Solving geocryology problems based on generalized Fourier theory for temperature waves in half-space

Background. Currently, in geocryology, to predict seasonal changes in the state of frozen rocks and soils, Fourier formulas obtained earlier are widely used, modeling temperature fluctuations in the surface layer of the earth’s crust caused by annual fluctuations in its surface temperature. A significant drawback of this approach to modeling is manifested in the fact that in reality the state of the medium is characterized not only by the temperature field, but also by the moisture content field, which Fourier theory does not contain. Aim. It is required to generalize the Fourier problem known in mathematical physics on fluctuations of the temperature field in half-space by introducing into consideration, along with the temperature field, the moisture content field and taking into account the phenomena of evaporation and condensation associated with this field. Methods. Within the framework of A.V. Lykov’s theory, a spatially one-dimensional mathematical model of the processes of heat and moisture propagation in a homogeneous half-space, the boundary of which is in a state of heat and mass exchange with an airless medium, has been developed. By the method of complex amplitudes, formulas are obtained for time-asymptotic fluctuations in temperature and moisture content in a material filling a half-space, provided that the air temperature changes according to a harmonic law, and water vapor, both near the surface of the material and outside the boundary layer, is in a state close to saturation. Results. According to the results obtained, the temperature field is represented by a superposition of two damped harmonic waves, which have the same frequency, but different attenuation coefficients and phase velocities. The moisture retention field has the same structure. For a material with clay characteristics and with specific values of all the process-defining quantities for each of the waves, the depth of penetration and the delay time of vibrations at a given depth relative to fluctuations in air temperature are calculated, and the results obtained are compared with experimental data. Conclusion. The proposed solution and the following conclusions from it are the development of Fourier studies known in the literature on fluctuations in the temperature field in the surface layer of the Earth’s crust and are valid only in a situation when the material does not contain moisture, and according to the harmonic law, the temperature of the surface of the material does not change. The results of the work can be used in geocryology as a theoretical tool for modeling seasonal fluctuations in the thermal state of frozen rocks and soils.