Gridded 300-hPa meridional wind data produced by the ECMWF reanalysis project were analyzed to document the seasonal and hemispheric variations in the properties of upper-tropospheric wave packets. The properties of the wave packets are mainly illustrated using time-lagged one-point correlation maps performed on υ′. Based on indices that show the coherence of wave propagation, as well as examination of correlation maps, schematic waveguides were constructed for the summer and winter seasons of both hemispheres along which waves preferentially propagate with greatest coherence. In the summers, the waveguides basically follow the position of the midlatitude jets. In the Northern Hemisphere winter, the primary waveguide follows the subtropical jet over southern Asia into the Pacific, but there is a secondary branch running across Russia, joining the primary waveguide near the entrance to the Pacific storm track. Over the Atlantic, the waveguide passes east-southeastward toward North Africa, then back to southern Asia. During the Southern Hemisphere winter, the primary waveguide splits in two around 70°E, with the primary (more coherent) branch deviating equatorward to join up with the subtropical waveguide, and a secondary branch spiraling poleward along with the subpolar jet and storm track maxima. Wave packet envelopes were also defined and group velocities of wave packets were computed based on correlations performed on packet envelopes. These group velocities were found to agree qualitatively with those defined previously based on wave activity fluxes.
By examining the wave coherence indices, as well as individual correlation maps and Hovmöller diagrams of correlations computed along the primary waveguides, it was concluded that wave propagation is least coherent in Northern Hemisphere summer, and that waves in Southern Hemisphere summer are not necessarily more coherent than those in Southern Hemisphere winter. Data from a GCM experiment were also analyzed and showed that wave packets in the GCM also display such a seasonal variation in coherence. Results from experiments using an idealized model suggest that coherence of wave packets depends not only on the baroclinicity of the large-scale flow, but also on the intensity of the Hadley circulation, which acts to tighten the upper-tropospheric potential vorticity gradient.
In Part I of this study (Chang and Yu 1999, hereafter CY99), the characteristics of wave packets in the wintertime Northern Hemisphere upper troposphere were studied in detail. Based on time-lagged one-point correlation maps of 300-hPa meridional wind variations computed over much of the hemisphere, a wave correlation index has been defined to display the geographical variations in wave propagation characteristics. Wave packet envelopes have also been defined by performing complex demodulation on the wind data, and basic characteristics of the waves such as wavelengths, phase speeds, and group velocities were computed. The results showed that during December–January–February (DJF) in the Northern Hemisphere, wave packets are most coherent along a band that extends from North Africa through south Asia into the Pacific storm track. In this paper, we extend the analyses of CY99 to the Southern Hemisphere and also the summer seasons to examine the seasonal and hemispheric dependence of wave packet characteristics.
In section 2 of this paper, we will briefly describe the data used and display the general characteristics of the storm tracks in the different seasons. Basic characteristics of the waves picked out by the correlation analysis will also be shown. The characteristics of wave packet propagation in the different seasons will be discussed in section 3 in terms of the downstream/upstream asymmetry index and wave coherence index defined in CY99. Sequences of lag-correlation maps will be shown to highlight propagation over selected regions. In section 4, we will examine the characteristics of wave packet propagation in models. Similar indices computed from a general circulation model (GCM) experiment will be displayed to show that models can basically reproduce the observed seasonal differences in wave packet propagation characteristics, and then results from experiments conducted with an idealized model will be examined in order to understand the factors giving rise to the seasonal variations. A conclusion will follow in section 5.
Since this paper is the second paper in a series, the introduction as well as some of the technical details are kept brief. For a more extended introduction as well as references to previous works, and more details about the methodology, statistical significance of the results, and the meaning of the indices shown here, please refer to CY99.
2. General characteristics of the basic state and wave packets
a. Brief description of the data
The data used in this study are the European Center for Medium-Range Weather Forecasting (ECMWF) reanalysis data. Twice daily (0000 and 1200 UTC) 300-hPa meridional winds on a 2.5° × 2.5° grid, for the northern winter/southern summer (DJF) and northern summer/southern winter (June–July–August, JJA) seasons, for the period June 1980 to January 1993, were analyzed. Hence, we have analyzed data from 13 DJF and JJA seasons. As is well known, over data-sparse regions, the analyses are heavily influenced by model extrapolation, both in space and time. The problem is expected to be especially severe in the Southern Hemisphere (SH), where data coverage is generally poor. In CY99, we compared the results computed from the ECMWF data with those from the NCEP–NCAR reanalysis data (Kalnay et al. 1996) and found very little differences in the Northern Hemisphere (NH). Unfortunately, the current edition of the NCEP–NCAR reanalysis dataset is contaminated by misplacement of PAOBS (SH surface pressure bogus data produced by Australia) for the years 1980–92; hence, comparisons similar to those performed in CY99 are not possible. Instead, we compared the two datasets by computing the spatial correlation of 300-hPa meridional winds, and for the month of December 1993, the average spatial correlation between the two datasets in the Northern Hemisphere midlatitudes (30°–70°N) is 0.98, while that in the Southern Hemisphere midlatitudes is 0.96. For the month of December 1980, the correlation is still 0.98 in the Northern Hemisphere, while the correlation in the Southern Hemisphere falls to 0.92, probably due to the PAOBS errors. The high correlation during December 1993 suggests that the analyses are rather tightly constrained by observations, even in the Southern Hemisphere, giving us confidence that features observed in the Southern Hemisphere are probably real.
As discussed in Chang (1993) and CY99) we have found that time filtering of the data distorts the temporal evolution of the waves. Hence, we will only analyze unfiltered data. To remove contributions from stationary waves, the 3-month mean from each of the individual seasons has been removed from the data, and the result is referred to as the meridional wind perturbations (υ′).
b. Basic characteristics of the storm tracks
The 300-hPa zonal wind, averaged over the analysis period, is shown in Fig. 1. In the Northern Hemisphere winter (Fig. 1a), the jet maximum extends from North Africa across southern Asia into the Pacific, with a slightly weaker maximum extending from the eastern United States into the Atlantic. The jet in the summer seasons is a lot more zonally symmetric, displaying only weak maxima. In the Southern Hemisphere winter (Fig. 1b), the upper-tropospheric jet splits into two over the Indian Ocean, with the stronger subtropical branch passing across Australia and a weaker branch passing south of New Zealand along 60°S. Such a splitting has also been shown in previous studies such as Trenberth (1991).
The standard deviation of 300-hPa υ′ over the entire analysis period is shown in Fig. 2. Figure 2a displays rms υ′ for DJF, while Fig. 2b displays it for JJA. For the northern winter season, we see a band of maximum variance extending from the Pacific across North America and the Atlantic into the Eurasian continent. Over Asia, the storm track appears to be split into two branches, a northern branch with the maximum near 60°N and a weak southern branch along 30°N. In CY99 we showed that this split is indeed real, with wave propagation splitting into two routes over Asia, with propagation along the southern branch being more coherent.
