Physical Oceanography, Vol.
NONLINEAR SURFACE WAVES IN A BASIN WITH FLOATING BROKEN ICE
A. E. Bukatov and A. A. Bukatov
The method of multiple scales is used to deduce equations for three nonlinear approximations of
a wave disturbance in a basin of constant depth covered with broken ice.
In deducing these equa-
tions, we take into account the space and time variability of the wave profile in the expression for
the velocity potential on the basin surface. These equations are used to construct uniformly suit-
able asymptotic expansions up to quantities of the third order of smallness for the liquid-velocity
potential and elevations of the basin surface formed by a periodic running wave of finite am-
plitude. We analyze the dependence of the amplitude-phase characteristics of elevations of the
basin surface on the thickness of ice, nonlinearity of its vertical acceleration, and the amplitude
and wavelength of the fundamental harmonic.
The theoretical analysis of low-amplitude wave disturbances in inviscid incompressible homogeneous li-
quid covered with ice was performed in [1–7]. The accumulated results show that the influence of ice on the
wave characteristics decreases as the period of oscillations increases, and long waves pass through the ice fields
practically without noticeable distortions. At the same time, some contradictions between the well-known theor-
etical results and the data of in-situ observations established in  do not allow us to make any definite conclu-
sion concerning the character of influence of ice cover on the propagation of long-period waves. Thus, it seems
reasonable to study the influence of ice on the basis of equations of the dynamics of nonlinear waves. This prob-
lem was solved for long waves in [9, 10] and for waves of arbitrary length in .
In the present work, we study the influence of floating ice cover on the propagation of finite-amplitude pe-
riodic waves by using equations for nonlinear approximations obtained by the method of multiple-scale asymp-
totic expansions with regard for the space and time variability of vertical displacements of the basin surface in
the expression for the velocity potential (appearing in deducing the kinematic and dynamic boundary conditions
for nonlinear approximations).
Statement of the Problem
Assume that an unbounded basin of finite depth
is filled with an inviscid incompressible homogeneous
liquid whose surface is covered with floating ice. We study the influence of ice on the propagation of periodic
waves of small but finite amplitude under the assumption that the oscillations are not separated and the sizes of
the ice blocks are small as compared with the wavelength of disturbances. Under these assumptions, the ice
blocks do not suffer bending. Therefore, studying oscillations, we may assume that gravity is a single restoring
force. Further, we assume that the motion of liquid is potential and pass to the dimensionless variables x
, and t
, where k is the wave number. As a result, the problem under consideration is reduced
to the solution of the Laplace equation
Marine Hydrophysical Institute, Ukrainian Academy of Sciences, Sevastopol. Translated from Morskoi Gidrofizicheskii Zhurnal,
34–46, September–October, 2002. Original article submitted March 23, 2001; revision submitted April 3, 2001.
266 0928-5105/02/1205–0266 $27.00 © 2002 Plenum Publishing Corporation