Changes in meandering of the Northern Hemisphere circulation

Strong waves in the mid-latitude circulation have been linked to extreme surface weather and thus changes in waviness could have serious consequences for society. Several theories have been proposed which could alter waviness, including tropical sea surface temperature anomalies or rapid climate change in the Arctic. However, so far it remains unclear whether any changes in waviness have actually occurred. Here we propose a novel meandering index which captures the maximum waviness in geopotential height contours at any given day, using all information of the full spatial position of each contour. Data are analysed on different time scale (from daily to 11 day running means) and both on hemispheric and regional scales. Using quantile regressions, we analyse how seasonal distributions of this index have changed over 1979–2015. The most robust changes are detected for autumn which has seen a pronounced increase in strongly meandering patterns at the hemispheric level as well as over the Eurasian sector. In summer for both the hemisphere and the Eurasian sector, significant downward trends in meandering are detected on daily timescales which is consistent with the recently reported decrease in summer storm track activity. The American sector shows the strongest increase in meandering in the warm season: in particular for 11 day running mean data, indicating enhanced amplitudes of quasi-stationary waves. Our findings have implications for both the occurrence of recent cold spells and persistent heat waves in the mid-latitudes.


Introduction
Recent studies have shown that high-amplitude Rossby waves in the mid-latitude flow are statistically linked to weather extremes at the surface [1,2]. Depending on geographical region, certain types of extreme weather are favoured during periods of strong wave amplitudes, including for example cold spells in eastern North America [2]. As an example, the recent 2013-2014 winter in the United States was characterised by several high-impact cold spells, which were associated with strongly meandering flow patterns (see figure 1).
Several mechanisms have been proposed which could affect Rossby wave characteristics. Classical studies have shown that anomalous sea surface temperatures in the tropical Pacific can generate quasistationary wave trains which can constructively interfere with and thereby alter the mid-latitude flow [3,4].
Recent work by Trenberth et al shows that some of the observed mid-latitude circulation anomalies in recent years might have arisen from a negative phase of the Pacific Decadal Oscillation and associated tropical rainfall anomalies, reporting that the strongest changes occur in winter [5]. Others have stressed the importance of rapid warming in the Arctic, one of the most striking climate change signals observed today which is associated with a rapid decline in sea-ice and changes in snow cover [6][7][8][9][10]. These cryosphere changes might affect the flow in the mid-latitudes and both tropospheric [11] and stratospheric [6,12,13] mechanisms have been proposed. Also internal dynamical mechanisms could create high-amplitude, quasi-stationary Rossby waves [14,15].
Irrespective of the underlying drivers, it is unclear whether Rossby wave characteristics have actually changed [7,[16][17][18]. Based on the minimum and maximum latitude of individual geopotential height contours (or isohypses), Francis and Vavrus [7] claimed that peaks of ridges have moved northward and that the North-South extent of Rossby waves has increased over the North American sector. However, Barnes [19] demonstrated that these findings are sensitive to the methodology used and especially to the selected isohypse(s), showing that the claimed increase in latitudinal wave extent rather reflected an upward migration of maximum wave activity. Further studies have reported both little [17] and pronounced [18] changes but these analyses were again based upon selecting one or a limited set of isohypses and thus it remains unclear how robust their findings are. Thus, so far no clear evidence exists on whether mid-latitude circulation has become wavier or not over recent decades [6,20,21].
Here we introduce a new meandering index which measures the waviness of the middle troposphere based on the full shape of each isohypse. Moreover, the index accounts for possible migration of wave-activity by searching for maximum waviness at any given time (following Barnes). We present results for different regions and time scales. Our results show significant changes in the meandering of isohypses in different regions and running mean, adding evidence that changes in the waviness of the middle troposphere have occurred in the recent decades. Figure 1 illustrates different meandering situations in the 500 mb geopotential height fields (Z500) over the Northern Hemisphere: figure 1(a) shows a typical zonally oriented flow pattern while figure 1(b) shows a strongly meandering one. The latter event occurred during the 2013-2014 winter, which saw several cold spells over the United States, as visible in the anomalously low geopotential heights over the northeastern US. The strongly meandering isohypse pattern simultaneously gave rise to extremely warm temperatures in California and Alaska [22].

