Generation Mechanism for Stratospheric Gravity Waves by Unbalanced Flow in Tropical Cyclones

The generation mechanism of stratospheric gravity waves (SGWs) in tropical cyclones (TCs) is investigated with an idealized experiment using numerical model. The results show that there are two peaks of SGW amplitude during the mature period of TC, one above the eyewall and the other above the rainband. The SGWs can only exist in unbalanced flow, so a simplified balance equation suitable for the special structure of the TC is proposed to investigate the SGWs. Diagnosis of this equation shows there is significant unbalanced flow around the eyewall and rainband caused by the convection and the vertical shear between outflow and compensating inflow. Diagnosis of the source function indicates that diabatic heating and mechanical oscillations are the main mechanisms generating the SGWs above the eyewall and rainband. The Kelvin–Helmholtz instability induced by vertical shear also makes a non‐negligible contribution to the generation of SGWs above the rainband.


Introduction
Stratospheric gravity waves (SGWs) are a common atmospheric phenomenon (Fritts & Alexander, 2003).They can be generated by many sources in the troposphere, among which tropical cyclones (TCs) are an important source (Hoffmann et al., 2018;Kim et al., 2012Kim et al., , 2014;;Wang et al., 2019).Wright (2019, hereafter W19) used data from three satellite limb sounders and concluded that TCs are associated with a 15% increase in SGW amplitudes over the background and a 25% increase in measured momentum fluxes.In general, TCs play an important role in the generation of SGWs.Wu et al. (2021) used a numerical model to find the lag relationship between the SGWs and the diabatic heating of the TCs.They argued that the SGWs were generated by diabatic heating mainly above the eyewall of the TCs.In the outflow layer, Nolan and Onderlinde (2022) (NO22, hereafter) found that the SGWs are generated by shear instabilities, in particular by Kelvin-Helmholtz instability (K-H instability).Cohen et al. (2017) found that there is unbalanced flow around the outflow layer that can generate SGWs by balance adjustment (Zhang, 2004).Therefore, an interesting question is what exactly is the SGW generation mechanism in TCs?
In this paper, we focus on the mechanisms of SWGs generated in TCs, especially those associated with the unbalanced flow using an idealized numerical experiment.The remainder of the paper is organized as follows: the second section introduces the radial distribution of the SGWs.The third section describes the generation mechanism of the SGWs, and we focus on the unbalanced flow.A summary and conclusions are presented in the last section.

Abstract
The generation mechanism of stratospheric gravity waves (SGWs) in tropical cyclones (TCs) is investigated with an idealized experiment using numerical model.The results show that there are two peaks of SGW amplitude during the mature period of TC, one above the eyewall and the other above the rainband.The SGWs can only exist in unbalanced flow, so a simplified balance equation suitable for the special structure of the TC is proposed to investigate the SGWs.Diagnosis of this equation shows there is significant unbalanced flow around the eyewall and rainband caused by the convection and the vertical shear between outflow and compensating inflow.Diagnosis of the source function indicates that diabatic heating and mechanical oscillations are the main mechanisms generating the SGWs above the eyewall and rainband.The Kelvin-Helmholtz instability induced by vertical shear also makes a non-negligible contribution to the generation of SGWs above the rainband.

Plain Language Summary
As intense convective systems, tropical cyclones (TCs) can generate intense stratospheric gravity waves (SGWs).However, TCs contain many unique structures such as the eyewall, spiral rainband, outflow layer, and inflow layer.How are the SGWs distributed in TCs and what effect does TC structure have on the generation of SGWs?Here we find there are two peaks of SGW amplitude during the mature period of TCs, one above the eyewall and one above the rainband.They appear mainly in the unbalanced flow regions, and the unbalanced flow in TCs is generated by convection around the eyewall and rainband and the vertical shear of the radial wind caused by outflow above the rainband.Thus, the convection and vertical shear play key roles in the generation of SGWs in TCs.Diabatic heating is the main mechanism generating SGWs above the eyewall and rainband by convection.Meanwhile, mechanical oscillation also generates convective SGWs above the eyewall.The Kelvin-Helmholtz instability induced by the vertical shear also makes a non-negligible contribution to the generation of SGWs above the rainband.

