The Yonsei CGWS scheme is described in detail by and here only a short summary is given. This analytical model assumes a diabatic forcing region in a three-layer atmosphere. This three-layer atmosphere has a linear wind shear increasing from U0 at the surface to Ut at an altitude lying between bottom height and top height of the diabatic forcing. Starting from that altitude level, the background wind is constant and equals Ut. Stability of this three-layer atmosphere is characterized by a piecewise function equaling N1 below the cloud top and N2 above the cloud top. Momentum flux due to gravity waves is calculated from the cloud top and can be presented as a function of horizontal phase speed: M‾(c)=-sgn(Ut-c)ρ02(2π)2LxLtgcpT0N122N2|Ut-c||X|2Θ(c).

Here c is the horizontal ground-based phase speed, ρ0 is the air density at cloud top, Lx and Lt are appropriate spatial and temporal scales, respectively, used for averaging, cp is the specific heat of air at constant pressure, T0 is the reference temperature at cloud top, |X|2 represents resonance between vertical harmonics of natural wave modes and diabatic forcing. |X|2 also represents gravity wave filtering by the background wind during vertical propagation up to the cloud top. Therefore, |X|2 is referred to as a wave-filtering and resonance factor. In the updated version of the CGWS scheme, later redefined N2|Ut-c||X|2 as the wave-filtering and resonance factor. Θ(c) is the diabatic source function, which is described as follows:

Θ(c)=2q02δxδxδt16π2π/21+(c-cq)2/c02, where q0 is the maximum magnitude of the diabatic forcing, cq presents the moving speed of the forcing, and c0=δx/δt, where δx and δt are spatial and temporal scales of the forcing, respectively. The function Θ(c) has a maximum at the phase speed equaling to cq and monotonically decreases as |c-cq| increases cf. Fig. 6a in.

The parameters q0 and cq as well as the cloud bottom and cloud top height needed to calculate the wave-filtering and resonance factor are determined from MERRA as follows: the vertical configuration of the heating is a 2nd order polynomial cf. Eq. 8 of . Based on that assumption, heating profiles provided by MERRA data are fitted using a second order fit. Cloud top and cloud bottom data from the MERRA data set are used as a first guess for the second order fitting. Results of this second order fit are then used to recalculate top and bottom of the forcing regions. Also, the maximum value of the fitted heating profile between recalculated forcing top and forcing bottom is defined as q0. The moving speed cq is determined as the horizontal background wind averaged below 700 mb .

The parameters δx and δt are free tunable parameters of this CGW source scheme. In this work we perform a systematic survey by running our simulations with different spatial and temporal scales. All the scales used for this survey are shown in the first two columns in Table . It should be noted that in previous studies the free input parameters δx and δt have been defined using educated guesses. In the current work, these parameters are defined using spectral information from observations. Therefore, we first keep an open mind and estimate δx and δt by adaption to the observed spectrum. A potential physical process related to these δx and δt values will be discussed later in the paper.

For computational efficiency the momentum flux spectrum M‾(c) is not sampled continuously, but up to 10 maxima in the phase speed range from -100 to 100 m s-1 are selected. These discrete values are used as input for the GW ray tracer, launched at the cloud top, and propagated away from the source. Shallow heating depths are not effective in exciting far-propagating GWs. Therefore, simulations are run only for heating depths equal or larger than 3.5 km. When coupling the GW ray tracer to the CGWS scheme, there is a further tuning potential to adapt the global distributions. We can reduce the launch amplitude by a factor of 1/α and simultaneously multiply the number of launched rays by a factor of α. In this way we retain the same total GWMF at launch but reduce the amplitude of the individual waves. This may be interpreted by spreading the same total GWMF over a larger area, i.e., assuming that the wave packet has a larger spatial extent. An important consequence is that by reducing the amplitude of the individual waves, saturation is reached at higher altitudes in the atmosphere.

