Physical Oceanography, Vol.
STATIONARY WAVES IN THE FLOW OF A HOMOGENEOUS FLUID
WITH VERTICAL SHEAR OF THE VELOCITY
M. V. Bilyunas
and S. F. Dotsenko
A plane problem of free stationary gravitational waves in a horizontal current with vertical shear of the
velocity is studied in the linear statement. The determination of the parameters of waves is reduced to
the solution of the Sturm–Liouville boundary-value problem. For some vertical distributions of current
velocity, we obtain analytic solutions. We propose a numerical algorithm for finding the parameters of
waves. On the basis of the performed analysis, we establish the possibility of existence of stationary
surface waves in currents for certain ranges of the Froude number. As the Froude number decreases, the
waves become shorter, which leads to a faster attenuation of waves disturbances with depth. Under the
actual conditions, the waves are short and suffer the influence of shear currents only in the subsurface
layer of the ocean.
Currents significantly affect the kinematics and dynamics of the surface gravitational waves in the oceans
and seas. Thus, the horizontal inhomogeneity of current velocities leads to the refraction of waves and, hence, to
the redistribution of energy of the wave field in the space . The refraction of wind-induced waves is well
visible in the zones of eddy formations and jet currents . It is also worth noting that the refraction of waves in
the zones of currents may lead to the focusing of waves and the formation of freak waves in the ocean . The
vertical variations of the velocities of horizontal flows (currents with vertical shear of the velocity) affect the
characteristics of the surface and internal waves and the vertical structure of the velocity field and, hence, may
lead to the instability of currents and strengthening of the process mixing in the ocean [4, 5]. Moreover, they
also affect the space structure of the wave field and the conditions of generation of forced waves [6–8]. Note
that the linear (constant shear of the velocity) or piecewise linear distributions of the horizontal current velocity
over the depth were specified in the major part of special cases.
In what follows, we study free surface gravitational waves in currents with vertical shear of the velocity,
which are assumed to be linear and stationary. We consider some general properties of waves, obtain analytic
solutions of the problem for some vertical distributions of current velocity, and describe the numerical procedure
aimed at the determination of the parameters and vertical structure of waves for any distribution of the unidirec-
tional horizontal velocity of the background current.
Mathematical Statement of the Problem
In the vertical plane
is the horizontal coordinate, and
is the vertical coordinate meas-
Marine Hydrophysical Institute, Ukrainian National Academy of Sciences, Sevastopol, Ukraine.
Corresponding author; e-mail: firstname.lastname@example.org.
Translated from Morskoi Gidrofizicheskii Zhurnal, No.
15–29, July–August, 2010. Original article submitted March 23, 2009.
0928–5105/10/2004–0251 © 2010 Springer Science+Business Media, Inc. 251