We adopt a broad spectral data analyzing method to derive the continuous altitude variability of inertial gravity wave (GW) parameter properties in the altitude range of 2–25 km from 11 year (1998–2008) radiosonde observations over 92 United States stations locating in the latitude range from 5°N to 75°N. To our knowledge, this is the first time presenting latitudinal and continuous altitudinal variability of lower atmospheric GW parameters. The presented latitudinal distribution of GW parameters indicates that the wave energy in the troposphere and lower stratosphere peaks, respectively, at the middle and lower latitudes; and at lower latitudes, GWs usually have larger ratios of wave intrinsic frequency to Coriolis parameter, smaller intrinsic frequencies, shorter vertical wavelengths, and longer horizontal wavelengths. Our analyses also revealed continuous altitudinal variations of GW parameters, most of which are closely related to those of the background temperature and wind fields, indicating the important role of background atmosphere in excitation and propagation of GWs. Moreover, our results suggested the profound climatological impacts of GWs on background atmosphere. The GW-induced force tends to decelerate the zonal jet at middle latitudes and produces a negative vertical shear in the northward wind closely above the tropopause altitude. The GW heat flux tends to cool the atmosphere around the tropospheric jet altitude and contributes significantly to the forming of tropospheric inversions at middle latitudes. Additionally, we demonstrated that GW energy densities, momentum, and heat fluxes have evident seasonal variations, especially at middle latitudes.
 Many efforts have been carried out on the climatology properties of gravity waves (GWs) in the lower atmosphere due to their significant impacts on atmospheric dynamical and thermal structures from the troposphere up to the lower thermosphere [Li et al., 2007; Dou et al., 2010; Li et al., 2010; Xue et al., 2012]. Among previous studies, multistations radiosonde [Wang and Geller, 2003; Wang et al., 2005; Zhang and Yi, 2007; Zhang et al., 2006, 2010] and satellite observations [Schmidt et al., 2008; Wang and Alexander, 2009, 2010; McDonald, et al., 2010; Hoffmann, et al., 2013] contributed greatly to our knowledge of climatology for lower atmospheric GWs in a wide latitude area. Since satellite observations in the lower atmosphere usually provide only the temperature, most studies of satellite observations are only concentrated on those GW parameters which can be derived from the measured temperature profiles (for instance, GW temperature perturbation and potential energy density) in the upper troposphere and stratosphere. Compared with satellite observations, the radiosonde sounding can measure relatively complete physical quantities (including horizontal wind, pressure, temperature, and humidity), which allows us to study complete GW parameters from radiosonde data. Recently, Zhang et al.  have pointed that the latitudinal distribution of total GW energy density revealed by multistations radiosonde observations has some obvious differences from that of the potential energy density revealed by satellite observations. Moreover, different from satellite observations, radiosonde sounding can measure atmospheric parameters not only in the lower stratosphere but also in the troposphere, which is believed to be the main source region of atmospheric GWs. Then, radiosonde observations are favorable for investigating altitude variation of complete GW parameters from the troposphere up to the lower stratosphere.
 GWs in the realistic lower atmosphere should consist of abundant spectral components due to their variety of excitation sources and complex propagation environment [Fritts and Alexander, 2003, and references therein], but in most radiosonde observation studies, a hodograph analysis method based on monochromatic GW theory was adopted to deduce monochromatic GW parameters [Tsuda et al., 1994; Vincent and Alexander, 2000; Yoshiki and Sato, 2000; Wang et al., 2005; Zhang and Yi, 2005, 2007; Ratnam et al., 2008; Zhang et al., 2008, 2010]. Moreover, in these traditional hodograph analyses, before the extraction of GW parameter, the background is typically removed by a polynomial fitting. For avoiding the influences of the extremely low temperature and extremely large zonal wind around the tropopause on this polynomial fitting and considering the evidently different stratification properties between the troposphere and stratosphere, above mentioned hodograph analysis based on quasi-monochromatic GW theory gave only averaged values of GW parameters in two separate segments rather than their altitude evolutions: a tropospheric segment from near ground to 10 km, and a low stratospheric segment from 18 to 25 km, while GW parameters at the intermediate altitude region between these two segments, roughly from 10 to 18 km, cannot be attained from the traditional hodograph analysis. In the real atmosphere, the temperature and wind of the background atmosphere in which GWs propagate vary continuously with altitude, then impact on GWs and thus lead to the altitudinal variation of GW properties. In order to comprehensively understand the altitudinal evolution of GWs as well as GW-background interaction, we need to seek a feasible approach to clearly display these altitudinal evolutions. On the other hand, most recent studies [Zhang et al., 2010, 2012] revealed that the intermediate region might be an important GW source region. Obviously, continuous altitude variation of GW parameters from near ground up to the lower stratosphere, including that in the intermediate region, is important for our understanding of the vertical propagation and evolution of GWs, interactions between GWs and background atmosphere, as well as locating the dominant source altitudes.
 The other limitation of most previous radiosonde observation studies is lack of vertical wind measurements. This limitation brings difficulties in directly calculating vertical wind disturbance-related important GW parameters, such as wave momentum flux and wave drag, which were usually indirectly derived in previous hodograph analyses [Vincent and Alexander, 2000]. Lane et al.  and Reeder et al.  assumed that the perturbations of the ascent rate of balloons might be derived by GW fluctuations. Furthermore, Gong and Geller  confirmed the reasonability of this assumption by investigating the tempo-spatial spectral characters of the vertical velocity perturbation calculated from the ascent rate. These efforts imply that the perturbations of the ascent rate of balloons could be treated as the vertical velocity component of GW perturbation. More importantly, although the altitudinal variations of GW momentum and heat fluxes have significant physical meaning because they can reflect wave-background interaction as well as GW momentum and energy deposition, most previous traditional hodograph analyses can only present these fluxes averaged over certain altitudinal regions (for instance, the tropospheric and lower stratospheric segments) instead of their continuous altitudinal variation.