Meandering index
We define a new, hemispheric meandering index 'M' which is based upon the most strongly meandering isohypse in each individual day. This index accounts for the full path of the isohypse, without modifying or simplifying its shape, such that also shapes during wave breaking events (with multiple latitudinal positions of the isohypse at one longitude) are correctly considered.
The meandering index m of a single isohypse i is defined as:   . When a sub-region is considered, 2π is replaced by λ, where λ is the longitudinal distance of the sub-region in radians.

Calculating arc length and normalising
The aim of our index is to find maximum waviness in the atmosphere and therefore it should be able to compare meandering isohypses at different latitudes in a consistent way. In other words, we are only interested in the shape of an isohypse and its North-South deviations, but we want to make sure that its mean latitudinal position does not affect the value of our index. Both the significance and magnitude of the reported trends are quantitatively similar when a referencelatitude of 50°or 70°is used (see SI, figures S11-S18) or when the original latitude is used (see SI, figures S29-31). This choice makes the index independent of the latitude of the isohypse and also avoids that a poleward shift would increase the meandering index due to a change in mean latitudinal position and not by a real change in the shape of the isohypse. This way, any latitudinal migration of the waviest isohypse over time will not affect M but only changes in its shape will. We do however track the isohypse height and original mean latitude of maximum meandering (M) for further analysis ( figure 5). The choice of θ ref =60°is justified as this is very close to the climatological mean latitude at which the waviest isohypses are found (see figure 5). The arc length of i, i.e. the walking-distance along isohypse i, is determined in the 2D grid, giving the meandering index m of a single isohypse as defined in equation (1). This way, the index accurately differentiates between isohypses with strong North-South deviations from those which follow a circle of constant latitude, which is similar in rationale as other studies [17,23,24]. In spherical coordinates, northward deviations could actually have a similar or even shorter distance than the straight East-West distance due to the reduced circumference at higher latitude. We therefore do not calculate the arc length in spherical coordinates as the meandering index should capture deviations from the straight East-West direction. The isohypses have to circle the full hemisphere and thus local closed contours due to blocks or cut-off lows are excluded from the calculations. Next, we calculate m for each isohypse in the range Z={4900, 4905, 4910, K, 6200 m} and define the meandering index M as those m values which satisfy equation (2).

Data
We calculate M for daily 500 mb and 300 mb geopotential height fields from the ERA Interim [25] and the NCEP/NCAR [26] reanalyses for the period 1979-2015. Due to the higher spatial resolution of ERA Interim (1.5°×1.5°) compared to NCEP/ NCAR (2.5°×2.5°) slightly larger meandering values are found for the former as meanders are measured to a finer scale. We calculate the meandering index both globally (i.e. around the hemisphere) and over the two continental sectors: the North American sector (180°W -40°W) and the Eurasian sector (40°W-180°E).
Next to daily data, 5 and 11 day running means are analysed. This way the influence of fast travelling free waves (i.e. the synoptic transients) and that of quasistationary waves can be separated. Synoptic transients have wave periods of 2-5 days [27] and thus will have a large influence when M is calculated on daily data but will be effectively filtered out when using 11 day running-means. Changes in the latter will thus reflect changes in quasi-stationary type waves.
For each of the three regions (global, America and Eurasia) and for all timescales, we statistically analyse the meandering index using quantile regressions to quantify changes in the full distribution of M rather than just changes in the mean. Any changes in upper quantiles (i.e. strongly meandering patterns) are particularly relevant as those have been linked to extreme weather [2]. Statistical significance of trends at α=5% is estimated using Monte Carlo analysis using 10 000 surrogate time series of shuffled data [28][29][30]. The shuffling algorithm accounts for the fact that running means present not truly independent data: Instead of considering 90 individual time series (one for each day of the season) with 37 values (one for each year) as done for daily data and perform shuffling of these series in time, when shuffling is applied on running means, blocks of 5 and 11 days (according to the respective running-mean) are shuffled.

