- 1Heat Transfer Laboratory, Research Centre of Physical and Thermal Engineering, Joint Institute for High Temperatures, Moscow, Russia
- 2Microhydrodynamic Technologies Laboratory, X-BIO Institute, University of Tyumen, Tyumen, Russia
- 3Department of Chemical Engineering, Biotechnology and Materials, Engineering Science Faculty, Ariel University, Ariel, Israel
Solar radiative heating and melting of lake and sea ice is a geophysical problem that has attracted the attention of researchers for many years. This problem is important in connection with the current global change of the climate. Physical and computational models of the process are suggested in the paper. Analytical solutions for the transfer of solar radiation in light-scattering snow cover and ice are combined with numerical calculations of heat transfer in a multilayer system. The thermal boundary conditions take into account convective heat losses to the ambient air and radiative cooling in the mid-infrared window of transparency of the cloudless atmosphere. The study begins with an anomalous spring melting of ice on the large high-mountain lakes of Tibet. It was found that a thick ice layer not covered with snow starts to melt at the ice-water interface due to volumetric solar heating of ice. The results of the calculations are in good agreement with the field observations. The computational analysis showed a dramatic change in the process when the ice is covered with snow. A qualitative change in the physical picture of the process occurs when the snow cover thickness increases to 20–30 cm. In this case, the snow melting precedes ice melting and water ponds are formed on the ice surface. This is typical for the Arctic Sea in polar summer. Known experimental data are used to estimate the melting of sea ice under the melt pond. Positive or negative feedback related to the specific optical and thermal properties of snow, ice, and water are discussed.
Introduction
This paper presents a general approach to solving geophysical problems related to solar heating and the melting of ice on the water surface. The spring melting of lake ice is a relatively simple problem and this will be considered in more detail. After that, a general view of the more complex problem of melting of sea ice during the polar summer will be discussed. Ice melting on the Arctic Sea surface is really important for global climate change. What is less well known is that the opening from the ice of high-mountain lakes in Tibet, sometimes referred to as the Earth’s third pole, significantly affects the climate of not only central Asia. Recall that the Tibetan Plateau is home to about a thousand large lakes with a total area of about 15,000 km2 (Su et al., 2020; Zhang and Duan, 2021). Interestingly, the early opening time of these lakes is an indicator of global warming (Su et al., 2019; Zhang and Duan, 2021).
It should be noted that the physical picture of heating and melting of the ice on the lake surface turns out to be quite different for the case when the ice is not covered with snow and in the case when a snow cover is present. However, the approach to solving problems of radiative transfer in snow and ice layers has a common methodological basis, and this can be used in theoretical studies of both lake and sea ice melting. In the spectral range of semitransparency of snow and ice, one should take into account the scattering of radiation either by ice grains in snow or by gas bubbles, which are usually contained in the ice. This means that one should focus on choosing a simple and sufficiently accurate method for solving the radiative transfer equation (RTE) (Dombrovsky and Baillis, 2010; Howell et al., 2021; Modest and Mazumder, 2021).
The main difficulties in radiative transfer modeling are caused by the RTE’s integral term, which takes into account the anisotropic scattering of radiation by particles or bubbles in the medium and contains a scattering phase function. Fortunately, we deal with multiple scattering of radiation in an optically thick medium when the transport approximation is quite sufficient. According to this approach, the scattering function is replaced by a sum of the isotropic and forward components. The resulting transport RTE looks like that for the hypothetic isotropic scattering but with the transport scattering coefficient (Dombrovsky, 2012; Dombrovsky, 2019). It should be recalled that the transport approximation has been successfully employed in diverse problems of radiative transfer in thermal engineering (Dombrovsky et al., 2017; Dombrovsky et al., 2020), biomedicine (Tuchin, 2007; Sandell and Zhu, 2011; Dombrovsky et al., 2012; Jacques, 2013; Eisel et al., 2018; Dombrovsky, 2022), and geophysics (Dombrovsky and Kokhanovsky, 2022; Dombrovsky and Kokhanovsky, 2023; Dombrovsky et al., 2019; Dombrovsky et al., 2022).
