Generation of Vertical Fine Structure by the Internal Waves with the Regard for Turbulent Viscosity and Diffusion

Abstract

Purpose. The aim is to study the mechanism of formation of a vertical fine structure due to the mass vertical transfer by the internal waves taking into account turbulent viscosity and diffusion as well as to investigate influence of the critical layers on the dispersion curves of internal waves. Methods and Results. In the Boussinesq approximation, the free inertia-gravity internal waves in a vertically inhomogeneous flow are considered with the regard for the horizontal turbulent viscosity and diffusion. The equation for the amplitude of vertical velocity of the internal waves contains a small parameter (in the dimensionless variables) proportional to the value of the horizontal turbulent viscosity. The solution of this equation is realized in a form of the asymptotic series of this parameter. In the zero approximation, the second-order homogeneous boundary value problem determined the vertical structure mode is solved numerically by the implicit third-order accuracy Adams scheme for real profiles of the Brent-Väisälä frequency and the current velocity. At the fixed wave frequency, the wave number is determined by the shooting method. In the first order with respect to the indicated parameter, the semi-homogeneous boundary value problem is also solved numerically according to the implicit Adams scheme of the third order of accuracy. A unique solution is found which is orthogonal to the solution of the corresponding homogeneous boundary value problem. The condition of this boundary value problem solvability yields the wave attenuation decrement. The dispersion curves of the first two modes are cut off in the lowfrequency region (the second mode is at a higher frequency), that is due to influence of the critical layers, where the wave frequency with the Doppler shift is inertial. It is shown that the mass vertical wave flux differs from zero and leads to correction (not oscillating on the wave time scale) of the average density, i. e. the internal wave generate fine structure that is of an irreversible character. Conclusions. When the horizontal turbulent viscosity and diffusion are taken into consideration, the mass vertical wave flux differs from zero and leads to generation of the vertical fine structure. The mass wave flux exceeds the turbulent one. The vertical scales of the generated vertical fine structure correspond to the actually observed ones.