Roles of Gravity Waves in Preconditioning of a Stratospheric Sudden Warming

As well as strong upward propagation of planetary waves from the troposphere, the state of the stratospheric mean flow has been recognized as a key factor for the occurrence of stratospheric sudden warmings (SSWs). The modification of the mean flow to a suitable state for an SSW occurrence is called “vortex preconditioning.” Recently, increasing attention has been paid to the role of gravity waves (GWs) in the preconditioning mechanism. However, because of the limited availability of data sets covering the whole neutral atmosphere, much uncertainty still exists in the role of GWs in the preconditioning. The aim of this study is to investigate the mechanism of modification of the mean flow in the stratosphere and mesosphere before SSWs from a climatological viewpoint and elucidate the role of GWs in it. We use two state‐of‐the‐art data sets covering the whole neutral atmosphere: a 17‐year medium‐resolution reanalysis data set and the output data from hindcast simulations performed with a GW‐permitting general circulation model. It is shown that the second principal component of the zonal‐mean zonal wind in the stratosphere and mesosphere tends to show a maximum prior to an SSW, characterizing preconditioning. GW forcing alters the structure of the upper part of the jet and contributes to the preconditioning along with planetary waves. Comparison of GW forcing between the reanalysis and GW‐permitting model suggests that the magnitude of parameterized GW forcing is approximately half that of the GW forcing in the polar upper stratosphere where the forcing is responsible for the preconditioning.


Introduction
A stratospheric sudden warming (SSW) is a rapid increase in temperature in the winter polar stratosphere accompanied by a disruption of the polar vortex.An SSW is mainly caused by strong negative (westward) zonalmean wave forcing due to planetary wave (PW) breaking in the stratosphere (Matsuno, 1971).Thus, anomalously large upward fluxes of PW activity from the troposphere may set off an SSW (e.g., Cohen & Jones, 2011;Garfinkel et al., 2010;Sun et al., 2012).For example, blocking events in the troposphere are possible SSW precursors (e.g., Martius et al., 2009;Quiroz, 1986).However, it is known that only a small number of such events are followed by SSWs (Birner & Albers, 2017;Martius et al., 2009).
Several previous studies showed that "vortex preconditioning" is also a key for the occurrence of SSWs.Vortex preconditioning is a modification of the polar vortex to a condition which concentrates rays of PWs and causes

10.1029/2023JD039881
Key Points: • Using high-top data sets from 17-year reanalysis and high-resolution hindcasts, the mechanism of vortex preconditioning is examined • The second empirical orthogonal function of the zonal-mean zonal wind in the winter middle atmosphere characterizes preconditioning • Gravity waves quickly reach and decelerate the top of the eastward jet, helping planetary waves converge in the polar upper stratosphere Supporting Information: Supporting Information may be found in the online version of this article.
PW breaking in the polar stratosphere (de la Cámara et al., 2019;Esler & Matthewman, 2011;Matthewman & Esler, 2011;Scott & Polvani, 2004, 2006).Thus, to clarify the mechanism of each SSW occurrence, it is necessary to examine not only anomalously large wave activity in the troposphere, but also vortex preconditioning in the middle atmosphere.Revealing the mechanism of SSWs leads to an extension of the lead time for forecasting an SSW event.
Vortex preconditioning is described as a poleward and downward shift of the polar vortex in the stratosphere (e.g., Limpasuvan et al., 2004).Previous studies showed that not only PWs but also gravity waves (GWs) contribute to the occurrence of vortex preconditioning (e.g., Albers & Birner, 2014;Scheffler et al., 2018).Performing composite analysis by using reanalysis data, the Japanese Meteorological Agency and Central Research Institute of Electrical Power Industry 25-year Reanalysis (JRA-25), Albers and Birner (2014) suggested that the polar vortex is tuned to structure that is suitable for PW resonance before a number of SSWs (Charlton & Polvani, 2007).In this mechanism, PWs are trapped in a "cavity" which is surrounded by three critical lines: two vertical lines in the mid-and high latitudes and one horizontal line in the upper stratosphere.PWs trapped in the cavity induce enhanced resonant responses there.However, the model top of JRA-25 is 0.4 hPa (an altitude of z ≈ 55 km) near the stratopause.
SSW events frequently follow a reversal of the zonal-mean zonal wind u in the polar lower mesosphere.Several observational studies reported the occurrence of the wind reversal about a week before SSWs, called a "mesospheric precursor" (e.g., Coy et al., 2011;Hoffmann et al., 2002Hoffmann et al., , 2007;;Kurihara et al., 2010;Manney et al., 2009).As well as the poleward shift of the stratospheric jet, changes in the zonal-mean zonal wind in the lower mesosphere are also likely to be important for strong convergence of PW fluxes to the polar upper stratosphere before an SSW.However, studies using general circulation models (GCMs) argued that mesospheric precursors are not statistically significant (e.g., Miller et al., 2013;Zülicke & Becker, 2013).
Recently, analysis data sets that cover the whole neutral atmosphere up to an altitude of z ≈ 100 km are becoming available.One of such data sets, reanalysis created by the Japanese Atmospheric General Circulation Model for Upper Atmosphere Research-Data Assimilation System (JAGUAR-DAS), has been developed by Koshin et al. (2020Koshin et al. ( , 2022)).The reanalysis agrees well with the Modern-Era Retrospective analysis for Research and Applications version 2 (MERRA-2) reanalysis data below 1 hPa (z ≈ 50 km) (Koshin et al., 2020).Compared with winds observed by meteor radars, the JAGUAR-DAS reanalysis showed a good skill in reproducing not only the zonal-mean fields but also disturbances, such as atmospheric tides, in the mesosphere and lower thermosphere (Koshin et al., 2022).McCormack et al. (2021) presented an intercomparison of four analysis data sets of MERRA-2, JAGUAR-DAS, the high-altitude version of the Navy Global Environmental Model (NAVGEM-HA), and the Whole Atmosphere Community Climate Model with thermosphere-ionosphere eXtension using the Data Assimilation Research Testbed (WACCMX + DART).They suggested that the data sets are in good agreement.The discrepancy is relatively large at low latitudes at z > 50 km among the three high-top data sets (i.e., JAGUAR-DAS, NAVGEM-HA, and WACCM-DART).They also showed that the JAGUAR-DAS reanalysis shares more similarities with the NAVGEM-HA than WACCMX + DART does.As of now, the JAGUAR-DAS reanalysis has the longest temporal coverage as an analysis data set covering the whole neutral atmosphere.This is because JAGUAR-DAS employs the four-dimensional local ensemble transform Kalman filter (4D-LETKF; Miyoshi & Yamane, 2007) which costs smaller computational resources than other schemes do.Such high-top data sets provide particularly useful means of analyzing variation of the dynamical structure in the upper stratosphere and mesosphere during vortex preconditioning.
Considering the importance of GWs for the dynamics of the upper stratosphere and mesosphere, quantitative research on GWs is needed to elucidate the mechanism of vortex preconditioning.Most GWs in analysis data are not resolved explicitly.Instead, GW parameterizations represent wave forcing due to GWs.However, most current GW parameterization schemes ignore lateral propagation (e.g., Sato et al., 2009Sato et al., , 2012) ) and in-situ generation of GWs in the middle atmosphere (e.g., Becker & Vadas, 2018;Vadas & Becker, 2018;Yasui et al., 2018).In addition, vertical group velocities of GWs are assumed to be infinitely large.These assumptions used in GW parameterization schemes are not necessarily valid, which causes difficulties for studies using hightop reanalysis data on the roles of GWs in dynamical processes.In reanalysis data, the impact of wave forcing due to GWs not reproduced by parameterizations may be partly compensated by assimilation increments (Sato & Hirano, 2019).While GW-permitting GCMs may not necessarily reproduce realistic GWs, it is still likely that they represent aspects of GWs ignored by GW parameterizations.We compare parameterized GW forcing and assimilation increments in the JAGUAR-DAS reanalysis with resolved GW forcing in hindcast simulations performed with a GW-permitting model.Such comparison will provide useful information about the impact of the assumptions on the dynamical roles of GWs.
The purpose of this study is to examine climatological characteristics and the mechanism of vortex preconditioning including variations of mesospheric zonal winds by using the JAGUAR-DAS reanalysis for 17 years from 2004/2005-2020/2021.The high-top reanalysis will be beneficial to examine modifications in the top part of the polar vortex associated with vortex preconditioning.We also use output data from hindcast simulations carried out with a high-top GW-permitting GCM to investigate differences and common features of parameterized and resolved GW forcings.By the combined use of these two high-top data sets, this study aims to provide a statistical and detailed picture of the typical variations in the intensity and location of the winter middle atmospheric jet before SSWs and of the contributions from waves including GWs to the mechanism.
Section 2 outlines the data from the reanalysis and hindcast simulations used in this study.In Section 3.1, we statistically analyze variation of the zonal-mean fields in the middle atmosphere and 10 major SSWs occurring in the Northern Hemisphere during 2004/2005-2020/2021.A case study on the SSW from late December 2018 to early January 2019 is performed by using both the reanalysis data set and output data from high-resolution hindcasts in Section 3.2.Comparison of parameterized GW forcing in the reanalysis data with resolved GW forcing in the GW-permitting GCM hindcasts is made in Section 3.3.In Section 4, the following topics are discussed: the case dependence of vortex preconditioning (Section 4.1), and the formation of regions where PWs cannot propagate in the polar upper stratosphere (Section 4.2).We also make remarks on GW parameterizations in Section 4.3.Section 5 includes a summary of this study and suggestions for further research.

