Gridded data produced by the ECMWF reanalysis project have been analyzed to document the properties of wave packets in the Northern Hemisphere winter midlatitude upper troposphere. Based on results from earlier investigations, 300-hPa meridional wind variations were chosen for analysis. Wave packet envelopes were also defined by performing complex demodulation on the wind data. The properties of the wave packets are mainly illustrated using time-lagged one-point correlation maps performed both on υ′ and wave packet envelopes.
The results show that, over most regions in the Northern Hemisphere winter, with the exception of the deep Tropics and near the Aleutian low, medium-scale waves (dominant wavenumber 5–8) exhibit the characteristics of downstream development and occur within wave trains that propagate with eastward group velocities much faster than the phase speeds of individual phases of the waves. Their group velocity is highly correlated with the local time mean 200–400-hPa wind, while the phase speed is well correlated with the 500–700-hPa flow.
A wave coherence index has been defined to show the geographical variations in the downstream development tendency of wave propagation. The results show that wave packets are most coherent along a band that extends from North Africa into southern Asia, toward the Pacific storm track, across North America, then over the central North Atlantic back toward North Africa. The maximum coherence occurs over southern Asia. This band can be regarded as the waveguide for upper-tropospheric wave packets in the Northern Hemisphere winter. Over this band, wave packets generally stay coherent significantly longer than individual troughs and ridges. There is also a secondary waveguide across Russia toward the Pacific, acting as a second source of waves that propagate across the Pacific storm track. Away from the primary waveguide, while wave packet coherence is less, the waves still show the characteristics of downstream development.
Observational studies (e.g., Namias and Clapp 1944;Cressman 1948; Hovmoeller 1949) have shown that downstream development of baroclinic waves occur in the mid- and upper troposphere. Numerical modeling studies (e.g., Simmons and Hoskins 1979) also showed that downstream development of baroclinic waves occur readily in model simulations starting with an initially localized perturbation in a baroclinically unstable atmosphere. Recently, Chang (1993), Lee and Held (1993), and Berbery and Vera (1996) showed that baroclinic waves in the midlatitude storm tracks tend to be organized in localized wave packets that clearly exhibit downstream development. In this paper and Part II (Chang 1999), we will try to study these wave packets in further detail.
In a series of papers, Blackmon et al. (1984a,b), Wallace et al. (1988), and Lim and Wallace (1991) documented the three-dimensional structure and temporal evolution of high-frequency fluctuations in the midlatitude areas of Northern Hemisphere winter, based on correlation analyses applied to high-pass filtered (half power point near 6.5 days) geopotential height data. However, it was later shown by Chang (1993) that by time filtering the data, the temporal evolution of wave propagation was distorted, leading to their erroneous conclusion that the baroclinic wave trains were simply advected by the 700-hPa flow field. Chang (1993) showed that over the Pacific and Atlantic storm track regions, baroclinic waves exhibit the distinct characteristics of downstream development and occur in wave packets that propagate with group velocities much faster than the phase speeds of individual waves. In this study, we would like to extend the study of Chang (1993) and follow in the footsteps of Blackmon, Wallace, and coworkers, to document the temporal evolution of baroclinic waves over the entire Northern Hemisphere midlatitudes.
Downstream development of baroclinic waves is related to the theory of spatial-temporal instability first developed in plasma physics (see, e.g., Briggs 1964) and brought into meteorology by Merkine (1977). In linear theory, a localized wave packet in an unstable medium will spread out toward both upstream and downstream directions in space, with the speed of the leading and trailing edge, the speed of the wave packet maximum, and the structure of waves in the wave packets all predicted by linear asymptotic theory (see also Pierrehumbert 1984). Lee and Held (1993) and Swanson and Pierrehumbert (1994) showed, using numerical experiments, that nonlinearity modifies the wave characteristics substantially, with the leading edge of the linear wave packet developing into a localized nonlinear wave packet, and the trailing end somehow dissipating, leaving just a localized downstream developing wave packet propagating with the speed of the leading edge as predicted by linear theory. However, what controls the length of the nonlinear wave packet is still unknown.
The dynamics of downstream-developing wave packets had been examined both in model simulations and observations. Orlanski and Chang (1993), Chang and Orlanski (1993), and Chang (1993) examined the energy budget of downstream-developing waves and found that growth of new perturbations toward the downstream side of existing waves are triggered by energy fluxes from existing upstream perturbations. Discussions of the dynamics as exemplified by potential vorticity advection can be found in Nielsen-Gammon (1995) and Nielsen-Gammon and Lefevre (1996). The above studies differentiate the life cycle of baroclinic waves characterized by their downstream development from the conventional one in which incipient disturbances amplify baroclinically as a normal mode and then decay barotropically (e.g., Simmons and Hoskins 1978). On the other hand, Lee and Feldstein (1996) showed that baroclinic waves in wave packets also exhibit the two distinct types of baroclinic wave life cycles described in Thorncroft et al. (1993).
In order to gain further understanding of these wave packets, we have analyzed gridded observational data to document the basic characteristics of these wave packets, and the geographical and seasonal variations in their characteristics. We have also conducted some simple numerical modeling experiments in order to understand some of the observed features. The results are presented in two papers; this paper, which is Part I of the series, will focus on the properties of wave packets in Northern Hemisphere winter. In Part II (Chang 1999), we will document the seasonal and hemispheric variations in the properties of wave packets by analyzing data from the Northern Hemisphere summer and Southern Hemisphere summer and winter to compare to the results in Part I.
In section 2 of this paper, we will describe the data used in our analyses, as well as the basic analysis techniques. In section 3, we will examine some basic properties of the waves, such as wavelengths, periods, phase speeds, and group velocities. In section 4, we will define a wave coherence index that summarizes the coherence of downstream development and examine the geographical variations of wave packet propagation. Correlation maps for wave propagating across selected locations will be examined in section 5. In section 6, we will return to the issue of time-filtered versus unfiltered data. Further discussion will be presented in section 7.
2. The data and basic analysis techniques
a. The data
The data we use in this study are the gridded data from the European Center for Medium-Range Weather Forecasting (ECMWF) reanalysis project.1 Following Chang (1993), we have chosen to analyze 300-hPa meridional wind (υ). Spectral analyses of eddy velocity (e.g., Kao and Wendell 1970) have shown that the meridional velocity in the midlatitude upper troposphere is dominated by zonal wavenumbers 5–7, which represent the typical scale of the baroclinic waves that we are interested in. Hence, we can use unfiltered data in our analyses. Chang (1993) showed that time filtering can alter the temporal evolution characteristics of wave packets, and the result was confirmed by the study of Berbery and Vera (1996). Hence, the use of unfiltered data is necessary in order to capture the temporal evolution and propagation characteristics of wave packets. We have also performed some analyses using time-filtered 300-hPa meridional wind, and 300- and 500-hPa geopotential height for comparison, and the results will be discussed in section 7.
