Stationary, long-lasting blocked weather patterns can lead to extreme conditions such as anomalously high temperatures or heavy rainfall. The exact locations of such extremes depend on the location of the vortices that form the block. There are two main types of blocking: (i) a

A blocking is a quasi-stationary, persistent large-scale atmospheric flow pattern that blocks the typical westerly flow and forces the jet and embedded pressure systems to bypass on its northern and southern sides

In order to classify blocking types, the point vortex model can be used. This idealized model gives one conceptual explanation of the quasi-stationarity of atmospheric blocking

It should be noted that the discrete point vortex perspective is contrary to the explanation of blocking based on Rossby waves as studied by, e.g.,

Application of point vortex theory to two distinct atmospheric blocking types:

The numerical prediction of blocking onset and persistence, i.e., the transition from zonal to blocked flow and vice versa, is still a challenge.
However, it is still important to study these transitions;

Due to their socio-economic relevance, it is of common interest to study and determine blocking climatologies and trends. It is also observed that the identification of blocking depends on the specific definition and method used (e.g., blocking indices). Blocking climatologies can, furthermore, differ with respect to the frequency and location of blocking

Do blocking occurrence probabilities undergo long-term changes? Do these changes depend on season or month?

Do onset (formation), decay (offset) or transition probabilities from one blocking type to another undergo long-term changes? Do these changes depend on season or month?

The work is structured as follows. First, we shortly describe the data set and variables in Sect.

We use the National Centers for Environmental Prediction–Department of Energy (NCEP–DOE) Reanalysis 2 data set

In the following we describe the six steps of the analysis, starting from the identification of blocked latitudes (Step 1; Sect.

Structure diagram of the individual steps of the evaluation as explained in Sect.

In the first step, we use the

Prior to the blocking classification, the time series of IBLs is post-processed in the following manner: first, we identify

Based on the blocking event list from Step 1, we now search for blocking patterns in the corresponding NCEP–DOE Reanalysis fields with the trapezoid method

The highs and lows are detected by an analysis of the kinematics of the flow. Therefore, we use the dimensionless kinematic vorticity number

For two-dimensional flow in spherical coordinates with longitude

For the calculation of

This means that each IBL is assigned with the maximum number of time steps (duration) at which this IBL is blocked. For each time step separately, each blocked IBL (either 0 or 1) is multiplied with its maximum duration and the associated longitude. This product is summed up and then divided by the sum of all IBL durations at this time step to get the duration-weighted IBL.

. Inside a box bounded by this duration-weighted IBLSimilar to the definition of the circulation centroid of a vortex system in point vortex theory, the circulation centroid of the real, extended high is defined as

Point vortex theory is the basis for this pattern-like identification. It states that a system of two or three vortices moves westward if the sum of their circulations is zero, the high lies poleward of the low(s), and the three-point vortex system forms an equilateral triangle

We inspect the relative vorticity in the area south of the circulation centroid of the high with coordinates (long

More precisely, the latitude lat

The middle box's width of 25

Note that we follow mostly the method described in

Labeling of experiments with different distance criteria that are used to estimate the uncertainty in the blocking identification method. Differences in longitude

Following Step 1 to 3, we are left with blocking events occurring between 90

Logistic regression is designed to model probabilities

Logistic regression is a special case of generalized linear models

The setting can be viewed as a generalization to standard linear models.
Let

Logistic regression describes the dependence of the blocking occurrence
probability

Rearranging Eq. (

In some cases, the influence of one covariate

For more than two states, the model can be extended to multinomial logistic regression. We next consider a multinomial random variable

We use Markov models with two and three states to describe transition probabilities between the states related to the different blocking types and the no-blocking state. For both cases, there is thus a discrete set of possible states:

two-state model, consisting of blocked (

three-state model, consisting of high-over-low (HoL), omega (

The system evolves along a discrete time axis

Let

The simplest Markov Chain is a Bernoulli process consisting of two states.
In our case, the two-state model with the states no blocking (nB) and blocking (

Homogeneous (time-independent) Markov chains can be illustrated using a
network diagram. Figure

A general example of a network diagram of a homogeneous Markov chain with two states: nB (no blocking) and

For a non-homogeneous (time-dependent) Markov process, we use logistic regression to estimate time-varying transition probabilities. For the two-state model with the two states blocking (

The results are divided into four subsections in order to answer the research questions posed in the introduction: (1) whether the blocking occurrence probabilities undergo long-term changes and (2) whether onset (formation), decay (offset) or transition probabilities between blocking types undergo long-term changes. In both cases, we ask if these long-term changes depend on season or month.
We start with an overview of blocking properties and explore the uncertainty in the identification algorithm in Sect.

