We used the Japanese Atmospheric GCM for Upper Atmosphere Research (Watanabe and Miyahara, 2009) as a forecast model, which we refer to as JAGUAR in this paper. This model has a high model top of approximately 150 km and is based on the T213L256 middle atmosphere GCM developed for the Kanto project (Watanabe et al., 2008) and the Kyushu-GCM (e.g., Yoshikawa and Miyahara, 2005). This model uses important physical parameterizations for the MLT region such as radiative transfer processes, including non-LTE and solar radiative heating due to molecular oxygen and ozone. The effects of ion drag, chemical heating, dissipation heating, and molecular diffusion are also parameterized in the model. In this study, a standard-resolution JAGUAR with a triangularly truncated spectral resolution of T42 corresponding to a horizontal resolution of about 300 km (a latitudinal interval of 2.8125∘) is used for the assimilation. The model has 124 vertical layers with a uniform vertical spacing of approximately 1 km in the middle atmosphere and 100–800 m in the troposphere (see Fig. A1 of Watanabe et al., 2015, for the vertical layers). Unlike a high-resolution JAGUAR, which resolves a certain portion of gravity waves (Watanabe and Miyahara, 2009), gravity waves are sub-grid-scale phenomena for a standard-resolution JAGUAR. For this reason, both orographic (McFarlane, 1987) and non-orographic (Hines, 1997) gravity wave parameterizations are used. The wave source distribution of non-orographic parameterization is given based on the results of a gravity-wave-resolving high-resolution GCM (Watanabe, 2008), and the intensity of the source is treated as one of the tuning parameters. Horizontal diffusion is set as an e-folding time of 0.9 d for the minimum resolved wave length in the troposphere and stratosphere, and it exponentially increases with increasing height over the MLT region. In this study, the vertical profile of horizontal diffusion above the stratopause is also treated as one of the tuning parameters. The monthly ozone mixing ratio from United Kingdom Universities Global Atmospheric Modeling Programme (UGAMP; Li and Shine, 1999) and monthly sea surface temperature and sea ice concentration from the Met Office Hadley Centre sea ice and sea surface temperature dataset (HadISST; Rayner et al., 2003) are linearly interpolated in time and used as boundary conditions.

The 4D-LETKF (Miyoshi and Yamane, 2007) is used as a data assimilation method. This method is an extension of the 3D-LETKF (Hunt et al., 2007), which includes the dimension of time (4D ensemble Kalman filter; Hunt et al., 2004). The base of the program used in this study has already been applied to many types of forecast models, such as the Global Spectral Model (GSM; Miyoshi and Sato, 2007), the Atmospheric GCM for Earth Simulator (AFES; Miyoshi et al., 2007), and the Non-hydrostatic Icosahedral Atmospheric Model (NICAM; Terasaki et al., 2015).

This section introduces the formulas used in the 4D-LETKF. The analyses, forecasts, and observations are denoted by xa, xf, and yo, respectively. The optimal value of xa is derived from xf and yo by the following equation: 1xa=xf+Kyo-Hxf=xf+Kd, where K is a weighting function, d(≡yo-Hxf) is the innovation, and H is an observation operator that converts the model space variables into observational space variables. For assimilating MLS retrievals, the observation operator includes the averaging kernel and the spatial operator. The second term on the right-hand side represents data assimilation corrections (i.e., increments). Using the differences from the true value (xt), δxa=xa-xt, δxf=xf-xt, and δyo=yo-Hxt, Eq. (1) can be rewritten as follows: 2δxa=δxf+Kδyo-Hδxf=I-KHδxf+Kδyo, where I is an identity matrix. The analysis error covariance is defined as 3Pa≡〈δxa(δxa)T〉=I-KHPfI-KHT+KRKT, where Pf≡〈δxf(δxf)T〉 is the forecast error covariance and R≡〈δyo(δyo)T〉 is the observation error covariance. The correlation between the forecast error and the observation error is supposed to be zero (〈δxf(δyo)T〉=0). The optimal xa should minimize the summation of the analysis error covariance (tr(Pa)). This means that 4∂∂KtrPa=0. Solving Eq. (4) with respect to the weight matrix K yields 5K=PfHTHPfHT+R-1, and the analysis xa is derived by Eq. (1). The weight matrix K is called the “Kalman gain”. Using K, the analysis error covariance Pa is rewritten as 6Pa=I-KHPf, which gives the relationship K=PaHTR-1.

