Braun J., van der Beek P., Batt G. Quantitative Thermochronology: Numerical Methods for the Interpretation of Thermochronological Data
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208 Forward models of fission-track annealing
The principle of equivalent time presumes that, at any moment, a track that has
been annealed to a certain degree behaves during further annealing independently
of the conditions which caused the previous annealing. Further annealing depends
only on the degree of annealing that has already occurred and the prevailing
t−T conditions. The order in which the isothermal steps occur is, therefore, not
important.
The isothermal steps are chosen to be of equal length and the associated
temperature is obtained by linear interpolation between input points of the thermal
history. The annealing history is traced for a set of representative tracks that
are formed at the beginning of each new time step. This procedure assumes
that the fission-track-production rate is constant through time. The half-life of
238
U is sufficiently long (445 ×10
9
yr; Fleischer et al. (1975)) to warrant this
assumption, for timescales less than about 500 only Myr. All newly formed
tracks share a similar thermal history with existing tracks from their formation
onwards. Therefore, the principle of equivalent time allows all of the reduced
track lengths to be calculated in a single pass through the isothermal steps in
reverse order. In this manner the evolution of the track-length reduction cannot
be traced but computing time is greatly reduced (e.g., by a factor of 595 for
100 time steps).
The calculated set of n reduced track lengths is transformed to a binned fission-
track-length distribution (FTLD) using a probability-density function fr. This
procedure is required because a given amount of annealing does not lead to a
single value of r but to a range in r, as a result of the anisotropy of annealing
as well as the range in chemical composition and annealing properties of apatite
in the sample (Donelick et al., 1999; Green et al., 1986). Lutz and Omar (1991)
proposed the function
fr =
1
n
n
j=1
K
r −r
j
%
(A1.1.5)
where is the standard deviation and Kx is a distribution function. A Gaussian
distribution is most applicable:
Kx =
&
1
2
e
x
2
/2
(A1.1.6)
The standard deviation in (A1.1.5) is itself a function of r (Green et al., 1986).
Different forms for this function have been suggested. Lutz and Omar (1991)
used a three-tier linear relationship with parameter values obtained from a linear
A1.1 Variable temperature history 209
regression of the data of Green et al. (1986). These values are, however, not well
constrained, with correlation coefficients well below 0.5:
=
253 for r ≤ 043
508−593r for 043 <r≤067
139−061r for 067 <r
(A1.1.7)
A histogram of the FTLD can now be constructed by integration of (A1.1.5)
over the range of r spanned by each histogram cell, and normalising with respect
to 100%.
Calculation of the fission-track age (FTA) is based on the observation of
Green (1988) that the contribution to the observed fission-track density, and
therefore FTA, of a particular track is dependent on the final length of that track.
For n representative tracks produced, the contribution of each track to the total
track density is calculated. The relationship between the reduced track density
d = /
0
and the reduced track length r has been estimated from the data of
Green (1988) by Lutz and Omar (1991):
d =
0 for r ≤035
215r −076 for 035 <r≤ 066
r for 066 <r
(A1.1.8)
The FTA is calculated by multiplying the mean reduced density by the modelled
time span t
a
:
FTA =
R
n
t
a
*
n
j=1
dr
j
n
(A1.1.9)
The factor R
n
renormalises the calculated FTA to calibrate it against fission-track-
age standards, which have a mean length of spontaneous tracks l
std
of about
145m; R
n
= l
0
/l
std
.
Appendix 2
Fortran routines provided with this textbook
Table A2.1. Stand-alone programs and utility subroutines provided with this
textbook
Program Functionality
Heat1D To solve the general transient one-dimensional heat-transport
equation
Pecube To solve the general transient three-dimensional heat-transport
equation
Blocking.f To determine age from a temperature–time history assuming a fixed
blocking temperature
Dodson.f90 To determine age from a temperature–time history using Dodson’s
method
Mad_He.f90 To determine age from a temperature–time history by solving the
solid-state diffusion equation by finite difference
MadTrax.f To determine the fission-track age and track-length distribution
from a temperature–time history
Biotite.f90 To determine the K–Ar biotite age from a temperature–time history
using Dodson’s method
Muscovite.f90 To determine the K–Ar muscovite age from a temperature–time
history using Dodson’s method
lagtimetoexhum.f Calculates exhumation rates from thermochronological ages using
the approach described in Brandon et al. (1998).
210
Appendix 3
One-dimensional conductive equilibrium with heat
production
A3.1 The problem
To understand the effect of heat-producing elements on the conductive heat bal-
ance within the continental lithosphere, we assume that a uniform heat production,
H, is limited to a crustal layer of thickness , the upper limit of which is at a
depth h. We assume that asthenospheric convection provides a constant heat flux
to the base of the lithosphere. We are going to calculate the temperature at the
base of the crust, L, as a function of H, h and .
