Three-dimensional evolution of ensemble forecast spread during the onset of a stratospheric sudden warming event in January 2006

Authors


Abstract

A singular value decomposition and a simple sensitivity analysis applied to an operational ensemble of monthly forecasts reveal that the prediction skill of a stratospheric sudden warming event observed in late January 2006 is particularly sensitive to uncertainty in the initial state in the vicinity of a developing synoptic-scale cyclone that was observed over the North Pacific more than two weeks prior to the peak of the event. For the first few days in the forecast, a local maximum of the forecast spread around the cyclone is translated eastward across the Pacific in association with observed downstream development of synoptic-scale disturbances. The spread then reaches the subpolar North Atlantic, where a blocking ridge was developing as the source of an upward-propagating Rossby-wave packet that gave rise to the deceleration of the stratospheric polar-night jet (PNJ). Following the observed wave packet, the maximum forecast spread is translated upward from the ridge and finally reaches the stratosphere, causing a large forecast spread in the PNJ deceleration. Copyright © 2010 Royal Meteorological Society

1. Introduction

During a stratospheric sudden warming (SSW) event, the stratospheric polar vortex warms up by several tens of degrees. A westerly polar-night jet (PNJ) concomitantly weakens and, in prominent events, reverses its direction within a few days. It is now established that SSW events, also regarded as manifestations of extreme negative events of the stratospheric Northern Annular Mode (NAM), act to turn the phase of the tropospheric NAM negative (Baldwin and Dunkerton, 1999, 2001; Limpasuvan et al.2004). Thus forecast errors of an SSW event may impact the forecast skill of the subsequent evolution of the tropospheric circulation, as suggested by ensemble-forecast experiments by Charlton et al. (2004, 2005) and by Mukougawa and Hirooka (2008).

In order to examine how the forecast skill of SSW events depends on the tropospheric circulation, a series of studies has recently been conducted using operational ensemble forecasts produced by the Japan Meteorological Agency (JMA). For example, Mukougawa et al. (2005) have shown that the formation of a blocking ridge over the North Atlantic was important for the enhancement of upward propagation of planetary-wave activity into the stratosphere and the warming of the polar stratosphere observed in December 2001. Mukougawa et al. (2007) have confirmed the above result through hindcast experiments with initial conditions in which the observed blocking-flow configuration was artificially perturbed. The JMA ensemble-forecast products were also used for examining the predictability of other prominent SSW events (Mukougawa and Hirooka, 2004; Hirooka et al., 2007).

In this paper, we present a case study on the predictability of a major SSW event observed in January 2006 (Manney et al., 2008). Prior to this event, the zonal mean PNJ had gradually weakened from late December 2005, associated with several events of enhanced upward propagation of planetary-wave activity from the troposphere. Nishii et al. (2009; hereafter NNM09) have found that a significant contribution to those events was made by zonally confined Rossby-wave packets propagating into the stratosphere from tropospheric quasi-stationary circulation anomalies. In one of these events that occurred just before the PNJ had become easterly in mid-January, a Rossby-wave packet that emanated from a tropospheric anticyclonic circulation anomaly over the subpolar North Atlantic was found to be the primary contributor to the enhanced injection of planetary-wave activity into the stratosphere. The tropospheric anticyclonic anomaly was amplified, with anomalous vorticity-flux divergence associated with synoptic-scale transient eddies along the Atlantic storm track. Their intensification over the Atlantic was found to follow the downstream development of synoptic disturbances from the North Pacific.

The aim of this study is to analyze the growth of initial errors in the individual members of the JMA operational ensemble forecasts over two weeks prior to the particular SSW event. By tracing the three-dimensional evolution of the ensemble-forecast spread and conducting a singular-value decomposition (SVD) analysis and a simple sensitivity analysis for the ensemble members, we show that large ensemble spread in the intensity of the stratospheric PNJ predicted in the SSW event arises largely from the forecast spread in the Rossby-wave packet emanating upward from the tropospheric anticyclonic anomaly over the Atlantic. We also show that the prediction skill of the anticyclonic anomaly is particularly sensitive to initial errors around the synoptic-scale cyclone developing over the North Pacific about two weeks before the peak of the SSW event. We confirm the relatively large analysis errors around this particular cyclone by investigating an experimental reanalysis data set that includes information on the accuracy of the analysis.

2. Data

The ensemble-forecast product utilized in this study was produced by the JMA Operational Monthly Forecast System (Japan Meteorological Agency, 2002). The resolution of the forecast model is T106 L40. The model has 13 vertical levels above the 100 hPa level, and the model top is placed at the 0.4 hPa level. Initial perturbation fields were constructed through a combination of the ‘breeding of growing modes’ (BGM) method (Toth and Kalnay, 1993) and the ‘lagged average forecasting’ (LAF) method (Hoffman and Kalnay, 1983). Routine forecasts are performed every Wednesday and Thursday. We use a particular set of the JMA forecasts with their initial fields for 11 or 12 January 2006. For each of the initial dates, six different (‘positive’) perturbation fields were generated by the BGM method. The ensemble members were doubled by adding the same perturbation fields but with their polarity reversed (‘negative’). With the unperturbed initial field included, the total number of ensemble members is thus 13 for each of the initial dates. Unless otherwise noted, the ensemble members initiated on 11 and 12 January are combined together in our analysis. This is because the members started on 11 January can be regarded as if they were initiated on 12 January with initial perturbations developing from those for 11 January. This interpretation is consistent with the basic idea of the LAF method. Forecast spread among the ensemble members is regarded as a measure of uncertainty in the forecast fields. The spread is defined at each grid point as the variance of a particular variable among the ensemble members about its ensemble mean.

