## 3 HKE Spectra and Spectral Energy Budgets
### Spectral HKE Budget Diagnosis
In this paper, however, as the limited-area domain we analyze has open lateral boundaries, the spectral flux integrated over the whole domain is not equal to zero, and the direction of energy cascade cannot be identified accurately; thus, we only discuss the distribution of \(t_{h}\). In the lower troposphere, the energy cascade makes a negative contribution at wavelengths (wavenumbers) around 300 km (\(2\pi\)/\(3\times 10^{-5}\) rad m\({}^{-1}\)) and makes a positive contribution for the rest scales (Figure 7a). In the midtroposphere, the effect of energy cascade is complicated and \(t_{h}\) crosses zero several times with the variation of wavelength. However, it makes a positive contribution at most scales (Figure 7c). When it comes to the upper troposphere, the role of \(t_{h}\) is weak, with negative contribution for \(400<\lambda_{h}<800\) km (\(\pi\)/\(4\times 10^{-5}<k_{h}<\pi\)/\(2\times 10^{-5}\) rad m\({}^{-1}\)) and \(200<\lambda_{h}<300\) km (\(2\pi\)/\(3\times 10^{-5}<k_{h}<2\pi\times 10^{-5}\) rad m\({}^{-1}\)), and positive contribution for the remainder (Figure 7e). The distribution of \(t_{h}\) is relatively clear in the lower stratosphere, which contributes positively for \(\lambda_{h}>500\) km (\(k_{h}<2\pi\)/\(5\times 10^{-5}\) rad m\({}^{-1}\)) and negatively at smaller scales (Figure 7g). The \(F_{h\uparrow}\) term is the HKE vertical flux, and the corresponding divergence term \(\partial_{z}F_{h\uparrow}\) reflects the direct contribution of vertical transportation of convection to HKE. It can be found from Figure 8b that, for most scales, the \(F_{h\uparrow}\) is consistently upward from the lower troposphere to the lower stratosphere, which suggests that convection within the troposphere transports the energy upward. Two prominent subranges of \(F_{h\uparrow}\) are collocated with those of the diabatic heating (Figure 2b), implying the close connection between the convection intensity and release of heating. There is also sporadic weak downward \(F_{h\uparrow}\) in the lower troposphere and midtroposphere for wavelengths (wavenumbers) around 100 km (\(2\pi\times 10^{-5}\) rad m\({}^{-1}\)) and below 60 km (above \(\pi\)/\(3\times 10^{-4}\) rad m\({}^{-1}\)), which is likely to be associated with the evaporation and the drag effect of precipitation particles. Reflected in the budget of HKE, the \(\partial_{z}F_{h\uparrow}\) term (red shaded lines in the left panels of Figure 7) in general makes a negative contribution in the lower troposphere (Figure 7a) and a positive contribution over the higher layers (Figure 7c,e,g). The 3D divergence term Div\({}_{h}\) and adiabatic nonconservation term \(J_{h}\) (blue solid and dashed lines in the right panels of Figure 7, respectively) stem from the compressibility and adiabatic nonconservation of the atmosphere, respectively. They are both significant in the midtroposphere and upper troposphere, and always play the balanced positive and negative contributions (Figure 7d,f). The diabatic term \(H_{h}\) (green solid lines in the right panels of Figure 7) represents the direct effect of diabatic processes on HKE, which is only relatively remarkable in the midtroposphere with negative contribution (Figure 7d). The dissipative term \(D_{h}\) (green dashed lines in the right panels of Figure 7) represents the effect of dissipative processes. Because of boundary-layer friction, its role is significant in the lower troposphere, with large negative contribution for \(\lambda_{h}>300\) km (\(k_{h}<2\pi\)/\(3\times 10^{-5}\) rad m\({}^{-1}\); Figure 7b). However, for higher layers, it plays weak role (Figure 7d,f,h). Overall, despite differences in magnitude and specific scales the effect are exerted on, the contributions of various budget terms at different heights during the intensification of Lekima are qualitatively consistent with those of the idealized TC in WZ2018, which suggests that the roles of various physical processes play are relatively steady.