The size of Pf and Pa is the square of the degree of freedom in the model. Thus, for systems with huge degrees of freedom, such as GCMs, the calculation of Pf and Pa requires a large computational cost. This problem is avoided by replacing the forecast, analysis, and each error with the mean and variance for m members of an ensemble. This is called the ensemble Kalman filter (EnKF; Evensen, 2003). The ensemble mean x‾ and background error covariance matrix P are written as follows: 7x‾=1m∑i=1mxi, 8P=〈δx(δx)T〉≈1m-1∑i=1m(xi-x‾)xi-x‾T=1m-1∑i=1mδxiδxiT. However, with a limited number of ensembles, the forecast error tends to be underestimated in a system with a large degree of freedom. A variety of methods have been proposed to overcome this problem (e.g., Whitaker et al., 2008). In our study, the forecast ensemble perturbation (δxf) is multiplied by the factor F, which is a little larger than 1 (F=1+Δ; Δ is called an “inflation factor”): 9δxif←1+Δδxif is employed, and the Δ value is optimized.

The Kalman gain is simply written by using E, which is the root of P. Using 10m-1E≡δx1|…|δxm, 11K=PfHTHPfHT+R-1=EfHEfTHEfHEfT+(m-1)R-1 is derived. Further manipulation yields another expression of K: 12K=Ef(m-1)I+HEfTR-1HEf-1HEfTR-1. To reduce the calculation cost, the inverse matrix of Eq. (11) or Eq. (12) with a smaller size is chosen. Usually, as the number of the ensemble is much smaller than the number of observations, Eq. (14) is used.

The LETKF treats the analysis error covariance matrix, 13P̃a≡(m-1)I+HEfTR-1HEf-1, in the ensemble space. The relationship between this matrix in the ensemble space and the analysis error covariance matrix in the model space is expressed as Pa=EfP̃a(Ef)T, and Ea=Ef(P̃a)12 is the ensemble update. In this way, the analysis xia is obtained as 14xia=x‾a+δxia=x‾f+δxia+EfP̃aHEfTR-1d. Using the shape of the N×m matrix, where N is the number of the variables, Eq. (14) is written as 15[x1a|…|xma]=[x‾f|…|x‾f]+EfW, where 16W=(P̃a)12+P̃aHEfTR-1d|⋯|P̃aHEfTR-1d. To avoid unrealistic correction caused by remote observations with the use of a limited ensemble size, a weighting function based on the distance from the analysis point is multiplied by the observation error. This method is called “localization”. The calculation is independently performed at each grid so it can be performed in parallel with high computational efficiency. The length of localization is also a setting parameter of the data assimilation system, and the sensitivity of the assimilation performance to this parameter is examined in Sect. 3.3.2.

When an analysis ensemble is derived, each ensemble takes its own time evolution calculated by the forecast model, and the forecast at the next step is derived by 17xi,t+1f=Mxi,ta. In this way, the forecast and analysis steps are repeated through the data assimilation cycles.

Here we extend to a 4D analysis. By the modification of the observation operator, the observation at any time (j2) can be assimilated as the information on time development from the target time (j1). One such assimilation is called the 4D-EnKF (Hunt et al., 2004).

The forecast at the time step j1 is written as a weighted mean of forecast ensembles: 18xj1f=[x1,j1f|⋯|xm,j1f]w≡Xj1fw. The weighting matrix w is unknown but is calculated by the pseudo-inverse matrix: 19w=(Xj1f)TXj1f-1Xj1fTxj1f. On the other hand, the (unknown) forecast at the time step j2 is also written as a weighted mean of forecast ensembles: 20xj2f=Xj2fw. Substituting Eq. (16) into this equation, the following formula is obtained: 21xj2f=Xj2fw=Xj2fXj1fTXj1f-1Xj1fTxj1f. Finally, the modified observation operator to assimilate the observation at time step j2 to the forecast at time step j1 is written as follows: 22H′=HXj2fXj1fTXj1f-1Xj1fT. Appendix A explains that directly assimilating the observation at a certain time step by the modified observation operator is the same as assimilating at the time of observation and then calculating the time evolution after the assimilation. Thus, this method is regarded as a kind of 4D assimilation including the information on the time development. Another advantage of this method is that future observations can be assimilated, as it is similar to the Kalman smoother. In this study, this extended LETKF with 4D assimilation is used. The time interval (called the “assimilation window”) between the observations and the analysis is one of the setting parameters.

