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Article

Water Temperature Evolution Driven by Solar Radiation in an Ice-Covered Lake: A Numerical Study and Observational Data

1
Northern Water Problems Institute of the Karelian Research Centre of the Russian Academy of Sciences, 185030 Petrozavodsk, Russia
2
Higher School of Applied Mathematics and Computational Physics, Peter the Great St. Petersburg Polytechnic University, 195251 St. Petersburg, Russia
*
Author to whom correspondence should be addressed.
Water 2022, 14(24), 4078; https://doi.org/10.3390/w14244078
Submission received: 10 November 2022 / Revised: 8 December 2022 / Accepted: 10 December 2022 / Published: 14 December 2022
(This article belongs to the Special Issue Hydrophysical Parameters and Gases in Ice-Covered Lakes)

Abstract

:
Until now, the phenomenon of radiatively driven convection (RDC) in ice-covered lakes has not been sufficiently studied, despite its important role in the functioning of aquatic ecosystems. There have been very few attempts to numerically simulate RDC due to the complexity of this process and the need to use powerful computing resources. The article presents the results of Large Eddy Simulations (LES) of RDC with periodic external energy pumping, which imitates the diurnal variations in solar radiation in the subglacial layer of lakes in spring. The research is aimed at numerically studying the initial stages in the formation and development of a convective mixed layer (CML). A numerical calculation was carried out for three variants of external energy pumping that differed in intensity. A diurnal acceleration and suppression of RDC due to a change in external pumping was revealed for all three variants. The results of numerical simulations provide estimates of such integral parameters of RDC development as the rate of deepening of the lower boundary of the CML, and the rate of water temperature rise within this layer. It was shown that as the cumulative heating of the CML increases over several days, daily increments in temperature and depth slowed down; that is, the dependence of the integral RDC parameters on external pumping was nonlinear. The LES results on RDC parameters were in good agreement with our observational data.

