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 The diurnal variation of gravity wave (GW) momentum flux is derived from the 5 years of meteor radar wind measurements at Maui, Hawaii. The amplitude and phase relationships between the GW forcing and the diurnal tide are analyzed by calculating their equivalent Rayleigh frictions. The results show that the GW momentum flux is clearly modulated by the diurnal tide. The forcing from the momentum flux convergence has strong effects on both the amplitude and phase of the diurnal tide. They can reach 80 ms − 1 day − 1 for the amplitude and 15 h day − 1 for the phase. The GW forcing tends to increase the diurnal tide amplitude above 90 km but has a small damping effect below 90 km. It tends to increase the phase of the diurnal tide throughout all altitudes. Seasonal variations of the GW forcing exist, which result in differences in their effects on the diurnal tide. The magnitudes of the forcing are in agreement with recent results from satellite observations but are much larger than values used in general circulation models. This work also demonstrates that meteor radar measurements can provide a valuable data set for the study of GW-tide interactions.
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 It is well recognized that gravity wave (GW) propagation from the lower atmosphere to the mesosphere and lower thermosphere (MLT) and the momentum deposition associated with wave dissipation in the MLT have profound dynamical effects on the zonal mean thermal structure, zonal wind, and meridional circulation. Recently, it has become increasingly clear that GWs also play important roles in the variability of this region [Fritts & Alexander, 2003]. In the MLT, atmospheric thermal tides and planetary waves can reach large amplitudes, and their interactions with GWs may result in variability with time scales varying from days to years [e.g., Kumar et al., 2008; Wu et al., 2008a; Wu et al., 2008b; Xu et al., 2009b; Iimura et al., 2010]. Although progress is being made in the understanding of the interactions between the diurnal tides and planetary waves through studies with both modelings [e.g., Chang et al., 2011; Suresh Babu et al., 2011] and observations [e.g., Thayaparan et al., 1995; Riggin et al., 2003; Palo et al., 2007; Smith et al., 2007; Hecht et al., 2010], the interactions between GWs and tides are less understood.
 The interactions between GWs and tides are complex. Although GW propagation and dissipation can be modulated by changing background associated with tidal oscillations, as shown in observational studies such as [Fritts & Vincent, 1987] and [Beldon & Mitchell, 2010], GWs also exert force on tides through their momentum deposition when they dissipate. The GW effects on modulating the tidal amplitude and phase are well appreciated and taken into account in numerous models [Hagan et al., 1995; Ortland & Alexander, 2006; Liu et al., 2008; Watanabe & Miyahara, 2009; Lu et al., 2012]. GW breaking also results in turbulence mixing that acts to damp tidal amplitudes, as suggested by some modeling studies [e.g., Miyahara & Forbes, 1991; Liu & Hagan, 1998].
 One major challenge in the study of GW-tide interactions is due to the large differences in spatial and temporal scales between tides and high-frequency GWs. For modeling studies, simulations of tides require a global model, in which GWs are often parameterized because of their small scales and computational constraint [e.g., Miyahara & Forbes, 1991; McLandress, 2002; Garcia et al., 2007]. The GW-tide interactions are sensitive to the parameterization schemes. For example, GWs parameterized with schemes by [Lindzen, 1981] and [Fritts & Lu, 1993] tend to suppress the diurnal tide, while with Hines's spectral scheme [Hines, 1997a; Hines, 1997b], they tend to increase the tidal amplitude [Miyahara and Forbes, 1991; Mayr et al., 1998; McLandress, 1998]. The roles that GWs play on the seasonal variation of the diurnal tide can also be distinct based on different models. [McLandress, 2002] found that the direct effects of GWs are not important for the seasonal variation of the diurnal tide, whereas [Mayr et al., 1998] stated that GWs are responsible for the equinox maxima and solstice minima of the tidal amplitudes. [Watanabe & Miyahara, 2009] used a GW-resolving general circulation model (GCM) to quantify GW forcing on the migrating diurnal tide, and found that the diurnal tide is amplified (suppressed) during equinoxes (solstices) by GWs.
