# Singular Phenomenon Analysis of Wind-Driven Circulation System Based on Galerkin Low-Order Model
## 4 Results and Analysis  
Based on the classic quasi-geostrophic vorticity equation, the governing equation of wind-driven ocean circulation is obtained, and the [PERSON] method is used to approach the solution of the circulation model. Using Equation (8), Equation (5) can be rewritten as follows, namely, the quasi-geotranslocation potential equation,  
\[\partial_{t}(\Delta\Psi-F\Psi)+J(\Psi,\Delta\Psi)+\beta\frac{\partial\Psi}{ \partial x}=\frac{1}{Re}\Delta^{2}\Psi-\mu\Delta\Psi+w_{f} \tag{11}\]  
where \(J(\Psi,\Delta\Psi)=\frac{\partial\Psi}{\partial x}\frac{\partial\Delta\Psi}{ \partial y}-\frac{\partial\Psi}{\partial y}\frac{\partial\Delta\Psi}{ \partial x}\) is the Jacobian determinant. [PERSON] [41] proposed to expand the quasi-geostrophic vorticity equation with the eigenfunction that is suitable for the specific problem in the \(x\) and \(y\) directions, so that the computation can become easy. Considering the long-term west boundary current in the ocean, the eigenfunction or mode that decays exponentially in the \(x\) direction and is sinusoidally distributed in the \(y\) direction is introduced in this study,  
\[\left\{G(x,y)=e^{-s\pi x}\sin\pi x\sin(2k\pi y),k=1,2,\cdots,+\infty\right\} \tag{12}\]  
Obviously, the eigenfunctions in Equation (12) keep the orthogonals to each other, and their set can form a complete space. The solution of the quasi-geostationary potential equation, Equation (11), can be projected onto the space. In addition, because of the exponential attenuation in the \(x\) direction, the eigenfunction will have nonlinear characteristics. As a projection operator, the selection of the attenuation factor, \(s\), only determines its attenuation rate and does not influence the result. In this study, the attenuation factor \(s=2\). Therefore, the expansion of the stream function can be expressed as  
\[\Psi(x,y,t)=\sum_{k=1}^{\infty}A_{k}(t)e^{-s\pi x}\sin\pi x\sin(2k\pi y) \tag{13}\]  
where \(A_{k}\) represents the amplitude of the \(k\)-th mode at time \(t\). It is clear that the stream function, namely Equation (13), satisfies the symmetry relationship, Equation (10), and the flow velocities \(u\) and \(v\) naturally satisfy this relationship. By using the [PERSON] method (see Appendix A), it can be concluded that \(A_{k}\) satisfies the following equation:  
\[\frac{dA_{k}}{dt}=-r_{k}A_{k}-\lambda_{k}\left(\sum_{ \begin{subarray}{c}i+j =k\\ i-j\
eq k\end{subarray} }jA_{i}A_{j}-\sum_{j-i=k}jA_{i}A_{j}\right)+(\sigma_{1,k}w_{1}+\sigma_{2,k}w_{2}),k=1,2,\cdots,+\infty \tag{14}\]  
Therefore, the modal amplitudes, \(A_{k}\), of the system can be obtained, and the stream function, \(\Psi\), and the velocities \(u\) and \(v\) also can be obtained. The set of the first-order differential equations can be solved by the Runge-Kutta methods with high precision. Theoretically, the truncation of eigenfunctions used in the Galerkin procedure affects the accuracy of the solution. However, this paper seeks to explore the nature of ocean circulation through the lens of bifurcation and catastrophe theories within the framework of nonlinear dynamics. As such, a low-order model can still qualitatively capture the essential characteristics of the circulation system to a certain extent. This section will focus on the low-order model of wind-driven circulation, specifically the reduced positive pressure quasi-geostrophic potential equation, referred to as Equation (11). The influence of wind stress intensity on circulation will be examined in detail. With the first two eigenfunction or modes,  
\[\Psi(x,y,t)=e^{-s\pi x}\sin\pi x[A_{1}(t)\sin(2\pi y)+A_{2}(t)\sin(4\pi y)] \tag{15}\]  
It is assumed that the wind stress intensity parameters \(w_{1},w_{2}\) are constant. Then, Equation (14) can be obtained as follows,  
\[\left\{ \begin{array}{rl}\frac{dA_{1}}{dt}&=-r_{1}A_{1}+\lambda_{1}A_{1}A_{2 }+\sigma_{1,1}w_{1}\\ \frac{dA_{2}}{dt}&=-r_{2}A_{2}-\lambda_{2}A_{1}^{2}+\sigma_{2,2}w_{2}\end{array} \right.