Generalization of the Fourier Problem on Fluctuations in the Temperature of the Earth’s Crust

The problem of asymptotic fluctuations of temperature and moisture content in a half-space is solved by the method of complex amplitudes. The material filling the half-space consists of a solid base (capillary-porous body) and water. The well-known Fourier solution for temperature fluctuations in half-space in the absence of moisture and under the boundary conditions of heat exchange of the first kind is generalized to the case of a wet material under the boundary conditions of Newton for temperature and Dalton for moisture content. The results of the work can be used in geocryology to model seasonal fluctuations in the thermophysical characteristics of frozen rocks and soils.


Introduction
The article deals with the classical problem of finding fluctuations in the temperature field in the upper layer of the earth's crust, provided that the temperature of the Earth's surface experiences diurnal or seasonal fluctuations. A partial solution of this problem, obtained by Fourier, can be found in the following work [1]. The Fourier formulas given in this work, as well as the Fourier laws based on them, are among the fundamental theoretical facts in permafrost science (geocryology) and are an important tool in solving problems of meteorology, climatology and environmental protection, as well as in the construction of buildings and the development of agriculture in the field of permafrost distribution [2,3]. A significant disadvantage of the Fourier formulas, which limit their application to the study of geocryology problems, is that they do not take into account the presence of moisture in the soil, its movement under the influence of temperature gradients and moisture content, and evaporation in the thickness of the material and from its surface. Applying the theory of heat and mass transfer of A.V. Lykov to the problem with periodically changing boundary conditions, we obtain formulas for the thermophysical state of the material, in which the movement of moisture and its transformations will be correctly taken into account.

Mathematical model of heat and mass transfer of a half-space with an air flow
Bearing in mind the problem of temperature and moisture content fluctuations in the surface layer of the earth's crust, let us consider the homogeneous half-space x>0 shown in Figure 1, the boundary of which x=0 is blown by air having a temperature T a and humidity φ outside the boundary layer. The material of the half-space will be considered to consist of a solid base (a capillary-porous body) and water. We also assume that the heat transfer intensity Q and the mass transfer intensity J of the surface x=0 with the air medium vary slightly along this surface, i.e. these values depend only on the time τ. In 2 the described situation, the distributions of temperature T and moisture content U will depend only on x and τ, i.e. the desired functions will be T(x, τ) and U(x, τ). The system of equations and boundary conditions for these functions will have the following form (1 -6): (1) (2) Formulas (1) and (2) represent the equations of heat and moisture propagation in the region occupied by the material; equations (3) and (4) define the boundary conditions at the boundary x=0; formulas (5) and (6) determine the intensities of heat and mass transfer at this boundary (heat transfer according to Newton's law and mass transfer according to Dalton's law). In the given relations: c, ρ, λ, γ, a m , δ -the thermophysical characteristics of the material, namely, the specific heat capacity, the density in the dry state, the coefficient of thermal conductivity, the evaporation criterion, the coefficient of moisture diffusion, the relative coefficient of thermal diffusion of moisture; a w =λ/(cρ)heat diffusion coefficient (coefficient of thermal conductivity); r -specific heat of water vaporization; a w and a m -coefficients of heat and mass transfer of the sample surface with the air medium; P(T)function of G. K. Filonenko, modeling the dependence of the relative partial pressure of saturated water vapor on its temperature T at general normal pressure; T 1 =238° C -constant.

