Two-Dimensional Periodic Waves in an Inviscid Continuously Stratified Fluid

Abstract

In this paper we study the propagation of two-dimensional periodic waves in an inviscid continuously stratified fluid with a free surface in the frequency range from ${{10}^{{ - 4}}}$ to $5\,\, \times \,\,{{10}^{2}}$ Hz. Dispersion relations, as well as expressions for phase and group velocities for surface and internal waves in physically observable variables, are given. The fluid is shown to behave as homogeneous when the wavelength reaches values on the order of the stratification scale. As the wave frequency approaches the buoyancy frequency, the energy transfer rate decreases: the group velocity of surface waves tends to zero, while the phase velocity tends to infinity. In the case of infinitesimal perturbations of a stratified fluid with a free surface, surface and internal waves exist in separated frequency intervals.

APPEDNIX A

In the Boussinesq approximation, the second equation of system (2) with allowance for accepted density distribution function (1) can be written as follows:

$${{\partial }_{t}}\mathbf{u} + \left( {\mathbf{u} \cdot \nabla } \right)\mathbf{u} = - \frac{{\nabla P}}{{{{\rho }_{{00}}}}} + \left( {1 + r\left( z \right) + s\left( {x,z,t} \right)} \right)g.$$

(A.1)

In the linear approximation, Eq. (A.1) in terms of components will be written as

$$\left\{ \begin{gathered} {{\partial }_{t}}u = - {{{{\partial }_{x}}P} \mathord{\left/ {\vphantom {{{{\partial }_{x}}P} {{{\rho }_{{00}}}}}} \right. \kern-0em} {{{\rho }_{{00}}}}} \hfill \\ {{\partial }_{t}}w = - {{{{\partial }_{z}}P} \mathord{\left/ {\vphantom {{{{\partial }_{z}}P} {{{\rho }_{{00}}}}}} \right. \kern-0em} {{{\rho }_{{00}}}}} - \left( {r\left( z \right) + s\left( {x,z,t} \right)} \right)g. \hfill \\ \end{gathered} \right.$$

(A.2)

It should be noted that, in expression (A.2), there are no pressure components responsible for the hydrostatic pressure of a liquid column with a constant density of ${{\rho }_{{00}}}$. We write the velocity components in terms of the $\psi$ stream function and differentiate the upper equation of system (A.2) with respect to the z coordinate and the lower equation with respect to the x coordinate:

$$\left\{ \begin{gathered} {{\partial }_{{tzz}}}\psi = - {{{{\partial }_{{xz}}}P} \mathord{\left/ {\vphantom {{{{\partial }_{{xz}}}P} {{{\rho }_{{00}}}}}} \right. \kern-0em} {{{\rho }_{{00}}}}} \hfill \\ - {{\partial }_{{txx}}}\psi = - {{{{\partial }_{{xz}}}P} \mathord{\left/ {\vphantom {{{{\partial }_{{xz}}}P} {{{\rho }_{{00}}}}}} \right. \kern-0em} {{{\rho }_{{00}}}}} - g{{\partial }_{x}}s. \hfill \\ \end{gathered} \right.$$

(A.3)

We subtract the lower equation from the upper equation and obtain an equation relating the $\psi$stream function and the $s$ density perturbation:

$$ - g{{\partial }_{x}}s + {{\partial }_{t}}\Delta \psi = 0.$$

(A.4)

The continuity equation with allowance for the accepted assumptions and notation will be rewritten in the form

$${{\rho }_{{00}}}{{\partial }_{t}}s + {{\rho }_{{00}}}\left( {1 + r + s} \right){\text{div}}{\mathbf{u}} + {\mathbf{u}}\nabla \rho = 0.$$

(A.5)

With allowance for the introduction of the stream function,

$${\text{div}}{\mathbf{u}} = {{\partial }_{x}}u + {{\partial }_{z}}w = {{\partial }_{{xz}}}\psi - {{\partial }_{{zx}}}\psi = 0,$$

(A.6)

the continuity equation will be rewritten:

$${{\partial }_{t}}s + {{\partial }_{z}}\psi {{\partial }_{x}}s{ } - {{\partial }_{x}}\psi \left( {{{\partial }_{z}}r + {{\partial }_{z}}s} \right) = 0.$$

(A.7)

In a linear approximation,

$${{\partial }_{t}}s{ } - {{\partial }_{x}}\psi {{\partial }_{z}}r = 0$$

(A.8)

is true.