Since we are interested in transient waves, we have removed the stationary component by removing the December–January–February mean from each winter season from the wind data. The data are otherwise unfiltered. This dataset, with the seasonal mean removed, will be referred to as υ′ in the following discussions. Data from 13 winters (winter is defined as the 90-day period starting from 1 December), from 1980/81 to 1992/93, available on a 2.5° × 2.5° grid, have been analyzed.
The standard deviation of υ′ over the 13 winters is shown in Fig. 1a. The maximum (highlighted region corresponds to >18 m s−1) corresponds to the midlatitude storm tracks, which extend from the central Pacific across North America and the Atlantic into Europe toward Russia. Over Asia, we can see a hint that the maxima in the eddy variances split into two branches, one passing north of the Caspian Sea with the maximum around 60°N, and a slightly weaker branch passing over south Asia between 20° and 30°N. A minimum in eddy velocity amplitude can be seen around Tibet. Unlike bandpass-filtered 500-hPa geopotential (e.g., Blackmon 1976), no obvious break can be seen between the Pacific and Atlantic storm tracks. However, it does not appear that the absence of the break is due to contributions from low-frequency perturbations, as analyses of the frequencies and phase speeds of the waves (to be shown later in Figs. 5c and 6a) do not show marked minima over the western North American region. It would be otherwise if heavy contamination from low-frequency perturbations were present. The geostrophic wind computed from the height perturbations do show such a break (see also Blackmon et al. 1977), but as discussed in Lau (1978), the winds in the eddies are strongly supergeostrophic over western North America, such that the variance in total υ′ does not display a significant minimum between the Pacific and Atlantic storm tracks.
b. One-point lag-correlation maps
The main analysis tool used here will be based on one-point lag-correlation maps. Blackmon et al. (1984a,b) used one-point lag-correlation maps to study the horizontal structure and time variation of 500-hPa height fluctuations. A slightly different regression technique was used by Lim and Wallace (1991) and Chang (1993) to study baroclinic waves in the Northern Hemisphere winter. The regression technique is useful in showing the relative amplitude of the wave disturbances at different altitudes. However, in this paper we will focus on the wave propagation characteristics in the upper troposphere and will concentrate on just one level (300 hPa), and one-point correlation maps are more useful in showing indications of differences in the coherence of wave propagation over different geographical regions, since correlations are normalized w.r.t. the local variance, while regressions are not and their values over different regions are highly dependent on the value of the local variance as well as on the coherence of wave propagation.
An example of one-point lag-correlation maps, based on the time series of υ′ at 40°N, 180°, over the middle of the Pacific storm track, correlating with υ′ at all points in the Northern Hemisphere, is shown in Figs. 2a–e. In appendix A, we will discuss the statistical significance of the correlation. Basically, we found that a correlation of 0.07 is significant at the 5% level. The maps show the basic characteristics of a downstream-developing wave train as discussed in Chang (1993). At lag 0 (Fig. 2c), we can see a wave train extending from east Asia all the way across the Pacific. Looking at time-lagged correlations from day −2 to day +2, we see that the waves have a phase speed of about 12 m s−1. When we follow the positive center that includes the base point on day 0 backward and forward in time with the phase speed, we see that on days −1 and −2, there is much higher correlation toward the upstream side of the reference center, while on days +1 and +2, the correlation toward the downstream side is stronger than that toward the upstream side. This is a clear indication of downstream development. In other words, we can see a wave packet propagating through individual phases of the waves with a group velocity much faster than the phase speed of the waves. From Figs. 2a–e, the group velocity of the wave packet can be estimated to be between 25 and 30 m s−1. Time-lagged correlation maps have been computed for every grid point on a 5° × 5° grid from 10° to 70°N and form the basic analysis tool that we will employ in this study.
c. Wave packet envelopes
While the examination of one-point lag-correlation maps of υ′ like those shown in Fig. 2 can give us a lot of information about the spatial structure and temporal evolution of wave packets, there are other questions that are difficult to answer based on analyses of these quantities alone. For example, it is difficult to estimate the group velocity of the wave packets from Fig. 2, since it is difficult to locate accurately the center of the wave packets at different time lags. In addition, if one wants to compile statistics on, for example, how long a typical wave packet lasts, whether there are preferred locations for the formation or dissipation of these wave packets, etc., one would need to follow and track individual wave packets. These tasks are complicated by the fact that it is quite difficult to objectively track a wave packet by examining time series of υ′, since, as seen above, the wave packets have a group velocity greater than the phase speed of individual waves, and the peak of the wave packet will reside in different phases of the wave at different times, making it difficult to follow the position of the peak.
Lee and Held (1993) used the method of complex demodulation in space (see, e.g., Bloomfield 1976) to study the spatial structure of wave packets seen in their modeling studies. In simple terms, the waves are assumed to exist in the form
where k is the wavenumber of a typical midlatitude baroclinic wave and A(x, t) is the envelope of the wave group and is slowly varying in space. The procedure of complex demodulation allows the envelope function A(x, t) to be retrieved given υ′(x, t). At each time and for each latitude circle, we divide υ′(x, t) by eikx, and then the resulting vector is spatially smoothed to retain just the low wavenumber components to get A(x, t)/2. In this procedure, we have to specify a carrier wavenumber, k. For this study, we have used a latitudinally varying wavenumber that is the zonal mean of the characteristic wavenumber that we find for these waves (to be shown in Fig. 5b).2 We have also performed demodulation using a single wavenumber (6 or 7) for all latitudes and found that the results are not sensitive to the precise value of k used except in high latitudes away from the main storm track. An example in which this procedure is applied to the data is shown in Fig. 3. In Fig. 3a, υ′ is plotted, while the corresponding envelope function |A(x, t)| (which will be referred to as υe in subsequent discussions) is plotted in Fig 3b. We see that υe basically picks out regions where υ′ has a large amplitude. We should note that while υ′ has phases with positive and negative values, υe is positive definite, and by examining the time sequence of maps of υe, we found that wave packets are much easier to track using maps of υe than following the time evolution of υ′.
In Fig. 1b, the time mean of υe over the 13 winter seasons is shown. Comparing that to Fig. 1a, we see that the spatial distribution of mean υe resembles that of the standard deviation in υ′, except that the storm track as depicted by Fig. 1b appears to be smoother.
Similar to what we have done to the time series of υ′, we can compute one-point lag-correlation maps of υe. The correlation maps based on the time series of υe at the base points 40°N, 180°, are shown in Figs. 4a–e. They should be compared to Fig. 2. Examining Fig. 4, we can see that the envelope of the wave packet propagates eastward at about 30 m s−1. Comparing Figs. 2 and 4, we see that the group velocity can be more easily estimated based on correlations of υe than from correlations of υ′.
