# Numerical Simulation of Radiatively Driven Convection in a Small Ice-Covered Lake with a Lateral Pressure Gradient

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{md}) [3,4,5]; at atmospheric pressure, the density of fresh water is at a maximum at a temperature of ~3.98 °C and decreases both with increases in temperature above the T

_{md}and with decreases in the range of 0–T

_{md}. In addition, T

_{md}decreases as the depth (pressure) increases at a rate of −0.021 °C/bar [3,4].

_{md}, radiatively driven convection (RDC) develops; the driving force for this type of convection is solar radiation fluxes that penetrate a certain depth and heat the surface layers of lakes in the temperature range of 0–T

_{md}[8,9,10,11]. RDC slows down when the water temperature of the surface layer of lakes rises above T

_{md.}

## 2. Problem Definition and Computational Aspects

_{0}= T

_{0}(z) is the initial linear temperature profile under hydrostatic equilibrium, α is the thermal diffusivity coefficient, ν is the kinematic viscosity coefficient, β is the thermal expansion coefficient, ∂I/∂z is the volumetric heat source, and gradP

_{0}is the external pressure gradient oriented along the x-axis.

_{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}, was the prescribed periodic function, formulated according to the experimental data [22,40].

^{−6}Pa/m to 3.28·10

^{−4}Pa/m. The values of the pressure gradient were calculated by the Darcy–Weisbach equation according to the characteristic mean velocity of the horizontal flow in the fully established turbulent regime of the flow in the 2D channel, whose size is equal to double the computational domain height (for the Darcy friction factor we use approximation $\lambda ={\left(1.8log\mathrm{Re}-1.64\right)}^{-2}$). Thus, the value 3.28·10

^{−6}Pa/m corresponds to about 1 mm/s, and the value 3.28·10

^{−4}Pa/m corresponds to several centimeters per second. The pressure gradient in all calculations was set as starting from 6 a.m. of the 4th day when the CML is fully formed, as shown in our previous study [40]. Thus, the calculations for the first 78 h were carried out with the original settings (without external pressure gradient).

## 3. Results and Discussion

#### 3.1. Radiatively Driven Convection in the Variant of Zero Lateral Pressure Gradient

_{ij}a

_{ji}/2 and third III = a

_{ij}a

_{jl}a

_{li}/3 invariants of the anisotropy tensor a

_{ij}= ⟨V

_{i}′V

_{j}′⟩/(2k) – δ

_{ij}/3 (V

_{i}′ is the pulsation of the i-th velocity component; angular brackets is time averaging; k is the turbulence kinetic energy; summation on repeated indexes henceforward). Each state of turbulence is represented by a point on the AIM, all physically acceptable states form the so-called Lumley triangle, where the upper bound is described by the equation II = 1/9 + 3III and corresponds to the 2D turbulence, the lateral bounds (described by the equation II = 3(III/2)

^{2/3}) represent the axisymmetric turbulence: “rod-like” or “disk-like” pulsations [48]. The point (0, 0) corresponds to the isotropic turbulence.

#### 3.2. Effect of a Lateral Pressure Gradient

_{0}equal to 3.28·10

^{−4}Pa/m, corresponding to the lateral flow velocity of several centimeters per second in an established regime. In the middle of the day (9 h after the start of the gradP

_{0}setting, at 3 p.m.), a decrease in the expression of large-scale convective cells was observed (Figure 5, Day 1). Here and below, the counting of days begins from the moment the pressure gradient is specified. Thus, 3 p.m. on the first day (see Figure 5) in total corresponds to 81 h of calculations (three days with the initial settings and 9 h with the gradP

_{0}settings). In the following days, further restructuring of the flow is observed with the formation of elongated “two-dimensional” rolls oriented along the superimposed horizontal flow (Figure 5, Day 2). Further development of convection in the layer, accompanied by an increase in longitudinal velocity, led to the disintegration of large-scale structures and a prevalence of small-scale turbulence (Figure 5, Day 3). The evolution of the flow obtained in the calculations with and without the pressure gradient in similar hours of 4–6 days of calculation can be traced by comparing the results shown in Figure 2 and Figure 5.