We differentiate Eq. (A.4) with respect to time, Eq. (A.8) with respect to the x variable, multiply the latter by g, and add the results:

$${{\partial }_{{tt}}}\Delta \psi - g{{\partial }_{{xx}}}\psi {{\partial }_{z}}r = 0.$$

(A.9)

For the case of exponential stratification, Eq. (A.9) can be written explicitly:

$${{\partial }_{{tt}}}\Delta \psi + {{\partial }_{{xx}}}\psi {\kern 1pt} \exp \left( { - z{\text{/}}\Lambda } \right){g \mathord{\left/ {\vphantom {g \Lambda }} \right. \kern-0em} \Lambda } = 0.$$

(A.10)

Up to the notation adopted in the paper, Eq. (A.10) coincides with Eq. (3) in the main text of the paper.

It can be shown that the solution of linearized problem (7) and (14) nullifies nonlinear term $\left( {{\mathbf{u}} \cdot \nabla } \right){\mathbf{u}}$ in the Euler equation in system (2). Indeed, substitution of both (7) and (14) into $\left( {{\mathbf{u}} \cdot \nabla } \right){\mathbf{u}}$ produces

$$\begin{gathered} u{{\partial }_{x}}u + w{{\partial }_{z}}u = {{\partial }_{z}}\psi {{\partial }_{{xz}}}\psi - {{\partial }_{x}}\psi {{\partial }_{{zz}}}\psi = 0, \\ u{{\partial }_{x}}w + w{{\partial }_{z}}w = - {{\partial }_{z}}\psi {{\partial }_{{xx}}}\psi + {{\partial }_{x}}\psi {{\partial }_{{xz}}}\psi = 0. \\ \end{gathered} $$

(A.11)

Therefore, it can be argued that the solution of the linearized problem satisfies system (2) in the given problem formulation.

APPEDNIX B

The first root of dispersion equation (13) takes the form

$${{k}_{z}} = \frac{{2 \times {{3}^{{1/3}}}\left( {1 - {{\omega }^{2}}} \right) + {{2}^{{1/3}}}{{\omega }^{2}}{{{\left( {\frac{{9\varepsilon {{{\left( {1 - {{\omega }^{2}}} \right)}}^{2}}}}{{{{\omega }^{2}}}} + \sqrt 3 \sqrt {\frac{{{{{\left( {{{\omega }^{2}} - 1} \right)}}^{3}}\left( {4 + 27{{\varepsilon }^{2}}{{\omega }^{2}}\left( {{{\omega }^{2}} - 1} \right)} \right)}}{{{{\omega }^{6}}}}} } \right)}}^{{2/3}}}}}{{{{6}^{{2/3}}}{{\omega }^{2}}{{{\left( {\frac{{9\varepsilon {{{\left( {1 - {{\omega }^{2}}} \right)}}^{2}}}}{{{{\omega }^{2}}}} + \sqrt 3 \sqrt {\frac{{{{{\left( {{{\omega }^{2}} - 1} \right)}}^{3}}\left( {4 + 27{{\varepsilon }^{2}}{{\omega }^{2}}\left( {{{\omega }^{2}} - 1} \right)} \right)}}{{{{\omega }^{6}}}}} } \right)}}^{{1/3}}}}}.$$

(B.1)

For surface waves, $\omega > 1$, the expression in the denominator is always positive. The first term in the numerator is negative while the second term is positive. If we neglect the small terms containing ε (which make a positive contribution), then we obtain an expression that coincides with the similar one for an ideal homogeneous liquid:

$$2 \times {{3}^{{1/3}}}\left( {1 - {{\omega }^{2}}} \right) + {{2}^{{1/3}}}{{\omega }^{2}}{{\left( {\sqrt 3 \sqrt {\frac{{{{{\left( {{{\omega }^{2}} - 1} \right)}}^{3}}4}}{{{{\omega }^{6}}}}} } \right)}^{{2/3}}}.$$

(B.2)

Comparing the negative and positive terms, we can see that they are equal to each other at ω = 1 only. Accounting for the stratification of the liquid leads to an increase in the modulus of the positive term only.