3. Basic properties of waves and wave packets
First, we will examine some basic characteristics of the waves and wave packets obtained using the correlation analysis. Using the one-point correlation maps with zero lag (e.g., Fig. 2c) applied on υ′, we can compute the wavelengths of the waves. As discussed earlier, we have done that for all points on a 5° × 5° grid between 10° and 70°N, and this wavelength is plotted in Fig. 5a. We see that north of 30°N, the waves that are picked out by the correlation analysis have wavelengths of between 4000 and 5000 km, the typical scale of midlatitude baroclinic waves. The wavelengths are quite a bit larger in the Tropics to the west of the date line. The corresponding zonal wavenumber is shown in Fig. 5b, which shows that over much of the midlatitudes, the signal is dominated by wavenumber-6–8 waves. By following the movement of the positive correlation maximum that passes the base grid point at time 0 from day −1 to day +1 (e.g., Figs. 2b and 2d), we can compute the phase speed of the waves (see next paragraph) and, hence, we can get the periods of the waves picked out by the analysis. This is shown in Fig. 5c. Over the midlatitude band, we see that the dominant period is between 4 and 8 days, while the period is longer over the Tropics because of the increase in wavelengths of the waves, and in the higher latitudes due to the decrease in phase speeds of the waves.
In Fig. 6a, the zonal component of the phase speed of the waves is displayed. The phase speed is computed from time-lagged correlation of υ′ by following the positive correlation center, which includes the base point at day 0 from day −1 to day +1. The spatial correlation between the zonal component of the phase speed with the time mean flow is best between 700 and 500 hPa, where the correlation is greater than 0.78. The 700-hPa zonal wind (u700) is shown in Fig. 6b. Comparing the two (and also the differences between the hemispheres, to be discussed in Part II), we see that over land, the zonal phase speed (cpx) is generally slightly larger than u700, while the reverse is true over oceans. Such a relationship suggests that the steering level of the waves is just above 700 hPa over land, and below 700 hPa over oceans. Wallace et al. (1988) also estimated phase speeds of baroclinic waves at the upper troposphere. Compared to their results, the zonal component of our phase speeds is generally weaker (which is not surprising since they analyzed high-pass filtered data), but the meridional component (not shown here) appears to be consistent.
One simple definition of the group velocity of perturbations is the speed of movement of wave packet envelopes [(1)]. This can be estimated by tracking the movement of correlation centers of υe (Fig. 4). In Figs. 6c and 6e, the zonal (cgx) and meridional (cgy) components of the group velocity, estimated from the displacement in the position of the maxima in the correlation of υe from time lag of day −1 to day +1, are shown. We have also computed the group velocity by following the correlation centers from day −0.5 to day +0.5, and the results are practically the same, except that when we used the shorter time period, the estimates of cgy become rather uncertain because of the finite grid size of the data and the generally small value of the meridional component of cg.
Examining Figs. 6c and 6e, we see that the group velocity is basically zonal, with a much weaker meridional component. The zonal component is largest near the jet maxima, which can be clearly seen when we compare the group velocity to the time mean 300-hPa zonal wind (u300) from the same period, which is shown in Fig. 6d. The spatial correlation between cgx and u300 is 0.88. We have also correlated cgx with zonal winds from other levels and found that the correlation is approximately the same from 400 through 200 hPa, basically the layer in the vicinity of the midlatitude tropopause, and the correlation decreases away from that layer. Over most areas, cgx is smaller than u300, and over the region 10°–70°N, the rms value of cgx is only 70% of that of u300. The magnitude of cgx is in fact closest to the zonal wind between 400 and 500 hPa. Comparing the two figures more carefully, it is apparent that the estimated value of cgx deviates most from u300 over the jet core region of the Pacific jet.
Comparing Figs. 6a and 6c, we see that over the storm track regions, the group velocity is between two and three times the magnitude of the phase speed, and both generally point eastward, which tells us that downstream development is a general characteristic of synoptic-scale waves in the midlatitudes.
How good are these estimates? Correlation amounts to averaging over many cases. One can imagine that if there exists a range in the meridional group velocity of the wave packets, the trajectories of wave packets that pass through each point will spread out meridionally as one moves away from the base point. Hence, by averaging the envelopes first before estimating the group velocity (essentially what is done here), one expects to have a bias against high group velocity cases, since those will be smoothed out much more than the low group velocity cases. To get an estimate of such a bias, we tracked wave packet envelopes across two base points (27.5°N, 60°E, and 35°N, 150°E) to estimate the average group velocity. For each of these points, a total of 18 and 22 cases, respectively, in which a wave packet can be unambiguously tracked for at least 3 days across the base points are found, and the mean velocity of the envelopes is found to be approximately 10% higher than that computed using the correlation analysis (Fig. 6c). But we also found that there is quite a large spread in envelope velocity (e.g., from 14 to 40 m s−1 for the first case, with a mean of 28.3 and a standard deviation of 7.5 m s−1), and the difference between the two estimates (3 m s−1) is not statistically significant at the 5% level.3 For these cases we have also estimated the wavelength of the waves, and the estimates varied between wavenumber 5 and 9. However, no relationship can be found between the wavelength of the waves and the envelope speed.4
Next, when we examine cgy in Fig. 6e, we see that wave packets to the north of the storm tracks generally have positive cgy, and those south of the storm tracks negative cgy. In fact, while cgy is generally smaller than υ300 (Fig. 6f), the zonal mean of cgy is nearly −2 m s−1 near 35°N, significantly larger than the equatorward values of υ300 (zonal mean >−0.5 m s−1). At 60°N, the zonal mean of cgy is over 1.5 m s−1, while that of υ300 is less than 0.3 m s−1. These suggest that the wave energy preferentially diverges away from the storm tracks, which act as wave energy sources. However, the magnitude of cgy is generally much smaller than that of cgx; hence, energy propagation is predominantly zonal.
Hoskins et al. (1983) defined an E vector, which qualitatively indicates the sense of the group velocity relative to the mean flow. Plumb (1986) derived an approximate wave activity conservation relationship for eddies in a time mean flow, with a total wave activity flux vector MT that is equal to the wave activity times the group velocity in the WKB limit. The divergence of MT is related to nonconservative wave activity sources and sinks. Plumb (1986) pointed out that if the flow is “pseudo-eastward,” the meridional component of MT is proportional to the meridional component of the E vector, while the angle between the group velocity vector relative to the mean flow (i.e., cg − u) and the x axis is one-half the angle between the E vector and the x axis. Anyway, one would expect the meridional group velocity to be related to the meridional component of either the E vector of Hoskins et al. or Plumb’s MT vector.
Comparing our estimates of the group velocity with Plumb’s MT vector (Fig. 7b of Plumb 1986), we find qualitative agreement in general. Both generally point eastward. In both cases, the vectors are predominantly northeastward over the Pacific and western Atlantic to the north of 45°N, and southeastward over the east Atlantic and Europe, eastern Siberia, and the central Pacific. With regard to the magnitude of the group velocity, the situation is less clear, since Plumb did not show plots of MT divided by the wave activity. The zonal component of Plumb’s MT vector is dominated by the advective flux, which would suggest that the zonal group velocity should be close to the local zonal wind speed, if the WKB limit holds. However, observed eddies are highly correlated all through the depth of the troposphere (Wallace et al. 1988; Lim and Wallace 1991); hence, one would expect the group velocity of actual perturbations to be a weighted mean of the group velocity computed from each individual level in the vertical, and values somewhat less than the upper-troposphere jet speed are not inconsistent.5 Anyway, it is comforting to see that the group velocities defined under different assumptions bear so much resemblance to each other, giving us more confidence in the possible validity of such a concept.
