GS 388 Handout on Thermal Isostasy
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SIMPLE THERMAL ISOSTASY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
COOLING OF SUB-OCEANIC LITHOSPHERE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
APPENDIX: BASIC HEAT FLOW EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Simple Thermal Isostasy
This section develops the very simplest view of thermal isostasy, and illustrates the
application of the coefficient of thermal expansion. The next section explains the application of one
dimensional heat diffusion to the cooling of the sub-oceanic lithosphere. The final section, as an
appendix, gives a brief derivation of the heat diffusion equation.
In the highly simplified diagram below the 1300° C isotherm is elevated by a vertical
distance ∆Z. The geotherms at the two locations 1 and 2 indicated in the section are shown below.
The geotherm is approximated as a simple linear conduction profile through the "lithosphere" and a
constant temperature in the "asthenosphere".
profile 1
profile 2
h
crust
Moho
1300°C
mantle
lithosphere
asthenosphere
∆Z = amount the
lithosphere is "thinned"
∆Z
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The differences in temperature between the two profiles imply a thermal expansion of the
material in the plateau section. This can be calculated for a small element, dZ, of a unit area vertical
column which has its temperature increased by ∆T, as follows:
dh = α ∆T dZ
or, integrating over the whole column,
z = Zo
h=
z=0
α T2 - T1 dZ
where T2and T1 are the two temperature profiles shown in the figure below, α is the coefficient of
thermal expansion and Ta is the temperature of the asthenosphere. The integration can be broken
up to two simpler ones, equivalent to computing the thermal expansion of the two profiles
separately relative to a zero thickness lithosphere (Ta at all depths), so that
z = Zo-∆Z
h=
z=0
α T2 dZ −
z = Zo
z=0
α T1 dZ
and evaluated to give
h = α ∆Z
(Ta -To)
2
profile 1
profile 2
T2(Z)
T1(Z)
∆Z
depth
depth
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Note that variations of α with depth are ignored. Estimates of α the upper mantle/crust of 3 x 10-5
cgs, Ta = 1300°C, and To = 0 give
h ~ (0.02) ∆Z
where h and ∆z are measured in the same units. Thus a 100 km rise in the 1300*C isotherm,
equivalent to a 100 km thinning of the lithosphere, implies an uplift of 2 km.
The basis for this very simple view of thermal isostasy is developed in the next section.
Cooling of Sub-oceanic Lithospher e
Sub-oceanic lithosphere begins to form as it moves away on either side of the spreading
ridge. The plate loses heat, cools and thickens as it moves away from the spreading ridge. The
cooling can be approximated by a very simple one dimensional model of a uniform half-space with
a fixed surface temperature. This one-dimensional model has only vertical heat flow and vertical
temperature variation. We start with the half space at a uniform temperature, Ta, and then expose
the suface to a lower temperature, To, which will then be held fixed in time. This starting
temperature throughout the medium will correspond to the asthenosphere temperature, while the
fixed surface temperature will approximate the temperature of the ocean bottom. We thus model the
cooling of a vertical column of material as it moves away from the oceanic ridge. The model
neglects the lateral flow of heat from one column to the next, which is assumed small in
comparison to the vertical flow.
spreading
ridge
T = To
T = To
t>0
t=0
T = Ta
T = T(z)
As the column cools, it contracts by an amount governed by the coefficient of thermal
expansion, α, of the material. The thermal contraction will deepen the ocean and thus increase the
thickness of the water column. The isostatic effect of the added water will further deepen the
GS 388 Handout on Thermal Isostasy
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ocean. The combined effects give a relation between ocean depth and the time since the column
started to cool, i.e. the age of the ocean floor.
Let T(z,t) = temperature as a function of depth, z, beneath the ocean floor, and time, t;.
To = T(z=0) = fixed surface temperature at t = 0 and all times later. Zero time is when the column
appeared along the ocean ridge system. Ta is the temperature of the asthenosphere, taken in the
simple half-space model to be the temperature throughout the half space at time t = 0. The
temperature is constant through the half space only at t = 0. The one-dimensional heat diffusion
equation must be solved to find out how the temperatures vary with depth and time after zero time
as a given column moves away from the ridge. The surface temperature To (< Ta) is suddenly
applied at zero time and then held fixed. The solution (see Turcotte and Schubert for derivation) for
T(z,t) is as follows:
T(z,t) = T0 + (Ta - T0) erf
z
2 βt
The "erf" function is an abbreviation for the error function (also called the probability
integral, although our problem has nothing to do with probability) and is defined as follows:
erf (x) =
u=x –u2
2
e du
π u=0
In the solution for temperature , the argument of the erf function is the combination
z
2 βt
which involves both z and t. The quantity in the denominator has the dimensions of z, i.e. depth,
and is a time-dependent scaling parameter for the plot of T versus z . Three profiles at times of 0, 5
and 50 MY are shown below (coefficient of thermal diffusivity, β, is taken as 25 km2/MY, a
reasonable value for the earth). The next figure is a normalized plot of the shape of the profile,
which is simply a plot of the "erf" function.