Reanalysis Data Set Created by JAGUAR-DAS
A reanalysis data set used in this study was created by a 4D-LETKF data assimilation system called JAGUAR-DAS (Koshin et al., 2020(Koshin et al., , 2022)).JAGUAR-DAS uses the standard-resolution JAGUAR model as the forecast model.This version of JAGUAR has a triangularly truncated spectral resolution of T42, corresponding to a horizontal resolution of ∼300 km (a latitudinal interval of 2.8125°).The vertical resolution is approximately 1 km.No sponge layers are used in the model, as the horizontal diffusion is enhanced with height in the mesosphere and lower thermosphere and is sufficiently strong at the model top to prevent spurious wave reflection.Orographic (McFarlane, 1987) and nonorographic GWs (Hines, 1997) are parameterized in this model.The distribution of nonorographic GW source is given based on the results from Watanabe (2008).Using the results from simulations performed with a GW-permitting GCM, they provided monthly climatology of the momentum flux and horizontal wind variance due to nonorographic GWs by excluding quasi-stationary components considered to be orographic waves.
JAGUAR-DAS assimilates the PREPBUFR global observation data set compiled by the National Centers for Environmental Prediction, temperature retrievals from Microwave Limb Sounder v.4.2 aboard the NASA's Aura satellite and the Sounding of the Atmosphere using Broadband Emission Radiometry aboard the Thermosphere Ionosphere Mesosphere Energetics and Dynamics satellite, and brightness temperatures from the Special Sensor Microwave Imager/Sounder instrument on the Defense Meteorological Satellite Program satellite.The incremental analysis updating (IAU; Bloom et al., 1996) method was introduced to JAGUAR-DAS in order to reduce the generation of spurious waves owing to increments.
We use this reanalysis data for the 17 boreal winters from December to February (DJF) of 2004/2005-2020/2021 for statistical analysis on the variation of the zonal-mean fields in the stratosphere and mesosphere.In total, 10 SSW events occurred in this period.The empirical orthogonal function (EOF) analysis is used to describe the evolutions of SSWs with a few essential variation patterns.In this paper, PWs are defined as waves having zonal wavenumbers s of 1-3.

