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  • View in gallery

    Silhouette values for every winter day when divided into (left) two, (middle) three, and (right) four groups using k-means clustering. The x axis shows the silhouette value for each day on the y axis. The y axis is also subdivided into groups so that the days in group 1 are located near the top of the y axis and the days in subsequent groups (as labeled) are located farther down the axis. Negative silhouette values indicate that the day may be mislabeled and silhouette values close to 1 indicate that the groups are well separated.

  • View in gallery

    Histogram of C1: the optimal linear combination of stratospheric measurements. The height of the histogram represents the number of winter days for each value. This is a bimodal distribution indicating two clear groups in the sample. The high C1 values (light gray) represent 10% of winter days (∼880 days) and the lower values (black) represent 90% (∼7133 days). The small region of overlap is shown in midgray. All previously defined major warmings lie in the high C1 group.

  • View in gallery

    Time series of vector C1: the optimal linear combination of stratospheric north polar values (black) and the 10-hPa NAM (gray). Previously defined major warmings are denoted by asterisks. Only data for October–March are employed; the tick marks labeled “1” denote the summer break points in the time series while the tick marks labeled with the year denote 1 Jan of that year.

  • View in gallery

    Height profiles of zonally averaged polar temperature anomalies (K) for each day at 80°–90°N for (middle) group 1 and (right) group 2. (left) The average profile for groups 1 (solid) and 2 (dashed).

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    Height profiles of zonally averaged zonal wind (m s−1) at 60°–70°N from (middle) group 1 and (right) group 2. (left) The average profile for group 1 (solid) and group 2 (dashed).

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    Height profile of (top) wave 1 and (bottom) wave 2 amplitude of geopotential height (m) at 60°N for each day in (middle) group 1 and (right) group 2. (left) The average profile for group 1 (solid) and group 2 (dashed). The values have been divided by the zonal mean at the appropriate height and thus are presented as a fraction of the whole.

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A Continuum of Sudden Stratospheric Warmings

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  • 1 Meteorology Department, University of Reading, Reading, United Kingdom
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Abstract

The k-means cluster technique is used to examine 43 yr of daily winter Northern Hemisphere (NH) polar stratospheric data from the 40-yr ECMWF Re-Analysis (ERA-40). The results show that the NH winter stratosphere exists in two natural well-separated states. In total, 10% of the analyzed days exhibit a warm disturbed state that is typical of sudden stratospheric warming events. The remaining 90% of the days are in a state typical of a colder undisturbed vortex. These states are determined objectively, with no preconceived notion of the groups. The two stratospheric states are described and compared with alternative indicators of the polar winter flow, such as the northern annular mode. It is shown that the zonally averaged zonal winds in the polar upper stratosphere at ∼7 hPa can best distinguish between the two states, using a threshold value of ∼4 m s−1, which is remarkably close to the standard WMO criterion for major warming events. The analysis also determines that there are no further divisions within the warm state, indicating that there is no well-designated threshold between major and minor warmings, nor between split and displaced vortex events. These different manifestations are simply members of a continuum of warming events.

Corresponding author address: Lesley J. Gray, Meteorology Dept., University of Reading, Earley Gate, P.O. Box 243, Reading RG6 6BB, United Kingdom. Email: l.j.gray@rdg.ac.uk

Abstract

The k-means cluster technique is used to examine 43 yr of daily winter Northern Hemisphere (NH) polar stratospheric data from the 40-yr ECMWF Re-Analysis (ERA-40). The results show that the NH winter stratosphere exists in two natural well-separated states. In total, 10% of the analyzed days exhibit a warm disturbed state that is typical of sudden stratospheric warming events. The remaining 90% of the days are in a state typical of a colder undisturbed vortex. These states are determined objectively, with no preconceived notion of the groups. The two stratospheric states are described and compared with alternative indicators of the polar winter flow, such as the northern annular mode. It is shown that the zonally averaged zonal winds in the polar upper stratosphere at ∼7 hPa can best distinguish between the two states, using a threshold value of ∼4 m s−1, which is remarkably close to the standard WMO criterion for major warming events. The analysis also determines that there are no further divisions within the warm state, indicating that there is no well-designated threshold between major and minor warmings, nor between split and displaced vortex events. These different manifestations are simply members of a continuum of warming events.

