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Vertical Mass Transport by Weakly Nonlinear Inertia-Gravity Internal Waves

  • A. A. Slepyshev
  • D. I. Vorotnikov
Conference paper
Part of the Springer Geology book series (SPRINGERGEOL)

Abstract

In the Boussinesq approximation, free inertia-gravity internal waves are considered in a two-dimensional vertically non-uniform flow. In the linear approximation was find vertical distribution of the amplitude of the vertical velocity and the dispersion relation. The boundary-value 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 non-oscillating 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

Inertia-gravity internal waves Stokes drift Wave fluxes of mass Critical layers 

1 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 small-scale 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 wave-scale 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 inertia-gravity internal waves (with allowance for the rotation of the Earth) vertical fluxes are non-zero and when turbulent viscosity and diffusion are ignored for vertically non-uniform two-dimensional 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 two-dimensional 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).

The system of equations of hydrodynamics for wave perturbations in the Boussinesq approximation has the form
$$\begin{aligned} \frac{D u}{D t} - fv + w\frac{d^{} U_0}{d z^{}} =-\frac{1}{\rho _0(0)}\frac{\partial P}{\partial x} \end{aligned}$$
(1)
$$\begin{aligned} \frac{D v}{D t} + fu + w\frac{d^{} V_0}{d z^{}} =-\frac{1}{\rho _0(0)}\frac{\partial P}{\partial y} \end{aligned}$$
(2)
$$\begin{aligned} \frac{D w}{D t} = -\frac{1}{\rho _0(0)} \frac{\partial P}{\partial z} - \frac{g\rho }{\rho _0(0)} \end{aligned}$$
(3)
$$\begin{aligned} \frac{D \rho }{D t} = -w \frac{d^{} \rho _0}{d z^{}} \end{aligned}$$
(4)
$$\begin{aligned} \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \end{aligned}$$
(5)
where xyz are two horizontal and vertical coordinates, the z-axis is directed vertically upwards, uvw are, respectively, two horizontal and vertical components of the wave flow velocity, \(\rho \) and P are wave perturbations of density and pressure, H is the depth of the sea, \(\rho _0(z)\) 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 free fall, the action of the operator \(\frac{D }{D t}\) is determined by the formula \(\frac{D }{D t} = \frac{\partial }{\partial t} + (u + U_0)\frac{\partial }{\partial x} + (v + V_0)\frac{\partial }{\partial y} + w\frac{\partial }{\partial z}\).
Boundary condition on the sea surface \((z\,{=}\,0)\) the condition of a “hard cover”, which filters out internal waves from surface waves [8]:
$$\begin{aligned} w(0) = 0 \end{aligned}$$
(6)
The boundary condition at the bottom is the condition of “non-flow”:
$$\begin{aligned} w(-H) = 0 \end{aligned}$$
(7)

