Biogeochemical Processes Including Oxygen Dynamics in a Deep Lake During the Spring Thermal Bar: A Numerical Experiment

Abstract

Biogeochemical processes, including the oxygen cycle, were investigated in Lake Baikal during the spring thermal bar using a coupled numerical model that takes into account the intraday variability of atmospheric parameters and contains the following variables: nitrate, ammonium, phosphate, oxygen, chlorophyll a, phytoplankton, zooplankton, and small and large detritus. Nitrification, photosynthesis, remineralization, and respiration processes describe the biochemical dynamics of oxygen in the model. As a study area, the deep-water cross-section of Lake Baikal, Boldakov River–Maloye More Strait, was considered using meteorological data for June 2024 at the lake surface. Numerical results show that the thermal bar can contribute to the transport of dissolved oxygen and phyto- and zooplankton to the deeper layers of the lake.

1. Introduction

Human activity negatively influences the water quality of Lake Baikal, primarily in the coastal zone [1,2,3], as the main source of anthropogenic and biogenic pollutants is the waters of its tributaries, which accumulate urban and industrial wastewater from the watershed basin. During periods of stratification breakdown, vertical mixing facilitates the transfer of nutrients from the deeper layers to the photic zone, which can trigger a spike in algal blooms [4,5,6].

Cross-shore water exchange in lakes is carried out through different mechanisms [7,8]. One is the thermal bar phenomenon—a sinking of maximum density waters in a narrow zone that appears in mid-latitude lakes during spring heating and autumnal cooling [9,10,11]. The thermal bar, acting as a barrier between littoral and pelagic waters (according to Tikhomirov [12], thermoactive and thermoinert regions, respectively), plays a large role in a lacustrine ecosystem. Cabbeling at the thermal bar can promote the propagation of pollutants and phytoplankton deep into the lake [13,14]. An understanding of the behavior of the thermal bar and its role in the ecological functioning of a lake has significance for limnology and water resources management. For example, during the existence of thermal bars in Lake Onega [9], the quality of water at the Petrozavodsk water intake facility deteriorates due to its unsuccessful location (to prevent this, the water intake should be located outside the thermal bar, in the open lake).

The springtime differential warming of littoral versus pelagic waters establishes a cross-shore density gradient that organizes circulation into the thermal bar. In Lake Baikal, there is an exchange between surface and deep waters due to along-the-slope circulations and density mixing at the thermal bar front [15,16]. Studies of the microbiological composition of Baikal’s waters during the development of the spring thermal bar showed high concentrations of heterotrophic bacteria and chlorophyll in the near-slope areas, as well as at the depths of 400–600 and 900–1100 m [17]. The differential warming of littoral versus pelagic waters by solar radiation in spring drives near-surface density contrasts and sets up the thermal bar circulation, as well as activates the growth of vegetation and many chemical and biological processes. Measurements regularly conducted in Baikal indicate that organic matter is unevenly distributed both in the photic zone, where it is consumed by algae, and in the deep-water parts [18].

When developing numerical models to simulate biogeochemical processes in freshwater lakes, it is very important to consider not only nitrogen and phosphorus—the nutrients that limit the growth of organisms—but also dissolved oxygen, the primary indicator of water quality. Its level rises significantly during intensive photosynthesis (due to the influx of large amounts of these nutrients) and drops sharply during excessive phytoplankton mortality. This work aims to describe a mathematical model for reproducing nutrient–plankton–ecosystem interactions in a freshwater lake. Also, we present here the first, to our knowledge, results of numerical modeling of biogeochemical processes, including the dissolved oxygen dynamics, during the thermal bar for a deep-water cross-section of Lake Baikal.

The paper is organized as follows. The reaction–convection–diffusion equations of the biogeochemical model with the initial and boundary conditions are described in Section 2. Then, the study area with model parameters is given in Section 3. And finally, in Section 4, we present the simulation results and their discussion.

2. Materials and Methods

The coupled model includes a 2.5D physical model and a biological model. The latter is developed with nitrogen [19], oxygen [20], and phosphorus [21] cycle models and contains the variables nitrate, ammonium, phosphate, oxygen, chlorophyll a, phyto- and zooplankton, and detritus of two types: small and large. Air–water O₂ exchange, biogeochemical processes, and physical transfer describe the dynamics of dissolved oxygen in a lake (Figure 1).

The 2.5D physical model [22,23,24] that transforms the problem into 2D (vertical cross-section of a lake), considering three components of the velocity vector (to take into account the Coriolis force), consists of the following governing equations [25]:

(a)

momentum equations

$${\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }{u}^2}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }uw}{\mathsf{\partial }z}}=-{\displaystyle \frac{1}{{\rho }_0}}{\displaystyle \frac{\mathsf{\partial }p}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left(K_x{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }x}}\right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\left(K_z{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }z}}\right)+2{\Omega }_{z}v-2{\Omega }_{y}w;$$

(1)

$${\displaystyle \frac{\mathsf{\partial }v}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }uv}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }wv}{\mathsf{\partial }z}}={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left(K_x{\displaystyle \frac{\mathsf{\partial }v}{\mathsf{\partial }x}}\right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\left(K_z{\displaystyle \frac{\mathsf{\partial }v}{\mathsf{\partial }z}}\right)+2{\Omega }_{x}w-2{\Omega }_{z}u;$$

