Lake Erken (59∘51′ N, 18∘36′ E) is a mesotrophic lake located in eastern central Sweden, with a maximum depth of 21 m, a mean depth of 9 m and a surface area of 23.7 km2. The lake is dimictic with summer stratification usually beginning in May–June and ending in August–September, while the onset of ice cover occurs between December and February and ice loss is in April–May (Persson and Jones, 2008). It is the lake's relatively shallow depth and large surface area which leads to large interannual variability in the timing and patterns of thermal stratification since heat can be readily transferred through the shallow water column by wind mixing (Magee and Wu, 2017), and since the lake has a relatively low heat storage and, therefore, responds more directly to short-term variations in weather. The lake has a retention time of approximately 7 years and shows annual variations in water level that are less the 1 m (Pierson et al., 1992; Moras et al., 2019).

General Ocean Turbulence Model (GOTM) is a one-dimensional water column model that simulates the most important hydrodynamic and thermodynamics processes related to vertical mixing in natural waters (Umlauf and Burchard, 2005). GOTM was developed by Burchard et al. (1999) for modelling turbulence in the oceans, but it has been recently adapted for use in hydrodynamic modelling of lakes (Sachse et al., 2014). The strength of GOTM is the vast number of well-tested turbulence models that have been implemented, spanning from simple prescribed expressions for the turbulent diffusivities to complex Reynolds stress models with several differential transport equations. Typically GOTM is used as a stand-alone model for investigating the dynamics of boundary layers in natural waters, but it can also be coupled to a biogeochemical model using the Framework for Aquatic Biogeochemical Models (FABM; Bruggeman and Bolding, 2014).

Local meteorological variables were collected either from a small island 500 m offshore from the Erken Laboratory or the Swedish Meteorological Hydrological Institute (SMHI) Svanberga station just behind the laboratory. The Malma Island meteorological station (59.83909∘ N, 18.629558∘ E) measured air temperature at 2 m above water surface, wind speed at 10 m above the water surface and short-wave radiation. These data were measured at 1 min intervals and saved as 60 min mean values. Mean sea level, pressure, relative humidity and precipitation were measured at the Svanberga meteorological station at 800 m from the Malma Island meteorological station (59.8321∘ N, 18.6348∘ E) with a frequency of 60 min. Hourly cloud cover was recorded from Svenska Högarna station (59.4445∘ N, 19.5059∘ E) at 69 km south-east of Lake Erken.

The measured hourly meteorological data were used to construct two additional data sets that would replicate the data resolution that could potentially be used to force the GOTM model with ISIMIP scenarios. First, to test running the model at a daily resolution, a daily data set was created by averaging the hourly one (except for precipitation which was summed). Second, this mean daily data set was disaggregated to form a synthetic hourly data set. Hourly estimations of air temperature, wind speed, relative humidity and short-wave radiation were estimated using the generalised regression artificial neural network (GRNN) methods described below. For atmospheric pressure and cloud cover, mean daily values were assumed to be constant over the day. Precipitation was disaggregated, assuming a uniform distribution of the daily total (Waichler and Wigmosta, 2003).

Since both of these data sets are based on the same measured hourly data, a comparison of model simulations of lake water temperature allows the importance of hourly vs daily temporal resolution in the forcing data to be evaluated, and also the improvements in model performance that can be obtained from daily data (as in the ISIMIP scenarios) when imposing a diurnal cycle on the mean daily data.

Water temperature data needed to calibrate the model was monitored from an automated floating station (59.84297∘ N, 18.635433∘ E). During ice-free conditions, measurements were made every 0.5 m from 0.5 m to a depth of 15 m. Measurements were made every minute, and a mean of these measurements was stored every 30 min.

The ISIMIP climate scenarios are bias-corrected global climate model (GCM; Hempel et al., 2013) data made available at daily temporal and 0.5∘ horizontal resolutions for the variables listed in Table 1. All data needed as input to the GOTM model are available in these climate scenarios with the exception of cloud cover, which was estimated from short-wave radiation (Martin and McCutcheon, 1999). Data from the grid box overlying Lake Erken were available from the GFDL–ESM2M, HadGEM2–ES, IPSL–CM5A–LR and MIROC5 GCM models that were each run under three emission scenarios. These included a scenario with historical levels of atmospheric CO2 between 1861 and 2005 and two future scenarios (RCP2.6 and RCP6.0) from 2006 to 2100. RCP2.6 is the strongest mitigation pathway that is expected to limit mean global warming to between 1.5 and 2 ∘C. RCP6.0 is an intermediate mitigation pathway where global warming is projected to rise to between 2.5 and 4 ∘C by the end of the century compared to the pre-industrial period (Frieler et al., 2017).

