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The generation of vertical fine structure by inertia-gravity internal waves in a two-dimensional stratified shear flow is investigated. In the linear approximation, the boundary value problem for the amplitude of the vertical velocity of internal waves has complex coefficients, the imaginary part of which is small. The wave frequency and the eigenfunction of the boundary problem for the internal waves are complex (and we show that a weak damping of the wave occurs). The phase shift between the fluctuations of density and vertical velocity differs from π/2; therefore, the wave-induced vertical mass flux is non-zero. It is shown that dispersion curves are cut off in the low-frequency domain due to the influence of critical layers, where the frequency of the wave with the Doppler shift is equal to the inertial one. The Stokes drift velocity is determined in the weakly nonlinear approximation, on the second order in the amplitude of the wave. The vertical component of the Stokes drift velocity is also non-zero and contributes to wave transfer. The summary wave mass flux exceeds the turbulent one and leads to irreversible deformation of the average density profile which can be interpreted like a fine structure generated by the wave. On the shelf, this deformation is more than in deep-water part of the Black Sea at the same amplitude of а wave. The vertical scale of the fine structure of Brunt-V äis äl ä frequency, generated by a wave, corresponds to really observed value.

The fine vertical structure of hydrophysical fields in the ocean was discovered in the second half of the last century after the creation of high - resolution sounding equipment [

The double diffusion mechanism is realized when the temperature and salinity simultaneously decrease (or increase) with depth [

In the areas of fronts, the “intrusive” mechanism is possible, when there is a mutual penetration of waters with different T , S characteristics (T-temperature, S-salinity). The fine structure of the “intrusion type” has temperature and salinity inversions with stable stratification [

The most typical situation is when the temperature decreases with depth and salinity increases. In this case, the “double diffusion” does not work and internal waves make a main contribution to the generation of vertical fine structure in the ocean.

Internal waves in the ocean play an important role in the dynamics of stratified deep layer. As a rule, the propagation of internal waves occurs at space-inhomogeneous flows, with the interaction of internal waves with currents. There are two effects that lead to wave energy dissipation. The first is to capture and focus the internal waves by a horizontally inhomogeneous picnocline [

Nonlinear effects at the propagation of packets of internal waves are manifested in the generation of average on a wave time scale current [

However, as shown below for inertia-gravity internal waves, the vertical wave mass flux is non-zero in the presence of a flow whose velocity component transverse to the wave propagation direction depends on the vertical coordinate. The vertical component of the Stokes drift velocity is non-zero too and contributes to the wave transfer. The presence of a vertical wave flux of mass leads to the generation of irreversible fine structure.

We will consider the free internal waves in an infinite pool of constant depth by using the Boussinesq approximation and taking into account the rotation of the Earth. The two components of the mean flow velocity depend on the vertical coordinate. In the linear approximation, the boundary value problem for the vertical velocity amplitude has complex coefficients; therefore, its solution is a complex function, and the wave frequency value is also complex (i.e., there occurs a weak attenuation of the wave). The Stokes drift velocity, the wave fluxes of mass, and the corrections to density that do not oscillate on the time scale of the wave are found in the second order in the amplitude of the wave.

The system of hydrodynamic equations for wave disturbances in the Boussinesq approximation has the form

D u D t − f v + w d U 0 d z = − 1 ρ 0 ( 0 ) ∂ P ∂ x (1)

D v D t + f u + w d V 0 d z = − 1 ρ 0 ( 0 ) ∂ P ∂ y (2)

D w D t = − 1 ρ 0 ( 0 ) ∂ P ∂ z − g ρ ρ 0 ( 0 ) (3)

D ρ D t = − w d ρ 0 d z (4)

∂ u ∂ x + ∂ v ∂ y + ∂ w ∂ z = 0 (5)

