# Singular Phenomenon Analysis of Wind-Driven Circulation System Based on Galerkin Low-Order Model
## 1 Introduction
From an ocean dynamics perspective, [PERSON] and [PERSON] (2023) extended [PERSON]'s wind-driven transport theory to the \(\beta\) plane by introducing pseudo-angular momentum, which simplifies the understanding of water column trajectories near the equator [25]. [PERSON] and [PERSON] (2021) emphasized the role of ocean topography in modulating Southern Ocean circulation responses to wind stress changes, demonstrating that stationary eddies play a dominant role in these responses [26]. Additionally, [PERSON] et al. (2021) found that eddy effects generally counteract Ekman circulation, with changes in air-sea heat flux influencing the strength of eddy cancellation. Specifically, as heat flux increases, eddy cancellation decreases, causing the flow to gradually approach classical Ekman circulation [27]. To explore the nature of ocean circulation evolution, scholars have developed an idealized multi-gyre model, grounded in the study of wind-driven circulation. In a confined fluid region, the system satisfies the governing equations of quasi-geostrophic (QG) or shallow water (SW) dynamics under a normalized gravity mode. When the system is forced by a wind field composed of multiple gyres, these gyres can represent subpolar cyclones and subtropical anticyclones, which correspond to ocean gyres in the real-world ocean environment. Consequently, such an idealized gyre model has been widely applied in studies of large-scale mid-latitude ocean circulation responses to wind forcing [28; 29; 30]. In conjunction with the wind-driven circulation model and actual observational data, statistical analysis methods have been extensively utilized to study complex ocean currents. Methods such as empirical orthogonal function (EOF) analysis [31] and nonlinear time series analysis [32; 33] have proven effective in climate change and oceanographic research. However, a limitation of these methods is the need for additional models to establish causal relationships between key physical quantities. Moreover, statistical integral quantities are often used to analyze interactions between mean physical quantities and perturbations [8]. Nonetheless, the complexity of most ocean circulation problems often results in significant deviations in statistical integrals, as observational or ocean models may not fully satisfy the necessary conditions. As discussed earlier, certain behaviors of ocean circulation exhibit dynamic properties, making it feasible to study these systems from a dynamical perspective. [PERSON] and [PERSON] identified that inter-annual phenomena in the ocean-atmosphere system's climate change process possess dynamic characteristics [34; 35]. In open ocean areas, [PERSON] analyzed gyres using adaptive harmonic decomposition and inverse stochastic modeling techniques, revealing that wind-driven ocean circulation exhibits quasi-periodic variations [28]. Indeed, understanding the evolution of ocean circulation by examining its sensitivity to various parameters using dynamic system theories is crucial. Since the 1990s, the dynamical system perspective has been introduced into oceanography, with researchers employing bifurcation theory from nonlinear dynamics to explain low-frequency periodic oscillations in ocean circulation [36]. Nonlinear dynamic theory suggests that low-frequency oscillations in the ocean are driven by attractors within the system, such as limit cycles, strange attractors, and homoclinic orbits [37; 38]. The results from quasi-geostrophic (QG) models have been instrumental in explaining the asymptotic behavior of the shallow water system. [PERSON]'s investigation of a low-order quasi-geostrophic ocean circulation model revealed the presence of multiple equilibrium solutions, periodic solutions, and quasi-periodic solutions within the circulation system [39]. Notably, it was found that the circulation system can exhibit multiple equilibrium states.