Meandering index and arctic oscillation (AO) index
Strongly meandering flows (i.e. large M) are expected to be linked to a negative Arctic Oscillation (AO) index, which is confirmed by cross-correlation analyses (see SI, figures S33-S35). The correlation has a peak of significant negative values near lag 0 for the warm seasons in all running means, while in the cold seasons the correlation has a peak of significant negative values near lag 5 for daily data and lag 0 for the 11 day running mean. A negative correlation implies that when M increases, AO decreases, linking strongly meandering isohypses to negative AO situations as expected. Figure 2 shows seasonal probability density distributions for daily data and quantile regressions for daily, 5 day and 11 day running means for the hemispheric meandering index M. A value M=1 is the minimum theoretically possible value (representing a straight East-West isohypse) and the climatological mean is close to double that (1.9) for all seasons and for annual analysis (figures 2(a)-(e)). Next to the mean, also the shapes of the seasonal distributions are similar, reaching maximum values of M∼3.

Hemispheric analyses
For illustrative purposes, we compare the daily probability density distributions of M for two periods: 1979-1999 and 2000-2015. In autumn, the distribution has shifted to larger meandering values (figure 2(d)), consistent with a significant upward trend in the mean (figure 2(i), upper panel). Upward trends for both upper and lower quantiles reflect a shift of the distribution toward larger values with little changes in variance (although only trend for the 5th, 10th and 25th quantiles are significant). The trend in the mean of ∼10 −3 yr -1 is detected (and even larger) for 5 day and 11 day running means as well (figure 2(i), lower panels), but these are not significant because the confidence intervals become larger as less independent data are available. This persistence of the trends when synoptic waves are filtered out thus suggests that in autumn the increasing trend is mostly due to changes in quasi-stationary waves. Daily data in winter have seen no significant changes in M in neither mean nor quantiles ( figure 2(f)).
In contrast to the cold seasons, daily M in summer shows a significant downward trend in the mean and in most quantiles. The observed trend in daily data (−10 −3 yr -1 ) becomes smaller (and non-significant) in 5 day and 11 day running means (figure 2(h)) suggesting that the trends observed in daily data are mostly due to changes in synoptic wave activity, which is filtered out when using running means. Spring and annual data have seen no significant changes in M at any timescale.

