1. Introduction
Recently, Kodera et al. (2008) revealed observational evidence of downward-propagating planetary waves from the stratosphere into the troposphere just after a stratospheric sudden warming (SSW) event in early March 2007. Based on a detailed analysis of the three-dimensional propagation of the planetary-scale wave packet, they showed that the wave packet incoming from the troposphere over Eurasia gradually deflects eastward and then propagates downward into the troposphere over the American–Atlantic sector. Based on several case studies, Kodera et al. (2013) also elucidated that the downward-propagating planetary wave packet promotes the formation of tropospheric blocking over the Pacific sector as well as a trough over eastern Canada. Thus, the downward-propagating stratospheric planetary wave exerts a significant zonally asymmetric influence on the tropospheric circulation. This contrasts with the zonally symmetric influence of a downward-migrating annular-mode anomaly indicated by Baldwin and Dunkerton (1999, 2001).
Although there is robust observational evidence of the downward propagation of stratospheric planetary waves, its occurrence mechanism and the related dynamics still remain controversial. By conducting a zonal wavenumber decomposition analysis on observed downward-propagating planetary waves in the stratosphere, Perlwitz and Harnik (2003) argued that the existence of a reflecting surface for vertical propagation at around 5 hPa is key to inducing the downward propagation. On the other hand, Tomikawa (2010) suggested that the overreflection of the zonal wavenumber (WN)-1 component at the critical surface is important in producing downward-propagating planetary waves.
However, since these arguments are based on the quasi-linear framework of wave and zonal-mean flow interaction, the applicability to downward-propagating events frequently observed just after SSWs with a highly distorted polar vortex is still uncertain. In fact, a limitation of the quasi-linear framework based on the zonally symmetric basic state was suggested by Noguchi et al. (2016), which assessed the predictability of an SSW event in January 2009 with vortex splitting. By conducting a series of ensemble reforecasts using an AGCM, they showed that predicted planetary waves are absorbed in the stratosphere for a group of ensemble members in predicting the vortex splitting, whereas they propagate downward for another group. However, the zonal-mean basic states of both groups in the upper stratosphere satisfy a preferable condition for the wave reflection. Hence, they insisted that zonal asymmetry of the basic state would be more relevant to the different propagating properties of the planetary wave during this event.
The predictability of the downward-propagation event of the planetary wave must also be revealed by taking account of its critical influence on surface weather (Kodera et al. 2008, 2013). So far, no attempts have been made to assess the practical predictability of the downward propagation of the planetary wave using sophisticated AGCMs. Hence, in this study, we have conducted a series of ensemble reforecast AGCM experiments using the same experimental setting of Noguchi et al. (2016) for the downward-propagation event of planetary waves in March 2007, for which the observed time evolution was revealed by Kodera et al. (2008). It is noteworthy that the downward propagation becomes most prominent on 5 March just after a displacement-type SSW event in which the North Pole temperature had attained local maxima on 24 February and 4 March (Kodera et al. 2008, 2013).
Analyses on all ensemble members also help us to speculate as to the occurrence mechanism of the targeted atmospheric event. In fact, atmospheric anomalies during some SSW events have been clarified in some studies using ensemble forecasts. Mukougawa et al. (2005) pointed out the importance of the persistence of tropospheric blocking over the Atlantic in the occurrence of a vortex-displacement-type SSW in December 2001, which has been also confirmed by a complementary numerical experiment using an AGCM (Mukougawa et al. 2007). Noguchi et al. (2016) detected the important role of a preexisting stratospheric circulation anomaly in the occurrence of vortex splitting during the SSW event in January 2009. Here, through analysis on an ensemble reforecast experiment, we aim to build a plausible hypothesis for the emergence mechanism of downward-propagating planetary waves in the stratosphere in March 2007.
Constructing a robust dynamical framework with which to understand the predictability change during the downward-propagating event in March 2007 is another important challenge to be addressed in this paper. To accomplish this formidable task, we must first confirm the correspondence between the practical and intrinsic predictability. Although the practical predictability of the real atmosphere is easily assessed by the ensemble-mean forecast error or spread of ensemble forecasts with many members, we cannot necessarily guarantee that the obtained practical predictability corresponds to the intrinsic predictability, which represents the instability of the atmospheric state trajectory in phase space. The intrinsic predictability is quantified by a local divergent rate of nearby trajectories in phase space based on a tangent linear equation describing the evolution of the infinitesimally perturbed trajectory in the vicinity of the basic trajectory, such as the Lorenz index (e.g., Mukougawa et al. 1991), which is given by singular values of the error matrix (Lorenz 1965) and the local Lyapunov exponent (e.g., Nese 1989). The Lyapunov exponents represent global characteristics of the intrinsic predictability averaged over the flow attractor. However, since the real atmospheric state trajectory is embedded in infinite-dimensional phase space, it is practically impossible to compute these indices for the real atmosphere. Therefore, we have to compromise by using simplified models with a rather coarse horizontal resolution representing a limited class of atmospheric motions to estimate the intrinsic predictability. For example, in the pioneering work of Kimoto et al. (1992), which revealed the predictability change during a tropospheric blocking event, they used a truncated barotropic model to estimate intrinsic predictability. However, they recognized that the time evolution of the computed Lorenz index had only weak correspondence with that of the practical predictability assessed from operational forecasts. Hence, the dynamics relevant to the predictability change during the blocking event has not been clarified.
