1 Introduction

Climate change induces global climate patterns that can lead to an increase in extreme weather events, such as droughts and torrential downpours. Droughts result in precipitation deficits, consequently diminishing river water levels. Intense downpours generate significant amounts of precipitation in a short time, resulting in a rapid rise in river water levels. Elevated river water temperatures, driven by climate change, can facilitate eutrophication and reduce oxygen concentrations in lakes and oceans, thereby causing anoxia.

River inflows act as the primary suppliers of nutrients, organic matter, and suspended sediments to oceans, lakes, and reservoirs (Ramón et al. 2020; Piton et al. 2022; May et al. 2023). These materials serve as nutrient sources in lake ecosystems, impacting biological growth and productivity. Additionally, river inflows influence the water quality of lakes and oceans, as river water exhibits distinct water quality characteristics, resulting in variability when mixed with lake water quality. Furthermore, river inflows impact water levels and temperatures in lakes. Variations in precipitation and snowmelt water alter river flows, which are subsequently reflected in lake levels and water temperatures. In particular, increased snowmelt water elevates the lake level and affects water temperature (Trikoilidou et al. 2017; Mavromati et al. 2018; Mentzafou and Dimitriou 2019; Ambrose-Igho et al. 2021; Alehu and Bitana 2023; Šarović and Klaić 2023).

Lake Biwa is Japan’s largest freshwater lake and serves as a crucial water resource for the Kansai region. It also plays a pivotal role in the ecosystems surrounding its shores and the local economy, making its maintenance and management essential for the sustainable development of the region. In recent years, the effects of climate change due to global warming have been widely discussed, and the influence of changes in precipitation and snowmelt water on the local water environment has also garnered attention.

Recent studies suggest that the characteristics and quality of river water flowing into Lake Biwa are influenced by factors such as local precipitation and topography. For instance, the relationship between precipitation and river water quality in the Yasu River basin, a major river in Lake Biwa, was investigated (Endoh et al. 2007). The results revealed a clear correlation between flow and water quality parameters, such as dissolved oxygen and turbidity, in the Yasu River. On the other hand, snowmelt entering rivers is affected by climate change. Gordon et al. (2022) examined the relationship between climate change and snowmelt in 537 watersheds in the United States. They identified three interconnected mechanisms: snow season mass and energy exchanges, the intensity of snow season liquid water inputs, and the synchrony of energy and water availability. These mechanisms reveal the interplay between snowpack and streamflow.

River plumes that flow into the lake remain near the surface if their density is lower than the surface density (Jirka 2008). However, if the density is high due to low temperatures and high levels of dissolved organic matter, such as snowmelt water, they flow as gravity currents downward along the bottom (Finger et al. 2006). In cases where the lake is stratified, the plumes may flow horizontally above the lake bottom along a layer with a density similar to that of river water (Ahlfeld et al. 2003). Most studies on plumes from riverine inflows have concentrated on the coastal ocean, where seawater and river water mix, and the diffusion of river plumes at the estuary is complicated by factors such as flow, salinity, tides, geographic shape, winds, and stratification strength (Arevalo et al. 2022; Clark and Mannino 2022). In comparison to large river systems, buoyant plumes originating from small river systems have smaller spatial scales and, consequently, shorter residence times of freshened water, typically measured in hours and days due to the relatively low volume of river discharge and its intense mixing with the ambient sea (Baldoni et al. 2022). Osadchiev and Sedakov (2019) and Basdurak and Largier (2022) demonstrated that the spreading pattern of small river plumes differs from that typical for large and medium-sized rivers and is primarily influenced by wind forcing. The influence of wind in Lake Biwa is discussed in Koue et al. (2023); in this study, I analyze how multiple river plumes affect the seasonal variability of mixing processes throughout the enclosed water body in Lake Biwa, using scenarios of large and small inflows and high and low inflow river temperatures.

The characteristics of river water entering lakes and the influence of snowmelt water are also closely tied to changes in precipitation in the region. However, detailed studies on the interaction between these factors are still limited. The aim of this study is to comprehensively investigate and analyze the effects of river water characteristics, the influence of snowmelt water, and climate change-induced alterations in precipitation on the stratification structure of Lake Biwa.

