Persistence is an important concept in meteorology. It refers to surface weather or the atmospheric circulation either remaining in approximately the same state (quasi-stationarity) or repeatedly occupying the same state (recurrence) over some prolonged period of time. Persistence can be found at many different timescales; however, sub-seasonal to seasonal (S2S) timescales are especially relevant in terms of impacts and atmospheric predictability. For these reasons, S2S persistence has been attracting increasing attention from the scientific community. The dynamics responsible for persistence and their potential evolution under climate change are a notable focus of active research. However, one important challenge facing the community is how to define persistence from both a qualitative and quantitative perspective. Despite a general agreement on the concept, many different definitions and perspectives have been proposed over the years, among which it is not always easy to find one's way. The purpose of this review is to present and discuss existing concepts of weather persistence, associated methodologies and physical interpretations. In particular, we call attention to the fact that persistence can be defined as a global or as a local property of a system, with important implications in terms of methods and impacts. We also highlight the importance of timescale and similarity metric selection and illustrate some of the concepts using the example of summertime atmospheric circulation over western Europe.

Surface weather persistence at sub-seasonal to seasonal (S2S) timescales can have severe impacts on human and natural systems. Long-lasting dry conditions, for instance, can lead to droughts and wildfires and can affect agriculture and energy production. Long-lasting wet spells may cause severe flooding and crop loss. Persistent surface weather can result either from quasi-stationary, long-lived atmospheric circulation conditions (quasi-stationarity) or from repeated, shorter-lived circulation features (recurrence). Recurrence refers to the repeated occurrence of similar large-scale circulation types or weather systems within some (S2S) time interval, usually with brief interruptions. Many recent high-impact weather and climate events were linked to persistent quasi-stationary or recurrent weather conditions. An example for recurrence are the western European floods in July 2021 that occurred at the end of an extreme wet spell in western Europe. The wet spell resulted from repeated atmospheric blocks and Rossby wave breaking episodes

S2S weather persistence offers the potential for improved predictability at the S2S timescale

Persistence is also an important aspect of climate model evaluation and climate projections. Whether global climate models are able to correctly simulate persistence is key to the robustness of long-term projections, especially of high-impact weather events – all the more so as climate projections suggest enhanced persistence

Characterizing weather persistence is therefore key to our understanding of the atmospheric circulation and its predictability and the associated hazards. Previous studies have focused on weather persistence from varied perspectives. Some assessed the persistence of specific weather systems or features, like atmospheric blocking

It is difficult to give a unique definition of weather persistence. Besides, it may not even be desirable as different interpretations are possible and useful, depending on the system and timescale of analysis and on the motivations and goals of the study. Our goal here is therefore to review existing concepts of weather persistence, associated methodologies and physical interpretations. We present and structure a wide variety of approaches, definitions, techniques and metrics that have been used to analyze these concepts and that allow for answering one or several of the following questions.

Is there persistence in the data?

What are the persistent timescales in the data?

In which specific periods does persistence occur?

What are the persistent locations in the state space?

Overview of the persistence methods discussed in this paper. Section numbers relative to each method are indicated in bold between brackets.

Persistence in a dynamical system (climate variable, atmospheric circulation field, etc.) arises from the repeated occurrence of the same value(s) or pattern(s) over a period of time. Successive occurrences can follow each other continuously – a situation we refer to as “quasi-stationary” – or in an interrupted sequence – in which case we speak of recurrence. Persistence therefore comes in two flavors, quasi-stationarity and recurrence, which we illustrate with the example of extreme warm conditions in a region north of the Black Sea in Fig.

The literature often uses persistence as a synonym for quasi-stationarity

Note that recurrence, as we define it here, is sometimes referred to as “temporal” or “serial clustering”

Illustrating quasi-stationarity and recurrence on S2S timescales.

Because quasi-stationarity and recurrence are two faces of the same coin, distinguishing one from the other may not be evident nor necessarily relevant.

First, the distinction often depends on the variable of interest. Recurrent weather systems can indeed result in quasi-stationary surface weather anomalies and vice versa. For example, the long-lived heat waves of the 2010 and 2021 summers in western Russia (Fig.

Second, the longer the timescale of analysis, the less obvious the difference between quasi-stationarity and recurrence becomes. Impacts, for example, often depend on anomalies of surface temperature or precipitation averaged or accumulated over several weeks to months (e.g., droughts). Weekly or monthly values may thus sometimes be preferred to daily ones, in which case synoptic-scale quasi-stationarity and recurrence would both result in large weekly or monthly anomalies that would result from simple “persistence”.