WANG ET AL.
• There is significant unbalanced flow that can generate stratospheric gravity waves around the eyewall and rainband of tropical cyclones • The unbalanced flow is generated by convection and vertical shear of the radial wind around the eyewall and outflow above the rainband • Diabatic heating, mechanical oscillation, and Kelvin-Helmholtz instability contribute to the generation of stratospheric gravity waves

Supporting Information:
Supporting Information may be found in the online version of this article.

Experimental Design
The Weather Research and Forecasting (WRF) 4.2 Model (Skamarock et al., 2019) is used to carry out the idealized simulation without the background wind.The vortex is initialized according to Rotunno and Emanuel (1987).
The 10-m maximum velocity (Vmax) is 16 m s −1 and the radius of maximum wind (inner-core size, RMW) is 80 km.The initial depth of the vortex is 15 km and the surface sea temperature is 28°C.The simulation domain is a 1,200 × 1,200 horizontal grid with 4-km resolution.The vertical domain extends from the surface to 30 km with a 5-km damping layer at the top of the model.The vertical resolution is about 300-400 m with 80 sigma levels.The simulation results are interpolated to 500-m resolution.The simulation time is 8 days corresponding to 192 hr and the data is outputted every hour.The physical parameterizations include the Yonsei University boundary layer scheme (Hong et al., 2006), the surface layer scheme of Dudhia et al. (2008), and Weather Research and Forecasting single-moment six-class microphysics (Hong & Lim, 2006), but neither the convective parameterization nor the radiation scheme is used.
The background fields and evolution of the characteristics of the TC are shown in Figure 1.The Brunt-Väisälä frequency of the background field increases from z = 13 km and the peak is around z = 19 km (Figure 1a).The tropopause is around z = 15 km.To reduce the influence of the troposphere and damping layer (z = 25-30 km), we analyze the SGWs at z = 22.5 km.The characters of the SGW amplitude at other altitude levels are similar and the results can be seen in Figure S1 in Supporting Information S1.Vmax varies steadily after 24 hr (the red smooth line in Figure 1b), which shows that the first 24 hr may be regarded as the spin-up time.From 40 to 64 hr (the first and second purple lines), Vmax increases from 22.0 to 37.6 m s −1 , at a rate of change exceeding 15.4 m/s/(24hr), which is the definition of TC rapid intensification used by the National Hurricane Center (Hoffmann et al., 2018).After 64 hr, the rapid increase of Vmax ceases and Vmax varies slowly, thus defining the TC slowly intensification period.Therefore, the TC evolution can be divided into two stages: the rapid intensification phase (40-64 hr; P-RI) and the slowly intensification phase (64-192 hr; P-SI).For consistency with the 24-hr length of P-RI, the first 24 hr of 64-192 hr (the second and third purple lines in Figure 1b) is used as P-SI, and in fact, there are also two peaks after 88 hr (Figure S2 in Supporting Information S1).The RMWs contracts from 50 km in P-RI to 35 km in P-SI (Figure 1c).And the characteristics of the SGWs are different.

Structure of the TCs and Distribution of SGWs
The most significant wave at every grid point is calculated using three-dimensional Stockwell transforms (3DST) (Hindley et al., 2016(Hindley et al., , 2019;;Wright et al., 2017Wright et al., , 2021) ) based on the Fourier transform and inversion.The detail description about the 3DST can be found in Text S1 in Supporting Information S1.The center of the TC is identified by the surface pressure centroid method, as in Nguyen et al. (2014).The data on Cartesian grids are bilinearly interpolated onto a polar grid with radial grid points at radius r = 4 km, 8 km, … to r = 1,000 km, as in NO22.
The azimuth-mean amplitude of the SGWs is shown in Figure 1d.
The amplitude of the SGWs decreases with the radius, which is similar to the results of W19.The radius of the maximum amplitude is around the eyewall, showing that the intense SGWs mainly appear above the eyewall.
The distribution of the 10-m wind is also shown in Figure 1d because the 10-m wind represents the bottom structure and intensity of the TC.The radial profile of the SGWs in P-RI is similar to that of the 10-m wind in the same period.However, the profile of the SGW amplitude in P-SI shows two peaks; one peak is similar to the 10-m wind showing the position of the eyewall of the TCs in the same period and other peak appears around r = 90-120 km.
Because diabatic heating is a mechanism that can generate SGWs (Lane et al., 2001;Wu et al., 2021), the diabatic heating profile is shown as the purple line in Figure 1d.There are two peaks of diabatic heating in P-SI, corresponding to the amplitude of the SGWs.However, the second peak is very weak and the second peak of the amplitude cannot be attributed to diabatic heating alone.The detailed investigation can be found in Section 3.2 and 3.3.
The structure of the TCs during the two periods is first shown in Figure 2. The classical wind structure of the TC can be seen for P-RI in Figure 2a and for P-SI in Figure 2b.The slant eyewall in P-SI is more significant.
From Figures 2c and 2d, the outflow layer is at z = 10-14 km in both P-RI and P-SI.The outflow in P-SI exceeds 14 m s −1 , whereas that in P-RI is no more than 8 m s −1 .There is a compensating inflow below the outflow layer in