(2)

$${\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }uw}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }{w}^2}{\mathsf{\partial }z}}=-{\displaystyle \frac{1}{{\rho }_0}}{\displaystyle \frac{\mathsf{\partial }p}{\mathsf{\partial }z}}+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left(K_x{\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }x}}\right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\left(K_z{\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }z}}\right)-{\displaystyle \frac{g\rho }{{\rho }_0}}+2{\Omega }_{y}u-2{\Omega }_{x}v;$$

(3)

(b)

continuity equation

$${\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }z}}=0;$$

(4)

(c)

energy equation

$${\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }uT}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }wT}{\mathsf{\partial }z}}={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left(D_x{\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }x}}\right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\left(D_z{\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }z}}\right)+{\displaystyle \frac{1}{{\rho }_0{c}_p}}{\displaystyle \frac{\mathsf{\partial }{H}_{sol}}{\mathsf{\partial }z}};$$

(5)

(d)

equations of salinity balance

$${\displaystyle \frac{\mathsf{\partial }S}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }uS}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }wS}{\mathsf{\partial }z}}={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left(D_x{\displaystyle \frac{\mathsf{\partial }S}{\mathsf{\partial }x}}\right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\left(D_z{\displaystyle \frac{\mathsf{\partial }S}{\mathsf{\partial }z}}\right),$$

(6)

where u, v are the horizontal velocity components along x and y axes, respectively, m/s; w is the vertical velocity component, m/s; T is the temperature, °C; S is the salinity, g/kg; Ω_x, Ω_y, and Ω_z are the vector components of the Earth’s rotation angular velocity, s⁻¹; K_x and K_z are the horizontal and vertical viscosity coefficients, respectively, m²/s; g is the acceleration of gravity (9.81 m/s²); c_p is the specific heat capacity, J kg⁻¹ °C⁻¹; p is the pressure, bar; ρ₀ is the water density at standard atmospheric pressure, kg/m³. The equation of Chen and Millero [26] connecting water density with temperature, salinity, and pressure, and valid within the range of 0 ≤ T ≤ 30 °C, 0 ≤ S ≤ 0.6 g/kg, and 0 ≤ p ≤ 180 bar, was taken as the state equation.

The parameterization of atmospheric oxygen input into the lake is implemented taking into account the solubility of oxygen in water [27] and wind speed on the lake surface [28]. Processes of nitrification, photosynthesis, remineralization, and respiration are considered in this model to simulate the biochemical dynamics of oxygen. A thermohydrodynamic model, which includes equations of continuity, momentum, energy, and turbulent characteristics [29], reproduces the physical transfer of dissolved oxygen in the body of water.

The reaction–convection–diffusion equations, which represent biochemical dynamics in the lake, are

$$\begin{gathered} {\displaystyle \frac{\mathsf{\partial }[N{O}_3]}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }u[N{O}_3]}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }w[N{O}_3]}{\mathsf{\partial }z}}={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left(D_x{\displaystyle \frac{\mathsf{\partial }[N{O}_3]}{\mathsf{\partial }x}}\right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\left(D_z{\displaystyle \frac{\mathsf{\partial }[N{O}_3]}{\mathsf{\partial }z}}\right) \\ -{\mu }_{\max }(T)f(I){\displaystyle \frac{L_{N{O}_3}}{L_N}}\min \left(L_N,L_P\right)[Phyto]+L_{O_2}n[N{H}_4]; \end{gathered}$$

(7)

$$\begin{gathered} {\displaystyle \frac{\mathsf{\partial }[N{H}_4]}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }u[N{H}_4]}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }w[N{H}_4]}{\mathsf{\partial }z}}={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left(D_x{\displaystyle \frac{\mathsf{\partial }[N{H}_4]}{\mathsf{\partial }x}}\right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\left(D_z{\displaystyle \frac{\mathsf{\partial }[N{H}_4]}{\mathsf{\partial }z}}\right) \\ -{\mu }_{\max }(T)f(I){\displaystyle \frac{L_{N{H}_4}}{L_N}}\min \left(L_N,L_P\right)[Phyto]-L_{O_2}n[N{H}_4] \\ +\left(l_{BM}+l_E{\displaystyle \frac{{[Phyto]}^2}{k_{Phyto}+{[Phyto]}^2}}\beta \right)[Zoo]+L_{O_2}\left(r_{SD}[SD]+r_{LD}[LD]\right); \end{gathered}$$

(8)

$$\begin{gathered} {\displaystyle \frac{\mathsf{\partial }[P{O}_4]}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }u[P{O}_4]}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }w[P{O}_4]}{\mathsf{\partial }z}}={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left(D_x{\displaystyle \frac{\mathsf{\partial }[P{O}_4]}{\mathsf{\partial }x}}\right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\left(D_z{\displaystyle \frac{\mathsf{\partial }[P{O}_4]}{\mathsf{\partial }z}}\right) \\ -{\theta }_{N:P}^{-1}{\mu }_{\max }(T)f(I)\min \left(L_N,L_P\right)[Phyto] \\ +{\theta }_{N:P}^{-1}\left(l_{BM}+l_E{\displaystyle \frac{{[Phyto]}^2}{k_{Phyto}+{[Phyto]}^2}}\beta \right)[Zoo]+{\theta }_{N:P}^{-1}{L}_{O_2}\left(r_{SD}[SD]+r_{LD}[LD]\right); \end{gathered}$$

