## 3 HKE Spectra and Spectral Energy Budgets
### Evolutions of the HKE Spectra
In [PERSON] and [PERSON] [32], they calculated the HKE spectra based on numbers of in situ observations and then classified the spectra according to storm intensity. Although they did not describe the hurricane spectra with expression similar to \"arc-like shape\" or \"swelling\", their conclusion is in essence consistent with ours. From their results (e.g., their Figure 7), the spectrum for tropical storms has a slope close to \(-5/3\) over the central mesoscale. As the storm intensifies from category-1 to category-5, the swelling of the spectrum becomes increasingly obvious, which extends to increasingly smaller wavelengths. This leads them to the conclusion that \"as storm intensity increases, spectral slope becomes steeper and the transition zone where spectral slope begins to steepen shifts downscale\". The variation of their hurricane spectrum with storm intensity is in perfect agreement with the evolution of our TC spectrum during the intensification of Lekima. Note that in our simulation, Lekima reaches a maximum intensity of category-2 and the swelling of the spectrum extends to wavelengths (wavenumbers) around 100 km (\(2\pi\times 10^{-5}\) rad m\({}^{-1}\)); while in WZ2018, their idealized TC reaches a maximum intensity of category-4 and the swelling extends to wavelengths (wavenumbers) around 50 km (\(2\pi/5\times 10^{-4}\) rad m\({}^{-1}\)). This also proves the relationship between TC intensification and the arc-like spectral shape. Figure 4: Time evolution of the 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, and zoomed in for \(100<\lambda_{h}<300\) km (\(2\pi/3\times 10^{-5}<k_{h}<2\pi\times 10^{-5}\) rad m\({}^{-1}\)). Spectra are plotted every 3 h. The reference lines correspond to the \(-3\) and \(-5/3\) slopes. Compared to that in the troposphere, the growth of HKE spectrum in the lower stratosphere shows different characteristics (Figure 5a-d). After the spinup adjustment of the model, the spectral shape keeps relatively stable throughout the intensification of Lekima. Differing from the canonical atmospheric spectrum, the lower-stratospheric HKE spectrum here has consistent \(-5/3\) slope for \(\lambda_{h}<1000\) km (\(k_{h}>2\pi\times 10^{-6}\) rad m\({}^{-1}\)), but not \(-3\) slope at the large-scale end. Meanwhile, the spectrum shows flat shape throughout the mesoscale but not arc-like shape over the central mesoscale as in the troposphere. These features are also identical to those found in WZ2018. To further understand the formation of the unique HKE spectra for TC, Figure 6 shows the rotational kinetic energy (RKE) and divergent kinetic energy (DKE) spectra at selected times. The RKE has spatiotemporal variable spectral shape. At 0600 UTC 6 August, the RKE spectrum in the troposphere is steeper at large scales and shallower at smaller scales, and the visible swelling of the spectrum appears for \(200<\lambda_{h}<400\) km (\(\pi/2\times 10^{-5}<k_{h}<\pi\times 10^{-5}\) rad m\({}^{-1}\); Figure 6a); while the RKE spectrum in the lower stratosphere is flat for \(\lambda_{h}<1000\) km (\(k_{h}>2\pi\times 10^{-6}\) rad m\({}^{-1}\)) with a slope approaching \(-5/3\) (Figure 6b). After the intensification of Lekima at 0000 UTC 8 August, the swelling of the RKE in the troposphere is more significant, which extends downscale to \(\lambda_{h}=100\) km (\(k_{h}=2\pi\times 10^{-5}\) rad m\({}^{-1}\); Figure 6c); while the lower-stratospheric RKE spectrum suffers much less change (Figure 6d). The DKE spectrum, in contrast, always shows a flat shape, which changes little with time and height. Figure 5: Time evolution of the HKE spectrum [\(E_{h}(k_{h})\)] averaged vertically over the lower stratosphere 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.