The vertical profile of latent heat release is taken from three-hourly MERRA (modern-era retrospective analysis for research and applications) assimilated data, which use a parameterization for convection. The chief aim of the parameterization is to capture the total amount of rain and the vertical redistribution of water. That should confine the vertical structure of latent heat release. This is the only explicit input we require from MERRA for our study. Spatial and temporal scales of the convection are formulated in the CGWs scheme as described above. More detailed information about MERRA data as well as convective parameterization in MERRA can be found, for example, in , and .

In the current work, propagation of GWs from convective GW sources into the middle atmosphere is performed using GROGRAT. Details about this ray tracer are presented in and . We here only give a brief description. GROGRAT is based on the full gravity wave dispersion relation, which includes both non-hydrostatic gravity waves and the Coriolis force. Wave packets are propagated according to the local group velocity of the wave depending on the wave vector and intrinsic frequency. The ray-tracing equations are solved using a 4th order Runge Kutta integrator. The integration comprises the calculation of refraction of the wave vector caused by gradients of the atmospheric background in both vertical and horizontal directions. Wave action is calculated along the wave trajectory accounting for dissipation, damping, and saturation processes. Amplitude damping caused by turbulence is calculated following the work of . Radiative damping due to the temperature difference between warm and cold phases of the wave is considered following . The saturated amplitude of the wave is limited using saturation criteria of . Moreover, MERRA winds and temperature are used as atmospheric background for the ray-tracing calculations.

For comparing modeled results with HIRDLS observations, a comprehensive observational filter for satellite infrared limb sounding of gravity waves was applied. Details of this observational filter are described in . The observational filter considers both the visibility of waves to an infrared limb sounder and a sophisticated representation of the observation geometry. The absolute GWMF simulated by applying this observational filter to the model results can be directly compared to the GWMF from observed temperature amplitudes : F=12ρλzλhgN2T^T2, where F denotes absolute GWMF, ρ is the background atmosphere density, λz and λh are vertical and horizontal wavelengths, respectively, g denotes the gravity acceleration, N is the buoyancy frequency, T is the background temperature and T^ is the temperature amplitude of the wave. Although Eq. () is based on a monochromatic wave assumption while CGWs are a multiscale problem, it was shown that in a statistical sense, spectra of CGWs obtained from 2-D Fourier transform can be well reproduced using a single-wave approach . Furthermore, our modeling approach is based on a number of discrete waves, which enhances compatibility between the two approaches. We thus expect that no larger biases will be introduced by the chosen approach.

The observational filter comprises four main processes: (1) visibility filter, (2) projection of the wavelength on the tangent-point track, (3) aliasing effect, and (4) calculation of the vertical observed wavelength. The first process (visibility filter) considers impacts caused by radiative transfer and retrieval . This visibility filter directly influences the temperature amplitude T^. From HIRDLS observations information is provided only along measurement track. The horizontal wavelength along measurement track is generally larger than the real horizontal wavelength. The second process of the observational filter determines this along-track wavelength and the associated reduction of GWMF by modifying λh in Eq. (). The aliasing effect (the third process) estimates the projection of waves towards much longer wavelengths by aliasing and the corresponding reduction of GWMF. Due to the aliasing effect, λh in Eq. () may have a larger value and accordingly, F may decrease. The calculation of the vertical observed wavelength (the fourth process) addresses effects of non-vertical altitude profiles. Due to this effect, not only the vertical structure of an observed wave is sampled, but also to some extent the horizontal structure. The vertical wavelength λz in Eq. () therefore should be recalculated by considering this effect. Finally, additional corrections are applied that are required for the real satellite data to remove dominant vertical oscillation of quasi-stationary planetary waves as well as to keep only those vertical wavelengths for which amplitudes can reliably be determined in the 10 km vertical window of the MEM/HA spectral analysis .

The purpose of this systematic survey is to find sets of free parameters δx and δt which describe spectra in terms of horizontal and vertical wave numbers observed by HIRDLS. For the systematic survey of the spatial and temporal scales we tested the whole set of combinations given by the surveyed scales in the first two columns in Table . These scales are selected on one hand to cover the whole potential ranges, on the other hand with the appropriate step width to minimize the number of different scales and therefore to improve the computational efficiency. An appropriate step width also helps to distinguish the changing of the spectrum in the base 10 logarithmic scale of the wave numbers.