1. Introduction

The hydrodynamics of ice-covered lakes still remains a poorly studied area of winter limnology [1,2,3], despite the growing interest in the winter period among biologists and ecologists [4,5]. Currently, as a result of years of measurements, it has been established that there are different types of motions in lakes throughout the ice-covered period—circulations, vortices, internal waves, etc. [1,2,6,7,8,9,10,11,12,13,14,15]. These motions play a significant role in the functioning of lake ecosystems during the winter, since they transfer dissolved and suspended substances and particles, algae cells [16,17,18,19,20], determine the gas regime [12,21,22,23,24], intensify the heat transfer inside the water and at interfaces with bottom sediments and ice, and accelerate ice melting [14,24,25,26,27,28].
The hydrodynamics of ice-covered lakes are under the influence of a number of factors, such as bathymetry, river discharge, ice fluctuations due to atmospheric effects, heat exchange with bottom sediments, and others [1]. A significant contribution to the hydrodynamics of ice-covered lakes comes from radiatively driven convection (RDC) [2,29,30,31], which develops due to the penetration of solar radiation into the water, and the non-uniform vertical heating of the water column. A detailed overview of how modern ideas about the causes and characteristics of RDC in ice-covered lakes is given in [31].
Depending on the thickness of the snow, the structure of the ice, the size of the lake, the temperature gradient of the water column, weather conditions, and a number of other factors, RDC in ice-covered temperate lakes can last for several weeks, and cover the entire column of water [10,29,32,33,34]; this can have a significant effect on the development of under-ice plankton [16,17,18,19,20]. Snow and white ice effectively reflect and disperse solar radiation and prevent its penetration into water [19,35,36,37]. Therefore, in temperate lakes, RDC reaches its maximum at the end of the winter and in spring, when the flux of solar radiation penetrating the water continuously increases as snow and white ice melt. In arid zones, where there is little snow on the ice surface and a layer of white ice is rarely formed, solar radiation can penetrate the ice and heat the water for several months. As a result, convective mixing often pervades the entire water column long before ice-off. The water temperature in arid ice-covered lakes can exceed 6–8 °C along the water column, which means they can be considered as “heat collectors” [24,38,39].
RDC develops in the temperature range from 0 °C to 3.98 °C (maximum density temperature of fresh water). During RDC, four layers are distinguished in a vertical temperature profile: an under-ice gradient layer, a convective mixed layer (CML), a layer of involvement, and the underlying stratified layer [29,30,31,32]. A specific feature of RDC is a gradual increase in the temperature of CML and deepening of its lower boundary [31]. After the CML temperature reaches the maximum density temperature of fresh water, further radiative warming leads to water column stabilization and the termination of RDC [40].
The factors determining the rate of deepening of the lower boundary of CML and the increase in its temperature are the flux of solar radiation at the bottom of the ice, water transparency, as well as the temperature gradient within the water column before the start of RDC. The values of the temperature gradient, in turn, depend on a number of factors, such as the depth of the lake, its surface area, water transparency, heat exchange with bottom sediments, retention time, wind load, and others [1,41,42]. The narrower the temperature gradient, the faster the lower boundary of the CML will go deeper, and the slower its temperature increases. For example, in deep transparent lakes such as Lake Onega [2], Lake Pääjärvi [1], and Babine Lake [29], the temperature gradient of the water before the start of RDC is approximately (10−2) °C/m. The rate of deepening of the CML lower boundary in such lakes can reach several meters per day, and the rate of increase in CML water temperature approximately (10−2) °C/day. In shallow turbid lakes with a water temperature gradient before the start of RDC at around (10−1) °C/m, the rate of deepening of the CML lower boundary rarely exceeds 0.5 m per day, but its temperature increases noticeably, up to several tenths of a degree per day [6,43].
Field studies of RDC in temperate ice-covered lakes involve a number of challenges. The main one is that RDC develops in the end of the winter, when snow and ice melting are intense, and the ice-carrying capacity is sharply reduced, which poses a danger to people and equipment. Furthermore, the highest possible spatial–temporal resolution of the measurements is required; that is, a large number of highly sensitive measuring sensors and long-term measurements are needed [2,29,33,38,39,43,44]. To overcome these difficulties, mathematical simulation methods can be applied, which opens up wide opportunities in the study of RDC in ice-covered lakes. An overview of RDC modeling projects carried out in 2001 is given in [31].
One of the first models of CML evolution was developed using long-term measurements of the water temperature in Babine Lake [29]. This model was based on considering a distributed source of buoyancy, and was aligned with a solution for a stable under-ice border layer. Farmer estimated the efficiency of entrainment, and found the scale of the speed related to kinetic energy production due to radiative heating.
To describe convection due to vertical heating of the water column, a simple parameterized model of the CML was developed using a convective scale of the length, speed, temperature, similarities for vertical temperature profiles, and the characteristics of turbulence [30,31]. The authors of this model analyzed the involvement modes, and showed that convection in ice-covered lakes can be satisfactorily described by the same equations of involvement as convection in atmospheric and oceanic boundary layers; they used convective proportions that take into account the volumetric absorption of solar radiation, instead of convective sizes of Deardorff, based on a flux in buoyancy through the surface of the liquid.
The application of large eddy simulation (LES) methods allows for exploration of the spatial structure of the CML. One of the most interesting LES results, arguably, is detection of convective cells within the CML [31,45,46]. Numerical simulation that resolves convective plumes, gravity currents, and secondary instabilities shows a possible effect of the faster heating of shallow waters compared to the interior on the evolution of the CML in lakes with significant shallow areas [11]. It was also shown that the dynamic response of the CML to these gradients depends sensitively on the size and latitude of the lake (the Earth’s rotation) and is controlled by the Rossby number [14].
Despite the large number of field studies and simulations of RDC in ice-covered lakes, many features of this process remain insufficiently studied. In particular, there is no understanding of how the rate of convection changes with a change in external energy pumping.
In this study we attempted to numerically simulate the development of RDC with periodic external energy pumping, which imitates the daily cycle of solar radiation entering the under-ice layer of lakes in spring. The computations were carried out using the in-house finite-volume «unstructured» code SINF/Flag-S developed at Peter the Great St. Petersburg Polytechnic University. The model calculations were verified using data from field measurements of the water temperature and solar radiation in the small shallow temperate Lake Vendyurskoe, in 1995–2012.
One of the main scientific questions that we raised in this article is the following: is there a relationship between the rate of RDC development in ice-covered lakes and the magnitude and periodicity of external energy pumping? The working hypothesis was formulated as follows: the rate of RDC development depends on the external energy pumping; the integral parameters of RDC (the temperature and depth of CML lower boundary increments) linearly depend on cumulative heating of the CML.

