Jet streams are upper-level fast currents of air that circulate and meander around the hemisphere and play a key role in the general circulation of the atmosphere as well as in generating weather conditions throughout the Northern Hemisphere midlatitudes. In general, we distinguish between two jet stream types in the troposphere: the subtropical jet stream (STJ) and the polar front jet stream or, simply, the polar jet stream (PFJ).

The STJ is located at the upper branch of the Hadley circulation and forms due to momentum conservation, when air moves poleward, and meridional contrasts in solar heating (Woollings et al., 2010). The PFJ is situated along the polar front and is driven by baroclinic eddies that evolve due to temperature gradients along the region of the polar front (Pena-Ortiz et al., 2013) and is therefore often referred to as an eddy-driven jet. Those transient eddies transport heat and vorticity and thereby accelerate the westerly winds (Woollings, 2010). The hemispheric north–south temperature gradient is strongest in winter and weakest in summer, and this can explain variations in the jet stream strength and position between seasons. In summer, the winds are weaker and the jets move farther polewards, whereas in winter the winds are stronger and the jets move farther equatorwards as the cold front extends into subtropical regions (Ahrens, 2012).

Jet streams are thus sensible to changes in temperature gradient and variability and hence also to climate change (Barnes and Polvani, 2013; Grise and Polvani, 2014; Solomon and Polvani, 2016). Large-scale undulations in the jets (Rossby waves) can sometimes become quasi-stationary (i.e., stagnant), which can lead to persistent weather conditions at the surface. Persistent weather can favor some types of extreme weather events (Coumou et al., 2014; Stadtherr et al., 2016). Petoukhov et al. (2013) proposed a mechanism that could provoke such weather extremes in the Northern Hemisphere midlatitudes. Quasi-stationary Rossby waves in summer are linked to persistent heat waves and severe floods (Kornhuber et al., 2016; Petoukhov et al., 2013, 2016). Likewise in winter, strongly meandering jets, driven by either anomalous tropical (Palmer, 2014; Trenberth et al., 2014) or extratropical (Peings and Magnusdottir, 2014) sea-surface temperatures or stratospheric variability (Cohen et al., 2014; Kretschmer et al., 2016), can lead to midlatitude cold spells.

Hence, jet streams play a key role in the general circulation and for generating midlatitude weather conditions and extremes.

Several schemes have been proposed to extract the jet stream positions from wind data, each one with advantages, but also limitations.

Rikus developed a detection method to analyze zonal-mean positions of the jet streams (Rikus, 2015) using the zonally averaged zonal wind in latitude–height space to identify local maxima as cores of the jet streams. This method thus cannot analyze the development of the jet stream in the longitudinal east–west direction.

A method for calculating the jet stream core in the latitude/longitude direction was developed by Archer and Caldeira (2008). They define the jet's latitudinal position for each longitude using mass flux weighted monthly mean wind speeds between 100 and 400hPa in the northern (15–70∘ N) and southern hemispheres (Southern Hemisphere jet (SHJ): 40–15∘ S; Southern Hemisphere Polar jet (SHP): 70–40∘ S).

Their algorithm detects only one jet position in the Northern Hemisphere and thus cannot distinguish between polar and subtropical jet streams. It is also not possible to capture omega-shaped jet patterns, since that method assigns only one latitude for each longitude.

Koch et al. (2006) classify so-called deep or shallow jet stream events. Their three-step algorithm first calculates the vertically averaged horizontal wind speed between two pressure levels (p1 = 100 hPa and p2 = 400 hPa) for each time instance and grid point. Next, a threshold of 30 m s-1 is applied to detect a so-called jet event in a grid cell. Further analysis over vertical layers classifies events into deep or shallow jet stream events but it neither extracts the actual stream core, nor distinguishes between polar and subtropical jet streams (Koch et al., 2006).

Gallego et al. (2005) developed a scheme using a geostrophic streamline of maximum daily averaged velocity at 200 hPa to find the jet stream in the southern hemisphere. It uses wind velocitiy threshold of 30 m s-1 and distinguishes between the subtropical and polar jet stream when the average latitudinal difference is greater than 15∘. The threshold was set by manual optimization (Gallego et al., 2005). This approach might work reasonably for the southern hemisphere jets; a fixed threshold approach is particularly problematic for the Northern Hemisphere polar jet, which can change drastically in strength on weekly timescales.

