Internal seiche detection

We have analyzed the formation of internal seiche in different thermal-stratified basins, from field campaigns to modeling results, including lakes and reservoirs with different shapes and sizes. Here we present a brief compilation of some internal seiche detected in thermally stratified basins, including background knowledge to understand internal seiche activity.

We have detected basin-scale internal waves in Vossoroca reservoir (de Carvalho Bueno and Bleninger, 2018; RBRH and de Carvalho Bueno et al., 2021; Environ. Fluid Mech.), located in a humid subtropical climate, presenting hot and humid summers with frequent frosts. The region presents a mean annual temperature of approximately 17 °C, with a stratification of 5 °C during springs. During summer the upper layer has temperature higher than 23 °C, present a linear stratification condition, with bottom temperature near 16 °C (Fig 1c). The reservoir has two long narrow arms 300 m wide (~ 2 km length) with a maximum depth of 17 m. The reservoir has irregular boundaries, presenting a shoreline development index of 4.2.

Given the complexity of the reservoir, the internal wave field is not easily identified through countourf plots. In addition, the thermal stratification is completely different from a simple two-layer approximation (Kanari, 1975; Limnol. Oceanogr.), presenting linear stratification from bottom to surface (Fig 1b). Using a thermistor chain we identified internal seiches with different modes, including fundamental and second vertical modes internal waves (de Carvalho Bueno et al., 2021; Environ. Fluid Mech.).

Although we can see some disturbances on thermal distributions, we cannot correlate easily these variations to internal wave activity (periodic disturbances within the stable interior). To highlight these oscillatory responses induced by the evolution of internal seiche, we use spectral analysis, such as Fourier transform or wavelet transform (Emery and Thomson, 1998; "Data Analysis Methods in Physical Oceanography"). These transformations, based on a mathematical basis, show to us the spectrum of a signal, highlighting if a time-series has periodic oscillations and in which frequencies those oscilations occur.

**Fig 1.** Time averaged thermal-structure profile of Vossoroca reservoir during a period of internal seiche activity (de Carvalho Bueno et al., 2021; Environ. Model. Softw.).

**Fig 2.** Power spectral density of 22 °C and 17.5 °C isotherms. The dashed lines indicate the mean red noise spectrum for the time series at a 95 % confidence level (de Carvalho Bueno et al., 2021; Environ. Model. Softw.).

The power spectral density, which is based on the Fourier principle (Fig. 2), was applied to two isotherms. As we can note both isotherms (17.5 °C and 22 °C isotherms) presented high spectral energy with a period of approximately 12 h, which means that the wave has a period of 12 h. As we can note, differently from surface waves, due to small density differences internal waves have periods in the order of hours to days, which depends on the lake size, stratification conditions, location (latitude), thermocline depth, number of vertical modes, and many other variables.

But the difference does not stop in the wave period. Different from surface seiche that occurs in a single interfacial system, internal seiche can have several vertical modes depending on the vertical structure of density stratification and wind dynamics (Munnich, et al., 1992; Limnol. Oceanogr. and LaZerte, et al., 1980; Limnol. Oceanogr.). Different modes of internal seiches can be excited, with opposing currents in a number of vertical layers with different densities.

Due to pressure distribution, each layer moves out-of-phase to the closest layers. Although higher vertical modes just can be excited in a system with higher continuous stratification), the number of layers does not necessarily guarantee the formation of higher vertical modes. Many studies have pointed out that higher vertical modes can be generated due to the resonance effect with the wind frequency (Vidal, et al., 2007; Limnol. Oceanogr.). One of our recent finding suggests that the lake bathymetry, meanly the slope of the bathymetry near the lakeshore, is also responsible to favor the formation of internal seiche of higher vertical modes.

To help with the identification of Internal waves, we also use models to predict the period of the wave (de Carvalho Bueno et al., 2021; Environ. Model. Softw.). The multi-layer model indicates the formation of a V2H1 baroclinic mode (de Carvalho Bueno et al., 2021; Environ. Fluid Mech.), corroborating with previous observations obtained from a two-layer models (de Carvalho Bueno and Bleninger, 2018; RBRH), which identified values much lower than that associated with the peak from the power spectral density (Fig. 2). Similar behavior and oscilatory modes have been observed also in Harp Lake (de Carvalho Bueno et al., 2021; Environ. Model. Softw.).