4. Propagation of wave packets as indicated by lag-correlation statistics
a. Downstream versus upstream development
In Fig. 2, we saw that if we examine the temporal evolution of the correlation maps of υ′, we can observe an asymmetry between upstream and downstream correlations for nonzero time lags. As we follow the positive correlation maximum that passes the base grid point at day 0, the correlation extrema toward its upstream are stronger than those toward its downstream for negative lags, whereas the opposite is the case for positive lags. Such an asymmetry indicates downstream development. To see whether this is a general characteristic of waves over the entire hemisphere, we define an index to indicate the degree of downstream versus upstream asymmetry. The index compares the relative magnitude of the upstream and downstream negative correlation centers flanking the reference positive center 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. Index values greater than 1 will indicate that downstream development is preferred, whereas those less than 1 will indicate upstream development. Values close to 1 will suggest that the wave packets are simply advected by the steering level flow (in other words, the waves are nondispersive). From Fig. 2, we can estimate the value of this index to be about 2 at 40°N, 180°. This downstream/upstream asymmetry index is plotted in Fig. 7a. We see that the index is larger than 1 over most of the hemisphere. In fact, over most of the area, the index is larger than 1.5. This result is consistent with the earlier observation (Fig. 6) that the eastward group velocity is larger than the phase speed over most regions, which necessarily requires downstream development with respect to the phase.
b. Coherence of wave packets and the schematic waveguide
While Fig. 7a shows that waves exhibit downstream development characteristics over most areas, it does not tell us how coherent the downstream-developing wave packets are. To show that, we define a wave coherence index (WCI2), based on the lag-correlation maps of υ′. The index is defined as the average magnitude of a pair of the strongest (either positive or negative) correlation values, one of which is observed at lag −2 upstream of the positive reference correlation center that passes the base grid point at lag 0, and the other observed downstream of it at lag +2. Note that if the maximum correlation occurs within the positive reference center, the value is disregarded, and only values observed at upstream centers at lag −2 or downstream centers at lag +2 are taken to compute the index value. As an example, for the point 40°N, 180°, from Fig. 2a, we see that on day −2, the maximum upstream correlation is about −0.44 in the upstream negative center near 130°E, and at day +2, the maximum correlation toward the downstream side is about −0.25 near 120°W (Fig. 2e); hence, the value of the index (WCI2) is approximately 0.35 for this base point. The geographical variation of WCI2 is shown in Fig. 7b. As an indication of the level of uncertainty of WCI2, we also computed the index using the NCEP–NCAR reanalysis data. The rms difference between the index based on the two datasets is about 0.02, showing that the pattern shown is independent of the details of the data analysis. We have also computed wave coherence indices using correlation maps with other time lags, and the spatial patterns are, in general, similar. For example, for a time lag of 4 days (not shown here), the maximum coherence still occurs along a band extending from southern Europe to the western Pacific, but the maximum index is just 0.35 over south and southeast Asia, instead of near 0.6 for WCI2.
To put the values we see in Fig. 7b into perspective, let us compare WCI2 to the coherence of individual phases (ridge/trough) of the waves. In the time-lagged correlation of υ′ as in Fig. 2, we follow the positive correlation center that passes the base grid point at lag 0 and average its maximum values at lags −2 and +2. For example, for the base point 40°N, 180° (Fig. 2), at day −2 the center corresponding to the same phase is located around 45°N, 155°E, with a correlation of about 0.25, while at day +2, the center is around 30°N, 160°W, with a correlation of about 0.23, and the average value of 0.24 is plotted at the base point in Fig. 7c. Basically, Fig. 7c tells us how well individual trough and ridge lines remain coherent in 2 days. We see that the coherence of individual phases are highest over south Asia, and low over the central North Pacific and North Atlantic storm track regions. To compare with WCI2, we can define a packet decorrelation time (τpac) based on WCI2 as follows:
Similarly, we can define a phase decorrelation time (τphase) based on same phase coherence (Fig. 7c). The ratio τpac/τphase is shown in Fig. 7d. We see that over south Asia and the midlatitude storm track regions, this ratio is mostly larger than 1, suggesting that over these regions, wave packets remain coherent entities longer than individual troughs and ridges. While the ratio is not very large, a 20%–40% increase in decorrelation time over south Asia and the northeastern Pacific is not negligible.
Examining Fig. 7b, we can see a band of maximum WCI2 extending all the way from northeastern Africa toward the central Pacific, suggesting that there is a“waveguide” over those regions in which wave packets preferentially propagate with high coherence. North of the main south Asian waveguide, we can see a weak band of maxima extending from northern Europe across Siberia, joining up with the main band of maxima near the entrance to the Pacific storm track. Over North America and the western Atlantic, we can observe another weak band of maxima. Fig. 7d shows that over these regions, wave packets remain coherent longer than individual troughs and ridges. Based on these two figures, as well as examination of temporal sequences of correlation maps (to be discussed in the next section), we have constructed a “schematic waveguide” (shown as the shaded band in Fig. 7e and subsequent figures) to depict the trajectory that wave packets preferentially take. This band runs along 25°N over south Asia, then passes east-northeastward across the Pacific, reaching about 40°N east of the date line, and continues eastward across North America. It then passes east-southeastward across the Atlantic and north Africa back toward southern Asia. As seen in Fig. 7b, coherence of wave packets are lower over the eastern Pacific and eastern Atlantic;hence, the waveguide is depicted by broken rather than continuous bands over those regions. A secondary branch running across Russia (again denoted by a broken band) joins up with the main wave guide near 140°E. We will come back to this secondary branch in section 5b.
The schematic waveguide is basically constructed following the band of maximum WCI2 shown in Fig. 7b. We will see later that temporal sequences of time-lagged correlation maps also show that wave packets do in fact propagate along such a track. From an alternative point of view, Ertel’s potential vorticity on the 330 K isentrope (IPV; see Hoskins et al. 1985) is also plotted in Fig. 7e. Since large-scale waves can be viewed as waves propagating on potential vorticity (PV) gradients, another possible definition of a waveguide would be regions of strong PV gradients (where waves can propagate) flanked by regions of weak or zero PV gradient (where waves cannot propagate). It is interesting to note that the most coherent portions of the schematic waveguide defined here do coincide with those regions with strong PV gradients flanked by regions with weak gradients. Plots of the quasigeostrophic refractive index (not shown here) also show a wave duct across the midlatitudes, but the exact position of the duct (e.g., whether waves can propagate across the eastern Atlantic) depends on the position of the critical latitude and, hence, the phase speed chosen, and it is not clear which value of phase speed is most representative. In fact, Fig. 6d shows that over the eastern Atlantic (near 30°N, 25°E) there is a region where the mean zonal wind is near or below 10 m s−1, suggesting that this region of weak zonal flow could act as a “low-pass filter,” where only waves with low phase speeds can pass through without being absorbed, leading to the relatively low frequency of waves observed over the subtropical wave guide across south Asia.