For the northern summer season (Fig. 2b), we see that the storm track, at least as depicted by the variance of 300-hPa υ′, is a lot more zonally symmetric, with two relatively weak maxima, one over the northeastern Pacific and the other over the Atlantic. The same is true for the Southern Hemisphere summer storm track (Fig. 2a), which basically consists of a band of maxima along 50°S and has a weak maximum near 90°E. The southern winter storm track (Fig. 2b) appears to be more complicated and consists of a poleward spiral, starting over Australia, running across the Pacific, South America, the southern Atlantic, passing south of the southernmost tip of Africa, continuing along the southern limits of the Indian Ocean, and then passing south of Australia and New Zealand, some 3000 km south of the northern end of the storm track. Such a spiral can also be seen in the standard deviation in relative vorticity (Berbery and Vera 1996) but is not as evident in the height variance (e.g., Trenberth 1991) since height variances are generally small in the low latitudes.
In this section, we have only discussed briefly those storm track characteristics that are directly relevant to the discussions below. Other properties of the storm tracks can be found in Nakamura (1992) and Wallace et al. (1988) for the NH winter, White (1982) for the NH summer, Trenberth (1991) for the SH seasons, and references within these papers.
c. Basic characteristics of the waves and wave packets
As in CY99, we examined the characteristics of wave packets by computing one-point lag-correlation maps, based on the correlation of the time series of 300-hPa υ′ at each grid point on a 5° × 5° grid between 10° and 70° latitude, with the time series of 300-hPa υ′ at every grid point on the globe, with or without time lag. (Examples of the correlation maps based on the time series at several different grid points can be seen in Figs. 10–14 in this paper, as well as in figures in CY99 for the NH winter.)
At lag 0 (e.g., Figs. 10c and 11c, etc.), the correlation maps typically show a wave train extending generally along the latitude circles, showing several positive and negative phases. Using these maps, we can estimate the dominant wavelength of the waves picked out by the correlation analysis on υ′ at each location. The results, displayed in terms of zonal wavenumber, are shown in Figs. 3a and 3b. We see that over the latitude band of the storm track maxima (see Fig. 2), the dominant wavenumbers picked out by the correlation analysis are, respectively, 5–8 for the Northern Hemisphere winter and Southern Hemisphere summer, 5–7 for the Southern Hemisphere winter, and 6–9 for the Northern Hemisphere summer, all of which basically correspond to medium-scale waves. The wavenumbers generally decrease toward the high latitudes and also show indications of decreasing toward the equator. In fact, in the Northern Hemisphere winter, the variations in the dominant wavenumber poleward of 40°N nearly correspond to a constant wavelength of 4000 km. South of 40°N, the wavelength increases gradually to around 6000 km close to the equator. Similar variations can be seen for the Southern Hemisphere winter, except that the wavelength of the waves appears to be slightly longer. In the summer hemispheres, the dominant wavenumbers are, in general, larger, corresponding to waves with shorter wavelengths, and the wavelength of the waves generally lies between 3000 and 5000 km. Unlike the winter seasons, the wavelengths do not increase appreciably until equatorward of 20°N and 15°S in the NH and SH summer, respectively.
The sequence of time-lagged correlation maps computed for each individual grid point (e.g., Figs. 10–14) also show a general eastward progression of each individual phase of the wave train, as well as an eastward propagation of the whole wave train. The characteristic phase speed of the waves picked out by the correlation analyses can be estimated by objectively tracking the movement of the maximum positive center from day −1 to day +1. The results, shown in Fig. 4, show that these waves typically have eastward phase speeds, except over the Tropics in summer, and over the eastern part of Siberia in the northern winter season, where the phase propagation of the waves is westward. As discussed in CY99, the zonal phase speed shows significant correlations with the time mean 700-hPa zonal wind (not shown). The correlations for the different seasons are 0.78 (NH winter), 0.82 (SH winter), 0.90 (SH summer), and 0.73 (NH summer), respectively. As discussed in Part I, comparing the magnitude of the phase speed to the 700-hPa flow, over the storm track regions, the phase speed is slightly larger than the 700-hPa flow over land, while it is generally less than the 700-hPa flow by about 3–6 m s−1 over the oceans, suggesting that the steering level of waves over land is near or slightly above 700 hPa, while that for waves over the oceans is lower and closer to 800 hPa. Note that Berbery and Vera (1996) also estimated the phase speed of waves in the SH winter, and our results for that season are quite similar to theirs.
The characteristic period of the waves can be found by dividing the wavelength by the phase speed. The results are shown in Fig. 5. We can see that in the midlatitudes, between 30° and 60° latitude, the periods of the waves are generally between 4 and 8 days, except in the Northern Hemisphere summer, where the periods range from 5 to 10 days between 40° and 60°N. Examination of Figs. 3 and 4 suggests that the longer periods of the waves observed in the Northern Hemisphere summer are associated with slower phase speeds rather than longer wavelengths, as the characteristic wavenumbers of the waves are higher than those in the Northern Hemisphere winter. The lower phase speed is consistent with the weaker strength of the basic-state flow.
In CY99 we defined the envelope of a wave packet (υe) based on complex demodulation (e.g., Bloomfield 1976) on υ′. It was shown in CY99 that wave packet propagation is tracked more easily by following υe than υ′. In fact, we have been able to subjectively track some wave packets for over 3 weeks in the Southern Hemisphere summer by tracking υe (details to be presented elsewhere).
We can also compute one-point lag correlations with υe just like we did for υ′. The group velocity of wave packets can be estimated by objectively tracking the movement of the centers of correlation of υe from day −1 to day +1 (see CY99). The results are shown in Figs. 6 and 7. We have also computed group velocities using the movement of correlation centers between day −0.5 and day +0.5, and the results are basically the same as those shown, with an rms difference between the estimated group velocities being less than 2 m s−1. Comparing Fig. 6 to Fig. 1, we can see that basically the group velocity is large where the 300-hPa flow is strong. In fact, the spatial correlations between the estimated zonal group velocity and the time mean 300-hPa zonal wind for the different seasons are 0.89 (NH winter), 0.87 (SH winter), 0.97 (SH summer), and 0.94 (NH summer), respectively, with the correlation generally higher in summer than in winter. By forming the difference between Fig. 6 and Fig. 1. (not shown), we can see that in summer, the group velocity is about 20% less than the 300-hPa zonal flow, whereas in winter, over the jet core of the subtropical jet stream, the estimated group velocity can be over 40% less than the 300-hPa flow, while over other regions, the difference is less, and on average, the rms value of the group velocity is about 70% that of the rms 300-hPa flow for both winter seasons.