High-Resolution JAGUAR Hindcasts
For quantitative investigation of the role of GWs in vortex preconditioning, we also analyzed the output data from the hindcast simulations of 2018/2019 boreal winter conducted using the high-resolution JAGUAR.The DJF of 2018/2019 includes an SSW event that occurred in January 2019.The details of the model are described by Watanabe and Miyahara (2009) and Okui et al. (2021Okui et al. ( , 2022)).Here we shortly present the configuration of the model and simulations.The horizontal resolution of the model is T639, whose minimum resolvable horizontal wavelength is ∼60 km and corresponding latitudinal interval is ∼0.1875°.The model is composed of 340 vertical layers from the surface to a geopotential height of ∼150 km with a log-pressure height interval of 300 m.No GW parameterization schemes were used.Each simulation consists of a 3-day spectral nudging and 4-day free run.These simulations for 7 days in total were performed every 4 days.We analyzed only the outputs from the free runs.The JAGUAR-DAS reanalysis data was used for spectral nudging of the total horizontal wavenumber (n) components of n = 0-15 above a pressure level of 100 hPa.The ERA5 global reanalysis (Hersbach et al., 2020) provided by the European Centre for Medium-Range Weather Forecasts is used for nudging the components having n = 0-40 in the model levels below 100 hPa in combination with the JAGUAR-DAS reanalysis data.The model simulations whose outputs are analyzed in this study used the result from the previous runs for their restart (see also Figure S1 in Supporting Information S1).In the sections of this paper showing the results from the high-resolution JAGUAR hindcasts, components having n = 21-639 are defined as GWs.

A Statistical View of Vortex Preconditioning
The progressions of SSWs and their preconditioning depend on the case.To identify common features shared by the complex events, we first reduced the dimensionality of the data by the empirical orthogonal function (EOF) analysis.The EOF analysis was performed on zonal-mean and daily averaged zonal wind u in a height region of z = 15-90 km at latitudes of 15°N-90°N for the 17 boreal winters from 2004/2005 to 2020/2021 using the JAGUAR-DAS reanalysis data.Before the EOF analysis was performed, seasonal variation was excluded by removing the daily averaged 17-year climatology that is smoothed by a low-pass filter with a 10-day cutoff.The region for the EOF analysis (i.e., z = 15-90 km, 15°N-90°N) was chosen so that the variation of the polar vortex can be examined not only in the stratosphere but also in the mesosphere and that direct influence of the phases of the quasi-biennial oscillation and semiannual oscillation can be excluded.
Figure 1 shows the latitude-height sections of u averaged over DJF of the 17 years and of the first, second, and third EOFs multiplied by the standard deviations of their respective principal components (PCs).The dimensional EOF patterns, or the factor loadings, are hereafter denoted as the EOF1, 2, and 3, while their scores normalized by the standard deviations are referred to as the PC1, 2, and 3.The EOF1, EOF2, and EOF3 account for 45.0%, 19.6%, and 12.0% of the total variance, respectively.Degeneracy between the EOFs due to sampling errors does not occur according to the criterion of North et al. (1982).The EOF1 has a maximum at ∼60°N, z ≈ 55 km.It appears that the EOF1 explains the variance due to the variation of the polar vortex intensity.The EOF2 has a large positive maximum at ∼70°N, z ≈ 40 km bounded by negative values at lower latitudes and higher altitudes.The latitudinal distribution of u in the stratosphere, namely positive at >∼45°N and negative at <∼45°N, is roughly opposite in sign to that in the middle and upper mesosphere at z> ∼70 km.The stratospheric pattern of u for the EOF2 is similar to the u distribution associated with a poleward and downward shift of the eastward jet during vortex preconditioning (e.g., Limpasuvan et al., 2004).The pattern of the EOF3 has positive and negative peaks at ∼45°N, z ≈ 60 km and ∼70°N, z ≈ 63 km.
Figure 2 displays the time series of the PCs in DJF of years with Arctic major SSW events that occurred during 2004/2005-2020/2021.Time series for boreal winters with no major SSW events are shown in Figure S2 in Supporting Information S1. Figure 2k shows the lead-lag composite of the PCs centered around the SSW onset dates.The gray shadings in Figures 2a-2j represent the periods of poleward shifts of the polar vortex just prior to the SSWs.We defined these periods as the time period from when the jet axis shifts to the north of 60°N at z = 45-55 km to when u at z = 50 km becomes westward.Such poleward shifts of the jet are typically observed during preconditioning (e.g., Limpasuvan et al., 2004).Small variations in these thresholds do not cause significant difference in the periods.It is notable that these periods accord well with positive peaks of the PC2.The SSW onsets roughly coincide with positive peaks of the PC1.Such time evolution of the PCs is clearly observed for most of the SSW events shown here, especially SSWs in 2009SSWs in , 2010SSWs in , 2013SSWs in , 2017SSWs in , 2019SSWs in , and 2021.While the composite of the PC1 is largely positive around the SSW onset, a positive peak of the PC2 before the onset appears to be weak and widely spread in the composite (Figure 2k).It is considered that the unclear PC2 peak in the composite is mainly due to the significant case dependence in the timing of preconditioning relative to the SSW onset.Difference between vortex displacement, split, and mixed-type SSWs is not clearly identified in the time series of the PCs.The temporal variations of the PC3 are diverse across SSW events, and the relation with SSW is not clear for the PC3.
Based on the good agreement between poleward jet shifts typical for preconditioning and periods with positive PC2 in most of the cases, we use the PC2 as a "preconditioning proxy" and refer the occurrence of a PC2 maximum before and closest to an SSW onset as the onset of preconditioning.To extract the characteristics of variations in the zonal-mean fields when the preconditioning proxy takes high positive values, we calculated their linear regressions on the normalized PC2 using the ordinary least squares method.Hereafter u EOF2 represents the linear regression of u in the 17 boreal winters on the PC2. Figure 3 shows the latitude-height sections of the linear regressions of u (i.e., u EOF2 ), the Eliassen-Palm (EP) flux and its divergence associated with PWs, and parameterized GW forcing on the PC2.The distribution of u EOF2 (Figure 3a) is the same as the pattern of the EOF2 in 15°N -90°N, z = 15-90 km (Figure 1c).Figures 3d-3f illustrate linear regressions onto the PC2 for the same variables but at a negative lag of 5 days.Contours in Figures 3b and 3c represent u EOF2 .
The EP fluxes with PWs projected on the PC2 are upward in the polar lower stratosphere, while they partially change their directions to equatorward in the upper stratosphere, as seen in Figure 3b.The regression of the EP flux divergence due to PWs, that is, PW forcing, is strongly negative (i.e., westward) in the upper stratosphere.Leading the PC2 by 5 days, PW forcing is positive at high latitudes and negative at mid-latitudes (Figure 3e).It is considered that these positive and negative PW forcings primarily contributes to the poleward jet shift in the stratosphere.PW forcing also seems to develop u EOF2 < 0 in 40°N-70°N, z = 70-85 km.In the mesosphere, strong negative PW forcing is observed at ∼55°N, z ≈ 77 km.
The parameterized GW forcing (GWF p ) is negative to the north and positive to the south of 45°N in z = 40-80 km at a negative lag of five days (Figure 3f).The negative and positive GW forcings are observed above the eastward and westward u, respectively.These characteristics can be consistently explained by the GW filtering by u in the stratosphere.The negative GW forcing contributes to the development of negative u EOF2 (Figure 3a) at mid-tohigh latitudes in the mesosphere together with PW forcing.Especially in the high-latitude (>60°N) mesosphere at z = 55-80 km, the GW forcing makes a major contribution.GW forcing is also negative above the eastward u EOF2 at lag 0 in Figure 3c.Since the negative GW forcing in the mesosphere locates below westward u EOF2 , it acts to move the mesospheric westward u EOF2 downward.
Overall, the results from the statistical analysis suggest that, as common characteristics shared by the majority of the 10 SSWs we examined, the vortex preconditioning manifests as the development of the EOF2 of u in the winter stratosphere and mesosphere.The preconditioning in the stratosphere occurs mainly owing to PW forcing, while the mesospheric precursor primarily results from GW forcing.GW forcing moves the mesospheric westward u EOF2 downward, which decelerates the upper part of the stratospheric eastward jet.

