Diffusion timescale:
t∼
d2
κ
Convection
• Convection arises because fluids expand and
decrease in density when heated
• The situation on the right is gravitationally
Cold - dense
Fluid
unstable – hot fluid will tend to rise
• But viscous forces oppose fluid motion, so
there is a competition between viscous and
(thermal) buoyancy forces
Hot - less dense
• So convection will only initiate if the buoyancy forces
are big enough
Peclet Number
• It would be nice to know whether we have to worry
about the advection of heat in a particular problem
• One way of doing this is to compare the relative
timescales of heat transport by conduction and advection:
t cond ~
L2
κ
t adv ~
L
u
Pe ~
uL
κ
• The ratio of these two timescales is called a dimensionless
number called the Peclet number Pe and tells us whether
advection is important
• High Pe means advection dominates diffusion, and v.v.*
• E.g. lava flow, u~1 m/s, L~10 m, Pe~107 ∴ advection is
important
* Often we can’t ignore diffusion even for large Pe due to stagnant boundary layers
Examples of Geological Flow
Mantle convection
Glaciers
Salt domes
~50km
Lava flows
Flow Mechanisms
• For flow to occur, grains must deform
• There are several ways by which they may do this,
depending on the driving stress
• All the mechanisms are very temperature-sensitive
Increasing stress / strain rate
Diffusion creep
(n=1, grain-size dependent)
Grain-boundary sliding
Dislocation creep
(n>1, grain-size dependent) (n>1, indep. of grain size)
Here n is an exponent which determines how sensitive to strain rate is to the applied stress.
Fluids with n=1 are called Newtonian.
Atomic Description
• Atoms have a (Boltzmann)
distribution of kinetic energies
Peak = kT/2
No. of particles
Mean= 3kT/2
• The distribution is skewed –
there is a long tail of highEnergy E
energy atoms
• The fraction of atoms with a kinetic energy greater
than a particular value E0 is:
⎛ E ⎞
f(E 0 ) = 2 ⎜ 0 ⎟ exp(−E 0 / kT)
⎜ πkT ⎟
⎝
⎠
• If E0 is the binding energy, then f is the fraction of
atoms able to move about in the lattice and promote
flow of the material
• So flow is very temperature-sensitive
Elasticity and Viscosity
• Elastic case – strain depends on stress (Young’s mod. E)
• Viscous case – strain rate depends on stress (viscosity η)
ε! = σ / η
• We define the (Newtonian) viscosity as the stress
required to cause a particular strain rate – units Pa s
• Typical values: water 10-3 Pa s, basaltic lava 104 Pa s, ice
1014 Pa s, mantle rock 1021 Pa s
• Viscosity is a macroscopic property of fluids which is
determined by their microscopic behaviour
Navier-Stokes
• We can write the general (3D) formula in a more compact
form given below – the Navier-Stokes equation
• The formula is really a mnemonic – it contains all the
physics you’re likely to need in a single equation
• The vector form given here is general (not just Cartesian)
⎡ ∂v
⎤
2
⎢
⎥
ρ
+ v ⋅ ∇ v = η∇ v − ∇P + Δρgẑ
⎢⎣ ∂t
⎥⎦
( )
Inertial term. Source
of turbulence
Zero for steadystate flows
Pressure
gradient
Diffusion-like viscosity
term. Complicated,
especially in nonCartesian geom.
Buoyancy force (e.g.
thermal or electromagnetic)
Reynolds number
⎡ ∂v
⎤
∂2 u
ρ ⎢ + v ⋅ ∇v⎥ = η 2 − ∇P + F
⎢⎣ ∂t
⎥⎦
∂z
• Is the inertial or viscous term more important?
• We can use a scaling argument to get the ratio Re:
Re =
ρvL
η
Here L is a characteristic
lengthscale of the problem
• Re is the Reynolds number and tells us whether a flow is
turbulent (inertial forces dominate) or not
• Fortunately, many geological situations allow us to neglect
inertial forces (Re<<1)
Convection equations
• There are two: one controlling the evolution of
temperature, the other the evolution of velocity
• They are coupled because temperature affects flow
(via buoyancy force) and flow affects temperature (via
the advective term)
NavierStokes
⎛ ∂2 v ∂2 v ⎞
ρ = η ⎜⎜ 2 + 2 ⎟⎟ − ∇P + ρgαT Buoyancy force
∂t
⎝ ∂x ∂z ⎠
Note that here the N-S equation is
∂v
neglecting the inertial term
Thermal
Evolution
⎛ ∂2 T ∂2 T ⎞
= K ⎜⎜ 2 + 2 ⎟⎟ − v ⋅ ∇T Advective term
∂t
∂z ⎠
⎝ ∂x
∂T
• It is this coupling that makes solving convection
problems hard
•
The Earth’s mantle can be described in a simple convective
model set up as a highly viscous layer cooled from above and
heated both internally and from below.
