Global static stability and its relation to gravity waves in the middle atmosphere

The global atmospheric static stability (N 2) in the middle atmosphere and its relation to gravity waves (GWs) were investigated by using the temperature profiles measured by the Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) instrument from 2002 to 2018. At low latitudes, a layer with enhanced N 2 occurs at an altitude of ~20 km and exhibits annual oscillations caused by tropopause inversion layers. Above an altitude of ~70 km, enhanced N 2 exhibits semiannual oscillations at low latitudes caused by the mesosphere inversion layers and annual oscillations at high latitudes resulting from the downward shift of the summer mesopause. The correlation coefficients between N 2 and GW amplitudes can be larger than 0.8 at latitudes poleward of ~40°N/S. This observation provides factual evidence that a large N 2 supports large‐amplitude GWs and indicates that N 2 plays a dominant role in maintaining GWs at least at high latitudes of the middle atmosphere. This evidence also partially explains the previous results regarding the phase changes of annual oscillations of GWs at high latitudes.


Introduction
In a stably stratified atmosphere, an air parcel oscillates adiabatically around its equilibrium because of the buoyancy force and gravitational force. The oscillation frequency is referred to as the buoyancy frequency (or Brunt-Väisälä frequency) and is denoted as N. The square of the buoyancy frequency (N 2 ) is a measure of the atmospheric static stability. The static stability is determined by the structure of the background atmosphere and has an important influence on the generation, propagation, and dissipation of atmospheric waves. In turn, the momentum and energy transported by atmospheric waves control the atmospheric structures in the mesosphere (Andrews et al., 1987;Holton, 2004). Thus, static stability acts as a bridge between the background at-mosphere and atmospheric waves.
In the troposphere and stratosphere, static stability is mainly controlled by the temperature lapse rate and the height of the tropopause. Generally, below (above) the tropopause, the temperature decreases (increases) with increasing height and results in lower (higher) values of static stability. This general scenario does not hold above the tropopause region because of the presence of tropopause inversion layers (TIL). The TIL is defined as a region with enhanced static stability just above the cold-point tropopause (Birner et al., 2002(Birner et al., , 2006. Recent studies have shown that the TIL and the associated static stability enhancement occur at all latitudes, as revealed by satellite and ground-based observations, as well as reanalysis data and model simulations (Grise et al., 2010;Gettelman et al., 2011;Zhang YH et al., 2015Liu HL, 2017). The enhanced static stability above the tropopause is closely related to the wind shear and gravity waves (GWs) in this region (Liu HL, 2007, 2017; Sunilkumar et al., 2015Sunilkumar et al., , 2017Zhang YH et al., 2015Pilch Kedzierski et al., 2016). Through model simula-tions, Liu HL (2017) showed that large wind shears exist in the TIL region. This is because enhanced static stability in the TIL allows wind shears to attain large values before the onset of instability (Liu HL, 2007). The wind shears can be attributed to the background atmosphere and atmospheric waves. Through a spatialscale decomposition, Liu HL (2017) showed that large wind shears are mainly caused by GWs and modulated by tidal waves (Yue J et al., 2010;Liu X et al., 2014). By analyzing high-resolution radiosonde measurements, Zhang YH et al. (2015 showed that upward-propagating GWs have maxima in the TIL, which is at ~1 km above the tropopause. These results indicated that the enhanced static stability in the TIL provides favorable conditions for GW propagation and thus for the large wind shears (Sunilkumar et al., 2015(Sunilkumar et al., , 2017Liu HL, 2017).
