Mesosphere-stratosphere-troposphere radar measurements have been used to estimate momentum flux of gravity waves in a 2–6 hour period in the lower atmosphere over Gadanki (13.5°N, 79.2°E), a tropical station. Different methods have been adopted to compute momentum flux profiles, and a comparative study shows that momentum flux estimates produced by a hybrid method (Worthington and Thomas, 1996) which measures the vertical wind with a vertical beam and the horizontal wind with a pair of radial beams and those obtained by direct computation of spatial covariances ( and ) show satisfactory agreement. The symmetric beam radar method of Vincent and Reid (1983) has the unique advantage since it does not require vertical beam measurement, and it is found to produce momentum flux estimates compatible with other methods except in high wind shear zones. It is observed that the result of momentum flux obtained by the symmetric beam method shows excellent matching with other methods when the average of vertical winds derived from E-W and N-S beams is used in the formula for both zonal and meridional fluxes.
 Gravity waves represent the most important means of momentum coupling in the middle atmosphere. These waves are supposed to be generated in the lower atmosphere, and as they propagate upward from their source region, they couple the lower and upper atmosphere by transferring their momentum and energy. Internal gravity waves are also supposed to play a major role in the mesopause semiannual oscillation forcing. Hamilton , while reviewing the current status of modeling of the circulation in the middle and upper atmosphere, pointed out that general circulation models will need to include a significant drag in the mesosphere associated with the effects of unresolvable small gravity waves in order to obtain a realistic simulation of the middle atmospheric general circulation, and the drag is most plausibly associated with gravity waves generated by various sources in the real troposphere. The gravity wave parameterization problem is one of the most significant issues confronting climate modelers. Measurements of vertical flux of horizontal momentum of gravity waves and its divergence are very important because of their role in the large-scale circulation and thermal structure. It is therefore important to understand how these waves affect upper atmospheric dynamics and to quantify them in their source region of the lower atmosphere.
 Different techniques have been employed to measure vertical flux of horizontal momentum. Alexander and Pfister  estimated lower stratospheric momentum fluxes using winds measured by an aircraft-mounted system and showed that gravity waves propagate away from deep convection. Nastrom and Fritts  used Global Atmospheric Sampling Program data collected by commercial aircraft to compare gravity wave variances and momentum fluxes measured near the mountains to those of a plain region. Using middle and upper atmosphere (MU) radar data at Shigaraki, Sato  showed that the mountains near the radar enhanced the momentum flux estimates. Murayama et al.  analyzed MU radar data and observed seasonal variation of zonal momentum flux, whereas no significant variation could be found in the meridional component. Worthington and Thomas  used VHF radar data located at Aberystwyth and estimated momentum fluxes by three different methods. They showed that a hybrid method which measures the vertical wind with a vertical beam and the horizontal wind with a pair of radial beams produces reliable estimates of momentum flux.
 There have been very few studies of gravity wave momentum flux in the tropical lower atmosphere using radar observations. The purpose of the present work is to derive momentum flux profiles of long-period gravity waves (2–6 hours) in the troposphere and lower stratosphere of a tropical station, Gadanki (13.5°N, 79.2°E), using mesosphere-stratosphere-troposphere (MST) radar wind data by different methods and to make a comparison to find a simple and robust method for the estimation of this important parameter.
 The paper is organized as follows. Section 2 gives a brief description of the radar system and data processing. We describe different methods of estimation of momentum flux in section 3. Results of the present study are discussed in section 4. Finally, the conclusions are presented in section 5.
2. Radar System and Data Description
 The MST radar at Gadanki is a high-power VHF radar and operates at a frequency of 53 MHz with an average power-aperture product of 7 × 108 W m2. The phased antenna array consists of 32 × 32 three-element Yagi antennas occupying an area of 130 m2. A detailed description of the radar is available in work by Rao et al. . Observations made by the radar in five beam directions (one zenith beam and four 10° off-zenith beams in the north, east, south, and west directions) continuously for 24 hours starting at 0950 (local time) on 15 July 2004 and ending at 0940 of the next day without any break, have been used for the present study. Spectral data collected by the radar with 16 μs coded pulse, 1 ms interpulse period, and a duty ratio of 1.6% have been processed both online and off-line [Dutta et al., 1999; Sasi et al., 1998] to obtain radial velocities and the wind components, namely, zonal (u), meridional (v), and vertical (w) with a height resolution of 150 m and time resolution of approximately 2 min 30 s. Radial velocities corresponding to poor SNR values (<−12 dB) have been discarded. Outliers have been removed by taking mean wind in 1-hour sections for each height and discarding values exceeding 1.7 times the standard deviation. Data gaps with respect to height are filled up by linear interpolation. The quality of the data was very high, and such outliers were found to be very rare, appearing above 14–15 km for a few profiles. The mean zonal, meridional, and vertical wind profiles averaged over 24 hours are shown in Figure 1. The maximum amplitudes of zonal, meridional, and vertical winds are ∼40 m s−1, 10 m s−1, and −0.14 to 0.04 m s−1, respectively. The wind velocities are quite normal for this period of the season with average level of convection. The background wind profile, which is not related to the mesoscale features of interest, has been removed from the data by subtracting the mean profile of the day. The time series of perturbation profiles have been filtered using a fourth-order Butterworth filter to retain oscillations between 2–6 hours. The detrended and filtered profiles of radial, zonal, meridional, and vertical velocities have been used to derive momentum fluxes of horizontal winds.
