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

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{−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].

## 2. Observational Study

^{2}, 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

^{2}). 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]).

## 3. Computational Problem Definition

_{0}is the temperature under hydrostatic equilibrium, and ∂I/∂z is the volumetric heat source.

_{1}·(T − T

_{md}), where b

_{1}= 1.65 × 10

^{−5}K

^{−2}, T

_{md}= 3.84 °C.

_{s}(t)[a

_{1}exp(−γ

_{1}z) + a

_{2}exp(−γ

_{2}z)]

_{s}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 I

_{s}is I

_{s}(t) = I

_{0}·max(sin(2πt/T*), 0), where T* = 24 h is the diurnal period (Figure 2).

_{0}= 1.9 × 10

^{−5}K·m/s, a

_{1}= a

_{2}= 0.5, γ

_{1}= 0.7 m

^{−1}, and γ

_{2}= 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

_{1}(9.6 m × 9.6 m × 6.4 m) and Γ

_{2}(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 Γ

_{1}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).

^{1/3}).

_{K}= (ν

^{3}/ε)

^{0.25}

^{1/3}/δ

_{K}almost always assumed values near one. The maximum ratio was about 10. The maximum value of the energy dissipation rate was 2 × 10

^{−3}mm

^{2}/s

^{3}inside the CML, whereas typical values of the energy dissipation rate in most of the CML were about 5 × 10

^{−4}mm

^{2}/s

^{3}. These values are in line with the previous estimations, derived directly from observational data [49].

_{K}= (ν/ε)

^{0.5}in the entire region for all the cases considered.

## 5. Results and Discussion

_{CML,0}>

_{CML,0}

_{CML,0}> and h

_{CML,0}correspond to the average temperature and depth of the CML at the time of its formation.

_{0}. 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.

_{T}is the temperature gradient in the stably stratified layer (in our case G

_{T}= 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

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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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 3.**Time evolution of the water temperature (

**a**) and vertical velocity component (

**b**) at different depths (Variant 1).

**Figure 4.**(

**a**,

**b**) Isosurfaces of the time-averaged vertical velocity component (|V

_{z}| = 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 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 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 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 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.

Cases | I_{0}, K·m/s |
---|---|

Variant 1 (initial) | 1.9 × 10^{−5} |

Variant 2 | 0.95 × 10^{−5} |

Variant 3 | 0.475 × 10^{−5} |

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**MDPI and ACS 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*, 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