where

( fm - 1)qg

*M*=

.

*X*o L

The time to complete melt is

*X*o L ln *f*m

*t*m =

.

(B10)

*q*g ( fm - 1)

This leads to the value of δ when melt is completed.

3*L * ln *f*m

*X*o

δo =

+

- 1 .

(B11)

*GC*u *f*m - 1

2

We define a linear temperature distribution that will have the same sensible heat when thaw is completed.

*T*i′ = *T*f + *G*′*x*

(B12)

*X*o

δo -

*G*′

3.

=

(B13)

δo + *X*o

*G*

where *G' *is the equivalent geothermal gradient. Finally, the new temperature distribution at the beginning

of freeze is

*T*i = *T*f + *b*o x + *c*o x 2

(B14)

where

**Table B1. Melt relations for Prudhoe Bay.**

*G*(δo - *X*o )

*GX*o

G′

*c*o =

*b*o =

,

.

*ln *f*m*

δο

(δo + *X*o )

*Melt time*

-1

δo + *X*o

2

~

fm - 1

f

*(m)*

*(years)*

fm

G

0.1

0.55

1.5584

7948.1

0.9064

108,926

This initial temperature distribution is shown as curve a

0.2

0.60

1.0118

5265.6

0.8636

85,654

in Figure 12. Table B1 shows some results for Prudhoe

0.3

0.65

0 .720

3833.5

0.8196

73,230

Bay. Note the long melt times even if *f *is as high as

0.4

0.70

0.5272

2887.3

0.7706

65,022

90%.

0.5

0.75

0.3863

2195.8

0.7139

59,023

0.6

0.80

0.2771

1659.9

0.646

54,373

0.8

0.90

0.1157

867.8

0.455

47,502

**Freeze of cooled soil**

*X*o = 600 m, *L *= 30.21 cal/cm3, *C*u = 0.6457 cal/cm3 C.

The freezing process is as discussed earlier except

*q*g = 1.35 106 cal/cm2 s, *G *= 0.0286C.

that the initial soil temperature is lowered as noted in

Figure 10. The basic equations used earlier are still valid

except that the coefficients of eq 6 and 7 change, owing to the new initial temperature given by eq B14.

The basic equation, replacing eq 9, is

1

1

*dF*σ

= *k*21 - 2 -

(B15)

*d*τ

σ

*g*

β2

β2 β 1 σ*S*o

1 1 ρ21

(β - 1)(β + 1)2

- - -

- *C*21σ

*F *= -

++

- *m*o

(B16)

6*g * 3 * S*T

6 3 2

6

3

2ρ (g - 1)β

[

]

β

- *k*21σ -β + 2*m*o (β + 1) + 2σ*S*o (β + 1) = 21

2

(B17)

*S*T

*g*

26