The initial temperature gradient, at *x *= *X*o, is

( Xo , 0) =

δo =

.

(D12)

*

δo - δ

δ -δ

t

δ

t/λ

δ

Note that this simplified form of δo gives virtually the same

0.01

0.301

502

0.0314

heat flux as the value from eq D1, when *x *= *X*o. Now

0.0297

0.3421

570

0.0440

0.1

0.540

899

0.1022

∞

δo - δ * T*s′ - *T*s*o*

1 + 2 ∑ (-1) e .

=

0.20

1.446

2408

0.20

(D13)

δo

∞

∞

0.212

--

* Calculated without eq D14 approximation.

The time for a given change in the gradient can be closely ap-

proximated as

.

4

2

= 2 ln

δo - δ *T*f - *T*s*o *

(D14)

λπ

1 -

δ *T*s′ - *T*s

Table D3 shows the results for Prudhoe Bay.

It would take about 490 years for the bottom growth to cease and 900 years for the bottom gradient to

change significantly. Thus, the approximations used in the derivation of eq D5 are acceptable.

The temperature in the frozen zone requires 1666 years for sensible heat adjustment, leaving ∆*t *= 15000

1666 = 13,334 years for bottom melt (the interglacial is 15,000 years long). The energy balance at the bottom

of the permafrost is

- *Aq*g .

= *k*f f

(D15)

where *A *is the fraction of the geothermal energy that goes into melting; it can exceed 1.0.

The heat flow from the thawed zone is greater than the geothermal heat flow, as we

have seen. For the example discussed here, *A *= 1.179 at the beginning of thaw and will

decline towards 1 as thaw proceeds. The permafrost thickness *X*f after ∆*t *years is given by

Xf

A

*X *+ *b*

1.179

591.0

= *a *+ *X*o

(D16)

*X*o + *b *

1.0895

605.5

1.0

620.0

where

*Final thickness

after surface tem-

perature increase,

,

,

Prudhoe Bay.

The final permafrost thickness is strongly dependent upon the value of *A*. Table D4 shows values for the

Prudhoe Bay example.

The heat flow from the thawed zone varies continuously during the thaw, denoted in eq D15 as Α(*t*) *q*g.

The heat flow at equilibrium is such that *A *= *A*e = 1.0. Thus, let *A*(*t*) be a linear function of *X*, given by

*y *- *y*e

+ *A*e

(D16a)

1 - *y*e

37

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