Solitary Waves in a Two-Layer Fluid with Piecewise Exponential Stratification

Abstract

The problem of internal stationary waves in a two-layer fluid with a density, which exponentially depends on the depth inside the layers and undergoes a jump at the interface, is considered. A nonlinear equation is derived in the second-order long-wave approximation, and asymptotic submodels describing solitary waves of finite amplitude are examined. The dispersion properties and wave-propagation modes depending on the dimensionless parameters of the background piecewise constant flow are studied.

APPENDIX

Presented below are expressions for the coefficients in Eq. (4.4) of the functions $P$ and $Q$ in Eq. (4.3). For the function $P$ linearly depending on the square of Froude numbers $F_{1}^{2}$ and $F_{2}^{2}$, the following presentation is valid:

$$P(\eta ;F) = {{P}_{0}}(\eta ) + \mu F_{1}^{2}{{P}_{1}}(\eta ) + \mu F_{2}^{2}{{P}_{2}}(\eta ),$$

where ${{P}_{j}}(\eta ) = {{p}_{{j0}}} + {{p}_{{j1}}}\eta + {{p}_{{j2}}}{{\eta }^{2}}$ ($j = 0,1,2$) are polynomials of the second power in $\eta$ with the trigonometric coefficients

$${{p}_{{00}}} = - \frac{\mu }{2}$$

$${{p}_{{01}}} = - \frac{{{{\sigma }_{1}}(1 + \mu )}}{{48{{{\sin }}^{2}}{{\alpha }_{1}}{{{\cos }}^{2}}({{\alpha }_{1}}{\text{/2}})}}(2{{\sigma }_{1}} - 2 + (4{{\sigma }_{1}} - 1)\cos {{\alpha }_{1}} + 2\cos 2{{\alpha }_{1}} + \cos 3{{\alpha }_{1}})$$

$$ - \;\frac{{r{{\sigma }_{2}}}}{{48{{{\sin }}^{2}}{{\alpha }_{2}}{{{\cos }}^{2}}({{\alpha }_{2}}{\text{/2}})}}(2{{\sigma }_{2}} + 2 + (4{{\sigma }_{2}} + 1)\cos {{\alpha }_{2}} - 2\cos 2{{\alpha }_{2}} - \cos 3{{\alpha }_{2}}),$$

$${{p}_{{02}}} = - \frac{{\sigma _{1}^{2}(1 + \mu )}}{{24{{{\sin }}^{2}}{{\alpha }_{1}}{{{\cos }}^{2}}({{\alpha }_{1}}{\text{/2}})}}(1 + 2\cos {{\alpha }_{1}}) + \frac{{r\sigma _{2}^{2}}}{{24{{{\sin }}^{2}}{{\alpha }_{2}}{{{\cos }}^{2}}({{\alpha }_{2}}{\kern 1pt} /{\kern 1pt} 2)}}(1 + 2\cos {{\alpha }_{2}}),$$

$${{p}_{{10}}} = \frac{1}{4}(2{{\lambda }_{1}}{\text{ctg}}{{\alpha }_{1}} + {{\sigma }_{1}}),$$

$${{p}_{{20}}} = \frac{1}{4}(2{{\lambda }_{2}}{\text{ctg}}{{\alpha }_{2}} + {{\sigma }_{2}}),$$

$${{p}_{{11}}} = - \frac{{{{\lambda }_{1}}{{\sigma }_{1}}}}{{144\sin ({{\alpha }_{1}}{\kern 1pt} /{\kern 1pt} 2){{{\cos }}^{3}}({{\alpha }_{1}}{\text{/2}})}}{{(1 + 2\cos {{\alpha }_{1}})}^{2}},$$

$${{p}_{{21}}} = - \frac{{r{{\lambda }_{2}}{{\sigma }_{2}}}}{{144\sin ({{\alpha }_{2}}/2)\cos 3({{\alpha }_{2}}{\text{/2}})}}{{(1 + 2\cos {{\alpha }_{2}})}^{2}},$$

$${{p}_{{12}}} = {{p}_{{22}}} = 0$$

Consequently, the function $Q$ can be presented as:

$$Q(\eta ;F) = \frac{{\mu F_{1}^{2}}}{{8{{\lambda }_{1}}}}\frac{{{{Q}_{1}}(\eta )}}{{{{{\sin }}^{5}}{{\alpha }_{1}}(\eta )}} + \frac{{\mu F_{2}^{2}}}{{8{{r}^{2}}{{\lambda }_{2}}}}\frac{{{{Q}_{2}}(\eta )}}{{{{{\sin }}^{5}}{{\alpha }_{2}}(\eta )}},$$

where ${{Q}_{j}}(\eta ) = {{q}_{{j0}}} + {{q}_{{j1}}}\eta + {{q}_{{j2}}}{{\eta }^{2}} + {{q}_{{j3}}}{{\eta }^{3}} + {{q}_{{j4}}}{{\eta }^{4}}$ with coefficients

$${{q}_{{10}}} = {{\sin }^{3}}{{\alpha }_{1}}(2{{\lambda }_{1}} - \sin 2{{\alpha }_{1}}),$$

$${{q}_{{20}}} = {{\sin }^{3}}{{\alpha }_{2}}(2{{\lambda }_{2}} - \sin 2{{\alpha }_{2}}),$$

$${{q}_{{11}}} = - \frac{1}{4}{{\lambda }_{1}}\sin {{\alpha }_{1}}(8\lambda _{1}^{2} - 7 + (8\lambda _{1}^{2} + 4)\cos 2{{\alpha }_{1}} + 3\cos 4{{\alpha }_{1}} + 4{{\lambda }_{1}}\sin 2{{\alpha }_{1}}),$$

$${{q}_{{21}}} = \frac{1}{4}r{{\lambda }_{2}}\sin {{\alpha }_{2}}(8\lambda _{2}^{2} - 7 + (8\lambda _{2}^{2} + 4)\cos 2{{\alpha }_{2}} + 3\cos 4{{\alpha }_{2}} + 4{{\lambda }_{2}}\sin 2{{\alpha }_{2}}),$$

$${{q}_{{12}}} = \frac{{\lambda _{1}^{2}}}{{16}}\left( {\left( {56\lambda _{1}^{2} - 18} \right)\cos {{\alpha }_{1}} + \left( {8\lambda _{1}^{2} + 15} \right)\cos 3{{\alpha }_{1}} + 3\cos 5{{\alpha }_{1}} - 44{{\lambda }_{1}}\sin \alpha _{1}^{{}} - 28{{\lambda }_{1}}\sin 3{{\alpha }_{1}}} \right),$$

$${{q}_{{22}}} = - \frac{{{{r}^{2}}\lambda _{2}^{2}}}{{16}}\left( {\left( {56\lambda _{2}^{2} - 18} \right)\cos {{\alpha }_{2}}{\kern 1pt} {\kern 1pt} + \left( {8\lambda _{2}^{2} + 15} \right)\cos 3{{\alpha }_{2}}{\kern 1pt} {\kern 1pt} + 3\cos 5{{\alpha }_{2}}{\kern 1pt} {\kern 1pt} - 44\lambda _{2}^{{}}\sin {{\alpha }_{2}}{\kern 1pt} {\kern 1pt} - 28{{\lambda }_{2}}\sin 3{{\alpha }_{2}}} \right),$$

$${{q}_{{13}}} = \frac{{\lambda _{1}^{3}}}{4}(28{{\lambda }_{1}}\cos {{\alpha }_{1}} + 4{{\lambda }_{1}}\cos 3{{\alpha }_{1}} - 7\sin {{\alpha }_{1}} - 3\sin 3{{\alpha }_{1}}),$$

$${{q}_{{23}}} = - \frac{{{{r}^{3}}\lambda _{2}^{3}}}{4}(28{{\lambda }_{2}}\cos {{\alpha }_{2}} + 4{{\lambda }_{2}}\cos 3{{\alpha }_{2}} - 7\sin {{\alpha }_{2}} - 3\sin 3{{\alpha }_{2}}),$$

$${{q}_{{14}}} = \frac{{\lambda _{1}^{4}}}{2}(7\cos {{\alpha }_{1}} + \cos 3{{\alpha }_{1}}),\quad {{q}_{{24}}} = \frac{{{{r}^{4}}\lambda _{2}^{4}}}{2}(7\cos {{\alpha }_{2}} + \cos 3{{\alpha }_{2}}).$$

In all the formulas presented above, ${{\alpha }_{1}} = {{\lambda }_{1}}(1 + \eta )$, ${{\alpha }_{2}} = {{\lambda }_{2}}(1 - r\eta )$.