(9)

$$\begin{gathered} {\displaystyle \frac{\mathsf{\partial }[O_2]}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }u[O_2]}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }w[O_2]}{\mathsf{\partial }z}}={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left(D_x{\displaystyle \frac{\mathsf{\partial }[O_2]}{\mathsf{\partial }x}}\right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\left(D_z{\displaystyle \frac{\mathsf{\partial }[O_2]}{\mathsf{\partial }z}}\right)+F_{air-lake}{d}^{-1} \\ +{\mu }_{\max }(T)f(I)\left({\displaystyle \frac{L_{N{O}_3}}{L_N}}{R}_{O_2:N{O}_3}+{\displaystyle \frac{L_{N{H}_4}}{L_N}}{R}_{O_2:N{H}_4}\right)\min \left(L_N,L_P\right)[Phyto]-2L_{O_2}n[N{H}_4] \\ -R_{O_2:N{H}_4}\left(l_{BM}+l_E{\displaystyle \frac{{[Phyto]}^2}{k_{Phyto}+{[Phyto]}^2}}\beta \right)[Zoo]-R_{O_2:N{H}_4}{L}_{O_2}\left(r_{SD}[SD]+r_{LD}[LD]\right); \end{gathered}$$

(10)

$$\begin{gathered} {\displaystyle \frac{\mathsf{\partial }[Chl]}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }u[Chl]}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }w[Chl]}{\mathsf{\partial }z}}={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left(D_x{\displaystyle \frac{\mathsf{\partial }[Chl]}{\mathsf{\partial }x}}\right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\left(D_z{\displaystyle \frac{\mathsf{\partial }[Chl]}{\mathsf{\partial }z}}\right) \\ +\left\{{\rho }_{Chl}{\mu }_{\max }(T)f(I)\min \left(L_N,L_P\right)-m_{Phyto}-\tau \left([SD]+[Phyto]\right)\right\}[Chl] \\ -g_{\max }{\displaystyle \frac{{[Phyto]}^2}{k_{Phyto}+{[Phyto]}^2}}[Zoo]{\displaystyle \frac{[Chl]}{[Phyto]}}; \end{gathered}$$

(11)

$$\begin{gathered} {\displaystyle \frac{\mathsf{\partial }[Phyto]}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }u[Phyto]}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }w[Phyto]}{\mathsf{\partial }z}}={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left(D_x{\displaystyle \frac{\mathsf{\partial }[Phyto]}{\mathsf{\partial }x}}\right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\left(D_z{\displaystyle \frac{\mathsf{\partial }[Phyto]}{\mathsf{\partial }z}}\right) \\ +\left\{{\mu }_{\max }(T)f(I)\min \left(L_N,L_P\right)-m_{Phyto}-\tau \left([SD]+[Phyto]\right)\right\}[Phyto] \\ -g_{\max }{\displaystyle \frac{{[Phyto]}^2}{k_{Phyto}+{[Phyto]}^2}}[Zoo]; \end{gathered}$$

(12)

$$\begin{gathered} {\displaystyle \frac{\mathsf{\partial }[Zoo]}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }u[Zoo]}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }w[Zoo]}{\mathsf{\partial }z}}={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left(D_x{\displaystyle \frac{\mathsf{\partial }[Zoo]}{\mathsf{\partial }x}}\right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\left(D_z{\displaystyle \frac{\mathsf{\partial }[Zoo]}{\mathsf{\partial }z}}\right) \\ +\left\{\left(g_{\max }-l_E\right){\displaystyle \frac{{[Phyto]}^2}{k_{Phyto}+{[Phyto]}^2}}\beta -l_{BM}\right\}[Zoo]-m_{Zoo}{[Zoo]}^2; \end{gathered}$$

(13)

$$\begin{gathered} {\displaystyle \frac{\mathsf{\partial }[SD]}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }u[SD]}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }w[SD]}{\mathsf{\partial }z}}={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left(D_x{\displaystyle \frac{\mathsf{\partial }[SD]}{\mathsf{\partial }x}}\right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\left(D_z{\displaystyle \frac{\mathsf{\partial }[SD]}{\mathsf{\partial }z}}\right) \\ +g_{\max }{\displaystyle \frac{{[Phyto]}^2}{k_{Phyto}+{[Phyto]}^2}}\left(1-\beta \right)[Zoo]+m_{Zoo}{[Zoo]}^2+m_{Phyto}[Phyto] \\ -\tau \left([SD]+[Phyto]\right)[SD]-L_{O_2}{r}_{SD}[SD]; \end{gathered}$$

(14)

$$\begin{gathered} {\displaystyle \frac{\mathsf{\partial }[LD]}{\mathsf{\partial }t}}+{\displaystyle \frac{\mathsf{\partial }u[LD]}{\mathsf{\partial }x}}+{\displaystyle \frac{\mathsf{\partial }w[LD]}{\mathsf{\partial }z}}={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }x}}\left(D_x{\displaystyle \frac{\mathsf{\partial }[LD]}{\mathsf{\partial }x}}\right)+{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }z}}\left(D_z{\displaystyle \frac{\mathsf{\partial }[LD]}{\mathsf{\partial }z}}\right) \\ +\tau {\left([SD]+[Phyto]\right)}^2-L_{O_2}{r}_{LD}[LD], \end{gathered}$$

(15)

where NO₃, NH₄, PO₄, O₂, Chl, Phyto, Zoo, SD, and LD are the concentrations of nitrates (mmol N m⁻³), ammonium (mmol N m⁻³), phosphate (mmol P m⁻³), dissolved oxygen (mmol O₂ m⁻³), chlorophyll a (mg m⁻³), phytoplankton (mmol N m⁻³), zooplankton (mmol N m⁻³), small detritus, and large detritus, respectively (mmol N m⁻³); and D_x and D_z are the diffusion coefficients (m²/s).