The first 3-D method (longitude, latitude, height) developed by Limbach at al. (2012), detects and tracks specific properties of atmospheric features as merging and splitting jet streams (via clustering of data points). Still, this method cannot distinguish between subtropical and polar jet streams and also requires the use of a wind velocity threshold (Limbach et al., 2012).

Another 3-D detection scheme was developed by Pena-Ortiz et al. (2013), which identifies local wind maxima in the zonal wind field by using a specified wind speed threshold. The algorithm distinguishes between the subtropical and polar jet stream via a specified threshold in latitude. A limitation of such an approach is that the values of such thresholds are not well defined. In particular the polar jet, which is our prime interest, can meander over large latitudinal ranges and experience strong variability in its strength (Pena-Ortiz et al., 2013).

To overcome these issues, we propose a new method which uses Dijkstra's shortest path algorithm to find the shortest path in a network of nodes and edges with an edge cost function, defined by any combination of relevant variables. We develop a 2-D detection scheme for both the PFJ and STJ core, and define our edge cost function using wind speed, wind direction and a latitudinal guidance parameter (which is not thresholded). This way, we are able to accurately differentiate between subtropical and polar jet.

In Sect. 2 we describe the data used in this algorithm. In Sect. 3 we explain the details of our detection scheme, parameter optimization process and its results. Afterwards (Sect. 4), we analyze jet stream positions from 1979 onward and calculate probabilistic maps for different seasons. In Sect. 5, we calculate trends in latitudinal position and wind strength for the STJ and the PFJ. We conclude with a summary and a discussion in Sect. 6.

In this study, we used ERA-Interim data (Dee et al., 2011) from the European Centre for Medium-Range Weather Forecasts (ECMWF). The ECMWF provides meridional and zonal wind velocity components with a 0.75 latitude–longitude grid resolution. We chose 11 vertical layers of the upper troposphere stretching from 500 to 150 mb and for four 6-hourly time steps per day (00:00, 06:00, 12:00, 18:00 UTC) for the years 1979–2014. From these data, we calculate 15-day running mean and vertically averaged (mass-weighted) wind velocity, which is used for all analysis in this paper.

In the following text, a “time period” denotes a 15-day mean centered on a given day.

Our jet stream core detection scheme is based on Dijkstra's shortest-path algorithm, which is a widely used method for finding the shortest path from a source to a destination within an edge-weighted graph (Dijkstra, 1959). We assume that the jet stream core is a closed path along the hemisphere, with source (most westerly point) and destination (most easterly point) at the same location.

We use wind data on a two-dimensional grid of the Northern Hemisphere, where each grid point is taken as a node in a network graph. Only geographically adjacent grid points (nodes) are connected via edges and thus no teleconnections are considered. The nodes within the most westerly column are copied after the end of the most easterly column to ensure that that the path found with Dijkstra's algorithm starts and ends at the same location. The path itself is not an injective function of longitude meaning that the path can pass the same longitudinal coordinates multiple times.

To avoid noise and reduce computational costs only those grid points where the wind velocity is greater than 10 % of the maximum wind velocity for the considered time period are connected.

In order to reduce computational costs, the spatial domain is reduced to the main region of interest, 0–75∘ N, for the subtropical jet stream on the Northern Hemisphere. The spatial domain for the polar jet stream is 0–90∘ N, since in some rare cases the polar jet stream could be occasionally close to the 30∘ N limit.

We define an edge cost function, Cj, based upon wind speed, wind direction and a latitudinal guidance function using the climatological mean latitudinal position of each jet: Cj=w1Xj+w2Yj+w3Zjw1+w2+w3=1. The variables Xj ,Yj and Zj, each normalized to the interval [0, 1], are the three terms for computation of the cost at edge ej and w1, w2 and w3 are the weights that control the contributions of the three cost terms. These weights are non-negative and their sum is equal to one.

The three terms and their respective factors are illustrated in Fig. 1a and b. Figure 1a shows all nodes and edges as well as the wind velocities of the considered node (blue arrows) in the grid. For each edge, ej, the cost is computed depending on the wind velocities (term Xj, length of blue arrows) and wind directions (term Yj, angle between blue arrow and black edge) at its two nodes, A and B, and from its latitude (term Zj, shown in Fig. 1b).

