# Internal wave focusing by a partial torus

Natalia Shmakova, Svetlana Kudryashova, Bruno Voisin

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Natalia Shmakova, Svetlana Kudryashova, Bruno Voisin. Internal wave focusing by a partial torus. 26th International Congress of Theoretical and Applied Mechanics, Aug 2024, Daegu, South Korea. hal-04683028

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# INTERNAL WAVE FOCUSING BY A PARTIAL TORUS

Natalia Shmakova¹*, Svetlana Kudryashova¹ & Bruno Voisin²

¹Lavrentyev Institute of Hydrodynamicas SB RAS, Novosibirsk, Russia

²LEGI, CNRS & Université Grenoble Alpes & Grenoble INP, Grenoble, France

*Summary* A joint experimental and theoretical study is presented for the focused internal waves generated by the horizontal oscillations of a torus segment in a linearly stratified fluid. The wave amplitude increases towards the focal zone for half- and quarter-tori, irrespective of the orientation of the segment with respect to the direction of oscillation, implying that horizontal curvature is the key ingredient for wave focusing. At low oscillation amplitude, good quantitative agreement is obtained between laboratory measurements using standard PIV techniques and a linear three-dimensional theory including viscous effects. At moderate and high oscillation amplitudes, asymmetric oscillations, such that the segment is oriented sideways with respect to the direction of oscillation, are seen to lead to horizontal momentum generation in the focal region.

## INTRODUCTION

Tidal internal waves, generated by barotropic tidal flow over ocean bottom topography, are believed to play an important role in the mixing of the ocean and thereby in the climate [1, 2, 3]. To model these waves in the laboratory, the reference frame of the tidal motion is generally used, i.e. the waves are generated by the oscillation of an object. Typical objects are cylinders, representing ocean ridges, and spheres, representing isolated seamounts. These have been studied in detail in [4, 5, 6, 7]. We look here at the effect of the curvature of the ridges, through the use of a torus of vertical axis. In contrast to the sphere or the cylinder, a torus generates not only diverging waves but also converging waves that increase in amplitude almost twice towards two biconical focal zones, one above and one below the torus [8, 9]. In these zones as well as close to the torus, the waves overturn at high oscillation amplitude leading to a Helmholtz 'fast' instability, and consequently to the mixing of the stratification [10]. Recent experimental studies of internal wave attractors have shown that the triadic instability, referred to as 'slow', also implies the mixing of the stratification [11]. In the focal zones of the torus, subharmonic waves form such resonant triad with the fundamental wave [12]. In this work we continue the investigation of the effect of the horizontal curvature of ocean ridges, by considering only a segment of torus.

## EXPERIMENTAL SETUP

The experiments are conducted in a plexiglas tank of horizontal section 43 × 43 cm² and height 47 cm, filled over a depth of 43 cm with a linearly salt-stratified fluid of buoyancy frequency N = 1.03 rad/s. The waves are generated by the horizontal oscillations of two torus segments (a half-torus and a quarter-torus) of vertical axis and aspect ratio ε = b/a = 5, with a = 1 cm and b = 5 cm the minor and major radii of the torus, respectively. For each experiment a specific segment is mounted on a frame attached to a connecting rod, of motion controlled by a step motor with a very small step. With this system, the segment performs oscillations of given frequency ω = 0.77 rad/s, such that the waves propagate at the angle θ = arccos(ω/N) ≈ 41° to the vertical, and varying amplitude A. Introducing a Cartesian coordinate system of origin at the centre of the full torus, horizontal x- and y-axes parallel and perpendicular to the direction of oscillation, respectively, and vertical z-axis pointing upwards, the oscillations we investigate are of either of two types: one, called 'symmetric', if the centre of the segment is located in the plane y = 0; and the other, called 'asymmetric', if the centre of the segment is located in the plane x = 0.

