The depth of the thermal influence zone of the surface in the snow cover

Galkin Aleksandr Fyodorovich ORCID: 0000-0002-5924-876X Doctor of Technical Science Chief Researcher; P.I. Melnikov Permafrost Institute SB RAS 677010, Russia, Yakutsk, Permafrost str., 36, IMZ SB RAS. Cryolithozone Geothermy Laboratory afgalkin@yandex.ru Other publications by this author     ZHirkov Aleksandr Fedotovich PhD in Technical Science Head of the Laboratory; Cryolithozone Geothermy Laboratory; P.I.Melnikov Permafrost Institute SB RAS 677010, Russia, RS(I) region, Yakutsk, Permafrost str., 36 zhirkov_af@mail.ru Pankov Vladimir Yur'evich PhD in Geology and Mineralogy Associate Professor, Department of Construction of Roads and Airfields, North-Eastern Federal University 677027, Russia, respublika Sakha(Yakutiya), g. Yakutsk, ul. Belinskogo, 58 pankov1956@gmail.ru Other publications by this author

Plotnikov Nikolay Afanasievich ORCID: 0000-0001-6013-931X Postgraduate student; P.I.Melnikov Institute of Permafrost Science SB RAS 677010, Russia, Republic of Sakha(Yakutia), Yakutsk, Permafrost str., 36 plotnikov-nikolay96@mail.ru Other publications by this author

Introduction. Snow cover plays an important role in the formation of the thermal regime of the active rock layer of the heliothermozone. Artificial intelligence (the Yandex neural network), for example, describes the effect of snow cover on the temperature regime of the soil in this way (in a popular form). "The snow cover has a significant effect on the temperature of the frozen ground. It performs several functions that help maintain a more stable and higher ground temperature compared to open areas of the surface. 1. Thermal insulation. Snow is a good thermal insulator, which prevents the rapid cooling of the soil. It traps the heat radiated by the earth's surface and prevents its loss to the atmosphere. This is especially important in cold climates, where temperatures can drop to very low values. 2. Wind protection. Snow also protects the ground from the effects of wind, which can increase surface cooling. It creates a barrier that reduces wind speed and reduces its cooling effect. 3. Heat distribution. Snow evenly distributes heat over the ground surface, preventing the formation of local hot or cold spots. This contributes to a more uniform distribution of temperature in the soil and reduces the likelihood of cracks and deformations. 4. Moistening of the soil. Melting snow can lead to moistening of the soil, which can also affect its temperature. Water has a high heat capacity, which means that it can absorb and release a large amount of heat without significantly changing the temperature. This can lead to cooling or heating of the soil, depending on environmental conditions. It is important to note that the effect of snow cover on the temperature of frozen ground may depend on many factors, such as the thickness of the snow cover, its density, soil moisture and climatic conditions. However, in general, it can be said that snow has a positive effect on the temperature of frozen ground, helping to maintain more stable conditions and preventing rapid cooling of the surface."

Much attention is paid to the study of the processes of formation and properties of snow cover in the scientific and engineering community ^[1-6]. An important direction in glaciology is the study of the thermophysical properties of snow and their variability in the snow cover of varying degrees of metomorphism, which have been actively studied since the 19th century ^[7-9]. One of the indicators of thermal efficiency is the index of attenuation (attenuation) of air temperature fluctuations in the snow cover, since it determines the degree of influence of the snow cover on the temperature regime of the underlying soil ^[10-15]. If we turn to the neural network again, we get the following conclusion of artificial intelligence, which, unlike the above, is already more similar to a scientific presentation. "Attenuation of temperature fluctuations in the snow cover is the process of a gradual decrease in the amplitude of temperature fluctuations within the snow mass over time. This process is due to several factors. 1. Thermal conductivity of snow. Snow has low thermal conductivity, which means that it does not conduct heat well. This leads to the fact that temperature fluctuations inside the snow cover fade faster than in other materials. 2.Insulation properties. Snow is a good insulator, which means that it prevents heat from entering from the outside and heat from escaping from the inside. This also contributes to the attenuation of temperature fluctuations. 3. Heat diffusion. The heat inside the snow cover can spread due to diffusion, that is, the movement of molecules. This process also helps to dampen temperature fluctuations. 4. Heat conduction. Conduction is the transfer of heat through direct contact of molecules. In the snow cover, this process occurs slowly, which also contributes to the attenuation of temperature fluctuations. 5. The heat capacity of snow. Snow has a high heat capacity, that is, a large amount of energy is required to heat or cool the snow mass. It also slows down the process of temperature changes inside the snow cover. Thus, the attenuation of temperature fluctuations in the snow cover is a complex process that depends on many factors. Studying this process is important for understanding climate change and its impact on the environment."