The weak correspondence between the practical and intrinsic predictability noted by Kimoto et al. (1992) is partially due to the fact that the barotropic model they used to estimate the intrinsic predictability does not represent energetic baroclinic instability in the troposphere. However, the barotropic model would be suitable to examine the intrinsic predictability of stratospheric circulation. This expectation is plausible since many important dynamical aspects of the stratospheric circulation have been revealed using the barotropic model (e.g., Matsuno and Hirota 1966; Hirota 1967; Juckes and McIntyre 1987; Manney et al. 1991; Yoden and Ishioka 1993) owing to the predominance of planetary-scale dynamics and the absence of baroclinic instability in the stratosphere. Hence, we try to pursue the possibility that the practical predictability change of the stratospheric circulation during the downward-propagating event in March 2007 can be described by the time evolution of the intrinsic predictability of the stratospheric circulation assessed by a nondivergent barotropic vorticity equation. This attempt could be an invaluable milestone in revealing the dynamics of the predictability change of the stratospheric circulation during this event.
The remainder of this paper is organized as follows. Section 2 will describe the reanalysis dataset along with the experimental setup of the conducted ensemble reforecast experiment. In section 3, we will show results of the regression analysis as well as the EOF analysis using all ensemble members to elucidate the precursory signal for the downward propagation of the stratospheric planetary wave in March 2007. Based on this examination, we will postulate the following working hypothesis to explain the emergence of the precursor: the barotropic instability inherent to the distorted zonally asymmetric polar vortex in the upper stratosphere by incoming planetary waves from the troposphere determines whether the planetary waves are eventually absorbed or emitted downward in the stratosphere. In section 4, the dynamical stability of the distorted polar vortex will be examined by eigenvalue analysis of a nondivergent barotropic vorticity equation on a sphere, and we will demonstrate the relevance of the working hypothesis to the dynamics and predictability of downward-propagating planetary waves observed in March 2007. Section 5 will conclude the paper.
2. Data and experimental setup
a. Data
The 6-hourly ERA-Interim dataset (Dee et al. 2011) was used for both the analysis and in constructing initial conditions for the ensemble reforecast experiments. The horizontal grid interval is 1.25° longitude by 1.25° latitude with 37 vertical pressure levels extending up to 1 hPa. To conduct the analysis, we first derived daily means from four 6-hourly values for each day.
b. Experimental setup of ensemble forecast
As in Noguchi et al. (2016), ensemble forecast experiments were conducted using the Meteorological Research Institute Ensemble Prediction System (MRI-EPS) (Yabu et al. 2014). An AGCM used in MRI-EPS is MRI-AGCM, which was developed based on the NWP model of the Japan Meteorological Agency (JMA) (Mizuta et al. 2006, 2012).
The MRI-AGCM has a horizontal resolution of TL159, corresponding to a horizontal grid interval of about 110 km, and has 60 vertical levels of which four levels are above 1 hPa with the top boundary at 0.1 hPa. For dynamical dissipative processes in the stratosphere, the MRI-AGCM implements an orographic gravity wave drag (Palmer et al. 1986; Iwasaki et al. 1989), Rayleigh friction above 50 hPa, and fourth-order horizontal diffusion (JMA 2013). The damping time scale of the horizontal wind due to the gravity wave drag and the Rayleigh friction is estimated to be about 10 and 3 days at 1 and 0.1 hPa, respectively (K. Yoshida, MRI, 2016, personal communication). The ozone concentration was given by the zonal-mean climatology, while the sea surface temperature (SST) was specified by the monthly mean climatology added to a fixed observed anomaly at the initial time of forecasts (see also Noguchi et al. 2016).
Initial perturbations necessary for the ensemble forecast were generated by the MRI-EPS using a breeding of growing mode (BGM) method (Toth and Kalnay 1993). Note that generated perturbations in MRI-EPS are approximated as baroclinically unstable modes in the troposphere, and their amplitude was artificially reduced by a factor proportional to pressure at each vertical level above 100 hPa (Yabu et al. 2014). With addition and subtraction of the generated perturbations to the ERA-Interim dataset, we prepared 24 perturbed initial conditions except for the unperturbed condition comprised only of the ERA-Interim. Then, we conducted each ensemble forecast of 25 members starting at 1200 UTC every day from 20 February to 5 March 2007. The forecast period is 60 days. Daily mean prediction data on 2.5° by 2.5° horizontal grids with 38 vertical pressure levels with a top at 0.4 hPa constructed from 6-hourly model outputs were examined.
