# Energy Flux Paths in Lakes and Reservoirs

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

^{2}). For the first time we use directly measured atmospheric momentum fluxes. The estimated energy fluxes and content agree well with results reported for larger lakes, suggesting that the energetics governing water motions in enclosed basins is similar, independent of basin size. The largest fraction of wind energy flux is transferred to surface waves and increases strongly nonlinearly for wind speeds exceeding 3 m s

^{−1}. The energy content is largest in basin-scale and high-frequency internal waves but shows seasonal variability and varies among aquatic systems. At one of the study sites, energy dissipation rates varied diurnally, suggesting biogenic turbulence, which appears to be a widespread phenomenon in lakes and reservoirs.

## 1. Introduction

^{2}) water bodies. The study sites, a lake and a reservoir, differ in surface area by one order of magnitude, but have comparable water depth (~10 m). Given the difference in surface area and the fact that the reservoir experiences water level variations, we expect the hydrodynamic processes in these two water bodies to be different. The selected lake is considered representative of a large number of small lakes, belonging to the most abundant size class which contributes 54% of the global lake surface area [22,23]. To overcome shortcomings of previous studies, we used direct measurements of momentum fluxes in the atmosphere above the water surface for the estimation of the wind energy flux into the lakes. We investigate the wind speed and fetch dependence on the surface wave characteristics based on measurements covering nearly a complete annual cycle. The predominant modes of basin-scale internal waves and the presence of high-frequency internal waves are identified and examined. The data are used to complement and to re-examine mean energy budgets of small lentic systems, their temporal dynamics, and their variation with water body size.

## 2. Materials and Methods

#### 2.1. Study Sites and Measurements

^{2}, volume: 39.2 × 10

^{6}m

^{3}, maximum depth: 12.2 m) and a small lake (Lake Dagow, surface area: 0.3 km

^{2}, volume: 1.2 × 10

^{6}m

^{3}, maximum depth: 9.5 m)—both situated in Germany. Bautzen Reservoir is a part of the dammed river Spree in southeastern Germany, with a mean water residence time of 164 days [24]. It can be classified as a small storage-type reservoir [25,26] with additional purposes of flood control and leisure activities. Besides, the reservoir is used to regulate the water supply for wetlands and power stations located downstream of the river. The outlet tower located near the dam regulates water discharge through the bottom of the reservoir. Major water withdrawal in summer is associated with a gradual decrease of water level [27]. The reservoir is often not persistently stratified throughout the summer due to a lack of shelter against strong winds and experiences several full mixing events [24,28]. Lake Dagow is a glacial lake in the Lake Stechlin area in northern Germany. It is a small eutrophic lake with a water residence time of 5 years [29]. The lake develops persistent density stratification every year leading to anoxic conditions in its hypolimnion [29]. Lake Dagow was impacted by wastewater, duck and carp farming in the 1960–1970s, with restoration activities in the 1980s.

#### 2.2. Energy Content

#### 2.2.1. Internal Waves

_{max}(Hz) [19,34]:

^{−2}) is gravitational acceleration, ρ

_{w}(kg m

^{−3}) is water density, ρ

_{w0}(kg m

^{−3}) is the mean water density, z (m) is the height above the bottom with positive direction upwards. The conversion from temperature to density was done based on the freshwater equation of state following [35].

^{−3})) from temporal variations of water density observed at a single mooring location [37,38]:

^{−3}to 6.1 × 10

^{−3}Hz was selected.

_{surf}is the surface area of the water body, H is the water depth, A is the depth-dependent cross-sectional area.

#### 2.2.2. Surface Waves

_{sig}(m) and energy content in surface waves E

_{wave}(J m

^{−2}) were calculated from pressure fluctuations measured by the wave recorders. The calculation was carried out following standard procedures based on linear wave theory [40] by using the “Ruskin” software provided by the manufacturer [41]. The calculations take into account the attenuation of wave-induced pressure fluctuations at the sampling depth of the sensor. Significant wave height is defined as the average height of the highest one third of the waves during each sampling interval. Mean wave energy was calculated from the variance of water surface elevation. Note that for Bautzen Reservoir we used only those wave measurements that corresponded to the acceptable wind directions (195–355°) to avoid the possible sheltering effect of the measurement platform (the sensor was deployed at the south-western corner).

