## Vertical Transfer of Momentum by Inertia-Gravity Internal Waves on a Two-Dimensional Shear Flow

*A. A. Slepyshev*

Marine Hydrophysical Institute, Russian Academy of Sciences, Sevastopol, Russian Federation

*e-mail: slep55@mail.ru*

### Abstract

*Purpose*. The paper is aimed at investigating the momentum vertical transfer by inertia-gravity internal waves on a two-dimensional flow with a vertical shear of velocity, and also at studying the Stokes drift of liquid particles and the mean current effect on it.

*Methods and Results*. Free internal waves in an infinite basin of constant depth are considered in the Boussinesq approximation with the regard for the Earth rotation. Two components of the mean current velocity depend on the vertical coordinate. The equation for the vertical velocity amplitude has complex coefficients; therefore the eigenfunction and the wave frequency are complex. The corresponding boundary value problem is solved numerically by the implicit Adams scheme of the third order of accuracy. The wave frequency at a fixed wavenumber was found by the shooting method. It was determined that the frequency imaginary part was small and could be either negative or positive depending on a wave number and a mode number. Thus, both weak attenuation and weak amplification of an internal wave are possible. The vertical wave momentum fluxes are nonzero and can exceed the corresponding turbulent fluxes. The Stokes drift velocity, transverse to the wave direction, is nonzero and less than the longitudinal velocity. The vertical component of the Stokes drift velocity is also nonzero and four orders of magnitude less than the longitudinal component. The signs of the vertical component of the Stokes drift velocity for the waves with the frequencies 10 and 16 cph are opposite, since the signs of their frequency imaginary parts are different; and the vertical component of the Stokes drift velocity is proportional to the wave frequency imaginary part.

*Conclusions*. The vertical momentum wave flux of inertia-gravity internal waves differs from zero in the presence of the current whose velocity component, transverse to the wave propagation direction, depends on the vertical coordinate. The component of the Stokes drift velocity, transverse to the wave propagation direction, is nonzero and less than the longitudinal one. The vertical component of the Stokes drift velocity is also nonzero and can contribute to formation of the vertical fine structure.

### Keywords

internal waves, imaginary correction to frequency, wave momentum flux, Stokes drift

### Acknowledgements

The author is grateful to N.O. Ankudinov for numerical calculations in solving the boundary problem. The study was carried out within the framework of the state task on theme No. 0827-2019-0003 “Fundamental investigations of oceanological processes conditioning state and evolution of marine environment being affected by natural and anthropogenic factors, based on the observation and modeling methods”.

### Original russian text

Original Russian Text © A. A. Slepyshev, 2021, published in MORSKOY GIDROFIZICHESKIY ZHURNAL, Vol. 37, Iss. 4, pp. 391-404 (2021)

### For citation

Slepyshev, A.A., 2021. Vertical Transfer of Momentum by Inertia-Gravity Internal Waves on a Two-Dimensional Shear Flow. *Physical Oceanography*, 28(4), pp. 363-375. doi:10.22449/1573-160X-2021-4-363-375

### DOI

10.22449/1573-160X-2021-4-363-375

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