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Fig. 1.10. Change in the ablimation intensity /a and in deposited layer thickness h(J with time q: 1 and 1' - ice-unsaturated and ice-saturated sand respectively; 2 - clay (polymineral, bentonite and kaolinite) 3 - ice; 4 - metal.

ceed. The moisture transfer proceeds in this case in vapour as well as in the liquid phase. At the same time /a turns out to be dependent on the parameters responsible for the diffusion intensity (i.e. on free soil porosity, snow cover, temperature, etc). This is well followed in Fig. 1.10, reflecting the dependences /a = /(t) and h{* = </> (t) obtained experimentally for sand, clay, ice and metal samples. The similarity of values of ice ablimation intensity for the moisture-saturated clays of different mineral composition depends upon the fact that their thermal conductivity does not vary much. The moisture-unsaturated sand, having a lower value of thermal conductivity, shows lower intensity of ablimation as compared with the moisture-saturated sand. In the case of moisture-unsaturated (for example, sandy) soil samples the moisture ablimation proceeds not only on the surface but also inside the ground. In clay soils in parallel with the processes of vapour transfer and ablimation the transfer of unfrozen water under the effect of grad t takes place, as was determined experimentally from the change in the initial distribution of the moisture content of the fine-grained soil samples. The process of water vapour ablimation in rudaceous rocks has its special features. Experimental investigations of metamorphic and igneous rock debris (detritus, 5-15 mm, and rock debris, 40-50 mm) have shown that the process of ablimation begins not throughout the whole ground surface but on the surfaces of the boulders which is then followed by a flux which propagates inward. A denser ablimation hoar is formed in this way on rock debris. The ablimation intensity is greater for larger debris components and this is likely to be associated with their higher heat conduction.

1.4 Freezing and thawing of ground

In Section 1.1 we considered temperature fields without regard for water freezing or thawing in the soil as the temperature went through 0°C. In other words, we analysed the processes of ground heating or cooling rather than freezing or thawing proper. Such cases are typical of soils which in effect contain no water. The zero isotherm does not differ fundamentally from any other isotherm of the temperature field in this case.

In actual conditions rather important thermal-physical and physico-chemical processes are responsible for soil properties, structure and composition as well as for the particular features of the soil temperature field in the moist soil system near or at the zero isotherm. The most important of these processes is the transition of water contained in the soil from liquid to solid state and vice versa, because release and absorption of a great amount of heat is associated with these phase transitions. As a consequence, additional heat sources or sinks appear in the region where the phase transitions proceed which affect fundamentally the ground temperature field and its dynamics. Thus, for example, heat of the order of 70 J is released in the course of freezing of the moisture contained in 1 g of the soil with 20% moisture content, when it is cooled below 0°C. Such an amount of heat is sufficient to heat this 1 g of rock by nearly 40°C. The amount of heat gph being released or absorbed in the course of complete freezing or thawing of a unit soil mass with the moisture (ice content) by weight JVW and a skeleton (dry) density ysk is determined from the expression:

where 334 Jg-1 is the heat of phase transition of water into ice and vice versa.

It is obvious that the temperature fall (below 0°C) of moist soils in the course of their freezing must proceed slower than their simple cooling. The layer of freezing soil must in this case protect the lower layers against intense cooling, because the zero isotherm cannot propagate deeper until freezing of all the free water occurs and the appropriate amount of crystallization heat has been released at this boundary. The zero isotherm serves in a given case as the interface between thawed and frozen portions of freezing soils and is termed the freezing boundary or front. And vice versa, the thawing of frozen soils slows down their heating, because the greater part of the heat received by the soil is expended in the phase transition of ice into water in the neighbourhood of the zero isotherm, which is in a given case the thawing boundary or front. This effect, of the moisture phase transitions proceeding at the zero isotherm, on the temperature field of freezing and thawing soils

Fig. 1.11. Dependence of values #an, £fr and £tha on the relationship between the mean annual temperature fmean, and the amplitude of temperature fluctuations on the surface A0: 1 - the depth of zero annual amplitude Han at fmea„4 < fmean3 < Lean,, < Lea^ < Lean,', 2 " the depth of seasonal ground thawing; 3 - the depth of seasonal freezing; 4 - the envelopes of the temperature curves: ffri at +fmeani; ¿frQ = ¿th() at fmean(); at -fmeai,3; seasonal freezing and thawing are absent at fmea„2 and tmnn^