3 Linear Approximation

We seek solutions of the linear approximation in the form:
$$ u_1 = u_{10}(z)Ae^{i\theta } + c.c., \ v_1 = v_{10}(z)Ae^{i\theta } + c.c., \ w_1 = w_{10}(z)Ae^{i\theta } + c.c. $$
$$\begin{aligned} P_1 = P_{10}(z)Ae^{i\theta } + c.c., \ \rho _1 = \rho _{10}(z)Ae^{i\theta } + c.c. \end{aligned}$$
(8)
where c.c. is complex conjugate term, A is the amplitude factor, \(\theta \) is phase of the wave; \(\frac{\partial \theta }{\partial x} = k, \frac{\partial \theta }{\partial t} = -\omega \), k is horizontal wave number, \(\omega \) 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 \(u_{10}, v_{10}, \rho _{10}, P_{10}\) with \(w_{10}\)
$$\begin{aligned} u_{10} = \frac{i}{k}\frac{d^{} w_{10}}{d z^{}}, \qquad \varOmega = \omega - k\cdot U_0 \end{aligned}$$
(9)
$$\begin{aligned} \frac{P_{10}}{\rho _{0}(0)} = \frac{i}{k}\left[ \frac{\varOmega }{k}\frac{d^{} w_{10}}{d z^{}} + \frac{d^{} U_0}{d z^{}}w_{10} + \frac{f}{\varOmega }\left( i\frac{d^{} V_0}{d z^{}}w_{10} - \frac{f}{k}\frac{d^{} w_{10}}{d z^{}}\right) \right] \end{aligned}$$
(10)
$$\begin{aligned} \rho _{10} = -\frac{i}{\varOmega }w_{10}\frac{d^{} \rho _0}{d z^{}}, \qquad v_{10} = \frac{1}{\varOmega }\left( \frac{f}{k}\frac{d^{} w_{10}}{d z^{}} - iw_{10}\frac{d^{} V_0}{d z^{}}\right) \end{aligned}$$
(11)
Function \(w_{10}\) satisfies the equation
$$\begin{aligned}&\frac{d^{2} w_{10}}{d z^{2}} + k \left[ \frac{if\frac{d^{} V_0}{d z^{}}}{\varOmega ^2-f^2}- \frac{f^2\frac{d^{} U_0}{d z^{}}}{\varOmega (\varOmega ^2-f^2)} \right] \frac{d^{} w_{10}}{d z^{}} + \nonumber \\&\qquad \qquad + kw_{10} \left[ \frac{k(N^2-\varOmega ^2) + \varOmega \frac{d^{2} U_0}{d z^{2}} + if\frac{d^{2} V_0}{d z^{2}}}{\varOmega ^2-f^2}+ \frac{ifk\frac{d^{} U_0}{d z^{}}\frac{d^{} V_0}{d z^{}}}{\varOmega (\varOmega ^{2}-f^{2})} \right] =0 \end{aligned}$$
(12)
where \(N^2 = -\frac{g}{\rho _0(0)}\frac{d^{} \rho _0}{d z^{}}\) is the square of Brunt-Vaisala frequency.
Boundary conditions for \(w_{10}\):
$$\begin{aligned} z = 0, \qquad \qquad \qquad \qquad w_{10} = 0 \end{aligned}$$
(13)
$$\begin{aligned} z = -H, \qquad \qquad \qquad \qquad w_{10} = 0 \end{aligned}$$
(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)
$$ z = Hz^\prime , \ t = t^\prime /\omega _*, \ w_{10}=w_{10}^\prime V_{0*}, \ V_0 = V_0^{\prime }V_{0*}, \ U_0=U_0^\prime V_{0*}, $$
$$\begin{aligned} \ k = k^\prime /H, \ f = f^\prime \omega _*, \ \omega = \omega ^\prime \omega _*, \ \varOmega = \varOmega ^\prime \omega _*, \end{aligned}$$
(15)
where \(\omega _*\) 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:
$$\begin{aligned}&\frac{d^{2} w_{10}^\prime }{d z{^\prime }^{2}} + k'\left[ \frac{i\varepsilon f'\frac{d^{} V_0^\prime }{d z{^\prime }^{}}}{\varOmega ^{\prime 2}-f^{\prime 2}}- \frac{\varepsilon f'^{2}\frac{d^{} U_0^\prime }{d z{^\prime }^{}}}{\varOmega '(\varOmega ^{\prime 2}-f^{\prime 2})} \right] \frac{d^{} w_{10}^\prime }{d z{^\prime }^{}}+\nonumber \\&\quad +k'w_{10}' \left[ \frac{k'(N'^{2}-\varOmega '^{2}) + \varepsilon \varOmega ' \frac{d^{2} U_0^\prime }{d z{^\prime }^{2}} + i\varepsilon f\frac{d^{2} V_0^\prime }{d z{^\prime }^{2}}}{\varOmega '^{2}-f'^{2}}+ \frac{i\varepsilon ^2f'k'\frac{d^{} U_0^\prime }{d z{^\prime }^{}}\frac{d^{} V_0^\prime }{d z{^\prime }^{}}}{\varOmega '(\varOmega '^{2}-f'^{2})} \right] =0 \end{aligned}$$
(16)
\(\varepsilon = V_{0*}/H\omega _{*}\) is a small parameter. The imaginary part of the coefficients in Eq. (16) is of the order of \(\varepsilon \), therefore the imaginary part of the solution \(w_{10}\) is also proportional \(\varepsilon \) i.e. the solution of Eq. (16) is represented in the form:
$$\begin{aligned} w_{10}^{\prime } = w_{0}^{\prime }(z^{\prime }) + \varepsilon iw_{1}^{\prime }(z^{\prime }) \end{aligned}$$
(17)
where \(w_0^\prime (z^\prime )\) and \(w_1^\prime (z^\prime )\) are real functions. The frequency is also expressed as a parameter expansion \(\varepsilon \)
$$\begin{aligned} \omega ^\prime = \omega _0^\prime + \varepsilon \sigma _1^\prime + \dots \end{aligned}$$
(18)
then \(\varOmega ^\prime =\varOmega _0^\prime +\varepsilon \sigma _1^\prime + \dots \). After substituting (17), (18) into (12), we obtain boundary value problems for \(w_0^\prime (z^\prime )\) and \(w_1^\prime (z^\prime )\). Function \(w_0^\prime (z^\prime )\) satisfies the equation (up to terms \(\sim \varepsilon \)):
$$\begin{aligned}&\frac{d^{2} w_0^\prime }{d z{^\prime }^{2}}-\epsilon k^\prime \frac{d^{} w_0^\prime }{d z{^\prime }^{}}\frac{d^{} U_0^\prime }{d z{^\prime }^{}}\frac{f^{\prime 2}}{\varOmega _0^\prime (\varOmega _0^{\prime 2}-f^{\prime 2})}+\nonumber \\&\qquad \qquad \qquad \qquad +\frac{k^\prime w_0^\prime }{(\varOmega _0^{\prime 2}-f^{\prime 2})}\left[ k^\prime (N^{\prime 2}-\varOmega _0^{\prime 2})+\varepsilon \varOmega _0^\prime \frac{d^{2} U_0^\prime }{d z{^\prime }^{2}}\right] =0 \end{aligned}$$
(19)
The boundary conditions for \(w_0^\prime \)
$$\begin{aligned} w_0^\prime (0) = 0, \qquad \qquad \qquad \qquad w_0^\prime (-1) = 0 \end{aligned}$$
(20)
Function \(w_1^\prime (z^\prime )\) satisfies the equation (up to terms \(\sim \varepsilon \)):
$$\begin{aligned}&\frac{d^{2} w_1^\prime }{d z{^\prime }^{2}}-\varepsilon k^\prime \frac{d^{} w_1^\prime }{d z{^\prime }^{}}\frac{d^{} U_0^\prime }{d z{^\prime }^{}}\frac{f^{\prime 2}}{\varOmega _0^\prime (\varOmega _0^{\prime 2}-f^{\prime 2})}+ \nonumber \\&\qquad \qquad \quad +\frac{k^\prime w_1^\prime }{(\varOmega _0^{\prime 2}-f^{\prime 2})}\left[ k^\prime (N^{\prime 2}-\varOmega _0^{\prime 2})+\varepsilon \varOmega _0^\prime \frac{d^{2} U_0^\prime }{d z{^\prime }^{2}}\right] =F^\prime (z^\prime ) \end{aligned}$$