Statement of the problem on the asymptotics of heat and mass transfer fields
We assume that at τ<0, the temperature of the material and its moisture content had constant values T 0 and U 0 over the entire half-space, the air temperature T a was equal to the material temperature T 0 , and the air humidity φ was equal to 1. We see that the system can be in this state indefinitely, because all the above equations are satisfied, and for the heat and mass transfer intensities we will have Q=0 and J=0. Let now, starting from the moment τ=0, the air temperature T a begins to make small fluctuations near the temperature T 0 . For small deviations of the surface temperature T(0, τ) and the air temperature T a from the fixed temperature T 0 , the dependence (6) can be linearized by decomposing the function P(T) into a Taylor series in the vicinity of the point T 0 . Having done this, and taking φ=1, instead of the formula (6) for the mass transfer intensity, we get an approximate formula for the mass transfer coefficient for the temperature difference: The representation of the function J(τ) in the form of (7) turns the system of equations we have introduced into a linear system.
Next, we will consider the case when small changes in air temperature occur according to the harmonic law: are the set values. We can offer a solution to the system (1)-(5), (7), (8) in the following form: where both the temperature and the moisture content of the material, as well as the air temperature, make small harmonic oscillations near the equilibrium values of T 0 and U 0 at each fixed x. Indeed, by calculating the difference using (9) and (8), and substituting it into formula (7), we get Thus, the function J(τ) turns out to be harmonic. Similarly, on the basis of the formula (5), the harmony of the function Q(τ) is proved. But after all, all the other terms in equations (1)-(4), in accordance with (9), will also be harmonic, and therefore you can try to satisfy these equations by properly selecting the dependencies in formulas (9): In making such a selection, we must take into account the obvious physical meaning of the condition at infinity: The formulated function selection problem (9) is one of the problems without initial data; its solution gives the asymptotics of the fields T and U at τ→∞.

Problem statement for harmonic field complexes
The functions of our problem, T(x, τ) and U(x, τ), are defined by formulas (9). We introduce instead the new desired functions, T*(x, τ) and U*(x, τ), and, turning to the method of complex amplitudes, we compare these new functions with their complexes     , according to the following rules: Obviously, the functions T* and U* will satisfy the same equations (1) and (2) as the functions T and U. Based on this, and using the rules for working with complexes, instead of equations (1) and (2) for T* and U*, we get the equations for their complexes U T   and : Here we have a system of two second-order ordinary differential equations with respect to two unknown functions. The system is linear, homogeneous, with constant coefficients, and indeterminate. To identify the only solution to the system, we turn to the remaining equations of the problem. In the domain of complexes, equations (3) and (4) will look like this: Forming complexes from both parts (10), and equating them, we get a given complex number Similarly, referring to the formula (5), for the heat transfer intensity complex we will have Substituting the obtained expressions in (14), after the transformations we obtain a system of two equations: Here is the effective heat transfer intensity. Having found the general solution of system (13), i.e., the general expressions for functions U T   and , we then, from system (15), which plays the role of boundary conditions, find the arbitrary constants included in these general expressions.

An asymptotic solution for a mathematical model of heat and mass transfer with δ=0 and γ=0
Here we will limit ourselves to calculations within the framework of one of the simplest mathematical models, in which we assume δ=0 (neglect thermal diffusion, i.e., the movement of moisture to the surface occurs only due to the difference in moisture content) and γ=0 (neglect internal vaporization, i.e., the transformation of water into steam occurs only on the surface). The conditions for the applicability of such a simplified model to the problems of the theory of heat and mass transfer and the analysis of the solutions obtained with the approximations made can be found in [7][8][9]. As can be seen from (1) and (2), the equations for temperature and moisture content in this case are independent (the relationship between the functions T and U is carried out in this model only through boundary conditions), which makes significant simplifications in the algorithm for studying the process. The equations for complexes (13) under the conditions of our problem can be represented as: Using the Euler method, we find general solutions to these equations that satisfy the condition at infinity (11). They will look like this: Now we must return from the complexes (images) to the original harmonic functions of time (originals). In the method of complex amplitudes, this stage corresponds to finding the inverse Fourier transform when solving physical problems by the spectral method. We will first find the original for the complex   x T  . We will first introduce a new designation. The constant µ 1 from formulas (17) is written in the form , where coefficient of attenuation of heat waves: The inverse value of Δ w =1/β w is called the depth of penetration of heat waves. These names are borrowed from the theory of electromagnetic waves. The validity of the use of such terms will be justified below. The transition to the original is based on the rule of formation of complexes and is carried out according to the following scheme: x The main difficulty here is the reduction to the exponential form of the complex constant C 1 , i.e., the calculation of the modulus of this constant 1 С and its argument arg C 1 . After performing such calculations, and adding, in accordance with (12), to the resulting harmonic function, the constant T 0 , we obtain the desired asymptotic solution for the temperature field: (21)