_{0}, the lower the lateral flow velocity, which leads to a later transition from large-scale cells to elongated “two-dimensional” rolls. For example, with gradP

_{0}equal to 3.28·10

^{−5}Pa/m, the transition to elongated structures occurred on the second day, while large-scale structures were still observed on the first day (Figure 6). For even lower values of gradP

_{0}equal to 6.56·10

^{−6}Pa/m and 3.28·10

^{−6}Pa/m, illustrated in Figure 7 and Figure 8, the transition to “two-dimensional” rolls happened on the fifth and ninth days, respectively. In the previous days, due to the relatively low lateral flow velocities, the flow structure in the CML was almost the same as in the case without an external lateral pressure gradient. The structure of the formed elongated rolls is similar to the well-known Langmuir circulation [2]. At this time, we cannot give a specific cause for the transition from the convective cells to the elongated 2D rolls. There are some considerations concerning the numerical formulation of the problem, where the ideal periodic conditions are specified on the lateral bounds: the posed non-uniformity (pressure gradient) leads to the appearance of a preferred direction. Thus, in the formulation under consideration, we expect a transition from cells to two-dimensional rolls even with a very small applied lateral pressure gradient. In real conditions, there should be a critical value of the lateral pressure gradient, below which the transition is not observed. Note that in variants with lower values of the pressure gradient, we did not observe the disintegration of the rolls and transition to small-scale turbulence as in the case with the highest pressure gradient. Perhaps the reason is that the model time was insufficient for variants with smaller pressure gradients, and if the calculation duration was longer, we would see the disintegration of the rolls at some point in time.

_{0}= 3.28·10

^{−5}Pa/m (fifth day from the setting of the external pressure gradient). We see that the horizontal flow velocity is maximal in the bottom stratified layer, reaching about 12 mm/s. In the CML, convective cells interact with superimposed lateral motion, which leads to a significant decrease in the horizontal velocity component. The vertical flow velocity reaches its maximum inside the CML; in the stratified layer, the vertical component is close to zero.

^{−5}Pa/m and 3.28·10

^{−4}Pa/m. In the case of gradP

_{0}= 3.28·10

^{−5}Pa/m and less, the temperature profile has a characteristic form of RDC with distinct CML boundaries. The large pressure gradient leads to significant changes in profile: shear stress between horizontal flow in a stable stratified layer and CML leads to strong turbulent mixing in the lower part of the CML. These changes in flow make it difficult to determine such characteristics of the CML as its depth and temperature. Thus, our simulation shows that at a relatively large pressure gradient, the structure of the flow and CML can significantly transform over time, and such effects need to be studied in detail in the future.

_{0}= 3.28·10

^{−5}Pa/m in the middle of the fifth day after setting the pressure gradient. We see that in the plane oriented along the x-axis (upper row in Figure 10), “two-dimensional” rolls are observed very clearly. The illustrations in the perpendicular plane (bottom row of Figure 10) are more inhomogeneous and show the distribution of elongated rolls (along the x-axis) in the direction of the y-axis.

^{−5}Pa/m. It can be seen that the imposition of a pressure gradient leads to an increase in the velocity fluctuations inside the CML for both depths under consideration. However, due to the inhomogeneity of their distributions, the ratio between the horizontal and vertical pulsations can vary significantly across depths. In particular, in the middle of the CML (depth 1.5 m, Figure 11), vertical pulsations prevail over horizontal ones both for the zero-gradient case and with a lateral gradient present. In Figure 12, which shows the distribution of pulsations near the CML lower boundary, we see the opposite correlation and a prevalence of horizontal pulsations for both variants of calculations.

