## 2 Materials and Methods
### Wavelet Analysis Methodology  
In this study, we use the Morlet wavelet, a complex continuous wavelet with zero mean, localized in both frequency and time [32]. The Morlet wavelet is defined as  
\[\psi_{0}(\tau)=\pi^{-1/4}e^{i\omega\tau}e^{-\tau^{2}/2} \tag{1}\]  
where \(\omega\) and \(\tau\) represent the dimensionless frequency and time, respectively. With a dimensionless frequency of \(\omega=6\), the Morlet wavelet provides an optimal balance between time and frequency localization, and its Fourier period, \(T_{\mathcal{F}}\), is close to its scale (\(T_{\mathcal{F}}=1.03\) s) [33]. To extract features from a time series, \(x_{n}\) (\(n=1,2,\cdots,N\)), the continuous wavelet-transform (CWT) is applied as a convolution of \(x_{n}\) with the scaled and normalized wavelet [34], expressed as  
\[W_{n}^{X}(s)=\sqrt{\Delta/s}{\sum_{n^{\prime}=1}^{N}x_{n^{\prime}}\psi_{0}} \big{[}(n^{\prime}-n)\Delta/s\big{]} \tag{2}\]  
The wavelet power spectrum (WPS), \(\big{|}W_{n}^{X}(s)\big{|}^{2}\), represents the power of the time series at each temporal period and its complex argument can be interpreted as the local phase. In order to find a dominant mode in a given time series, we use the global wavelet spectrum (GWS) as an analog of the Fourier transform spectrum, which is obtained by averaging the WPS estimates over the time duration [35], that is,  
\[\overline{W}^{2}(s)=\frac{1}{N}{\sum_{n=1}^{N}}\big{|}W_{n}^{X}(s)\big{|}^{2} \tag{3}\]  
Figure 3: A time-depth map of temperature anomalies, \(\eta(z,t)\), obtained during the glider observation period within the anticyclonic eddy. where \(N\) denotes the time duration. For evaluating the degree of correlation between two oscillating time series \(\{x_{n},y_{n}\}\), we compute the cross-wavelet transform (XWT), defined as \(W_{n}^{XY}(s)=W_{n}^{X}(s)\cdot W_{n}^{Y*}(s)\), where \(*\) denotes a complex conjugation [33; 34]. Further, by normalizing the XWT, we define the wavelet coherence squared (WCS) of two time series [36], which is expressed as  
\[R^{2}(n,s)=\frac{\left|S\big{(}s^{-1}W_{n}^{XY}(s)\big{)}\right|^{2}}{S\Big{(} s^{-1}\big{|}W_{n}^{X}(s)\big{|}^{2}\Big{)}\cdot S\Big{(}s^{-1}\big{|}W_{n}^{Y}(s) \big{|}^{2}\Big{)}} \tag{4}\]  
where \(S\) is a smoothing operator [33]; the smoothing operator for the Morlet wavelet is given by [PERSON] and [PERSON] [36]. The WCS indicates the coherence of XWT in time-frequency space and ranges from 0 to 1. In this work, a pair-wise WCS is used to construct a network from multivariate temperature anomaly time series through statistical significance testing based on surrogate data methods. In order to perform statistical significance testing on WCS estimates through Monte Carlo methods, we construct 100 ensembles of the surrogate time series generated via the iAAWT algorithm, according to [PERSON] and [PERSON] [30], which consists of the following steps:  
1. Generation of a Gaussian white noise time series to match the length of the original data. 2. Calculation of the continuous wavelet (Fourier) transform of the noise to extract its phase. 3. Combination of the randomized phase and the wavelet transform modulus of the original signal to produce a surrogate time-frequency distribution. 4. Reconstruction of a non-stationary surrogate time series by taking the real part of the inverse wavelet transform. 5. Rescaling of the surrogate to match the distribution of the original time series. The WCS estimates are calculated from each surrogate ensemble, and all the WCS ensembles at specific times and frequencies are used to obtain the corresponding critical value satisfying a 95% significance level; that is, one critical value is determined at every lattice point of time and frequency. It is important to note that a wavelet analysis introduces an inherent limitation known as the cone of influence (COI) [32; 33], referring to the region in which edge effects can influence the wavelet coefficients. In this study, we identified COI gaps for two specific scales: the semidiurnal and diurnal scales.