One of the particular problems, the solution of which we did not find in an analytical review of literary sources, is to determine the depth at which the amplitude of temperature fluctuations in the snow cover can be considered equal to zero.

The purpose of the work is to determine the depth of the zone of thermal influence of the surface in the snow cover. That is, the determination of the attenuation zone of temperature fluctuations (daily, decadal) in the snow cover when the temperature of the atmospheric air changes.

Method.

The depth of the zone of thermal influence of the surface in a solid can be determined by the formula of T. Goodman ^[16]

H = sqrt(12aT) (1)

H is the depth of the zone of thermal influence, m; a is the thermal conductivity of the material, m2/s; T is the time, s.

The depth of the zone of thermal influence in the translated literature is also sometimes called the "depth of penetration".

The coefficient of thermal conductivity of the material (in this case snow) is determined by a well-known formula derived from the Fourier thermal conductivity equation

a=λ/Cρ (2)

Here and further it is assumed that: C is the specific heat capacity of snow, J/kgK; λ is the coefficient of thermal conductivity of snow, W/ mK; p is the density of snow, kg/m3.

The coefficient of thermal conductivity of snow (coefficient of thermal diffusion) determines the rate of passage and attenuation of temperature fluctuations with the depth of the snow cover ^[10].

With a known thickness of snow cover (H), it is easy to determine from formula (1) the time during which the complete attenuation of temperature fluctuations occurs

T=H^2/12a=0,083H^2/a (3)

To determine the value of the coefficient of thermal diffusion (thermal conductivity), it is necessary to determine the values of the coefficient of thermal conductivity and specific heat capacity included in formula (2). The study of the coefficient of thermal conductivity of snow and its variability from various parameters has received great attention in the scientific community. The generalized opinion of which is the postulate that it is impossible to establish a single law for determining this coefficient, due to the unlimited (huge) number of parameters on which it depends (geographical location, climate, time of year, air temperature, humidity, wind speed, etc.). Some generally recognized universal parameter by which the coefficient of thermal conductivity can be determined snow, without resorting to direct measurements, is the density of snow. There are a large number of formulas for determining the value of the thermal conductivity coefficient depending on the density of the snow cover ^{[10,17,18,19]}. In ^[17], an attempt was made to generalize these formulas and obtain a kind of universal dependence, which, according to research ^[20], can be written as the following parabolic function

λ=(9-4j +3j^2)/100. j=ρ/100 (4)

The disadvantage of this formula is that it is valid, according to the authors, in a limited range of variation of the parameter "j" (2.0 ≤ j ≤ 4.0). Therefore, to find the thermal conductivity coefficient as a function of snow density, we use the classical Abels formula ^[8,9], which is more universal. The dependence between the coefficient of thermal conductivity and the density of snow, obtained by G.F. Abels as a result of processing experimental research data, can be written as follows:

G.F. Abels also obtained the dependence of the coefficient of thermal diffusion (thermal conductivity) of snow on density, which has a linear form and, taking into account the designations we have adopted, can be written as

a=(0,13)10^-6 (6)

The specific heat capacity of snow is also a function of snow density and can be approximately determined by the formulas

C= Ca +Ki(Ca-Ci) (7)

Here, the index "i" refers to ice, and the index "a" refers to air; the parameter "" is the concentration of ice in the snow cover, which is equal to the ratio "". Parameters: – this is the mass of ice and air per unit volume, kg.

Considering that the mass of ice in the volume is significantly greater than the mass of air( ), we can assume that the parameter "". In this case, as is customary in glaciology, the specific heat capacity of snow is equal to the specific heat capacity of ice ^{[7, 8, 10]}. Naturally, such a statement cannot relate to the period of snow thawing. In this case, a term should be added to equation (7) that takes into account the heat capacity of water, the concentration of which during this period can reach 22-25% by volume ^[7]. In our case, just like G.F. Abels, we will assume that the specific heat capacity of snow is equal to the specific heat capacity of ice - 2.1 kJ/kgK.

The formulas obtained do not take into account the dependence of snow density on the thickness of the snow cover. In ^[21], a comparison of various calculation formulas for determining density as a function of thickness was carried out and it was concluded that all the formulas considered give approximately the same results and any of them can be used in practical calculations. Let's use the classical Abe formula, which we write down in the following form

j≈1,8+3,2H (8)

The average value of the snow cover density of a given thickness is defined as the average integral density value, i.e.

j≈1,8+1,6H (9)

Using the obtained dependencies (4), (5), (6) and (9), from (2) we obtain formulas for determining the quantitative values of the thermal conductivity coefficient of snow cover of various thicknesses. If we are interested in the change in the coefficient of thermal conductivity over the depth of the snow cover, then instead of formula (9), formula (8) should be used in (2).