 Most recently, Zhang et al.  proposed a broad spectral data analyzing method and extracted the GW associated vertical velocity perturbation from the vertical ascent rate of balloon as suggested by Lane et al.  and Reeder et al. . Furthermore, Zhang et al.  applied above mentioned broad spectral and vertical wind perturbation extraction methods to an 11 year (1998–2008) radiosonde data set at a midlatitude station, Miramar Nas (32.87°N, 117.15°W) CA to reveal continuous altitude variation of complete GW parameters, including wave energy, intrinsic frequency, wavelengths, polar factor, and directly calculated horizontal momentum flux as well as the heat flux. The quantitative comparisons [Zhang et al., 2012] with conventional hodograph method [Vincent and Alexander, 2000; Wang et al. 2005; Zhang and Yi, 2005, 2007; Geller and Gong, 2010; Zhang et al., 2010] exhibited considerable consistence, confirming the validation of this broad spectral method.
 Radiosonde, satellite, and ground-based instrument observations have revealed that lower atmospheric GW parameters have evident latitudinal distribution. And, it is well known that convection, orography, and subtropical jet are important GW sources in the troposphere, while at different geophysical regions, the contributions of these potential GW sources may be different. Therefore, it is necessary to study the latitudinal dependence of GW parameters and investigate their links with different sources.
 In this study, we extend the study by Zhang et al.  to a wider geography area. We follow the broad spectral data analyzing method to derive inertial GW parameter properties from 11 years (1998–2008) radiosonde observations over 92 United States stations in the Northern Hemisphere. The aim of this paper is to study latitudinal and continuous altitudinal variability of GW parameters from the troposphere up to the lower stratosphere. The presented paper is laid out as follows: an introduction of the utilized data set and the data processing method is described in the following section. The primary scenarios of the background winds and temperature are given in section 3. The results of GW parameters are presented in section 4. In the last section, we give a brief summary on our observations.
2 Data Description and Analysis Approach
 In the present study, we use the high-resolution United States radiosonde data in 1998–2008 from National Oceanic and Atmospheric Administration (NOAA) National Climatic Data Center. These data can be freely accessed through the Stratospheric Processes and Their Role in Climate Data Center (http://www.sparc.sunysb.edu/). There are 92 stations located in the Northern Hemisphere while only one station located in the Southern Hemisphere. Therefore, data from 92 radiosonde stations located in the Northern Hemisphere were used in this paper. The longitudinal and latitudinal coverage of these 92 stations are [170.22°W, 171.38°E] and [6.97°N, 71.30°N], respectively. The detailed geographical locations and available data accumulations of these 92 stations can be found from Figure 1 in Wang and Geller  and Table 1 in Zhang et al. .
 The routine radiosondes are usually launched twice daily at 0000 and 1200 UT. In each sounding, accompanying with the ascent of the balloon, pressure, temperature, ascent rate, relative humidity, and horizontal winds are measured at irregular height resolutions, ranging from ten to hundreds meters. For convenience, in this paper, the raw data were processed to have an even height resolution (50 m) by a cubic spline interpolation. The sounding altitude range is from the surface up to the burst altitude of balloons, typically in the lower stratosphere (25–30 km). Since too small elevation angle at the heights close to the surface may lead to some uncertainties in the measurement of horizontal winds, in our analyses, we choose 2 km as the lowest height. And, we chose the altitude of 25 km as the upper height limit of our analysis to ensure that most balloons can reach this upper limit.
 We adopt the broad spectral analysis method as proposed by Zhang et al.  to extract GW relative parameters. This method was introduced in detail in Zhang et al. . Here we summarize briefly as follows. We suppose that the observed parameters X(X = [u, v, AR, T, ρ, P], where u, v, AR, T, ρ, and P are, respectively, zonal wind, meridional wind, vertical ascent rate, temperature, mass density, and pressure) mainly consist of background () and GW perturbation components (X ′), i.e., , where the over bar and prime denote, respectively, the background and GW perturbation components. Here by adopting the assumption proposed by Reeder et al.  and Lane et al. , we treat the perturbation component of ascent rate AR′ as the vertical velocity fluctuation w′. Additionally, with respect to the exponential decrease with altitude of mass density and pressure, in our analyses, they and their GW perturbations are normalized by the background density (calculated from the background pressure and temperature), i.e., in the following processing, and In the broad spectral analysis method, one of the most important issues is a correct separation of background and GW disturbance components from each measurement [Zhang et al., 2012]. Thus, first, a monthly averaged value is removed as the majority of background structure from the raw data. Then, considering that the inertial GWs usually have smaller vertical wavelengths (no larger than 10 km), a high-pass filter with a cutoff wavelength of 10 km is applied to the residual components of the raw data. The background profile is then defined as the sum of the monthly averaged value plus the large vertical scale components, e.g., the components filtered out by the high-pass filter. The filtered profiles can almost be regarded as inertial GW perturbations. Further, in order to remove fluctuations due to smaller-scale effects such as measurement errors, variation of drag coefficients of the balloon, etc., and to suppress the small-scale noise, which may be introduced by the interpolation process, the filtered profile was smoothed by a Hanning window. We selected the smoothing kernel of the Hanning window as
The smoothing is applied in the altitude domain as a weighted average centered at each sampled altitude. For example, the smoothed value at location z is
Since the observed vertical wavelength of GWs in the lower atmosphere is usually larger than 1 km, we settled on a altitude half width α of 0.7 km with a RMS width about 250 m [Zhang et al., 2012]. Finally, the smoothed profile is then taken as the superposition of broad spectral GW components.