2 Hydrodynamic Model in Lake Biwa

2.1 Simulation Domain

This research focused on Lake Biwa, which is located in Shiga prefecture in Japan. Figure 1 depicts the simulation domain and water depth in the Lake Biwa hydrodynamic model. With a horizontal resolution of 500 m, the horizontal domain is 36 km × 65.5 km. The vertical domain is made up of 86 layers that extend from the surface of the lake to a depth of 107.5 m. The vertical grid size starts at 0.5 m and gradually increases up to 2.5 m from the surface to a depth of 20 m. In the summer, a thermocline forms at depths ranging from 10 to 30 m. This thermocline prevents the vertical transport of momentum and heat.

Fig. 1
figure 1

Simulation domain in Lake Biwa: a horizontal domain, b south-north and (c) west-east vertical cross sections through Imazu-oki point

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2.2 Constitutive Equation

In the present investigation, the three-dimensional hydrodynamic model, as established by Koue et al. (2018), was employed to conduct an in-depth examination of the thermocline and vorticity. The verification of this model’s ability to faithfully replicate the thermal stratification and flow patterns in Lake Biwa was conducted through a meticulous comparison with observed data. It is pertinent to note that the coordinate system employed in this study is referenced to the southwestern boundary of the spatial domain within the horizontal plane. The x and y axes were aligned with the west–east and south–north directions, respectively, while the z-axis was oriented in an upward direction.

Momentum equations (x, y direction) are written by

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}-fv=-\frac{1}{{\rho }_{0}}\frac{\partial p}{\partial x}+{\nu }_{h}\frac{{\partial }^{2}u}{\partial {x}^{2}}+{\nu }_{h}\frac{{\partial }^{2}u}{\partial {y}^{2}}+{\nu }_{z}\frac{{\partial }^{2}u}{\partial {z}^{2}}-\frac{g}{{\rho }_{0}}{\int }_{z}^{0}\frac{\partial \rho }{\partial x}dz$$
(1)
$$\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}+fu=-\frac{1}{{\rho }_{0}}\frac{\partial p}{\partial y}+{\nu }_{h}\frac{{\partial }^{2}v}{\partial {x}^{2}}+{\nu }_{h}\frac{{\partial }^{2}v}{\partial {y}^{2}}+{\nu }_{z}\frac{{\partial }^{2}v}{\partial {z}^{2}}-\frac{g}{{\rho }_{0}}{\int }_{z}^{0}\frac{\partial \rho }{\partial y}dz$$
(2)

Hydrostatic equation is given by

$$0=-\frac{1}{{\rho }_{0}}\frac{\partial p}{\partial z}-\frac{\rho }{{\rho }_{0}}g$$
(3)

Continuity Equation is described by

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{\partial z}=0$$
(4)

Conservation equation for water temperature is written by

$$\frac{\partial T}{\partial t}+u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}+w\frac{\partial T}{\partial z}={\kappa }_{h}\frac{{\partial }^{2}T}{\partial {x}^{2}}+{\kappa }_{h}\frac{{\partial }^{2}T}{\partial {y}^{2}}+{\kappa }_{z}\frac{{\partial }^{2}T}{\partial {z}^{2}}$$
(5)

Density of water is calculated as follows (Fofonoff and Millard 1983)

$$\rho=1000.07-0.00469(T-273.15)^{2}-0.0035(T-273.15)$$
(6)

where \(u\), \(v\), and \(w\) are the \(x\), \(y\), and \(z\) components of current velocity (\(\text{m} \ \text{s}^{-1}\)), \(T\) is the water temperature (\(\text{K}\)), \(p\) is the pressure (\(\text{N} \ {\text{m}}^{-2}\)), \(\rho\) is the density of water (\(\text{kg} \ {\text{m}}^{-3}\)), \({\rho }_{0}\) is the reference density of water (= \({10}^{3} \ {\text{kg}} \ {\text{m}}^{-3}\)), \(\text{g}\) is the acceleration due to gravity (= \(9.8 \ \text{m} \ {\text{s}}^{-2}\)), \(f\) is the Coriolis parameter (= \(8.34\times 1{0}^{-5} \ {\text{s}}^{-1}\) corresponding to 35° N), \({\nu }_{h}\) is the horizontal eddy viscosity coefficient for momentum equations (=\(1.0 \ {\text{m}}^{2} \ {\text{s}}^{-1}\)), \({\kappa }_{h}\) is the horizontal eddy diffusivity coefficient (=\(\ 1.0 \ {\text{m}}^{2} \ {\text{s}}^{-1}\)), \({\nu }_{z}\) is the vertical eddy viscosity coefficient (\({\text{m}}^{2} \ {\text{s}}^{-1}\)) for momentum equations, and \({\kappa }_{z}\) is the vertical eddy diffusivity coefficient (\({\text{m}}^{2} \ {\text{s}}^{-1}\)). The horizontal eddy viscosity and diffusivity coefficients are chosen \(1.0 \ {\text{m}}^{2} \ {\text{s}}^{-1}\) as constant values.

