## 4 Results
### Temporal Features of Local Energy Cascade and Intermittency  
To quantitatively measure the energy transfer process and exhibit the scale correlations, a number of multiscale IMF modes is estimated by means of an LIM approach. Local energy peaks and absolute values of phase difference of 8 (K = 8) neighboring IMF components are shown in Figure 6, where the top panels denote local intermittency measure peaks at two adjacent time scales, and the bottom panels refer to the absolute values of phase difference at two neighboring modal components. Through computing the corresponding average time scale of IMFs using a wavelet transform method, it is shown that the time scales are increasing with the number of the IMF modes (Figure 7), and the number of IMF modes has no effects on analyzing multiscale properties in the energy transfer process. By comparing the overlapped LIM energy peaks and corresponding phase differences of all the IMF modes, quantitative correlations of multiscale modes and properties of turbulent cascade and local intermittency are identified. The overlapped local energy peaks and the occurrence of a simultaneous phase between adjacent scales indicate an energy transfer between two adjacent time scales. When the LIM energy peaks of the intermittent  
Figure 5: Shear power spectra density (PSD) in wavenumber domain. The red and blue lines denote the raw shear spectra and the cleaned spectra using the cross-spectrum method. The red and blue dashed curves denote the standard Nasmyth spectra, and the two vertical dotted lines denote the Kolmogorov cutoff wavenumber.  
events in the dissipated process overlap, the phase difference between two neighboring scales can be nearly negligible; this phenomenon suggests that turbulent energy is being transferred between two neighboring scales, which have been highlighted in blue dotted lines (Figure 6). It is worth noting that the LIM peaks in the whole turbulent energy evolution process are discontinuous, indicating that this energy transfer process in a dissipated range occurs intermittently. Take IMF2 and IMF3 as an example. The absolute value of the phase difference between IMF2 and IMF3 can be negligible when their LIM energy peaks overlap twice between two neighboring scales at 0.54 s and 1.16 s (Figure 6b). It is illustrated that the overlapped energy peaks and the phases between IMF2 and IMF3 occur synchronously. For the last panel (Figure 6g), there are no simultaneous LIM peaks between two nonadjacent larger scales at 45 s and 180 s, indicating that the process of local energy cascade does not occur at two larger adjacent scales. In other words, the intermittent energy cascade can be easily found in smaller scales than in larger scales. In conclusion, phase synchronization is obvious at two neighboring smaller scales, and the simultaneous occurrence of cascade structures at neighboring pairs of smaller time scales is an effective indication of the local energy transfer process.  
Figure 6: \(Cont\).  
Figure 6: Phase correlations between IMF1 and IMF8 for profile1. (**a-g**) Top panels represent the local energy peaks at two neighboring scales; bottom panels represent the phase difference at two neighboring IMFs. The black line and red line respectively represent the LIM energy peaks of IMFi and IMF i + 1, where \(i\) = 1, 2, 3..., K, K is the number of modal components in VMD decomposition. The blue dotted lines denote that the phase difference is zero when LIM peaks are overlapping.  
Figure 7: The correlations between the time scales and the number of modes K for four profiles.