 It is well accepted that the background atmospheric structure has important impacts on the excitation and propagation of GWs. Then, we calculated the latitudinal variation of mean background temperature, zonal, and meridional winds in the altitude range from 2 to 25 km and plotted them in Figure 1. The mean background components are calculated by an average from all background profiles defined as that in section 2 over the whole period from 1998 to 2008. In our calculations, the extracted background and GW perturbation were divided into 5° bins in latitude. And, we would like to remind that due to the limited geographic distribution of the adopted data set, the background quantities shown in Figure 1 are the mean value over the North America region. The tropospheric temperature decreases with altitude, and the cold point tropopause moves downward with the increasing latitude. At the lowest latitude 5°N, the tropopause is at 17 km, while it moves down to 10 km at 75°N. Above the cold point tropopause, the temperature increases with altitude. The mean zonal wind poleward from 15°N is eastward at almost all altitudes. A prominent tropospheric jet with an 11 year averaged value larger than 24 m s−1 can be observed around 12 km at middle latitudes (30°N–40°N). With latitude increasing, the value and altitude of the maximum eastward zonal wind in the troposphere decrease. At the polar region (poleward from 60°N), a polar jet above 20 km with mean eastward wind larger than 15 m s−1 is obviously displayed in Figure 1b. Over the tropical stations, the mean zonal wind is westward at the lower troposphere and stratosphere, with the largest westward wind about 14 m s−1 at 25 km. Compared with the zonal wind, the mean meridional wind is rather weak, usually no larger than 10 m s−1. At 5°N–20°N and 35°N–45°N, the meridional wind is southward at most altitudes, while at other latitudes, it is northward, particularly at high latitudes.
4.1 GW Energy
 Subsequently, we investigate the latitudinal and altitudinal variations of GW energy. The zonal kinetic energy density (EKu), meridional kinetic energy density (EKv), vertical fluctuation energy (EKw), and potential energy density (EP)can be computed from the extracted GW perturbation components:
Where, g and are, respectively, the gravitational acceleration and background temperature.
 The 11 year (1998–2008) averaged latitudinal and altitudinal variations of the potential energy density, zonal and meridional kinetic energy densities, and vertical fluctuation energy density are illustrated in Figure 2. To our knowledge, this is the first time presenting a continuous altitude variation of GW energy from near ground up to 25 km from radiosonde soundings in a broad latitudinal region. Among these energy densities, the potential energy density has the largest value, with the maximum energy density as large as 13.5 J/m3; the zonal and meridional kinetic energy densities are comparable with each other, with the maximum value about 6.7 and 7.1 J/m3, respectively, which are smaller than the peak in the potential energy density, while the sum of the zonal and meridional kinetic energy densities, i.e., the horizontal kinetic energy density EKh is generally larger than the potential energy density. The vertical fluctuation has the smallest value, with the maximum density no larger than 8.7 × 10- 2J/m3. Additionally, in most cases, the energy densities in the troposphere are larger than those in the lower stratosphere, which is attributed to the background wind absorption of downwind propagating GWs at the upper troposphere.
 These energy densities have basically similar latitudinal and altitudinal variations. Generally, the peak altitudes for all energy densities move downward with the increase of latitude, similar to those of the tropopause and maximum eastward wind heights, implying the potential effects of the background thermal structure and wind on GW activity. As reported by Zhang et al.  by applying the traditional hodograph analysis method on the same data set, all energy densities have prominent peaks at middle latitudes in the troposphere, indicating the validity of the presented broad spectrum analysis method in extracting GW parameters.
 The most prominent feature in the lower stratosphere is those energy densities generally decrease poleward. This poleward decrease trend in potential energy density has been discovered by many satellite [Tsuda, et al., 2000; Venkat Ratnam et al., 2004; Wang and Alexander, 2010] and multistation radiosonde observations [Wang and Geller, 2003; Wang et al., 2005; Zhang et al., 2010], and was suggested to be attributed to the latitudinal dependence of wave source [Zhang and Yi, 2005, 2007; Zhang et al., 2010], background wind filtering [Zhang and Yi, 2005, 2007; Zhang et al., 2010], and Coriolis parameter f [Alexander et al., 2002], i.e., the lowest limit of GW intrinsic frequency. In more details, at lower latitudes, strong convection activity will excite stronger GW activity; smaller Coriolis parameter implies that more low frequency GW components can be captured by radiosonde sounding; and weak background wind induces weak GW filtering effects. Consequently, the GWs in the lower stratosphere at low latitudes have the largest energy densities. At middle latitudes, although strong tropospheric jet here may excite stronger GWs, the jet can also significantly filter out more GWs. While at high latitudes, due to the lack of strong excitation sources and larger Coriolis parameter, the wave energy densities here are the smallest. In the investigation of latitudinal variation of GW features, we should be careful that the derived latitudinal variation of GW energy densities may be affected by the limited and uneven geographic distribution of the adopted data set and possible stationary wave patterns. The consistence between the presented latitudinal variation in GW potential energy density and that revealed by satellite observations suggests that the primary latitudinal variation of GW features over North America can be well exhibited by the adopted data set.
 Additionally, above 15 km, besides the primary energy density peaks at the lowest latitude (5°N), a second but prominent peak at middle latitudes (30°N–40°N) can also be obviously observed in the horizontal kinetic and vertical fluctuation energy densities, especially in the zonal kinetic energy density, which has been reported by Zhang et al.  by using the traditional hodograph analysis method. This secondary peak was suggested to be due to strong zonal tropospheric jet excitation of GWs at extratropic latitudes [Plougonven et al., 2003; Zhang and Yi, 2005, 2007, Zhang et al., 2008, 2010, 2012]. Due to the lack of lower atmosphere wind field measurements, this secondary peak in lower stratosphere at middle latitudes has seldom been shown by satellite observations except Venkat Ratnam et al. , in which a second maximum in EP at midlatitudes during winter was confirmed.