Passaúna reservoir (Brazil)

No significant peaks that could be related to internal seiche activity were onserved in the power spectra of horizontal flow velocity in the lake interior from simulated (Fig 3a) and field measurements (de Carvalho Bueno et al., 2023; ; Environ. Fluid Mech.), corroborating with previous observations (Ishikawa et al., 2021; Inland Waters). The low-frequency peaks (with period of 24 h and 12 h) identified in the power spectral density are more energetic near the water surface, indicating that the observed oscillatory response is related to the diurnal and semidiurnal components of the wind forcing that accelerate the water surface periodically. Particularly near the water surface, the spectra of the simulated horizontal velocity indicates the presence of oscillatory flows at a frequency of 4 × 10^-4 Hz (Fig. 3a). Since the model is not capable of resolving high-frequency internal waves due to its coarse spatial resolution, the peak can likely be assigned to surface (baroclinic) seiches of different horizontal modes, which were hidden by other unresolved motions in the measured data.

**Fig 3.** a) Power spectral density of simulated time series of horizontal flow velocity at two different depths. For low-frequencies (10^-6 to 10^-4 Hz), the spectra were calculated with a window size of 3 d, while in the high-frequency range (up to 10^-4 Hz), we used a window size of 12 h. The dashed lines show the 95 % confidence limit of the mean red noise spectrum estimated for each time series. The vertical black dashed line marks the buoyancy frequency (N = 0.02 Hz), and the vertical solid black line shows a 24-h period. b)

The theoretical period of the fundamental surface seiche in Passauna reservoir, assuming a reservoir-shaped basin, is around 42 min, which matches the frequency of the spectral peak at 3.8 × 10^-4 Hz, corresponding to period of 43 min (Fig. 3a).

From the temporal dynamics of the vertical profiles of horizontal flow velocities near the center of the reservoir, we can see that there is not a pattern that could indicate the occurance of persistent oscillatory motion, but we can idenatify uppwelling events after persistent wind events (de Carvalho Bueno et al., 2023; ; Environ. Fluid Mech.).

New observations have suggested that upwelling events are followed by a strong energy dissipation due to the interaction between interfacial wave and lake bathymetry, indicating that the oscillatory motion is quickly damped.

Milada Lake (Czech Republic)

The study analyzed the temperature distribution (Fig 4a) and internal wave dynamics in Lake Milada. The Interwave Analyzer software was used to analyze the temperature profiles and wind data. Four different isotherms (20°C, 18°C, 15°C, and 13°C) were selected to represent the temperature range in the lake. Spectral analysis revealed significant peaks at 4.5×10^-5 Hz and 9.5×10^-5 Hz, corresponding to internal seiche modes (Fig 4b). The highest spectral energy was observed at the 15°C isotherm, consistent with the thermocline depth. The analysis showed that internal seiche of the second vertical mode was not generated during the studied period (Fig 5a and Fig 5b). Bandpass filtering of the isothermal depth time series identified five periods of significant internal seiche activity (Fig 5a). The analysis also indicated that the internal seiches in Lake Milada were triggered by strong wind events, and the vertical displacement varied between different locations in the lake (Fig 5c). The study highlighted the potential biological consequences of thermocline shifts caused by seiches and the importance of studying ecosystem responses to internal seiche dynamics.

**Fig 4.** Analysis of the time series of isothermal depths at the sampling stations. a) Time series of the 15 ° C isotherm. b) Power spectral density (PSD) of isotherm displacements at the west station. The vertical green, blue, and red bars in the spectrum mark the theoretical frequencies of internal seiches for the first, second, and third vertical modes, respectively. The dashed lines indicate the confidence level.

**Fig 5.** The occurrence and potential periods indicated by dimensionless physical parameters. a) Time series of the isothermal depth bandpass filtered with cutoff frequencies of theoretical fundamental mode. The vertical gray bar indicates the period of isothermal fluctuations highlighted in Figure 5b. c) Time series of the Wedderburn number and the wind shear stress. The blue color indicates the period of potential internal seiche activity estimated by the Interwave Analyzer (de Carvalho Bueno et al., 2021; Environ. Model. Softw.) based on constant wind events that can generate seiche. The software only considers wind events longer than ¼ of the theoretical period of the internal seiche and act in a constant direction.

Reavealing internal waves in thermally stratified lakes

Are you a curious student interested in understand more about the hydrodynamic of thermally stratified lakes?

Main tasks include:

Collaborate with the team to gather data from various thermally stratified lakes worldwide;
Utilize advanced analytical techniques to identify and characterize internal wave patterns in the collected data;
Conduct a comparative study by analyzing internal waves across different lakes, identifying common patterns;
Document your findings, prepare reports, and contribute to scientific publications disseminate the project's outcomes;