Observational fields for the verification have been provided by the JMA Climate Data Assimilation System (JCDAS), as a continuation from the Japanese 25-year Reanalysis (JRA-25: Onogi et al., 2007). The ensemble prediction system operated around the SSW event of interest used the same forecast model as in the JRA-25 assimilation system, which is reported to have a cold bias in the lower and middle stratosphere (Onogi et al., 2007). While the cold bias in the assimilation system does not emerge in the JRA-25 and JCDAS data as long as observational data are available in the stratosphere, the ensemble forecasts may suffer from it in a significant manner.

3. Uncertainty in the predicted PNJ

Figure 1(a) shows the time series of 20 hPa zonal-mean zonal wind velocity for the individual ensemble-forecast members operated by the JMA with their initial conditions taken from the observations on 11 or 12 January 2006 (red and blue lines). The spread of the predicted wind velocity among the members increases rapidly with time in the period between 21 and 28 January, during which the observed 20 hPa PNJ (a dashed black line) underwent the most rapid deceleration during the SSW event. Correspondingly, the forecast spread is also large in the zonally averaged meridional-eddy heat flux at the 100 hPa level (Figure 1(b)). The heat flux is equivalent to a vertical component of the Eliassen and Palm (E–P) flux based on the quasi-geostrophic, beta-plane formulation (Andrews et al., 1987). The flux can therefore be regarded as a measure of the upward injection of planetary-wave activity from the troposphere that caused the PNJ deceleration. Many of the members can predict the amplification of the flux until around 17 January. While the observed heat flux remains as large as 20 K m s−1 for the next several days, however, most of the ensemble members apparently underestimate the flux, especially after 20 January. Presumably this underestimation causes the misforecast of the PNJ deceleration after 21 January in most of the ensemble members. One may notice that the heat flux predicted in several ensemble members is greater than its observational counterpart (Figure 1(b)), but they nevertheless underestimate the PNJ deceleration (Figure 1(a)). As will be discussed later, this could be due to the stratospheric cold bias in the numerical model used for the JMA ensemble forecasts we analyze in this study.

Figure 1.

(a) Time series of 20 hPa zonal-mean zonal wind (m s−1) averaged over 50°N–80°N. Solid lines without any symbols denote the individual ensemble members starting on either 11 or 12 January 2006. The thick dashed line denotes the observation based on the reanalysis data (JCDAS). The line with circles denotes the ensemble average, which is shown only in this panel. (b) The same as in (a), but for zonal-mean 100 hPa eddy heat flux averaged over 50°N–80°N, where eddy components of meridional wind velocity and temperature are defined as deviations from their zonal averages. (c) 10 400 m isolines of 250 hPa geopotential height predicted in the individual ensemble members for 14 January (solid lines) and the corresponding observation (dashed black line). (d) The same as in (c), but for 16 January. In each panel, individual ensemble members are classified into two groups based on a cluster analysis of 250 hPa geopotential height on 14 January over the region (180°–300°E, 20°N–60°N) as shown in (c). The clusters indicated by red (cluster A) and blue (cluster B) lines include 7 and 19 members, respectively. Thick, red dashed lines in each panel highlight the two members that apparently succeeded in the prediction of PNJ deceleration in (a).