A Case Study on the 2018/2019 SSW
In this section, we show the results from a case study on a major SSW event occurring in early January 2019.The aim of this section is to further examine the time evolution of the preconditioning.Since this SSW is classified as a mixed-type SSW transitioning from a displacement-type event to a split-type event (Rao et al., 2019), the event is not very "typical" as a sample of displacement-type or split-type events.However, the event was focused on in this section for the following reasons.First, there is a clear peak of the PC2 on 22 December 2018 corresponding to a poleward jet shift prior to the 2018/2019 SSW (Figure 2i).Regarding the relation between the PC1 and major warming, while the onset of the major warming was on 1 January 2019 by its definition, the anomaly of polar temperature at 10 hPa reached its maximum on 29 December 2018, a few days after a maximum of the PC1.Second, this event is characterized by the second largest temperature anomaly averaged over 60°N-90°N at 10 hPa from 2004/2005 to 2020/2021 behind the 2009 SSW (not shown).In addition, Rao et al. (2019) reported that the predictability of the 2019 SSW is higher in models with higher model tops.Closer examination of this event may lead to a better understanding on the high predictability of SSWs in high-top models, which is the third reason for choosing the event.Based on the above reasons, we consider that it is worthwhile to conduct a case study on the 2018/2019 SSW event.
Figure 4 shows the latitude-pressure sections of u, PW forcing, parameterized GW forcing (GWF p ), the sum of GWF p and assimilation increments for u (Δu t ), and difference of daily averaged u from that on the previous day from the JAGUAR reanalysis data set on 20, 21, and 23 December 2018.The assimilation increment Δu t is the difference of u between analysis and forecast.The sum GWF p + Δu t reflects subgrid-scale processes not reproduced accurately by parameterizations and other model deficiencies such as deficiencies in radiative processes and numerical diffusion.
Figures 4a-4e present the results for 20 December, which corresponds to the day when the PC2 is in a slightly positive and increasing phase before the SSW occurrence.As can be seen in Figure 4a, there are two maxima of the eastward jet, one of which is located at 58°N, z = 51 km in the high latitudes of the upper stratosphere and the other is at 32°N, z = 72 km in the mid-latitudes of the mesosphere.PWs propagate equatorward around the stratospheric jet axis (Figure 4b).The PWs cause strong negative PW forcing in the mid-latitude stratosphere.The EP flux associated with PWs are similar to those observed in the regressions on the PC2 (Figure 3b).GWF p is negative above the two jet axes (Figure 4c).A funnel-shaped region of GWF p < 0 is observed on the equatorial side of the jet whose axis is located at ∼55°N, z ≈ 52 km.The maximum of this negative GWF p ranges from 20 m s 1 d 1 to 50 m s 1 d 1 .By nearly balancing with Coriolis force associated with meridional flow and altering the structure of temperature and wind, the GWF p acts to decelerate the eastward u in the midlatitude upper stratosphere (Figure 4e) along with the negative PW forcing.
There are some notable differences between GWF p and GWF p + Δu t .While GWF p at low latitudes below the mesospheric eastward jet is weakly positive (Figure 4c a wider latitude region of 0°-40°N at z = 55-75 km (Figure 4d).Positive GWF p + Δu t is also observed to the north of the stratospheric jet axis at >55°N, z ≈ 30-55 km, though the positive forcing is hardly seen in GWF p alone.Poleward of ∼70°N in the mesosphere, the negative forcing due to GWF p + Δu t is more pronounced than that caused by GWF p alone.The difference between GWF p and GWF p + Δu t are further discussed in Section 4.3 through a comparison with resolved GW forcing in the GW-permitting GCM.
The results for the next day, 21 December 2018, are shown in Figures 4f-4j.The onset of the preconditioning occurred on 22 December.As seen in Figure 4f, the stratospheric jet axis has moved from 58°N to 65°N.The two jet axes are disconnected more clearly, and the stratospheric jet axis is located at a higher latitude than those on 20 December.The region where the jet disconnection occurs from 20 to 21 December (Figure 4j), specifically at 40°N -50°N in z = 50-80 km, roughly accords with the region where both the negative PW forcing and funnelshaped GWF p are observed on 20 December.On 21 December, PWs propagate equatorward in the upper stratosphere and converge on the equatorial side of the polar stratospheric jet (Figure 4g).In the polar region, PW forcing is still positive in z = 25-45 km and 55-75 km.The positive PW forcing acts to shift the jet axis poleward, which is consistent with the positive PW forcing projected on the PC2 at a negative lag of 5 days (Figure 3e).Along with the poleward shift of the stratospheric jet, the negative GWF p above the jet axis extends further to the polar region (Figure 4h).While GWF p in the polar mesosphere ranges from 5 to 20 m s 1 d 1 , GWF p + Δu t has larger negative values from 20 to 50 m s 1 d 1 at >70°N in z = 60-85 km (Figure 4i).It is noteworthy that both positive PW forcing below and negative GW forcing above the poleward-shifted stratospheric jet work together to lower the jet core.
By 23 December 2018, one day after the onset of the preconditioning, the stratospheric jet core has descended and u in the polar mesosphere has become westward (Figure 4k).The deceleration and reversal of u in the polar mesosphere (Figure 4o) appear primarily owing to negative GW forcing observed on 21 December (Figures 4h  and 4i).Eventually, PW forcing is strongly negative in the polar upper stratosphere as seen in Figure 4l.The negative PW forcing is considered to be the direct cause of the major SSW.It is also interesting that positive PW forcing is observed in the lower mesosphere to the north of ∼45°N, which is discussed briefly in Section 4.4.The negative GWF p to the north of 50°N in the mesosphere has become weak (Figure 4m).
The paths of the PW propagation in the stratosphere have changed remarkably before and after the statistical preconditioning on 22 December 2018.On 20 and 21 December 2018, PWs propagate equatorward in the upper stratosphere (Figures 4b and 4f).The EP fluxes associated with PWs converge at low latitudes at z ≈ ≈ 50 km with weak u which is located on the high-latitude side of the equatorial region with u < 0 at z ≈ 50 km (Figures 4a and  4f).On 23 December, upward propagation of PWs became dominant and strongly negative PW forcing is observed in the polar region in the upper stratosphere (Figure 4l).Again, this negative PW forcing leads to the occurrence of the SSW.
To investigate the factors altering the PW paths further, the Rossby-wave refractive index squared (e.g., Karoly & Hoskins, 1982;Matsuno, 1970) was examined.We used the refractive index squared n 2 s for stationary waves in a spherical coordinate system as follows: where is the latitudinal gradient of zonal-mean quasi-geostrophic potential vorticity q; f is the Coriolis parameter; a is the Earth's radius; ρ 0 is the basic density; ε 2 ≡ f 2 /N 2 , where N 2 is the Brunt-Väisälä frequency squared; z is the logpressure height; H is the scale height; and subscripts φ denote derivatives with respect to the latitude φ.The formulation of n 2 s is derived by Weinberger et al. (2021) taking into account the z dependence of N. Waves tend to be refracted toward a region with large positive values of n 2 s , while they cannot propagate in a region with negative n 2 s .
Figure 5 shows nondimensionalized n 2 s , that is, n 2 s multiplied by a 2 for stationary PWs with s = 1 in the latitudepressure section.We consider here that PWs primarily propagate from the troposphere.Originating from stationary wave sources such as the continent-ocean thermal contrast, a substantial portion of the PWs are likely to have small phase velocities.We calculated n 2 s using u and N 2 averaged over 3 days.The period of averaging was set considering that vertical group velocities of PWs are of the order of several kilometers per day to a dozen kilometers per day.
During 19-21 December 2018, n 2 s exhibits significantly positive values at low latitudes (Figure 5a).There is a weak positive peak extending from ∼50°N, z ≈ 25 km to ∼55°N, z ≈ 45 km along the jet axis.A cavity is found in the region of polar upper stratospheric jets (z ≈ ≈ 40 km, ∼75°N) with areas of negative n 2 s in the southern and northern flanks and upper boundary of the cavity, namely in (a) z = 42-57 km, 58°N-67°N; (b) z = 53-65 km, >60°N; (c) z = 39-65 km, >78°N (Figure 5b).This n 2 s structure is similar to the suitable condition for PW resonance discussed by Albers and Birner (2014) who focused on split-type SSWs.Another noticeable change after the preconditioning onset is that, during 22-24 December, a region with relatively high n 2 s along the stratospheric jet axis tilts poleward, distributing from ∼50°N, z ≈ 25 km to ∼70°N, z ≈ 35 km (Figure 5b).The poleward-tilting high n 2 s and three-sided cavity seem to be responsible for the convergence of the EP fluxes associated with PWs (Figure 4l).