To check the accuracy of the group velocity estimated using one-point lag-correlation statistics, we have performed subjective tracking of wave packets in the Southern Hemisphere summer, where the storm track is more or less zonally symmetric such that wave packets are easiest to track. From Fig. 6a, the zonal average of the zonal group velocity at 50°S (at the storm track maximum) is about 22 m s−1, implying that wave packets should propagate around the latitude circle in about 14 days. We have tracked all the wave packets in one season (DJF 1984/85) and found that they took between 12 and 16 days to propagate around the latitude circle once, suggesting that the estimates shown in Fig. 6a are more or less accurate. In CY99 we also tracked all wave packets that passed selected points during the Northern Hemisphere winter season and found that the subjectively determined group velocity was about 10% higher than those shown in Fig. 6a, but the differences are not statistically significant.
Berbery and Vera (1996) also estimated the group velocity of waves in the Southern Hemisphere winter storm track using a different method. Comparing our results (Fig. 6b) to theirs (Fig. 9c in their paper), we see that the values estimated here are significantly lower than those obtained by Berbery and Vera over most regions. For example, near 45°S, 0°E, our estimate is 23 m s−1, whereas the estimate given by Berbery and Vera is over 32 m s−1. In order to assess which estimate is better, we tried to subjectively track υe for all cases in which wave packets can be unambiguously identified to propagate across this point. Twenty cases were found in the dataset, and the average propagation speed of the envelope is found to be 23 m s−1, with a standard deviation of 8 m s−1. This value agrees very well with the estimate computed from correlation of υe shown in Fig. 6b. In fact, Berbery and Vera had assumed that on days −1 and +1, the wave packet centers are located over the negative correlation centers of the one-point lag-correlation maps of υ′. With such a definition, the group velocity is exactly equal to the phase speed plus Lx/(2 days), where Lx is the wavelength of the wave. Such a relationship between the group velocity and phase speed cannot be generally true. Hence, we believe that the group velocities obtained here are better estimates than those shown in Berbery and Vera (1996).
The meridional group velocity estimated from the correlation analysis is shown in Fig. 7. Apart from NH summer, in which no significant pattern can be seen, during the other seasons, the group velocity is found to be generally equatorward on the equatorward side of the storm tracks and poleward on the poleward side, signifying divergence of wave energy away from the storm tracks.
The correspondence between our estimates of the group velocity in NH winter with the E vector of Hoskins et al. (1983) and the MT vector of Plumb (1986) was discussed thoroughly in CY99. Trenberth (1986) defined a localized Ellassen–Palm (E–P) flux vector (Eu), which is a three-dimensional extension of the E–P flux discussed in Edmon et al. (1980), but with the ageostrophic terms fully included. In the WKB limit, Trenberth’s E–P flux is parallel to the group velocity relative to the mean flow. Trenberth (1991, Fig. 18) computed the localized E–P flux for 2–8-day eddies for January and July in the Southern Hemisphere, using the ECMWF operational analyses for the period 1979–89. Since Trenberth’s flux indicates relative group velocity, it is difficult to compare the zonal component to the group velocity estimated here. Still, it is reasonable to compare their meridional components since the mean flow is basically zonal. During the Southern Hemisphere summer (Fig. 7a), our estimates show equatorward group velocity equatorward of 50°S, maximizing between 30° and 50°S, in general agreement with Trenberth’s results for January. Figure 7a also shows generally poleward group velocity south of 60°S, with the largest negative values of meridional group velocity (cgy) near 150°E, again in agreement with Trenberth’s results.
For the Southern Hemisphere winter, Fig. 7b shows that the largest equatorward group velocity lies around 45°S, between 40° and 140°E. Near 150°W, the meridional component is near zero, and near 60°W, the group velocity is equatorward even south of 60°S, while it is poleward near 60°S, 120°E. All these results are again in close agreement with Trenberth’s results. The consistencies between group velocities defined and computed in different ways shown here and in CY99 suggest that the group velocity is a useful concept even though some of the underlying WKB assumptions may not necessarily be valid over all regions.
3. Seasonal variations in coherence of wave propagation
In this section, we will discuss the gross differences in wave propagation characteristics among the different seasons. First, we will examine indices (introduced in CY99) that show the asymmetry between upstream and downstream development and the coherence of wave propagation. Then we will examine lag-correlation maps over selected grid points to show the characteristics of wave propagation in a bit more detail.
a. Downstream/upstream asymmetry
The downstream/upstream asymmetry index compares the relative magnitude of the upstream and downstream negative correlation centers flanking the reference positive center1 at days +1 and −1. It is equal to the sum of the maximum negative correlation upstream at day −1 and that downstream at day +1, divided by the sum of the maximum downstream negative correlation at day −1 and that upstream at day +1. It effectively measures the tendency of downstream propagation of the wave group (relative to the phase) versus the tendency of upstream propagation. A ratio greater than 1 indicates that downstream development is preferred, whereas values smaller than 1 indicate upstream development, and values close to 1 suggest that the wave train is simply being advected by the steering level flow. This index is shown in Fig. 8. Areas where the index is less than 1 and greater than 1.5 are shaded heavily and lightly, respectively. Examining Fig. 8, it is clear that over most regions in the different seasons, the value of the index is over 1.5, and only small regions in the NH Tropics have an index of less than 1, suggesting that downstream development is strongly favored over upstream development nearly everywhere for waves in the upper troposphere. This is consistent with the phase speeds and group velocities shown in Figs. 4 and 6, which showed that the group velocity is larger than the phase speed nearly everywhere, which necessarily means the downstream development of waves. These results show that downstream development is prevalent in the upper troposphere, even over regions away from the storm tracks, extending the conclusions of previous findings that downstream development occurred in individual cases (e.g. Cressman 1948; Hovmöller 1949; and others) and within the storm tracks of NH winter (Chang 1993), SH summer (Lee and Held 1993), and SH winter (Berbery and Vera 1996).
b. Wave coherence index
The wave coherence index at each base point represents the average of the maximum (positive or negative) correlation upstream on day −2 and downstream on day +2. As an example, for the southern winter season, for the grid point 45°S, 85°E (Fig. 11), the maximum upstream correlation at day −2 (Fig. 11b) is just over (−)0.43 near 50°S, 25°E; the maximum downstream correlation at day +2 (Fig. 11d) is about 0.42 near 25°S, 150°E; and the value of the wave coherence index plotted in Fig. 8b at 45°S, 85°E is simply the average of the magnitude of the two values (∼0.42). The index basically indicates how well the waves at each location correlate with upstream waves 2 days earlier and downstream waves 2 days later, thus showing the tendency of the downstream development of waves. Note that if the maximum correlation occurs within the same phase as the reference positive center that passes the base grid point at day 0 (see, e.g., Fig. 13b), that value is ignored, and only upstream phases are considered on day −2 and downstream phases on day +2. For further details, please refer to CY99.