This is an important problem to solve since it is going to tell us how the
temperature distribution within the crust is influenced by the heat produced by
radioactive elements such as K, Th and U, which are known to be relatively
abundant within the upper crust. An interesting application of this problem is the
effect on the Moho temperature (and lithospheric strength) of the presence of a
strongly heat-producing layer at or near the surface.
A3.2 The basic equation
The one-dimensional equation governing the steady-state transport of heat by
conduction can be written as
z
k
T
z
+H =0 (A3.2.1)
where T is the temperature, z is the vertical spatial coordinate, k is the thermal
conductivity, is the density and H is the heat production. This is an elliptic,
second-order partial differential equation that requires two boundary conditions.
We assume that the surface is kept at a constant, zero temperature:
Tz = 0 = 0 (A3.2.2)
211
212 1D conductive equilibrium with heat production
whereas the base is characterised by a fixed heat flux from the underlying astheno-
sphere:
k
T
z
= q
m
(A3.2.3)
A3.3 The dimensionless form
This equation can be expressed in terms of a dimensionless depth, z
′
= z/L, and
temperature, T
′
= Tk/q
m
L. The temperature scale is derived from the basal
boundary condition. This leads to
2
T
′
z
′2
+H
′
= 0
T
′
= 0atz
′
= 0 (A3.3.1)
T
′
z
′
= 1atz
′
= 1
where H
′
= HL/q
m
.
A3.4 Analytical solution
The crustal column ranging between z
′
=0 and z
′
=1 must be divided into three
sections. In each section the solution takes a different form. Where there is no
heat production, the temperature varies linearly with depth; where there is heat
production, the temperature varies quadratically with depth:
T
′
=
a
1
z
′
+b
1
for z
′
≤ h
′
−
H
′
2
z
′2
+a
2
z
′
+b
2
for h
′
<z
′
≤ h
′
+
′
a
3
z
′
+b
3
for z
′
>h
′
+
′
(A3.4.1)
where h
′
=h/L and
′
=/L. The surface boundary condition implies that b
1
=0
whereas the basal boundary condition leads to a
3
= 1. To find the values of the
other parameters, one imposes that both temperature and heat flux are conserved at
the two interfaces z
′
=h
′
and z
′
=h
′
+
′
. This leads to the following expressions
for the dimensionless temperature field:
T
′
=
1+H
′
′
z
′
for z
′
≤ h
′
−
H
′
2
z
′2
+
1+H
′
h
′
+
′
z
′
−
H
′
h
′2
2
for h
′
<z
′
≤ h
′
+
′
z
′
+H
′
′
h
′
+
′
/2 for z
′
>h
′
+
′
(A3.4.2)
A3.6 The relationship between heat flux and heat production 213
A3.5 Temperature at the base of the crust
The dimensionless temperature at the base of the crust z
′
= 1 is given by
T
′
1 = 1 +H
′
′
h
′
+
′
/2 (A3.5.1)
or, defining
¯
h
′
as the depth of the centre of the heat-producing layer,
T
′
1 = 1 +H
′
′
¯
h
′
(A3.5.2)
which shows that the Moho temperature is a linear function of (a) the amount of
heat production, (b) the thickness of the heat-producing layer and (c) the depth of
the layer. It is thus important to notice that it is not only the vertically integrated
heat production H
′
′
that determines the basal temperature but also the vertical
distribution of heat sources.
Note, however, that the dimensionless surface heat flux is given by
T
′
z
′
z
′
=0
= 1 +H
′
′
(A3.5.3)
It is proportional to the integrated heat production, H
′
′
, but independent of its
vertical distribution, h
′
. This clearly demonstrates that one cannot deduce the
temperature structure within the lithosphere/crust from the value of the heat flux
(or temperature gradient) at the surface. There is no simple relationship between
surface heat flow and the Moho temperature, unless one assumes that all heat
sources have the same vertical distribution within the crust.
A3.6 The relationship between heat flux and heat production
By integrating the basic partial differential equation (A3.2.1) over the thickness
of the crust, one obtains
k
L
0
2
T
z
2
dz +
L
0
H dz = 0
k
T
z
z=L
−k
T
z
z=0
+
L
0
H dz = 0 (A3.6.1)
−q
0
=−q
m
+
L
0
H dz
This is another, more direct way to show that the surface heat flow q
0
is simply the
sum of the mantle heat flow and the vertically integrated crustal heat production,
regardless of its vertical distribution.
Appendix 4
One-dimensional conductive equilibrium with
anomalous conductivity
A4.1 The problem
Here we wish to investigate the effect of a non-uniform conductivity distribution.
This situation is commonly encountered where sedimentary layers are deposited
at the surface of the Earth and, potentially, buried to finite depths. How does
this affect the crustal temperature structure, especially the Moho temperature and
surface heat flux?
