0

'

Ts

Tf

Ts

lem. Obviously, after infinite time the new temperature profile is

t=∞

as shown. However, the temperature will adjust to near the new

equilibrium in a relatively short time. This is a linear problem in

X0

t=0

non-phase-change conduction and has been solved by Lachen-

bruch et al. (1982). The transient temperature is

∞

(

)

1 *M *

2 o

+ * T*s - *T*s′ ∑

* X*o

where

2

,

.

4α f

4*t*c

(

) Xxo

(D6)

.

(D7)

The change in sensible heat, going from the state at *t *= 0 to the state at *t *= ∞, is

(

) X2o .

(D8)

The change in sensible heat at any time *t *is

∞

[

]

(

)

(T - *T*o )dx = *Q*s ∞ + 2 Ts - *T*s′ *X*o ∑ 2 1 - (-1)n .

2*C*f * o*

∫

(D9)

π

o

Thus, the relative change in sensible heat is

4 ∞ e *M*

[

]

= 2∑

1 - (-1)n .

(D10)

π n =1 n

2

Note that this quantity does not depend upon the surface temperatures. The relative change is shown in

Table D2. The sensible heat change attains 93% of its ultimate value at *t*/λ = 1.0 (*t *= 1666 years) and 99% at

years. From this time on, the bottom of the permafrost would slowly melt.

The time required for the temperature gradient, in the frozen zone, to

change is important since this quantity will determine the rate of change of the

Qs ∞ - Qs

t/λ

permafrost bottom depth. The gradient at the depth *X*o can be found from eq

Qs ∞

D5 and is

0.25

0.438

0.50

0.236

∞

1.0

0.069

(

)

+

δ=

=

(D11)

1.50

0.020

1.78

0.010

2.0

0.006

36