## 1 Introduction  
Mesoscale eddies are circular water currents that are ubiquitous in the ocean, extending tens to hundreds of kilometres over tens to hundreds of days, and serve as a key bridge in the energy cascade between large-scale and small-scale variability in the ocean [1]. Eddies, which can be classified as cyclonic or anticyclonic depending on their direction of rotation, are important for understanding long-term climate change as they play a major role in the transport of freshwater, heat, dissolved carbon, and other biogeochemical tracers in the ocean [2, 3], effectively influencing ocean circulation [4], large-scale water distribution, and biology [5]. Mesoscale eddies play a pivotal role in ocean mixing and drive global water circulation [6] with profound effects on climate and ecosystems [7]. Enhanced mixing induced by mesoscale eddies has been observed in regions of the strong Antarctic Circumpolar Current and the West Boundary Current, such as the Kuroshio Current [6]. The role of eddies in these areas is a long-standing and observationally challenging topic and eddy science has therefore become a central theme of modern oceanography in recent decades. The thermal structure of the eddies is critical for heat transport [2, 8] and air-sea interactions [9, 10], including changes in the intensity of tropical cyclones [11, 12]. In particular, eddy-induced heat transport is critical for maintaining the world's ocean and climate states [4], and researches have illustrated that eddies can induce heat fluxes by trapping and transporting water with certain temperature in their cores [2, 8]. The vertical temperature anomalies are crucial for heat transport estimation and related research analysis. With the progress of the global Argo project, fast, accurate, and numerous temperature profiles can be accessed worldwide [13]. The temperature anomaly (TA) profiles derived from Argo floats surfaced in eddies have been widely used to study the vertical structure of mesoscale eddies [14; 15]. These TA profiles have a complex structure and typically exhibit one or more peaks, with fluctuations in adjacent depth layers. [PERSON] et al. [16] analysed the evolution of a single long-lived Agulhas Ring using an eddy tracking algorithm to uncover its hydrological properties. The track of the single eddy was made possible by the large extent (100 to 400 km diameter) and intense altimetry signal of Agulhas Rings [16]. More studies use the composite method to construct the averaged vertical structures [17; 18; 19; 4; 8; 15], which are conducted by compositing sufficient Argo hydrographic profiles that are concurrently captured by sea surface eddies in certain regions. As an example, [PERSON] et al. [15] constructed the mean eddy vertical structure of the eastern South Pacific Ocean by compositing the average of 420 (526) profiles acquired by Argo floats that surfaced into cyclonic (anticyclonic) mesoscale eddies. Also, the 3-D structures of composite cyclonic eddies (CEs) and anticyclonic eddies (AEs) were analysed using collocated altimetry sea surface height anomalies (SSHA) and Argo profiles in the Kuroshio Extension (KE) region [19], and analysis showed that at the centre of the composite eddy, a cooling of \(-2.008\,^{\circ}\)C was observed at a depth of 360 m for the CE, while for the AE, a warming of 1.788 \({}^{\circ}\)C was observed at a depth of 410 m.  
Regional ocean climate can be estimated more objectively using individual cluster analysis rather than a fixed region approach. [PERSON] et al. [20] divided the KE region into 9 grids to study the spatial characteristics of the corresponding composite eddies. However, the coexistence of eddies with different vertical structures in the same grid may affect the accuracy of the characterisation of the composite eddies. [PERSON] et al. [21] grouped eight classes of vertically coherent heat patterns in the North Atlantic by using an unsupervised classification Gaussian Mixture Modelling (GMM) method for Argo temperature profiles.