Exact solution and instability for geophysical edge waves

Abstract

<p style='text-indent:20px;'>We present an exact solution to the nonlinear governing equations in the <inline-formula><tex-math id="M1">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>-plane approximation for geophysical edge waves at an arbitrary latitude. Such an exact solution is derived in the Lagrange framework, which describes trapped waves propagating eastward or westward along a sloping beach with a shoreline parallel to the latitude line. Using the short-wavelength instability method, we establish a criterion for the instability of such waves.</p>

Free surface equatorial two-layer flows in the $\beta$-plane approximation with discontinuous stratification

“…Based on this work, there has been a plenty of researches about the exact azimuthal equatorial flows with continuous stratification, such as [15,18] in the cylindrical coordinates, [16,17] in the spherical coordinates, and [26] in the β-plane approximation. Moreover, we introduce some excellent studies related to the exact solutions to the geophysical fluid dynamics equations in the β-plane, see [14,21,24,25,28]. For exact azimuthal solutions for equatorial flows with discontinuous stratification, see [22] for the case of a discontinuous density that varies with depth, and [23] for a discontinuous density that varies with both depth and latitude.…”