•
In such a model the earth’s surface and the core-mantle
boundary can be represented by impermeable boundaries.
From the theory of thermal convection in viscous fluids it is
known that under certain conditions heat transport in such
model systems takes place predominantly by thermal
convection in the interior of the fluid layer.
•
The convective heat transport increases when the viscosity of
the fluid is decreased and also when the temperature
contrast between the cooler top surface and the hotter
bottom surface is increased.
•
In this model heat is transported conductively through the
thermal boundary layers that develop at the top and bottom
boundary. This process, known as Rayleigh-Benard convection
was investigated theoretically by Rayleigh (1916), who showed
in a linear stability analysis that two regimes can exist
depending on the value of the non-dimensional Rayleigh
number
•
Rayleigh’s linear stability analysis shows that the fluid is at
rest for subcritical values of the Rayleigh number, Ra < Rac,
and thermal convection sets in for supercritical Rayleigh
number values, Ra > Rac. The critical value Rac is a so called
bifurcation point of the heat transport model
Convection
• Convective behaviour is governed by the Rayleigh
number Ra
• Higher Ra means more vigorous convection, higher heat
flux, thinner stagnant lid
• As the mantle cools, η increases, Ra decreases, rate of
cooling decreases -> self-regulating system
Stagnant lid (cold, rigid)
Plume (upwelling, hot)
Sinking blob (cold)
The number of upwellings and
downwellings depends on the
balance between internal heating
and bottom heating of the
mantle
Image courtesy Walter Kiefer, Ra=3.7x106, Mars
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
•
•
The cells are 2-D cylinders that rotate about their horizontal axes
•
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.
When density difference becomes large enough, lateral variations appear
and convection begins
With more heating, these cells become unstable by themselves and a
second, perpendicular set forms
Fabio Romanelli
Heat
Initiation of Convection
Top temperature T0
d
• Recall buoyancy forces favour
motion, viscous forces oppose it
• Another way of looking at the
problem is there are two competing
timescales
d
Incipient upwelling
Hot layer
Bottom temp. T1
• Whether or not convection occurs is governed by the
dimensionless (Rayleigh) number Ra:
Ra =
ρgα(T1 − T 0 )d3
κη
• Convection only occurs if Ra is greater than the critical
Rayleigh number, ~ 1000 (depends a bit on geometry)
Constant viscosity convection
• Convection results in hot
and cold boundary layers
and an isothermal interior
• In constant-viscosity
convection, top and bottom
b.l. have same thickness
cold T0
T0 (T0+T1)/2
δ
Isothermal
interior
d
δ
hot T1
• Heat is conducted across boundary layers
T1
Fconv = k
(T1 −T0 )
Fcond = k
(T1 −T0 )
2δ
d
• The Nusselt number defines the convective efficiency:
F conv
d
Nu =
=
Fcond 2δ
•
The critical phenomenon of the
onset of thermal convection is
illustrated. Nusselt number values
of steady state Rayleigh-Benard
convection, derived from
numerical modelling calculations,
are plotted against the Rayleigh
number.
•
For high Rayleigh numbers Ra ∼
106 this figure suggests that
convective heat transport is
more than an order of magnitude
more effective than purely
conductive heat transport.
Boundary layer thickness
d
• We can balance the timescale for
conductive thickening of the cold
boundary layer against the timescale
for the cold blob to descend to obtain
an expression for the b.l. thickness δ:
δ ~ d ⋅ Ra
δ
δ
−1/3
• So the boundary layer gets thinner as convection
becomes more vigorous
• Also note that δ is independent of d.
