The polar and subtropical jet streams are strong upper-level winds with a crucial influence on weather throughout the Northern Hemisphere midlatitudes. In particular, the polar jet is located between cold arctic air to the north and warmer subtropical air to the south. Strongly meandering states therefore often lead to extreme surface weather.

Some algorithms exist which can detect the 2-D (latitude and longitude) jets' core around the hemisphere, but all of them use a minimal threshold to determine the subtropical and polar jet stream. This is particularly problematic for the polar jet stream, whose wind velocities can change rapidly from very weak to very high values and vice versa.

We develop a network-based scheme using Dijkstra's shortest-path algorithm to detect the polar and subtropical jet stream core. This algorithm not only considers the commonly used wind strength for core detection but also takes wind direction and climatological latitudinal position into account. Furthermore, it distinguishes between polar and subtropical jet, and between separate and merged jet states.

The parameter values of the detection scheme are optimized using simulated annealing and a skill function that accounts for the zonal-mean jet stream position (Rikus, 2015). After the successful optimization process, we apply our scheme to reanalysis data covering 1979–2015 and calculate seasonal-mean probabilistic maps and trends in wind strength and position of jet streams.

We present longitudinally defined probability distributions of the positions
for both jets for all on the Northern Hemisphere seasons. This shows that
winter is characterized by two well-separated jets over Europe and Asia (ca.
20

With this algorithm it is possible to investigate the position of the jets' cores around the hemisphere and it is therefore very suitable to analyze jet stream patterns in observations and models, enabling more advanced model-validation.

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

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 (

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

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.

Definition of edge costs:

In order to reduce computational costs, the spatial domain is reduced to the
main region of interest, 0–75

We define an edge cost function,

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,

Calibration scheme. Before calculating the shortest path with
Dijkstra's algorithm, the cost of each edge has to be calculated according to
the three terms

The first term,

The second term

Rikus' scheme. In

The third term,

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

Naturally, there are other slightly different ways to define wind strength,
wind direction and latitudinal dependence for the edges of the network. For
example,

After calculating the edge cost for each edge according to Eq. (1), our
algorithm returns from the set of all possible paths

The optimal weights

Panel

Rikus' algorithm is a closed-contour object identification scheme
(Rikus, 2015). It operates on a zonal-mean zonal wind and treats
the two-dimensional (pressure height and latitude) zonal-mean

Figure 3 shows the scheme of Rikus' algorithm.
First a local maximum or minimum filter is applied to the original zonal-mean

This way, the fields

In a second step Rikus' algorithm examines for each grid cell whether

We applied Rikus' algorithm to the zonal-mean zonal wind field of each time
period (i.e., 15-days running mean ERA-Interim data; Dee
et al., 2011) to identify the zonal-mean jet stream latitude for all levels
and latitudes in the domain 150–430 mb and 50–70

Simulated annealing (Kirkpatrick, 1984) is an optimization
method that approximates the global minimum of a high-dimensional skill
score function. We use the multi-run simulation environment SimEnv
(Flechsig et al., 2013) to calibrate the weights

We expect the mean of all latitudinal positions calculated by our algorithm
to be close to the zonal-mean jet position found by Rikus' algorithm and
thus define our zonal-mean skill function accordingly:

The reason for tuning our spatially resolved tool to a zonal-mean approach is that the characteristics of the jet stream such as the zonal-mean latitude position should be ultimately the same. The mean latitude detected by our algorithm should be very close to the maxima in zonal-mean zonal wind.

We determined the wind direction weight

As starting point for our automatic optimization scheme, the parameters
(

Start and optimized jet stream parameters used for the edge cost function.

With the zonal-mean subtropical and polar jet stream latitudes found by
Rikus' algorithm we optimized the parameters

The results of our automatic optimization scheme are listed in Table 1. The
jet stream guidance parameter

The climatological mean latitude

We would like to emphasize that all terms are important even though

The zonal-mean latitudinal difference between Dijkstra (a longitudinally
resolved latitude) and Rikus (a zonal-mean latitude) for the subtropical jet
stream (

Fifteen-day running mean around 13 January 2010. Jet stream cores calculated with Dijkstra's algorithm using optimized weights (compare with Fig. 2).