The wave coherence index (WCI2) for the different seasons is shown in Fig. 9. It is obvious that the Southern Hemisphere summer exhibits the largest degree of zonal symmetry, with a nearly continuous band with WCI2 > 0.4 around 50°S latitude, with a maximum of over 0.5 near 30°E. The zonal asymmetries in the other seasons are much more marked. Comparing the hemispheres, it is obvious that the Southern Hemisphere shows much bigger areas with significant values of WCI2, suggesting that downstream wave propagation is more coherent in the Southern Hemisphere. As for seasonal variations within each hemisphere, in the Northern Hemisphere, WCI2 is larger in winter than in summer over Asia, where the NH winter waveguide is most coherent. However, across North America, wave propagation appears to be slightly more coherent in summer than in winter. By examining individual correlation maps as well as Hovmöller diagrams of time-lagged correlations (similar to Fig. 8 of Part I, not shown here), one forms the impression that wave propagation is overall least coherent in NH summer. For the Southern Hemisphere, both seasons exhibit a nearly continuous band of WCI2 with values over 0.4 and maxima over 0.5. However, in the SH summer the band is located along the 50°S latitude circle, with relatively minor latitudinal excursions, whereas in the SH winter the band is centered near 45°S around 0° but appears to split into two bands east of 120°E, much as the upper-tropospheric jet is split into two near the same location (Fig. 1b). Anyway, by simply examining WCI2, it is not obvious that wave packets in the SH summer propagate more coherently than those in the SH winter. Moreover, it appears that wave propagation is least coherent in NH summer, the season in which baroclinicity (as measured by the Eady growth rate; see, e.g., Hoskins and Valdes 1990) is weakest.
Lee and Held (1993) conducted a series of numerical experiments using a two-layer quasigeostrophic channel model on a β plane and concluded that wave packets are more coherent when the baroclinicity of the basic state is weaker. They also examined the SH summer and winter ECMWF analyses. Based on Hovmöller diagrams of one-point lag correlations for υ′ at 45°S, they concluded that their data analyses also supported their modeling results. Our results are obviously inconsistent with their conclusions. The different conclusions drawn from the data analyses are not surprising, since they have analyzed only data from 45°S but here we have examined data from all latitudes. Forming correlations using data only from 45°S is probably sufficient for the SH summer. However, from Fig. 2b, we see that the SH winter storm track is definitely not along a single latitude circle. In fact, from 120°E to 120°W, the storm track is split into two, with a storm track relative minimum at 45°S. Figure 9b also suggests that wave propagation is also split into two branches from 120°E to 60°W, with maximum coherence near 30° and 60°S, but relative minimum near 45°S. Hence, it is not surprising that by using data only from 45°S, Lee and Held found much less coherence in wave propagation in the SH winter than we did here. In section 4, we will conduct further experiments to try to reconcile our current findings with the modeling results of Lee and Held.
c. Schematic wave guide and characteristics of the individual seasons
1) Northern Hemisphere winter
The characteristics of wave propagation in the Northern Hemisphere winter are discussed in detail in CY99. Here we will just briefly summarize the main results.
Examining WCI2 (Fig. 9a) for the NH winter, it is obvious that wave propagation is most coherent along a band extending from North Africa through southern Asia into the central North Pacific. There is also a slightly less coherent band of maxima extending across North America, in which waves over the eastern North Pacific propagate downstream to seed waves near the entrance to the Atlantic storm track. Examination of lag-correlation maps of υ′ at different locations (see CY99) suggests that wave packets propagate out from Asia into the Pacific, then over North America, then turn slightly southeast across the Atlantic toward North Africa back to the subtropical waveguide over south Asia, rather than following the axis of the Atlantic storm track maximum toward northern Europe. CY99 also suggested that some waves in the Pacific storm track originated from northern Asia. That pathway, which appears to be less coherent than the one across southern Asia, appears as secondary maxima in WCI2 near 50°–60°N in Fig. 9a. The schematic waveguide where wave propagation is most coherent is depicted by shaded bands in figures in CY99. For more details, including time sequences of lag-correlation maps at selected locations, please refer to CY99.
2) Southern Hemisphere winter
As discussed earlier, the SH winter storm track (see Fig. 2b) appears to spiral from Australia eastward and poleward, returning to the same longitude around 30° to the south of its northern end. This poleward spiral appears to follow the spiral of the climatological-mean jet (Fig. 1b) as well as the maxima in the baroclinicity (not shown). One obvious question is whether wave packets also propagate along such a spiral.
To investigate this, we will examine time-lagged correlation maps at a few strategic locations. The first location is 25°S, 165°E, just east of Australia, near the middle of the subtropical jet maximum (Fig. 1b), close to the upstream end of the band of maximum variance (Fig. 2b), as well as within the region where WCI2 (Fig. 9b) is large. The lag-correlation maps, with time lags of −4, −2, 0, +2, and +4 days, based on the time series of υ′ at this point, are shown in Fig. 10. The shaded bands in this (and subsequent) figure represent the schematic waveguide, which will be described later. First, when we examine the downstream direction at positive lags (Figs. 10d,e), we can indeed see an eastward and poleward propagation of the wave packet, roughly following the axis of the storm track maximum shown in Fig. 2b. We can see that wave propagation over this region is very coherent. At day +4, the correlation at the downstream negative center near 40°S, 90°W (the second negative downstream maximum from the reference wave), is still as high as −0.27, and the correlation at the positive center farther downstream is 0.24.
Now let us look in the upstream direction at negative lags. Examining Figs. 10a,b, we see that while the base point is located close to the apparent entrance of the subtropical Pacific storm track, the wave trains propagating through this location do not seem to have arisen there but appear to have propagated in from the southwest. At day −4, the maximum correlation is about 0.22 near 40°S, 90°E, and −0.24 near 50°S, 55°E. It appears that the waves over the subtropical jet maximum in SH winter are a continuation of upstream wave packets propagating along the main body of the storm track near 45°S.
To investigate this further, in Fig. 11 we display the time-lagged correlation maps for the base point 45°S, 85°E. This point is located within the storm track upstream from the “split.” First, looking at the correlations at positive lags (Fig. 11c–e), we see that, indeed, toward the downstream direction of this base point the wave train appears to split into two, one branch propagating northeastward into the subtropical jet stream, the other branch southeastward into the subpolar jet. The wave train appears to be slightly more coherent over the subtropical branch, as shown by higher correlations near 25°S than near 50°S in days +2 and +4. Even if we had taken the base point further toward the south (as far south as 60°S, not shown here) but at the same longitude, the split into two branches is still apparent, and the day +4 correlation at the subtropical branch is still higher than the correlation for the branch near 60°S. Only base grid points to the south of 60°S have the property that the correlations for the higher-latitude branch exceed those for the subtropical branch at day +4, but those locations are already well south of the main body of the storm track, and the correlations are much lower than those at 45°S. Hence, we see that waves upstream of the split of the storm track do not just continue to spiral poleward, but split into two branches, with a stronger tendency to propagate toward the subtropical branch than toward the higher latitudes.