Kathib and Elmernreich (2015) proposed a generalised regression artificial neural network (GRNN) model for predicting hourly variations in short-wave radiation from daily average measurements. Using the GRNN model to predict hourly solar radiation required 10 geographical and climatic variables as input including hour, day, month, latitude, longitude, daily average short-wave radiation, daily precipitation, the solar elevation associated with the hour, and time of sunrise and sunset. Precipitation was used to define wet and dry status that affected atmospheric attenuation (Waichler and Wigmosta, 2003).

There are also empirical models developed for calculating hourly air temperature, wind speed and relative humidity. Parton and Logan (1981) proposed a model for predicting diurnal variations in air temperature. Daylight air temperature was modelled using a sine wave with the minimum value at sunrise, maximum value at solar noon and mean value at sunset. Night-time air temperature was modelled as a linear interpolation between air temperature of the previous day and sunrise air temperature of the following day. Guo et al. (2016) generated hourly values of wind speed by computing a cosine function dependent on the mean daily wind speed, the maximum daily wind speed and the hour of the day when the wind speed is maximum. Waichler and Wigmosta (2003) estimated hourly values of relative humidity from daily maximum and minimum air temperature and daily maximum and minimum relative humidity. Using these studies as a guide, we developed GRNN models to predict hourly air temperature, wind speed and relative humidity. The input parameters for GRNN models were geographical variables and meteorological variables. The geographical variables include longitude, latitude, solar elevation associated with the hour, time of sunrise and sunset, hour, day and month, while the meteorological variables include average, maximum and minimum daily air temperature, daily wind speed, daily relative humidity, and daily precipitation.

The GRNN models were constructed using 8 years of data. From this whole set of data, the first 5 years, from 2008 to 2012, were used for training, and the final 3 years of data, from 2013 to 2015, were used for validating the results. The accuracy of the trained network was assessed by comparing the simulated output with actual observed hourly data. The performance index for training and validating sets of GRNN models is given in terms of mean bias error (MBE), root mean squared error (RMSE) and Nash–Sutcliffe efficiency (NSE; Nash and Sutcliffe, 1970). More detailed descriptions of the GRNN methods and models are given in Sect. S1 of the Supplement. The GRNN models were used to disaggregate the mean daily measured data, used to evaluate the necessity of disaggregation (Sect. 3.2), and also for all GCM scenarios (Sect. 3.3) to further evaluate the effects of disaggregation on the results of simulations of future changes in lake thermal structure.

The GOTM model version 5.1 was used in this study. The meteorological parameters for running the model were air temperature (∘C), wind speed (m s-1), short-wave radiation (W m-2), cloud cover (dimensionless, 0–1), relative humidity (%), atmospheric pressure (hPa), and precipitation (mm d-1 or mm h-1). Inflows and outflows were not included in this study, and water level was considered fixed in the simulations. This version of GOTM did not have the ability to simulate lake ice, so for this study the inverse stratification period was not analysed. Moras et al. (2019) have shown that, despite this limitation, the model is able to accurately simulate water temperature and the phenology of thermal stratification during the remainder of the year. The initial conditions for water temperature were derived from a measured vertical profile. GOTM was run at an hourly model computational time step, and simulated water temperature was saved as daily mean values each 0.5 m (42 layers).

Calibration of the GOTM model was conducted to adjust the model parameters within their feasible range in order to minimise the error between measured and modelled temperature (Huang and Liu, 2010). A period of 9 years was selected for the calibration of GOTM, namely 2006–2014 (including 1 year spin-up time followed by 8 years for calibration). The model parameters that were calibrated were surface heat flux factor (shf_factor), short-wave radiation factor (swr_factor), wind factor (wind_factor), minimum turbulent kinetic energy (k_min), and e-folding depth for visible fraction of light (g2). The programme used to calibrate the model was Auto-Calibration Python (ACPy), developed by Bolding and Bruggeman (https://bolding-bruggeman.com/portfolio/acpy/, last access: 12 June 2018); it uses a differential evolution algorithm which calculates a log likelihood function based on comparing the modelled and measured water temperature (Storn and Price, 1997). The validation period was 2 years (2015–2016).