3. Ensemble forecast experiment
a. Predictability of the event
First, we examine the predictability of the downward-propagation event of stratospheric planetary waves observed in early March 2007 using the conducted ensemble forecast experiments. Figure 1 shows the meridional profile of zonal-mean zonal wind 
Meridional distribution of zonal-mean zonal wind (contours; m s−1) and E-P flux (arrows; kg s−2) on day 0 (5 Mar 2007). (a) The analysis (ERA-Interim). (b) The ensemble-mean prediction starting from day −10 (23 Feb 2007). Contour interval is 5 m s−1. Positive (negative) values are indicated by reddish (bluish) colors. E-P flux vectors above 125 hPa are shown and their magnitude is scaled by the inverse of the pressure. Arrows at the right bottom indicate the scale of vertical and meridional components of E-P flux at 1000 hPa. The thick horizontal line at 100 hPa indicates the latitude band used for the definition of EPz100.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0330.1
Time evolution of EPz100 (105 kg s−2). Blue lines indicate the analysis, black lines the prediction of each ensemble member, and red lines the corresponding ensemble mean. The date corresponding to day 0 (5 Mar 2007) is denoted by the vertical line in each panel and the initial date of the forecast is shown in the left-bottom corner of each panel.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0330.1
The predictability of the downward-propagating event is assessed by the time evolution of the predicted EPz100 (Fig. 2). The predicted ensemble mean (red line) of EPz100 on 5 March, when the observed EPz100 (blue line) attains its largest negative value of this period, becomes negative for forecasts starting after 25 February (Fig. 2d). Hereafter, 5 March 2007 is referred to as day 0. Moreover, ensemble spread of EPz100 on day 0 gradually decreases for forecasts starting from day −7 and day −6 (Figs. 2e and 2f). Hence, the predictable period of the downward-propagating event is roughly estimated as 1 week, which is quite short compared with that of a displacement type SSW event in December 2001 (Mukougawa et al. 2005) but is comparable with that of a vortex-splitting-type SSW event in January 2009 (Noguchi et al. 2016).
b. Regression analysis
To reveal a precursory signal for the downward-propagating event in March 2007, we conducted regression analyses with respect to the predicted EPz100 on day 0 using all (25) ensemble members run from the initial condition of day −10 (23 February 2007). In this analysis, the 25 different ensemble-member fields for different days are regressed against EPz100 on day 0. Figure 2b shows that the number of ensemble members predicting a positive EPz100 on day 0 is almost the same as that predicting a negative EPz100 for this ensemble forecast. Hence, we could obtain unbiased regression by using these ensemble members. Note that the polarity of the obtained regression maps in the following figures is reversed to emphasize the downward propagation with a negative EPz100.
First, we examined regression of the predicted E-P flux and its divergence with respect to EPz100 on day 0. The positively (negatively) composited fields of the ensemble mean and the regression (with the opposite sign) from day −3 to day 0 are shown in the top (middle) row of Fig. 3. Hence, the top (middle) row represents the time evolution of ensemble members that predict downward (upward)-propagating planetary waves in the stratosphere on day 0. On day −3, it is found that both composite fields of E-P flux and its divergence are almost the same, and the E-P flux vectors are directed upward and equatorward in the stratosphere. Hence, it could be suggested that a strong divergent region of E-P flux in the lower stratosphere poleward of 50°N does not play an important role for the occurrence of the subsequent downward-propagating planetary waves. In fact, the regressed E-P flux divergence in the lower stratosphere has no significant relationship with EPz100 on day 0 as shown in the bottom panel in Fig. 3a. On day −2, we also detect another lower-stratospheric divergent region of E-P flux below 50 hPa around 50°–60°N for both composites (top and middle panels in Fig. 3b), which gradually shifts equatorward and upward. Hence, the downward propagation of stratospheric planetary waves on day 0 has no significant relationship with this lower-stratospheric divergence, too (see bottom panel in Fig. 3b). On the other hand, a region with divergence of E-P flux anomaly that has significant relationship with the downward propagation on day 0 appears on day −2 in the middle and upper stratosphere above 50 hPa around 60°–70°N (bottom panel in Fig. 3b). The downward E-P flux anomaly also emerges there although its magnitude is tiny in the upper stratosphere (bottom panel in Fig. 3b). Then, the significant region gradually extends over the whole stratosphere in high latitudes (bottom panels in Figs. 3c and 3d). We can also detect a large difference in E-P flux divergence field in the middle and upper stratosphere poleward of 60°N between both composites (top and middle panels in Figs. 3c and 3d): in the positive (negative) composite, divergence (convergence) of E-P flux prevails there. Thus, in the following, we will focus on the generating mechanism of the precursory signal detected by the E-P flux diagnosis residing in the stratosphere above 50 hPa before day −2, which is significantly related to the occurrence of the downward propagation of stratospheric planetary waves on day 0.