Corresponding author address: Lesley J. Gray, Meteorology Dept., University of Reading, Earley Gate, P.O. Box 243, Reading RG6 6BB, United Kingdom. Email: l.j.gray@rdg.ac.uk

1. Introduction

The climatological zonal winds in the stratosphere are generally westerly and their strength increases with height. These winds form the “polar night jet” vortex. In concert with these winds, the climatological temperatures decrease toward the winter pole along pressure surfaces. However, at times this zonal-mean configuration is dramatically disturbed. The polar temperatures can increase at a spectacular rate of ∼10 K day−1 or more and in less than a week the normal latitudinal temperature gradient can be reversed (Muench 1965; Tung and Lindzen 1979; Andrews et al. 1987; Limpasuvan et al. 2004), with the pole becoming warmer than the midlatitude temperatures. On occasion this is accompanied by a reversal of the zonal-mean winds at the pole, from mean westerly to mean easterly. These events are called sudden stratospheric warmings (SSWs).

In the past, these phenomena have been arbitrarily defined using a variety of different criteria involving winds, temperatures, and measures of the vortex shape. A major SSW is defined by Andrews et al. (1987, see p. 259) as when “at 10 mb or below the zonal-mean temperature increases pole-ward from 60° latitude and the zonal-mean zonal wind reverses. If the temperature gradient reverses but the circulation does not, it is defined to be a minor warming.” The World Meteorological Organization (WMO) has a slightly different definition, requiring the latitudinal gradient of the 10-hPa zonal-mean temperatures between 60° and 85°N to be positive for more than 5 days. The WMO also uses 65°N, instead of 60°N, as their latitude to reference the wind changes. Other variants on these definitions using polar temperatures, temperature gradients, and wind reversal have also been used (McInturff 1978; Labitzke 1982). In addition to definitions of major and minor SSWs, the WMO also defines a category of SSWs known as Canadian warmings, which occur early in the winter, and final warmings, which mark the transition from winter to summer circulation (see, e.g., Labitzke and van Loon 1999).

SSWs have also been characterized by the structure of the polar vortex (Randel 1988; Waugh 1997; Waugh and Randel 1999; O’Neill 2003; Charlton and Polvani 2007). Fourier or elliptic diagnostics can be used to quantify the shape of the vortex. In this way, the temporal evolution of the vortex is tracked in order to determine when it becomes disturbed, which is an indication of a SSW. The SSWs have therefore been further divided in terms of the dominant wavenumber of the disturbed vortex (e.g., a wave 1 or wave 2 warming; as in Yoden et al. 1999) or alternatively (using the terminology of O’Neill 2003), the SSWs are further divided into events where the vortex is displaced off the pole (dominant wave 1 events) or split vortex events (dominant wave 2 events).

These studies and others, have thus arbitrarily categorized SSWs in terms of amplitude (temperatures or winds), timing (early winter versus midwinter versus late winter) and structure (wave 1 or wave 2 SSWs). Using thresholds can be a powerful and useful way to understand variability and the mechanisms that may cause it. The question that this study addresses is whether there are indeed categories that represent distinct and naturally separated stratospheric winter states. If so, then we would also like to know what is the best measure that defines the boundary between one state and the next and whether the measures already in place and described above are the optimal ones to employ. If there are no distinct states, perhaps we can determine which measure best describes the variability seen over the pole. Based upon the previous definitions one might expect these measures to be combinations of more than one variable. For example, the WMO definition uses a combination of the latitudinal temperature gradient, the wind velocity, and an indicator of the time for which the changes were sustained.

The approach employed in this paper to address these questions is k-means clustering, which is a method of identifying different states in a completely objective manner with no preconceived notion of the groups and no preselection on the basis of known influencing factors. This is in contrast, for example, to the recent approach of Camp and Tung (2007) who identified four states in terms of a preselection based on the phase of the quasi-biennial oscillation (QBO), the El Niño–Southern Oscillation (ENSO) index, and the 11-yr solar cycle.

In the k-means method winter stratospheric days are grouped according to the similarity of measurements in north polar regions. Once these groups are defined, silhouette values are used (see section 3d) as a quantitative way to measure the separation of the groups. From this it is possible to determine if the groups are well separated and if so, how many distinct groups exist and how they are characterized. We can then investigate what measure best distinguishes the stratospheric groups. We propose that the k-means clustering technique is more useful than hierarchical clustering for these stratospheric investigations because, as we will demonstrate, k-means clustering easily allows for uneven groups, whereas hierarchical clustering tends to determine groups of similar size.

