zbMATH — the first resource for mathematics

Uni-modal destabilization of a viscoelastic floating ice layer by wind stress. (English) Zbl 1103.74025
Summary: We study a destabilization by wind stress of a homogeneous viscoelastic ice layer of infinite horizontal extent and finite thickness floating on a water layer of finite depth. The water is assumed to be weakly compressible; the viscous dissipation in the water layer is shown to be negligible compared to that in the ice layer. In the model, we assume that a homogeneous wind shear stress is applied to the upper surface of the ice layer at near field, and the compression within the ice layer is fixed below the maximum admissible value, i.e. below the value above which ice can no longer be treated as an elastic material. The effect of viscosity is shown to stabilize the acoustic mode and all the unstable seismic modes that in a purely elastic model, treated by L. Brevdo and A. Il’ichev [Cold Reg. Sci. Technol. 33, 77–89 (2001)], possess unbounded growth rates for a growing wave number. The buckling mode is unstable in the domain of parameters considered. In all the cases treated, the model is marginally absolutely stable. The localized unstable disturbances propagate against the wind. The spatially amplifying waves in the model amplify in the direction opposite to the wind direction. The stability results are well approximated by those for a Kirchoff–Love thin ice plate model.

74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74D05 Linear constitutive equations for materials with memory
74H55 Stability of dynamical problems in solid mechanics
86A40 Glaciology
Full Text: DOI
[1] Andreas, E.L., 1996. The Atmospheric Boundary Layer over Polar Marine Surfaces. Monogr. 96-2. U.S. Army Cold Reg. and Engin. Lab. Hanover, NH, 38 p
[2] Balmforth, N.J.; Craster, R.V., Ocean waves and ice sheets, J. fluid mech., 395, 89-124, (1999) · Zbl 0957.76009
[3] Bers, A., 1973. Theory of absolute and convective instabilities. In: Auer, G., Cap, F. (Eds.), International Congress on Waves and Instabilities in Plasmas, Innsbruck, Austria, pp. B1-B52
[4] Bogorodsky, V.V.; Gavrilo, V.P.; Nedoshivin, O.A., Ice destruction: methods and technology (glaciology and quaternary geology), (1987), Kluwer
[5] Brevdo, L., A study of absolute and convective instabilities with an application to the eady model, Geophys. astrophys. fluid dynam., 40, 1-92, (1988) · Zbl 0726.76041
[6] Brevdo, L., Spatially amplifying waves in plane Poiseuille flow, Z. angew. math. mech., 72, 163-174, (1992) · Zbl 0765.76023
[7] Brevdo, L., Wave packets, signalling and resonances in a homogeneous waveguide, J. elasticity, 49, 3, 201-237, (1998) · Zbl 0922.73010
[8] Brevdo, L., Resonant destabilization of a floating homogeneous ice layer, Z. angew. math. phys., 52, 397-420, (2001) · Zbl 1006.74029
[9] Brevdo, L., Neutral stability and resonant destabilization of the Earth’s crust, Proc. roy. soc. London ser. A, 457, 1951-1971, (2001) · Zbl 0993.86003
[10] Brevdo, L., Neutral stability of, and resonances in, a vertically stratified floating ice layer, Eur. J. mech. A solids, 22, 119-137, (2002) · Zbl 1107.74305
[11] Brevdo, L., A dynamical system approach to the absolute instability of spatially developing localized open flows and media, Proc. roy. soc. London ser. A, 458, 1375-1397, (2002) · Zbl 1056.76034
[12] Brevdo, L., Global and absolute instabilities of spatially developing open flows and media with algebraically decaying tails, Proc. roy. soc. London ser. A, 459, 1403-1425, (2003) · Zbl 1068.76028
[13] Brevdo, L.; Bridges, T.J., Absolute and convective instabilities of spatially periodic flows, Philos. trans. roy. soc. London ser. A, 354, 1027-1064, (1996) · Zbl 0871.76032
[14] Brevdo, L.; Il’ichev, A., Exponential neutral stability of a floating ice layer, Z. angew. math. phys., 49, 401-419, (1998) · Zbl 0909.73022
[15] Brevdo, L.; Il’ichev, A., Multi-modal destabilization of a floating ice layer by wind stress, Cold reg. sci. technol., 33, 77-89, (2001)