During the summer season, an examination of observed water temperature profiles revealed the presence of a thermocline at water depths ranging from 10 to 30 m. This thermocline effectively impedes vertical momentum and heat fluxes. To account for this effect, the values of the vertical eddy viscosity and diffusivity coefficients are estimated based on the Richardson number (\(Ri\)), which is computed as follows:

$$Ri\text{= }-\frac{g}{{\rho }_{0}}\frac{\frac{\partial \rho }{\partial z}}{{\left(\frac{\partial {U}_{w}}{\partial z}\right)}^{2}}$$
(7)

where \({U}_{w}=\sqrt{{u}^{2}+{v}^{2}}\) is the horizontal current velocity (\({\text{m s}}^{-1}\)). The Richardson number is the dimensionless number that indicates the proportion of the buoyancy term to the flow gradient term as followed (Munk and Anderson 1948; Webb 1970).

The vertical eddy viscosity is given by:

$${\nu }_{z}\text{= }\frac{0.0001}{1.0+5.2Ri}$$
(8)

The vertical eddy diffusivity is given by:

$${\kappa }_{z}\text{= }\frac{0.0001}{(1.0+\frac{10}{3}\times Ri{)}^{\frac{3}{2}}}$$
(9)

2.3 Initial Conditions

The initial condition was set to 0 m/s for the current velocity. The initial water temperature on April 1st, 2006 was calculated using linear interpolation of data collected on March 20th and April 10th, 2006. The Lake Biwa Environmental Research Institute conducted the observations of water temperature twice a month at the monitoring point Imazu-oki (35°23’41” N., 136°07’57” E.), with depths of 0.5 m, 5 m, 10 m, 15 m, 20 m, 30 m, 40 m, 60 m, 80 m, and approximately 90 m, which is deepest observed point of Lake Biwa.

2.4 Boundary Conditions

The formula, which takes wind stress into account, gives the boundary conditions for the current velocity at the surface of the lake. Short-wave solar radiation (S↓)), latent heat flux (Ql), sensible heat flux (Qs), and net long-wave radiation (\({L}_{net}\)) comprise the heat flux through the sea surface.

2.4.1 Boundary Conditions at the Surface of the Lake

At the surface of the lake, considering the effect of the wind, \({\tau }_{x} \ \text{and} \ {\tau }_{y}\) are the surface wind frictional stresses which can be described by bulk formula:

$$\begin{array}{c}{\tau }_{x}={\rho }_{a} {C}_{f} {U}_{\text{ax}}{ U}_{a} \\ {\tau }_{y}={\rho }_{a}{ C}_{f} {U}_{\text{ay}}{ U}_{a}\end{array}$$
(10)
$${C}_{f}=(1.0+0.07{U}_{a})\times 1{0}^{-3}$$
(11)

where \({\rho }_{a}\) is the density of air (\({\text{kg m}}^{-3}\)), \({C}_{f}\) is the wind frictional constant, \({U}_{ax}\) and \({U}_{ay}\) are the \(x\) and \(y\) components of wind velocity (\({\text{m s}}^{-1}\)), \({U}_{a}\) is the wind speed from a height of 10 m above the surface (\({\text{m s}}^{-1}\))

The boundary conditions for the velocity at the surface of the lake are given by the following formula, taking the wind stress into account:

$$\begin{array}{c}{\upsilon }_{z}\frac{\partial u}{\partial z}=\frac{{\tau }_{x}}{{\rho }_{0}} \\ {\upsilon }_{z}\frac{\partial v}{\partial z}=\frac{{\tau }_{y}}{{\rho }_{0}}\end{array}$$
(12)

The heat flux through the water surface consists of short-wave solar radiation (\(S\downarrow\)), latent heat flux (Ql), sensible heat flux (Qs) and net long-wave radiation (Lnet). The heat balance equation on the water surface is given by:

$$\rho{\text{ C}}_{p} {\kappa }_{z}\frac{\partial T}{\partial z}=\text{S}\downarrow +{\text{Q}}_{l}+{\text{Q}}_{s}+{\text{L}}_{\text{net}}$$
(13)

where \({C}_{p}\) is specific heat of water (=\(4.2\times 1{0}^{3}\) (\(\text{J kg}^{-1} \ {\text{K}}^{-1}\))).