3. Methods of Analysis
3.1. Direct Approach (DA)
 MST radars have the unique capability of measuring directly the vertical winds with sufficient accuracy [Gage, 1983]. Simultaneous observations of horizontal and vertical wind velocities with time can be suitably filtered to isolate a particular frequency or a frequency band of interest. The products of the perturbation winds (u′w′ and v′w′) averaged over a sufficient length of time provide the estimation of vertical fluxes of zonal and meridional momentum per unit mass. The momentum flux values have been multiplied with atmospheric density to obtain density-weighted momentum flux estimates. This direct measurement of momentum flux will be termed the direct approach.
3.2. Symmetric Beam Method (SBM)
Vincent and Reid  devised the symmetric beam radar method to calculate momentum flux in a vertical plane using two radial radar beams in that plane pointing at symmetric zenith angles (θ), provided that the atmospheric motions are horizontally homogeneous:
and similarly for the N-S beam:
where vE, vW, vN, and vS represent the radial perturbation velocities in the east, west, north, and south beams, respectively.
3.3. Hybrid Method (HM)
Worthington and Thomas  have used a method which measures the perturbations of vertical wind with a vertical beam and the horizontal winds using a pair of radial beams. The vertical flux of horizontal momentum is estimated as
4. Results and Discussion
 Momentum fluxes of gravity waves with period between 2–6 hours have been estimated by different methods using the wind fluctuations profiles as mentioned in section 2. Mean momentum flux profiles computed by different methods have been smoothed by taking a three-point running mean with weighted average and are shown in Figure 2 for zonal and meridional components with their respective standard deviations. Mean density normalized flux values are found to range between ±0.01 kg m−1 s−2 for both zonal and meridional components. Chang et al.  used ST radar data collected over Christmas Island in 1994 and estimated tropospheric momentum fluxes to range between ±0.02 and ±0.01 kg m−1 s−2 for waves with period shorter than 8 hours. The standard deviations obtained were larger than their momentum flux values.
Prichard and Thomas  estimated vertical flux of horizontal momentum for waves with periods less than 6 hours in the troposphere using VHF radar data located at Aberystwyth. Their zonal and meridional momentum flux estimates per unit mass ranged between ±0.01 m2 s−2. Murayama et al.  analyzed the seasonal variation of gravity wave activity for 4 years using MU data. Their estimates of vertical flux of zonal momentum per unit mass at 6.5–8 km ranged between −0.15 and 0.25 m2 s−2, and no significant seasonal variation could be found. Mean momentum flux per unit mass obtained in the present study are found to vary between ±0.03 m2 s−2 and ±0.02 m2 s−2 for zonal and meridional winds, respectively. A comparison of momentum fluxes (density weighted) obtained by three methods is shown in Figure 3 for zonal and meridional components with standard deviations at a few heights on direct approach. Since the mean values of momentum fluxes obtained are usually comparable with the standard deviation at each height, confidence in the estimated flux values is not high.
 Meaningful estimation of momentum flux requires integration over a very long period since geophysical noise introduces uncertainty in the fluxes [Kudeki and Franke, 1998; Thorsen et al., 2000]. Riggin et al.  carried out a careful error analysis and observed that the stationarity assumption is likely to be violated during integration periods approaching or exceeding a day. Their Figure 4 shows this evidence of day-to-day variability. As a result, they suggest about 1 day of integration to get momentum flux estimates larger than the standard deviations. Thorsen et al.  have examined the statistical estimation error for the dual coplanar beam momentum flux estimates in the presence of geophysical noise. They showed that there is an optimal beam separation that minimizes the estimation error, and for typical mesopause parameters this beam separation is approximately ±13°. The off-vertical beam angles of ±10° and 24 hours continuous data used in the present study may yield momentum flux estimates with reduced error. Reid  examined the basic assumptions made when a Doppler radar is used to measure the mean and fluctuating components of the wind field in the middle atmosphere with various beam configurations. He has considered the dependence of normalized kinetic energy (equation 18) on θ for a gravity wave of 60 min period and has shown that the error is greater for small tilt angle (θ = 5°) compared with θ = 11.6° and 15°. At any of these tilt angles, the shortest scale that could be measured correctly would be about 150 km at an altitude of 84 km. The error in the estimate of is found to be about 15% for most scales for a period of 60 min. The error increases with decreasing period until it becomes quite substantial at T = 12 min. Similar analysis for spatial covariance () shows that unlike , longer-period waves are also affected. However, it is found that there will be substantial errors in () for the scales less than about 150 km, but that periods greater than 180 min will lead to the best measurement of momentum flux. Since the present study has concentrated on gravity waves of periods between 2–6 hours, we consider the results to be meaningful and acceptable.
