# Towards Turbulent Stresses Estimates by Special Geometric Adjustment of Two ADCPs

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{i}′ (I = 1, … n; n is the number of beams); this procedure, known as the variance method, is widely used. For a Janus device (n = 4, the beam pairs 1, 3 and 2, 4 are opposite in the horizontal projection), the explicit expression for one of the shear stresses can be represented as [1,8]):

_{0}is the angle of deflection of the beam from the vertical (slant angle), ${u}_{x}^{\prime}$ and ${u}_{z}^{\prime}$ are pulsations of the velocity Cartesian components in the plane XY, which includes beams 1 and 3, the angle brackets < > denote the time averaging.

_{i}′

^{2}> of beam velocities in terms of the required stresses; and the number of explicit expressions for the stresses is restricted. For a device with three beams, an explicit expression cannot be obtained for any element of the stress tensor; for a device with Janus configuration (n = 4), explicit expressions are available only for two off-diagonal terms (see Equation (1)). In such a situation, the calculation of the pulsation intensities (diagonal elements of the stress matrix) can only be carried out using special assumptions, for example, concerning the smallness of vertical pulsations compared to horizontal ones.

_{i}′

^{2}>. This system can be solved in the least-squares sense. At the same time, as was shown by Vermeulen et al. [9], this approach fails if the devices’ axes are collinear: in this case only five independent equations for turbulent stresses are derived from eight beams. To overcome the system singularity, one of the devices (slave) should be slanted. Moreover, the tilt of the “slave” ADCP has to be considerable (20° or more), otherwise, the determinant of the equation system is close to zero and the system becomes unsolvable. The problem, however, is that ADCP manufacturers advice against tilting of the device axis. Besides, such tilting enlarges the beam spread notably and, hence, makes the horizontal homogeneity assumption less safe and leads to an increase in error.

## 2. Method Description and Observational Setup

_{0}= 25° were considered.

_{0}and h = l ctgα

_{0,}respectively. The geometric configuration of beams for the version Double is presented in Figure 1a; beams 2, 6 and 3, 5 intersect at the points C and D, at the same depth.

_{0}, M is the 6 × 7 matrix of coefficients with the elements determined by the angles α

_{0}and γ, B

_{j}is presented by six intensities of the beams velocities and velocity correlations of intersecting beans (see Appendix B).

_{i}is the standard deviation for the component i of the vector R, n

_{2}is the number of measurements used in ensemble averaging. Following the turnover time estimation procedure presented in [11], we chose the value 150 for the parameter n

_{2}; it corresponds to one-hour averaging. In most cases, however, the calculated stress values became “saturated” even with an averaging period of 30 min.

^{T}V matrix, which, in turn, depends not only on the device parameter α

_{0}, but also on the angle γ, which is determined by the experiment details. The calculated dependence of D on γ is shown in Figure 3.

_{1}O

_{2}) and the corresponding shear stress (off-diagonal component). In turn, in each of these two planes the beam correlations, e.g., $<{b}_{2}^{\prime}{}^{2}>,{b}_{6}^{\prime}{}^{2},{b}_{2}^{\prime}{b}_{6}^{\prime}$ for the plane of beams 2 and 6, are available directly from observational data. On the other hand, each of these correlations can be expressed in terms of three unknown stresses. This yields a closed system of three equations for three unknowns. It should be also stressed that the horizontal homogeneity assumption is not applied in deriving this system: all the three mentioned beam correlations belong to the same point (point B for the upper and point J for the lower depth).

**M**) for the upper and the lower depths (points B and J) are

_{0}= 25°, the depth range contaminated by sidelobe relection constitutes over 9% of the scanned area; this percentage may be sufficiently enhanced by tilt variability [14]. Taking all these issues into account, we used the value 15° for the angle γ. The beam intersection points were calculated using Equations (A3) and (A4); they occurred at depths of 35 and 140 cm from the bottom.

^{−8}–10

^{−6}) W/kg for the kinetic energy dissipation rate ε for shear-induced convection during open water period [15], the range of the Kolmogorov time scale t

_{d}= (ν/ε can be estimated as (1–10) s. So, the expected value of t

_{d}exceeded the time interval between pings (0.5 s) by up to an order of magnitude, and the active burst interval (8 s) was close to t

_{d}. In view of this fact, we used the procedure of preliminarily averaging the signal within each burst (16 samples) in order to reduce the instrumental errors. The second step for calculating the mean values of the parameters includes the averaging over 1-h interval (150 samples).

^{3}/s (300 m

^{3}/s at the peak of the spring flood) [16]. More information about Petrozavodsk Bay, including bathymetry, is given in [16]. The experiment was designed to check the estimates of turbulent stresses obtained in convectively mixed layer (CML) of ice-covered lakes previously [16,17,18], as well as to supplement them by comparing the results for two different depths within CML.