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WANG ET AL. 10.1029/2023GL104907 3 of 10 P-SI, which is absent in P-RI.The vertical shear of the radial speed along the interface is ∼7 m s −1 km −1 in P-SI, which is larger than that in P-RI (<3 m s −1 km −1 ), and the large shear may generate the GWs (Dong et al., 2023;Fritts, 1982;Fritts & David, 1984).The azimuth-mean profiles of the hourly precipitation during each period are shown as green lines in Figures 2c and 2d.The most intense precipitation can be found around the eyewall during both periods, and there is also a secondary peak in precipitation during P-SI at r = 90-120 km that is absent during P-RI.The second peak during P-SI is the so-called spiral rainband, which is consistent with the second peak of the diabatic heating and amplitude of the SGWs in Figure 1d.
Figures 2e and 2f show the time mean absolute vertical velocity during each period.There is intense convection below z = 15 km, while the vertical velocity shows a wave-like distribution (Figure S3 in Supporting

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WANG ET AL.
10.1029/2023GL104907 5 of 10 Information S1) representing SGWs above z = 15 km during the two periods.The most intense SGWs at 52 hr are above the eyewall (the dashed box in Figure 2e), whereas those at 76 hr are above both the eyewall and rainband (the solid and dashed box in Figure 2f).An interesting question is why the intense SGWs are in different areas.

Balance Equation for a TC
GWs cannot occur and propagate in balanced non-divergent flow.Therefore, diagnosis of the unbalanced flow can be used to investigate the generation of SGWs.Zhang (2004) used the residual of the balance equation (ΔNBE) based on the traditional balance equation to investigate GWs, and the terms for the vertical shear and vertical velocity in this equation are ignored.However, the vertical velocity and its shear cannot be ignored in TCs (Figure 2), and the total divergence Equation 1 must be considered: here, D is the divergence; u, v, w are the zonal, meridional and vertical velocities; f is the Coriolis force parameter (5 × 10 −5 s −1 ); β 0 is the Rossby parameter (5 × 10 −11 m −1 s −1 ); ζ is the vertical vorticity; α is the specific volume; and ) and ∇ h is the horizontal gradient operator.The six terms on the right-hand side of Equation 1 are respectively horizontal shear, beta effect, relative vorticity, pressure gradient, vertical shear (VS), and baroclinic influence.A detailed description of each term may be found in Zhang et al. (2000).After calculation of the magnitude of each term using the TC simulation, the magnitude of three terms (beta effect, relative vorticity and baroclinic influence) is at least one order smaller than that of the other three terms.The calculation is carried out on the Cartesian coordinates and then it is interpolated into the polar coordinates.
The divergence is absent in the balance equation so the balance equation for TCs may be written as follows: To reveal the unbalanced flow in the TCs, the three significant terms of Equation 2 during P-SI are given in Figure 3.It should be noted that the time-mean during each period is carried after the calculation of the corresponding physical variables.
There is intense horizontal shear in the center and eyewall of the TC (Figure 3a), and that term can increase the divergence in the eye and decrease the divergence around the eyewall.There is an intense pressure gradient at the same position (Figure 3b), which can decrease (increase) the divergence in the eye (eyewall).Note that the trends of the above two terms are opposite, and the sum of the two terms is small (Figure 3c).Therefore, the divergence is dominated by VS.In this way, the balance equation for a TC may be further simplified as VS = 0.After converting to polar coordinates, the balance equation for a TC may be written as follows: here, u r and u ϑ are the radial and tangential wind, respectively, and λ is the azimuth.Equation 3indicates that the emergence of divergence is mainly attributed to the horizontal inhomogeneity of vertical velocity and the vertical shear of horizontal wind in TC.