#### 2.2.3. Schmidt Stability

^{−2}) describes the integrated potential energy in vertical density stratification of the entire basin. It is equivalent to the work required for vertical mixing, i.e., the energy required to move the vertical coordinate of the center of mass of all water in the basin to the corresponding center of volume z

_{v}:

#### 2.3. Energy Fluxes

#### 2.3.1. Wind Energy Flux and Rate of Working

_{10}(W m

^{−2}) is equivalent to the vertical shear stress multiplied by the horizontal wind velocity [42]:

^{−1}s

^{−1}) is the shear stress, ${u}_{\ast \mathrm{a}}$ (m s

^{−1}) is the atmospheric friction velocity and ρ

_{a}(kg m

^{−3}) is air density. In our analysis, we estimated ${u}_{\ast \mathrm{a}}$ from measurements of turbulent velocity fluctuations in the atmospheric boundary layer using the eddy-covariance (EC) method [43]. The mean wind speed measured at a height of 1.8 m (Bautzen Reservoir) and 1.97 m (Lake Dagow) above the water surface was corrected to a standard height of 10 m (${U}_{10}$ (m s

^{−1})) by considering atmospheric stability [44,45]. We calculated the fraction of the wind energy that is transferred to the water as the total rate of working RW (W m

^{−2}) following [16]:

_{DN10}was first determined as ${C}_{\mathrm{D}10}={u}_{\ast}^{2}/{U}_{10}^{2}$ (-) and then corrected to its neutral counterpart ${C}_{\mathrm{DN}10}={C}_{\mathrm{D}10}{\left(1+{C}_{\mathrm{D}10}^{1/2}{\kappa}^{-1}\psi \left(10/L\right)\right)}^{-2}$, where κ = 0.4 (-) is the von Kármán constant, L is the Monin-Obukhov length scale and ψ are the stability functions described in [44]. Note that, we used the first acceptable measurement of flow velocity below the surface: for Bautzen Reservoir, it corresponds to ~1 m depth, for Lake Dagow ~0.6 m (first deployment), ~1.3 m (second and third deployment).

#### 2.3.2. Surface Wave Energy Flux

_{wave}(W m

^{−1})) was calculated as the product of the wave energy and the wave group velocity. The group velocity is a function of wave period, which we assign to the period corresponding to the maximum in the wave spectrum. The estimation of the wave energy flux proceeds in the same way as in [46]. To compare P

_{wave}with the wind energy flux P

_{10}, we considered the ratio P

_{wave}/(P

_{10}× F) 100 (%), where F (m) is the wind fetch at the wave measurement location. The wind fetch is interpolated from distances obtained from the map corresponding to the standard grid of wind direction. Note that, as in Section 2.2.2, we disregarded data with unacceptable wind directions for Bautzen Reservoir.

#### 2.3.3. Surface Heat Flux and Buoyancy Flux

_{net}(W m

^{−2}) is expressed as the sum of net shortwave radiation Q

_{SW}, longwave radiation Q

_{LW}, and latent and sensible heat fluxes H

_{L}, H

_{S}(W m

^{−2}). Latent and sensible heat fluxes were calculated following the standard EC methodology using the “Eddy Pro Version 6.2.1” software (LI-COR, Inc., Lincoln, NE, USA).

_{BO}(W m

^{−2}) was calculated as:

^{−1}) is the temperature-dependent thermal expansion coefficient of water and c

_{p}(J kg

^{-1}K

^{−1}) is the specific heat capacity of water.

#### 2.3.4. Energy Flux to Basin-Scale Internal Waves

#### 2.4. Dissipation Rates

^{−1}) were estimated following two methods: inertial subrange fitting (ISF) [47] and second-order structure function (SF) [48]. Both methods have been widely applied and validated for obtaining dissipation rates from velocity data measured by ADCP [49,50,51,52]. Under the assumption of isotropic turbulence, the theoretical power spectrum of turbulent velocity fluctuations S (m

^{3}s

^{−2}rad

^{−1}) is expressed as:

_{K}= 1.5 (-) is the universal Kolmogorov constant, k (rad m

^{−1}) is the spatial wavenumber and C

_{1}(-) is the isotropy constant, which depends on the direction of the velocity component 18/55 ≤ C

_{1}≤ 4/3 × 18/55. We used a constant value of C

_{1}= 18/55 as we used beam-averaged velocity spectra from the ADCP, which measures along-beam velocity fluctuations without directional information [53]. Power spectra estimated from measurements were fitted to Eq. 10 to estimate the dissipation rates. The upper wavenumber limit for the fit was found as a breakpoint where the power spectral density became smaller than the level of noise. The noise level was determined as the logarithmically averaged high-frequency end of the spectra at frequencies higher than 0.2 Hz. To find the lower frequency limit for inertial subrange fitting, we used the optimization procedure described in [54]. We used three criteria for quality assurance for calculated dissipation rates: validity of Taylor’s frozen turbulence hypothesis, coefficient of determination of the fit (for both—see [47]) and the length of observed inertial subrange (set to a minimum of 10/8 of decade). The application of these quality criteria led to significant reduction in dissipation rate estimated using ISF (~70% of the data were removed).