• We can therefore calculate the convective heat flux:
1/3
Fconv
(T1 − T 0 )
(T1 − T 0 ) 1/3 k ⎛ ρgα ⎞
⎟
=k
~k
Ra = ⎜⎜
⎟
2δ
2d
2 ⎝ κη ⎠
(T − T )
1
0
4/3
Example - Earth
1/3
4/3
k ⎛ ρgα ⎞
⎟ T −T
Fconv = ⎜⎜
⎟
1
0
2 ⎝ κη ⎠
• Plug in some parameters for the terrestrial mantle:
(
)
ρ=3000 kg m-3, g=10 ms-2, α=3x10-5 K-1, κ=10-6 m2s-1,
η=3x1021 Pa s, k=3 W m-1K-1, (T1-T0) =1500 K
• We get a convective heat flux of 170 mWm-2
• This is about a factor of 2 larger than the actual
terrestrial heat flux (~80 mWm-2) – not bad!
• NB for other planets (lacking plate tectonics), δ tends to
be bigger than these simple calculations would predict,
and the convective heat flux smaller
• Given the heat flux, we can calculate thermal evolution
Thermodynamics & Adiabat
• A packet of convecting material is often moving fast
enough that it exchanges no energy with its surroundings
• What factors control whether this is true?
• As the convecting material rises, it will expand (due to
reduced pressure) and thus do work (W = P dV)
• This work must come from the internal energy of the
material, so it cools
• The resulting change in temperature as a function of
pressure (dT/dP) is called an adiabat
• Adiabats explain e.g. why mountains are cooler than
valleys
Adiabatic Gradient (1)
• If no energy is added or taken away, the entropy of
the system stays constant. Entropy S is defined by:
⎛ ∂S ⎞
dQ ⎛ ∂S ⎞
dS =
= ⎜⎜ ⎟⎟ dT + ⎜⎜ ⎟⎟ dP
T
⎝ ∂T ⎠P
⎝ ∂P ⎠T
• What we want is
⎛ ∂T ⎞
⎜ ⎟
⎜ ⎟
⎝ ∂P ⎠S
• We need some definitions:
⎛ ∂S ⎞ C
⎜ ⎟ = P
⎜ ⎟
⎝ ∂T ⎠P T
Specific heat capacity
(at constant P)
⎛ ∂S ⎞ ⎛ ∂V ⎞
⎜ ⎟ =⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎝ ∂P ⎠T ⎝ ∂T ⎠P
1 ⎛ ∂V ⎞
α = ⎜⎜ ⎟⎟
V ⎝ ∂T ⎠P
Maxwell’s identity
Thermal expansivity
Adiabatic Gradient (2)
approximations)
• We can assemble these pieces to get the
diabatic temperature gradient we would expect convection.
adiabatic
temperature gradient:
transfers heat more
efficiently than conduction
T=0
T=Tm
T=0
convection
T=Tm
steep surface gradient
convection
z
hot blob
boundary layers
dT
dP
=
uction (if no heat sources) get linear gradient as function of depth.
αT
ρC p
f up to the surface - changes geotherm
• An often more useful expression can be
ins how we mayobtained
avoid having the wholeby
mantle converting
melting, as the conduction geotherm
pressure to depth
tion geotherm is an average over the convection cell at each depth. Note: This is why
wrong! At the surface a convection geotherm looks like a young conduction geotherm resetting the geotherm by bringing up hot material.
T
conduction
z
convection
dT
dz
z
=
αgT
Cp
adiabat
T
•
For a complete description of the Earth’s interior
we need to know it’s chemical composition, temperature and pressure.
•
Once the internal pressure distribution is known,
sharp transitions or discontinuities in the material
properties can be identified with mineral phase
transitions and as such they can be related to the
mineral (P,T) phase diagram of candidate mantle
silicate materials in order to estimate the
temperature in the Earth’s interior.
•
Such phase diagrams are determined from
experimental (HPT) and theoretical work in mineral
physics
Anchor points
•
A sharp transition at a pressure Pt in the PREM model can
then be located at the corresponding pressure in the phase
diagram by the intersection of the Pt isobar with the diagram
phase boundaries. The (possibly multiple) intersection points
define the corresponding transition temperature Tt.
•
The pressure-temperature point located in the phase diagram
defines an ‘anchor point’ that constrains the geotherm. In this
procedure the phase transition is used as a mantle/core
thermometer.