Figure  provides an overview of the results for a reduced set of filtered, simulated GW spectra. In Fig. , for a better visualization, only spectra corresponding to the values given in bold in Table  are shown. Spectra in Fig.  are shown for July 2006 at the altitude of 25 km and averaged over the same regions defined as deep convection (DC) regions in . The location of these regions is indicated in Fig. ; regions for boreal summer (Northern Hemisphere) are marked in red, regions for austral summer (Southern Hemisphere) are marked in green. The spectra in Fig.  are generated by binning absolute GWMF from ray-tracing calculations according to horizontal and vertical wave numbers (kh and m) using a technique similar to that of . The base 10 logarithmic scale is employed here, i.e., the x axis shows kh̃=log⁡10(1/λh) while the y axis shows m̃=log⁡10(1/λz). Here λh and λz denote the horizontal and vertical wavelengths, respectively. The sizes of each bin in x and y direction are δkh̃=0.1 and δm̃=0.1.

In Fig.  the color shading represents the GWMF spectra simulated by the Yonsei CGWS, propagated to observation altitude by GROGRAT and filtered according to the comprehensive observational filter. The dashed contour lines indicate the spectrum observed by HIRDLS for regions of deep convection . For the model spectra, spatial scale increases from the right column to the left column in Fig. . The temporal scale increases from the bottom to the top in Fig. . As shown by Fig. , the horizontal wavelength of the spectral peak increases as the spatial scale δx of the convective system increases. Due to the resonance effect formulated in the CGWS scheme, the phase speeds and hence the vertical wavelength of the spectral peak depends only weakly on the temporal scale δt of the convective system. We hence use the observed spectra to (a) gain information on the horizontal scale, which is not determined by the model conception, and (b) to validate the vertical wavelength spectrum. This is a corroboration of the model assumptions. The comparison of the spectrum hence is a confirmation that the basic assumptions of the model describe reality well, at least in a statistical sense.

It can also be seen in Fig.  that GWs forced by convective systems with very short spatial and temporal scales (e.g., Fig. n, o, s, t) are strongly suppressed in HIRDLS observations due to the observational effect. Some of the spectra (e.g., Fig. g, h, l) show a spectral peak, which locates closely to the observed spectral peak. To complement the survey grid with a parameter set providing a close match with the observed spectral peak, we calculated additional spectra with δx=200 km and δt=150 min for January 2006 and δx=160 km and δt=100 min for July 2006.

As mentioned in the introduction, the large-scale convective system studies indicate that CGW forcing is a multi-scale problem with major contributions by a few dominant scales. For this reason we combine several spectra from the systematic survey in order to obtain a best fit to the observed spectra. Spectra are super-imposed by minimizing the following function: η=Mobs-∑iζiMi/∑iζi,

where Mi is a single spectrum from the systematic survey, ζi is the respective intermittency factor, and Mobs is the observed GWMF spectrum. Combinations of Mi with respective ζi, which give the best fit to the observed spectrum, are chosen from minimization of η and shown in Fig. c for January and Fig. d for July 2006. For computational efficiency, we limit the maximum number of combined spectra to 4 and only largest ζi are selected. The selected spatial scale δx, temporal scale δt as well as corresponding intermittency factor ζ are shown in Table  for January and July 2006. As shown by Fig. c and d, the filtered simulated spectrum matches the observed spectrum very well in both shape and location of the spectral peak for both conditions of January and July 2006. The scales selected by our approach quantitatively agree well with those found in the convective system studies, which were mentioned in the introduction.

Moreover, it is noteworthy that different combinations of scale sets are needed for January and July (cf. Table ). This leads to a question: is the difference between January and July 2006 caused by a persistent difference between the two hemispheres or is it caused by the temporal variability of convective source processes? In order to investigate this question we have considered spectra for 3 consecutive years. This is described in Appendix A. In summary, we find a general tendency for longer horizontal wavelengths in the Southern Hemisphere caused by different source characteristics. We also find a modulation of the GW spectrum at 25 km altitude likely due to the QBO which can be reproduced with the same set of model parameters. This indicates that the QBO modulation is caused primarily by different propagation conditions of the GWs.