The first term, Xj, captures the magnitude of the wind field at the nodes A and B. Jet streams are strong upper-level winds and hence the jet stream core should be where the wind strength is maximal: Xj=1-uA2+vA2+uB2+vB22maxk=1nuk2+vk2, where uA,uB,vA and vA are the zonal and meridional wind speeds at nodes A and B connected by edge j and maxk=1n(uk2+vk2) is the maximum wind speed found at the considered time period for any node k (see also Fig. 1a). The second term in Eq. (2) is thus always smaller than or equal to 1. We subtract this value from 1, and thus low values of Xj refer to high wind speeds because Dijkstra's algorithm will minimize the edge cost of the path (i.e., find the shortest path).

The second term Yj weights each edge ej according to the angle between the normal vector of the edge and the wind direction: Yj=1-VA⋅ej2. Here |VA| is the normalized vector of the wind direction in node A and |ej| is the normalized vector of the edge direction (see also Fig. 1a).

The third term, Zj, is used to differentiate between polar and subtropical jet streams. Basically, it favors pathways that are close to the climatological mean latitude of polar and subtropical jets but still allows free movement within a latitudinal belt of roughly ±20 % of the climatological mean. Outside this latitudinal belt, Z rapidly grows according to Zj=ϕj-ϕclim4maxϕclim,90-ϕclim4. Here, ϕj and ϕclim are the latitude of the edge and of the climatological mean latitude, respectively.

The reason for taking the difference between the latitudes raised to the fourth power is to give flexibility to the detected path to move almost freely in the vicinity of the desired latitude, but a strongly increasing weight farther away. This is also illustrated in Fig. 1, where the condition Zj for the STJ and PFJ is shown.

Naturally, there are other slightly different ways to define wind strength, wind direction and latitudinal dependence for the edges of the network. For example, Xj and Yj could be merged to a term which considers the wind projection along the edge unitary vector. In addition, it is possible to use a lower- or higher-ordered function for Eq. (4), e.g., a linear function or a function with the order of 8. However, a lower order means less free movement within the latitudinal belt centered around ϕclim. A higher order has negligible effects since Eq. (4) already gives values close to zero within the central latitudinal belt .

After calculating the edge cost for each edge according to Eq. (1), our algorithm returns from the set of all possible paths Pi with total edge costs of the path TCi the path Pmin with minimal total edge cost TCmin: TCMin=MinTCi=Min∑j=0nCj, where n is the number of edges in the path Pi.

In this section we present some results of the analysis of the jet stream paths that were detected by our algorithm.

Figures 10–13 show probabilistic jet stream positions for different seasons with brown dashed contour lines representing the subtropical jet and black solid contour lines representing the polar jet.

The seasonal cycle of the STJ is clearly seen with winter latitudes between 20 and 40∘ latitudes and summer latitudes further north. Moreover, in summer the probability that the jets merge in the western hemisphere is higher, whereas in winter the probability that they are clearly separated over almost all longitudes is higher.

In addition, the probability frequency of the PFJ is much broader than the probability of the STJ and no clear latitudinal shift between seasons is observed. In particular, in summer the PFJ distribution is smeared out (indicating large fluctuations in its position), whereas in winter it is more confined.

This strong meandering of the eddy-driven PFJ is explainable due to the nature of wave-mean flow feedbacks (Harnik et al., 2014). The PFJ cores always lie between 40–80∘ N; only in longitudinal direction is there a seasonal dependence. Over Asia the probability of a high-latitude PFJ is larger in summer than in winter. Over Europe the probability of a low-latitude PFJ is higher in summer. This is also observable for eastern Pacific and North America , but less pronounced; instead there seem to be, in spring and summer, two preferable states: a merged jet state with a jet at ca. 50∘ N and a second state with two jets at respectively ca. 50 and ca. 70∘ N.

In general, the probability of PFJ at low latitude is small over the European sector compared to other regions and therefore double jet states occur in every season here. In North America such a clearly separated STJ and PFJ is only observed in winter.