_{0}= 3.28·10

^{−4}Pa/m, we also evaluate horizontally averaged AIMs. In Figure 13, AIMs for different days are presented. One can see that on the first day after involving the pressure gradient, the AIM becomes different from the one shown in Figure 4 for the zero-gradient case, but this difference is minor: in most of the CML, the turbulence anisotropy is similar for both cases. In the next days, however, the AIM differs significantly; the red line corresponds to anisotropy with a prevalence of one (horizontal) pulsation in the middle of the CML and a tendency towards 2D turbulence near the top and bottom of the CML (Figure 13a). A similar pattern of 2D-3D transformation is observed for other values of the pressure gradient (Figure 13b,c). Previously, we discovered changes in the turbulence regime in the CML of Lake Vendyurskoe, when multiple 2D-3D transitions recurred every day [34]. These previous studies, however, did not detect changes in the structure of turbulence in the CML with irreversible transition from cells to rolls. As it was mentioned earlier, such a transition, even in the case of a relatively small pressure gradient, can be associated with an “ideal” numerical problem formulation with the imposition of periodic boundary conditions.

_{CML,0}>

_{CML,0}

_{CML,0}and <T

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

^{−5}Pa/m, we did not observe significant changes in the evolution of the depth of the lower boundary of the CML and its temperature. As mentioned above, at gradP

_{0}= 3.28·10

^{−4}Pa/m there is a noticeable change in the temperature profile, so the CML is not expressed so clearly; therefore, these data are not shown in Figure 14.

