Seasonality of density currents induced by differential cooling

When lakes experience surface cooling, the shallow littoral region cools faster than the deep pelagic waters. The 10 lateral density gradient resulting from this differential cooling can trigger a cold downslope density current that intrudes at the base of the mixed layer during stratified conditions. This process is known as a thermal siphon (TS). A TS flushes the littoral region and increases water exchange between nearshore and pelagic zones and may thus potentially impact the lake ecosystem. Past observations of TS in lakes are limited to specific cooling events. Here, we focus on the seasonality of TS-induced lateral transport and investigate how seasonally varying forcing conditions control the occurrence and intensity of TS. This research 15 interprets one-year-long TS observations from Rotsee (Switzerland), a small wind-sheltered temperate lake with an elongated shallow region. We demonstrate that a TS occurs for more than 50 % of the days from late summer to winter and efficiently flushes the littoral region within ~10 hours. We further quantify the occurrence, intensity and timing of the TS over seasonal time scales. The conditions for TS formation become optimal in autumn when the duration of the cooling phase is longer than the time necessary to initiate a TS. The decrease in surface cooling by one order of magnitude from summer to winter reduces 20 the lateral transport by a factor of two. We interpret this transport seasonality with scaling relationships relating the daily averaged cross-shore velocity, unit-width discharge and flushing timescale to the surface buoyancy flux, mixed layer depth and lake bathymetry. The timing and duration of diurnal flushing by TS relate to daily heating and cooling phases. The longer cooling phase in autumn increases the flushing duration and delays the time of maximal flushing relative to the summer diurnal cycle. Given their scalability, the results reported here can be used to assess the relevance of TS in other lakes and reservoirs. 25

shallow region. We demonstrate that a TS occurs for more than 50 % of the days from late summer to winter and efficiently flushes the littoral region within ~10 hours. We further quantify the occurrence, intensity and timing of the TS over seasonal time scales. The conditions for TS formation become optimal in autumn when the duration of the cooling phase is longer than the time necessary to initiate a TS. The decrease in surface cooling by one order of magnitude from summer to winter reduces 20 the lateral transport by a factor of two. We interpret this transport seasonality with scaling relationships relating the daily averaged cross-shore velocity, unit-width discharge and flushing timescale to the surface buoyancy flux, mixed layer depth and lake bathymetry. The timing and duration of diurnal flushing by TS relate to daily heating and cooling phases. The longer cooling phase in autumn increases the flushing duration and delays the time of maximal flushing relative to the summer diurnal cycle. Given their scalability, the results reported here can be used to assess the relevance of TS in other lakes and reservoirs. 25

Introduction
As lateral boundaries in lakes, littoral regions exchange matter and energy with the surrounding watershed. The littoral region receives both sediment and chemicals from the terrestrial environment, including anthropogenic contaminants (Rao and Schwab, 2007). Particulate matter accumulates in the littoral zone (Cyr, 2017) and this biologically active region can host unique and intensified biogeochemical reactions (Wetzel, 2001). Cross-shore flows, which transport water laterally, connect a 30 lake's littoral region to its pelagic region and thereby facilitate critical ecosystem processes (MacIntyre and Melack, 1995).
They control the residence time of nearshore waters and redistribute heat, dissolved and particulate compounds throughout the lake. Representing lakes as isolated vertical water columns fails to account for the effects of cross-shore flows on lake biogeochemistry (Effler et al., 2010;Hofmann, 2013).
Various mechanisms drive horizontal exchange flows. The pioneer limnologist already elucidated the effect of large-35 scale surface and internal gravity waves on horizontal transport (Forel, 1895;Wedderburn, 1907;Mortimer, 1952). In addition to serving as the primary energy source for gravity waves, wind is also responsible for fuelling lateral flows via wind-driven cross-shore circulation, cross-shore Ekman transport in rotating environments and differential deepening (Imberger and Patterson, 1989;Brink, 2016). Buoyancy-driven lateral flows can be induced by horizontal temperature gradients due to differential heating or cooling. Such gradients could form when heat fluxes are spatially heterogeneous, due to large-scale 40 differences of meteorological forcing (Verburg et al., 2011), variable geothermal heating (Roget et al., 1993), shading from vegetation (Lövstedt and Bengtsson, 2008), wind sheltering (Schlatter et al., 1997) or variation in turbidity (MacIntyre et al., 2002). Horizontal temperature gradients also occur when waters at different depths experience a spatially uniform heat flux (Horsch and Stefan, 1988;Monismith et al., 1990). For a given surface area, the volume of water in the shallow littoral region is less than its offshore counterpart. Littoral waters will thereby cool or warm faster than pelagic waters. Monismith et al. 45 (1990) designated cross-shore circulation resulting from a bathymetrically induced temperature gradient as a thermal siphon.
The fact that all lakes have sloping boundaries makes thermal siphons potentially ubiquitous.
This study focusses on thermal siphons driven by differential cooling, wherein lake temperature exceeds the temperature of maximum density (Fig. 1). The study does not consider heating-driven thermal siphons because induced lateral exchange remains weaker and more sensitive to wind disturbance than cooling-driven thermal siphons (Monismith et al., 1990;James 50 et al., 1994). Atmospheric forcing affects lakes at diurnal, seasonal or synoptic scales. This includes periods of net cooling when the net surface heat flux is directed towards the atmosphere (Bouffard and Wüest, 2019). This diel or seasonal surface cooling mixes and deepens the surface layer by penetrative convection. Once the shallow littoral region becomes fully mixed, it cools down faster than the deep pelagic region. Littoral waters become denser and plunge as a downslope density current, which intrudes at the base of the mixed layer during stratified conditions. At the surface, a reverse flow brings water from the 55 pelagic to the littoral region (Horsch and Stefan, 1988;Monismith et al., 1990). We will hereafter refer to the density current induced by differential cooling simply as the thermal siphon (TS).