The maximum phytoplankton growth rate is calculated by Eppley [30]

$${\mu }_{\max }(T)={\mu }_0\cdot {1.066}^T.$$

(16)

The factors $L_{N{O}_3}, L_{N{H}_4}, L_{O_2}, L_N$, and $L_P$ [31] are responsible for the limitation of lake productivity through major nutrients (

Table 1. Limitation factors.

Symbol	Description	Equation
$L_{N{O}_3}$	Nitrate limitation factor	${\displaystyle \frac{[N{O}_3]}{k_{N{O}_3}+[N{O}_3]}}\cdot {\displaystyle \frac{1}{1+[N{H}_4]/k_{N{H}_4}}}$
$L_{N{H}_4}$	Ammonium limitation factor	${\displaystyle \frac{[N{H}_4]}{k_{N{H}_4}+[N{H}_4]}}$
$L_{O_2}$	Oxygen limitation factor	$\max \left[\left({\displaystyle \frac{O_2-O_2^{th}}{k_{O_2}+O_2-O_2^{th}}}\right),0\right]$
$L_N$	Nutrient limitation factor for nitrogen	$L_{N{O}_3}+L_{N{H}_4}$
$L_P$	Nutrient limitation factor for phosphorus	${\displaystyle \frac{[P{O}_4]}{k_{P{O}_4}+[P{O}_4]}}$

).

Oxygen exchange across the air-lake interface is parameterized as

$$F_{air-lake}=\nu k_{O_2}\left(O_2^{\ast }-O_2\right).$$

(17)

To calculate the saturation concentration of oxygen, $O_2^{\ast }$, an approximation of empirical tabular data within 0–20 °C [32] is used

$$O_2^{\ast }=456.96-12.86\cdot T_C+0.2771\cdot T_C^2-0.0033\cdot T_C^3,$$

(18)

where T_C is the water temperature in °C.

The gas transfer velocity for oxygen is parameterized as [28]

$$\nu k_{O_2}=0.31U_{10}^2\sqrt{{\displaystyle \frac{S{c}_{C{O}_2}}{S{c}_{O_2}}}},$$

(19)

where $U_{10}=\sqrt{u_{10}^2+v_{10}^2}$ is the wind speed 10 m above the lake surface, and $S{c}_{C{O}_2}$ and $S{c}_{O_2}$ are the Schmidt numbers for $C{O}_2$ and $O_2$, respectively. For fresh water,

$$S{c}_{O_2}=1800.6-120.1\cdot T_C+3.7818\cdot T_C^2-0.047608\cdot T_C^3;$$

(20)

$$S{c}_{C{O}_2}=1911.1-118.11\cdot T_C+3.4527\cdot T_C^2-0.04132\cdot T_C^3.$$

(21)

The photosynthesis–light relationship is described by the function [33]:

$$f(I)={\displaystyle \frac{\alpha I}{\sqrt{{\mu }_{\max }^2+{\alpha }^2{I}^2}}},$$

(22)

where

$$I=I_s\cdot PAR\cdot \exp \left\{-d\left(k_{water}+k_{chl}{\displaystyle \underset{d}{\overset{L_z}{\int }}Chl(z)dz}\right)\right\},$$

(23)

here I_S is the incoming light at the surface of the lake; k_water and k_chl are the light attenuation for water (=0.04 m⁻¹) and by chlorophyll (=0.025 (mg Chl)⁻¹ m⁻²), respectively; and PAR is the fraction of light available for photosynthesis (=0.43) [19].

The fraction of phytoplankton growth devoted to chlorophyll synthesis is given by

$${\rho }_{Chl}={\displaystyle \frac{{\theta }_{\max }{\mu }_{\max }(T)f(I)\min \left(L_N,L_P\right)[Phyto]}{\alpha I[Chl]}}.$$

(24)

The nitrification rate is determined as [34]:

$$n=n_{\max }\cdot \left(1-\max \left[0,{\displaystyle \frac{I-I_0}{k_I+I-I_0}}\right]\right).$$

(25)

Values for the model parameters [19,31] are presented in

Table 2. Model parameters.

Symbol	Description	Value
Nutrients
I0	Threshold for light-inhibition of nitrification	0.0095 W m−2
kI	Light intensity at which the inhibition of nitrification is half-saturated	0.1 W m−2
nmax	Maximum nitrification rate	0.05 d−1
θ N:P	Nitrogen to phosphorus ratio	16 mmol N/mmol P
Oxygen
$k_{O_2}$	Oxygen half-saturation concentration	3.0 mmol O2 m−3
$O_2^{th}$	Oxygen threshold below which no aerobic respiration or nitrification occurs	6.0 mmol O2 m−3
$R_{O_2:N{O}_3}$	Stoichiometric ratio corresponding to the oxygen produced per mol of nitrate assimilated during photosynthetic production of organic matter	8.5 mmol O2/mmol NH4
$R_{O_2:N{H}_4}$	Stoichiometric ratio corresponding to the oxygen produced per mol of ammonium assimilated during photosynthetic production of organic matter	6.625 mmol O2/mmol NH4
Phytoplankton
$k_{N{O}_3}$	Half-saturation concentration for the uptake of nitrate	0.8 mmol N m−3
$k_{N{H}_4}$	Half-saturation concentration for the uptake of ammonium	0.8 mmol N m−3
$k_{P{O}_4}$	Half-saturation concentration for the uptake of phosphate	0.06 mmol P m−3
mPhyto	Phytoplankton mortality	0.15 d−1
α	Initial slope of the P-I curve	0.1 mg C (mg Chl W m−2 d)−1
θmax	Maximum chlorophyll to phytoplankton ratio	0.0535 mg Chl (mg C)−1
μ0	Phytoplankton growth rate at 0 °C	0.69 d−1
τ	Aggregation rate of phytoplankton and small detritus	0.08 (mmol N m−3)−1 d−1
Zooplankton
gmax	Maximum grazing rate	0.5 d−1
kPhyto	Half-saturation concentration of phytoplankton ingestion	2 (mmol N m−3)2
lBM	Excretion rate due to basal metabolism	0.1 d−1
lE	Maximum rate of assimilation-related excretion	0.1 d−1
mZoo	Zooplankton mortality	0.025 (mmol N m−3)−1 d−1
β	Assimilation efficiency	0.75
Detritus
rLD	Remineralization rate of large detritus	0.01 d−1
rSD	Remineralization rate of small detritus	0.03 d−1