2. Observational Study

Lake Vendyurskoe is a small shallow temperate lake located in northwest Russia (62°10′ N, 33°20′ E). The surface area of the lake is 10.4 km2, its mean and maximal depths are 5.3 and 13.4 m. The maximal length of the lake is 7 km, and the average width is 1.5 km. The bathymetric map of this lake and some additional information about hydrology and thermal and oxygen regimes of this lake is given in [6,7,21,34,47,48,49]. The attenuation coefficient of solar radiation in water changes 0.5–2.8 m−1 in different seasons [50]. Ice-on occurs by late November to early December, and ice-off occurs by the end of April to mid-May. Every year, at the end of the ice-covered period, RDC develops and lasts for 3 to 7 weeks, depending on the thickness and structure of the snow-ice cover and the weather conditions [6,34,47,48,49,50,51,52].

Field Measurements of Water Temperature and Solar Radiation

Measurements of solar radiation fluxes at the lower boundary of ice were performed in the spring seasons of 1995–2012 using a pyranometer placed directly under the ice (M-80m universal pyranometer, made in Russia, accuracy 1 W/m2). The pyranometer measured the solar radiation flux every minute for several days (measurement periods and characteristic values of under-ice radiation are given in Table S3 of the Supplementary Materials [34]).
Vertical temperature profiles were obtained for a full water column with a 3–5 cm step using a TCD-profiler manufactured at the Northern Water Problems Institute, Petrozavodsk, Russia (accuracy ±0.05 °C, resolution ±0.01 °C) [6], and a CTD-90m Sea and Sun Technology probe (accuracy ±0.005 °C, resolution ±0.001 °C). The upper and lower boundaries of the CML and its temperature were determined from the TCD-profiler and CTD-90m data. Vertical soundings were carried out at several-day intervals to be able to trace the evolution of the CML.

3. Computational Problem Definition

Numerical simulation of the turbulent RDC was performed in 19.2 m × 19.2 m × 6.4 m and 9.6 m × 9.6 m × 6.4 m rectangular domains, which are periodical in the horizontal x and y directions; see Figure 1.
Fixed 0 °C temperature was specified at the upper boundary, and constant heat flux was specified at the bottom (so that the temperature gradient was equal to 0.4 °C/m). Periodic conditions were set on lateral surfaces, top and bottom surfaces were the walls with no-slip condition. The initial fields were set corresponding to the equilibrium state: velocity was zero; linear profile from top to bottom was specified for the temperature (according to the boundary conditions).
Turbulent heat and mass transfer are described by the following Navier–Stokes equations for an incompressible fluid using the Boussinesq approximation for the buoyancy force coupled with the energy equations:
V = 0
V t + ( V ) V = 1 ρ p + β ( T 0 T ) g + ν 2 V
T t + ( V ) T = a 2 T + I z
where V = ( V x , V y , V z ) is the velocity field; t is the time, p is the pressure, T is the temperature, ρ is the medium density, a is the thermal diffusivity coefficient, ν is the kinematic viscosity, β is the thermal expansion coefficient, g is the gravitational acceleration, T0 is the temperature under hydrostatic equilibrium, and ∂I/∂z is the volumetric heat source.
The thermophysical parameters of the fluid were taken for pure water at 2 °C. The quadratic equation of fresh water state was used for calculating the thermal expansion coefficient, β = b1·(TTmd), where b1 = 1.65 × 10−5 K−2, Tmd = 3.84 °C.
We used a two-band approximation of the decay law for radiative heating:
I(z, t) = Is(t)[a1exp(−γ1z) + a2exp(−γ2z)]
Here, I(z, t) is a kinematic flux in solar radiation, i.e., the radiation heat flux divided by the water density and specific heat. In our calculations, the volumetric heat source is added to the energy equation for the incompressible fluid (∂I/∂z). The radiation heat flux at the ice-water interface Is is a periodic function modeled by approximation of the observational data. These data were obtained during measurements in Lake Vendyurskoe in the spring of 2020. The approximation law for Is is Is(t) = I0·max(sin(2πt/T*), 0), where T* = 24 h is the diurnal period (Figure 2).
The effect of the radiative intensity on the rate of RDC development was studied (Table 1). The initial variant corresponded to parameters from the field observations: I0 = 1.9 × 10−5 K·m/s, a1 = a2 = 0.5, γ1 = 0.7 m−1, and γ2 = 2.7 m−1 [31]. The second and third variants corresponded to the radiative intensity divided by 2 (Variant 2) and by 4 (Variant 3), respectively.