Thermal siphons have often been evoked to explain cold anomalies observed in lakes. These include tilted isotherms above the sloping sides (Talling, 1963;Eccles, 1974;MacIntyre and Melack, 1995;Woodward et al., 2017) or cold water intrusions (Wells and Sherman, 2001;Peeters et al., 2003;Rueda et al., 2007;Forrest et al., 2008;Ambrosetti et al., 2010). 60 However, only a few studies have provided detailed in situ descriptions of the process. The first extensive studies of TS assessed power plant heat disposal within cooling lakes (Adams and Wells, 1984). Warm water discharged into these reservoirs flows onshore at the surface, cools and induces a TS in sidearms. These observations are however not representative of naturally formed TS. Cooling lakes constantly receive heat from power plants, which increases the intensity and duration of surface cooling compared to that experienced in other lakes and reservoirs (Horsch and Stefan, 1988). An extensive review of 65 literature suggests that Monismith et al. (1990) provided the first in situ observations of a diurnal cycle associated with a naturally formed TS. The authors studied a sidearm of Wellington Reservoir (Australia) in summer and described the complete diurnal cycle of daytime differential heating and nighttime differential cooling. In both cases they observed cross-shore flows at velocities of ~2 cm s -1 . Cross-shore circulation was characterized by inertia with a delay of several hours between the shift in forcing conditions and the flow reversal. Other studies have reported the presence of TS along the thalweg of narrow 70 reservoir embayments (James and Barko, 1991a, b;James et al., 1994;Rogowski et al., 2019) but also along the sloping sides of lakes (Sturman et al., 1999;Thorpe et al., 1999;Fer et al., 2002a, b;Pálmarsson and Schladow, 2008) and between different basins (Roget et al., 1993). In most of these examples, TS characteristics comported with the description given in Monismith et al. (1990). These included thermal stratification above the sloping bottom, inertial effects and a cross-shore velocity on the order of 1 cm s -1 . The work by Fer et al. (2002b) on the sloping sides of Lake Geneva (Switzerland) represents one of the most 75 extensive TS studies. The authors collected continuous measurements over two winters and identified density currents initiated in the middle of the night and lasting until the late morning. The research detected increasing discharge with distance downslope and found that one TS event could flush the littoral region almost twice.
The lake stratification and the intensity and duration of the daily cooling period vary seasonally (Bouffard and Wüest, 2019). These changes in forcing conditions may affect the occurrence and magnitude of TS over time. Previous studies have 80 focused on specific TS events and did not monitor the process over different seasons. A three-year-long dataset on Lake Anna (VA, USA; Adams and Wells, 1984) addressed a cooling lake and thus did not reflect the seasonality of naturally formed TS.
Although the time series collected by Roget et al. (1993) between two lobes of Lake Banyoles (Spain) spans eight months (October-May), these authors do not address seasonal variability in the observed TS. Geothermal fluxes between the lobes enhance differential cooling in Lake Banyoles, which may modify the seasonality of TS. James and Barko (1991b) performed 85 six tracer experiments between June and September. These authors obtained estimates of TS velocity that varied between the experiments and were higher for days with stronger lateral temperature gradients. While the experiments did not identify a clear seasonal trend, thermistor arrays detected an increase in the occurrence of differential cooling from May to August suggesting a higher frequency of TS events in late summer. This hypothesis was not verified due to the lack of long-term velocity measurements. 90 Thus, the seasonality of TS occurrence and intensity has not been systematically investigated in lakes. Dimensional analysis and laboratory experiments have provided relationships linking forcing and background conditions (surface cooling, stratification) to cross-shore transport by TS have been derived from dimensional analysis and laboratory experiments (Harashima and Watanabe, 1986), but these lack validation in the field. The comprehension of the TS seasonality would improve the prediction of TS events in lakes and their contribution to exchanges between littoral and pelagic regions. The 95 present research monitored TS in a small temperate lake over a one-year period. The study establishes three characteristics of TS seasonality including its occurrence (Sect. 3.3), intensity (Sect 3.4) and diurnal dynamics (Sect 3.5). The observed TS seasonality clearly relates to forcing parameters and thus pinpoints key parameters for predicting TS events in lakes (Sect. 4).
The observation and interpretation of TS events across seasonal timescales provides a comprehensive understanding of the conditions required to form TS in lakes. 100 Figure 1: Data-based schematic of the cooling-driven thermal siphon. The schematic shows the plateau, littoral and mixed regions, the seasonality of the forcing and the variables used for the transport scaling. Here, the littoral zone is the region upslope of MT, where the current velocity is measured and transport variables are calculated. The cross-shore temperature field is linearly interpolated from a transect of CTD (Conductivity-Temperature-Depth) profiles collected in the morning on 22 August 2019 (08:20-105 08:50 UTC), from x = 225 m (white dashed line) to x = 714 m. Black dots on the x-axis show the location of the profiles. The green dashed line in the seasonality diagram corresponds to the transition period between the mixing period (winter) and the stratified period (summer), when the mixed layer is not well defined.