Here we use the coordinate system x , y , z where the z-axis is directed upwards; u , v , w are respectively two horizontal and vertical components of the wave flow velocity, ρ and P are wave perturbations of density and pressure, H is the depth of the sea, ρ 0 ( z ) is the average density profile, f is the Coriolis parameter, U 0 ( z ) , V 0 ( z ) are two components of the mean velocity flow, g is

acceleration of gravity and the action of the operator D D t is determined by the formula

D D t = ∂ ∂ t + ( u + U 0 ) ∂ ∂ x + ( v + V 0 ) ∂ ∂ y + w ∂ ∂ z

At the sea surface (z = 0), we use a rigid-lid boundary condition which filters the internal waves out from the surface waves [

w ( 0 ) = 0. (6)

At the bottom, we satisfy the non-flow condition

w ( − H ) = 0. (7)

In the linear approximation, the solutions can be written in the form

u 1 = u 10 ( z ) A e i θ + c .c . , v 1 = v 10 ( z ) A e i θ + c .c . , w 1 = w 10 ( z ) A e i θ + c .c .

P 1 = P 10 ( z ) A e i θ + c .c . , ρ 1 = ρ 10 ( z ) A e i θ + c .c . , (8)

where c .c . is a complex conjugate term, A is the amplitude factor, θ is a phase of the wave; ∂ θ / ∂ x = k , ∂ θ / ∂ t = − ω ; k is horizontal wave number, ω is wave frequency. It is assumed that the wave propagates along the x-axis.

After substituting (8) into system (1) - (5), follows coupling of the amplitude functions u 10 , v 10 , ρ 10 , P 10 with w 10

u 10 = i k d w 10 d z , Ω = ω − k ⋅ U 0 , (9)

v 10 = 1 Ω ( f k d w 10 d z − i w 10 d V 0 d z ) , ρ 10 = − i Ω w 10 d ρ 0 d z , (10)

P 10 ρ 0 ( 0 ) = i k [ Ω k d w 10 d z + d U 0 d z w 10 + f Ω ( i d V 0 d z w 10 − f k d w 10 d z ) ] , (11)

function w 10 satisfies the equation

d 2 w 10 d z 2 + k Ω 2 − f 2 [ i f d V 0 d z − d U 0 d z f 2 Ω ] d w 10 d z + k w 10 Ω 2 − f 2 [ k ( N 2 − Ω 2 ) + Ω d 2 U 0 d z 2 + i f d 2 V 0 d z 2 + i f k Ω d U 0 d z d V 0 d z ] = 0 , (12)

where N 2 = − g ρ 0 ( 0 ) d ρ 0 d z is the square of Brunt-Väisälä frequency.

Boundary conditions for w 10

z = 0 , w 10 = 0 (13)

z = − H , w 10 = 0 (14)

Equation (12) has complex coefficients, the imaginary part of which is small, so let us turn to dimensionless variables (the dashed line denotes dimensionless physical quantities)

z = H z / , t = t / / ω * , w 10 = w 10 / V 0 * , V 0 = V 0 / V 0 * , U 0 = U 0 / V 0 * , k = k / / H , f = f / ω * , ω = ω / ω * , N = N / ω * , Ω = Ω / ω * , (15)

where ω * is characteristic wave frequency, V 0 * is a characteristic value of the flow velocity which is transverse to the wave propagation direction.

Equation (12) then takes the form:

d 2 w 10 / d z / 2 + k / [ i ε f / d V 0 / d z Ω / 2 − f / 2 − ε f / 2 d U 0 / d z Ω / ( Ω / 2 − f / 2 ) ] d w 10 / d z / + k / w 10 / [ k / ( N / 2 − Ω / 2 ) + ε Ω / d 2 U 0 / d z / 2 + i ε f / d 2 V 0 / d z 2 Ω / 2 − f / 2 + i ε 2 f / k / d U 0 / d z d V 0 / d z Ω / ( Ω / 2 − f / 2 ) ] = 0 , (16)

ε = V 0 * / H ω * is a small parameter. The imaginary part of the coefficients in Equation (16) is the order of

where

then

The boundary conditions for

Function

where

The boundary conditions for

After the transition to dimensional variables, Equation (19) takes the form:

where

Equation (23) should be supplemented by boundary conditions:

The boundary-value problem (23), (24) in the absence of flow

Let

Equation (25) leads to a selfadjoint form, multiplying both sides of the equation by

here

After the transition to dimensional variables, Equation (21) is transformed into the form

where

The boundary conditions for the function

We multiply both sides of the linear inhomogeneous Equation (27) by the function

where

The solvability condition for the boundary value problem (28), (29) [

Hence the expression for

where

The value

Stokes drift velocity of fluid particles is determined by the formula [

where

The vertical component of Stokes drift velocity determined by the formula [

where

In the presence of an average flow in which component of velocity

The vertical wave mass flux is determined by the formula

The presence of a vertical wave flux of mass leads to an irreversible deformation of the density field, which can be considered as a vertical fine structure generated by a wave. The equation for the non-oscillating on the time scale of the correction to the average density

hence

Integrate Equation (35) in time

Substituting

where

Passing to the limit in (37) for

The

We calculate the mass wave flux for the internal waves observed during the full-scale experiment in the third stage of the 44th voyage of the research vessel Mikhail Lomonosov on the North-Western 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 of the second mode.

The vertical profiles of the two components of the flow velocity are shown in

The normalizing factor

this implies

Thus, the amplitude of vertical displacements is proportional to^{−4} rad/s, for the second mode 3.49 × 10^{−}^{4} rad/s (for comparison we point out that the Coriolis frequency is equal to 1.048 × 10^{−}^{4} rad/s). The dispersion curves are cut off due to the influence of critical layers where the frequency of the wave with Doppler shift is equal to the inertial one.

In the deep-water area of the North-Western part of the Black Sea, above the continental slope, there is a jet of the Main Black sea current (MBC). The Brunt-Väisälä frequency profile, the two components of the flow velocity and the eigenfunction of the fifteen-minute internal wave of the second mode are shown in

The dispersion curves of the first two modes are shown in

The boundary value problem for determining the function

Vertical wave fluxes of mass

The total vertical wave mass flux consists of the flux

determined by the formula

turbulent exchange is estimated by the formula ^{2}/s,

The vertical density profile is shown in

The non-oscillating correction to the average density (38), normalized to the square of the wave amplitude, is shown in

A comparison of the non-oscillating corrections to the density

Non-oscillating corrections to the Brunt-Väisälä frequency

The vertical wave mass flux is different from zero for inertia-gravity internal waves in a two-dimensional vertically non-uniform flow, when the component of the flow velocity transverse to the direction of wave propagation depends on the vertical coordinate. The vertical component of the Stokes drift velocity is also nonzero and makes a main contribution to the wave transfer.

Total vertical wave mass flux exceeds the turbulent ones. The wave mass flux leads to a non-oscillating on the time scale of the wave correction to the average density, which is irreversible, i.e. to the fine structure generated by the wave. In shallow water, this correction is greater than in the deep-sea part of the sea with the same wave amplitude. The vertical scale of the Brunt-Väisälä frequency fine structure generated by the wave corresponds to the observed one. It is shown that dispersion curves are cut off in the low-frequency domain due to the influence of critical layers, where the frequency of the wave with the Doppler shift is equal to the inertial one.

The authors are grateful to Morozov A.N. for the presented experimental material on the deep-sea part. The work was carried out as part of the state assignment on the subject No. 0827-2019-0003 “Fundamental studies of oceanological processes determining the state and evolution of the marine environment under the influence of natural and anthropogenic factors, based on observation and modeling methods” (code “Oceanological processes”).

The authors declare no conflicts of interest regarding the publication of this paper.

Slepyshev, A.A. and Vorotnikov, D.I. (2019) Inertia-Gravity Internal Waves, Stokes Drift, Wave Fluxes of Mass, Vertical Fine Structure, Critical Layers. Open Journal of Fluid Dynamics, 9, 140-157. https://doi.org/10.4236/ojfd.2019.92010