In addition, in order to demonstrate effects of the observational filter, we show unfiltered combined spectra of the same selected scale sets for January 2006 in Fig. a and for July 2006 in Fig. b. Comparison of Fig. a, b and Fig. c, d shows that the observational filter not only reduces the magnitude, but also significantly changes the shape of the spectrum. In particular, contributions of short horizontal wavelength waves as well as short vertical wavelength waves strongly decrease and are also partly shifted to longer horizontal wavelengths. More details about effects of the observational filter on GW spectrum can be found in .

In Sect.  the free parameters of the convective source scheme were estimated. In this section, we apply these parameters to global-scale simulations in order to estimate the effect of CGWs on the global distribution of GWMF and GW drag. There are limitations to our approach: the selected scales are determined in regions that are likely dominated by GWs that are excited by deep convection (see Fig.  for regions of deep convection). However, in regions which are not dominated by one source process, we cannot sort the observed waves according to sources, at least not based on current limb sounding observations. Comparison between observed and modeled spectra thus does not provide a meaningful constraint on the CGW source scheme parameters. Therefore the constraint from the regions of deep convection is the only observational guidance we have and which we consider preferential to a guess. Accordingly, in order to study the importance of CGWs on a global scale, the selected sets of δx and δt are used globally for the convective sources for January and July, respectively.

Zonal averages of GWMF and its vertical gradient are calculated and shown in Fig.  for January and in Fig.  for July 2006. In all panels of Figs.  and , contour lines indicate zonal average zonal wind for the respective month. In order to generate zonal averages of absolute GWMF, the values from ray-tracing calculations are first binned onto a three-dimensional grid with bin sizes of 10∘ in longitude, 2∘ in latitude and 1 km in altitude. Results are normalized by total number of rays and the zonal averages are calculated.

Figures a and a show simulated absolute GWMF without observational filter effects for January and July, respectively. Unfiltered GWMF shown in Figs. a and a is calculated as Fph=Fpx2+Fpy2, where Fpx and Fpy are GWMF components in zonal and meridional direction, respectively, as determined from the ray-tracing calculations. Both Figs. a and a show a main maximum of GWMF in the summer subtropics spreading from the equator to about 25∘ S (Fig. a) or to about 25∘ N (Fig. a), which is consistent with the latitude band of deep convection e.g.. The width and magnitude of this maximum decreases with altitude due to wave dissipation, wave breaking and wind filtering. For January, this decrease is significant and strongly related to wind filtering at the altitude of about 20 km, where GWs encounter a wind reversal.

Figures b and b show filtered, simulated absolute GWMF for January and July, respectively. Due to the observational filter the magnitude of GWMF is reduced by about half an order of magnitude.

Observed GWMF from HIRDLS for January and July are shown in Figs. c and c, respectively. It should be noted that observations from HIRDLS shown in Figs.  and contain global data of all regions, not only the regions of deep convection. For comparing filtered, simulated GWMF (Figs. b, b) with observed GWMF (Figs. c, c) it is very important to keep in mind that the model results show GWMF only from convective sources, while observations from HIRDLS contain GWMF from a variety of sources. As convection is believed to dominate the summer subtropics, we expect model and observations to better match at low latitudes. At mid and high latitudes HIRDLS observations indicate an enhancement, likely due to other sources. Considering these facts, the simulated maximum in Fig. b and the secondary maximum in Fig. c in the summer subtropics match well. The same agreement can be seen by comparing Fig. b and c. In particular, centers of these maxima are both located at about 15∘ S for austral summer (Fig. b, c) or at about 15∘ N for boreal summer (Fig. b, c). Also, the structure of these maxima in the tropics from about 20 to about 40 km altitude in both simulations and observations follow the contour line of the wind reversal. However, the magnitude of the simulated maximum is somewhat lower than the observed one. Moreover, the width of the simulated maximum is slightly narrower than the observed one. The weaker and narrower peak in the simulations could indicate that deep convection is spread over a wider latitude range than simulated in MERRA or that we need to revisit the forcing efficiency. However, there are also indications from radiosonde and ground-based measurements e.g. and modeling e.g. that in the regions of deep convection GWs are excited around tropopause altitude. Details of that GW excitation are not fully understood, but they are presumably associated with the shear and strong buoyancy change present at these altitudes. These waves seem to be less focused to the convective regions and produce a substantial background outside the selected convection regions used for the spectra (cf. Figs. c and c) and thus potentially become important in the zonal means. At high winter latitudes, CGWs contribute only a minor part of the total GWMF. This can be seen from the unfiltered GWMF values as well.