Study site and field measurements 110
Rotsee is a small peri-alpine monomictic lake near Lucerne, Switzerland (Fig. 2a). It is located at 419 m above sea level and has a surface area of 0.5 km 2 with a mean and maximum depth of 9 m and 16 m, respectively. The main in-and outflows are located at the south-western and north-eastern ends of the lake, respectively, and have a low discharge of ∼ 0.1 m 3 s -1 . Rotsee is an elongated lake (2.5 km long, 0.2 km wide), with steep sides (slope of 15°) across its longitudinal axis and more gradual slopes (1.5°) at its two ends. The north-eastern end of the lake finishes with a 200 m long plateau region of approximately 1 m 115 depth that can trigger TS. Moreover, Rotsee is located in a depression and sheltered from wind. It is thereby famous for international rowing championships and an ideal field-scale laboratory where convective processes are distinctively observable.

125
This study focuses on quantifying TS originating from the north-eastern plateau region (Fig. 2a). Because of the elongated shape of Rotsee, we use the 2D (x, z) framework shown in Fig. 1 by orienting the x-axis along the thalweg. We assume that TS flows preferentially along the x-axis, and we neglect the influence of perpendicular flows. We will now refer to the north-eastern end of the lake as the "shore" and call the direction of the x-axis the "offshore direction".
We monitored the background stratification at the deepest location ("background mooring" MB, approx. 16 m deep) as 130 well as the dynamics of TS off the plateau region ("TS mooring" MT, approx. 4 m deep), from March 2019 to March 2020 ( Fig. 2a). Mooring MT was located along the thalweg, at the beginning of the sloping region. This shallow water column is already vertically mixed in summer by the action of surface cooling. We briefly describe the moorings below, and provide more detailed information on the specifications and setup of the instruments in Appendix A. Each mooring included an array of thermistors (Fig. 2b). The evolution of the background temperature was monitored at MB with Vemco Minilog II-T loggers 135 (accuracy: 0.1 °C). High-resolution thermistors RBR TR-1050 and RBR Duet TD (accuracy: 0.002 °C) were used at MT to track finer-scale temperature fluctuations near the sloping boundaries. The thermistors were installed from 1 to 13 m above the bottom with 1 m vertical resolution at MB and from 0.25 to 2 m above the bottom with 0.25 m resolution at MT. The evolution of the near-surface temperature (0.2 to 0.3 m below the surface) was monitored with Vemco loggers attached to surface buoys at both locations. An upward-looking Acoustic Doppler Current Profiler (ADCP, Nortek Aquadopp Profiler 1 140 MHz) was deployed at MT for fine-scale measurements of the bottom currents (Fig. 2b). It collected velocity profiles with a vertical bin resolution of 0.05 m from 0.25 to 3 m above the bottom. Velocity was obtained by averaging 512 burst measurements every 15 min. A week-long maintenance of the moorings was conducted every 40 days on average. The measurement periods by the ADCP were shorter in December and January due to low battery (Appendix A). The monitoring was stopped in May 2019 to avoid any hazard during the European Rowing Championships. 145 To capture the spatial variability of TS, cross-shore transects of Conductivity-Temperature-Depth (CTD) profiles (Sea&Sun CTD 60M, sampling interval of 0.4 s) were performed along the x-axis. The chosen CTD lowering speed of 15 cm s -1 offered an optimal 6 cm vertical resolution while completing an entire transect of 11 profiles in ~45 min (i.e. short time compared to the duration of flushing by TS, Sect. 3.1). The 11 profiles were distributed along the sloping region of the lake, between 2 and 14 m depth (Fig. 2a). Figure 1 shows a CTD transect collected in the morning on 22 August 2019, as an example. 150 A WxPRO Campbell weather station was installed at the lake shore, near the plateau region, in September 2019 ( Fig.   2a). It measured atmospheric pressure, air temperature, wind speed, incoming shortwave radiation and relative humidity every 10 min. In addition, incoming longwave radiation was obtained from the Lucerne weather station (47°02'12'' N, 8°18'04'' E; source MeteoSwiss), ~4 km from Rotsee. The Lucerne station also provided meteorological forcing before September 2019 or to fill data gaps from the reference Rotsee weather station. In these cases, the Lucerne dataset was adjusted based on the Rotsee 155 observations from September 2019 to June 2020. Pressure, air temperature and shortwave radiation were linearly corrected ( 2 > 0.9). This simple method did not allow reliable correction to the relative humidity and wind speed. Instead, we applied a Neural Network Fitting approach to correct these two variables. The shallow neural network consisted of one hidden layer with 50 neurons and was trained by a Levenberg-Marquardt algorithm. 70 % of the reference period September 2019-June 2020 was used to train the neural network, and the remaining percentage was equally distributed for validation and testing. All 160 the meteorological variables measured in Lucerne were used as inputs to the network and the procedure was repeated for each of the two target variables. The network performance was satisfactory, with a coefficient of determination 2 and a root mean square error RMS between observations and estimates of 2 ≈ 0.72 and RMS ≈ 0.67 m s -1 for wind speed, and 2 ≈ 0.94 and RMS ≈ 4.7% for relative humidity. The uncertainty in wind speed and relative humidity estimates leads to an average uncertainty of 5 % and 3 % of the surface heat fluxes (Sect. 2.2), respectively. 165 We captured the spatial variability of TS in the morning on 22 August 2019 ( Fig. 1) by collecting 11 Conductivity-Temperature-Depth (CTD) profiles (Sea&Sun CTD 60M) along the x-axis, between 2 and 14 m depth (Fig. 2a).

Heat and buoyancy fluxes
Heat fluxes were estimated from meteorological data and lake surface temperature at MB. We assumed that the meteorological conditions and the heat fluxes were spatially uniform over the lake surface (0.5 km 2 ). Heat fluxes were defined positive in the 170 upward direction (lake cooling) and negative in the downward direction (lake heating). The non-penetrative surface heat flux Q 0 is defined as where C and E are the sensible and latent heat fluxes, and LW,in and LW,out are the incoming and outgoing longwave radiation, respectively. The net surface heat flux 0,net includes the shortwave radiative flux at the surface SW,0 : 175 (2) The different terms of the heat budget were calculated similarly to Fink et al. (2014). Cloudiness modulates the proportion of direct and diffuse shortwave radiation and was derived from daily clear-sky irradiance as in Meyers and Dale (1983). We modified the empirical wind and air temperature-based calibration function, , in the sensible and latent heat fluxes estimates of Fink et al. (2014) to take the lake fetch into account, as proposed in McJannet et al. (2012). 180 The non-penetrative surface buoyancy flux 0 was inferred from Q 0 under the assumption that only heat fluxes modify the potential energy near the surface (Bouffard and Wüest, 2019): In Eq.
(3), = −(1/ ) / is the thermal expansivity of water [°C -1 ], is the gravitational acceleration [m s -2 ], is the water density [kg m -3 ] and p,w is the specific heat of water [J °C -1 kg -1 ]. The thermal expansivity was estimated from the 185 surface temperature using the relationship reported in Bouffard and Wüest (2019). To calculate water density, we used the equation of state from Chen and Millero (1986), with a measured average salinity of = 0.2 g kg -1 for Rotsee. Our analysis assumed a constant salinity over the year since the seasonal changes in surface density are controlled by temperature fluctuations rather than variations in salinity.
Similarly, a radiative surface buoyancy flux ,0 was obtained as 190 The net buoyancy flux at the surface is 0,net = 0 + SW,0 . 0,net > 0 indicates a destabilizing buoyancy flux (net cooling) whereas 0,net < 0 indicates a stabilizing buoyancy flux (net heating). We used 0,net to identify the daily cooling and heating phases (Sect. 3.1) and we averaged 0 over the daily cooling phase to quantify the intensity of convection (Sect. 2.6).

Mixed layer depth 195
We used hourly averaged temperature data from MB to calculate the thermocline depth (ℎ t ) and the mixed layer depth (ℎ ML ).
Based on the review of Gray et al. (2020), we estimated ℎ ML from the temperature-gradient method with a threshold of / = 0.05 °C m -1 . The water column was defined as fully mixed when / < 0.05°C m -1 at all depths. In this case, ℎ ML was set to the depth of the bottom temperature sensor. For stratified conditions, the thermocline depth was defined as the depth of maximum gradient in / . Starting from ℎ t and moving upward, the lower end of the mixed layer was reached 200 when the local temperature gradient dropped to / < 0.05 °C m -1 . When / > 0.05°C m -1 at all depths, the entire column was stratified and ℎ ML was zero.

Occurrence of thermal siphons
We used the velocity data from the Aquadopp profiler at MT to estimate the cross-shore transport over the bottom 3 m of the sloping region. Quality control of the velocity was performed by discarding values with beam correlation below 50 % or signal 205 amplitude weaker than 6 dB above noise floor. The horizontal velocity was projected onto the x-axis (angle of 56° from north), which crosses the isobath at MT perpendicularly (Fig. 2a). Following the 2D framework of Fig. 1, we will now call x the "cross-shore velocity".
The time series of x was then divided into 24 h subsets, starting and ending at 17:00 UTC. Each 24 h subset was analysed separately to identify TS events. We decided to focus on periods when MT was located in the sloping mixed region, 210 with TS flowing at the bottom of the water column ( Fig. 1). This condition allowed us to relate our current measurements to scaling formulae of downslope density currents (Sect. 3.4). A downslope TS event was detected on a specific day if the three following conditions were met: 1. significant cross-shore flow: depth-averaged velocity of the current x ��� > 0.5 cm s -1 for at least two hours; 2. weak wind during the selected event: | MO,avg |/ℎ ML,avg < 0.5, with MO,avg and ℎ ML,avg the averaged Monin-215 Obukhov length scale (Wüest and Lorke, 2003) and mixed layer depth during the cross-shore flow event, respectively; 3. mixed water column at MT before the onset of the flow: / < 0.05°C m -1 at all depths.
Cross-shore flows that did not meet conditions #2 and #3 were noted as "wind-driven circulation" and "stratified flows", respectively. The term "stratified flows" refers here to any cross-shore flow occurring when the mixed layer depth was 220 shallower than the lake depth at MT. This includes basin-scale internal waves and intruding TS. Further justifications of the above criteria are provided in Appendix D.