(top) Composited fields of the ensemble mean and the regression of the predicted E-P flux (arrows; kg s−2) and its divergence (contours; 10−7 kg s−2 m−2) from day −3 to day 0 with respect to EPz100 on day 0. (middle) As in (top), but for composited fields of the ensemble mean and the regression of the predicted E-P flux and its divergence with the opposite sign. (bottom) As in (top), but for regressed anomalies of the predicted E-P flux and its divergence. The polarity of the regressed field is reversed. Regressed E-P flux vectors are shown if either the meridional or vertical component of the regressed E-P flux is statistically significant at the 95% level. Regions where the regressed divergence anomaly has a statistical significance at the 95% (99%) level have lighter (darker) shading. Arrows at the right bottom of each row indicate the scale of vertical and meridional components of E-P flux. Contours are drawn at ±1, ±5, ±10, ±50, and, ±100 (×10−7 kg s−2 m−2) for all panels. E-P flux vectors above 150 (200) hPa are shown for (top) and (middle) [(bottom)].
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0330.1
Regressed anomalies of the zonally asymmetric geopotential height field averaged over 60°–70°N shown in Fig. 4 also support the importance of the precursory signal in the upper stratosphere in exciting the downward-propagating planetary waves. The ensemble-averaged eddy field of the forecast starting from day −10, as shown by contours in Fig. 4, has a westward phase tilt with altitude in the Eastern Hemisphere, corresponding to the upward propagation of the wave packet, as in the observation (Kodera et al. 2008). Then, the wave packet propagates eastward in the upper stratosphere in accordance with the gradual amplification of a ridge around the date line and attenuation of the trough over Europe in the stratosphere. In contradiction to the observation, the quasi-stationary ridge at 180°E has an equivalent barotropic structure with no vertical phase tilt. Hence, the apparent downward propagation of the wave activity is absent for the ensemble-mean prediction on day 0 in the stratosphere, as shown in Fig. 1b. Note that the WN-1 component with maximum amplitude around 5 hPa dominates the ensemble-mean eddy field in the stratosphere. On the other hand, the WN-2 structure prevails over regressed anomalies of eddy components with respect to EPz100 on day 0, especially in the stratosphere of the Western Hemisphere after day −2, as shown by color shading in Fig. 4. A quasi-stationary positive anomaly centered around 240°E and with two negative centers located around 150° and 330°E is conspicuous in the stratosphere. Note that the positive anomaly is located just on the node of the ensemble-averaged eddy field. Moreover, the regressed eddy anomaly has an equivalent barotropic structure in vertical and attains its maximum amplitude in the upper stratosphere as the ensemble-mean eddy field. Before day −3, planetary-scale regressed anomalies with small amplitude are seen in the upper stratosphere, but eddy anomalies with a significant relationship to EPz100 on day 0 are apparently absent in the troposphere. A significant upward component of anomalous E-P flux at the tropopause is also absent during this period (bottom panel of Fig. 3a). Hence, these facts also support our perception that the dynamical origin of the downward-propagating planetary waves resides in the stratosphere.
Longitude–height cross section of the ensemble mean (contours; m) and the regressed anomaly (color shading; m) of the predicted eddy geopotential height averaged over 60°–70°N with respect to EPz100 on day 0. The arrows indicate vertical and zonal components of the anomalous 3D Plumb flux above 300 hPa (m2 s−2) associated with the composited (ensemble mean plus regressed anomaly) eddy height field. Contour interval is 100 m. Regions where the regressed anomaly has a statistical significance at the 95% level are shaded with different tones designated by the bottom color bar. The 3D Plumb flux is scaled by the inverse of the pressure. Arrows at the right bottom indicate the scale of longitudinal and vertical components of 3D Plumb flux at 1000 hPa.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0330.1
The vertical-propagating characteristic of stratospheric planetary waves is easily understood by the superposition of the regressed eddy anomaly on the ensemble-mean field owing to their almost 90° out-of-phase relationship. Figure 4 indicates that when the positive (negative) anomaly is located downstream of the stratospheric ridge of the ensemble-mean field, the phase line of the composited field tends to have eastward (westward) tilt with altitude in the Western Hemisphere, and the wave activity tends to propagate downward (upward) in the stratosphere. In fact, the resultant 3D Plumb flux (Plumb 1985) anomaly shown by arrows in Fig. 4, which is defined by the difference between the 3D Plumb flux computed for the composited field and that for the ensemble-mean field, tends to be directed downward in the Western Hemisphere. Hence, the emergence of the barotropic anomaly with dominant WN-2 structure is key for the downward propagation.
The polar stereographic map of the 5-hPa geopotential height field shown in Fig. 5 also reveals an amplification of the regressed WN-2 anomaly (color shading) in the upper stratosphere. After day −4 (Fig. 5b), the three anomaly centers along 60°N at 150°, 240°, and 330°E more or less uniformly increase in magnitude with an almost geographically fixed horizontal structure. Moreover, there is no apparent signal of the horizontal energy propagation, in contrast to the ensemble-mean field (contours), which is characterized by an extensively distorted polar vortex. In addition, since the anomaly centers are located at the same high latitudes as the ensemble-mean field, interference between the two fields occurs effectively. Such horizontally uniform amplification of the 5-hPa regressed anomaly with an almost barotropic structure in vertical (Fig. 4) suggests that dynamical instability of the upper-stratospheric circulation, rather than meridionally or vertically propagating waves from elsewhere, is the most likely cause of its amplification.