While we have defined the schematic waveguide following the maximum wave coherence (Fig. 7b), we want to point out that this picture is largely consistent with the picture seen from the E vector of Hoskins et al. (1983) and Plumb’s MT fluxes. The E vector at 250 hPa [Fig. 6a of Hoskins et al. (1983)] shows “propagation” from the Pacific storm track across North America over the Atlantic. Then, to the east of the Greenwich meridian, the vectors can be seen to point southeastward toward the Mediterranean and North Africa, then propagating across south Asia. The MT fluxes shown in Plumb (1986, Fig. 7b) also show a similar picture. Moreover, Plumb’s fluxes give a clearer picture about what happens across the Atlantic. The wave activity flux apparently splits into two branches over the Atlantic, with the northern branch pointing northeastward toward northern Europe, showing strong convergence and hence dissipation, and the southern branch running across the Mediterranean and North Africa with more constant amplitude and hence indicating propagation. East of the Greenwich meridian, the maximum flux can be seen to lie south of 40°N, consistent with our main waveguide across North Africa and south Asia. The main discrepancy between our waveguide and the wave activity fluxes occurs over southeast Asia, where the MT flux does not show significant propagation from south Asia into the Pacific. One possible reason is that a relatively narrow bandpass filter (2.5–6 days) was used to compute the eddy fluxes. From Fig. 5c, we see that the waves over south Asia near 25°N have characteristic periods of around 8–10 days; hence, fluxes associated with those waves could have been filtered out by the filter.6
c. Longitudinal variations in wave propagation characteristics along the waveguide
Before examining temporal sequence of time-lagged correlation maps to show the propagation of wave packets over different regions, in Fig. 8 we show Hovmöller (longitude–time) diagrams of the time-lagged correlation to show how waves propagate along the schematic waveguide. At each longitude, we averaged υ′ over a 20° latitude band centered on the schematic waveguide (shaded band in Fig. 7e). Over Asia, where we have a split waveguide, the average was performed following the more coherent southern branch. We then computed one-point lag-correlation statistics based on the time series at different longitudes. The Hovmöller diagrams for the correlation based on the time series at six different longitudes are shown in Fig. 8. Note that each of the panels has been shifted such that the base point is at the center of the longitude axis. For each of the longitudes shown, we clearly see the signal of a downstream-developing wave train. We can also see that the wave groups maintain their coherence best around 60°E (Fig. 8b), where the associated wave packet can be tracked from west of 90°W to near the date line.
Comparing Fig. 8d (base point at the date line) with Fig. 3 in Chang (1993), which was computed using υ′ averaged between 30° and 60°N, we see that by following the latitudinal excursion of the waveguide, we are now able to pick up signals of the wave packet over longer periods of time, propagating across a wider longitude range. In fact, Fig. 7b shows that around 45°N, 60°E, there is a minimum in wave coherence. Hence, averaging from 30° to 60°N over that region would mean averaging across the minimum, explaining why Fig. 3 in Chang (1993) failed to pick up the strong upstream coherence of the Pacific storm track, which shows up much clearer in Fig. 8d here.
5. Characteristics of wave propagation over selected regions
a. Propagation across south Asia
Examining Figs. 7b and 8, we can see that wave propagation is most coherent across south Asia. In Fig. 9, we show the maps for the base point 25°N, 65°E, located near the maximum of WCI2. Figures 9a–e show the correlation with time lags of −4, −2, 0, +2, and +4 days, respectively. On day −4, we see that the maximum correlation is located over western Europe, with maximum correlations near 0.3. On day −2, the wave packet has propagated eastward and somewhat toward the south into North Africa and southwest Asia, reaching south Asia on day 0. After that, the eastward propagation continues, but the wave packet appears to turn in a more east-northeast direction toward east Asia, and by day +4, the maximum correlation is located over the west Pacific. This series of correlation maps shows significant correlations between waves over Europe and waves in the western North Pacific over a period of 8 days. From Fig. 9, we can estimate that over the 8 days, the center of the wave train has traveled nearly 180° in longitude, translating into an average group velocity of about 25 m s−1, consistent with the estimates shown in Fig. 6c. Note that apart from propagating eastward, the wave train also shows tendencies to follow the north–south displacement of the waveguide defined in the preceding section.
Figure 9 shows that waves over south Asia are significantly correlated with waves over the western Pacific. If we examine Fig. 5c, we see that waves over south Asia near 25°N have characteristic periods of around 10 days or more, while waves over the western Pacific have periods close to 6 days. This is consistent with differences both in the phase speeds (Fig. 6a) and wavelengths (Fig. 5a) of the waves over the two regions. The change in phase speed can be seen clearly in Figs. 8c and 8d, both of which show a significant increase in the phase speed of the waves near 120°E. In section 6, we will discuss whether one should regard waves over these two regions having different periods to be truly related, or whether one should separate the data into low-frequency and high-frequency perturbations and regard them as different types of perturbations to be treated separately.
b. Entrance to the Pacific storm track
In Fig. 7b, we see that over Asia, there are two branches of relatively high WCI2, the stronger branch lying along 25°N, and a weaker one just south of 60°N, separated by a region of much lower WCI2 near 45°N. To examine what happens there, let us look at the correlation near the entrance of the Pacific storm track, at 40°N, 140°E. Figures 10a,b show the correlation maps for υ′ at days −2 and +2. We see that for day +2 (Fig. 10b), there is significant downstream correlation, showing propagation of the wave packet from the storm track entrance region across the Pacific. However, for the correlation on day −2, we see that the upstream waves appear to be split into two wave packets, one traveling across southern Asia along the subtropical jet stream, the other traveling across Siberia. Correlation maps computed at the two upstream maxima around 50°N, 95°E, and 30°N, 80°E, are shown in Figs. 10c–f. These figures show that while waves over either branch have a strong tendency to develop downstream into waves in the Pacific storm track, the two pathways are not mutually correlated; that is, wave packets in the two branches that propagate toward the Pacific do not necessarily arrive at the storm track entrance simultaneously. Figure 10 also shows that the propagation along the southern path appears to be more coherent. In summary, Fig. 10 shows that waves in the Pacific storm track have two upstream sources—a northern branch across Siberia and a southern branch with waves propagating across southern Asia along the subtropical jet stream. This is the reason why we have added a secondary branch for the schematic waveguide shown in Fig. 7 and subsequent figures. We have also computed maps similar to those shown in Fig. 10 using υe rather than υ′. The results (not shown here) also show that the wave packets propagate via two pathways across Asia. While WCI2 over the northern branch is not very high, we believe that it should be considered as a legitimate secondary branch of the waveguide. Joung and Hitchman (1982) also found downstream-developing wave trains propagating across Siberia. We have examined correlation maps for other base points and found that the Pacific storm track entrance is the only region where there are two apparent upstream sources for the wave trains.
c. Wave propagation across other regions
In Figs. 9 and 10, we examined wave trains propagating across Asia into the Pacific. Wave propagation across the Pacific storm track was examined in detail in Chang (1993) and also discussed briefly above when we examined Figs. 2 and 4. Here, we will examine propagation of wave trains over other regions along the waveguide.
