Physical Oceanography, Vol. 18, No. 3, 2008
THERMOHYDRODYNAMICS OF THE OCEAN
ON THE NONLINEAR INTERACTIONS IN TRIADS OF EDGE WAVES
ON THE SEA SHELF
V. A. Dubinina, A. A. Kurkin, and O. E. Polukhina
We study nonlinear three–wave interactions between edge waves propagating in the same direc-
tion over the shelf step. The conditions of synchronism are determined and the coefficient of in-
teraction is computed for the cases where the waves of the five lowest modes participate in the
interaction. The space-time dynamics is studied by analyzing, as an example, a single triad of
edge waves. The possibility of interaction of edge waves in the regions with actual topography is
Trapped waves with relatively high frequencies propagating along the coast and rapidly decaying toward
the open ocean are usually called edge waves. At present, there are numerous facts corroborating the existence
of edge waves in the wave field of the coastal zone of the ocean (see, e.g., [1–3]). According to the contempor-
ary ideas, edge waves play an important role in the dynamics of the coastal zone and lead to the appearance of
specific phenomena and effects such as, e.g., edge festoons and rip currents [4–8].
Within the framework of the linear model, the edge-wave solution is constructed in the form of the super-
position of modes with orthogonal structure specified by the characteristic parameters of the shelf and continen-
tal slope. If the nonlinear terms in the equations of motion are taken into account, then we observe the redistri-
bution of energy over the spectrum of the waves, which may lead to a relatively strong growth of some modes in
the case of resonance interaction. The investigations in this direction are performed for more than three decades.
Thus, the existence of resonant triads of Stokes edge waves over a slope with constant inclination was proved in
 and then confirmed in the subsequent works [10–12]. For edge waves, three-wave resonances also appear
over the underwater features of the bottom topography of complex shapes [13–15]. The present work is devoted
to the analysis of possible nonlinear effects accompanying the interaction of triads of edge waves propagating
over a shelf step.
The principal equations used to describe long-wave motions of an ideal nonstratified liquid in a basin with
cylindrical bottom topography specified by the depth profile h
) have the form
Nizhegorodskii State Technical University, Nizhnii Novgorod, Russia.
Translated from Morskoi Gidrofizicheskii Zhurnal, No.
3–19, May–June, 2008. Original article submitted September 29, 2006.
0928-5105/08/1803–0117 © 2008 Springer Science+Business Media, Inc. 117