The paper is organized as follows: in sections 2 and 3 the data and methods are described in detail. The main results are presented in section 4 and summarized in section 5.

2. Data

The data employed are from the 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40) project (Uppala et al. 2005). Daily winter data (October–March) for the stratosphere are used. No SSWs occur in October, so the analysis does not change substantially when only November to March data are used. In the results presented here, all available data from the winter of 1958/59 to 2001/02 are used. There is no noticeable change in the definition of the groups and their time series when only data from 1979 onward are used.

The polar stratospheric measurements included in this analysis are temperature (at three different latitudes), time change of temperature, latitudinal temperature gradients, time change of latitudinal temperature gradients, zonal winds (at two different latitudes), time change of zonal winds, and wave 1 and wave 2 components of geopotential height at 60°N (i.e., a total of 14 different variables). Data at 11 stratospheric pressure levels between 1 and 100 hPa were employed. In total, 154 measurements, many of which are highly correlated with one another, are used in the initial determination of groups. Apart from the geopotential height data, all quantities are zonally averaged. All changes in time are simply calculated as the difference between the given day and 7 days before. The latitudinal gradients are calculated as differences between the 80°–90°N average value and the 50°–60°N average value. Polar temperatures are averaged into three bins: 60°–70°, 70°–80°, and 80°–90°N. Winds are averaged into 60°–70° and 70°–80°N bins.

Temperature anomalies are employed. These are obtained by subtracting the corresponding daily climatological value, which was derived by extrapolating the monthly averaged temperature onto a daily time series. This latter method has been used in order to create a smooth climatology. We note that it is important to use temperature anomalies, instead of the actual temperature on each day, in order to de-emphasize the annual cycle. On the other hand, in order to preserve the asymmetry in the winds, the mean is not subtracted from the zonal winds. The asymmetry may be important because theoretically one expects different dynamical behavior for easterly and westerly winds (Charney and Drazin 1961).

Wave amplitudes were calculated using geopotential heights at 60°N projected onto wavenumbers 1 and 2 and then normalized by the zonal mean at 60°N. This latitude was chosen because the largest wave amplitudes occur at this latitude (Matsuno 1970; Andrews et al. 1987). Tests using wave amplitudes calculated at different latitudes showed little difference to the resulting groups.

The variance of the measurements is important in determining whether or not the polar vortex is disturbed. Without normalizing the measurements by the standard deviation, there is a correlation between the variance and the mean. Particularly for wind speeds, the variance increases with the mean, which implies that for stratospheric levels that have a large mean wind speed, there is also a large spread of wind speed values, which may skew the resulting cluster routine. It was found that normalizing the data changes the number of days in each group by only a small amount. A number of checks were made to test whether substantially different results were achieved depending on whether the data were normalized or not. Even when all the measurements were normalized, the resulting time series remained relatively robust and unchanged compared with results from the nonnormalized analysis, with only a few more days on either side of each warming event identified as part of the warming group. In the following analysis, we choose the narrower grouping achieved without normalizing, which gives us a sharper group definition.

3. Methods

The main issue to be addressed is whether the polar stratosphere exists in distinct states and if so, how many. Second, we would like to know which measurement (or combination of measurements) can be used to distinguish between these groups. In this paper these questions are addressed using k-means clustering. The first part of the question is addressed using silhouette values to determine the optimum number of groups and an f test to determine how well separated these groups are. Furthermore, we try to distinguish the important factors in forming these groups using a MANOVA algorithm (Krzanowski 1988) in MATLAB. The details of the methodology are described below. Further detail and discussion of this type of analysis can be found in Lachenbruch (1975), Colley and Lohnes (1971), or in StatSoft’s online textbook (StatSoft Inc. 2007).

a. The k-means cluster analysis

K-means is a sorting algorithm that uses the criteria of minimum distance in order to group data. Initially, each winter day in our dataset is randomly assigned a group. The mean of each randomly defined group are calculated and two distances are defined. The “within-group” distance is the average distance between the days in a group and the mean of the group. The “between-group” distance is the distance between the means of the groups. One-by-one the days are randomly given the opportunity to change groups. If the within-group distance decreases when the day changes group then the change is made; otherwise, the group assignment remains unchanged. The number of initial groups is preset. The procedure was carried out with different numbers of initial groups, ranging from two to five, to test the optimum number of natural groupings of the data.