^{2}s

^{−2}) is the mean squared velocity difference at two locations separated by the distance r (m), C

_{2}= 2.1 (-) is a constant, C

_{3}(-) is a coefficient describing wave orbital motion and N

_{m}(m

^{2}s

^{−2}) is the measurement noise. C

_{3}, C

_{2}ε

^{2/3}and N

_{m}were determined using least square fits of measured along-beam velocity fluctuations to Equation (11). We used fixed numbers of ADCP bins for the fitting. First, we applied the procedure with 5 bins (for the purpose of the calculations the number of bins should be odd). We noticed that noise could be negative in cases when the theoretical structure function was not long enough to reach its “plateau”. Therefore, we used the procedure with seven bins and replaced the values of dissipation rates from the previous step for cases when the noise was negative. We disregarded fits, if either N

_{m}, C

_{2}ε

^{2/3}, the difference between the first point of the structure function and the noise, or the difference between the second and first point of the structure function were negative. The application of these criteria led to ~51% and ~30% reduction of the dissipation rate estimates for Bautzen Reservoir and Lake Dagow, respectively.

^{−8}W kg

^{−1}. The final dissipation rates combined both estimates using ISF and SF techniques considering the ISF calculations as default value and gap-filling with estimates from SF.

^{−1}) at the measurement depth (ADCP bin) closest to the bottom using the law of the wall [56]:

^{−1}) is the bottom friction velocity. Following [56], the bottom drag coefficient ${C}_{\mathrm{Db}}$ was corrected for the distance from the bottom at which the flow speed was measured, using a standard value of 1.5 × 10

^{−3}at 1 m height. Resulting dissipation rate profiles were integrated over depth as in Equation (3) and multiplied by density of the water ρ

_{w}to obtain areal estimates of depth-integrated dissipation rates (in W m

^{−2}).

## 3. Results

#### 3.1. Overview of the Measurements

^{−1}. These mixing events are consistent with observations at this reservoir reported in previous studies [24,28,57]. The maximum temperature at the water surface was 29.2 °C (4 August) in Bautzen Reservoir and 18 °C (11 September) in Lake Dagow. The maximum temperature difference between surface and bottom was 15.2 °C (10 June) and 8 °C (12 September) in Bautzen Reservoir and Lake Dagow, respectively. In Lake Dagow, the thermocline was located close to the bottom and the thickness of the hypolimnion was only ~0.7 m. Average Schmidt stability was 4.3 and 41.8 J m

^{−2}and maximum values were 21 J m

^{−2}(17 September) and 178 J m

^{−2}(01 June) in Lake Dagow and Bautzen Reservoir, respectively. The heat and buoyancy fluxes varied between −155 and 1113 W m

^{−2}(−5.1 × 10

^{−4}and 4.2 × 10

^{−3}W m

^{−2}) in Bautzen Reservoir and between −130 and 763 W m

^{−2}(−2.6 × 10

^{−4}and 3.1 × 10

^{−3}W m

^{−2}) in Lake Dagow (buoyancy flux in parenthesis, Table S1, Figure S8, Supplementary Material).

^{−1}(here and further, ± denotes standard deviation) with maximum values of 13.7 and 10.1 m s

^{−1}in Bautzen Reservoir and Lake Dagow, respectively. West-northwestern (280–300°) and south-western (220–240°) wind directions were predominant for Bautzen Reservoir and Lake Dagow, respectively. The water level continuously declined throughout the study period from 11.2 m to 6.8 m at the platform location in Bautzen Reservoir, while in Lake Dagow it remained constant (~8.3 m at the ADCP location). Water discharge at the inflow and at the outlet tower varied between 0.6 and 3.9 m

