ISIMIP – the Inter-Sectoral Impact Model Intercomparison Project – is a framework for consistently projecting the impacts of climate change across affected sectors and spatial scales (https://www.isimip.org/, last access: 20 February 2025; ). The ISIMIP Lake Sector considers the impact of global warming on two categories of lakes: “local lakes” and “global lakes” . The local lakes were used for this study: 73 lakes for which observed in situ water temperature data and hypsographic information are available (Table S1 in the Supplement). The resolution (vertical and temporal) of the observed data and the detail of the hypsograph varied for each lake. For all but two lakes, data covered a period of at least 1 year; for 75 % of the cases, they covered at least 5 years. Profiles (three unique depths or more) were provided for all but four lakes, and all lakes had more than 100 unique observations (Fig. S1 in the Supplement). A link to the observed data and hypsographs is provided in the Code and data availability statement. No inflow or outflow data are available, so we assumed a constant water level throughout the simulation.

For the forcing of the models, we used the GSWP3-W5E5 reanalysis dataset, which combines the GSWP and the W5E5 datasets . The meteorological forcing, available at daily resolution, for each lake was extracted by the ISIMIP organizational team for the grid cells (at a spatial resolution of 0.5° × 0.5°) in which each lake was located . The following meteorological variables were used to drive the lake simulations: air temperature, relative humidity, precipitation, shortwave radiation, longwave radiation, surface air pressure, and wind speed.

Initial conditions were estimated from observed water temperatures. Therefore, all available data in a period of days (depending on data availability) before and after the start date of the simulation were taken and averaged to set the initial temperature profile. All simulations used a spin-up period of 1 year.

In order to analyze the impact of lake characteristics (Table S2) on the model performance and parameter sensitivity, we used K-means clustering to group the 73 lakes (Fig. ). Prior to clustering, we log-transformed the elevation, mean depth, maximum depth, and lake area and then applied a z-score transformation. We created a silhouette plot to determine the optimal number of clusters, which was two. However, we decided to use five clusters instead, as this gave a more meaningful representation of different lake types (Fig. S2).

In this study, four vertical lake temperature models with varying algorithms and calculations regarding vertical temperature and heat transport were used to explore the model sensitivity around climate change projections in lakes: the two-layer (0.5D) model FLake , the 1D integral energy model GLM version 3.1.0 , and the 1D turbulence-based models GOTM lake-branch version 5.4.0 and Simstrat version 2.4.1 . The models were set up and run using the LakeEnsemblR R package to standardize the approach. We refer to for detailed information regarding general concepts in water temperature modeling. In this section, we provide a summarized overview of the main differences in process description between these four models. The models were applied in an identical way, with the exceptions that FLake was used to simulate up to the mean depth instead of the maximum depth (in line with assumptions in the model) and that cloud cover was calculated from the meteorological variables using the LakeEnsemblR functions for the GOTM model.

The vertical 0.5D (i.e., a box model but with two separate boxes for upper and lower water layers) lake model FLake was originally designed for weather prediction studies, in which a large-scale climate model is coupled to multiple small-scale lake models. To achieve computational efficiency, FLake simulates the temperature dynamics of an upper completely mixed layer and a thermocline layer (commonly also known as metalimnion), while neglecting temperature dynamics below the latter layer . The vertical temperature evolution itself is parameterized based on the self-similarity concept of the vertical temperature profile . This observed and theoretically explained concept states that a dimensionless temperature profile in the thermocline can be replicated using a “universal” function of the dimensionless depth ζ: 1θs(t)-θ(z,t)Δθ(t)=Φθ(ζ), where t and z are the dimensions over time and depth, respectively. In Eq. (), θs is the absolute temperature of the upper completely mixed layer, Δθ(t) is the absolute temperature gradient across the thermocline layer, and Φθ is the universal function of the dimensionless depth. The dimensionless depth can be parameterized as follows: 2ζ=z-h(t)Δh(t), where h(t) is the depth of the upper completely mixed layer and Δh(t) is the depth difference between the mixed-layer depth and the bottom of the metalimnion. Note that, in this study, we set the bottom metalimnion depth to each lake's mean depth. Applying this concept to temperature evolution, FLake parameterizes both layers (upper completely mixed and thermocline layer) as follows: 3Θ=θs,if0<z<hθs-θs-θbΦ(ζ),ifh≤z≤Dlake, where Dlake is the maximum depth . Similar to the other models, the upper completely mixed layer receives the energy fluxes from the atmosphere: 4hdθsdt=1ρwcwQs+Is-Qh-I(h), where ρw is water density, cw is heat capacity, Qs is the turbulent heat flux at the surface, Is is the surface shortwave radiative flux, Qh is the heat flux from the bottom to the upper layer, and I(h) is the radiative shortwave flux through the water column . We can state the sum of these individual heat fluxes as the net heat flux exchange (Hnet). Although FLake's numerical implementation combines empirical formulations with physical processes, it has demonstrated good performance for surface water temperature modeling as well as ice phenology investigations (e.g., ) and is commonly applied to global studies .