 For observational studies, the large differences in scales between GWs and tides also pose a challenge. The global structure of tidal oscillations is best obtained with satellite measurements. However, satellite observations in the MLT cannot resolve small-scale GWs at high temporal resolution, and it often takes about 2 months to cover the full 24 hour local time needed to fully characterize diurnal variations. Ground-based instruments with high temporal and spatial resolutions can do much better in resolving GWs and diurnal variations at a single site but cannot distinguish different tidal components and modes. The early study by [Fritts & Vincent, 1987] is an observational study that examined the relationships between GW momentum flux and diurnal tide. During an 8-day measurement of zonal wind from Doppler radar near Adelaide, Australia, they observed strong tidal modulation of GW activities. They also proposed a GW-tides interaction model that generally agrees with the observation. Another example is the 9-day continuous Na lidar observations made by [She et al., 2004] at Ft. Collins, Colorado. With high-resolution measurements of temperature and horizontal winds, both GWs and local tidal variation were characterized. Such high-resolution observations usually cover short periods of time because of observational challenges. Due to large variabilities of GWs and tides, case studies based on short-term observations may not represent the typical behavior of GW-tide interactions in an average sense. Long-term observations thus facilitate examination of these interactions and how they change on a seasonal scale.
 In this study, we use meteor radar observations at Maui, Hawaii (20.7 ∘ N, 156.3 ∘ W), to study the relationships between GW momentum flux and the diurnal tide and their seasonal variations. Although other tidal harmonics, such as semidiurnal and terdiurnal tides, are also observed at Maui, they are much weaker than the diurnal component [e.g., Figure 2 in Jiang et al., 2009]. The focus of this study is to quantify the climatological relationship between GW momentum flux and the diurnal tide. The GW momentum flux is calculated by using a technique proposed by [Hocking, 2005]. The 24-hour, year-round continuous measurements of meteor radar thus make it possible to examine the diurnal variations of GW momentum flux and their relationships with tidal oscillations and their seasonal variations. Because the diurnal tide is dominated by the migrating mode with wave number one (DW1) in low latitudes, and the latitude of Maui is the region where the meridional wind in DW1 reaches maximum [see, e.g., Wu et al., 2008a; Lieberman et al., 2010; Lu et al., 2011], the results can be considered representing mainly the interactions between the migrating diurnal tide DW1 and GWs. The characteristics of the diurnal tide at Maui has been analyzed by [Lu et al., 2011] based on the same meteor radar data. These results are used in this study to examine the relationship between the diurnal tides and the derived GW momentum flux.
 We will first discuss the method and data in section 2. The results of this analysis are presented in section 3, which includes both the mean relationship and seasonal variations. The implications of our results are discussed in section 4, followed with conclusions in section 5.
2 Method and Data
2.1 Momentum Flux Calculation and Uncertainty
 Meteor radars are now widely used to measure the horizontal winds in the 80 to 100 km region on a 24 hour basis, typically with 1 hour temporal and 2 to 3 km vertical resolutions. The meteor radars measure the locations and line-of-sight drift velocities of meteor trails based on interferometry and Doppler shift [Hocking et al., 2001]. The horizontal winds are derived by fitting the mean horizontal wind to all detected meteor trail velocities within certain height range and time interval. The measurements are ideal for studying long-period oscillations such as tides and planetary waves because of the continuity of the measurements. The Maui meteor radar instrument and verification of the wind measurements are described in detail by [Franke et al., 2005]. The horizontal wind data were used by [Lu et al., 2011] to study the seasonal variation of the diurnal tide.
 Recently, [Hocking, 2005] proposed a method for estimating GW variances and momentum fluxes through a least squares fitting of the meteor trail velocity variance. It is a generalization of the dual-beam method used by [Vincent & Reid, 1983]. This method has since been applied successfully by several researchers to study GWs. [Antonita et al., 2008] used 3-year data near the equator (8.5 ∘ N, 76.9 ∘ E) to examine the GW momentum flux and its forcing on the mesosphere semiannual oscillation. [Placke et al., 2011] analyzed the seasonal variation of the momentum flux at a high latitude station (51.3 ∘ N, 13.0 ∘ E) in Germany based on 5 years of meteor radar data. [Clemesha et al., 2009] studied the seasonal variation of GW activities in three locations in Brazil. In this study, we examine the diurnal variation of the GW momentum flux and its relationship with the diurnal tide. This is done by constructing a composite day using the entire 5 years of data to look at the long-term mean and constructing composite days for every calendar month to examine the seasonal variation of this relationship.