Lag-correlation maps for the base point 45°N, 110°W, with time lags of −2 and +2 days are shown in Figs. 11a and 11b, respectively. The maps show a downstream-developing wave train propagating from the eastern end of the Pacific storm track across North America toward the Atlantic, suggesting that wave developments near the entrance of the Atlantic storm track are influenced by waves from the Pacific. This result is consistent with the studies of Nielsen-Gammon and Lefevre (1996) and Orlanski and Sheldon (1995). These studies examined two separate cases of trough formation over the east coast of North America, and both concluded that the formation of the trough was primarily due to downstream propagation of Rossby wave energy from disturbances over the northeast Pacific.
In Figs. 11c and 11d, lag-correlation maps for the base point 40°N, 65°W, near the entrance to the Atlantic storm track, are shown. Figure 11c shows significant correlation on day −2 toward the upstream side, consistent with the discussions in the preceding paragraph. On day +2 (Fig. 11d), the correlations toward the downstream side are quite low (the maximum correlation of over 0.3 occurs within the positive center, which corresponds to the phase that contains the base point at day 0). In fact, while the PV gradient (Fig. 7e) is relatively weak over the eastern Atlantic, eddy amplitudes in terms of υ′ (Fig. 1) or IPV perturbations (not shown) are still relatively strong, implying that the waves are more nonlinear and exhibit less of the dispersive wave packet type of behavior seen in the other regions along the wave guide. Figure 11e, which shows the day −2 correlation for the base point 35°N, 0°, confirms that wave packets are not very coherent across the Atlantic. However, as discussed earlier, the wave activity flux MT of Plumb (1986) and the E vector of Hoskins et al. (1983) both show significant southeastward propagation across the Atlantic toward the Mediterranean and North Africa, and Figs. 8b and 8c also show signs of upstream correlation connecting perturbations over the Atlantic to the south Asian waveguide; hence, we have connected the wave guide over North America with the North Africa/South Asia waveguide (albeit with a broken instead of continuous band) to complete the waveguide around the globe. Note that even though the magnitude of the correlations are not high over this region, the preference for downstream over upstream development still shows up clearly, both on the correlation maps (Figs. 11d, e) and in the downstream/upstream asymmetry index shown earlier in Fig. 7a. Finally, Fig. 11f, which shows the day +2 correlation for the base point 35°N, 0°, displays significant downstream correlation across North Africa toward south Asia.
6. Time-filtered versus unfiltered data
In Fig. 5c, we saw that using unfiltered data, the correlation analysis picks out waves characterized by different periods over different regions of the globe. One obvious question is whether waves with different periods should be treated together, or whether we should somehow separate the waves into different frequency domains, as was done in previous analyses (e.g. Blackmon et al. 1984a,b). This question had been addressed briefly by Chang (1993) and Berbery and Vera (1996), both studies concluding that wave propagation showed up much better in unfiltered versus time-filtered data. Here we will examine this issue in a bit more detail.
Chang (1993) and Berbery and Vera (1996) compared correlations computed using unfiltered and filtered data and suggested that downstream development shows up much more clearly in correlations computed using unfiltered data. Let us first examine what filtered and unfiltered data look like and then examine how different the correlation statistics are. In Fig. 12a, a Hovmöller diagram of unfiltered υ′, averaged over a 20° latitude band centered on the schematic waveguide defined above, is plotted. the period covers days 10–80 (i.e., 10 December through 18 February) of the 1985/86 winter season. In the unfiltered data, we can clearly identify wave packets propagating eastward with group velocity much faster than the phase speed.7
In Figs. 12b and 12c, corresponding plots of high-passed (HP; period less than 10 days) and medium-passed (MP; period between 10 and 20 days, obtained by subtracting HP and 20-day running mean data from the total wind) data are shown, respectively. Compared to the total wind (Fig. 12a), the HP data show weaker wave packet characteristics. Some wave packets can be seen (e.g., between days 15 and 25), but they appear to be less coherent than those seen in the total wind. In addition, individual phases of the waves appear to last significantly longer in the HP data than in the total wind. On the other hand, the MP data display rather coherent wave packet characteristics. However, the wave packets seen in the MP data alone appear to have much slower phase speed and group velocity than those seen in the total wind data. In other words, the wave packet characteristics displayed by the total wind (Fig. 12a) appear to be significantly different from those displayed by either the HP or MP wind data alone.
Next, we will examine how the HP and MP data both contribute to define the wave packet. In Fig. 13a, the day −2 time-lagged covariance8 computed for the base point 37.5°N, 140°E, using unfiltered wind is shown. At day −2, the position of the reference positive center is just east of 120°E, and we can observe a clear upstream/downstream asymmetry, with strong covariance toward the upstream side and nearly zero covariance toward the downstream side. The corresponding figures for the covariances computed using HP and MP data are shown in Figs. 13b and 13c. For the HP data (Fig. 13b), the positive reference center at day −2 is near 110°E, due to faster phase speed imposed by time filtering. Looking toward the upstream and downstream sides of the reference wave, one can only see a weak upstream/downstream asymmetry. In fact, the asymmetry is not obvious until one compares the third covariance center from the reference wave on the two sides, with the upstream one being stronger. For the MP data (Fig. 13c), the asymmetry is again weak. However, when we compare Figs. 13b and 13c carefully, we see that the contributions from the two frequency bands are of opposite signs east of 120°E, toward the downstream side of the reference wave, whereas the two contributions are in phase near 70°E. This is due to the fact that the HP wave has higher phase speed and shorter wavelength, while the MP wave has slower phase speed and longer wavelength. When we add the two together (Fig. 13d), the day −2 covariances on the upstream side interfere constructively, whereas those on the downstream side destructively, giving rise to a strong upstream/downstream asymmetry. Such interferences also lead to much higher group velocity in the composite signal. Examining the covariances computed using the HP or MP data alone, one would estimate that at day −2, the center of the wave packet is near 90°–100°E. In the composite data (Fig. 13d), one sees that the center is near 70°E. Hence, over this region, wave packets observed with the composite data exhibit a significantly faster group velocity compared to wave groups seen using either the MP or HP data alone.
We have examined covariance plots computed using other base points and found similar results to those shown in Fig. 13. In fact, we have computed the downstream/upstream asymmetry index shown earlier in Fig. 7a using the HP and MP data (not shown here), and we find that in either case, the index is less than 1.25 over most regions, with large areas with an index less than 1, whereas for the composite data, the index is greater than 1.5 over most regions (Fig. 7a). We see that wave packets and their downstream dispersion are only well defined if the data contain waves with a range of wavelengths and phase speeds, and it appears that in time filtering, even using filters as wide as a 10-day high-pass filter, the waves retained are insufficient to define accurately a wave group that is localized both in time and space.