•
This way several (P,T) ‘anchor points’ of the geotherm have
been determined, related to the solid state phase transition
near 660 km depth and the solid/liquid inner/outer core
boundary at 1220 km from the Earth’s centre.
•
16/01
28
Starting from these anchor points the temperature is then extrapolated
Starting
from these
anchor
pointsmantle
the temperature
is then extrapolated
sides For
to thethis
from both
sides
to the
core
boundary
at 2900 from
km both
depth.
core mantle boundary at 2900 km depth. For this temperature extrapolation assumptions have to
be made about
the dominant heat
transport mechanism
this made
case it isabout
assumed the
that heat
temperature
extrapolation
assumptions
haveand
toinbe
transport operates mainly through thermal convection. This will be further investigated in later
dominant
heat
transport
mechanism
andmantle.
in this case it is assumed that
sections
dealing
with heat transport
in the Earth’s
heat transport operates mainly through thermal convection.
Schematic radialFigure
temperature
distribution
in the mantle
and core,
by core,
majorconstrained
phase transitions
(Boehler, 1996),
7: Schematic
radial temperature
distribution
in constrained
the mantle and
by major phase
The temperature oftransitions
the upper/lower
boundary mantle,
is constrained
by mantle,
the γ-spinel
to postspinel
phase
transition
(Boehler, mantle
1996), (UM-upper
LM lower
OC outer
core, IC inner
core).
The at 660 km
depth. The temperature
at the
inner/outer
boundary
km depth
(radius
1220
km) is constrained
by the melting
temperature
of the
upper/lowercore
mantle
boundaryatis5150
constrained
by the
-spinel
to postspinel
phase transition
temperature of the
hypothetical
‘Fe-O-S’ alloy.
right hand
shows
a schematic
temperature
at 660
km depth. core
The temperature
at theThe
inner/outer
coreframe
boundary
at 5150
km depth core
(radius
1220 km) is distribution
(geotherm) labeledconstrained
‘CORE ADIABAT’
in thetemperature
liquid outer
core
versus pressure
and alloy.
the melting
curve
of athe core ‘Feby the melting
of the
hypothetical
core ‘Fe-O-S’
The right
hand (liquidus)
frame shows
O-S’
alloy. (CMB
core-mantle
boundary,
ICB inner
core boundary).
schematic core
temperature
distribution
(geotherm)
labeled ‘CORE
ADIABAT’
in the liquid outer core versus
pressure
and
the
melting
curve
(liquidus)
of
the
core
‘Fe-O-S’
alloy.
(CMB
core-mantle
boundary,
ICBmoves
inner outward as
The ICB is determined by the intersection of the liquidus and the geotherm. During core cooling
the ICB
core boundary). The ICB isthe
determined
by the
intersection
of the liquidus and the geotherm. During core
inner core
grows
by crystallisation.
•
The ‘head’ of the extrapolated outer core adiabat is at a
temperature of approximately 4000 K and the ‘foot’ of the lower
mantle adiabat at approximately 2700 K. This result indicates a large
temperature contrast of about 1300 K across the CMB.
•
How can such a large contrast be explained physically? This can be
explained by interpreting the CMB as a boundary between two
separately convecting fluid layers, each with a thermal boundary
layer where the main heat transport mechanism shifts from
convection in the interior of the fluid layers, to conduction near the
boundary interface, where vertical convective transport vanishes
with the flow velocity component normal to the boundary.
•
Separately convecting layers are in agreement with the large density
contrast across the CMB where the density almost doubles,. The
resulting strong temperature contrast across the CMB is consistent
with a lower mantle in a state of vigorous thermal 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
Fabio Romanelli
Heat
Mineralogy
In the upper mantle, the
model’s major mineral
component is olivine. Such a
composition satisfies the
density and bulk sound speed
data and is consistent with the
observed seismic anisotropy .
The transition zone corresponds
to a series of solid state phase
changes. Olivine undergoes
several transformations before
converting to a perovskite
structure in the lower mantle.
Because of the predicted
predominance of perovskite
(~70%) in the voluminous lower
mantle, perovskite is the most
abundant material in the earth.
Phase changes
An important factor for the velocity
structure is that some phase
transformations happen gradually
over a range of depths (Fig. 3.8-11).
A simple univariant phase change, in
which material of a single
composition changes completely from
one phase to another as pressure
increases, causes a sharp
discontinuity in velocity.
