Strong bottom currents in large, deep Lake Geneva generated by higher vertical-mode Poincaré waves

Although internal seiches are ubiquitous in large, deep lakes, little is known about the effect of higher vertical-mode seiches on deepwater dynamics. Here, by combining entire summer season current and temperature observations and 3D numerical modeling, we demonstrate that previously undetected vertical mode-two and mode-three Poincaré waves in 309-meter deep Lake Geneva (Switzerland/France) generate bottom-boundary layer currents up to 4 cm s−1. Poincaré wave amphidromic patterns revealed three strong cells excited simultaneously. Weak hypolimnetic stratification (N2 ≈ 10−6 s−2), typical of deep lakes, significantly modified the wave structure by shifting the lower vertical node in the lake’s center from ~75-meter depth (without stratification) to ~150-meter depth (with stratification). This shift induces shear in the middle of the hypolimnion and strengthens bottom currents, with important implications for hypolimnetic mixing and sediment-water exchange. Our findings demonstrate that classical concepts based on constant temperature layers cannot correctly characterize higher vertical-mode Poincaré seiches in deep lakes.


Supplementary Text 1. Estimation of the V2 and V3 Poincaré wave periods in Lake Geneva
In the main text, vertical mode-two (V2) and vertical mode-three (V3) Poincaré waves in stratified Lake Geneva are investigated with field observations and 3D numerical modeling; wave periods around ~14-15.5 h were determined.Below, the V2 and V3 Poincaré wave periods are estimated from classical concepts provided in the literature for a typical summer stratification using a linear, analytical model for a flat-bottomed, elliptical basin, with the non-rotational phase speed obtained by solving the Taylor-Goldstein equation.Following Antenucci and Imberger 1 , the dispersion relation for rotationally-modified internal seiches (Kelvin waves and Poincaré waves) in such basins can be approximated as: where  is the angular wave frequency and  is the latitude-dependent Coriolis parameter, which is O(10 −4  −1 ) at the latitude of Lake Geneva. =  () ⁄ is the Burger number, with the nonrotating phase speed  and a characteristic length scale , taken as half the length of the basin's major axis.The constants  0 to  3 depend on the wave's sense of rotation, the Burger number, the horizontal wave mode, and the horizontal aspect ratio of the basin (see Table 1 in Antenucci and Imberger 1 ).From the dispersion relation in Eq. ( 1), the internal wave period  is obtained as: ( The phase speed of a non-rotating seiche of vertical mode n,   , for a given stratification profile can be estimated by solving the Taylor-Goldstein equation: where () denotes the vertical structure or streamfunction,  � the background horizontal current, Water density as a function of temperature was computed with the 25-term equation-of-state of McDougall et al. 4 with salinity kept constant at 0.03 psu; salinity plays a minor role in determining water density in Lake Geneva.
From Eq. ( 3), the non-rotating V2 and V3 phase speeds for the realistic temperature profile, Tr, are  2 ≈ 21 cm s −1 and  3 ≈ 13.5 cm s −1 , respectively.At 100-m depth, the main basin has a length of ~40 km and a width of ~10 km, which gives an aspect ratio of ~ 0.25.
Finally  1b, c and 4a).These results suggest that previously undetected V2 and V3 Poincaré waves caused the dominant current signal in the nearinertial band in the lake's deepest layers.
Supplementary Table 1.Details of the mooring deployed at ~305-m depth in the center of Lake Geneva (for location, see Figure 1a).For the present study, measurements from 1 June to 1 September in 2021 and 2022 are used.
the phase speed of vertical mode n, and  the horizontal wavenumber. 2 () = − 0 −1 (  ⁄ ) is the squared buoyancy frequency and  0 a reference density.At the lake bottom and surface, ( = 0) = ( = −) = 0. Equation (3) was solved for a typical mean summer temperature (stratification) profile, referred to as the realistic temperature profile, Tr (Figure 1j in the main text), using the MATLAB code provided by Smyth 2 , with  � = 0 (no background current).The horizontal wavenumber,   , corresponding to horizontal mode , was set to   =    ⁄ , where L is the basin length 3 .

Supplementary Figure 1 .
Nortek Signature 1000 (1000 kHz) Acoustic Doppler Current Profiler (ADCP): 29 bins of 1 m, 7 min ensemble interval, 1200 pings per ensemble.Note: Due to low backscatter in the deepest layers, the last ~5-10 m above the lakebed often showed poor signal quality and were not used.Teledyne RDI Sentinel V (500 kHz) ADCP: 32 bins of 2 m, 20 min ensemble interval, 300 pings per ensemble.294 Upward-looking Teledyne RDI Workhorse Quartermaster (150 kHz) ADCP: 38 bins of 8 m, ~8 min ensemble interval, 130 pings per ensemble.Close-ups of the near-inertial band of the measured (black solid) and modeled (realistic simulations; red solid) clockwise rotary current spectra at the mooring location (depth-averaged over the lowest ~20 m) from 1 June to 1 September (a) 2021 and (b) 2022.The full frequency range is shown in Figure 1b, c in the main text.Vertical lines (left to right): Black dotted line: Inertial period (Ti = 16.5 h); Magenta solid line: V3 Poincaré period (TV3 = 15.0 h); Green solid line: V2Poincaré period (TV2 = 14.3 h); Black dash-dotted line: V1 Poincaré period (TV1 = 10.5 h, as reported in Lemmin et al.5 ) and Red dash-dotted line: V1 Poincaré period (TV1 = 9.3 h, as determined in this study based on the idealized modeling results for summer 2022; see also Figure4a).The 95% confidence interval is given in (a).Note that the V1, V2 and V3 Poincaré wave periods reported here were determined based on the idealized modeling results for summer 2022 as described in section, V1, V2 and V3 Poincaré wave features revealed by idealized simulations in the main text.These wave periods change depending on the stratification and, thus, can vary between different (summer) months and years.The latter explains the "mismatch" between the location of the near-inertial peaks in the current spectra for summer 2021 (panel a) and the V2 and V3 wave periods determined for summer 2022 (compare near-inertial peaks and green and magenta lines in panels a and b; see also Figure1c, b).