Physical Oceanography, Vol.
4, November, 2011 (Ukrainian Original No.
4, July–August, 2011)
INFLUENCE OF ICE COMPRESSION ON COMPONENTS OF THE VELOCITY OF
MOTION OF LIQUID UNDER THE ICE COVER IN A TRAVELING PERIODIC
FLEXURAL GRAVITY WAVE OF FINITE AMPLITUDE
Ant. A. Bukatov
and O. M. Bukatova
By the method of multiple scales, we obtain (to within the third order of smallness) the asymptotic ex-
pansions for the components of the velocity of motion of liquid under a floating ice cover in the process
of propagation of periodic surface flexural gravity waves of finite amplitude under the conditions of ice
compression. We study the dependences of the distributions of the velocity components along the wave
profile on the compressive forces and the parameters of the initial harmonic. It is shown that the ampli-
tude values of the velocity components decrease and the phase shift of oscillations increases as the com-
pressive forces increase.
Keywords: waves of finite amplitude, flexural gravity waves, motion of liquid particles.
The investigation of the Stokes drift velocity  in the direction of propagation of waves of finite amplitude
in the absence of ice was carried out in [2–6]. The influence of floating broken ice on the velocity of motion of
liquid in periodic running nonlinear waves was studied in . In the presence of a continuous elastic ice cover,
the analysis of the dependence of the distribution of components the velocity of motion of liquid particles along
the profile of the nonlinear wave on the thickness and elasticity modulus of ice was performed in . The pre-
sent work is devoted to the analysis of the influence of ice compression on the components of the velocity of
motion of liquid under a floating ice cover in the process of propagation of periodic surface flexural gravity
waves of finite amplitude.
Statement of the Problem
Assume that a continuous elastic ice cover floats on the surface of a homogeneous ideal incompressible liq-
uid of finite depth
. We study the influence of ice compression on the components of the velocity of motion
of liquid under ice in the process of propagation of periodic flexural gravity waves of finite amplitude. The ice
cover is modeled by a thin elastic longitudinally compressed plate [9, 10]. The motion of liquid is regarded as
potential and the oscillations of the plate as nonseparating. In the dimensionless variables
x = kx
z = kz
t = kgt
, the potential
ϕ(x, z, t)
satisfies the Laplace equation
Δϕ = 0
−∞ < x <∞
−H ≤ z ≤ζ
Marine Hydrophysical Institute, Ukrainian National Academy of Sciences, Sevastopol, Ukraine.
Translated from Morskoi Gidrofizicheskii Zhurnal, No.
28–35, July–August, 2011. Original article submitted February 22, 2010.
0928–5105/11/2104–0245 © 2011 Springer Science+Business Media, Inc. 245