Horizontal distribution of the ensemble mean (contours; m) and the regressed anomaly (color shading; m) of the predicted 5-hPa geopotential height with respect to EPz100 on day 0. Contour interval is 100 m. Regions where the regressed anomaly has a statistical significance at the 95% level are shaded with different tones designated by the bottom color bar.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0330.1
c. Analysis on ensemble spread
We further examined the time evolution of the ensemble spread in order to explore the relevance of dynamical instability to the growing anomaly. The spread was measured by the root-mean-squared (rms) differences between each ensemble member and the ensemble mean for the predicted 5-hPa geopotential height field northward of 30°N. First, Fig. 6 shows the time evolution of the ensemble spread for each forecast. It is found that spreads for forecasts starting from 21 to 23 February have similar growth rates during the period from 28 February to 3 March; these are larger than those of the other forecasts. This is recognized as the steeper gradient of the spread curves for these forecasts in early March. Moreover, the amplification rate of the spread in the first 24 h of the forecast, which is assessed by the ratio of the spread at the 1-day forecast to the initial spread, attains a maximum of about 2.8 on 28 February, which also implies the quick divergence of nearby trajectories in phase space during this period. Such an enhanced growth of the spread would suggest existence of a “predictability barrier” for the prediction of the upper-stratospheric circulation.
Time evolution of the rms ensemble spread (m) for the 5-hPa geopotential height field northward of 30°N for each forecast (black lines). The scale is shown by the left ordinate. The abscissa denotes the initial date of each forecast. The red line corresponds to the forecast starting from day −10. The blue line indicates initial growth rate of spread plotted on the initial date of each forecast, which is assessed by the ratio of the spread of the 1-day forecast to the initial spread. Its scale is shown by the right ordinate.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0330.1
An EOF analysis on difference fields of each ensemble member from the ensemble mean reveals the pattern of greatest spread among members. Figure 7 shows the first EOFs of the difference field for the predicted 5-hPa geopotential height poleward of 30°N during a period from day −5 to day 0 for the ensemble forecast starting from day −10. The resemblance between Figs. 5 and 7 after day −3 is striking in that the first EOFs possess the predominant WN-2 structure as well as the 90° out-of-phase relationship with the ensemble-mean field. The geographical loci of centers of action in the first EOF are also fixed after day −2, which is similar to the quasi stationarity of the regressed anomaly seen in Fig. 5. Moreover, the first EOFs explain about 70% of the total variance of the field among ensemble members during this period. The associated percentage variance explained is indicated by the numeral at the top-right corner of each panel in Fig. 7. Hence, the analyses of the ensemble spread also enforce our supposition that the regressed anomaly shown in Fig. 5 is spontaneously developed as a result of the instability inherent to distorted polar vortex in the upper stratosphere.
The first EOFs of the difference field of each ensemble member from the ensemble mean for the predicted 5-hPa geopotential height northward of 30°N (m) on each prediction date of the forecast starting from day −10. Contour interval is 100 m. The corresponding variance explained is shown at the top right of each panel.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0330.1
4. Stability analysis using a barotropic model
a. Model and methodology
b. Eigenvalue problem for ensemble-mean field
The eigenvalues were obtained using a version of the model with a triangular truncation retaining N = 63 wavenumbers (T63), of which the spatial resolution is high enough for the convergence of obtained eigenvalues (Anderson 1991). The hyperviscosity coefficient 
Figure 8 shows growth rates 
Growth rate of unstable modes computed for the basic flow composed of the T21 truncated 5-hPa streamfunction of the ensemble-mean field on each prediction date of the forecast starting from day −10 (day−1). Red (black) circles correspond to stationary (transient) modes with a zero (nonzero) imaginary component of the eigenvalue.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0330.1
The fact that the spread has a somewhat larger growth rate than the fastest-growing mode can be understood by taking account of the contribution of the optimal excitation mechanism to the spread growth during a finite time interval (Mukougawa and Ikeda 1994; Hartmann et al. 1996). It is also interesting to note that the unstable modes obtained for ensemble-mean fields during this analysis period have much larger growth rates than those reported in previous literature, such as Frederiksen (1982) for the basic state of three-dimensional distorted stratospheric polar vortices in the Northern Hemisphere and Manney et al. (1991) for the Southern Hemisphere. Moreover, their growth rates are even larger than those of baroclinically unstable modes in the troposphere (Hirota 1968; Frederiksen 1983). It should be also noted that stationary modes prevail over transient modes during the maximum period of growth rates in early March (Fig. 8).