 The differences among the latitudinal and altitudinal variations of these energy densities are also evident as well as their similarities. In the troposphere, the exact peak values for different energy density components occur at different latitudes and altitudes. For instance, the peak of the zonal kinetic energy density in the troposphere occurs at 15 km over 35°N, exactly corresponding to where the maximum tropospheric eastward wind occurs as shown in Figure 1. While the peak values for the potential and meridional kinetic energy densities occur at lower altitudes (around 10 km or lower altitude) and higher latitudes (45°N–50°N). By applying the traditional hodograph analysis method, Zhang and Yi  and Zhang et al.  have also reported the differences between the zonal and meridional kinetic energy densities and pointed out that around the strong zonal jet, the GWs usually have larger zonal kinetic energy. Zhang and Yi  and Zhang et al.  suggested this discrepancy could be explained by that since the troposphere is the main source region of atmospheric GWs, GWs in the troposphere are probably within or in the vicinity of the source region (or freshly generated) and far from the status of free propagation. Then, the zonal jet excited GW disturbances might occur first in the zonal wind component. Furthermore, a simulation study by Huang et al.  indicated that, when a GW was newly excited by a zonal momentum source, large zonal wind disturbances could happen before the occurrences of the other GW disturbance components. The vertical fluctuation energy density in the troposphere peaks at 6 km over 35°N, and a weaker peak can be observed at the lower troposphere at 50°N–60°N. These differences may due to the fact that these energy densities are sensitive to different GW components. Geller and Gong  have suggested that the horizontal kinetic energy density and vertical fluctuation energy density are more sensitive to low- and high-frequency GWs, respectively. In the lower stratosphere, compared with the zonal kinetic energy density, the secondary peaks in the other energy densities over middle latitudes are rather weak. And, the altitude decrease in the vertical fluctuation energy density is not as evident as in the other energy densities. Above 20 km at the middle latitudes, it even increases with altitude. It can be explained by that the vertical fluctuation is more possible to penetrate the midlatitudinal tropospheric jet due to its higher sensitivity to high-frequency GWs [Gong and Geller, 2010; Geller and Gong, 2010].
 Besides latitudinal and altitudinal variations, GW energy densities also have clear seasonal cycle [Zhang et al., 2010]. Here Figure 3 illustrates the monthly averaged total GW energy density ET(ET = EP + EKu + EKv + EKw) at low (5°N–15°N), middle (30°N–40°N), and high (65°N–75°N) latitudes. As shown in Figure 3a, at low latitudes, evident semiannual oscillation of GW energy density can be observed above 10 km, with maximum values at around 16 km in winter and summer; while below 10 km, the seasonal variation of GW energy density is rather weak. At middle latitudes (Figure 3b), the GW energy density in all sounding altitudes (from 2 up to 25 km) has obvious annual variation, with maximum values at about 12 km in winter, which is closely linked with strong tropospheric jet in winter at middle latitudes. At high latitudes, above 12 km, the seasonal variation of GW activity is dominated by annual oscillation, with stronger GW activity in winter at 25 km. Below 12 km, the GW energy density is dominated by semiannual variation, with larger values at 8 km in winter and summer. Venkat Ratnam et al.  have suggested that the strong GW activity at high latitudes in winter months might be in connection with a strong PW activity leading to sudden stratospheric warming; however, due to the uneven geophysical distribution of the radiosonde stations, it is difficult to resolve planetary waves from the radiosonde data alone.
 To clarify the contribution of convection to the seasonal and latitudinal variations of GW activity, we take the outgoing radiation (OLR) as a proxy of the deep convection and calculate the correlation coefficients between the monthly averaged OLR and the GW horizontal kinetic energy density EKh(EKh = EKu + EKv), potential energy density EKP, and vertical fluctuation energy density EKw at their corresponding latitudes in Figure 4. The outgoing longwave radiation data at the top of the atmosphere are observed from the Advanced Very High Resolution Radiometer instrument aboard the NOAA polar orbiting spacecraft. The applied OLR data were interpolated into 2.5° in latitude and 2.5° in longitude bins. And, considering the uneven geographical distribution of the radiosonde stations, only the OLR data in those bins in which the radiosonde stations locating are adopted in the correlation coefficients calculation. The lower OLR indicates strong convection; thus, the negative correlation between GW energy density and OLR implies that the observed GWs may be excited by deep convection. For the horizontal kinetic and potential energy densities, their correlation coefficients with OLR (CEkh−OLR and CEP−OLR, respectively) have rather similar latitudinal and altitudinal variations. At the altitude range of 10–20 km over 10°N, both correlation coefficients have significant negative values, with maximum magnitudes about −0.5, indicating in low latitudes convection activity might have contribution to the excitation of GWs. Except the significant negative correlation, there are also obvious positive correlations between the OLR and GW horizontal kinetic and potential energies. The positive correlation cannot suggest the link between convection and GWs but simply indicate they have similar seasonal variations at the corresponding altitudes. On the other hand, in the lower atmosphere where is thought to be the main source region of GWs, different GW disturbance components may be sensitive to different sources, then the horizontal wind and temperature disturbances might not be so sensitive to convection as the vertical fluctuation does. In fact, Figure 4c demonstrates that the correlation coefficient between the OLR and vertical fluctuation energy density (CEkw−OLR) is negative at almost all altitudes over low latitudes equatorward from 15°N, with largest negative values (in magnitude) about −0.8 at 12 and 22 km, suggesting the vertical fluctuation energy density at low latitudes is closely linked with the convection. At most altitudes in higher latitudes poleward from 30°N, the correlation coefficients are negative except around the tropopause at latitudes poleward from 50°N, the most prominent negative correlation coefficients with magnitude larger than 0.9 occur around 7 km at the latitudes 35°N–40°N, indicating the important role of convection in excitation of GWs. At latitudes northward from 30°N, the correlation coefficient CEkw−OLR is all negative above 10 km. The obvious different latitudinal and altitudinal variations of CEkw−OLR from those of CEkh−OLR and CEP−OLR can be accounted from that compared with the horizontal kinetic energy density; the vertical fluctuation energy density is more sensitive to convection activity [Geller and Gong, 2010].
