FLake is a bulk model capable of predicting the vertical temperature structure and mixing conditions in lakes of various depths on timescales from hours to years . The model is based on a two-layer parametric representation of the evolving temperature profile in the water and on the integral budgets of energy for the layers in question. Bottom sediments and the thermodynamics of the ice and snow on ice layers are treated separately. FLake depends on prescribed lake depth information. The prognostic and diagnostic variables of HIRLAM FLake together with the analysed lake surface variables in HIRLAM are listed in the Appendix (Table ).

At each time step during the HIRLAM forecast, FLake is driven by the atmospheric radiative and turbulent fluxes as well as the predicted snowfall, provided by the physical parametrizations in HIRLAM. This couples the atmospheric variables over lakes with the lake surface properties as provided by FLake parametrization. Most importantly, FLake provides HIRLAM with the evolving lake surface (water, ice, snow) temperature and radiative properties that influence the HIRLAM forecast of the grid-average near-surface temperatures, humidity and wind.

Implementation of the FLake model as a parametrization scheme in HIRLAM was based on the experiments described by . Compared to the reference version of FLake , minor modifications were introduced, namely use of a constant snow density of 300 kg m-3, molecular heat conductivity of 1 J m-1 s-1 K-1, constant albedos of dry snow of 0.75 and ice of 0.5. Bottom sediment calculations were excluded. The Global Lake Database (GLDB v.2; ) was used for derivation of mean lake depth in each grid square. Fraction of lake was taken from HIRLAM physiography database, where it originates from GLCC .

Lake surface temperature is diagnosed from the mixed layer temperature for the unfrozen lake grid points and from the ice or snow-on-ice temperature for the frozen points. In FLake, ice starts to grow from an assumed value of 1 mm when temperature reaches the freezing point. The whole lake tile in a grid square is considered by FLake either frozen or unfrozen. Snow on ice is accumulated from the model's snowfall at each time step during the numerical integration.

A comprehensive description of the optimal interpolation (OI) of the LSWT observations in HIRLAM is given by . Shortly, LSWT analysis is obtained by correcting the FLake forecast at each grid point by using the weighted average of the deviations of observations from their background values. Prescribed statistical information about the observation and background error variance as well as the distance-dependent autocorrelation between the locations (observations and grid points) are applied. The real-time observations entering the HIRLAM surface analysis system are subject to quality control in two phases. First, the observations are compared to the background, provided by the FLake short forecast. Second, optimal interpolation is done at each observation location, using the neighbouring observations only (excluding the current observation) and comparing the result to the observed value at the station.

A specific feature of the lake surface temperature OI is that the interpolation is performed not only within the (large) lakes but also across the lakes: within a statistically pre-defined radius, the observations affect all grid points containing a fraction of lake. This ensures that the analysed LSWT on lakes without own observations may also be influenced by observations from neighbouring lakes, not only by the first guess provided by FLake forecast.

The relations between the OI analysis and the prognostic FLake in HIRLAM are schematically illustrated in Fig. . Within the present HIRLAM setup, the background for the analysis is provided by the short (+6 h) FLake forecast, but the next forecast is not initialized from the analysis. Instead, FLake continues running from the previous forecast, driven by the atmospheric state given by HIRLAM at each time step. This means that FLake does not benefit from the result of OI analysis, but the analysis has remained as an extra diagnostic field, to some extent independent of the LSWT forecast. However, FLake background has a large influence on the analysis, especially over distant lakes where neighbouring observations are not available. The diagnostic LSWT analysis, available at every grid point of HIRLAM, might be useful for hydrological, agricultural or road weather applications.

Missing LSWT observations in spring and early winter are interpreted to represent the presence of ice and given a flag value of -1.2 ∘C. If, however, the results of the statistical LSWT model , provided by SYKE along with the real-time observations, indicate unfrozen conditions, the observations are considered missing. This prevents the appearance of ice in summer if observations are missing but leads to a misinterpretation of data in spring if the SYKE model indicates too early melting. In the analysis, fraction of ice is diagnosed from the LSWT field in a simple way. The lake surface within a grid square is assumed fully ice covered when LSWT falls below -0.5 ∘C and fully ice free when LSWT is above 0 ∘C. Between these temperature thresholds, the fraction of ice changes linearly .