While it is clear that wave packets cannot be well defined unless one retains a wide enough spectrum of wave, it is not clear whether wave packets composed of waves having different frequencies should be considered physically related. For example, we showed that waves over south Asia, which have characteristic periods of around 10 days, are significantly correlated to waves over the entrance to the Pacific storm track (Figs. 9 and 10), which have periods closer to 6 days (Fig. 5c). One can ask whether the development of synoptic timescale waves over the Pacific storm track region is physically related to the propagation of lower-frequency waves across south Asia, or whether only the lower-frequency waves are correlated while the synoptic-scale waves develop independently, and the interference between HP and MP data to define a wave packet is entirely accidental rather than physical.
In a WKB analysis of wave propagation in a time mean basic state that is stable (e.g., ray tracing; see Karoly and Hoskins 1982), one would expect frequencies of waves to be unchanged while their wavelengths change as waves propagate in a three-dimensional time mean flow; hence, propagation of lower-frequency waves will not affect the development of higher-frequency waves. However, the situation could be different if the basic state were unstable over some region. One can imagine a situation in which the unstable region acts as a selective amplifier of higher-frequency waves while leaving the lower frequency waves unmodified. Hence, if a wave packet propagates over the region, the character of the waves will be changed and the observed frequency of the waves can increase. In Fig. 12a, there are several instances where waves propagating across Asia into the Pacific appear to experience an increase in phase speed and frequency as they pass 120°E (e.g., the packets near 0°E on days 17 and 28).
To show that the above hypothesis is indeed possible, we conducted a simple experiment using a nondimensionalized linear barotropic model in a β channel. For the control experiment, the basic state is a jet, with a profile such that the absolute vorticity gradient is zero over much of the channel (i.e., u = βy2/2 for the southern half of the channel and symmetric about the central latitude) except near the jet core, where there is a large positive vorticity gradient.9 Since absolute vorticity gradient is everywhere either zero or positive, this jet profile is stable. For the perturbation experiment, the basic state is the same as in the control experiment, except over a small region near x = 2π, where a narrower jet profile is imposed such that the basic state is barotropically unstable over the region (Fig. 14a). An initial value problem is conducted, with a wave packet centered near x = 0 at T = 0. For the control experiment, in the absence of the unstable region, the wave packet propagates eastward with a group velocity equal to the zonal wind speed at the jet core without change in amplitude (see solid and dotted contours in Fig. 14b), because dω/dk is nearly independent of k for such a basic state (see appendix B for more details).
For the perturbation experiment, when the unstable region is added to the basic state, the wave packet grew as it entered the unstable region. For the example shown in Fig. 14, the initial wave packet has a carrier wave that has longer wavelength and lower phase speed than the fastest growing mode of the unstable region. As the wave packet moved across the unstable region, we see that the character of the observed waves changed to take on properties closer to the most unstable normal mode, that is, an increase both k and the phase speed. For this example, both k and c increased by nearly 30%, and ω increased by approximately 60%.
Note that we have used this simple model for illustrative purposes only. In the real atmosphere, we expect the unstable region near the entrance to the Pacific storm track to be baroclinically rather than barotropically unstable. However, we envisage that such selective amplification of the higher-frequency component leading to changes in observed wave frequency is possible regardless of the instability mechanism. Whether this is what actually happens near the Pacific storm track entrance remains to be investigated by detailed case studies. We do want to note here that Nielsen-Gammon (J. W. Nielsen-Gammon 1996, personal communication), applying the method discussed in Nielsen-Gammon and Lefevre (1996) to study trough formation over east Asia, did find cases of mobile trough formation with contributions from downstream development from upstream waves propagating along the subtropical jet stream across south Asia.
7. Summary and discussion
In this paper, we have examined in detail the characteristics of upper-tropospheric wave packets in the Northern Hemisphere winter. We used complex demodulation to define wave packet envelopes, and by tracking the movement of correlation centers on time-lagged correlation maps of υe, we estimated the average group velocity of wave packets. We also showed that our results are consistent with previous estimation of the group velocities using wave activity fluxes.
By examining time-lagged correlation maps for υ′ over much of the hemisphere, we defined a wave coherence index (Fig. 7b) that characterizes the coherence of wave propagation across each grid point. We found that wave packets appear to be most coherent across southern Asia, with weaker but still significant coherence extending all the way across the Pacific, into North America, across the mid-Atlantic, across North Africa, and back toward southern Asia. We may regard this band as the primary wave guide for synoptic-scale wave packets. There is also a secondary branch leading from Russia into the Pacific, acting as a second source of waves for the Pacific storm track. We have shown this wave guide schematically as a shaded band in Fig. 7e and subsequent figures.
If we compare our schematic wave guide to the idealized wave guide suggested by Wallace et al. (1988, see their Fig. 19b), we see substantial agreement but also differences. Their figure shows a main wave guide extending southeastward from Siberia into the Pacific, then passing eastward across the northern United States into the Atlantic, and then east-northeastward toward Europe. There is also a separate wave guide extending only across the Middle East, corresponding to part of our main wave guide across northern Africa and southern Asia. Our analyses here differ from those of Wallace et al. in that we have used unfiltered data whereas they used high-pass filtered data, and we used υ′ at the 300-hPa level instead of z′ at 500 hPa. In order to see what has led to the somewhat different conclusions, we have computed wave coherence indices similar to that shown in Fig. 7b using filtered υ′ and z′, and the results are shown in Fig. 15.
In Fig. 15a, WCI2 computed using high-pass (10-day low-frequency cutoff) filtered 300-hPa υ′ is shown. Compared to Fig. 7b, which is computed using unfiltered data, we see that the main effect of time filtering on WCI2 is a uniform reduction of the index, but the largest coherence still lies across south Asia and extends into the Pacific. Next, WCI2 computed using high-pass filtered 300-hPa z′ is shown in Fig. 15b. Compared to Fig. 15a, the picture is not too different, except that the coherence over Siberia has become a bit higher, and the two branches over Asia are more apparent with filtered z′ than with υ′. The largest differences are seen when we examine WCI2 computed using high-pass filtered 500-hPa z′ (Fig. 15c). Now the wave guide across south Asia along the subtropical jet is no longer apparent, and the most coherent region in the midlatitudes extends across Siberia southeastward into the Pacific, in agreement with the idealized wave guide of Wallace et al. We believe that the suppression of the subtropical portion of the wave guide in the 500-hPa data is due to the presence of high terrain over south Asia, which extends to above 500 hPa over some areas. Another interesting point to note is that in the Tropics, the analyses using filtered 500-hPa z′ pick up waves that have large westward phase speed (up to 30° per day) that exhibit eastward group propagation (notice the shaded bands of relatively large WCI2 south of 25°N from just east of the date line to 20°E, and from 80°E to just west of the date line in Fig. 15c). These waves are not as apparent higher up at the 300-hPa level. Part of this tropical signal may be similar to the tropical disturbances discussed in Lau and Lau (1990), except that winter instead of summer data are examined here. Since our focus is on midlatitude waves, the characteristics of these tropical waves will not be pursued further here.