In the mesosphere and lower thermosphere region, static stability has a similar configuration as that around the tropopause. Above the mesopause, the temperature increases with height. This results in enhanced static stability, which can sustain large winds and wind shears. Gravity waves are the main contributors to the large winds and wind shears (Liu HL, 2007, 2017Liu X et al., 2014), forming a positive feedback process among the static stability, GWs, large winds, and wind shears. In the mesosphere, there is a layer with a thickness of ~10 km at ~65 km and 85 km, in which the temperature enhancement can be up to ~20 to 50 K, called the mesosphere inversion layer (MIL; Leblanc and Hauchecorne, 1997;Meriwether and Gerrard, 2004;Xu JY et al., 2007b;Fechine et al., 2008;Gan Q et al., 2012). In the MIL, the temperature increases with height and results in enhanced static stability. For example, Gan Q et al. (2012) showed, by using temperature data derived from the Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) instrument (Russell III et al., 1999) onboard the Thermosphere Ionosphere Mesosphere Energetics and Dynamics (TIMED) satellite, that the monthly mean amplitude of the lower MILs exhibits semiannual oscillations (SAOs) with peaks in spring and autumn at an altitude range of 76 to 83 km at low latitudes. At the middle latitudes, the monthly mean amplitude of the lower MILs exhibits annual oscillations (AOs) with peaks in winter and at an altitude range of ~70 to 79 km. The increasing (decreasing) temperature above (below) the bottom of the MIL produces a sharp increment (decrement) of static stability in the mesosphere. As a consequence, the seasonal and latitudinal variations of the MIL, such as its amplitude and height, will induce corresponding variations in the static stability. It can be speculated that the positive feedback between the static stability and GWs would be similar in the MIL.
Using temperature profiles measured by the SABER instrument, Liu X et al. (2017) showed that the AO of the monthly zonal mean GW potential energy undergoes a clear phase shift at latitudes poleward of 40°N/S (Fig. 3 of their paper). The peaks of the GW potential energy shift from winters below an altitude (hereafter, referred to as z) of ~70 km to summers above z ~80 km, whereas at z 70-80 km, the SAO is dominant. The global GW potential energy ( Fig. 2b of their paper) exhibits a much weaker peak at z ~75 km, which corresponds to the altitudes of the phase shifts of the AO and SAO. We suggest that when the altitudes of the MIL (Gan Q et al., 2012) are compared with phase shifts of the AO and SAO in the zonal mean temperature (Shepherd et al., 2004;Xu JY et al., 2007b), N 2 might link the phase shifts of the interannual oscillations of GWs with the MILs, as well as the interannual oscillations of the zonal mean temperature. The purpose of this study is to illustrate the global climatology of the static stability and its relation to GWs.
Understanding the global static stability and its seasonal, latitudinal, and altitude variations may have important implications for global GW variations. In this study, we examined the SABER temperature profiles to characterize the global distribution and intraannual oscillations of the static stability. In addition, we examined the relations between the static stability and GWs, which were also derived from the SABER temperature profiles.

Data and Method
The temperature profiles (V2.0) measured by the SABER instrument (Russell et al., 1999(Russell et al., ) over a 17-year period (2002(Russell et al., -2018 were used in this study. These profiles cover an altitude range of 15-110 km and a latitude range of 53°S-83°N or 83°S-53°N. The temperature accuracy is 1-3 K from 30 to 80 km and 5-10 K from 90 to 100 km, as reported on the SABER website (http://saber. gats-inc.com/; Remsberg et al., 2008).