The wave propagation is visualized using the standard 2D2C Particle Image Velocimetry (PIV) technique in vertical or horizontal planes through the focal zone, using vertical or horizontal laser sheets, respectively, illuminating 50 µm polyamide particles. A long image series (over 200 oscillation periods) is taken by a Tucsen Dhyana 400BSI V3 sCMOS camera with a high resolution of 2048 × 2048 pixels and 16-bit pixel depth. The recorded images are treated using the PIVlab toolbox for MATLAB [13], providing two in-plane velocity components (u and w for the vertical cut, u and v for the horizontal cut). Further, the harmonic analysis is performed using Hilbert transforms [14].

## THEORETICAL APPROACH

The theory is three-dimensional and generalizes that for the full torus, given in [15] and corrected in [9]. It assumes that the aspect ratio is large, such that $ε \gg 1$, the oscillation amplitude small, such that $Ke = A/a \ll 1$, and the viscosity weak, such that $St = ωa^2/ν \gg 1$, with $ν = 0.01 \text{ cm}^2/\text{s}$ the kinematic viscosity. The waves are expressed in a system of cylindrical coordinates $(r, φ, z)$, with $x = r \cos φ$ and $y = r \sin φ$, and these coordinates are made non-dimensional as $(R, Z) = (r, z)/a$, the time $t$ as $T = ωt$ and the vertical velocity $w$ as $W = w/(ωA)$. The torus segment occupies the sector $Φ - ΔΦ < φ < Φ + ΔΦ$, and its oscillation is such that the centre of the full torus would be at the position

<footnote>*) Corresponding author. E-mail: shmakova@hydro.nsc.ru</footnote>
$X_0 = iA \exp(-iT)$ in complex notation. The velocity is obtained as

$$W = \varepsilon \cos^2 \theta e^{-i(T-\theta)} \operatorname{sign} z \int_0^\infty dK K J_1 \left( K - i \frac{K^3}{2St} \right) \exp \left( -\frac{K^3 |Z|}{2St \sin \theta} \right) \exp(-iK|Z| \sin \theta) \sum_{n=0}^\infty \frac{\varepsilon_n}{2} J_n(KR \cos \theta) \\ \times \left\{ \frac{\sin[(n-1)\Delta\Phi]}{(n-1)} J_{n-1}(K\varepsilon \cos \theta) \cos[n\varphi - (n-1)\Phi] - \frac{\sin[(n+1)\Delta\Phi]}{(n+1)} J_{n+1}(K\varepsilon \cos \theta) \cos[n\varphi - (n+1)\Phi] \right\},$$

where for $n = 0$ the ratio $\sin(n\Delta\Phi)/n$ is replaced by $\Delta\Phi$, and $\varepsilon_n = 1$ for $n = 0$ and 2 otherwise is Neumann's factor. The integrals are evaluated numerically with Mathematica, keeping up to 50 terms in the summation.

## RESULTS AND DISCUSSION

The internal waves generated by a torus segment amplify towards the focal zone as shown in figure 1 for symmetric and asymmetric oscillations of the half- and quarter-tori. The experimental results obtained at small oscillation amplitude $Ke = 0.2$ appear to be in good agreement with the linear theory. When the segment performs asymmetric oscillations of moderate amplitude, an additional horizontal current is generated in the focal zone propagating away from the object. At high oscillation amplitude the frequency spectrum reveals subharmonic waves implying triadic resonance instability. In oceanic applications the results suggest a possibility of wave focusing and thus mixing near any curved underwater ridge, consistent with the wave amplification observed in laboratory and numerical models of the effect of the slightly curved ridges in the Luzon Strait of the South China Sea [16, 17].

<figure><img src="image_2.png" /><figcaption>Figure 1. Vertical velocity amplitude in the plane $Y = 0$ for the waves generated by the horizontal oscillations of a half-torus in (a) symmetric and (b) asymmetric configurations, and a quarter-torus in (c) symmetric and (d) asymmetric configurations, calculated using the three-dimensional linear theory.</figcaption></figure>

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