In this paper, we have investigated the characteristics of wave packet propagation in the Northern Hemisphere winter. In Part II (Chang 1999), we will examine the seasonal variation and hemispheric differences in wave propagation characteristics. In these two studies, we have focused on some of the statistical properties of wave packets. As discussed earlier, we have found that we can track wave packets relatively easily by following the temporal evolution of wave packet envelopes (υe). By tracking the wave packets, we can study the dynamics of the packets and the waves associated with them. We have performed some preliminary studies and confirmed that downstream development (as opposed to other mechanisms such as baroclinic or barotropic instability) is indeed the physical mechanism responsible for development of waves associated with coherent wave packets. These results will be presented elsewhere.
The authors would like to thank Y. Chen and the Scientific Computing Division at NCAR for assistance in accessing the ECMWF and NCEP–NCAR reanalyses data, and J. Sloman for editorial comments. Thanks are also due to NCEP, NCAR, and ECMWF for making the data available. We would also like to thank the anonymous reviewers for helpful comments that led to substantial reorganization of the paper. The figures have been produced using GrADS developed by Brian Doty of COLA. This research is supported by NSF Grant ATM-9510008.
Statistical Significance of Correlations and Wave Coherence Indices
The statistical significance of the correlations (e.g., Figs. 2, 4, 9, etc.) have been assessed in two ways. For the first method, assuming a random variable with Gaussian distribution, and that the 13 yr are independent of each other, the correlations were calculated for each of the 13 yr, and then the mean and the standard deviations were estimated from the 13 independent estimates of the correlations. Using that, we found that a correlation of about 0.07 (for both υ′ and υe) is statistically significant at the 5% level using the two-tailed t test.
As another estimate, we assume that cross-hemispheric correlations are zero among midlatitude disturbances.10 Hence, for each Northern Hemisphere grid point, correlation maps for the Southern Hemisphere are computed and the correlations thus obtained are then assumed to have arisen from pure chance. A total of about 107 (not all independent, since there is spatial correlation between the points) such correlations have been produced, and we found that for υ′, slightly fewer than 5% of the point correlations have values >0.07, consistent with the estimate using the first method, and fewer than 0.02% of these cross-hemispheric correlations attain a value >0.2. None of the 107 correlations is larger than 0.32. For υe, 4% attain correlations >0.07, 0.01% attain values >0.2, and none is larger than 0.25.
Apart from the significance of point correlations—since each correlation map contains a total of over 5000 (not all independent) grid points—we have to assess also the probability that a center with high correlation could have arisen by chance due to multiplicity. To account for that, one needs to estimate the effective number of degrees of freedom in the spatial domain, which is difficult. A simple estimate of the field significance is motivated by Livezey and Chen (1983). Again, assuming that true cross-hemispheric correlations are zero, we computed correlation maps for Southern Hemisphere midlatitudes using base points from the Northern Hemisphere and vice versa and assume all nonzero correlations to have occurred by chance. We then find the maximum value of the correlation for each of the maps and then compute the probability that the maximum correlation is above a certain value. Using this method, the spatial and temporal correlation structure of the data is retained. We found that 2% of the maps contain at least one point with correlation >0.225, 0.5% with correlation >0.25, and 0.1% of the maps have a maximum correlation >0.30. These represent the significance levels for the wave coherence indices (Figs. 7b and 15).
Analytic Dispersion Relation for Simple Barotropic Model
On the beta plane, the basic state
implies that the absolute vorticity gradient is zero everywhere within the domain except at y = ½, where there is a vorticity jump. The dispersion relation for linear waves in this basic state can be solved easily by writing the barotropic vorticity equation into a standard eigenvalue problem, with the boundary values ψ′ = 0 at y = 0 and 1. The dispersion relation is as follows:
where yc = ½ is the center of the channel and Δζ is the absolute vorticity jump at y = yc. For U defined by (B1), U(yc) = β/8, and Δζ = β. The waves are all neutral since the absolute vorticity gradient is either zero or positive everywhere.
The group velocity of the waves is
It can be seen that for short waves (k > 3) the group velocity is approximately constant and is equal to the mean wind at the center of the channel, whereas for long waves, the group velocity is less, and the group velocity equals zero when k = 1.76 and is negative for even smaller k. Note that if there are no side boundaries at y = 0 and 1, the dispersion relation for a basic state with a vorticity jump at y = yc is
* Additional affiliation: Physics Department, Massachusetts Institute of Technology, Cambridge, Massachusetts.
Corresponding author address: Dr. Edmund K. Chang, Program in Atmospheres, Oceans, and Climate, Massachusetts Institute of Technology, Room 54-1614, Cambridge, MA 02139.
We have also performed similar analyses using the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis product (Kalnay et al. 1996) for the years 1980–87, and the results for Northern Hemisphere winter are nearly identical to those from the ECMWF data. We did not use the NCEP–NCAR reanalysis as our primary data source because of the problems they have with misplacing PAOBS (bogus surface pressure data produced by Australia) in the Southern Hemisphere.
When the zonal average of k is not an integer, the demodulation is performed for the two adjacent wavenumbers that are integers, and a weighted mean is taken as the final envelope function.
Another point to note about the group velocity follows. We have also attempted to estimate a group velocity using the Hovmöller diagrams shown later (Fig. 8) by joining up the correlation centers of each of the phases. The estimates came out to be significantly higher than those shown in Fig. 6c. However, another way of estimating a group velocity from Fig. 8 is by first locating the position of maximum correlation at each time lag and then following the motion of such a position. By doing so, we obtained group velocities less than those shown in Fig. 6c. We believe that either way of estimation from the Hovmöller diagrams has biases, and the results shown in Fig. 6c, being bracketed by those two estimates, are consistent with both.
The correlation between the wavelength and phase speed comes out to be −0.11, which is not statistically significant.
However, see the discussions in Chang and Orlanski (1994), which show that for a highly baroclinic system where the WKB approximation is not valid in the vertical, even a vertical mean of MT may not bear a close relationship with the group velocity of baroclinic waves.
Plumb’s low-pass (LP with period greater than 10 days) flux (his Fig. 8) does indicate significant propagation from south Asia into the Pacific, but it is not clear whether the main contribution comes from waves with periods near 10 days or from much lower-frequency waves.
Wave packets seen in Fig. 12a appear to have significantly higher group velocities than the estimates shown in Fig. 6c. However, as discussed in section 3, by tracking individual wave packets, we found a large spread in envelope velocity, and those in Fig. 12a lie within the upper range. What contributes to the variations in group velocity is still under investigation.
Here covariance instead of correlation is shown because covariances can be easily split up into contributions from different frequency bands, whereas correlations are not simply additive.
Such a jet profile is chosen so that the model can be easily verified by comparison with analytic solutions. See appendix B.
This is probably not true for tropical disturbances. Anyway, any non-zero correlation would only render the following estimate to err on the conservative side.