In this section we show horizontal distributions of simulated convective GWMF as well as spectra of GWMF in terms of zonal phase speed and latitude. In Figs.  and data for January, and in Figs.  and data for July are shown. Figures  and give values for 25 km altitude, while Figs.  and give values for 40 km altitude. In each of these figures, the left column contains global maps of GWMF: panel a the unfiltered, simulated GWMF, panel c the filtered, simulated GWMF and panel e the observed GWMF. The right column shows zonal GWMF as a function of zonal phase speed and latitude (hereafter referred to as phase speed spectra) in bins of 4 m s-1 and 4∘. For both maps and spectra the color scales indicate the base 10 logarithm of GWMF. As for phase speed spectra GWMF with negative (westward) phase velocities is negative (westward GWMF), these values were multiplied by -1 before applying the logarithm. HIRDLS does not provide a propagation direction and accordingly phase speed cannot be deduced from the measurements.

In the global maps (panels a, c, e), the rectangle indicated by the magenta dashed line shows the low-latitude area, where convection is assumed to dominate. Blank areas in panels a, c, e refer to values of GWMF, which are out of the shown value range (<-5.0(log⁡10 Pa) or >-1.0(log⁡10 Pa)). In panels b, d, the gray thick line indicates zonal mean of zonal wind at the considered altitude level, magenta plus line shows maximal zonal mean of zonal wind in the altitude range from cloud top to the considered altitude level, and magenta dot line shows minimal zonal mean of zonal wind in the same altitude range.

For January 2006, at 25 km altitude, high values of unfiltered GMWF are found over central south America, south Africa, a strip spreading from Madagascar to Indonesia, Indonesia and north Australia, and a strip over the Pacific ocean at around 20∘ S latitude spreading from 180 to 120∘ W (Fig. a). After applying the observational filter, GWMF decreases by about half an order of magnitude (Fig. c). In addition, the observational filter also changes the distribution of GWMF at some locations. For example, over the Indonesian region, a band of high GWMF at about 10∘ N spreading from the Philippines to 180∘ E, is decreased stronger in comparison with the band of high GWMF at about 0–10∘ S. Figure e shows observed GWMF from HIRDLS. Comparison of filtered GWMF (Fig. c) and observed GWMF (Fig. e) shows quite a good agreement in location of GWMF peaks in the summer subtropics. In particular, the maxima over central south America, south Africa, Indonesia and north Australia are reproduced. The magnitude of filtered GWMF is lower in comparison with observed GWMF which, as discussed above, can be explained by a lack of other sources than convection.

The phase speed spectrum for January 2006 at 25 km altitude (Fig. b) shows a major peak in the tropics with eastward phase speed from several ms-1 to about 25 ms-1, with the center of the peak at about 5 ms-1. At higher phase speeds (beyond 40–50 ms-1), two minor peaks in both eastward and westward directions are found. The GWMF associated with these minor peaks is about 2 orders of magnitude lower in comparison with the main peak. In mid- and high-latitude regions, high GWMF values are mainly associated with westward phase speeds varying from several ms-1 to about 40 ms-1. Figure d shows the phase speed spectrum of filtered GWMF. The observational filter in this case reduces GWMF magnitude but almost does not impact the spectrum structure.

