## M (1.36)

where

mean

mean^ + WJph where C is the soil volumetric thermal capacity; T is a period, equal to 1 year for the seasonal processes of freezing (thawing); Ç is the unknown freezing (thawing) depth; r^ean is the mean annual ground temperature at the depth A0 is the amplitude of the temperature sine-wave fluctuation at the ground surface. The above expressions hold good for determining the depths of seasonal freezing as well as of seasonal thawing of ground in the case when the thermal-physical characteristics of soils in the frozen and unfrozen state are equal. Kudryavtsev has also obtained the solutions for the case of various thermal-physical characteristics; however, this adds complexity to the calculations. Therefore they are averaged out and the reduced characteristics are used in practice. Thus,

while

where zunf and zfr are the duration of the thawed and frozen states, respectively, in the ground.

The reduced formula for determining the freezing (thawing) depth represents the quadratic equation in The roots of this equation are real and opposite in sign, if the condition A0>|t|,ean| required for the seasonal freezing (thawing) process to proceed is fulfilled. From physical considerations the negative root is neglected. At the same time the advantage of

Kudryavtsev's formulae over all the others discussed above for the case when Qph is small should be noted here. As already noted, all the previous formulae were derived on the assumption that Qph is large (2Ph»C|tLan |), and lose their meaning at Qph —» 0, as can be easily verified. Koudryavtsev's formula works very well in this case, because at Q h = 0:

-exactly in line with the Fourier solution for a medium without phase transition. The value £ represents the depth of penetration of the zero isotherm or the depth at which A{ = | t^ean |. Thus Koudryavtsev's formula transforms to Fourier's formula in the limiting case of Qph 0. The fact that this transforms to the Stefan formula in the other limiting case (at great Qph and the zero initial conditions, i.e. with the assumptions for which the Stefan formula has been derived), as shown by V.Ye. Romanovskiy, is a further remarkable property of this formula. Thus, Kudryavtsev's formula not only is in line with the data which are obtained in the course of permafrost surveys as far as its basic parameters are concerned, but also is applicable at practically all the values of Qph. This is a unique property for the whole family of approximate formulae for calculating m.

Although Kudryavtsev's formula is the quadratic equation in the coefficients in this equation are very complex and it is rather difficult to carry out any calculations using this formula without special computers (minicomputers). Therefore nomograms, presented widely in literature, allowing determination of £ from the predetermined t^ean,^0,Afr,Aunf,Cfr,Cunf,Q were calculated on the basis of this formula. However such a form for presenting the calculation results has proved too cumbersome (three sets of nomograms have been constructed, with each of them including a set of seven nomograms) and involves the difficulty of interpolating the values Cfr, Cunf and Qph. Therefore Romanovskiy conducted a new nomogram through identical manipulation of Kudryavtsev's formula as a starting point. It turned out that this formula gave the more simple expression:

where a dimensionless value depends only on two dimensionless variables: Fig. 1.17. V.Ye. Romanovskiy's nomogram for determining the depth of the seasonal ground freezing (or thawing) £ from Kudryavtsev's formula.

QJ2C r QJ2C

Such manipulation allows calculations with Kudryavtsev's formula that take 25-30 s using a microcalculator. The range of a and /? changes in the new nomogram (Fig. 1.17) which includes all the possible combinations of values A0, tiean,Qph,A{r and 2unf, Cfr and Cunf, for which the previous (Kou-dryavtsev's) nomograms were constructed. Given the mean perennial temperature fper on the surface and the amplitude of the perennial cycle of temperature fluctuations Aper, Koudryavtsev's formula can be used for calculating the depth <!;per of the perennial ground freezing also.

The problem is complicated by the fact that the mean perennial temperature at the depth £per(i*per), the value of which enters into Kudryavtsev's formula for calculating the value £per, is in its turn linearly dependent on £per. The value of the geothermal gradient g serves as a proportional constant in this case:

 (2^mean£2C + Qphz \6ph 2CW2C\/ [XT 7tC mean 1 ^^ i where