Forcing Exerted by Resolved GWs in the High-Resolution Model
By using the results from hindcast simulations performed with the high-resolution JAGUAR model, we compare GWF p in the reanalysis with GW forcing explicitly resolved in the high-resolution JAGUAR (GWF r ). Figure 6 shows the EP flux and its divergence due to PWs and resolved GWs (i.e., waves having n = 21-639) obtained from the high-resolution GCM outputs.PW forcing shown in Figures 6a-6c generally accords well with the results from the JAGUAR-DAS reanalysis data set (Figures 4b,4g,and 4l).PWs propagate into the polar upper stratosphere and exert negative wave forcing there on 23 December.The refractive index for stationary PWs with s = 1 exhibited characteristics mostly consistent with that from the reanalysis when interpolated to the grid for the reanalysis (not shown).
GWF r is strongly negative above the stratospheric jet, ranging from 20 to 50 m s 1 d 1 on 20 and 21 December 2018 (Figures 6d and 6e).A u maximum of ∼40 m s 1 in the high-latitude stratosphere on 23 December (shown by contours in Figures 6c and 6f) is much smaller than that on 21 December with a value of ∼70 m s 1 (Figures 6b  and 6e).The negative GWF r above the stratospheric jet also becomes weak on 23 December (Figure 6c).Another interesting feature seen in Figures 6d-6f is positive GWF r below the mesospheric eastward jet to the south of 40°N in z = 55-70 km.Comparison of the negative and positive GWF r with GWF p and GWF p + Δu t shown in Figure 4 is made in Section 4.3.