Looking toward the upstream side of 45°S, 85°E (Figs. 11a,b), we can track the wave train upstream toward the South Atlantic and South America. To complete the picture, we next examine the correlation maps for the base point 40°S, 70°W, which are shown in Fig. 12. This point is located in the main body of the storm track, close to midway between the previous two points. From Figs. 12a–e, we can see that wave trains passing through this area have propagated from the subtropical jet stream (in agreement with Figs. 10c–e). Downstream of this point, the wave train appears to split into two paths, with one branch proceeding equatorward toward the southern part of Africa and the second branch following the poleward spiral in the storm track. At day +4, the maximum values of the correlations for the tropical path (0.26) appears to be slightly higher than that for the midlatitude path (0.20). However, examination of lag-correlation maps based on other grid points suggests that waves over the tropical path simply disappear, whereas those on the midlatitude path continue to propagate eastward; hence, we regard the midlatitude path to be the primary waveguide over this region.
Next, let us examine waves over the higher-latitude branch of the split jet. Figure 9b suggests that wave propagation over this branch is slightly less coherent than waves propagating along the subtropical jet stream. In Fig. 13, time-lagged correlation maps for the base point 60°S, 145°W, are shown. Over this region, the orientation of the wave trains appears to be basically along the axis of the storm track, with a slight northwest-southeast tilt corresponding to the poleward spiral discussed above.2 As discussed earlier when we examined Fig. 11, further upstream the waves over this region constitute the poleward branch of waves that had split off from the main storm track at around 70°E. In the downstream direction, we see rather coherent propagation almost all the way to 0° longitude. Inspection of the correlation maps suggests that coherent wave propagation over the high-latitude branch of the storm track apparently stops there, with waves near 60°S, 0° having only very weak tendencies to develop further in the downstream direction. Consistent with this, waves east of 0° on the main storm track show only weak upstream correlations to the high-latitude wave trains and appear to have mainly come from the subtropical branch of the split storm track.
Based on examination of WCI2 (Fig. 9b) and the paths taken by the wave packets as shown in Figs. 10–13 and other correlation maps not shown here, we have defined a schematic waveguide along which wave propagation is seen to be most coherent. This waveguide is shown as shaded bands in Figs. 10–13. The primary waveguide runs from Australia east-southeastward, following the poleward spiral in the upper-troposphere jet and storm track across the Pacific and Atlantic (Figs. 1b and 2b). Then, over the Indian Ocean the waveguide splits into two, with the main branch deviating northeastward back toward the subtropical waveguide over Australia. There is a secondary waveguide that continues to follow the poleward spiral in the storm track, all the way across the southern fringe of the Pacific and the Atlantic. Over the leeward side of the Andes, there are also indications that some waves tend to deviate northward toward the equator. Comparison of this band with Fig. 9b shows that the waveguide defined here basically follows the maxima in WCI2, and Figs. 10–13 show that wave packets do actually follow the latitudinal excursion of this waveguide as they propagate eastward.
If we compare the orientation of the waveguide to the estimates of the meridional group velocity shown in Fig. 7b, we see that toward the southwest of Australia, where the subtropical waveguide deviates equatorward, the group velocity is also estimated to be equatorward. However, east of the date line where the schematic waveguide deviates poleward, the group velocity is estimated to be only weakly negative (magnitude less than 1 m s−1). Similar discrepancies are also apparent for the NH winter subtropical waveguide across southern Asia, where cgy also depicts the equatorward deviation of the waveguide near the Mediterranean much better than its poleward deviation over China. Two points should be noted here. The magnitude of the poleward group velocity, which is consistent with the orientation of the storm track, comes out to be about 2 m s−1. The difference between this and the group velocity estimated in Fig. 7 is probably within the error bounds of the cgy estimates. In addition, as mentioned earlier when we discussed Fig. 7, cgy apparently depicts predominantly equatorward propagation over the mid- and low-latitudes, and poleward propagation in the high latitudes, consistent with meridional dispersion of waves away from the storm tracks. We believe that cgy estimated by following υe not only tracks the coherent propagation of waves along the waveguide (e.g., Fig. 10), but also the less coherent equatorward dispersion of waves over the region; thus, the cgy estimates show equatorward “biases” when compared to the orientation of the waveguide.
The results shown here in Figs. 10–13 are largely consistent with the results of Berbery and Vera (1996). Their results also suggested a split of the southern winter waveguide near 120°E, and that wave propagation is most coherent along the subtropical waveguide near the date line. However, by following the wave packets from day −4 to day +4 (Berbery and Vera only computed correlations from day −2 to day +2) and defining and displaying WCI2, we are able to show a clearer picture of the spatial structure of the waveguide and demonstrate that coherent wave packets do tend to follow the waveguide as they propagate downstream.
3) Southern Hemisphere summer
In contrast to the storm tracks and waveguides in the NH and SH winter, the SH summer counter part appears to be much simpler. The jet and maximum variance in υ′ both lie on a band centered near 50°S, with the jet maximum [and maximum in baroclinicity and potential vorticity (PV) gradients] located near 30°E, and the maximum in variance in υ′ near 160°E. Zonal asymmetries are much weaker than during the winter seasons. Accordingly, the wave coherence index (Fig. 9a) also shows only weak zonal variations, indicating slightly stronger coherence near 30°E than near the date line.
Time-lagged correlation maps for the base point 50°S, 30°E, are displayed in Fig. 14. From day −4 to day +4, we basically see a wave train propagating from near 75°W to near 150°E, along the 50°S latitude circle. The maximum correlations at days −4 and +4 are both around 0.25. Correlation maps for other base points along the shaded band in Fig. 14 (the schematic waveguide for SH summer) are very similar, except that the maximum correlation is generally slightly lower than those shown in Fig. 14. Because of the simple geometry of the SH summer waveguide, and the near zonal symmetry of the wave amplitudes, we have found that wave packets are particularly easy to track in this season.
4) Northern Hemisphere summer
Like the SH summer, the storm track in the NH summer also shows only weak zonal variations, with maximum variance over the northeastern Pacific and the Atlantic, just downstream of the weak jet maxima. Baroclinicity is much weaker than in the other three seasons, and the PV gradient is also weaker. Figure 9b shows that the most coherent wave propagation occurs near 40°N over Asia, with another band of relative maxima in WCI2 located over North America near 45°N. Compared to the other seasons, waves in NH summer have smaller spatial scale and lower frequency, while wave packets are generally less coherent and propagate with slower group velocity. However, Fig. 8b shows that just as in the other seasons, waves in NH summer midlatitudes also show strong downstream development characteristics (as opposed to simple advection or upstream development), even though coherence of the wave packets may not be as high as in the other seasons (Fig. 9b). Correlation maps for wave packets in NH summer are not as interesting as those for the other seasons and are not shown here due to space limitations.