The HIRLAM surface data assimilation system produces comprehensive feedback information from every analysis–forecast cycle. The feedback consists of the observed value and its deviations from the background and from the final analysis at the observation point. Bilinear interpolation of the analysed and forecast values is done to the observation location from the nearest grid points that contain a fraction of lake. In addition, information about the quality check and usage of observations is provided. Fractions of land and lake in the model grid as well as the weights, which were used to interpolate grid point values to the observation location, are given. This information is the basis of the present study (see Sects.  and ).

FMI operational HIRLAM is based on the last reference version (v.7.4), implemented in spring 2012 (, and references therein). FLake was introduced into this version. After that, further development of the HIRLAM model was stopped. Thus, during the years of the present comparison, the FMI operational HIRLAM system remains unmodified, which offers a clean time series of data for the model–observation intercomparison. The general properties of the system are summarized in Table .

In this study we used three different types of SYKE lake observations: LSWT, freeze-up and break-up dates, and ice thickness and snow depth on lake ice. In total, observations on 45 lakes listed in the Appendix (Table ) were included as detailed in the following. The lake depths and surface areas given in Table  are based on the updated lake list of GLDB v.3 (Margarita Choulga, personal communication, 2018).

Figure  shows the frequency distribution of LSWT according to FLake forecast and SYKE observations. It is evident that the amount of data in the class of temperatures which represents frozen conditions (LSWT flag value 272 K) was underestimated by the forecast (Fig. a). When subzero LSWT values were excluded from the comparison (Fig. b), underestimation in the colder temperature classes and overestimation in the warmer classes still remain.

LSWT analysis (Fig. ) improved the distribution somewhat but the basic features remain. This is due to the dominance of FLake forecast via the background of the analysis. In Sect. , we will show time series illustrating the physics behind these LSWT statistics.

Table  confirms the warm bias by FLake in the unfrozen conditions. Similar results were obtained for all stations together and also for our example lakes Lappajärvi and Kilpisjärvi, to be discussed in detail in Sect. . There were three lakes with negative LSWT bias according to FLake forecast, namely the large lakes Saimaa and Päijänne and the smaller Ala-Rieveli. After the correction by objective analysis, a small positive bias converted to negative over six additional lakes, among them the large lakes Lappajärvi in the west and Inari in the north. The mean absolute error decreased from forecast to analysis on every lake.

In the frequency distributions, the warm temperatures were evidently related to summer. For FLake, the overestimation of maximum temperatures, especially in shallow lakes, is a known feature (e.g. ). It is related to the difficulty of forecasting the mixed layer thermodynamics under strong solar heating and possibly to the effect of the direct radiative heating of the bottom sediments. Cold and subzero temperatures occurred in spring and autumn. In a few large lakes like Saimaa, Haukivesi and Pielinen, LSWT tended to be slightly underestimated in autumn both according to FLake and the analysis (not shown). The cold left-hand side columns in the frequency distributions (Figs. a and a) are mainly related to spring, when HIRLAM tended to melt the lakes significantly too early (Sect.  and ).

There are problems, especially in the analysed LSWT, over (small) lakes of irregular form that fit the HIRLAM grid poorly and where the measurements may represent more the local than the mean or typical conditions over the lake. These are the only ones where an underestimation of summer LSWT was seen. Cases occurred where FLake results differ so much from the observations that the HIRLAM quality control against background values rejected the observations, forcing also the analysis to follow the incorrect forecast (not shown).

In this section the freeze-up and break-up dates from HIRLAM are verified against corresponding observed dates over 45 lakes (Appendix Table ). In the following, “LSWT an” refers to the LIDs estimated from analysed LSWT and “IceD fc” to those estimated from the forecast ice thickness by FLake. The time period contains six freezing periods (from autumn 2012 to autumn 2017) and seven melting periods (from spring 2012 to spring 2018). Due to some missing data, the number of freeze-up cases was 233 and break-up cases 258. The “IceD fc” data for the first melting period in spring 2012 were missing. The overall statistics of the error in freeze-up and break-up dates are shown in Table . In most cases the difference in error between the dates based on forecast and analysis was small. This is natural as the first guess of the LSWT analysis is the forecast LSWT by FLake. We will discuss next the freeze-up and then the break-up dates.