Vertical Propagation of PWs and Resolved GWs
GW forcing projected on the PC2 is negative (positive) above u EOF2 > (<) 0. It is considered that the GW forcing, owing to GWs filtered by u in the stratosphere, is responsible for the modification of the structure of u in the upper stratosphere and mesosphere.To examine how fast GWs in the mesosphere respond to the change in the stratospheric u, their vertical group velocities c (z)  g are examined.In this section, GWs are defined as disturbances having total horizontal wavenumbers of 21-639.The dispersion relation of hydrostatic inertia-GWs can be written as (e.g., Andrews et al., 1987): Here, n is the total horizontal wavenumber, ω is the intrinsic frequency, ω is the ground-based frequency, and U is the component of the background wind in the direction of the horizontal wavenumber vector.For sake of simplicity, we assume n ≈ k (k is the dimensional zonal wavenumber, i.e., k = s/(a cos φ)) and U ≈ u. Figure 7 shows momentum flux spectra in the k-ω space that are the real part of the product of the zonal wavenumberfrequency Fourier components of zonal (u′(k,ω)) and vertical wind fluctuations (w′(k,ω)) (i.e., Re[u′w′ * ] , where A* is the complex conjugate of A) at 60°N, z = 30 km (Figure 7a) and 70 km (Figure 7b) during 19-22 December.The reddish-colored curves in Figure 7 represent constant values of c (z) g drawn based on Equation 3.
At z = 30 km, negative Re[u′w′ * ] for k > 0 is observed for waves having slower zonal phase speed than 10 m s 1 , whose c (z)  g is roughly 20-50 km d 1 (Figure 7a).For k < 0, negative Re[u′w′ * ] is most prominent for GWs with c (z)  g ≲ 100 km d 1 .At z = 70 km in the mesosphere, vertical group velocities of waves having large negative Re[u′w′ * ] range widely from zero to c (z) g ≈ 200 km d 1 (Figure 7b).This result indicates that GWs with negative Re[u′w′ * ] have large c (z)  g and propagate from the stratosphere to the mesosphere in one to several days.
Regarding PWs, c (z) g is derived from the dispersion relation of Rossby waves as: where β is the latitude gradient of the Coriolis parameter and l represents the meridional wavenumber when a sinusoidal solution is assumed in the meridional direction (e.g., Andrews et al., 1987).We also assumed here that the meridional gradient of relative vorticity is much smaller than β and that m 2 ≫ 1/4H 2 in the right-most side of the equation.Table 1 displays c (z) g of for stationary (c = 0) PWs at a latitude of 60°when N ≈ 2 × 10 2 s 1 , which is a typical N value for the stratosphere.In this estimation, the value of l was assumed to be π × 10 7 m 1 .It is obvious that the time required for vertical propagation of PWs is several to several tens of times longer than that for GWs, though this estimate is a rough approximation.Thus, GWs from the troposphere can reach the mesosphere much faster than PWs from the troposphere.Therefore, it is likely that fast propagation of GWs through the stratospheric jet results in the almost instantaneous response of GW forcing to the jet shift in the stratosphere.The rapid response of GW forcing seems to be the cause of the nearly simultaneous occurrence of preconditioning (i.e., development of the EOF2 pattern in u) in both the stratosphere and mesosphere.

Case Dependence of the Preconditioning
The results from the statistical analysis using the high-top reanalysis data set showed that the PC2 of u in the region of z = 15-90 km at 15°N-90°N has a positive peak in most of the SSW events during the 17 years just prior to the onset of a major warming (Figure 2).However, the process of the preconditioning is case-dependent.For example, the PC2 peaks before the SSWs are less distinct in January 2006 (Figure 2a), February 2008 (Figure 2c), and February 2018 (Figure 2h).The time lags from the PC2 peaks to the subsequent SSW onsets also vary from event to event.Thus, it seems impractical to directly use the PC2 for SSW prediction.On the other hand, this proxy of  SSW preconditioning is still useful as an indicator for SSW likelihood based on conditions in the stratosphere and mesosphere.
Regarding the time lags from the preconditioning onset to the SSW onset, there are two probable causes of the case dependence: one is the calendar day of the SSW occurrence and the other is the amplitudes and zonal wavenumbers of PWs entering from the troposphere.The former is related to the seasonal variability of the polar vortex.The onsets of the January events in 2006, 2009, 2010, 2013, 2019, and 2021 occur in relatively long time from the PC2 maximum (∼9 days on average) compared with the February events in 2007, 2008, 2017, and 2018 (∼4 days on average).Since the eastward jet in the stratosphere is stronger in January, it is expected that more westward momentum deposit is required to decelerate a strong winter jet in January (Monnin et al., 2022).The latter is related to the difference in paths and group velocities of dominant waves.PWs with s = 2 are less likely to propagate upward in the polar region than PWs with s = 1.Vertical group velocities also depend on dominant wavenumber.The difference in the dominant wavenumber seems responsible at least partly to different time evolutions of displacement-type and split-type SSWs.Note that SSW events are always distinctly classified.The 2018/2019 major SSW, for which a case study was made in Section 3.2, is a mixed (displacement to split) type, for example,

Mechanism of the Formation of a Cavity
In this section, we discuss further into the mechanism of the formation of the walls of a cavity surrounded by negative refractive index n 2 s .For this purpose, each term in Equations 1 and 2 comprising n 2 s is examined separately for 22-24 December 2018.Figures 8a-8d shows the latitude-pressure sections of f φ / (ua) s 2 / a 2 cos 2 φ), ερ 0.5 0 ερ 0.5 0 ) zz , (ua) 1 [(a cos φ) 1 (u cos φ) φ ] φ , and u 1 ρ 1 0 ρ 0 ε 2 u z ) z , respectively.All values in Figure 8 are nondimensionalized, that is, multiplied by a 2 .
The term associated with u φφ (Figure 8c) is largely positive in an elongated region from ∼50°N, z ≈ 25 km to the jet axis and a region at ∼75°N, z = 40-60 km in the jet core.The meridional shear of the background zonal wind u φ is positive equatorward and negative poleward of the jet axis.As a result, u φφ is negative and thus the contribution of the term with u φφ to n 2 s is positive.This positive term is responsible for the poleward-tilted region along the jet and the internal area of the cavity with positive n 2 s .The polar wall of the cavity is mainly attributable to negative values in f φ / (ua) s 2 / a 2 cos 2 φ) as seen in Figure 8a.This is due to small f φ and cosφ at high latitudes.The term with u zz is negative above the jet core (Figure 8d).The negative term associated with u zz in the polar upper stratosphere is responsible for the negative n 2 s composing equatorial wall and upper lid of the cavity.Therefore, the vertical shear in the upper part of the jet is likely to be of crucial importance for strong PW breaking and thus SSW occurrence.
As the PC2 rises, GW forcing becomes negative above the poleward-shifted jet (Figures 4h,4i,and 7e).PW forcing is positive in the lower and poleward part of the jet (Figures 4g and 7b).Then, negative u z becomes larger between the negative GW and positive PW forcings.The negative u z is associated with negative u zz around the jet axis and positive u zz around the top of the jet.As a result, the term with u zz in the equation of n 2 s became negative in the regions covering the top of the stratospheric jet (Figure 8d).These facts suggest that both PW forcing and GW forcing contributed to the formation of the three-sided cavity.