(21)
where
$$\begin{aligned} F'(z')&=-k'\frac{d^{} w_0^\prime }{d z{^\prime }^{}}\frac{d^{} V_0^\prime }{d z{^\prime }^{}}\frac{f'}{(\varOmega _0^{'2}-f^{'2})}+ ik'\frac{d^{} w_0^\prime }{d z{^\prime }^{}}\frac{d^{} U_0^\prime }{d z{^\prime }^{}}\frac{\sigma _1^\prime f^{'2}(3\varOmega _0^{'2}-f^{'2})}{\varOmega _0^{'2}(\varOmega _0^{'2}-f^{'2})^2}-\\&-\frac{k'w_0^\prime }{(\varOmega _0^{'2}-f^{'2})} \left[ k'\frac{2i\varOmega _0^\prime \sigma _1^\prime (N^{'2}-f^{'2})}{(\varOmega _0^{'2}-f^{'2})}+\varepsilon \frac{d^{2} U_0^\prime }{d z{^\prime }^{2}}\frac{i\sigma _1^\prime (\varOmega _0^{'2}+f^{'2})}{(\varOmega _0^{'2}-f^{'2})}+\right. \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \left. +f'\frac{d^{2} V_0^\prime }{d z{^\prime }^{2}}+\varepsilon \frac{f'k'}{\varOmega _0^\prime }\frac{d^{} U_0^\prime }{d z{^\prime }^{}}\frac{d^{} V_0^\prime }{d z{^\prime }^{}}\right] \end{aligned}$$
The boundary conditions for \(w_1'\)
$$\begin{aligned} w_1^\prime (0) = 0, \qquad \qquad \qquad \qquad w_1^\prime (-1) = 0 \end{aligned}$$
(22)
After the transition to dimensional variables, Eq. (19) takes the form:
$$\begin{aligned}&\frac{d^{2} w_0}{d z^{2}}-k\frac{d^{} w_0}{d z^{}}\frac{d^{} U_0}{d z^{}}\frac{f^2}{\varOmega _0(\varOmega _0^{2}-f^{2})}+ \nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad +\frac{k w_0}{(\varOmega _0^{2}-f^{2})}\left[ k(N^{2}-\varOmega _0^{2})+\varOmega _0\frac{d^{2} U_0}{d z^{2}}\right] =0 \end{aligned}$$
(23)
where \(\varOmega _0=\omega -kU_0\) is wave frequency with Doppler shift.
Equation (23) should be supplemented by boundary conditions:
$$\begin{aligned} w_0(0) = 0, \qquad \qquad \qquad \qquad w_0(-H) = 0 \end{aligned}$$
(24)
The boundary-value problem (23), (24) in the absence of flow for \(U_0=0\) has a countable set of eigenfunctions, a set of modes. Moreover, to each value of the wavenumber k corresponds to a certain frequency value \(\omega _0<max(N)\) corresponding to the given mode. When \(U_0\ne 0\) the discrete spectrum of real eigenfrequencies may not exist [9]. This is connected with the singularities in Eq. (23) with \(\varOmega _0=0\) and \(\varOmega _0=\pm f\) (hydrodynamically stable flows are considered). In the presence of singular \(\varOmega _0=0\), there is a critical layer [10], where the phase velocity of the wave is equal to the flow velocity. With allowance for the rotation of the Earth, the singularity shifts to a level where \(\varOmega _0 = f\) [11]. The effect of this singularity on the dispersion curves is illustrated by the calculations given below.