The GW extraction method was similar to that by Liu X et al. (2017Liu X et al. ( , 2019, and references therein). First, the daily SABER temperature profiles, , in a latitude band of 5° were selected. Second, at each altitude, the selected data were fitted by harmonics with zonal wavenumbers ranging from 0 to 6 to obtain the background temperature, , and residual temperature, . The component of wavenumber 0 was the zonal mean temperature, . Third, a band-pass filter with a cutoff wavelength of 5-25 km was applied on the residual profiles to obtain the GW perturbation profiles, . The above-mentioned three steps were applied on the ascending and descending nodes, respectively, such that they minimized the influence of tides on GWs (Preusse et al., 2009;Yamashita et al., 2013). Finally, the static stability was calculated according to the method of Holton (2004), where g is the gravitational acceleration and is the specific heat for dry air at a constant pressure. If , the atmosphere is statically stable and the air parcel will oscillate around its equilibrium level for a period of . Thus, is also referred to as the buoyancy frequency or Brunt-Väisälä frequency. If , the atmosphere is statically neutral and the air parcel will remain at a neutral equilibrium level. If , the atmosphere is statically unstable and the air parcel will be displaced far from its equilibrium level. The strength of a GW is measured by its amplitude, . We did not use the GW potential energy ( ) to measure GWs because it contains N 2 , which affects the correlations between static stability and GWs. In each latitude band, we obtained one N 2 profile and one daily zonal mean profile, . The latitude bands had centers from 50°S to 50°N, with a step of 2.5° and an overlap of 2.5°. Figure 1 shows time-height sections of the monthly zonal mean N 2 and temperature at four latitude bands of the northern hemisphere (left) and southern hemisphere (right). Figure 1 provides a general picture of the altitude, latitude, and seasonal variations of N 2 and temperature. The N 2 is smaller in the lower mesosphere (z 50-70 km) than at the other altitudes. The largest N 2 occurs above the mesopause (blue dots) because of the sharp increase in the lower thermospheric temperature. At 5°N/S, the quasi-biennial oscillations of N 2 can be seen below z ~45 km because of the quasi-biennial oscillations in the lower stratosphere (Baldwin et al., 2001). The enhanced N 2 can be seen in the bottom of the stratosphere (z ~20 km) as compared with that above z ~25 km in the stratosphere. Moreover, the enhanced N 2 can be seen in the middle mesosphere (z ~80 km) as compared with that at other heights of the mesosphere.

General Features of the Global Static Stability
The altitude variations of N 2 also depend on the latitude and season. At 5°N/S and 20°N/S, a layer with enhanced N 2 occurs at z ~20 km. This layer is just above the cold-point tropopause and is associated with the TIL (Birner et al., 2002(Birner et al., , 2006Gettelman et al., 2011). Below the TIL, N 2 decreases sharply with decreasing height. At 35°N/S, an enhanced N 2 associated with the TIL can still be seen, but it is much weaker than that at lower latitudes. The TIL revealed here coincides with previous results based on observations from satellite (Grise et al., 2010), radiosonde (Zhang YH et al., 2015, and reanalysis data (Pilch Kedzierski et al., 2016), as well as model simulations (Liu HL, 2007). At 50°N/S, the enhanced N 2 associated with the TIL cannot be seen. This is because the height of the tropopause decreases from z ~17 km at lower latit-  udes to z ~10 km at high latitudes (Grise et al., 2010). The altitude coverage of the SABER measurements can be as low as z ~15 km, although uncertainties in the measurements are relatively greater below z ~20 km than above z ~20 km. Through comparisons between Figure 1 and the previous results, we found that the enhanced N 2 associated with the TIL could be reasonably revealed by the SABER measurements at latitudes equatorward of 35°N/S. The enhanced N 2 associated with the TIL exhibits AO with peaks during January and February in the lower latitudes (i.e., 5°N/S and 20°N/S). At 35°N (35°S), the enhanced N 2 associated with the TIL is much weaker than that in the lower latitudes, and it exhibits AO with peaks during June and July (January and February). The amplitudes and phases of the AO and SAO in N 2 associated with the TIL were quantified through harmonic fitting, as presented in the next subsection.
Another layer with enhanced N 2 occurs in the upper mesosphere (>70 km). This layer is associated with the MIL (e.g., Meriwether and Gerrard, 2004). The layer of enhanced N 2 associated with the MIL extends from the tropical region to high latitudes, but its seasonal variations depend on the latitudes. At 5°N/S and 20°N/S, the enhanced N 2 at z ~75-85 km exhibits SAO with peaks in spring and autumn. This oscillation corresponds to the tropical MIL, in which the temperature increases with height. Moreover, the amplitude of the SAO at 5°N/S is larger than that at 20°N/S. This is because the MIL has peaks in spring and autumn in the tropical region (Xu JY et al., 2007b) and the tropical MIL is mainly induced by diurnal migrating tides (Xu JY et al., 2009;Gan Q et al., 2012). At 35°N/S and 50°N/S, the enhanced N 2 above z ~85 km might be associated with both the MIL during winter (Gan Q et al., 2012) and the downward shift of the mesopause during summer (Xu JY et al., 2007a). The enhancement of N 2 above z ~85 km is more prominent at 50°N/S than at 35°N/S in summer. This is because the temperature increases sharply with height in the lower thermosphere, which begins at a lower height (~85 km) because of the downward shift of the mesopause (blue dots) in summer (Xu JY et al., 2007a).