This coexistence of the STJ and PFJ in the eastern hemisphere, compared to more frequent merged jet states in the western hemisphere, is well documented in the literature, but has never been shown in probabilistic plots as presented here (Eichelberger and Hartmann, 2007; Li and Wettstein, 2012; Son and Lee, 2005; Woollings et al., 2010). Those different jet stream states occur since the processes which lead to their existence operate and interact in nonlinear ways (Harnik et al., 2016; Lee and Kim, 2003). In the North Atlantic, STJ and PFJ are separated because the region of strongest baroclinicity is located relatively far poleward. In contrast, the region of strongest baroclinicity in the North Pacific is located near the latitude of maximum zonal wind, favoring a merged jet (Lee and Kim, 2003; Li and Wettstein, 2012). Such a merged jet stream is also called the eddy-thermally driven jet because of the two different genesis mechanisms. In special cases, there is the possibility that this eddy-thermally driven jet stream also appears over the North Atlantic (Harnik et al., 2014). This happens if the tropical forcing strengthens or the midlatitude baroclinicity weakens.

In addition, Fig. 10–13b give probabilities of the zonal-mean latitude of both jets, showing enhanced variability of the PFJ compared to the STJ. The range of overlapping latitudes between STJ and PFJ is larger in summer than in winter because of the poleward shift of the STJ. The latitudinal variability in STJ is lower in summer and winter than in spring and autumn, whereas the variability in the PFJ is similar between seasons. However, the location of the maximum in the PFJ histogram changes per season: in winter, the maximum is at ca. 55∘ N, whereas in summer there are two maxima at 50 and at 70∘ N. These two maxima probably reflect the different behaviour in western and eastern hemisphere in the PFJ. In spring, there is no clear maximum visible (between 40–60∘ N), and in autumn it is again close to 55∘ N.

To quantify those merged and separated states further, one could use the latitudinal difference between STJ and PFJ, for all longitudes, and this way create the probability density distributions of merged and separated jets. The presented results (Figs. 10–13) might in principle also be the result of clearly separated jets which displace latitudinally over time to create the overlapping probability density.

For verification, we compare the probabilistic jet fields with seasonal climatological wind fields (panels c). In general, all probability density functions (PDFs) of the jet stream cores in their respective seasons coincide well with the wind fields. In summer, the wind field magnitude is very low and more homogeneously spread over the hemisphere. In summer the jet stream cores are farther north than in winter due to the weaker temperature gradient in summer. In general, the gradient of the wind velocities, as well as the strength of the velocities, in summer is weaker than in winter.

Figure 14 shows trends in the latitudinal position and wind velocity for summer and winter as well as annual data derived from our Dijkstra jet detection scheme. Table 2 summarizes the results giving linear trends in mean jet stream latitude and mean wind velocity with bold values indicating statistical significance (p < 0.05).

In order to compare our results with literature results, we calculated mean jet stream latitude and mean wind velocity trends, which are shown in Table 2. Bold values indicate statistical significance (p < 0.05). We used Monte Carlo analysis with 10 000 surrogate time series of shuffled data to determine significance (Di Capua and Coumou, 2016; Pollard and Lakhani, 1987; Schreiber and Schmitz, 2000). To account for the fact that running means present not data that is truly independent data, we shuffle blocks of 15 days in this method.

In general, we observe a northward trend for the STJ (except for SON) which is significant for winter and annual time series. T latitudinal position of the PFJ shows more mixed behavior with different signs for different seasons. A pronounced and significant equatorward trend is detected for the PFJ in winter. Wind velocities have generally weakened for both STJ and PFJ, something which is significant for summer, in agreement with Coumou et al. (2015) and Lehmann and Coumou (2015).

Overall these reported trends are in good agreement with previous studies, though it is somewhat difficult to make direct comparisons as different studies have analyzed different aspects of the flow field. For example, Pena-Ortiz et al. (2013) did not calculate separate trends for the STJ and PFJ, but instead for different ranges of latitudes: for winter 15–40∘, for spring and autumn 10–70∘ and for summer 30–60∘. Since STJ winds are in general stronger, we assume that, at least for spring, summer and autumn, their reported trends reflect trends of the STJ. Similarly, Archer and Caldeira (2008) considered only trends in Northern Hemisphere jet stream between 15–70∘ N, where we again expect that this mostly reflects the behavior of the STJ. Rikus (2015) calculated trends for one northern jet stream core within 20–54∘ N, so we can assume that the trend most probably describes the trend of the STJ. The findings of those studies can thus be best compared to our STJ findings. The annual poleward trend in latitudinal position of the STJ, detected with our method, is consistent with the results of Rikus (2015) and Archer and Caldeira (2008). Also, the latitudinal trend in summer calculated by our method has the same sign and order of magnitude as in Rikus and Pena-Ortiz et al., but the trend in winter is greater in our and Rikus' method compared to that of Pena-Ortiz et al. The trends for spring and autumn agree in sign with the analysis of Pena-Ortiz et al. using 20th century data, but they are weaker and even change sign for the NCEP/NCAR data set in autumn.