## 4. Conclusions

_{0}value less than or equal to 3.28·10

^{−5}Pa/m. As the pressure gradient increases and, as a consequence, the horizontal velocity increases, large-scale rolls are suppressed, and the CML changes, which leads to difficulties in detecting its lower boundary.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Farmer, D.M.; Carmack, E. Wind mixing and restratification in a lake near the temperature of maximum density. J. Phys. Oceanogr.
**1981**, 11, 1516–1533. [Google Scholar] [CrossRef] - Wüest, A.; Lorke, A. Small-scale hydrodynamics in lakes. Annu. Rev. Fluid Mech.
**2003**, 35, 373–412. [Google Scholar] [CrossRef] - Bouffard, D.; Wüest, A. Convection in lakes. Annu. Rev. Fluid Mech.
**2019**, 51, 189–215. [Google Scholar] [CrossRef] - Weiss, R.; Carmack, E.; Koropalov, V. Deep-water renewal and biological production in Lake Baikal. Nature
**1991**, 349, 665–669. [Google Scholar] [CrossRef] - González-Salgado, D.; Noya, E.G.; Lomba, E. Simulation and theoretical analysis of the origin of the temperature of maximum density of water. Fluid Phase Equilibria
**2022**, 560, 113515. [Google Scholar] [CrossRef] - Jonas, T.; Stips, A.; Eugster, W.; Wüest, A. Observations of a quasi shear-free lacustrine convective boundary layer: Stratification and its implications on turbulence. J. Geophys. Res.
**2003**, 108, 3328. [Google Scholar] [CrossRef] - Ghane, A.; Boegman, L. Turnover in a small Canadian shield lake. Limnol. Oceanogr.
**2021**, 66, 3356–3373. [Google Scholar] [CrossRef] - Farmer, D.M. Penetrative convection in the absence of mean shear. Q. J. R. Meteorol. Soc.
**1975**, 101, 869–891. [Google Scholar] [CrossRef] - Jonas, T.; Terzhevik, A.Y.; Mironov, D.V.; Wüest, A. Radiatively driven convection in an ice-covered lake investigated by using temperature microstructure technique. J. Geophys. Res.
**2003**, 108, 3183. [Google Scholar] [CrossRef] - Austin, J.A. Observations of radiatively driven convection in a deep lake. Limnol. Oceanogr.
**2019**, 64, 2152–2160. [Google Scholar] [CrossRef] - Cannon, D.J.; Troy, C.D.; Liao, Q.; Bootsma, H.A. Ice-Free Radiative Convection Drives Spring Mixing in a Large Lake. GRL
**2019**, 46, 6811–6820. [Google Scholar] [CrossRef] - Carmack, E.C.; Weiss, R.F. Convection in Lake Baikal: An Example of Thermobaric Instability; Chu, P.C., Gascard, J.C., Eds.; Elsevier Oceanography Series; Elsevier: Amsterdam, The Netherlands, 1991; Volume 57, pp. 215–228. [Google Scholar] [CrossRef]
- Boehrer, B.; Golmen, L.; Løvik, J.E.; Rahn, K.; Klaveness, D. Thermobaric stratification in very deep Norwegian freshwater lakes. J. Great Lakes Res.
**2013**, 39, 690–695. [Google Scholar] [CrossRef] - Carmack, E.; Vagle, S. Thermobaric processes both drive and constrain seasonal ventilation in deep Great Slave Lake, Canada. J. Geophys. Res. Earth Surf.
**2021**, 126, e2021JF006288. [Google Scholar] [CrossRef] - Austin, J.; Hill, C.; Fredrickson, J.; Weber, G.; Weiss, K. Characterizing temporal and spatial scales of radiatively driven convection in a deep, ice-free lake. Limnol. Oceanogr.
**2022**, 67, 2296–2308. [Google Scholar] [CrossRef] - Kelley, D. Convection in ice-covered lakes: Effects on algal suspension. J. Plankton Res.
**1997**, 19, 1859–1880. [Google Scholar] [CrossRef] - Pernica, P.; North, R.L.; Baulch, H.M. In the cold light of day: The potential importance of under-ice convective mixed layers to primary producers. Inland Waters
**2017**, 7, 138–150. [Google Scholar] [CrossRef] - Huang, W.; Zhang, Z.; Li, Z.; Lepparanta, M.; Arvola, L.; Song, S.; Huotari, J.; Lin, Z. Under-ice dissolved oxygen and metabolism dynamics in a shallow lake: The critical role of ice and snow. Water Resour. Res.
**2021**, 57, e2020WR027990. [Google Scholar] [CrossRef] - Bouffard, D.; Zdorovennova, G.; Bogdanov, S.; Efremova, T.; Lavanchy, S.; Palshin, N.; Terzhevik, A.; Vinnå, L.R.; Volkov, S.; Wüest, A.; et al. Under-ice convection dynamics in a boreal lake. Inland Waters
**2019**, 9, 142–161. [Google Scholar] [CrossRef] - Suarez, E.L.; Tiffay, M.-C.; Kalinkina, N.; Tchekryzheva, T.; Sharov, A.; Tekanova, E.; Syarki, M.; Zdorovennov, R.E.; Makarova, E.; Mantzouki, E.; et al. Diurnal variation in the convection-driven vertical distribution of phytoplankton under ice and after ice-off in large Lake Onego (Russia). Inland Waters
**2019**, 9, 193–204. [Google Scholar] [CrossRef] - Yang, B.; Wells, M.G.; Li, J.; Young, J. Mixing, stratification, and plankton under lake-ice during winter in a large lake: Implications for spring dissolved oxygen levels. Limnol. Oceanogr.
**2020**, 65, 2713–2729. [Google Scholar] [CrossRef] - Mironov, D.; Terzhevik, A.; Kirillin, G.; Jonas, T.; Malm, J.; Farmer, D. Radiatively driven convection in ice-covered lakes: Observations, scaling, and a mixed layer model. J. Geophys. Res.
**2002**, 107, 1–16. [Google Scholar] [CrossRef] - Kirillin, G.; Terzhevik, A. Thermal instability in freshwater lakes under ice: Effect of salt gradients or solar radiation? Cold Reg. Sci. Tech.
**2011**, 65, 184–190. [Google Scholar] [CrossRef] - Mironov, D.V.; Terzhevik, A.Y. Spring Convection in Ice-Covered Freshwater Lakes. Izv. Atmos. Ocean. Phys.
**2000**, 36, 627–634. [Google Scholar] - Mironov, D.V.; Danilov, S.D.; Olbers, D.J. Large-eddy simulation of radiatively-driven convection in ice covered lakes. In Proceedings of the Sixth Workshop on Physical Processes in Natural Waters, Girona, Spain, 27–29 June 2001; Casamitjana, X., Ed.; University of Girona: Girona, Spain, 2001; pp. 71–75. [Google Scholar]
- Ulloa, H.N.; Winters, K.B.; Wüest, A.; Bouffard, D. Differential heating drives downslope flows that accelerate mixed-layer warming in ice-covered waters. Geophys. Res. Lett.
