Existence and uniqueness results for a second order differential equation for the ocean flow in arctic gyres

In this paper, existence and uniqueness of solutions for the nonlinear and linear models of the arctic gyres over 84∘\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$84^{\circ }$$\end{document} north latitude are studied by using the stereographic projection, which represents the streamline and no jet at the outside boundary. By using the fixed point technique, we prove the existence and uniqueness of the local solution of the nonlinear model. Next, we present the existence and uniqueness of the solution in the semi-infinite interval under the suitable asymptotic conditions. In the case of linear vorticity function, we give the explicit solutions by adopt the idea for linear ODEs.