Cross-shore transport by thermal siphons
The cross-shore transport was calculated for each identified TS event (Sect. 2.4). We defined for each event sub-periods with continuous positive x (further details in Appendix D). We calculated the unit-width volume of water transported over each 225 of these sub-periods: where 0,i and f,i are the initial and final times of the sub-period , respectively, and x is the unit-width discharge defined as In Eq. (6), bot and top are the bottom and top interfaces of the region over which x > 0 at time . For each day, we then 230 defined the flushing period of TS as the sub-period with the largest volume transported x,i .
Four daily quantities characterizing the cross-shore transport were calculated over the flushing period of each day ( 0 ≤ ≤ f ): the average cross-shore velocity avg and unit-width discharge avg , the maximum cross-shore velocity max (both in time and depth) and the flushing timescale F = lit / avg with lit ≈ 500 m 2 the unit-width volume of the littoral region upslope of MT (Fig. 1). avg and avg were obtained as follows: 235

Scaling formulae
We relate the transport quantities introduced in Sect. 2.5 to the forcing parameters by using scaling formulae from theoretical and laboratory studies. 240 The horizontal velocity scale of TS is ∼ ( 0 ) 1/3 where 0 is the destabilizing surface buoyancy flux and is a horizontal length scale (Phillips, 1966). Following Wells and Sherman (2001), we used the horizontal length scale ML based on the mixed layer depth: with U a proportionality coefficient. For each value of ℎ ML , we calculated ML as the distance between the north-eastern edge 245 of the lake and the location where the mixed layer intersects the lake sloping bottom (Fig. 1).
The unit-width discharge is scaled by ∼ ( 0 ) 1/3 , where is a vertical length scale. We used the average depth of the littoral region upslope of MT (ℎ lit = lit / lit , Fig. 1) as : with q a proportionality coefficient. We define the flushing timescale from Eq. (10) as 250 with F = 1/ q . The flushing timescale represents the time that TS needs to flush the littoral region of volume lit .
The daily cooling phase of duration c is defined by 0,net > 0. After the onset of surface cooling, TS does not form immediately. Vertical convective mixing takes place before the horizontal density gradient starts to drive TS (Wells and Sherman, 2001;Ulloa et al., 2022). The period between the start of the cooling phase and the formation of TS is quantified by 255 an initiation timescale ( ini ). If the littoral region is initially mixed vertically, ini equals the transition time scale t derived by Ulloa et al. (2022) under constant surface cooling: with p ≈ 170 m and ℎ p ≈ 1 m the length and depth of the plateau region of Rotsee, respectively. This timescale is based on a three-way momentum balance between the lateral pressure gradient due to differential cooling and the inertial terms (Finnigan 260 and Ivey, 1999).
In Eqs. (9)-(12), 0 and ML are averaged over c every day. We discuss the choice of the length scales ML and ℎ lit in more detail in Sect. 4.3. The key parameters used for the scaling formulae and TS dynamics are listed in Table 1.

Diurnal cycle
The magnitude of the net surface buoyancy flux 0,net oscillates daily and the intensity of the diel fluctuation is modulated at 270 a seasonal timescale (Fig. 3a). We illustrate the diurnal cycle with an example on 9-10 September 2019 . This summertime period is characterized by a strong day-night variability, which is ideal to elucidate how changes in forcing conditions affect the formation and destruction of TS. The net surface buoyancy flux 0,net varies from −2 × 10 −7 W kg -1 during daytime to 1 × 10 −7 W kg -1 during nighttime (Fig. 3b). In contrast to 0,net , 0 remains positive over 24 hours, indicating a continuous cooling at the air-water interface. In late summer, 0 is ~10 −7 W kg -1 with limited variability of ± 5 × 275 10 −8 W kg -1 between day and night. The radiative forcing SW,0 is the main driver of the diurnal variability, with diel variations that are one order of magnitude larger than 0 during cloud-free summer days like 9-10 September 2019. We observe differential cooling at night, with lateral temperature gradients of ~5 × 10 −4 °C m -1 in late summer (Fig. 1).
We divide the diurnal cycle into a net cooling phase during the night ( 0 > | SW,0 |) and a net heating phase during the day ( 0 < | SW,0 |). Starting the period at 17:00 (UTC), the water column is initially thermally stratified by the radiative 280 heating of the daytime period (Fig. 3d). During the first part of the night, convective cooling mixes and deepens the surface layer (period #1 in Fig. 3d). The water column becomes entirely mixed at MT around 00:00 (see the vertical isotherms starting around 00:00 in Fig. 3d). The convective plumes intensify in the second part of the night to reach vertical velocities of ~5 mm s -1 (Fig. 3c). Three hours after complete vertical mixing at MT, a cross-shore circulation is initiated (period #3 in Fig. 3d), with a downslope flow near the bottom (positive x ) and an opposite flow above (negative x ). The velocity magnitude of ~1 cm 285 s -1 is typical for TS. We will hereafter refer to the period during which TS flows as the flushing period. The interface between the two flows oscillates vertically by 1-2 m (~100 % of the TS thickness), suggesting that the transport is limited by the vertical mixing of convective plumes (Fig. 3c). At the end of the cooling phase, the water column is distinctly stratified near the bottom, indicating the presence of persistent colder water flowing downslope. A striking observation is the intensification of TS in phase with the weakening of vertical convection at the onset of radiative heating. The cross-shore velocity increases above 2 290 cm s -1 , and the induced bottom stratification reaches ~0.1 °C m -1 at the beginning of the heating phase. This strong flushing lasts for several hours, while radiative heating increases and re-stratifies the water above the density current. The cross-shore flow weakens in the late morning, yet continues until noon.