Short-wave solar radiation is written by:

$$S\downarrow ={S}_{0}\downarrow (1-\text{ref})$$
(14)

where \({S}_{0}\downarrow\)(\({W \ m}^{-2}\)) is the observed global solar radiation. ref stands for albedo. ref was assumed to be 0.07. Some of short-wave solar radiation are absorbed by the surface water, and the others intrude into deep waters.

Assuming that the intensity of the short-wave-radiation decreases exponentially with the distance from the water surface, the intensity of the short-wave radiation at the water depth of z (m) can be described as follows:

$$S\downarrow \left(z\right)={S}_{0}\downarrow (z=0)\cdot {exp}(-\frac{1.7}{4.5}z)$$
(15)

where \(z\) is water depth (m), \({S}_{0}\downarrow (z=0)\) is short-wave solar radiation at the water surface (\({\text{W m}}^{-2}\)), and \(S\downarrow \left(z\right)\) is the intensity of the short-wave solar radiation at the depth of \(Z\) (m) (\({\text{W m}}^{-2}\))

The latent heat flux (Ql) is estimated by:

$${Q}_{l}=E\times {\rho }_{a}{C}_{E}({q}_{s}-{q}_{a}){U}_{a}$$
(16)

where \({C}_{E}\) is bulk constant coefficient of latent heat, which is chosen \(1.2\times 1{0}^{-3}\) as the constant value, 𝐸 is latent heat of evaporation (2.45 × 106). \({q}_{s}\) is saturation specific humidity, \({q}_{a}\) is specific humidity.

The sensible heat flux (Qs) is estimated by:

$${Q}_{s}={\rho }_{a}{ C}_{pa}{ C}_{H}({T}_{a}-{T}_{w}){U}_{a}$$
(17)

where \({C}_{pa}\) is the specific heat of air (=\(1.01\times 1{0}^{3}\) (\(\text{J kg}^{-1} \ {\text{K}}^{-1}\))), \({C}_{H}\) is the bulk constant coefficient of sensible heat (=\(1.2\times 1{0}^{-3}\)), \({T}_{a}\) is atmospheric temperature (K), and \({T}_{w}\) is water temperature (K) at the surface of the lake.

The net long-wave radiation (Lnet) is written as:

$${L}_{net}=-\varepsilon \sigma {T}^{4}+\left(1.0-Rr\right)\times (1.0+0.17n{)}^{2}\times\beta\times\sigma{\text{T}}_{a}^{4}$$
(18)

where it is assumed that emissivity (\(\varepsilon\)) is \(0.98\), \(\sigma\) is Stefan-Boltzmann constant (=\(5.67\times 1{0}^{-8} \ {\text{W m}}^{-2} \ {\text{K}}^{-4}\)), \(Rr\) (surface reflectivity) is \(0.2\), \(n\) is the amount of cloud in the range from 0 to 100 (%), and \(\beta\) is a correction coefficient (=\(0.937\times 1{0}^{-5}\times {T}^{2}\)).

At the bottom, a no-slip condition is imposed. A no-slip condition is also placed on the side walls. Normal temperature gradients of water are also zero, resulting in thermal insulation.

In the Lake Biwa flow field model, the meteorological data for 2006 and 2007 (Grid Point Value (GPV) data) from the Japan Meteorological Agency were used for the heat balance at the lake surface and wind-induced shear stress on the water surface. The data items used were air temperature (K), atmospheric pressure (Pa), wind speed (m/s) at 10 m above ground level, and relative humidity (%). The GPV data were initially set to (00, 03, 06, 09, 12, 15, 18, 21 UTC) for a total of 8 times, and 15-hour forecast values (1-hour intervals for ground data and 3-hour intervals for each pressure plane data) were handled. In order to reflect the GPV data appropriately for each surface mesh, interpolated values were obtained by weighted averaging inversely proportional to the square of the distance, and used as meteorological data. The geographic resolution of GPV MSM data is 0.0625° (longitude) 0.05° (latitude) (approximately 5 km) and the temporal resolution was one hour. The data were horizontally interpolated into the hydrodynamic model’s surface meshes. Hourly observations at the Hikone local meteorological observatory (35°16’30” N., 136°14’36” E.) were used to calculate solar radiation. The solar radiation on the lake was considered to be horizontally uniform. This meteorological data was updated at hourly intervals and used in the flow field model calculations. The hourly updates enabled a more detailed reproduction of the heat balance and flow at the lake surface.