 Subsequently, we also investigate the contributions of topography and tropospheric jet to the excitation of GWs. In each latitudinal bin, we take the monthly averaged ground surface flow and vertical shear of the background horizontal wind around the tropospheric jet as the proxies of the topography and tropospheric jet sources. The ground surface flow is specified by , where, and are the background zonal and meridional winds at the ground surface, respectively. The tropospheric jet-induced vertical shear of horizontal wind S here is represented by the mean vertical shear of background wind around the tropospheric jet, which is calculated from the averaged vertical shear of background horizontal wind over the altitude range from 2.5 km below the jet altitude (defined as the altitude of the maximum eastward wind in the troposphere) up to 2.5 km above the jet. At the ith sampling altitude, the vertical shear of the background horizontal wind Si is calculated from , where Δz = 50 mis the altitude resolution of our data set. Except moderate negative values above 10 km at latitudes 5°N–10°N, both the correlation coefficients (shown in Figure 5) between the monthly total GW energy density with ground surface flow (CET−GF) and tropospheric jet-induced vertical shear of the horizontal wind (CET−S) are positive in almost all presented latitudinal and altitudinal regions. For CET−GF shown in Figure 5a, it exhibits prominent positive value as large as 0.9 below 10 km in latitudinal range of 30°N–60°N, indicating the airflow over orography is one of the most important local sources for GWs in the mid-high latitudes. The tropospheric jet excitation of GWs is mainly at the middle latitudes of 25°N–40°N, where the correlation coefficient CET−S shown in Figure 5b is larger than 0.8 at almost all altitudinal range, indicating at the middle latitudes, the tropospheric jet-induced vertical shear is one of the most important GW sources [Zhang and Yi, 2005, 2007; Zhang et al., 2010, 2012].
4.2 Intrinsic Frequency
 According to the GW theory, the GW intrinsic frequency can be expressed in terms of different wave energy density components [Geller and Gong, 2010]. As pointed by Geller and Gong  and Zhang et al. , the horizontal kinetic energy is more sensitive to the low-frequency GWs, i.e., inertial GWs. Therefore, in this study, we deduce the ratio of GW intrinsic frequency Ω to Coriolis parameter f in terms of horizontal kinetic energy density EKh and potential energy density EP [Geller and Gong, 2010]:
The altitudinal and altitudinal variations of the ratio of and Ω are demonstrated in Figure 6. The calculated intrinsic frequencies do exhibit inertial GW characters, varying in a small value range from 1.6 to 3.1 times of the inertial frequency. The ratio has evident altitude variability at latitudes northward from 15°N. From 2 km up to about 5 km, it increases with altitude with a peak value of about 2.7 at 5 km. Above this altitude, the ratio decreases rapidly with altitude, with a minimum of about 1.6 at the highest altitude and latitude. The ratio has only slight latitudinal variation in the troposphere. While in the lower stratosphere, the ratio peaks at lower latitudes and generally decreases with latitude, which has also been revealed by GPS radio occultation observations [Wang and Alexander, 2010]. Besides this overall poleward decrease variation, there are several weak peaks of the ratio that occur at 15°N, 40°N, and 60°N, respectively. Additionally, the ratio has only weak seasonal variation (not presented), with slightly larger values in winter, which is attributed to slightly larger buoyancy frequency in winter [Zhang and Yi, 2005; Zhang et al., 2012].
 The variation of the intrinsic frequency is not as complex as that of the ratio. Due to the increase of the Coriolis parameter f with latitude, the intrinsic frequency monochromatically increases with latitude. As to the altitude variation, an obvious peak occurs at 4 km, and above this altitude, the frequency decreases with altitude.
4.3 Wavelengths and Propagation Directions
 In this subsection, in order to investigate the propagation features of inertial GWs, we will calculate their wave numbers, which satisfy following dispersion equation [Fritts and Alexander, 2003]:
where kh, k, l, and m are, respectively, the horizontal, zonal, meridional, and vertical wave numbers. Adopting the method proposed by Zhang et al. , we can specify three-dimensional wave vector. First, we calculate the vertical wave number m from
where the positive and negative signs of m denote the downward and upward energy propagation, respectively. Further, Zhang et al.  deduced the relation between the zonal and meridional wave numbers from the polarization relation between the zonal and meridional wind disturbances:
The over bar in above equation denotes an average over a wavelength scale, while the average was realized by a low-pass filtering with a cutoff vertical wavelength of 10 km in our calculation. Then, the magnitudes of zonal and meridional wave numbers can be resolved by combining equations ((2)) and ((4)). As suggested by Zhang et al. , the signs of these horizontal wave numbers are specified from those of the vertical wave number and mean horizontal momentum fluxes conversely.
