## 3 HKE Spectra and Spectral Energy Budgets
### Spectral HKE Budget Diagnosis
With the reduction of height from 15 to 6 km, the magnitude of \(F_{p\uparrow}\) increases, which suggests that the downward-propagating waves are not only generated at the tropopause, but also generated throughout the upper troposphere. The \(F_{p\uparrow}\) achieves its local maximum around \(z=6\) km and decreases from 6 to 4 km, indicating that the IGW energy are deposited into this height range during the downward propagation of the waves. It then increases again and achieves another local maximum around \(z=2.5\) km. Obviously, a secondary wave source exists in the midtroposphere for \(z=2.5\)-4 km. At lower heights, the \(F_{p\uparrow}\) gradually decreases, and the IGW energy are powerfully deposited into the lower troposphere. The vertical distribution of \(F_{p\uparrow}\) for \(\lambda_{h}<100\) km (\(k_{h}>2\pi\times 10^{-5}\) rad m\({}^{-1}\)) is similar to that for \(400<\lambda_{h}<1000\) km (\(2\pi\times 10^{-6}<k_{h}<\pi/2\times 10^{-5}\) rad m\({}^{-1}\)), except that the boundary height between positive and negative \(F_{p\uparrow}\) reduces to \(z=13\) km; namely, the highest source region of the waves drops in height. For wavelengths (wavenumbers) slightly longer (smaller) than 100 km (\(2\pi\times 10^{-5}\) rad m\({}^{-1}\)); however, the distribution is not quite the same. Firstly, weak upward \(F_{p\uparrow}\) appears for heights below 4 km, which can be attributed to upward-propagating IGWs excited in the boundary layer. Secondly, the boundary between positive and negative \(F_{p\uparrow}\) in the upper troposphere drops rapidly to \(z=6\) km. In other words, the upper troposphere is not an effective source region for downward-propagating waves at these scales. Figure 7: Spectral HKE budget terms averaged vertically over the (**a**,**b**) lower troposphere, (**c**,**d**) midtroposphere, (**e**,**f**) upper troposphere, and (**g**,**h**) lower stratosphere, and in time over the intensification period. The legend for the left panels is given in (**a**) and that for the right panels in (**b**). The asterisk * represents any one of the budget terms. The spectra are multiplied by \(k_{h}\) to preserve area in log–linear coordinates. Although there is weak upward \(F_{p\uparrow}\) at wavelengths (wavenumbers) slightly greater (smaller) than 100 km (\(2\pi\times 10^{-5}\) rad m\({}^{-1}\)), the lower troposphere is primarily the net inflow layer of downward \(F_{p\uparrow}\). Meanwhile, the upper troposphere is always the net outflow layer of \(F_{p\uparrow}\) and the lower stratosphere is the net inflow layer of upward \(F_{p\uparrow}\). Consequently, the \(\partial_{z}F_{p\uparrow}\) term (black shaded lines in the left panels of Figure 8) is the net inflow layer of downward \(F_{p\uparrow}\). Figure 8: Horizontal wavelength–height distribution of (**a**) pressure vertical flux and (**b**) HKE vertical flux averaged in time over the intensification period (m\({}^{2}\) s\({}^{-3}\) kg m\({}^{-2}\)). Figure 7) makes clear positive, negative, and positive contribution in the lower troposphere (Figure 7a), upper troposphere (Figure 7e), and lower stratosphere (Figure 7g), respectively. The midtroposphere, however, is both the inflow and the outflow layer of \(F_{p\uparrow}\), depending on the wavelength; thus the contribution of \(\partial_{z}F_{p\uparrow}\) with wavelength over this layer is complicated (Figure 7c). The spectral transfer term \(t_{h}\) (red solid lines in the left panels of Figure 7) represents the transfer of HKE among different horizontal scales, which is caused by nonlinear interactions. It actually reflects the role of the so called \"energy cascade\" on budget of HKE at different scales. By constructing nonlinear spectral flux of HKE using \(t_{h}\), one can identify the direction of energy cascade, which is a key topic of research on atmospheric energy (WZ2018).