Fig. 1.11. Dependence of values #an, £fr and £tha on the relationship between the mean annual temperature fmean, and the amplitude of temperature fluctuations on the surface A0: 1 - the depth of zero annual amplitude Han at fmea„4 < fmean3 < Lean,, < Lea^ < Lean,', 2 " the depth of seasonal ground thawing; 3 - the depth of seasonal freezing; 4 - the envelopes of the temperature curves: ffri at +fmeani; ¿frQ = ¿th() at fmean(); at -fmeai,3; seasonal freezing and thawing are absent at fmea„2 and tmnn^

was termed the'zero curtain by M.I. Sumgin.

In line with this phenomenon the depth of penetration of the harmonic (sine-wave) temperature fluctuations into the ground in which the moisture phase transitions proceed turns out to be essentially less than that in dry soils, where the processes of moisture freezing and thawing are absent. In the case when the amplitude of the temperature fluctuations on the surface proves to be smaller than the mean annual ground temperature {A0 < | fmean |) the moisture phase transitions are absent and the depth of annual temperature fluctuations penetration turns out to be greatest (Fig. 1.11). Otherwise (A0> |fmean|), and within the upper part of the section either a layer of seasonal thawing (fmean<0) underlain by the perennially frozen ground or a layer of seasonal freezing (fmean>0) underlain by unfrozen soils is developed. Presence of the seasonal freezing (or thawing) layer causes a reduction of thickness of the layer with annual temperature fluctuations (see Fig. 1.11). This is associated with the fact that the essential part of annual heat storage which could go into heating or cooling of ground occurring below the layer of seasonal freezing or thawing is expended within this layer for phase transitions of water into ice and vice versa. Therefore the temperature wave penetrates to a smaller depth. At the same time the greater the thickness of layer £tha or £fr, the less is the thickness of the layer with annual temperature fluctuations i/an. And finally, at fmean = 0 there is the case where the layer with annual temperature fluctuations is the same in thickness as the layer of seasonal freezing or thawing (£ = Han) (see Fig.

1.11), i.e. all the heat storage proceeds in the layer of freezing or thawing only, while the layers below are not subject to cooling or heating. We can observe the process of seasonal freezing as well as of seasonal thawing in this case depending on the state (perennially frozen or not frozen) of the underlying ground. At the same time it should be stressed that the thickness of the layer of seasonal freezing (thawing) at tmean = 0°C is proved to be maximal decreasing with increasing absolute values of tmean.

Thus, given the process of seasonal freezing or thawing, the depth of penetration of annual fluctuations of temperature will be equal to:

Within the layer ht occurring between the base of the seasonal freezing (or thawing) layer and the depth at which the annual temperature fluctuations are damped out (see Fig. 1.11), moisture phase transitions are practically absent in the ground and thus Fourier's laws discussed above (see Section 1.1) turn out to be true for this layer. With the formula for calculating the depth of penetration of temperature fluctuations (1.8) we can calculate the thickness of layer hv

Given a layer of seasonal freezing or seasonal thawing, the change of mean annual ground temperature with depth differs from that when the phase transitions are absent. By and large, when the basic factors affecting the formation of the ground temperature field under a periodically steady-state temperature regime are considered, the mean annual temperature change throughout the depth (tmean (z)) can be represented by a broken, linear relationship consisting of a number of straight portions. Let us consider this curve separately for the cases of seasonal freezing and seasonal thawing (Fig. 1.12), beginning with the mean annual air temperature t™an, which usually is measured by the weather stations at the height of 2 m.

Within the layer of air adjacent to the Earth's surface (of 0 to 2 m in height) a fall of mean annual temperature with distance away from the surface is observed, i.e. CeL > Cean where t^eran is the mean annual surface temperature of the cover. This is associated with the fact that it is the ground (day, or active) surface absorbing the solar radiation, which heats the air (and not the reverse). In the case when there is no cover on the ground surface, the mean annual air temperature turns out to be higher than the mean annual ground temperature t°ean by the value of a so-called radiative correction (AtR) which does not exceeding 1 °C usually. Given the various kinds of covers on the soil surface, the temperature change from tj^an to the mean annual ground temperature t°ean turns out to be more complex because the covers can exert warming and cooling effects.

