In this paper, existence and uniqueness of solutions for the nonlinear and linear models of the arctic gyres over $$84^{\circ }$$ 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.

中文翻译：

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北极环流海洋流动二阶微分方程的存在唯一性结果

本文利用立体投影研究了北纬$$84^{\circ }$$北极涡旋非线性和线性模型解的存在唯一性，立体投影表示的是流线，在外边界没有急流。 . 利用不动点技术，证明了非线性模型局部解的存在唯一性。接下来，我们给出了在合适的渐近条件下半无限区间解的存在唯一性。在线性涡度函数的情况下，我们采用线性常微分方程的思想给出了显式解。