Regional analyses
Regional trends, as presented in figures 3 and 4, can typically be a factor 2 to 3 larger than those observed for the full hemisphere. The Eurasian sector (figure 3) is found to be dominant in determining the behaviour of the hemispheric index. The daily mean values are close to hemispheric ones (∼1.9), but the distribution is slightly broader with maximum values of M∼3.3 (figures 3(a)-(e)). Comparing distributions of the two periods, 1979-1999 and 2000-2015, the same tendencies as for the hemispheric index are found. Autumn shows a pronounced and significant upward trend in both the mean and all quantiles, with the stronger trend for the 95th quantile (0.005 yr -1 , figure 3(i), top panel). Similar to the hemispheric index, the trend in the mean (≈0.002 yr -1 ) can be detected for daily, 5 day and 11 day running means, suggesting that changes in stationary wave activity are the most important contributors ( figure 3(i)). Summer shows a significant downward trend in both mean (≈−0.0023 yr -1 ) and most quantiles similar to the hemispheric index. Winter, spring and annual data show minor upward trends but none is significant.
Presented in figure 4, the American sector has seen different changes as compared to the hemispheric and Eurasian analyses. Though the climatological mean values are close to the hemispheric mean M values (∼1.9), the upper tail of the distribution stretches further with maximum values of M∼3.5. Autumn shows no significant changes neither in daily data nor in running-means. The warm seasons (AMJ-JAS) show generally upward trends with the largest magnitudes of trends observed in 11 day running means. This suggests that increased meandering of quasi-stationary waves has played the most important role in the warm seasons. In particular, in summer a non-significant upward trend appears in the 5 day running mean analysis, which becomes larger (0.003 yr -1 ) and significant for 11 day running means (figure 4(h), lower panel). Both spring and summer 11-day running mean analyses show that the upper quantiles have seen the strongest trends, whereas the lower quantiles have seen essentially no trend, indicating a broadening of the distribution (figures 4(g)-(h), lower panels). Winter has seen no trend in the mean but a pronounced and significant upward trend (∼0.007 yr -1 ) in the 95th quantile for daily data. The upward trends in high quantiles seen in most seasons can also be detected in the annual analysis showing significant increases in the 95th quantile for all time scales ( figure 4(j)). Figure 5 shows climatological mean values and linear trends for the seasonal and annual mean height and latitude where M is detected, i.e. where maximum waviness occurs. In the hemispheric analysis, maximum waviness is usually found between 5200 and 5300 m in the cold seasons and between 5500 and 5600 m in the warm seasons ( figure 5(a)). There are significant upward trends in this height by about 5-10 m/decade for all seasons and for annual data ( figure 5(a)), indicating an upward migration of the waviest isohypse, likely reflecting warming of the lower troposphere. Mean latitudes are found between 60°N and 62°N, placing the waviest isohypse near the position of the climatological Polar jet ( figure 5(d)). Autumn shows a significant poleward trend in the mean latitude of about 0.3°/decade. The other seasons and annual data show no trend in latitude ( figure 5(d)).
Regionally, Eurasian maximum waviness tends to be at somewhat lower latitude and higher 500 mb geopotential heights, while in the American sector a slightly opposite tendency is found. However, values for both regions remain close to the hemispheric values. Note that for a specific day, maximum meandering in the different regions can occur at different isohypses, and hence different latitude and heights. Still only small differences are detected when the region is changed, as expected. At regional scale, a similar tendency of the most meandering isohypse to migrate toward higher heights is found, even though magnitude and significance of the trends can slightly change between different seasons and regions (figures 5(b) and (c)). In summer, the two regions show opposite behaviour in the latitudinal trends, with a significant southward trend in Eurasia and a significant but northward trend for the American sector. In autumn, Eurasia shows a significant northward trend, with the same tendency of the hemispheric latitudinal trend for this season. We repeated all our analyses using 300 hPa geopotential heights fields (SI, figures S7-S10) and for the NCEP/NCAR reanalysis (SI, figures S3-S6), which showed that our findings are robust.