The bias in the error of freeze-up dates was small according to both “IceD fc” and “LSWT an”, -0.3 and -3.5 d, respectively. The minimum and maximum errors were large in both cases: the maximum freeze-up date occurred about 2 months too late, the minimum about 1.5 months too early. However, as will be shown later, the largest errors mostly occurred on a few problematic lakes while in most cases the errors were reasonable.

Figure a shows the frequency distribution of the error of freeze-up dates. Forecast freeze-up dates occurred slightly more often in the unbiased class (error between -5 and +5 d) compared to the estimated dates from the analysis. Of all cases, 48%/40% (percentages here and in the following are given as “IceD fc”/“LSWT an”) fell into this class. In 20%/26% of cases the freeze-up occurred, more than 5 d too late, and only in 11%/9% cases more than 2 weeks too late. In case of “IceD fc”, the class of freeze-up more than 15 d too late comprised 25 cases distributed over 15 lakes, thus mostly one or two events per lake. This suggests that the error was related more to individual years than to systematically problematic lakes. It is worth noting that of the eight cases where the error was over 45 d, six cases were due to a single lake, Lake Kevojärvi. This lake is situated in the northernmost region of Finland. It is very small and narrow, with an area of 1 km2, and located in a steep canyon. Therefore, it is poorly represented by the HIRLAM grid (grid square almost 50 km2) and the results seem unreliable.

Concerning too early freezing, in 33%/44% of the cases freeze-up occurred more than 5 d too early and in 15%/19% more than 2 weeks too early. According to the forecast, these 15 % (34 cases) were distributed over 19 lakes. Each of the five large lakes, Pielinen, Kallavesi, Haukivesi, Päijänne and Inari, occurred in this category three times while all other lakes together shared the remaining 19 cases during the six winters.

The break-up dates (Table ) show a large negative bias, about 2 (“LSWT an”) or 3 weeks (“IceD fc”), indicating that lake ice break-up was systematically forecast to occur too early. However, the standard deviation of the error was only about half of that of the error of freeze-up dates and there were no long tails in the distribution (Fig. b). Hence the distribution is strongly skewed towards too early break-up, but much narrower than that of freeze-up (Fig. a). The large bias was most probably due to missing snow over lake ice in this HIRLAM version (see Sect. ). The maximum frequency (47 %) was in the class -24 to -15 d for “IceD fc”, while in case of “LSWT an” the maximum frequency (52 %) occurred in the class -14 to -5 d. FLake forecast “IceD fc” suggested only three cases in the unbiased class -4 to +5, while according to “LSWT an” there were 12 cases in this class. Hence, the break-up dates derived from analysed LSWT corresponded to the observations better than those derived from FLake ice thickness forecast.

Note that this kind of method of verifying LID compares two different types of data. The observations by SYKE are visual observations from the shore of the lake (see Sect. ), while the freeze-up and break-up dates from HIRLAM are based on single-grid-point values of LSWT or ice thickness (see Sect. ). In addition, the resulting freeze-up and break-up dates from HIRLAM are somewhat sensitive to the definition of the freezing and melting thresholds. Here we used 1 mm for the forecast ice thickness and the freezing point for the LSWT analysis as the critical values.

In conclusion, the validation statistics show that HIRLAM succeeded rather well in predicting freezing of Finnish lakes. In almost half of the cases the error was less than ±5 d. Some bias towards too early freeze-up can be seen both in forecast and in the analysis. Melting was more difficult. FLake predicted lake ice break-up always too early, with a mean error of over 2 weeks, and the analysis mostly followed it.

In this section we present LSWT and LID time series for two representative lakes, Kilpisjärvi in the north and Lappajärvi in the west (see the map in Fig. ). Observed and forecast ice and snow thicknesses are discussed, using also additional data from Lake Simpelejärvi in south-eastern Finland.

Lake Kilpisjärvi is an Arctic lake at an elevation of 473 m, surrounded by fells. The lake occupies 40 % of the area of the HIRLAM grid square covering it (the mean elevation of the grid square is 614 m). The average and maximum depths of the lake are 19.5 and 57 m and the surface area is 37.3 km2. The heat balance as well as the ice and snow conditions on Lake Kilpisjärvi has been subject to several studies . Typically, the ice season there lasts 7 months from November to May. Lake Lappajärvi is formed from a 23 km wide meteorite impact crater, which is estimated to be 76 million years old. It is Europe's largest crater lake with a surface area of 145.5 km2 and average and maximum depths of 6.9 and 36 m. Here the climatological ice season is shorter, typically about 5 months from December to April. The average and maximum depths of Lake Simpelejärvi are 8.7 and 34.4 m and the surface area is 88.2 km2. This lake is located at the border between Finland and Russia and belongs to the catchment area of Europe's largest lake, Lake Ladoga in Russia.