4. Computational Aspects

Numerical simulations were performed using in-house finite-volume code SINF/Flag-S developed at Peter the Great St. Petersburg Polytechnic University (Saint-Petersburg, Russia); the code can operate with unstructured grids [53]. The SIMPLEC algorithm with second-order accuracy advancing in time was used. The third-order QUICK scheme was used for spatial discretization of the convective fluxes. The diffusion terms were approximated using the central-difference scheme of second-order accuracy. In our calculations we used the Implicit LES (ILES) method, in which the sub-grid turbulent viscosity is not introduced, and the role of physical viscosity on sub-grid scales is replaced by dissipative properties of a proper numerical scheme.
The computational grids used here consisted of hexagonal elements. The number of elements varied from 0.5 to 27 mln cells. The grids were clustered to the top wall. Moreover, to study the influence of the sizes of the computational domain on the resulting solution, the calculations were performed for two domains Γ1 (9.6 m × 9.6 m × 6.4 m) and Γ2 (19.2 m × 19.2 m × 6.4 m) on grids with 4.5 and 18 mln cells, respectively. A series of calculations to study the influence of the domain size and the computational grid were carried out for Variant 1. Based on the analysis of the velocity and temperature fields, as well as the fluctuations in the velocity components in the CML, the domain Γ1 was chosen for the series of parametric calculations. A grid-independent solution was obtained on a grid of 27 mln cells (300 × 300 × 300 cells).
To assess the quality of the calculations, it was necessary to compare the characteristic size of the grid cells with the Kolmogorov scale of the turbulent flow. We analyzed the fields of the Kolmogorov scale computed by the relation relative to the cubic root of the computational cell volume (Vol1/3).
δK = (ν3/ε)0.25
Here, ε is the dissipation rate of kinetic energy, as shown below:
ε = ν V j x k V j x k ¯
where V j is the fluctuation of the j-th velocity component. Having performed the calculations, we found that throughout the convection region the ratio Vol1/3K almost always assumed values near one. The maximum ratio was about 10. The maximum value of the energy dissipation rate was 2 × 10−3 mm2/s3 inside the CML, whereas typical values of the energy dissipation rate in most of the CML were about 5 × 10−4 mm2/s3. These values are in line with the previous estimations, derived directly from observational data [49].
The time step was 2.5 s, and was noticeably smaller than the Kolmogorov time scale tK = (ν/ε)0.5 in the entire region for all the cases considered.

