# Energy Spectra of Ensemble of Nonlinear Capillary Waves on a Fluid

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{0}, k

_{1}), where energy input and dissipation are balanced; (V) locality of interactions in k-space (only waves with wavelengths of the same order do interact; (VI) interactions are locally isotropic (no dependence on direction); (VII) at initial moment energy is distributed among an infinite number of modes. Under these and other assumptions, wave kinetic equations have stationary solutions in the form of energy power spectra E

_{k}~k

^{−}

^{ν}, ν > 0 [11], with k being the length of the wave vector

**k**. These spectra are called kinetic spectra or K-spectra.

## 2. Three-Wave Interactions of Capillary Waves

#### 2.1. Space Dimension and Wavevector’s Coordinates

#### 2.2. Resonance Width

#### 2.3. Wavenumbers and Angles

#### 2.4. Norms of Wavevectors

#### 2.5. Resonance Curves

**k**∈ R can be produced by stretch and rotation of our unit vector, and its resonance curve is obtained by stretching the curve of the unit vector (with the same coefficient) and rotation (by the same angle). If two vectors interact resonantly, then each of them lies on the resonance curve of another (Figure 4b). We may conclude with confidence that conditions for a three-wave kinetic regime to occur are decidedly violated.

## 3. Dynamic Energy Cascade of Capillary Waves

## 4. Discussion

## 5. Conclusions

- (1)
- A K-cascade among capillary waves cannot be formed by three-wave resonant interactions; four-wave resonant interactions should be regarded instead. Accordingly, a K-cascade of capillary waves is formed at the time scale $1/{\mathsf{\epsilon}}^{4}$ with $\mathsf{\epsilon}\text{}~\text{}{10}^{-2}$.
- (2)
- (3)
- The absence of three-wave resonances of capillary waves has also been noticed in numerical simulations [19,20] and was coined by the term “frozen turbulence”. This has been attributed to the interplay of two facts: the discretization of the numerical scheme and the absence of exact three-wave resonances among capillary waves with integer wave numbers, as first proven in [14]. It was observed that capillary waves demonstrate “fluxless modes, there is virtually no energy absorption associated with high-wavenumbers damping in this case” ([20], p. 107).
- (4)
- Speaking very generally, if dispersion function $\hspace{0.17em}\mathsf{\omega}(k)$ has a decay type, this only means that three-wave resonance conditions$$\hspace{0.17em}\mathsf{\omega}({k}_{1})+\mathsf{\omega}({k}_{2})=\mathsf{\omega}({k}_{3}),\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{k}_{1}+{k}_{2}={k}_{3}$$

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Ermakov, S.A.; Pelinovsky, E.N. Variation of the spectrum of wind ripple on coastal waters under the action of internal waves. Dyn. Atmos. Oceans
**1984**, 8, 95–100. [Google Scholar] [CrossRef] - Pelinovsky, E.; Talanov, V. Ocean Subsurface Layer: Physical Processes and Remote Sensing; Institute of Applied Physics Press: Nizhny Novgorod, Russia, 1999. [Google Scholar]
- Jackson, C.R. An Atlas of Internal Solitary-like Waves and Their Properties; Global Ocean Associates: Alexandria, VA, USA, 2004. [Google Scholar]
- Shats, M.; Punzmann, H.; Xia, H. Capillary rogue waves. Phys. Rev. Lett.
**2010**, 104, 104503. [Google Scholar] [CrossRef] [PubMed] - Abdurakhimov, L.V.; Brazhnikov, M.Y.; Kolmakov, G.V.; Levchenko, A.A. Study of high-frequency edge of turbulent cascade on the surface of He-II. J. Phys. Conf. Ser.
**2009**, 150, 032001. [Google Scholar] [CrossRef] - Xia, H.; Shats, M.; Punzmann, H. Modulation instability and capillary wave turbulence. EPL
**2010**, 91, 14002. [Google Scholar] [CrossRef][Green Version] - Deike, L.; Berhanu, M.; Falcon, E. Decay of capillary wave turbulence. Phys. Rev. E
**2012**, 85, 066311. [Google Scholar] [CrossRef][Green Version] - Slavchov, R.I.; Peychev, B.; Ismail, A.S. Chracterization of capiullary waves: A review and a new optical method. Phys. Fluids
**2021**, 33, 101303. [Google Scholar] [CrossRef] - Falcon, F.; Mordant, N. Experiments in surface gravity-capillary wave turbulence. Annu. Rev. Fluid Mech. Annu. Rev.
**2022**, 54, 1–25. [Google Scholar] [CrossRef] - Zakharov, V.E.; Filonenko, N.N. Weak turbulence of capillary waves. J. Appl. Mech. Tech. Phys.
**1967**, 4, 500. [Google Scholar] [CrossRef] - Zakharov, V.E.; L’Vov, V.S.; Falkovich, G. Kolmogorov Spectra of Turbulence; Springer: Berlin/Heidelberg, Germany; New York, NY, USA, 1992. [Google Scholar]
- Kartashova, E. Nonlinear Resonance Analysis; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Kartashova, E.; Shugan, I.V. Dynamical cascade generation as a basic mechanism of Benjamin-Feir instability. EPL
**2011**, 95, 30003. [Google Scholar] [CrossRef] - Kartashova, E. Energy spectra of 2D gravity and capillary waves with narrow frequency band excitation. EPL
**2012**, 97, 30004. [Google Scholar] [CrossRef] - Kartashova, E. Partitioning of ensembles of weakly interacting dispersing waves in resonators into disjoint classes. Phys. D
**1990**, 46, 43. [Google Scholar] [CrossRef] - Kartashova, E. Discrete wave turbulence. EPL
**2009**, 87, 44001. [Google Scholar] [CrossRef][Green Version] - Zakharov, V.E.; Korotkevich, A.O.; Pushkarev, A.N.; Dyachenko, A.I. Mesoscopic wave turbulence. JETP Lett.
**2005**, 82, 487–491. [Google Scholar] [CrossRef] - Kartashova, E. Time scales and structures of wave interaction exemplified with water waves. EPL
**2013**, 102, 44005. [Google Scholar] [CrossRef][Green Version] - Pushkarev, A.N. On the Kolmogorov and frozen turbulence in numerical simulation of capillary waves. Eur. J. Mech. B/Fluids
**1999**, 18, 345. [Google Scholar] [CrossRef] - Pushkarev, A.N.; Zakharov, V.E. Turbulence of capillary waves—theory and numerical simulation. Phys. D
**2000**, 135, 98–116. [Google Scholar] [CrossRef]

