Multiple solutions of a nonlinear boundary layer problem (Q1319034)

Summary: We study a mathematical model for large-scale ocean circulation. Under reasonable physical assumptions, the governing partial differential equations reduce to a nonlinear third order ordinary differential equation \(\varphi'''+\lambda (\varphi \varphi''-\varphi'{}^ 2)+1- \varphi=0\) with stress-free conditions \(\varphi (0)=\varphi''(0)=0\) and boundary condition \(\varphi (\infty)=1\). We prove, by a topological shooting method, that in the range of the parameter \(\lambda>(27/4)^{1/3}\) there are at least two solutions. These solutions can be used further to construct the whole flow pattern in the ocean.