Page 193 - Physical Principles of Sedimentary Basin Analysis

6.18 Periodic heating of the surface 175

The skin depth for daily variations of the surface temperature is 0.17 m and for yearly
variations 3.2m.
The temperature with units follows directly from the dimensionless solution. We just
ˆ
insert the definitions for T , ˆz and ˆ t in equation (6.272) and get
ω ω

T (z, t) = T exp − z cos ωt − z . (6.273)
2κ 2κ
The solution in this form shows that variations with high frequencies die out with depth
much faster than variations with low frequencies. Depth therefore acts as a filter that
removes high-frequency variations in the surface temperature.
We have already seen that we can add temperature solutions to obtain a new solution,
because the temperature equation is linear. We can for instance add the stationary solution
az + b, given by a constant thermal gradient a and constant average surface temperature
b. Another possibility is to add together the solutions with different amplitude T i and
frequency ω i ,

ω i ω i

T (z, t) = T i exp − z cos ω i t − z . (6.274)
2κ 2κ
i
This solution solves the problem with periodic surface temperature variations of the form

f (t) = T i cos(ω i t) (6.275)
i
and such a general Fourier series can be used to approximate almost any kind of periodic
surface temperature.
Figure 6.36 shows observations of the temperature at the depths 1.2 m, 4 m and 9 m
over a time span of 2 years and 4 months (Isaksen and Sollid, 2002). The data set is from a
site with permafrost on the island of Svaldbard. We notice that there are weekly variations
in the temperature data at 1.2 m depth, and that these high frequency variations have died
out at 4 m where the temperature signal is smooth. We have that l 0 ≤ 0.6 m for variations
with a period up to 1 month, and these variations are reduced to nearly zero (by a factor
2 −1
1/100) at the depth 6l 0 = 3.9 m. The model (6.273) is plotted for κ = 1 · 10 −6 m s ,
◦
T = 9.2 C, and a constant offset −6.3 C has been added. The model does not match
◦
the data at 4 m and 9 m, although weekly variations are filtered away, because there are
variations in the observations with a period longer than a year.
Note 6.10 The temperature equation (6.270) with boundary conditions (6.271)issolved
by the separation of variables, T (ˆz, ˆ t) = U(ˆ t)V (ˆz). When the product U(ˆ t)V (ˆz) is inserted
ˆ
into the temperature equation (6.270) we get
U V
= = a + ib. (6.276)
U V

These two ratios have to be equal to a constant (a+ib), because U /U is only a function of

ˆ t and V /V is only a function of ˆz. The constant is the complex number a+ib where a and