# Singular Phenomenon Analysis of Wind-Driven Circulation System Based on Galerkin Low-Order Model
## Appendix B Catastrophe Behaviors of Reduced Models of Circulation System  
Theoretically, the truncation of the eigenfunction used in the Galerkin procedure influences the accuracy of the solution. However, this paper aims to study the nature of ocean circulation from the viewpoint of bifurcation and catastrophe theories in nonlinear dynamics, so the low-order model can also qualitatively describe the nature of the circulation to a certain degree. This section will consider the low-order model of wind-driven circulation, i.e., the reduced positive pressure quasi-geostationary potential equation, namely Equation (11). And the influence of wind stress intensity on circulation is studied in detail. With the first two eigenfunction or modes,  
\[\Psi(x,y,t)=e^{-s\tau x}\sin\pi x[A_{1}(t)\sin(2\pi y)+A_{2}(t)\sin(4\pi y)]\] (A12)  
It is assumed that the wind stress intensity parameters \(w_{1},w_{2}\) are constant. Then, Equation (14) can be obtained as follows,  
\[\left\{ \begin{array}{l}\frac{dA_{1}}{dt}=-r_{1}A_{1}+\lambda_{1}A_{1}A_{2}+ \sigma_{1,1}w_{1}\\ \frac{dA_{2}}{dt}=-r_{2}A_{2}-\lambda_{2}A_{1}^{2}+\sigma_{2,2}w_{2}\end{array} \right.\] (A13)  
Following Equation (A10), the coefficients of the coupling terms can be given as \(\lambda_{1}=1.2973,\lambda_{2}=0.5897\) in Equation (A13). Considering that the coefficient of horizontal momentum exchange, \(A_{H}\), is relatively large, it might as well be given as \(Re=20\). Then the coefficient of the linear term is \(r_{1}=0.4500,r_{2}=0.5045\). For the constant term, \(w_{1}\) and \(w_{2}\) are variable parameters, and the coefficient is taken as \(\sigma_{1,1}=0.3643,\sigma_{2,2}=0.8015\). Following the definition, the equilibrium position of system Equation (A13) satisfies \(dA_{1}/dt=dA_{2}/dt=0\). Thus the equilibrium position can be governed by the following equations,  
\[\bar{A}_{1}^{3}+\frac{r_{1}r_{2}-\lambda_{1}\sigma_{2,2}w_{2}}{\lambda_{1} \lambda_{2}}\bar{A}_{1}-\frac{r_{2}\sigma_{1,1}w_{1}}{\lambda_{1}\lambda_{2}}=0\] (A14)\[\tilde{A}_{2}=\left(\sigma_{2,2}w_{2}-\lambda_{2}\tilde{A}_{1}^{2}\right)/r_{2} \tag{111}\]  
For Equation (110), set  
\[p=\frac{r_{1}r_{2}-\lambda_{1}\sigma_{2,2}w_{2}}{\lambda_{1}\lambda_{2}},q=- \frac{r_{2}\sigma_{1,1}w_{1}}{\lambda_{1}\lambda_{2}} \tag{112}\]  
So the discriminant can be obtained,  
\[\Delta=(q/2)^{2}+(p/3)^{3}=\left(-\frac{r_{2}\sigma_{1,1}w_{1}}{2\lambda_{1} \lambda_{2}}\right)^{2}+\left(\frac{r_{1}r_{2}-\lambda_{1}\sigma_{2,2}w_{2}}{ 3\lambda_{1}\lambda_{2}}\right)^{3} \tag{113}\]  
As \(\Delta>0\),  
\[\tilde{A}_{1}=\left(\frac{2\sqrt{\Delta}-q}{2}\right)^{1/3}-\frac{p}{3}\left( \frac{2\sqrt{\Delta}-q}{2}\right)^{-1/3}. \tag{114}\]  
As \(\Delta=0\),  
\[\tilde{A}_{1}^{1}=\sqrt{-p/3},\tilde{A}_{1}^{2}=-2\sqrt{-p/3}. \tag{115}\]  
As \(\Delta<0\),  
\[\tilde{A}_{1}^{i}=2\sqrt[3]{R}\cos\left(\frac{\theta+2i\pi}{3}\right),i=1,2,3. \tag{116}\]  
where \(R=\sqrt{-(p/3)^{3}},\theta=\frac{1}{3}\arccos\left(-\frac{q}{2R}\right)\). Therefore, as parameters \(w_{1}\) and \(w_{2}\) are determined, the equilibrium position \((\tilde{A}_{1},\tilde{A}_{2})\) can be obtained according to Equation (111). The derivative operator of system Equation (110) at the equilibrium position is given by the Jacobian matrix,  
\[\left[ \begin{array}{cc}-r_{1}+\lambda_{1}\tilde{A}_{2}&\lambda_{1}\tilde{A} _{1}\\ -2\lambda_{2}\tilde{A}_{1}&-r_{2}\end{array} \right] \tag{117}\]  
The eigenvalues, \(\kappa\), of the matrix are given by the following quadratic equation,  
\[\kappa^{2}+(r_{1}+r_{2}-\lambda_{1}\tilde{A}_{2})\kappa+2\lambda_{1}\lambda_{ 2}\tilde{A}_{1}^{2}+r_{2}(r_{1}-\lambda_{1}\tilde{A}_{2})=0 \tag{118}\]  
Note that Equation (118) is defined in the complex domain, and the type of equilibrium position depends on the eigenvalues. Since the equilibrium position is governed by the one-dimensional cubic equation, namely, Equation (110) with two parameters \((w_{1},w_{2})\), it is clear that such an equation is an extension of the governing equation for the system with a cusp catastrophe.