Gravity Wave Generated by Initial Axisymmetric Disturbance at the Surface of an Ice-covered Ocean with Porous Bed

Abstract

This paper is concerned with the generation of gravity waves due to prescribed initial axisymmetric disturbances created at the surface of an ice sheet covering the ocean with a porous bottom. The ice cover is modeled as a thin elastic plate, and the bottom porosity is described by a real parameter. Using linear theory, the problem is formulated as an initial value problem for the velocity potential describing the motion. In the mathematical analysis, the Laplace and Hankel transform techniques have been used to obtain the depression of the ice-covered surface in the form of a multiple infinite integral. This integral is evaluated asymptotically by the method of stationary phase twice for a long time and a large distance from the origin. Simple numerical computations are performed to illustrate the effect of the ice-covered surface and bottom porosity on the surface elevation, phase velocity, and group velocity of the surface gravity waves. Mainly the far-field behavior of the progressive waves is observed in two different cases, namely initial depression and an impulse concentrated at the origin. From graphical representations, it is clearly visible that the presence of ice cover and a porous bottom decreases the wave amplitude. Due to the porous bottom, the amplitude of phase velocity decreases, whereas the amplitude of group velocity increases.

Keywords

Axisymmetric disturbances and ice-covered ocean / Porous bottom / Laplace and Hankel transform / Method of stationary phase / Free surface depression / Phase velocity and group velocity

References

[1] Banerjea S, Rakshit P, Maiti P. On the waves generated due to a line source present in an ocean with an ice cover and a small bottom undulation. Fluid Dyn Res, 2011, 43(2): 384-410

[2] Bhattacharjee J, Sahoo T. Flexural gravity wave generation by initial disturbances in the presence of current. J Marine Sci Tech, 2008, 13: 138-146

[3] Chung H, Fox C. Calculation of wave-ice interaction using the Wiener-Hopf technique. New Zealand J Math, 2002, 31(1): 1-18

[4] Das L, Mohapatra S. Effects of flexible bottom on radiation of water waves by a sphere submerged beneath an ice-cover. Meccanica, 2019, 54: 985-999

[5] Fox C, Squire VA (1994) On the oblique reflexion and transmission of ocean waves at shore fast sea-ice. Phil Trans R Soc Lond, A 347;185–218.  https://doi.org/10.1098/rsta.1994.0044 .

[6] Gayen R, Islam N. Effect of a floating elastic plate/membrane on the motion due to a ring source in water with porous bed Indian. J Pure Appl Math, 2018, 49(2): 239-256

[7] Khuntia S, Mohapatra S. Effects of ice-floe on surface wave interaction with an irregular flexible seabed. European J Mech, 2020, 84: 357-366

[8] Kranzer HC, Keller JB. Water waves produced by explosions. J of Appl Phys, 1959, 30(3): 398-407

[9] Kundu P, Mandal BN. Generation of surface waves due to initial axisymmetric surface disturbance in water with a porous bottom. Int J of Appl Mech and Eng, 2019, 24: 625-644

[10] Lamb H. Hydrodynamics, 1945, NewYork: Dover, 351-469

[11] Maiti P, Mandal BN. Water waves generated due to initial axisymmetric disturbances in water with an ice cover. Arch Appl Mech, 2005, 74: 629-636

[12] Maiti P, Mandal BN. Water wave scattering by an elastic plate floating in an ocean with a porous bed. Appl Ocean Res, 2014, 47: 73-84

[13] Mandal BN, Mukherjee S. Water waves generated at an inertial surface by an axisymmetric initial surface disturbance. Int J Math Educ Sci Technol, 1989, 20: 743-747

[14] Meylan MH, Squire VA. Response of a circular ice floe to ocean waves. J Geophys Res, 1996, 101: 8869-8884

[15] Mohapatra S. Scattering of surface waves by the edge of a small undulation on a porous bed in an ocean with ice-cover. J Marine Sci Appl, 2014, 13: 167-172

[16] Mohapatra S, Bora SN. Propagation of oblique waves over small bottom undulation in an ice-covered two-layer fluid. Geo Phy Astro Phy Fluid Dyn, 2009, 103(5): 347-374

[17] Paul S, De S. Wave scattering by porous bottom undulation in a two layered channel. J Marine Sci and Appl, 2014, 13(4): 355-361

[18] Paul S, De S. Wave scattering by uneven porous bottom in a three layered channel. J Marine Sci and Tech, 2017, 22(3): 533-545

[19] Rhodes-Robinson PF. On the generation of water waves at an inertial surface. J Austral Math Soc Ser b, 1984, 25: 366-383

[20] Squire V. Review of ocean waves and sea-ice revisited. Cold Regions Sci Tech, 2007, 49(2): 110-133

[21] Stoker JJ (1957) Water waves, the mathematical theory with applications. Pure and Applied Mathematics. Interscience Publishers, New York, Vol IV, 149–196.

[22] Sturova IV. Unsteady three-dimensional sources in deep water with elastic cover and their application. J Fluid Mech, 2013, 730: 392-418

[23] Tsai C, Chen H, Lee F. Wave transformation over submerged permeable breakwater on porous bottom. Ocean Eng, 2006, 33: 1623-1643

[24] Weitz M, Keller JB. Reflection of water waves floating ice in water of finite depth. Comm Pure Appl Math, 1950, 3: 305-318

[25] Wen SL. A note on water waves created by surface disturbances. Int J Sci and Math Edu, 1982, 13: 55-58