. Even though several biological model parameters are adopted from marine studies [19,21], we can assume that they are suitable for oligotrophic lakes because our coupled model takes into account the dependence of the plankton biomass on the nutrient concentrations, the incoming solar radiation, and the water temperature and salinity.

Boundary conditions are

(a)

at the surface of the lake

$$\begin{array}{c}{K}_z{\displaystyle \frac{\mathsf{\partial }u}{\mathsf{\partial }z}}={\displaystyle \frac{{\tau }_{surf}^u}{{\rho }_0}};K_z{\displaystyle \frac{\mathsf{\partial }v}{\mathsf{\partial }z}}={\displaystyle \frac{{\tau }_{surf}^v}{{\rho }_0}};w=0;D_z{\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }z}}={\displaystyle \frac{H_{net}}{{\rho }_0\cdot c_p}};{\displaystyle \frac{\mathsf{\partial }S}{\mathsf{\partial }z}}=0;{\displaystyle \frac{\mathsf{\partial }\left[N{O}_3\right]}{\mathsf{\partial }z}}=0;{\displaystyle \frac{\mathsf{\partial }\left[N{H}_4\right]}{\mathsf{\partial }z}}=0;\hfill \\ {\displaystyle \frac{\mathsf{\partial }\left[P{O}_4\right]}{\mathsf{\partial }z}}=0;{\displaystyle \frac{\mathsf{\partial }\left[Chl\right]}{\mathsf{\partial }z}}=0;{\displaystyle \frac{\mathsf{\partial }\left[Phyto\right]}{\mathsf{\partial }z}}=0;{\displaystyle \frac{\mathsf{\partial }\left[Zoo\right]}{\mathsf{\partial }z}}=0;{\displaystyle \frac{\mathsf{\partial }[SD]}{\mathsf{\partial }z}}=0;{\displaystyle \frac{\mathsf{\partial }[LD]}{\mathsf{\partial }z}}=0,\hfill \end{array}$$

where H_net is the heat flux of longwave radiation, latent, and sensible heat; ${\tau }_{surf}^u$ and ${\tau }_{surf}^v$ describe the wind shear stress on the lake surface;

(b)

in the river mouth

$$\begin{array}{c}u=u_R;v=0;w=0;T=T_R;S=S_R;\left[N{O}_3\right]={\left[N{O}_3\right]}_R;\left[N{H}_4\right]={\left[N{H}_4\right]}_R;\hfill \\ \left[P{O}_4\right]={\left[P{O}_4\right]}_R;\left[O_2\right]={\left[O_2\right]}_R;\left[Chl\right]={\left[Chl\right]}_R;\left[Phyto\right]={\left[Phyto\right]}_R;\left[Zoo\right]={\left[Zoo\right]}_R;\hfill \\ \left[SD\right]={\left[SD\right]}_R;\left[LD\right]={\left[LD\right]}_R,\hfill \end{array}$$

where ${\left[N{O}_3\right]}_R,{\left[N{H}_4\right]}_R,{\left[P{O}_4\right]}_R,{\left[O_2\right]}_R,{\left[Chl\right]}_R,{\left[Phyto\right]}_R,{\left[Zoo\right]}_R,{\left[SD\right]}_R,$ and ${\left[LD\right]}_R$ are the nitrates, ammonium, phosphates, dissolved oxygen, chlorophyll a, phyto-, zooplankton, small and large detritus at the river–lake boundary, respectively;

(c)

at the solid boundaries

$$\begin{array}{c}u=0;v=0;w=0;D_z{\displaystyle \frac{\mathsf{\partial }T}{\mathsf{\partial }n}}=-{\displaystyle \frac{H_{geo}}{{\rho }_0{c}_p}};{\displaystyle \frac{\mathsf{\partial }S}{\mathsf{\partial }n}}=0;{\displaystyle \frac{\mathsf{\partial }\left[N{O}_3\right]}{\mathsf{\partial }n}}=0;{\displaystyle \frac{\mathsf{\partial }\left[N{H}_4\right]}{\mathsf{\partial }n}}=0;{\displaystyle \frac{\mathsf{\partial }\left[P{O}_4\right]}{\mathsf{\partial }n}}=0;\hfill \\ {\displaystyle \frac{\mathsf{\partial }\left[O_2\right]}{\mathsf{\partial }n}}=0;{\displaystyle \frac{\mathsf{\partial }\left[Chl\right]}{\mathsf{\partial }n}}=0;{\displaystyle \frac{\mathsf{\partial }\left[Phyto\right]}{\mathsf{\partial }n}}=0;{\displaystyle \frac{\mathsf{\partial }\left[Zoo\right]}{\mathsf{\partial }n}}=0;{\displaystyle \frac{\mathsf{\partial }[SD]}{\mathsf{\partial }n}}=0;{\displaystyle \frac{\mathsf{\partial }[LD]}{\mathsf{\partial }n}}=0,\hfill \end{array}$$

where H_geo is the geothermal heat flux, W/m²; n is the direction of the outward normal to the domain; and

