Physical Oceanography, Vol.
1, May, 2011 (Ukrainian Original No.
1, January–February, 2011)
VELOCITIES OF MOTION OF LIQUID PARTICLES UNDER THE FLOATING ICE COVER
IN THE CASE OF PROPAGATION OF PERIODIC WAVES OF FINITE AMPLITUDE
Ant. A. Bukatov
and And. A. Bukatov
By using the velocity potential obtained by the method of multiscale asymptotic expansions to within the
quantities of the third order of smallness, we study the dependences of the components of the velocity of
motion of a homogeneous liquid under the floating ice cover on the thickness of the cover and its
modulus of elasticity in the process of propagation of periodic waves of finite amplitude. It is shown
that the presence of broken ice leads to a decrease in the moduli of components of the velocity of liquid
particles and the phase delay of generated oscillations. The effect of the elasticity of ice becomes more
pronounced as the wavelength of the initial harmonic decreases and manifests itself in the increase in the
maximum values of the components of velocity and in the phase shift of oscillations in the direction of
propagation of waves.
Keywords: waves of finite amplitude, flexural gravitational waves, motion of liquid particles.
In the linear statement, the influence of broken ice on the velocity of wave currents in a homogeneous liquid
was studied in . The velocity of the translational motion of liquid in the direction of propagation of waves of
finite amplitude predicted by the Stokes theory  was analyzed in [3–5] and [6–8] for basins with free surface
of infinite and finite depths, respectively. In , the same problem was solved for water with floating broken ice
without quantitative analysis of the distributions of components of the velocity of liquid particles over the wave-
length. The dependences of the components of the orbital velocity of motion of particles of homogeneous liquid
with open surface on the wave number and steepness of running periodic waves with finite amplitude were in-
vestigated in .
In the present work, on the basis of the asymptotic expansion of the velocity potential obtained in  by
the method of multiple scales  to within the quantities of third order smallness in the steepness of waves, we
analyze the dependences of the distributions of components of the velocity of motion of liquid particles over the
wavelength on the characteristics of the ice cover.
Statement of the Problem
Assume that the surface of a homogeneous ideal incompressible liquid placed in an unbounded basin of
is covered with floating ice. Consider the influence of the ice cover on the orbital velocities
of motion of liquid particles formed by running periodic waves with finite amplitude under the assumption that
the motion of liquid is potential and the oscillations of the ice cover are nonseparating. In the dimensionless
Marine Hydrophysical Institute, Ukrainian National Academy of Sciences, Sevastopol, Ukraine.
Translated from Morskoi Gidrofizicheskii Zhurnal, No.
15–24, January–February, 2011. Original article submitted November 24,
2009; revision submitted December 21, 2009.
0928–5105/11/2101–0013 © 2011 Springer Science+Business Media, Inc. 13