One of the main benefits of this method is that the resulting group sizes can be unequal. This is an important factor when studying the stratosphere as we expect there to be only a small number of warming events compared to the majority of winter days where there is an undisturbed vortex. There are, however, some issues that need to be addressed regarding this method. The first one is the measure of distance used to define the groups. There are various types of measurements for each day and it is not immediately obvious which of these is the most sensible measure in this multivariate space. The second issue is that of multiple minima. Especially when there are many groups to choose from, the end group determination may depend on the order in which the days are given the chance to switch groups, or on the initial random distribution of groups. These two issues are explored further in the following two sections.

b. Measure of distance

The choice of what measure of distance to use is not obvious, since there are different types of variables employed in the analysis. When one is comparing variables of similar type (e.g., all length scales) it is easier to determine an appropriate distance measure, but in this case our variables include temperature, velocity, and length scales. Generally, standardizing variables can be helpful, so that variables with a high variance can be downweighted compared to others. However, this is not always adequate, for example, if the variance is nonnormal or dependent on the mean; absolute values may also be pertinent and nonlinear relationships may be important which would not be accounted for in a general standardization.

The results presented below are calculated using 1 minus the correlation as a measure of distance. If the measurements on 2 days are well correlated then 1 minus the correlation will be a small number and the days will be considered close to one another in phase space. Other distance measures were tested (city block, Mahalanobis, Hamming, squared Euclidean, and cosine) and the optimum number of groups (determined by silhouette values as described below) for each measure of distance was the same, although not all days were categorized in exactly the same groups using these different distance measures. In the results presented, correlation was used as the distance measure because the measure was easier to interpret physically for this case: 2 days that are in the same group will have correlated measurements.

c. Local minima

The issue of local minima is of particular concern when there are more than two initial groups and the solution to this problem is to perform multiple iterations. In each iteration the initial distribution of groups was randomly changed, as was the order. This gave the days the chance to swap groups. The within-group distance for each iteration was calculated and the iteration with the smallest within-group distance was selected. For five or fewer initial groups, eight iterations were used in the analysis, although for only two initial groups, which the results showed was the optimal number of initial groups (see next section), this was not needed.

d. Statistical tests

Silhouette values are a measure of the within-group and between-group differences. These values indicate how similar one point is to other points in its own cluster compared to the similarities that it has with points in other clusters. Mathematically, it is defined as
i1520-0469-66-2-531-eq1
where ai is the average distance between point i and other points within its same cluster and bi(k) is the average distance between point i and points in other clusters k. When S(i) is negative, there is an overlap between groups and there is a possibility that point i has been placed in the wrong group. When S(i) is positive and close to 1 then the groups are well separated. By comparing silhouette values, the cluster parameters that are able to give the best separation of the data can be determined.

Figure 1 shows the silhouette values of the k-means clusters (using one minus the correlation as a measure of distance) for two, three, and four initial groups. When the days are clustered into two groups (left panel), most of the silhouette values are large and close to 1, with only a few points potentially mislabeled (i.e., negative). The mean silhouette value is 0.73 (cf. 0.59 and 0.39 for the clustering into three and four groups) indicating that the data best separates into two groups. Note that the choice of three or four groups also increases the number of mislabeled days. When different measures of distance were used, the actual silhouette values differ but the comparison remains the same (i.e., the silhouette value for two groups is always greater than for more groups).

4. Results

a. The groups

Using k-means clustering and the silhouette test described above, it was found that the north polar winter stratosphere optimally separates into two natural groups. Once the optimum group for each winter day from 1958 to 2002 was found, the character of each state could be characterized. To test whether these two groups are different from one another, the variance of the observations within the groups can be compared to the total variance. If the means of the two groups are far from one another (i.e., the groups are well separated), then the total variance will be much larger than the variance of each individual group. This total variance can be partitioned into the within-group variance (called the error variance because this is the variance that cannot be explained by the grouping) and the between-group variance (this is the variability that is explained by our choice of grouping). The significance is determined by a comparison of the matrix of explained variance and the matrix of unexplained variance. The test of significance is then determined by an f test. For our grouping, the Wilk’s lambda parameter is 0.3 and the mean difference between groups 1 and 2 is highly significant with an f statistic of 17 089. This value indicates that we are 99.99% confident that the two groups are well separated.