For both calibration and validation, daily average water temperatures were simulated when GOTM was forced using the three meteorological data sets described above, namely measured average daily, measured average hourly and synthetic hourly data. Model simulated profiles of mean daily water temperature were then compared to mean daily measured water temperature. Three separate model calibrations were made based on simulations forced with the different meteorological data sets. During calibration the model was run approximately 10 000 times to obtain a stable solution specifying the optimal parameter set. The details of the feasible range of model parameters and the parameters associated with each calibration are given in Table 2.

Model performance was evaluated by comparing average daily modelled and measured temperature profiles and other metrics describing the lake thermal structure (surface and bottom temperature, volumetrically weighted averaged whole-lake temperature, Schmidt stability, thermocline depth, duration, and onset and loss of thermal stratification). The coefficients used to quantify the strength of model fit were MBE, RMSE and NSE.

A range of thermal metrics, namely surface temperature (shallowest observation), bottom temperature (deepest observation) and thermocline depth (depth of the maximum density gradient), were derived on a daily basis from the daily simulated lake temperature profiles (temperature data with a vertical resolution of 0.5 m from 0.5 to 15 m depth). Also, Schmidt stability (resistance to mechanical mixing due to the potential energy inherent in the density stratification of the water column; Schmidt, 1928; Idso, 1973) and whole-lake temperature (volumetric weighted mean whole-lake temperature) were estimated from these profiles using Lake Analyzer (Read et al., 2011). The duration of thermal stratification was calculated as the longest continuous period when the water column density difference from the bottom to the surface of the lake was greater than 0.1 kg m-3 (according to ISIMIP2b lake sector protocol). The date of the onset and loss of the thermal stratification was defined as the first time that this density difference persisted for more than 5 d or was absent for at least 5 d (Kraemer et al., 2015).

Anomalies were calculated to further evaluate the impacts on lake water temperature and thermal stratification. The anomalies were computed for each GCM by taking the difference between the annual average of each year (2011–2100) from RCP2.6 and 6.0 scenarios and the average for the entire period 1981–2010 from the historical scenario. These average values were calculated using the months between April and September. The slope of the significant trends was evaluated by least squares linear regression, except when the residuals did not follow a normal distribution. Then the non-parametric Mann–Kendall test for the significance of trends and the Theil–Sen method (Theil, 1950; Sen, 1968) to estimate the slope of the significant trends were used instead. The Student t mean difference test was used to compare average values of each of the thermal indices. Distribution normality and variance homoscedasticity were assessed by the Shapiro–Wilk test and Fisher's F test respectively. When thermal indices time series did not follow a normal distribution the non-parametric Mann–Whitney U test (equal variances) or Kolmogorov–Smirnov test (different variances) were used instead. The statistical analysis was carried out using R version 3.4.4. The progress of climate-related impacts on the thermal stratification of the lake over the 21st century was assessed as the difference in mean conditions between the reference period (1981–2010) and both the mid-century (2041–2070) and the late century (2071–2100). Climate model data followed the same statistical analysis.

There was a close agreement between GRNN model predictions and measured meteorological data as shown in Fig. 1, for 1 year, and in Sect. S1. For air temperature, short-wave radiation and humidity, the statistics of model performance always suggested a strong model fit in the training data sets and also remained strong, but somewhat lower, in the validation data sets (Table 3). NSEs were 1.00 for the training data sets and ranged from 0.69 to 0.94 for the validation data sets. Estimates of bias were very small. Wind speed was the variable showing the poorest performance with a NSE of 0.78 and 0.58 and RMSE values of 1.06 and 1.37 m s-1 for the training and validate data sets. In general, the GRNN model tended to overestimate wind speed (MBE of 0.63±0.92 m s-1) when the observed wind speed was lower than or equal to 3.84 m s-1, whereas projected wind speed tended to be underestimated (MBE of -0.78±1.17 m s-1) when the observations were greater than 3.84 m s-1.

Temperature observations and simulations for the three configurations of meteorological forcing data for both calibration and validation periods are shown in Fig. 2 and Sect. S2. Model performance metrics associated with these simulations are provided in Table 4. These data demonstrate that GOTM was able to accurately reproduce the measured temperature profiles. For an average of all three calibration data sets, a RMSE of 0.95 ∘C and NSE of 0.94 were obtained. Temperature simulations in the shorter and less variable validation period (RMSE of 0.66 ∘C and NSE of 0.97) were more accurate than for the calibration period, but in both periods the model performance was considered strong. For a full-profile temperature, the maximum RMSE value was 1.04 ∘C and the minimum NSE was 0.93. Bottom temperature was least accurately simulated, with RMSE and NSE values reaching 1.33 ∘C and 0.83 respectively.

