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 Gravity wave forcing (GWF), which is the primary driver of the mesospheric and lower thermospheric (MLT) circulation, is difficult to measure directly. In this work, the zonal mean GWF at extratropical MLT is deduced from measured winds using momentum balance. With the GWF dominating in the MLT, the zonally averaged zonal momentum equation can be simplified to a balance relation between the GWF and the Coriolis force in the extratropics. The meridional advection of zonal momentum makes a higher order contribution to the momentum balance, especially at places where the GWF maximizes. This method is tested with WACCM3 model and preliminary results are obtained from wind measurements by CSU Na lidar and TIMED/TIDI.
 Gravity wave forcing (GWF) plays a key role in the mesosphere and lower thermosphere (MLT) by controlling the circulation and the wind, thermal and compositional structures of this region [e.g., Garcia and Solomon, 1985]. In spite of their importance, direct measurement or inference of the gravity wave (GW) impacts, namely the mean flow acceleration and eddy mixing from GW breaking, have been difficult. There were a few significant research efforts in this respect. Alexander and Rosenlof  calculated the residual forcing due to GWs in the stratosphere from UARS data and UKMO results. For that calculation, the diabatic heating rate is needed as an input for deriving residual circulation velocities and the total zonal momentum force. The force from dissipation of resolved waves is determined from UKMO fields and then substracted from the total zonal momentum to obtain the force of unresolved GWs. Khattatov et al.  derived an effective eddy diffusion coefficient by fitting the diurnal tide derived from UARS/HRDI measurements to a linear tidal model. More recently, J. Xu et al. (Estimation of the equivalent Rayleigh friction in MLT region from the migrating diurnal tides observed by TIMED, submitted to Journal of Geophysical Research, 2009) deduced an effective drag coefficient from temperature measurements by the Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) instrument and zonal and meridional wind measurements by the TIMED Doppler Interferometer (TIDI) instrument on board the NASA Thermosphere Ionosphere Mesosphere Energetics and Dynamics (TIMED) satellite. Zhu et al.  developed a diagnostic method to derive all the dynamical and chemical tracer fields from temperature and ozone measurements, based on a globally balanced 2D diagnostic model. The method uses the linear saturation theory to relate GWF and eddy diffusion, and the net heating from GW breaking is also considered. The primary balance relationship used therein is a balance between the meridional gradient of GWF and the vertical gradient of total heating.
 In this study we examine the momentum balance in the MLT region using scale analysis and model simulations, and the feasibility of applying this balance relation to deduce GWF from ground-based and satellite wind measurements. It is well known that the gradient wind approximation [Randel, 1987] in the meridional direction still holds well in the MLT at middle and high latitudes [Lieberman, 1999; Oberheide et al., 2002], so that the zonal wind can be calculated from the latitudinal gradient of geopotential (thus temperature). In the zonal direction the GWF becomes increasingly important in the MLT, and its relative significance compared with other terms is studied here in the zonal mean sense based on scale analysis, and then evaluated using results from the NCAR Whole Atmosphere Community Climate Model (WACCM3). The applicability of the balance relation for deriving GWF from wind measurements from both ground-based and satellite measurements is demonstrated. WACCM3 solves the primitive equations from the Earth surface to the lower thermosphere (5.1 × 10−6 hPa, approximately 140 km), and is thus appropriate for this study. GWF in WACCM3 is parameterized based on the linear saturation theory [Lindzen, 1981]. Detailed information of WACCM3 can be found by Garcia et al. . The ground-based wind observations used are from the monthly mean winds from the two-beam Na lidar at Colorado State University (41°N, 105°W), which has been observing full diurnal cycles of the mesopause region temperature and horizontal wind in campaign mode since May 2002, weather permitting [She et al., 2004; Yuan et al., 2008]. TIDI zonal mean winds are also used to infer GWF. The TIDI zonal mean winds are obtained by removing the tidal components and then averaging the residual winds over all longitudes. Details of the TIDI instrument and TIDI tidal analysis are given by Killeen et al.  and Wu et al. .