The very large growth rates of unstable modes for the upper stratosphere can also be seen in Fig. 9, which shows growth rates of unstable modes at each pressure level of the predicted ensemble-mean flow on day −3 of the forecast starting from day −10. Growth rates of unstable modes for the upper-stratospheric flow are more than twice those for the tropospheric and lower-stratospheric flows. In addition, the corresponding e-folding times of unstable modes in the upper stratosphere are much shorter than the time scale of the dissipative processes incorporated in MRI-AGCM (see section 2b). Hence, these unstable modes play an invaluable role in the time evolution of the forecast spread of the upper-stratospheric flow field.
As in Fig. 8, but for the basic flow composed of the T21 truncated streamfunction field on day −3 at each pressure level for the ensemble-mean prediction starting from day −10.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0330.1
Horizontal structures of the obtained unstable modes shown in Fig. 10 also support the relevance of these modes to the forecast spread and the regressed anomaly. The top panels show 5-hPa ensemble-mean flows specified as the basic flow 
(top) Horizontal structure of the basic flow given by the T21 truncated 5-hPa streamfunction field (107 m2 s−1) of the ensemble-mean prediction on days (a) −9, (b) −5, (c) −4, and (d) −3 for the forecast starting from day −10. (middle),(bottom) Streamfunction field for the first and second unstable modes computed for the basic flow. The first and second numbers in parentheses above each panel indicate the growth rate (day−1) and the period (days) of the unstable mode, respectively. Stationary modes with a zero imaginary component of the eigenvalue are designated by the period of the infinity.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0330.1
Thus, the characteristics of the obtained unstable modes for the 5-hPa streamfunction field of the ensemble-mean forecast, characterized by a highly distorted polar vortex, strongly support our hypothesis that the barotropic instability inherent to the upper-stratospheric flow plays a significant role in producing the precursory disturbance that promotes the subsequent downward-propagating planetary waves.
5. Summary and discussion
To reveal the predictability of a downward-propagating event of planetary waves in the stratosphere that was observed in early March 2007, we conducted a series of ensemble forecasts starting every day using the MRI-AGCM. It was found that the predictable period of the downward-propagation event is about 7 days. Regression analysis using all members of an ensemble forecast starting 10 days before the occurrence of the downward propagation revealed that amplifying quasi-stationary planetary-scale anomalies with a predominant WN-2 barotropic structure in polar regions of the upper stratosphere are significantly related to the subsequent downward-propagating event. Moreover, the upper-stratospheric anomalies are 90° out of phase with the ensemble-mean field. On the other hand, there is no significant precursory signal in the troposphere. Thus, the regressed upper-stratospheric anomalies decisively determine whether planetary waves incoming from the troposphere eventually propagate upward or downward by changing their vertical structure, which depends on the polarity of the regression. Namely, when a positive height anomaly of the regressed field lies to the east (west) of the anticyclonic vortex in the ensemble-mean field, planetary-scale stratospheric disturbances in the composited field of the regressed anomaly and ensemble-mean have an eastward (westward) phase tilt with altitude and propagate downward (upward).
Furthermore, an EOF analysis on difference fields of each ensemble member from the ensemble mean of the predicted 5-hPa geopotential height indicated that the growing quasi-stationary planetary-scale disturbances seen in the regressed anomaly have a similar horizontal structure to the pattern of greatest spread among nearby trajectories representing the time evolution of the upper-stratospheric circulation in phase space around the ensemble-mean forecast. In addition, the growth rate of the ensemble spread in the upper stratosphere reaches a maximum prior to the occurrence of the downward-propagation event. The upsurge of the spread growth during this period also suggests the existence of a predictability barrier for the prediction of the upper-stratospheric flow. Hence, we hypothesized that the amplifying barotropic precursory disturbance with quasi-stationary properties emerges as a result of the barotropic instability inherent to the upper-stratospheric circulation.
In fact, the eigenvalue analysis for basic states constituting the zonally varying 5-hPa streamfunction field of the ensemble-mean forecast using a nondivergent barotropic vorticity equation on a sphere supports our hypothesis—that is, the obtained most unstable mode attains a local maximum in growth rate with an e-folding time of about 1.3 days and becomes stationary prior to the downward-propagation event. In addition, the growth rate is almost comparable to the amplification rate of the spread during the initial forecast period. Hence, the predictability barrier is understood to be the upsurge of the dynamical instability of the distorted polar vortex. Moreover, the fastest-growing unstable stationary modes with large growth rates prior to the downward-propagation event have predominant WN-1 and WN-2 components in polar regions, the structure of which has a resemblance to the regressed upper-stratospheric anomaly fields in that they satisfy the 90° out-of-phase relationship with the ensemble-mean field. Thus, in early March 2007, the barotropic instability inherent to the distorted polar vortex in the upper stratosphere, which is forced by incoming planetary waves from the troposphere, determines whether the planetary waves are eventually absorbed or emitted downward in the stratosphere.