**2019**, 46, 13872–13882. [Google Scholar] [CrossRef] - Ramón, C.L.; Ulloa, H.N.; Doda, T.; Winters, K.B.; Bouffard, D. Bathymetry and latitude modify lake warming under ice. Hydrol. Earth Syst. Sci.
**2021**, 25, 1813–1825. [Google Scholar] [CrossRef] - Kirillin, G.; Aslamov, I.; Leppäranta, M.; Lindgren, E. Turbulent mixing and heat fluxes under lake ice: The role of seiche oscillations. Hydrol. Earth Syst. Sci.
**2018**, 22, 6493–6504. [Google Scholar] [CrossRef] - Kiili, M.; Pulkkanen, M.; Salonen, K. Distribution and development of under-ice phytoplankton in 90-m deep water column of Lake Päijänne (Finland) during spring convection. Aquat. Ecol.
**2009**, 43, 707–713. [Google Scholar] [CrossRef] - Vehmaa, A.; Salonen, K. Development of phytoplankton in Lake Pääjärvi (Finland) during under-ice convective mixing period. Aquat. Ecol.
**2009**, 43, 693–705. [Google Scholar] [CrossRef] - Jansen, J.; MacIntyre, S.; Barrett, D.C.; Chin, Y.-P.; Cortés, A.; Forrest, A.L.; Hrycik, A.R.; Martin, R.; McMeans, B.C.; Rautio, M.; et al. Winter limnology: How do hydrodynamics and biogeochemistry shape ecosystems under ice? J. Geophys. Res. Biogeosci.
**2021**, 126, e2020JG006237. [Google Scholar] [CrossRef] - Wüest, A.; Pasche, N.; Ibelings, B.; Sharma, S.; Filatov, N. Life under ice in Lake Onego (Russia)—An interdisciplinary winter limnology study. Inland Waters
**2019**, 9, 125–129. [Google Scholar] [CrossRef] - Bogdanov, S.; Zdorovennova, G.; Volkov, S.; Zdorovennov, R.; Palshin, N.; Efremova, T.; Terzhevik, A.; Bouffard, D. Structure and dynamics of convective mixing in Lake Onego under ice-covered conditions. Inland Waters
**2019**, 9, 177–192. [Google Scholar] [CrossRef] - Bogdanov, S.; Maksimov, I.; Zdorovennov, R.; Palshin, N.; Zdorovennova, G.; Smirnovsky, A.; Smirnov, S.; Efremova, T.; Terzhevik, A. Anisotropic Turbulence in the Radiatively Driven Convective Layer in a Small Shallow Ice-Covered Lake: An Observational Study. Bound.-Layer Meteorol.
**2023**, 187, 295–310. [Google Scholar] [CrossRef] - Martinat, G.; Xu, Y.; Grosch, C.E.; Tejada-Martínez, A.E. LES of turbulent surface shear stress and pressure-gradient-driven flow on shallow continental shelves. Ocean Dyn.
**2011**, 61, 1369–1390. [Google Scholar] [CrossRef] - Crosman, E.T.; Horel, J.D. Idealized Large-Eddy Simulations of Sea and Lake Breezes: Sensitivity to Lake Diameter, Heat Flux and Stability. Bound.-Lay. Meteorol.
**2012**, 144, 309–328. [Google Scholar] [CrossRef] - Santo, M.A.; Toffolon, M.; Zanier, G.; Giovannini, L.; Armenio, V. Large eddy simulation (LES) of wind-driven circulation in a peri-alpine lake: Detection of turbulent structures and implications of a complex surrounding orography. J. Geophys. Res. Oceans
**2017**, 122, 4704–4722. [Google Scholar] [CrossRef] - Zhang, Y.; Huang, Q.; Ma, Y.; Luo, J.; Wang, C.; Li, Z.; Chou, Y. Large eddy simulation of boundary-layer turbulence over the heterogeneous surface in the source region of the Yellow River. Atmos. Chem. Phys.
**2021**, 21, 15949–15968. [Google Scholar] [CrossRef] - Grace, A.P.; Stastna, M.; Lamb, K.G.; Scott, K.A. Numerical simulations of the three-dimensionalization of a shear flow in radiatively forced cold water below the density maximum. Phys. Rev. Fluids
**2022**, 7, 023501. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - Smirnovsky, A.A.; Smirnov, S.I.; Bogdanov, S.R.; Palshin, N.I.; Zdorovennov, R.E.; Zdorovennova, G.E. Numerical Simulation of Turbulent Mixing in a Shallow Lake for Periods of Under-Ice Convection. Water Resour.
**2023**, 50, 622–632. [Google Scholar] [CrossRef] - Chang, Y.; Scotti, A. Characteristic scales during the onset of radiatively driven convection: Linear analysis and simulations. J. Fluid Mech.
**2023**, 973, A14. [Google Scholar] [CrossRef] - Noto, D.; Ulloa, H.; Yanagisawa, T.; Tasaka, Y. Stratified horizontal convection. J. Fluid Mech.
**2023**, 970, A21. [Google Scholar] [CrossRef] - Smirnov, S.; Smirnovsky, A.; Bogdanov, S. The Emergence and Identification of Large-Scale Coherent Structures in Free Convective Flows of the Rayleigh-Bénard Type. Fluids
**2021**, 6, 431. [Google Scholar] [CrossRef] - Lumley, J.L. Computational modeling of turbulent flows. Adv. Appl. Mech.
**1978**, 18, 123–176. [Google Scholar] [CrossRef] - Choi, K.-S.; Lumley, J.L. The return to isotropy of homogeneous turbulence. J. Fluid Mech.
**2001**, 436, 59–84. [Google Scholar] [CrossRef] - Penna, N.; Coscarella, F.; D’Ippolito, A.; Gaudio, R. Anisotropy in the Free Stream Region of Turbulent Flows through Emergent Rigid Vegetation on Rough Beds. Water
**2020**, 12, 2464. [Google Scholar] [CrossRef] - Simonsen, A.J.; Krogstad, P.-A. Turbulent stress invariant analysis: Clarification of existing terminology. Phys. Fluids
**2005**, 17, 088103. [Google Scholar] [CrossRef] - Cortés, A.; MacIntyre, S. Mixing processes in small arctic lakes during spring. Limnol. Oceanogr.
**2020**, 65, 260–288. [Google Scholar] [CrossRef] - Salonen, K.; Pulkkanen, M.; Salmi, P.; Griffiths, R.W. Interannual variability of circulation under spring ice in a boreal lake. Limnol. Oceanogr.
**2014**, 56, 2121–2132. [Google Scholar] [CrossRef]

