Jökull - 01.12.1957, Side 8

model, that is, a one dimensional temperature
field in a semi-infinite solid which is subjected
to erosion or sedimentation and variable tem-
perature at the boundary. The heat conduc-
tivity, k, and the diffusivity, a, are assumed
constant.
The differential equation for the temperature
T’ in a fixed coordinate systern is
aT’xx = T’t (1)
with the boundary y = 0 at T = 0 and the
initial temperature a linear function of the
depth. In this case f(t) = 0 and h(y) = gy,
where g is a constant. The integrals in equa-
tion (6) are easily computed. A transformation
back to T by the second equation in (3) gives
the final solution
where x is the coordinate in the solid and t'
the time.
This equation has to be solved with the
boundary conditions
x-vt’ = 0, T’ = f (t),
t’ = 0, T’ = h (x), ( ^
where v is tfie velocity of erosion (or sedimenta-
tion where v is negative) which will be assumed
constant.
Equation (1) is solved by the means of the
transformation
y = x — vt’, t = t’ (3)
T = T’ = u (y, t) exp (— vy/2a — v2 t/4a),
which transform the equation (1) into
auxx = ut (4)
and the boundary conditions into
y = 0, u = f (t) exp (v2 t/4a),
t = 0, u = h (y) exp (vy/2a).
The solution of the problem presented by (4)
and (5) is given by Carslaw (10)
f V2 f^ V*2
u - J/?/yJe2a ðffáyizjetz + J/'fzJeJio 02(/,y;zJc/z
'o Jn
(6)
where Gi (t, y; z) and G2 (t, y; z) are defined
, / / . fe+yj*
(7)
G»fr.y;zJ =

y
— e Jett-I)
Special case: Period of erosion with the initial
condition h(y) = gy where g is a constant. An
interesting special case is presented when the
erosion of the velocity v is started at t = 0
The derivative of (8) gives the temperature
gradient at the surface
The temperature gradient at the surface de-
pends therefore on the term r/2 a/ at, where
r is the portion eroded in the time t, that is,
r = vt.
Approximation methods in tlie case of two
or more short periods of erosion. The problem
of estimating the effect of more than one period
of erosion will be encountered in the following.
In principle, this problem is solved by the ap-
plication of the general solution (6) to each
period. The temperature distribution found at
the end of a period serves as the initial distribu-
tion for the subsequent period. The parameter
v is zero for periods of no erosion.
The computations involved in this procedure
become quite lengthy when several periods with
different external parameters are encountered.
In the present case, we are mainly interested
in the erosion during the Pleistocene glacial
stages. As will be mentioned below the indica-
tions are that each of the 4 major glacial stages
were relatively short compared to the total
length of the Pleistocene. Each stage may have
covered less than 10% of the total length of
the Pleistocene.
Let At be the length of a period of erosion
and t the tíme that has passed since the middle
of the period. In the case where t > > At the
effects of the erosion at the time t may be
approximated on the basis that the total erosion
occurred instantaneously at the middle of the
period.
Let the initial conditions before the erosion
be T = gy, where g is a constant gradient, and
furthermore f(t) = 0. If r is the total thickness
6