The EnKF initial condition is obtained using the time-lagged method as follows. First, a 6-month free run is performed from a climatological restart file for 1 June. The results from the free run over about 10 d with a center of 1 January are used as the initial condition for each ensemble member on 1 January. For the runs with 30 ensemble members, 30 initial conditions at a time interval of 6 h are used. For runs with 90 and 200 ensemble members, the time intervals for the initial conditions are taken as 4 and 2 h, respectively. The analysis data for the first 10 d of the assimilation are regarded as a spin-up and are hence not used to examine the assimilation performance.

As already mentioned, the parameter set of data assimilation usually made for the troposphere and stratosphere is not necessarily appropriate for the analysis when the MLT region is included. This is because the dominant physical processes and scales of motions could be different (e.g., Shepherd et al., 2000; Watanabe et al., 2008). This section describes the parameters that should be optimized for the data assimilation system for the whole neutral atmosphere from the troposphere up to the lower thermosphere. The relevance criteria of the data assimilation for each parameter are also described.

The parameters included in the data assimilation system are divided into two categories. The first category includes two parameters describing the GCM: the horizontal diffusion coefficient and the factor of gravity wave source intensity in the gravity wave parameterization. The second category includes five parameters related to the data assimilation: the degree of gross error check, the localization length, the inflation factor, the length of assimilation window, and the number of ensembles. The sensitivity of the performance of assimilation is tested by changing one parameter among the standard set of the parameters as shown in Table 1. Finally, the performance of the assimilation with the best set of parameters is confirmed.

The criteria used for the evaluation of the data assimilation for each parameter setting are observation minus forecast (OmF) and observation minus analysis (OmA) in the observational space. One more criterion for examining the quality of data assimilation is χ2, which was introduced by Ménard and Chang (2000): χ2=trYYT,Y=1my-Hx‾fHEfHEfT+R-12. The parameter χ2 describes the consistency between the innovation with the covariance matrices for the model forecast and the observations. The χ2 values should be close to 1 if the background and observation errors are properly specified in the assimilation system. The χ2 values higher (lower) than 1 mean that the background or observation error has been underestimated (overestimated) against the innovation in the observational space.

To reduce the model bias in the mesosphere, the vertical profile of the horizontal diffusion coefficient and the gravity wave source intensity in the non-orographic gravity wave parameterization are examined by comparing observations in the summertime Antarctic mesosphere. Here, the zonal wind observed by an MST radar called the PANSY radar in the Antarctic (Sato et al., 2014) is used as a reference of the mesospheric wind. Note that the temporal and longitudinal variation of the dynamical field is relatively small in January and February in the summertime Antarctic mesosphere. The model performance may depend on the parameters describing the MLT processes, although we used default values of the model for this study. For example, climatological concentrations of chemical species are used for the calculation of the radiative heating rate, although the O3 and NO concentrations are affected by the solar activity in a short timescale. The effects of ion drag are neglected because it is important mainly above the height of ∼ 200 km. The chemical heating caused by the recombination of atomic oxygen is incorporated using a global mean vertical profile of its density, and we neglected spatial and temporal changes.

According to the Aura MLS data quality document (Livesey et al., 2018), the MLS temperature data have a bias compared to the SABER ones as a function of the pressure, which is -5 to +1 K for pressure levels of 1–0.1 hPa, -3 to 0 K for 0.1–0.01 hPa, and -10 K for 0.001 hPa. Thus, before performing the data assimilation, the MLS observation bias was removed as much as possible. Previous studies treated the bias correction in various ways: Hoppel et al. (2008) used bias-corrected SABER (v.1.06) data with the MLS data (v.2.2). Eckermann et al. (2009) used SABER (v.1.07) and MLS (v.2.2) temperatures for their assimilation in which the SABER (v.1.07) temperatures at altitudes below 2.7 hPa were bias-corrected using the mean difference from MLS (v.2.2), and the MLS temperatures above 2 hPa were bias-corrected using the mean difference relative to other satellite, suborbital, and analysis temperatures. Note that these studies used different versions of the SABER data (v.1.0X) from that we used (v.2.0). Pedatella et al. (2014b) did not perform the bias correction. Pedatella et al. (2016) adjusted the SABER temperatures to account for the bias between SABER (v.2.0) and MLS (v.2.2) temperatures. Eckermann et al. (2018) performed a bias correction for the MLS temperatures (v.4.0) above 5 hPa using the mean difference between MLS and SABER (v.2.0) temperatures, as well as the SABER temperatures for the pressure levels from 68 to 5 hPa using the mean difference between MLS and SABER. McCormack et al. (2017) and Pedatella et al. (2018) do not state explicitly whether or not a bias correction is applied, so it is not clear which bias correction was performed in their studies. We confirmed that the bias estimated in this study is similar to the globally averaged mean differences between MLS temperature and other correlative datasets shown in the data quality document (Liversey et al., 2018) at each height.

