## 3 Methodologies  
### The VMD Decomposition  
Currently, the empirical mode decomposition (EMD) method [17; 18] has been widely applied to the physical oceanography fields, such as experimental turbulence records. It is especially effective to decompose the nonlinear and nonstationary time series. The essence of the EMD method is that it decomposes the original signals into several IMF modal components with different scales and frequencies, and these different local features on different time scales can identify the structures of the turbulent cascade and intermittency through the dissipative power spectrum. However, EMD decomposition has several disadvantages, such as mode mixing and an endpoint effect. To reduce the effect on the mode mixing problem, [PERSON] et al. [19] proposed a new non-recursive variational mode decomposition (VMD) method, which decomposed the original signal into K sparse amplitude-frequency modulation components by solving the optimal solution of the variational problem [20]. The variational model can be expressed as follows:  
\[\left\{ \begin{array}{c}argmin\bigg{\{}\sum\limits_{k=1}^{K}\|\partial_{t} \Big{[}\Big{(}\delta(t)+\frac{j}{\pi t}\Big{)}\times u_{k}(t)\Big{]}e^{-i \omega_{k}t}\|_{2}^{2}\\ \hskip 142.26378 pts.t.\sum\limits_{k=1}^{K}u_{k}(t)=f(t)\end{array} \right. \tag{1}\]  
where \(u_{k}(t)=A_{k}(t)\cos(\varphi_{k}(t))\) denotes the intrinsic mode functions (IMF) components. \(\{\omega_{k}\}=\{\omega_{1},\omega_{2}\ldots\omega_{K}\}\) represents the frequency centers of each \(u_{k}(t)\), and K represents the total number of IMF components. In order to solve the optimal solves in Equation (1), two parameters of an extended Lagrange operator \(\lambda(t)\) and a second penalty factor \(\alpha\) are developed to transform the constrained variational model into a non-constrained variational problem, and the Lagrange operator \(L(u_{k},\omega_{k},\lambda_{k})\) is depicted as follows:  
\[L(u_{k},\omega_{k},\lambda_{k})=\alpha\sum\limits_{k=1}^{K}\|\partial_{t}\bigg{[} \bigg{(}\delta(t)+\frac{j}{\pi t}\bigg{)}\times u_{k}(t)\bigg{]}e^{-i\omega_{k }t}\|_{2}^{2}+\|f(t)-\sum\limits_{k=1}^{K}u_{K}(t)\|_{2}^{2}+\bigg{(}\lambda(t ),f(t)-\sum\limits_{k=1}^{K}u_{k}(t)\bigg{)} \tag{2}\]  
\[\left\{ \begin{array}{c}\hat{u}_{k}^{n+1}(\omega)=\frac{f(\omega)-\sum_{i<k} \hat{u}_{i}(\omega)-\sum_{j<k}\hat{u}_{i}^{n}(\omega)+\hat{\lambda}(\omega)/2} {1+2\left(\omega-\omega_{k}\right)^{2}}\\ \omega_{k}^{n+1}=\frac{\int_{0}^{\infty}|\omega|_{2}^{n+1}(\omega)|_{2}d\omega }{\int_{0}^{\infty}|\omega|_{2}^{n+1}(\omega)|_{2}d\omega}\\ \hat{\lambda}^{n+1}(\omega)=\hat{\lambda}^{n}(\omega)+\tau\bigg{(}\hat{f}( \omega)-\sum\limits_{k}\hat{u}_{i}^{n+1}(\omega)\bigg{)}\end{array} \right. \tag{3}\]  
The method of alternating multipliers is used to update weights \(\hat{u}_{k}\), \(w_{k}\), and \(\hat{\lambda}\) to find the optimal solution. In Equation (3), \(\hat{u}_{k}\) and \(\omega_{k}\) represent the modal component and center frequency, respectively. For the iteration process, the variational solution process will stop when the iteration condition \(\sum_{k}(\|u_{k}^{n+1}-u_{k}^{n}\|_{2}^{2}/\|u_{k}^{n}\|_{2}^{2})\)\(<\)\(\varepsilon\) is satisfied, and K modal components with limited bandwidth are output. Figure 2: Deployment site, the location of the vertical profiler in SCS (red square). Compared with the traditional EMD decomposition, the VMD method cannot completely remove the confusion of the modes mixing, but it has an inhibitory effect on mode mixing. Actually, the VMD method with its mathematical theoretic foundation outperforms the EMD in dealing with mode aliasing problems. Additionally, the VMD method can effectively eliminate the cumulative errors of envelope estimation caused by recursive mode decomposition. However, two parameters, the number of modal components and the penalty parameters in VMD need to be optimized by the optimization algorithm to find the optimal solution in future work.