The linearity of the RTE makes possible another significant simplification of the problem. In the case of direct solar irradiation of a scattering medium, the radiation intensity can be represented in the form of two additive components: direct radiation and a diffuse component formed due to the scattering. In highly scattering media, the diffuse component of radiation intensity can be calculated using one of the simplest differential approximations: either the known P1 approximation of the spherical harmonics method or the two-flux approximation. These approximations make it sufficient to solve a boundary-value problem for a second-order ordinary differential equation instead of the RTE. The choice between these approximations is determined by the problem statement. Comparison with exact numerical solutions (Dombrovsky and Baillis, 2010) has shown that the two-flux method is preferable when solving one-dimensional problems typical of solar heating. In this case, the error of P1 is larger because this method does not take into account the discontinuity in the angular dependence of the radiation intensity on the illuminated surface of the medium.
Transient heat transfer model
Generally speaking, there are various thermal processes in a snowpack or scattering ice sheet and these processes should be involved in a complete heat transfer model. As an example, one can recall the ice sublimation and diffusion of water vapor through a snow layer. This may be important for the snow microstructure but the related effects are insignificant in our problem.
The 1D transient energy equation for temperature
where
It is usually difficult to choose a realistic initial profile of temperature,
Solution for ice-covered lake
In the case of ice not covered by snow, sunlight penetrates through the ice layer and leads to water heating. In limnology, the transfer of solar radiation in ice and water is described using the exponential Bouguer law, which is applicable only in the case of direct incident radiation and single scattering of light in the medium. This methodological drawback remained even in recent papers (Kirillin et al., 2012; Leppäranta, 2015; Kirillin et al., 2021). The error of this simplified model for the radiative transfer was partially compensated for by the selection of an extinction coefficient, which provided a satisfactory agreement between the calculations and the field measurements. This semiempirical approach to radiative transfer is incorrect. The multiple scattering of short-wave solar radiation by microcracks and gas bubbles leads to stronger solar heating of ice. In addition, the radiation transmitted through the ice layer contains both direct and diffuse components.
According to (Hale and Querry, 1973), the absorption index of water increases with the wavelength from
The influence of uncertainty in the experimental values of the ice absorption index on the computational data for solar heating of snow and ice has been analyzed by Dombrovsky and Kokhanovsky (2022), Dombrovsky and Kokhanovsky (2023). It was shown that the discrepancy between the data of (Warren and Brandt, 2008; Picard et al., 2016) in the
The gas bubbles in ice are assumed to be spherical and the radii of these bubbles,
where
According to (Kirillin et al., 2012), the gas bubbles are usually not uniformly distributed in the ice layer: there are more of them in the lower part of the layer. This can be taken into account in the calculations. However, the simplest model including only two bubble parameters is chosen in the present study: the average values of volume fraction and radius of bubbles. According to (Dombrovsky and Kokhanovsky, 2020b), the scattering parameter
where
Following the recent study by Dombrovsky and Kokhanovsky (2023), the present paper is focused primarily on modeling the thermal regime of mountain lakes like those of the Qinghai–Tibet Plateau, located at a height of more than 4 km above sea level, for which the problem is somewhat easier: it can be assumed that the sky is clear and only a small part of the solar light is scattered in the atmosphere. As a result, it is sufficient to consider the transfer of direct solar radiation, the intensity and spectral composition of which change with the zenith angle of the Sun. Minor atmospheric precipitation and strong winds (Wang et al., 2022) result in the absence of snow cover on the ice surface. This leads to a significant solar heating of the ice and water in the lake.