Let \(a(z)=-\frac{f^2k}{\varOmega _0(\varOmega _0^2-f^2)}\frac{d^{} U_0}{d z^{}}\), \(b(z)=\frac{k}{(\varOmega _0^2-f^2)}\left[ k(N^2-\varOmega _0^2)+\varOmega _0\frac{d^{2} U_0}{d z^{2}}\right] \), then Eq. (23) can be written in the form:
$$\begin{aligned} \frac{d^{2} w_0}{d z^{2}}+a(z)\frac{d^{} w_0}{d z^{}}+b(z)w_0=0 \end{aligned}$$
(25)
Equation (25) leads to a self-adjoint form, multiplying both sides of the equation by \(p(z)=\exp (\int a(z)\, dz)\):
$$\begin{aligned} \frac{d}{dz}\left( p(z)\frac{d^{} w_0}{d z^{}}\right) -q(z)w_0=0 \end{aligned}$$
(26)
here \(q(z)=-b(z)p(z)\).
After the transition to dimensional variables, Eq. (21) is transformed to the form
$$\begin{aligned} \frac{d^{2} w_1}{d z^{2}}+a(z)\frac{d^{} w_1}{d z^{}}+b(z)w_1=F_1(z) \end{aligned}$$
(27)
where
$$\begin{aligned} F(z)&=-k\frac{d^{} w_0}{d z^{}}\frac{d^{} V_0}{d z^{}}\frac{f}{(\varOmega _0^{2}-f^{2})}+ ik\frac{d^{} w_0}{d z^{}}\frac{d^{} U_0}{d z^{}}\frac{\sigma _1 f^{2}(3\varOmega _0^{2}-f^{2})}{\varOmega _0^{2}(\varOmega _0^{2}-f^{2})^2}-\\&\quad -\frac{kw_0}{(\varOmega _0^{2}-f^{2})} \left[ k\frac{2i\varOmega _0\sigma _1(N^{2}-f^{2})}{(\varOmega _0^{2}-f^{2})}+i\frac{d^{2} U_0}{d z^{2}}\frac{\sigma _1(\varOmega _0^{2}+f^{2})}{(\varOmega _0^{2}-f^{2})}+\right. \\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left. +f\frac{d^{2} V_0}{d z^{2}}+\frac{fk}{\varOmega _0}\frac{d^{} U_0}{d z^{}}\frac{d^{} V_0}{d z^{}}\right] \end{aligned}$$
The boundary conditions for the function \(w_1\):
$$\begin{aligned} w_1(0) = 0, \qquad \qquad \qquad \qquad w_1(-H) = 0 \end{aligned}$$
(28)
We multiply both sides of the linear inhomogeneous Eq. (27) by the function p(z) we obtain on the left-hand side a self-adjoint operator, the same as in the linear homogeneous Eq. (26):
$$\begin{aligned} \frac{d}{dz}\left( p(z)\frac{d^{} w_1}{d z^{}}\right) -q(z)w_1=F_1(z) \end{aligned}$$
(29)
where \(F_1(z)=p(z)F(z)\).
The solvability condition for the boundary value problem (28), (29) [12]:
$$\begin{aligned} \int _{-H}^{0} F_1w_0\, dz = 0 \end{aligned}$$
(30)
Hence the expression for \(\sigma _1\):
$$\begin{aligned} \sigma _1 = \frac{a}{b} \end{aligned}$$
where
$$\begin{aligned} a = ifk\int _{-H}^{0} \frac{pw_0}{(\varOmega _0^2-f^2)}\left( \frac{d^{} w_0\frac{d^{} V_0}{d z^{}}}{d z^{}}+w_0\frac{k}{\varOmega _0}\frac{d^{} U_0}{d z^{}}\frac{d^{} V_0}{d z^{}}\right) \, dz, \end{aligned}$$
$$\begin{aligned} b&= \int _{-H}^{0} \frac{pkw_0}{(\varOmega _0^2-f^2)^2}\left[ w_0\left( 2k\varOmega _0(N^2-f^2)+\frac{d^{2} U_0}{d z^{2}}(\varOmega _0^2+f^2)\right) -\right. \nonumber \\&\qquad \qquad \qquad \qquad \qquad \left. -f^2\frac{d^{} w_0}{d z^{}}\frac{d^{} U_0}{d z^{}}\frac{(3\varOmega _0^2-f^2)}{\varOmega _0^2}\right] \, dz \end{aligned}$$
(31)