When comparing the metrics of model fit associated with simulations forced with the three different input data sets the simulations forced with mean daily input were slightly less accurate than those forced with either the measured or synthetic hourly input. As an example, full-profile RMSE values for the calibration period ranged from 0.88 to 1.04 ∘C, with the lower error levels associated with simulations driven by hourly meteorological data sets, whereas for the validation period the RMSEs were comparable for all data sets. The MBE values of the full-profile temperatures indicated a slight cold temperature bias (average MBE of -0.05 ∘C). The model performance predicting just surface temperatures was similar for all of the three calibrations (average RMSE of 0.67 ∘C and NSE of 0.97) and was more accurate than the estimations of the full-profile temperatures. The MBE showed that for surface temperature GOTM also tended to produce a small cold temperature bias (average MBE of -0.10 ∘C). The simulations of bottom temperature were slightly less accurate, having average RMSE of 0.96 ∘C and NSE of 0.90, and also showed a tendency have a slightly higher RMSE values for calibrations forced with daily input. Also, the bottom temperature showed lower RMSE values for the validation period (average RMSE of 0.67 ∘C) than the calibration period (average RMSE of 1.25 ∘C), but, in contrast to the surface temperature, there was a slight warm temperature bias (average MBE of 0.06 ∘C). When evaluating the simulations of volumetrically weighted averaged whole-lake temperatures, we found that model errors were of a similar magnitude for all simulations in both the calibration and validation periods with an average RMSE of 0.53 ∘C and NSE of 0.98, tending to a slight cold temperature bias (average MBE of -0.08 ∘C).

The calculation of Schmidt stability was also well simulated using all three data sets (average RMSE of 17.24 J m-2 and NSE of 0.88). The lowest RMSE values were for the validation period (average RMSE of 13.34 J m-2), whereas during the calibration period the values were slightly greater (average RMSE of 21.14 J m-2). Thermocline depth was the parameter with the poorest performance (average RMSE of 3 m). The MBE values (average MBE of 0.80 m) indicate a bias towards an underprediction of thermocline depth (shallower thermocline depths). The RMSE associated with the prediction of the duration of stratification was, on average, 10.43 d, with lower RMSE values for the validation period (average RMSE of 8.04 d) than the calibration period (average RMSE 12.81 d). The simulations of the onset of the stratification were more accurate, having average RMSE of 2.64 d, but, in contrast, predictions of the loss of stratification were less accurate (average RMSE of 7.99 d).

The lake model simulations undertaken here were forced by four climate model projections (GFDL–ESM2M, HadGEM2–ES, IPSL–CM5A–LR and MIROC5) that were in turn forced with three emissions scenarios (historical, RCP2.6 and RCP6.0). Average annual air temperature of the climate model ensemble for the reference period (1981–2010) was 11.88 ∘C. Disaggregation of the climate input to an hourly time step resulted in a slightly warmer temperature (12.05 ∘C)

Results based on the hourly disaggregated data are always shown in parenthesis.

Lake model simulations were made using both the original daily resolution of the ISIMIP–GCM scenarios and also at hourly resolution using disaggregated data developed using the GRNN models. Simulated water temperatures for the historical, RCP2.6 and 6.0 scenarios under daily IPSL–CM5A–LR projections are presented as temperature isopleths in Fig. 3 and Sect. S4. These were created by averaging the daily temperature profiles for all years associated with each of the emission scenarios. These mean scenario temperature isopleths provide a clear visualisation of how, for future scenarios, surface and bottom water temperatures are projected to increase with stronger and shallower stratification, an earlier stratification onset, a later fall overturn and, consequently, a longer stratification period.

In Figs. 4–5 we show the long-term trends in the anomalies in lake thermal metrics simulated to occur over the RCP2.6 and 6.0 emission scenarios. Trends in whole-lake temperature calculated for over a period of 90 years (2011–2100) were projected to increase except in the case of GFDL–ESM2M, which showed weaker or non-significant changes for all measures of thermal stratification (Table 5 and Sect. S4). Under RCP2.6, significant trends ranged from 0.07 to 0.10 ∘C per decade (0.05 to 0.08 ∘C per decade), but most of the change occurred in the first half of the century. For RCP6.0, the projected rate of change ranged from 0.14 to 0.26 ∘C per decade (0.10 to 0.19 ∘C per decade). By the late century, the mean projected increase in whole-lake temperature was 1.34 ∘C (1.00 ∘C) for RCP2.6 and 2.39 ∘C (1.75 ∘C) for RCP6.0, with a negligible change after the mid-century under RCP2.6 (Fig. 6, Table 6 and Sect. S4).