The problem can be simplified because the refractive indices of water and ice differ only slightly in the wavelength range of
It is assumed that optical properties of ice do not change along the horizontal ice surface and the radiative transfer along the surface of an ice-covered lake may not be considered. The 1D model for the propagation of obliquely incident radiation in an ice layer containing gas bubbles has been described recently by Dombrovsky and Kokhanovsky (2022). It is assumed that there are no any bubbles in water, and scattering by a small amount of plankton is insignificant. In this case, the problem of radiative transfer in the ice layer can be solved independently of the propagation of light in water. It is convenient to write the transport RTE and the boundary conditions at an oblique illumination of the ice layer with thickness
where
The intensity of radiation and the spectral irradiation are presented as follows:
The resulting equations for the diffuse component
The two-flux method gives the following boundary-value problem for the irradiation
where
The calculations made after publication of paper by Dombrovsky and Kokhanovsky (2023) showed that the reflection of sunlight from the illuminated ice surface is negligible and can be disregarded without any significant loss of calculation accuracy. At the same time, the refraction of light at an oblique illumination of ice should be taken into account and
The relation between
The radiative transfer problem in water is as follows:
where
According to Eq. 10, the intensity of light in water at
The solar radiation power absorbed in ice and water can be calculated as follows:
The functions in the right-hand side of the second of these equations are defined as:
Obviously,
Profiles of absorbed radiation power in ice and water
The spectral radiative flux at the surface of Ngoring Lake at the end of March (the time of the beginning of ice melting) at different values of the Sun’s zenith angle has been calculated in (Dombrovsky and Kokhanovsky, 2023). The results obtained are used below in calculating profiles of absorbed radiation power in ice and water. Typical profiles of this radiation power at the zenith angle 60o for two values of the scattering parameter and different thicknesses of the ice layer are shown in Figure 1. A significant attenuation of light in the ice layer draws attention. Increasing the scattering parameter leads to an increase in the absorption of solar radiation in the upper part of the ice layer and a significant decrease in the absorption in the lake water. When ice melts and the thickness of the ice becomes smaller, the radiation absorption in ice decreases considerably and the absorption in water increases very strongly. The latter is the physical cause of an observation by Lazhu et al. (2021) for several lakes in the Tibetan Plateau where the water temperature at some distance from the ice-water interface increased rapidly during ice melting.
Temperature stratification of water in a deep lake
The temperature field in the lake water is largely determined by the non-monotonic dependence of the water density on temperature with a maximum density
Field observations of Ngoring Lake and some other lakes in the Qinghai–Tibet Plateau have shown an interesting thermal regime, named “anomalous winter” by Kirillin et al. (2021), when at a depth of 1.5–3 m under the ice layer the water is heated by solar radiation to a temperature
Averaging thermal boundary conditions for thick ice layers
The heat transfer on the illuminated ice surface changes significantly during the day but this does not affect the temperature of ice at some distance from the surface because of the large heat capacity of the ice layer. A simple estimate based on the Fourier criterion confirms that a thermal relaxation time for the 0.5 m thick ice is about 10 days. This allows using a steady-state model with constant heat transfer parameters, varying according to weather changes from week to week.
The ice melting at Ngoring Lake in March is very slow, and only in April does ice melting accelerate, completed by April 16 (Kirillin et al., 2021). This result is clear from the observations by Zhou et al. (2022): the air temperature in March is almost constant and does not exceed
The boundary-value problem for the quasi-steady temperature profile in the ice layer is as follows:
where
Here
The radiative cooling of ice compensates the radiative flux to the ice surface in the opacity range and the infrared solar heating cannot lead to the surface ice melting.
The condition for the beginning of ice melting can be obtained using the analytical solution to the problem (13a)–(13b):
The ice melting on its lower surface begins when
The calculated temperature profiles are shown in Figure 2. As one might expect, scattering plays a significant role in the case of a thick ice layer and the assumption of uniform distribution of gas bubbles in the ice is acceptable only for ice layers less than 0.5 m thick. One can also determine the temperature of the illuminated ice surface, at which melting begins at the ice-water interface:
Figure 2. Temperature profiles in ice layers of different thicknesses at the onset of melting on the ice-water interface.
Typical dependences of
Effect of a snow layer on lake ice melting
In the absence of snow cover on the lake ice surface, solar radiation penetrates through the semitransparent ice and already in the beginning of spring significantly heats up the water under the ice. Interestingly, the lake ice, even with a thickness of about 1 m, begins to melt from the lower surface. Calculations have shown that this is not due to heating of the ice by the warmer water, but almost exclusively due to solar heating of the ice. This is so because the upper layer of ice is continuously cooled by the colder air, as well as by the thermal radiation in the middle-infrared window of transparency of the cloudless atmosphere. This cooling is not compensated by the daytime heating of the ice surface by the infrared radiation of the Sun but does not prevent heating of the lower part of the ice layer and the beginning of spring ice melting.