After concluding that the two groups are distinguishable and naturally separated, we are interested in examining the characteristics of each of the groups and the variables (or a combination of the 14 variables) that are most influential in determining into which group the day is assigned. A canonical correlation is formed to determine the linear combination of variables that provides the most overall discrimination between groups. In this case, only one eigenvector, denoted C1, is needed, which is a linear combination of the measurements that best separate the groups. We note here that C1 is a vector in the 154-dimensional phase space that is described by the measurements used in the clustering algorithm. When picturing the phase space in which we are distinguishing the groups, it is useful to think of the vector that separates the groups. We can then look at the projection of each day on to the vector to give us a time series that shows how the stratosphere moves between these states.

A histogram of the C1 values for each winter day is plotted in Fig. 2. The height of the histogram represents the number of winter days for each value of C1. The bimodal distribution indicating two clear groups in the sample is evident, and the means of each group are well separated, as described above. The high C1 values shown in gray, which we refer to as group 1, includes approximately 10% of the winter days (i.e., 719). The lower values of C1 shown in black represents approximately 90% of the days (i.e., 7294) and are referred to as group 2.

b. Time series

Figure 3 shows the time series of C1, (black line), the vector that optimally separates the groups. Only winter (October–March) values are shown. Days in group 1 have large values of C1and days in group 2 have small values of C1. The asterisks show the previously identified major SSWs, as listed in Charlton and Polvani (2007) and the gray line shows the northern annular mode (NAM) at 10 hPa (Baldwin and Dunkerton 1999). The previously documented major warmings are all captured in group 1, which is therefore referred to as the stratospheric warming group, as opposed to group 2, which contains by far the largest number of days (see Fig. 2) and represents the normal, relatively undisturbed days.

The transition from a normal winter day (group 2) into a warming (group 1) and then back again can be clearly seen in Fig. 3. First we notice that the transitions from one group to the other are often quite sudden. In particular, the transition from group 2 to 1 is often quick but then there is a much slower transition back into group 2. Good examples of this behavior are present in 1970, 1984, and 1985. This indicates that perhaps the stratospheric vortex breaks up quite quickly, but in some situations takes a relatively longer time to recover. This is consistent with the idea that the initial disturbance over the pole is a dynamical effect while the recovery of the vortex is more influenced by the longer radiative time scales. Limpasuvan et al. (2004) found a similar situation when they composited stratospheric events according to the first EOF of winds at 50 hPa. We note that since each day is individually optimized in our analysis to be in either group 1 or 2, the only time dependence or autocorrelation is that within the data itself.

If the C1 time series is compared with the time series of the NAM at 10 hPa (gray line), we see that they generally capture the same events. The correlation coefficient between the two is significant at 0.6. This inherently makes sense because, in broad strokes, both C1 and the NAM at 10 hPa are dominated by zonal winds in the upper stratosphere. A closer comparison of the two time series shows that the individual warming events are actually smoothed over in the NAM time series and are much more jagged, or abrupt, in C1. This is because the NAM is a time series that describes only the change in one particular large-scale pattern while the C1 time series from the clustering algorithm includes various characteristics of each day individually and determines the optimum group for that day. This means that C1 is able to cleanly capture the day-to-day variability.

c. Characteristics of the groups

To see the character of each group, the wind, temperature, and wave amplitude profiles of group 1 can be compared with those of group 2. Figure 4 shows the polar temperatures of group 1 versus those of group 2. The polar temperature height profile is plotted in a so-called spaghetti diagram for group 1 (middle panel) and group 2 (right panel). On the left, the average polar temperatures for each group are plotted. The polar temperatures of group 1 tend to be warmer than those of group 2, which is as expected, since group 1 contains the previously documented major warming events. Although group 1 clearly contains the warmest events and the temperatures in group 1 are, on average, much warmer than group 2, there is still quite a lot of overlap and group 2 still contains days where the stratospheric polar temperatures are very warm.

Figure 5 shows the corresponding height profiles of zonally averaged zonal winds at 60°–70°N for groups 1 and 2. These latitudes were chosen for comparison with the WMO-type definition of major SSWs. In this case the distinction between the groups is much clearer, especially in the upper stratosphere at around 2 hPa where the group 1 profiles tend to have a minimum. The kneelike profiles of group 2 winds are generally facing the opposite way with an average maximum at slightly higher levels just above the stratopause.