 Momentum flux estimated by direct approach and the hybrid method uses vertical velocity directly measured by the radar. VHF wind profiling radars are considered to provide continuous measurements of the vertical wind with good height and time resolution. However, the reliability of these measurements has often been questioned. Any systematic tilt of the array axis from the horizontal will cause the beam to be displaced relative to the zenith, and consequently, the horizontal winds will not be fully removed from the vertical estimates [Vincent and Reid, 1983]. Gage  presented an overview of the measurements of atmospheric vertical motion using MST radar technique and observed consistency with the magnitudes of vertical motion obtained by other methods. It is now well accepted that vertical velocities measured by radars do provide velocity estimates to a useful precision.
 The symmetric beam method, on the other hand, uses the velocity variances measured by both radial beams, which can be split into horizontal and vertical velocity terms as follows:
where the second terms in the brackets represent perturbations in the vertical winds. A comparison of the vertical winds derived using E-W (Wew) and N-S (Wns) beams show a lot of discrepancy with direct radar measurements of vertical wind, but the average of derived vertical velocities (Wewns) is found to be in good agreement with the measured one (Figure 4). Momentum flux profiles were recalculated by the symmetric beam method using the averages of vertical winds derived from E-W and N-S (Wewns) beams as
 Profiles obtained by using equations (1) and (7) for zonal wind and equations (2) and (8) for meridional wind, averaged over the whole observational day, are illustrated in Figure 5 along with the previous results of Figure 3. It can be seen that the results obtained by the symmetric beam method using averages of derived vertical wind perturbations from E-W and N-S beams (SBM II) agree very well with the results of other methods. Figure 6 depicts a comparison of standard deviation profiles of different methods, which confirms the same result. Riggin et al.  advocate use of the same symmetric coplanar beams to form the horizontal and vertical velocities to avoid mixing of horizontal (u,v) momentum flux terms with the desired vertical flux of horizontal momentum. However, our results show that vertical velocities derived from E-W and N-S beams are significantly different, particularly in high wind shear regions, and hence momentum flux estimates yielded by the symmetric beam method which uses these indirect values of vertical velocities show some differences with those produced by other methods. Gadanki is a tropical station surrounded by small hills, and the zonal winds are normally stronger compared with meridional ones. So it is likely that the vertical wind derived from E-W beam measurements will be more contaminated. Furthermore, stationary waves may be prevalent at the site, restricting the usage of symmetric beam technique. It can still be observed that this unique and popular method yields momentum flux estimates of the same order as those of other methods with similar range of uncertainties when the average of the derived vertical winds is used for both zonal and meridional components. We plan to make similar studies of momentum flux of gravity waves in different wind conditions. Future work will also emphasize the study of momentum flux of short-period gravity waves with periods less than 2 hours and horizontal wavelengths less than a few hundred kilometers since they carry about 70% of the upward flux of horizontal momentum [Reid and Vincent, 1987; Fritts and Vincent, 1987].
5. Concluding Remarks
 Momentum flux profiles of gravity waves between 2–6 hours have been derived in the troposphere and lower stratosphere using different methods: direct approach (DA), symmetric beam method (SBM), and hybrid method (HM). An intercomparison of the methods reveals that results obtained by direct approach and the hybrid method have closer agreement, whereas the symmetric beam method produces little larger values of momentum flux with larger standard deviations in the upper troposphere and lower stratosphere where SNR values are poor and vertical shear in horizontal wind is high. Moreover, the vertical velocity terms, implicit in the formula of symmetric beam technique, are derived from E-W and N-S radial velocities and are found to disagree with direct vertical beam measurements. Worthington and Thomas  pointed out that errors occur in momentum flux when a large but reliably measured horizontal wind component is multiplied by a less accurate estimate of vertical wind. It should also be noted that this tropical radar site, surrounded by small hills, may generate Lee waves, and hence the symmetric beam method could not be readily applied. However, we observe that the key advantages of this method of not requiring the vertical beam measurement and remaining un-affected by beam separation can be exploited, and reliable estimates of gravity wave momentum flux can be inferred if the average of derived vertical winds is used in the computation.
 The Indian MST Radar Project was funded jointly by DOE, DOS, DOEn, DST, DRDO, and CSIR with DOS as the nodal agency. The authors are grateful to the Indian Space Research Organization (RESPOND), government of India, for providing financial assistance to run the project. The authors also wish to thank the UGC-SVU Centre for MST Radar Applications, S.V. University, for providing necessary facilities to carry out the research work. The program (atmospheric data processing) provided by V. K. Anandan for analysis of MST radar data is gratefully acknowledged. The authors would like to thank the chairman and the secretary of Anwarul-uloom College for their active support and kind encouragement. The authors express their deep gratitude to the anonymous referees, whose comments have substantially improved the quality of the paper.