4 Nonlinear Effects

The velocity of the Stokes drift of the liquid particles is determined by the formula [13]:
$$\begin{aligned} \mathbf{u}_s = \overline{\int _0^t \mathbf{u}d\tau \triangledown \mathbf{u}} \, , \end{aligned}$$
(32)
where \(\mathbf{u}\) is the field of wave Euler velocities, the bar above denotes averaging over the wave period.
The vertical component of the Stokes drift velocity is determined by the formula:
$$\begin{aligned} w_s = iA_1A_1^{*}\left( \frac{1}{\omega }-\frac{1}{\omega ^{*}}\right) \frac{d}{dz}(w_{10}w_{10}^{*}) \end{aligned}$$
(33)
where \(A_1 = A\exp (\delta \omega \cdot t)\), \(\delta \omega = \sigma /i\) is the damping decrement of the wave, the value of \(\sigma _1\) is purely imaginary, \(A_1=A\) at the initial time at \(t = 0\).

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.

The vertical wave mass flux is determined by the formula:
$$\begin{aligned} \overline{\rho w}/\left| A_1\right| ^2 = -w_{10}w_{10}^{*}\left( \frac{i}{\varOmega }-\frac{i}{\varOmega ^{*}}\right) \frac{d^{} \rho _0}{d z^{}} \end{aligned}$$
(34)
The presence of a vertical wave mass flux leads to an irreversible deformation of the density field, which can be regarded as a vertical fine structure generated by a wave. The equations for the nonoscillating on a time scale, the correction to the mean density \(\overline{\rho }\):
$$\begin{aligned} \frac{D \overline{\rho }}{D t}+\frac{\partial \overline{\rho u}}{\partial x}+\frac{\partial \overline{\rho v}}{\partial y}+\frac{\partial \overline{\rho w}}{\partial z}+w_s\frac{d^{} \rho _0}{d z^{}}=0 \end{aligned}$$
from here we have:
$$\begin{aligned} \frac{\partial \overline{\rho }}{\partial t}+\frac{\partial \overline{\rho w}}{\partial z}+w_s\frac{d^{} \rho _0}{d z^{}}=0 \end{aligned}$$
(35)
Integrate Eq. (35) in time
$$\begin{aligned} \overline{\rho (t)}=\overline{\rho (0)}-\int _0^t \left( \frac{\partial \overline{\rho w}}{\partial z}+w_s\frac{d^{} \rho _0}{d z^{}}\right) \, dt' \end{aligned}$$
(36)
Substituting \(\overline{\rho w}\) (34) and the vertical component of the Stokes drift velocity \(w_s\) (33), we obtain in (36) after integration
$$\begin{aligned} \overline{\varDelta \rho }=\overline{\rho (t)}-\rho (0)=\left[ \frac{\partial \overline{\rho w_0}}{\partial z}+w_{s0}\frac{d^{} \rho _0}{d z^{}}\right] \cdot \frac{1}{2\delta \omega }\left( 1-\exp ^{2\delta \omega \cdot t}\right) , \end{aligned}$$
(37)
where
$$\begin{aligned}\begin{gathered} \overline{\rho w_0} = i\left| A\right| ^2 w_{10}w_{10}^{*}\left( \frac{1}{\varOmega ^{*}}-\frac{1}{\varOmega }\right) \frac{d^{} \rho _0}{d z^{}}, \qquad w_{s0} = i\left| A\right| ^2\left( \frac{1}{\omega }-\frac{1}{\omega ^{*}}\right) \frac{d}{dz}(w_{10}w_{10}^{*}) \end{gathered}\end{aligned}$$
Passing to the limit in (37) for \(t\rightarrow \infty \) taking into account that \(\delta \omega <0\) we find \(\overline{\varDelta \rho }\)
$$\begin{aligned} \overline{\varDelta \rho } = \left[ \frac{\partial \overline{\rho w_0}}{\partial z}+w_{s0}\frac{d^{} \rho _0}{d z^{}}\right] \cdot \frac{1}{2\delta \omega } \end{aligned}$$
(38)

5 Results of Calculations

Wave mass fluxes are calculated for internal waves that were observed during the full-scale experiment in the third stage of the 44th voyage of the research vessel “Mikhail Lomonosov” on the northwestern shelf of the Black Sea.

At the Fig. 1 presents four realizations of elevations of temperature isolines obtained from GRAD instruments (gradient-distributed temperature sensors) [14].
Fig. 1.