 The uncertainty of momentum flux calculated using this technique is largely dependent on the number of meteor detections used for the least squares fitting. [Vincent et al., 2010] analyzed this uncertainty and found that although an hourly rate of 6 to 10 meteors is required to obtain a reliable mean wind estimate, significantly more meteors are needed to obtain reliable GW variance and momentum flux estimates. [Fritts et al., 2012] did a comparative study on this uncertainty with actual meteor detections obtained from five meteor radars with various power and antenna designs. They found that conventional meteor radars, with relatively low transmitting power, can be used to obtain reasonable hourly GW momentum flux estimates of a monthly composite day in the altitudes when the meteor counts are high. The Maui meteor radar is a SKiYMET radar with a single three-element Yagi antenna and 6 kW peak power [Franke et al., 2005]. It is similar to the meteor radars at Socorro, New Mexico, and Yellowknife, Canada, two conventional meteor radars compared in [Fritts et al., 2012], who showed that momentum flux estimated with these radar measurements are reasonable between 85 and 90 km for a monthly composite day. In our analysis, to further increase the detections used for momentum flux calculation, we use all data from multiple years (2002–2007) in the same calendar month to construct the composite day, and use 3-km and 2-hour bin sizes. At this resolution, diurnal variation and vertical structure of the momentum flux can still be adequately resolved, and we can extend the vertical range of reliable momentum flux estimates. To avoid large measurement errors at large and small zenith angles, we only use meteors detected with zenith angles between 15 ∘ and 50 ∘, which are about half of the total meteors detected. We also discard detections where the horizontal wind perturbations are larger than 30 ms − 1, to avoid outliers skewing the least squares fitting results. Figure 1 shows the total meteor detections within this zenith angle range, at 3 km, 2 hour bins for a composite day for all 12 calendar months. These are the counts of meteor detections that are used for momentum flux calculation for individual months. Strong seasonal and diurnal variations of meteor counts are associated with the variation of meteor input rate. The meteor counts generally peak around June and 16 to 18 UT (6 am to 8 am local time at Maui). In the vertical, the counts peak at about 90 km. At most altitude and hour bins, the counts are above 500. These counts are comparable or larger than those obtained in Yellowknife as described in [Fritts et al., 2012] and should provide reasonably accurate estimate of momentum flux. We also apply a bootstrap method [Efron, 1979] to quantify the uncertainty of the calculated momentum flux. This method randomly resamples all the meteor detections, and different momentum fluxes are obtained from different samplings. The standard deviation of all momentum flux estimates provides a measure of the uncertainty. In this analysis, we use 100 random samplings for this uncertainty calculation.
 In addition to temporal and vertical variations, the meteor detections also vary with azimuth angles, as shown in Figure 2. This anisotropic distribution is caused by the fact that the antennas used for transmission and reception consist of a single dipole and reflector, which are more sensitive to targets located north and south of the radar. This results in more detections in the meridional than in the zonal direction, and smaller (larger) uncertainties in the meridional (zonal) momentum flux calculation.
 Notably, for the mean momentum flux estimate, another important uncertainty arises from the geophysical variabilities of horizontal and vertical winds caused by GW disturbances. Because of the inherent correlation time of this variability, enough independent sampling (therefore, long time integration) is needed to reduce this uncertainty. [Kudeki & Franke, 1998] concluded that for stratospheric momentum flux, at least 16 days of measurement is needed. [Gardner & Liu, 2007] provided quantitative measures of these uncertainties based on 400 hours of lidar measurement of horizontal and vertical winds at Starfire Optical Range (their Tables 3 and 4). The amplitude uncertainties of their annual and semiannual components of momentum flux are about 2 to 3 m 2 s − 2. For comparison, in our analysis, the momentum flux at every 2 hour period is estimated from measurements in a calendar month from all 5 years of data, which is equivalent to about 2 × 30 × 5 = 300 hours. This is more than the lidar data used for each season by [Gardner & Liu, 2007] (∼ 100 hours per season). Based on this, we expect that the uncertainty of our momentum flux caused by geophysical variability is no larger than 2 to 3 m 2 s − 2.