GLM, GOTM, and Simstrat are vertical 1D lake models in which temperature evolution is quantified at every time step over a vertical grid. Conceptually, the models differ regarding how the vertical grid is configured: GLM applies a flexible structure, whereas the others use a fixed grid with the possibility of refinements. Nonetheless, all three models are based on the vertical water temperature equation, which – in its general form – can be stated as follows: 5∂T∂t=-1ρcp∂I∂z︸(1)+HsedAρcp∂A∂z︸(2)+STρcp︸(3)+1A∂∂zADzT∂T∂z︸(4). Here, the change in temperature T over time depends on four terms on the right-hand side: (1) the internal heat generation due to shortwave solar radiation I, (2) a geothermal heat flux Hsed that acts over an area A, (3) an internal heat source term ST, and (4) a turbulent diffusive term that includes the eddy-diffusivity coefficient DzT . The layer adjacent to the atmosphere–water interface receives a net heat flux exchange similar to the one described in Eq. (), where Hnet is the sum of radiative and turbulent heat fluxes: 6ρwcpDzT∂Tz∂z|z=s=Hnet. Here, Tz is the water temperature of the layer adjacent to the atmosphere–water interface at the surface depth s.

The main difference between GLM and both GOTM and Simstrat is how they simulate the turbulent diffusive transport. GLM applies a combination of empirical and physical relationships that use the available external turbulent kinetic energy (TKE) to calculate the thickness of a completely mixed surface layer (for general information about integral energy models, see ). For this, mixing in a surface mixed layer is calculated by comparing the available external energy to the potential energy of the water column that is needed to lift up denser water from below a completely mixed layer into a newly formed mixed layer until the TKE is no longer sufficient for further mixing . Below the depth of this surface mixed layer, a parameterization for the eddy-diffusivity coefficient in relation to water column stability is used to calculate diffusive transport: 7DzT=CHYPεTKEN2+0.6kTKE2u*2, where CHYP is a constant coefficient for the mixing efficiency (later referred to as the calibration parameter coef_mix_hyp), εTKE is a simplified approximation of the turbulent dissipation rate based on the dissipation by inflows and wind, N2 is the squared buoyancy frequency, kTKE is the turbulence wave number, and u* is the wind shear velocity . The buoyancy frequency (Brunt–Väisälä frequency) quantifies local stability to vertical displacements as follows: 8N=gρ∂ρ∂z, where g is gravitational acceleration.

Simstrat and GOTM are turbulence-based models that apply a two-equation turbulence model to compute the quantities of the production, transport, and dissipation rates of TKE. Here, we highlight the k-ε two-equation turbulence model which is implemented in both models : 9∂k∂t=1A∂∂zADzk∂k∂z+P+B-ε,10∂ε∂t=1A∂∂zADzε∂ε∂z+εkcε,1P+cε,3B-cε,2ε, where Dzk and Dzε are the turbulent diffusivities of TKE and TKE dissipation, respectively; P is the TKE production due to shear; and B is the production and dissipation of TKE related to buoyancy . cε,1, cε,2, and cε,3 are empirical constants. In GOTM, whenever the simulated TKE is lower than the calibration parameter k_min , it is set to the value of k_min. We can compute the eddy-diffusivity coefficient DzT as a function of the turbulence kinetic energy k and the dissipation rate ε: 11DzT=cμσtk2ε, where cμ is an empirical coefficient and σt is the turbulent Prandtl number.

Simstrat further employs an empirical seiche excitation and damping model to improve the representation of internal seiches in transport processes . Here, seiche movement can produce additional TKE, Eseiche, inside the water column with the intention to provide a more realistic simulation of vertical transport due to bottom boundary mixing as seiche motion damping acts as an energy source below the mixed layer: 12dEseichedt=αA0ρairc10u102+v1023/2︸PW-CDeffA0V-3/2ρ0-1/2Eseiche3/2︸LS, where PW is energy production, LS is energy loss, α is a model parameter to describe the wind energy fraction that is transferred to the seiche motion (later referred to as the calibration parameter a_seiche), c10 is the drag coefficient, u10 and v10 are velocity components of wind speed measured at 10 m above water surface, and CDeff is the effective bottom friction coefficient . We note that similar algorithms, designed to improve vertical mixing dynamics below the epilimnion, also exist in other models, including integral energy models, i.e., the turbulent benthic boundary layer mixing algorithm by , but are not – to the best of our knowledge – implemented in GLM and GOTM.

