## Ar dT d dT

I 0m(Pc)m + 0i(Pc)i + 0w(Pc)w + 0w PwL ~ I ~ = ~ I Keff ~ I. (39)

In this equation 0 is the total porosity, and the indices m, i, and w denote rock matrix, ice, and fluid, respectively, while Keff is the effective thermal conductivity of the matrix filled by water and/or ice (Keff = 0mKm+ 0wKw+ 0Ki). Because ice has a thermal conductivity 3-4 times higher than water, the thawing of the active layer in summer creates progressively thickening lower-conductivity layer between the ground surface and permafrost.

Phase change at temperature T* (for water, usually 0°C) is characterized by the latent heat L, and the moisture content is controlled by a function r(T)

0m = 1 - 4,0w = 4 x r, 0, = 4 - 0w ^ 4m + 4w + 4> = 1. (40)

The quantity r in this equation denotes the fraction of fluid in the pore space of the rock, while the latter constraint implies that the pore space is saturated. Analytical as well as approximate solutions using different functions r have been discussed in numerous works. Recent studies apply numerical approach to the solution of freezing/thawing equations and adopt a half-Gaussian function for r that secures smooth boundaries and facilitates the nonlinear convergence (Lunardini, 1991)

Parameter w is usually > 1K. As in other similar problems, surface boundary condition is T(0,i) = T0(i), and there are two boundary conditions at the phase boundary at the base of permafrost, namely the equivalence of temperatures and the heat flux between permafrost and underlying unfrozen soil. These boundary conditions ensure the continuity of the temperature at the phase boundary as well as a balance of heat conducted to the permafrost boundary, heat conducted away from it, and heat absorbed/released by the phase change.

Above equations belong to a class of so-called Stefan20 problems (moving boundary problems; Carslaw and Jaeger, 1962). Their solutions for permafrost were obtained for various basic field situations such as freezing of the half-space with initial temperature gradient, freezing/thawing of the subsurface layer from initial freezing temperature, etc., using approximate methods as well as more exact finite-difference/finite-element approaches. Trial calculations revealed significant deviations of temperature-depth profiles including phase change from that obtained by solution of Eq. (4), which does not take into account effects of emission or absorption of heat. The GST histories inferred from synthetic data have emphasized an importance of including latent heat effect (Mottaghy and Rath, 2006). Figure 70 compares the GST histories reconstructed from synthetic temperature-depth profile simulated using boxcar event scheme and freezing/thawing effects. As shown, inversion performed by means of the above-described

100000 10000 1000 100 TIME, year B.P.

Fig. 70. Inversion of synthetic temperature profile with and without latent heat. Inclusion of the freezing effects appears to be crucial in exact evaluation of the true GST history. (Redrawn from Mottaghy and Rath, 2006.)

100000 10000 1000 100 TIME, year B.P.

Fig. 70. Inversion of synthetic temperature profile with and without latent heat. Inclusion of the freezing effects appears to be crucial in exact evaluation of the true GST history. (Redrawn from Mottaghy and Rath, 2006.)

20A large class of problems containing a free or moving boundary is called "Stefan problems" after Austrian scientist Josef Stefan (1835-1893). The original Stefan problem has treated the formation of ice in the polar seas. Nowadays there is a growing interest in Stefan problems.

technique including latent heat effect provides almost perfect coincidence of the true and the reconstructed GST histories. Exclusion of the latent heat effect gives significantly slowed down GST increase with the glaciation maximum shifted to the present by —10000 years. It should be mentioned, however, that to enhance latent heat effect the authors have assigned rather high porosity to the medium. Effects of freezing will be reasonably smaller or even negligible for the consolidated rocks.

This and former numerical modeling experiments have shown that relatively small changes in surface climate can result in significant changes in permafrost temperatures and that permafrost is highly susceptible to the long-term warming (Lachenbruch and Marshall, 1986; Lachenbruch et al., 1988; Beltrami and Taylor, 1994, 1995; Majorowicz and Judge, 1994; Kukkonen and Safanda, 2001; Safanda et al., 2004). These studies have proved that temperature variations measured in this layer can be used as a powerful indicator of the long-term climate and/or surface energy balance variability.

Because heat transfer within thick permafrost occurs almost exclusively by conduction, it is affected primarily by the long-term temperature changes. There are two timescales that are important from the point of view of permafrost response to the surface temperature variations. The first one characterizes the time required for the adjustment of the underground temperatures to new surface conditions after sudden surface temperature change. This time can be estimated as At—D2/4k, where D is the permafrost thickness and k its thermal diffusivity (Lachenbruch et al., 1988). For terrestrial permafrost conditions this time can vary from a few years to thousands of years. Table 5 presents the depth of propagation of a temperature wave on frozen ground. As shown, typical moderate depth boreholes are able to record signal of the last glacial. The second timescale characterizes the time that the permafrost thickness needs to respond to the changes in surface temperature. In the case of the step increase of the surface temperature to higher constant value, and after initial response with the characteristic time of At, the thawing of the permafrost base can be expected. Its rate depends on the heat balance at the phase change boundary and on the ice content in permafrost. Calculations by Lachenbruch et al. (1988) and Ostercamp and Gosink (1991) have shown that the thawing rate is about 1-15 mm/year. Thus, again thousands of years may be required for adjustment of the permafrost thickness to the changed surface temperatures. Such a long characteristic time represents a significant portion of the time intervals between glacial periods of the last million years.

Kukkonen et al. (1998) have detected extremely low temperature gradients in a suite of 250-750m deep boreholes in eastern Karelia (Russia) ranging between 0.8 and 3.7K/km.

Table 5. Depth of propagation of a temperature wave in frozen ground (thermal diffusivity of the frozen strata is 1.3-1.8 X 10—6m2/s)

Time (years)

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