# Singular Phenomenon Analysis of Wind-Driven Circulation System Based on Galerkin Low-Order Model
## 3 Governing Equations of Ocean Circulation  
In geophysical fluid dynamics, [PERSON] and [PERSON] proposed the shallow water (SW) equations, and simplified the equations with the different environments, having a profound impact on the quantitative analysis [8,9]. This paper mainly studies the ocean circulation driven by wind. The Rossby number corresponding to the SW model is very small, and the topological change at the bottom of the calculation area is also very small. In this case, the SW model can be simplified by assuming a small Rossby number. A single-layer quasi-geostrophic vorticity equation in a square region is obtained, using the equalization dimensionalization of characteristic flow rates and characteristic scales with the quasi-geostrophic theory. \[\left[\frac{\partial}{\partial t}+u\frac{\partial}{\partial x}+v\frac{ \partial}{\partial y}\right]\left[\zeta-F\Psi+\beta y\right]=\frac{1}{Re} \
abla^{2}\zeta-\mu\zeta+\alpha_{QG}\left(\frac{\partial\tau^{y}}{\partial x} -\frac{\partial\tau^{x}}{\partial y}\right) \tag{5}\]  
\[\zeta=\
abla^{2}\Psi \tag{6}\]  
where \(u\) and \(v\) denote the horizontal and vertical velocities, respectively. \(\Psi\) is the flow function, and \(\zeta=\partial v/\partial x-\partial u/\partial y\) is the vertical relative vorticity. \(\tau^{x}\) and \(\tau^{y}\) represent the horizontal and vertical wind stresses, respectively. In addition, Equation (7) shows the dimensionless parameters in Equation (5). \[ \begin{split}\beta&=\frac{\beta_{0}L^{2}}{U};F=\frac{ \beta_{0}^{2}L^{2}}{\beta^{2}D}\\ Re=\frac{U_{H}}{U};\mu&=\frac{U_{H}}{U}\sqrt{A_{V}f_{0 }};\alpha_{QG}=\frac{\tau_{0}L}{\rho_{0}DU^{2}}\\ \delta_{I}=L^{-1}\sqrt{\frac{U}{\beta_{0}}};\delta_{S}& =L^{-1}\frac{\sqrt{A_{V}f_{0}}}{2\rho_{0}};\delta_{M}=L^{-1}\Big{(}\frac{A_{H }}{\rho_{0}}\Big{)}^{1/3}\end{split}  \tag{7}\]  
where \(\beta\) is the Planetary vorticity gradient strength, \(F\) is the Rotating Froude number, \(Re\) is the Reynolds number, \(\mu\) is the bottom friction coefficient, \(\alpha_{QG}\) is the wind stress intensity, \(\delta_{I}\) is the relative inertial boundary layer thickness, \(\delta_{M}\) is the relative horizontal viscous boundary layer thickness, and \(\delta_{S}\) is the relative vertical viscous boundary layer thickness. The dimensional parameters are listed in Table 1, and the values of these parameters refer to the settings in reference [17]. The normalized gravitational acceleration can be obtained by \(g^{\prime}=gd\rho/\rho_{0}\). The value \(g^{\prime}=2.5\times 10^{-2}\) m \(\cdot\) s\({}^{-2}\) in Ref. [17] is used, and the corresponding dimensionless parameters are given in Table 2, and the values of these parameters also refer to the settings in reference [17]. It is worth noting that the velocity field of flow is expressed as a stream function in the field of geophysical fluid dynamics,  
\[u=-\partial\Psi/\partial y;v=\partial\Psi/\partial x \tag{8}\]  
In the square region \([0,1]\times[0,1]\), Equation (5) satisfies the non-slip boundary condition and no penetration boundary condition, that is,  
\[ \begin{split} x&=0,1:\Psi=0,u=-\partial\Psi/ \partial y=0,\partial u/\partial y=-\partial^{2}\Psi/\partial y^{2}=0\\ y&=0,1:\Psi=0,v=\partial\Psi/\partial x=0,\partial v/ \partial x=\partial^{2}\Psi/\partial x^{2}=0\end{split}  \tag{9}\]