Nevertheless, the distinction between quasi-stationarity and recurrence remains highly relevant for several reasons: from a methodological perspective (Sects.

Before reviewing how persistence can be apprehended and quantified, let us begin with some basic notation and definitions. In the following, we denote the dynamical system under analysis with

Characterizing a system as “persistent” can mean different things. Consequently, it is important to always specify the perspective that is taken to avoid confusion. First, there are different flavors to persistence (global, state or episodic persistence; see Sect.

Persistence (whether quasi-stationarity or recurrence) is a broad concept that covers different kinds of behavior in dynamical systems. We might say, for instance, that temperature is more persistent than precipitation because temperature evolves, on average, over longer timescales than precipitation. In this sense, persistence quantifies the system's inertia. However, if we qualify last summer's weather as particularly persistent, we mean something different: namely, last summer's weather varied much less than what one could have reasonably expected in a normal summer. Here, persistence refers to some unusual behavior of the system over a particular period. Saying that zonal jets or atmospheric blocks are persistent again means something else, i.e., that these particular states of the circulation tend to be more long-lived or recurrent than other states.

This leads us to make the distinction between three types of persistence: global, state and episodic persistence.

While it helps to capture the various interpretations of persistence, this classification is not perfect, and there is some overlap between categories. In practice, the state or episodic perspectives can also be used to quantify global persistence (by averaging persistence metrics across systems states or time intervals) and global persistence metrics can be computed on subsets of the data to identify persistent periods. Some methods, like recurrence plots (Sect.

The three types of persistence are also not independent from one another. Global persistence, for instance, can emerge from repeated occurrences in one or a handful of system states only, while the rest of the trajectory, if analyzed separately, may not be qualified as persistent. There are also strong relationships between state and episodic persistence. The most common system states to occur during persistent periods are indeed likely to be persistent states. Correspondingly, persistent states, when they occur, are likely to be associated with persistent periods. Persistent states can thus be uncovered from the knowledge of persistent periods, for instance with pattern recognition or clustering algorithms applied to system values during persistent periods, or simply by averaging system values during persistent periods

However, state persistence only characterizes the average behavior of system states – it does not imply that all occurrences of a persistent state will necessarily be persistent. Consequently, there is no one-to-one relationship between persistent states and persistent periods. Some system states can behave in a persistent way under certain conditions but not under others. The lifetime and travel speed of atmospheric blocks, for instance, is affected by land–atmosphere feedbacks or upstream latent heating

However, this classification is useful to illustrate the different methodological ways that weather persistence can be tackled, and we rely on it to structure the description of methods in Sects.

Weather persistence is most often analyzed from an Eulerian perspective, i.e., persistence of the same weather pattern or conditions at a fixed location in space. By contrast, in the Lagrangian perspective the focus is on the persistence of a given weather pattern in time. In the Eulerian framework, the system

Both the Eulerian and the Lagrangian perspectives are relevant for impact and risk assessment. The former links persistence to impacts at a given location, and the latter highlights impacts along the trajectory of a weather pattern (system) during its lifetime. Indeed, the same weather system can produce hazardous weather over large areas, putting strain on the resources of insurance companies or governments. The two perspectives can be brought together by considering the translation speed and lifetime of the tracked weather systems. Systems with long lifetimes and low translation speed lead to both Eulerian and Lagrangian persistence. By contrast, long-lived systems that travel quickly are persistent from a Lagrangian perspective only. Likewise, slow-moving but short-lived systems are not Lagrangian persistent, but Eulerian persistence can still be detected if several such systems occur over the same area in close succession.

Assessing persistence typically requires quantifying the self-similarity of system values

Categorical metrics focus on specific features of the system, such as the presence of a given weather pattern or the occurrence of a specific event. They require the set of all

Continuous metrics measure the degree of similarity between any pair of system values in a continuous way. Common examples of continuous metrics include the Euclidean distance

In comparison to categorical metrics, continuous metrics offer the advantage that they do not require specifying features or events of interest beforehand. They are also more flexible insofar as similarity can be quantified with respect to any system value as reference and not just representative values for each category. Since they do not require simplifying the state space, continuous metrics may also be able to pick up rare persistent patterns that are missed by dimension reduction methods that focus on the most common patterns in a series. These advantages come at a cost: working with continuous metrics, especially complex ones, can be more computationally intensive and sometimes more difficult to interpret physically.