Changes in lake surface temperature were, as expected, greater than for whole-lake temperature. For the reference period, the mean April–September surface temperature was on average 13.61 ∘C (13.84 ∘C) warming up significantly over the 21st century, so that by the late century the average projected increase was 1.79 ∘C (1.35 ∘C) for RCP2.6 and 3.08 ∘C (2.31 ∘C) for RCP6.0. From 2011 to 2100 there was a significant long-term trend for RCP2.6 surface temperature, which increased at a rate from 0.07 to 0.15 ∘C per decade (0.06 to 0.13 ∘C per decade). Under RCP6.0 the mean surface temperature increase of the ensemble was 0.30 ∘C per decade (0.23 ∘C per decade) ranging between 0.16 and 0.38 ∘C per decade (0.12 to 0.29 ∘C per decade). The projected increase in bottom temperature was not as marked as it was for the other metrics of lake temperature. On average, the bottom temperature increased from 9.20 ∘C (9.67 ∘C) in the reference period to 9.77 ∘C (9.99 ∘C) and 10.32 ∘C (10.34 ∘C) by the late century for RCP2.6 and 6.0 respectively. Significant rates of change in bottom temperature were not predicted during the RCP2.6 scenario, but for the RCP6.0 scenario bottom temperature did undergo significant warming rates for HadGEM2–ES and MIROC5 projections were 0.06 ∘C per decade and 0.11 ∘C per decade (0.09 ∘C per decade) respectively.

There were also projected changes in the resistance to mechanical mixing. For the reference period, an average of 68.65 J m-2 (65.56 J m-2) was required to completely mix the water column, while by the late century it increased by 29.08 J m-2 (22.74 J m-2) for RCP2.6 and 49.22 J m-2 (38.07 J m-2) for RCP6.0 (Fig. 4, Table 6 and Sect. S4). This greater stability also corresponds to a longer duration of stratification. From 2011 to 2100, a significant increase in the duration stratification was projected for both future scenarios RCP2.6 and 6.0, ranging from 1.13 to 1.70 d per decade (0.87 to 1.30 d per decade) for RCP2.6 and 2.45 to 3.56 d per decade (2.00 to 3.08 d per decade) for RCP6.0 (Fig. 5, Table 5 and Sect. S4), which led to a mean change of 13 d (11 d) and 22 d (18 d) for RCP2.6 and 6.0 respectively (Fig. 7, Table 6 and Sect. S4). The longer period of stratification resulted from both an earlier onset of thermal stratification and a later loss of thermal stratification (Figs. 5 and 7, Tables 5–6 and Sect. S4). Mean annual thermocline depth was simulated to be shallower under future conditions. By the late century, the reduction in thermocline depth was projected to be 0.38 m (0.41 m) for RCP2.6 and 0.49 m (0.57 m) for RCP6.0, although a significant trend in the decline was only found for the later scenario.

The trends in Figs. 4–5 are quite variable from year to year, and as would be expected, there is no direct correspondence in the temporal variations of one GCM relative to another. To provide an alternative method of comparing the changes simulated by the future climate scenarios shown in Figs. 4–5, the daily anomalies for each trend line are also presented as frequency distributions in the Figs. 6–7 for the simulations made under the RPC6.0 scenario. These show that for all metrics there is a clear shift in the lake thermal conditions that are consistent with a warmer climate, but also that there are extremes in the distributions that can lead to unrepresentative results when, for example, future conditions briefly return to historical levels, or when the effects of warming are much greater that would be expected on average. This later case can cause important changes in lake ecology if the extreme conditions result in a change in lake state by the passing of a tipping point. Figures 6–7 also clearly show the differences in simulations forced by different GCMs. Most obvious is the difference in the results from GFDL–ESM2M, which consistently simulated smaller changes in lake thermal structure during the mid- and late-century periods despite having a data distribution that was similar to the other models during the historical period.

Future changes in thermal metrics based on both RCP2.6 and RCP6.0 were slightly greater when the GOTM model was forced at daily resolutions (Tables 5–6 and Sect. S4) than at an hourly resolution. This included changes in mean surface temperatures and also in the annual average whole-lake temperature (Sect. S5). However, under RCP2.6 non-significant differences were found for bottom temperature, Schmidt stability, thermocline depth, the duration, and onset and loss of stratification. In all cases where differences were found between the simulations forced with daily vs hourly data there were no changes in direction and only minor changes in the magnitude of the change suggested by the simulations (Sects. S4–S5).