The extremely large growth rate of the obtained barotropically unstable mode is due to the distorted polar vortex in the upper stratosphere. Figure 11 shows the horizontal structures of unstable modes with the two largest growth rates for the basic flow composed of 5-hPa zonally symmetric components (Fig. 11a) of the T21 truncated ensemble-mean prediction field on day −3. Each mode is respectively composed of transient WN-2 and WN-3 components and is centered around 50°–60°N, corresponding to a region with a negative absolute vorticity gradient (Fig. 11a). Hence, these are conventional barotropic unstable modes for zonal flows (Ishioka and Yoden 1992). The growth rates of these modes (0.07 and 0.03 day−1) are less than one-tenth those of the unstable modes for the zonally varying basic flow (Fig. 8) and even smaller than the damping time scale of at most several days for the imposed Rayleigh friction in the MRI-AGCM (cf. section 2). Thus, the conventional barotropic instability of stratospheric zonal winds is totally irrelevant to the dynamics, as already noted by Hirota (1967). However, the zonally varying component of the polar vortex is essential to the vigorous barotropic instability of the upper-stratospheric flow during this period.
(a) Meridional profile of the 5-hPa zonal-mean zonal wind (black line; m s−1) and the corresponding meridional gradient of absolute vorticity (red line; 10−11 m−1 s−1) on day −3 for the ensemble-mean forecast starting from day −10. Horizontal structure for (b) the first and (c) the second unstable modes of the zonally symmetric basic state are also shown. The first and second numbers in parentheses at the top right of (b) and (c) indicate the growth rate (day−1) and the period (days) of the unstable mode, respectively.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0330.1
Next, we argue the relevance of the predicted zonally varying ensemble-mean field to the basic state in the stability analysis. The conventional instability theory has been utilized to reveal the origin of observed fluctuations in the circulations of the real atmosphere. Hence, the eigenvalue analyses have been conducted for the basic flows without fluctuations (Pedlosky 1987), including a highly idealized zonal flow (e.g., Charney 1947; Eady 1949). Although adequately time-averaged zonally varying flows such as climatological flows have been also adopted as the basic state (e.g., Simmons et al. 1983; Anderson 1991), we should realize that the observed fluctuations cannot be attributed to an instability process of such time-averaged states because those states are inevitably affected by the presence of the targeted unstable fluctuations if they exist (Pedlosky 1987). However, the relevance of the ensemble-mean field to the basic state is not of concern in this study. This is because we utilize an instability theory to explain not the dynamical origin for observed fluctuations, but that for the direction of most spreading among the ensemble members. In this framework, the ensemble-mean field is the most natural choice for the basic state with which to conduct the stability analysis. We have indeed succeeded in understanding the presence of the predictability barrier and the origin of the regressed anomaly in the upper stratosphere within the framework of the barotropic instability of the ensemble-mean field representing the distorted vortex.
It is also tempting to conduct a similar stability analysis by specifying time-averaged observed flows as the basic state to explain the dynamics of the upper-stratospheric flow prior to the downward-propagation event. This is because we could predict the forecast skill of the upper-stratospheric circulation without conducting any ensemble forecasts if the time-averaged observed flows have a similar stability property to the ensemble-mean field. Figure 12 shows the growth rate of the fastest-growing modes for the 3-day running-mean observed 5-hPa streamfunction field with the T21 truncation. The time averaging was conducted to diminish transient disturbances with high frequency. We detect a local maximum of the growth rate during a period from day −7 to day −5. However, the peak is attained 3 days earlier and with a magnitude slightly smaller than that in Fig. 8 for the ensemble-mean field. Hence, the predictability barrier is qualitatively detectable as the upsurge of the dynamical instability of the observed polar vortex.
As in Fig. 8, but for the basic flow composed of the T21 truncated 3-day running-average 5-hPa streamfunction field of the ERA-Interim data.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0330.1
Unstable modes for the observed flows shown in Fig. 13 also have some horizontal structures similar to those for the ensemble-mean field (Fig. 10). On day −9, the two fastest-growing modes (Fig. 13a) are also characterized by zonally localized wave disturbances along the vortex edge similar to those in Fig. 10a because both basic flows are almost identical owing to a short forecast period of 1 day. During a period from day −5 to day −3, the fastest-growing modes become stationary or quasi stationary and their maximum amplitude occurs in polar regions as shown in Fig. 13; this is in contrast to those on day −9. These properties of unstable modes for the observed flows are also qualitatively similar to those for the ensemble-mean field. However, the detailed wave structure and phase relationship with basic flow are not identical in both kinds of unstable modes, indicating the sensitivity of unstable modes to the basic flow. Thus, the precognition of the predictability barrier by the stability analysis for the observed flows is promising but remains qualitative.
As in Fig. 10, but for the basic flow given by the 5-hPa streamfunction field of the ERA-Interim data.
