# On (n,1) Wave Attractors: Coordinates and Saturation Time

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*Symmetry*)

## Abstract

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## 1. Introduction

## 2. Dynamics of Internal and Inertial Wave Attractors

## 3. The Algorithm for Calculation of the Coordinates of an (n,1) Attractor

## 4. Direct Numerical Simulation

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A domain filled either with linearly stratified fluid or homogeneous-density, uniformly -rotating rotating fluid. The blue dashed line gives the result of ray-tracing of a ray emitted from the left lower corner, as indicated by the blue arrow. The solid blue line displays the wave attractor and red dots correspond to coordinates of its boundary reflections.

**Figure 2.**Areas of the existence of attractors (n,1) according to Equation (7) in the $(d,\tau )$ plane. Blue triangles of (n,1) attractors are depicted over the grey scale image, where lighter grey tones correspond to greater Lyapunov exponents, defining the rate of conversion to limit trajectories as in [7].

**Figure 3.**Typical behaviour of the vertical component of velocity in the vertical $x,y$ plane with red color for positive and blue for negative values (

**left**), and oscillations of the total kinetic energy (

**right**). Dimensions correspond to the quasi-2D laboratory experiments. The saturation time is 24.8 ${T}_{0}$.

**Figure 6.**Examples of (2,1) and (3,1) attractors computed according to Equation (6). Attractor (2,1) is symmetric with respect to the central vertical line, as well as all the attractors with n even. Attractor (3,1) has rotational symmetry with period $\pi $, as all the (n,1) attractors with n odd. Dashed lines correspond to the geometries of the domain, for which all the ray analysis is also applicable, with Lyapunov exponents (9) being doubled.

**Figure 7.**Direct numerical simulation of the (1,1) attractor. On the left: the instantaneous vertical component of velocity (red is upward, blue-downward), on the right: oscillations of the total kinetic energy. The time of saturation is $96{T}_{0}$.

**Figure 8.**Direct numerical simulation of the (2,1) attractor. On the left: the instantaneous vertical component of velocity, on the right: oscillations of the total kinetic energy. The time of saturation is 154 ${T}_{0}$.

**Figure 9.**The instantaneous vertical component of velocity after direct numerical simulation of the (6,1) attractor. The time of saturation is 367 ${T}_{0}$.

**Figure 10.**Linear approximation of saturation time in ${T}_{S}$, given in terms of oscillation periods ${T}_{0}$, needed to establish a stationary wave regime of an (n,1) attractor, on number of cells, n.

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**MDPI and ACS Style**

Sibgatullin, I.; Petrov, A.; Xu, X.; Maas, L.
On (n,1) Wave Attractors: Coordinates and Saturation Time. *Symmetry* **2022**, *14*, 319.
https://doi.org/10.3390/sym14020319

**AMA Style**

Sibgatullin I, Petrov A, Xu X, Maas L.
On (n,1) Wave Attractors: Coordinates and Saturation Time. *Symmetry*. 2022; 14(2):319.
https://doi.org/10.3390/sym14020319

**Chicago/Turabian Style**

Sibgatullin, Ilias, Alexandr Petrov, Xiulin Xu, and Leo Maas.
2022. "On (n,1) Wave Attractors: Coordinates and Saturation Time" *Symmetry* 14, no. 2: 319.
https://doi.org/10.3390/sym14020319