The effect of wind filtering can also be found in the phase speed spectrum. This effect occurs when a wave encounters the critical level, i.e., where the background wind equals the phase speed of the wave. In this case, the intrinsic phase speed and thus the vertical wavelength approach zero causing saturation and the release of GWMF. In Fig. b and d, GWMF are mainly found in the areas where GW phase speed is larger than maximal zonal mean of zonal wind (magenta plus line) or lower than minimal zonal mean of zonal wind (magenta dotted line). A small amount of GWMF still can be found in between these two lines because of the following: (a) these lines only indicate the zonal mean of the zonal wind and thus waves may be able to propagate due to local variations and (b) many waves also have a meridional component. Still the findings indicate that filtering is dominated by the variation of the zonal wind.

For January 2006 at 40 km altitude, the horizontal band of high GWMF values in the summer subtropics becomes narrower (more concentrated around latitude of ∼ 15∘ S) and is slightly shifted poleward. The magnitude of GWMF decreases strongly with altitude, as can be seen by comparing Figs. a and a. This is also in agreement with the vertical cross section of absolute GWMF shown in Fig. a. A wind reversal at about 30 km altitude is likely the main reason for the reduced GWMF values close to the equator, or in other words, for narrowing the horizontal band of high GWMF values. The location of simulated and observed GWMF hot spots agrees well (cf. Fig. a, b, c). At 40 km altitude the measurements indicate that GWMF in the hot-spot regions, where the model results suggest deep convection as the dominant source, is enhanced by about 1 order of magnitude compared to the background regions. This is a stronger enhancement than at 25 km altitude (about half an order of magnitude) and indicates that the non-convective background is relatively less important at 40 km.

Also the phase speed spectrum changes with altitude: the main peak of the spectrum at 40 km does not stretch over the equator as for 25 km but is limited only to the Southern Hemisphere. This change is according to the change of the maximal zonal wind, which is indicated by the magenta plus line. This line surrounds the main peak in the tropics. Another major difference of the phase speed spectrum at 40 km in comparison with the one at 25 km is the absence of the peak at mid- and high latitudes in the Southern Hemisphere at westward phase speeds (20 to 40∘ S). This absence is again likely related to wind filtering as indicated by the magenta dotted lines for the difference in minimum zonal winds in Figs. b, d and b, d, respectively.

For July 2006 at 25 km altitude, unfiltered GWMF (Fig. a) shows high values over the Caribbean sea, central Africa and the Asian Monsoon region. Figure c shows filtered GWMF with the magnitude reduced significantly due to the observational filter. Comparison of filtered GWMF (Fig. c) and observed GWMF (Fig. e) shows quite a good agreement in locations of GMWF maxima. As seen before in the zonal means, however, the observed GWMF maxima are wider, i.e., extend further to the north. In addition, the model results also show GW excitation following the warm water currents of the Gulf Stream and the Kuroshio.

Similar to January 2006 at 25 km altitude, the phase speed spectra (Fig. b and d) show a main peak in the summer subtropics with eastward phase speeds from several ms-1 to ∼ 30 ms-1. Moreover, two secondary peaks in the summer subtropics at high phase speeds (larger than 40–50 ms-1), which are ∼ 2.5 orders of magnitude smaller in comparison with the main peak, are also found. The observational filter decreases the GWMF magnitude but almost does not alter the structure of the phase speed spectrum. This can be seen by comparing the spectrum structure shown Fig. b and d.

Similar to the January case, the magnitude of GWMF decreases with altitude (Fig. ). This decrease, however, is weaker than in January. In particular, there is no strong reduction of GWMF in the tropics, which narrows the region of strong GWMF between 25 and 40 km, such as for January.

Concerning the phase speed spectrum at 40 km altitude (Fig. b, d), a major difference in comparison with the spectrum at 25 km is much lower GWMF at mid and high latitudes in the Northern Hemisphere (from 40–80∘ N). This can be seen by comparing Figs. b, d with b, d. This reduction is likely related to the change of minimal zonal wind at this latitude range, which can be seen by comparing minimum zonal mean of zonal wind in Figs. b, d and b, d, respectively.