Remarks on GW Parameterizations
To estimate the contribution of GW forcing not represented by GW parameterizations to the preconditioning, we compared GWs parameterized in the reanalysis data with those resolved in the GW-permitting model.The distribution of the GWF p around the stratospheric jet, that is, at >40°N, z = 30-80 km in Figures 4c, 4h, and 4m, showed good qualitative agreement with GWF r in Figures 6d, 6e, and 6f, respectively.This fact supports the validity of qualitative discussions on the role of GWs in vortex preconditioning by using the JAGUAR-DAS reanalysis data set.However, the magnitude of negative GWF p located to the north of 70°N in z = 60-80 km (Figures 4c and 4h) is about half as large as that of GWF r on 20 and 21 December (Figures 6c and 6f).In this region, GWF p + Δu t (Figures 4d and 4i) agrees quantitatively with the GWF r .This fact indicates that the westward acceleration due to GW forcing cannot be fully reproduced by GW parameterizations.Nevertheless, the reanalysis reproduces the distribution and variation of the zonal winds by compensating the deficiency in the parameterizations with assimilation increments.Similar deficiencies in the GW parameterizations are observed in the low-latitude (i.e., 0-30°N) lower mesosphere (e.g., Figures 4c, 4d, and 6d) where positive GWF r is prominent.It is worth noting that this region corresponds to that with the relatively large model discrepancy in the zonal-mean zonal wind in the intercomparison of the four analysis data sets of the whole middle atmosphere (McCormack et al., 2021).
As seen in Figure 8, strong momentum fluxes are observed both sides of the black dashed line representing a phase velocity that is the same as the background wind.This fact suggests that GWs propagate from other latitudes into the height region above their critical lines at this latitude and/or they are generated above their critical lines.It is also noteworthy that some GWs accompanying strong momentum fluxes have fast phase velocities.Further analysis on GW characteristics using the GW-permitting GCM seems to be useful to elucidate the causes of discrepancies between GWF p and GWF r .

Possibility of PW Amplification in the Polar Upper Stratosphere and Lower Mesosphere
Using the MERRA2 reanalysis data, Song et al. (2020) indicated that PWs generated by GW forcing in the polar upper stratosphere and lower mesosphere propagated downward and contributed to the occurrence of the SSW event in January 2009.Similarly, Okui et al. (2021) suggested that the barotropic and/or baroclinic (BT/BC) instability induced by GWF caused PW generation in the winter polar mesosphere ∼10 days before the 2018/2019 SSW onset.Then, here we briefly discuss the possibility of amplification and downward propagation of PWs in the upper stratosphere and lower mesosphere in terms of n 2 s shown in this paper.
During 22-24 December 2018, n 2 s becomes negative above the upper end of the waveguide along the jet axis (Figure 5b).PWs become external in this region.Above the region of n 2 s < 0, largely positive n 2 s is distributed to the north of 55°N in z = 70-85 km.This positive n 2 s mainly results from negative u and positive u zz just above the wind reversal in the polar mesosphere.The wind reversal is mainly due to negative GW forcing.The critical line of stationary PWs, where u ≈ 0, lies within the region of q φ < 0 at z ≈ 70 km.The distributions of n 2 s , u, and q φ are consistent with the condition in which Rossby waves can be overreflected (i.e., amplified when reflected), as illustrated in Figure 1 of Lindzen and Tung (1978).They showed that the necessary conditions for the BT/BC instability (e.g., Kuo, 1949;Rayleigh, 1879) are sufficient conditions for Rossby wave overreflection.In that sense, our results do not exclude the possibility of PW overreflection.However, note that the contribution of overreflection is not crucial to this SSW event even if possible, because there is no obvious downward EP flux associated with PWs.

