Physical parameters of solitary wave packets in shallow basins under ice cover

A. T. Il'ichev

Teoretičeskaâ i matematičeskaâ fizika, Tome 201 (2019) no. 3, pp. 347-360

Résumé

We determine the velocities and lengths of solitary envelope waves whose velocity is located in a left half-neighborhood of the phase velocity minimum in the dispersion relation for shallow basins under ice cover. The ice cover is modeled as an elastic Kirchhoff–Love ice plate. The Euler equation for the liquid layer (water) includes an additional pressure from the plate, which floats freely on the liquid surface. We consider the case of weakly nonlinear waves in the limit of long wavelengths and small amplitudes where the initial dimensionless stress in the ice cover does not exceed one third. These waves are described by a fifth-order Kawahara equation. We then compare the obtained results with the parameters found using a strongly nonlinear description. The comparison yields very good results for shallow depths of the considered basin. This phenomenon is explained by the properties of the lowest nonlinearity coefficient in the equations describing the solitary envelope waves branching from the phase velocity minimum on the dispersion curve. We discuss possible applications of the obtained results to experimental wave measurements under an ice cover.

Détail

Keywords: ice cover, solitary envelope wave, nonlinear Schrödinger equation
Mots-clés : bifurcation, equation in quasinormal form.

Bibliographie

[1] A. V. Marchenko, “O dlinnykh volnakh v melkoi zhidkosti pod ledyanym pokrovom”, PMM, 52:2 (1988), 230–234 | DOI | Zbl

[2] R. Grimshaw, B. Malomed, E. Benilov, “Solitary waves with damped oscillatory tails: an analysis of the fifth-order Korteweg–de Vries equation”, Phys. D, 77:4 (1994), 473–485 | DOI | MR

[3] A. Korobkin, E. I. Părău, J.-M. Vanden-Broeck, “The mathematical challenges and modelling of hydroelasticity”, Phil. Trans. Roy. Soc. London Ser. A, 369:1947 (2011), 2803–2812 | DOI | MR

[4] L. K. Forbes, “Surface waves of large amplitude beneath an elastic sheet. Part 1. High-order series solution”, J. Fluid Mech., 169 (1986), 409–428 | DOI | MR

[5] L. K. Forbes, “Surface waves of large amplitude beneath an elastic sheet. Part 2. Galerkin solutions”, J. Fluid Mech., 188 (1988), 491–508 | DOI | MR

[6] G. Iooss, M. Adelmeyer, Topics in Bifurcation Theory and Applications, Advanced Series in Nonlinear Dynamics, 3, World Sci., Singapore, 1992 | DOI | MR

[7] K. Kirchgässner, “Wave-solutions of reversible systems and applications”, J. Diff. Equ., 45:1 (1982), 113–127 | DOI | MR

[8] A. Mielke, “Reduction of quasilinear elliptic equations in cylindrical domains with applications”, Math. Methods Appl. Sci., 10:1 (1988), 51–66 | DOI | MR

[9] A. T. Ilichev, “Uedinennye volny s dispersiei i dissipatsiei (obzor)”, Izv. RAN. MZhG, 2000, no. 2, 3–27 | MR

[10] E. I. Părău, F. Dias, “Nonlinear effects in the response of a floating ice plate to a moving load”, J. Fluid Mech., 460 (2002), 281–305 | DOI | MR

[11] P. A. Milewskii, J.-M. Vanden-Broeck, Z. Wang, “Hydroelastic solitary waves in deep water”, J. Fluid Mech., 2011, 628–640 | DOI | MR

[12] J.-M. Vanden-Broeck, E. I. Părău, “Two-dimensional generalized solitary waves and periodic waves under an ice sheet”, Phil. Trans. Roy. Soc. London Ser. A, 369:1947 (2011), 2957–2972 | DOI | MR

[13] E. I. Părău, J.-M. Vanden-Broeck, “Three-dimensional waves beneath an ice sheet due to a steadily moving pressure”, Phil. Trans. Roy. Soc. London Ser. A, 369:1947 (2011), 2973–2988 | DOI | MR

[14] A. T. Ilichev, “Solitonopodobnye struktury na poverkhnosti razdela voda–led”, UMN, 70:6(426) (2015), 85–138 | DOI | DOI | MR | Zbl

[15] P. I. Plotnikov, J. F. Toland, “Modelling nonlinear hydroelastic waves”, Phil. Trans. Roy. Soc. London Ser. A, 369:1947 (2011), 2942–2956 | DOI | MR

[16] A. Müller, R. Ettema, “Dynamic response of an ice-breaker hull to ice breaking”, Proceedings of the 7th IAHR International Symposium on Ice, v. II, Hamburg, W. Germany, 1984, 287–296

[17] P. Guyenne, E. I. Părău, “Computations of fully nonlinear hydroelastic solitary waves on deep water”, J. Fluid Mech., 713 (2012), 307–329 | DOI | MR

[18] Z. Wang, J.-M. Vanden-Boeck, P. A. Milevski, “Two-dimensional flexural-gravity waves of finite amplitude in deep water”, IMA J. Appl. Math., 78:4 (2013), 750–761 | DOI | MR

[19] P. Guyenne, E. I. Părău, “Finite-depth effects on solitary waves in a floating ice sheet”, J. Fluids Struct., 49 (2014), 242–262 | DOI

[20] Z. Wang, E. I. Părău, P. A. Milewski, J.-M. Vanden-Broeck, “Numerical study of interfacial solitary waves propagating under an elastic sheet”, Proc. Roy. Soc. London Ser. A, 470:2168 (2014), 20140111, 17 pp. | DOI | MR

[21] E. I. Părău, “Solitary interfacial hydroelastic waves”, Phil. Trans. Roy. Soc. A, 376:2111 (2017), 20170099, 11 pp. | DOI

[22] T. Gao, J.-M. Vanden-Broeck, Z. Wang, “Numerical computations of two-dimensional flexural-gravity solitary waves on water of arbitrary depth”, IMA J. Appl. Math., 83:3, 436–450 | DOI | MR

[23] O. Trichtchenko, E. I. Părău, J.-M. Vanden-Broeck, P. A. Milewskii, “Solitary flexural, gravity waves in three dimensions”, Phil. Trans. Roy. Soc. A, 376:2129 (2018), 20170345, 14 pp. | DOI | MR

[24] A. T. Il'ichev, V. Ja. Tomashpolskii, “Characteristic parameters of nonlinear surface envelope waves beneath an ice cover under pre-stress”, Wave Motion, 86 (2019), 11–20 | DOI | MR

[25] T. Kawahara, “Oscillatory solitary waves in dispersive media”, J. Phys. Soc. Japan, 33 (1972), 260–264 | DOI

[26] F. Dias, E. A. Kuznetsov, “On the nonlinear stability of solitary wave solutions of the fifth-order Korteweg–de Vries equation”, Phys. Lett. A, 263:1–2 (1999), 98–104 | DOI | MR

[27] G. Iooss, M. C. Pérouème, “Perturbed homoclinic solutions in reversible 1:1 resonance vector fields”, J. Differ. Equ., 102:1 (1993), 62–88 | DOI | MR

[28] F. Dias, G. Iooss, “Capillary-gravity solitary waves with damped oscillations”, Phys. D, 65:4 (1993), 399–423 | DOI | MR