4. Three-dimensional evolution of the forecast spread

On 16 January, just before the initiation of the rapid PNJ deceleration (Figure 1(a)), the lower-stratospheric polar vortex had already been weakened and split (Figure 2(a)). In fact, anticyclonic anomalies at the 50 hPa level were observed over the Arctic and the North American continent (contoured in Figure 2(d)). Here, anomalies are defined as local deviations from the climatology defined for each calender day based on the JRA-25 data for the period 1980–2003. Shading in Figure 2(d) indicates the horizontal distribution of the ensemble-forecast spread for 16 January in the lower stratosphere. Because its magnitude increases with time, the local spread for a given forecast day has been normalized by its instantaneous maximum within the domain (poleward of 20°N). It is apparent in Figure 2(d) that the large spread is confined to the subpolar North Atlantic. Although the area of the large forecast spread in 50 hPa height gradually expands horizontally, the spread maximizes over the subpolar North Atlantic throughout the period of rapid PNJ deceleration from 20–28 January (Figure 2(e) and (f)). As shaded in Figure 3, the ensemble-forecast spread over the North Atlantic and Europe (60°W–60°E) undergoes upward extension from the troposphere into the stratosphere at the initiation stage of the PNJ deceleration (note that the spread and anomaly are normalized by the pressure in the figure). On 16 January (Figure 3(a)), the particularly large spread (plotted with heavy shading) is confined to the vicinity of a tropospheric anticyclonic anomaly around 40°W associated with a prominent blocking-flow configuration over the North Atlantic (Figure 4(b)). The anomaly acted as the source of the Rossby-wave packet that propagated into the stratosphere (NNM09). In fact, as the tropospheric anticyclonic anomaly weakened, the stratospheric cyclonic anomaly over western Europe (around the Greenwich meridian) was amplified, with phase lines tilting westward with height (Figure 3(a) and (b)). These features are indicative of upward group velocity propagation of a Rossby-wave packet. Interestingly, unlike in the troposphere, the ensemble spread in the stratosphere is not maximized near the centre of the cyclonic anomaly at each level. Rather, it is maximized around the node lines of the height anomalies on the upstream and downstream side of the centre (Figure 3(b) and (c)). Furthermore, the cyclonic anomaly centre over Europe for each of the ensemble members, which is defined as the strongest negative deviation of 50 hPa height from the observed climatology and plotted with a black dot in Figure 2(d) and (e), tends to displace further from the observed cyclonic anomaly centre as the forecast spread extends upward into the stratosphere (from 16–20 January). We thus conjecture that the forecast spread extends into the stratosphere in association with the Rossby-wave packet that contributed to the SSW event, probably reflecting inconsistencies in its magnitude, group velocity and/or wavelengths simulated among the ensemble members. In fact, Figure 1(b) indicates that as the forecast spread extends into the stratosphere after 16 January, underestimation of the wave-activity injection into the stratosphere becomes apparent in many of the ensemble members.

Figure 2.

Top: 50 hPa geopotential height (contour interval 200 m) observed on (a) 16, (b) 20 and (c) 28 January 2006. Bottom: local spread of 50 hPa geopotential height predicted for (d) 16, (e) 20 and (f) 28 January 2006 (shaded lightly and heavily for 0.3–0.6 and values greater than 0.6, respectively), superimposed on observed 50 hPa height anomalies (contoured for ±80, ±240, ±400, ±560 m; dashed for negative values). The local spread has been normalized by its instantaneous maximum within the domain poleward of 20°N. In (d) and (e), black dots indicate the centres of 50 hPa cyclonic anomalies over Europe predicted in the individual ensemble members.

Figure 3.

Zonal cross-sections for 50°N of the local ensemble spread of geopotential height predicted for (a) 16, (b) 18 and (c) 20 January 2006, shaded lightly and heavily for 0.03–0.3 and values greater than 0.3, respectively. Superimposed with contours are observed geopotential height anomalies (±30, ±90, ±150, ±210, ±270 m; dashed for negative). At each pressure (p) level, the spread and anomaly are multiplied by (p/1000 hPa) and its square root, respectively.

Figure 4.

Top: as in Figure 2(a), but for 250 hPa height for (a) 12 and (b) 16 January 2006. Bottom: as in Figure 2(c), but for 250 hPa height anomaly for (c) 12 and (d) 16 January 2006.

As NNM09 pointed out, the tropospheric blocking ridge and associated anticyclonic anomaly over the subpolar North Atlantic (Figure 4(b) and (d)), from which the ensemble spread extends into the stratosphere (Figure 3), developed by 16 January in association with the downstream development of synoptic-scale disturbances from the North Pacific into the North Atlantic. The downstream development can be confirmed in the 250 hPa geopotential height and squared meridional wind fields shown in Figure 5. The height contours over the Pacific were almost zonal on 11 January (Figures 5(a)). The wave packet associated with meandering westerlies emerged on 12 January over the central Pacific and then propagated downstream until blocked by the developing anticyclonic ridge over the Atlantic around 15 January. Observed maxima of the squared 250 hPa meridional wind velocity associated with this wave packet (Figure 5(b)) exhibits a signature of group velocity propagation across the North Pacific and Atlantic from 12–16 January. As indicated in Figure 5, crude estimations of the group velocity (rectangle with solid line) and phase speed (dashed line) based on Figure 5(b) are about 31° and 7° in longitude per day, respectively. The former appears to correspond to the zonal group velocity of a baroclinic wave packet (Chang, 1993), while the latter is in agreement with the typical zonal phase speed of synoptic-scale baroclinic waves (Wallace et al., 1988).

Figure 5.

(a) Time sequence of 250 hPa geopotential height observed from (top) 11 to (bottom) 17 January 2006 over a domain (20°N–70°N). Contour intervals are 300 m and the thick contour corresponds to 10 400 m. (b) As in (a), but for 250 hPa meridional wind velocity squared. Contour intervals are 400 m2 s−2. (c) As in (a), but for the spread of predicted 250 hPa meridional wind velocity among the ensemble members for the initial date of 11 January. Contour intervals are 0.2. The spread has been normalized by its instantaneous maximum within the domain.