Summary and Concluding Remarks
A study on vortex preconditioning during Arctic SSW events was made by utilizing the two state-of-the-art data sets covering the whole neutral atmosphere: reanalysis created by JAGUAR-DAS over 17 boreal winters of 2004/ 2005-2020/2021 and output data from hindcast simulations performed with a GW-permitting GCM, the highresolution version of JAGUAR, for December 2018.
First, we conducted the EOF analysis on u time series in the region of 15°N-90°N in the stratosphere and mesosphere using the JAGUAR-DAS reanalysis.The analyzed 17 boreal winters include 10 major SSWs.The results indicated that the vortex preconditioning characterized by a poleward and downward shift of the eastward jet can be captured as a development of the u pattern of the positive EOF2 (u EOF2 ).The PC1 is maximized around the onsets of major warmings.We considered the PC2 as a "preconditioning proxy" and defined the onset of preconditioning as the PC2 maximum just prior to a major SSW onset.The statistical characteristics of the EOF2 was analyzed by calculating the linear regressions on the PC2.Simultaneously with the poleward shift of the jet in the stratosphere, GW forcing acts to decelerate eastward winds in the upper part of the stratospheric jet.In this way, the state of preconditioning is formed.
To obtain more detailed information about the preconditioning, we also conducted a case study on the preconditioning before the SSW in January 2019 with a significant increase in the polar stratospheric temperatures.The onset of the preconditioning occurred on 22 December 2018, 10 days before the onset of major warming on 1 January 2019.An elucidated scenario of the preconditioning is as follows.First, the eastward jet in the middle atmosphere splits into two jets in the polar stratosphere and low-latitude mesosphere and the stratospheric jet shifts poleward.This evolution of the eastward jets is caused by negative PW forcing (e.g., Limpasuvan et al., 2004) and GW forcing at midlatitudes in the upper stratosphere and mesosphere.Second, eastward GWs from the lower atmosphere are more likely to be filtered out by the poleward-shifted jet in the polar stratosphere.Consequently, westward GWs come to account for a higher proportion to waves reaching the polar upper stratosphere and mesosphere and exert negative forcing there.The deceleration of u in the upper part of the jet owing to GW forcing leads to the formation of the equatorial wall and top lid of the cavity for PWs.The PWs exert negative forcing in the cavity in the polar upper stratosphere, which is the direct cause of the SSW occurrence.
New findings of the present study are summarized as follows: First, it was shown that the pattern of the SSW preconditioning is not a special pattern, but a common pattern of zonal mean zonal wind fluctuations in the stratosphere and mesosphere that is obtained as the second mode of the EOF analysis.Next, a quantitative analysis of GW forcing has revealed that the contribution of GWs is as important as that of PWs when the second EOF mode is enhanced as the SSW preconditioning.A key process of the preconditioning formation is related to the difference in the vertical group velocity c (z)  g between the GWs and PWs.Because of larger c (z) g of GWs, the forcing by GWs in the upper stratosphere and mesosphere is modified almost instantaneously in response to the poleward shift of the stratospheric jet, creating an environment (i.e., preconditioning) for PWs that propagate upward from the lower atmosphere more slowly because of their smaller c (z)  g .
In this paper, we also aimed to estimate the contribution of GWs not conforming to the assumptions used in GW parameterizations to the preconditioning.To achieve this, parameterized GW forcing (GWF p ) from the JAGUAR-DAS reanalysis was compared with resolved GW forcing (GWF r ) in the GW-permitting GCM during the preconditioning of the 2018/2019 SSW.Albeit only in a qualitative manner, the results suggested that the contribution of GWs to the wave forcing responsible for the preconditioning is reproduced by basic GW parameterization schemes.From a quantitative perspective, however, the GW parameterizations underestimate the forcing by up to ∼50% in the polar upper stratosphere and mesosphere, given that the GWpermitting GCM reproduces GW forcing accurately.We confirmed that the assimilation increments supply deficiencies of subgrid-scale wave forcing in the GCM with GW parameterizations.This result indicates that improvements in GW parameterization schemes are important particularly for more accurate prediction of the time evolution of the eastward jets in the middle atmosphere related to the SSW preconditioning by numerical models.
An informative by-product of the comparison made in this study is a suggestion that underestimation of GW forcing by GW parameterizations may be one of the reasons for the relatively large discrepancy in the horizontal winds in the equatorial region near the stratopause between high-top analysis data sets (Kawatani et al., 2020;McCormack et al., 2021).Further research on GWs using high-top data sets would provide useful information for improving SSW forecasts and deepening our understanding about dynamical roles of GWs in the middle atmosphere.

Figure 1 .
Figure 1.Latitude-pressure sections of (a) zonal-mean zonal wind u averaged over the boreal winters (DJF) from 2004/2005 to 2020/2021 and the spatial patterns of the (b) EOF1, (c) EOF2, and (d) EOF3 of u in 15°N-90°N, z = 15-90 km in DJF multiplied by the standard deviation of the respective principal components.The contour intervals are 10 m s 1 in panel (a) and 2.5 m s 1 in panelss (b) and (c).The percentages shown on the top of panels (b)-(d) represent the proportions of the contribution of each component to the total variance.

Figure 2 .
Figure 2. (a-j) Time series of the normalized empirical orthogonal function scores (PC1, PC2, and PC3) during 10 boreal winters with major SSWs.Vertical lines denote the onset dates of the major SSWs.Gray shadings indicate the periods of the poleward jet shift (see the main text for the definition).(k) Lead-lag composite time series of the principal components from the SSW onset dates.

Figure 3 .
Figure 3. Latitude-pressure sections of the linear regressions on the PC2 (i.e., normalized EOF2 score).Panels (a)-(c) display the results for (a) u (i.e., u EOF2 ), (b) Eliassen-Palm (EP) flux (vectors) and its divergence (colors) of PWs, and (c) parameterized gravity wave forcing, respectively.For ease of visibility, the EP flux denoted by the vectors is divided by the square root of the pressure, which is the same in the following figures of EP fluxes.Contours in panels (b) and (c) show u EOF2 with the same interval (2.5 m s 1 ) as panel (a).Panels (d)-(f) show the same as panels (a)-(c) but for 5 days prior to the PC2.
), GWF p + Δu t takes large positive values and spreads over Journal of Geophysical Research: Atmospheres 10.1029/2023JD039881 OKUI ET AL.

Figure 4 .
Figure 4. Latitude-pressure sections of (a, f, k) u, (b, g, l) the Eliassen-Palm flux of PWs (vectors) and its divergence (PWF; colors), (c, h, m) parameterized gravity wave (GW) forcing, (d, i, n) the sum of GW forcing and assimilation increment of u acceleration and (e, j, o) difference in daily averaged u from the previous day to the respective day.Panels (a)-(e), (f)-(j), and (k-o) show the results for 20, 21, and 23 December 2018, respectively.Contours show u on each day with a contour interval of 10 m s 1 .

Figure 5 .
Figure 5. Latitude-pressure sections of nondimensionalized refractive index squared a 2 n 2 s for stationary waves having zonal wavenumber 1 averaged over (a) 19-21 and (b) 22-24 December 2018.Contours show u averaged over the same periods with a contour interval of 10 m s 1 .

Figure 6 .
Figure 6.Latitude-pressure sections of Eliassen-Palm (EP) flux and its divergence associated with (a)-(c) PWs and (d)-(f) resolved gravity waves (GWs) (GWF r ) in the GW-permitting JAGUAR model.Panels (a, d), (b, e), and (c, f) show the results for 20, 21, and 23 December 2018, respectively.Contours show u on each day with a contour interval of 10 m s 1 .Vectors in panels (a)-(c), and (d)-(f) represent the EP flux for PWs and GWs, respectively.

Figure 7 .
Figure 7.The momentum flux spectra in the k and ω space (Re[u′w′*]) at 60°N, (a) z = 30 km and (b) 70 km for 19-22 December 2018.Colored curves represent contours of c (z) g of gravity waves having respective k and ω.The values of c (z) g are shown by numerals near the contours.The u and N values used as the background wind U and Brunt-Väisälä frequency N for the calculation of c (z) g are u = 23.0 m s 1 and N = 2.04 × 10 2 s 1 for panel (a) and u = 13.1 m s 1 and N = 1.77 × 10 2 s 1 for panel (b), obtained for the same period and region.The black dashed line indicates the critical level (i.e., u = c).The gray dashed line in panel (a) shows c = 10 m s 1 .