Formula (4) is not very convenient in calculations. It is advisable to present it in the form λ =, where p is the coefficient of approximation of the dependence (4). The performed estimated calculations show that with sufficient accuracy for engineering calculations, this coefficient can be assumed to be equal to 0.025. It is easy to show that the maximum discrepancy between the results of calculations of the coefficient of thermal conductivity of snow according to the Abels formula and the approximating Osokin formula does not exceed 14%, and, on average, fits into the range [-10.0 ≤ (e,%) ≤ +10.0]. That is, to achieve our goal, you can use any of them. A detailed analysis of the degree of disagreement between the results of calculating the coefficient of thermal conductivity of snow according to the formulas of Osokin and Abels is given in ^[19]. The general formula for calculating the coefficient of thermal conductivity of snow cover will be as follows

(10)

Where the parameter "j" is determined by formula (8). Comparing formulas (10) and formula (6), we see that with a constant value of snow cover density, the discrepancy between the results does not exceed 8%, which is quite acceptable in engineering practice.

The degree of discrepancy between the results (relative percentage error) of the calculation of the thermal conductivity coefficient using formulas (4) and (5) can be determined using the well-known dependence ^[22]:

e=100(1-a1/a2) (11)

Taking into account the simplifications and additions made, the calculation formula for determining the time of complete attenuation of the amplitude of temperature fluctuations in the snow cover at a depth of "H" will look like

(12)

If you use the time in hours (which is more convenient) instead of seconds as an estimate, then the calculation formula will look like

(13)

It is of interest to determine the degree of discrepancy between the results of calculating the attenuation time of temperature fluctuations at a given depth of snow cover without taking into account changes in density in depth. In this case, the calculation formula will look like

(14)

Here, the parameter "j" is determined by the formula (5). From this dependence, it is easy to determine the depth of the zone of thermal influence of the surface in the snow cover (the depth at which the temperature gradient is zero)

(15)

It is known that a change in the density of snow cover (snow reclamation) plays an essential role in the formation of the thermal regime of soils, since during compaction the thermal resistance decreases in proportion to the square of the compaction coefficient. (The compaction coefficient is a value numerically equal to the ratio of the depths of the snow cover before and after compaction). The property of snow cover to significantly change its thermal resistance during compaction is used to control the thermal and humidity conditions of the active soil layer ^[2-6]. In particular, it is the basis for new ways to combat the negative manifestations of cryogenesis and restore agricultural lands disturbed by cryogenesis in the permafrost zone ^[23-25].

The degree of change in the time to reach zero amplitude at depth "H" during the compaction of the snow cover can be expressed by the formula

(16)

Here k is the compaction coefficient equal to "N 1/N 2"; N 1, N 2 is the depth of the snow cover before and after compaction, m.

If we use formula (14), then

(17)

This expression means that the duration of the period (time) of reaching zero temperature amplitude at a given depth, when compacting the snow cover, is inversely proportional to the compaction coefficient. This pattern allows you to quickly, without resorting to calculations, determine, for example, the necessary degree of compaction of the snow cover, excluding the influence of fluctuations in air temperature on the thermal regime of the soil. In addition, this pattern is of purely scientific interest, since it was obtained for the first time and was not previously found in the literature known to the authors.

Results and discussion. According to the obtained formulas, multivariate calculations were carried out, the results of which, for clarity, are presented in the form of 2D and 3D graphs. Figure 1 shows a graph of the dependence of the time of total attenuation of the temperature amplitude at depth depending on the coefficient of thermal diffusion (coefficient of thermal conductivity) of snow.

Fig.1. Duration of the attenuation period of the temperature amplitude according to the depth of the snow cover, depending on the value of the coefficient of thermal conductivity of snow

It follows from the graph that the lower the thermal conductivity of snow, the longer it takes to reach a depth at which the temperature gradient is zero. Figure 2 shows graphs of the change in the duration of the attenuation period of the temperature amplitude in depth at different snow cover densities, determined by formula (14) and formula (13), which takes into account the change in density in depth.:

Fig.2. The duration of the attenuation period of the temperature amplitude in depth at different snow cover densities, determined by the formula (14):

1- at j= 2; 2- at j=3; 3- at j= 4; 4- at j= 5; 5 - when calculated using the formula (13).

The analysis of the graphs in the figure shows that, without taking into account the change in the density of the snow cover in depth, we can make a significant mistake. Especially for freshly fallen snow and snow after initial metamorphism (j ≤ 2.0). In addition, the discrepancy between the results of the two formulas increases with increasing duration of the considered period of temperature fluctuations. At the same time, for dense, formed snow cover (j ≥ 3.0), the results of calculations using two formulas, especially for periods of daily fluctuations, are in good agreement.

