Physical Oceanography, Vol. 19, No. 5, 2009
THERMOHYDRODYNAMICS OF THE OCEAN
NONLINEAR EFFECTS IN THE PROCESS OF PROPAGATION OF INTERNAL
WAVES WITH REGARD FOR THE TURBULENT VISCOSITY AND DIFFUSION
A. A. Slepyshev
and I. S. Martynova
By the method of asymptotic multiscale expansions in the Boussinesq approximation, we study
nonlinear effects observed in the process of propagation of internal waves with regard for the
turbulent viscosity and diffusion. We determine the decrement of attenuation of waves and the
boundary-layer solutions at the bottom and on the free surface. The wave-induced mean current
is found in the second order of smallness in the wave steepness. The coefficients of the nonlinear
Schrödinger equation are obtained for the envelope of the wave packet. It is shown that a weakly
nonlinear plane wave is stable under longitudinal modulation in the long-wave limit. If the wave-
length is smaller than a certain critical value, then the wave is unstable under modulation.
The nonlinear effects appearing in the process of propagation of internal waves manifest themselves in the
generation of currents intermediate on the wave scale [1, 2]. From the physical point of view, this phenomenon
is explained by the presence of nonzero wave stresses caused by the dependence of the envelope of wave packet
on the space-and-time coordinates [3, 4]. The envelope of the narrow spectral packet of internal waves satisfies
the nonlinear Schrödinger equation . As a rule, internal waves propagate in the form trains, i.e., wave packets
localized in the space. As the physical causes of alternation of the wave field, we can mention, on the one hand,
large distances between the sources and sinks of energy and, on the other hand, the modulation instability of in-
ternal waves responsible for the complicated evolution of the envelope of wave packet .
The theory of nonstationary weakly nonlinear wave packets of internal waves in the absence of turbulent
viscosity and diffusion was proposed in [1, 2]. The mean currents and nonoscillating corrections to the mean
density induced by the waves are found in the second order of smallness in the wave steepness. The boundary-
layer solutions for the surface waves, as well as the mean currents induced by the waves due to nonlinearity, are
described in . In the present work, we determine the mean currents induced by internal waves with regard for
the turbulent viscosity and diffusion, the coefficients of the nonlinear Schrödinger equation for the envelope, the
decrement of attenuation of waves, and the boundary-layer solutions at the bottom and on the free surface. It is
shown that internal waves are unstable under modulation.
Statement of the Problem
We consider free internal waves with regard for the turbulent viscosity and diffusion by applying the
asymptotic method of multiscale expansions aimed at the analysis of nonlinear effects in the presence of a sink
Marine Hydrophysical Institute, Ukrainian Academy of Sciences, Sevastopol, Ukraine.
Division of the Lomonosov Moscow State University, Sevastopol, Ukraine.
Translated from Morskoi Gidrofizicheskii Zhurnal, No.
3–22, September–October, 2009. Original article submitted March 13,
2008; revision submitted April 14, 2008.
0928-5105/09/1905–0267 © 2009 Springer Science+Business Media, Inc. 267