AO and SAO of the Global Static Stability
To quantify the AO and SAO of N 2 , we applied harmonic fitting to the time series of the monthly zonal mean N 2 at each height and latitude. The fitting function is expressed as where A 0 is the mean N 2 over the 17 years, and A AO and A SAO (θ AO and θ SAO ) are the amplitudes (phases) of AO and SAO, respectively. The phases correspond to the peaks of oscillations. Figure 2 shows the latitude-height sections of the amplitudes and phases of the AO and SAO.
From Figure 2, we can see that at latitudes poleward of ~35°S, the AO is prominent at z ~55-70 km with the peak in summer (June-July-August) and at z ~85-105 km with the peak in winter (December-January-February) in the southern hemisphere. In contrast, at latitudes poleward of ~30°N, the AO is prominent at z~6 0-75 km with the peak in winter and at z ~82-105 km with the peak in summer in the northern hemisphere. Between these two regions with prominent AO, the AO is much weaker in both hemispheres at high latitudes. The phase shift crossing this thin layer is mainly induced by the downward shift of the cold summer mesopause (shown as blue dots in Figure 1). During the summer months of each hemisphere and above the cold summer mesopause (~82-85 km), the temperature increases with height and results in a larger value of N 2 (Xu JY et al., 2007a). Moreover, the downward shift of the cold summer mesopause makes the vertical temperature gradient more negative because the height of the stratopause (shown as magenta dots in Figure 1) does not change as much as that of the mesopause. The hemispheric asymmetry of the AO amplitude, which is larger in the northern hemisphere than in the southern hemisphere at high latitudes, results from the hemispheric asymmetry of the mesopause (Xu JY et al., 2007a).
At low latitudes, the AO is prominent around the tropopause region, which is separated by a thin layer of weak AO at z ~18 km. The peak of the AO shifts from August below z ~18 km to March above z ~18 km. This phase shift is induced by the AO of the TIL and the height of the stratopause, as reported by Grise et al. (2010). Those authors analyzed more than 6 years of global positioning system radio occultation temperature profiles and showed that the tropical TIL and tropopause are at a greater height from December to May than from July to September (Fig. 5 of Grise et al., 2010). A similar situation occurs at z ~70-80 km, and a thin layer of weak AO occurs at z ~75 km. The peak of AO shifts from below z ~75 km in September to above z ~75 km in April. This phase shift might be induced by the MIL and the enhanced diurnal tides around the spring equinox above z ~75 km (Xu JY et al., 2007b;Gan Q et al., 2012).
A prominent SAO occurs at z ~75-82 km in the tropical region and extends to the middle latitudes (~35°N/S) with the phase in April/October. The SAO has another peak at z ~70-75 km with the phase in June/December. These two peaks in the tropical regions appear to be separated by a thin layer at z ~75 km. This is a consequence of the SAO in the background temperature and because the bottom of the MIL is at z ~75 km in the tropical region in spring and autumn (Shepherd et al., 2004;Xu JY et al., 2007b;Gan Q et al., 2012). The SAO has peaks at the latitudes poleward of 45°N/S and in the height ranges of 90-100 km in January/June. The latitudinal dependencies and the phase shifts of the AO and SAO of N 2 are summarized below. The AO is prominent at high latitudes. Specifically, at southern high latitudes, the phase of the AO shifts from summer at z ~55-70 km to winter at z ~85-105 km. At northern high latitudes, the phase of the AO shifts from winter at z 60-75 km to summer at z ~82-105 km. The SAO is prominent at z ~70-82 km in the tropical region. The phase shifts from z 70-75 km in June/December to z ~75-82 km in April/October.

