# Elliptical Structures of Gravity Waves Produced by Typhoon Soudelor in 2015 near Taiwan
## 2 Description of the Model and Methods of Analysis
The wave energy densities were computed from the mean squared perturbations of potential temperature and horizontal wind to measure GW activity [45; 46] as follows:  
\[\begin{array}{l}E_{p}=\frac{1}{2}\left(\frac{\partial}{\partial}\right)^{2} \left(\frac{\partial^{\prime}}{\partial}\right)^{2}\quad\text{with}\quad N^{2 }=\frac{\delta}{\partial}\left(\frac{d\overline{\theta}}{dz}\right)\\ E_{k}=E_{u}+E_{v}+E_{w}=\frac{1}{2}\left(\overline{u^{\prime\,2}}+\overline{v^{ \prime\,2}}+\overline{w^{\prime\,2}}\right)\\ E where \(\overline{\theta}\) and \(\theta^{\prime}\) are the mean and perturbations of the potential temperature, respectively; \(u^{\prime}\), \(v^{\prime}\) and \(w^{\prime}\) are the perturbations of the zonal, meridional, and vertical wind, respectively; \(g\) is the acceleration of gravity; \(N\) is the Brunt-Vaisala frequency; and \(E_{t}\), \(E_{p}\), \(E_{k}\), \(E_{u}\), \(E_{v}\) and \(E_{w}\) are the total, potential, kinetic, zonal, meridional and vertical energy densities, respectively. The overbar corresponds to the average height. The use of the potential temperature in Equation (1) enabled us to remove, fairly-well, the potential temperature perturbations near the tropopause.  
Then, hodographs of the horizontal wind perturbations were drawn as a function of altitude to identify elliptically polarized structures of low-frequency GWs in the troposphere and LS. The intrinsic frequency \(\omega\) resulted from the axis ratio of the ellipse using the variance method of analysis, as described in Reference [24]. The one-dimensional fast Fourier transform (FFT) was used to characterize the vertical wavelength \(\lambda_{\mathrm{p}}\) of the vertical profiles of the wind perturbations. The horizontal wavelength \(\lambda_{h}\) was estimated from the simplified dispersion relation of the GW linear theory described by the Equation (8.4.19) in Reference [47]:  
\[\lambda_{h}=\left(\frac{N^{2}-\omega^{2}}{\omega^{2}-f^{2}}\right)^{\frac{1}{2 }}\lambda_{\mathrm{p}}\quad\mathrm{with}\quad f<\omega<N \tag{2}\]  
Figure 1: (a) Best track of Typhoon Soudelor (black) from 5 August 00:00 UTC to 8 August 18:00 UTC in 6 h intervals, and the WRF (Weather Research and Forecasting) TC track (gray) from CTL from 5 August 00:00 UTC to 8 August 00:00 UTC in 1 h intervals. Dotted rectangles visualize parent and nested domains D1, D2 and D3 of the simulation. Shaded contours (every 500 m derived from 2 arc min gridded global relief data Etopo2) and the colorbar (m) display the Central Mountain Range heights. (b) Initial SST field (\({}^{\circ}\)C) on 5 August 00:00 UTC and terrain elevation (m) with 30 s resolution used for CTL. (c) Vertical distribution of 130 eta levels (circle) and corresponding vertical step height (plus) up to 27 km in the lower stratosphere (LS). where \(f\) is the inertial frequency. The continuous wavelet transform (CWT) with the complex Morlet wavelet was used to analyze the time evolution of GW periods in the LS when TC Soudelor was approaching Taiwan. To focus on dominant GWs, spectral lines were derived from the CWT maxima. The method is detailed in Reference [48]. The combined conventional method (CCM) was based on two techniques commonly used to extract spectral characteristics of GW dominant modes from high-resolution radiosonde data: the SPARC (Stratosphere-troposphere processes and their role in climate) and the Stokes parameter methods [24]. Vertical profiles of temperature and wind were broken into parts to improve the extraction of perturbations and to avoid the detrending problem near the tropopause arising from strong changes in the temperature gradients. In particular, a fraction of upward energy (\(Fup\)) \(>\) 50% (\(<\)50%) indicated that GWs with upward (downward) energy propagation were dominant. The degree of polarization (\(d\)) measured the ratio of completely polarized power to total power [45; 49]. Thus, a value of 0% (100%) indicated a random or irregular (monochromatic) wave. A mean value \(>\)40% suggested a significant degree of polarization.