For clarity, Figure 3 shows a 3D graph in the form of two characteristic planes reflecting the dependence of the duration of the attenuation period of the temperature amplitude in depth at different snow cover densities, determined by two formulas.

Fig.3. Duration of the attenuation period of the temperature amplitude in depth at different snow cover densities:

1 - when calculating by formula (13); 2 - when calculating by formula (14).

As can be seen from the graphs, the degree of disagreement between the results decreases significantly with increasing snow density. Of particular interest is the study of the degree of attenuation of daily fluctuations in outdoor air. Figure 4 shows graphs characterizing the change in the depth of the zone of thermal influence depending on the density of snow cover with daily fluctuations in outdoor air. The area of the most characteristic snow cover densities is highlighted in blue. As follows from the graphs, when the density is doubled, for example from 1 to 2, the depth of the zone of thermal influence increases approximately 1.41 times during daily fluctuations with a period of 12 hours. With daily fluctuations with a period duration of 6 hours, this figure does not change, that is, it also increases by 1.41 times. Just as this ratio does not change with an increase in snow density by the same two times, but in the range from 2.5 to 5.0. The following conclusion can be drawn: the degree of change in the depth of the zone of thermal influence does not depend on the range of changes in snow cover density and the duration of the period of daily fluctuations, provided the same (constant) multiplicity of density changes snow.

Fig.4. The depth of the zone of thermal influence depending on the density of snow cover with daily fluctuations in surface temperature:

1-the duration of the period is 6 hours; 2-the duration of the period is 12 hours.

It is interesting to note that the data obtained coincide well with the results of the studies given in the "Glaciological Dictionary" [Glaciological Dictionary, p.455], where the following is noted. "In dry freshly fallen snow, daily temperature fluctuations fade at a depth of 30-40 cm, in older snow at a density of 300 kg/ m3 – at a depth of 50 cm." As can be seen from the graphs, the estimated depth of the zone of daily fluctuations (highlighted in blue) for the characteristic range of changes in snow density does not exceed 43 cm. This confirms the possibility of using the obtained formulas to estimate the depth of the zone of thermal influence in snow cover of various densities.

Figure 5 shows a three-dimensional graph of the dependence of the depth of the zone of thermal influence of daily temperature fluctuations with different duration of the oscillation period for snow cover with a reduced density from 0.5 (freshly fallen, loose snow) to 6.0 (packed, firm snow).

Fig.6. The depth of the zone of thermal influence depending on the density of snow cover for different duration of the period of daily temperature fluctuations on the surface.

The color differentiation of the plane in the figure confirms the quantitative patterns formulated earlier. At the same time, it also becomes obvious that with an increase in snow density, regardless of the duration of the period of daily fluctuations, the depth of the zone of thermal influence increases. And the degree of this increase is stronger than the degree of increase in the depth of the zone of thermal influence from the duration of the period of daily temperature fluctuations.

Further research should be aimed at studying such an important indicator of the thermal efficiency of snow cover as thermal stability. We have not found any dependencies for determining this indicator in the analysis of literary sources.

Conclusion.

Formulas have been obtained for determining the duration of the attenuation period of surface temperature fluctuations at a certain depth of snow cover, depending on the density of snow. Two calculation formulas are compared: one that takes into account and one that does not take into account the change in density by depth of snow cover. The comparison showed that for engineering calculations, a simpler formula can be used with a constant value of snow density equal to the average integral value of density in depth. To assess the effect of snow reclamation, a formula is proposed that allows us to determine the degree of change in the duration of the period of complete attenuation of temperature in depth during compaction of snow cover, depending on the compaction coefficient. For the first time, a pattern has been established: the duration of the period (time) of reaching zero temperature amplitude at a given depth, when compacting snow cover, is inversely proportional to the compaction coefficient. A dependence has been obtained linking the depth of the zone of thermal influence with the duration of the period of daily temperature fluctuations on the surface of the snow cover and its density. It is shown that with increasing snow density, regardless of the duration of the period of daily fluctuations, the depth of the zone of thermal influence in the snow cover increases. So, for example, for an average snow density of 200 kg/^m3, the depth of complete attenuation of daily fluctuations is 33 cm. And, with a snow density twice as high (400 kg/^m3) – 48 cm. Comparison of the calculated data according to the obtained formulas with the data on the depth of attenuation of daily temperature fluctuations in snow cover with different snow densities, given in the literature, showed good convergence of the results. For example, according to experimental data given in the Glaciological Dictionary, the depth of complete attenuation of daily temperature fluctuations at a snow density of 300 kg/^m3 is no more than 30-40 cm. The calculation based on the obtained formulas gives a value equal to 43 cm. This allows us to recommend the obtained formulas for practical use in assessing the process of formation of the thermal regime of snow cover.