Discussion
The meandering index M as defined here is different from indices defined previously like for example the DayMaxMin index [19], which captures the maximum North-South amplitude of isohypses but not their actual shape. These two indices therefore cannot be directly compared: the arc length of an isohypse can change even when its maximum North-South amplitude remains unaffected. The meandering index captures the total waviness in the atmosphere, which can be pronounced even when the North-South wave extent is moderate (i.e. when high wave numbers dominate). These two indices thus measure different aspects of the mid-latitude flow and therefore cannot be compared one-to-one. Other studies used spectral analysis to decompose isohypse patterns into their different wave number contributions and analysed changes in amplitude of these wave numbers [17]. Spectral methods cannot handle multiple latitudinal positions of an isohypse for a single longitude (i.e. during Rossby wave breaking events) and thus they cut isohypses short in such situations. While this approximation is unlikely to affect isohypses with small M, it could substantially modify the shape of isohypses that are curling over (for example during wave breaking), forcing them into a 'less wavy' shape. Our approach is different in that we measure total waviness rather than focus on individual wave numbers and our meandering definition also correctly retains and measures events that are curling over. This is important since we are particularly interested in such strongly meandering flow patterns. To differentiate between transient and quasi-stationary waves (the latter normally have lower wave-numbers) we apply running mean filtering.
More generally, following Barnes [19], we argue that quantifying changes in maximum waviness, rather than in some pre-selected isohypse(s), provides more stable and insightful results. Results of the latter are very sensitive to the chosen isohypse(s). Seasonal distributions and trends in M are very similar in the two reanalyses and at different pressure levels, whereas they can differ substantially when analysing single isohypses (figure S32). Our method also illustrates the usefulness of quantile regressions to estimate changes in the full distribution of circulation patterns. Forcing of the polar jet, irrespective of the nature of the forcing, is likely to favour only some type of circulation patterns but this might have little or no effect on the mean circulation [31]. Using quantile regressions, changes in the frequency of rare, strongly meandering patterns can be quantified.
We have reported significant trends in maximum waviness, with an overall tendency towards more strongly meandering circulation patterns. The detected trends appear to be rather moderate in magnitude, on the order of 0.1% yr -1 or about 4% change over 1979-2015, but they can nevertheless have pronounced effects on the occurrence-frequency of strongly meandering flow patterns. To illustrate this, we estimate the 90th percentile of M in 1979-1999 and calculate how often this threshold is exceeded in 2000-2015. Autumn has seen significant upward trends in the hemispheric analysis (figure 2(i)) resulting in the 90th percentile (M=2.12) to be exceeded in ∼15% of autumn days in the post-2000 period, representing a ∼50% increase compared to 1979-1999. Likewise, the Eurasian sector has seen a similar ∼55% increase in meandering patterns exceeding the 90th percentile and even a ∼74% increase of those exceeding the 95th percentile. The American sector has seen the strongest changes in summer for the 11 day running means, with an increase in events exceeding the 90th (95th) percentile of 50% (34%).
Our findings indicate that meandering in autumnearly winter (OND) has increased (significant trends in hemispheric and Eurasian analysis) and that this has led to an increase in occurrence-frequency of strongly meandering flow patterns. This might have favoured the occurrence of cold spells in recent autumns (though overall, cold seasons tend to become milder due to long-term warming trends). Drivers behind these autumn circulation changes are not analysed here, but previous studies have proposed anomalous tropical Pacific sea surface temperatures [5], anomalous extra-tropical sea-surface temperatures [32] or rapid warming in the Arctic [6]. Also internal atmospheric variability might well play a role [33,34].
The warm season (JAS) has seen a more diverse behaviour among different seasons, regions and timescales. At the hemispheric scale, a significant downward trend is observed for daily data but this trend is not seen in 5 day or 11 day running means (figure 2(h)), suggesting that the observed daily trends are due to changes in synoptic wave activity. This is consistent with the recently reported hemispheric weakening of summer synoptic activity [35,36,37]. Weakening synoptic activity implies reduced synoptic pressure anomalies and hence the associated isohypse deviations from a straight line become smaller. Moreover, using spectral analysis of daily data of meridional wind fields, Coumou et al [35] show that, in summer, wave amplitudes have become smaller. The regional analysis shows that the Eurasian sector largely follows these hemispheric-wide changes but that the American sector has seen pronounced changes in the quasistationary wave. Here, we detect significant upward trends in the 11 day running mean data for JAS (and similar patterns for AMJ) indicating increased quasistationary wave amplitudes. This is particularly relevant for recent extreme weather events as the high quantiles have seen the most pronounced trends, as also detectable in the annual analysis (figures 4(g), (h), (j) lower panels). The drivers behind this enhanced meandering might include western north-Pacific seasurface temperatures [38], changes in the Arctic cryosphere [39], or internal atmosphere-dynamical mechanisms [40,41]. Irrespective of the underlying drivers, the tendency towards more-pronounced quasi-stationary meandering conditions during the American warm season might contribute to morepersistent weather and prolonged heat and drought conditions.
The aim of this study was to analyse maximum waviness in the Northern Hemisphere atmosphere via an index which is not affected by vertical or latitudinal migration of maximum waviness. We reported some robust and significant changes in this index which are relevant for mid-latitudes extremes associated with strongly meandering flow patterns. Future work should analyse possible drivers behind these changes.