Seasonal variability of the forcing parameters
Two key parameters for the formation of TS are 0 and ℎ ML (Sect. 2.6), indirectly related to 0,net . Hereafter, we refer to these parameters as "forcing parameters" of TS. The daily intensity of convective cooling is estimated based on the convective velocity scale * = ( 0 ℎ ML ) 1/3 (Deardorff, 1970), where 0 and ℎ ML are daily averages during the cooling phase at MB.
These two parameters are both seasonally dependent and their relationship is characterised by an annual hysteresis cycle (Fig.  315   4a). Although the surface heat flux Q 0 increases from winter to summer, the strong seasonal variability of 0 (one order of magnitude larger in summer than in winter) is mostly due to seasonal changes in the thermal expansivity .
From spring to summer, ℎ ML remains shallow (< 2 m) and 0 increases from ~10 −9 W kg -1 in March to ~10 −7 W kg -1 in August. The latter results in an increase of the convective velocity scale from * ≈ 3.8 ± 0.7 mm s -1 to * ≈ 6.7 ± 0.9 mm s -1 over the same period (Figs. 4a, b). The lake undergoes net daily heating over this period ( 0,net < 0), with a cooling 320 phase that remains shorter than 15 h (Fig. 4b). In August, 0,net ≈ 0 and daily heating is balanced by daily cooling. At this period, the intensity of convective mixing reaches its maximum and the diurnal heating-cooling cycle is pronounced (Fig. 3a).
From September to January, the lake undergoes every day a net cooling ( 0,net > 0) and the mixed layer deepens by convection with an average rate of ~0.1 m d -1 , leading to a complete mixing in mid-December. Over the same period, 0 continuously decreases due to the drop in with colder temperatures. This decrease in 0 balances the deepening of ℎ ML and 325 * remains constant around * ≈ 7 ± 1 mm s -1 from August to November. The duration of the cooling phase increases in autumn to reach its maximum in November ( c ≈ 21 ± 2 h). Convective cooling is occurring almost continuously at that time. In winter, the convective velocity * drops to * ≈ 3.5 ± 1 mm s -1 and the effect of the cooling phase is again balanced by the heating phase ( 0,net ≈ 0, c ≈ 15 ± 3 h in February). The differences between the cooling and heating phases of the diurnal cycle are reduced in winter, with 0,net ≈ 0 over the entire day (Fig. 3a). Note that winter 2019-2020 was warm and 330 Rotsee remained mixed from December to February (no inverse stratification, surface temperature > 4°C).

Seasonal occurrence of thermal siphons
We apply our detection algorithm for cross-shore flows (Sect. 2.4) on 227 days with continuous measurements at MT from 340 March 2019 to February 2020. We identify 156 days with a significant cross-shore flow (69 % of the days with observations), over which 85 days (37 %) were reported as TS events. The remaining days with identified cross-shore flow were either associated with wind-driven circulation for 13 days (6 %) or stratified flows for 58 days (26 %). We estimate the percentage of occurrence of TS relative to the number of days with measurements ( TS ) for each month (Fig. 5). Before late July, the mixed layer is usually shallower than 4 m (Fig. 4a) and stratified flows prevail at MT (Fig. 5). The first events identified as TS 345 occur on 28 July and 31 July ( TS = 12.5 % in July). Yet, it is rather in August when TS reaches a significant percentage of occurrence of ~50 %. The occurrence increases in autumn to reach 87 % in November. In winter, TS is less common: it occurs for only 38 % of the days in December, two days in early January (not shown in Fig. 5 due to the few days measured during this month) and is not observed later in winter.
The 15-day moving average in Fig. 5 reveals the short-term variability due to synoptic changes in the meteorological 350 conditions that naturally modulate the seasonal pattern. Although the monthly averaged occurrence increases from August to November, the bi-weekly percentage of occurrence drops over shorter periods. These periods are associated with strong heating that re-stratifies the water column at MT, reduces surface cooling at night and prevents the formation of TS (Appendix B, Fig.   B1). The 15-day averaged occurrence reaches 100 % in November, when TS is present over 25 consecutive days.
Although TS becomes more frequent in autumn, the percentage of occurrence of cross-shore flows ( CS ) remains 355 constant around 80-90 % from July to November. Lateral flows observed at MT in spring and summer primarily result from direct wind forcing or subsequent stratified mixing. These flows may also include TS intruding before reaching MT and thereby not counted as downslope TS (further details in Appendix D). These different processes become less frequent in autumn due to the deeper mixed layer and are replaced by downslope TS. Overall, there seems to be an effect of TS on the number of cross-shore flows if we compare the period from July to December ( CS ≈ 80 ± 14 %) with the rest of the year ( CS ≈ 47 ± 360 12 %), suggesting that TS is the main mechanism connecting the littoral and pelagic regions in Rotsee.

Scaling the cross-shore transport
We compare our field-based transport estimates for the TS events identified in Fig. 5 with the laboratory-based scaling 370 formulae introduced in Sect. 2.6 (Fig. 6). The daily average and maximum cross-shore velocities are linearly correlated with the horizontal velocity scale from Eq. (9) ( 2 > 0.5, val,F < 0.01; Figs. 6a, b). The daily average velocity avg ≈ 0.9 ± 0.2 cm s -1 is three times lower than the daily maximum velocity max ≈ 2.6 ± 0.7 cm s -1 that occurs in the morning (Fig. 3).
The best linear fits (with zero intercept) are given by avg = 0.33 ⋅ ( 0 ML ) 1/3 and max = 0.99 ⋅ ( 0 ML ) 1/3 . The 95 % confidence interval of the proportionality coefficient is 0.32 ≤ U ≤ 0.34 for avg and 0.95 ≤ U ≤ 1.02 for max . Despite 375 the natural variability, a seasonal trend is distinguishable, with a decrease of avg and max by a factor of two from July to December. This is consistent with the weakening of convective cooling observed in Fig. 4.
As observed for the cross-shore velocity, the cross-shore transport weakens from summer to winter: avg decreases and F 385 increases by a factor of two. The flushing timescale is F ≈ 7 h in summer but reaches F ≈ 20 h in winter. Yet, due to longer TS events in winter than in summer, the volume flushed by a single TS event is independent of the season and remains larger than the volume of the littoral region (Appendix C).

6). The equation of the linear regressions with zero intercept, the coefficient of determination (R 2 ) and the p-value of an F-test (
, ) are indicated.