Fifteen-day running mean around 2 March 1979. The right panel shows three
maxima (30, 50 and 75

Fifteen-day running mean around 12 May 1979. The right panel shows only a maximum in the wind field in the region between
0 and 100

Improvements in the detected jet stream core positions due to the optimization process, relative to the positions, found by the untuned algorithm (Fig. 4; parameters are given in Table 1) are illustrated in Fig. 5. Here, the left panels show the zonal-mean latitude of the jet stream core calculated with Dijkstra's algorithm (light-blue lines) and that computed by Rikus' algorithm (blue circles). The black solid (dashed) lines are the borders of the PFJ (STJ) core latitudinal positions as detected with Dijkstra's algorithm around the hemisphere.

After tuning, the zonal-mean latitude of the polar jet stream core detected with Dijkstra's algorithm is close to the latitude computed by Rikus' algorithm (compare Fig. 5 with Fig. 4). Moreover, visual inspection of the right panel of Fig. 5 illustrates that our algorithm now correctly finds the polar jet around the hemisphere.

The mean latitude calculated with Dijkstra's algorithm does not always match perfectly with the mean latitude computed by Rikus' algorithm because the first is a 2-D algorithm in longitude and latitude and the latter is a 2-D algorithm in latitude and height. Rikus' algorithm therefore does not capture the undulations of the jet stream.

Often any such differences are related to the existence of not one but two
zonal-mean PFJ maxima. For example, in Fig. 6 there
exists a zonal-mean maximum at latitude

In other cases, a zonal-mean maximum found by Rikus' algorithm exists only in
one longitudinal range. For example, in Fig. 7 the
maximum of the pressure–height latitude plot exists mainly because of the
region between 0 and 100

In Fig. 8 the differences between the zonal-mean polar jet stream cores
calculated by Rikus' algorithm and with Dijkstra's algorithm are shown in
two different subplots. Figure 8a shows a day–year plot depicting, in blue,
days for which Rikus' algorithm finds a polar jet stream in agreement with
the range of jet stream core latitudes detected with Dijkstra's algorithm.
In yellow are those days where Rikus' polar jet stream core position is not
between the minimum and maximum latitude of the polar jet stream path
detected with Dijkstra's algorithm. These are 199 of 3122 data points which
are equivalent to 6.4 %. Figure 8b shows the difference between the mean
latitude calculated by Rikus' and the mean latitude calculated with
Dijkstra's algorithm. The mean of the difference is 5

The day–year plot of the subtropical jet stream in Fig. 9 shows that, for every single time
period, Rikus' latitude position is within the range of latitudes found with
Dijkstra's algorithm. Figure 9b indicates the
difference between the mean latitude calculated by Rikus' and the mean
latitude calculated with Dijkstra's algorithm, which is very small. The mean
is 2

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

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.

Probability analysis for spring months (MAM): Panel

Probability analysis for summer months (JJA; compare with Fig. 10).

Probability analysis for autumn months (SON; compare with Fig. 10).

Probability analysis for winter months (DJF; compare with Fig. 10).

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

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

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 (

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 (

Slope parameter for the latitude and velocity trends of the jet stream
cores. Bold values indicate statistical significance (

Annual, DJF, and JJA: mean latitudinal trends and mean wind velocity trends of the STJ and PFJ cores.

In general, we observe a northward trend for the STJ (except for SON) which
is significant for winter and annual time series.

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

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

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,

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.

Sonja Molnos, Tarek Mamdouh, Stefan Petri, Thomas Nocke, Tino Weinkauf and Dim Coumou.

Sonja Molnos, Tarek Mamdouh, Tino Weinkauf and Dim Coumou developed the study conception. Tino Weinkauf, Tarek Mamdouh and Thomas Nocke developed the analysis method. Sonja Molnos, Tarek Mamdouh and Stefan Petri developed the model code and performed the simulations. Sonja Molnos and Dim Coumou analyzed and interpreted the data. Sonja Molnos prepared the paper with contributions from all co-authors.

The authors declare that they have no conflict of interest.

We thank ECMWF for making the ERA-Interim available. The work was supported by the German Federal Ministry of Education and Research, grant no. 01LN1304A. (Sonja Molnos, Dim Coumou). The authors gratefully acknowledge the European Regional Development Fund (ERDF), the German Federal Ministry of Education and Research and the state of Brandenburg for supporting this project by providing resources on the high-performance computer system at the Potsdam Institute for Climate Impact Research. Edited by: R. A. P. Perdigão Reviewed by: L. Rikus, C. Pires, and one anonymous referee