Downstream extension of the ensemble-forecast spread maximum in the troposphere has been reported by Anwender et al. (2008) and Harr et al. (2008), and is also observed in our analysis associated with the downstream development of migratory synoptic-scale disturbances. In fact, the local maximum of the ensemble spread in 250 hPa meridional wind velocity appears to translate zonally across the Northeastern Pacific rather slowly (about 8° per day; dashed line in Figure 5(c)). In addition, the downstream extension of the large forecast spread is hinted at, as indicated by a rectangle with a solid line in Figure 5(c). The speed of this downstream extension into the Atlantic is about 26° in longitude per day, which is similar to but slightly slower than the group velocity of the observed wave packet as estimated in Figure 5(b). Eastward translation of forecast errors across the North American continent associated with the downstream development of synoptic-scale disturbances has been reported by Langland et al. (2002). In the case they analyzed, the eastward propagation of the errors was faster than the phase speed of synoptic-scale troughs and ridges and slightly slower than wave-packet propagation, consistent with our analysis. Our analysis is also consistent with the result of Rabier et al. (1996), who showed that forecast errors propagate downstream from the North Pacific or North America as far as into Europe, in a comparison of forecasts started from routinely constructed analyses with those from initial values improved by the adjoint method.

On 12 January, one of the initial dates for the ensemble forecasts we analyzed, local maxima of the ensemble spread in 250 hPa geopotential height are found not only over the North Pacific but also over North Africa and South Asia (Figure 4(c)). However, the growth of the latter two spread maxima is less than that of the North Pacific maximum located just downstream of the particular cyclone that appears to be the origin of the ‘downstream development’ of the ensemble spread. The corresponding spread maximum is also observed in SLP over the North Pacific around the particular surface cyclone (not shown).

5. Cluster analysis among the ensemble members

The ‘group-velocity propagation’ of the forecast spread shown in the preceding section (Figure 5(c)) may represent inconsistency among the ensemble members in the phase and/or amplitude of eddy components developing downstream. To confirm this, a cluster analysis based on the Ward method (Ward, 1963) was applied to 250 hPa geopotential height over the Pacific predicted by the ensemble members for 14 January (Figure 1(c)). The analysis domain is (180°E–300°E and 20°N–60°N), and pattern proximity between a pair of clusters is measured by Euclidean distance.

After several trials, we found the ensemble members to be categorized into two clusters, one (Cluster A) characterized by relatively fast phase speed and small amplitudes of synoptic-scale disturbances (red contours in Figure 1(c)), and the other (Cluster B) by slower phase speed and larger amplitudes (blue contours). It is noteworthy that in Cluster A the blocking high over the North Atlantic, which is regarded as the source for the upward-propagating wave packet (Figures 3 and 4(b)), tends to develop more strongly with more pronounced poleward meanders of the westerlies than in Cluster B and in the observations as well (Figure 1(d)). This feature is particularly obvious in two members of Cluster A for which the PNJ deceleration is successfully predicted (red heavy dashed lines in Figure 1(a)) and the tropospheric westerlies are predicted with pronounced poleward meanders over the Atlantic (red heavy dashed contours in Figure 1(d)). These two members predict the upward wave-activity flux that is stronger than in the observations (Figure 1(b)), although the predicted PNJ deceleration is nevertheless comparable to the observations (Figure 1(a)). The fact that the ensemble members that predict realistic PNJ deceleration tend to overestimate the blocking intensity in the JMA forecast model suggests that an unrealistically strong upward flux of planetary-wave activity may be necessary in the forecast model for decelerating the PNJ as much as in the real atmosphere associated with the SSW event. This may be due to the cold bias in the forecast model (Onogi et al., 2007), which can introduce an artificial tendency for stratospheric temperatures to resist the warming induced by incoming planetary waves from the troposphere. These results imply that mechanisms for the amplification and propagation of the planetary waves in ensemble members that predict an SSW event successfully may not necessarily be the same as in the real atmosphere, if the forecast model has a non-negligible bias.

6. SVD analysis

A singular-value decomposition (SVD) analysis was applied to all 26 ensemble-forecast members, in order to confirm the relationship between the development of the Atlantic anticyclonic anomaly (Figure 4(b)) and initial errors or other forecast fields. An SVD analysis is often applied to time-varying fields of two variables, in order to extract their dominant covariability based on their temporal covariance matrix (Bretherton et al., 1992). In our application, SVD is applied to a matrix, the elements of which are the covariances between deviations of a given variable among the 26 members from its ensemble mean at a given location for a particular forecast time and the corresponding deviations of any variable at any location for any forecast time. In each of our applications, every field has been normalized by its standard deviation among the ensemble members, and therefore the matrix is eventually a cross-correlation matrix. We focus on the leading SVD mode that has the largest singular value and therefore explains the largest fraction (more than 40% in our applications) of the spatially integrated squared covariance.

