Physical Oceanography, Vol.
2, July, 2011 (Ukrainian Original No.
2, March–April, 2011)
NONLINEAR EFFECTS IN THE PROCESS OF PROPAGATION OF INTERNAL WAVES
IN THE PRESENCE OF TURBULENCE
A. A. Slepyshev,
A. N. Alieva,
and N. V. Laktionova
In the Boussinesq approximation, we study the nonlinear effects observed in the process of propagation
of internal waves in the presence of turbulence. The space damping factor of the waves is evaluated.
The Stokes drift velocity and the Euler velocity of the mean current induced by waves due to the pres-
ence of nonlinearity are determined. It is shown that the principal contribution to the wave transfer is
made by the horizontal velocity of the induced current. The Stokes drift is significant only near the bot-
tom. The vertical component of the Stokes drift velocity obtained with regard for the turbulent viscosity
Keywords: internal waves, turbulent viscosity, Stokes drift.
Nonlinear effects in the process of propagation of internal waves manifest themselves in the generation of
currents averaged over the scale of waves [1, 2]. This effect has the second order of smallness in the amplitude
of waves and is caused by the space inhomogeneity of wave stresses in the region of the wave packet. The Euler
velocity of the mean current induced by waves due to the presence of nonlinearity should be distinguished from
the velocity of Stokes drift also observed in weakly nonlinear plane waves [3, 4]. The total velocity of the drift
of particles of the liquid is obtained as the sum the Euler velocity of the mean current and the Stokes drift veloc-
ity [4, 5].
As one of the factors responsible for the sink of energy of internal waves, we can mention their dissipation
in the presence of small-scale turbulence. The parametrization of the action of turbulence upon internal waves is
performed via the introduction of the coefficients of turbulent exchange . The analysis of the influence of
small-scale turbulence on internal waves is carried out in [6, 7]. This problem is solved in the linear statement,
and the authors determine the damping factor of waves in the course of time caused by the dissipation of wave
energy in the presence of turbulent viscosity. The analysis of the damping factor of long waves is performed in
[8, 9]. We emphasize that the authors of the cited works study the process of decay of the amplitude of waves in
the course of time.
However, the waves propagating in turbulent media also suffer space damping . Hence, in the present
work, we consider the phenomenon of space damping of weakly nonlinear internal waves observed in the pres-
ence of turbulent viscosity and determine the space damping factor of waves, the Stokes drift velocity of the par-
ticles of liquid, and the velocity of the Euler mean current induced by the waves due to the presence of nonline-
Marine Hydrophysical Institute, Ukrainian Academy of Sciences, Sevastopol, Ukraine.
Sevastopol Branch of the Lomonosov Moscow State University, Sevastopol, Ukraine.
Translated from Morskoi Gidrofizicheskii Zhurnal, No.
13–23, March–April, 2011. Original article submitted December 1, 2009;
revision submitted December 24, 2009.
0928–5105/11/2102–0085 © 2011 Springer Science+Business Media, Inc. 85