(d)

at the open boundary

$$\begin{array}{c}{\displaystyle \frac{\mathsf{\partial }\varphi }{\mathsf{\partial }t}}+c_{\varphi }{\displaystyle \frac{\mathsf{\partial }\varphi }{\mathsf{\partial }x}}=0;{\displaystyle \frac{\mathsf{\partial }w}{\mathsf{\partial }x}}=0\hfill \\ \left(\varphi =u,v,T,S,\left[N{O}_3\right],\left[N{H}_4\right],\left[P{O}_4\right],\left[O_2\right],\left[Chl\right],\left[Phyto\right],\left[Zoo\right],[SD],[LD]\right)\hfill \end{array}$$

where c_ϕ is the phase velocity [35]. While the main objective of this study was to reproduce the oxygen and zooplankton transport at the thermal bar due to cabbeling instability, we did not consider oxygen depletion from the sediment and zooplankton diel vertical migration. Concerning the vertical transport of zooplankton, it is assumed that its vertical propagation at the thermal bar is much more intense than its diel vertical migration. Oxygen exchange at the lake surface is parameterized by (17).

To parameterize radiative and turbulent heat fluxes entering the water surface, the following calculation formulas [36] were used:

• $H_{lw}={\epsilon }_w{\epsilon }_a\sigma \left(1+0.17C^2\right)T_A^4-{\epsilon }_w\sigma T^4,$
  where T and T_A are the air and water temperature (K), respectively; C is the cloud amount (from 0 to 1); σ = 5.669 × 10⁻⁸ W/m²/K⁴ is the Stefan–Boltzmann constant; and ε_w and ε_a are the water atmospheric emissivity, respectively;
• $H_L=f_u\left(e_A-e_w\right); f_u=6.9+0.345\cdot U_{10}^2; e_w=6.112\exp \left({\displaystyle \frac{17.67\cdot T_A^C}{T_A^C+243.5}}\right)$,
  where e_w is the pressure of saturated water vapor (hPa); e_A = 0.01·RH·e_w is the pressure of water vapor in the atmosphere (hPa); RH is the relative humidity (%); f_u is the heat-transfer coefficient (W/m²/hPa); $U_{10}=\sqrt{u_{10}^2+v_{10}^2}$ is the wind speed (m/s); and T_A^C is the air temperature (°C);
• $H_S=\beta \cdot f_u\left(T_A-T\right)$,
  where β = 0.61 hPa/K; and
• $H_{Ssol,0}=\left\{\begin{array}{c}{S}_0\cdot \left(a_g-a_w\right)\cdot \cos \zeta \left[a\left(C\right)+b\left(C\right)\ln \left(\cos \zeta \right)\right],if\cos \zeta >0;\\ 0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} if\cos \zeta \le 0,\hfill \end{array}\right.$
  where S₀ ≈ 1367 W/m² is the solar constant; a(C) and b(C) are the empirical coefficients [37]; $\zeta$ is the solar zenith angle; and the empirical functions a_g and a_w represent, respectively, molecular scattering and absorption by permanent gases [38].

The numerical method for solving the coupled differential equations follows [25,29]. Validation of the physical model that has been applied to Lake Baikal is presented in [39,40]. Simulated temperature, dissolved oxygen, and plankton distributions at the initial stage of the thermal bar evolution quantitatively correspond to the data measured in 1993 on the Boldakov River (52.59° N and 107.28° E)–Maloye More Strait (53.00° N and 106.94° E) cross-section of Lake Baikal [41,42]. Unfortunately, this study is limited by a lack of observational data on temperature, dissolved oxygen, and plankton dynamics for this cross-section in 2024. This makes it difficult to quantitatively compare results between different years.

3. Study Area

The calculations were performed for the cross-section of the Boldakov River–Maloe More Strait of Lake Baikal (Figure 2a) over a computational domain of 10,000 m × 1200 m (Figure 2b). The bathymetry data corresponding to the specified cross-section were taken from [41]. The computational domain is covered by a uniform rectangular grid with steps h_x = 50 m and h_z = 5 m. The time step is 30 s. A grid convergence study was conducted in [14]. Comparison of grid resolution results for the cross-section under consideration was discussed in a previous work [40].

The initial and boundary conditions for the biological model were assessed based on available sources on field observations [18,42,43,44,45,46,47]. The initial concentrations of nitrates, phosphates, ammonium, and oxygen were set at 1.0 mmol N m⁻³, 0.05 mmol P m⁻³, 1.0 mmol N m⁻³, and 312 mmol O₂ m⁻³, respectively. The concentrations of all detritus components and zooplankton at the beginning of simulations were 0.1 and 0.2 mmol N m⁻³, respectively. Chlorophyll a concentration was set based on the ratio of 1.59 for [Chl]/[Phyto] [21]. In the river mouth area, [O₂]_R = 375 mmol O₂ m⁻³, [PO₄]_R = 0.15 mmol P m⁻³, [Phyto]_R = 0.6 mmol N m⁻³, and boundary conditions for the remaining variables of the biogeochemical model matched the initial conditions.