 Another issue to consider is the day-to-day and year-to-year variations of the tidal amplitude and phase, as shown in [Lu et al., 2011]. Because the wind perturbations are obtained by subtracting the hourly mean wind from the total wind field, tidal components in individual days are removed, and thus the effects of daily variation of tidal amplitude are also removed. However, when constructing a composite day according to the local or universal time, data from different days at the same hour may correspond to different phases of the diurnal tide. To consistently represent the relationship between GWs and diurnal tides, we can perform the compositing according to the tidal phase at each day rather than the local time. Meteors detected in the same tidal phase on different days are used together to calculate the momentum flux for this particular tidal phase. This should better represent the modulation of the GW momentum flux by the diurnal tide. In practice, there are times when the tidal amplitude is very small and its phase is unreliable, so data at these times have to be discarded. This does not have much adverse effect on the momentum flux estimate for the 5-year composite day because of the large number of meteor detections, but does increase the uncertainty of monthly momentum flux estimates. Therefore, we decide to still use local time to bin the data for monthly momentum flux calculation, but use a reference tidal phase to bin the data for the 5-year composite day. This is reasonable considering the magnitude of the seasonal variation of the tidal phase is larger than that within each month. The diurnal tidal phase was obtained in [Lu et al., 2011] by fitting a 24 hour sinusoidal function within a 5-day sliding window. The reference tidal phase for each day is chosen to be the tidal phase of the meridional wind at 90 km, where the radar detection rate is the highest. Observations show that the monthly mean phase differences between the zonal and meridional winds are nearly constant [Lu et al., 2011]. The diurnal variation of the derived momentum flux for the 5-year composite day is thus relative to the phase of the diurnal tide in the meridional wind at 90 km.
2.2 GW Forcing and Equivalent Rayleigh Friction
 Because the calculations of GW forcing and equivalent Rayleigh friction (ERF) are the same for the zonal and meridional components, we describe only the procedures in terms of the zonal components. The acceleration of the horizontal wind caused by GW momentum flux divergence is calculated as:
where denotes the vertical of the zonal momentum, and X is the corresponding acceleration. The overbar denotes ensemble average. ρ is the background atmospheric density, which can be written as ρ = ρ0e− zH , where H = RT/g is the density scale height and ρ0 is the density at the surface. R = 287 Jkg− 1K− 1 is the gas constant of the atmosphere, and g is gravity. We calculated H using monthly mean temperature from the MSIS00 [Hedin, 1991]. The scale height varies from 5 to 6 km. The diurnal components of the momentum flux and the acceleration are obtained by fitting with a 24-hour period sinusoidal function. The fitted function of X can be expressed as , where is the complex amplitude representing both the amplitude and phase of the diurnal variation of the GW acceleration, and ω = 2π/24 h.
 The effects of GW forcing on the diurnal tide are assessed by calculating the ERF, a complex quantity that describes effects on both the amplitude and the phase of the diurnal tide. The ERF was introduced by [Miyahara & Forbes, 1991] and [Forbes et al., 1991], and adopted by many studies to calculate the forcing on the diurnal tide [e.g., McLandress, 2002; Ortland & Alexander, 2006; Watanabe & Miyahara, 2009; Xu et al., 2009a; Lieberman et al., 2010; Chang et al., 2011; Lu et al., 2012]. If we express the complex amplitude of the diurnal tide as , where a(t) is the real amplitude and ϕ(t) is the phase, the ERF for the zonal component is defined as
For GW forcing, , so the ERF is calculated as
The effects on the amplitude and the phase of the diurnal tide are then
The real part of ERF describes the relative amplitude change of the diurnal tide, and the imaginary part describes the phase change.
 Typically, the complex tidal amplitude used in calculating ERF in equation (3) is based on the measured diurnal tide. When the GW effects are large, this observed diurnal tide is significantly different from the original diurnal tide in the absence of GWs. Therefore, the ERF calculated using equation (3) does not represent the GW effects on the original tide. We can, however, modify the ERF with the following approach. Assume is the complex amplitude of this original tide u0 generated by a forcing F without GWs, which has a complex amplitude ; that is, and . We can write
Similarly, the measured tide satisfies
Subtracting equation (5) from (6) and solving for , we get
For monthly or longer-term time average, the rate of change of the diurnal tide amplitude is much smaller than the frequency of the diurnal tide. By ignoring the first term in equation (8), we get
The ERF with respect to the tide without GW forcing is then
Note that when the ERF is small compared with the diurnal frequency, |γ| ≪ ω, the GW effects on the diurnal tide is small and, as expected, γ0 ≈ γ. We will calculate both γ and γ0 using equations (3) and (10), respectively, and show their differences in the following section.