The boundary conditions for the water volume and temperature related with inflow from each river and outflow from the lake were calculated by the hydrological model (Shrestha and Kondo 2015).

2.4.2 River Condition

For taking into account the effect of the river inflow to Lake Biwa, the simulated data and not the observed data are given as boundary conditions. The water temperature and the water volume which are calculated by the hydrological model are interpolated each day for 56 rivers. The position of 56 inflow rivers and one outflow river (setagawa), and the annual mean river inflow are shown in Fig. 2. The only one outflowing river is located at the southern-most part of Lake Biwa. Furthermore, the grid size of the area of the simulation is 1 km, and the total number of mesh cells is 7557. The data of the river inflow is updated every day for detailed analysis.

Fig. 2
figure 2

The position of 56 inflow rivers and one outflow river, and annual mean river inflow

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2.5 Simulation Cases

The baseline simulation (BASE) utilized authentic meteorological data spanning from April 1st, 2006 to March 31st, 2009. This temporal framework encompassed a spin-up phase commencing on April 1st, 2006, and concluding on March 31st, 2007. Furthermore, we conducted experimental simulations for this identical time period employing meteorological data featuring modified temperature and precipitation volume, as well as alterations in river characteristics, commencing from April 1st, 2007. The objective underlying these simulations was the comprehensive exploration of the influence of temperature, precipitation volume, and river conditions on stratification dynamics.

For the numerical experiments, Table 1 shows the annual streamflow data for the six main rivers in the baseline case, and Table 2 shows the annual mean water temperature data for these rivers. Numerical simulations were performed for each condition, adjusting for precipitation and river inflow water temperature. The simulation case, referred to as the “R_double_O” or “R_half_O” was run to evaluate a scenario in which both precipitation and river flow were doubled or halved compared to the baseline conditions shown in Table 1; Fig. 3, while water temperature levels were held constant. The simulation case, referred to as the “R _O_+5℃” or “R _O_-5℃” was run to evaluate a scenario in which both rainfall water temperature and river temperature were increased and decreased by 5.0 ℃ compared to the baseline conditions presented in Table 2. Table 3 shows the nomenclature of each simulation case based on rainfall temperature and precipitation and river characteristics.

Table 1 Mean annual river flow rate (m3/s)
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Table 2 Mean annual river temperature (℃)
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Fig. 3
figure 3

Hourly precipitation and annual total river inflow rate used in the baseline simulation at Hikone local meteorological observatory from April 1st, 2007 to March 31st, 2009

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Table 3 Precipitation and river conditions in numerical experiments
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The total amount of river water, groundwater, and direct precipitation that flows into Lake Biwa is estimated to be about 5.89 billion tons per year. Since the storage capacity of Lake Biwa is 27.5 billion tons, the storage period is approximately 4.7 years. Given the large amount of river inflow and precipitation relative to Lake Biwa’s water storage capacity, it is necessary to comprehensively evaluate the fluctuations in river inflow and precipitation in analyzing the circulation pattern from the lake surface to the deepest part of the lake.

Assuming that the heat content of Lake Biwa’s water storage is 18.5 × 105GJ/kg/K and the heat content from precipitation and river inflow is 4.0 × 105 GJ/kg/K, the R_double_O and R_half_O cases are 8.0 × 105 GJ/kg/K and 2.0 × 105 GJ/kg/K, respectively, and the R _ O _+ 5°C and R _ O _-5°C cases result in 5.2 × 105 GJ/kg/K and 2.7 × 105 GJ/kg/K, respectively, of heat provided by precipitation and river inflow. In other words, 43% of the total heat storage in Lake Biwa is provided by precipitation and river inflow in R_double_O, 11% in R_half_O, 28% in R _ O _+5°C, and 15% in R _ O _ -5°C.

2.6 Sensitivity Analysis of the Stratification by Using the Brunt–Väisala Frequency

In the context of a continuous stratified fluid, the behavior of an object suspended in a state of neutral buoyancy, where its density matches that of the surrounding fluid, exhibits distinct characteristics. When this object is displaced upward, it experiences an increased density relative to the surrounding fluid, whereas when it is pushed downward, it encounters a decreased density compared to the surrounding fluid. Consequently, a restorative force, seeking to return the object to its initial position, comes into play. The magnitude of this restorative force is directly proportional to the vertical displacement of the object. Under conditions where the fluid is undisturbed by the motion of the object and where viscosity can be disregarded, this object undergoes simple harmonic oscillation. The frequency of this oscillation is known as the Brunt–Väisala frequency. The Brunt–Väisala frequency is directly proportional to the square root of the vertical variation in density, making it a crucial parameter for quantifying the strength of stratification in the fluid.