The initial water temperature in the numerical experiments had a vertically non-uniform distribution that corresponded to the average temperature in June in the Central Baikal [46]. At the river mouth, the temperature was 11 °C initially and grew by 0.4 °C per day. Water mineralization in the lake and in the river was 96 mg/L [46] and 128.2 mg/L [47], respectively. The latter corresponds to average mineralization across all tributaries. The river flow velocity at the point of entry into the lake was 0.5 cm/s, which approximately corresponds to the value measured at the Srednyaya arm of the Selenga River [48]. The open boundary at the river inflow is 50 m deep. The horizontal turbulent diffusion coefficient was taken as D_x = 5.0 m²/s [49,50]; more detailed information on horizontal diffusivity diagnostics for Lake Baikal is contained in [51]. Bottom heat flux was set as constant conductive flux, H_geo = 0.138 W/m² [52]. Effects of geothermal heating in Lake Baikal were discussed in depth in [53]. Atmospheric parameters were based on meteorological data from the Sukhaya weather station from 1 June 2024 to 30 June 2024 [54].

4. Results and Discussion

In June 2024, wind speed (Figure 3a) predominantly ranged from 1 to 7 m/s. Gusts of northeastern wind ≥ 8 m/s were observed on the 4th and 18th days. Wind direction changed constantly throughout the month, settling only briefly (Figure 3b). On days 1, 3–4, 7–9, 16, 18–19, 23–26, and 30, prevailing winds were from the north, north-northeast, and northeast. On days 6, 14–15, 22, and 27, wind was from the west, while on days 4–5, 12–15, and 20, winds were from the south and southeast.

The lake warmed due to shortwave radiation and sensible heat flux (Figure 3c). The minimum peaks of shortwave radiation flux (360–420 W/m²) occurred on days 5 and 20–22 due to high cloud cover. The maximum shortwave radiation flux was 749 W/m², though on 8 out of 30 days (5, 8, 10–11, 19–22) it did not exceed 600 W/m². Sensible heat flux ranged from −2 to 90 W/m². Negative contributions to the thermal balance of the lake came from latent heat fluxes (from −88 W/m² to 0 W/m²) and, to a significant extent, from longwave radiation (from −93 W/m² to 56 W/m²).

The water in the thermoactive region rose to 8 °C on day 2 (Figure 4(a1)) and 10 °C on day 5 (Figure 4(a2)). These two days most informatively demonstrated the dynamics of the riverine thermal bar under prescribed conditions of simulation. The thermoinert region (especially the upper 200 m layer of the lake at a distance of 3000 m and beyond) remained cold relative to the remainder of the cross-section.

The 4 °C isotherm on day 5 extended from a point 2800 m away from the shore on the lake surface to a point 70 m deep at the estuary area. Dissolved oxygen ranged from 314 mmol O₂ m⁻³ to 360 mmol O₂ m⁻³, with maximum concentrations at the lake surface near the river mouth (Figure 4b). Oxygen transport to the deeper area of the lake was observed at the thermal bar: waters with a high oxygen content (>314 mmol O₂ m⁻³) sank to 390 m on day 2 (Figure 4(b1)), and to 480 m on day 5 (Figure 4(b2)). In the open lake, changes in oxygen concentration were only in the upper 200 m layer. Phytoplankton quantity (Figure 4c) in the thermoactive region varied from 0.24 mmol N m⁻³ to 0.41 mmol N m⁻³ on day 2 (to 0.47 mmol N m⁻³ on day 5). In the remainder of the section under study, phytoplankton concentration dropped to 0.18 mmol N m⁻³ on day 5. Vertical flows formed at the thermal bar carried phytoplankton biomass into the deeper layers. Over time, the concentration of zooplankton tended to decrease: In the open lake on day 2, it was below 0.1995 mmol N m⁻³ (Figure 4(d1)), and on day 5 it was below 0.1885 mmol N m⁻³ (Figure 4(d2)). Phyto- and zooplankton descended to the deep-water area due to downwelling generated by the thermal bar (Figure 5).

Note that the chemical analysis of water samples from the cross-section Kharauz arm (Selenga River)–Cape Krasnyj Yar of Lake Baikal in 2002–2007 [2] showed that the level of nutrients and phytoplankton in the thermoactive region was higher than in the thermoinert region. The vertical transport of oxygen (Figure 4(b1)) and plankton (Figure 4(c1,d1)) along the underwater slope at the initial stage of thermal bar development was consistent with the observation data [41,42]. As the thermal bar moved toward the open lake (this is displayed in the positions of the 4 °C isotherm at the lake surface in Figure 4(a1,a2)), the exchange of upper and deep waters with a higher content of dissolved oxygen occurred to depths of 500–600 m due to instability at the thermal bar position, ~2800 m from the river mouth (Figure 4(b2)).

Wind forcing orientations illustrated in Figure 3b demonstrated that wind direction relative to the coastline (Figure 2a) was highly changeable during June 2024. The position of the 4 °C isotherm at the lake surface and the vector velocity field (Figure 5) showed that the sinking of waters at the thermal bar reached a depth of 500 m on day 5. Interestingly, the thermohydrodynamic picture on day 2 (Figure 5a) was consistent with a classical structure of a thermal bar as a zone of water convergence at the temperature of maximum density [12,55], but, on day 5, there was a continuous subsurface flow (Figure 5b) resulting from changing the wind direction from the southwest to the southeast (Figure 3b) and increasing the wind speed from 1 m/s to 3 m/s (Figure 3a). This effect was especially strong in the thermoinert region (Figure 5b).