Correlations Between N 2 and GWs
For freely propagating GWs in a stable wind environment, a larger (smaller) N 2 sustains GWs reaching a larger (smaller) amplitude before breaking because of dynamic or static instability. Because GWs are the dominant contribution to wind shear (Liu HL, 2007, 2017, the local Richardson number (Ri) is defined by Fritts and Rastogi (1985) as where and are zonal and meridional horizontal velocity perturbations attributable to GWs. The Ri is measured as the ratio between N 2 and GW-induced wind shear. In this work, only the GW-induced temperature is derived from SABER observations. A relation should exist between GW-induced wind shear and temperature based on the polarization for medium-and low-frequency GWs. The relations among and are (Fritts and Rastogi, 1985;Eckermann et al., 1995;Gubenko et al., 2008) f ω where and are inertial and intrinsic frequencies of GW, respect-ively. Substituting Equation (4) into (3), we have where is the vertical wavenumber of the GW. According to the threshold for dynamic instability ( ), we have (6) shows that N 2 is proportional to the product of and . Here, we concentrate on the correlations between N 2 and , and the correlations between N 2 and will be examined in a future work.
The results in the previous section illustrate that the amplitudes and phases of the AO and SAO of N 2 depend on latitude and height. This might influence the corresponding variations of GWs. Previous studies have shown that the amplitudes and phases of AO (SAO) of GWs are also dependent on latitude and height (Krebsbach and Preusse, 2007

508
Earth and Planetary Physics doi: 10.26464/epp2020047 Liu X et al., 2017). To explore the possible relations between N 2 and GWs, Figure 3 shows the time series of the normalized trimonthly running mean N 2 (black) and (red) at two heights in three representative latitudes (Figure 3a-3c) and the latitude-height distributions of the correlation coefficients (CC, Figure 3d) between N 2 and . The representative time series of N 2 at 50°N/S have prominent AO with an antiphase between z = 90 km and z = 65 km. The representative time series of N 2 at the equator have prominent SAO with an antiphase between z = 80 km and z = 65 km (Figure 2). Figures 3a1 and 3a2, we can see that the temporal variation of at 50°N is almost in phase with that of N 2 and that their CC are 0.96 and 0.87 at 90 km and 65 km, respectively. The temporal variation of N 2 and at 90 km is almost antiphase with that at 65 km. At 50°S (Figures 3b1 and 3b2), the CC between N 2 and are 0.89 and 0.81 at 90 km and 65 km, respectively. The CC at 50°S are slightly lower than those at 50°N because of the presence of the SAO in at 90 km and in N 2 at 65 km. At 50°N/S, the phases of N 2 and are nearly identical and shift from winter at 65 km to summer at 90 km. At the equator (Figures  3c1 and 3c2), the CC between N 2 and are 0.33 and 0.65 at 80 km and 70 km, respectively. However, the time series of N 2 and consist of more spectra and exhibit larger variability with time than those at 50°N/S. From Figure 3d, at low latitudes, al- though there are regions with CC ≥ 0.5 at z ~65-70 km and z 85-100 km, the relations between N 2 and are complex, as shown in Figures 3c1 and 3c2 and as discussed in the next paragraph. At high latitudes, the time series of N 2 and are nearly quasi-monochromatic and have CC ≥ 0.8 in some regions. The regions having large CC (≥ 0.8) are generally at latitudes poleward of 40°N/S and are separated by a layer with CC well below 0.5 at z 70-85 km (z ~75-82 km) over the high latitudes of the southern hemisphere (northern hemisphere). An exception is that the region having large CC (≥ 0.8) extends to a wider range and from 20°N to 50°N at z ~85-105 km compared with that in the southern hemisphere.