Persistence is linked to a notion of timescale during which the system continuously remains in the same state (for quasi-stationarity) or occupies that same state repeatedly (for recurrence). There are three common ways to approach persistence timescales.

The first option is to choose a single, fixed timescale for analysis. This choice can be guided by impact and forecasting considerations, by physical knowledge of the underlying system, or by observations of persistent events

Recurrence can similarly be assessed by calculating the number of times that a particular system state or event occurs during

The second option is to select an analysis method that explores a range of timescales and pinpoints the relevant persistence timescales, sometimes accompanied by some notion of statistical significance. This approach only works for global persistence. Autocorrelation analysis, for instance, detects the timescales at which the system exhibits significant lagged memory (Sect.

Finally, the third option is to characterize persistence not by a single timescale but by a distribution of timescales. To assess quasi-stationarity, one can typically work with the distribution of persistent event durations. Persistent events are periods during which the system satisfies a persistence criterion: for quasi-stationarity, successive system values must be similar, and for recurrence the same event must occur multiple times, with each occurrence being separated by at most

The diversity of perspectives on persistence translates into a wide range of methods, of which we give an overview in the following two sections dedicated, respectively, to quasi-stationarity and recurrence. Following the distinctions introduced in Sect.

We begin with several methods that quantify global quasi-stationarity in one-dimensional time series. They characterize quasi-stationarity in the series as a whole but are generally unable to identify quasi-stationary states or periods. However, they often allow the user to characterize the timescales of variability and persistence in the data and are hence relevant for system predictability and process understanding.

Autocorrelation is a frequently used measure of quasi-stationarity in weather and climate science. If

Several summary metrics for autocorrelation exist, like the autocorrelation timescale:

The absolute value on

Both

Autocorrelation has many advantages: it has a simple definition, the ease of interpretation (

Example applications include

We show the characteristic time for daily rescaled 500 hPa geopotential height over Europe during summer in Fig.

Persistence in a system is associated with memory effects that lead to variability being concentrated at long rather than short timescales. How variability in the series is distributed across timescales is therefore an important indicator of global persistence and can point to relevant persistence timescales. Specifically, persistence often translates into scaling laws: in the time series variability as a function of frequency or timescale (for quasi-stationarity)

A first scaling law can be obtained by considering how autocorrelation decreases as a function of time lag. The more quasi-stationary a series, the less rapidly its autocorrelation should decrease. Conceptually, time series can be broadly divided between short-range and long-range dependence series. For a short-range dependent series (like autoregressive models), the autocorrelation decreases rapidly with the time lag, eventually reaching 0 after a certain lag or decaying exponentially.

Like autocorrelation,

The Hurst coefficient (or exponent)

Like the Hurst exponent, spectral analysis also characterizes how a time series' variability is distributed across timescales. The power spectrum is commonly defined as the Fourier transform of the autocorrelation function

The larger

Schematic diagram of the scaling regimes of continental European rainfall obtained by spectral analysis, along with the hypothesized meteorological interpretations of the various regimes. Reproduced from

Note that, as with the autocorrelation, the definition of the power spectrum can be extended to third-order statistics to yield the bispectrum (the Fourier transform of the bi-correlation function)

We now turn to methods that focus on the quasi-stationarity of specific system states. Unlike global methods, which take one-dimensional time series as input, several of the following methods are directly applicable to multidimensional data. We begin with methods that identify quasi-stationary states from the system trajectory (Sects.

We begin with an identification method for quasi-stationary states based on the concept of weather regimes

There are many ways to calculate weather regimes (see

Regimes can also be identified as local maxima in the (multidimensional) probability distribution function (PDF) of the target field, obtained empirically through e.g., kernel smoothing

The main drawback of the weather regime approach is that it is biased toward preferred or frequent states. Quasi-stationary but rare flow patterns that fall outside the range of the major regimes may therefore be missed. Additionally, certain patterns may fall in between regimes and are not classified or are misclassified. Despite this, weather regimes are very useful because they transform complex multidimensional systems into categorized, one-dimensional series (according to which regime the system is closest to at each time step).

