We investigated the internal seiche amplitude by examining temperature profiles obtained from field monitoring at Vossoroca Reservoir and Harp Lake. The results were then compared with theoretical amplitudes reported in the literature (Stevens and Lawrence. 1996; Aquatic Sciences).

Our study revealed the presence of internal seiches with higher vertical modes in both Vossoroca Reservoir and Harp Lake. The majority of interna waves in Vossoroca Reservoir occured during the winter season, while in Harp Lake, they were more prevalent during autumn. These findings indicate that periods of weaker stratification coincide with period of higher internal seiche amplitudes (de Carvalho Bueno et al., 2021; Environ. Fluid Mech.), which aligns with previous theoretical outcomes (see Fig 1).

Internal waves are often predicted by the Wedderburn number (W), which describes the ratio between the gradient of pressure stablished by stratification and wind friction

\[W=\frac{g'~h_e^2}{L~u^2},\]

in which g' is the reduced gravity, h_e is the upper mixed layer, L is the basin length, and u is the wind friction velocity.

Although the internal seiche pattern is well described by previous theories (Stevens and Lawrence. 1996; Aquatic Sciences), the estimation fails to predict the internal seiche amplitude when the stratification deviates from the idealized two-layer system (Fig 1).

**Fig 2.** The longitudinal temperature profile during the wind event (3 h of simulation time) showing the strong erosion of thermocline at downwind region. The white dashed line indicates the unperturbed thermocline depth and the blue dashed line the theoretical displacement according to linear theory

We observed a much smaller amplitude for small Wedderburn number (de Carvalho Bueno et al., 2021; Environ. Fluid Mech.). To understand why the internal seiche was attenuated in some thermally stratified basins, we ran 32 simulations to analyze the lake response to a wind event (varying the Wedderburn number from 0.1 to 10).

Numerical observations have indicated that the amplitude of internal seiche is influenced by various factors beyond the Wedderburn number. These factors include the additional contribution of the h_e/H ratio (ratio between the epilimnion theckness and total water depth), wind-wave resonance effects, and turbulence production near the wave crest (de Carvalho Bueno et al., 2021; Environ. Fluid Mech.). We observed from measured results (which include also observation from the wide literature) that previously established theories sometimes fail to predict internal seiche amplitude.

The internal seiche amplitude is directly associated with the daily Wedderburn number of the analyzed period. The overall result, more clear for V1H1 internal seiche mode observed in Vossoroca reservoir, shows an inversely proportional relationship between internal wave amplitude and the Wedderburn number, with higher variation for different h_e/H.

We observed that for a low Wedderburn number (W<1), instabilities due to shear interactions reduce the internal wave amplitude, and the wave amplitude does not follow the previous mathematical formulation. This observation was also evidenced through three-dimensional numerical simulations.

For a shallow upper layer, the three-dimensional model also indicated that the increase of pressure at the upper layer, leads to a higher internal seiche compared with previous theories (Fig. 2).

Parameterization of Internal seiche amplitude

**Fig 3.** The longitudinal temperature profile during the wind event (3 h of simulation time) showing the strong erosion of thermocline at downwind region. The white dashed line indicates the unperturbed thermocline depth and the blue dashed line the theoretical displacement according to linear theory

Based on numerical data, and supported by field observations, we proposed a new parameterization to better describe the excitation of basin-scale internal waves in thermally stratified lakes and reservoirs:

\[\frac{\zeta_o}{h_e} = A\exp\Bigg(\frac{-(W-k)^2}{2~f^2}\Bigg)\]

where W is the Wedderburn number, k = 6 and A = 0.1 are universal constants, and f is a function of he/H, which determines an higher order influence of h_e/H:

\[f(h_e/H) = g(h_e/H)\exp\Bigg(\frac{(h_e/H)^2}{B}\Bigg)\]

where B = 0.125 is a universal constant (found empirically) and the function g is a function of he/H that was found empirically from the numerical simulations and is defined as:

\[g(h_e/H) = 12.156\Bigg(\frac{h_e}{H}\Bigg)^3- 15.714\Bigg(\frac{h_e}{H}\Bigg)^2 + 2.843 \frac{h_e}{H} + 2.085\]

The vertical displacement of upweeling events described by the above equations does not provide a definitive method for calculating the wave amplitude. However, it serves as a valuable tool for comprehending the physical processes that impact the development of internal seiches in thermally stratified lakes, which are not accounted in simplified model schemes.