A number of experiments with different distance criteria between the high-pressure centroids of subsequent time steps allow us to estimate the sensitivity of the identification method with respect to the distance criterion (see Table

Furthermore, we take a closer look at some general blocking properties such as frequency and duration and their sensitivity to the distance criterion.
Generally, the total number of blocks as well as the mean blocked days per year is lower for stricter distance criteria and increases almost linearly with relaxation of the criterion (Fig.

Temporal development of

In the 30-year period from 1990 to 2019 in the region 90

The temporal development of the annual blocking frequency and duration for the larger and the smaller Euro-Atlantic region is shown in Fig.

Note that block numbers can only take integer values.

. Considering the fraction of blocked time steps per year (green, Fig.Blocking probability over time for the full

The mean duration of blocking events shows a small inter-annual and inter-experimental variability compared to the maximum duration (see red and blue crosses and bars in Fig.

We investigate the inter-annual variability in blocking occurrence probability taking only two states into account: blocking and no blocking; both omega and high-over-low blocks are for now considered to be blocking. Starting with annual probabilities, we resolve the blocking occurrence seasonally and monthly in further steps.

We study annual blocking occurrence probability with logistic regression as described in Sect.

The colored lines in Fig.

In the Euro-Atlantic region, the occurrence probability increase is weaker in summer (JJA) (

We now break down trends in occurrence probability in the Euro-Atlantic sector to a monthly resolution using a categorical term for the month

Blocking probability over time for individual

Figure

The consistency between seasonally and monthly resolved results gives confidence in the analysis. Furthermore, it seems worth looking at both resolutions as strong monthly resolved signals can average out when aggregated to the season as in winter, or weak but consistent monthly signals can add to a stronger seasonal signal as in summer. The monthly resolution now allows us to postulate that in the Euro-Atlantic region there are signs for blocking occurrence probability increases at the beginning of the year (JFM) and decreases towards the end of the year (SOND); inter-annual variation in between is comparably small.

We now additionally distinguish between the two blocking types high-over-low and omega, considering occurrence probabilities for three states in a multinomial logistic model.

Blocking probability estimated for individual months for blocking in general as well as separately for high-over-low and omega.

Taking a look at the annual cycle of blocking probability reveals that the colder months from September to March are characterized by blocking probabilities of about 22 % (

Having now three states with distinct high-over-low and omega blocks, we use multinomial logistic regression (Sect.

Blocking probability over time for the full

Figure

A total of 8 (omega) and 11 (high-over-low) out of 11 experiments agree on the trend in DJF, and confidence intervals are not compatible with a constant occurrence or transition probability.

, leading to almost constant overall blocking in that season (Fig.A total of 10 out of 11 experiments agree on the increasing trend of the omega blocks in JJA, and confidence intervals are not compatible with a constant occurrence or transition probability.

A total of 9 (high-over-low) and 5 (omega) out of 11 experiments agree on the trend in SON, and confidence intervals are not compatible with a constant occurrence or transition probability.

Blocking probability over time for individual

Graph representation of the transition matrix estimated for a Markov process

We now estimate monthly resolved trends in blocking occurrence probabilities for the three-state model in Eq. (

We conceive the dynamics between different blocking and no-blocking states as a stochastic process with Markov properties (see Sect.

Within the framework of homogeneous Markov processes, we estimate stationary transition probabilities. Figure

Considering the three-state model, we can now break down the total blocking onset probability into the onset of high-over-low (

Note that for blocking onset, both probabilities (

Probability estimates

Probability estimates

Transition probabilities analogous to the matrix of a three-state Markov process (Eq.

We describe the change in transition probabilities with time (years) using logistic and multinomial regression on the annual timescale and broken down into seasons. For the two-state Markov model (Eq.

Figure

Transition probabilities analogous to the matrix of a three-state Markov process (Eq.