4. Seasonal variations of wave propagation in models
The results discussed in section 3 suggest that wave packets in NH summer are not as coherent as those in the other seasons, even though the baroclinicity of the time mean flow is weakest during that season. Furthermore, wave packets in SH winter are just about as coherent as those in SH summer. These results are inconsistent with the conclusions of Lee and Held (1993), who found that wave packets are more coherent when the baroclinicity of the basic state is weaker. In this section, we will examine wave packets in numerical modeling experiments to try to understand why wave packets are not more coherent in summer. Before we examine results from idealized experiments, we will first examine the properties of wave packets in a GCM experiment to see whether the model reproduces the observed seasonal variation in wave propagation characteristics.
a. Wave packets as simulated by a GCM
The GCM experiment examined in this study was conducted with the standard version of the spectral atmospheric GCM developed and maintained by the Climate Dynamics Project at the Geophysical Fluid Dynamics Laboratory (GFDL). The model is basically the same as the one described in Ting and Lau (1993), except that the experiments shown here are performed at higher horizontal (R30, with an equivalent Gaussian grid of 3.75° in longitude and approximately 2.25° in latitude) and vertical (14 sigma levels) resolutions. The model is forced by prescribed climatological solar forcing, sea surface temperature, and sea ice, with the full seasonal cycle, but without any interannual variations in the forcing. A total of 17 yr (16 DJF, 17 JJA) of once-daily 205-hPa meridional wind data have been analyzed, using the same techniques that we used to study the ECMWF reanalysis data. Even though the model and observed data are located at different vertical levels, previous results of Lim and Wallace (1991) and Chang (1993) showed that in the upper troposphere the wave trains are highly correlated vertically; hence, we do not expect the vertical displacement to give rise to any significant differences.
The properties of wave packets in the GFDL GCM experiment can be summarized by the downstream/upstream asymmetry index and the wave coherence index, which are shown in Figs. 15 and 16, respectively. These figures should be compared to Figs. 8 and 9, which show the same indices computed using the ECMWF reanalysis data. First, let us examine the downstream/upstream asymmetry index (Figs. 15 and 8). Recall that a value larger than 1 suggests that downstream development is favored over upstream development. In the GCM simulations, as well as in observation, the value of the index is smaller than 1 only in isolated small areas and is larger than 1.5 across most regions during all seasons examined. Comparing the GCM simulations to observation, downstream/upstream asymmetry is slightly stronger in the GCM in both NH and SH summer, but weaker over Asia in NH winter. Nevertheless, Figs. 15 and 8 show a high degree of agreement between GCM and observation—that upper-tropospheric waves in both tend to exhibit a high degree of downstream development characteristics.
Figures 16 and 9 show the wave coherence index (WCI2) for waves in the GCM and observation, respectively. The first thing to note is that wave packets are slightly more coherent in the GCM than in observation, as WCI2 for the GCM is generally larger than that in observation by about 0.05. In the NH winter, the GCM correctly simulates a split in the waveguide across Asia. However, unlike in observation where the coherence in the subtropical branch is considerably higher, in the GCM, the coherence in the two branches is about the same. WCI2 computed from the GCM also suggests a more continuous waveguide across the NH winter midlatitudes than that observed. In the SH winter, the GCM only shows hints of a split waveguide near the date line and again underestimates the coherence in the subtropical branch relative to the high-latitude branch.
While the GFDL GCM failed to simulate exactly the observed behavior of wave packets, especially over the winter subtropical waveguides, it did manage to capture the gross seasonal variation in WCI2. Results shown in Fig. 16 clearly show that wave packets are least coherent in NH summer, and that wave packets in SH winter are about as coherent as those in SH summer, consistent with the conclusions drawn from observation as previously discussed. Such agreements between the model and observations give us confidence that we can use modeling studies to investigate the mechanisms behind the observed seasonal variations, which we will turn to in the next paragraphs.
b. Wave propagation in idealized model experiments
In order to understand the observed seasonal variations in the coherence of wave packets, and the apparent disagreement between our conclusions with the modeling results of Lee and Held (1993), we have conducted a series of experiments using an idealized model. The model used in this study is a primitive equation spectral model programmed by Held (Held and Suarez 1994). The model includes nonlinear dynamics, but diabatic processes are simply represented by Newtonian relaxation of the temperature field to a zonally symmetric state to represent heating, and Rayleigh damping of low-level winds to represent boundary layer friction. The experimental setup, with the exception of the radiative equilibrium temperature profiles used here, is exactly the same as that described in Held and Suarez (1994). The experiments described below have been conducted with a horizontal truncation of T30 (approximately 3.75° × 3.75° equivalent Gaussian grid), and 10 equally spaced sigma levels in the vertical. All forcing is zonally symmetric, and no orography is present in the experiments. Each experiment was initialized from a state of rest apart from random noise and integrated for 800 days. The model climate (as indicated by the total eddy energy) generally reached a state of statistical equilibrium after about 100–150 days, and data from the last 600 days of each experiment were analyzed.
For the control experiment the equilibrium temperature profile is similar to the one used in Held and Suarez (i.e., a T0 − ΔT sin2ϕ profile at the surface, with ΔT equal to 60 K, and ϕ denotes the latitude) except that our vertical profile is more stable by 1 K km−1 everywhere in the troposphere.3 With the added stability (and lower resolution), eddy variances are about 30% less than those shown in Held and Suarez, and the zonal mean jets are displaced poleward by approximately 5° in latitude because of the reduction in the intensity of the Hadley circulation due to increased stability (see, e.g., Chang 1995).
To examine the coherence of wave packets in the model, daily fields of the simulated 250-hPa meridional winds were analyzed. First, we averaged υ′ over a 20° latitude band centered about the model storm track. One-point lag correlation was then computed using this data. The result is displayed in Fig. 17a in the form of a Hovmöller diagram from day −5 to day +5. In the figure we see clearly that the time-lagged correlation pattern displays a downstream-developing wave train.
To see how the coherence of wave packets changes with variations in baroclinicity, two other experiments were performed, in which the baroclinicity of the basic state is changed to one with weaker baroclinicity (ΔT equals 45 K) and another with stronger baroclinicity (ΔT = 80 K). In terms of wave propagation, the results for these two experiments are shown in Figs. 17b and 17c, respectively. Comparing these two figures to Fig. 17a, we see that as the baroclinicity is increased, wave packets in the model become progressively less coherent, in agreement with the modeling results of Lee and Held (1993). We have also conducted experiments in which the temperature profile in one hemisphere has a larger ΔT than that in the other hemisphere, but as long as the maximum equilibrium temperature remains on the equator, the results are similar to those shown in Figs. 17b and 17c.