The discussed peculiarity of the thermal regime of ice on the lake surface and its melting under the action of spring solar heating changes radically in the presence of even a thin layer of snow on the ice surface. The effect of snow is due to two main factors: firstly, snow significantly reduces the solar radiative flux on the ice surface due to strong scattering of radiation and, secondly, the thermal conductivity of snow is so small that even a thin layer of snow protects the ice surface from convective and radiative cooling. The physical model of snow’s effect on ice melting should include not only the transfer of solar radiation in the snow and ice layer but also the heat conduction process. If we do not take into account changes in the structure and optical properties of snow during its heating (before the melting), the above physical problems can be solved sequentially: first to calculate radiative transfer, and then to solve the heat transfer problem taking into account the absorption of solar radiation both on the snow surface and in the volume of snow and ice. Of course, in the thermal part of the computational model, it is necessary to consider the convective heat transfer with the surrounding air, as well as the mid-infrared radiative cooling of the snow surface.
Let us first consider the propagation of direct solar radiation through the snow layer. In order not to complicate the solution, we will not take into account the scattered radiation from a cloudless sky. Nor will we take into account that there is ice under the snow, which also scatters sunlight. The ice with air bubbles scatters light much more weakly than snow, and only a small fraction of the light transmitted through the snow is scattered by the ice in the direction of the snow. In the case of a very thin snow layer, the scattering of radiation in the ice layer has some influence on the radiative transfer in the snow layer, but this does not affect the main results of the calculations presented below.
As usual, when calculating radiative transfer in a medium with multiple scattering, the transport approximation is used. In addition, the radiation intensity is represented in the form of two additive components: direct radiation and a diffuse component formed due to the scattering of radiation in the medium. The irradiation is also additive:
Here
This problem is a little simpler than that for the ice layer since the sunlight is not reflected and refracted on the snow surface. For a snow layer with constant values of
In the limit of
The optical properties of snow were calculated in the same way as in (Dombrovsky et al., 2019). The analytical solution for single spherical ice grains obtained by Kokhanovsky and Zege (1995) in the geometrical optics approximation was used. As usual, it was assumed that the ice grains have a mean radius
The real parameters of the problem may differ from the accepted values. However, the effect of this choice on the subsequent physical analysis is expected to be insignificant. The possible contamination of snow by particles suspended in the atmosphere is not considered. This effect was considered, for example, by He et al. (2018) and Dombrovsky and Kokhanovsky (2020a).
The effect of snow layer thickness on the directional-hemispherical transmittance of snow,
Consider now the heat transfer problem for solar heating and possible snow melting on the ice surface, taking into account the conductive heating of the ice layer. The temperature of the lower surface of the ice (at the ice-water interface) is 0°C. This boundary condition allows one not to consider the heat transfer under the ice, which is necessary when the ice layer is located, for example, on bare ground. At the same time, when analyzing the thermal state of the snow layer illuminated by the Sun, it is necessary to take into account the heat conducted away from the snow into the ice layer. In other words, the energy equation should be solved in the computational domain including both the snow cover and the ice layer:
where
The initial temperature profile was assumed to be as follows:
When modeling the effect of the snow layer on the opening of the lake from ice, it would be incorrect to consider the solar heating conditions typical of the high-mountain lakes of northeastern Tibet, when, due to low precipitation and constantly strong winds, the ice on these lakes is not covered by snow. In addition, at low altitudes, unlike in high mountains, not only direct solar radiation but also diffuse radiation from the light-scattering atmosphere should be taken into account. In this problem, we use the same data as in the papers by Dombrovsky et al. (2019) and Dombrovsky and Kokhanovsky (2022) for the summer solstice during the Arctic polar summer at latitude 70o.
The motion of the Sun across the sky during the day was calculated in the same way as in (Dombrovsky et al., 2019). At the same time, taking into account the diffuse atmospheric radiation requires additional relations, which are obtained by solving the following boundary-value problem for spectral irradiation:
The analytical solution to this problem is given by:
In the limit of
To exclude the influence of the initial temperature profile, we consider the calculated temperature profiles on day 14 from
Figure 5. Temperature profiles in the snow and ice layers underneath at different times of the day: (A)–at noon and midnight, (B)–in the afternoon.