The square of the buoyancy frequency is described as follows:

$${N}^{2}=-\frac{g}{\rho }\cdot \frac{\partial \rho }{\partial z}$$
(19)

where

$$\rho=1000.07-0.0469(T-273.15)^{2}-0.0035(T-273.15)$$
(20)

The stratification is statically stable when N2 > 0, and statically unstable when N2 < 0.

3 Results and Discussion

3.1 Effect of Change in Volume of Precipitation and River Inflow on Stratification

Figure 4 illustrates the seasonal averages of the vertical distribution of water temperature and the differences in Brunt–Väisala frequency between the BASE case and each alternative scenario at Imazu-oki. When the precipitation and river inflow into Lake Biwa are doubled, river water flowing from the surface to depths shallower than the thermocline cools the surface layer from April to September. This cooling results in a negative and unstable Brunt–Väisala frequency. Near the thermocline, the water temperature increases by 0.3 to 0.8 °C, indicating the occurrence of vertical mixing with the surface layer. During the stratified period from April to June, the stratification becomes unstable at depths ranging from 30 to 50 m due to the increased river inflow, in contrast to the July to August period, when stratification is stronger.

Fig. 4
figure 4figure 4

Seasonal average of vertical distribution of Water Temperature and Brunt–Väisara frequency differences between BASE and each case at Imazu-oki in case of: a-d R_double_O, R_half_O and (e)-(h) R_O_+5℃, R _O_-5℃

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During the cooling period spanning from October to December, the impact of increased river water extends deeper into the thermocline. The water temperature increases by approximately 0.4 °C from 60 m to 70 m in depth, signifying the influence of the river inflow in accelerating the descent of the thermocline. In the period from January to March, encompassing the full-layer circulation phase, the river inflow contributes to diffusion to deeper layers.

In the scenario where precipitation and river inflow into Lake Biwa are halved, the Brunt–Väisala frequency becomes negative and unstable from April to June, spanning from the surface to the thermocline. From July to September, it turns positive and stable near the surface. In both October to December and January to March periods, the water temperature changes are less than 0.4 °C, and the stratification remains stable across all layers.

3.2 Effect of Change in Temperature of Precipitation and River Inflow on Stratification

When the water temperature of the rivers flowing into Lake Biwa is increased by 5 °C, the water temperature in the lake rises by 0.4–0.8 °C from the surface to near the thermocline during the period from April to June. Additionally, there is an increase of about 0.2 °C in the bottom layer. From October to December, warmer river water enters the surface layer, causing an elevation of the water temperature in the lake by approximately 0.4 °C. These temperature increases are observed across all layers. During January to March, when full-layer circulation takes place, water temperature increases in the range from 0.2 to 0.4 °C across all layers.

Conversely, if the river water temperature decreases by 5 °C, a significant decrease in water temperature is noted from the surface to a depth of approximately 50 m during the April to June period when stratification begins, with a maximum decrease of 0.8 °C. This indicates unstable stratification and active vertical mixing. From July to September, when stratification is stronger, water temperature decreases up to 0.8 °C, extending toward the vicinity of the thermocline, suggesting active vertical mixing from the surface to the vicinity of the thermocline. In the October to December period, when the temperature decreases, there is a negative fluctuation in the Brunt–Väisala frequency from the surface to a depth of 70 m, indicating unstable stratification and a greater likelihood of vertical diffusion. Even during January to March, which encompasses full-layer circulation, fluctuations in river water temperature contribute to vertical mixing.

When the temperature of a river is either increased or decreased by 5 °C, it is observed that the fluctuations in the temperature of a lake are more pronounced compared to the scenario where the river flow is doubled or halved. As a result, the doubling or halving of the river flow rate causes greater fluctuations in lake water temperature and affects stratification intensity than a 5 °C increase or decrease in river water temperature, even though twice as much heat is added to the water.