GS 388 Handout on Thermal Isostasy
0
200
400
5
Temperature, °C
600
800
1000
Zo = 0 km
Ta = 1300°C
1200
t = 0 MY
t = 5 MY
Zo = 25.3 km
t = 50 MY
T = 0.89 Ta
depth, km
50
Zo = 79.9 km
100
150
GS 388 Handout on Thermal Isostasy
0.0
0.0
6
0.5
1.0
normalized temperature = (T-To)/(Ta-To)
0.89
0.5
linear approximation
1.0
1.13
error function solution
1.5
normalized depth =
z
2 βt
2.0
Suppose we define the thermal "thickness" of the lithosphere, Zo, as the depth at which
the temperature reaches a certain large fraction of the temperature of the asthenosphere (Ta), say
89%; that is, T(z=Zo)/(Ta-To)=0.89. This means that the argument of the erf function must have
the specific value 1.13. There is nothing magic about 89% in terms of the mechanical properties of
the material, but it will allow us to make a simplification of the temperature profile that preserves its
essential characteristics, as discussed below. The transition from the strong material carried along
with the lithosphere and the weak, fluid-like material in the asthenosphere is thought to occur at a
temperature which is a large fraction of Ta, but the exact value is not very well known. 89% is
probably not a bad guess. The main point is that by choosing a particular value for this ratio, we
then set the value for the argument of the erf function and so define a particular relationship
between Zo and t. Zo will thus be a measure of the "thickness of the lithosphere" as the thickness
of the layer whose bottom reaches 89% of the asthenospheric temperature, Ta. Thus,
Zo = 1.13 2
βt = 2.26
βt
gives a convenient measure of the "thickness" of the lithosphere, Zo, as a function of time, t.
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The thermal contraction due to the cooling during the time t is obtained by integrating the
effect in a small depth range, dz, from the surface on down through the column. We did this in the
handout on "thermal isostasy". The change of the height of an element of the column, dh', due to
a change in temperature in the cooled section compared to the initial section will be given by:
dh' = [Ta-T(z,t)] • (α) • (dz)
and
h' =
z=∞
z=0
(Ta - T(z,t)) α dz
where α = coefficient of thermal expansion and h' is the decrease in height of the column due to
thermal contraction alone. Now we can simplify this integration if we replace the curvilinear
form of the erf function by a simple linear function as shown in in the figure. This replacement
preserves certain important characteristics of the problem, even though it over simplifies the exact
shape of the temperature profile (note that the linear approximation is not a bad one, though). If we
define Zo as we did above to give an isotherm at the "base of the lithosphere" at 89% of the
asthenosphere temperature, and draw the linear profile as in the figure above so that it reaches the
asthenosphere temperature at the depth Zo, then the integration of this linear function will give the
exact same answer for h' as we would get by integrating the "right" expression for T(z) with the
erf function. This is in fact the rationale for choosing the seemingly abitrary value of 89%. It
provides a simple but quantitatively accurate way to replace the erf function with a simple linear
profile, one that is easy to plot and think about, as demonstrated in the handout on thermal
isostasy. The linear relationship between temperature and depth is given by
T(z) = To + (z/Zo)(Ta-To)
T(z) = Ta
for z < Zo, and
for z > Zo
and the integral is easy to do, giving the contraction, h', as
h' = α (Ta - To) (Zo/2)
Note that the information from the erf function solution is contained in the definition of Zo, which
also relates depth, time and the coefficient of thermal diffusivity, β, so the essential physics is
preserved in this approach.
A result of the thermal contraction, however, is that additional water is added to the
column. Let h* be the height change due only to the isostatic response of the added water, and let
Do be the depth of the ocean at the oceanic ridge ( i.e., the column at t = 0 ). The mass (per unit
area) added to a column located off the ridge at a depth D > Do is (D-Do)ρw. This added mass
must be compensated by removal of an appropriate thickness, h*, of asthenospheric material,
resulting in an isostatic subsidence h*. The mass balance gives
h* • ρm = (D-Do) • ρw.
where ρm is the density of the asthenosphere and ρw is the density of the ocean water. The depth
of the ocean is thus given by the combination of h', the thermal contraction effect, and h*, the
isostatic effect of the extra water:
D-Do = h' + h* = h' + (D-Do)(ρw/ρm) , or, solving for D-Do,
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D - Do =
h'
=
ρw
1ρm
α (Ta - To)
1-
ρw
ρm
Zo
2
This is the equation which gives ocean depth, D, as a function of age; remember that Zo has the
square root of time in it. Estimates of the parameters which are reasonable from what we know
about the physical properties of mantle material, and which together fit the observed relationship
between ocean depth versus age, are as follows: To = 0 °C, Ta=1300 °C, β=25 km2 /MY,
ρm=3.3 gm/cm3 , Do=2.5 km, and α = 0.00003 cgs. The situation looks like the figure below.
spreading ridge
Do
ρw
D
zo
ρm
Curves of lithospheric thickness and ocean depth for these values are shown below as functions of
time intepreted as age of the lithosphere.
