# Modulation Instability of Surface Waves in the Model with the Uniform Wind Profile

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Problem

## 3. Modulation Instability of Capillary-Gravity Waves on the Tangential Discontinuity of Wind Speed

#### 3.1. Problem Formulation

#### 3.2. Derivation of Nonlinear Schrödinger Equation for Interfacial Waves

## 4. Analysis of Modulation Instability at the Air-Water Interface

#### 4.1. NLS Equation and Modulation Instability on the Lower Branch of the Dispersion Relation

#### 4.2. NLS Equation and Modulation Instability on the Upper Branch of the Dispersion Relation

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Landau, L.D.; Lifshits, E.M. Fluid Mechanics; Butterworth-Heinemann: Burlington, MA, USA, 1987. [Google Scholar]
- Fabrikant, A.L.; Stepanyants, Y.A. Propagation of Waves in Shear Flows; World Scientific Publishing Company: Singapore, 1998. [Google Scholar]
- Ostrovski, L.A.; Rybak, S.A.; Tsimring, L.S. Negative energy waves in hydrodynamics. Sov. Phys. Uspekhi
**1986**, 29, 1040. [Google Scholar] [CrossRef] - Ezersky, A.B.; Ostrovsky, L.A.; Stepanyants, Y.A. Wave-induced flows and their contribution to the energy of wave motions in a fluid. Izv. Acad. Sci. USSR Atmos. Ocean. Phys.
**1981**, 17, 890–895. [Google Scholar] - Miles, J.W. On the reflection of sound at an interface of relative motion. J. Acoust. Soc. Am.
**1957**, 29, 226–228. [Google Scholar] [CrossRef] - Ribner, H.S. Reflection, transmission and amplification of sound by a moving medium. J. Acoust. Soc. Am.
**1957**, 29, 435–441. [Google Scholar] [CrossRef] - Jeffreys, H. On the formation of water waves by wind. Proc. R. Soc. Lond. Ser. A
**1925**, 107, 189–206. [Google Scholar] - Benjamin, T.B. The threefold classification of unstable disturbances in flexible surfaces bounding inviscid flows. J. Fluid Mech.
**1963**, 16, 436–450. [Google Scholar] [CrossRef] - Gelash, A.; Agafontsev, D.; Zakharov, V.; El, G.; Randoux, S.; Suret, P. Bound state soliton gas dynamics underlying the noise-induced modulational instability. Phys. Rev. Lett.
**2019**, 123, 890–895. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ablowitz, M.J.; Segur, H. On the evolution of packets of water waves. J. Fluid Mech.
**1979**, 92, 691–715. [Google Scholar] [CrossRef] - Djordjevic, V.D.; Redekopp, L.G. On two-dimensional packets of capillary–gravity waves. J. Fluid Mech.
**1977**, 79, 703–714. [Google Scholar] [CrossRef] - Cairns, R.A. The role of negative energy waves in some instabilities of parallel flows. J. Fluid Mech.
**1979**, 92, 1–14. [Google Scholar] [CrossRef] - Nezlin, M.V. Negative-energy waves and the anomalous Doppler effect. Sov. Phys. Uspekhi
**1976**, 19, 946. [Google Scholar] [CrossRef] - Abourabia, A.M.; Mahmoud, M.A.; Khedr, G.M. Solutions of nonlinear Schrödinger equation for interfacial waves propagating between two ideal fluids. Can. J. Phys.
**2009**, 87, 675–684. [Google Scholar] [CrossRef] - Peregrine, D.H. Water waves, nonlinear Schrödinger equations and their solutions. Anziam J.
**1983**, 25, 16–43. [Google Scholar] [CrossRef] [Green Version] - Stiassnie, M. Note on the modified nonlinear Schrödinger equation for deep water waves. Wave Motion
**1984**, 6, 431–433. [Google Scholar] [CrossRef] - Yuen, H.C.; Lake, B.M. Nonlinear dynamics of deep-water gravity waves. In Advances in Applied Mechanics; Elsevier: Amsterdam, The Netherlands, 1982; Volume 22, pp. 67–229. [Google Scholar]
- Ablowitz, M.J.; Segur, H. Solitons and the Inverse Scattering Transform; SIAM: Philadelphia, PA, USA, 1981. [Google Scholar]
- Lighthill, M.J. Contributions to the theory of waves in nonlinear dispersive systems. IMA J. Appl. Math.
**1965**, 1, 269–306. [Google Scholar] [CrossRef] - Lighthill, M.J. Waves in Fluids; Cambridge University Press: Cambridge, UK, 1978. [Google Scholar]
- Ostrovsky, L.A.; Potapov, A.I. Modulated Waves: Theory and Applications; The Johns Hopkins University Press: London, UK, 1999. [Google Scholar]
- Kharif, C.; Pelinovsky, E.; Slunyaev, A. Rogue Waves in the Ocean; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Slunyaev, A.; Sergeeva, A.; Pelinovsky, E. Wave amplification in the framework of forced nonlinear Schrödinger equation: The rogue wave context. Phys. D
**2015**, 303, 18–27. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Sketch of fluid flow in two layers of infinitely deep fluids with an immovable lower layer and tangential discontinuity of velocity in upper layers.