Unbalanced Flow in a TC
The simplified balance equation for a TC has been proposed as Equation 3, and there will be unbalanced flow in the TCs if VS don't vanish.To reveal the generation mechanism of the unbalanced flow in the TC, every term of VS is shown in Figure 4.Because VS is mainly positive in Figure 3d, only the positive parts of VS are shown.
It should be noted that the negative part in Figure 4d is associated with weak vertical shear of the radial wind in Figure 4e so the product of them is smaller in Figure 4f.
In Figure 4, the second term on the left-hand side of Equation 3 (Figure 4c) is much smaller than the first term (Figure 4f) because the TC is almost axisymmetric and the tangential variation in vertical wind is very small (Figure 4a).Therefore, the term that contributes to the unbalanced flow is the first term of VS (Figures 4d-4f).

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WANG ET AL. 10.1029/2023GL104907 6 of 10 Calculation of the first term of VS shows that there is intense    inside the eyewall and rainband (Figure 4d) and intense     at the rainband (Figure 4e).Therefore, the appearance of the unbalanced flow around the eyewall and rainband is attributed to       ≠ 0 (Figure 4f).Results from diagnosing the TC balance equation suggest that the contribution to the unbalanced flow is from the radial shear of vertical wind and vertical shear of radial wind in TCs.The continuous generation of unbalanced flow from the TC will lead to the continuous emission of SGWs, as described in Zhang (2004).Therefore, the mechanism by which the unbalanced flow generates the GWs in TCs and the roles of

Generation Mechanism of SWGs by Unbalanced Flow
To reveal the possible processes by which the unbalanced flow in TCs generates the SGWs, a comparison of Figures 2f and 4d indicates that the intense unbalanced flow is associated with intense convection around both the eyewall and rainband.Convection is a very common mechanism for generating SGWs in TCs.Lane et al. (2001) proposed that convection can generate the SGWs mainly through three mechanisms: mechanical oscillation, diabatic heating, and the obstacle or moving mountain effect, and they also suggested a source function to diagnose the three mechanisms.The source function (Choi & Chun, 2014;Lane et al., 2001) can be shown as following: where U(z) and u' are the basic-state and perturbation zonal winds, respectively; V(z) and v' are the basic-state and perturbation meridional winds, respectively; w' and θ' are the perturbations of the vertical wind and potential temperature respectively;   and   are the basic-state density and potential temperature, respectively; g is the gravitational acceleration.The forcing terms F u , F v , F w , F θ , and Q are expressed as follows: where H and C represent the diabatic heating and cooling, respectively.In the right hand of Equation 4, the term A represent the nonlinear advection mechanism, the term B represent diabatic heating mechanism, and term C represent shear mechanism.
Figure 5 shows the diagnosis of the source function during P-SI in a TC.
The diabatic heating dominates around the eyewall and the rainband (Figure 5a), corresponding to the areas of strong convection and heavy precipitation (Figures 1d and 2d).Advection is still significant around the eyewall, and this term dominates the generation of the SGWs between z = 11 and z = 15 km (tropopause), where the diabatic heating is absent.Because the shear term is very small (Figure 5c), the influence of the obstacle effect on the SGWs is excluded.
In addition to the diabatic heating and mechanical oscillation resulting from convection around the eyewall and rainband, the role of radial wind vertical shear (     ) caused by the TC outflow layer should not be ignored above the rainband because     ≠ 0 is a necessary condition for K-H instability.The threshold of the K-H instability is that the Richardson number should be larger than 0 but less than 0.25 (Dong et al., 2023).The areas where there is always K-H instability at each hour throughout P-SI are shown by green shading in Figure 5c.There is K-H instability in the intense shear areas (Figure 4e) below the outflow layer, especially at z = 5.5-10 km (Figure 5c).There are also some areas with K-H instability in the eyewall.Therefore, the vertical shear of the radial wind can generate the SGWs by K-H instability.The K-H instability may not be as important as convection, but it cannot  In general, the first peak of the SGW amplitude around the eyewall is generated by the diabatic heating, mechanical oscillation, and K-H instability, whereas the second peak of SGW amplitude above the rainband is generated by diabatic heating and K-H instability.Therefore, the sources of the SGWs above the rainband can be from three parts as the convection in eyewall and rainband and the K-H instability in the outflow above the rainband.
During P-RI of the TC (Figure S4 in Supporting Information S1), because the outflow layer and the rainband are very weak (Figure 2c),     is absent and the role of K-H instability can be ignored.The SGWs above the eyewall are generated by diabatic heating and mechanical oscillation during P-RI.