 The derivation of the momentum balance in the presence of strong GWF and the test of the balance using WACCM3 are presented in Section 2. The application of the balance relation to deducing GWF from wind measurements is discussed in Section 3. Conclusions are given in Section 4.
2. Momentum Balance in the Presence of Strong Gravity Wave Forcing
 The full momentum equation in the zonal direction is:
where D/Dt is the Lagrangian time derivative, u and v the zonal and meridional winds, f the Coriolis parameter, r the Earth radius, ϕ the latitude, λ the longitude, Fx the total body force, and ν the total viscosity (eddy plus molecular) coefficient. From standard scale analysis [e.g., Holton, 2004], the Coriolis force and the geopotential gradient terms in (1) are ∼10−3 ms−2 and dominate over other terms in the lower atmosphere. In the MLT region, however, the zonal mean zonal GWF is on the same order of magnitude (∼100 ms−1 d−1 or 10−3 ms−2), although locally the GWF may or may not be comparable to the Coriolis force given the large spatial and temporal variability of GWF. With ν ∼ 100 m2 s−1 and ∂2u/∂z2 ∼ 5 × 10−7 m−1 s−1 (assuming a 50 ms−1 change within 10 km) in the MLT, the viscous term is thus ∼5 × 10−5 ms−2. It is much smaller than the zonal mean GWF and will be ignored in the following discussion. Therefore the primary force balance in the zonal direction in the MLT is between the zonal mean GWF and the Coriolis force (the geopotential gradient term becomes zero when integrated along the longitude circle):
with the assumption that ion drag is not important below 100 km so that x ≈ xGW. This simple force balance allows the determination of zonal mean GWF solely from the zonal mean winds. It should be noted that the advective terms in the full zonal mean momentum equation
are from mean and eddy advection, and their impact on the mean flow is generally secondary compared to the GWF in the MLT region, unlike that in the stratosphere. This will be further examined using numerical model results.
 To test this momentum balance and to better understand the relative significance of the terms in (3), WACCM3 simulation results under December solstice conditions are examined. The simulation results first confirm that the zonal mean of the gradient wind is in good agreement with the zonal mean of the actual zonal wind below 100 km in the extratropics, with the maximum difference less than 5 ms−1 (10%) (not shown). Because of this good agreement, is interchangeable with the gradient wind in the following analysis. Figures 1a and 1b show the parameterized zonal mean GWF in WACCM and the zonal mean GWF obtained from (2), respectively. It is seen that the estimated GWF reproduces the general morphology of the parameterized GWF in both hemispheres in the MLT region. The two are in good quantitative agreement except at places where the parameterized GWF maximizes (80–85 km around 50° in the summer hemisphere and around 70 and 80 km at mid to low latitudes in the winter hemisphere). The inferred GWF there is weaker than the maximum GWF. This discrepancy at the maximum GWF is reduced if the meridional advection term is included:
Figure 1c shows excellent agreement with Figure 1a. Therefore, by comparing to WACCM simulations, (2) is a good first-order approximation to the momentum balance in the MLT region, and the meridional advection term can provide correction to this approximation. The advection term includes both advection by the mean flow and the eddy advection term . It is worth noting that the gradient wind balance could still be reasonable in the zonal direction for the eddy components [Oberheide et al., 2002]. Therefore, the eddy advection term can be approximated using both zonal and meridional gradient winds. Equations (2) and (4) allow us to infer zonal mean GWF from meridional wind measurement and zonal wind or temperature measurement in the MLT region in the extratropics. This is valuable given the importance of this quantity in the MLT and the difficulty of measuring it directly.