Flushing period
The diurnal cycle described in Sect. 3.1 varies at seasonal scale, as the forcing parameters change. To assess the effects on the flushing period (period #3 in Fig. 3d), we average the diurnal cycle monthly between August and December (Fig. 7). The 395 cooling and heating phases are illustrated with the diurnal cycle of 0,net . The duration of the cooling phase (blue shading in Fig. 7) markedly increases from August to December, as already observed in Figs. 3a and 4b. The histograms of | 0,net max | show that the magnitude and variability of 0,net decrease over the same period (Fig. 7b, d, f, h, j).
These changes of the forcing parameters have a direct effect on the cross-shore transport. The unit-width discharge in Eq. (6) during the flushing period 0 ≤ ≤ f and x ( ) = 0 for < 0 or > f . In August and September (Fig. 7a, b), x increases at night to reach its maximum at the beginning of the heating phase. The flow generally stops in the late morning, depicted by a drop of x around 10:00. This cycle corresponds to the example of Fig. 3. The daily peak of flushing is also observed in October and November but it is more spread with a continuous increase of cross-shore transport at night, starting in the evening already (Figs. 7e, g). The average flushing duration of TS is thus longer in late autumn (see for example the 24 405 h long TS event in Appendix B, Fig. B2). In December, the peak of discharge is reduced and the cross-shore transport is nearly continuous over 24 h (Fig. 7i). The time of maximal x (vertical black dashed line in Fig. 7) is delayed from August to December. We associate this maximal x to the heating phase as it always occurs ~1-2 h after 0,net becomes negative (except in December where the peak is less pronounced).
The diurnal cycle of the cross-shore transport is also described by the flow geometry parameter introduced by Ulloa et 410 al. (2022) as the ratio between the root mean square (RMS) of the cross-shore velocity and the RMS of the vertical velocity: where <. . . > denotes a depth-average. G informs on the relative magnitude of TS with respect to the surface convection. We expect G to be close to unity when surface convection is the only process acting on the water column. A deviation from unity towards larger values of G (i.e., G ≫ 1) implies that the flow is anisotropic, with a stronger cross-shore component, as TS 415 develops. A sharp peak of the monthly averaged G is visible at the beginning of the heating phase, from August to November (Fig. 7). It matches the time of maximal x and results from the combined effect of TS-induced flushing (increase of x ) and convection weakening (decrease of z ).
To give a theoretical estimation of the starting time of the flushing, we estimate the initiation time scale of TS from the monthly averaged transition timescale ini ≈ t (Eq. (12)). The assumption of constant surface cooling is reasonable for 420 nighttime cooling in Rotsee (Fig. 3b). The starting time depicted by a vertical pink dashed line in Fig. 7 corresponds to ( sunset + t ), with sunset the time of sunset when SW,0 drops to zero. The increase of t from t ≈ 3 h in late summer to t ≈ 20 h in winter leads to a later theoretical onset of TS in winter, despite the earlier time of sunset. Based on this seasonal variability of t , TS is expected to start in the evening in summer but in the morning in winter. This theoretical starting time, however, can be improved by considering the initial stratification at the beginning of the cooling phase, as further discussed 425 in Sect. 4.2 and Sect. 4.4.

Seasonality of thermal siphons
Our one-year-long measurements show that both the occurrence and intensity of TS vary seasonally in lakes with shallow littoral zones comparable to Rotsee. TS occurs regularly from late summer to early winter, with a maximum frequency in November, and is absent the rest of the year (Sect. 3.3). While the frequency of TS increases in autumn, the intensity of the 440 net cross-shore transport decreases compared to the summer period (Sect. 3.4). Yet, the daily averaged volume flushed from the littoral region increases from summer to autumn (Appendix C). The diurnal cycle is well defined in summer and divided into a cooling phase at night and a heating phase during daytime (Sect. 3.1). TS forms during the second part of the night and lasts until late morning, reaching a maximal flushing at the beginning of the heating phase. In autumn, the flushing duration increases, and the time of maximal flushing is shifted later in the day (Sect 3.5). 445 To explain the seasonality of TS, we need to relate it to the seasonal variability of the forcing parameters. The duration of the cooling phase ( c ) is a key parameter to understand the occurrence of TS and the diurnal dynamics, while the intensity of surface cooling ( 0 ) and the stratification (ℎ ML ) parametrize the cross-shore transport. We discuss the effects of the forcing parameters on the seasonality of TS in the following sections.

Effects of the forcing parameters on the occurrence of thermal siphons 450
The occurrence of TS can be predicted by estimating the initiation timescale ini and comparing it with the duration of the cooling phase ( c ). TS will occur if the cooling phase is long enough, i.e. c > ini . In initially stratified surface waters, the initiation period is divided into (1) a stratified period during which the mixed layer deepens and (2) a transition period during which the littoral region is vertically mixed (Figs. 3d and 8a). We express the initiation time scale as ini = mix + t , with mix the time needed to vertically mix the littoral region and t the transition time scale given by Eq. (12). To estimate mix , 455 we use the deepening rate of the mixed layer expressed as (Zilitinkevǐc, 1991): where 2 = −( / ̅ )( / ) [s -2 ] is the squared buoyancy frequency below the mixed layer, ̅ is the depth-averaged water density and ≈ 0.2 is an empirical coefficient. The model of Eq. (14) assumes that the mixed layer deepens by convection only, without any wind contribution. This assumption is valid for calm conditions, which prevail in Rotsee due to wind 460 sheltering (Zimmermann et al., 2021). The average duration required for mixing the water column at MT, assuming a constant surface cooling 0 and an initial mixed layer depth ℎ ML,ini , can be derived from Eq. (14) as: In Eq. (15), the buoyancy frequency squared is approximated . The depth ℎ MT corresponds to the maximum depth of the littoral region upslope of MT (Fig. 1), which has to be mixed 465 to observe downslope TS at MT. From the seasonality of the forcing parameters (Fig. 4), we predict the optimal period for the occurrence of TS based 475 on ini (Fig. 8b). In summer, the shallow mixed layer (ℎ ML ≈ 2 m) and strong nighttime surface cooling ( 0 ≈ 10 −7 W kg -1 ) lead to a short transition timescale of t ≈ 2 ± 0.5 h. However, the strong stratification near the surface limits the deepening of the mixed layer and causes mix to be too large for the water column to mix at MT (orange period in Fig. 8b). The littoral region remains stratified for most of the nights (ℎ ML < ℎ MT ) and the occurrence of downslope TS at MT is low ( ini > c on average). Downslope TS starts to be observed in late summer when the mixed layer reaches ℎ MT before the end of the night. 480 Starting in October, the monthly averaged mixed layer is deeper than ℎ MT and mix ≈ 0. At the same time, t increases because of the deepening of the mixed layer and the decrease of 0 . The transition timescale becomes longer than c in winter (purple period in Fig. 8b), reaching its maximum of t ≈ 23 ± 7 h in January. This prevents TS to form in winter, except for days with continuous surface cooling (i.e., c > 24 h). The transition timescale decreases in spring but mix increases simultaneously due to the re-stratification of the littoral region, which prevents TS to occur. The conditions to observe 485 downslope TS at MT are thus optimal between September and December, when ini ≪ c (grey shaded period in Fig. 8b). This period coincides well with our frequent observations of TS in autumn. The limiting factor for the formation of TS in spring and summer is mix and it is t in winter.
During the optimal period of occurrence, the variability of forcing conditions ( 0,net ) between days can modify the occurrence condition ini < c at a shorter time scale than the seasonal scale. Days with TS are characterized by higher 0,net 490 than days without TS (Appendix B, Fig. B1). This results in the short-term variability of the biweekly averaged occurrence in Fig. 5. Other mechanisms affect the occurrence of TS on a daily basis, such as wind events. Direct wind forcing either enhances or suppresses TS. We identified several wind events from September to December that prevented any flow in the offshore direction (i.e., no significant flow detected by the algorithm) or, on the opposite, generated a strong cross-shore circulation (wind-driven circulation in Fig. 5). 495 Although the occurrence of TS over short periods may vary from one year to another, we expect the seasonal trend previously described to be repeated every year in Rotsee. We also suggest that a similar trend should be common in other temperate lakes. Most of the TS events reported for other systems were observed in late summer or autumn, corresponding to the occurrence period in Rotsee (Monismith et al., 1990;James and Barko, 1991a, b;James et al., 1994;Rogowski et al., 2019). James and Barko (1991b) showed that the percentage of days with differential cooling in Eau Galle Reservoir (WI, 500 USA) increased from 31 % in May to 95 % in August. However, these high values do not correspond to the percentage of occurrence of TS since differential cooling can happen without necessarily forming a cross-shore circulation. Pálmarsson and Schladow (2008) also found that differential cooling was rare in May and June in Clear Lake (CA, USA). Only two nights were cold enough to drive cooling-driven TS. In winter, TS events are not frequent in Rotsee but they were reported for larger systems such as Lake Geneva in January (Thorpe et al., 1999;Fer et al., 2002a, b). A different bathymetry modifies t (Eq. 505 (12)), which becomes shorter if ℎ p /ℎ ML is small or the slope is high, leading to a more frequent occurrence of TS in winter.
In addition, convective cooling stops earlier in small lakes like Rotsee that reach the temperature of maximum density before larger and deeper lakes.