In this study, the MLS observation bias is first estimated as a function of the calendar day at each latitude and each pressure in a range of 177.838 to 0.001 hPa. As the reference for the correction, we used the TIMED SABER temperature data (v.2.0), which are considered to have little observation bias, at least in the altitude range from 85 to 100 km, as confirmed by Xu et al. (2006), who used data from the sodium lidar at Colorado State University, providing the absolute value of the temperature. Xu et al. (2006) attributed the disagreement below 85 km to high photon noise contaminating the lidar observations. Thus, we used the SABER temperature data for the Aura MLS bias estimation in the height range of 10–100 km.

The observation view of SABER is altered every ∼ 60 d by switching between northward (50∘ S–82∘ N) and southward (82∘ S–50∘ N) view modes. Thus, the latitudinal coverage of the SABER data is narrow compared to the MLS data. However, both datasets overlap for a long time period sufficient for statistical comparison between them.

The bias is estimated using data from 2005 to 2015. The original vertical resolution of MLS (∼ 1–5 km) is coarser than that of SABER (∼ 0.5 km). First, the vertical mean of the SABER data corresponding to the vertical resolution at each of the 55 height levels of MLS data is obtained. Next, by linear interpolation, data at the grid of 25∘ (longitude) × 5∘ (latitude) at a time interval of 3 h are made for MLS and SABER data. The MLS and SABER data made at a 3 h interval have a lot of missing values because observations are sparse. These missing values are not used for the bias calculation. This means that the bias was estimated using MLS and SABER data at nearly the same local time. Whereas the longitudinal variation of the bias is small (1 K at the most), there is large annual variation in addition to the dependence on latitude and height. Thus, the MLS bias Tbias is obtained as a function of the latitude (y), height (z), and calendar day (t): Tbiasy,z,t=Tmeany,z+Ay,zcos⁡2πt365+By,zsin⁡2πt365, where Tmean represents the annual mean, and the coefficients A and B are estimated by the least-squares method.

Figure 5 shows the estimated MLS bias as a function of calendar day and latitude at 10, 1, 0.1, and 0.01 hPa. The MLS bias shows a strong dependence on the altitude and latitude and has an annual cycle with amplitudes of 2 to 4 K. Figure 6 shows the vertical profiles of global mean Aura MLS bias along with the bias reported in the data quality document (Liversey et al., 2018). For example, the global mean MLS biases estimated by the present study are -1.6±0.7 K at 56.2 hPa, 2.0±1.4 K at 1 hPa, and -3.8±1.1 K at 0.316 hPa. These are comparable to the biases described in the data quality document, which are -2 to 0 K at 56.2 hPa, 0 to 5 K at 1 hPa, and -7 to 4 K at 0.316. This consistency indicates the validity of using the SABER data as a reference to estimate the MLS bias.

To evaluate the effect of the bias correction, data assimilation was performed using the MLS data with and without bias correction. Figure 7 compares the latitude and pressure section of the zonal mean temperature and zonal wind between the two analyses. The difference in zonal mean temperature between the two (Fig. 7c) resembles the corrected bias (Fig. 7d). In contrast, the difference in the zonal mean zonal wind is not very large (Fig. 7g). This is because the latitudinal difference in the bias, which largely affects the zonal mean zonal wind through the thermal wind balance, is not large compared to the vertical difference. In our study, the bias correction of the MLS data is made before the data assimilation, as in standard assimilation parameter setting.

A series of sensitivity tests were performed to obtain the best values of each parameter in the data assimilation system with 30 ensemble members. This number of members is practical for the data assimilation up to the lower thermosphere with current supercomputer technology. The examined assimilation parameters are the gross error coefficient, localization length, inflation coefficient, and assimilation window length. The best parameter set obtained by the sensitivity tests is denoted by Ctrl in Table 1. There are six assimilation parameters to be examined. We performed an assimilation run with almost all combinations of the parameters. Several parameter settings did not work due to computational instability. We found a parameter set that provides the best assimilation results in our system. This best parameter set is placed as the control setting (Ctrl), and the parameter dependence of the assimilation performance is examined by using the results in which one of the parameters is changed from the Ctrl set. Section 3.3.5 gives a short summary of the data assimilation setting optimization for 30 ensemble members.