^{3}s

^{−1}, with mean values of 1.2 and 1.9 m

^{3}s

^{−1}, respectively (Figure S2, Supplementary Material). In both water bodies, flow velocities were relatively small for most of the time (~0.01–0.02 m s

^{−1}). The maximum flow speed was 0.1 m s

^{−1}in Bautzen Reservoir and 0.07 m s

^{−1}in Lake Dagow. The mean significant wave height H

_{sig}was (3.9 ± 9.6) × 10

^{−3}m and (1.9 ± 2.7) × 10

^{−2}m at shore sampling locations (Point B in Bautzen Reservoir, Point G in Lake Dagow, see Figure 1) and (7.4 ± 9.9) × 10

^{−2}m at the platform in Bautzen Reservoir. The maximum value of significant wave height was 0.1 and 0.2 m for the shore sampling locations in Lake Dagow and Bautzen Reservoir, respectively, and 0.5 m at the platform in Bautzen Reservoir.

#### 3.2. Wind Energy Flux and Rate of Working

_{10}) to the rate of working associated with shear stress in the surface layer of the water column (RW) provides an estimate of the efficiency of the energy transfer from wind to water. We analyzed the ratio separately for mixed and for stratified conditions, but we did not find significant differences between both conditions (Figure 3). We used a fixed, although arbitrary, threshold for Schmidt stability (Sc = 5 J m

^{−2}) to distinguish between mixed and stratified conditions. Distributions of the ratio of RW to P

_{10}for periods when the Schmidt stability was smaller or larger than 5 J m

^{−2}were in close agreement (Figure S3, Supplementary Material). The median values of the ratio were 1.8 × 10

^{−3}and 1.6 × 10

^{−3}in Bautzen Reservoir, and 1.7 × 10

^{−3}and 0.7 × 10

^{−3}in Lake Dagow for non-stratified and stratified conditions, respectively. We applied linear regressions of RW as a function of P

_{10}considering P

_{10}less than 2 W m

^{−2}, as most of the data belonged to this interval. Different values of the threshold in Schmidt stability to distinguish stratified and mixed conditions did not result in significant changes in slopes. However, we noticed that the slope of the regression was sensitive to the inclusion of few high-magnitude values. The slope coefficient for all data, which describes the efficiency of energy transfer is equal to (1.3 ± 0.1) × 10

^{−3}and (2.61 ± 0.05) × 10

^{−3}(± here denotes standard error for the slope) for Bautzen Reservoir and Lake Dagow, respectively. Data from Bautzen Reservoir were additionally filtered based on the same wind directions as for surface waves (Section 2.2.2) to avoid potential sheltering by the measurement platform. These values were comparable to the mean efficiency of 1.3 × 10

^{−3}estimated under mixed conditions in a lake by [16].

#### 3.3. Surface Waves

^{−2}m) with weak dependence on wind speed for winds below ~3 m s

^{−1}, while wave heights strongly increased for wind speeds exceeding 3–4 m s

^{−1}. Significant wave heights measured at the shore locations were generally smaller in amplitude and showed a weaker increase with increasing wind compared to open-water measurements (Figure 4a). This could be related to the interference with waves reflected from the shore and shallow water depth at the sampling locations.

^{−1}and underestimated wave heights for higher wind speeds (see blue lines, Figure 4a). This supported the earlier finding of [18]; however, their “breakpoint” was at around 6 m s

^{−1}and the JONSWAP predictions agreed well with the measurements at higher wind speed. The authors of [18] note that the possible reasons for underestimation at low wind speeds could either be insufficient sensor accuracy to resolve the small amplitude of waves, or a failure to compute the wind fetch correctly as the variability in wind direction at low wind speeds is very high. The wind fetch indeed varied considerably due to the variations in wind direction; however, smoothing of the calculated fetch prior to applying the wave model did not significantly reduce the predicted values at low wind speeds. The JONSWAP model overestimated the significant wave height for both measurements at shore locations.

^{−4}and 9.1 × 10

^{2}J m

^{−2}with a log-averaged value of 0.3 J m

^{−2}for the measurements at the platform in Bautzen Reservoir. In Lake Dagow, it varied between 1.6 × 10

^{−4}and 1.1 × 10

^{1}J m

^{−2}with a log-averaged value of 1.5 × 10

^{−3}J m

^{−2}. Wave energy measured at the shore in Bautzen Reservoir (1.3 × 10

^{−4}–2.1 × 10

^{1}J m

^{−2}with log-averaged value 2 × 10

^{−2}J m

^{−2}) was comparable in magnitude to the shore measurements in Lake Dagow. Wave energy showed strong dependence on the wind speed exceeding 3 m s

^{−1}with a power-law exponent of ~8–9 (Figure S4, Supplementary Material).