Citation: Journal of the Atmospheric Sciences 74, 11; 10.1175/JAS-D-16-0330.1
The above argument on the basic state should be taken into account when evaluating the pioneering work of Kimoto et al. (1992) in predicting forecast skill during a tropospheric blocking event by computing the error growth rate based on a tangent linear barotropic equation about the observed tropospheric flows. The limited correspondence of the computed error growth rate to the spread of the operational prediction reported in their study would be eased by adopting the ensemble-mean field as the basic state when computing the error growth rate. A more important factor for limiting such correspondence results from the fact that besides the existence of a predominant baroclinic instability in the troposphere, the instability property of the troposphere is different from that of the upper stratosphere, as shown in Fig. 9. Figure 9 reveals that a few unstable modes with large growth rates dominate in the upper stratosphere, whereas many unstable modes with comparable growth rates exit in the troposphere. Hence, it is implied that in comparison with the upper-stratospheric trajectory, the tropospheric trajectory is embedded in a much-higher-dimensional phase space. Such low dimensionality of the upper-stratospheric dynamics would make it possible to describe the spread growth by a few fastest-growing modes in this study. It is also interesting to note that the implication of the difference in dimensionality between the troposphere and stratosphere is also in accordance with the relative importance of the stochastic (deterministic) error growth over the deterministic (stochastic) error growth in the troposphere (stratosphere), as revealed by Inatsu et al. (2013, 2015).
The relevance of the obtained unstable modes to the time evolution of the spread would be quantitatively assured by obtaining singular vectors, which are usually examined in predictability studies (e.g., Hartmann et al. 1996). However, we intentionally did not conduct singular vector calculations based on a tangent linear equation along the basic trajectory in this study. Instead, we conducted eigenvalue analysis about the instantaneous basic state. This is because the former analysis includes an artificial parameter of a time interval to assess the error growth, and the obtained growth rate depends on the assumed initial perturbations. Along with the fact that the singular vectors grow during a finite time interval even in a dynamically stable basic state (Mukougawa and Ikeda 1994), these artificial factors obscure the dynamical property inherent to the basic state and make it difficult to understand the amplifying mechanism of the ensemble spread. It is also noteworthy that in the context of the predictability study, the veracity of the stability analysis for the instantaneous basic flow is always assured regardless of the growth rate of the obtained unstable modes. Moreover, whether or not the basic flow is a stationary solution of the system does not impair the validity of the analysis since we aim to obtain the spreading directions of nearby trajectories around a point in phase space, which is given by the basic flow.
The relevance of the upper-stratospheric circulation anomaly to the lower-stratospheric circulation should also be quantitatively examined in further studies. For this purpose, it is necessary to conduct eigenvalue analysis and examine the evolution of the regressed upper-stratospheric anomaly using a multilayer model extending from the tropopause to the upper stratosphere. If the obtained 3D unstable modes have a barotropic planetary-scale structure with a considerable vertical extent that is similar to the regressed anomaly shown in Figs. 4 and 5, we can readily stretch the credibility of the barotropic instability of the upper-stratospheric circulation for the generation of downward-propagating planetary waves in the stratosphere. Reforecast experiments starting from initial conditions composed of the superposition of the ensemble-mean field and the regressed upper-stratospheric anomaly will describe the nonlinear evolution of the upper-stratospheric anomaly and its quantitative influence on the lower stratosphere and troposphere. A similar approach was used by Mukougawa et al. (2007) to examine the quantitative impact of Atlantic blocking in the troposphere on the warming of the polar stratospheric region associated with the subsequent SSW event.
Finally, stability analyses on distorted polar vortices in the stratosphere prior to the mature phase of SSW events using a nondivergent barotropic vorticity equation will reveal the important role of the dynamical stability of stratospheric circulation in determining the predictable period of SSW events, which considerably varies from several days to 2 weeks depending on the event (Tripathi et al. 2015; Ichimaru et al. 2016). For example, an SSW event with vortex displacement in December 2001 had a prolonged predictability of 2 weeks (Mukougawa et al. 2005), whereas an event with vortex splitting in January 2009 had a short predictable period of 6 days (Noguchi et al. 2016). The analysis of the spread of the conducted ensemble forecast can easily dictate the predictable period of each event but does not give us any dynamical interpretation as to what factor limits the predictability of each event and why the predictable period differs from one event to another. Hence, if the eigenvalue analysis on the stratospheric basic flow comprising predicted ensemble mean or observed field detects the presence of a predictability barrier with an upsurge of growth rates as in Figs. 8 and 12, we can expect that the in situ barotropic instability of the distorted polar vortex limits the predictable period of the SSW, except for the predictability of tropospheric planetary waves providing the wave forcing to set up the SSW. Our preliminary analysis on the barotropic instability of the polar vortex prior to SSW events in 2001 (2009) provides promising results in which such a predictability barrier is absent (present) prior to the former (latter) SSW event. These results will be presented in a separate paper.
Acknowledgments
A spectral transformation library ISPACK (http://www.gfd-dennou.org/arch/ispack/) developed by Prof. K. Ishioka was used to solve the eigenvalue problem for the nondivergent barotropic vorticity equation. The authors thank his kind instruction on how to use the ISPACK library. We would like to thank K. Yoshida for kindly providing information on the dissipative process of the MRI-AGCM. Thanks are also extended to three anonymous reviewers for their thoughtful comments. Figures were drawn using GFD-DENNOU Library. This study was partly supported by Grants-in-Aid for Scientific Research 24224011 and 26287115 funded by Japan Society of the Promotion of Science.
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