According to the linear GW theory presented at the beginning of this section, a possible explanation for the correlations between N 2 and is presented below. The propagation of GWs is influenced not only by the N 2 but also by the wind structure (Jones, 1969) and GW source. If the GW amplitudes are large enough to reach dynamic instability, then N 2 and should be positively correlated based on Equation (6). Otherwise, the GW amplitude is too small and is not prohibited solely by the N 2 . Thus, any relations between N 2 and are possible. At lower latitudes, the main GW source is the tropical convection, which is quite spontaneous and intermittent (Fritts and Alexander, 2003). At low latitudes, the wind reversal varies frequently with time and height (Fig. 6 of Swinbank and Ortland, 2003;Fig. 4 of Smith et al., 2017). For example, Swinbank and Ortland (2003, Fig. 6) showed that the zonal wind exhibits SAO and changes from eastward to westward and then eastward at z ~50-80 km (~1.000-0.001 hPa) over the equator. These frequent wind reversals change or prohibit the upward propagation of GWs and contribute to the temporal and height variations of GWs together with N 2 . Moreover, according to ray-tracing theory (Jones, 1969), the great variability in wind speed results in a large magnitude of (where is the background wind) and then variations in k z . In summary, it is the spontaneousness and intermittency of GW sources and the highly variable wind environment in which GWs propagate that induce poor linear relations between N 2 and at low latitudes. At high latitudes, the wind direction is almost invariant with height for a half-year period (Fig. 5 of Swinbank and Ortland, 2003). The wind provides an environment in which GWs propagate; thus, it induces the variations in k z . This will decrease the correlation between N 2 and to some extent, based on Equation (6). Thus, the temporal variations of GWs are influenced mainly by N 2 and less so by the background wind. This leads to good linear relations between N 2 and at high latitudes.
|T ′ | /T 0 Figure 3d shows that the CC are well below 0.5 at z ~70-85 km (z 80 km) over the southern hemisphere (northern hemisphere) high latitudes. Below and above these regions, the CC between N 2 and are large and can be used to explain the phase changes of AO in the GWs at high latitudes. Liu X et al. (2017, Fig. 3) showed the monthly zonal-mean relative GW potential energy, whose peak occurs below z ~70 km in winter and above z ~80 km in summer at ~40-50°N/S. The feature that the GW peak changes with height has been revealed by observations from lidar, radar, and rocketsondes at the middle and high latitudes (Manson and Meek, 1988;Senft and Gardner, 1991;Wilson et al., 1991;Tsuda et al., 1994;Nakamura et al., 1996). We now propose that the GW peak changes with height are mainly caused by the corresponding variations of the background N 2 at high latitudes. According to the necessary condition for dynamical instability (Ri ≤ 0.25), the enhanced N 2 can sustain GWs having large amplitudes in winter below z ~70 km and in summer above z ~80 km. The enhanced N 2 in summer above z ~80 km is due to the downward shift of the summer mesopause. The downward shift of the summer mesopause reduces the vertical extent of the mesosphere and thus increases the vertical gradient of the background temperature in summer. In turn, this reduces the N 2 in the summer mesosphere. On the contrary, the higher mesopause in winter increases the vertical extent of the mesosphere and reduces the vertical gradient of the background temperature. Thus, the peak of N 2 occurs below z ~70 km in winter.
The altitude variations of N 2 depend on latitude and season. At low latitudes, a layer with enhanced N 2 occurs at z ~20 km, which is associated with the TIL and coincides with the results from satellite, radiosonde, and reanalysis data, as well as model simulations. The N 2 in this layer exhibits AO, with the peak shifting from below z ~18 km in August to above z ~18 km in March. Another layer with enhanced N 2 occurs in the upper mesosphere. This layer is associated with the MIL at low latitudes and a downward shift in the cold summer mesopause at high latitudes. The N 2 in this layer exhibits SAO at z ~70-82 km at low latitudes, with phase shifting from z ~70-75 km in June/December to z ~75-82 km in April/October. Moreover, the N 2 in this layer exhibits prominent AO at high latitudes, with a phase shift from below z ~70 km in winter to above z ~85 km in summer in each hemisphere.
The correlations between N 2 and are large at latitudes poleward of ~40°N/S in general. A possible mechanism behind this is that the wind direction is almost invariant with height for a half-year period at high latitudes. Thus, the temporal variations of GWs are influenced mainly by N 2 and less by wind. The CC between N 2 and are large and can be partially used to explain the phase changes of AO in the GWs at high latitudes.