3. Gravity Wave Forcing Inferred From Observations
 The mean zonal and meridional wind climatology between 83–100 km above Fort Collins is obtained from multiple years of 24-hour Na-lidar measurements at CSU [Yuan et al., 2008]. Afforded by the 24-hour measurements of this lidar system, tidal waves have been removed in processing the data to obtain the mean winds. The mean winds can still have contributions from planetary waves (PWs), though the magnitudes of the traveling PWs should be significantly reduced after averaging over each month from multiple years. The quasi-stationary PWs (QSPWs) are usually not very large above 80 km. Therefore, the monthly climatological winds obtained should be approximately equal to the zonal mean winds, and (2) can be applied to derive the zonal GWF climatology. It should be noted that the method can introduce large error if applied during periods with enhanced QSPWs (e.g. stratospheric sudden warming), because of the large difference between temporal mean and zonal mean winds.
Figures 2a and 2b are the monthly climatological zonal and meridional winds, respectively, from the CSU lidar measurements, and Figure 2c is the zonal GWF derived from these winds using (2). The GWF is eastward during most of the summer months (April–October) between 83–100 km, and is westward in part or all of that height range during the winter months. The eastward GWF during the summer months is generally stronger than the westward GWF during the winter months. The maximum eastward GWF occurs in May at ∼87 km with a magnitude of ∼130 ms−1 d−1, and the maximum westward GWF is found in January at 83 km with a magnitude of ∼100 ms−1 d−1. The GWF changes rapidly from April to May within the observed altitude range. As argued by Liu and Roble , a change of GWF is a likely cause of the atomic oxygen spring equinox transition, and this derived GWF lends support to that argument. By comparing the GWF with the monthly zonal wind in Figure 2a, it is clear that the GWF generally corresponds well to the vertical wind shear around the wind reversal. The rapid change in the mean zonal wind after spring equinox is also consistent with the rapid increase in GWF.
 The derived GWF also supports model studies of GWF, which are mostly parameterized and tuned in global models so that the zonal winds match observations [e.g., Garcia and Solomon, 1985] (also Figure 1). The model results suggest that the GWF in the summer hemisphere is strong and has a prominent peak around the mesopause region, while in the winter hemisphere the GWF is relatively weak and more uniformly distributed over altitude. The magnitudes of the GWF around June and December/January shown in Figure 2c are similar to those from WACCM (Figure 1a) at 41°S and 41°N, respectively.
 At 41° latitude, a 1 ms−1 error in the mean meridional wind translates to an error of 9.5 × 10−5 ms−2 or 8.2 ms−1 d−1 of mean GWF, according to (2). The error bar of the monthly mean meridional wind from the lidar measurement is ±2 ms−1 between 85–95 km, mainly due to geophysical variability, and increases to ±5 ms−1 at 100 km with increasing measurement errors [Yuan et al., 2008, Figure 2]. Therefore, the error of GWF between 85–95 km is ±16.4 ms−1 d−1 and increases to ±41 ms−1 d−1 at 100 km. An additional error comes from neglecting the meridional advection term, especially at altitudes where the GWF maximizes, as discussed in the previous section. Calculation of this term requires knowledge of the meridional shear of the zonal wind. This information may be derived from networks of ground based measurements at different latitudes, or the ability to make wind measurements at multiple azimuth angles at appropriate accuracy.
Equation (2) can also be applied to wind measurements from satellite to estimate zonal mean GWF and its latitude distribution. Figures 3a and 3b are the zonal mean zonal and meridional wind, respectively, calculated from TIMED/TIDI measurements. These are 61-day average zonal mean winds centered on day 352 of 2005 [Wu et al., 2008]. The zonal mean zonal wind has been compared with the zonal mean geostrophic zonal wind derived from TIMED/SABER temperature measurements for the same time period. It is found that the two are in general agreement at middle to high latitudes, though the TIDI zonal wind displays a westward bias of about 10 ms−1 compared with the geostrophic zonal wind (not shown). Figure 3c is the zonal mean GWF derived from the winds using (2). The GWF in the southern (summer) hemisphere is eastward between 90–100 km (mid-latitudes) or 85–100 km (poleward of 40°S), and the maximum eastward GWF at southern mid-latitudes is ∼75 ms−1 d−1. In the northern (winter) hemisphere mid-latitudes the GWF is westward between 85–95 km and becomes weakly eastward above, while westward above 85 km poleward of 50°N. At mid-latitudes the westward GWF is relatively weak (below 50 ms−1 d−1), and it reaches larger values (about 100 ms−1 d−1) at higher latitudes. The values and the vertical variation of GWF at northern mid-latitudes are in general agreement with those obtained from CSU lidar measurements (Figure 2c) for winter (December and January). The values at southern mid-latitudes above 85 km are weaker than those from WACCM (Figure 1a). The GWF below 85 km in the southern hemisphere is significantly different from the model climatology, which could result from geophysical variability and/or bias in TIDI analysis.