-2.0
-2.5
OCEAN DEPTH
VERSUS TIME
-3.0
-3.5
-4.0
-4.5
ocean depth, km
-5.0
-5.5
time, MY
-6.0
0
10
20
30
40
50
60
70
80
90
100
GS 388 Handout on Thermal Isostasy
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120
110
lithospheric
thickness,
km
100
90
80
70
60
50
LITHOSPHERE THICKNESS
VERSUS AGE
40
30
20
10
0
age, MY
0
10
20
30
40
50
60
70
80
90
100
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Appendix: Basic Heat Flow equation
Consider a small box with sides dx, dy and dz. Heat flows parallel to the z axis only, and
temperatures vary only as a function of z.
Qin = q(z 1) dxdydt
z = z1
z = z2
Note: positive q is pointing
downwards towards the
positive z direction
Qout = q(z 2) dxdydt
The net heat gain in the box during the time dt is the difference between the amount of heat flowing
into the box and the amount flowing out of the box. The inflow is
Qin = q(z1) dx dy dt,
where q(z1) is the heat flow per unit area per unit time in the positive z direction, and taken across
a unit area that is oriented perpendicular to the z axis and is located at z = z1. Likewise, the amount
of heat flowing out of the box is
Qout = q(z2) dx dy dt.
The net gain in heat will be related to an increase in temperature ∆T given by the expression
{heat gain per unit mass per unit temp.} • {mass per unit volume} • {volume} • {∆T}
or
{C}
•
{ρ}
• {dx dy dz} • {(∂T/∂t) dt}
where C is the specific heat capacity in calories per gram per degree, ρ is the density, and T is the
GS 388 Handout on Thermal Isostasy
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temperature. Thus
Qin - Qout = [q(z1) - q(z2)] dx dy dt = Cρ (∂T/∂t) dx dy dz dt
or
q(z1) - q(z2) = Cρ (∂T/∂t) dz
Now q(z2) can be evaluated from the differential expression
q(z2) = q(z1) + (∂q/∂z)dz,
and then the difference q(z1) - q(z2) substituted for in the preceding equation to give, after
cancelling the dz, dx and dy terms,
- ∂q/∂z = Cρ (∂T/∂t).
Now we need to add a further relation between heat flow and temperature. This is the basic
conduction relation for one dimensional heat flow,
q = - κ (∂T/∂z),
where κ is now the coefficient of thermal conduction (don't confuse this with the other parameters
that we have considered and unfortunately denoted by the same greek letter!). κ, like C and ρ, are
material constants. This equation is the basis of heat flow deteminations: measurements of
temperature as a function of depth in a piston core, borehole or mine are used to estimate ∂T/∂z,
and κ is measured for samples of the rock in which the temperature measurements were made.
Substituting for q, we have now the basic heat flow equation
κ
∂2T
∂z
2
=Cρ
∂T
∂t
or
β
∂2T
∂z
2
=
∂T
∂t
where
β=
κ
Cρ
β is called the coefficient of thermal diffusivity, a composite material constant with
dimensions of length squared divided by time. The square root of this constant scales the relation
between time and space in the movement or diffusion of heat. For the seismic wave equation we
had a constant whose square root scaled the movement of stress or strain as distance proportional
to time (the seismic wave velocity). Here, the square root of β gives us distance proportional to the
square root of time, which is the fundamental characteristic of diffusion. Typical values of β
estimated for mantle material are 25-40 km2/MY where MY = million years (it so turns out that
GS 388 Handout on Thermal Isostasy
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these are convenient units). Thus for a time of 1 MY a characteristic distance of heat diffusion is
about 5-6 km; for 10 MY, 16-20 km; for 100 MY, 50-60 km; and for a billion years, only 160-200
km! Conductive heat flow in the earth is a very sluggish process.
The heat flow equation developed above does not include sources of heat within the rock
itself. This can be included with a parameter, A, which is the amount of heat generated per unit
volume per unit time. Heat is generated by radioactive decay of certain elements found in rocks,
and is especially important in the upper crust of continental regions where the radioactive isotopes
are concentrated. We put this heat producing factor on the left side of the heat flow equation as
another addition to the rate of heat addition to our little unit volume of material. This source of heat,
together with the difference between the heat flowing in and the heat flowing out, produces the
change in temperature seen on the right side of the equation:
A+κ
∂2T
∂z
2
=Cρ
∂T
∂t
Note that the terms each give heat accumulated per unit volume per unit time. This equation can be
rewritten in terms of only κ and β as
A ∂2T 1 ∂T
+
=
κ ∂z 2 β ∂t
If there is no change of temperature with time – a "steady state" situation – the right side is zero,
and we have the very simple differential equation
d 2T
dz
2
=-
A
κ
Note that all this discussion is for temperature varying only in the z direction, i.e. the one
dimensional heat flow problem.