**Figure 1.**Two-dimensional wavevectors ${k}_{1},\hspace{0.17em}{k}_{2}$ satisfying Equation (2). The X- and Y-coordinate axes denote the coordinates of the wave vector m and n, respectively, so each vector is represented by a point on the plane (m, n). (

**a**) Wavevectors with non-negative coordinates that interact with vectors with arbitrary (positive or negative) coordinates. Computation domain $-50<{m}_{1},{n}_{1}<50$; (

**b**) All wavevectors interacting with those shown in the previous panel; same computation domain; (

**c**) Both wavevectors have non-negative coordinates. Computation domain $0<{m}_{1},{n}_{1}<100.$ Wavevectors from the lower triangle interact only with vectors from the upper triangle and vice versa.

**Figure 2.**Dependence of the ratio of interacting vectors’ norms (longer to shorter) on the angle between vectors. (

**a**) Complete data in computation domain $0\le {m}_{j},{n}_{j}\le 100$ are presented; (

**b**) Zoomed presentation of the initial interval (ratio ≤ 10) of the left panel (

**a**). Axes X and Y denote angles (in grad) and ratios correspondingly.

**Figure 3.**Resonance curve of vector (0, 1) in k-space for dispersion function $\mathsf{\omega}\text{}~\text{}{k}^{3/2}$. (

**a**) The initial segment of the curve: $m<<1\Rightarrow n\text{}~\text{}{m}^{3/2}$; (

**b**) The overall view of the curve for $m>>1\Rightarrow n\text{}~\text{}{m}^{1/2}$. The X- and Y-coordinate axes denote the coordinates of the wave vector m and n, respectively, so each vector is represented by a point on the plane (m, n).

**Figure 4.**Resonance curves in k-space (schematic representation). (

**a**) For the vector ${k}_{1}=(0,\hspace{0.17em}1)$, all vectors $\hspace{0.17em}{k}_{2}$ lie on the interaction curve shown; (

**b**) Two interacting vectors lie on each other’s resonance curves reciprocally. The resonance curve of the rotated vector is shown by the dashed line.

**Figure 5.**Shapes of functions $\hspace{0.17em}{\mathsf{\gamma}}_{1}\cdot {b}^{-x}$ and $\hspace{0.17em}{\mathsf{\gamma}}_{2}\cdot {x}^{-a}$ for various choices of parameters $\hspace{0.17em}a,\hspace{0.17em}b,\hspace{0.17em}{\mathsf{\gamma}}_{1},\hspace{0.17em}{\mathsf{\gamma}}_{2}$. In both panels, function $\hspace{0.17em}{x}^{-1.5}$ is shown by bold black line, and function $\hspace{0.17em}{b}^{-x}$ is shown by dashed lines of various colors. (

**a**) Function $\hspace{0.17em}{b}^{-x}$ is shown for b = 1.4, 1.6 and 2.3. (

**b**) Zoomed presentation of the interaction point of the functions $\hspace{0.17em}{x}^{-1.5}$ and $\hspace{0.17em}{1.6}^{-x}$.

Property | $\hspace{0.17em}{\mathit{E}}_{\mathit{k}}$ | $\hspace{0.17em}{\mathit{E}}_{\mathit{n}}$ |
---|---|---|

Coherent phases | no | yes |

Dependence on the excitation parameters | no | yes |

Local interactions | yes | no |

Existence of inertial interval | yes | not important |

Small parameter | $~{10}^{-2}$ | $~{10}^{-1}$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Tobisch, E.; Kartashov, A.
Energy Spectra of Ensemble of Nonlinear Capillary Waves on a Fluid. *J. Mar. Sci. Eng.* **2021**, *9*, 1422.
https://doi.org/10.3390/jmse9121422

**AMA Style**

Tobisch E, Kartashov A.
Energy Spectra of Ensemble of Nonlinear Capillary Waves on a Fluid. *Journal of Marine Science and Engineering*. 2021; 9(12):1422.
https://doi.org/10.3390/jmse9121422

**Chicago/Turabian Style**

Tobisch, Elena, and Alexey Kartashov.
2021. "Energy Spectra of Ensemble of Nonlinear Capillary Waves on a Fluid" *Journal of Marine Science and Engineering* 9, no. 12: 1422.
https://doi.org/10.3390/jmse9121422