5. Results and Discussion

Figure 3a shows the water temperature evolution at different monitoring points located at different depths, h, in Variant 1. In the first few hours, the near-ice water layer warmed up, and convective motion arose due to gravitational instability. At this stage, a CML with intensive turbulent motion began to develop; its temperature was almost spatially constant. The CML depth and temperature increased over time until the end of daily radiative heating (first 12 h). When heating stopped (12–24 h), the convective motion also nearly stopped. The water closest to ice cooled down, and the upper part of the CML re-stratified. Figure 3a clearly illustrates a temperature decline at 0.1 and 0.3 m depths in the upper part of the CML during the “night” hours; at the same time, the temperature at other depths within the CML did not decrease. With the beginning of radiation heating the next day, the temperature at these two upper depths rose and became “re-involved” in the CML. As can be clearly seen from Figure 3a, this cycle recurred every night. It appears that the thickness of the under-ice gradient layer increased when the external energy pump was removed, and decreased when the pump returned. A similar pattern of water temperature change during night hours was observed at the boundary of the under-ice gradient layer and the CML in Lake Onega in the spring of 2017 [42]. These diurnal changes can be interpreted as a balance between vertical diffusion and the buoyancy flux [40,42].
The temperature of the CML reached 1.2 °C, and the depth of its lower boundary reached 3 m in 120 h (about 5 days) (Figure 3a). The average temperature of the CML increased by 0.25 °C/day, and the depth of its lower boundary increased by 0.6 m/day. That is, the simulated rates of RDC development were in good agreement with the measurement results for small temperate lakes [6,43].
The turbulent velocity fluctuations at the different depths are illustrated by Figure 3b. One can see that the intensity of the turbulent motion increased, day by day. The turbulent motion led to temperature fluctuations observed during the daytime (Figure 3a). Note that at depths where stratification was stable (see, for example, dark blue and red lines in Figure 3a), we observed some temperature fluctuations during the daytime that were possibly related to RDC. One of the possible causes of high-frequency temperature fluctuations in the stratified layer below the CML in ice-covered lakes can be internal waves. High-frequency internal waves arose in the stratified layer below the CML in the afternoon, and were observed until night in Lake Onega during RDC; their appearance was presumably associated with both the lateral inflow of dense water from warmer shallow regions and convective cells [54]. However, since the depth of the computational domain was the same over its entire area and the radiation flux at the upper boundary was also uniform, the influence of internal waves can be excluded, and the appearance of high-frequency oscillations at the lower boundaries of the computational domain was most likely due to either numerical instability or convective cells. This issue will be studied elsewhere.
Previously, it was suggested that a community of convective cells may exist inside a CML [45,46,55]. In our simulations, we also observed similar large-scale structures within the CML. To better visualize such structures, the isosurfaces of the positive and negative values of the vertical velocity component are shown in Figure 4a,b for two different days of Variant 2. It is evident that the upstream motion concentrated inside cells, whereas the downstream motion dispersed around these cells. Taking into account the length of the computational domain (9.6 m), the horizontal size of these cells can be several meters (about 3 m). It can also be seen that these cells are long-lived, and can travel in the horizontal direction, become disrupted, and re-emerge again. Similar behavior of large-scale coherent structures was reported in [56,57] in the context of Rayleigh–Bénard-type free convection.
Figure 4c,d show the distributions of the time-averaged (by one hour) vertical velocity component, as well as the streamlines. One can see that these large-scale cells span the entire depth of the CML; thus, we can conclude in the considered case that free convective motion of the Rayleigh–Bénard-type takes place. Similar large-scale convective cells have been observed in our observational studies (e.g., see [55,57]).
Depthwise temperature profiles in different days and hours are shown in Figure 5. The evolution of the CML is clearly seen (see Figure 5b). The evolution of horizontally averaged temperature profiles during one day heating is shown in Figure 5a. Note that during the nighttime, convective motion also takes place, but its intensity is significantly lower. Several regions can be clearly distinguished in the figures: the near-ice layer, with a strong temperature gradient; the CML; and the stably stratified layer, where the temperature profile is almost linear and fluid motion is absent. Averaged temperature profiles were used to determine the depth of the CML: a depth was taken at which the temperature began to differ noticeably (by more than 1%) from the depthwise averaged temperature inside the CML.
The influence of the radiative intensity on CML evolution can be seen in Figure 6, where depthwise temperature profiles are shown for the same day and time, and for the different variants of energy pumping. One can see that in Variant 3 (with the lowest energy pumping intensity) the CML has the smallest depth, and its temperature profile is noticeably non-monotonous; similar profiles were observed before the emergence of the CML—such profiles are gravitationally unstable. The enlarged temperature profiles in Figure 6 show also that the temperature deviations inside the CML (due to the turbulent motion) are relatively small (for Variant 1 on day 4 they are less than 0.02 °C).
Horizontally averaged temperature profiles for the different times in Variant 2 are shown in Figure 7a. The fastest rise in the temperature of the CML occurred in the middle of the day. It is also worth mentioning that below the CML, near the stably stratified layer, a minor change in the average temperature gradient occurred (see Figure 7b): the temperature gradient just below the CML was slightly greater than in the underlying stratified layer.
To compare between numerical simulation and filed observations, we introduced integral parameters of the CML: cumulative integral heating (Q), CML temperature increment (TCML), and CML depth increment (HCML), which were estimated as follows:
Q = I 0 ρ C p o t s i n ( 2 π t T * ) d t
TCML = <T> − <TCML,0>
HCML = hhCML,0
where <TCML,0> and hCML,0 correspond to the average temperature and depth of the CML at the time of its formation.
As follows from the energy balance, the heating radiative energy is utilized to raise the CML heat content (enthalpy) [31], which, in turn, is proportional to the product of the temperature TCML and depth HCML. Thus, one may expect that CML enthalpy will grow linearly with Q. The dependence of the increments in the temperature TCML and depth HCML on cumulative heating Q is more sophisticated. The corresponding results of computations are presented in Figure 8. As the curves demonstrate, both the TCML and HCML correlate with cumulative heating Q; moreover, the dependence of both these parameters on Q is practically the same for different values of radiative intensity I0. It is noteworthy that this dependence is not linear: when cumulative heating Q increased, the growth rate of the CML temperature and the deepening of its lower boundary decreased. The field observations are also presented in Figure 8; they demonstrate good correlation with the simulations.
Observational and simulated cumulative values can be compared with the debated correlations between the CML depth and its temperature. One such correlation is the encroachment approximation, most applicable for the early stages of convection development [25]:
d h d t = 1 G T d T d t
where GT is the temperature gradient in the stably stratified layer (in our case GT = 0.4 °C/m). The comparison of the simulated and observational data with the correlation is given in Figure 9. One can see relatively good agreement between the observational data and the correlation. Numerical simulation showed good agreement at the onset of the process, and a small discrepancy in the late stage. This discrepancy can arise from the growing error of the numerical method as the CML develops.