A4.2 The basic equation
We assume that the crustal conductivity is uniform, kz = k
0
, except in a layer
of thickness of which the top boundary is at a depth h, where the conductivity
is k
0
/. Sediments are usually characterised by a lower thermal conductivity than
that of crystalline basement, thus is usually greater than 1. Neglecting heat
production, the basic partial differential equation and boundary conditions are
z
kz
T
z
= 0
T = 0atz = 0 (A4.2.1)
k
0
T
z
= q
m
at z = L
where
kz =
k
0
for z ≤ h
k
0
/ for h<z≤ h +
k
0
for z>h+
(A4.2.2)
214
A4.5 Temperature at the base of the crust 215
A4.3 The dimensionless form
This equation can be expressed in terms of a dimensionless depth, z
′
= z/L, and
temperature, T
′
= Tk/q
m
L. The temperature scale is derived from the basal
boundary condition. This leads to
z
′
kz
k
0
T
′
z
′
= 0
T
′
= 0atz
′
= 0 (A4.3.1)
T
′
z
′
= 1atz
′
= 1
A4.4 Analytical solution
Because we neglect heat production, the solution is made up of three linear
temperature distributions:
T
′
z
′
=
a
1
z
′
+b
1
for z
′
≤ h
′
a
2
z
′
+b
2
for h
′
<z
′
≤ h
′
+
′
a
3
z
′
+b
3
for z
′
>h
′
+
′
(A4.4.1)
where h
′
= h/L and
′
= /L. The various coefficients can be derived from
the boundary conditions and the continuity of temperature and heat flux across
the two interfaces defining the anomalous conductivity layer. This leads to the
following expressions for the temperature field in each of the three regions:
T
′
z
′
=
z
′
for z
′
≤ h
′
z
′
+h
′
1− for h
′
<z
′
≤ h
′
+
′
z
′
+
′
−1 for z
′
>h
′
+
′
(A4.4.2)
A4.5 Temperature at the base of the crust
The dimensionless basal temperature is given by
T
′
1 = 1 +
′
−1 (A4.5.1)
It is a function of the conductivity anomaly and the thickness of the anomalous
layer but not its position. Furthermore, the surface heat flux is unaffected by the
presence of the anomalous layer. This demonstrates that the presence of low-
conductivity sediments can strongly affect the temperature at the base of the
crust. The depth of burial of the sediments, however, does not affect the Moho
temperature and thus the strength of the lithosphere.
Appendix 5
One-dimensional transient conductive heat transport
A5.1 The problem
To illustrate the transient behaviour of conductive systems, we solve the problem
of the temporal evolution of the temperature within a crustal layer of thickness
L subjected to an instantaneous increase in basal temperature (from T
0
to T
1
).
This may represent the rapid change in basal crustal temperature accompanying
mantle delamination in regions of highly thickened lithosphere such as the Tibetan
Plateau. It has been shown that, when the lithosphere is thickened, the lower part
of the lithosphere becomes gravitationally unstable and can be rapidly removed by
convective asthenospheric flow (Houseman et al., 1981). Here we will assume that
this delamination is instantaneous and leads to a step increase in basal temperature.
A5.2 The basic equation
Neglecting heat production by radioactive elements, the one-dimensional equation
governing the transient transport of heat by conduction can be written as
c
T
t
=
z
k
T
z
(A5.2.1)
where T is temperature, t is time, z is the vertical spatial coordinate, k is the
thermal conductivity, is the density and c is the specific heat. This is a parabolic,
second-order partial differential equation, which requires two boundary conditions,
Tt z = 0 = 0
Tt z = L = T
1
(A5.2.2)
and an initial temperature distribution,
Tt = 0z= T
0
z/L (A5.2.3)
216
A5.4 Analytical solution 217
where the ratio
T
0
/T
1
= ≤ 1 (A5.2.4)
represents the assumed jump in temperature imposed for times t>0.
Assuming that the conductivity, density and specific heat are spatially uniform,
this equation can be simplified to
T
t
=
2
T
z
2
(A5.2.5)
where = k/c is the thermal diffusivity.
A5.3 The dimensionless form
To illustrate the importance of each term in the equation, one usually re-writes it
using the dimensionless variables z
′
= z/L and T
′
= T/T
1
:
T
′
t
=
L
2
2
T
′
z
′2
(A5.3.1)
which leads to the introduction of the conductive timescale,
c
= L
2
/, and the
further reduction of the equation to its fully dimensionless form:
T
′
t
′
=
2
T
′
z
′2
(A5.3.2)
where t
′
= t/
c
. The boundary and initial conditions become
T
′
t
′
z
′
= 0 = 0
T
′
t
′
z
′
= 1 = 1
T
′
0z
′
= z
′
(A5.3.3)
A5.4 Analytical solution
First, we redefine the temperature to simplify the basal boundary condition. If we
define the new function as = T
′
−z
′
, it is trivial to see that it obeys the same
partial differential equation as T
′
,
t
′
=
2
z
′2
(A5.4.1)
but now both boundary conditions are homogeneous:
t
′
z
′
= 0 = 0
t
′
z
′
= 1 = 0
(A5.4.2)