The above computational study gives the following general picture of solar heating of the ice layer covered by snow. When the upper surface of ice on a lake is illuminated by the spring Sun, the thick ice layer begins to melt even in very cold air and melting occurs from the lower surface of the ice layer. But if there is a thin layer of snow on the surface of the ice (even less than 1 cm thick) the snow does not allow much of the sunlight to penetrate into ice and makes melting of the ice surface underneath impossible. At the same time, the snow itself on the ice surface does not heat up because it scatters a significant part of the visible solar radiation, and the absorbed solar heat is almost immediately transferred to the relatively thick layer of ice. Snow melting can only begin when the snow layer is thicker than
An estimate of the effect of melt pond on melting of ice
Meltwater, formed initially by snowmelt, does not cover the ice surface uniformly. This is evidenced by numerous observations both for ice-covered lakes and for the much better-studied Arctic Sea ice. Note that the abnormally strong melting of sea ice in the Arctic during the polar summer is an important process that has been intensively analyzed over the last decade. One should name a number of studies that are specifically devoted to the formation and evolution of melt ponds on the Arctic Sea ice. In chronological order, these are experimental and analytical works by Polashenski et al. (2012), Hudson et al. (2013), Schröder et al. (2014), Popović et al. (2018), Malinka et al. (2018), Ma et al. (2019), König and Oppelt (2020), Perovich et al. (2021), Sterlin et al. (2021), Lei et al. (2022), and Rosenburg et al. (2023). Photographs in the literature show numerous melt ponds and the change in the ice surface area occupied by these ponds in the polar summer. Note that similar melt ponds are also observed on glaciers (Rockström and Gaffney, 2021). In most of the above-mentioned works on sea ice, the authors are focused on positive feedback that leads to an enormous increase in the total area of the melt ponds. The point is that the albedo of snow in the visible range of the spectrum is very high, and when snow melts in polar summer, forming melt ponds, the reflection of solar radiation is radically reduced. As a result, sea ice receives much more solar heat, causing it to melt and increasing the area occupied by melt ponds. This positive feedback over a large surface area of the Arctic Sea is accompanied by an increase in the water vapor content of the atmosphere and is important for the continuation of global warming.
Even visual observations allow us to distinguish between two types of melt ponds: the so-called bright ponds and dark ponds. The ice on the bottom of the bright pond is mostly smooth and dense but with small cracks and highly light-scattering porous areas with fine pores. The bottom of the dark pond is more heterogeneous and has relatively large cracks and voids (König and Oppelt, 2020). This is related to the different structure of the first-year ice and multiyear ice (Li et al., 2020). The experimental work by König and Oppelt (2020) confirmed the qualitative results of the analytical study of melt pond reflectivity by Malinka et al. (2018). Note that the work by Malinka et al. (2018) was based on the same methodological framework as the present paper, including the transport approximation and the two-flux model for radiative transfer.
For simplicity, we will focus on a more simple melting problem for the first-year ice. The formation of a layer of water on the ice surface and the subsequent melting of ice are complex processes that deserve special modeling. Nevertheless, it is possible to suggest a simple physical model for ice melting under a meltwater layer. It is obvious that on both surfaces of the ice layer, the temperature is constant and equal to 0°C. Of course, the total radiation power absorbed in the ice volume leads not to increase the ice temperature, but to its melting. The heat flux from water under the ice, as was shown above, is insignificant. Indeed, there is a stable temperature stratification of water directly under the ice, and the thermal conductivity of water is rather small. On the contrary, the natural convection of water in a melt pond can give a noticeably larger contribution to ice melting, and this should be taken into consideration. This effect is more significant in the case of a small ice layer thickness.
The temperature profile along the depth of the melt pond is non-monotonic and has a maximum at some depth below the surface. Indeed, the heating of the water surface by infrared solar radiation is compensated by convective cooling and radiative cooling in the middle-infrared transparency window of the cloudless atmosphere. Even at temporary heating of the surface layer of water to the temperature
Small reflection of sunlight from water can be neglected. In subsequent calculations, the cosine of the refraction angle is assumed constant and equal to
To calculate the transfer of solar radiation in the ice layer under the melt pond, the above suggested method with the spectral irradiation
The following approximation of the radiation power absorbed in the ice layer is used:
According to Figure 1, at the scattering parameter
Of course, the dependences of
The integral radiative flux absorbed in a melt pond is given by the following equation:
It is assumed that
Using the approximate value of
The value
The suggested model can be considered only as a simple physical assessment. The real process needs a more sophisticated analysis which is beyond the scope of the present paper. However, it is clear that a relatively simple approach to solving the radiative transfer problem may be useful in the physical modeling of one of the stages of ice melting and the complex evolution of melt ponds.