3 Observations and Results
3.1 Tidal Modulation of GW Momentum Flux
 We first examine the mean diurnal variation of the momentum flux from all 5 years of data. The momentum flux for the composite day is fitted with a 24-hour sinusoidal function at all altitudes to obtain the diurnal component, which is shown in Figure 3. The uncertainty of the momentum flux is also shown with shading. The uncertainty is the standard deviation of the diurnal component of the momentum flux calculated from 100 random samplings using the bootstrap method discussed earlier. Note that the horizontal axis is not universal time, rather the time relative to the phase of the diurnal tide in the meridional wind at 90 km (see section 2.1). The uncertainty values are less than 1 m 2 s − 2 at all times and altitudes, with slightly smaller values in the meridional component because of more detections in the north-south direction (see Figure 2). The amplitudes of the diurnal variation vary from 2 to 8 m 2 s − 2 throughout this altitude range, much larger than the uncertainties in most regions. Therefore, the diurnal variations shown in Figure 3 are robust features. In the meridional component, and above 90 km in the zonal component, there is a clear tilt in the vertical structure of the momentum flux. This is clear evidence that the GW momentum flux is strongly modulated by the diurnal tide. The fact that this feature is present in the composite day from 5 years of data also indicates that this modulation is likely persistent throughout the year.
3.2 GW Forcing on the Diurnal Tide
 The acceleration rates are calculated based on the GW momentum flux according to equation (1). The momentum flux is first smoothed in altitude with a sliding 8 km full-width Hamming window to remove small vertical scale variations that are not related to the diurnal tide. The Hamming window is defined as
where z0 is the center of the window and z is altitude in kilometers. It is typically used to minimize the maximum side lobe in a Fourier transform. Here it is used as a low-pass filter for vertical smoothing.
 The calculated acceleration is again fitted with a 24-hour sinusoidal function to obtain the diurnal component, which is shown in Figure 4, together with the diurnal component of the horizontal wind. The diurnal variation of acceleration reaches maximum values of about 100 ms − 1day − 1 in the zonal component and approximately 150 ms − 1day − 1 in the meridional component. These values are much larger than the 15 ms − 1day − 1 obtained by [Watanabe & Miyahara, 2009] with a GW resolving GCM. In the zonal component, the positive (eastward) acceleration generally corresponds to the positive tidal wind and vice versa, which implies that GW acceleration is in-phase with the diurnal tide. Therefore, the zonal component of the diurnal tide at Maui is amplified by GWs. In the meridional component, the acceleration and the diurnal tide are in-phase above 92 km but are about 90 ∘ out of phase below. This 90 ∘ phase difference below 92 km implies that GWs tend to change the phase of the diurnal tide but have little effect on its amplitude. The vertical tilts of the horizontal wind and momentum flux structures are different, indicating that the GW effects on the tide are altitude dependent.
 The overall effects of GW forcing on the diurnal tide depend on the relative phase between the forcing and the tide [McLandress, 2002; Ortland & Alexander, 2006]. This is quantified by the ERF as described in section 2.2. Figure 5 shows the calculated ERFs based on the accelerations and mean tidal winds shown in Figure 4. Because the maxima of the ERF values are large (compared with the diurnal tide frequency of about 7.3 × 10− 5 s − 1), we also calculate the modified ERF using equation (10), shown in Figure 6. The main differences in the corrected ERFs are smaller values in the real parts below 85 km and more uniform imaginary parts at different altitudes.
 Figure 6 shows that the magnitudes of the real ERFs are less than 10 − 5 s − 1 below 90 km. They increase markably with altitude above 90 km, reaching maximum values of about 7 × 10 − 5s− 1 and 3 × 10 − 5 s − 1 in zonal and meridional directions, respectively. The negative values indicate that the GW forcing acts to increase the diurnal tide amplitude. The imaginary values of the ERFs are mostly positive for both zonal and meridional directions, indicating that the GW forcing acts to increase the phase of the diurnal tide.