Figures  and show the frequency distributions of LSWT according to forecast vs. observation and analysis vs. observation for Lappajärvi and Kilpisjärvi. Features similar to the results averaged over all lakes (Sect. , Figs.  and ) are seen, i.e. underestimation of the amount of cold temperature cases and overestimation of the warmer temperatures by the forecast and analysis. On Lake Lappajärvi, only the number of below-freezing temperatures was clearly underestimated, otherwise the distributions look quite balanced. According to the observations on Lake Kilpisjärvi, ice-covered days dominated during the period from November to May. According to both LSWT analysis and forecast, the number of these days was clearly smaller in HIRLAM.

Yearly time series of the observed, forecast and analysed LSWT, with the observed LID marked, are shown in Figs.  and . In the absence of observations, the HIRLAM analysis followed the forecast. Missing data in the time series close to freeze-up and break-up are due to missing observations, hence missing information in the feedback files (see Sect. ). Differences between the years due to the different prevailing weather conditions are seen in the temperature variations.

Generally, FLake tended to melt the lakes too early in spring, as already indicated by the LID statistics (Sect. ). The too early break-up and too warm LSWT in summer show up clearly in Kilpisjärvi (Fig. ). In Lappajärvi (Fig. ), the model and analysis were able to follow even quite large and quick variations in LSWT in summer but tended to somewhat overestimate the maximum temperatures. Overestimation of the maximum temperatures by FLake was still more prominent in shallow lakes (not shown). In autumn over lakes Lappajärvi and Kilpisjärvi, the forecasts and analyses followed closely the LSWT observations and reproduced the freeze-up dates within a few days, which was also typical for the majority of lakes.

Figure  shows a comparison of forecast and observed evolution of ice thickness and snow depth on Lappajärvi, Kilpisjärvi and Simpelejärvi in winter 2012–2013, typical also for the other lakes and years studied. The most striking feature is that there was no snow in the HIRLAM forecast.

On all three lakes, the ice thickness started to grow after freeze-up both according to the forecast and the observations. In the beginning HIRLAM ice grew faster than observed. However, according to the forecast ice thickness started to decrease in March of every year but according to the observations only a month or two later.

The too early break-up of lake ice in the absence of snow can be explained by the wrong absorption of the solar energy in the model. In reality, the main factor of snow and ice melt in spring is the increase in daily solar radiation. In HIRLAM, the downwelling shortwave irradiance at the surface is known to be reasonable, with some overestimation of the largest clear-sky fluxes and all cloudy fluxes . Over lakes, HIRLAM uses constant values for the snow and ice shortwave reflection, with albedo values of 0.75 and 0.5, correspondingly. When there was no snow, the lake surface was thus assumed too dark. A 25 % more absorption of an assumed maximum solar irradiance of 500 W m-2 (valid for the latitude of Lappajärvi at the end of March) would mean availability of an extra 125 W m-2 for melting of the ice, which corresponds to the magnitude of increase in available maximum solar energy within a month at the same latitude.

The forecast of too thick ice can also be explained by the absence of snow in the model. When there is no insulation by the snow layer, the longwave cooling of the ice surface in clear-sky conditions is more intensive and leads to faster growth of ice compared to the situation of snow-covered ice. In nature, ice growth can also be due to the snow transformation, a process whose parametrization in the models is demanding .

Also the downwelling longwave radiation plays a role in the surface energy balance. We may expect values from 150 to 400 W m-2 in the Nordic spring conditions, with the largest values related to cloudy situations and the smallest to clear-sky situations. The standard deviation of the downwelling longwave radiation fluxes predicted by HIRLAM has been shown to be of the order of 20 W m-2, with a positive systematic error of a few watts per square metre (W m-2) . Compared to the systematic effects related to absorption of the solar radiation, the impact of the longwave radiation variations on lake ice evolution is presumably small.