Figure 6 shows the wave 1 and 2 amplitudes for the two groups. In both cases the wave amplitudes of group 1 tend to be larger than those of group 2, although the differences in profiles are not large in either case. Interestingly, wave 1 amplitudes peak lower and larger, at around 10 hPa, within group 1 than group 2. This is consistent with Fig. 4 where we see that the upper-stratospheric winds are more easterly. By simple wave theory (Charney and Drazin 1961), vertically propagating waves are inhibited from penetrating into easterly winds and only propagate in low-speed westerly backgrounds. This means that the waves in group 1 cannot propagate as high into the stratosphere.

From these comparisons, we note that days in group 1 are warmer over the winter pole, with slower wind speeds and larger wave amplitudes in the stratosphere than those in group 2. Thus, group 1 has the general characteristics of a disturbed polar vortex.

d. Backward stepwise reduction of dimensions

The optimum vector C1 is important in defining the two groups. However, it is a linear combination of 154 variables, many of which are highly correlated with one another. It would be much more useful if this combination could be reduced to only one or two variables that give a similar level of information. Because the observations are correlated, it is unwise to consider just the largest coefficients of the C1 vector. Correlated time series of the measurements compensate for one another, causing unrealistically large magnitude coefficients. A backward time-stepping algorithm was therefore used to eliminate variables that are less important, leaving only those that are most important for discriminating between the groups. Note that forward stepwise discriminant analysis, like forward stepwise regression analysis, is much less stable than the backward form because the combination of correlated variables cannot be taken into consideration when only one variable is included in the model at a time.

In this approach, all the variables are initially used to discriminate between the groups and then one by one the variables that contribute least to defining the groups are removed. During each iteration, a canonical correlation of the remaining variables is calculated to find the vector which best separates the groups. An f test compares the subsequent means between groups and the variable that contributes the least to the prediction of group membership (as determined by the f test) is eliminated. This process continues until the remaining variables can no longer significantly distinguish (at a 95% level) between the groups.

The final result of this process was that only one variable was needed to distinguish between the groups. Using the zonal wind at 6.44 hPa, the two groups could still be discriminated and the means were significantly different from one another. The time series of the ∼7-hPa winds averaged between 70° and 80°N is almost identical to the time series of C1 in Fig. 3. This means that just this one measurement enables us to determine whether any given day is in the warm state or in a nondisturbed state. For group 1, the cutoff value for zonal wind speeds at 7 hPa was 4 m s−1. It is interesting how close this definition is to the WMO definition for major SSWs of less than 0 m s−1 at 10 hPa at 65°N. On the other hand, we note that the analysis does not result in a further natural separation, for example between major and minor SSWs or between displaced (wave 1) or split vortex (wave 2) SSWs. This suggests that physically there is no difference between each of these manifestations, and they are simply part of a continuum of disturbed states.

5. Summary and discussion

The k-means clustering technique has been used to study the nature of NH winter SSW events in the ERA-40 dataset. The technique involves randomly assigning each day into one of a number of groups. The within-group variance is assessed from a combination of parameters including high-latitude temperatures and winds and their various gradients, and is then compared with the between-group variance. One by one the days are selected in a random order and given the opportunity to change groups in order to minimize the within-group variance and maximize the between-group variance, thus organizing the data into natural groupings based on these parameters. This method has no preconceived notion of the groups and there is no preselection of winters based on factors, such as the QBO or solar cycle, which influence the occurrence of sudden warmings, as in Camp and Tung (2007).

The data were found to be optimally separated into only two natural, well-separated states (see Fig. 2), a warm disturbed group containing approximately 10% of the days (group 1) and a cold undisturbed group containing the remaining 90% of the days (group 2). These were compared with previous studies of stratospheric warming events (Charlton and Polvani 2007) and confirmed that all previously identified major warmings were included in group 1. The time series of the results also compare well with the evolution of the NAM (Baldwin and Dunkerton 1999) and it was shown that the technique is able to cleanly capture the day-to-day variability and provides a less smoothed evolution than the NAM index.

One of the main benefits of the k-means methods is that the resulting group sizes can be unequal, as we have shown, with a split of the data into 10% and 90% of the total number of days. Despite extensive tests that permitted greater numbers of groupings than two (up to five), no further groupings were identified. We note that there is no contradiction between these results and those of Camp and Tung (2007) who obtained four separate states, because they distinguished their states by their physical causes: easterly QBO, solar maximum conditions, and warm ENSO in addition to the unperturbed state. The first three of their states are characterized by SSWs and hence all fall into group 1.