 The precision of the TIDI zonal mean wind is 1.5 ms−1 [Wu et al., 2008], which translates to 3.3 ms−1 d−1 at 10° and 17.8 ms−1 d−1 at 70°. In addition to the error from neglecting meridional advection, there could also be a systematic bias in meridional wind (as well as zonal wind) due to uncertainty of the zero wind. Given the dependence of GWF on the mean meridional wind, future studies will seek better methods to determine the zero meridional wind, so that the latitudinal distribution and temporal variability of GWF can be better quantified.
 In this work we have shown that in the extratropical MLT the primary zonal balance is between the zonal mean GWF and the zonal mean Coriolis force. The meridional advection of zonal momentum provides higher order correction to this balance relationship, especially at places where the GWF maximizes. This is probably because maximum GWF corresponds to a large shear of zonal wind and zonal wind reversal, where PWs are likely to interact with the mean wind. This simple balance relation affords the deduction of GWF, at least to leading order, from wind measurements in the MLT at extratropical latitudes. This is in contrast to that in the stratosphere, where the forcing by the PWs is most significant and the calculation of GWF requires knowledge of these PWs and the residual circulation [e.g., Alexander and Rosenlof, 2003].
 The feasibility of this method is demonstrated using climatological zonal and meridional winds from CSU Na lidar and wind measurements from TIMED/TIDI. The estimated zonal GWF from these measurements show weak westward forcing during winter and strong eastward forcing during summer between 85–100 km, and their peak values correspond to large vertical shear and reversal of mean zonal wind. The GWF estimated from the CSU wind climatology also displays rapid change from westward to eastward around April.
 It is important to extract and remove the tides and PWs when applying this method. For ground-based measurements, the removal of these waves is essential for the temporal mean to be a good approximation of the zonal mean. It is thus important to make 24-hour measurements for tidal removal, and to either have networks of ground-based measurements to extract PWs or perform long time averaging at a single site to reduce PWs. Further, simultaneous measurements of the meridional gradient of zonal wind, either from multiple sites or by wind measurements at multiple azimuth angles, will enable calculation of meridional advection of zonal momentum and thus refinement of the GWF calculation. It is thus desirable for future studies to apply this method to more ground-based facilities that are capable of 24-hour wind measurements (e.g. lidar, meteor radar) to better quantify GWF at extratropical latitudes.
 Global wind measurements from TIDI certainly have the advantage of making it possible to deduce GWF over extended extratropical latitudes, although the slow precessing rate of the satellite orbit requires multiple days (∼60 days) of data for complete local time coverage to separate tides, PWs and zonal mean winds. The extracted wind tides and PWs from TIDI and gradient wind components derived from SABER will be used to calculate the meridional advection of zonal momentum for higher order correction of the GWF in future studies. The key problem in applying this method to TIDI wind measurement, however, is that systematic bias may stem from uncertainty in determining the zero wind. Removal of the systematic bias requires careful cross-validation with ground-based measurements.
 We thank Rolando Garcia for helpful discussions. HLL's effort is supported in part by the Office of Naval Research (N00014-07-C-0209). QW's effort is supported by a NASA grant NAG5-5334 to NCAR. The National Center for Atmospheric Research is operated by the University Corporation for Atmospheric Research under sponsorship of the National Science Foundation.