The Neumann solution is a special case that occurs when the initial temperature gradient *G *is zero.

Thus, it is the conduction solution that will predict the minimum time needed to form a given thickness of

frozen material. The exact solution is well known as

(19)

γN is a function of *S*T , φ and the property ratios and is known (Lunardini 1981). Equation 19 can also be

written as

σ N = 2γ N τ .

(20)

Unlike the Neumann solution, the permafrost zone for the general case with a non-zero geothermal gra-

dient reaches an equilibrium value. The equilibrium value is the thickness of permafrost that will form if in-

finite time is available for growth. At steady state, the net heat flux at the phase change interface will be

zero so that (*dX*/*dt*)∞ = 0. Then the temperature in region 1 is

∆*T*1x

+ *T*s .

(21)

At the solidification interface

= *k*2G .

(22)

Thus

∆*T*1

(23)

or

1

σ∞ =

.

(24)

The previous discussion assumed that the soil was initially unfrozen, with a temperature distribution giv-

en by eq 2c, as shown in Figure 9. However, the cyclic warming and cooling that has taken place over the

past million years will tend to cyclically lower the ground temperature as compared to the previous initial

temperature distribution. If the temperature is lower than that of eq 2c, the permafrost growth will be accel-

erated. Figure 10 shows the thermal state of permafrost at equilibrium with a geothermal heat flow--the

solid line. The approach to the equilibrium state of the per-

mafrost is affected by the initial temperature of the soil in

that the time to reach equilibrium will be greatly changed

but the final equilibrium thickness is not affected by the

initial assumption (see eq 23).

Let us assume that surface warming to *T*f occurs and

the permafrost eventually reaches a stage where the entire

frozen zone is at *T*f . The system starts to thaw in this con-

dition and we ignore the time needed to reach this state

(small compared to melt time scales) and also neglect the

bottom freezethaw during this initial warming time. Fig-

ure 11 shows the thermal state as the bottom thaw com-

mences, at *x *= *X*o. The geothermal heat flow is used to

melt permafrost and also to warm the soil in the region *X*

11