A set of SVD analyses was performed between the ensemble of 250 hPa geopotential height fields forecast for 16 January over the subpolar North Atlantic (270°E–357.5°E, 40°N–85°N), where prominent blocking was observed, and the same or other forecast fields over the entire extratropical Northern Hemisphere (northward of 20°N). Each panel of Figure 6 shows a hemispheric map of the heterogeneous regression coefficient of a given variable for a particular forecast time calculated with the normalized expansion coefficients of the first SVD mode for the 26 members of the 250 hPa height over the North Atlantic forecast for 16 January. In each of the panels (a), (b), (d) and (e), we can identify regions where the forecast spread or initial error is strongly related to the spread of the blocking signature over the subpolar North Atlantic forecast for 16 January. In panels (f) and (g) of Figure 6, we can also identify regions where the spread of the blocking signature forecast for 16 January influences the forecast spread at a later forecast time most sensitively. The corresponding heterogeneous regression map (Figure 6(c)) of 250 hPa height forecast over the North Atlantic for 16 January with the expansion coefficients of the first SVD mode represents uncertainty in the forecast intensities of the blocking and concomitant upstream wave signal.

Figure 6.

Results of SVD analysis based on the JMA ensemble forecasts applied to the spread of 250 hPa height over the subpolar North Atlantic forecast for 16 January 2006. Panels (a)–(c): forecast spread of hemispheric 250 hPa height (a) for 12 January (contour interval (CI) is 10 m and zero contours are omitted), (b) for 14 January (CI 20 m) and (c) for 16 January (CI 100 m). Dashed thin contours are for negative values. Heavy and light shading denotes positive and negative correlations, respectively, the absolute values of which exceed 0.4. Bold lines in (a)–(c) indicate specific values of 250 hPa geopotential height observed on (a) 12 (10 100 m), (b) 14 (10 400 m) and (c) 16 (10 400 m) January. Panels (d)–(g): as in (a)–(c), but for the forecast spread of (d) 1000 hPa height for 12 January (CI 10 m), (e) 250 hPa meridional wind for 12 January (CI 10 m), (f) 100 hPa upward wave-activity flux for 18 January (CI 0.01 m2 s−2) and (g) 50 hPa height for 20 January (CI 50 m). Thick solid and dashed contours denote (d) 1000 hPa heights of 200 m and 0 m, respectively, and (e) 250 hPa meridional wind velocity of 15 m s−1 and − 15 m s−1, respectively, both observed on 12 January. The thick contour in (f) denotes an observed upward 100 hPa wave-activity flux of 0.06 m2 s−2 for 18 January. Panels (a)–(g) show heterogeneous regression maps that represent typical local deviations from the ensemble mean state in an ensemble member that predicts the North Atlantic blocking as stronger than the ensemble mean for 16 January. (h) As in (a) but for a heterogeneous regression map of 250 hPa geopotential height (CI 10 m) for 12 January with the normalized expansion coefficient of 50 hPa geopotential height forecast for 28 January, based on SVD analysis applied to the hemispheric forecast spread.

Figure 6(a) indicates that the large forecast spread in 250 hPa height over the North Atlantic for 16 January is particularly sensitive to initial errors in 250 hPa height over the North Pacific on 12 January. Specifically, a positive signal in Figure 6(a) is located slightly upstream of an anticyclonic ridge observed near the date line and downstream of the pressure trough that has been identified as the origin of the downstream development of synoptic disturbances (Figure 5(a) and (b)). The corresponding maximum sensitivity in 1000 hPa geopotential height on 12 January is located over the central North Pacific (Figure 6(d)). Again, this positive signal is located slightly upstream of a surface pressure ridge and downstream of the surface cyclone, as represented by heavy solid and dashed contours, respectively, in Figure 6(d).

This SVD result indicates that the synoptic-scale cyclone over the North Pacific is one of those circulation systems in the initial state to which the amplification of the North Atlantic blocking ridge in the forecast for 16 January is most sensitive. More specifically, the positive height signal just upstream of the pressure ridge suggests that the synoptic wave packet that consists of the pressure ridge and trough over the North Pacific tends to be shorter in zonal wavelength for the ensemble members with stronger development of the North Atlantic blocking on 16 January, and vice versa. A wave packet with shorter wavelength tends to accompany stronger fluctuations in meridional wind velocity and thus a stronger zonal component of wave-activity flux than one with longer wavelength. In fact, the corresponding positive signal of 250 hPa meridional wind is found over the observed southerlies over the North Pacific (Figure 6(e)). The enhanced meridional wind velocity yields stronger downstream development across the Pacific and North America, which can lead to enhanced development of the Atlantic blocking.

The same SVD analysis as above but for the hemispheric field of 250 hPa height forecast for 14 January (Figure 6(b)) reveals that significant positive and negative signals of geopotential height along the east coast of North America forecast for 14 January are sensitively related to the strength of the North Atlantic blocking forecast for 16 January. These positive and negative signals are located slightly upstream and downstream of an observed pressure trough off the east coast of Canada. A comparison with Figure 1(c) indicates that these signals are a manifestation of the sensitivity of the blocking ridge development to the longitudinal position of the pressure trough to its upstream that constitutes the wave packet propagating from the Pacific.