**Figure 2.**(Color online.) (

**a**) Dispersion relations for water waves in the cases when U = 0 (line 1) and U = U

_{c1}= 6.676 m/s (line 2); (

**b**) magnified fragments of the dispersion relations in the vicinity of the critical point at different velocities: U = 6.679 m/s (line 3), U = U

_{KH}= 6.68 m/s (line 4), and U = 6.7 m/s (line 5). The figures are plotted for a = 0.0012 and T = 0.073 N/m.

**Figure 3.**(Color online.) Dispersion curves of the KH model. Lines 1 and 1${}^{\prime}$ pertain to the case when $U=0$; lines 2 and 2${}^{\prime}$ to ${U}_{c1}$; and lines 3 and 3${}^{\prime}$ to $U>{U}_{c1}$. The dashed portion of line 3 corresponds to NEWs, and its counterpart with $\omega >0$ is shown by the solid line in the right half-plane for $k>0$.

**Figure 4.**(Color online.) Dispersion curves for waves at the air–water interface in the ‘mean-mass reference frame’ with (

**a**) U = 0.275 m/s and (

**b**) U = 6.679 m/s. The portions of the dispersion curves corresponding to the retarded waves that can be a subject of instability and are highlighted in red.

**Figure 5.**(Color online.) The dispersion coefficient $P\left(k\right)$ in the NLS equation as a function of the wavenumber k of the lower branch of the dispersion relation for three different values of the speed of the upper layer fluid with $U=0$ (line 1), $U={U}_{c1}=6.676$ m/s (line 2), and $U={U}_{KH}=6.68$ m/s (line 3).

**Figure 6.**(Color online.) The nonlinear coefficient $Q\left(k\right)$ in the NLS equation as a function of the wavenumber k for the lower branch of the dispersion relation for three different values of the speed of the upper layer fluid with $U=0$ (line 1), $U={U}_{c1}=6.676$ m/s (line 2), and $U={U}_{KH}=6.68$ m/s (line 3). Lines 2 and 3 are practically indistinguishable on the left of the dashed vertical line.

**Figure 7.**Zones of modulation stability (S) and instability ($US$) in the $(k,U)$ plane. The dashed line on the top represents the critical velocity ${U}_{KH}=6.681$ m/s when the KH instability arises. The bifurcation points in the diagram are denoted by ${B}_{1}$ and ${B}_{2}$ for ${U}_{m}=3.84$ m/s, and two other bifurcation points ${B}_{3}$ and ${B}_{4}$ are shown for higher values of U.

**Figure 8.**(Color online.) The dispersion coefficient $P\left(k\right)$ in the NLS equation as a function of the wavenumber k for the upper branch of the dispersion relation for three different values of the speed of upper layer fluid with $U=0$ (line 1), $U={U}_{c1}=6.676$ m/s (line 2), and $U={U}_{KH}=6.68$ m/s (line 3).

**Figure 9.**(Color online.) The nonlinear coefficient $Q\left(k\right)$ in the NLS equation as a function of the wavenumber k for the upper branch of the dispersion relation at three different values of the speed of upper layer fluid, $U=0$ (line 1), $U={U}_{c1}=6.676$ m/s (line 2), and $U={U}_{KH}=6.68$ m/s (line 3). Lines 2 and 3 are practically indistinguishable on the left of the dashed vertical line.

**Figure 10.**Zones of modulation stability (S) and instability ($US$) in the $(k,U)$ plane. The dashed line on the top represents the critical velocity ${U}_{KH}=6.681$ m/s when the KH instability arises. The threshold value of ${U}_{c2}=0.23$ m/s, at which NEWs arise, is shown by the lower dashed line. The bifurcation points for ${U}_{m}=4.32$ m/s are marked by ${B}_{1}$, ${B}_{2}$ in the diagram. Two other bifurcation points, ${B}_{3}$ and ${B}_{4}$, are shown for the greater values of U.

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**

Boral, S.; Sahoo, T.; Stepanyants, Y.
Modulation Instability of Surface Waves in the Model with the Uniform Wind Profile. *Symmetry* **2021**, *13*, 651.
https://doi.org/10.3390/sym13040651

**AMA Style**

Boral S, Sahoo T, Stepanyants Y.
Modulation Instability of Surface Waves in the Model with the Uniform Wind Profile. *Symmetry*. 2021; 13(4):651.
https://doi.org/10.3390/sym13040651

**Chicago/Turabian Style**

Boral, Susam, Trilochan Sahoo, and Yury Stepanyants.
2021. "Modulation Instability of Surface Waves in the Model with the Uniform Wind Profile" *Symmetry* 13, no. 4: 651.
https://doi.org/10.3390/sym13040651