We now use the model given in Eqs. (

Inclusion of the trend in years broken down to seasons is – according to the likelihood-ratio test – a significant improvement over a homogeneous Markov process for summer – JJA, (

Point vortex theory and the kinematic vorticity number allow us to automatically classify atmospheric blocking into high-over-low and omega blocking; the two types are distinguished by the position of the associated low-pressure system(s). These positions are of general importance as the associated blocking types can affect and impact different regions due to their different structures. A key element is the Lagrangian framework, within which we require the high-pressure system to remain the same vortex over the whole lifetime of the system. We do, however, allow for local variations in and replacements of the low(s). This distinguishes our approach from other studies that rather focus on blocking as a large-scale weather regime within a defined region (Eulerian perspective). We thus refine the method developed by

Investigating blocking without the distinction of omega and high-over-low, we see a slight increase in annual occurrence probability in both regions (Fig.

In their work on observed blocking trends during the time period 1980–2012,

Distinguishing between omega and high-over-low blocking, we find occurrence probabilities of omega blocking being larger than of high-over-low (Figs.

Within the two-state model, the probability of remaining in a blocked state for the next 6 h is

The transition probabilities for the three-state model (Table

The likelihood-ratio test suggests a significant improvement when including the covariate

Number of total blocked time steps for omega and high-over-low blocks (columns) and percentage of omega (solid red line) and high-over-low (dashed red line) blocks with respect to total blocked time steps. Note that this analysis is based on

Characterizing blocking and blocking types is to a large degree dependent on time resolution and methods. The 6-hourly time resolution yields a larger data basis compared to daily means but potentially contains also more uninteresting variability (noise). Our identification and classification strategy starts with the well-accepted instantaneous blocked longitudes (IBLs) after

However, we want to point out that a major benefit of our method is the identification and location of each single vortex – the high-pressure area as well as the one or two low-pressure areas – forming the high-over-low and the omega block, respectively. This allows us to distinguish between the blocking types within and for each single blocking period separately. This gives more detail compared to averaging over multiple blocking periods as is typically done to derive composites; e.g., analyzing composites of the blocking onset in the time period June to August,

Blocked weather situations are usually analyzed with respect to the persistent high-pressure system. This quasi-stationary high can for example lead to droughts with devastating consequences. Here, we additionally consider the position(s) of the low-pressure system(s) in a blocking identification method. This is a novel approach and provides possibilities for further studies, for example on the impact of the steady low-pressure systems such as heavy rainfall and flooding events.

This novel strategy is based on point vortex theory to identify and classify blocking. Combined with logistic regression and Markov processes, this allows a fresh view on blocking occurrence and transition dynamics. We consider the time period 1990–2019 in the Euro-Atlantic sector (40

Conclusions with respect to the two research questions posed in the introduction are given in the following.

Our strategy to distinguish omega and high-over-low blocking with subsequent logistic regression involving Markov models can provide the basis for future studies to investigate the dependence of onset and decay of blocking on, for example, the North Atlantic Oscillation (NAO) index,
temporal gradients in mid-latitude wind speeds, or the speed and location of the jet that could influence the blocking process in Europe; see e.g.,

We finally conclude that distinguishing blocking types and describing their occurrence and transition probabilities with logistic regression combined with Markov models give valuable insight into the dynamics of atmospheric blocking and their changes for the Euro-Atlantic and potentially also for other regions.

The NCEP/DOE Reanalysis 2 data set can be accessed via the Research
Data Archive at the National Center for Atmospheric Research,
Computational and Information Systems Laboratory (

The supplement related to this article is available online at:

AM, PN and HR designed the study. CD did the statistical analysis and visualized the results, mainly at the FU Berlin. LS wrote and adapted the trapezoid method and blocking type decision method and performed further uncertainty experiments. HR and CD wrote the statistical chapters and discussed the related results. CD, HR, AM and LS continuously wrote the paper draft and discussed the results. All authors discussed and finalized the paper together.

The authors declare that they have no conflict of interest.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank Igor Kröner for critical discussions and reading the manuscript. We would like to thank the two anonymous reviewers for their input, questions and comments that helped improve this work. Moreover, we thank George Pacey and Edmund Meredith for helpful comments. This research has been partially funded by the Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114 “Scaling Cascades in Complex Systems”, project number 235221301, projects A01 “Coupling a multiscale stochastic precipitation model to large scale atmospheric flow dynamics” and C06 “Multiscale structure of atmospheric vortices”.

This research has partially been supported by the Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114 “Scaling Cascades in Complex Systems”, project number 235221301.

This paper was edited by Peter Knippertz and reviewed by two anonymous referees.