The same SVD analysis with the forecast 250 hPa height over the subpolar North Atlantic was also applied to the upward fluxes of Rossby-wave activity (Plumb, 1985) evaluated for 18 January at the 100 hPa level based on the individual forecast members (Figure 6(f)). The analysis indicates that the wave-activity flux into the stratosphere from the tropospheric anticyclonic anomaly over the North Atlantic tends to increase with the amplitude of the anomaly (Figure 6(f)). Taking into consideration the fact that the upward wave-activity flux formulated by Plumb (1985), if zonally averaged, is equivalent to the upward E–P flux, this result is in agreement with the tendency observed in Figure 1 that the stronger meander of the tropospheric westerlies over the North Atlantic is more favourable for stronger upward wave-activity propagation into the stratosphere. Our SVD analysis further demonstrates that the large forecast spread in 50 hPa height over the North Atlantic for 20 January (Figure 2(e)) is most sensitive to the amplitude of the tropospheric blocking ridge forecast a few days earlier (Figure 6(g)), in a manner consistent with its sensitivity to the magnitude of the upward wave-activity flux (Figure 6(f)). The strongest sensitivity and maximum forecast spread in 50 hPa height for 20 January are identified around the node of the circulation anomalies actually observed (Figure 2(e)), suggesting that the large uncertainty in the position of the 50 hPa cyclonic anomaly centre among the ensemble members for 20 January (Figure 2(e)) probably arises from the forecast errors in the strength of the tropospheric blocking ridge over the North Atlantic. Our SVD analysis shown in Figure 6(a)–(g) has revealed the relationship between forecast spread in the intensity of the blocking ridge over the North Atlantic on 16 January and that in other fields on the days before and after 16 January. The covariability of the leading SVD mode for each of the pairs explains more than 40% of the total covariance squared.

Finally, a similar SVD analysis was applied to hemispheric fields between 250 hPa height on 12 January and 50 hPa height on 28 January, when the observed PNJ was most strongly westward (Figure 1(a)). In a heterogeneous regression map (Figure 6(h)), a positive signal in 250 hPa height is again found for 12 January over the North Pacific, as in Figure 6(a) but slightly downstream. The covariability of the particular leading mode, however, explains only 24% of the total covariance, because of the weaker correlation between the two fields reflecting the longer lead time.

7. Simple sensitivity analysis

To substantiate the aforementioned results further, we have conducted a simple sensitivity analysis as introduced by Enomoto et al. (2006, 2007). The analysis utilizes a result of an ensemble forecast to identify initial perturbations that can grow optimally into a given verification region over a given forecast period. We first give a brief explanation of the analysis, following Enomoto et al. (2006).

Suppose that the time evolution of the ith ensemble member (i = 1, 2, ···, m) may be expressed as

equation image(1)

where equation image and equation image are initial and forecast perturbations, respectively, and M denotes a mapping operator. Note that the perturbations here are defined as deviations from the unperturbed member but not from the ensemble average. With matrices Y and Z that consist of equation image and equation image, respectively, as their columns, linear combinations of the initial and forecast perturbations can be expressed as

equation image(2)

and

equation image(3)

respectively, with a vector equation image that consists of the coefficients pi. Then one can find a particular equation image that maximizes the norm of equation image (equation image) in the verification region under the constraint that the norm of equation image (equation image) equals unity. Here, equation image and equation image are defined as

equation image(4)

and

equation image(5)

where equation image and equation image are positive definite symmetric matrices. In our application, the norm of an arbitrary perturbation equation image is defined in terms of dry total energy as

equation image(6)

where u′, v′, T′ and equation image denote perturbations in the zonal and meridional wind velocities, temperature and surface pressure, respectively. In Eq. (6), Cp and R denote the specific heat at constant pressure and the gas constant, respectively, of dry air, Tr and pr signify the reference values of temperature and pressure, respectively, at the surface, A denotes the area of the domain specified for analysis and F is an operator that symbolically represents the particular energy norm defined for the domain. In our analysis, the operator matrix equation image in Eq. (4) was chosen in such a manner that the dry total energy of the initial perturbations was integrated horizontally over the Northern Hemisphere poleward of 30°N and vertically between the 1000 and 100 hPa levels for each of the initial dates of the forecast (11 or 12 January). Likewise, the operator equation image was defined for expressing the dry total energy of the forecast field over a given verification domain as specified below.

To find the equation image that maximizes equation image, the Lagrange multiplier method was used for finding extrema of the following function:

equation image(7)

Since equation image and equation image are symmetric, differentiating f with respect to equation image,

equation image(8)

leads to a generalized eigenvalue problem

equation image(9)

By substituting the equation image as obtained as a solution of Eq. (9) into Eq. (2), we can identify initial perturbations that will evolve into the most developed perturbation over the given forecast within the verification domain. We used only the first eigenvector for each of our analyses as discussed below, the eigenvalue of which accounts for nearly 40% of the sum of all the eigenvalues. Thus one may infer that any local areas with particularly large initial perturbations in terms of dry total energy (Eq. (6), but without horizontal integration) are the most sensitive domains for the forecast over the given verification domain. In Figure 7, areas with particularly large values of dry total energy associated with the particular initial perturbations thus obtained are highlighted with shading.