 The time rates of change of the tidal amplitude and phase caused by GW forcing are shown in Figure 7. They are calculated with the corrected ERFs and equation (4). The effect on the amplitude is most significant above 90 km, where it can reach more than 80 ms − 1day − 1. Below 90 km, the effect is much weaker, with values varying between -15 and 10 ms − 1day − 1 for the zonal wind and between -2 and -6 ms − 1day − 1 for the meridional wind. For the phase, the rate of change is around 10 to 15 h day − 1 at most altitudes, except in the middle range where it is small for the zonal wind.
3.3 Seasonal Variation of the Meridional GW Forcing
 The same analysis is done on a monthly basis, with the exception that the compositing is done based on universal time instead of tidal phase, as described in section 2.1. The results are in Figure 8, which shows the same quantities as in Figure 3, but for 12 calendar months. For this calculation, we show only the meridional component because the meteor detections in the zonal direction at Maui are not enough for a reliable estimate. This is due to the anisotropic distribution of meteor detections explained in section 2.1 and shown in Figure 2. Although the uncertainties of the monthly mean meridional momentum flux estimates are larger than the 5 year mean, they are generally smaller than the amplitudes of the diurnal variation for all months. April has the largest uncertainty, whereas June has the smallest uncertainty.
 The most common feature of the monthly mean composite GW momentum flux is the downward phase progression, which reaffirms the tidal modulation of GW activities. This downward phase progression of the momentum flux was also reported by [Fritts & Vincent, 1987] based on the dual-beam radar observations. The diurnal components of GW momentum fluxes are characterized by a noticeable seasonal variation with relatively larger values in summer and smaller ones in winter. The maximum amplitude of the diurnal momentum flux reaches 12 m 2s − 2 in July and the minimum amplitude is less than 6 m 2s − 2 in November to January months.
 This seasonal characteristics of GW momentum flux are different from that of the diurnal tide. [Lu et al., 2011] have shown that the diurnal tide at Maui is strongest at equinoxes and weakest at solstices. This suggests that although the diurnal tide clearly modulates GW momentum flux by inducing its diurnal variation, the amplitude of the diurnal variation in momentum flux is not directly related to the amplitude of the diurnal tide. The amplitude of the diurnal variation of the momentum flux is likely related to the overall intensity of GWs, which, in turn, is strongly influenced by wave sources or background filtering effects, or both.
 The diurnal variations of the GW forcing and the diurnal tide in the meridional direction are shown in Figure 9. From March through November, the relationship between the forcing and the tidal wind is similar to that in the 5-year composite with a near 90 ∘ phase shift in the lower part of the layer. In the upper part, there are various degrees of shift in their relative phases. The relationship we observed in the 5-year composite can also be identified in most months, indicating that it is a persistent feature.
 The seasonal variation of the GW forcing on the diurnal tide is further illustrated in Figure 10 as a function of altitudes for the 12 calendar months. These are the same quantities shown in Figure 7 but for individual months. Below 90 km, there are significant variations of GW effect on the tidal amplitude. GWs tend to reduce the tidal amplitude in February, March, April, August, and November, and increase in other months. On average, the effect is slightly negative below 90 km, as shown in Figure 7. Above 90 km, the GW forcing on the diurnal tide amplitude is positive in most months, consistent with the 5-year composite. The GW effect on the tidal phase is uniformly positive; that is, GWs tend to increase the phase at all months throughout all altitudes.
 There has been an ongoing debate on the effects of GWs on the tides in the MLT. This analysis provides some observational evidence to help further understand this issue. The results show that GW forcing on the diurnal tide amplitude is almost always positive above 90 km and slightly negative below 90 km with month-to-month variations. The enhancement of the diurnal tide above 90 km is consistent with models that use Hines's GW parameterization scheme [Mayr et al., 1998; McLandress, 1998; England et al., 2006]. It is also consistent with [Liu et al., 2008], who used a compressible, nonlinear, two-dimensional GW model and reported that GW breaking not only accelerates the mean winds, but also increases the tidal amplitudes at various altitudes. On the other hand, the damping of the diurnal tide below 90 km is consistent with models using Lindzen's scheme [Meyer, 1999; Chang et al., 2011; Lu et al., 2012]. This suggests that there are differences in GW breaking characteristics at different altitudes, and they are only partly captured by different parameterization schemes.