**Figure 2.**Isosurfaces of the time-averaged vertical velocity component presented for six consecutive days for variant of zero lateral pressure gradient (|<V

_{z}>| = 0.6 mm/s, blue structures correspond to descending currents, red structures correspond to ascending ones), the averaging was carried out from 2 p.m. to 3 p.m. of each day.

**Figure 3.**The fields of pulsation of the velocity components in the central vertical sections: the top and the bottom rows correspond to planes xz and yz respectively; the averaging was carried out on the eighth day from 2 p.m. to 3 p.m. The points indicated by black rectangles represent the corresponding states on AIM in Figure 4.

**Figure 4.**AIM: distribution at vertical coordinates averaged over the horizontal sections (eighth day, 3 p.m.). The black points correspond to the different depths presented in Figure 3; the point

**A**corresponds isotropic turbulence,

**B**—two component axisymmetric,

**C**—“1D” turbulence; the black line

**BC**corresponds “2D” turbulence,

**AC**—axisymmetric “rod-like”,

**AB**—axisymmetric “disk-like”.

**Figure 5.**Isosurfaces of the time-averaged vertical velocity component presented for three consecutive days (|<V

_{z}>| = 0.4 mm/s, blue structures correspond to descending currents, yellow structures correspond to ascending ones); gradP

_{0}= 3.28·10

^{−4}Pa/m. Here and in Figure 6, Figure 7 and Figure 8 the averaging was carried out from 2 p.m. to 3 p.m. of each day.

