Three Dimensional Fully Localized Waves on Ice-covered Ocean

We have recently shown [1] that fully-localized threedimensional wave envelopes (so-called dromions) can exist and propagate on the surface of ice-covered waters. Here we show that the inertia of the ice can play an important role in the size, direction and speed of propagation of these structures. We use multiple-scale perturbation technique to derive governing equations for the weakly nonlinear envelope of monochromatic waves propagating over the ice-covered seas. We show that the governing equations simplify to a coupled set of one equation for the envelope amplitude and one equation for the underlying mean current. This set of nonlinear equations can be further simplified to fall in the category of Davey-Stewartson equations [2]. We then use a numerical scheme initialized with the analytical dromion solution of DSI (i.e. shallow-water and surface-tension dominated regimes of Davey-Stewartson equation) to look for dromion solution of our equations. Dromions can travel over long distances and can transport mass, momentum and energy from the ice-edge deep into the solid ice-cover that can result in the ice cracking/breaking and also in posing dangers to icebreaker ships. INTRODUCTION Two-dimensional solitary waves were first observed by John Scott Russell [3, 4]. About half a century later Korteweg and de Veries derived the nonlinear governing equations and found analytical form of 2D solitary waves. The profile of a twodimensional solitary wavesimilar to the one Russell observeddecays exponentially fast in all horizontal directions except along a ray. Later it was shown that governing equations for two-dimensional weakly nonlinear envelope of monochromatic waves reduces to the Nonlinear Schroedinger equation [5] and it too admits soliton solutions. Extension of KdV equation to three-dimension is obtained by Kadomtsev and Petviashivili [6] (long waves and slow transverse dependence) and that of NLS equation by Davey and Stewartson [2]. On water dominated by surface tension(i.e., Bond number> 1/3), the KP equation (KPI) admits three dimensional fully localized structures named lumps [7, 8] and the long wave limit of DS equation (DSI) is found to admit dromions [9–12]. Both lumps and dromions are capable of propagating on water with constant speeds without changing their forms. The difference is that dromions decay exponentially in space while lumps algebraically. Also dromions form at the intersection of line-solitary mean-flow tracks and therefore their underlying structure to the leading order extends to infinity or finite boundaries. In polar area, waves can propagate on the surface of icecovered waters. For these waves to exist bending of the ice must be taken into account and therefore these waves are often called flexural-gravity waves. Many studies have been done on flexural gravity waves based on two dimensional model or/and linear wave theory [13–15]. Due to the flexural rigidity of ice, dromions can exist on water of depth much larger than that for capillary-gravity waves [1]. This study is motivated by observations of (relatively) large amplitude localized waves deep inside the icepack in polar waters. For instance 560km from the ice edge at Weddell Sea observations of breakup of an ice pack due Proceedings of the ASME 2013 32nd International Conference on Ocean, Offshore and Arctic Engineering OMAE2013 June 9-14, 2013, Nantes, France