In the atmosphere, seasonal variation in heating involves changes not only in the equator-to-pole temperature difference, but also in the position of the thermal equator. Lindzen and Hou (1988) showed that displacement of the thermal equator toward the summer hemisphere results in a substantial strengthening of the winter Hadley circulation and tightening of the subtropical potential vorticity gradient. Since upper-tropospheric waves basically propagate along bands of strong PV gradients, we expect that changes in the forcing of the PV gradient should affect wave propagation. Hence, we conducted a “solstitial” experiment in which the maximum heating is displaced away from the equator toward the summer hemisphere. The equilibrium temperature profile at the surface for this experiment is T0 − ΔT(sinϕ − sinϕ0)2. With a ΔT of 60 K, and ϕ0 of 6° latitude, the equator-to-pole equilibrium temperature difference works out to be 48 K for the “summer” hemisphere and 73 K for the “winter” hemisphere. The results for this experiment are shown in Figs. 18a and 18b. Compared to the control experiment (Fig. 17a), wave packets in both the winter and summer hemisphere appear to be slightly less coherent, even though the baroclinicity in the summer hemisphere is much weaker. For this case, if we consider the coherence of wave packets at a fixed time lag (e.g., 2-day lag), the correlation is slightly larger in the summer hemisphere, but the difference between summer and winter hemispheres has become much less than that suggested by Figs. 17b and 17c.
The results from the solstitial experiment suggest that forcing by the more intense Hadley circulation in the winter hemisphere, as well as the weakened circulation in the summer, counteracts the effects of changes in baroclinicity on wave coherence. Such a conclusion requires that the coherence of wave packets should increase along with an increase in Hadley circulation intensity. To test this hypothesis, we conducted two other experiments. The equilibrium temperature profiles for these two experiments were modified from the control experiment, with heating symmetric about the equator, and a ΔT of 60 K. For the “stronger Hadley forcing” experiment, we prescribed an additional heating perturbation with amplitude 1 K day−1 centered over the equator,4 and cooling perturbations with amplitude −0.5 K day−1 centered at 20°N and 20°S such that the net heating perturbation integrated over the globe was zero. This additional heating perturbation acted to increase the intensity of the Hadley circulation without changing the baroclinicity of the model climate substantially. For the “weaker Hadley forcing” experiment, we started with the control equilibrium temperature profile and then set the equilibrium temperature between 20°N and 20°S to be equal to the value at 20°N. As a result, the Hadley circulation in this experiment was weakened, and the baroclinicity of the model climate was slightly reduced. The properties of the wave packets in this two experiments are shown in Figs. 17e and 17f, while those for the control are reproduced in Fig. 17d for comparison. Comparing Figs. 17d–f, we see that the coherence of wave packets indeed increases progressively along with an increase in the Hadley intensity. In summary, Figs. 17a–c show that when changes in the Hadley intensity are small, coherence of wave packets decreases with an increase in the baroclinicity [consistent with the results of Lee and Held (1993)], while Figs. 17d–f show that when changes in the baroclinicity are small, coherence of wave packets increases along with an increase in the Hadley intensity. The similarity of wave packet coherence shown in Figs. 18a and 18b is due to these two effects largely canceling each other out.
While results from the solstitial experiment suggest that poleward displacement of the thermal equator can offset the effects of differences in baroclinicity between summer and winter, it is still not clear why wave packets are more coherent in NH and SH winter than in NH summer. One possibility is that even in the solstitial experiment, the intensity of the winter Hadley circulation is still only about one-third of that observed. Hou and Lindzen (1992) suggested that the discrepancies between the Hadley intensity predicted by idealized models and that observed could be due to the fact that heating within the Tropics is concentrated within a narrow band around the intertropical convergence zone (ITCZ), which can act to increase the Hadley intensity in the winter hemisphere even further. Diabatic heating “diagnosed” from observed thermal and dynamic fields [e.g., the climatological diabatic heating fields in the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis CD-ROM; Kalnay et al. 1996] does show strong convective heating over the thermal equator, flanked by radiative cooling in the subtropics, qualitatively similar to the concentrated heating profiles suggested by Hou and Lindzen (1992).
The diabatic heating fields from the NCEP–NCAR reanalysis project suggest that in JJA, the cooling in the winter (SH) subtropics is much stronger than that in the summer hemisphere (NH), while in DJF, cooling in the summer (SH) subtropics is only slightly weaker than that in winter (NH). Motivated by this observation, we conducted two additional “concentrated heating” experiments. For both experiments, the equilibrium temperature profile was based on the solstitial experiment, with the thermal equator at 6° latitude in the summer hemisphere. For case I, a heating perturbation with an amplitude of 1 K day−1 centered over the thermal equator was added, and a cooling perturbation with the same amplitude was added in the winter subtropics, while for case II, we added in the same heating perturbation over the thermal equator, but the cooling was divided between the winter and summer hemispheres, with 60% of the cooling in the winter subtropics and 40% of the cooling in the summer hemisphere. The discussions above suggest that case I is qualitatively similar to the JJA season, while the DJF season is more closely represented by case II. The intensities of the Hadley circulation in these two experiments are much closer to those observed, when compared to that in the solstitial experiment.
The results from these two concentrated heating experiments are shown in Figs. 18c–f. In case I (Figs. 18c and 18d), with additional heating over the thermal equator and cooling entirely in the winter subtropics, the winter Hadley circulation is greatly intensified, while the summer Hadley intensity is not significantly changed. As a result, wave packets are much more coherent in the winter hemisphere (Fig. 18c) as compared to the solstitial experiment (Fig. 18a), while the coherence in summer are not significantly affected. For case I, wave packets in winter are significantly more coherent that those in summer, qualitatively similar to what is observed in the JJA season.
In case II, with the cooling partitioned between the winter and summer subtropics, Hadley intensity in both hemispheres are stronger than those in the solstitial experiment. As a result, coherence of wave packets in both hemispheres (Figs. 18e,f) are higher than those in the solstitial experiment (Figs. 18a,b), and the coherence in summer is now slightly higher than that in winter. This result is qualitatively similar to what is observed in the DJF season.
In Table 1, we have listed several quantities that are representative of the climates of the various experiments. Relating the parameters listed in Table 1 to the results shown in Figs. 17 and 18, we see that, as discussed above, baroclinicity alone (as indicated by the equator-to-pole temperature difference observed at the 850-hPa level) definitely does not determine the coherence in wave propagation. To indicate the strength of the Hadley forcing, two quantities are listed in Table I—the absolute vorticity gradient at 250 hPa, as well as the PV gradient computed on the 330 K isentropic surface. In most sets of experiments, the coherence of wave propagation is stronger when the absolute vorticity gradient at 250 hPa is stronger, except for the solstitial experiment in which the wave coherence in the two hemispheres is quite similar, indicating a cancellation in the effects of stronger Hadley forcing by the large difference in baroclinicity between the hemispheres. Overall, as an indicator of wave coherence, the PV gradient at 330 K appears to do better than the baroclinicity, but not as well as the absolute vorticity gradient at 250 hPa. However, as the vertical resolution of the model near the tropopause is relatively coarse, there could be inaccuracies in the estimation of the PV gradients that depend strongly on vertical derivatives.
