This paper investigates systematic changes in the global atmospheric circulation statistics during Eurasian heat waves (HWs). The investigation of Rossby wave energy anomalies during HWs is based on the time series of Hough expansion coefficients representing Rossby waves with the troposphere–barotropic structures through the extended boreal summer in the European ERA5, ERA-Interim, Japanese 55-year Reanalysis (JRA-55) and Modern-Era Retrospective analysis for Research and Applications (MERRA). The climatological Rossby wave energy distribution is shown to follow a

The applied multivariate decomposition reveals signatures of the Eurasian HWs in the probability density functions (PDFs) of the Rossby wave energy across scales. Changes in the PDFs are consistent with changes in the intramonthly variance during HWs. For the zonal-mean state (the zonal wavenumber

Heat waves, periods with the daily maximum temperatures exceeding the climatological conditions by certain thresholds, have been increasing in number and magnitude, especially over Eurasia

In contrast to previous studies investigating particular aspects of HWs, our research aims to identify changes in the global Rossby wave energy statistics during Eurasian HWs and to couple them with the observed circulation. While a number of studies addressed particular aspects of HWs over Eurasia

The distributions of atmospheric fields are in general known to be non-Gaussian

Under the independence of components or variables in a high-dimensional system, one can consider their time series to be uncorrelated. The identity of distributions of summing components can be regarded in terms of their mean and variances being equal.

As we demonstrate, the distributions of anomalies in atmospheric energy can appear visually close to the normal distribution due to the central limit theorem. However, the energy anomaly distributions are still skewed, which can be considered an inherited property from energy (Advanced statistical methods are common tools in the research of extreme weather events. For example,

A more common tool for the examination of midlatitude circulation during heat waves is the Fourier series analysis of single-variable data along the latitude circles. This approach identifies anomalies in the planetary- and synoptic-scale Rossby waves during extreme events in terms of the Fourier amplitudes and phases of temperature, geopotential or wind variables at different levels. For example,

Our heuristic approach to spectral analysis of HWs considers the horizontal and vertical scales simultaneously by using the normal-mode function (NMF) decomposition to project daily circulation fields onto Rossby and non-Rossby components

The real-time decomposition of the ECMWF circulation in Rossby and non-Rossby components is available on the MODES web page at

Previous applications of the NMF decomposition showed that modal analysis complements other methods of analysing global circulation by providing scale- and dynamical-regime-dependent information on the variability and by quantifying it in wavenumber space

Our goal is to investigate whether and how surface heat waves during boreal summer over Eurasia affect the global atmospheric variability spectrum. While it is not evident a priori that regional HWs have their signatures in the global Rossby wave spectra, we show that this is, in fact, the case. First, we demonstrate statistically significant changes in the global total energy anomalies probability density functions (PDFs) during HWs. Then, we interpret the dynamics of the planetary Rossby waves through the change in active degrees of freedom and in temporal variance on intramonthly scales. At last, we provide an overall picture of the changes in atmospheric circulation across scales.

The paper is organised as follows. The 3D decomposition method, statistical analysis and the heat wave identification algorithm are explained in Sect.

In this section we describe our research method that makes use of the NMF decomposition and the MODES software

The NMF decomposition is carried out in the terrain-following global coordinate system

The projection of discrete data consists of two steps. In the first step, the data vector

In the second step, the horizontal nondimensional motions are projected onto a series of Hough harmonics

We use both “modes” and “waves” interchangeably but the latter refers to the case without the zonal-mean state (

It is the inverse of Eqs. (

MODES is applied to the four modern reanalyses: European ERA5

We are interested in the balanced circulation with the troposphere–barotropic vertical structure that characterises the midlatitude weather during HWs. This is taken into account by selecting a subset of the VSFs that do not change their signs within the tropopause. In the NMF decomposition, the rigid lid is at zero pressure, just like in the models used for reanalyses. The 43-level datasets extend vertically up to about 0.5

Vertical structure functions (VSFs) for the first seven vertical modes. VSFs are derived for 43

The study area is the Eurasian region limited by the Ural mountains (35–65

Heat waves in Eurasia during May–September 1980–2019.

Our statistics make use of Rossby wave energy anomalies during HWs in comparison to the climatology. We compute the energy time series, their anomalies and standard deviations used for normalisation, followed by combining normalised time series for all troposphere–barotropic modes and statistical analysis. In the first step, the total energy (the kinetic energy plus the available potential energy) is computed for every circulation mode

The time series of the daily total energy,

The next step is to split the normalised energy anomalies of the single Rossby modes into planetary (

Our presentation of the results starts by showing that the selected Rossby modes from the NMF decomposition and the applied HW detection method correspond to the circulation patterns typical for the HW events. After demonstrating our methodology, we continue with the statistical analysis of the Eurasian HWs in global spectra and wrap up by coupling statistical properties with the circulation changes during HWs. But first we demonstrate in Fig.

The Greek letter

Atmospheric energy distribution for the Rossby wave with the zonal wavenumber

Now we demonstrate that the selected subset of vertical modes is suitable for the statistical analysis of HWs by showing the climatological state and two selected events.
Figure

The circulation during the Eurasian HWs is commonly associated with the blocking and can be in the NMF-filtered circulation during two recent HW events: the Russian heat wave in 2010

Our next step is to investigate how the Eurasian HWs affect the global spatial variability spectrum indicating their impact on global circulation. Here, the term global variability spectrum refers to the PDFs of the normalised anomalies in global energy, whereas the effects (or signatures) of HWs imply significant changes in the distribution of energy anomalies. The climatological PDFs are analysed for zonal wavenumbers corresponding to three ranges as described in Sect.