It is especially interesting that the method did not further divide the 10% grouping in order to distinguish between major and minor SSWs or to distinguish between displaced and split vortex (waves 1 and 2) SSWs. This suggests that there is not a well-designated threshold between these types of SSWs and they are simply different manifestations of a continuum of SSWs. This result is consistent with a study of SSWs in a perpetual January GCM simulation by Yoden et al. (1999), who found a continuous distribution between flow characterized by the amplitude of the wave 1 and 2 geopotential height anomalies. They interpreted this in terms of vortex breakup. If the vortex breaks up into vortices of similar size, the wave 2 will be dominant but if they have different sizes then wave 1 will dominate. However, if the actual break up of the vortex occurs with an arbitrary ratio, it will result in a continuous probability distribution between the two wavenumbers.

The method was extended further to investigate whether the two groups could be distinguished using only one (or a small number of) parameters, instead of the full suite of parameters supplied (see section 2). It was shown that the zonally averaged zonal wind in the polar upper stratosphere averaged over 70–80°N near 7 hPa can be used, with a threshold value of ∼4 m s−1. This is similar to the WMO definition for major SSWs of less than 0 m s−1 at 65°N and 10 hPa, thus confirming the WMO definition as an appropriate measure of the two different states.

Acknowledgments

The authors are grateful to Fiona Underwood of the Statistics Department, University of Reading, and K. K. Tung, University of Washington, for their very helpful comments on the manuscript and to Andy Heaps, University of Reading, for his technical assistance. The study was funded by the U.K. Natural Environment Research Council.

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Fig. 1.
Fig. 1.

Silhouette values for every winter day when divided into (left) two, (middle) three, and (right) four groups using k-means clustering. The x axis shows the silhouette value for each day on the y axis. The y axis is also subdivided into groups so that the days in group 1 are located near the top of the y axis and the days in subsequent groups (as labeled) are located farther down the axis. Negative silhouette values indicate that the day may be mislabeled and silhouette values close to 1 indicate that the groups are well separated.

Citation: Journal of the Atmospheric Sciences 66, 2; 10.1175/2008JAS2792.1

Fig. 2.
Fig. 2.

Histogram of C1: the optimal linear combination of stratospheric measurements. The height of the histogram represents the number of winter days for each value. This is a bimodal distribution indicating two clear groups in the sample. The high C1 values (light gray) represent 10% of winter days (∼880 days) and the lower values (black) represent 90% (∼7133 days). The small region of overlap is shown in midgray. All previously defined major warmings lie in the high C1 group.

Citation: Journal of the Atmospheric Sciences 66, 2; 10.1175/2008JAS2792.1

Fig. 3.
Fig. 3.

Time series of vector C1: the optimal linear combination of stratospheric north polar values (black) and the 10-hPa NAM (gray). Previously defined major warmings are denoted by asterisks. Only data for October–March are employed; the tick marks labeled “1” denote the summer break points in the time series while the tick marks labeled with the year denote 1 Jan of that year.

Citation: Journal of the Atmospheric Sciences 66, 2; 10.1175/2008JAS2792.1

Fig. 4.
Fig. 4.

Height profiles of zonally averaged polar temperature anomalies (K) for each day at 80°–90°N for (middle) group 1 and (right) group 2. (left) The average profile for groups 1 (solid) and 2 (dashed).

Citation: Journal of the Atmospheric Sciences 66, 2; 10.1175/2008JAS2792.1

Fig. 5.
Fig. 5.

Height profiles of zonally averaged zonal wind (m s−1) at 60°–70°N from (middle) group 1 and (right) group 2. (left) The average profile for group 1 (solid) and group 2 (dashed).

Citation: Journal of the Atmospheric Sciences 66, 2; 10.1175/2008JAS2792.1

Fig. 6.
Fig. 6.

Height profile of (top) wave 1 and (bottom) wave 2 amplitude of geopotential height (m) at 60°N for each day in (middle) group 1 and (right) group 2. (left) The average profile for group 1 (solid) and group 2 (dashed). The values have been divided by the zonal mean at the appropriate height and thus are presented as a fraction of the whole.

Citation: Journal of the Atmospheric Sciences 66, 2; 10.1175/2008JAS2792.1

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