Figure 7.

Results of a sensitivity analysis based on the JMA ensemble forecast. (a) Initial perturbations in the troposphere for 11 January (shaded lightly and heavily for 0.3–0.6 and values greater than 0.6, respectively) to which the intensity of the blocking ridge over the subpolar North Atlantic (310°E–340°E, 50°N–65°N) forecast for 16 January is particularly sensitive. The intensities of the blocking anomaly and initial perturbations are measured as dry total energy integrated between the 1000 and 250 hPa levels and then normalized by its instantaneous maximum within the entire Northern Hemisphere poleward of 20°N. Superimposed with contours is sea-level pressure (every 10 hPa: dashed for less then 1010 hPa, thick solid contours for 1010 hPa) observed on 11 January. (b) Same as in (a) but initial perturbations for 12 January 2006 to which the North Atlantic blocking forecast from 16 January is sensitive. (c)–(d) Same as in (a) but for initial tropospheric perturbations for (c) 11 and (d) 12 January 2006 to which the lower stratospheric (100–50 hPa) field over the entire domain poleward of 50°N forecast for 28 January is particularly sensitive.

Ideally, the analysis requires a large number of independently perturbed forecast members. As noted in section 2, however, only six members have been independently perturbed out of the 12 perturbed members for each of the initial dates in the JMA ensemble forecast we utilize. Still, there are 64 ( = 26) combinations in total that can be formed for our sensitivity analysis by assigning the polarity (either ‘positive’ or ‘negative’) of the six independently perturbed members. For each of the 64 combinations, we evaluated the vertically integrated total energy locally (based on Eq. (6) without horizontal integration) by solving Eq. (9) for the first eigenvector before taking their average.

As the first exercise of our sensitivity analysis, the verification time and domain were set to be 16 January and a region over the subpolar North Atlantic (310°E–340°E and 50°N–65°N), respectively. In that domain, the forecast spread for that day is largest (Figure 4(b)), associated with the developing blocking anticyclone. The matrix equation image in Eq. (5) was determined to represent dry total energy integrated horizontally over the subpolar North Atlantic and vertically between the 1000 and 100 hPa levels. As shown in Figure 7(a) and (b), the forecast blocking high intensity is particularly sensitive to the initial perturbations for either 11 or 12 January in the vicinity of the surface cyclone migrating eastward over the North Pacific, as discussed in the preceding sections.

The same analysis was repeated but with the matrix equation image in Eq. (5) determined for the entire stratospheric polar and subpolar domain (100 to 10 hPa levels and poleward of 50°N) on 28 January, when the observed PNJ was most strongly westward (Figure 1(a)). Even for a forecast period longer than two weeks (15–16 days in this case), the maximum sensitivity of the stratospheric forecast to the initial perturbations is again found around the cyclone over the central North Pacific (Figure 7(c) and (d)). In fact, our analysis in section 4 has suggested that the particular surface cyclone acted as the origin of the ‘downstream development’ of the ensemble spread. Figure 7(c) and (d) suggest that observational errors around the particular cyclone are one of the factors that induce large discrepancies in the SSW prediction among the ensemble members. The maximum sensitivity over the Northwestern Pacific is consistent with Buizza and Palmer (1995), who showed that this particular region is one of the most dynamically unstable areas, as indicated by singular vectors.

8. Evolution of analysis spread

In the preceding sections, we have suggested that the SSW event in January 2006 is particularly sensitive to initial errors around the cyclone over the North Pacific. In order to examine whether the large uncertainty of the initial field is caused by an unstable flow configuration around the cyclone, we used the daily circulation data derived from the AFES–LETKF experimental ensemble reanalysis (ALERA: Miyoshi and Yamane, 2007; Miyoshi et al., 2007; available at http://www.jamstec.go.jp/esc/afes/alera/), where data assimilation is based on the Local Ensemble Transform Kalman Filter (LETKF: Hunt et al., 2007) performed on the AGCM for the Earth Simulator (AFES: Ohfuchi et al., 2004, 2007). This reanalysis provides us with analysis and uncertainty in time-evolving flow fields in terms of spread among the ensemble members. Miyoshi et al. (2007) showed that the accuracy of ALERA in the troposphere is overall comparable to the NCEP/NCAR reanalysis (Kalnay et al., 1996) except over Antarctica and the southern ocean. Since ALERA does not assimilate satellite observations except satellite-based sea-surface wind data, it tends to be less accurate over the ocean than over land (Miyoshi and Yamane, 2007). Over maritime regions, uncertainty as measured by ALERA-based spread tends to be larger than the corresponding uncertainty in the JMA operational analysis in which radiance observations by satellites are included.

