## 3 HKE Spectra and Spectral Energy Budgets
### Evolutions of the HKE Spectra  
Time evolutions of the vertically averaged HKE spectra over the troposphere (\(z=0\)-12 km) and lower stratosphere (\(z=17\)-19 km) are shown in Figures 3-5, respectively. In the troposphere, it can be found from the distribution of HKE spectrum as a function of horizontal wavelength \(\lambda_{h}\) (wavenumber \(k_{h}\)), which, with the decrease (increase) of \(\lambda_{h}\) (\(k_{h}\)), the magnitude of HKE in general reduces sharply at the initial time (0000 UTC 6 August; Figure 3a), especially for \(\lambda_{h}<200\) km (\(k_{h}>\pi\times 10^{-5}\) rad m\({}^{-1}\)). It suggests that although the ECMWF ERA5 reanalysis data reach a horizontal resolution of 0.25\({}^{\circ}\), the atmospheric information they contain at mesoscale is still limited. The model then takes about 6 h to complete the spinup adjustment, which reflects in the variation of the HKE spectrum. The spectrum experiences a significant lift for \(\lambda_{h}<400\) km (\(k_{h}>\pi/2\times 10^{-5}\) rad m\({}^{-1}\)) during the first 3 h and a continued lift for \(\lambda_{h}<80\) km (\(k_{h}>\pi/4\times 10^{-4}\) rad m\({}^{-1}\)) during the second 3 h. Obviously, compared with larger scales, the spinup adjustment exerts more influence on \(\lambda_{h}<100\) km (\(k_{h}>2\pi\times 10^{-5}\) rad m\({}^{-1}\)). Accompanied with the intensification of Lekima, the HKE in the troposphere then increases at various horizontal scales in the following time (Figure 3b-d). Note that the HKE spectrum has already reproduced well the \(-3\) slope at the synoptic scale and the \(-5/3\) slope at the mesoscale of the real atmosphere just after the completion of spinup (0600 UTC 6 August; Figure 3a). Thus, it would have kept this canonical spectral shape if energy at various scales could increase comparatively in the log coordinate, whereas it is not the case here. The HKE spectrum can be roughly divided into three wavelength regions: \(\lambda_{h}>300\) km (\(k_{h}<2\pi/3\times 10^{-5}\) rad m\({}^{-1}\)), \(100<\lambda_{h}<300\) km (\(2\pi/3\times 10^{-5}<k_{h}<2\pi\times 10^{-5}\) rad m\({}^{-1}\)), and \(\lambda_{h}<100\) km (\(k_{h}>2\pi\times 10^{-5}\) rad m\({}^{-1}\)). For \(\lambda_{h}>300\) km (\(k_{h}<2\pi/3\times 10^{-5}\) rad m\({}^{-1}\)) and \(\lambda_{h}<100\) km (\(k_{h}>2\pi\times 10^{-5}\) rad m\({}^{-1}\)), the spectrum in general appears as lifting consistently; thus, the spectral shape over this two wavelength regions remains stable. For \(100<\lambda_{h}<300\) km (\(2\pi/3\times 10^{-5}<k_{h}<2\pi\times 10^{-5}\) rad m\({}^{-1}\); see Figure 4a-d for a zoomed-in view); however, the significant growth of energy firstly occurs at larger scales, and then extends to smaller scales. Consequently, the spectrum appears to be swelling during this wavelength region and shows an arc-like spectral shape. Figure 3: Time evolution of the horizontal kinetic energy (HKE) spectrum [\(E_{h}(k_{h})\)] averaged vertically over the troposphere for (**a**) 0–12 h, (**b**) 12–24 h, (**c**) 24–36 h, and (**d**) 36–48 h. Spectra are plotted every 3 h. The reference lines correspond to the \(-3\) and \(-5/3\) slopes. The arc-like shape of HKE spectrum in the troposphere over the central mesoscale is an interesting phenomenon, which has also been revealed by the idealized TC simulation of WZ2018 and the observational hurricane study of [PERSON] and [PERSON] [32]. Beyond that, this phenomenon has not been identified by any other studies related to atmospheric energy spectrum. Therefore, it can be treated as a unique feature for TC. For the case in this study, it can be found from the evolution of the spectrum that before the intensification of Lekima, the HKE already possesses canonical spectrum, which is flat over the central mesoscale with a slope of approximately \(-5/3\); while, with the rapid intensification of Lekima, the HKE spectrum gradually swells. In other words, the formation and development of this unique spectral feature is associated with the intensification of TC, which can be demonstrated by the observations.