GS388 Handout: Radial density distribution via the Adams-Williamson equation
TABLE OF CONTENTS
ADIABATIC COMPRESSION: THE ADAMS WILLIAMSON EQUATION.......................................................................1
EFFECT OF NON-ADIABATIC TEMPERATURE GRADIENT..............................................................................................3
SUMMARY OF RADIAL VARIATIONS OF DENSITY, GRAVITY, AND PRESSURE IN EARTH.................................4
DENSITY ...................................................................................................................................................................................... 4
PRESSURE AND GRAVITY....................................................................................................................................................... 5
ELASTIC MODULI .........................................................................................................................................................................6
Adiabatic compression: the Adams Williamson equation
Density changes with depth in the earth because of the effects of changes in (1) pressure, (2)
temperature, (3) composition, and (4) crystalline structure. In much of the earth, the last two factors do
not change with depth, as appears to be the case within the lower mantle, or the outer core, or the inner
core. Within these regions the density changes with depth mainly because the increase in pressure
compresses the material. Temperature plays a secondary role within these regions, because the
temperature does not change very much within those compositionally more homogeneous parts of the
earth. Within those regions the temperature change as a function of depth may be close to adiabatic.
A way to imagine the adiabatic temperature gradient is a follows: assemble the earth with its
present distribution of material but without gravity. The material is uncompressed and there is no
pressure increase with depth. Set the initial temperature everywhere to the earth's surface temperature.
Now turn gravity back on. The gravitational pressure causes the material to contract, with material
compressing more at greater depths because of the greater pressure. The temperature will also increase
because of the compression. If this is done such that no heat is gained or lost by any given piece of the
material, the temperature increase for any parcel of matter will be adiabatic, and the temperature
increase with depth will thus be adiabatic.
Another more realistic way to have an approximate adiabatic temperature distribution is to have
material convect heat from the hotter interior to the cooler exterior. The heat is carried upwards by the
upwards movement or flow of material, while material cooled near the surface descends. If the
temperature gradient (increase of temperature with depth) is adiabatic, then upwards movement of a
parcel of material will not result in a temperature difference of the parcel with respect to the surrounding
material. However, if the temperature gradient is greater than adiabatic (super-adiabatic), the
temperature of an upwards moving parcel will only decrease by the adiabatic gradient, and so will be
greater than that of its surroundings. This is shown in the following figure.
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GS388 Handout: Radial density distribution via the Adams-Williamson equation
temperature
temperature excess of
parcel at shallower depth
adiabatic gradient
depth
in situ geotherm:
super adiabatic
temperature gradient
upward movement
of parcel with
adiabatic gradient
Because the temperature is greater than the surroundings, the material will be less dense,
assuming that mantle is chemically homogeneous and no phase changes occur. The parcel will therefore
experience an upwards directed bouyancy force, i.e., the lighter parcel will tend to rise in the superadiabatic temperature field.The upwards directed bouyancy can thus lead to upwards motion of
material, thereby transporting heat by convection. Convection is a much more rapid means for moving
heat out of the earth than conduction. In the steady state the heat flowing out will equal the heat
generated within, and the temperature gradient will tend to be just slightly greater than adiabatic by the
amount that is required to maintain the convection. The amount required depends upon the resistance to
flow in the mantle, i.e. its viscosity.
Throughout most of the mantle (except within the lithosphere) the mantle is convecting, so within
those regions it is reasonable to assume that the temperature gradients are not far from adiabatic. We
can therefore consider the case where density as a function of pressure and temperature under adiabatic
conditions. The way to do this is to choose, instead of temperature, another thermodynamic variable
called entropy, denoted by the symbol, S, which is defined such that the gain or loss of heat divided by
the temperature is equal to the entropy change. Entropy is a constant for the adiabatic situation, because
no heat is lost or gained. So, instead of considering density as a function of pressure and temperature, as
one often does in thermodynamic calculations, we can consider density as a function of pressure and
entropy, ρ(P,S). The advantage of this is that the adiabatic and non-adiabatic effects are made explicit.
Density will vary as P and S are changed, and both P and S might change with radius, R. In the
adiabatic case, S will not change, which simplifies things. Thus in general we can write the change of
density with change in radius as follows:
d ρ  ∂ρ  dP  ∂ρ  dS
=
+
dR  ∂P  S dR  ∂S  P dR
If S is constant because the change with depth is adiabatic, then dS/dR = 0, and the last term
drops out. Thus we have the simple Adams-Williamson equation by substituting dP/dR = - ρg (simple
hydrostatic pressure in a self gravitating fluid) and use the definition of the bulk modulus, κ, to substitute
for the other derivative.
 ∂P 
ρ
 =κ
 ∂ρ  S
2
GS388 Handout: Radial density distribution via the Adams-Williamson equation
Here κ must is the bulk modulus under adiabatic conditions. This is what seismic waves
measure, because the compressions and dilations of the waves are too rapid for any appreciable amount
of heat to flow in or out during the passage of the waves. From seismic wave velocities we can get the
ratio κ/ρ = φ by calculating
φ = Vp 2 - 4 Vs2
3
which follows from the definitions of the seismic wave velocities. We thus have the simple AdamsWilliamson equation for adiabatic conditions,
dρ
ρg
=φ
dR
This is the basic equation for calculating density as a function of depth with regions of the earth
that are homogeneous and approximately adiabatic: the upper mantle below the lithosphere, the lower
mantle, the outer core, and the inner core.
Effect of non-adiabatic temperature gradient
Note that the adiabatic equation above does not assume constant temperature, but only that the
temperature gradient is adiabatic. The second term in the original equation deals with the effect of a
temperature gradient that is not adiabatic. Let us explicitly consider this by viewing the temperature
gradient (dT/dR)insitu to consist of an adiabatic part (dT/dR)ad plus a non-adiabatic part (dT/dR)nad.
We can thus subtract the adiabatic part from the total temperature gradient to get the non-adiabatic part
as follows (where the gradients are taken with respect to R and are thus negative), where the first term
on the right side of the equation is the adiabatic gradient:
 dT 
 ∂T  dP  dT 
= 
−
 dR 