The EnKF statistically estimates the forecast error covariance using ensembles. A large ensemble size (i.e., a large number of ensemble members) is favorable because it reduces the sampling error of the covariance and improves the quality of the analysis. However, the ensemble size has a practical limit related to the allowable computational resources. An ensemble size of ∼ 30 is usually used in operational weather forecasting. Here, the minimum optimal ensemble size is estimated by performing additional experiments with 90 (M90) and 200 (M200) members and comparing them with the Ctrl experiment of 30 members (M30, Ctrl). No assimilation parameters, except for the ensemble size, are changed in the M90 and M200 experiments (see Table 1). Note that the best values of the assimilation parameters for the larger ensemble sizes may be different from those for Ctrl. For example, a larger ensemble size may allow for a larger localization length. However, further investigation was not made because a high computational cost is required.

Figure 16 shows the vertical profiles of the mean OmF, OmA, and χ2 for the M30 (Ctrl), M90, and M200. The OmFs for the M90 and M200 are significantly smaller than for the M30, particularly below 0.1 hPa (by ∼ 50 %), while the OmAs are comparable for the three runs. The difference between the OmFs of the M90 and M200 runs is small. Although the χ2 values are ∼ 6–17 for M30, they are ∼ 3–4 for M90 and ∼ 1–2 for M200, which are close to the optimal values of χ2. The reduced values of χ2 by increasing the ensemble size are remarkable compared with those by optimizing the other parameters (Sect. 3.3). However, the M90 and M200 both require such large computational costs, as already stated, that they are not available for long-term reanalysis calculations. The time series obtained from the M90 and M200 are also shown in Fig. 15. Both time series agree well with the MERRA-2 time series and do not have even a slight warm bias like that observed at 0.1 hPa in the Ctrl time series.

In the following, an attempt is made to estimate the minimum number of ensemble members as a function of height using forecasts of ensemble members from the M200 experiment because 90 is a sufficient number for high-quality assimilation, judged from the similarity of the performances of the M90 and M200 runs. Figure 17 shows the correlation coefficient of forecasts at each longitude for a reference point of 180∘ E for 40∘ N at 10 and 0.01 hPa that are computed using 12, 25, 50, and 100 members randomly chosen from the M200 forecasts at 06:00 UTC, 21 January 2017. The longitudinal profile of the correlation coefficient for 200 members is also shown. The correlation is generally reduced as the distance from the reference point increases, with large ripples for the results of ensemble sizes smaller than 50, although the correlation profile near the reference point is expressed for all the members. Large ripples reaching ±0.6 observed for 12 members are considered spurious correlations caused by the under-sampling. In contrast, the correlation coefficients for 100 and 200 ensemble members are generally smaller than 0.2 except for a meaningfully high correlation region around the reference point. We performed the same analysis for other latitudes and heights and obtained similar results (not shown).

The RMS of the spurious correlation in the region outside the meaningful correlation region is used to estimate the minimum optimal number of ensemble members. The RMS is examined as a function of the number of ensemble members. Each edge longitude of the meaningful correlation region is determined where the correlation falls below 0.1 nearest the reference point, which are 171.6∘ E and 171.6∘ W for 10 hPa and 149.1∘ E and 146.2∘ W for 0.01 hPa for the case shown in Fig. 17. Note that the threshold value 0.1 to determine the edge longitude is arbitrary and is used as one of several possible appropriate values. The RMS of the spurious correlation is calculated by taking each longitude and latitude as a reference point. Figure 18 shows the results at 40∘ N and 0.01 hPa as a function of the number of ensemble members as an example. Different curves denoted by the same thin line show the results for different longitudes. The thick curve shows mean RMS for all longitudes. Similar results were obtained at other latitudes (not shown). The mean RMS decreases as the number of ensemble members increases and falls to 0.1 for 91 ensemble members. Again, the choice of 0.1 is arbitrary, but from this result, we can estimate the minimum optimal ensemble size at 91. In a similar way, the minimum number of optimal ensemble members is estimated as a function of the latitude and height.

Figure 19 shows the minimum optimal number of ensemble members as a function of the height for 40∘ S, the Equator, and 40∘ N. Roughly speaking, 100 members are sufficient below 80 km for all displayed latitudes except for 50 km at the Equator. The minimum optimal number of ensemble members above 80 km is larger than 100 and close to 150. From this result, more than 150 ensemble members likely give an optimal estimation of the forecast error covariance for the middle atmosphere. However, it is important to note that even if the number of ensemble members is smaller than 150, using the localization as examined in Sect. 3.3.2 will minimize the effect of the spurious error covariance due to under-sampling, as understood from the good performance of the Ctrl assimilation using 30 ensemble members.