**Figure 7.**Isosurfaces of the time-averaged vertical velocity component presented for three consecutive days (|<V

_{z}>| = 0.6 mm/s); gradP

_{0}= 6.56·10

^{−6}Pa/m.

**Figure 8.**Isosurfaces of the time-averaged vertical velocity component presented for three consecutive days (|<V

_{z}>| = 0.6 mm/s); gradP

_{0}= 3.28·10

^{−6}Pa/m.

**Figure 9.**Time-averaged horizontal velocity (

**a**) and vertical velocity (

**b**) profiles along the central vertical line (averaging was carried out on the fifth day from 2 p.m. to 3 p.m.); (

**c**) instantaneous horizontally averaged temperature profile on the fifth day at 3 p.m.; the black curves correspond to gradP

_{0}= 3.28·10

^{−5}Pa/m, the red curve corresponds to gradP

_{0}= 3.28·10

^{−4}Pa/m.

**Figure 10.**The fields of the pulsations of the velocity components in the central vertical sections: the top and the bottom rows correspond to planes xz and yz respectively; gradP

_{0}= 3.28·10

^{−5}Pa/m, averaging was carried out on the fifth day from 2 p.m. to 3 p.m. The points indicated by black rectangles represent the corresponding states on AIM in Figure 13.

**Figure 11.**Evolution of horizontal and vertical turbulent pulsations at a depth of 1.5 m; the averaging was carried out over time within 1 h and over space in horizontal directions. The color tab corresponds to the fields of the pulsations shown in Figure 3 and Figure 10 (the averaging was carried out from 176 to 177 h).

**Figure 12.**Evolution of horizontal and vertical turbulent pulsations at a depth of 3.5 m; the averaging was carried out over time within 1 h and over space in horizontal directions. The color tab corresponds to the fields of the pulsations shown in Figure 3 and Figure 10 (the averaging was carried out from 176 to 177 h).

**Figure 13.**AIM: distribution at vertical coordinates averaged over the horizontal sections: (

**a**) gradP

_{0}= 3.28·10

^{−}

^{4}Pa/m (day 1–3), (

**b**) gradP

_{0}= 3.28·10

^{−}

^{5}Pa/m (fifth day; black points correspond to the different cross-sections in Figure 10), (

**c**) gradP

_{0}= 6.56·10

^{−}

^{6}Pa/m (fifth day); the averaging was carried out from 2 p.m. to 3 p.m. on the corresponding day. The meaning of the black lines is the same as in Figure 4.

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**MDPI and ACS Style**

Smirnov, S.; Smirnovsky, A.; Zdorovennova, G.; Zdorovennov, R.; Efremova, T.; Palshin, N.; Bogdanov, S.
Numerical Simulation of Radiatively Driven Convection in a Small Ice-Covered Lake with a Lateral Pressure Gradient. *Water* **2023**, *15*, 3953.
https://doi.org/10.3390/w15223953

**AMA Style**

Smirnov S, Smirnovsky A, Zdorovennova G, Zdorovennov R, Efremova T, Palshin N, Bogdanov S.
Numerical Simulation of Radiatively Driven Convection in a Small Ice-Covered Lake with a Lateral Pressure Gradient. *Water*. 2023; 15(22):3953.
https://doi.org/10.3390/w15223953

**Chicago/Turabian Style**

Smirnov, Sergei, Alexander Smirnovsky, Galina Zdorovennova, Roman Zdorovennov, Tatiana Efremova, Nikolay Palshin, and Sergey Bogdanov.
2023. "Numerical Simulation of Radiatively Driven Convection in a Small Ice-Covered Lake with a Lateral Pressure Gradient" *Water* 15, no. 22: 3953.
https://doi.org/10.3390/w15223953