## 3 HKE Spectra and Spectral Energy Budgets
### Spectral HKE Budget Diagnosis  
To determine the roles of different physical processes play during the intensification of Lekima, here we investigate the spectral budget of HKE using Equation (3). The 12-h period from 0000 UTC 7 August to 1200 UTC 7 is chosen for analysis. The earlier period is not chosen to avoid the impact of initial spinup adjustment of the model on the results. The later period is not chosen mainly for two reasons: (1) this period involves the mature stage of Lekima, and (2) Lekima in this period is too close to some islands, which leads to the local absence of physical quantities and affects the implementation of 2D DCT over the lower layers. Considering the particularity of the boundary layer and the vertical distribution of diabatic heating, the details of dynamics and energetics of HKE should vary with height in the troposphere. Thus, the troposphere is further divided into the lower troposphere (\(z\) = 0-1.5 km), midtroposphere (\(z\) = 2.5-5 km), and upper troposphere (\(z\) = 6-12 km). Figure 7 shows the various spectral budget terms averaged in time over the intensification of Lekima and vertically over different layers. For the energy conversion term \(C_{A\to h}\) (black solid lines  
Figure 6: Horizontal wavenumber spectra of HKE (black line), rotational kinetic energy (RKE) (red line), and divergent kinetic energy (DKE) (blue line) averaged vertically over the (**a,c**) troposphere and (**b,d**) lower stratosphere, at (**a,b**) 0600 UTC 6 August and (**c,d**) 0000 UTC 8 August. The reference lines correspond to the \(-\)3 and \(-\)5/3 slopes. in the left panels of Figure 7), a positive value means a conversion induced by the buoyancy effect from APE to HKE and vice versa. In the lower troposphere, the magnitude of \(C_{A\to h}\) is relatively weak at various horizontal scales (Figure 7a); while with the increase of height, the positive buoyancy dominates the troposphere and the importance of \(C_{A\to h}\) becomes prominent. In the midtroposphere, except for sporadic scales such as wavelengths (wavenumbers) around 100 and 400 km (\(2\pi\times 10^{-5}\) and \(\pi/2\times 10^{-5}\) rad m\({}^{-1}\)), it mainly makes a positive contribution, implying a dominant conversion from APE to HKE (Figure 7c). The sporadic negative contribution comes from the local-scale negative buoyancy forcing induced by the increase of air density, which originates from the evaporation and melting processes. In the upper troposphere, it becomes entirely positive, and develops two peaks at wavelengths (wavenumbers) of 300 and 1000 km (\(2\pi/3\times 10^{-5}\) and \(2\pi\times 10^{-6}\) rad m\({}^{-1}\)), respectively (Figure 7e). In general, a level of neutral buoyancy which is characterized as a transition zone from positive to negative buoyancy always exists near the tropopause. Thus, differing from the troposphere, the lower stratosphere is dominated by negative buoyancy forcing, and the \(C_{A\to h}\) term always makes a negative contribution at all scales, implying a conversion from HKE to APE (Figure 7g). The pressure vertical flux \(F_{p\uparrow}\) corresponds to the flux of IGW energy at lowest order; thus, the divergence term \(\partial_{z}F_{p\uparrow}\) can reflect the effect of vertical transportation of IGWs on the budget of HKE. The wavelength-height distribution of \(F_{p\uparrow}\) is shown in Figure 8a, and note that a positive (negative) \(F_{p\uparrow}\) means a downward (upward) flux. On the whole, \(F_{p\uparrow}\) is upward in the lower stratosphere and downward in the troposphere at various scales, implying the domination of IGWs which occur in the upper troposphere and then propagate to higher and lower layers. There are also nonnegligible differences at different scales. For \(400<\lambda_{h}<1000\) km (\(2\pi\times 10^{-6}<k_{h}\)\(<\pi/2\times 10^{-5}\) rad m\({}^{-1}\)), the boundary between upward and downward \(F_{p\uparrow}\) is around \(z=15\) km; namely, the highest source region of the waves can reach the tropopause.