An additional structural difference between the models is their process description of the treatment of the attenuation of shortwave radiation, especially the nonvisible near-infrared light (NIR) and the visible parts of shortwave radiation. FLake does not distinguish between these parts of the light spectrum, and it applies the Beer–Lambert law for light attenuation with depth (see also , for a more detailed analysis), although the model can be parameterized to consider a set of different wavelength bands with variable attenuation coefficients . GLM has the option to apply the Beer–Lambert law for only the photosynthetically active fraction (PAR), while the NIR and ultraviolet bandwidths are attenuated directly in the layer adjacent to the atmosphere–water interface . A second option in GLM uses the algorithm by to simulate light penetration of individual bandwidth fractions. However, in this study, the first option was applied; this option treats 45 % of the incoming shortwave radiation as PAR which is subsequently attenuated in the layers below the atmosphere–water interface. Similarly, GOTM was configured to have a separate depth-specific attenuation for the visible and nonvisible light fractions. In this study, the incoming shortwave radiation was split into the nonvisible (55 %) and visible (45 %) fractions. The light extinction coefficient for nonvisible light was set to 2 m⁻¹. Although Simstrat does not split light into separate fractions, it uses a parameter to absorb a fixed fraction of shortwave radiation (set to 30 % in this study) in the uppermost water layer, eventually resulting in a similar impact of fast absorption of a part of the solar energy near the atmosphere–water interface (see also ). This highlights that more heat potentially gets absorbed in the layer adjacent to the atmosphere–water interface in the GLM, GOTM, and Simstrat simulations than in the FLake simulations.

The workflow (Fig. ) to calibrate the models for the 73 lakes is described in the following section. For each lake, we gathered the available data from ISIMIP: observed water temperatures, lake hypsography, lake location (elevation, coordinates), and light extinction (or Secchi disk depth data to derive light extinction). Observed water temperature data with subdaily resolution were averaged to daily mean values. If no data on the light extinction were available, we estimated it from Secchi disk depth . If no Secchi disk depth was available, we estimated it from the maximum lake depth . We then formatted the ISIMIP data to a pre-defined standard format, from which the LakeEnsemblR package generated model-specific forcing and configuration files. We used four lake models included in LakeEnsemblR (GLM, GOTM, Simstrat, and FLake) which are described in Sect. .

Finally, we ran the calibration using a Latin hypercube approach (see, e.g., ). Here, we chose six parameters for each model: three model-specific parameters and three scaling factors (for wind speed, incoming shortwave radiation, and the estimated light extinction coefficient, respectively; Table ). For the model-specific parameters, we chose parameters that are commonly used to calibrate these models, based on the literature (see ) and discussions on the parameter range held by the Lake Modelling working group at the GLEON All Hands' Meeting in 2020 and 2021 . We sampled and ran the four models for 2000 parameter sets, and we calculated four performance metrics over all water temperature observations for each of the parameter sets: root-mean-square error (RMSE), Nash–Sutcliffe model efficiency (NSE), Pearson correlation coefficient (R), and mean error (bias).

Based on the sampled parameter sets and the calculated performance metrics, we performed a delta moment-independent sensitivity analysis for each performance metric per lake per model, using the SALib Python library . The analysis calculates two sensitivity measures, the moment-independent δ and variance-based Sobol' S1. The delta moment-independent measure δ considers the entire distribution of the model output instead of a particular moment (e.g., variance) by calculating the difference between the unconditional and conditional cumulative distribution functions of the simulated model output, whereas the variance-based first-order Sobol' index S1 calculates a parameter's influence on the variance of the simulated model output . As this study was interested in identifying the most important parameters (i.e., factor prioritization setting), we followed the recommendations of and used both variance-based and moment-independent measures to increase the robustness when inferring which parameters are most important when simulating water temperatures. In addition to the six calibrated parameters, we included a dummy parameter that had no influence on the model output in the sensitivity analysis, which we sampled from a uniform distribution ranging from zero to one. In theory, this dummy variable should have a sensitivity of zero; however, due to the numerical approximation of the sensitivity measures, it can have small nonzero values. This can be used to approximate the error related to estimating sensitivity indices and thereby avoid classifying non-influential parameters as influential. This approach has been used in previous studies (e.g., ). A resample size of 100 was used to compute confidence intervals on both sensitivity analysis metrics. To provide an estimate of potential parameter interactions, we additionally calculated the interaction indicator Sinteraction that describes the fraction of model output variation apportioned by interactions: 13Sinteraction=1-∑i=1kSi, where (Si) is the first-order variance-based sensitivity measure (S1) of parameter i out of k tested parameters.