 All wavelengths exhibit distinct latitude and altitude variations. The vertical wavelength shown in Figure 7 varies in the range of 0.3–6.2 km, with smallest value no larger than 1.2 km at the lowest latitudes. With latitude increasing, the vertical wavelength increases and can reach as large as 6 km at 7 km over 75°N. According to the inertial GW dispersion equation, the poleward increase of the vertical wavelength is consistent with the poleward increase of the intrinsic frequency as shown in Figure 6 and has been reported by Wang and Alexander  from satellite observations. As to the altitude variation, the vertical wavelength increases with altitude from ground up to the middle troposphere and then decreases with altitude, thus forms a maximum value around the middle troposphere. And this maximum altitude shows a decrease trend with increasing latitude. The vertical wavelength in the lower stratosphere is usually shorter than that in the troposphere, which has been known as a typical feature of inertial GWs in the lower atmosphere due to the Doppler shifting of the tropospheric jet and higher buoyancy frequency in the stratosphere. Similar to that in previous observations [Wang et al., 2005; Zhang and Yi, 2005, 2007; Zhang et al., 2012], the fraction of upward propagation shown in Figure 7b is larger in the lower stratosphere than that in the troposphere. Below 10 km, the fraction is around 50% or smaller, while above 15 km, the fraction is generally larger than 50%, suggesting most GWs are generated at the altitudes below 15 km. The maximum fraction is larger than 65% and occurs at 15 km over 10°N. Over different latitudes, the maximum upward propagation fraction decreases with the increasing latitude, and the maximum fraction altitude is likely to decrease poleward. It seems that the altitude range between the maximum (13–17 km ) and minimum (3–7 km ) fraction region is the important source regions for lower atmospheric GWs, which has been emphasized by Zhang et al. [2010, 2012].
 The latitudinal and altitudinal variations of the zonal and meridional wavelengths shown in Figure 8 are similar to each other. Both of them vary in the range of 150–500 km, which are at least one order larger than the vertical wavelength, indicating the observed GWs were propagating in a very shallow angle to the horizon. In two latitudinal regions, i.e., a lowest latitude region of 5°N–20°N and a highest latitude region of 70°N–75°N, both the zonal and meridional wavelengths have longer values, with maxima larger than 420 km. The longer horizontal wavelength at the tropical region has also been reported by recent radiosonde [Leena et al., 2012] and satellite [Wang and Alexander, 2010] observations. There are also two large horizontal wavelength altitude regions, one is from 5 to 10 km in the middle troposphere, and the other is from the tropopause up to about 23 km in the lower stratosphere. Shorter zonal and meridional wavelengths are observed in the lower troposphere (below 5 km) over almost all latitudes and above 23 km over middle and high latitudes, where the wavelengths are no longer than 240 km.
 Observing the horizontal propagation directions, we can find that in most cases, the eastward and northward propagation fractions are around 50%, indicating there is no obvious prevailing horizontal propagation direction, i.e., the horizontal propagation is generally isotropic. Extremely large eastward and northward propagation fractions (with maxima larger than 60%) only occur at lower troposphere over latitudes from 40°N to 75°N, which may be related to tropospheric sources in the mid-high latitudes.
 In addition, compared with the latitudinal and altitudinal variations, the seasonal variations (not presented) of wavelengths and propagation directions are relatively weak.
4.4 Momentum Flux and Wave Drag
 GW momentum flux is an important parameter in describing the wave momentum transportation, and its altitudinal variation can be applied to quantitatively address the wave momentum deposition and wave-induced drag on the background atmosphere, which is one of the most important wave-mean flow interaction mechanisms. In this paper, we apply the vertical fluctuation wind to directly calculate the continuous altitude variation of mean horizontal momentum fluxes, i.e., and , which were usually indirectly derived from the GW polarization equations due to the lack of vertical wind perturbation and was given only in term of an altitude averaged value [Vincent and Alexander, 2000; Zhang and Yi, 2007]. The zonal momentum flux has different features from that of the meridional momentum flux. The mean zonal momentum flux shown in Figure 9 varies in a range of −6.1–4.7 m2s−2, while the meridional momentum flux varies in a smaller range of −3.1–3.0 m2s−2. The negative zonal momentum flux concentrates in the troposphere over the middle latitudes, with the maximum (in magnitude) negative flux appearing at 6 km over 35°N, where is below the strongest troposphere jet. At middle latitudes, closely above the strong negative zonal momentum flux region, in the lower stratosphere, is the strong upward momentum flux region, and the positive flux increases with altitude. The intensive negative and positive zonal momentum fluxes occurring, respectively, below and above the jet indicate that the strong tropospheric jet at middle latitudes is an important inertial GW source. At the tropical and high (poleward from 55°N) latitudes, the zonal momentum flux is almost positive at all altitudes. For the meridional momentum flux, the boundary between the positive and negative values in the latitude-altitude plane decreases almost linearly poleward at most latitudinal range. Above and below the boundary, there are negative and positive meridional momentum flux regions, respectively. At 15°N, the boundary is at the altitude of 17 km, and with the increase of latitude, the boundary altitude linearly decreases to about 7 km at 60°N. At the high latitudes poleward from 60°N, the negative flux region is from 8 to 17 km, closely above the tropopause. Moreover, both the maximum positive and negative meridional momentum fluxes occur at 45°N, which is located at 7 and 16 km, respectively.