A more complicated multivariant
phase change involving a system of
variable compositions causes two or
more phases to coexist over a
broad region of pressure, and so
produces a velocity gradient. Thus
seismological studies that better
define the velocity structure of the
transition zone improve our
understanding of its composition.
Left: If the core is homogeneous, the solidus should be continuous across the inner and outer cores,
so the gradient of the geotherm must be shallower than that of the solidus for the inner core to be
solid and the outer core to be liquid.
Right: If the inner and cores are chemically different, the solidus can differ between them, allowing
a steeper gradient for the geotherm.
Thus, only in the inner core does the
geotherm lie below the solidus and result
in a solid phase.
Figure 3.8-14 illustrates this idea, assuming
that the light element in the core is
sulfur. In this phase diagram for the Fe–
FeS system extrapolated to core conditions,
sulphur significantly lowers the melting
temperature of iron.
Cooling a liquid iron mixture with 12%
sulphur, corresponding to 33% FeS, causes
solid Fe to freeze out, leaving the liquid
richer in FeS.
In this analogy, the outer core corresponds
to the FeS-rich liquid, and the inner core
to the denser Fe solid.
Seismic tomography
Mean density
Moment of inertia
Seismological
model
Vp , Vs
ρ , P
K , μ
Equation of state
A-W equation
Birch’s law
Bullen parameter
Bullen’s parameter
• Adams-Williamson equation
dρ(r)
dr
=−
ρ(r)g(r)
Φ(r)
• Bullen parameter
dρ(r)
ρ(r)g(r) dr
=
Composition and structure of Earth’s interior
dρ(r)
dr
d! !2 g should
dT be one for
By definition, the Bullen parameter
" !"th
¼
:
(17:13)
dz KT
dz
regions where density is determined by compression in the
This relation means that if the density changes through
adiabatic temperature gradient.
compression and thermal expansion, then the actual
In contrast, in regions of super-adiabatic
gradient,
the
depth variation in density
depends on
the depth variation in temperature.
density increases with depth less
than in the case of
Let us consider a case where the temperature–depth
relation
follows parameter
the adiabatic gradient,
adiabatic gradient. Therefore the
Bullen
will benamely,
ðdT=dzÞad ¼ Tg"th =CP . Inserting this relation into
less than one in such a case. (17.13), one obtains,
When the density increases with
more than adiabatic
! "depth
d!
!2 g
Tg"th
¼
"
!"thor the change in
compression due to phase transformations
CP
dz ad KT
!
"
(17:14)
2
2
T"
!
g
!2 g !g
chemical composition (i.e., increase ¼in Fe1 "content
with
th KT
%
¼ 2
!CP
K
KS
V#
depth), then the Bullen parameter willT be larger
than
one.
where V#2 & VP2 " 43VS2 ¼ K!S and use has been made
#
$
of the relation KT ¼ KS 1 " ð"2th KT T=!CV Þ % KS '
#
$
1 " ð"2th KT T=!CP Þ (equation (4.21)).
Although the concept of adiabatic temperature
/
dρ(r)
dr
ad
313
2.50
2.00
Bullen parameter
ηB =
Φ(r)
1.50
1.00
0.50
0.00
–0.50
0
1000
2000
3000 4000
Depth (km)
5000
6000
FIGURE 17.3 Distribution of the Bullen parameter in the Earth.
that the Bullen parameter is nearly one in most of the
Vp Vs
ρ P
K μ
Seismological
model
Seismic tomography
Mean density
Moment of inertia
Equation of state
A-W equation
Birch’s law
Bullen parameter
Heat flux
Anchoring points
Mantle viscosity
Adiabatic gradient
Thermal
model
T
Bulk composition
(T,P) phase diagrams
Vp Vs
ρ P
K μ
Seismological
model
Seismic tomography
Mean density
Moment of inertia
Equation of state
A-W equation
Birch’s law
Bullen parameter
Compositional
model
Mineralogy
Assemblage
Chemistry
Bulk composition
Mineralogy
Assemblage
Chemistry
Compositional
model
(T,P) phase diagrams
Vp Vs
ρ P
K μ
Seismological
model
Seismic tomography
Mean density
Moment of inertia
Equation of state
A-W equation
Birch’s law
Mantle viscosity
Heat flux
Bullen parameter
Adiabatic gradient
Anchoring points
Thermal
model
T