 In another study, [Ortland & Alexander, 2006] reproduced the observed structure of the migrating diurnal tide in a mechanistic model by including GW-tide interaction. They found that whether GW forcing enhances or damps the tide highly depends on the source spectrum, which determines the altitude of wave breaking, momentum profile, and relative phase of GW forcing to the tide. The amplitude of the tide is usually increased when GW forcing is weak and decreased when it is strong. This leads to an enhancing effect on tidal amplitude at lower altitudes and damping effect at higher altitudes where GW forcing becomes larger. This, however, appears to be opposite to our observational results. Both their model results and our observation show that the GW forcing and meridional wind are nearly in quadrature, but the phase difference is opposite. This observation shows that GW forcing lags the diurnal tide, but in their model it is leading the diurnal tide.
 Recently, [Lieberman et al., 2010] analyzed the effects of GWs on migrating diurnal tides by using measurements of temperature, wind, and geopotential height from the TIMED satellite in the 80 to 100 km region. The effects of GWs are estimated as a residual (wave drag) in the momentum equations after all other large-scale terms, the momentum tendency, Coriolis force, pressure gradient, and advection are calculated directly from measurements. An important finding is that the zonal momentum residual leads the zonal wind by one-quarter cycle, and the meridional momentum residual is in antiphase with the meridional wind. Here the phase relationships are again different from our results, which show both zonal and meridional GW forcing lag the diurnal tide by about quarter cycle. However, it is important to note, as [Lieberman et al., 2010] pointed out, that their residual forcing includes not only GWs, but also other mechanisms that are not resolved by the satellite measurements, including nonlinear wave interactions and advection by the meridional wind.
 The nominal value of our calculated ERF is around 5 × 10 − 5 s − 1 = 4.3 day − 1 (Figure 5) and is in good agreement with those obtained by [Khattatov et al., 1997], [Xu et al., 2009a], and [Lieberman et al., 2010] from HRDI and SABER satellite observations. These values are, however, one order of magnitude larger than that in the whole atmosphere community climate model (WACCM) [Lu et al., 2012] and even larger than the ERF applied in the global scale wave model (GSWM) [Hagan et al., 1999]. Furthermore, the large values of the imaginary part of ERF indicate that the relative phase between GW forcing and the diurnal tide is an important factor that should be taken into account in prescribing GW forcing in model simulation.
 Based on 5 years of meteor radar observation at Maui, Hawaii, we have derived for the first time a climatological relationship between the diurnal tide and the diurnal variation of GW momentum flux and forcing in the mesopause region. Both the 5-year composite and monthly composite show clear indication of tidal modulation of GW forcing. The 5-year composite ERF shows strong GW effects on both the amplitude and the phase of the diurnal tide. The magnitude of the forcing is in agreement with those derived indirectly from satellite measurements such as in [Lieberman et al., 2010]. The magnitude and phase relationships also vary with altitude and season. GWs tend to enhance the diurnal tide above 90 km and slightly damp the tide below 90 km. Throughout all altitudes between 80–100 km, GWs tend to increase the phase of the diurnal tide with varying magnitude, which is different from most model predictions.
 Large discrepancies exist in the magnitude of GW forcing when compared with those used in WACCM and GSWM. In GCMs, GWs parameterized with Lindzen's scheme tend to damp the tide, whereas with Hines's scheme, they tend to enhance the tide. The vertical variation of GW forcing in our observation suggests that different GW parameterization schemes may capture only part of the GW dissipation process and are suitable for only part of the region or time periods.
 For most GCMs, GW parameterizations are often tuned to match observed zonal mean winds or tidal structure. Parameterization schemes with different assumptions on the source spectra and how GWs are filtered and dissipate after saturation or breaking can all be tuned to match zonal mean observations reasonably well. However, because of differences in the description of GW breaking process, the effects of GW interactions with tides are different and can result in different diurnal variation of GW forcing. Comparison of model results with observations in this aspect is an effective way to check whether the GW dissipation process is properly represented in models.
 This study also shows that meteor radar is very useful for observational study of GW-tide interactions. Its long-term, continuous observations can provide robust estimate of such interactions. Although 5 years of meteor radar from Maui is used in this study, newer powerful meteor radars that can detect more meteor trails can accumulate enough data within a shorter time period for such investigation. It is also important to perform such studies at different geographic locations under different tidal and GW conditions.
 The authors thank the two anonymous reviewers for their valuable comments that helped improve the manuscript. This work was supported by National Science Foundation grant AGS-1110199. X.L. is supported by NSF grant AGS-0737656.