 The time series of monthly averaged zonal and meridional momentum fluxes at different latitudinal areas are presented in Figures 10 and 11, respectively. Both the monthly averaged zonal and meridional momentum fluxes vary in the range of −0.15–0.21 m2s−2. At low latitudes (5°N–15°N), extremely large positive and negative zonal momentum fluxes occur in the altitude range of 10–20 km, with maximum positive (negative) fluxes usually but not always appearing in summer (winter). Similar to zonal momentum flux, at low latitudes, strong positive and negative values of the meridional momentum flux occur usually at 10–20 km, but its seasonal variation is rather irregular. At high latitudes (65°N–75°N), both the zonal and meridional momentum fluxes have complex and irregular seasonal variations and have peak values below 15 km. At middle latitudes (30°N–40°N), the momentum fluxes have obvious annual cycle. The zonal momentum flux below 15 km is negative or weakly positive, with maximum negative values occurring at 5–10 km at vernal equinox. Above 15 km, the zonal momentum flux is almost all negative in all seasons, with prominent positive values as large as 0.15 m2s−2 in winter. While for the meridional momentum at middle latitudes, at altitudes below 15 km and above 20 km, it is positive and has maximum values in winter; at altitudes 15–20 km, it is negative and usually peaks also in winter.
 The GW drags (shown in Figure 12) are calculated from the vertical gradient of the GW horizontal momentum fluxes, i.e., where Fu and Fv are the zonal and meridional wave drags, respectively. The wave-induced zonal force Fu shown in Figure 13a is negative at middle latitudes. For example, at 35°N, one can observe a prominent negative value of about −0.8 m/s/day at 14 km, where is closely above the jet altitude, implying that above the eastward jet, most GWs are propagating westward due to the jet absorption of eastward and upward propagating waves from below. It is noted that the maximum negative zonal force is around the tropospheric jet altitude at middle latitudes, suggesting the GW-induced zonal force tends to decelerate the jet as well as the jet-induced vertical shear. There are also several positive force latitude-altitude regions, the strongest one is at (15 km, 55°N), with a maximum acceleration of 0.65 m/s/day; the other two regions are located at low latitudes, which are (20 km, 10°N) and (10 km, 20°N), respectively.
 As that in the momentum flux, the latitude and altitude distribution of GW-induced force in the meridional direction is more regular than that in the zonal direction. The positive (positive northward) region displays a tilt band shape. The band is higher at the lower latitudes, with the maximum and minimum altitudes at 17 km at 5°N and 7 km at 75°N, respectively, which is close to the latitudinal variation of the cold point tropopause as shown in Figure 1. The strongest northward acceleration appears at (10 km, 45°N), with the maximum value of 0.9 m/s/day. Above and below the acceleration region are the deceleration regions, with the maximum deceleration of about −0.7 m/s/day at (17 km, 45°N), closely above the maximum acceleration region, indicating the inertial GWs will yield a negative vertical shear in the background meridional wind.
4.5 Heat Flux
 Accompanying with the propagation, GWs also transport and deposit heat and thus impact background atmospheric thermal structure. Recent radiosonde observations have confirmed GW-induced heat transportation can impact on the background thermal structure, change the tropopause temperature and altitude [Zhang et al., 2008; Huang et al., 2009], and yield tropospheric inversion [Zhang et al., 2009, 2011]. Here we calculate the averaged GW heat flux to investigate the GW heat transportation and possible impacts on atmospheric thermal structure, where θ′ is the perturbation component of potential temperature θ. Potential temperature can be specified as , where P0 is a constant pressure 1000 hpa; R is the gas constant (R = 287 J kg- 1 K- 1); and cP is the specific heat at constant pressure (cP = 1005 J kg- 1 K- 1). Figure 13 illustrates that at the lower latitudes equatorward from 25°N, the heat flux is positive at almost all altitudes. At higher latitudes poleward from 60°N, the negative and positive heat fluxes concentrate in the troposphere and lower stratosphere, respectively. The strongest negative (downward) heat flux occurs in the troposphere from 65°N to 75°N, with a prominent peak downward hear flux of −0.12 K m/s at (10 km, 70°N). While in the middle latitudes from 25°N to 60°N, the positive flux is located at lower stratosphere, the negative flux is located between these two positive flux altitudes, i.e., 5–10 km . And a secondary maximum negative heat flux of about −0.04 K m/s appears at (11 km, 45°N). The upper boundary of the downward heat flux increases slightly poleward. At 20°N, the upper boundary is at 10 km, and it increases to 17 km at 75°N. Since above the upper boundary is the upward heat flux region, the upper boundary may imply the altitudes of wave source or wave reflection due to the strong background wind. While, the bottom boundary descends obviously with the increase of latitude, it descends from 10 km at 20°N to near ground at 75°N. We want to note that below the lower boundary of the negative heat flux region is the positive heat flux region. This vertical heat flux gradient leads to heat accumulation and then yields tropospheric inversion at the lower troposphere in middle latitudes, which is consistent with previous radiosonde observation of tropospheric inversion at middle latitudes [Zhang et al., 2009, 2011].
 Figure 14 demonstrates the GW associated heat flux also has prominent seasonal variation. At low latitudes (5°N–15°N), the heat flux is positive at most altitudes and in most months; the negative heat flux concentrates in 15–20 km altitude and persists usually only about 1 month in summer. At middle latitudes (30°N–40°N), the negative heat flux occurs at the altitude range of 5–15 km and persists usually from October to April with maximum negative flux appearing in January. Below 5 km, the heat flux is positive. The negative vertical gradient of the heat flux at the lower edge of the negative heat flux region implies the GWs may induce temperature enhancement at 5–10 km, especially in winter, which is consistent previous tropospheric inversion layers observations [Zhang et al., 2009, 2011]. Above 15 up to 25 km, the heat flux is positive with maximum value occurring in winter. The positive vertical gradient of the heat flux at the upper edge of the negative heat flux region indicates the GWs may lead to the cooling of atmosphere around the tropopause. The altitudinal and seasonal variations of the heat flux at high latitudes (65°N–75°N) are similar to those at middle latitudes, except that the heat flux at high latitudes has larger magnitudes and strong negative heat flux at 5–15 km can also be observed in summer. Several reasons can account for the different GW parameter features at high latitudes: (1) Different wave excitations. At high latitudes, although there are lacks of strong convection activity and tropospheric jet, the polar night jet at stratosphere and sudden stratosphere warming may contribute significantly to the generation and enhancement of GW activity [Venkat Ratnam et al., 2004]. (2) The background atmosphere in which GWs are propagating at high latitudes has different features. For example, stratospheric jet, lower tropopause altitude, and larger Coriolis parameter, et al. These different background atmosphere parameters will impact GW features. (3) The uneven geographical distribution of the radiosonde stations may also influence the statistical results. For instance, in the latitudinal bin of 60°N–65°N, there are five stations, while in the bins of 65°N–70°N and 70°N–75°N, there is only one station in each bin.