Concluding Remarks
In this paper, the mechanism by which unbalanced flow in TC generates SGWs is investigated via an idealized experiment using the WRF model.Two stages can be defined according to the changes in TC intensity and structure: the period of rapid intensification (P-RI) and the period of maturity (P-SI).The SGW amplitude has different characteristics in the two periods.There are two peaks in the radial distribution of the amplitude of the SGWs in P-SI but only one in P-RI.The structure of the TC during the two periods is different.The TC during P-SI has a mature rainband and intense outflow and compensating inflow, which are all absent during P-RI.
To investigate the cause of the peaks in SGW amplitude, considering that GWs do not exist in a balanced nondivergent flow, the divergence equation has been diagnosed.Through scaling analysis of the diagnostic results, a simplified balance equation suitable for TCs is obtained.This equation demonstrates that there is intense unbalanced flow around the eyewall (the first peak of SGW amplitude) and rainband (the second peak of SGW amplitude) associated with the intense horizontal shear of the vertical wind.The intense vertical shear of the radial wind induced by the outflow layer above the rainband also contributes to the generation by the unbalanced flow.To further reveal the mechanism by which the unbalanced flow generates SGWs in a TC, the relationship between the unbalanced flow and TC structure is analyzed.The peak of SGW amplitude is found over the area of strong convection in the eyewall and rainband.The diagnostic results for the source function indicate that diabatic heating and mechanical oscillations are the main generation mechanisms for the two peaks of the SGW amplitude above the eyewall and rainband, while the obstacle effect is negligible.The vertical shear around the outflow layer can generate SGWs through the K-H instability mechanism above the rainband, and there are also some areas of K-H instability in the eyewall.When the TCs are in P-RI, the SGWs above the eyewall are only generated by diabatic heating and mechanical oscillations around the eyewall, and there is no diabatic heating and K-H instability around the rainband, so there is only one peak of SGW amplitude above the eyewall.
Note that the background wind shear can enhance the obstacle effect (Lane et al., 2001) and decrease the intensity of the TCs so that the diabatic heating and mechanical oscillation may decrease.Experiments with many kinds of background wind should be considered to further investigate the generation mechanism of the SGWs by unbalanced flow in TCs.

Figure 1 .
Figure 1.(a) Vertical profile of the background Brunt-Väisälä frequency (s −2 ), (b) the time series of Vmax (m s −1 ) and (c) RMW (km).(d) Radial distribution of azimuth-mean SGW amplitude (m s −1 , black lines), normalized 10-m wind fields (red lines), and the normalized mean diabatic heating over z = 0-15 km (purple lines) during P-RI (dashed lines) and P-SI (solid lines).The inset figures in (d) are the partially enlarged figure of the amplitude.Purple vertical lines in (b) and (c) show 40, 64, and 88 hr.Black (red) solid lines in (b) and (c) are the raw (smoothed) time series.

Figure 2 .
Figure 2. Azimuth and time mean radius-altitude profiles of the full horizontal wind (a, b, m s −1 ), radial wind (c, d, m s −1 ) and absolute vertical velocity during P-RI (a, c, e) and P-SI (b, d, f).Green lines in panels (c) and (d) are the azimuth-mean profiles of the hourly precipitation (mm/h, right axis).

Figure 3 .
Figure 3. Radius-altitude profiles of the significant terms in the divergence equation (10 −7 s −2 ) during P-SI: (a) horizontal shear; (b) pressure gradient; (c) sum of horizontal shear and pressure gradient; and (d) vertical shear.

Figure 4 .
Figure 4. Radius-altitude profiles of each term (s −1 ) on the left-hand side of Equation 3 during P-SI.

Figure 5 .
Figure 5. Radius-altitude profiles of diabatic heating (a), advection (b), and shear (c) in the source function during P-SI (10 −13 m −1 s −3 ).Green shading in (c) shows the areas where there is always K-H instability throughout P-SI.
it is always associated with the intense vertical shear at each hour during P-SI (Figure5c), especially around the outflow layer above the rainband.It is a common phenomenon during the P-SI of the TC.