Vertical Mass Transport by Weakly Nonlinear InertiaGravity Internal Waves
Abstract
In the Boussinesq approximation, free inertiagravity internal waves are considered in a twodimensional vertically nonuniform flow. In the linear approximation was find vertical distribution of the amplitude of the vertical velocity and the dispersion relation. The boundaryvalue problem for internal waves has complex coefficients when the flow velocity component, transverse to the wave propagation direction depends on the vertical coordinate. Therefore, the eigenfunction and frequency of the wave are complex (it is shown that there is a weak damping of the wave). Vertical wave mass fluxes are nonzero. The vertical component of the Stokes drift velocity also differs from zero and contributes to the wave transport. A nonoscillating on a time scale of the wave correction to the average density, which is interpreted as an irreversible vertical fine structure generated by a wave, is determined on the second order of amplitude.
Keywords
Inertiagravity internal waves Stokes drift Wave fluxes of mass Critical layers1 Introduction
Vertical exchange in the marine environment is of key importance in the functioning of the ecosystem. The supply of oxygen to the deep layers of the sea is due to metabolic processes. Traditionally, vertical transfer is associated with smallscale turbulence, which is intermittent in the stratified layers of the sea. Therefore, one can speak of an “effective” turbulent exchange coefficient. When the turbulent viscosity and diffusion are taken into account, the internal waves decay [1, 2, 3]. In the pycnocline the processes of water mixing are strongly suppressed, and therefore it seems relevant to study the contribution of internal waves to vertical exchange. Nonlinear effects at the propagation of packets of internal waves are manifested in the generation of average wavescale currents and corrections to the density, which is treated as a vertical fine structure generated by a wave [4, 5]. This correction is proportional to the square of the current wave amplitude and after the wave packet passes the unperturbed profile the density field is restored.
Vertical wave fluxes of heat, salt and momentum, when turbulent viscosity and diffusion are taken into account, are nonzero [6, 7], but for inertiagravity internal waves (with allowance for the rotation of the Earth) vertical fluxes are nonzero and when turbulent viscosity and diffusion are ignored for vertically nonuniform twodimensional flow. It is of interest to find vertical wave fluxes of mass and to investigate their contribution to the formation of a vertical fine structure.
2 Problem Definition
In Boussinesq approximation, free internal waves are considered in a stratified twodimensional flow with rotation. Two components of the average flow velocity depend on the vertical coordinate. In the linear approximation, the boundary value problem for the amplitude of the vertical velocity has complex coefficients, so its solution is a complex function, the wave frequency is also a complex quantity (it is shown that there is a weak damping of the wave).
3 Linear Approximation
It is assumed that the wave propagates along the xaxis.
4 Nonlinear Effects
In the presence of an average flow, in which the velocity component transverse to the direction of propagation of the wave \(V_0\) depends on the vertical coordinate, the value of \(w_s\) is distinct from zero.
5 Results of Calculations
Wave mass fluxes are calculated for internal waves that were observed during the fullscale experiment in the third stage of the 44th voyage of the research vessel “Mikhail Lomonosov” on the northwestern shelf of the Black Sea.
The first device was located in the 5–15 m layer, the second in the layer 15–25 m, the third in the layer 25–35 m, the fourth in the layer 35–60 m. It is easy to see that powerful oscillations with a period of 15 min in the 25–60 m layer are in antiphase with oscillations in the 15–25 m layer, which indicates the presence oscillation of the second mode. The vertical profiles of the two components of the flow velocity are shown in Fig. 2, the BrentVaisala frequency is shown in Fig. 3a. The boundary value problem (23), (24) for internal waves is solved numerically according to the implicit Adams scheme of the third order of accuracy. The wavenumber is found by the method of adjusting from the necessity of performing boundary conditions (24). The eigenfunction of the 15min internal waves of the second mode is shown in Fig. 3b.
A boundary value problem for the definition of a function \(w_1\) (28), (29) solve numerically according to the implicit Adams scheme of the third order of accuracy, we find the unique solution orthogonal \(w_0\) and wave damping decrement \(\delta \omega \). For the 15min internal waves of the second mode, the attenuation decrement is \(\delta \omega =1.15\cdot 10^{5}\) rad/s. Attenuation decrement is two orders of magnitude smaller than the wave frequency. The dependence of the attenuation decrement on the wavenumber for the first two modes is shown in Fig. 5. Difference in behavior \(\delta \omega \) in the lowfrequency area is due to the cutoff of the dispersion curves in the vicinity of the inertial frequency.
The total vertical wave mass flux is added from the stream \(\overline{\rho w}\) (34) and the flux due to the vertical component of the Stokes drift velocity \(J_{\rho s} = \rho _0(z)w_s\). A comparison of the total fluxes for the first two modes with the corresponding turbulent flux \(\overline{\rho 'w'}\) is shown in Fig. 7. The turbulent flux is determined by the formula \(\overline{\rho 'w'} = M_z\frac{d^{} \rho _0}{d z^{}}\). The coefficient of vertical turbulent diffusion is estimated from formula \(M_z \cong 0.93\cdot 10^{4}N_c^{1}\, m^2/s\), \(N_c\) corresponds to the BruntVaisala frequency in the cycle/hour [15]. The first mode dominates in the upper 40m layer, deeper these fluxes are comparable in magnitude (Fig. 7). The wave fluxes exceed the turbulent flux in absolute value.
6 Conclusions

The vertical component of the Stokes drift velocity of internal waves is different from zero and makes a decisive contribution to the wave transport of the mass.

The vertical wave mass flux leads to an irreversible deformation of the average density profilethe fine structure generated by the wave.

The wave flux of the first mode in the upper 40meter layer exceeds the flux of the second mode. Wave fluxes dominate over turbulent fluxes.
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