^{−1}.

#### 3.4. Internal Waves

^{−2}, while it remained nearly constant below 1 J m

^{−2}during most of the time. Its average value for 13 analyzed internal wave events was 1.2 ± 1.0 J m

^{−2}. Note that the periods of the internal waves in the two last events (October and November, see Figure 6a) were not supported by the predictions of the Internal Wave Analyzer; however, we visually observed internal waves as a peak in the velocity spectrum. The average APE in high-frequency internal waves evaluated for 210 events reached its maximum value of 0.45 J m

^{-2}in summer during the strongest stratification (end of June–beginning of September). The average value over the entire measurement period was 0.06 ± 0.05 J m

^{−2}.

^{−2}, which was comparable to the APE (note that the difference between them is greater during April to August, probably because velocity measurements cover only the upper part of the water column). In Lake Dagow, we identified only one event with basin-scale internal wave activity and the corresponding APE of 0.05 J m

^{−2}was two orders of magnitude smaller than in Bautzen Reservoir because the thermocline was close to the bottom. Following [14], we used the double value of APE as a measure of the total energy in internal waves.

#### 3.5. Dissipation Rate in Surface and Bottom Boundary Layers

^{−8}W kg

^{−1}) in Bautzen Reservoir and Lake Dagow (Figure 7). They tended to increase towards the water surface, while remaining nearly constant in the middle of the water column. Estimates of dissipation rate in the bottom boundary layer (Equation (12)) were on average one (Bautzen Reservoir) and two (Lake Dagow) orders of magnitude smaller than dissipation rates in the interior and the surface layer.

^{−1}in Bautzen Reservoir (Figure S7). We did not find this relationship in the data from Lake Dagow, where the wind energy flux reached comparable magnitude, but integrated dissipation rates remained below the high values observed in Bautzen Reservoir. However, rare events with high integrated energy dissipation rates did not contribute to mean conditions. The average ratio of total integrated dissipation rate and wind energy flux was 0.23% and 0.5% for Bautzen Reservoir and Lake Dagow, respectively. Similarly, the mean values of depth-integrated dissipation rates were a factor of two lower in Bautzen (1.7 × 10

^{−5}W m

^{−2}) compared to Lake Dagow (3.4 × 10

^{−5}W m

^{−2}).

## 4. Discussion

#### 4.1. Overall Energy Budget

#### 4.2. Energy Transfer Efficiency

_{10}), which were slightly higher but of the same order of magnitude as RW (0.14–0.27% of P

_{10}). Generally, this agreement supports the magnitudes of the energy fluxes compiled in Figure 8.

#### 4.3. Surface and Internal Waves

^{2}) lake in Switzerland. However, this fraction strongly increased with the wind speed exceeding 3 m s

^{−1}and depended on the sampling location. We demonstrated that the JONSWAP model for the estimation of the significant wave height may not be an appropriate approach for estimating significant wave heights in smaller lakes and reservoirs as it significantly overestimated wave height at low wind speed. At high wind speed, we found an extremely strong increase of wave energy with a power-law coefficient of ~8–9 with wave height exceeding the JONSWAP predictions. More wave observations in different lakes and reservoirs and further detailed investigation of the relationship between the wave characteristics and wind speed are needed to improve predictions of wave height and wave energy fluxes in lakes and reservoirs.

^{−2}versus 22 ± 3 J m

^{−2}) and slightly higher than the values in [15] (10

^{−2}–1 J m

^{−2}). We assume that this difference can be related to the strength of stratification and lake depth. The alpine lakes studied in [14] and [15] showed persistent and large-amplitude internal seiching, which occurred rather sporadically and with smaller amplitudes in Bautzen and Dagow. In addition, the energy content in basin-scale internal waves varied with season and was on average five-fold higher in spring than for the remaining sampling period in Bautzen Reservoir. This can be explained by the deepening of the thermocline and the way of calculation of the energy content with the thermocline depth being the upper limit for vertical integration. Also, lake bathymetry can affect the seasonal variation of the internal waves [64]. The energy flux to the basin-scale internal waves can be up to 0.1% of the wind energy flux but is on average two orders of magnitude smaller than that reported for the alpine lake (0.04% versus 1% in [13]). Energy content in high-frequency internal waves was on average one order of magnitude smaller than in basin-scale internal waves (Bautzen Reservoir) and comparable with basin-scale internal waves in Lake Dagow. During the stratified season, high-frequency waves can contain on average twice as much energy than during the remaining period.