The two remaining roots are complex conjugate and are as follows:

$$\begin{gathered} {{k}_{z}} = \frac{{2 \times {{3}^{{1/3}}}\left( {{{\omega }^{2}} - 1} \right) - {{2}^{{1/3}}}{{\omega }^{2}}{{{\left( {\frac{{9\varepsilon {{{\left( {1 - {{\omega }^{2}}} \right)}}^{2}}}}{{{{\omega }^{2}}}} + \sqrt 3 \sqrt {\frac{{{{{\left( {{{\omega }^{2}} - 1} \right)}}^{3}}\left( {4 + 27{{\varepsilon }^{2}}{{\omega }^{2}}\left( {{{\omega }^{2}} - 1} \right)} \right)}}{{{{\omega }^{6}}}}} } \right)}}^{{2/3}}}}}{{2 \times {{6}^{{2/3}}}{{\omega }^{2}}{{{\left( {\frac{{9\varepsilon {{{\left( {1 - {{\omega }^{2}}} \right)}}^{2}}}}{{{{\omega }^{2}}}} + \sqrt 3 \sqrt {\frac{{{{{\left( {{{\omega }^{2}} - 1} \right)}}^{3}}\left( {4 + 27{{\varepsilon }^{2}}{{\omega }^{2}}\left( {{{\omega }^{2}} - 1} \right)} \right)}}{{{{\omega }^{6}}}}} } \right)}}^{{1/3}}}}} \\ \pm \,\,i\frac{{2 \times {{3}^{{5/6}}}\left( {{{\omega }^{2}} - 1} \right) + {{2}^{{1/3}}}\sqrt 3 {{\omega }^{2}}{{{\left( {\frac{{9\varepsilon {{{\left( {1 - {{\omega }^{2}}} \right)}}^{2}}}}{{{{\omega }^{2}}}} + \sqrt 3 \sqrt {\frac{{{{{\left( {{{\omega }^{2}} - 1} \right)}}^{3}}\left( {4 + 27{{\varepsilon }^{2}}{{\omega }^{2}}\left( {{{\omega }^{2}} - 1} \right)} \right)}}{{{{\omega }^{6}}}}} } \right)}}^{{2/3}}}}}{{2 \times {{6}^{{2/3}}}{{\omega }^{2}}{{{\left( {\frac{{9\varepsilon {{{\left( {1 - {{\omega }^{2}}} \right)}}^{2}}}}{{{{\omega }^{2}}}} + \sqrt 3 \sqrt {\frac{{{{{\left( {{{\omega }^{2}} - 1} \right)}}^{3}}\left( {4 + 27{{\varepsilon }^{2}}{{\omega }^{2}}\left( {{{\omega }^{2}} - 1} \right)} \right)}}{{{{\omega }^{6}}}}} } \right)}}^{{1/3}}}}}. \\ \end{gathered} $$

(B.3)

As can be seen, the real part of the complex conjugate roots coincides with the first root up to a factor of $- \frac{1}{2}$.

The condition for the physical realization of the roots of the dispersion relations arises due to the need for motion attenuation with depth,

$$\operatorname{Re} \left( {{{k}_{z}}} \right) > 0,$$

(B.4)

and, for the existence of components in the wave vector responsible for the wave motion,

$$\operatorname{Im} \left( {{{k}_{z}}} \right) \ne 0,$$

(B.5)

and/or

$$\operatorname{Re} \left( {{{k}_{x}}} \right) \ne 0.$$

(B.6)

Condition (B.4) is satisfied only by root (B.1). In this case, for surface waves, this condition is satisfied both for waves with a frequency greater than the buoyancy frequency (in dimensionless variables, $\omega > 1$) and for waves with a frequency less than the buoyancy frequency ($\omega < 1$).

For surface waves, the relation between the ${{k}_{x}}$ and ${{k}_{z}}$ components in a dimensionless form is as follows:

$${{k}_{x}} = {{k}_{z}}\sqrt {\frac{{{{\omega }^{2}}}}{{{{\omega }^{2}} - 1}}} ;$$

(B.7)

for waves with a frequency greater than the buoyancy frequency,

$$\operatorname{Re} \left( {{{k}_{x}}} \right) = \frac{\omega }{{\sqrt {{{\omega }^{2}} - 1} }}\operatorname{Re} \left( {{{k}_{z}}} \right) > 0,$$

(B.8)

and for waves with a frequency less than the buoyancy frequency,

$$\operatorname{Re} \left( {{{k}_{x}}} \right) = \frac{\omega }{{\sqrt {1 - {{\omega }^{2}}} }}\operatorname{Im} \left( {{{k}_{z}}} \right)$$

(B.9)

is true.

For real liquids, $\varepsilon \ll 1$, and if we expand expression (B.1) in a series in terms of a small parameter ε, then no imaginary component is detected in the ${{k}_{z}}$ wavenumber component at least up to terms of the 20th order of smallness. Based on expressions (B.9) and the conditions for the physical realization of the roots of dispersion equation (B.4)–(B.6), it can be argued that, for surface waves, only one root (B.1) is physically realizable in the frequency range greater than the frequency buoyancy.