The heat flows at the base of the permafrost layer determine the rate of movement of the permafrost bot-

tom. Examine Figure D1 to clarify the concepts involved. At any instant of time, an amount of energy,

the thawed zone 2, *q*u. It is important to realize that the constant geothermal heat flow *q*g = *k*uG will only

equal *q*u(*t*) at equilibrium. During movement of the permafrost base, *q*u(*t*) can be greater or less than *q*g. We

can examine the transient behavior of these terms. Let

Ts

Tf

x=0

=

(D1)

X (t)

Frozen, 1

qf (t)

.

(D2)

x = X (t),

Bottom of

In terms of dimensionless parameters

Permafrost at t

qu (t)

1

(D3)

δ (t)

Thawed, 2

α (σ + φ)

σ*k*21

+ 1

21

[

]

β σ(β + 2) + 2φ

x = X + δ,

Depth for

qu = qg

σ(β + 2) + 2φ

qg = ku G

(D4)

σβ

Thawed, 3

where we assume that *k*u = *k*2, i.e., the thermal conductivity of the thawed zone is constant. The results of

example 2 for Prudhoe Bay are tabulated in Table D1. Clearly, the heat flow from the deep layer greatly ex-

ceeds the geothermal heat flow for much of the permafrost formation period. Also, note how rapidly the

heat flow out of the frozen zone *Q*f drops to slightly more than the flow from the thawed layer *Q*u.

The temperature in the frozen zone adjusts very quickly toward the equilibrium result of a linear temper-

ature distribution; however, the thermal zone approaches equilibrium very slowly.

Suppose the permafrost has thickness *X*o,

is growing and the surface temperature ini-

∆*T*1 = 12.69C

φ=0

tially increases by a certain amount and is

held constant for several thousand years.

α21 = 0.3915

What is the effect on the permafrost? The

q*f*

1

bottom of the permafrost will continue to

=

α 21

q fe

grow for several hundred years before start-

+1

β

+ 2)

β(β

Qu

Qf

ing to thaw, but Table D1 assures us that

this will be negligible. At *t *= 225,000 years,

1

4.44

1.9060

2.423

159.057

0.9251

350

79.9

1.6717

2.196

8.986

0.9405

3500

219.3

2.3380

1.855

3.352

0.9629

will only be 27.5 cm during the next 1000

35,000

461.4

4.8154

1.4153

1.6351

0.9826

years. Thus, we can assume that the perma-

100,000

567.8

7.7133

1.2593

1.3375

--

225,000

626.5

11.1633

1.1792

1.2153

--

frost thickness remains essentially constant

775,000

687.7

20.3168

1.0984

1.1090

--

while the temperature in the frozen zone

35