# Detuned Resonances

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## Abstract

**:**

## 1. Introduction

**NR**-diagram [9]. An

**NR**-diagram allows one to represent uniquely the dynamical system on ${A}_{\mathbf{k}}\left(T\right)$ which describes the time evolution of each cluster at the slow time scale $T$. In this context, $t$ is often referred to as the linear (or fast) time. All polynomial conservation laws can be written out explicitly for an arbitrary cluster [10]. An important example of geophysical application of this theoretical approach can be found in [11].

**NR**- diagram (which provides a one-to-one correspondence between a cluster and its dynamical system) but a topological presentation of a cluster [9]. The topological presentation of a cluster has been introduced for constructing all the isolated components of a given set of triads. It gives no information about cluster dynamics; the dynamical system describing a cluster cannot be derived from its topological presentation. This drawback is of course known to researchers in the area [14]: “In order to better understand these issues, we believe that it is important to move beyond the kinematic picture of resonance broadening and attempt to devise methods of studying these effects dynamically”. A first step in this direction has been taken in [15], where the dynamics of a particular cluster formed by two triads with two joint modes is studied while investigating the case “the linear wave time scales are comparable to the time scales of nonlinear oscillations”. In the pioneering work of [20], the study of dynamic equations for resonances in the famous Fermi–Pasta–Ulam problem made it possible to establish that the first nontrivial resonances correspond to six-wave interactions and thus to determine the time scale for establishing equipartition.

## 2. Notion of a Detuned Triad

## 3. The Choice of Detuning

#### 3.1. Detuning on a Lattice

#### 3.2. Detuning Outside the Lattice

## 4. Zero-Frequency Mode

## 5. Clusters of Detuned Triads

## 6. Brief Discussion and Conclusions

**NR**-diagrams, etc., is quite developed, the theory of approximate resonances is only in its infancy. To the best of our knowledge, this is the first attempt to introduce some systematic terminology to the field and give illustrative examples confirming the importance of understanding what resonance detuning is, why it cannot be arbitrary, what properties of a dynamic system cannot be attributed to the original PDE, etc. Both exact and detuned resonances play important roles—at different time scales—in energy transport in (weakly) nonlinear dispersive wave systems. They may be studied within at least three different mathematical frameworks: (a) the initial PDE (full evolution equation), with chosen boundary conditions; (b) wavenumber resonance conditions (kinematics); (c) a dynamical system deduced as a reduction of the initial PDE (dynamics).

- 1
- Detuning cannot be arbitrary small.

- 2
- Detuned and exact resonances are not necessarily close in the Fourier space.

- 3
- Zero-frequency modes participate in detuned resonances.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Color online. Triads on an integer lattice. Each mode has an integer length in both directions, $\mathbf{k}=\left({k}_{x},{k}_{y}\right)=\left(m,n\right)$. Each triad satisfies ${\mathbf{k}}_{1}+{\mathbf{k}}_{2}={\mathbf{k}}_{3}$ and therefore forms a triangle (the location of the triangle on this grid has no significance). The triads shown represent a resonant triad (black, solid) and a quasi-resonant triad (blue, dashed) in its one-vicinity (also called a near-resonance); a general detuned triad (red, dashed); and a zonal triad (teal, dot-dashed), which is a (detuned) triad that includes a zonal mode (has ${k}_{x}=m=0$ and is therefore vertical).

**Figure 2.**Color online. Scatter plot showing, for each detuned triad, the detuning $\delta $ versus the distance $d$ to the closest resonant triad. The color of each point indicates ${n}_{max}$, the largest $n$ found amongst all the six modes in the detuned triad and its closest resonant triad. The plot can be thought of as a layer of each color, with layers corresponding to low ${n}_{max}$, appearing in front of layers corresponding to high ${n}_{max}$. The number of layers is limited by the domain size $N=21$.

**Figure 3.**Color online. A resonant triad (black, solid) is shown, as in Figure 1, and an off-grid near-triad (violet, dashed) in its $\delta $—neighborhood (yellow disc, hatched); the two labelled modes do not lie on the lattice, i.e., do not satisfy the original system of PDE and boundary conditions.

**Figure 4.**Color online. The fluctuation range of the energy ${E}_{3}$ for the triad $\left(4,12\right)\left(5,14\right)\left(9,13\right)$ which is an isolated resonant train in the truncation $N=21$, [11]: (

**a**) as a function of (artificial) detuning $\delta $ and initial phase $\varphi $ (at time $t=0$; two $2\pi $ periods in $\varphi \left(0\right)$ are shown); (

**b**) as a function of $\delta $ at $\varphi \left(0\right)=0$. The initial energies are ${E}_{1}\left(0\right)=0.4,{E}_{2}\left(0\right)=0.4,{E}_{3}\left(0\right)=0.2,$ which is case (

**b**) of [16].

**Figure 5.**Color online. Triads on an integer lattice 2. Here, each lattice point represents a particular mode $({k}_{x},{k}_{y})=\left(m,n\right)$. The resonant triad (black, solid) and the zonal triad (teal, dot-dashed) from Figure 1 are shown. The zonal mode has ${k}_{x}=m=0$ and therefore lies on the ${k}_{y}$ axis.

**Figure 6.**Color online. Each panel shows the time evolution of (top) the amplitude of the modes in the resonant triad; (middle) the amplitude of the modes in the detuned triad; (bottom) the energy E of the resonant triad, the energy F of the detuned triad and the total energy. The panels are as follows: (

**a**) uncoupled with physical detuning $\delta =0.000396825$; (

**b**) coupled with physical detuning; (

**c**) uncoupled with zero detuning; (

**d**) coupled with zero detuning.

**Figure 7.**Color online. Time evolution of the amplitude (

**top**) and energies (

**bottom**) of the modes in the zonal triad (0; 2); (1; 3) $\to $ (1; 4) with physical detuning $\delta =0.033333333$ and $Z=7.82.$

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**MDPI and ACS Style**

Colyer, G.; Asahi, Y.; Tobisch, E.
Detuned Resonances. *Fluids* **2022**, *7*, 297.
https://doi.org/10.3390/fluids7090297

**AMA Style**

Colyer G, Asahi Y, Tobisch E.
Detuned Resonances. *Fluids*. 2022; 7(9):297.
https://doi.org/10.3390/fluids7090297

**Chicago/Turabian Style**

Colyer, Greg, Yuuichi Asahi, and Elena Tobisch.
2022. "Detuned Resonances" *Fluids* 7, no. 9: 297.
https://doi.org/10.3390/fluids7090297