In this section, we conducted a series of experiments to try to understand the seasonal variations of wave packet coherence. Our results suggest that coherence of wave packets not only depends on the baroclinicity, as suggested by Lee and Held (1993), but also depends on the intensity of the Hadley circulation, which acts as a restoring force to the subtropical vorticity and PV gradients against the mixing action of the waves. Our results suggest that the strong asymmetry between the intensities of the winter and summer Hadley circulations could explain why wave packets are not significantly more coherent in summer, and depending on the details in the distribution of diabatic cooling in the subtropics, wave packets can even be more coherent in winter than in summer. However, our model results do show at least one significant difference from observation, in that the correlations shown in Figs. 17 and 18 are significantly higher than those observed. Figure 16 also shows that wave packets in the GFDL GCM are more coherent than observed wave packets. It is not clear what determines the absolute magnitude of packet coherence. We have performed sensitivity studies by varying model resolution and diffusion and found that coherence of wave packets decreases slightly when the resolution is increased to T42, and coherence is also reduced when the strength of diffusion is reduced. We expect that other factors, such as interactions between moist and dry physics, dynamics of the surface boundary layer, and asymmetry in the basic-state flow, may also affect the absolute magnitude of wave packet coherence. However, our sensitivity experiments suggest that even with different resolution and diffusion, the relative magnitude of wave packet coherence is still dependent upon both the baroclinicity and the Hadley intensity, and we believe that the conclusions reached in this section do not depend on model details.
5. Summary and conclusions
In CY99 and in this paper, we have documented the basic characteristics of wave packets in the winter and summer seasons of both hemispheres, mainly through examination of time-lagged correlation maps of υ′. We have also defined indices to quantify the degree of downstream/upstream asymmetry as well as wave packet coherence (Figs. 8 and 9) to highlight regions where wave propagation is most coherent. Based on these indices as well as individual correlation maps, we have constructed schematic waveguides for the different seasons that demarcate the pathways in which waves preferentially propagate with highest coherence.
During the summers, the waveguides basically follow the position of the midlatitude jets (see Fig. 14 for the SH summer waveguide). In the two winter seasons, things are a bit more complicated. In the Northern Hemisphere winter (see CY99), the primary waveguide follows the subtropical jet over Asia, but there is a secondary branch running across Russia, joining the primary waveguide near the entrance to the Pacific storm track. Over the Atlantic, instead of following the Atlantic storm track maximum toward the northeast, the waveguide passes east-southeastward toward North Africa and southern Asia. During the Southern Hemisphere winter (see Figs. 10–13), the waveguide splits into two around 70°E, with the primary (more coherent) branch deviating equatorward to join up with the subtropical waveguide, and a slightly less coherent branch spiraling poleward along with the upper-troposphere wind and storm track maxima.
By examining the wave coherence indices, as well as individual correlation maps and Hovmöller diagrams of correlations computed along the primary waveguides (not shown here), we arrived at the conclusion that wave propagation is least coherent in the NH summer, and that wave packets in the SH summer are not necessarily more coherent than those in the SH winter, in contrast to what was suggested by Lee and Held (1993). In section 4, we examined data generated by the GFDL GCM and found that the coherence of wave packets in the GCM also displays similar seasonal variations. We have conducted a series of modeling studies using an idealized model and found that the coherence of wave packets not only depends on the baroclinicity of the basic-state flow, but also on the intensity of the Hadley circulation, which acts to restore the PV gradients against the mixing action of the waves. In winter, while the baroclinicity is high, the intensity of the Hadley circulation is also much stronger than that in summer, and these two factors can offset each other with the result that wave packets are not necessarily more coherent in summer. However, we also found that wave packets in the idealized models are usually significantly more coherent than those observed, suggesting that there are some factors that contribute to determining the coherence of wave packets that are missing in the idealized model experiments. We are currently conducting other experiments to investigate this.
In this paper as well as in CY99, we used complex demodulation to define wave packet envelopes and estimated the average group velocity of wave packets by tracking correlation centers of wave packet envelopes. We have compared our group velocities with group velocities defined based on fluxes of wave activity, including the E vector of Hoskins et al. (1983) and the MT vector of Plumb (1986) for NH winter, and Eu vector of Trenberth (1991) for SH, and found strong qualitative agreement between our group velocity and these fluxes. A more quantitative comparison cannot be made since these works only presented the fluxes without normalizing them with the wave activity, and they only showed fluxes at a single level, whereas we believe that the physical group velocity should involve a vertical average as observed waves in the upper troposphere generally have significant amplitudes from below 500 hPa to above 200 hPa (Lim and Wallace 1991).
In Part I and this paper, we have shown that upper-tropospheric waves over the midlatitudes of NH and SH generally display the characteristics of downstream development, in summer as well as in winter. Much of the properties and dynamics of these downstream-developing wave packets are still not well understood and require further studies. One exciting prospect is that our correlation analysis suggests that a lot of cases of wave development in the upper troposphere appear to be connected to downstream-developing wave trains that can sometimes be tracked for weeks. Whether such a connection can be utilized to benefit forecasting in any way remains to be investigated.
The author would like to thank D. B. Yu for helping with the data analyses; J. Sloman for editorial comments; Y. Chen and the Scientific Computing Division at NCAR for assistance in accessing the ECMWF and NCEP–NCAR reanalysis data; and NCEP, NCAR, and ECMWF for making the data available. Thanks are also due to P. Phillips of GFDL for providing the GCM data, and I. Held of GFDL for providing the model used in this study. The author would also like to thank the anonymous reviewers, whose comments on the original manuscript resulted in significant revisions of this paper and Part I. The figures have been produced using GRADS developed by Brian Doty of COLA. This research is supported by NSF Grant ATM-9510008.
Corresponding author address: Dr. Edmund K. Chang, Program in Atmospheres, Oceans, and Climate, Massachusetts Institute of Technology, Room 54-1614, Cambridge, MA 02139.
The reference positive center is the correlation center that passes the base grid point at lag 0.
The NE–SW phase tilt of the waves also suggest poleward propagation (see Hoskins et al. 1985).
We added this extra stability to the equilibrium temperature profile because if we do not, eddy activity becomes unrealistically high for the experiments in which the heating is not centered at the equator.
The latitudinal profile of the heating perturbation is a Gaussian with an e-folding width of 10°. Vertically, the heating perturbation is a half-wavelength sine profile extending from 900 to 100 hPa.