#### 4.4. Energy Dissipation Rates

^{−9}W kg

^{−1}). More energy was dissipated with increasing wind energy flux. Although in Bautzen Reservoir at high wind speeds almost all of the wind energy flux was dissipated, this was not observed at Lake Dagow, which may be related to the smaller size of the lake and the sheltering effect of the surrounding forest. On average, a similar percentage of the wind energy flux was dissipated in both water bodies. However, the dissipation rates estimated for the bottom boundary layer were on average one (Bautzen Reservoir) and two (Lake Dagow) orders of magnitude smaller than those calculated for the remaining water column. This can potentially be explained by the fact that flow velocities were generally very low, and the boundary layer may not be observed within the ADCP profiling range, making an underestimation of the dissipation rates possible.

^{−3}have been shown to enhance dissipation rates by one order of magnitude [67]. The role of biogenic turbulence in marine in inland waters has been widely discussed and analyzed in the past decade (see reviews in [68,69]). The main conclusion was that although small swimmers may generate additional flow and energy dissipation, they are unlikely to contribute to vertical mixing. Small swimmers generate flow disturbances at the scale of some multiple body length [70]. The dissipation rate estimates from ADCP are limited by the relatively large size of the sampling volume (bin size) and are theoretically based on turbulent energy transfer from large to small scales. These limitations may challenge the measurement of energy dissipation rates with ADCP in the presence of small swimmers. The increasing reporting of diurnal patterns in energy dissipation rates in relation to acoustic backscatter in recent studies calls for careful validation of these estimates using alternative methods for estimating dissipation rates, such as microstructure profiling and particle image velocimetry.

#### 4.5. Study Limitations

## 5. Conclusions

^{−1}. Existing parametrizations of wave height as a function of wind speed and fetch length fail to reproduce observed wave amplitudes in small water bodies.

^{−1}. We observed a pronounced diurnal pattern in dissipation rates at one of our study sites, which is most likely related to vertically migrating organisms. The reliability of commonly applied measurement and analysis procedures for estimating energy dissipation rates in the presence of swimming organisms needs to be confirmed in future studies.

## Supplementary Materials

^{−2}(area shown in gray, corresponding to non-stratified conditions) and Sc ≥ 5 J m

^{−2}(area shown in red, corresponding to stratified conditions) for (a) Bautzen Reservoir. The median values are 1.8 × 10

^{−3}and 1.6 × 10

^{−3}, the average values (±standard deviation) are (2 ± 4)·10

^{−3}and (0.6 ± 21.6)·10

^{−2}, for non-stratified and stratified conditions, respectively. (b) Lake Dagow. Figure S4: Surface wave energy versus wind speed at 10 m height at the platform location in Bautzen Reservoir (gray dots). Wave energy shows strong dependence on wind speed exceeding 2–3 m s

^{−1}. The black line shows bin-average data, the red line represents a power-law relationship with an exponent of nine. The latter was obtained from a linear regression of log-transformed data. Figure S5: (a) Dissipation rates of turbulent kinetic energy averaged over night and over daytime during the first ADCP deployment in Lake Dagow. (b) Acoustic backscatter strength recorded by the ADCP (upper panel), vertical flow velocity (middle panel), dissipation rate (lower panel). Figure S6: Depth-integrated dissipation rate (including surface and bottom boundary layers and interior of the water bodies) versus the vertical wind energy flux above the water surface in (a) Bautzen Reservoir; (b) Lake Dagow. Figure S7: Dissipation rate integrated over the bottom boundary layer (the thickness of 2 m, light gray dots) and over the rest of the water column where the ADCP measurements are available (dark gray dots) using data from (a) Bautzen Reservoir; (b) Lake Dagow. The gray solid line represents a 1:1 relationship. Figure S8: Temporal dynamics of wind energy flux (black line, upper panel), dissipation rates integrated over the water depth (red dots, upper panel) and buoyancy flux (lower panel) for data measured in (a) Bautzen Reservoir; (b) Lake Dagow. Note the pronounced diurnal pattern in integrated energy dissipation rates in Lake Dagow during the first ADCP deployment (cf. Figure S4). Table S1: Energy content and energy fluxes.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Bathymetric maps of the study sites: (