The total inflow of river water, groundwater contributions, and direct surface precipitation into Lake Biwa is estimated to be approximately 5.89 billion m3 per year. Given that the storage capacity of Lake Biwa is 27.5 billion m3, the calculated retention time stands at approximately 4.7 years. The substantial volume of river inflow and precipitation, relative to Lake Biwa’s storage capacity, underscores the need to assess their variations comprehensively, as they play a pivotal role in the analysis of circulation patterns from the lake’s surface to its deepest layers. When the river water temperatures are low, it leads to a downstream movement of the plume as it enters the lake, resulting in a significant alteration of the structure of stratification in the lake. This effect intensifies the descent of the thermocline, both during the heating season, where it descends from the surface towards the thermocline temperature, and during the cooling season, where it sinks from a depth below the thermocline towards the bottom layer. This phenomenon promotes vertical mixing and enhances the oxygenation of the bottom layer, particularly in cases where the bottom layer becomes anoxic. In Lake Biwa, observations have also shown that snowmelt water contributes to the dissolved oxygen supply at the bottom of Lake Biwa, proving this phenomenon (Fukushima et al. 2019). Furthermore, near the mouth of the Tama River in central Tokyo Bay, it has been observed that continuous strong southerly winds, dense seawater intrusion, and extreme freshwater upwelling of the Tama River due to typhoons also contribute to DO recovery (Karim et al. 2022).

3.3 Effects of Changes in Volume of Precipitation and River Inflow on Vertical Velocity and Water Temperature

Figure 5 shows the vertical velocity in cross sections in the east-west and north-south directions offshore of Imazu, along with the velocities in the surface layer, when the precipitation and river inflow to Lake Biwa are doubled and halved, respectively. Figure 6 depicts cross-sectional profiles through Imazu-oki in both the east-west and north-south directions, along with the distribution of water temperatures in the surface layer when the precipitation and river inflow into Lake Biwa are doubled or halved. When precipitation and river flow are doubled, the velocities near the surface in Fig. 5a and c show that the velocities are stronger near the river inflow. As shown in Fig. 6a to d, in May and August, when precipitation and river flow are increased by a factor of 2, the plume from the river inflow flows into the stratification layer deeper than when precipitation and river flow are decreased by a factor of 1/2, destroying the stratification layer deeper than the water temperature dynamic layer and promoting vertical mixing in the deep layer. As Fig. 5c shows, near-surface current velocities are stronger near the inflow river in November, and vertical current velocities are also comparatively higher in the downward direction of the inflow. Figure 6c shows that the influence of the incoming river extends to the bottom layer, destroying the stratification of the bottom layer and increasing vertical mixing, resulting in a uniform water temperature in the bottom layer. As Fig. 6g to h show, in February during the winter season, when precipitation and river flow are increased by a factor of 2 and decreased by a factor of 1/2, respectively, the flow near the surface layer does not change that much, but a comparison of vertical velocities confirms that the flow is flowing in along the slope of the lake bottom.

Fig. 5
figure 5figure 5

Seasonal change in vertical distribution of simulated water temperature through Imazu-oki (Color contour: Temperature (◦C)): a, c, e, g R_double_O; and (b), (d), (f), (h) R_half_O)

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Fig. 6
figure 6figure 6

Seasonal change in vertical distribution of simulated water temperature through Imazu-oki (Color contour: Temperature (◦C)): a, c, e, g R_double_O; and (b), (d), (f), (h) R_half_O)

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3.4 Effects of Changes in Temperature of Precipitation and River Inflow on Vertical Velocity and Water Temperature

Figure 7 shows the velocity distribution in the surface layer and the vertical velocity cross sections in the east-west and north-south directions off Imazu when the river water temperature is increased or decreased by 5 °C. Figure 8 depicts cross-sectional profiles through Imazu-oki in both the east-west and north-south directions, along with the distribution of water temperatures in the surface layer when the river water temperature is increased or decreased by 5 °C. As shown in Fig. 7a and b, in May, when the river temperature is increased by 5 °C, the surface current velocity is higher than when the river temperature is decreased by 5 °C. As a result, as shown in Fig. 8a and b, there is an approximately 2 °C difference in the distribution of surface water temperatures, indicating the influence of fluctuations in river water temperature extending to depths below the thermocline. Notably, the difference in water temperature is more pronounced in the southern part of the lake compared to the northern part, and it is also more significant in the coastal area than in the offshore region. The northern part of the lake experiences a lesser impact from variations in water temperature compared to the coastal area around the center, off the coast of Imazu.

