# Singular Phenomenon Analysis of Wind-Driven Circulation System Based on Galerkin Low-Order Model
## Appendix A Approaching Circulation Model Using the Galerkin Method  
Based on the classic quasi-geostrophic vorticity equation, the governing equation of wind-driven ocean circulation is obtained, and the Galerkin method is used to approach the solution of the circulation model. Using Equation (8), Equation (5) can be rewritten as following, namely, the quasi-geotranslocation potential equation Equation (11). [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 sinusoidally distributed in the \(y\) direction, is introduced in this study as Equation (12) Obviously, the eigenfunctions in Equation (12) keep orthogonal 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 \(s=2\). 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  
\[\Psi(x,y,t)=\sum_{k=1}^{\infty}A_{k}(t)e^{-s\pi x}\sin\pi x\sin(2k\pi y) \tag{14}\]  
where \(A_{k}\) represents the amplitude of the \(k\)-th mode at time \(t\). It is clear that the stream function, namely Equation (14), satisfies the symmetry relationship, Equation (10), and the flow velocities \(u\) and \(v\) naturally satisfy this relationship. Let the complementary function be  
\[F(x,y,\Psi)=\partial_{t}(\Delta\Psi-F\Psi)+J(\Psi,\Delta\Psi)+\mu\Delta\Psi- \frac{1}{Re}\Delta^{2}\Psi-w_{f} \tag{15}\]  
Using the Galerkin method, the following equations are obtained,  
\[(F,G)_{0}=\int_{0}^{1}\int_{0}^{1}F(x,y,\Psi)\cdot G(x,y)dxdy=0,k=1,2,\cdots,+\infty \tag{16}\]  
where \((,)_{0}\) is the inner product. With the [PERSON] method, the unsteady term can be obtained,  
\[\left(\partial_{t}(\Delta\Psi-F\Psi),G(x,y)\right)_{0}=\sum_{k=1}^{\infty}- \frac{1-e^{-2s\pi}}{8\pi s}\bigg{[}1+\frac{4k^{2}+F}{1+s^{2}}\bigg{]}A_{k}^{\prime} \tag{17}\]  
Further, the convection term can be obtained,  
\[\left(J(\Psi,\Delta\Psi),G(x,y)\right)_{0}=\sum_{k=1}^{\infty}-\frac{3s\left( 1+e^{-3s\pi}\right)}{\pi(1+9s^{2})}\left(\sum_{ \begin{subarray}{c}i+j\to k \\ i-j\to k\end{subarray} }jA_{i}A_{j}-\sum_{j-i=k}jA_{i}A_{j}\right) \tag{18}\]  
Also, the dissipation term can be obtained,  
\[ \begin{split}&\left(\mu\
abla^{2}\Psi-\frac{1}{Re}\Delta^{2} \Psi,G(x,y)\right)_{0}\\ &=\sum_{k=1}^{\infty}\frac{1-e^{-2\pi s}}{8\pi s}\bigg{\{}\mu \Big{(}1+\frac{4k^{2}}{1+s^{2}}\Big{)}+\frac{1}{Re}\bigg{[}1+\frac{\left(5s^ {2}-8k^{2}\right)\left(1+s^{2}\right)+16k^{4}}{1+s^{2}}\bigg{]}\bigg{\}}A_{k} \end{split}  \tag{19}\]  
The wind stress term should satisfy the antisymmetric relationship \(\Re_{\text{QG}}\), which can be expressed as  
\[w_{f}(t,y)=\sum_{k=1}^{\infty}-w_{k}(t)\sin 2k\pi y,k=1,2,\cdots,\infty \tag{20}\]  
where \(w_{k}(t)\) is the intensity parameter relevant to the antisymmetric wind stress. In this study, the first two modes are used.