**a**) Bautzen Reservoir; (

**b**) Lake Dagow. The plots were created based on the published maps in [24,29]. Black lines show isolines with equal elevation in meters above sea level (m a.s.l.). Small panels on the right show wind roses with wind directions and windspeed. The locations of the instruments are indicated by triangles and circles labeled with capital letters. Points A and E (triangles) mark the locations of the platforms for micrometeorological measurements and the acoustic Doppler current profiler (ADCP). The ADCPs were deployed at the bottom, at a distance of approximately 10 m from the platforms. Points A and F mark the locations of the thermistor chains. A, B and G mark the location of the surface wave observations (pressure sensors).

**Figure 2.**Overview of wind forcing and hydrodynamic conditions in (

**a**) Bautzen Reservoir and (

**b**) Lake Dagow. From the top to the bottom: (1) wind speed corrected to a height of 10 m; (2) significant wave height; (3) temperature profile (color denotes temperature, lines show isothermal depths); (4) flow velocity profiles (velocity magnitude). The black lines mark the location of the water surface. All data are shown at 30 min resolution.

**Figure 3.**Potential energy in stratification and efficiency of energy transfer from wind to water in (

**a**) Bautzen Reservoir and (

**b**) in Lake Dagow. From top to bottom: (1) the black line shows the time series of Schmidt stability (Sc). The horizontal red lines mark the threshold value (5 J m

^{−2}) to separate mixed and stratified conditions. (2) Relationship between rate of working RW and wind energy input P

_{10}. Gray dots show all data. Red and blue lines represent bin-averages for two selected cases: Sc ≥ 5 and Sc < 5 J m

^{−2}—indicating stratified and mixed conditions, respectively. A minimum of 5 data points was considered for the bin-averaging. The black line shows a linear regression for all data with P

_{10}< 2 W m

^{−2}. Inset graphs in the lower panels show a detailed view of the data at small energy fluxes. The slope coefficient, i.e., the efficiency of energy transfer from wind to water is equal to (1.3 ± 0.1) × 10

^{−3}and (2.61 ± 0.05) × 10

^{−3}for Bautzen Reservoir and Lake Dagow, respectively.

**Figure 4.**(

**a**) Significant wave H

_{sig}as a function of wind speed ${U}_{10}$. Lines with markers represent measurements: black color shows data from [18]. Blue, green and red colors represent the bin-averaged data from the present study: Bautzen Reservoir A (platform), B (shore) and Dagow Lake G (shore). Solid lines show a commonly applied empirical parameterization of the significant wave height based on the wind speed and fetch length (JONSWAP). The parameterization was applied to the observed wind speed and direction with a resolution of 30 min. (

**b**) Boxplots showing the percentages of the ratio of wave energy flux per unit length of wave crest (P

_{wave}) to the fetch-integrated wind energy flux (approximated as P

_{10}multiplied by the fetch length) for three measurement locations. The central horizontal line in boxes indicates the median; the bottom and top edges of the boxes denote the 25th and 75th percentiles; the whiskers extend to the largest data points which are not considered outliers.

**Figure 5.**(

**a**) Sample data from Bautzen Reservoir illustrating high-frequency internal waves with a period of 6 min (upper panel) and basin-scale internal waves with the period of 9 h (lower panel). The upper and lower panels show vertical (w) and horizontal (u) velocity components, respectively. Black lines show temperature isotherms; blue lines show the distance above the bed for which the velocity power spectra are shown in panel (

**b**). (

**b**) Power spectral density estimates for flow velocity components (m

^{2}s

^{−1}) and isotherms (m

^{2}s). Blue, light blue and red solid lines show velocity spectra; gray, black and light red show isotherm spectra for Bautzen Reservoir and Lake Dagow, respectively. Internal waves are associated with distinct spectral peaks and vertical dashed lines denote major internal wave periods.

**Figure 6.**(

**a**) Average available potential energy (APE) in basin-scale internal waves (upper panel) and in high-frequency internal waves (lower panel) throughout the measurement period in Bautzen Reservoir. The red-shaded areas correspond to the presence of the basin-scale waves with 8–9 h period, the green-shaded area shows the presence of a wave with 21 h period. The width of the bars is not to scale but is proportional to the event duration (i.e., narrow bars indicate shorter events where waves are present, and wider bars indicate longer events). The black line in the lower panel (

**a**) shows a moving average of the APE. (

**b**) Sample data demonstrating the transfer of wind energy to basin-scale internal waves. The upper panel shows the time series of P

_{10}and APE averaged over one wave cycle. After the wind event stopped, APE grows first and then decays after three wave cycles. The lower panel shows isothermal depths with the event duration marked by red vertical lines.