On 11 and 12 January (Figure 8) the ALERA-based spread, as a measure of uncertainty of the analysis data, over the North Pacific is maximized in the vicinity of the particular developing cyclone discussed above. This is probably because in the course of the data assimilation in ALERA, uncertainty in the six-hourly forecasts due to an unstable flow configuration associated with the developing cyclone could not be reduced due to the lack of enough free-tropospheric observations over the ocean. This result supports our hypothesis that large uncertainty in the intensity and/or the central position of the cyclone adds some difficulties to that particular SSW forecast. In fact, Figure 9 shows that the ALERA-based spread maxima identified around the cyclone translate downstream across the North Pacific as far as 120°W with a speed of about 8° in longitude per day, following the migration of the particular low-pressure system. This relatively slow development of uncertainty following the particular cyclone can also be seen in the JMA ensemble-forecast spread (dashed line in Figure 5(c)). Our result is consistent with Moteki et al. (2007) and Inoue et al. (2009), who showed that ALERA-based spread develops horizontally and vertically from local domains with relatively poor observations. In Figure 9, the analysis spread diminishes over the North American continent, where more observations are available for ALERA than over the Pacific. This may mask the fast downstream extension of spread maxima in ALERA following the group velocity propagation of the observed wave packet (solid lines in Figure 5(a) and (b)). Nevertheless, the ALERA-based spread in the troposphere over the subpolar North Atlantic suddenly increases around 15 January (Figure 9), concomitant with the development of the prominent anticyclonic anomaly observed on the arrival of the wave packet from the Pacific. Correspondingly, ALERA-based spread in the stratosphere around 16 January was also maximized over the North Atlantic (not shown), in agreement with the JMA forecast spread (Figures 5 and 6).

Figure 8.

Sea-level pressure on (a) 11 and (b) 12 January 2006 produced by ALERA (contoured for every 10 hPa). Shaded lightly and heavily where the ensemble spread is between 1 and 2 (hPa) and greater than 2 (hPa), respectively.

Figure 9.

As in Figure 5(c), but for an ALERA-based spread of 250 hPa meridional wind velocity (contoured for every 6 m2 s−2).

9. Concluding remarks

Utilizing a product of the JMA monthly ensemble-forecast system, we have examined the time evolution of forecast spread among the ensemble members during the development of a major SSW event observed in late January 2006. As the source of particularly large ensemble-forecast spread (i.e. uncertainty in the forecast) for the SSW event, we have identified errors in the initial state in the vicinity of a synoptic-scale cyclone developing over the North Pacific about two weeks before the SSW event. After growing as forecast errors for the following several days, the initial errors are translated eastward into a blocking ridge over the subpolar Atlantic and then upward into the stratosphere to cause the large forecast spread in the PNJ deceleration. This three-dimensional propagation of the forecast errors is associated with the downstream development of synoptic-scale disturbances observed in the troposphere and then with the upward propagation of a Rossby-wave packet, both of which have been identified as important dynamical processes for the occurrence of the particular SSW event (NNM09). We have found the time evolution of the forecast spread to be consistent with the result of our sensitivity analysis, which shows that predictions of the SSW event and its precursory development of the tropospheric blocking ridge over the North Atlantic both tend to be particularly sensitive to local errors in the vicinity of a particular cyclone developing over the North Pacific in the initial field for the forecast. We have confirmed these results through our SVD analysis applied to the JMA ensemble forecast and our examination of the ALERA-analyzed spread, the development of which is similar to the JMA forecast spread.

At the time of January 2006, a set of ensemble forecast was conducted in the JMA monthly forecast system only once a week with a limited number of independent ensemble members. Furthermore, the forecast model at that time is known to suffer from a cold bias in the stratosphere. Thus, more frequent ensemble forecasts with a larger ensemble size based on a less biased forecast model are needed to confirm our findings in the present study. We are planning to conduct a set of ensemble hindcast integrations for the SSW event we analyzed with initial fields taken from the ALERA system.

While Mukougawa and Hirooka (2007) have suggested that an improvement in SSW prediction does not necessarily yield a better extended forecast of the tropospheric circulation, Mukougawa et al. (2009) have shown that the prediction skill of the tropospheric NAM is higher when the stratospheric NAM is negative. Since those studies are, however, based only on a particular SSW event, or those only over five winters, more SSW events must be analyzed to assess their influence on the predictability of the tropospheric circulation system.

Acknowledgements

The authors are particularly grateful to the two anonymous referees for their sound criticism and constructive comments. They are also grateful to Dr T. Enomoto for his useful advice not only on the sensitivity analysis but also on the ALERA data. Discussion with Dr Y. J. Orsolini was invaluable. The ensemble-forecast data used in this study were provided by way of the ‘Meteorological Research Consortium’, a framework for research cooperation between the Meteorological Society of Japan and JMA. The JRA-25 dataset is provided by JMA and the Central Research Institute of Electric Power Industry (CRIEPI). This study is supported in part by the Japanese Ministry of Education, Culture, Sports, Science and Technology under the Grant-in-Aid for Scientific Research (A) #18204044. The Grid Analysis and Display System (GrADS) was used for drawing the figures.

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