Fig. 7
figure 7figure 7

Seasonal change in vertical distribution of simulated water temperature through Imazu-oki (Color contour: Temperature (°C)): a, c, e, g R_O_+5 °C; and (b), (d), (f), (h) R _O_-5 °C)

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Fig. 8
figure 8figure 8

Seasonal change in vertical distribution of simulated water temperature through Imazu-oki (Color contour: Temperature (°C)): a, c, e, g R_O_+5 °C; and (b), (d), (f), (h) R _O_-5 °C)

Full size image

According to Fig. 7a to h and 8a to h, the effect of the river flow and water temperature difference on the bottom layer remains consistent throughout the entire period. During the cooling period, vertical velocities along the slope, due to inflow from the river, can be seen from Fig. 7e to h. In November, when the river water temperature decreases by 5 °C, the water temperature near the surface is 2 °C lower from nearshore to offshore compared to when the river water temperature increases by 5 °C, resulting in a more pronounced thermocline. Bottom water temperatures, in the case of a 5 °C decrease in river water temperature, are warmer than those in the case of a 5 °C increase in river water temperature, suggesting that vertical mixing occurs near the bottom layer.

In February, when the river temperature decreases by 5 °C, the water temperature near the surface decreases by 1 to 2 °C compared to when the river temperature increases by 5 °C, also indicating the occurrence of vertical mixing near the bottom layer. This decrease in water temperature is more noticeable in the coastal area than in the offshore region, suggesting that water temperature changes are more prominent on the east coast than on the west coast.

The fluctuation in water temperature from the deep layer to the bottom layer suggests that the plume of river water flows as a gravity current downward along the bottom, which is consistent with observations in previous studies (Finger et al. 2006; Ahlfeld et al. 2003). This phenomenon contributes to the overall circulation observed in February, encompassing the entire water column.

4 Conclusions

The impact of precipitation volume and temperature, as well as river flow variations, on the structure of stratification in Lake Biwa was investigated through simulations. Increasing precipitation and river water flow by a factor of 2 in Lake Biwa cools the surface layer from April to September, resulting in a negative and unstable Brunt–Väisala frequency. This leads to vertical mixing with a temperature increase near the thermocline. During the stratification period from April to June, increased river inflow results in unstable stratification at depths ranging from 30 to 50 m, while July to August exhibits stronger stratification. In the cooling period from October to December, increased river water affects the thermocline, causing a temperature increase of approximately 0.4 °C at depths between 60 and 70 m. River inflow facilitates the descent of the thermocline during January to March, promoting diffusion to deeper layers. Decreasing precipitation and river water flow by a factor of 2 leads to a negative and unstable Brunt–Väisala frequency near the surface and thermocline from April to June. From July to September, the frequency becomes positive and stable near the surface, while from October to December and January to March, water temperature changes are less than 0.4 °C, indicating stable stratification across all layers.

Increasing river water temperature by 5 °C raises water temperature near the thermocline and in the bottom layer from April to June. In October to December, warm river water raises the surface temperature of the lake by approximately 0.4 °C. Similar temperature increases are observed in all layers from January to March. Decreasing river water temperature by 5 °C results in significant temperature decreases from the surface to around 50 m depth from April to June, indicating unstable stratification and vertical mixing. From July to September, temperature decreases are observed near the thermocline, indicating active vertical mixing. Fluctuations in the Brunt–Väisala frequency suggest unstable stratification and increased vertical diffusion from the surface to a depth of 70 m during the cooling period from October to December. River temperature fluctuations also contribute to vertical mixing during full-layer circulation in January to March.

The influence of river water temperature fluctuations on surface water temperature distribution shows greater differences in the southern and coastal areas of Lake Biwa. This effect is more pronounced on the east coast compared to the west coast. Decreasing river water temperature by 5 °C leads to lower surface water temperatures nearshore, with a more pronounced thermocline and warmer bottom water temperatures, indicating vertical mixing near the bottom layer. River temperature changes have a more significant impact on lake temperature fluctuations and stratification strength compared to doubling or halving the river flow rate.

Overall, this study demonstrates that precipitation, river water flow, and river water temperature have a significant impact on the thermal structure, stratification, and vertical mixing patterns of Lake Biwa. Changes in river water temperature, in particular, provide greater stratification variability than river water flow and play a crucial role in the circulation of materials to the bottom layer in lakes and oceans. Given the recent increase in eutrophication and anaerobic conditions in oceans and lakes, it is essential to evaluate the influence of river plumes on material circulation in oceans and lakes. This is necessary to understand the fluctuations in stratification structure from the surface to near the thermocline during spring and summer and from the deep layer to the bottom layer during autumn and winter.