Figure

PDFs of the normalised energy anomalies in the global balanced (Rossby mode) circulation for

Focusing on the skewness and kurtosis of the PDFs, Fig.

Box plots for the

Now we compare PDFs during the observed HWs over Eurasia with the climatology in terms of the skewness and excess kurtosis that diagnose the changes in shape, especially in the tails of distributions.

The PDFs of the normalised energy anomalies in Fig.

As in Fig.

How do the skewness and the excess kurtosis change during the Eurasian HWs? An increase (decrease) in skewness hints to fewer (more) active degrees of freedom, which can be interpreted as less (more) independent modes contributing to the variability. This can be caused by both a change in the number of contributing modes and a change in temporal coherence between different modes contributing. An increase in excess kurtosis reflects a rise in the probability of extreme values.

Together with the climatology, Fig.

The change in skewness allows for the estimation of the change in the active degrees of freedom during the HWs compared to the climatology. For the estimation, we use the exact relation for the skewness of the

Finally, we make a note of the fact that the changes in PDFs during the Eurasian HWs apply to the global atmosphere. Our Rossby modes consist of symmetrical (

The changes in the PDFs for different scales can be physically interpreted by filtering selected Rossby waves to physical space, similar to what has been done in Fig.

Figure

Planetary-scale troposphere–barotropic Rossby waves (

So far, we discussed signatures of HWs in spatial variance (energy). Now we investigate related changes in temporal variance on intramonthly scales. The temporal variance and its square root, variability, are usually studied at single points or using the time series of atmospheric indices such as the North Atlantic Oscillation. The global intraseasonal variance was analysed by

The unbiased variance (J kg

Intramonthly variance is computed for all months and averaged to create the climatological variance spectrum,

The global intramonthly Rossby wave variance spectrum is shown in Fig.

The blue shading around the variance spectra in Fig.

A more detailed view of the changes in the global intramonthly variance during HWs is provided in Fig.

Zonal-mean zonal wind in the Northern Hemisphere troposphere in 1980–2019, May–September ERA5 data.

Other features of the HWs seen in Fig.

Extreme events such as surface HWs are accompanied by changes in atmospheric circulation across many scales. Our study shows that Eurasian HWs have signatures in the global balanced circulation. The changes in global statistics of the Rossby-wave variance are made evident by analysing the four modern reanalyses: the ERA5, ERA-Interim, JRA-55 and MERRA datasets. The Rossby waves are identified by a multivariate projection of the global horizontal winds and geopotential height on the eigensolutions of the linearised primitive equations on the sphere with a basic state at rest (the so-called normal-mode functions). A complete projection basis provides global statistics of Rossby waves as a function of the zonal wavenumber, the meridional mode index and the vertical modes associated with the vertical structure functions. The method includes scale-selective multivariate Rossby-wave filtering in physical space offering an attractive global complement to the single-variable, single-level Fourier analysis.

Our analysis focuses on the Rossby waves with the barotropic structure within the troposphere that is characteristic of the midlatitude circulation during HWs. The reconstructed physical space picture of the Eurasian HWs is in agreement with previous studies

The statistical analysis is carried out on the complex time series of the Hough expansion coefficients representing Rossby modes across many horizontal scales with the troposphere–barotropic vertical structure. We demonstrate that the energy distribution of a single mode follows a

During the Eurasian HWs, the skewness in planetary-scale Rossby waves increases, while the opposite occurs in the zonal-mean state. The increase in skewness for planetary-scale waves reveals the decrease in the number of active degrees of freedom during HWs. This aligns with the results of

Consistent changes in wavenumber space are found in the intramonthly variance. Eurasian HWs are characterised by a statistically significant increase of about 5 % in the intramonthly variance at synoptic scales

Despite the uncertainties due to the limited sample size, our results provide the following overall picture, consistent with previous studies. During HWs, the planetary-scale Rossby waves (primarily

The ERA-Interim datasets are available via

All authors contributed to the study conception and design. IS developed the algorithm, performed the data analysis and wrote a first draft of the article. All authors participated in data interpretation and revised previous versions of the article. All authors read and approved the final article.

The contact author has declared that none of the authors has any competing interests.

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This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC 2037 “CLICCS – Climate, Climatic Change, and Society” (CLICCS, A6) – project number 390683824, contribution to the Center for Earth System Research and Sustainability (CEN) of Universität Hamburg. We thank the former MODES group members at the University of Ljubljana, Damjan Jelić and Khalil Karami, for the MODES decomposition of the four reanalysis datasets, Žiga Zaplotnik for his advice on processing the Hough coefficients in Python, Qiyun Ma for the algorithm for the heat wave identification, and Frank Sielmann for the variance analysis and technical support. We would also like to thank Valerio Lucarini for the discussion, as well as two anonymous reviewers and the editor Gwendal Rivière for their constructive comments on the article.

This research has been supported by the Deutsche Forschungsgemeinschaft (grant no. 390683824).

This paper was edited by Gwendal Rivière and reviewed by two anonymous referees.