As a result of the strong melting of sea ice during the polar summer, significant areas of the Arctic Sea may become ice-free. Most likely, this will not prevent the restoration of the ice cover during the polar winter, which would be an extremely undesirable effect of global warming. The fact is that the surface layer of water in the Arctic Sea cannot heat up above the melting point of ice due to the natural convection of water in its unstable temperature stratification. Thus, the unique physical properties of water allow us to count on serious negative feedback and retain some optimism about the rate of global climate change accompanied by intensive seasonal melting of polar ice.
Conclusion
Physical and computational models of solar radiative heating and melting of ice on water surface were developed. In the semitransparency range of ice and snow, the multiple scattering of radiation in the ice which contains gas bubbles and in the snow cover that may be present on the ice surface is described using the transport approximation for the scattering phase function and two-flux model for the diffuse component of the spectral radiation intensity. Analytical solution derived for the radiative transfer are coupled with a numerical solution for the transient energy equation. The heat transfer model takes into account convective heat transfer with the atmospheric air, infrared solar radiation absorbed at the illuminated surface, and radiative cooling in the mid-infrared transparency window of the cloudless atmosphere.
It was shown that a thick layer of ice not covered with snow begins to melt at the ice-water interface due to solar heating of ice. The computational data are in good agreement with the field observations for Ngoring Lake in the Qinghai–Tibet Plateau. The theoretical analysis showed a dramatic change in the process if there is a layer of snow on the ice. Even in the case of a thin snow layer less than 1 cm thick, the snow does not transmit most of the sunlight and makes ice melting impossible. At the same time, the snow itself on the ice surface does not heat up because it scatters a significant part of visible solar radiation, and the absorbed solar heat is transferred almost instantly to the relatively thick layer of ice.
Snow melting can only begin when the snow layer is about 15 mm thick, and only when the snow layer is about 200 mm thick does melting become significant. If the initial thickness of the snow layer exceeds 300 mm, snow melting near the sunlit surface leads to the formation of a melt pond on the ice surface. Such melt pools are regularly observed in polar summer on the ice of the Arctic Sea. Abnormally strong melting of sea ice in the Arctic during polar summer is an extremely important process that has been intensively studied in recent years. The last part of the paper provided an estimate of ice melting under a melt pond. The results obtained are in qualitative agreement with in-situ observations.
The general approach and particular solutions suggested for approximate calculations of solar heating and melting of lake and sea ice with possible snow cover on the ice surface can be used as the basis of more specific computational models for a variety of combined heat transfer problems in geophysics and other research areas.
Data availability statement
The original contributions presented in the study are included in the article/Supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
LD: Conceptualization, Investigation, Writing–original draft, Writing–review and editing.
Funding
The author(s) declare financial support was received for the research, authorship, and/or publication of this article. The work was financially supported by the Ministry of Science and Higher Education of the Russian Federation (project no. FEWZ-2023-0002).
Acknowledgments
The author is grateful to Professor Alexander Kokhanovsky for useful discussions.
Conflict of interest
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The author(s) declared that they were an editorial board member of Frontiers, at the time of submission. This had no impact on the peer review process and the final decision.
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Nomenclature
Keywords: heat transfer, solar radiation, ice, snow, scattering, melting, lake, sea
Citation: Dombrovsky LA (2024) An effect of a snow cover on solar heating and melting of lake or sea ice. Front. Therm. Eng. 3:1354265. doi: 10.3389/fther.2023.1354265
Received: 12 December 2023; Accepted: 28 December 2023;
Published: 15 January 2024.
Edited by:
Cun-Hai Wang, University of Science and Technology Beijing, ChinaReviewed by:
Junming Zhao, Harbin Institute of Technology, ChinaYuge Han, Nanjing University of Science and Technology, China
Copyright © 2024 Dombrovsky. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Leonid A. Dombrovsky, ldombr4887@gmail.com