6. Conclusions

This paper presented the results of both observational and numerical investigations of radiatively driven convection (RDC), which takes place in ice-covered temperate lakes at the end of winter. The tasks were to conduct LES of the initial stages of RDC development under varied external energy pumping. The computations were carried out using the in-house finite-volume «unstructured» code SINF/Flag-S developed at Peter the Great St. Petersburg Polytechnic University. The simulations were carried out for periodic external energy pumping that imitated diurnal variations in solar radiation in the subglacial layer of lakes in spring; three variants differing in the intensity of external energy pumping were considered. A diurnal acceleration and suppression of convection due to a change in external pumping was revealed for all three variants. A non-linear relationship was revealed between an increase in the cumulative heating of the CML over several days and the integral parameters of the RDC-CML temperature and thickness increments: the convection rate slowed down as heat accumulated in the under-ice layer. In general, however, the question is still open, and generates some of the following challenges for future studies: (i) Is this result valid only for the initial phase (first few days) of CML development? (ii) Will this result persist for other types of heat flux dependence on time?
The LES quite also accurately reproduced some specific features observed in ice-covered lakes during RDC, in particular, the evolution of the water temperature at the upper part of the CML at night. In the numerical simulation, we clearly discerned large-scale convective cells, which occupied the whole CML depth, and had horizontal dimensions of several meters. The existence of such large-scale cells was confirmed by other authors, both in numerical and observational studies. The typical simulated values of the kinetic energy dissipation rate in most of the CML were in line with previous estimations that derived directly from observational data.
The good agreement between the performed LES and field data suggests that the model can be considered as a usable tool for studying RDC in ice-covered lakes. Further numerical experiments will be aimed at revealing some important aspects of the structure and parameters of under-ice turbulence, at studying its nature and anisotropy properties, the difference between the spectra of vertical and horizontal fluctuations, energy transfer features, etc.

Author Contributions

Conceptualization, A.S., S.S., S.B., A.T. and G.Z.; methodology, A.S., S.S., S.B., A.T. and G.Z.; software, S.S. and A.S.; validation, S.S. and A.S.; formal analysis, S.B., S.S., G.Z., I.N. and A.S.; investigation, R.Z., N.P., I.N. and G.Z.; resources, S.S. and A.S.; data curation, R.Z., N.P., I.N., G.Z., S.S. and A.S.; writing—original draft preparation, S.B., S.S., G.Z. and A.S.; writing—review and editing, S.B., S.S., G.Z., N.P., A.T., R.Z., I.N. and A.S.; visualization, S.S. and A.S.; supervision, S.B.; project administration, G.Z.; funding acquisition, G.Z. All authors have read and agreed to the published version of the manuscript.

Funding

The study was funded by the Russian Science Foundation grant #21-17-00262 “Mixing in boreal lakes: mechanisms and its efficiency”.

Data Availability Statement

The data are available at https://aero.spbstu.ru/cloud/index.php/s/ariJoyxkQZkqBoT (accessed on 31 October 2022).

Acknowledgments

The computations were performed using resources of the St. Petersburg Polytechnic University Supercomputer Center, http://www.scc.spbstu.ru/ (accessed on 13 December 2022).

