# Singular Phenomenon Analysis of Wind-Driven Circulation System Based on Galerkin Low-Order Model
## 1 Introduction  
Ocean circulation is a fundamental component of Earth's climate system, influencing weather patterns and contributing to the development of extreme climate events. Currently, the nature of ocean circulation and its evolution on time scales ranging from several months to several years are critical issues in oceanography [1]. Ocean circulation exhibits several distinct features. For instance, the thickness of the boundary layer formed by the atmosphere on the sea surface is greater due to the significantly lower density of wind-driven circulation compared with seawater [2]. Under the influence of wind stress, atmospheric momentum is transferred to the water body, causing the water to move along with the wind. Notable examples of this phenomenon are the Kuroshio in the North Pacific and the Gulf Stream in the North Atlantic, which represent the major western boundary currents in the Northern Hemisphere [1, 3, 4]. Historically, much of our understanding of ocean circulation was derived from observations. In 1898, [PERSON], a Norwegian scientist, was likely the first to qualitatively describe the formation process of wind-driven circulation. According to his theories, wind stress actsupon the sea surface, transferring kinetic energy into the water body and causing surface water movement. He also observed that, in the Northern Hemisphere, the flow shifts to the right of the wind direction due to the Coriolis force. Subsequent theoretical research can be traced back to [PERSON], who quantitatively described wind-driven circulation and studied the circulation structure within the sea surface boundary layer [5]. [PERSON]'s theory suggests that velocity within the boundary layer exhibits a spiral distribution, with the flow direction deviating to the right of the wind stress. This boundary layer has since been termed the Ekman layer. However, [PERSON]'s model assumes that wind stress on seawater acts solely as a frictional force, which does not fully align with actual observations. In 1947, building on [PERSON]'s quantitative model, [PERSON] considered the horizontal pressure gradient force in the wind-driven circulation model, establishing the relationship between basin-wide circulation and the wind stress curl [6]. Although [PERSON] innovatively proposed the formation process of wind-driven ocean circulation, he did not account for the western boundary effect (such as the formation of the Atlantic Gulf Stream, which includes the western boundary effect). Based on previous theoretical frameworks and actual wind field characteristics, [PERSON] introduced lateral friction in his model to represent the western boundary effect [3]. This model was later refined by [PERSON], who developed the inertial western boundary layer theory [7]. The series of studies conducted in the mid-20 th century significantly deepened our understanding of ocean circulation. However, the spatial distribution of wind-driven circulation and the evolution of ocean currents were not accurately characterized due to the limitations of computing technology at the time. [PERSON] and [PERSON] synthesized the results of previous research on geophysical flows, starting from the three-dimensional spherical rotation equations, and simplified the barotropic quasi-geostrophic model and the shallow water model [8; 9]. Subsequent research on geophysical fluid dynamics has since been based heavily on their foundational work. In the atmosphere-ocean system, the ocean serves as the largest reservoir of heat and water, with the movement of ocean currents exerting a significant impact on the climate of coastal areas, particularly on temperature and precipitation. The most direct meteorological manifestation of changes in ocean circulation is short-term climate oscillation. The study of this phenomenon is crucial, as common weather models, constrained by computational resources, are often inadequate for predicting climate changes over longer time scales. This limitation can lead to large-scale natural disasters when extreme climate events occur [10].