The wind velocity trends are positive in the publication of Pena-Ortiz et al., whereas we observed a negative trend like that of Rikus (2015) (except summer) and Archer and Caldeira (2008). With our more advanced approach which is able to differentiate between subtropical and polar jet, we detect stronger (and mostly significant) weakening compared to the other studies.

We have proposed a novel and objective method to detect the subtropical and polar jet stream cores which overcomes some limitations of previous studies. Our method uses a graphical approach employing Dijkstra's shortest path algorithm. With this method we are able to describe both spatially separated and merged jet stream cores. If the subtropical and polar jets merge, the two detected jet stream core positions become very close to each other.

We used three terms to define the edge costs: wind magnitude, wind direction and a jet stream latitudinal guidance term.

Based on those three terms, the algorithm finds the jet stream core as a closed path. Parameters entering this detection scheme were optimized using simulated annealing and comparing our spatially resolved scheme with a zonal-mean detection scheme to avoid unrealistic results. Here we discuss some possible improvements to our scheme.

Instead of using the wind direction and wind strength, it is also possible to merge both conditions and consider only the wind projection along the edge unitary vector. However, with two terms we have more flexibility regarding the weights of the terms.

In addition, the jet stream latitudinal guidance term, which is in our case a fourth-order function of latitude, could be a lower- or higher ordered function like a linear function or a function with the order of 8. A lower order means less freedom for the path to move away from the climatological latitude, whereas a higher order has only little effect, since the cost of a fourth-order function is already small in the latitudinal belt.

As a result the latitudinal guidance term seemed the most important factor. This large value of w3 is admissible, since Eq. (4), which describes the latitudinal guidance, gives values close to zero within the central latitudinal belt. Hence the current choice still allows free movement of roughly ±20 % of the climatological mean.

We calculate the probabilities of the northern STJ and PFJ core and show that the probability of two clearly separated jet streams is very high over the east Atlantic and Eurasia and very low over the Pacific and America. This is consistent with previous studies (Li and Wettstein, 2012; Son and Lee, 2005). The underlying reason is the different location of strongest baroclinicity between the North Pacific and the North Atlantic. In the former, the strongest baroclinicity is located near the latitude of the maximum zonal wind, and in the latter it is located relatively far poleward. The histograms of STJ and PFJ density for different seasons and for the annual mean show that the latitudinal variability of the PFJ is much larger than the variability of the STJ. This much larger variability is due to the nature of wave-mean flow-feedbacks (Harnik et al., 2014).

We reported the zonal-mean jet stream properties and trends of the mean latitude and wind velocity and show them to be in good agreement with other studies. Differences between studies can largely be explained by different data sets, time periods, pressure level and/or methodology (Pena-Ortiz et al., 2013; Rikus, 2015).

For future work we plan to extend the algorithm to three dimensions and apply it to the southern hemisphere. Parameters for the third dimension could be optimized in a similar way as done for latitude, but using pressure heights.

In addition, to account for splitting of the STJ and PFJ, we plan to calculate not two but four (or even more) jet stream cores with different climatological mean latitudes, ϕclim. In cases where only one path exists, the found jet stream cores would be combined to one path, (based on their similarities to each other) and in other cases where two paths exist, they would split.

Furthermore, we intend to analyze the influence and impacts of the jet stream to extreme events using cluster analysis. This way, we can examine the link of particular cluster patterns on extreme weather events and determine which jet stream patterns have a higher probability for extremes. In addition, we plan to find possible drivers which lead to those jet stream patterns, using causal effect networks (Kretschmer et al., 2016).

Another possibility is to apply our method to model data such as CMIP5 in order to analyze whether models can reproduce the jet accurately.

All input data were downloaded from public archives. Code and data are stored in Potsdam Institute for Climate Impact Research's long-term archive and are made available to interested parties on request.