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Computational domain and thermal boundary conditions.
Figure 1. Computational domain and thermal boundary conditions.
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Figure 2. The radiation heat flux at the ice-water interface for the initial variant (blue line) and measured in Lake Vendyurskoe during the spring of 2020 (grey line).
Figure 2. The radiation heat flux at the ice-water interface for the initial variant (blue line) and measured in Lake Vendyurskoe during the spring of 2020 (grey line).
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Figure 3. Time evolution of the water temperature (a) and vertical velocity component (b) at different depths (Variant 1).
Figure 3. Time evolution of the water temperature (a) and vertical velocity component (b) at different depths (Variant 1).
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Figure 4. (a,b) Isosurfaces of the time-averaged vertical velocity component (|Vz| = 0.5 mm/s, red structures correspond to ascending currents, blue structures correspond to descending currents); (c,d) the fields of the time-averaged vertical velocity component in the central vertical section, Variant 2; (a,c) 4th day, 3 p.m., (b,d) 5th day, 3 p.m. X, Y—horizontal coordinates, Z—vertical coordinate.
Figure 4. (a,b) Isosurfaces of the time-averaged vertical velocity component (|Vz| = 0.5 mm/s, red structures correspond to ascending currents, blue structures correspond to descending currents); (c,d) the fields of the time-averaged vertical velocity component in the central vertical section, Variant 2; (a,c) 4th day, 3 p.m., (b,d) 5th day, 3 p.m. X, Y—horizontal coordinates, Z—vertical coordinate.
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Figure 5. Depthwise temperature profiles: (a) horizontally averaged profiles at different hours of the 3rd day of calculations (Variant 1); (b) instant temperature profiles at 3 p.m. (Variant 1) for days 3 to 8.
Figure 5. Depthwise temperature profiles: (a) horizontally averaged profiles at different hours of the 3rd day of calculations (Variant 1); (b) instant temperature profiles at 3 p.m. (Variant 1) for days 3 to 8.
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Figure 6. Temperature profiles for the different variants (4th day, 3 p.m.).
Figure 6. Temperature profiles for the different variants (4th day, 3 p.m.).
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Figure 7. (a) Horizontally averaged temperature profiles for Variant 2 at different hours of the 5th day: green curves are 12 p.m., black—3 p.m., red—6 p.m. (b) Profiles of the averaged temperature gradients at the same time moments.
Figure 7. (a) Horizontally averaged temperature profiles for Variant 2 at different hours of the 5th day: green curves are 12 p.m., black—3 p.m., red—6 p.m. (b) Profiles of the averaged temperature gradients at the same time moments.
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Figure 8. Dependence of the CML on lower boundary depth HCML (a), temperature TCML and (b) increments in cumulative heating Q: black symbols—Variant 1, blue—Variant 2, purple—Variant 3, red symbols—observational data.
Figure 8. Dependence of the CML on lower boundary depth HCML (a), temperature TCML and (b) increments in cumulative heating Q: black symbols—Variant 1, blue—Variant 2, purple—Variant 3, red symbols—observational data.
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Figure 9. CML temperature as a function of the CML lower boundary depth: comparison of the simulation and observational data with the correlation [Equation (10)]: black symbols—Variant 1, blue—Variant 2, purple—Variant 3, red symbols—observational data, black line—correlation.
Figure 9. CML temperature as a function of the CML lower boundary depth: comparison of the simulation and observational data with the correlation [Equation (10)]: black symbols—Variant 1, blue—Variant 2, purple—Variant 3, red symbols—observational data, black line—correlation.
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Table 1. Considered variants of radiative intensity.
Table 1. Considered variants of radiative intensity.
CasesI0, K·m/s
Variant 1 (initial)1.9 × 10−5
Variant 20.95 × 10−5
Variant 30.475 × 10−5
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Smirnov, S.; Smirnovsky, A.; Zdorovennova, G.; Zdorovennov, R.; Palshin, N.; Novikova, I.; Terzhevik, A.; Bogdanov, S. Water Temperature Evolution Driven by Solar Radiation in an Ice-Covered Lake: A Numerical Study and Observational Data. Water 2022, 14, 4078. https://doi.org/10.3390/w14244078

AMA Style

Smirnov S, Smirnovsky A, Zdorovennova G, Zdorovennov R, Palshin N, Novikova I, Terzhevik A, Bogdanov S. Water Temperature Evolution Driven by Solar Radiation in an Ice-Covered Lake: A Numerical Study and Observational Data. Water. 2022; 14(24):4078. https://doi.org/10.3390/w14244078

Chicago/Turabian Style

Smirnov, Sergei, Alexander Smirnovsky, Galina Zdorovennova, Roman Zdorovennov, Nikolay Palshin, Iuliia Novikova, Arkady Terzhevik, and Sergey Bogdanov. 2022. "Water Temperature Evolution Driven by Solar Radiation in an Ice-Covered Lake: A Numerical Study and Observational Data" Water 14, no. 24: 4078. https://doi.org/10.3390/w14244078

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