**Figure 7.**Vertical profiles of log-averaged energy dissipation rates (ε) for all available data (gray and red shaded areas show the 5th to 95th range of data): (

**a**) combined data with dissipation rates calculated using the structure function and inertial subrange fitting methods (black circles, the platform-mounted ADCP facing downward) and using bottom boundary layer approach (see Section 2.4, red circles, the ADCP deployed at the bottom) in Bautzen Reservoir. The vertical axis is split into two subaxes with identical scaling: The lower axis corresponds to the distance from the bottom, the upper—to the distance from the surface. Thus, we avoid averaging over the entire water column because the water level change was significant throughout the measurements. Data are based on measurements in Bautzen Reservoir. (

**b**) Dissipation rate calculated using the structure function method and the BBL approach based on measurements in Lake Dagow (separately for the first and for the following two ADCP deployment periods).

**Figure 8.**Scheme showing mean energy fluxes in (W m

^{−2}) and mean energy content (J m

^{−2}) for the two studied water bodies. Numbers written before/after a slash correspond to the values obtained from Bautzen Reservoir and Lake Dagow measurements, respectively. Solid and dashed lines illustrate the motion and equilibrium positions of vertical layers of water with the same density.

**Table 1.**Instrumentation and resolution for the water-side and atmospheric measurements conducted in Bautzen Reservoir and Lake Dagow.

Type of Measurements | Water Body | Instrument | Resolution | Location on the Map (Figure 1) |
---|---|---|---|---|

Flow velocity | Bautzen Reservoir | (a) ADCP RDI Workhorse 600 kHz (range: 1.4–10 m); (b) Workhorse 1200 kHz (range: 0.8–4.7 m) | (a) 10 min with 200 pings with 0.25 m bin size; (b) 1 s with 0.1 m bin size | (a) Bottom deployment (facing upward) ~10 m from southern corner of the platform; (b) platform deployment (facing down, southwest corner) |

Lake Dagow | ADCP RDI Workhorse 600 kHz (3 deployments, range: (1) 0.5–6.8 m; (2–3) 0.8–7.1 m) | 5 s with 12 pings with 0.1 m bin size | 3 deployments: (1) platform (facing down, west corner), point E; (2–3) ~6–7 m from northern corner of the platform (facing upward) | |

Water temperature | Bautzen Reservoir | (a) Thermocouples (type T, Copper/Constantan) | 10 min averages from measurements in 30 s intervals | Platform, point A |

Lake Dagow | RBR solo | 10 s | Point F | |

Wave measurements | Bautzen Reservoir | RBR duet | 10 min with 512 measurements of 16 Hz | Platform, point A; shore, point B |

Lake Dagow | Shore, point G | |||

Wind speed | Bautzen Reservoir | Campbell Scientific, CSAT3 (1.8 m) | 20 Hz, as well as 10 min and 30 min averages | Platform, point A |

Lake Dagow | Gill Instruments HS-50 (1.97 m) | 20 Hz | Platform, point E | |

Radiation | Bautzen Reservoir | Kipp and Zonen, CNR1 | 10 and 30 min averages from measurements in 30 s intervals | Platform, point A |

Lake Dagow | Kipp and Zonen, CNR4 | measured at 1 Hz, logged at 1 min averages | Platform, point E |

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**MDPI and ACS Style**

Guseva, S.; Casper, P.; Sachs, T.; Spank, U.; Lorke, A.
Energy Flux Paths in Lakes and Reservoirs. *Water* **2021**, *13*, 3270.
https://doi.org/10.3390/w13223270

**AMA Style**

Guseva S, Casper P, Sachs T, Spank U, Lorke A.
Energy Flux Paths in Lakes and Reservoirs. *Water*. 2021; 13(22):3270.
https://doi.org/10.3390/w13223270

**Chicago/Turabian Style**

Guseva, Sofya, Peter Casper, Torsten Sachs, Uwe Spank, and Andreas Lorke.
2021. "Energy Flux Paths in Lakes and Reservoirs" *Water* 13, no. 22: 3270.
https://doi.org/10.3390/w13223270