Stratified equatorial flows in the β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-plane approximation with a free surface

We investigate the exact solutions to the governing equations for the equatorial flows with the associated free surface and rigid bottom boundary conditions in the β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-plane approximation which incorporates two considerations of the density stratification. Compared to the spherical coordinates and the cylindrical coordinates, the employment of the β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}-plane approximation admits that the density can be provided generally. Utilizing the implicit theorem, we present the Bernoulli relation between the pressure imposed on the free surface and the resulting distortion of the surface and we obtain that this relation exhibits the expected monotonicity properties. Finally, we prove that certain flows established by the exact solutions are stable via the short-wavelength stability method and the specific assumption of the density distribution.