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Calculation of the depths of the points C and D where the beams intersect.**

_{1}of the point C:

_{2}of the point D:

## Appendix B

**Derivation of a matrix relating stress tensor components to beam velocity variances.**

_{i}and R

_{i}, can be written in a compact form (summation is assumed over repeated indices):

_{0}, M is the 6 × 7 matrix of coefficients with its elements determined by the design angle α

_{0}, as well as the angle of rotation γ:

_{i}. The explicit expression for the matrix M in this case is:

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**Figure 1.**(

**a**) Geometric configuration of beams for coupled devices, version Double (axes X

_{I}and X

_{II}are opposite), plan view. Emitters A and B are separated by a distance l = AB. (

**b**) Beams configuration after rotation of X-axes by an angle γ.

**Figure 2.**Dependences of the depths of beam intersection points (h) on the angle γ and the distance between the emitters l. The legend gives the values of l in centimeters. The vertical dashed lines correspond to the values of γ used in the experiments on Lake Vendyurskoe (γ = 15°) and on Petrozavodsk Bay (γ = 20°).

**Figure 3.**Dependence of the determinant of the over determined system of equations (A7) on the rotation angle γ.

**Figure 4.**(

**a**) Beam configuration for γ ≠ 0, plan view. Points O

_{1}and O

_{2}correspond to the emitter locations. (

**b**) Beam spread areas for the two depths where pairs of beams intersect. Capital Latin letters are used to designate those points at different depths where measurements of the beam velocity components were taken.

**Figure 5.**Sketches of experiment designs with coupled ADCPs. (

**a**) 16–18 November 2021, Lake Vendyurskoe. (

**b**) 14–18 April 2022, Petrozavodsk Bay of Lake Onega.

**Figure 6.**The dynamics of Cartesian components of mean velocity in the bottom layer of Lake Vendyurskoe: (

**a**) the upper beam intersection point, 140 cm above the bottom, (

**b**) the lower beam intersection point, 35 cm above the bottom.

**Figure 7.**The dynamics of Cartesian components of mean velocity in the under-ice layer on upper (

**a**) and lower (

**b**) beam intersection points. Petrozavodsk Bay of Lake Onega.

**Figure 8.**Intensities of turbulent pulsations (components 11, 22 and 33 of the stress matrix) in the upper beam intersection point 140 cm above the bottom. The thick line represents the results of alternative calculations of the pulsation intensity <u

_{x}′

^{2}> performed using the exact Equation (4). Lake Vendyurskoe data.

**Figure 9.**Intensity of horizontal velocity component pulsations (components 11, 22 of the stress matrix) in the lower beam intersection point located 35 cm above the bottom. The shaded area represents the significance interval for the pulsations $<{u}_{x}^{\prime}{}^{2}>$ Lake Vendyurskoe data.

**Figure 10.**Stress matrix components in the horizontal (

**a**) and vertical (

**b**) planes. Petrozavodsk Bay data. Distance from the lower ice surface is 58 cm.

**Figure 11.**Intensities of horizontal velocity component pulsations in the convectively mixed layer in the upper and lower beam intersection points at distances of 58 and 265 cm from the lower ice surface.

**Table 1.**The correlation coefficients for components of mean velocity calculated independently from the data of each ADCPs.

Vendyurskoe Lake Upper Beam Intersection Point | Vendyurskoe Lake Lower Beam Intersection Point | Onega Lake Upper Beam Intersection Point | Onega Lake Lower Beam Intersection Point | |
---|---|---|---|---|

U_{x1}–U_{x2} | 0.96158025 | 0.84117983 | 0.99221171 | 0.97880755 |

U_{y1}–U_{y2} | 0.99899851 | 0.99874128 | 0.99723331 | 0.99399168 |

U_{z1}–U_{z2} | 0.90556808 | 0.94104025 | 0.92498406 | 0.95267151 |

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**MDPI and ACS Style**

Bogdanov, S.; Maksimov, I.; Zdorovennova, G.; Zdorovennov, R.; Palshin, N.; Mitrokhov, A. Towards Turbulent Stresses Estimates by Special Geometric Adjustment of Two ADCPs. *Water* **2023**, *15*, 28.
https://doi.org/10.3390/w15010028

**AMA Style**

Bogdanov S, Maksimov I, Zdorovennova G, Zdorovennov R, Palshin N, Mitrokhov A. Towards Turbulent Stresses Estimates by Special Geometric Adjustment of Two ADCPs. *Water*. 2023; 15(1):28.
https://doi.org/10.3390/w15010028

**Chicago/Turabian Style**

Bogdanov, Sergey, Igor Maksimov, Galina Zdorovennova, Roman Zdorovennov, Nikolai Palshin, and Andrey Mitrokhov. 2023. "Towards Turbulent Stresses Estimates by Special Geometric Adjustment of Two ADCPs" *Water* 15, no. 1: 28.
https://doi.org/10.3390/w15010028