Download Heat in the Earth

Heat in the Earth
 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.
 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
 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
 Conduction
 Transfer of heat through a material by atomic or molecular
interaction within the material
 Radiation
 Direct transfer of heat as electromagnetic radiation
 Convection
 Transfer of heat by the movement of the molecules themselves
 Advection is a special case of convection
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
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
 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 
 T T T 

T  
,
,
 x y z 
2
2
2
    2  2  2
x
y
z
2
 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.
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
A
 0, and 2  
t
z
k
2
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:
T
Az

 c1
z
k
 Use the second bc to
constrain c1
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
c2  0
constrain c2
 Substitute for c2:
 Link to spreadsheet
Az 2 Q0 z
T 

2k
k
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
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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
4
and are characterized by a “nona
gd
Q

Ad


dimensional” number called the
Ra 
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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
kn
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
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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