Summer weather regimes computed by

One major disadvantage of weather regimes is that they strongly simplify the state space. They also focus on the patterns that account for most of the variability in the data, regardless of their persistence. They may thus miss rare yet impact-relevant persistent patterns. Several other techniques exist to directly extract quasi-stationary patterns from the system trajectory.

Quasi-stationary circulation patterns can be seen as mathematically quasi-stationary solutions of the atmosphere's equations of evolution

It is important to note that even if

Figure

Five quasi-stationary states of JJA 5 d averaged rescaled Z500.

The technique of optimally persistent patterns (OPPs) was introduced by

OPPs are especially relevant for forecasting since they correspond to the patterns with the most low-frequency variability. However, like EOFs and quasi-stationary states, OPPs are mathematical objects that do not necessarily correspond to “real” patterns ever attained by the system.

The pattern associated with the leading eigenvector maximizing the squared decorrelation time for daily anomaly fields of 500 hPa geopotential height for the 1950–1999 period (data from NCEP/NCAR reanalysis; units in

Other methods have been proposed to identify persistent patterns in a spatiotemporal field. One consists of minimizing the one-step-ahead forecast error, where the forecast is obtained by projecting the observed field

The quasi-stationary state method only focuses on the most quasi-stationary states of a system, and OPPs characterize quasi-stationarity for a handful of possible patterns only. By contrast, dynamical systems theory provides a convenient framework to describe the quasi-stationarity of any system state

In extreme value statistics,

The advantage of the method is that it is grounded in mathematical theory and, unlike approaches based on, e.g., transition probabilities, it does not require categorizing the data. It also provides an easily interpretable index of quasi-stationarity for each system state (the average residence time of the system's trajectory around that state).

Most persistent Z500 anomaly patterns during JJA, obtained by clustering the top 3 % (115) daily patterns with the highest quasi-stationarity index

A common way to quantify quasi-stationarity in time series relies on the concept of “residence time”. Residence times extend the dynamical systems approach of the previous section. For a continuous series, the residence time

The residence time approach can characterize episodic, state and global quasi-stationarity and is particularly useful for predictability and risk assessment

In the state space, the quasi-stationarity of any state

Finally, averaging

PDF of the North Atlantic jet latitude index (solid) together with the weighted Gaussian PDFs from the HMM: the southern (dashed), northern (dotted) and central (dashed–dotted) regimes. Regime duration curves (southern (black), northern (red) and central (blue)), expressed as the frequency of occurrences lasting at least

Residence times characterize how long the system remains in the same state

It is often convenient to assume that the distribution of system states at

Another possibility to investigate state quasi-stationarity is to fit a statistical model to the data that links successive system values with each other:

The running window approach identifies persistent periods of a given fixed length in continuous or categorical data. It assesses the degree of quasi-stationarity over fixed time intervals by means of a “similarity index”. Given a time interval

Averaging

The running window approach is well suited to compute temporal trends in quasi-stationarity, since the similarity index can be defined at all time steps

While we try to give a comprehensive view of common methods used in quasi-stationarity analysis, we could not include all the methods that exist in the literature.

Overview of quasi-stationarity methods: method name, corresponding section, type of persistence it applies to, a brief definition, main features and main limitations.

We now discuss methods to capture recurrence in atmospheric data (see Table

Because most recurrence methods work directly with binary time series, point processes are a convenient theoretical tool in recurrence analysis. Point processes are a class of probability models for the random occurrence of points in a space (one- or multi-dimensional space). The most simple hypothesis that can be made is that of complete serial randomness, i.e., points occur completely independently of one another. In this case, recurrence occurs by chance only. The homogeneous Poisson point process is a simple model without memory for binary series in which events occur without correlation and with a constant intensity. In a homogeneous Poisson process, the number of points in individual time intervals are independent, and the number of points in an interval of length

Separating distinct occurrences of a state, pattern or event is critical to distinguish recurrence from quasi-stationarity. Over short periods, like a season, it is in principle possible to visually separate multiple systems occurring in close succession, for instance with Hovmöller diagrams

A simple way to identify recurrent periods is to look for periods with high event counts. One possibility is to set a time window and require the number of event counts during this window to exceed some threshold (two or higher; Fig.