This paper gives the first results of the 4D-LETKF applied to the GCM that include the MLT region. Thus, to examine the performance of our assimilation system, the best result obtained from the M200 run among the assimilation experiments is compared with MERRA-2 as one of the standard reanalysis datasets. First, we calculated the spatial correlation of the geopotential height anomaly from the zonal mean as a function of the pressure level and time (Fig. 20). The correlation is higher than 0.9 between ∼900 and ∼1 hPa. The top height with the high correlation varies with time. This time variation may be related to the model predictability, although such a detailed analysis is beyond the scope of the present paper. The reduction of correlation near the ground is likely due to the difference in the resolution of topography.

Next, the zonal mean zonal wind and temperature in the latitude–height section from our analysis and MERRA-2 are shown for the time periods before the major warming onset (i.e., 21–25 January) and after (i.e., 1–5 February) (Fig. 21). A thick horizontal bar shows the 0.1 hPa level up to which MERRA-2 pressure level data are provided. The overall structures of the two datasets are similar: the stratopause in the northern polar region is located at a height of ∼ 50 km (∼ 40 km) in the early (later) period. In the Northern Hemisphere, an eastward jet is observed at ∼ 63∘ N in the wide height range of 20–53 km in the early period. A characteristic westward jet associated with the SSW is observed in the later period in both datasets. The spatial structure and magnitude of the westward jet are both quite similar. In the Southern Hemisphere, a summer westward jet is clearly observed in both sets of data. The poleward tilt with height and a maximum of ∼-70 m s-1 of the jet also accord well. A relatively large difference is observed around 55 km in the equatorial region. The eastward shear with height seems much stronger in MERRA-2 than in our analysis. As the geostrophic balance does not hold in the equatorial region, it may be difficult to reproduce wind by assimilation of only temperature data. This may be the reason for the large discrepancy observed in the equatorial upper stratosphere between the two datasets.

The winds obtained from the M200 assimilation experiment are also compared with wind observations by meteor radars at Longyearbyen (78.2∘ N, 16.0∘ E; Hall et al., 2002) at 91 km and Kototabang (0.2∘ S, 100.3∘ E; Batubara et al., 2011) at 92 km, as well as by the PANSY radar at Syowa Station (69.0∘ S, 39.6∘ E) at 85 km. Note that these radar observations were not assimilated and can thus be used for validation as independent reference data. Table 4 gives a brief description of these data. Figure 22 shows the time series of zonal wind and meridional wind observed at each site (black) and corresponding 6-hourly data from our data assimilation analysis (red). A 6 h running mean was made for the time series of the radar data, although the time intervals of original data are much shorter.

Strong fluctuations with time-varying amplitudes are observed for both zonal and meridional radar winds for each station. The dominant time period is longer at Kototabang in the equatorial region than that of ∼ 12 h at Longyearbyen in the Arctic. The amplitudes of the meridional wind fluctuations there are larger than those of zonal wind fluctuations at Kototabang. These characteristics are consistent with the wind fluctuations estimated by our assimilation system. However, there are significant differences: the time variation of the amplitudes do not accord well with observations. Differences in the phases of the oscillations between observations and estimations are sometimes small and sometimes large.

In contrast, some consistency is observed for relatively long-period variations (periods longer than several days). At Longyearbyen, the slowly varying zonal wind component is slightly positive before 31 January and significantly positive from 1 to 6 February, while the slowly varying meridional wind tends to be significantly negative in the time period of 28–31 January. At Kototabang, the slowly varying zonal wind tends to be slightly negative before 29 January and almost zero afterward, while the slowly varying meridional wind is almost zero throughout the displayed time period. At Syowa Station, the slowly varying zonal wind tended to be negative from 23 to 30 January and after 2–5 February, while the slowly varying meridional wind tends to be positive from 24 to 31 January.

There are several plausible causes for the discrepancy in short-period variations. First, there may be room to improve the model performance to reproduce such short-period variations. Second, the Aura MLS provides data along a sun-synchronous orbits, and hence fluctuations associated with migrating tides may be hard for it to capture. Third, a large local increment added by the assimilation of the MLS data may cause spurious waves in the model. Fourth, there may be inertia–gravity waves with large amplitudes in the upper mesosphere (e.g., Sato et al., 2017; Shibuya et al., 2017), which cannot be captured with the relatively low-resolution GCM.