Fig. 1.12. The mean annual temperature change with depth for seasonal ground thawing (a) and freezing (b): tair , isurf , t4 are the mean annual temperatures of

° v ' o v / mean' mean' mean air, at the surface and at the base of the seasonal thawing (freezing) layer; Clha and Cfr are the depths of seasonal thawing, freezing, respectively; Han, is the depth of annual temperature fluctuations; HBP is the permafrost thickness.

Fig. 1.12. The mean annual temperature change with depth for seasonal ground thawing (a) and freezing (b): tair , isurf , t4 are the mean annual temperatures of

° v ' o v / mean' mean' mean air, at the surface and at the base of the seasonal thawing (freezing) layer; Clha and Cfr are the depths of seasonal thawing, freezing, respectively; Han, is the depth of annual temperature fluctuations; HBP is the permafrost thickness.

The mean annual temperature is not constant in the layer of seasonal freezing (thawing) either. It decreases with depth from the first inflection of the temperature curve on the ground surface at z = 0 (see Fig. 1.12) The main reason for such a decrease of tmean in the layer is the difference between the values of heat conductivity of this layer £ when unfrozen Aunf and when frozen Afr, this being in its turn the reason for formulating the so-called temperature offset Atx in the layer, defined as the difference between the mean annual temperature at the base of the seasonal freezing (thawing) layer ¿mean- and *he mean annual temperature on the ground surface in conditions of a periodically steady-state temperature regime:

Of great importance in this case is the fact that there exists a heat balance at any depth of the annual temperature fluctuations, i.e. the ground receives as much heat for the half-period of heating as it loses for the half-period of cooling, otherwise the periodically steady-state regime could not exist. For example, if the ground received more heat during the period of heating than that being lost by it during the period of cooling, heating of the ground would proceed and the mean ground temperature would rise with time. The temperature offset is absent and t^ean, when there is a balance of

'incoming' and 'outgoing' heat in ground in which there are uniform thermal

Lyl k 'mean lm mean mean

physical characteristics (of heat conductivity Aun{ = Afr especially) in the unfrozen and in the frozen state. If Ath Afr such an offset exists. At the same time numerous calculations and simulation results have shown that the other thermal physical characteristics such as thermal capacity Cvol and heat of phase transition Qph do not effect the value of this offset. According to calculations and experimental data, Atx varies usually in the range 0.5 — 1.5°C.

The ground temperature below the layer of seasonal freezing (thawing) increases with depth on account of the geothermal gradient, resulting in the second inflection of the temperature curve at the base of the seasonal freezing or thawing layer i„ean. At the same time if f^ean < 0, there is perennially frozen ground in the profile, with the base of which one more inflection (the third) of the given temperature curve is associated (see Fig. 1.12a). At this point, unfrozen and frozen materials with different thermal physical properties (as a rule Afr > ^unf) are in contact on the lower boundary of the perennially frozen layer. However if this boundary does not move the thermal fluxes must be equal (gft = gunf) in frozen and unfrozen zones, i.e. 2.it grad ifr = /unf grad iunf. If Afr >\nf, grad iunf > grad ifr, i.e. the temperature increases with depth more rapidly in the thawed zone than in the frozen zone. As a result the temperature curve has a point of inflection at the interface. If f|,ean > 0, i.e. there is no perennially frozen ground in the profile, the temperature curve is straight below the layer of seasonal freezing (see Fig. 1.12b).

Thus though the solution of the heat conduction problem without considering the phase transitions formulated in Section 1.1 in the form of Fourier's laws (1.6) - (1.8), is of great importance for understanding the processes of thermal conduction in the ground generally, it is suitable only for a very restricted class of natural objects when rigorous quantitative description of these processes is required. Among such objects are rock masses, for which the processes of phase transitions of water can be neglected. Among them are the masses of monolithic or drained, faulted hard rocks, of completely drained loose rocks (which occur rarely), and unfrozen or frozen masses in which phase transitions are absent for the time of observation. It is impossible in the majority of cases to define quantitatively and adequately, temperature fields of rock masses and their change with time without regard for phase transitions.

The processes of water migration in soil are associated in a particular way with phase transitions over a range of negative temperatures, which, on the one hand, affect the character of the moisture field (ice content) formed in the ground, and on the other hand, exert the reverse effect on the process of

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