# Short Standing and Propagating Internal Waves in an Ice-Covered Shallow Lake

^{*}

## Abstract

**:**

## 1. Introduction

^{2}in stratified lakes during the open water period [1], which is only a minor part (1% or less, according to the initial estimates of Lorenz [7]) of the characteristic values of the potential energy (Schmidt stability parameter).

- -
- planning of experiments to reveal the presence of standing and propagating short IWs in an ice-covered lake;
- -
- detection of propagating short IWs and specific features of their generation and propagation;
- -
- assessment of the key parameters of IWs (phase and group velocities, wave numbers) that are essential for studying the intensification of vertical mixing.

## 2. Material and Methods

^{2}, the surface area is 10.4 km

^{2}, the water volume is ~5.5·10

^{7}m

^{3}, the average depth is 5.3, and the maximum depth is 13.4 m. The basin of the lake is elongated from the southwest to the northeast; its length is 7 km, and its width reaches 1.5–2 km. Ice-on occurs from early November to mid-December in different years; ice-off occurs from the end of April to the second ten days of May.

^{−2}rad/s. Long-term temperature measurements [9] indicate significant wave activity in the whole water column throughout the winter season. During the entire ice-covered period, the water temperature spectrum shows peaks corresponding to 7 and 27 min [9]. These peaks are very close to the periods of the first modes of the transverse and longitudinal barotropic seiches of the lake [38,47].

**k**, $\lambda $ is the wavelength, $\omega $ is the frequency; $\omega $ is determined by the main seiche frequencies or their higher harmonics. In the case of standing IWs, m = $\pi $n/H (n is the number of the vertical mode) and relation (1) can be written as

**k**. The same straight lines set the direction of the group velocity and the beams along which the wave trains travel and energy is transferred. For a wave train traveling in the XZ plane, the group velocity components u

_{g}are determined by the relations [48,52]

**k**= (k, m). It should be noted that the angle $\mathsf{\vartheta}$ and, accordingly, the angle of inclination of the “beams” in accordance with (3) is uniquely determined only by the base frequencies N, $\omega $ and does not depend on the mode numbers. A result of this feature of IWs is that, under certain conditions, energy flows are concentrated in narrow bands adjacent to the beams [53]. It is this circumstance that can enhance the role of IWs in the mixing process: in the zones adjacent to the beams, the velocity shear can reach sufficiently large values and, accordingly, when the threshold value of the Richardson parameter Ri is reached, instability and intensification of local mixing may occur.

## 3. Results

_{12}of temperature fluctuations ΔT at different depths:${f}_{12}\left(\u2206z\right)=<\u2206{T}_{1}\left({z}_{1}\right)\u2206{T}_{2}\left({z}_{2}\right)>/{(<\u2206{T}_{1}^{2}\left({z}_{1}\right)><\u2206{T}_{2}^{2}\left({z}_{2}\right)>)}^{1/2}$, where Δz = z

_{2}− z

_{1}.

_{1}. For both years, the calculated curves have an oscillating character, despite the rather significant averaging period. This fact can be regarded as indirect evidence of the stability of the multimode structure of the water column. At the same time, the vertical mode of the fifth order was predominant in 2014, while the third-order mode dominated in 2016, which is consistent with the visual image of the ΔT field shown in Figure 3.

_{gx}of the group velocity was estimated by the analysis of some surges. For different surges, the value of c

_{gx}varied in the range of a few mm/s, depending on depths. For the surge shown in Figure 10, the value c

_{gx}≈ 0.65 cm/s was obtained for shallow depths. Since the direction of the horizontal projection of the wave vector

**k**is, strictly speaking, not known, this value can be considered as a lower estimate of the value of c

_{gx}. Substituting the values H ~ 6 m, N ~ 10

^{−2}rad/s, and ω/N ~ 1/3 in Formula (3), the above estimate of c

_{gx}allows us to identify the disturbance under consideration as a vertical mode of a sufficiently high (n = 3 or 4) order. Taking these vertical mode numbers into account, the horizontal wavenumber k is derived directly from (2). Finally, we obtain the estimate λ ~ (9–12) m for horizontal wavelength.

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Location of Lake Vendyurskoe (red star) and of measurement stations (yellow rectangle) in 2014 and 2016. The position of measurement stations relative to each other is shown in the insets.

**Figure 2.**Welch’s spectra of temperature pulsations at different depths for TR-chain data at station 3 in 2016 (

**a**) and 2014 (

**b**).

**Figure 3.**Dynamics of ΔT in the water column as an indicator of the presence of short standing IWs. The asterisks on the left-hand side of the panels indicates the depths of the temperature sensors. (

**a**)—at station 1 on 3 February 2014, (

**b**)—at station 1, on 4 February 2014, (

**c**)—at station 1 on 31 January–1 February 2016.

**Figure 4.**Correlation functions of ΔT and profiles of temperature variance $\langle {\left(\u2206T\right)}^{2}\rangle $ for TR-chains installed at station 1 in 2014 (

**a**), at station 1 in 2016 (

**b**), at station 2 in 2014 (

**c**), and at station 2 in 2016 (

**d**).

**Figure 5.**Dynamics of ΔT at a depth of 4 m at stations 1 and 3 in 2014. Dots are the rough data, and lines are the 27-min moving averages.

**Figure 6.**Examples of in-phase ((

**a**), depth 1 m) and anti-phase ((

**b**), depth 5.5 m) oscillations of ΔT for TR-chains 1–4 on 30 January 2016.

**Figure 7.**Fragments of propagating IWs (green rectangles). The set of lines represents the temperature deviations from the average over the selected 2 h time interval at different depths for TR-chains at stations 1 (

**a**), 2 (

**b**), and 4 (

**c**) on 4 February 2014. For clarity, each curve is shifted upward from the previous one by 0.04 °C.

**Figure 8.**Temperature trend dynamics in several layers adjacent to the bottom TR-chain at station 4, 2016. For clarity, the curves are sequentially shifted vertically by 0.15 °C.

**Figure 9.**Dynamics of ΔT in the near-bottom layers of four TR-chains from 23:30 28 January 2016 to 12:30 1 February 2016 (

**a**). For clarity, the temperature graphs are shifted vertically by 0.10 °C. Panel (

**b**) shows a fragment for all TR-chains from 09:30 to 15:15 on 29 January 2016.

**Figure 10.**Temperature trend field disturbances recorded by sensors of all four TR-chains on 30 January 2016. For clarity, the graphs for different depths are sequentially shifted relative to each other by 0.05 °C. The inset illustrates the delay of the signal as it propagates in the horizontal direction. TR-chains at stations 1, 2, 3, and 4 correspond to the red, green, blue, and cyan colors, respectively.

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**MDPI and ACS Style**

Bogdanov, S.; Zdorovennov, R.; Palshin, N.; Efremova, T.; Zdorovennova, G.
Short Standing and Propagating Internal Waves in an Ice-Covered Shallow Lake. *Water* **2023**, *15*, 2628.
https://doi.org/10.3390/w15142628

**AMA Style**

Bogdanov S, Zdorovennov R, Palshin N, Efremova T, Zdorovennova G.
Short Standing and Propagating Internal Waves in an Ice-Covered Shallow Lake. *Water*. 2023; 15(14):2628.
https://doi.org/10.3390/w15142628

**Chicago/Turabian Style**

Bogdanov, Sergey, Roman Zdorovennov, Nikolai Palshin, Tatiana Efremova, and Galina Zdorovennova.
2023. "Short Standing and Propagating Internal Waves in an Ice-Covered Shallow Lake" *Water* 15, no. 14: 2628.
https://doi.org/10.3390/w15142628