[16] Brevdo, L.; Il’ichev, A., Absolute instability of a thin visco-elastic plate in an air flow, Eur. J. mech. A solids, 23, 1069-1084, (2004) · Zbl 1064.74096
[17] Brevdo, L.; Laure, P.; Dias, F.; Bridges, T.J., Linear pulse structure and signalling in a film flow on an inclined plane, J. fluid mech., 396, 37-71, (1999) · Zbl 0982.76037
[18] Bridges, T.J.; Morris, P.J., Differential eigenvalue problem in which the parameter appears nonlinearly, J. comput. phys., 55, 437-460, (1984) · Zbl 0543.65063
[19] Briggs, R.J., Electron-stream interaction with plasmas, (1964), MIT Press Boston, MA
[20] Duffy, D.G., The response of floating ice to a moving, vibrating load, Cold reg. sci. technol., 20, 51-64, (1991)
[21] Duffy, D.G., On the generation of internal waves beneath sea ice by a moving load, Cold reg. sci. technol., 24, 29-39, (1996)
[22] Forbes, L.K., Surface waves of large amplitude beneath an elastic sheet. part 1. high order series expansion, J. fluid mech., 169, 409-428, (1986) · Zbl 0607.76015
[23] Forbes, L.K., Surface waves of large amplitude beneath an elastic sheet. part 2. Galerkin series expansion, J. fluid mech., 188, 491-508, (1988) · Zbl 0643.76013
[24] Fox, C.; Squire, V.A., Strain in Shore fast ice due to incoming Ocean waves and swell, J. geophys. res., 96, C3, 4531-4547, (1991)
[25] Fox, C.; Squire, V.A., On the oblique reflexion and transmission of Ocean waves at Shore fast sea ice, Philos. trans. roy. soc. London ser. A, 347, 185-218, (1994) · Zbl 0816.73009
[26] Garrison, T., Oceanography, (1996), Wadsworth Publishing Company Belmont
[27] Gavrilo, V.P.; Gusev, A.V.; Mal’kov, B.N.; Poliakov, A.P, Vnutrennee trenie v polikristallicheskom l’de v diapazone chastot 5-400 Hertz, Fisika zemli, 6, 82-85, (1971)
[28] Gottlieb, D.; Hussaini, M.Y.; Orszag, S.A., Theory and applications of spectral methods, (), 1-54
[29] Hobbs, P.V., Ice physics, (1974), Clarendon Press Oxford
[30] Il’ichev, A., Solitary waves in media with dispersion and dissipation, Fluid dynam., 35, 157-176, (2000) · Zbl 0995.76013
[31] Il’ichev, A.; Marchenko, A.V., The formation of nonlinear waveguides in the resonant interaction of three surface waves, J. appl. math. mech., 61, 183-193, (1997)
[32] Lepparanta, M., The dynamics of sea ice, (), 305-342
[33] Lifshitz, E.M.; Landau, L.D., Theory of elasticity, (1986), Pergamon Press Oxford
[34] Lingwood, R.J., Absolute instability of the boundary layer on a rotating disk, J. fluid mech., 299, 17-33, (1995) · Zbl 0868.76028
[35] Marchenko, A.V., Long waves in a shallow liquid under an ice cover, J. appl. math. mech., 52, 180-183, (1988) · Zbl 0691.76009
[36] Müller, A., Ettema, R., 1984. Dynamic response of an ice-breaker hull to ice breaking. In: Proc. IHAR Ice Symp., II, Hamburg, pp. 287-296
[37] Orszag, S.A., Accurate solution of the orr – sommerfeld stability equation, J. fluid mech., 50, 689-703, (1971) · Zbl 0237.76027
[38] Pearlstein, A.J.; Goussis, D.A., Efficient transformation of certain singular polynomial matrix eigenvalue problems, J. comput. phys., 78, 305-312, (1988) · Zbl 0665.15009
[39] Peyret, P., 1986. Introduction to Spectral Methods. Von Karman Institute Lecture Series 1986-04, Rhode-Saint Genese, Belgium
[40] Schulkes, R.M.S.M.; Hosking, R.J.; Sneyd, A.D., Waves due to a steadily moving source on a floating ice plate. part 2, J. fluid mech., 180, 297-318, (1987) · Zbl 0624.76124
[41] Squire, V.A., On the critical angle for Ocean waves entering Shore fast ice, Cold reg. sci. tech., 10, 59-68, (1984)
[42] Squire, V.A.; Hosking, R.J.; Kerr, A.D.; Langhorne, P.J., Moving loads on ice plates, (1996), Kluwer
[43] Stocker, J.J., Water waves, (1957), Wiley-Interscience
[44] Strathdee, J.; Robinson, W.H.; Haines, E.M., Moving loads on ice plates of finite thickness, J. fluid mech., 226, 37-61, (1991) · Zbl 0718.76030
[45] Tuck, E.O., An inviscid theory for sliding flexible sheets, J. austral. math. soc. B, 23, 403-415, (1982) · Zbl 0497.76029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.