Coriolis effects on the cross-shore circulation are important during the thermal bar in temperate lakes [22,56,57]. Under these effects, the thermal bar acts as a barrier between alongshore flows in opposing directions, specifically strong cyclonic (inshore of the thermal bar) and weaker anticyclonic (offshore of the thermal bar) circulations [4,58]. At the eastern shore of Lake Baikal, the general cyclonic round-the-Baikal circulation is directed along the shoreline to the north [59,60,61]. The cyclonic circulation in the shallow water of the Selenga River and Lake Baikal can reach a speed of 1.5 cm/s [40]. It is possible to assume, therefore, that lateral gas transport to the thermal front can occur by convective circulation [62], and vertical gas transport to greater depths by thermal bar-driven downwelling. Thermal bar circulations are expected to dominate in deep temperate lakes during spring and autumn transitions.

In Lake Baikal, nutrients and organic matter are distributed very unevenly [44,63]. Nitrate depletion and phytoplankton concentrations can vary significantly over a 2-week period during the formation of thermal stratification [44]. The spring phytoplankton development differs from year to year; in particular, the “Melosira year” has a tenfold increase in productivity of the lake [43,47]. In addition, an unstable wind situation is typical of Lake Baikal, which leads to changeable water circulations [64].

Under the influence of northeastern and north-northeastern winds, the position of maximum density temperature on the lake surface (Figure 6a) quickly moved away from the river mouth on day 4 (from 1800 m to 4200 m) and on day 7 (from 2900 m to 4500 m). Due to southwesterly winds, this position moved toward the river mouth, compensating for the past rapid advance. Speed and direction of wind affect the formation and duration period of thermal bars [23,65], and sometimes active wind forcing can destroy vertical flows induced by a thermal bar [39].

The phosphate concentration at the position of maximum density temperature (Figure 6b) decreased from 0.07 mmol P m⁻³ to 0.058 mmol P m⁻³ over 7 days, with diurnal fluctuations exhibiting a diminishing amplitude (from 0.008 mmol P m⁻³ to 0.0015 mmol P m⁻³). The ammonium levels also decreased (from 1.0 mmol N m⁻³ to 0.98 mmol Nm⁻³), while the nitrate levels increased (from 1.0 mmol N m⁻³ to 1.03 mmol N m⁻³). Oxygen concentration in this position (Figure 6c) increased from 320 mmol O₂ m⁻³ to 337 mmol O₂ m⁻³ (with a maximum on day 7). Zooplankton biomass decreased from 0.2 mmol N m⁻³ to 0.185 mmol N m⁻³, and phytoplankton biomass decreased from 0.275 mmol N m⁻³ to 0.215 mmol N m⁻³. Phytoplankton concentration also fluctuated daily (similar to the phosphate dynamics) with a decreasing amplitude (from 0.03 mmol N m⁻³ to 0.01 mmol N m⁻³).

Oxygen at the thermal bar tended to grow when it moved toward the center of the lake (Figure 6c), similar to the case for the shallow section of Lake Baikal [66]. However, due to the shift in the thermal bar front to the river mouth under the opposite wind forcing, the reverse trend was observed. The monotonous nature of the change in the nitrate, ammonium, and zooplankton concentrations at the thermal bar (Figure 6b,c) coincided with the simulation results performed for Barguzin Bay of Lake Baikal [67].

Previous studies show that the behavior of the thermal bar is highly sensitive to wind, cloudiness, and diurnal variability of solar radiation [14,25,36,39]. Hence, the distribution of oxygen and plankton can also depend on the atmospheric factors. Note that global climate changes, documented since the middle of the 20th century, are reflected in the thermal structure of large temperate lakes [68,69,70].

In Lake Baikal, the phytoplankton concentrations between a high and a low production year can vary substantially [47,71]. In these circumstances, numerical modeling of the biogeochemical processes during the riverine thermal bar requires more detailed initial and boundary conditions. Due to the continuous interaction between river and lake waters, the selection of appropriate conditions presents a problem for simulations [4]. In developing aquatic biogeochemical models, it is also important to consider the adopted model complexity and the space-time scale in accordance with the objectives of the study [72].

Thermally driven cross-shore flows affect the biogeochemical functioning of lake ecosystems through the redistribution of water masses between littoral and pelagic areas [7]. Our study was aimed at understanding the effect of the thermal bar on the oxygen and plankton propagation. In this case, it is crucial to analyze first the qualitative picture. Quantitative indicators are also important; however, that is the subject for future work.

5. Conclusions

The mathematical model described here allows us to simulate hydrobiological processes in a deep lake during the spring thermal bar development. It incorporates both biochemical and physical dynamics of dissolved oxygen.

The results of numerical modeling conducted for the cross-section of Lake Baikal, Boldakov River–Maloye More Strait, which is characterized by great depths, show high concentrations of oxygen, phytoplankton, and zooplankton in two areas:

(i)

the upper layer of the lake—in the littoral zone;

(ii)

the deep-water column of the lake—over the underwater slope.

Increased oxygen content was also noted in the surface layer of the thermoinert region, which is related to the influx of atmospheric oxygen into the lake under the influence of wind.

As the spring thermal bar develops in the Boldakov River–Maloe More cross-section under the 2024 meteorological conditions, a tendency was observed at the thermal bar for:

(i)

increased concentration of nitrates and dissolved oxygen;

(ii)

decreased ammonium, phosphates, phyto- and zooplankton.

This study revealed that by contributing to the vertical transfer of dissolved oxygen, thermal bars can play a determining role in natural mechanisms of deep-water renewal in Lake Baikal.