Effects of the forcing parameters on cross-shore transport
The seasonality of the cross-shore transport induced by TS can be examined by using the scaling for the cross-shore velocity 510 x (Eq. (9)) and the unit-width discharge x (Eq. (10)). The timescale F is also a key parameter to assess the seasonality of the littoral region flushing (Eq. (11)). We showed that these scaling formulae reproduce the observed seasonality in Rotsee (Fig. 6). The weakening of the transport from summer to winter is explained by the decrease of 0 by a factor of 10, which overcomes the increase of ML due to the deepening of the mixed layer (Fig. 4). Yet, F remains shorter than the duration of the cooling phase (Fig. 8b), and the littoral region upslope of MT ( lit ≈ 500 m 2 ) is entirely flushed by a single event 515 (Appendix C). The use of the constant length scale ℎ lit in Eq. (10) implies that the daily averaged thickness of TS ℎ TS,avg = avg / avg should not show any seasonal variability. This is what we observe in Rotsee: ℎ TS,avg remains close to its yearly average of 1.8 ± 0.2 m, without any seasonal trend.
We further compare our scaling for x , avg and F with previous studies on buoyancy-driven flows. The velocity scale (1966) is commonly used, with defined as the horizontal length scale along which a lateral 520 density gradient is set. This length scale varies with the experimental configuration and basin geometry (Table 2). In laboratory experiments with a localized buoyancy loss, is the length along which the destabilizing buoyancy flux applies (Harashima and Watanabe, 1986;Sturman et al., 1996;Sturman and Ivey, 1998). For more realistic systems undergoing uniform surface cooling, is the length of the littoral region, which can be defined as the plateau zone (Ulloa et al., 2022) or the region vertically mixed by convection (Wells and Sherman, 2001). In the first case, = p is a constant whereas = ML varies with the 525 stratification in the second case. We chose to use ML because it corresponds to the region affected by differential cooling. We expect that a deeper mixed layer will increase the temperature difference between the littoral and pelagic regions, resulting in a higher velocity x . In Eqs. (10) and (11), the transport is expressed as a function of the velocity scale and the size of the littoral region of length lit and depth ℎ lit (Fig. 1). We defined the littoral region based on the location of MT since we measured the discharge out of this region. This definition of ℎ lit in Eq. (10) is consistent with the fact that x varies spatially, 530 increasing with distance from the shore (Fer et al., 2002b). The vertical length scale ℎ lit can also be chosen as the depth of the plateau (ℎ p ) (Sturman and Ivey, 1998;Wells and Sherman, 2001;Fer et al., 2002b;Ulloa et al., 2022) to predict the discharge from the plateau region only. Despite these different choices of length scales, the coefficient q ≈ 0.34 is close to other estimates from the literature (Table 2). Harashima and Watanabe (1986) found that q is increasing with the flux Reynolds number defined as f = 0 1/3 ℎ lit 2 / ML 2/3 , with ≈ 1.5 × 10 −6 m 2 s -1 the kinematic viscosity of water. The empirical 535 relationship q ≈ 0.43 − 1.67/ f 1/2 was obtained for f > 50. For the days where TS is observed in Rotsee, f varies from 250 in summer to 50 in winter, with an average value of ~140 which leads to q ≈ 0.29, close to our estimate of q ≈ 0.34.

Effects of the forcing parameters on the flushing period
The seasonality of the forcing parameters does not only affect the occurrence and magnitude of TS but also the diurnal 545 dynamics of the littoral flushing (Fig. 7). The initiation of TS always occurs several hours after the beginning of the cooling phase. This delay is modulated by τ ini and is consistent with previous studies reporting a formation of TS at night (Monismith et al., 1990;James et al., 1994;Fer et al., 2002b;Pálmarsson and Schladow, 2008) or in the morning (Rogowski et al., 2019;Sturman et al., 1999). t is a good estimate of the initiation period when the water column is initially mixed ( mix = 0). In summer and warm autumn days, however, the initiation of TS occurs later than predicted by t , due to the initial stratification 550 at the beginning of the cooling phase ( mix ≠ 0). This leads to a delay in the increase of x and G (Fig. 7a, c). In autumn, TS events lasting for more than 24 h lead to monthly averaged x > 0 before the starting time predicted by t (Fig. 7e, g). In winter, t → 24 h implies that the rare TS events are all characterized by a continuous flow (Fig. 7i), without any diurnal cycle of flushing anymore.
When a diurnal cycle is present in summer and autumn, the flushing period ends several hours after the beginning of 555 the heating phase (Fig. 3, Fig. 7). This inertia was already reported by Monismith et al. (1990), who observed TS until the midafternoon, despite the opposite pressure gradients due to differential heating. In addition to the delay between the end of the cooling and the end of the flow, we also showed that a peak of flushing occurred a few hours after the beginning of the heating phase (Fig. 3). The time of maximal flushing is shifted later in the day from summer to winter, as the heating phase starts later (Fig. 7). The maximal flushing seems related to the transition from the cooling phase to the heating phase. This finding suggests 560 that convective plumes eroding the flow control the weak initial transport during the cooling phase. A weakening of convection reduces this vertical mixing and finally enables the flushing to reach its maximum intensity. Further investigations are required to better understand this process.