The single best-performing model (out of the four applied models) for each lake reproduced observed water temperatures well for all 73 lakes, with a median RMSE of 1.2 °C and a median R of 0.98 (Fig. ). The variation in error metrics between the best- and worst-performing model for each lake was rather small, e.g., a standard deviation of 0.5 °C or less in the RMSE (Fig. S3). Simstrat performed the best in most lakes in terms of the RMSE, R, and NSE, while GLM performed best in most lakes with respect to the bias (Fig. ). However, all four models outperformed the others in at least some of the lakes. In over 90 % of all lakes, at least two different models performed best for different metrics.

Following the cluster analysis, we classified the lakes into five clusters. We visually compared the characteristics of the clusters (Fig. S4) and characterized them according to their most noticeable features: “deep” (n = 3), “medium temperate” (n = 25), “small temperate” (n = 32), “large shallow” (n = 4), and “warm” lakes (n = 9) (Fig. ). Model performance was comparable among the clusters, although the deep lakes had a lower RMSE, whereas medium, small temperate, and large shallow lakes performed best in terms of the NSE and R (Fig. S5). When considering the four models separately, the overall better performance of Simstrat was mostly due to its better performance in the deep and medium temperate lakes, compared with the other models. In the other three clusters, the four models performed similarly (Fig. S6).

We calculated the ensemble mean by taking the arithmetic mean of the four models for each time step and depth individually. We then tested this as an additional predictor for water temperature and calculated its performance in terms of the RMSE. For the majority of lakes, the ensemble mean performed better than any single model (Fig. S7). This is especially visible in the medium temperate, small temperate, and large shallow lakes, where the ensemble mean performed best for the majority of lakes. The cases where the ensemble mean did not perform better than each single model were often lakes in which a single model performed notably better or worse than the other three models.

Looking at the distribution of the model error in terms of the RMSE over the water column depth (Fig. a), we can see that Simstrat performed better over all depths for the medium temperate lakes. In the deep lakes, FLake performed considerable worse than the other three models, especially at intermediate depths. For the other three models, the error increased towards the surface. For all four models in the large shallow lakes, the error was larger at the surface, while the error was largest at intermediate depths for the warm lakes.

From the observed water temperatures, we calculated the thermocline depth and then chose the simulation–observation pairs closest to that depth to estimate the RMSE at the thermocline temperature (Fig. b). For the large shallow lakes, no thermocline could be calculated. Simstrat performed best at the thermocline depth for deep, medium temperate, and warm lakes: its performance for deep and medium temperate lakes was about 0.5 °C better, while it was only about 0.1 °C lower than the next best model for warm lakes. For small temperate lakes GLM performed better, with a median RMSE that was about 0.3 °C lower than the next best model. FLake performed most poorly at the thermocline depth for all lake clusters.

From the calibration runs using the Latin hypercube approach, we calculated the moment-independent measure δ and the variance-based first-order measure S1 for each combination of models, performance metrics, and lakes (Fig. ). We saw a similar ranking of the most influential model parameters on most combinations of models, performance metrics, and lakes for δ and S1. For almost all lakes, the same three to four parameters were classified as sensitive: the scaling factors for wind speed, shortwave radiation, and light extinction as well as k_min for GOTM. Moreover, one or two of these parameters accounted for more than 75 % of the sum of the sensitivity measures for most lakes (Fig. S8). Most often, these were meteorological scaling factors, which are not model-specific, with the exception of GOTM, in which k_min was most sensitive. Additionally, the light extinction coefficient and other model-specific parameters appeared to be sensitive in a couple of lakes (Fig. ), although to a lesser degree.

For most models and performance metrics, interaction effects accounted for less than 20 % of the variation in model performance, although interactions were relevant for specific models and lake groups (Fig. ). For instance, interactions were relevant for GLM modeling deep lakes and, to a lesser degree, for GLM and Simstrat modeling large shallow lakes. In contrast, increased parameter interactions were observed for FLake, especially for the NSE and RMSE, for all lake clusters except deep lakes. We highlight that, especially in lakes with shorter time series of observed water temperature data, the interaction measure was larger (Fig. S9). Interactions were low for the bias for all models and lake clusters.

Looking at the parameter values from the best-performing parameter sets, the optimal meteorological scaling factors differed between models. Especially GOTM showed a different behavior from the other models, with lower wind speed scaling factors and a higher shortwave radiation scaling factor (Fig. ). The lake clusters also differed with respect to the optimal scaling factors, although their effects seemed model-specific. Differences in extinction factor scaling were less clear than the meteorological scaling factors, but GLM preferred a higher extinction factor in large shallow lakes, whereas FLake preferred a higher extinction factor in deep and medium temperate lakes. Most model-specific parameters had a low sensitivity, but some still showed markedly different behavior among clusters (Fig. S10). The single model-specific parameter with high sensitivity, GOTM's k_min, had distinctly lower values in small temperate lakes. For both the scaling factors and the model-specific parameters, we saw that the outcome was different depending on which performance metric was used to select the best parameter set (see Fig. S11).