 non _ ad  ∂P  S dR  dR in _ situ
The second term in the previous equation for the density gradient can be reduced to the following
form:
 ∂ρ  dS  dT 
 ∂S  dR −  dR 
 P

 non _ ad
where α is the coefficient of thermal expansion,
1  ∂ρ 
α=−  
ρ  ∂T  P
Thus the complete equation with both adiabatic and non-adiabatic effects is
dρ
ρg
 dT 
=−
+ αρ 

dR
ϕ
 dR non_ ad
Note that the non-adiabatic temperature term is opposite that of the adiabatic term. For a temperature
gradient that increases with depth at a rate greater than adiabatic (-dT/dR exceeds the adiabatic
gradient) the effect is to reduce the increase in density with depth that would be calculated with the
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GS388 Handout: Radial density distribution via the Adams-Williamson equation
4
adiabatic compression term alone. This makes sense- the effect of increased temperature is to expand
the material or reduce the density.
Summary of radial variations of density, gravity, and pressure in Earth
The following two pages show plots of density, gravity and pressure in the earth as
functions of radius. These plots are from an Excel table of the Preliminary Reference Earth Model
(PREM) given by Masters and Shearer, “Seismic Models of the Earth: Elastic and Anelastic”, in Global
Earth Physics, A Handbook of Physical Constants, T.J. Aherns, editor, American Geophysical
Union, 1995, p. 88-103. The curves represent a modern best fit solution to a large set of body wave
and surface wave seismic data in addition to satisfying constraints of total mass and moment of inertia of
Earth. The Excel spreadsheet is available as Z:\classes\Class_geo388\notes\prem.xls.
density
density
6000
5000
Radius,
meters
4000
3000
2000
1000
0
0
2000
4000
6000
8000
density, kg/m^3
10000
12000
14000
GS388 Handout: Radial density distribution via the Adams-Williamson equation
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Pressure and gravity
6000
5000
pressure
4000
3000
radius,
meters
2000
1000
0
0
50
100
150
200
250
pressure,GPa
300
350
400
6000
5000
gravity
4000
radius,
meters
3000
2000
1000
0
0
2
4
6
meters/sec
8
10
12
GS388 Handout: Radial density distribution via the Adams-Williamson equation
6
Elastic moduli
With both seismic velocities and density determined, the equations for seismic velocity can then
be used to calculate the moduli of incompressibility and rigidity and Poisson’s Ratio. The last was
shown already together with the seismic velocities. The two moduli are plotted below:
6000
5000
radius,
meters
4000
modulus of incompressibility
modulus of rigidity
3000
2000
1000
0
0
200
400
600
800
stress/strain,GPa
1000
1200
1400