5 Summary and Remarks
Zhang et al.  have proposed a broad spectral analysis method for GW parameter extraction from radiosonde data. The present work extends the study by Zhang et al.  to a wide geographical area, from 5°N to 75°N, to study the latitude and continuous altitude variability of GW parameters in the altitude range 2–25 km by analyzing 11 year (1998–2008) radiosonde data over 92 United States stations in the Northern Hemisphere. To our knowledge, this is the first time presenting latitudinal and continuous altitudinal variability of lower atmospheric GW parameters.
 The primary statistical features for the latitudinal and altitudinal distributions for GW parameters are generally consistent with previous radiosonde observations by using traditional hodograph analysis method based on monochromatic GW theory [Wang et al., 2005; Zhang et al., 2010] and satellite observations [Tsuda, et al., 2000; Wang and Alexander, 2010]. For instance, the wave energy densities in the troposphere and lower stratosphere peak, respectively, at the middle and lower latitudes. Moreover, in the lower stratosphere, besides the prominent wave energy density peak at lower latitude, there is also a secondary peak in the kinetic energy at middle latitudes, which has seldom been revealed by satellite observation due to the lack of lower atmospheric wind measurements. Basically, at lower latitudes, GWs usually have larger ratio of wave intrinsic frequency to Coriolis parameter, smaller wave intrinsic frequency, shorter vertical wavelength, and longer horizontal wavelength.
 More interesting, our analyses can reveal continuous altitude variations of GW parameters in altitude range from 2 km up to 25 km, allowing us to study the GW properties in the altitude range from 10 to 18 km, which was suggested to be important in generating GWs and impacting GW features [Zhang et al., 2010, 2012] but cannot be resolved from traditional hodograph analysis. Our analyses revealed that most latitudinal and altitudinal variations of GW parameters are closely related to the background temperature and wind fields. For instance, the extreme value altitudes for most GW parameters, i.e., wave energy density, vertical wavelength, upward propagation fraction, momentum fluxes, and wave-induced drag and heat flux, show poleward descending trend, and this trend is closely connected with latitudinal variations of the altitudes for the zonal wind jet and tropopause, implying the impact of background dynamic and thermal structures on GW properties.
 Besides the impacts of atmospheric background on the inertial GWs, our results also revealed the profound climatological impacts of GWs on background atmosphere. The calculated wave-induced zonal force tends to decelerate the zonal jet at middle latitudes, while the wave-induced meridional force will produce a negative vertical shear in the northward wind closely above the tropopause altitude. The altitudinal variation of GW heat flux indicates a cooling effect on the atmosphere around the tropospheric jet altitude. Moreover, the negative altitude gradient of the GW heat flux below 10 km at middle latitudes suggests that GW heat flux contributes significantly to the forming of frequently observed tropospheric inversions at middle latitudes.
 We also investigated the seasonal variations of inertial GW parameters at different latitudinal regions. GW energy densities, horizontal momentum fluxes, and heat flux have evident seasonal variations, while the other GW parameters (not presented), such as wave intrinsic frequency and wavelengths, have only weak and irregular seasonal variations. GW energy density above 10 km at low latitudes has maximum value in summer. At middle latitudes, GW activity is the strongest in winter. At high latitudes, GW energy density below 12 km exhibits semiannual variation with maxima in summer and winter, while above 12 km, the GW activity is dominated by annual variation, with larger value in summer. Correlation analyses indicate that the convection activity contributes significantly to the GW vertical fluctuation energy in the lower stratosphere; the topograph plays an important role of GW excitation at latitudes 30°N–60°N; the tropospheric jet is one of the most important sources of GWs at 30°N–40°N, where the tropospheric jet is very strong in winter. Moreover, the horizontal momentum fluxes also have evident seasonal variations, especially at middle latitudes, where prominent positive fluxes usually occur in winter.
 Another sort of data products, i.e., the assimilated reanalysis data, for instance, the NCEP/NCAR and ECMWF data, has also been extensively applied to study lower atmospheric dynamics [Mihalikova and Kirkwood, 2013; Pena-Ortizet al., 2013] due to their long-term accumulation, global coverage, and high accuracy. However compared with the radiosonde data, these reanalysis data have larger height resolution, usually larger than 1 km, while the derived inertial GW vertical wavelengths have very short values, especially at low latitudes, where the vertical wavelengths are as smaller as 1–2 km . Moreover, it is hard to derive the continuous altitudinal variation of GW parameters (for example, momentum flux) from the reanalysis data due to the lack of effective vertical wind. Therefore, until now, it is difficult to derive inertial GW parameters from those reanalysis data in the presented manner.
 We wish to thank the anonymous reviewers for their valuable suggestions on this paper. This work was jointly supported by the National Basic Research Program of China (grant 2012CB825605), the National Natural Science Foundation of China (through grants 40825013, 41221003, and 41174126), and the Ocean Public Welfare Scientific Research Project of the State Oceanic Administration of the People's Republic of China (201005017).