Physical Oceanography, Vol.
THERMOHYDRODYNAMICS OF THE OCEAN
TRANSPORT PROPERTIES OF TRAPPED TOPOGRAPHIC WAVES
A. A. Slepyshev
In the Boussinesq approximation, we study weakly nonlinear topographic waves trapped by a
flat slope of arbitrary orientation. We compute the mean currents induced by the waves due to
the nonlinearity in the quadratic approximation with respect to the wave amplitude in the pre-
sence of dissipation of the wave energy into the turbulent motion. In the diffusion approxima-
tion, we determine the vertical distribution of the concentration of wave-suspended sediments. It
is shown that the consumption of sediments across the isobaths is directed downward along the
slope. At the same time, the consumption of sediments along the isobaths has the same direction
as the projection of the horizontal wave vector.
At present, the analysis of the phenomena of suspension and resettling of bottom sediments becomes a fair-
ly urgent problem in connection with the development of the shelf resources and construction and exploitation of
the bottom communications and pipelines. An important contribution to the dynamics of the bottom layer is
made by the wave processes in the shelf zone and over the continental slope. Wind waves is also an important
factor in the accumulation or washout of the sediments directly in the coastal zone of the sea [1, 2]. The influ-
ence of the surface waves is traced, as a rule, down to depths comparable with the half length of the wave .
At large depths, the influence of internal and topographic waves is predominant. The nonlinear effects in the
propagation of both surface and internal waves are connected with the generation of currents averaged over the
time scale of the wave and induced by the action of wave stresses in a weakly nonlinear packet [4–7]. In the
bottom layer of the sea in the shelf zone and over the continental slope, there exists an important class of trapped
topographic Kelvin-type waves for which the component of the orbital wave velocity normal to the bottom is
equal to zero [8–12]. Most likely, the bottom waves make a significant contribution to the transport of sedi-
ments in the shelf zone.
If the turbulent tangential stresses at the bottom exceed the critical values corresponding to the onset of mo-
tion of the sediments, then the wave roils bottom sediments and initiates their horizontal transport by the mean
currents induced by the bottom topographic waves. In this connection, it is of interest to determine the mean
currents caused by the bottom waves due to the presence of nonlinear effects in turbulent viscosity and diffusion
over a slope of arbitrary orientation. The original nonlinear equations of hydrodynamics for wave perturbations
are solved in the weakly nonlinear approximation by the perturbation method : In the first order of smallness
with respect to the wave amplitude, we find solutions in the linear approximation and the dispersion relation. In
the second order of smallness, the mean currents induced by the waves are found after averaging of the original
equations over the wave period.
The horizontal bottom is defined as a plane perpendicular to the vector of gravitational acceleration and
parallel to the free undisturbed ocean surface. The plane tangential to the Earth’s surface and parallel to the hor-
izontal bottom is denoted by
R. The plane
corresponding to the sloping bottom is obtained from the plane
as a result of its rotation by an angle
about the line of intersection of the planes
X-axis). We as-
Marine Hydrophysical Institute, Ukrainian Academy of Sciences, Sevastopol. Translated from Morskoi Gidrofizicheskii Zhurnal,
3–13, January–February, 2004. Original article submitted July 11, 2002; revision submitted July 26, 2002.
0928-5105/04/1401–0001 © 2004 Plenum Publishing Corporation 1