# Singular Phenomenon Analysis of Wind-Driven Circulation System Based on Galerkin Low-Order Model
## 1 Introduction
Two of the most significant examples are the El Nino Southern Oscillation in the equatorial Pacific and the Atlantic multi-decadal oscillation in the North Atlantic [11; 12]. Indeed, the study of the El Nino-Southern Oscillation (ENSO) has long been a central topic in ocean and atmospheric sciences, with particular focus on the relationship between oceanic and atmospheric variables, their spatial structures, and temporal evolution. By analyzing the singular frequency spectrum of various global average temperature records, it has been shown that the ocean surface temperature in the North Atlantic exhibits inter-annual variability [11]. Through spectral analysis, [PERSON] and [PERSON] identified distinct spectral peaks within the 50-70 year frequency band [13]. This short-term climate oscillation in the Atlantic Ocean is termed the Atlantic multi-decadal oscillation (AMO), with the AMO index defined as the 10-year running average of ocean surface temperature, exhibiting an anomalous trend north of the equator. Interestingly, results indicate a strong negative correlation between AMO and precipitation across the continental United States. Specifically, during periods of a high (low) AMO index, precipitation is reduced (increased) across much of central United States [14; 15]. [PERSON]'s research demonstrates that the evolution of ocean circulation between the subtropical and subpolar zones can influence the development of AMO, and a negative AMO has been closely linked to the accelerated sea level rise along the northeast coast of the United States [12]. Consequently, some researchers suggest that inter-annual periodic climate changes in the atmosphere-ocean system are highly correlated with the influence of external atmospheric wind fields on ocean circulation [16]. In addition, the marginal seas, such as the South China Sea, exhibit evident characteristics of low-frequency periodic oscillation. [PERSON] and [PERSON] studied the climate evolution characteristics of the South China Sea using a two-layer quasi-geostrophic model, and found that the periodic behavior of the local climate is closely related to monsoonal patterns. They concluded that the monsoon first influences the evolution of ocean circulation, thereby inducing periodic climate oscillations [17]. The Kuroshio Extension (KE), an extension of the Kuroshio warm current in the North Pacific Ocean, is the region with the highest gyre kinetic energy in the North Pacific [4]. Observations indicate that the large-scale periodic oscillation of the KE can transport high-temperature, high-salinity seawater from the Pacific Ocean to coastal areas, affecting the meteorology and hydrology of North America's coastal regions, leading to abnormally high temperatures [18; 19]. The broader impact of wind on climate systems and ocean circulation has also been a focal point of recent studies. [PERSON] examined how low- and mid-level wind couplings and topography affect mesoscale convective systems (MCSs) in Northwest China, discovering that low-level anticyclonic circulation impacts the distribution of high humidity zones [20]. [PERSON] et al. analyzed Arctic cyclones, identifying asymmetries in atmospheric and sea ice structures that resemble mid-latitude cyclones with distinct warm and cold sectors [21]. [PERSON] and [PERSON] assessed the impact of Aeolus wind data on near-surface wind forecasts, noting positive effects in high-latitude regions, particularly with extended forecast time steps [22]. [PERSON] et al. investigated the effects of wind-driven ocean circulation on global warming, finding that these changes amplified the warming rate by 17% from 1979 to 2014, underscoring the importance of atmospheric circulation changes for accurate climate predictions [23]. [PERSON] et al. (2023) explored tropical cyclone (TC) wind forcing within a fully coupled climate model, demonstrating that TCs increase ocean heat content through effects on vertical mixing, air-sea enthalpy fluxes, and cloud cover [24].