Time course of vertical displacements of temperature isolines

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 Brent-Vaisala 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 15-min internal waves of the second mode is shown in Fig. 3b.

The wavenumber is 0.032 rad/m. The normalizing factor \(A_1\) we find from the known value the maximum amplitude of the vertical displacements. To do this, we express the vertical displacement \(\zeta \) using the relation \(\frac{d^{} \zeta }{d t^{}}=w\):
$$\begin{aligned} \zeta = \frac{iw_0}{\varOmega _0}A_1\exp (ikx-i\omega _0t)+c.c. \end{aligned}$$
This implies:
$$\begin{aligned} A_1 = \frac{\max \zeta }{2\max \left| w_0/\varOmega _0\right| } \end{aligned}$$
Fig. 2.

Vertical profiles of flow velocity components \(U_0\) (1), \(V_0\) (2)

Fig. 3.

Vertical profiles of Brunt-Vaisala frequency (a) and eigenfunction of the 15-min internal waves of the second mode (b)

Thus, the amplitude of vertical displacements is proportional to \(w_0\). Extremums of the function \(w_0\) correspond to the maximum vertical displacements from the experimental data (Figs. 1 and 3b) i.e. in the experiment, the second mode was observed. The wavelength of fifteen-minute internal waves of the second mode is 196 m. Dispersion curves of the first two modes are shown in Fig. 4. If the flow were not taken into account, the dispersion curves in the low-frequency area would begin with a minimum frequency equal to the inertial one. When the flow is taken into account, because of the effect of the singularity \(\varOmega _0 = f\) the dispersion curves are cut off at low frequencies. The minimum frequency of the first mode corresponds to \(1.13\cdot 10^{-4}\) rad/s, for the second mode \(3.49\cdot 10^{-4}\) rad/s (for comparison we indicate that the Coriolis frequency is \(1.048\cdot 10^{-4}\) rad/s). Cutoff dispersion curves occurs due to the influence of critical layers, where the frequency of the wave with the Doppler shift is equal to the inertial one.
Fig. 4.

Dispersion curves of the first (1) and second (2) modes

Fig. 5.

Dependence of the wave attenuation decrement on the wavenumber for the first (1) and second (2) modes

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 15-min 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 low-frequency area is due to the cutoff of the dispersion curves in the vicinity of the inertial frequency.

Wave mass flux \(\overline{\rho w}\) (34) are compared for the first two modes at the same wave amplitude in Fig. 6. The wave flux of the first mode dominates in the upper 30-m layer. Deeper these flows are comparable in magnitude.
Fig. 6.

Vertical wave mass fluxes \(\overline{\rho w}\) for the first (1) and second (2) modes

Fig. 7.

A comparison of the total wave and turbulent (3) mass fluxes for the first (1) and second (2) modes

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 40-m layer, deeper these fluxes are comparable in magnitude (Fig. 7). The wave fluxes exceed the turbulent flux in absolute value.

The vertical density profile is shown in Fig. 8a. A nonoscillating wave on a time scale, the corrections to the mean density \(\overline{\varDelta \rho }\) (37) contains the quantity \(|A^2|\) which is exactly equal to \(|A_1^2|\) at the initial time. The nonoscillating correction to the average density is shown in Fig. 8b and is a vertical fine structure generated by a wave which is irreversible, with no inversions in the field of average density.
Fig. 8.

Vertical profiles of medium density (a) and nonoscillating wave-scale corrections to the density (b) for the first (1) and second (2) modes

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 profile-the fine structure generated by the wave.

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

Notes and Comments. The work was carried out within the framework of the state task on the topic № 0827-2014-0010 “Complex interdisciplinary studies of oceanological processes determining the functioning and evolution of the systems of the Black and Azov Seas on the basis of modern methods for monitoring the state of the marine environment and grid technologies”.

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Marine Hydrophysical InstituteRussian Academy of SciencesSevastopolRussia
  2. 2.Moscow State UniversityMoscowRussia

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