Practical recommendations to predict and measure thermal siphons in other lakes
Rotsee is an ideal field-scale laboratory to investigate TS, due to wind sheltering and its elongated shape minimizing complex 565 recirculation. To assess the seasonality of TS in other lakes, long-term velocity and temperature measurements in the sloping region are required. The developed algorithm to detect TS events (Sect. 2.4) and calculate the cross-shore transport (Sect. 2.5) fulfilled our requirements but remained lake specific. The limitations of the algorithm are discussed in more detail in Appendix D. The general structure of the algorithm (Sect. 2.4) can serve as a basis for detecting TS in lakes, by adapting the lake-specific criteria to other systems. The 2D framework of TS requires specific validation in more complex nearshore systems and large 570 lakes, where the topography, large-scale circulation and Coriolis may also affect the TS dynamics (Fer et al., 2002b). In these systems, the along-shore velocity component of TS must be considered in the cross-shore transport analysis. Further development is needed to build a more robust algorithm with lake independent physically based criteria.
The effects of a different bathymetry can be predicted from the scaling discussed in the previous sections. A shallower nearshore plateau region or a steeper sloping region would decrease the transition timescale t (Eq. (12)) causing the initiation 575 of TS to happen earlier and more often. Higher slopes would also decrease the length of the mixed region ML and reduce the horizontal velocity of TS (Eq. (9)). Past observations of TS reported horizontal velocities ranging from ~0.1 cm s -1 (James and Barko, 1991a, b) to ~10 cm s -1 (Roget et al., 1993;Fer et al., 2002b), with x ∼ 1 cm s -1 in most cases (Monismith et al., 1990;Sturman et al., 1999;Pálmarsson and Schladow, 2008;Rogowski et al., 2019). The cross-shore transport x and flushing timescale F are strongly dependent on the size and depth of the littoral region considered (Eqs. (10) and (11)). A deeper littoral 580 region (larger ℎ lit ) would lead to a stronger discharge and a longer littoral region (larger lit ) would take more time to flush.
Values of TS thickness and discharge can be one order of magnitude larger in deeper lakes than Rotsee (Thorpe et al., 1999;Fer et al., 2002a;Rogowski et al., 2019). The effects of more complex bathymetries, departing from our 2D framework, could be further investigated with 3D numerical simulations.
Information on lake bathymetry and thermal structure are needed to predict the occurrence ( ini < c ) and intensity of 585 TS ( x , x , F ) under specific forcing. This approach does not consider wind effects, which deserve further investigation.
Windier conditions would hinder TS by locally enhancing and reducing the cross-shore transport. In this case, the scaling formulae (9), (10) and (11) should be modified to take wind shear into account. The littoral region would also be mixed faster under windy conditions and the effects of wind mixing on mix should be included in Eq. (15). These questions could be addressed by performing numerical simulations with varying wind speed. 590 The seasonality of TS may evolve in a changing climate, which also needs to be investigated. Changes in heat fluxes, summer stratification and surface temperature would affect both the intensity ( 0 ) and occurrence ( ini ) of TS.

Conclusions
The flushing of the littoral region by cross-shore flows increases the exchange between nearshore and pelagic waters. In this study, we investigated one of the processes that enhance the renewal of littoral waters, the so-called thermal siphon (TS) driven 595 by differential cooling. From a one-year-long monitoring of TS in a small temperate wind-sheltered lake, we quantified the seasonality of the cross-shore transport induced by TS. This seasonality is related to the intensity of surface cooling (surface buoyancy flux 0 ) and the lake stratification (mixed layer depth ℎ ML ). Three aspects of the seasonality of TS are highlighted.
First, TS is a recurring process from late summer to winter (when c > ini ), occurring on ~80 % of the autumn days. Second, the seasonal changes in the TS-induced transport are well reproduced by the scaling avg ≈ 0.33 ⋅ ( 0 ML ) 1/3 , avg ≈ 0.34 ⋅ 600 ( 0 ML ) 1/3 ℎ lit and F = lit / avg ≈ 2.99 ⋅ lit /( 0 ML ) 1/3 , with ML the length of the region mixed by convection and lit = lit ℎ lit the unit-width volume of the littoral region. This study provides a field validation of this laboratory and theoretically based scaling. Third, the diurnal dynamics of the flushing follow the seasonal changes of the cooling and heating phases, by evolving from a well-defined diurnal cycle in summer to a more continuous flow in winter.
Our results demonstrate that TS significantly contributes to the flushing of the nearshore waters. This process occurs 605 frequently during the cooling season (> 50 % of the time), each time flushing the entire littoral region. We stress that this buoyancy-driven transport is, perhaps counterintuitively, stronger in summer and in the morning. Such a timing has implications for the transport of dissolved compounds, with, for instance, maximal exchange between littoral and pelagic waters at a time of high primary production (summer and daytime). Overall, this study provides a solid framework to integrate the role of TS in the lake ecosystem dynamics. 610 Appendix A: Instrumentation  3,4,5,6,7,8,9,10,11,12,13,14,15 .  gravitational adjustment triggered by differential cooling. They could be wind-driven or result from free surface convection.
In Rotsee, cross-shore flows driven by internal waves are expected to be short and to be discarded by the algorithm ( V1H1 /2 ≈ 2.5 h in summer with V1H1 the period of V1H1 internal waves). Other types of data could be included to detect TS, such as 675 lateral temperature gradients, near-surface velocity (return flow) or vertical velocity (convective plumes). Using other techniques, like machine learning algorithms, could also be a useful approach to better identify TS events.
The filter used to discard days with wind-driven circulation is based on the criterion | MO |/ℎ ML = −1 | * / * | 3 > 0.5, with = 0.41 the von Kármán constant and * the friction velocity. The ratio * / * expresses the relative importance of wind shear compared to convection. The threshold value of 0.5 was successfully tested on wind-driven circulation events observed 680 in Rotsee in winter. This filter discards the strongest wind events but cross-shore flows with | MO |/ℎ ML < 0.5 can still be affected by a wind peak during the flushing period (e.g., interaction between wind and TS). Being more conservative and decreasing the threshold value from 0.5 to 0.1 would discard TS events mainly in December (when * is low) but would not significantly modify the occurrence of TS during the other periods (Fig. D2). Additional filters could be implemented to distinguish between TS and wind-driven cross-shore flows, based for example on high-resolution vertical velocity 685 measurements, observed oscillations of the thermocline (e.g., wavelet analysis), estimates of the period of internal waves and identification of upwelling events (e.g., Wedderburn and Lake numbers) (Imberger and Patterson, 1989). By discarding days with stratified conditions at MT, the second filter does not consider a TS that intrudes before reaching MT and appears as an interflow. The occurrence of TS in summer is thereby underestimated. In Fig. 5, the higher percentage of occurrence of stratified flows in July (81 %) compared to spring (around 40-50 %) suggests a possible contribution of intruding TS. From visual inspection of each event, we estimated that intruding TS could represent almost half 695 Love Råman Vinnå, Edgar Hédouin, Josquin Dami and Alois Zwyssig for their assistance in the field. Discussions with Mathew Wells, Oscar Sepúlveda Steiner and Love Råman Vinnå helped to improve the data analysis and the quality of the manuscript. The meteorological data from the Lucerne weather station has been provided by MeteoSwiss, the Swiss Federal Office of Meteorology and Climatology. We gratefully acknowledge the three anonymous reviewers for their helpful comments and suggestions. 725

Financial support
This study was financed by the Swiss National Science Foundation (project "Buoyancy driven nearshore transport in lakes", HYPOlimnetic THErmal SIphonS, HYPOTHESIS, grant no. 175919).