Dispersion metrics characterize the distribution of event occurrences in a time series. In a series with no tendency to clustering, high event counts over a given duration should be much less frequent than in a series where clustering is prevalent. If events occur with the same average probability in the two series, then in the clustered series, sequences with no events should mechanically be more frequent, and the variance in event counts will be higher. In a homogeneous Poisson process (see above)

Similarly, the Allan factor (AF) is defined as the variance of successive event counts over an interval of length

The statistical significance of

Since its introduction by

Alternative metrics that characterize the distribution of event counts have been proposed in the literature. For example,

Given a system state

Effect of European–Atlantic zonal regime occurrence (corresponding 500 hPa geopotential mean and anomalies are shown in the left panel) on the temporal clustering of extreme precipitation at the 3-week timescale during winter (measured as a probability multiplicative factor). Reproduced from

Quasi-stationarity in a given system state

We have so far discussed diagnostic methods that characterize recurrence in binary time series. A different approach consists of fitting point processes to the series. A point process is a random process representing the occurrence times of specific events. It can be characterized by event times

The point process approach has mainly been used to relate recurrence in extreme events to atmospheric or climate variability. To that end,

All previously discussed methods share the same limitation: they require binary time series as input, meaning that they are designed to characterize recurrence only in a given system state. It is not possible with such methods to explore recurrence in an “unsupervised” way, i.e., to automatically detect recurrent system states or sequences of states and associated recurrent periods. Recurrence plots (RPs) may offer a solution to this problem.

An RP is a graphical representation of a system's self-similarity with time

Detecting recurrence with recurrence plots.

Two major 3 d recurring sequences of atmospheric circulation (Z500) over Europe during summer obtained by clustering the

The same as Table

Before calculating

RPs are a convenient way to visualize a system's trajectory, especially for multi-dimensional systems (like circulation fields). RPs capture persistent behavior: vertical lines indicate quasi-stationarity, while diagonal lines (outside of the main diagonal) indicate recurrence. RPs can thus highlight periods when the trajectory of a system roughly visits the same sequence of states or parts of the state space.

Several measures have been proposed to quantify the presence of specific patterns in RPs. They are known collectively as “recurrence quantification analysis”

RPs can also detect local recurrence in time. With RPs, one can extend the counting approach introduced in Sect.

RPs have recently gained attention in a range of disciplines that deal with complex systems

Weather persistence on S2S timescales has been a topic of research since the early days of meteorology. Quasi-stationary or recurrent behavior are common features of weather dynamics and are strongly related to fundamental physical processes, weather predictability and surface weather impacts. Studying weather persistence is therefore important for theoretical and practical reasons. One challenge is that persistence remains a very broad concept that relates to different behaviors in dynamical systems. We propose a typology/structure for the broad concepts related to persistence. Namely that persistence is used to describe the average behavior of a system across its whole trajectory (global persistence), to refer to specific segments of this trajectory (episodic persistence) or to qualify the behavior of particular system states (state persistence).

A wide range of methods has been introduced in the literature to describe persistence in weather and climate series, and several exist for each type of persistence. They offer many distinct and often complementary perspectives on persistence. Some methods quantify persistence in a statistical framework, which can be very relevant for weather forecasting. Others focus instead on persistent periods, including statistical flukes; this is a useful approach for risk assessment, but it is one that says nothing about the overall behavior of the series. Other methods still aim to identify which weather patterns tend to be more persistent than others.

The diversity of existing methods presented in this review reflects the fact that persistence is a multi-faceted concept. While we can agree on a general definition of the concept, many options exist when it comes to actually quantify persistence in real-world data. Though different methods may be related, each sheds light on a particular aspect of persistence. What is meant by persistence cannot be dissociated from the metric used to quantify it. The choice of method should be guided by the end goal, whether it is process understanding, risk and impact assessment, or predictability. Future research should nevertheless perhaps consider testing the robustness of results to the choice of persistence metric more systematically. This could be particularly important for better characterizing potential trends in persistence under climate change and their associated impacts. As a final note, while we centered our review on S2S persistence, most of the methods we discussed apply in principle to other timescales, be they sub-daily or multi-decadal. What really differs is how to define the system to analyze (daily time steps may not be relevant for inter-annual variability, for instance) and interpret persistence.

R code implementing several of the methods presented in this paper is available at

We illustrate some of the methods (Figs.

AT: conceptualization, formal analysis, visualization, software, writing – original draft. OM: conceptualization, funding acquisition, supervision, writing – review and editing.

At least one of the (co-)authors is a member of the editorial board of

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This research has been supported by the Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (grant no. 178751).

This paper was edited by Christian Franzke and reviewed by Abdel Hannachi and two anonymous referees.